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Full text of "The Theory Of Electric And Magnetic Susceptibilities"

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OSMANIA UNIVERSITY LIBRARY 
Call N*.3 / * * r Accession No. /of 73 



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Thi&JDOOK shcmld be returned on or before^he date 
last marked belo\\\^ / ^ < ^//'/g/c7itfe * . 



THE 

INTERNATIONAL SERIES 

OF 

MONOGRAPHS ON PHYSICS 

GENERAL EDITORS 

R. H. FOWLER AND P. KAPITZA 



OXFORD UNIVERSITY PRESS 

AMEN HOUSE, B.C. 4 

LONDON EDINBURGH GLASGOW 

LEIPZIG NEW YORK TORONTO 

MELBOURNE CAPETOWN BOMBAY 

CALCUTTA MADRAS SHANGHAI 

HUMrHKEY MILFOBD 

PUBLISHER TO THE 

UNIVERSITY 



PHINTED IN GREAT BRITAIN 



THE 

THEORY 

OE 

ELECTRIC AND MAGNETIC 
SUSCEPTIBILITIES 



BY 

J. II. VAN VLECK 



PROFESSOR OF THEORETICAL PHYSICS IN 
THE UNIVERSITY OF WISCONSIN 



OXFORD 

AT THE CLARENDON PRESS 
1932 



TO.MY FATHER 
EDWARD BURR VAN VLECK 

EMERITUS PROFESSOR OF MATHEMATICS 
THE UNIVERSITY OF WISCONSIN 



PREFACE 

THE new quantum mechanics is perhaps most noted for its triumphs 
in the field of spectroscopy, but its less heralded successes in the 
theory of electric and magnetic susceptibilities must be regarded as one 
of its great achievements. At the same time the accomplishments of 
classical mechanics in this field must not be overlooked, and so the first 
four chapters are devoted to purely classical theory. Most of the com- 
parison with experiment regarding dielectric constants is included in 
one of these (Chap. III). This can be done without making the com- 
parison obsolete because the new quantum mechanics has restored the 
validity of many classical theorems violated in the old quantum theory. 
On the other hand, the analysis of experimental magnetic suscepti- 
bilities cannot be attempted until the quantum chapters, since the 
numerical values of magnetic susceptibilities are inextricably connected 
with the quantization of angular momentum. At the outset 1 intended 
to include only gaseous media, but the number of paramagnetic gases 
is so very limited that any treatment of magnetism not applicable to 
solids would be rather unfruitful. Therefore, salts of the rare earth and 
iron groups are examined in considerable detail. A theory is developed 
to explain why, as conjectured by Stoner, inter-atomic forces obliterate 
the contribution of the orbital angular momentum to the magnetic 
moment in the iron group. Chapter XII includes the aspects of ferro- 
magnetism so far amenable to the Heisenberg theory, which has at last 
divested the Weiss molecular field of its mystery. This means that here 
the discussion is centred on the thermal behaviour of the saturation, 
rather than on hysteresis and retentivity. As far as practicable, I have 
striven throughout the volume to avoid duplication of the existing 
literature, especially Debye's Polar Molecules and Stoner 's Magnetism 
and Atomic Structure. 

In the preface to a book on theoretical physics it is customary for 
the author to express the laudable but, alas, usually unwarranted hope 
that the volume will prove simultaneously rigorous to mathematical 
readers and intelligible to the non-mathematical, at least provided the 
latter omit the particular sections where the density of equations is 
excessive. At any rate this has been the aim of the present volume, and 
I hope that it has not fallen too far short. A detailed knowledge of 
quantum mechanics or of spectroscopic nomenclature has not been pre- 
supposed only an elementary acquaintance with the SchrMinger wave 



viii PREFACE 

equation. The necessary perturbation theory and theorems of spectro- 
scopic stability are developed in Chapter VI. Here I have tried to 
correlate and intermingle the use of wave functions and of matrices, 
rather than relying exclusively on the one or the other, as is too often 
done. It is hoped that this chapter may be helpful as a presentation 
of the perturbation machinery of quantum mechanics, quite irrespective 
of the magnetic applications. 

I am much indebted to the Guggenheim Memorial Foundation for a 
travelling fellowship which enabled me to visit many European institutes 
for theoretical physics. I wish to take this occasion to thank the staffs 
of these institutes for their cordiality and helpful discussions. The list 
is rather extensive Cambridge, Leipzig, Munich, Gottingen, Berlin, 
Zurich, Copenhagen, Leiden, Utrecht, Groningen, Bristol, Paris. I am 
also indebted to the University of Wisconsin for extension of leave which 
permitted me to attend the sixth Solvay Congress, devoted to magnet- 
ism, and to Professors W. Weaver and J . W. Williams of this university 
for valuable criticisms on Chapters I and III respectively. I also wish 
to thank Miss A. Frank and Mr. R. Serber for assistance in some 
of the computations and in proof -reading. 

J. H. V. V. 

DEPARTMENT OF PHYSICS, 
UNIVERSITY OF WISCONSIN, 

June, 1931. 



CONTENTS 

I. CLASSICAL FOUNDATIONS 

1. The Macroscopic versus Microscopic Field Equations . . 1 

2. Correlation of the Microscopic and Macroscopic Equations . 3 

3. Proof of the Preceding Correlation Formulae .... 7 

4. Relation between the Index of Refraction and Dielectric Con- 

stant 13 

5. The Local Field 14 

6. The Force Equation . . . . . . . .17 

7. The Lagrangian and Hainiltonian Functions . . . .19 

8. Larmor's Theorem ........ 22 

0. The Fundamental Theorem of Statistical Mechanics . . 24 

II. CLASSICAL THEORY OF THE LANOEV1N-DEBYE FORMULA 

10. Polar versus Non -polar Molecules ...... 27 

11. Rudimentary Proof of the Langeviii-Debye Formula . . 30 

12. More Complete Derivation of the Langevhi-Debyo Formula . 32 

13. Derivation of a Generalized Laiigevin-Debye Formula . 37 

III. DIELECTRIC CONSTANTS, REFRACTION, AND THE 

MEASUREMENT OF ELECTRIC MOMENTS 

14. Relation of Polarity to the Extrapolated Refractive Index . 42 

15. Effect of Infra-Red Vibration Bands 45 

16. Independence of Temperature of the Index of Refraction. . 4 

17. Dispersion, at Radio Frequencies ...... 54 

18. The Dielectric Constants of Solutions ..... 56 

19. Numerical Values of the Electric Moments of Various Molecules. 

Comparison of the Different Methods . . . .60 

20. Dielectric Constants and Molecular Structure . . . .70 

21. Optical Rcfraetivities and Molecular Structure . . .82 

22. Saturation Effects in Electric Polarization .... 85 

IV. THE CLASSICAL THEORY OF MACJNETIC SUSCEPTIBILI- 

TIES 

23. Conventional Derivation of the Langevin Formulae for Para- 

ancl Diamagtictism ........ 89 

24. Absence of Magnetism with Pure Classical Statistics . . 94 

25. Alternative Proof of Miss Van Lceuweii's Theorem . . 97 

26. Absence of Diamagnetism from Free Electrons in Classical 

Theory 100 

27. Inapplicability of Classical Statistics to any Real Atomic System 102 

V. SUSCEPTIBILITIES IN THE OLD QUANTUM THEORY, CON- 

TRASTED WITH THE NEW 

28. Historical Survey 105 

29. Weak and Strong Spacial Quantization . . . . .108 

30. Spectroscopic Stability in the New Quantum Mechanics . .111 

31. Effect of a Magnetic Field oil the Dielectric Constant . .113 



x CONTENTS 

VI. QUANTUM-MECHANICAL FOUNDATIONS 

32. The Schrodiiiger Wave Equation 122 

33. Construction of the Heisenberg Matrix Elements by Use of the 

Wave Functions . . . . . . . .124 

34. Perturbation Theory 131 

35. Matrix Elements of a Perturbed System. Proof of Spectroscopic 

Stability 137 

36. Formulae for the Electric and Magnetic Moments of a Stationary 

State 143 

37. The Rotating Dipole in an Electric Field .... 147 

38. The Electron Spin 155 

39. Orbital and Spin Angular Momentum Matrices . . .159 

40. Russell-Saunders Coiipling ; Spectroscopic Notation . . 162 

41. Classical Analogues of the Angular Momentum Matrices, and the 

Correspondence Principle . . . . . .169 

42. The Anomalous Zeeman Effect in Atomic Spectra . . .172 

43. The Diamagrietic Second-Order Zeeman Term . . .178 

VII. QUANTUM-MECHANICAL DERIVATION OF THE LANGE- 

VLN-DEBYE FORMULA 

44. First Stages of Calculation . . . . . . .181 

45. Derivation of the Langevin-Debye Formula with Special Models 183 

46. General Derivation of the Langevin-Debye Formula . . 186 

47. Limit of Accuracy of the Langevin-Debye Formula . .197 

VIII. THE DIELECTRIC CONSTANTS AND DIAMAGNET1C SUS- 
CEPTIBILITIES OF ATOMS AND MONATOMIC IONS 

48. The Dielectric Constant of Atomic Hydrogen and Helium . 203 

49. The Diai Magnetism of Atoms, especially Hydrogen and Helium 206 

50. Adaptation to Other Atoms by Screening Constants . . 209 

51. Polarizability of the Atom-Core from Spectroscopic Quantum 

Defect 215 

52. Ionic Refractivities and Diamagnetic Susceptibilities . . 220 

IX. THE PARAMAGNETISM OF FREE ATOMS AND RARE 

EARTH IONS 

53. Adaptation of Proof of Luiigeviii-Debye Formula given in 46 226 

54. Multiple! Intervals Small compared to kT .... 229 

55. Multiple! Intervals Large compared to kT .... 232 

56. Multiple! Intervals comparable to kT ..... 235 

57. Susceptibilities of Alkali Vapours ...... 238 

58. Susceptibilities of tho Rare Earths 239 

59. The Special Cases of Europeum and Samarium . . . 245 

60. Temperature Variation, in the Rare Earths. The Gyromagiictic 

Ratio 249 

61. Saturation Effects 257 

62. Lack of Influence of Nuclear Spin ...... 259 

X. THE PARA- AND DTAMAGNETISM OF FREE MOLECULES 

63. Spectral Notation and Quantization in Diatomic Moloculen . 262 

64. Multiplet Intervals Small compared to kT . . . 264 

65. Multiplet. Intervals Large compared to kT .... 265 



CONTENTS xi 

66. The Oxygen Molecule 266 

67. The Nitric Oxide Molecule 269 

68. Polyatomic Molecules 272 

69. , The Diamagnetism of Molecules ...... 276 

70. Absence of Magneto -Electric Directive Effects . . . 279 

XI. THE PARAMAGNETISM OF SOLIDS, ESPECIALLY SALTS 

OF THE IRON GROUP 

71. Delineation of Various Cases . . . . . .282 

72. Salts and Solutions Involving the Iron Group . . . 284 

73. Quenching of Orbital Magnetic Moment by Asymmetrical Ex- 

ternal Fields 287 

74. Further Discussion of Salts of the Iron Group . . . 297 

75. The Palladium, Platinum, and Uranium Groups . . .311 

XII. HEISENBERG'S THEORY OF FERROMAGNETTSM ; FUR- 

THER TOPICS IN SOLIDS 

76. The Heisonberg Exchange Effect 316 

77. Heisenborg's Theory of Ferromagnetism .... 322 

78. Proof of Formulae for Mean and Mean Square Energy . . 340 

79. Magneto -caloric and Magnetostrictive Effects .... 343 

80. Feeble Paramagnetism ....... 347 

81. Tho Diamagnetism of Free Electrons in Quantum Mechanics . 353 

XIII. BRIEF SURVEY OF SOME RELATED OPTICAL PHE- 
NOMENA 

82. The Kramers Dispersion Formula . . . . .361 

83. The Kerr Effect 366 

84. Tho Faraday Effect 367 

INDEX OF AUTHORS 375 

SUBJECT INDEX 380 



SYLLABUS OF NOTATION 

BESIDES symbols which are standard usage, such as e, h, m, E, H, the following 
notation commonly occurs in the present volume : 

a = constant term in the Langevin-Debye formula % N (a + Q ^r Usually a 

\ OrC-L 1 

arises from induced polarization or diamagnetic induction. 

j8 = Bolir magneton 0-9174 x 10" 20 e.m.u. ( = 4-95 Weiss magnetons). (The so- 
called 'molar' Bohr magneton number is Lj8 = 5564 e.m.u.) 

e it m t = arbitrary charge or mass, whereas e - 4-77 x 1()" 10 e.s.u.,w = 9-04 X 10~ 28 gm. 

Jf = Hamiltonian function (to be distinguished from the magnetic field H). 

K = 'molar polarizability ' 4:nLP/3NE. 

L Avogadro number 6-004 x 10 23 . (Occasionally L is also used for the Lagraii- 
gian function or for the azimuthal quantum number.) 

ni H (or m z ) = component of an individual molecule's total magnetic moment, 
inclusive of both induced and permanent parts, in direction of the applied 
field U (or of the z axis). 

M NWiji = magnetic moment per unit-volume. (B -H -\-knM.} 

p, permanent moment of the molecule. (On pp. 1-17 only, \L instead denotes 
tlie magnetic permeability.) 

/^ofl ~ 'effective Bohr magneton number", defined in terms of the susceptibility 
by the relation p^ = V(3A;7 T ^/iVj8 2 ). We throughout use Bohr rather than 
Weiss magneton numbers because of the former's more elemental physical 
significance. Note that the empirical number /z^ has no connexion with 
the permanent moment /z except when Curie's law is obeyed. 

N -~ number of molecules per unit-volume ( 2-706:: 10 19 per e.o. at C., 
76 cm.). 

p E = component of an individual molecule's total electric moment in direction 
of the applied field K. 

P Np =^ electric moment per unit-volume. (D E -\-4i7rP.) 

fS f/ = spin quantum number for entire crystal (used 111 Chap. XII) to be distin- 
guished from spin 8 of a single atom. 

^-susceptibility per unit-volume (electric or magnetic). Xmnl~ ^X/-^ sus " 

ceptibihty per gramme znol. 
Expressions in bold-face typo are vectors. Single bars denote time average for 

a single molecule. Double bars denote) statistical average over a very largo number 

of molecules. Equations involving entire matrices are numbered in angular 

parentheses, e.g. Eq. ^12>. A dotted equality such as p^ = mJi/^TT moans that 

p<k is a diagonal matrix whose characteristic values are //< / /?/27T, i.e. 
p$(n;n') - 8(w;H y ) w z^/ 2 7r. 

For explanation of spectroscopic nomenclature and quantum numbers see 40 

(atoms) and 63 (molecules). 



I 

CLASSICAL FOUNDATIONS 

1 . The Macroscopic versus Microscopic Field Equations 

The conventional Maxwell equations are 

crlE=-if, curlH^i + f), (,) 

together with div D =. - 4?rp, div B == 0. (2) 

We shall term these the 'macroscopic field equations' as they do not 
aim to take direct cognizance of the atomicity of matter or electricity. 
Throughout the volume all expressions printed in bold-face type are 
vectors. Between the four field vectors there exist the so-called con- 
stitutive relations ^ n 

*J & /<3 , 

E = e, H = M) (3) 

which may be regarded as defining the dielectric constant e and the 
permeability /i. 1 The ratios and p are, except for ferromagnetic media, 

1 The logically minded will immediately object that Eqs. (3) do not really define t 
and fjL, inasmuch as the solutions of Kqs. (1) and (2) and hence the left sides of (3) are 
not per se unique, because (1 ), (2) involve four unknown vectors E, H, D, B rather than 
two as in vacuo. Tho solutions of (1 ), (2) become unique as soon as wo know something 
about the nature of the ratios (3), but this is clearly arguing in a circle, and (3) cannot 
serve simultaneously as an auxiliary relation and as a definition. This inability to give 
a simple and rigorous definition of e and fi is inherent in the macroscopic field equations, 
but is a purely academic difficulty, as from a practical standpoint there never seems to 
have been any particular ambiguity in knowing in simple cases what is meant by a 
dielectric constant or magnetic permeability. Two ways of avoiding the looseness in 
definition immediately suggest themselves. One is to assume, as one always does, that 
e and fi are independent of position in homogeneous media, and also of time in static or 
in monochromatic phenomena. For electromagnetic waves of given frequency the ex- 
pressions c and /u. are then constants of the homogeneous body, which are not calculable 
from (1), (2) but which can be determined once for all by observing once through experi- 
ment which particular values of the constants are true experimentally i.e. verifying 
which values of the otherwise indeterminate ratios B/H and D/E are actually realized. 
Wo then regard e and /u, at a given point in a non-homogeneous body to be the same as they 
would in a homogeneous body of the same density and material throughout as at the 
givrii point. If the electromagnetic waves are not monochromatic wo would have to make 
a harmonic analysis into the various Fourier components, and knowing the e and /* for 
each component, find the total solution by the principle of superposition. Tho other way 
of avoiding the looseness is to appeal to the microscopic point of view and define and 
ft by means of (8), (11), and (12). Although this is more rigorous from a postulational 
standpoint, it docs not seem as desirablo to follow, since most physicists have felt in the 
past, and still feel, that the task of the microscopic theory is to explain dielectric con- 
stants and magnetic permeabilities already measured macroscopically rather than to 
define something not already known. We therefore aim in iho present chapter to analyse 
or dissect tho macroscopic equations from the microscopic standpoint, rather than to 
synthesize from microscopic to macroscopic phenomena. 

3595.3 



2 CLASSICAL FOUNDATIONS I, 1 

independent of the field strength for sufficiently small fields, and in 
general we must have such an independence, or at least a known 
dependence on the field strength, before Eqs. ( 1 ), ( 2) become unambiguous 
enough to be useful. We suppose throughout the volume that the 
medium is isotropic; in crystalline media directional effects make it 
necessary to use six dielectric constants or permeabilities instead of 
one, and D, B cease in general to be parallel to E, H respectively as 
presupposed by (3). 

Of course e and p depend on many factors, notably on the tempera- 
ture, density, chemical constitution, and frequency, as well as on the 
field strength if great. The theoretical description of their modes of 
dependence is the main aim of the present volume. This description is 
accomplished by means of the molecular theory of matter, and especially 
by means of the dynamics governing the electrons within each atom or 
molecule. The dawn of the twentieth century brought to light the 
electrical origin of matter, unknown to Maxwell when he developed his 
macroscopic equations in 1861-73. This electrical origin implied that 
by probing down to sub-atomic distances it should be possible to 
formulate the equations of electrodynamics in terms of charges in vacua 
without the introduction of ponderable dielectric and magnetic media. 
H. A. Lorentz 2 therefore proposed and studied what we shall term the 
'microscopic field equations' 



8 

div e -- 47T/>', div h = 0, (5) 

which are similar in structure to the macroscopic equations in vacuo 
(where, of course, B H, D E), except that instead of the ponderable 
current density i Lorentz introduced the convection current density 
p\ due to motion of the charge density p with the vector velocity v. 
The microscopic fields e, h and charge p are not the same as the macro- 
scopic fields or charge, and have therefore been printed in small letters 
or else designated by a prime. Eqs. (4), (5) are more fundamental than 
(1), (2), (3), as (4), (5) are supposed to hold at every point cither inside or 
outside the molecule, whereas (1), (2), (3) are essentially statistical in 
nature, and the expressions E, D, H, B, p which they involve must be 
correlated in some way with averages of microscopic fields and charges 
over a large number of molecules. How this correlation is achieved will 
be discussed in the two following sections. 

2 Cf., for instance, H. A. Lorontz, The Theory of Electrons (Leipzig, 1916). His original 
papers were published considerably earlier in the Proceedings of the Amsterdam Academy. 



1, 2 CLASSICAL FOUNDATIONS 3 

2. Correlation of the Microscopic and Macroscopic Equations. 

The Fundamental Lemma on the Significance of Molecular 
Moments 

Let e and h denote the averages of e and h over a 'physically small' 
element of volume; i.e. an element too small to be accessible to ordinary 
methods of measurement but nevertheless large enough to contain a 
very great number of molecules. Throughout the volume we shall use 
double bars to designate statistical averages involving a large number 
of molecules, to avoid confusion with time averages for a single mole- 
cule, for which a single bar is used. It turns out (see 3) that E and B 
are identical with the microscopic fields averaged over such a volume 

element, so that ^ = c 

*/ e, r> n. (6) 

It is to be noted that the electric and magnetic cases are not entirely 
parallel, as by analogy with the electric one we should expect H rather 
than B to enter in (6). 3 

In order to describe the statistical significance of D and H or of the 
constitutive relations (3) in tcnns of the microscopic theory, let us, as 
customary, write D = E+4wF> B-.H+^M. (7) 

The expressions P and M so defined are called respectively the electric 
and magnetic polarizations (or intensity of magnetization), while the 

quotients r> i */r i 

P e 1 _ M //, 1 

E ^ ~47r~ = Xt " H = ~4^~ ^ Xm (8) 

are the electric and magnetic susceptibilities X G an< ^ Xm- Let, us form 

the expressions /// , _ 

1 'dv, (9) 



and P= j I j p'rdv, m ^ ~ \ \ \ p'[rx v\dv, (10) 

I/I/ 1M 

in which the integration is to include only the charge which belongs to 
a single molecule, as indicated by the subscript 1Ar . In general molecules 
may overlap each other, but we are to suppose that the charges per- 
taining to individual molecules can still be identified as such. The origin 
for the radius vector r is to be taken at the centre of gravity of the 

3 Tho appearance of B rather than II in (6) shows that B rather than H is the funda- 
mental field vector, so that it would seem preferable to write the microscopic equations 
with the notation b instead of h, and to retain B rather than H to denote the common 
value of B and H in vacuo. However, we do not make these changes, in order to conform 
more closely to most of the existing literature, which regards H as the fundamental 
magnetic field vector. 

B2 



4 CLASSICAL FOUNDATIONS I, 2 

molecule, whose velocity we shall suppose negligible. If the molecule 
is electrically neutral, the integral in (9) vanishes and the origin for r is 
then, as a matter of fact, immaterial in the first integral of (10). The 
expressions p and m defined by (10) are called the electric and magnetic 
moments of the molecule. The integrands of (9), (10), of course, vanish 
except where the element of integration falls inside an electron or 
nucleus of the molecule. In the conventional electron theory it is 
customary to think of the dimensions of the electrons and nuclei as 
negligible. Then the integrations may be replaced by a summation over 
all the discrete charges c t constituting the molecule, making 



Throughout the volume e L is used to denote a discrete charge of undeter- 
mined sign and magnitude, while e, without a subscript is used for the 
numerical magnitude 4-770 XlO~ 10 e.s.u. of the charge of an electron. 
Thus e t is equal either to e or -]~Ze according as the discrete particle 
is an electron or a nucleus of atomic number Z. The contribution of 
a particle to the magnetic moment is seen to differ from its angular 
momentum w/[r xv f ] only by a factor e,-/2w A c equal to half the ratio 
of its charge e t to the product of its mass m t and the velocity of light c. 
We shall later see (end of 33) that in many respects the time average 
of the electronic distribution in the new quantum mechanics can be 
treated like a classical 'smeared out' or continuously distributed charge 
pervading all space. Consequently the use of the integration (9), (10) in 
place of the summation (11) no longer seems an abstract academic 
refinement, as it did prior to Schrodinger's work. 

It will be proved in 3 that the expressions P and M defined in (7) 
are equal respectively to the average electric and magnetic moments 
p and m per molecule multiplied by the number N of molecules per 
c.c,sothat 



This immediately furnishes the desired correlation formulae for D and 
H, as by (6), (7), and (12), 

D e+47T#, H fi 



By the term average moment per molecule we mean the molecular 
moment averaged over all the molecules in a 'physically small' element 
of volume. This is equivalent to the time average moment for an 
individual molecule if all the molecules are alike except for phase. If 
they are of several different classes, i.e. form a chemical mixture, the 



I, 2 CLASSICAL FOUNDATIONS 5 

average denoted by the bars in (12) can be regarded as the mean of 
the time averages for single molecules of the various classes, weighted 
according to their relative abundance. In non-homogeneous media the 
term 'number of molecules per c.c.' is to be understood to mean the 
' number-density', i.e. the number of molecules which there would be 
in unit volume if they were distributed throughout a unit volume with 
the same density as that with which they actually are distributed in the 
immediate vicinity of the point at which the polarization is being com- 
puted. The information conveyed by Eqs. (12) must be regarded as 
a very important lemma on the physical or macroscopic significance of 
the mean molecular moments, as it interprets the distinction between 
D and E or between B and H in terms of simple properties (10) of the 
molecules. Because of their simple connexion (12) with the average 
moments of individual molecules, the polarizations P and M are often 
called the specific moments per unit volume. 

Eqs. (12) underlie all theories of dielectric or magnetic media, and hence 
are fundamental to the rest of the book. This concept of the polariza- 
tion of the molecule as the cause of the departures of e and //, from 
unity is by no means a purely twentieth-century concept, and was 
intimated by Faraday. In 1830 Mossotti 4 pictured the molecule as a 
conducting sphere of radius a, on which the charge would, of course, 
readjust or 'polarize' itself under the influence of an applied field, thus 
making the molecular moment different from zero. If the electric 
susceptibility X G ^ small compared to unity, he thereby showed that 
X e = Na^. It seems almost too hackneyed to mention that the values 
of a obtained from this simple equation (together with the observed N 
and Xe) are comparable in magnitude with the molecular radii in kinetic 
theory. This is illustrated by the following table, taken from Jeans's 
Electricity and Maynetism : 

Molecule Ho If a O 2 Ar N 2 CO CO., N a O C^ 

a (Mossotti) 060 0-92 1-17 1-18 1-20 1-26 1-40 146 160 X 10 - 8 cm. 

a (Kinetic Theory) 1-12 1-35 1-82 1-83 1-91 1-90 2-30 231 2-78 xJO- 8 cm. 

The agreement is remarkably good in view of the crude nature of both 
values of a, but some similarity in orders of magnitude is perhaps not 
so startling after all because a freely circulating swarm of electrons 
probably readjust themselves somewhat like the charge on a conductor, 
and the rigorous quantum theory formula to be developed later proves 
to involve the atomic diameter dimensionally in the same way as 

4 O. P. Mossotti, tfur les forces qui regissent la constitution intime des corps (Turin 1836). 
An account of his theory is given in Jeans's Electricity and Magnetism, p. 127. 



6 CLASSICAL FOUNDATIONS I, 2 

Mossotti's formula. Passing now to magnetism, Ampere's picture of 
a magnetic molecule as containing a continuous circulating current 
(instead of the more modern electron circumnavigating the nucleus) is 
well known, and Weber in 1854 was able to elaborate this Amperian 
concept to give the beginnings of a molecular theory of magnetic media 
just as did Mossotti for the electric case. We, however, shall prove 
Eqs. (12) with the aid of the more modern Lorcntz electron theory, 
even though these relations were suspected and to a certain extent 
established at earlier dates. 

To complete the correlation of the macroscopic and microscopic equa- 
tions we must state how the macroscopic and microscopic currents and 
charges are connected. We assume that the velocity of the centre of 
gravity of a molecule is negligible. The convection current then arises 
entirely from the migration of conduction electrons, rather than of 
molecular ions, and the current and charge densities are given by the 

expressions 

i - ~N c ev c , p = Nl^-Nj, (13) 

where N c is the number of conduction electrons per c.c., and v c is their 
velocity. Similarly N denotes the number of molecules per c.c., and 
l mol is the mean value of their net charge (9). The term Ne m(A could, 
of course, be written equally well as JV ion g ion , if now N ioa denote the 
number of ions (exclusive of free electrons) per c.c., and e ion be their 
average charge, which is much larger than the average molecular charge 
as most molecules are neutral. We thus regard the conduction electrons 
as distinct entities from the molecules. There is actually probably no 
such sharp cleavage between free and bound electrons, and the con- 
duction electrons may in reality be itinerant valence electrons which 
migrate from atom to atom, making transient stops at each. The use of 
such idealizations as perfectly free conduction electrons and stationary 
molecular ions does no harm as far as our investigations of dielectric 
and magnetic media are concerned. As a matter of fact it is possible 
to establish a statistical connexion between the macroscopic and micro- 
scopic equations even when the centres of the molecules are in motion. 
This has, indeed, been done by Lorentz. 5 It is, however, then necessary 
to complicate the macroscopic equations by the addition of 'convection 
terms' arising from the mass motion of the ponderable magnetic or 
dielectric media, and such considerations of the electrodynamics of 
moving media are unnecessary for our purposes. 

6 H. A. Lorentz, Encyklopedie der mathematiachen Wittsenschajten, Band V2, Hoft 1 , 
p. 200 ff. 



I, 3 CLASSICAL FOUNDATIONS 7 

3. Proof of the Preceding Correlation Formulae 6 ' 7 

This proof is probably most easily given by using the macroscopic scalar 
and vector potentials, <E> and A respectively, together with the analogous 
microscopic potentials <j> and a. From these potentials the electric and 
magnetic vectors are derivable by means of the formulae 

E--gradO-- , B^curlA; 

C 8t (14) 

1 da 

e ~ grad< --- , h = curl a. 
c ot 

The differential equations for determining the potentials are 

=-4 w [p-divP], nA==~~[i+ -1-ccurlMl; 

^ (15) 

= 477/3', 



together with the auxiliary conditions that 

. (16) 



, 

c dt C dl 

The symbol a denotes the d'Alembertiaii operator 



To prove (15) one substitutes (14) in the field equations (1 ), (2) or (4), (5). 
One finds that the first set of equations in (1) or (4) is identically 
satisfied by the substitution (14), while the second set and (2) or (5) 
yield (15) by a well-known procedure (viz. taking the curl of the equa- 
tions and using (16) and the identity curl curl A graddiv A V 2 A). 

fl The formulation and proof of the statistical correlation of tho macroscopic and 
microscopic equations is duo originally to Loreiitz. See, for instance, tho preceding 
reference. Our method of proof is, however, somewhat different from his, although both 
invoke tho aid of the scalar and vector potentials in the fashion (21 ). Wo use the Taylor's 
expansion (24) rather than a somewhat artificial comparison of positive and negative 
charge elements at tho same point. In this particular respect our treatment resembles 
that in Mason and Weaver, The Electromagnetic Field, though obtained independently. 
For still other proofs of the correlation see Abraham, Theorie der Elektrizitat, 4th cd., 
vol. ii, pp. 22438; Swann, Electrodynamics of Moving Media, pp. 44-.54J Fronkel, Lehr- 
btich der Elektrodynamik, ii, p. 10. 

7 Throughout the present section in considering the macroscopic equations wo do not 
include surface phenomena such as surfaces of discontinuity between two media, con- 
ducting surfaces, surface charges, &c. The surface terms could, of course, be added, 
but would only make the equations more cumbersome, and their omission involves no 
loss of generality, as surface discontinuities can always be regarded as limiting cases of 
continuous volume changes. 



8 CLASSICAL FOUNDATIONS I, 3 

The solutions of (15) are 

-ccur 



(I> = 

III K ' III t.H 

(17) 



where the brackets { } enclose functions that are to be evaluated at the 
retarded time tR/c, and where E denotes the distance from the ele- 
ment of integration dv to the point at which the potentials are being 
calculated. 

Proof of the Solutions (17). Tlio scalar equations and scalar components of the 
vector equations in (15) are all of the type form f~~10 - 47rq(x t y,z,t). Now an 
equation of this type form is identically satisfied by JJJte}/^^- Without 
giving a rigorous proof of this solution, we may note with Joans 8 that it becomes 
quite evident when we observe that i/t Q(tR/c)/.R is a solution of I lift -- 0, 
and corresponds to a point charge q(t) at the origin, as i/t becomes infinite there 
like Q(t)jB s the retardation effects disappearing on account of H 0. Similarly 
the solution corresponding to a series of charges Q t at various points distant R t 
from the point of observation is IS Qi(t-Ki/ c )/tti> ai i ( l passage from discrete point 
singularities to a continuous charge distribution yields the desired integral 
formula. 

It remains to show that the solutions (17) fulfil the auxiliary conditions (16). 
Wo shall consider only the microscopic case, as the macroscopic is analogous. 
We must prove that 

JJ7 -*171*+ J/J *?*- 

where the subscript P means that the differentiations involved in taking the 
divergence are with respect to the coordinates of the terminal point of the vector 
jK drawn from Q at dv to P, the point of observation. Similarly, the subscript Q 
will denote differentiation at the initial point. Now at P the expression {p'v/R} 
involves the coordinates of the terminal point P only through the denominator R 
and implicitly through R in the retarded time tE/c. On the other hand, this 
expression involves the coordinates x, y, z of the initial point Q through pv' as 
well as through M in the two fashions just described. Hence 



div,[<>] -div^J+^iv,^}, (U) 

where the brackets outside the div mean that the retarded value of the time is 
to be substituted after, rather than before, the differentiation. Now the equation 
of continuity or indestructibility of charge is 

(20) 

When wo substitute (19) in (18) and use (20) all terms cancel except that coming 
from the first right-hand member of (19), and this integrates to zero, since by 
Green's theorem the volume integral over all space vanishes if the integrand is 
the divergence of a vector which vanishes properly at infinity. 

8 J. H. Jeans, Electricity and Magnetism, 4th ed., pp. 571-2. 



I, 3 CLASSICAL FOUNDATIONS 9 

The desired statistical correlation of the microscopic and macroscopic 
theories is obtained by assuming that the macroscopic potentials equal 
the microscopic ones averaged over a 'physically small' (cf . p. 3) element 
of volume. This means that 

0--=?, A -a. " (21) 

Formulae (6) are direct consequences of (21) and (14), as it is easily 
established that the order of averaging and of space or time differentia- 
tion is interchangeable. 

The Intel-changeability of the averaging and time differentiation is obvious 
since the space and time coordinates are independent. To prove 9 the inter- 
chaiigeability with space differentiation take the 'physically small' region over 
which the average is evaluated as a sphere of volume with centre, at X Q , i/ , Z . 
Let x', y' 9 z' be coordinates with origin at this centre and let f(jc,y,z,t) be any 
function (including the components of a vector) which wo are interested in averag- 
ing. Now the macroscopic differentiations, i.e. differentiations after averaging, 
involve small virtual displacements of the centre of the sphere without changing 
the range of values of the coordinates x', y f , z' relative to the centre. Hence 



^rf^W::' while =1 f f f - *rV*'fc'. 

tAr J dx J J J dx' 



dx dr 9 J J J J J 6 

The identity of the two expressions is now an immediate consequence of the fact 
that the function/ is of the form/(# +^2/o+2/'~<H ~'0- 

It remains to show that (12) and (13) also follow from (21). The first 
step in doing this is to transform the formulae for the macroscopic 
potentials by means of the two following vector identities : 10 > n 



(22) 

fff/i i\ rrr M rrnxM 

JJJ^curlM-Mxgracl^^ = JJJcurl-^-=JJ _ W, 

in which n denotes a unit vector along the exterior normal to the 
surface clement d/S. The surface integrals disappear, as we may sup- 
pose the magnetic and dielectric matter bounded in extent, so that 
P M on the surface of a sufficiently great sphere. Thus the first 
part of (17) becomes 






9 This proof is taken from Fronkel, Lekrbuch dcr Elektrodynatnik, ii, p. 4. 

10 For proof of the second identity of (22), which is not a particularly common one, 
see Abraham, Theorie der Elektrizitat, 4th od., vol. i, p. 76. 

11 All the differentiations in l3q. (22) et seq. are to be taken at the element of integra- 
tion dv, but for brevity wo no longer write in the subscript Q used in (19), (20). 



10 CLASSICAL FOUNDATIONS I, 3 

We must now throw the microscopic formulae for < and a into a form 
somewhat analogous to (23). Let us consider the portion of the micro- 
scopic potentials which results from integration over a single molecule, 
indicated by the subscript 1M . The radius vector R from Q, the position 
of an element (of integration) of the molecule to the point of observa- 
tion P, is the vector-difference of the radius vector R from the centre 
of gravity of the molecule to P and the radius vector r ~ ix+ly+Va 
from tin's centre to the given element. As r/R is small, we may develop 
l/R in a Taylor's series: 



Thuby(17) /v I (24 > 



Using the definitions (9) and (10), and the fact that E Q is constant with 
respect to the integration, we now see that 

A convenient formula for a analogous to (26) is obtained only after 
a certain amount of juggling involved in using the relations 

dP ff f V J f ff/V / N 7 fff / 7 /<VT X 

^ _L_rw> (divpv)rai; pvdv, (27a) 

M J J J dt JJJ JJJ 

= J J I (div/o'vjr/r-grad \dv 

r 1 1 r r r r / i \ / i\i 

2c inxgrad-- = \p vlr-grad rlvgrad \\dv 9 (27c) 
I. AJ JJJ I \ ^V \ ^o/J 

where k -.--. [[ ( p'rfr- grad \ rfw, (28) 

and where p and m are defined as in (10). The velocity v in (10), to 
be sure, was defined as relative to the centre of gravity of the molecule 
rather than relative to a fixed system of reference as in (17), but this 
distinction is of no consequence, as we have supposed the velocities of 
the centres of gravity of the molecules to be negligible. The inter- 
mediate forms of (27 a, b) follow from the equation of continuity (20), 



I, 3 CLASSICAL FOUNDATIONS 11 

and the final forms by partial integration. 12 Eq. (27 c) is obtained from 
the second expression of (10) by using the vector identity 

Ax[BxC] = B(A-C)-C(A-B). 
After use of (27 a, b, c), the second equation of (25) may be written 



We have so far considered only the contribution of one molecule. 
Actually we desire the total contribution of all molecules in a 'physically 
small' volume element dv. This total contribution is the average con- 
tribution of one molecule multiplied by the number Ndv of molecules 
in dv. The term 'average' as here used means the space average over 
the different molecules in dv, but if the various molecules differ only 
in phase, it is the time average for a single molecule over a very long 
time interval. If there are several classes of molecules it may be con- 
sidered the weighted mean of the time averages for the different mole- 
cules. When such an average is made, the last term in (29) becomes 
negligible, for 0k/fl=[k]/( 2 y, and this is exceedingly small if 
J 2 *i is made very large and if k remains bounded in magnitude, as it 
will in virtue of the definition (28) (certainly at least in periodic or 
multiply periodic phenomena). Similar considerations would also per- 
mit the omission of the first term of (29), but this is unnecessary for 
the establishment of the correlation, and if the assumption of haphazard 
phases is not fulfilled the omission of the first term of (29) would be 
a more serious offence than that of the third, as it is of lower order in 
I/ R. Having found the contribution from dv, we must next sum over 
all the 'physically small' volume elements dv. Without appreciable 
error this summation may be replaced by a macroscopic integration, 
to be carefully distinguished from the previous microscopic integration 

12 The formulae obtained by partial integration seem to be most easily verified by 
writing out the components of the integrand in scalar notation rather than by manipula- 
ting the integrand by the appropriate vector identities, which would bo rather compli- 
cated. The simplicity in the scalar method results from the fact that the components of 
r are merely x t y, z. The integrand of the x component of (27 a) before the partial integra- 

tion for instance, is # ( -^~ + - Vy + (* M , and the final form is obtained by integrating 

the first, second, and third terms of this integrand partially with respect to x, y, z 
respectively. The surface integrals in the partial integration of course disappear for the 
same reason as in (22). The necessity for the partial integrations in (27 a, b) and also 
in (18) can bo avoided if one uses point singularities rather than continuous distributions, 
i.e. (11) rather than (9), (10). If this is done, it is convenient in performing tho time 
differentiation to travel with a particle rather than remain at a fixed element dv, so that 
the charge is to bo regarded as independent of t, whereas R is to be considered as varying 
with t in the fashion \-dRfdt. 



12 CLASSICAL FOUNDATIONS I, 3 

over one molecule. In this new integration, expressions such as jR , p, m 
may be considered to vary continuously with the macroscopic co- 
ordinates fixing dv, even though these expressions were previously 
defined in a discrete fashion. 13 As all parts of molecules in dv are, from 
a macroscopic standpoint, located at virtually the same distance from 
the point of observation P, the subscript may be omitted from R . At 
the same time that we perform this macroscopic integration we must 
add in the contribution of the conduction electrons, for which the 
Taylor's expansion is unnecessary, as their dimensions, and hence their 
moments, 14 may be considered negligible. In this way we find that 



dv + ^. gr ad- cfe, 



f - / JJ 

1= JJJ ( ---^/* 1 * +JJJ (*Sxg4) fe. (30) 

The desired results (12) and (13) now follow on using (21) and comparing 
the structure of (23) and (30). 

The reader has perhaps noticed that all the foregoing correlation of 
the microscopic and macroscopic electromagnetic theories is only 
approximate, due to neglect of terms beyond l/R 2 in the Taylor's 
expansion. Such terms arc usually unimportant 15 because molecular 
radii are small compared to distances of observation. Otherwise the 
ordinary macroscopic equations would presumably not be found to be 
valid experimentally. The omitted terms are sometimes characterized 
as representing 'multipoles'. The omitted term in <f> of lowest order, 
for instance, is readily shown to be Nq/E*, where q is the 'quadrupole 
moment' 



13 Ono is perhaps a bit solicitous about tho error incurred in treating JK, p, m, &c. 
as continuous functions in viow of tho fact that molecules are discretely and irregularly 
distributed, especially in a gas. Statistical theory shows that the root mean square 
deviation of the number of molecules in dv from its mean value Ndv is (Ndv)l. Tho 
relative error involved in ironing out tho fluctuations and substituting a continuum is 
thus at most of the order (IfNdu)*, and requires that dv be not too small or the medium 
too rarefied. For this reason dv cannot well be reduced beyond a certain value dv , but 
then there is a relative error of the order (dv )k/R in substituting an. integral for the sum. 
As N is of tho order 10 lfl in gases, arid as R is 1 cm. or greater in most macroscopic 
problems of interest, both (l[Ndv)i and (dv)bfR may be made small by taking dv about 
10- 12 . For further discussion of the legitimacy of tho substitution of macroscopic integra- 
tions for summations see Mason and Weaver, The Klectromaynctic Field, Chap. I, Part III. 

14 Wo do not introduce tho spinning electron, with its finite magnetic moment, until 
the quantum chapters. 

15 Retention of these higher order terms is, however, vital in molecular theories of 
the equation of state. 



I, 3 CLASSICAL FOUNDATIONS 13 

6 being the angle between r and R. The terms of highest order in l/R 
which have been retained are of order 1/JK 2 , and are the 'dipole' ones, 
as the potential due to a discrete dipole of vector moment P is 

P-gradi. (31) 

Instead of tracing through the details of the connexion with the micro- 
scopic theory, it is seen by direct comparison of (23) and (31) that the 
term in O which is the correction for dielectric action can be interpreted 
as due to a continuous space distribution of dipoles of specific moment 
P per unit volume. This explains why the existence of molecular 
dipoles in dielectrics could be inferred in the nineteenth century before 
the advent of the Lorentz electron theory. Similar remarks apply to 
the magnetic case, as the vector potential of a discrete magnetic dipole 
of moment M is Mxgrad(l/jR). 

4. Relation between Index of Refraction and Dielectric Constant 

It is a matter of common knowledge that the index of refraction n is 
connected with the dielectric constant e and magnetic permeability \L 
according to the relation ^ 2 _ /32) 

provided we use the term dielectric constant, as we already tacitly have 
throughout the chapter, in a generalized sense to mean the ratio of 
D to E in periodic fields rather than in the restricted sense to denote 
just this ratio in static phenomena. Now gases of a high refractivity 
usually show very little magnetic polarization, so that without much 
error we may take /*= 1. In other words, in gases the magnetic sus- 
ceptibility is usually small compared to the electric, and then (32) 

becomes % a = . (33) 

Discussion of the experimental confirmation of (33) will be deferred until 
Chapter 111. The theoretical validity of (32) can be seen as follows. Let us sup- 
pose that the radiation is monochromatic, and that the medium is homogeneous 
and infinite in extent. This is somewhat of an idealization, as a dielectric or 
magnetic medium never extends to infinity, but in a sufficiently large homogeneous 
medium the velocity of propagation of disturbances is virtually the same as though 
the medium extended indefinitely. In such an ideal medium e and /A may be 
regarded as constants independent of x, y, z, t, so that 

dV/dt = edE/dt, curl B - /z curl H, &e. 

Jf wo make the substitutions E' = *E, H' - /^H, p = p/e*, i' ^ i/e*, which may 
bo regarded as changing the scale of measurement of unit magnetic and electric 
poles, the field equations (1, 2) reduce to 

curlE'- --/?-', curlH' -= if^rri' I ^-X divE' - 4izy>', divH' - 0, 
c dt c \ dt ) r 



14 CLASSICAL FOUNDATIONS I, 4 

provided c' = c/fe//,)*. These are equations of the same type form as for a vacuum 
except for the change of scale just noted, and for the fact that the effective 
velocity of propagation is c/(e/Lt)*. Now the index of refraction n is defined as the 
quotient of the velocity of propagation c in vacuo to that c' in the material 
medium hence Eq. (32). 

5. The Local Field 

It is desirable to know the effective average field to which a molecule 
is subjected when a macroscopic field E is applied. This effective field 
is not the same, even in the mean, as the macroscopic E despite the 
fact that by (6) the vector E is the space average of the microscopic 
field e over a 'physically small' volume element. The explanation of 
this paradox is that the effective field in which we are interested is that 
in the interior of a molecule, whereas the space averaging presupposed 
in the relation E = e is over regions both exterior and interior to 
molecules. The effective field within a molecule may be resolved into 
two parts: first, the internal field exerted by other charges within the 
same molecule, and second, the remainder due not only to the applied 
electric field but also to the attractions and repulsions by other mole- 
cules, usually polarized under the influence of the external field. The 
first part is not our present concern, and could, of course, be calculated 
from the Coulomb law immediately we knew the configurations of the 
charges in the molecule. The second part we shall term the local field. 
Under certain simplifying assumptions the local field can be shown to be 

e loo =E+^ P . (34) 

The expression (34) is sometimes called the Olausius-Mossotti formula 
for the local field. We shall not prove (34), as this is tedious and is 
frequently done in the literature. 16 The usual derivations assume that 
the moleciile in question can be considered to be located at the centre 
of a spherical cavity in the dielectric, and that further the diameter of 
the cavity is large compared to the size of a molecule. The reader will 
recall that the field within a cavity is a function of the shape of the 
cavity. It has the value (34) at the centre of a spherical cavity, whereas 
it equals E within a needle-shaped cavity whose axis is parallel to E, 
and D E+4?rP within a slab-shaj)ed cavity whose surfaces are per- 
pendicular to E. The term 4?rP/3 in (34) is sometimes spoken of as the 
* inter-molecular field', and is a correction for the fact that the other 
molecules of the dielectric exert an average force on the given molecule 
when the dielectric is subject to an electric field. This is due, of course, 
16 Of., for instance, H. A. Lorentz, The Theory of Electrons, section 117 and note 54. 



I, 5 CLASSICAL FOUNDATIONS 15 

to their acquisition of an electric polarity, so that the resultant field 
which they exert at the centre of a spherical cavity no longer vanishes 
on the average as it would by symmetry (at least in isotropic media) 
without an external field to create a preferred direction. We here, as 
elsewhere in the book, assume that there is no residual polarization 
when the applied field vanishes. Otherwise it would be necessary to 
add a constant term to the right side of (34). Residual or permanent 
polarization of a dielectric, which would be the electric analogue of 
'hard magnetism', is not an unknown phenomenon, but is usually feeble 
and found only in complicated solids, a discussion of which is beyond 
the scope of the present volume. 

An important and simple experimental confirmation of (34) is furn- 
ished by the variation of the index of refraction with density in gases. 
At ordinary field strengths the average electrical moment may be taken 
proportional to the local field: 



p = e, 00 = p+ J, (35) 

where a is a constant independent of the density. On elimination of 
p, P by means of (8), (12) and use of (33) the relation (35) becomes 



The number N of molecules per c.c. is proportional to the density p. 

Hence IB ,_J ^ Nn ,.,,., 

- ---=- , a constant independent of density. (36) 
p n -\-2 3/o 

This is the so-called Lorenz-Lorentz formula, as it was proposed in- 
dependently by L. Lorenz and H. A. Lorentz, both in 1880. 17 The most 
thorough experimental test of (36) has perhaps been made by Magri, 18 
who varied the density by using pressures of from 1 to 200 atmospheres. 
According to Magri's data, the left side of (30) has the following values 
at different densities: 

Density 1 14-84 42-13 69-24 96-16 123-04 149-53 176-27 

l^^XlO 7 1953 1947 1959 1961 1961 1956 1956 1953 
p n 2 -|-2 

-7-XlO 7 (1953) (1953) (1976) (1988) (1998) (2005) (2015) (2023) 
P > 

17 H. A. Lorentz, Ann. Phya. Chcm. 9, 641 (1880); L. Lorenz, ibid. 11, 70 (1880). 

18 L. Magri, Pliys. Zeitft. 6, 629 (1905). The various measurements given in the 
above table wore all made at 14-15 C. except that the reading at unit density is at C. 
Thus practically all the change in density is from varying the pressure rather than 
temperature. 



16 CLASSICAL FOUNDATIONS I, 5 

The unit of density is the density of air under standard conditions. The 
values in parentheses are those which would he obtained by taking 
the local field to be identical with the macroscopic field E, thereby 
making the denominator 3 instead of n 2 +2. It is seen that the rigorous 
or first formula yields values which are much more nearly constant than 
with the second formula, thus confirming the correction term 4?rP/3 for 
the inter-molecular field in (35). Formula (36) as applied to dielectric 
constants (i.e. with n 2 replaced by e) has also been verified for several 
gases up to 100 atmospheres by Tangl and others. 19 

A much more severe, in fact unreasonably hard, test of (36) is 
obtained by examination of whether its left side has the same value in 
the liquid and vapour states. The agreement in the two states is sur- 
prisingly good in view of the fact that (36) cannot be expected to hold 
accurately in liquids because of association effects and the like. Tn 
water, for instance, it is found that at optical wave-lengths the left side 
of (36) changes by only about 10 per cent, in passing from the liquid 
to the vapour state, whereas the density changes by a factor over 1,000. 
In the static case of infinite wave -lengths, however, all traces of agree- 
ment between the two states are lost in polar materials like water. For 
non-polar substances the change in (36) between the two states is 
practically negligible (see p. 59 for numerical data). 

Our primary concern is gaseous media, and at any ordinary pressures 
(up to 10 atmospheres or more) the term n?-\-2 in the denominator of 
(36) can be equated to 3 in gases, thereby making the left side of (36) 
become ^TrxJSp (cf .Eq. (8) ). Consequently we shall, for simplicity, hence- 
forth make this approximation throughout the volume, unless otherwise 
stated. If, however, one should wish at any time to make the above 
'Clausius-Mossotti' correction resulting from the difference between the 
macroscopic and local fields, one has only to substitute 3(e -l)/4?r(e f-2) 
for the susceptibility XG m * ne formulae for susceptibilities given 
throughout the book. This remark applies to all formulae except those 
which relate to saturation effects and which thus do not take the 
polarization as proportional to field strength. In the study of liquids 
the correction for the difference between the local and macroscopic 
fields is important. In some instances a formula such as (36) seems to 
be applicable to liquids, but our feelings must not be hurt when we 
discover that (36) fails completely, at least in the static case, in the 

19 Tangl, Ami. der Physik, 10, 748 (1903); 23, 559 (1907); 26, 59 (1908); also Occhia- 
lini, Phys. Zeits. 6, 669 (1905) ; Waibel, Ann. der Physik, 72, 160 (1923) ; Ocehialini and 
Bodareu, Ann. der Physik, 42, 67 (1913); K. Wolf, Phys. Zeits. 27, 688 and 830 (1926). 



I, 5 CLASSICAL FOUNDATIONS 17 

so-called polar liquids where agglomeration or association effects arc 
important. The formula (34), on which Eq. (36) is based, is derived 
on the assumption that the arrangement of molecules is haphazard and 
that the inter-molecular distances are large compared to molecular 
diameters. These assumptions are clearly not legitimate in such polar 
liquids. 

We have so far considered only the local electric fields. Under 
the same assumptions as are made in the usual derivation of (34), the 
local magnetic field can be shown to be h local -^ H+(47rM/3). The 
expression (/u, l)//>(ju,+2) should then be independent of the density. 
However, in gases the permeability is so nearly unity, and magnetic 
measurements are so difficult that there is no experimental material 
adequate to test whether ju-+2 should occur in place of 3 in the de- 
nominator. On the other hand, in solid magnetic bodies local fields are 
apparently encountered which are tremendously larger than H + (47rM/3) ; 
otherwise we would never have ferromagnetism, and the cause for this 
will be discussed in Chapter XII. Thus in the magnetic case a correction 
of the type considered above in the electric case is either negligible or 
inapplicable. Consequently we shall not go to the refinement of making 
this correction and writing 3(//, l)/47r(j(x+ 2) m place of x m m ^ ne formulae 
for the paramagnetic and diamagiietic susceptibilities given later in 
the book. 

6. The Force Equation 

To formulate a dynamics governing the motion of the charges within 
the molecule it is first necessary to have an expression for the forces 
exerted upon them. This is furnished by the fundamental 'force 
equation' 

F, = e,e+*[v ( xh], (37) 

C 

postulated by the electron theory as the vector force F 4 - exerted on 
a charge (electron or nucleus) moving with the velocity v^ and subject 
to electric and magnetic fields e and h respectively. We here and for 
the balance of the chapter regard the dimensions of the electrons and 
nuclei as negligible, so that they can be treated as point charges. The 
total fields e and h may be resolved into two parts, viz. the internal or 
'intra-molecular' portions exerted by other charges in the same mole- 
cule, and the remaining or external parts exerted by the rest of the 
universe. 

If we neglect retarded time effects, the internal fields at the point 

3595.3 r 



18 CLASSICAL FOUNDATIONS I, 6 

occupied by the charge e i can be derived in the usual fashion (14) from 
the scalar and vector potentials 



(cf. Eq. (17)), where r^ denotes the distance between the charges e t and 
j. It is seen from Eqs. (14) and (37) that the terms in the force which 
arise from the internal vector potential are of order 1/c 2 in the velocity 
of light c, even though the vector potential itself is of order 1/c. Terms 
of this order are usually very small, and, in fact, are comparable with 
the relativity corrections and retardation effects in the scalar potential 
^, so that for our purposes we are fully warranted in dropping them, 
and it would be deceptive to include them without including the other 
effects of the same order. With their neglect the internal forces become 
entirely electrical in nature and can be expressed in the simple fashion 

F^-grad/F (38) 

in terms of the electrostatic potential energy function 



(39) 

as e t .grad<^ int grad^ V. Here the subscript in grad t indicates that the 
differentiation in the gradient operator is with respect to the coordinates 
x l9 y it z i of the charge e^ Use of V instead of the </ llt has the great 
advantage that it is necessary to use only one potential function for 
the whole molecule rather than one for each charge. 

The external part of the field will fluctuate with the approach and 
recession of other molecules, especially at the time of the so-called 
collisions of kinetic theory. However, we have mentioned in the pre- 
ceding section that in gases the average values of the external fields 
are equal to the expressions for the local fields given in (34) (and an 
analogous magnetic expression). As gases usually have small polariza- 
tions, we agreed to neglect the differences between the local fields and 
the macroscopic fields. The combined internal and external force can 
therefore be considered to be 

F = -grad,F-e,grad*,-l' ^+%<XcurlA t .J, 

C Ot C 

where O , A^ are the macroscopic scalar and vector potentials evaluated 
at the point x { , y iy z if at which the charge e i is located. If we equate 
force to mass times acceleration, the explicit formula in scalar notation 



I, 6 CLASSICAL FOUNDATIONS 19 

for the x component of motion is, for instance, 



to, to c a 

(40) 
7. The Lagrangian and Hamiltonian Functions 

The differential equations of motion (40), together with the analogous 
y and z equations, are equivalent to the Lagrangian system of equations 

d8L_8L_ 

~ 



provided we take the Lagrangian function to be 



The number of coordinates q k is, of course, three times the number 77 of 
particles constituting the atom or molecule. The identity of (40) and 
(41) can be established immediately by specializing the q k to be Car- 
tesian coordinates JT I ,..., a* 7y , y ,,..., y^ Z L ,..., z 1} \ then 



(42) 



With these relations Eqs. (41) follow directly from (40). The last part 
of (42) expresses the fact that in the total differentiation with respect 
to /, the vector potential A irmst be siipposcd to involve the time not 
only explicitly through t but also through the position x, y, z. As the 
Lagrangian form of the equations of motion is preserved under any 
point transformation, Eqs. (41) will also be valid in any system of 
generalized coordinates since they are valid for Cartesian coordinates. 
If we introduce the generalized momenta 

8L tA ~^ 

**=^' (43) 

and if we express the function 



in terms of the q k and p k rather than the q k and q k , then Jf is called 
the Hamiltonian function. To avoid confusion with the magnetic field 
H, we throughout print the Hamiltonian function in script type. It 

C2 



20 CLASSICAL FOUNDATIONS I, 7 

is immediately seen that in Cartesian coordinates 



<Pi (44) 
as in the Cartesian system by (41 a), (43) 

#r, = "W+*:f4r.' ( 45 ) 

o 

General dynamical theory 20 tells us that the 817 second-order equations 
(41) are equivalent to the 677 first-order Hamiltonian equations 

rf& itJt dp_ t Mf , 

dt~dp k ' ',11 >,' ( > 

Without appealing to the general theorems of dynamics, the identity 
of (46) with (40) or (41) can immediately be established by writing out 
the equations (40) explicitly in Cartesian coordinates, where <& lias the 
form (44). The first set of the Eqs. (46) then give us the definitions 
(43) of the momenta. When these definitions are substituted in the 
second set of Eqs. (4(>), we immediately obtain (41) if in the term 
dpy.Jdt^m^x^dt^-^e^IA^Jcdt we remember to introduce the total I 
derivative after the fashion (42). 

The preceding equations are rather more general than needed for 
most of our work, as they apply to variable and non-homogeneous 
impressed fields, which can be simultaneously electric and magnetic, 
whereas our main interest is in constant fields, if we assume that the 
applied electric and magnetic fields both have the z direction and have 
constant magnitudes E and II independent both of time and position, 
then 21 

<D--#z, A x ^ -\Hy, A^lHjc, J^-0, (47) 
and (44) becomes 




... 
(48) 

20 E. T. Whittaker, Analytical Dynamics, Chap. x. 

21 The choice of potentials in (47) is not the only one which will yield the doairod con- 
stant fields E, H. We could, for instance, take A x = 0, A v = fix, A z ^= 0. The latter 
choice would, however, make the Hamiltonian function havo a more complicated and 
less illuminating form, although it would still equal the energy (51). The choice (47) is 
the simplest, probably because it makes the scalar magnitude of A independent of a 



I, 7 CLASSICAL FOUNDATIONS 21 

That (47) gives the potentials appropriate to this special case can im- 
mediately be verified by substitution in (14). Even the case of parallel 
coexistent constant electric and magnetic fields is more general than 
usually needed, as we usually deal with entirely electric or entirely 
magnetic applied fields. Tt is, however, convenient to include both 
cases in a single equation. The Hamiltonian function appropriate to 
a purely magnetic constant field // can, of course, be obtained by 
setting E in (4H), and vice versa. 

Our main interest throughout the book is in the electric and magnetic 
moments p, m defined in Eqs. (11), and especially in their components 
p E , m H in the direction of the applied field. 22 Fortunately these are 
obtainable from the Hamiltonian function by simple differentiations, as 

a# a# 

f*=-7jp "H"?;//' (49) 

The first of these relations is immediately obvious from (48), as by (11) 
PK ^efii' The second is only a trille more difficult to establish, as 
in Cartesian coordinates 



= 2 to ^ ^~^ ^ w * (50) 



by (45), (47), (11). The relations (49) will apply not merely in Cartesian 
coordinates, but also in any set of 'canonical coordinates' that preserve 
the Hamiltonian form (46), provided the formulae of transformation 
from the Cartesian to the new coordinates and momenta do not involve 
the fields E or //. This proviso is necessary in order that holding fast 
the Cartesian coordinates and momenta in the*, differentiation be 
equivalent to holding fast the new ones. It is met in the case of the 
usual 'positional coordinates' and momenta defined by an 'extended 

rotation around tho z axis. The lack of uniqueness in the definition of A, <D, arid. so of 
the Hauiiltonian function in electric and magnetic Holds occasions 110 real diih'culty, as 
tho difi'nront choices inoroly yield different fully equivalent canonical forms which can 
bo obtained from each other by a contact transformation (and in somo cases a shift of the 
origin for tho energy). 

22 Wo use the notation p K , m H rather than p z , w 2 for the components of tho moments 
p, m in tho direction of tho applied fields in order to avoid confusion with the Cartesian 
momenta p e . Tn writing our Hamiltonian functions wo use a scalar rather than vector 
notation for tho electric and magnetic fields. This is something of a change from the 
vector notation used in the preceding sections, but tho various vector relations are usually 
of no particular interest in our use of tho TTamiltonian technique, and so it seems simpler 
to restrict E and H to directions along the r. axis rather than to complicate our Hamtl- 
tonian functions with scalar and vector products. 



22 CLASSICAL FOUNDATIONS I, 7 

point transformation' 



as such a transformation obviously does not depend on the fields E or 
IT in any way. 

The numerical value of the Hamiltonian function (44) or (48) is the 
same as that of the ordinary total, i.e. kinetic plus scalar potential 
energy T+ ,, + ^ ^ = j ^ w . v , +r+ ,^, (51) 

as can he verified hy substitution of (45) in (44). The reason why the 
energy formula (51) has no terms containing the magnetic field H is, 
of course, that magnetic forces do no work, so that the constancy of 
energy can be secured without the addition of such terms. This may 
at first seem contrary to the fact that the Zeemaii effect shows that 
the energy of an atom is not the same in a magnetic field as in its 
absence, but the solution of the paradox is that during the creation of 
the field, when dH/dt -/-- 0, the field equations (1) demand that there be 
a concomitant electric field, which does work just sufficient to account 
for the energy displacements in the field. It may also seem strange that 
the field H is involved in (48) but not in (51). This difference is due to 
the fact that in a magnetic field the canonical momenta necessary to 
preserve the Hamiltonian form are f)L/^q k and not dT/dq k , i.e. in Car- 
tesian coordinates are w^-j-e^^/c and not fw^ (cf. Eq. (45)). This 
distinction is nicely illustrated in Larmor's theorem. 

8. Larmor 's Theorem 23 

This theorem states that for a monatomic molecule in a magnetic field 
the motion of the electrons is, to a first approximation in //, the same 
as a motion in the absence of // except for superposition of a common 
precession of frequency He/47rmc. This may be proved as follows. Let 
us specify the Z electrons of the atom by cylindrical coordinates p^ z ?> 
</)}, whose axis is the direction of H and whose origin is at the centre of 
gravity. We suppose the nucleus to have such a large mass that it can 
be regarded as coinciding with the centre of gravity. The Lagrangian 
function is by (41 a) and (47), supposing E = 0, 



j^ 

i-l 
In place of the </>'s, let us introduce new coordinates: 

7l = <l>l> 72 = ^2 0J 73 = ^3 ^ 

23 J. Larmor, Aether and Matter, p. 341. 



I, 8 CLASSICAL FOUNDATIONS , 23 

The Hamiltonian function is then 



since all the electrons have the same value e/m of ejm it and since 



(54) 

Now the potential energy F(p 1 ,...,2J 1 ,...,y 2 ,...) does not involve y 1? as it 
depends only on the relative positions of the coordinates. Thus dJ^l^y l 
vanishes, making y x an 'ignorable coordinate', andy x hence has a con- 
stant value independent of t (cf . Eq. (46) ). If we disregard terms in H 2 , 
then with any given value of p yi the equations of motion for the other 
3Z 1 pairs of canonical variables are precisely the same as in the 
absence of the field. Hence, neglecting quadratic terms in H, there 
exists a solution in which p v ...,z l ,...,y 2 ,... are the same functions of the 
time as in the absence of the field. However, the angular velocity 

y x = dJf/dp yi is now ---- - t \p Yl ^ p Yt \ + -^ . The addition of the term 
m iPi\ fr{ / ^ nc 

involving // means that with a given value of p yi and a given solution 
for the other 3Z 1 pairs of canonical variables, the value of the angular 
velocity of electron 1 is augmented by an amoimt He/2mc. As the 
angular displacements y 2 ,... of the other electrons are measured relative 
to an apse line containing electron 1, it follows that the entire system 
processes about the axis of the field with an angular velocity He/2mc. 
The corresponding frequency of precession is, of course, Hej^irmc. Thus 
Larmor's theorem has been established, but it has been necessary to 
(a) disregard squares of the field H, (b) neglect the motion of the nucleus, 
and (c) suppose the molecule to be moiiatomic, as it was assumed in 
(53) that all particles not on the coordinate axis have a common value 
ft/ m f e i/ m i- J^ i g seen t na t t ne Larmor precession is essentially a 
correction for the different significance of the canonical momenta in 
terms of velocities with and without the field. It can be shown 24 that 
if the magnetic field is applied slowly and uniformly, then p yi does not 

24 P. A. M. Dirac, Proc. Cambr. Phil. Soc. 33, 69 (1926). 



24 CLASSICAL FOUNDATIONS I, 8 

change, i.e. is an 'adiabatic invariant 5 during the creation of the field, 
whereas the ordinary angular momentum 2 m ipl<l>ii which differs from 
(54), is not. When the field is thus adiabatically applied, the motion in 
the field can be proved 24 ' 25 the same (neglecting II 2 ) as the actual 
motion before the existence of the field, except for the superposition 
of the Larmor precession. We proved above merely that it resembled, 
except for this precession, some dynamically permissible motion for 
H = 0, not necessarily the actual original motion. 

9. The Fundamental Theorem of Statistical Mechanics 

The theoretical formulae for the electric or magnetic susceptibilities, 
which result from substitution of (12) in (8), involve the average 
moments p or in of a large number of molecules not necessarily all in 
a similar condition. Such averages can be calculated if we know the 
statistical 'distribution function' or probability that the molecule have 
any particular configuration. To determine such a distribution function 
is a problem in statistical mechanics. A probe into the foundations of 
classical statistical mechanics 26 is beyond our scope, and we shall be 
content to give the fundamental result without proof. Let q l9 ... t q^ 
Pi,,pf be any set of 'canonical' coordinates and momenta, i.e. a set 
which satisfies Hamilton's equations (46) and which suffices to determine 
the positions and velocities of the various particles composing the mole- 
cule. Then the probability that a molecule be in the configuration 



is Ce-^Uq^.dq f dp v ..dp 1 , (55) 

where (q v q^+dq^) means that q l falls between q l and q-\-dq^ &c., and 
where M is the Hamiltonian function. This is the so-called Boltzmann 
distribution formula. The value of the constant C is determined by the 
requirement that the total probability be unity, and hence 

i = J... J e-Wdq v ..dq f d V i...d Pl . (56) 

An immediate corollary of (55) is that the average value of any function 
/ of the p's and q's is 

/- C J... J/e-W d^.^dp^dp,. (57) 

Most often we shall want the function / to be either the electric or 
magnetic moment defined in Eq. (11). Thus by (8), (12), and (49) the 

25 J. H. Van Vleck, Quantum Principles and Line Spectra, p. 303. 

26 See, for instance, R. II. Fowler, Statistical Mechanics, Chap. II. 



I, 9 CLASSICAL FOUNDATIONS 25 

electric susceptibility is 

P Np E NC C CdJt MkT , , , KQN 

*^E = ~f - ~ E J " j M e ! dq -~ dp f> (58) 
with an analogous formula for the magnetic susceptibility. This result 
may also be written 

?i ...dp A . (59) 



The expression Z is called by Darwin and Fowler the 'partition func- 
tion', and is also sometimes termed the 'sum of states', as a translation 
of the name 'Zustandssumme' which it is given by German writers. 

A feature particularly to be emphasized in connexion with Eqs. (55)- 
(58) is that it is immaterial how we choose the coordinates and momenta 
#!,..., Pf as long as they be a 'canonical set' satisfying Hamilton's equa- 
tions (46). They might, for example, be Cartesian, polar, or parabolic 
coordinates and momenta, or the more general 'action and angle 
variables' used in perturbation theory. This indifference of (55)-(58) to 
the choice of coordinates is due to the well-known fact that any contact 
transformation, 27 i.e. any transformation which preserves the validity 
of the Hamiltonian form (40), has unit functional determinant. Hence 
dq 1 ...dp f = dQ l ...dP /9 (60) 

where p, q and P, Q are respectively the old and new variables. 

It can be objected that the derivations of the Boltzmann distribution 
formula (55) given in statistical mechanics are not without loopholes 
if one looks hard enough for them. The fundamental assumption made 
to obtain (55) is that the 'picking out' of numerical values for co- 
ordinates and momenta for any particular molecule is a lottery or 
random proposition, subject only to the requirement that the total 
energy of an assembly of molecules (e.g. box of gas) has a given value. 
On this basis it can be shown that the Boltzmann distribution is not 
merely the most probable, but that it is infinitely the most probable, 
and hence is what Jeans terms a 'normal property' of the system. 28 
Many of the common brief treatments of statistical mechanics, to be 
sure, merely calculate the most probable configuration without de- 
monstrating its overwhelming probability or normal property, but 

27 W. Gibbs, Elementary Principles in Statistical Mechanics, Chap. I (extended point 
transformations only); A. Szarvassi, Ann. dcr Physik, 60, 501 (1919). 

28 According to tho so-called 'orgodic hypothesis', which supposes that all assemblies 
having tho saino energy have the samo life-history, tho Boltzmann formula would be 
inescapable. Unfortunately, Planchnrel and also Roscnthal have demonstrated the 
impossibility of orgodic systems (Ann. dcr Pliyaik, 42, 796 and 1061, 1913). 



26 CLASSICAL FOUNDATIONS I, 9 

recent very complete and elegant work of Darwin and Fowler 29 shows 
very convincingly that the Boltzmann distribution is always a normal 
property. Thus this distribution is inescapable unless we assume that 
nature has a peculiar preference for configurations which would be 
abnormally improbable in the lottery described above. 30 As long as we 
retain a classical theory with continuous distributions rather than use 
a discrete quantum mechanics, we can therefore feel reasonably safe in 
resting our calculations of the susceptibility on (59). 

29 C. G. Darwin and R. H. Fowler, Phil. Mag. 44, 450 and 823 (1922) and several later 
papers, especially by Fowler. A detailed account is given in Fowler's book Statistical 
Mechanics. 

30 The Pauli exclusion principle and Fermi-Dirac statistics in the now quantum 
mochanicfi do indeed show that only configurations col-responding to antisymmetric wave 
functions are realized physically despite the fact that they oftentimes represent but 
a small fraction of the classical phase space. This restriction, however, is usually un- 
important for our purposes, as we shall deal mainly with media sufficiently rarefied to 
make the Fermi-Dirao interference effects unimportant, and so wo use Boltzmann 
statistics throughout except in 80-1. 



II 

CLASSICAL THEORY OF THE LANGEVIN-DEBYE FORMULA 

10. Polar versus Non- Polar Molecules 

In the study of dielectric phenomena, molecules are commonly divided 
into two categories called 'polar' and f non -polar', also sometimes 
termed 'homopolar' and 'heteropolar'. A molecule may be defined as 
polar if it has a permanent electrical moment, i.e. an electrical moment 
which is on the time average different from zero even in the absence 
of external fields. A molecule without such a permanent moment is 
termed non-polar. To find the permanent moment one must retain out 
of the total moment 2 e t r i on ly ^ ne constant part which remains on 
averaging over the 'internal' motions of the electrons relative to the 
nuclei. 1 All atoms and molecules contain instantaneous moments which 
fluctuate with the positions of the electrons. Hence the mean square 
of the electric moment is never zero, but on the other hand the square 
of the mean may vanish, and if it does the molecule is non -polar. 

As a simple example of this classification, we may mention that all 
diatomic molecules composed of two identical atoms, such as N 2 , O 2 , 
&c., are non-polar. This is quite obvious from symmetry considerations. 
Diatomic molecules composed of unlike atoms, e.g. HC1, are in general 
polar, as the charge therein will not distribute itself symmetrically with 
respect to the two ends of the molecule. Triatomic molecules are 
ordinarily polar. A few exceptions such as CO 2 will be discussed in 20. 
Molecules with more than three atoms may be non-polar if highly 
symmetrical, as in e.g. CC1 4 , CH 4 (see 19-20). Monatomic molecules 
are always non-polar, as on the average the electrons are symmetrically 
located with respect to any plane containing the single nucleus. 

We have not yet mentioned how to decide experimentally whether 
or not a molecule is polar, instead of appealing to our preconceived ideas 
of a molecular model. There arc many experiments, both physical and 
chemical, which furnish tests for the existence of permanent moments, 
but probably the best of these consists in the measurement of dielectric 
constants. If the field is not so extremely strong that it requires con- 
sideration of 'saturation effects', i.e. if we can treat the dielectric 
constant as independent of field strength, then it will be shown that 

1 This permanent moment is to be measured relative to axes fixed in the molecule, 
for the * external * rotations of a molecule as a whole make the axis of molecular polarity 
continually point in a different direction relative to axes fixed in space. 



28 CLASSICAL THEORY OF THE LANGEVIN-DEBYE FORMULA II, 10 

the electric susceptibility Xe == (* l)/4rr is given by the expression 



=*(*+ ) 



* 

which we shall call the Langevin-Dcbye formula. Here N is the number 
of molecules per c.c., T is the absolute temperature, and a is a constant. 
The permanent electrical moment will throughout be denoted by /i, and 
hence the existence of a noil- vanishing second term in (1) is a criterion 
for a polar molecule. We shall henceforth omit the subscript e or m 
from x, as ordinarily it will be clear from the context whether we are 
studying an electric or magnetic susceptibility. Eq. (1) assumes that 
X is small compared to unity, as it ordinarily is in gases; otherwise x 
should be replaced by 3(e l)/4ir(e+ 2) in accordance with the correction 
for the local field given in 5. 

The second term of (1) disappears if the molecule has no permanent 
moment. Hence the dielectric constant of a non-polar gas should be 
independent of temperature provided the density is maintained con- 
stant. It is often convenient to work at constant pressure rather than 
constant density when T is varied, and then the number N of molecules 
per c.c. is a function of T. The part of the temperature variation of the 
dielectric constant diie merely to changes in density may be eliminated 
by using the quotient xl^, which is the 'susceptibility per molecule', 
or else the expression Lx/N, which is the 'susceptibility per gram mol' 
or ' molar susceptibility'. Here L has been used to denote the Avogadro 
number. One therefore has the following fundamental result: in a non- 
polar gas the molar susceptibility should be independent of T, while in 
a polar gas it should be a linear function a-{-b(\/T) of l/T. This agrees 
well with experiment. From the experimental values of the coefficient 
b together with the relation b = .Lju, 2 /3fc following from (1), it is possible 
to deduce quantitatively the permanent electrical moment of the mole- 
cule. The numerical values of the moments so obtained will be con- 
sidered in 19. 

Before proceeding to the mathematical derivation of (1) it is perhaps 
illuminating first to discuss qualitatively the physical nature of the 
phenomena responsible for each of the two terms of (1). The existence 
of a susceptibility requires that the molecules have an average electrical 
moment in the direction of the applied field. This does not require an 
average or permanent moment before the field is applied, as the intro- 
duction of the field may distort or 'polarize 5 the molecule, since the 
positive charges are attracted in the direction of the field, and the 
negative ones repelled. The electrical centres of gravity of the positive 



II, 10 CLASSICAL THEORY OF THE LANGEVIN-DEBYE FORMULA 29 

and negative charges will then no longer coincide, as they did before 
the field was applied if the molecule is non-polar. Due to this deforma- 
tion effect, the molecule thus acquires what is termed an 'induced 
moment' or 'elastic polarization'. In addition there will be a contribu- 
tion to the susceptibility resulting from the permanent moment of the 
molecule, if it is polar. Even the mean electrical moment of an assembly 
of polar molecules is zero in the absence of external fields, since in the 
absence of orienting influences the molecular axes will have a random 
spacial distribution. To a given number of molecules oriented in one 
direction there will be an equal number oriented in the diametrically 
opposite direction. Hence, on averaging over a very large number of 
molecules the net moment is zero. Not so after the field is applied. 
Each polar molecule will be subject to a torque which tends to aline 
its axis of polarity parallel to the field. All the permanent molecular 
moments would aline themselves perfectly parallel to the field, were 
they not prevented by the centrifugal or gyroscopic forces due to mole- 
cular rotation, and also perhaps by the disturbing influences of molecular 
collisions. Both these influences which prevent perfect alinement be- 
come more potent with increasing temperature, and will usually be 
referred to as 'temperature agitation'. Fortunately, an exact examina- 
tion of the forces arising from the temperature agitation is unnecessary, 
as the great beauty of the method of statistical mechanics is that it 
yields the state of equilibrium between the orienting influence of the 
field and the temperature agitation without specializing the mechanism 
by which the equilibrium is secured. Although there is perfect aline- 
ment only at T 0, at any other temperature there is a predominance of 
the parallel over the anti-parallel aliiiemciits of the axes of polarity with 
respect to the field, as the former have a lower potential energy and hence 
a larger Boltzmann probability factor. Thus the orientation effect always 
enhances the susceptibility if the molecule has a permanent moment. 

From the foregoing discussion one immediately suspects that the first 
term of (1), which is independent of the temperature and which is found 
in both polar and non-polar molecules, is due to the induced polariza- 
tion; and that the second term, which disappears at T = oo and which 
is found only in polar molecules, is due to the orientation of the per- 
manent dipoles. This indeed turns out to be the case. Incidently, 
Eq. (1) would make it appear that the susceptibility became infinite in 
polar molecules at the absolute zero, but this is not really the case, as 
at low temperatures higher powers of the field and the resulting satura- 
tion effects not included in (1) must be considered. 



30 CLASSICAL THEORY OF THE LANGEVIN-DEBYE FORMULA II, 10 

The idea of induced polarization is an old one, as it was even used, 
for instance, in Mossotti's theory discussed in 2. The suggestion that 
part of the electric susceptibility might be due to alinement of per- 
manent moments, resisted by temperature agitation, does not appear 
to have been made until 1912, by Debye. 2 A magnetic susceptibility 
due entirely to orientation of permanent moments was suggested some 
time previously, in 1905, by Langevin, 3 and the second term of (1) is 
thus an adaptation to the electric case of Langevin 's magnetic formula. 
Hence we term (1) the Langevin-Debye formula. (In the electric case, 
a formula such as (1) is commonly called just the Debye formula, but 
we use the compound title Langevin-Debye in order to emphasize that 
the mathematical methods which we use to derive the second term of 
(1) apply equally well to magnetic or electric dipoles.) Despite the 
earlier appearance of the orientation theory in the magnetic case, we 
nevertheless first present this theory in the present chapter on electric 
polarization. We do this in order to defer until a later chapter certain 
rather delicate points in the classical theory of magnetic susceptibilities. 

11. Rudimentary Proof of the Langevin-Debye Formula 

We shall first give the conventional, rather crude derivation of (1). To 
obtain the first term of (1) we may follow the usual line of least resistance 
in classical theory and regard each charge in the atom or molecule as 
having a position of static equilibrium and subject to an isotropic linear 
restoring force when displaced therefrom. This naive depicture of an 
atom or molecule as a collection of harmonic oscillators is not in agree- 
ment with modern views of atomic structure as exemplified in the 
Rutherford atom, but yields surprisingly fruitful results. The restoring 
force on a harmonically bound charge e t is c^fo r fo ), where a t is 
a constant and r t r iQ is its vector displacement from the equilibrium 
position. The force on the charge in an electric field E is e E. Hence 
for static equilibrium between the internal restoring and external 
electric forces, r r,- equals e E/a^, and the induced electrical moment 
e^ r <0 ) is Eef/tf^. The total induced moment in the molecule is then 
E 2 e i/ a i> where the summation extends over all the charges in the 
molecule capable of displacement from equilibrium positions. Since the 
susceptibility is the total moment per c.c. divided by field strength 
(Eqs. (8) and (12), Chap. I) the susceptibility arising from the induced 
polarization of the elastically bound charges is Not, where a is an 

2 P. Debye, Phys. Zeits. 13, 97 (1912). 

3 P. Langevin, J. de Physique, 4, 678 (1905); Annales de Chimie et Physique, 5, 70 
(1906). 



II, 11 CLASSICAL THEORY OF THE LANGEVTN-DEBYE FORMULA 31 

abbreviation for 2 e f /%> a constant which is obviously independent of 
temperature. We have thus obtained an expression of the same form 
as the first term of (1). 

To obtain the second term we suppose that the molecule has also 
a permanent moment p, or in other words an electrical polarity even 
when the charges are at their equilibrium positions, If be the angle 
between the axis of polarity (i.e. direction of the permanent moment) 
and the applied field, then the part of the susceptibility arising from 
the permanent moment is clearly 



(2) 

where the double bar means an average over all the molecules in a 
physically small element of volume (cf . Chap. I). In the absence of the 
field, positive and negative values of cos0 are equally probable, as 
already mentioned, and the expression (2) vanishes. Let us boldly omit 
the kinetic energy of the molecule and regard the Hamiltonian function 
as consisting merely of the potential energy pE cos 8 of the permanent 
moment of the molecule in the applied field E. Then by a rather loose 
application of the Boltzmann distribution formula (55), Chap. I, the 
probability that a dipole axis fall within an element of solid angle 
dl=z&mQ(Wd(}> is proportional to e/* BcOB * lkT dQ, } and (2) becomes (cf. 
Eqs. (56) and (57), Chap. 1) 



Here on the right side we have developed the exponents as power series 
in E, and have carried the expansion only far enough to include the 
first term which does not vanish on integration. As the average value 
of cos0 over a sphere is zero, the quotient of the two integrals on the 
right side of (3) is, except for a factor fiE/kT, merely the average of 
cos 2 over a sphere. This average is 1/3, and hence the approximate 
value of (3) is Np?/3kT t which is the second term of (1), thus finishing 
the proof. 

As a matter of fact, the integrals in (3) are readily evaluated in closed 
form without resorting to a series development. If we make the sub- 
stitution q = iiEcoaO/kT and cancel the trivial integral over < from 
numerator and denominator, the left side of (3) becomes 


flxjtfdq 




32 CLASSICAL THEORY OF THE LANGEVIN-DEBYE FORMULA II, 11 

where coth# = (e x +e- x )/(e x e- x ) and where x is an abbreviation for 
the dimensionless ratio ^EjkT. The expression enclosed in square 
brackets is often called a Langevin function and commonly denoted by 
L(x). The series expansion of L(x) for small values of x is \x &?+... 
and hence (4) agrees with the second part of (1) when we keep only the 
first term in this development. On the other hand, for very large values 
of x, the asymptotic value of cothz, and hence of L(x), is unity. It 
thus follows that the polarization due to orientation, viz. EX = N^iL(x), 
becomes Np if the field strength is enormously great or if the tempera- 
ture is exceedingly low. This limiting or 'saturation' value Np, is what 
we should expect, as under such extreme conditions the dipole axes all 

aline themselves parallel to the field, 
making the cosine factor unity in (2). 
A graph of the Langevin function is 
shown in Fig. 1. Near the origin the 
curve is a straight line, so that the 
polarization is linear in the field 
strength if not too great. This is the 

portion of the curve in which only the first term in the expansion need 
be considered, and in which (4) can be replaced by the second part of 
(1). On the other hand, the fact that for large x the curve is asymptotic 
to a horizontal line shows that the polarization by orientation cannot 
exceed the saturation value NJJL. 

12. More Complete Derivation of the Langevin -Debye Formula 4 

The preceding proof of the Laiigcvin-Debye formula cannot be regarded 
as satisfactory, since in forming the Boltzmann distribution factor 
e -#lkT t h e ki ne tic energies of rotation and vibration were entirely 
omitted from the Hamiltonian function &. In dealing with the induced 
polarization, the charges were supposed in static equilibrium between 
the forces of restitution and the applied field. On the other hand, in 
order to make possible the Boltzmann distribution in the orientation, 
it was necessary to suppose a 'kinetic statistical equilibrium' between 
the impressed electrical forces and the forces of temperature agitation. 
The only position of static equilibrium for the permanent dipoles would 
be when they are all parallel to the applied field; hence the assumption 

4 In writing the present section the writer has been aided by the discussion of distribu- 
tion functions and Gans's theories of magnetism given by Wills in Theories of Magnetism 
(Bulletin No. 1 8 of the National Research Council). Gans's mathematical transformations 
and use of momentoids show considerable resemblance to the devices which we employ, 
but he treated rather different problems and assumed more symmetry in the molecule. 



II, 12 CLASSICAL THEORY OF THE LANGEVIN-DEBYE FORMULA 33 

of static equilibrium would yield the saturation moment Np even for 
infinitesimal fields, an absurdity. It is clearly illogical to include the 
temperature agitation in the rotational problem connected with the 
permanent dipoles, as one must of necessity, but to neglect this agita- 
tion in the vibrational problem connected with the induced polarization. 
Furthermore, although the temperature agitation was implicitly recog- 
nized in the orientation or rotational problem, it was explicitly neglected 
in omitting the kinetic energy of molecular rotation from the Hamil- 
tonian function. This omission is made in much of the literature, 
including Langevin's original paper, and it is fortunate that the 
Langevin-Debye formula (1) is obtainable regardless of whether one is 
painstaking enough to include the kinetic energy terms in the Hamil- 
tonian function. 

We shall therefore begin afresh and re-derive (1), taking cognizance 
of the kinetic energy. Also we shall free ourselves from the restriction 
that the forces binding the charges to their equilibrium configurations 
be isotropic. This restriction was a bad one, as few, if any, molecules 
have such perfect symmetry that displacements in all directions can be 
regarded as equivalent. Let the small vibrations of the charges about 
their equilibrium configurations be specified by a set of normal co- 
ordinates f ,, 2 ,..., 8 , equal in number, of course, to the number of 
degrees of freedom of the elastic vibrations. We may suppose the 
electrical moments p y , p^, p^ along the principal axes of inertia x', y', z' 
to be linear functions of the normal coordinates, so that 



1^ 

with analogous formulae for the moments along the y' and z' directions. 
Here p.^, juy, /v> which are the terms remaining when the displacements 
,. vanish, are, of course, the components of the permanent moment 
along the three principal axes. The other terms, which are inside the 
summation sign, represent polarization acquired in virtue of the elastic 
vibrations. 

Let 0, <, ifj be a set of Eulerian angles specifying the position of the 
principal axes of inertia x', y' ', z' which are fixed in the molecule, relative 
to another set of axes x, y, z which are fixed in space. Here 6 is the 
angle between the z and z' axes, while <, are respectively the angles 
between the nodal line (i.e. intersection of the xy and x'y' planes) and 
the x and x' axes respectively. We suppose the field in the z direction. 
The coordinate z is connected with x', y', z' according to the relation 
z = sin sin x' + sin cos i/j y' + cos Oz r . (6) 



34 CLASSICAL THEORY OF THE LANGEVIN-DEBYE FORMULA II, 12 

The kinetic energy of rotation of the molecule regarded as a rigid body is 



where Aj, A y ., A z , are the principal moments of inertia, and l x >, Q y >, O^ 
are the x' y y', z' components of angular velocity. The expressions for 
these components of angular velocity in terms of the Eulerian angles 
and related velocities are as follows: 5 

Q^==0cosi/r4-^sin0sini/r, Q,^ = ^ cos $ sin 6 6 sin ^, \, ^^+<^cos0. 

(8) 
The Hamiltonian function is 

.a- (9) 

sin 6 cos $ + 



Q = sin j/rjty + cos i/j cosec (^ cos 6 p^), R ~ p^. 



The first term of (9) is the kinetic energy of rotation of the molecule 
regarded as a rigid body, sometimes dubbed the 'asymmetrical top'. 
It is obtained in the usual way by first expressing (7) in terms of 6, <f>, \fs, 
Q, </>, ?/r by means of (8), and then passing from 6, </>, $ to the canonical 
momenta p d , p^, p^ defined by the relations p e = 3T Toi /dd y &c. The 
next two terms are the kinetic and potential energies of the small 
vibrations, and of course consist entirely of the sum of squares. The 
remainder of (9) is the potential energy 

^[p^cos(a; f ,)+p |/ .cos(y',2)+p s .cos(2',2)] (11) 

due to the applied electrical field. In writing this electrical potential 
energy in (9) we have utilized (5) and have expressed the direction 
cosines involved in (11) in terms of the Eulerian angles by means of (6). 

Even the rather formidable expression (9) is not a rigorous Hamiltonian func- 
tion, as it assumes that the vibrational and rotational kinetic energies enter in 
a strictly additive fashion, which is not true. Actually the molecular distortion 
accompanying the small vibrations will constantly cause slight variations in 
the moments of inertia A x >, A v - y A*, so that they are not really constants of 
the molecule. Also, reciprocally, the centrifugal forces from molecular rotation 
ought really to be added to the linear restoring forces acting on the coordinates 
fi> ^2>'" A rigorous Hamiltonian would, among other things, involve * cross 
products ' in the momenta of the form d^p^p^ where p^ denotes any of the rota- 
tional momenta p^ p^, pfa and where the coefficients d iat are functions of the various 
positional coordinates. However, all these corrections for the mutual interaction 

5 Cf., for instance, Born, Atommechanik, p. 30, or Webster, Dynamics, p. 275. The 
notation of these writers differs from ours in an interchange of the angles denoted by ^, ift. 



II, 12 CLASSICAL THEORY OF THE LANUEV1N-DEBYE FORMULA 35 
of the rotation and vibration are small if the stiffness coefficients O L for the normal 
vibrations are so large that the amplitudes of these vibrations are small. Hence 
(9) involves no serious error if the elastic vibrations are supposed small, and with- 
out such a supposition the dynamical problem would, of course, be unmanageable. 

Eq. (59) of the preceding chapter shows that the susceptibility is 
determined as soon as we evaluate the 'partition function' 



= IP 



--^^ 



It is convenient to change three of the variables of integration from 
Pd> Pfr P*/, to new variables P t Q, R, which are given in terms of p$ y p^, 
pj, in (10). The expressions P, Q, E are sometimes called 'momentoids'. 6 
They are not canonically conjugate to 0, <, ?/r, but equal the instanta- 
neous angular momenta about the three principal axes of the molecule. 
For this reason the rotational kinetic energy takes the exceedingly 
simple form given in the first term of (9) when expressed in terms of 
P, Q, R, whereas (10) shows that it would be exceedingly cumbersome 
if expressed explicitly in terms of the true canonical momenta. The 
functional determinant of the transformation to the momentoids is 
most easily evaluated by calculating its reciprocal 7 

cos0 sun/rcosee0 sin i/j cot 

3 sin0 cosiAcosecfl cos cot ~ : -. 
r sm0 

I 

The expression (9) for the energy is of the form 

/( l\ G, R, Pfr-)+9(0, f </<> i,., K). U3) 

It is this separability into two parts, depending respectively on the 
momentoids together with the vibrational momenta, and on the co- 
ordinates, which makes the momentoids so useful. If the true momenta 
Pe> Pfa Pili were introduced as variables in place of P, Q, E, this separation 
would not be possible, as (9) shows that then 6, (f>, i/j would be involved 
in the kinetic energy. The partition function now factors as follows: 

rr rr r/ r/ 

Zj ~-~~- J -i ZJ o , Zj -t ^^ 



= J ... J 



6 C'f. JtmTis, The Dynamical Theory of Cases, 3rd od., p. 97. 

7 The transformation equations (10) involve the variables 9, ^, as well as 
Pd* P$t P$* l' Q H ^ ut & i s iminodiatoly verified that tho complete sixth onlor 
Jacobian dip^pj, P^ 0, <f>, $)ld(l\ Q, H, 0, <f>, A) i the same as the third order one 
d(ptp^p^)IO(P 9 Q 9 R) in view of the fact that tho old and new 9, <, $ are identical. 
This makes a third order under-determinant in one corner of the sixth order one equal 
to zero, and tho sixth order determinant factors into two third order ones equal 
respectively to unity and sin 0. 



36 CLASSICAL THEORY OF THE LANGEVIN-DEBYE FORMULA II, 12 

The important thing is that the factor Z A is independent of E, and so 
makes no contribution to the susceptibility, for Eq. (59), Chap. I, is 

l ' 



E dE ' E dE ' 

The factor Z 1 can henceforth be omitted entirely. The physical mean- 
ing of this result is that the susceptibility is the same as though the 
kinetic energy were omitted entirely, provided we retain the 'weight 
function' sin0 in the integrand of Z 2 . In the present problem this 
weight function arises from transformation of the kinetic part of the 
problem, as in the original system of canonical variables the volume 
element was dpgdp < f > dp l f t dpg i ...d6d(f>di/jdt; v .. ) whereas after the transforma- 
tion it becomes sm6dPdQdBdpg i ...d6d<f>difjdt; l ... . In the conventional 
Langevin theory (11) this weight function is inserted because an 
element of solid angle is of the form dQ sin 6 ddd</) (cf . Eq. (3) ). Ffow- 
ever, rigorous statistical mechanics do not allow us to proceed in such 
a fashion without justification, as the statistical theory tells us that the 
weight function is unity (or some other constant) in the complete 
macroscopic phase space involving both the position and momentum 
variables, but tells us nothing about distributions in a space of half as 
many dimensions involving only the coordinates of position. There is 
thus quite an amusing and illuminating contrast between the way the 
factor sin0 makes its appearance in the present section and in the 
preceding one (11). 

In order to evaluate the term in the moment which is proportional 
to E, we must expand Z 2 to terms in E 2 . To this approximation we have 



f. 






by (9) and (11). Here we have used A^.-, A^,, Ay as abbreviations for the 
three direction cosines in (11). The integration over the Eulcriaii angles 
6, <f>, i/j is readily performed, either by substituting the explicit formulae 
for A^.', Aj,', Ay in terms of 6, <, i/j by means of (6) or without doing this 
by observing that the mean square of any direction cosine is 1/3, while 
the mean of the first power of a direction cosine or of the product of 
two different direction cosines is zero. Furthermore, 



2)3 tf -?#, = e-#,. (16) 



II, 12 CLASSICAL THEOKY OF THE LANUEVIN-DKBYE FORMULA 37 

The first of these relations is obvious from the fact that its integrand 
is an odd function of f f , while the second is obtained by partial integra- 
tion. On taking ft a-jZkT, formulae (10) show that the integral over 
the ^ of p|> = [p> X '~\- 2 c x'ii\ 2 i g identical with the integral of 

Hence (15) becomes 



X J...J e -^'tf/^'sinflc/fl/tytyr/f!... . (17) 

It is unnecessary to evaluate the integrals in (17), as they are indepen- 
dent of K and hence make no contribution to (14). The disappearance 
of the linear terms in E from (17) could have been predicted, as the 
susceptibility and hence by (14) the partition function must depend on 
the magnitude but not on the sign of E, so that Z 2 can involve only 
even powers of E. From (14) and (17) it follows that the susceptibility 
is 2yNkT/( 1 +yE 2 ) , where 1 +y E 2 is an abbreviation for the factor outside 
the integral sign in (17). We may neglect yE 2 in comparison with 1 in 
the denominator, as we have agreed to retain only the part of the 
susceptibility which is independent of field strength, and have already 
made approximations of this order. Hence 



This is the desired result, the same as (1), since p% -\-ptf-\-pi is the 
square of the magnitude of the permanent moment p, of the molecule, 
and since the summation in the first part of (18) consists solely of terms 
independent of T, and so is a constant, say a, independent of tem- 
perature. If in particular the oscillations are due to isotropically bound 
charges e t , then c^ = c u > t =- c z , t e t , and the expression for Not is 
N 2 il<i*i, the same as given in the first paragraph of 11. 

13. Derivation of a Generalized Langevin-Debye Formula 8 

We shall now deduce a very general form of the Langeviii-Debye 
formula. This proof is intended, and should be very easy, for readers 
familiar with the use of 'action and angle variables' in dynamics and 
perturbation theory. 9 Suppose that we have a multiply periodic dyna- 

8 J. H. Van Vleck, Phys. Rev. 30, GO (1927). 

9 For details on. the dynamical technique involved in the use of angle and action 
variables see Bom's Atommechanik or Chap. XI of the author's Quantum Principles and 
Line Spectra. 



38 CLASSICAL THEORY OF THE L ANGEVIN-DEB YE FORMULA II, 13 
mical system with / degrees of freedom, specified by 2/ canonical 
variables wj,..., w$, JJ,..., Jf. We assume further that the w's and J's 
are respectively true angle and action variables for the system in the 
absence of the field E. When E = the w$'s are thus linear functions 
vj^+cjj of the time t, while the J's are constants (which incidentally 
in the old quantum theory would be equated to integral multiples of h). 
The 2-component of electrical moment will be a Fourier series: 10 

Pz -- 2 ?/~W(-\ (19) 

T 

We use the same notation as in Born's Atommeclianik (i, p. 86, &c.). 
Thus (rw) is an abbreviation for T 1 w 1 +T 2 w 2 +-" -\~TjWj, and the subscript 
T i T 2--- T / * s abbreviated to r. The summation is /-fold, and extends over 
all positive and negative values of the integers TJ...T/. The complex 
amplitudes p ( ^ are, of course, functions of the J's as well as the r's, 
and p^ T is the conjugate of p^. Classical statistical mechanics (e.g. 
Eq. (57) or (58), Chap. I) shows that if an electric field E is applied 
along the z direction, the susceptibility is ll 



where dJdw Q denotes an element of volume dJ ( l...dJ^dw\...dwJ of the 
'phase space'. Also M denotes the Hamiltonian function in the field 
E, and equals J Ep 3 (of. Eq. (48), Chap. T), where p s is given by 
(19), and c# is the Hamiltonian for E 0. The 'unperturbed' Hamil- 
tonian # is a function only of the ./'s, whereas M involves also the 
w's through p z . It is to bo clearly understood that we are keeping the 
original canonical variables w},..., Wj, */?,..., Jf defined in the same way 
as for E 0. Since the transformation from a Cartesian to the W Q , / 

10 Wo henceforth use ordinary italic typo p for the components of electric moment. 
In the preceding section ( 12) a special type p was employed to avoid confusion with 
the generalized momenta p (/ , &c. 

11 Limits of I nlegration in Eq#. (20), (21). If we integrated over the entire phase space in 
the J , w system the limits of integration for each of the w's in (20) or (21) would he 
from co to +o> as all of the w^s may increase without limit. However, since the 
system is cyclic in each of the w's with unit period, it is clear that wo will obtain the 
correct statistical average if we take the limits of integration for each of the w's as zero 
and unity. Another way of saying the same thing is that Cartesian variables are multiple 
valued in the ty's, so that the entire Cartesian phase space corresponds to only one 
period for the u )0 's. 

The limits of integration for the J^'s can usually be taken as and -}- oo by proper 
choice of the fundamental periods. (The limits, however, are oo anil -f co for J's 
associated with both left- and right-handed rotations, as, for instance, in the case of 
a J associated with an axial component of angular momentum.) The precise form of 
the limits for the e/'s is immaterial for us, as the bars in (22) automatically denote an 
average over the entire J space. 



II, 13 CLASSICAL THEORY OF THE LANGEVIN-DEBYE FORMULA 39 

system is thus not modified to take account of the field E, the w's and 
J's will cease to be true angle and action variables, i.e. cease to be 
respectively linear in t and constant, after the field E is applied. The 
w's and J's will, however, remain canonically conjugate, and we can 
apply (20) because it is a fundamental theorem in statistical mechanics 
that the a priori probability is proportional to the volume occupied in 
the phase space, regardless of what 2/ variables we choose as con- 
stituting the coordinates of this space provided only they are canonical; 
this is because all c contact transformations' have unit functional deter- 
minant, as already explained in 9. 

Now e~^ kT = e-^- K ^ kT ==e-^ kT (l+Ep z /kT+ ...), and hence, if 
we keep only the part of the susceptibility which is independent of E y 
Eq. (20) becomes 



yf f rffl 

( ' 



kT $... 

Here we have assumed that the polarization vanishes when 7 0, i.e. 
that the numerator of (20) vanishes when Jf is replaced by Jf . Other- 
wise the body would exhibit a 'permanent' or 'hard* polarization, a 
phenomenon not usually encountered in dielectrics, at least not to an 
appreciable degree, although it is quite common in ferromagnetism. 
Ordinarily this assumption of no residual polarization is fulfilled by 
reason of symmetry, as in a gaseous medium there is no preferred 
direction in the absence of external fields. 

Now since the square of two multiple Fourier series is itself such 
a series, pi may be expressed as a multiple Fourier series in the w's. 
On integrating over the W Q part of the phase space, the contributions 
of all terms in this multiple Fourier development of pi vanish except 
the constant term (pf ) > for integrals of the periodic terms in the w's 
taken over a period are zero. By the rules for multiplying together two 
Fourier series term by term, (pl) Q equals Jl^-P-r' which i g > f course, 

T 

a function only of the J's. Eq. (21) now becomes 



X ^ ~ 7 /**' ~dJ* " kT 



where pi denotes the statistical mean square of pi in the absence of 
the field E, i.e. the average over only the J part of the phase space, 
weighted according to the Boltzmann factor, of the time-average value 
of pi for a molecule having given values of the t/'s. Now if the applied 
electric field E is the only external field, all spacial orientations will be 
equally probable when E = 0, and the mean squares of the #, t/, and z 



40 CLASSICAL THEORY OF THE LANGEVIN-DEBYE FOIIMULA II, 13 

components of moment will be equal. This will also be true even when 
there are other external fields (e.g. a magnetic field) besides the given 
electric field provided, as is usually the case, these other fields dc^not 
greatly affect the spacial distribution. We may hence replace p* by 
one -third the statistical mean square of the vector moment p of the 
molecule. Thus we have 

(22) 



which is a sort of generalized Langevin-Debye formula. It is much more 
comprehensive than the ordinary Eq. (1), since the statistical mean 
square moment in (22) must not be confused with the time average for 
an individual molecule, and is in general a function of the temperature. 
Eq. (22) gives the same temperature dependence as the ordinary 
Langevin-Debye formula (1) only if we make certain specializing 
assumptions concerning the nature of the multiply periodic dynamical 
system, such that the mean square moment p 2 becomes a linear function 
AT-\-E of the temperature. Such a specialization can be achieved by 
assuming, as in 12, that the molecule can be represented by a model 
consisting of harmonic oscillators mounted on a rigid, freely rotating 
framework of moments of inertia A^ A y >, A z ,. Then the instantaneous 
moment p is a linear function of the normal coordinates ,-, so that the 
components of p along the principal axes of the molecule are given by 
expressions of the form (5). With the model of 12 the molecule is 
supposed to have a permanent moment of invariant magnitude ^, so 
that there is no distinction between ju, 2 and /u, 2 . Furthermore, 

IT-O, CW=0, Kl-^-T. (23) 

The first two of these relations are obvious since positive and negative 
values of the displacements of the oscillators from their equilibrium 
positions are equally probable. The third relation of (23) is the well- 
known classical equipartition theorem that the mean of the potential 
energy ia^f of each normal vibration is \kT. Eqs. (5) and (23) show 
that pi' p,*'-}-kT ^ c x'tl (t i> & c -> an( l so (22) becomes identical with 
(18), the desired result. It is seen that the present method, although 
requiring more dynamical background, furnishes a much briefer means 
of deriving (18) than that given in the preceding section ( 12). The 
present method is equally rigorous, as the kinetic energy terms are 
included in the Hamiltonian function, although it is not even necessary 
to write out their explicit form. The rotational part N^/SkT follows 
particularly easily from (22), as the calculation is not a bit more com- 
plicated for the most unsymmetrical molecule, with three unequal 



II, 13 CLASSICAL THEORY OF THK LANGEVIN-DEBYE FORMULA 41 

moments of inertia, than for symmetrical ones. We have seen fit to 
include both methods of deriving (18), partly because they furnish an 
interesting contrast, and partly because some readers may not be 
familiar with action and angle variables. The method of these variables 
is exceedingly compact and general, but for this very reason perhaps 
does not furnish as much physical insight as the more explicit and 
longer representation by means of the ordinary Eulerian and normal 
'positional' coordinates and momenta used in 12. 

A model such as we have used, in which the electronic motions are 
represented by harmonic oscillators, is not compatible with modern 
knowledge of atomic structure. We know that actually the electrons 
are subject to inverse square rather than linear restoring forces, and 
move in approximately Keplerian orbits instead of executing simple 
harmonic vibrations about positions of static equilibrium. In fact Earn- 
shaw's theorem in electrostatics tells us that there are no such positions 
for all the charges. In actual molecules, to be sure, the motions of the 
nuclei, in distinction from electrons, can be regarded as approximately 
simple harmonic motions about equilibrium, as the nuclei are sluggish 
because of their large masses, but for this very reason the amplitudes 
of their vibrations are so small that the contribution of these oscillations 
to the susceptibility is usually small, though not always negligible. This 
will be shown in 15. Hence the part of the molecular motion which 
is really simple harmonic is of secondary importance for susceptibilities. 
Inasmuch as we have deduced a generalized Langevin-Debye formula 
for any multiply periodic system, the question naturally arises whether 
Eq. (22) cannot be specialized in a fashion appropriate to a real E/uther- 
ford atom instead of to a fictitious system of oscillators mounted 011 
a rigid rotating framework. This, however, is not possible. The reason 
is that in classical statistics the energy ordinarily ranges from to oo, 
whereas in a Rutherford atom it ranges from (the value for infinitely 
loosely bound electrons) to oo (the value for infinitely tight binding). 
The numerical value of the energy is the same as that of the Hamil- 
tonian function J, and hence the Boltzmami probability factor Ce~^ kT 
increases without limit as the energy approaches oo, so that the total 
integrated probability C J... J e~^ kT duPdJ cannot equal unity, as 
required by (56), Chap. I, unless (7 0. Thus, although (22) sum- 
marizes rather elegantly all the results of classical statistics applied to 
susceptibilities, the practical advantages of the increased generality of 
(22) as compared to (1) or (18) are somewhat restricted because of the 
inherent limitations in classical theory. 



Ill 

DIELECTRIC CONSTANTS, REFRACTION, AND THE 
MEASUREMENT OP ELECTRIC MOMENTS 

14. Relation of Polarity to the Extrapolated Refractive Index 

In the present chapter we shall examine the experimental confirmation 
of the Langevin-Debye formula derived in Chapter II, but it will first 
be necessary in 14-18 to discuss some related topics in the theory of 
refractive indices. 

In 4 we saw that tho static dielectric constant should equal the 
extrapolation of n 2 , the square of the index of refraction, to infinite 
wave-length. That n is really a function of frequency or wave-length 
can be seen from the following very elementary form of the Drude 
dispersion theory. 1 Suppose that a particle of charge e t and mass m i is 
harmonically and isotropically bound to a position of equilibrium. If 
and r^ rj denote respectively the coefficient of restitution and the 
vector displacement from equilibrium, the particle's equation of motion 
when subject to an incident electromagnetic wave of frequency i> is 

,72 r 

/^V + a *( r *- r ?) = *E COS tonrj, (1) 

where E is the vector amplitude of the electric field of the wave. 
Eq. (1) is merely that of a forced harmonic vibrator, and as the natural 
oscillation frequency of the particle is v i = a^/27Tm\ 9 the solution of (1) 
may be written e 



The total induced polarization of the molecule isp 2 e i( r i~ r ?)> where 
the summation extends over all its charges. Hence the index of refrac- 
tion is given by 

(2) 

' V ' 



_ 

E - E^T&ntf ~~ , 

Eq. (2) yields a dispersion curve of the familiar Sellmeier form charac- 
teristic of classical theory. It is well known that by proper adjustment 
of the natural frequencies v t and of the effective charges and masses 
e t , w , Eq. (2) can be made to represent quite well the observed variation 
of refraction with frequency. There is, however, the difficulty that the 
values of e t and m l which must be assumed are not the true values of 
either electronic or nuclear charges and masses. We shall see in 82 

1 Cf ., for instance, Drude, Theory of Optics, Chaps. V-V1I. 



IIT, 14 MEASUREMENT OF ELECTRIC MOMENTS 43 

that the quantum mechanics also yields a dispersion curve of the 
Sellmeier type, but without this difficulty in the interpretation of the 
constants. Kq. (2) might be generalized by assuming that the restoring 
forces are non-isotropic, somewhat along the lines of the model used in 
12. This generalization has commonly been made in the literature, 1 
and is, of course, necessary for optically anisotropic media. It will, 
however, be omitted here to avoid devotion of too much space to 
antique models based on positions of static equilibrium for the electrons. 
We are interested especially in the behaviour of Eq. (2) in the region 
of infinitely long incident waves. Here v - 0, and (2) becomes 



On comparing with the value of CY derived for the isotropic model in 
the first paragraph of 11, we see that n^ 1 47rNoi. Hence, according 
to (2) the square of the index of refraction extrapolates for infinite 
wave-lengths to the part I -\-47rNoc of the dielectric constant arising 
from the induced polarization rather than to the complete dielectric 
constant I + lTrNfa+pP/SkT). The reason is, of course, that in the pre- 
ceding paragraph we have neglected entirely all polarization by orienta- 
tion. We ought therefore to add to the right side of (2) a term 0(fi, v , T) 
representing the orientation effect, and reducing to ^irNfj^/^kT when 
i' --- 0. We shall not give the explicit calculation of O, as this is more 
difficult in the periodic as distinct from the static case owing to the 
necessity of having a statistical theory of non-conservative systems. 
Even without such a calculation it is quite apparent that the term O 
must be negligible at ordinary optical frequencies, since the electro- 
magnetic forces associated with visible light oscillate so rapidly in sign 
that they do not act in any direction long enough to orient the molecules 
in that direction. 2 The orientation term would first become significant 

2 Simple dimensional considerations show that the orienting effect of the field on the 
permanent dipoles is negligible at optical wave-lengths. Optical frequencies are very 
large compared to the frequencies of collisions or of molecular rotations. Hence the 
temperature cannot enter into the contribution of the orientation to the optical index 
of refraction. In other words the forces resisting the orientation of dipoles by fields of 
high frequency are mortial rather than statistical. If / and /t denote respectively the 
moment of inertia and electrical moment of the molecule, the part n? ot 1 of the refraction 
due to the orientation effect will involve the arguments /, p, N, V Q . As the dimensions 
of these expressions are respectively wZ 2 , 771 1 / 2 l^jt, J- 3 , t- 1 and as ?i 2 t 1 is dimensionless 
and is linear in N, it must be given by an expression of the form const. Afyi a //i?g. This 
is also quite evident by comparison with (2), as /A, / are the rotational analogues of the 
expressions e, m found in rectilinear problems. The contribution of the permanent 
dipoles to tho static dielectric constant is of the order N^/kT. The orientation effect 
is therefore smaller in tho periodic than in the static case by a ratio of tho order fcT/Ivg, 
or 10-* as kT~ IG'^, /-lO' 39 , > ~10 15 . 



44 DIELECTRIC CONSTANTS, REFRACTION, AND THE III, 14 

in gases when the incident frequency is comparable with the rotation 
frequencies of the molecule. This requires a frequency far in the infra- 
red (10 to 100 microns). Hence a series of measurements of indices of 
refraction made at optical frequencies will reveal practically no trace 
of the orientation term, and so the extrapolation of such data to infinite 
wave-length will yield only the part of the dielectric constant exclusive 
of the orientation effect. 




We have attempted to illustrate this graphically in Fig. 2. The curves 1 
and IT represent respectively the values of n?~-l exclusive and inclusive 
of the orientation effect. The theoretical values of the intersection of 
these two curves with the axis of zero frequency are respectively Nn 
and N(oL+fji 2 l3kT). If the molecule is polar the curve TE will lie con- 
siderably above curve i in the region of very small frequencies, as it 
contains a large contribution from orientation. This contribution is, 
however, rapidly blotted out when the frequency is increased, so that 
the two curves arc sensibly the same when the incident frequency 
exceeds the natural molecular rotation frequencies. The actual disper- 
sion of the material is given by curve II rather than curve I, but any 
one attempting to extrapolate to zero frequency a series of measure- 
ments made in the optical region, represented by crosses in the figure, 
would naturally follow curve I, since the measurements do not sensibly 
reveal the rise in the actual curve near the origin V Q =^ 0. Jn drawing 
curves I and II of Fig. 2 we have supposed that the molecule's natural 
frequencies all lie in the ultra-violet, in order not to complicate the 
curve with 'resonance catastrophes' in which some of the denominators 
in (2) become zero. 



Ill, 14 MEASUREMENT OF ELECTRIC MOMENTS 45 

The fact that the natural extrapolation of optical refractivities follows 
curve I rather than curve II furnishes a second method of experi- 
mentally determining electrical moments which is an alternative to the 
first method, that of temperature variation of the dielectric constant 
already mentioned in 10. Let n^ be the extrapolation of the square 
of the index of refraction to zero frequency, exclusive of the orientation 
term which is undctectable in the optical region. Let e be the measured 
static dielectric constant, which of course includes the polarization 
arising from orientation of permanent dipoles as well as the induced 

polarization. Then A __ _ 

., 4-rrNiJi 2 ,-. 

e == 3 



as the Langcvin-Debye formula (1), Chap. 11, shows that 
is the part of e arising from the orientation effect. Hence, knowing 
n^, e and the temperature and density, the electric moment can be 
determined without the necessity of varying the temperature. The 
numerical values of the electric moments obtained by the two methods 
will be compared in 19. 

15. Effect of Infra-red Vibration Bands. The 'Atomic Polariza- 
tion'. 

There is sometimes considerable discrepancy between the values of the 
electric moments deduced by the two methods, or, what is the same 
thing, the value of w'4 1 deduced by extrapolation does not agree with 
the value of 4?rJVc\ deduced from the temperature variation of the 
dielectric constant. To account for such disagreements it has often been 
suggested 3 that the dielectric constant is appreciably influenced by 
infra-red absorption bands associated with vibrations of the nuclei and 
revealed in the ordinary 'vibration spectrum' of molecules such as HC1. 
Suppose, for instance, that the molecule is diatomic. Let ?/? cff be the 
effective mass 7 1 m 2 /(m 1 +w 2 ) of the nuclei, v vib be the frequency of 
vibration of the nuclei along their line of centres, and e cir be the corre- 
sponding effective charge, which is defined as the rate of change dpjdr 
of the magnitude of the electrical moment p, with the inter-nuclear 
distance r. The contribution of this vibration to n 2 is 

c ^- . (4) 

(v; lb -v5) 

3 Cf., for instance, Dobyo, Hamlbuch der Padiologic, vi. 620, Tolar Molecules', p. 48; 
L. Ebert, Die Natitrwisscnschaftcn, 14, 919 (1926); H. A. Stuart, Zcits. f. Physik, 51, 
499 (1928); Singer and Steiger, Ilcl. PHys. Acta 2, 144 (1929). 



46 DIELECTRIC CONSTANTS, RKFRACT1ON, AND THE 111, 13 

This can be seen from Eq. (2); the factor -J-, however, must be inserted 
because the vibration under consideration has only one degree of free- 
dom rather than three as does the isotropic oscillator assumed in 
Eqs, (1), (2). The result (4), iiicidently, remains valid with quantum 
mechanics, as the classical and quantum theories yield identical results 
for the harmonic oscillator, and as there is no need of improving our 
model of the nuclear oscillations in the new theory inasmuch as the 
nuclei, in distinction from electrons, really do have positions of static 
equilibrium. 4 By (4) the contribution of the infra-red vibration to the 
static dielectric constant, corresponding to v ^0, is Ne^/^Trm^v^. 
Because of their large masses, the nuclei vibrate slowly, so that the 
frequency v vib is in the infra-red, and very small compared to the 
incident freqiiency v if y is in the visible region. By (4) the contribu- 
tion of the nuclear vibration to ri 2 in the latter region is approximately 
^ r off/^ 7r '"'ett i; o ail( l hence negligible in comparison with its contribution 
to the static dielectric constant. Thus the apparent extrapolation of 
measured optical refractivities to zero frequency will not include the 
effect of the infra-red vibration band, besides also, of course, omitting 
the polarization by orientation of permanent dipoles, as previously 
mentioned. This is illustrated by curve III of Fig. 2, p. 44. This curve 
is drawn inclusive of the contribution of this band, while the other 
curves are exclusive. Optical measurements indicated by crosses would 
clearly not reveal the difference between curve TTI and 1 or IT. The 
curve 111, of course, shows a discontinuity at the resonance point 
VQ v vil) . The part of the polarization arising from the difference be- 
tween curves TTI and IT is sometimes termed 'the atomic polarization', 
as it is due to oscillations in the positions of the atomic masses (or more 
accurately nuclei) within the molecule, in contrast to the 'electronic 
polarization.' due to changes in the electronic positions without appre- 
ciable motion of the nuclei. 

The foregoing shows that because of the infra-red nuclear vibration, 
the term 47rNa in the static dielectric constant should exceed n^l by 

an amount A/VJ 

. (5) 



The predicted sign of this difference is that found experimentally in the 
majority of cases. In HOI, for instance, Zahn 5 finds 4irNoi = 0-001040 

4 Strictly spouking this statement is not true, as there arc always rapidly varying 
instantaneous forties on the nuclei due to the continual changes in the positions of the 
electrons. Such forces, however, vanish on averaging over the electronic periods of 
motion, which are very short, and hence are inconsequential. 

5 C. T. Zahn, Phys. Rev. 24, 400 (1924). 



Ill, 15 MEASUREMENT OF ELECTRIC MOMENTS 47 

at 0., 76 cm., whereas extrapolation of 0. and M. Cuthbertson's 6 
dispersion data gives the smaller value ri^ I = 0-000871. 

Determination of e cfi from Absorption Intensities. Tn order to evaluate 
(5) it is necessary to know the magnitude of e cfl , which need not be at 
all like the charge of an electron or nucleus, as it is by definition not 
the charge of a single particle, but rather the differential coefficient 
dn/dr of the total molecular moment by inter-nuclear distance. For 
a non-polar diatomic molecule, for instance, e efl is zero. One method 
of determination of e Ga is by measurement of the absolute intensities of 
infra-red vibration bands, usually studied in absorption rather than 
emission. The absorption coefficient is proportional to (/* /x ) 2 , and 
hence very approximately to e* n (r~rl). T ne mos t accurate infra-red 
intensity measurements are probably those of Bourgiii 7 for HC1, who 
finds that here e off 0-86 x 10~ 10 e.s.u. Introducing this value of e cjl and 
the values m off -- 1-62 X ]0~ 24 , v vib -= 8-82 x 10 13 of the effective mass 
and vihrational frequency of HC1, we find that the right side of (5) is 
only l5x!0~ 6 , whereas we have seen that the experimental value of 
the left side is 1-7 X 10~ 4 . Values of the effective charge have also been 
calculated by Dennison 8 for HBr, CO, CO 2 , NH 3 , CH 4 from various 
intensity measurements. These values are all less than one-fifth the 
charge 4-77 x 10~ 10 of an electron; correspondingly, the expression (5) 
should be of the order 10 6 , and hence negligible. 9 Of course, absolute 
intensities and hence values of the effective charge are hard to measure 
with precision, but to account for a discrepancy between 4irN<\ and 
n^l as large as that 1-7x10 4 in HC1, for instance, the effective 
charge would have to be about 8-7 x 10 ' 10 instead of 0-86 X 10 10 c.s.u. 
Since the absorption coefficient varies as e* fl , the measurement of the 
absorption coefficient would have to be in error by a factor 1 00. 

As an instance of the difficulty of making accurate absolute, as distinct from 
relative, intensity measurements, we may cite the inability to deduce reliable 
values of the electric moment from the absolute intensities of 'pure rotation' 
absorption lines in the far infra-red (not to be confused with the vibration lines 

6 C. and M. CuthbortHon, Phil. Tratut. Roy. tfoc. 213A, 1 (1913). 

7 D. G. Bourgiii , Phys. Hev. 29, 704 (1927). Doimiscm deduces from Bourgiii's absorp- 
tion data the effective charge 0-949 X 10' 10 e.s.u. (Phys. Rev. 31, 501, 1928). Ho claims 
this to be more accurate than Bourgin's original value 828 ,c 10~ l . Still later Bourgiii 
raised slightly his own estimate to 0-86 X 10' 10 (Phys. jRcv. 32, 237, 1928). For our pur- 
poses it makes no appreciable difference which value is used. 

8 D. M. Dennison, Phil. Mag. 1, 195 (192G). 

9 This quantitative calculation of the contribution of the infra -red vibrations to the 
dielectric constant by means of tho effective charges yielded by absorption measurements 
was first made by tho writer, Phys. Rev. 30, 43 (1927). Tho difficulty of tho negligible 
contributions thus obtained appears to be too commonly overlooked in the literature. 



48 DIELECTRIC CONSTANTS, REFRACTION, AND TIJK III, 15 

in the nearer infra-red). Tho amount of absorption or emission in the pure rota- 
tional spectrum is proportional to the square of the electric moment /x, (rather 
than of e ctt ) and so should yield the numerical value of p, if absolute absorption 
measurements can be made. An attempt to determine fj, in this fashion was first 
made by Tolman and Badger, 10 using Czerny's absorption data on HC1. 11 
(Explicitly they calculated the intensity from the moment rather than the 
moment from the intensity, but tho two calculations aro simple converses.) 
Tho value thus found for the electrical moment of the HC1 molecule is less than 
one-third the standard value 1-03 X 10~ 18 obtained from Zahn's measurements of 
the temperature variation of tho dielectric constant. Subsequently Badger 12 
repeated Czerny's experiments in tho hope of removing this discrepancy, but 
instead increased it, as he foimcl an absorption only one-half as great as Czerny's. 
Thus the absolute measurements of absorption intensities in the pure rotation 
spectrum are apparently in error by a factor about 10 to 20. The determinations 
of absorption in tho vibration spectrum which are used in calculating the effective 
charge are presumably much more reliable, as they are in a much easier spectral 
region loss far out in tho infra-red. Even the vibration intensities, however, aro 
difficult to measure as exemplified by the fact that Balms' and Burmoister's early 
intensity data on. HC1 yielded according to Dennisoii's calculations* an absorption 
coefficient only one-sixth as great as that furnished by Bourgin's recent work. 
Tho latter is presumably much more accurate, and Bourgin himself explicitly 
states 13 that he does 110+ /think that them can be anything like enough error to 
permit an appreciable vibrational contribution to the dielectric constant in 1IC1. 

Thus the measurements on absorption intensities, if at all accurate, 
show that in molecules such as the hydrogen halides, the polarization 
due to infra-red vibration is too small to have any bearing on the 
discrepancy between the extrapolated square of the refractive index 
and the part of the dielectric constant due to induced polarization. 

Determination of e ott from Infra- Red Dispersion. A series of dispersion 
measurements in the infra-red should definitely settle whether the 
atomic polarization does really give an appreciable contribution to the 
dielectric constants. That is to say, such measurements would enable 
one to calculate the 'effective charge' by means of formula (4) and the 
values of e cfl thus obtained would presumably be much more reliable 
than those deduced from absorption coefficients. Unfortunately the 
available determinations of refraction sufficiently far in the infra-red are 
rather limited in number. Koch 14 measured the refractive indices of 
2 , H 2 , CO, 00 2 , and CH 4 at 6-70tyx and at 8-078/x, while Statescu 15 

10 Tolman and Badgor, Phys. Rev. 27, 383 (1926). In reading this paper, also rof. 12, 
the electrical moment should bo calculated by means of Eq. ( 1 ), Chapter II, rather than 
by a formula of the old quantum theory which they give. 

11 Czerny, Zeits.f. Physik, 34, 227 (1926). 

12 Badger, Proc. Nat. Acad. 13, 408 (1926). 

13 D. G. Bourgin, Phys. Rev. 32, 249 (1928). 

" J. Koch, Nova Acta Soc. Upsala, 2, No. 5 (1909). 
15 J. Statescu, Phil. May. 30, 737 (1915). 



Ill, 16 MEASUREMENT OF ELECTRIC MOMENTS 49 

even succeeded in measuring CO 2 at a wave-length as long as 13-19/4, 
and in addition supplemented Koch's data for CO 2 by various measure- 
ments between 1 and 11/4. The data are thus much more complete for 
CO 2 than for any other gas. 

Koch's work on H 2 and O 2 failed to reveal any anomalies in the 
infra-red, as was undoubtedly to be expected since non-polar diatomic 
molecules have no pure vibration spectra. The value 1-000332 which 
he found for the index of refraction n of CO at both 6-7/4 and 8-7/4 
was slightly lower than that in the optical region (e.g. 1-000335 at 
0-589/4) and agreed quite well with the value 1-000327 which would be 
obtained by extrapolation of optical data with neglect of vibrational 
resonance points. This is particularly significant since the fundamental 
band of CO is at 4-65/4, so that Koch's measurements extended beyond 
the vibrational singularity. The slight discrepancy between 1-000332 
and 1-000327 may be merely experimental error or perhaps indicate 
that there is a very small contribution 0-000005 of the atomic polariza- 
tion to n 1 or 0-00001 to n 2 1. Such a contribution is of no con- 
sequence for our studies of dipole moments, as it is smaller than the 
precision with which dielectric constants can be measured experi- 
mentally. Even such a very small contribution, if real, would demand 
an effective charge of the order 0-9e, whereas Dennison 8 estimated 
0-13e from the absorption data of Burmeister 16 and of Coblentz. 17 

A more striking result is obtained in carbon dioxide. The following 
values of the index of refraction are found by Koch or by Statescu at 
various wave-lengths: 

A- 1-0 20 3-0 4-0 5-0 6-7 8-7 11 13-19 ^ 

(n I)xl0 4 -^ 4-41 4-34 4-18 2-89 6-32 4-84 4-58 447 4-00 

The behaviour is thus different from that given by an ordinary optical 
dispersion formula which takes no cognizance of infra-red resonance 
points, and which predicts a steady and very gradual decrease of n from 
its value 1-000449 at optical wave-lengths (NaD lines) to a value 
1-000441 at infinite wave-lengths. 18 The anomalies shown by the table 
at 4-0 and 5-0/4 are due to the influence of the vibration band at 4-3/4. 
The abnormally low value 1-000400 of n at the longest wave-length 

18 B. Burmeister, Verh. d. D. Phys. Ges. 15, 689 (1913). 

17 W. W. Coblentz, Investigations of Infra-red Spectra, Part I, Carnegie Institute of 
Washington, 1906. 

18 Here, and also in the preceding discussion of CO, we make the extrapolations by 
means of the dispersion formulae given for CO 2 and for CO by C. and M. Cuthbertson, 
Proc. Boy. Soc. 9?A, 162 (1920). These formulae are typical of those based only on 
measurements in the visible and ultra-violet regions. 

3595.3 



50 DIELECTRIC CONSTANTS, REFRACTION, AND THE III, 15 

13-19jLt is undoubtedly because of another known vibration band at 
14-9/i. Reference to curve III, p. 44, shows that the index of refraction 
should be abnormally low on the short wave-length or high-frequency 
side of a resonance point, but high on the long wave-length side. Hence 
if measurements were available beyond 14-9/x, they would record a 
value of n considerably larger than the value 1-00044 given by an 
ordinary optical dispersion formula. Fuchs 19 has made a very careful 
comparison of the existing dispersion data for CO 2 , and has proposed 
a dispersion formula which represents the experimental points in the 
infra-red as well as optical region. The characteristic feature of his 
formula is that, besides the ordinary terms due to resonance with visible 
or ultra-violet frequencies, it contains two terms of the form (4) in which 
the resonance wave-lengths are taken as respectively 4- 31 /A and 14- 9 I/A, 
and in which the effective charges are taken to be 2-28e and 0-6 le, 
where e = 4-77 X 10" 10 e.s.u. Pcimison 8 deduced an effective charge of 
only 0-09e for the 14-9^ vibration from the measurements of absorption 
intensities by Cobleiitz 17 and by Rubens and Aschkinass. 20 Since deter- 
minations of absorption coefficients are probably much more difficult 
to put on a quantitative basis than those on dispersion, and since the 
absorption coefficient varies as the square of the effective charge it thus 
appears that existing measurements on absorption coefficients for the 
14-9 band in CO 2 are too small by a factor no less than (0-61/0-09) 2 or 
almost fifty, despite the fact that the data of Coblentz 17 and of Rubens 
and Aschkinass, 20 according to Dennison, 8 agree with each other to within 
20 per cent. Evidence that Fuchs's larger values of the effective charge 
are correct is furnished by the behaviour of the dispersion formulae 
at infinite wave-lengths. An ordinary dispersion formula such as 
that of Cuthbertson, which includes no atomic polarization, yields 
n ^_ 1^1-000882, whereas Fuchs's formula with the two infra-red 
resonance points yields ri^l = 1-000975. The value which Zahn 21 
finds for the dielectric constant of C0 2 under standard conditions is 
1-000968, while Stuart 22 finds 1-000987. Hence, according to the Cuth- 
bertson formula, the expression (3) is appreciably different from zero, 

19 O. Fuchs, Zeits.f. Phyaik, 46, 519 (1927). We interpret Fuchs's formula in terms 
of an effective charge different from e rather than in terms of an 'effective number of 
dispersion electrons' p t per vibration, which lias no real physical significance, as p l is 
not an integer. Our effective charge is connected with his number p t according to the 
relation (e e /e) 2 = 3p|. The factor 3 appears in this relation because he assumed the 
vibrations have three degrees of freedom rather than one. 

20 Rubens and Aschkinass, Ann. der Physik u. Chem. 64, 584 (1898). 
C. T. Zahn, Phys. Rev. 27, 455 (1926). 

22 H, A. Stuart, Zeits.f. Physik, 47, 457 (1928). 



Ill, 15 MEASUREMENT OF ELECTRIC MOMENTS 51 

and yields an electric moment 0-18 x 10~ 18 e.s.u. for the C0 2 molecule, 
whereas, according to the Fuchs formula, the expression (3) is zero 
within the experimental error and then CO 2 has no electric moment. 
This point was first noted by Wolf. 23 It is the consensus of opinion that 
the carbon dioxide molecule is without an electric moment, as this is 
shown, for one thing, by Stuart's 22 investigation of the temperature 
variation of the dielectric constant. The large effective charge assumed 
by Fuchs is, as we have seen, in nice quantitative agreement with 
this view. 

It may be noted that the Fuchs's dispersion formula has only two infra-red 
resonance points. It is well known that the OO a molecule has numerous other 
infra-red vibration bands besides those at 4-3/>t and 14-9/z.' 24 The fact that tho disper- 
sion measurements can be fitted quite well with only two resonance points must 
mean that these other vibrations have very low effective charges, or, much more 
probably, have low amplitudes 011 account of being 'combinations' or 'harmonics' 
rather than fundamentals. It is particularly noteworthy that the measurements 
at 2fji and 3/z can be fitted without including any term due to resonance with the 
quite pronounced absorption band at 2-72/*. 

In his dissertation (Upsala, 1924), not available to tho writer, Torston Wetter- 
blatt is reported to have explored tho dispersion in the vicinity of 2-72/z, and to 
find only a very slight anomaly when very close to this band, thus indicating pretty 
clearly that it is a harmonic or combination rather than a fundamental. In general 
a triatomic molecule has three fundamental modes of vibration, but the third 
fundamental may not show up in dispersion because it is an 'inactive* or 'sym- 
metrical' mode of vibration which gives rise to no oscillating electric moment. 
As mentioned by Wolf, the absence of a third intense resonance point in the 
infra-red dispersion lends considerable weight to Kucken's suggestion 24 that CO 2 
lias an * inactive' fundamental vibration at about 8/*. Inactive fundamentals 
are still allowed as Banian lines, and this 8/z, vibration does indeed play a lead- 
ing part in the Raman effect of OO 2 although the behaviour is irregular because 
of a complicated 'perturbation* by the harmonic of another vibration. 25 

The tremendous discrepancy between the effective charges deduced 
from absorption and from dispersion measurements for the 14-9 band 
of CO 2 makes one sceptical whether any information about the order of 
magnitude of the atomic polarization can be deduced from existing 
absorption data. Perhaps the best appraisal is that the effect of the 
atomic polarization on the dielectric constant is negligible in stable 
diatomic molecules, but not necessarily in molecules with more than 
two atoms. Our grounds for suggesting a smaller atomic polarization 
for diatomic than for polyatomic molecules are that: (a) there are no 

23 K. L. Wolf, Zeits.f. Phys. Chetn. 131, 90 (1927). 

24 See, for instance, the analysts by A. Euckon, Kelts, f. Physik, 37, 714 (1926) ; based 
on absorption curves by Schaofor and Phillips, ibid. 36, 641 (1926). 

26 Seo E. Fermi, Zeits.f. Physik, 71, 250 (1931). 

E2 



52 DIELECTRIC CONSTANTS, REFRACTION, AND THE III, 15 

very glaring discrepancies for diatomic molecules between the electric 
moments deduced from (3) and from the temperature variation of the 
dielectric constant (see table, 19); (6) Bourgin's recent determinations 
of absorption intensities in HC1 are probably more accurate than the 
early work of other investigators on C0 2 ; (c) the dispersion measure- 
ments reveal a considerably smaller effective charge for CO than for 
the 4-3/A vibration of CO 2 ; (d) one of the various fundamental vibrations 
in a polyatomic molecule usually has a longer wave-length and hence 
gives a smaller denominator in (5) than the sole vibration in a diatomic 
molecule. For the latter reason the 14'9/x vibration makes almost as 
large a contribution to the atomic polarization of CO 2 in the Fuchs 
formula at infinite wave-lengths as does the 4*3/z vibration, despite the 
fact that the latter has a considerably larger effective charge. 

Evidence that the atomic polarization is appreciable in polyatomic molecules 
is not confined to CO 2 , but is also revealed by the limited number of infra-red 
dispersion measurements available for methane (CH 4 ), viz. n = 1-000419 at 
6-557fA and n 1-000450 at 8-678/z. These measurements are, of course, insufficient 
to disclose the proper dispersion formula, but if tho anomalies which they exhibit 
are attributed to the influence of the vibration band at 7-7^, tho effective charge 
must be roughly 0-20e, 26 again larger than the effective charge 0-095e deduced by 
Dennison 8 from the absorption measurements by Coblentz. An effective charge 
0-20e for this vibration, will remove about one-tenth of the discrepancy between 
ganger's value 27 0-00096 for c 1 and the value 0-00086 for (n 2 !) obtained 
by extrapolation of optical dispersion data without considering the atomic 
polarization. As danger's investigation of the temperature variation of the dielec- 
tric constant of CU 4 shows that it has 110 electric moment, the discrepancy should 
disappear completely when proper corrections are made, and the other nine- 
tenths of tho discrepancy may be either experimental error or due to additional 
infra-red resonance points besides that at 7-7/z. 

It may be noted that in diatomic molecules such as HC1, often the discrepancy 
between n', 1 and ^rrNoL is only a fraction of ^TrNoc and that a itself is often small 
compared to {j?/3kT. Then either a small experimental error in the absolute 
value of the total dielectric constant or else in the electric moment, i.e. in the 
temperature coof licient of e, will suffice to explain away the discrepancy between 
^TrNa and n* 1. In HC1, for instance, an increase in the moment from Zahn's 
value 5 1-034 to 1-06 X 10~ 18 e.s.u. (which corresponds to an error of 6 per cent, in 
the temperature coefficient of e/N) will increase the contribution of the permanent 
dipoles to the dielectric constant enough so that tho remainder 4vrNoL to be ascribed 
to the induced polarization is decreased to a value 0-00087 1 in accord with optical 
data. The discrepancy is also removed if, instead of changing /x, we assume that 
the correct value of at 273, 76cm. is 1-00399 rather than 1-00416. In ammonia 
the polarization due to the permanent moments so far overshadows the induced 

26 To generalize the vibration formula (4) to molecules with more than two atoms, in 
particular CH 4 , it is necessary to replace w { by Wi^o/^i)/^ 1 ?* whore ff t and g fgi are 
dissymmetry and statistical weight factors explained in Dennisoii's paper. 8 

" K. Sanger, Phya. Zeits. 27, 656 (1926). 



Ill, 15 MEASUREMENT OF ELECTRIC MOMENTS 53 

polarization that an increase of only 0-3 per cent, in the moment, a change clearly 
within the experimental error, will diminish Zahn's 21 value 0-000768 for 47rA/ r a to 
a value 0-000729 in accord with the Cuthbertson 8 dispersion data. 

Often improvement in experimental technique in the temperature variation 
method has increased the values of the electrical moment and hence decreased 
the apparent excess of ^irNtx. over ri^ 1. A rather extreme example is the case 
of ethyl ether. From a study of old data by various experimenters on the tempera- 
ture variation of its dielectric constant, Debye concluded in the Handbuch der 
Radioloyie (vi, p. 625) that its electric moment was JLI 0-84 x 10" 18 , and that its 
value of 47rLa/3 was 38 cm" 3 . In order to make closer connexion with the usage 
in the literature, we here give the value of 47ra/3 where L is the Avogadro 
number, instead of 47TJVa. The expression 47rLoL/3 is called by Debye the induced 
molar polarization, or better, polarizability, and will be denoted by the letter /c . 28 
It differs from 4r7rNoi only by a factor L/'3N depending solely ori density, and has 
the advantage of being a molar quantity not requiring the specification of pressure 
or temperature. The value of (n^ l)L/3N obtained by extrapolation of disper- 
sion data is about 22. (Debyo originally gave 22-8, but Stuart suggests that a more 
accurate value is 22-0. 29 ) The discrepancy between 22 and 38 was so groat that 
elsewhere the writer considered it too great to attribute to experimental error. 9 
The possibility of sufficient experimental error seemed particularly unlikely 
because in ethyl ether the polarization by orientation is only a little over half the 
total polarization, so that a should be relatively insensitive to an error in the 
electric moment /z. However, careful recent experiments on the temperature 
variation of the dielectric constant of ethyl ether have recently been made by 
Stuart 30 arid by Hanger and Steiger. 31 Stuart iinds p, = 1 -Hi 0-03 X 10' 18 , 
K O -== 25-9, while Stinger iinds p -- 1 15 0-0 1 X 1 0" 18 , /c 26-1 in close agreement 
with him. If one uses these results the discrepancy between the values of K 
obtained from dielectric constants and from extrapolation, of dispersion data is 
only 3-9 or 4-1 as compared to 16 with the old data. Both Hanger and Stuart 
consider that even a discrepancy 3-9 is larger than the experimental error in K O . 
which they consider to be about 1-5. They therefore make the traditional sugges- 
tion that the refractive extrapolation is in error because of infra-red absorption 
bands. One cannot, nevertheless, help but wonder whether still further improve- 
ments in experimental refinement might remove all the discrepancy between the 
static and optical values of K O . This is unlikely in view of the excellent agreement 
between Stuart and Sanger, especially as their apparatus represents a high degree 
of experimental refinement, in marked contrast to the earlier work. 

28 Wo use the letters K, K O in place of Dcbyo's P, P n to denote respect i\ ply the total 
ami induced molar polarizabilities, as wo reserve the letter P for the electrodynamical 
polarization vector defined by tho rotation D E }-47rP. Wo shall refer to *, * as 
'molar polarizabilities ' rather than 'molar polarizations'. This change from Dobyo's 
usage seoms advisable since those quantities measure tho specific ability of tho material 
to acquire polarization, rather than tho total polarization, which depends on field 
strength. 

29 .Recent measurements of tho dispersion of ethyl ether by H. Lowory (Proc. Lond. 
Phys. Soc. 40, 23, 1928) give a value of n^l about 1 per cont. lower even than that 
used by Stuart. 

30 H. A. Stuart, Zcits.f. Physik, 51, 490 (1928). 

31 R. Sanger and O. Steiger, Helv. Phys. Acta 2, 136 (1929); also especially revision 
given by Sanger in Phys. Zeits. 31 , 306 (1930). 



64 DIELECTRIC CONSTANTS, REFRACTION, AND THE III, 16 

16. Independence of Temperature of the Index of Refraction 

Since we have seen that at visible frequencies the refraction results 
practically entirely from induced rather than permanent molecular 
moments, the optical index of refraction should not vary with the 
temperature except through the density. Such invariance is demanded 
by Eq. (2) and is also obviously to bo expected by analogy to the 
temperature behaviour of the static dielectric constants of molecules 
devoid of permanent dipolc moments, as in the optical region the per- 
manent polarity of a molecule is ineffective. The most thorough 
examination of the temperature variation of the index of refraction 
appears to have been made by Cheney. 32 He measured the refractive 
indices of air, N 2 , NH 3 , CO 2 , and SO 2 over a temperature range 0-300 C. 
and found that over this range the temperature coefficient of ri* 1 
(or of n 1, as n 2 I is approximately 2(ra 1) ) did not differ from the 
temperature coefficient of the density within the experimental error of 
1 or 2 per cent, in n 1. In other words, if v denotes the specific volume, 
the product v(n 1) remained constant with respect to temperature. 
The constancy of this product is sometimes spoken of as the Dale- 
Gladstone law. 

Slight departures from the Pale-Gladstone law are to be expected if the struc- 
ture of the molecule changes somewhat with temperature, as, for instance, due 
to dissociation, centrifugal expansion, &c. Meggers and Peters 33 find, for instance, 
that in the wave-length region 7500- 8700 A, the temperature coefficient of n I 
for air is exactly the samo as that 0-000367 of the density, but that the former 
coefficient increases in numerical magnitude to -0-000387 when the wave-length 
is diminished to 2500 A. These measurements are probably very accurate, though 
made over the very limited temperature range 30 C. Tlio departures from the 
Dale-Gladstone law which they find at 2500 A arc perhaps because air has an 
absorption band in the ultra-violet. Changes in temperature) will alter the distribu- 
tion of molecules among different quantized, rotational speeds, arid hence shift 
slightly the position of the maximum intensity in an absorption band, as the small 
molecular rotation frequencies are superposed on the electronic frequencies. 
A very small change in the location of such a maximum will, of course, materially 
affect the dispersion near resonance. On this view anomalies such as found by 
Meggers and Peters would have been absent if they had worked on monatomic 
vapours, devoid of the molecular rotation. 

17. Dispersion at Radio Frequencies 

We have treated only the two limiting cases of fields which are either 
static or else too rapid for orientation effects, without considering the 
gradual transition between the two cases. As already mentioned, the 

32 E. W. Cheney, Phys. Rev. 29, 292 (1927). 

33 Meggers and Petors, Bulletin of the Bureau of Stawlards, 14, 7 35 (1917). 



Ill, 17 MEASUREMENT OF ELECTRIC MOMENTS 55 

transition takes place in gases in the region of the natural molecular 
rotation frequencies, located in the far infra-red. Formulae for disper- 
sion in this region, based on the Kramers theory ( 82) and quantum 
mechanics, have been given by Debye, 34 but unfortunately there is not 
yet any experimental data adequate to test them. The most interesting 
feature is that the refractive index should display abrupt discontinuities 
when the incident frequency is resonant to any of the molecular rotation 
frequencies, which, because of the quantization, assume a discrete rather 
than continuous range of values. These discontinuities have, for sim- 
plicity, been ironed out in drawing curve III of Fig. 2. 

More stimulating and fruitful in experimental confirmation is the 
dispersion of liquids and solids at low frequencies. A classic theory of 
this has been developed by Debye 35 (not to be confused with his work 
on gases just mentioned). He assumes that the resistance to the orienta- 
tion of molecules by impressed fields arises primarily from a viscous 
force which, it is to be especially noted, is taken proportional to the 
angular velocity rather than angular acceleration, and which is probably 
a convenient approximate mathematical embodiment of the resisting 
effect of collisions. This viscous force is supposed more important than 
the inertial or acceleration reactions of the individual molecules, such 
as centrifugal force, which would be present even without collision 
phenomena. Because of this viscous retarding force, there is a definite 
upper limit to the rate at which a, field can orient a molecule, just as 
in mobility theory there is a maximum velocity of migration of ions, 
since the resistance is proportional to velocity rather than acceleration. 
Because of the large amount of viscous resistance, Debye finds that an 
incident field would not have an appreciable orienting influence on 
molecules in a liquid unless the incident wave-length were so very long 
as to be in the short radio rather than far infra-red region. His theory 
is very elegant, but would take us too far afield into liquids for the 
present volume, and also would require us to enter into the statistical 
theory of the Brownian movement, or its equivalent. Debye 's theory 
accounts nicely for the critical maxima of the absorption and of the 
dispersion dn/dX in a certain frequency region, and especially for the 
variation of the maxima with temperature. These phenomena permit 
the calculation of the * relaxation time' in which the molecules would 
de-orient themselves if a static field were suddenly removed. Debye 

34 P. Dobye, Polar Molecules, Chap. X; also further unpublished work by Manncback. 

35 P. Debyo, Vcrh. d. D. Phys. Gcs. 15, 777J^1913) f Polar Molecules, Chap. V; J. H. 
Tummers, Dissertation, Utrecht, 1914. ~ 



56 DIELECTRIC CONSTANTS, REFRACTION, AND THE III, 17 

also has extended his frictional theory to apply to solids, thereby 
explaining some of Errera's 36 interesting experiments on the anomalous 
dispersion of solids for waves of very long radio frequencies. Although 
many of the dielectric phenomena are explicable by treating the solid 
as a liquid of very high inner friction and viscosity, such a picture of 
a solid does not seem consonant with modern views of crystalline 
structure, and so Debye 37 modifies his frictional theory to allow the 
molecules to take up only certain particular orientations in solids. On 
the latter view the electric polarization of ice, for instance, is due to the 
fact that one H 2 molecule in five million in the ice crystal 'turns over' 
when an electric field of one volt/cm, is applied. 

It is rather striking to compare the orders of magnitude of the regions 
of anomalous dispersion due to interaction with molecular rotations in 
gases, liquids, and solids. The critical wave-lengths in the three cases 
are measured respectively in microns, centimetres, and kilometres. The 
corresponding values of the relaxation times for the liquids and solids 
are of the order 10~ 10 and 10~ 5 sec. respectively. It is clearly to be 
understood that we are here discussing only the effect of the molecular 
rotation. The anomalies in dispersion due to resonance with nuclear 
vibrations and electronic motions are, of course, in the near infra-red 
and ultra-violet. 

18. The Dielectric Constants of Solutions 

A pure polar liquid cannot in general be treated by the standard 
Langevin-Debye theory. One reason for this is that in such a liquid 
the local field e local is not at all the same as the macroscopic field E 
or even the Clausius-Mossotti expression E+4-7rP/3. Liquids have such 
high densities that the polarization P may be much larger than E, and 
hence the difference between e local and E is very great. Thus, until 
an adequate theoretical expression is available for the local field in 
dense media in which the inter-molecular distances are comparable with 
the molecular diameters, any attempts to determine quantitatively the 
intrinsic molecular polarity by measurements on pure liquids will be 
deceptive. An attempt, to be sure, to derive a theoretical expression 
for the local field in liquids has been made by Gans, 38 somewhat by 
analogy with the Weiss-Gans theory of magnetization. The faultiness 
of the underlying assumptions is evidenced by the fact that the electric 

36 J. Errera, J. de Physique, 5, 304 (1924); Polarisation Ditlectrique, pp. 127-30. 

37 P. Dobye, Polar Molecules, p. 102. 

38 R. Gans, Ann. der Physik, 50, 163 (1916) ; R. Gans and H. Isiiardi, Phys. Zeits. 22, 
230 (1921); H. Isnardi, Zeits. f. Physik, 9, 153 (1922). 



Ill, 18 MEASUREMENT OF ELECTRIC MOMENTS 57 

moments deduced by various experiments from pure liquids with the 
aid of the Gans theory are very frequently at variance with the values 
deduced by other more reliable methods, and hence should be guarded 
against in appraising the literature. 39 A particular complication in polar 
liquids is the probable existence of association or clustering, whereby 
several molecules combine to form a temporary unit very likely having 
a resultant moment quite different from that of a single molecule. 

Much valuable information on electric moments can, however, be 
derived by studying dilute solutions of polar molecules in non-polar 
solvents, as first suggested by Debye. Such a solvent is assumed not 
to contribute to aggregation effects, and to influence the local field only 
by adding a term 47^/3, in accordance with the Clausius-Mossotti 
relation (Eq. (34), Chap. I). We shall suppose the solute so dilutely 
dissolved that its contribution to the local field is also given by this 
relation. The total local field is then 



. (6) 

Here the subscripts 1 and 2 refer to the solvent and solute respectively. 
Since the solvent is supposed non-polar and the solute polar, the 
Langevin-Debye theory gives for the total polarization 

o( , ll , (7) 

provided we neglect saturation terms. If we introduce the 'mol frac- 
turns' A = NJW -\-N 2 ), f t = NJW+NJ, then 



where L is the Avogadro number, p is the density, and Jf l5 M 2 are the 
molecular weights of the two constituents. On using (6), (8) and the 
definitive relation P/E (e 1)/4?7, we find that (7) becomes 



with the abbreviations 



KI = '"p , * 2 = "- 1 ,+ _TL | =- 2-64 X 



(10) 

39 For instance, the very full compendium on electric moments given by O. Bliih in 
Phya. Zeits. 27, 226 (1926) does not emphasize which values of the moments tabulated 
therein are unreliable on account of being deduced from measurements 011 pure liquids. 



58 DIELECTRIC CONSTANTS, REFRACTION, AND THE III, 18 

The left side of (9) is termed the molar polarizability of the mixture, 
and will be denoted by * 12 . As f : = 1 / 2 , Eq. (9) demands that /c 1>2 
be a linear function of the concentration of the solute when the latter 
is varied. Actually this is usually not the case, as shown, for instance, 
by the following graphs taken from Debye's Polar Molecules. M In each 
instance benzene is utilized as the non-polar solvent. In only one of 
the three cases, viz. ethyl ether, is the experimental curve the straight 
line demanded by (9). The reason for the departures from linearity is, 
of course, simply because with high concentrations of the polar solute 




Ethyl alcohol in benzene. 



f--0 Z 4 .6 .8 H 

Ethyl other in hoir/cno. 
FIG. 3. 



f=0 ^ A 6 8 M 

.Nitrobenzene in benzene. 



the contribution of the latter to the local field cannot be calculated by 
the Clausius-Mossotti relation. 

From the various polarizability-concentration curves for the different 
materials valuable information can be obtained on the processes of 
association present in a polar liquid. 41 Any discussion of this subject 
is clearly beyond the scope of the present volume. The cases in which 
the curves are concave upwards and concave downwards evidently indi- 
cate quite different types of association. If one imputes all the curva- 
ture to the solute rather than the solvent, then the graphs will enable 
one to determine the molar susceptibility of the solute as a function 
of the concentration. Instead of being independent of the concentra- 
tion, as the simple theory would demand, it is found in some cases to 
increase, some to decrease, and in some instances to increase and then 
decrease as the concentration of the solute is gradually increased from 
zero to unity. The theoretical interpretation of such differences is at 
present a little obscure, but they should be valuable cJues to future 
theoretical investigation. 

It is interesting to note that for non-polar substances the molar 

40 P. Dobye, Polar Molecules, pp. 46-7. 

41 Cf., for further details, P. Debye in Marx's HanrJbuch der Radiologie, vi. 663 ; 
L. Ebert, Zeits.f. Phys. Chew. 113, 1 ; 114, 430 (1924) 



Ill, 18 MEASUREMENT OF ELECTRIC MOMENTS 59 

polarizability (e -l)Jf/(e-f2)p is almost identical in the liquid and 
vapour states, whereas for polar materials it has widely different values 
in the two states, presumably because of association in the liquid. 
Zahn, for instance, finds that it equals 3-869 and 4-395 respectively 
for O 2 and N 2 gas, while the corresponding values in the liquid state 
are 3-878 and 4-390, respectively determined by Werner and Keesom 
and by Gerold. 42 As an example of the great difference between the 
molar polarizabilities in the two states in the case of polar materials, 
we may cite that (e - l)M/(+2)p equals 4 and 1 8 respectively for water 
in the vapour and liquid states. The discrepancy between the values 
of (n 2 l)MI(n 2 -}~2)p for water in the two states is, nevertheless, only 
about 10 per cent, at sodium wave-lengths, as already mentioned in 
5, which clearly shows that association effects are unimportant at 
optical frequencies. 

If the departures from linearity in Fig. 3 are attributed solely to 
characteristic polarity effects, the curves should be straight lines in two 
cases: (a) binary mixtures of two non-polar materials, (b) binary 
mixtures of either polar or non-polar materials in which the optical 
refractivity rather than static dielectric constant is investigated, and 
at sufficiently short wave-lengths to suppress orientation effects. In 
this latter case a formula analogous to (9) should be applicable, except 
that e is replaced by n 2 , and that the theoretical expressions for the 
/c's are no longer (10) but are instead proportional to the expressions 
(2). The predicted linearity for case (a) is well confirmed experimentally, 
as, for instance, in Krchma and Williams's 43 work on mixtures of benzene 
and carbon tetrachloride. As regards case (6), refractive data for various 
binary mixtures show that the experimental values of (w 2 l)J//(w a +2)p 
for these mixtures usually do not differ by more than a few parts in a 
thousand from the values calculated on the basis of linearity. 44 

Reverting now to the dielectric constants of polar solutes in non-polar 
solvents, it is only at very low concentrations of the polar material, 
i.e. the extreme left portions of the graphs in Fig. 3, that there is any 
approach to gas-like conditions and that formulae such as (9) should 
be applicable. However, the asymptotic behaviour at zero concentra- 

42 For references see Zahn and Miles, Phys. Rev. 32, 502 (1928). Tho good agreement 
in the two states was apparently first noted by Ebert and Keesom, Proc. Amsterdam 
Acad. 29, 1888 (192tt). The value quoted for liquid N a is determined from refractive 
data rather than from, the static dielectric constant. 

43 Krchma and Williams, J. Amer. Chem. Soc. 49, 2408 (1927) ; cf. also Grutzmacher, 
Zeits.f. Physik, 28, 342 (1924). 

44 Cf . for instance, Schubt, Zeits.f. Phys. Ghem. 9, 349 ; Hubbard, ibid. 74, 207 (1910) ; 
also especially Hojendahl, Dissertation (Copenhagen, 1928), p. 27. 



60 DIELECTRIC CONSTANTS, REFRACTION, AND THE III, 18 
tion should agree with (9). Hence, if a straight line is drawn tangent 
to the experimental curve at the origin, its equation should be 



The intercept of the extrapolated tangent line with the right-hand axis 
f = 0, / 2 = 1 gives the value of * 2 . Hence, by determining the rate of 
change of the dielectric constant when small amounts of a polar material 
are dissolved in a non-polar solvent, one can find the molar polarizability 
of this material. To determine the electric moment one must isolate 
the two terms of K% representing the induced and dipole polarization. 
This can be done in either of two ways: either by measuring the tem- 
perature coefficient of the dielectric constant of the weak solution, or 
else by extrapolation of refractive data for the polar material, which, 
as explained in 14, enables one to determine the induced molar 
polarizability 47rLa 2 /3. Because aggregation effects are not important 
at optical wave-lengths, the refractive measurements need not be made 
in solution, but instead can be made on the pure polar liquid or, better 
still, on its vapour. 

Li some cases it may happen tluit no refractive data are available for tho mate- 
rial in question. In such cases tho contribution AnNa of the induced polarization 
is sometimes determined by one of the two following approximate methods: 
(1) calculation of the refraction of tho material from that of its constituent atoms 
or radicals by tho additivity method, highly elaborated to allow for the different 
kinds of chemical bonds (see 21) ; or (2) assumption that tho dielectric constant 
c BO iid in the solid state, if available, is the same as n'^. The theoretical work of 
Debye 35 and the experiments of Krrera and Wintsch 36 show that in a truly 
static field e 80]1(i is much larger than wj, unless tho temperature is much lower than 
the melting-poin t . hi the case of ice, for instance, the < lielcctric constant is near tho 
melting-point about the same as that of water, or about 80, so that the molar 
polarizability ( ft>M l)M/p(c Holld + 2) is about IS, whereas (/i 2 1) W JH //)(/, -\- 2) -= 4. 
This difficulty can, however, be at least partially overcome by measuring the 
dielectric constant of tho solid well below the freezing-point and at radio fre- 
quencies, which are large compared to the 'relaxation frequency ' of the solid, and 
hence too great to permit alinement of the di poles in tho solid. The dielectric 
constant of ice, for instance, is only 4-6 at 2 0. when the wave-length is 8 kilo- 
metres. The corresponding molar polarizability nevertheless still has an excessively 
high value 10, as -2 C. is too near the melting-point to freeze in tho dipolos 
completely. 

19. Numerical Values of the Electric Moments of Various 
Molecules. Comparison of the Different Methods 

The material thus far presented has revealed four methods, I gas , IT gas , 
I gol , II sol , for the quantitative determination of a molecule's electric 
moment. These methods are: 

I gas . This consists in measurement of the dielectric constant of a gas 



Ill, 19 MEASUREMENT OF ELECTRIC MOMENTS 61 

or vapour over a range of temperatures. If the measurements of the 
dielectric constant at various temperatures, when reduced to a standard 
density, say that at 76 cm., 27 3 K., can be represented by a formula 
of the form 



then it follows immediately from the Langevin-Debye formula (1), 
Chap. II, that the electric moment is given by 



IL. na . In the second method the dielectric constant need be measured 

fetlH 

at only one temperature. The electric moment is then deduced from 
comparison, with extrapolated refractive data, through the aid of Eq. (3). 
This method is precise only if the 'atomic polarization' due to the infra- 
red bands is negligible, or in the rare event that dispersion measure- 
ments are available which include the effect of these infra-red vibrations. 

I Hol , II sol . The third and fourth methods are similar to the methods 
I Kas , II gas except that the measurements are made in the fashion ex- 
plained in IS on dilute solutions of the material in a non-polar solvent 
instead of in the pure gaseous or vapour state. 

All these methods were originally suggested and stimulated by Debye. 
The method I gas has recently been used extensively by Zahn, 45 
Sanger , 46 > 47 > 48 > 4D Stuart, 50 ' 51 ' 52 Braimmiihl, 53 and others. Method ll gas 
has been most comprehensively applied by Hojendahl, 54 using measure- 
ments of dielectric constants of various gases made by Pohrt in 1913. 55 
In the table we have tried to supplement Hojendahl's calculations by 
applying the method to some of the more recent determinations of 
dielectric constants. Method I gol has been employed by Miss Lange 56 

45 C. T. Zahn, Phys. Rev. 24, 400 (1924) (HC1, HBr, HI, H 2 , O 2 ); 27, 455 (1926) 
(CO 2 , NH 3 , SO 2 , N 2 ); Zahn and Miles, ibid., 32, 497 (1928) (CO, COS, CS 2 , H,S); Zahn, 
ibt I. 35, 1047 (acetic acid) ; 35, 848 (1930) (revision for CS a ). 
li. Siinger, Phys. Zeits. 27, 656 (1926). 

R. Sanger and O. Steiger, Ilelv. Phys. Acta 1, 369 (1928); 2, 136 (1929). 
R. Sanger, Dipolmoment und cliemixchc Struktur (Loipziger Vortrage, 1929), p. 1. 
R. Sanger, Phys. Zeits. 31, 306 (1930). 
H. A. Stuart, Zeits. f. Physik, 47, 457 (1928). 
H. A. Stuart, Zeits. f. Physik, 51, 490 (1928). 

H. A. Stuart, Phys. Zeits, 31, 80 (1930). This article quotes unpublished measure- 
ments by Fuohs. 

63 H. v. Braunmuhl, Phys. Zeits. 28, 141 (1927). 

54 K. Hojondahl, Studies of Dipole-Moment, Copenhagen, 1929; brief summaries in 
Phys. Zeits. 30, 391 (1929); Nature, 117, 892 (1926). 

65 G. Pohrt, Ann. der Physik, 42, 569 (1913). 

66 L. Lange, Zeits. f. Physik, 33, 169 (1925). Most of Miss Lange's determinations 
are really a hybrid of methods I and II. 



62 DIELECTRIC CONSTANTS, REFRACTION, AND THE III, 19 

and especially by Smyth 57 and associates, while II Bol has been utilized 
for a very large number of substances by Williams 58 59 60 and co-workers. 

It is not the purpose of the present volume to discuss the technique 
of experimental methods, but we may nevertheless mention that prac- 
tically all the recent observations of dielectric constants are made by 
a 'heterodyne' method, in which the periods of two oscillating circuits 
are adjusted to be virtually identical. One circuit contains only known 
resistances, inductances, and capacities, while one unit in the other 
circuit is a condenser containing the gas or solution whose dielectric 
constant is desired. The beat phenomenon enables one to determine 
when the periods of the two circuits approach equality. The period of 
the first circuit can be calculated from its known constants, while con- 
versely from the thus determined period, the hitherto unknown capacity 
of the condenser in the second circuit, and hence the desired dielectric 
constant, may be found. The use of the vacuum tube is the cornerstone 
to the successful application of the heterodyne method. Most measure- 
ments of dielectric constants, and especially their temperature varia- 
tions, made prior to 1920, before perfection of the technique of the 
vacuum-tube circuits, are not as a rule very reliable. 61 Hence in our 
table we have not included the results of Badeker, 62 Jon a, 63 and others, 
although the importance of their pioneer work must not be overlooked. 
Pohrt's 55 measurements of the dielectric constants of gases at mainly 
one temperature are possibly somewhat more accurate than usual for 
early work, although the resulting moments are very often somewhat 
high, perhaps because method II neglects the 'atomic polarization'. 

The four methods described above are at present the most dependable 
ways of determining quantitatively the dipole moments of molecules. 

57 C. P. Smyth ami S. O. Morgan, J. Amer. Chcm. Soc. 49, 1030; 50, 1547 (1928); 
C. P. Smyth and W. N. Stoops, ibid. 50, 1883; C. P. Smyth, S. O. Morgan and J. C. 
Boyco, ibid. 50, 1536 (1928). 

58 J. W. Williams and Krchma, J. Amcr. Chcm. tioc. 49, 1676, 2408 (1927); Phys. 
Zi'its. 29, 204 (1928); Williams and Allgeior, J. Amer. Chem. tioc. 49, 2416 (1927); 
Williams and Ogg, ibid. 50, 94 (1928); Williams and Schwingel, ibid. 50, 362 (1928); 
summary and references to the appropriate individual papers for each mixture in Phys. 
Zeits. 29, 174, 683 (1928), or the following reference 59 . 

59 J. W. Williams, Molekulare Dlpolmotnente und ihre Bedeutung fur die chemiche 
Forschung. This is Band 20, Heft 5 of the series Fortschritte der Chemie, Physik, und 
physikalischen Chemie. 

60 C. H. Schwingel and J. W. Williams, Phys. Rev. 35, 855 (1930). 

61 For a description of typical experimental arrangements, see, for instance, Williams, 
I.e. 59 or Estormann's and Sack's articles in Ergebnisse der exakten Naturwissenschaften, 
VIII. 

62 K. Badeker, Zeits. f. Phys. Chem. 36, 305 (1901). 

63 M. Jona, Phys. Zeits. 20, 14 (1919). 



Ill, 19 MEASUREMENT OF ELECTRIC MOMENTS 63 

It must not, however, be inferred that there are not other experiments 
which should in principle permit the numerical determination of these 
moments. We have already mentioned on pp. 47-8 that the electric 
moment can be directly calculated from absorption coefficients for the 
pure rotation spectra in the far infra-red if these coefficients can be 
measured with quantitative precision, but there is apparently some 
enormous unknown systematic error which has as yet prevented this. 
Raman and Krishnan 64 have met with some success in deducing dipole 
moments from a combination of data on the Kerr effect and on the 
depolarization of light, but the complete theory of these effects is 
extremely complicated (cf. 83), and it is hard to say whether the 
moment can accurately be deduced from the experimental measure- 
ments in as simple a manner as implied by their formulae, although the 
latter are doubtless approximately correct. In some cases the electric 
moments have been deduced from the amount of electrostriction. 65 
Determinations of electrostriction are, in fact, merely one way of 
measuring the dielectric constant. Saturation effects have been used 
to calculate the dipole moment (see 22), but they are far too small to 
measure with precision, and furthermore they yield the moment only 
if one assumes that the induced polarization is a strictly linear function 
of field strength so that saturation is evidenced only in the orientation 
term. Attempts 66 have been made to calculate the molecular moments 
from the potential differences at interfaces between two materials, on 
the assumption that this difference is due entirely to a surface-layer of 
dipoles. The results thus obtained are not quantitatively reliable, and 
this is not surprising, as the molecules may not be 100 per cent, oriented 
as assumed in the simple theory, and especially there may well be at 
the interfaces a tremendous amount of molecular distortion and induced 
polarization. 67 Accurate measurement and analysis of the Stark effect 
or, what is more or less equivalent, of the electrical Stern-Gerlach 
effect, should in principle permit the calculation of the electric moment 
provided one can resolve the contributions of the induced and per- 

M Raman and Krishnan, Phil. Mag. 3, 713 (1927). They deduce the moments 
1-04 X 10- 18 and 1-66 X 10' 18 for HC1 and CH 3 C1 respectively. Reference to the table shows 
that the agreement with values obtained by the standard methods is much better for 
HC1 than for CH 3 C1. 

63 O. E. Frivold, Phys. Zeits. 22, 603 (1921); O. E. Frivold and O. Hassel, ibid. 
24, 82 (1923); Kliofoth, Zeils. f. Physik, 39, 402 (1926). Kliefoth finds no electric 
moment for O 2 and N 2 , and the values 1-7 X 10" 18 and 0-20 X 10' 18 for SO a and CO 2 
respectively. 

66 Rideal, Surface Chemistry, pp. 236-7, Cambridge University Press, 1926. 

" Frumkin and Williams, Proc. Nat. Acad. 15, 400 (1929). 



64 DIELECTRIC CONSTANTS, REFRACTION, AND THE III, 19 

manent polarizations. 68 The Born-Lertes rotation effect has also been 
used to calculate the electric moment, but the results have not been 
particularly successful. 69 

It would clearly be an unnecessary duplication to tabulate all the 
molecular electric moments which have been determined by any of the 
four main methods, as very complete tables, up to date at the time of 
this writing, have been given by Debye in the German edition of his 
Polar Molecules with a subsequent supplement published in 1930, 70 by 
Hojendahl in his dissertation, 54 by Williams in his monograph Dipol- 
momente und ihre Bedeuluny fur die chemische Forschung, 59 and by 
Estermann and by Sack in Band VIII of Ergebnisse der exakten Natur- 
wissenschaften (1929). 71 In the accompanying table we have attempted 
to include only the common inorganic molecules which have been 
measured and a selected group of organic ones. In making the selection 
for the latter we have aimed to list the molecules whose moments have 
been determined by the greatest number of different observers, and 
especially by as many of the four methods as possible. It is hoped that 
our placing in juxtaposition the results of the various methods in a 
single table rather than in separate ones will enable the reader to 
estimate more quickly the accuracy and consistency of the different 
types of observations. Attempts have sometimes been made to give 
the moments to one more significant figure than given in the table, but 
the light of experience, especially as revealed in the continual incon- 
sistencies between the different observations, seems to show that very 
often the experimenters underestimate their errors, and a determination 
of the moment to within 5 per cent, must be regarded as quite satis- 
factory. We have appended question-marks to some of the values which 

158 For further discussion of the Stark effect in relation to molecular structure see the 
end of 37. The attempt of R. J. Clark (Proc. Roy. 8oc. 124A, 689 (1929)) to deduce 
electric moments quantitatively from his measurements on the electric Stern-Gerlach 
effect appears erroneous to the writer, as he assumes the dipoles are alined either parallel 
or antiparallol with respect to the field. Actually the theory of the Stark effect for non- 
monatomic molecules shows that at any ordinary field strength the orienting effect of 
the field is very small because of the molecular rotation ; this is evidenced by the fact 
that (71), Chap. VI has J in the denominator. 

69 P. Lertes, Zciis. f. Physik, 6, 56 (1921). 

70 The very complete table of moments prepared by Sangor for the Gorman translation 
(Polare Mokkeln, pp. 191-8) was unfortunately not ready for the original English edition. 
The two 'Nachtrags' to Ihe table are sold separately. A full table of moments has just 
appeared in Smyth's now book, Dielectric Constant and Molecular Structure. 

71 Besides those references we may mention that recent measurements on the dipole 
moments of certain organic molecules and interesting discussions of the relation of 
dipole moment to chemical and physical properties are given by various writers in 
Dipolmoment und chemische Struktur (Leipziger Vortrage, 1929). 



Ill, 19 MEASUREMENT OF ELECTRIC MOMENTS 65 

seem particularly doubtful, and asterisks to values which are probably 
zero within the experimental error. The various investigators differ 
considerably in their usage in giving explicitly numerical moments 
which are virtually zero, and in some cases the observations which we 
have listed as exactly zero would yield moments about as large as those 
with asterisks if an attempt is made to calculate small moments literally 
from their data. In such instances the molecules are in all likelihood 
non-polar. 

Let us turn now to some of the specific items in the table. The 
vanishing electric moments reported for A, H 2 , N 2 , O 2 are to be expected, 
since monatomic molecules and diatomic molecules composed of two 
identical atoms are theoretically non-polar. For this reason the finite 
moments recorded for Br 2 and I 2 are hard to believe. The moment 
0-40 X 1 0' 18 found by Miss Anderson 72 for bromine is based on measure- 
ments in the pure liquid rather than gaseous state, and readings were 
taken only over the very limited temperature interval 0-30. For these 
reasons her results do not seem very conclusive. Even if the tem- 
perature variation of the susceptibility per molecule for Br 2 is real, it 
need not imply an electric moment if the induced polarization changes 
with temperature. 73 Such a change is not allowed for in the usual 
simple theory, but in relatively unstable molecules such as the halogens 
it is not inconceivable that there be a change due to the centrifugal 
expansion with increasing temperature, to say nothing of the possibility 
of a small amount of dissociation. It is significant that Miiller and 
Sack 74 find that the moment of the iodine molecule becomes zero when 
hexane is used as a solvent, so that the apparent non-vanishing moment 
found with benzene as the solvent is doubtless due to some sort of 
spurious association effect. The true moment of the I 2 molecule is thus 
zero in all probability, and this is hence also presumably true of F 2 , 
C1 2) Br 2 . 

Except for the figures given in bold-face, no attempt is made to 
include the atomic polarization in using the methods II gas and II B01 . It 
is seen that on the whole the figures in the second and fourth columns 
agree quite well with those in the first and third, sometimes about as 
well as the different observations by the same method. Thus the cal- 
culation of the electric polarization does not seem ordinarily to be very 

72 Annie I. Anderson, Proc. Lon. Phys. Soc. 40, 62 (1928). 

73 The possibility of temperature variations duo to other causes than a permanent 
dipole moment is discussed at length by L. Ebert in Dlpolmoment und chemische Struktur 
(Leipziger Vortrage, 1 929), although primarily for large, complicated molecules, 

74 H. Miiller and H. Sack, Phys. Zeits. 31, 815 (1930). 

3595.3 F 



66 



DIELECTRIC CONSTANTS, REFRACTION, AND THE III, 19 



ELECTRIC MOMENTS OF SOME REPRESENTATIVE MOLECULES IN E.S.TJ, 
(All values to be multiplied by 10~ 18 ) 

[Superscripts givo footnote reference to the observer] 





Method 


Method 


Method 


Method 


Solvent used in 


Molecule 


*v 


"gas 


^SOI 


^BOl 


solution methods 


Argon 


Q53 










Hydrogen (H a ) 


Q53 


O' 8 








Nitrogen (N 2 ) 


45,75 


() 76 


. . 


Q77 


Pure N 3 


Oxygen (O a ) 


Q 75 


o 78 




77 


l^uro O 2 


Bromine (Br 2 ) 






0-40'2 


10? 78 


Pure Br 2 


Iodine (T 2 ) 








1-2? 88 


Benzene 


,, 








1-0? 74 





,, 








O 74 


Hexano 


Hydrochloric acid (HC1) 


1-03 


1-06 78 








Hydrobromic acid (HBr) 


0-79 


0-80 78 








Hydriodic acid (HI) 


0-38 


0-41 78 








Carbon monoxide (CO) 


0-12 53 


0-14 78 












(0-12) 











0-12 75 


V79 








,, 


O10 45 


0-13 78 












(Oil) 








Carbon dioxide (CO 2 ) 


0-14? 53 






m 




,, 


0-06* 45 


0-17 78 (0) 








,, 


0-00 5 


0-18 78 (0) 








Nitrous oxide (N 2 O) 


Q 80 


0-13 78 








,, 


GU 


0-17 78 








Carbon disulphide (CS 2 ) 


80,45 






0. 06 * 58 


Benzene 


f . 


Q60 


0-28 78 




0-08* 58 


Hexane 


Sulphur dioxide (SOo) 


1-61 45 


1-63 78 








Water (H 2 O) 


1-84 49 


1-81 81 


. . 


1-81 82 


Benzene 


Hydrogen sulphide (H 2 S) 


MO 53 


092 78 











0-93 


0-94 78 








Ammonia (NH 3 ) 


1.4888 


1-47 78 








,, 


1-44 45 


1-44 78 








Acetylene (C 2 H 2 ) 


()84 


m 








Ethylene (C 2 H 4 ) 


Q84 










Ethane (C 2 H fl ) 


o 8 * 










Methane (CH 4 ) 


Q4C 


0-18 78 








Methylchloride (CH 3 C1) 


1-69 85 ? 


1-89 86 











1-86" 


1-90 78 








,, 


1-89 M 










Methylene chloride 


1-59 6 






1-55 74 


Benzene 


(CH 2 C1 2 ) 












,, 


1-61 87 










Chloroform (CHC1 3 ) 


0-95 


1-1 ?V 


1-05 57 




Hoxane 





1-0585 






1-10 58 


Benzene 











i-jr> 58 


CC1 4 


Carbon tetrachloride (CC1 4 ) 


46 






o 58 


Benzene 


Tin iodide (SnT 4 ) 








O 58 


tt 


Ethyl ether (C 2 H 5 ) 2 O 








1-22 88 


Benzene 


,, 


1-14" 


1-21 78 


1-15 90 


j.22 56 ' 58 


n 





1-15 






1-24 58 


CC1 4 





1-12 M 











Ill, 19 



MEASUREMENT OF ELECTRIC MOMENTS 



67 



Molecule 


Method 

/gaa 


Method 


Method 

/BOl 


Method 

n^ 


Solvent used in 
solution methods 


Methyl ether (CH 3 ) a O 


1-29 51 


1-33 78 











1-32 47 


1-37 78 








ft 




1-23 86 








Propyl other (C 3 H 7 ) 2 O 


0-86 47 


1-24 86 




t 




Acetone (CH 3 COCH 3 ) 


2-94 51 


2-88 51 




2-7(> M 


CC1 4 







2-97 8 




2-72 89 




Benzene (C 6 H 6 ) 


. . 


0-33 ? 8 




0-06 


CC1 4 


,, 






o' 9 ' 1 




Pure benzene 


,, 








0-10 58 


CS 2 


tt 








0-08 M 


Hoxatio 


Fluorbenzene (C 6 H 5 F) 








1-39 92 


Benzene 


Chlorobonzene (C 6 H 5 C1) 






1-52" 


1-55 58 


Hexano 


,, 








1-52 58 


CS 2 


,, 








1-55 58 


Benzene 


)t 








1-56 93 


tt 


ft 


, 


m 




1-57 74 


tt 


tt 








1-61 57 


t) 












Hoxano 


Bromobenzene (C 6 H & Br) 








1-56" 


Beiizono 


>f 








1-51 5t 


>} 











1-52 74 





lodobeiizene (C 6 H 5 I) 




;; 




1-30 9S 


" 


tf 








1-25" 


ff 


Nitrobenzene (C 6 H 5 NO 2 ) 








3-89 58 


Hexane 


,, 








3-89 58 


cs a 











390 58 


Bonzono 


)f 


, 






3-84 M 


ft 


Hexane (C 6 H 14 ) 


. . 






0-10 94 


}J 


,, 








0-08 9 * 


CC1 4 


,, 






0" 




Pure Hexane 


Kthyl alcohol (C 2 H 6 OH) 


1-70 05 


1-72 86 




.74 96 


Benzene 


,, 




. . 




63 58 


CC1 4 


Methyl alcohol (CH 8 OH) 


1-68 95 


1-73 8 




64 96 


Benzeno 











. 67 96 


CC1 4 


n-Propyl alcohol (C 3 H 7 OH) 


1-66 95 






53 5 


Benzene 


iso-Propyl alcohol 








.7590 


Benzene 


iso-Arnyl alcohol (C 5 H 1X OH) 








-85 58 


CC1 4 


" 








62 96 


Bonzono 



75 Magdalena Forro, Zefat.f. Phyvik, 47, 430 (1928). 

79 To show how exactly e and ri^ agree for H 2 , O 2 , N 2 , and hence how precisely method 
Ilgaa shows that these gasos must bo non-polar, wo can here give some of the measure- 
ments of n-^j and e. For hydrogen the value of n^ yielded by various dispersion measure- 
ments, including the infra-red data of Koch, is 1-000273 ; tho corresponding values of the 
dielectric constant are 1-000273 (Tangl, Ann. dcr Phyaik, 23, 559, 26, 59 (1907-8), 
1-000263 (Fritts, Phy*. Rev. 23, 345, 1924), 1-000259 (Braummihl), 53 1-000265 (Zahn). 45 
For nitrogen, tho value of ri^ ranges from 1-000580 to 1-000589 according to the 
observer, while Fritts finds e 1-000555, and Zahii obtains e 1-000581. For oxygen 
n^ is very approximately 1-000530 (Lowery 29 ), while -= 1-000507 (Fritts), 1-000518 
(Zahn) 45 . 

77 That method II BO i gives zero moment for N a and O a just as well as method II gM is 

F2 



68 DIELECTRIC CONSTANTS, REFRACTION, AND THE III, 19 

greatly impaired by the omission of the atomic polarization in method 
II, except when a high degree of accuracy is desired. In particular in 
the halogen hydrides and ammonia, the good agreement between the 
two methods shows that in these molecules the atomic polarization must 
be small, as also indicated by absorption measurements (15). In the 
case of propyl ether, on the other hand, the atomic polarization must 
be enormous if the observations in both columns are dependable, which 
is doubtful. In a few instances, the moments recorded by method II 
are actually smaller than those given by method I, and in such cases 
the discrepancy cannot be blamed on the atomic polarization. More 
often, however, method II seems to give slightly larger moments than 
method I, thus furnishing evidence for a certain amount of real 'atomic 

shown by the fact that N 2 and O 2 have almost exactly tho same molar polarizabilities in 
the liquid and gaseous states. See p. 59. 

78 Calculated by the writer from comparison of extrapolated dispersion data with the 
measurements of tho dielectric constant made by the observer listed in the column 
directly to the left in tho same row. Those calculated values are very often only approxi- 
mate. The values of n 2 , have usually been obtained from dispersion formulae given 
by C. and M. Cuthbertson (Proc. Hoy. Soc. 83, 171 (1909) (SO 2 , H 2 S); 97, 152 (CO, 
C0 2 , CH 4 ), Phil Trans. Roy. Soc. 213, 1 (1914) (Br a , HC1, HBr, HI, N 2 O, NH 3 ). As 
long as infra-red resonance points are not included, it would make little difference if 
we used the visible dispersion data of other observers (e.g. the more modern data for 
CHC1 3 , ethyl ether, methyl ether, and acetone given by Lowery 29 ) as the discrepancy 
between tho different refractive measurements is usually small compared to the error 
in the measurements of dielectric constants. More refined calculations appear useless as 
long as the amount of atomic polarization is uncertain, but tho results which are tabulated 
suffice to show that in any case this polarization cannot be very largo (except perhaps in 
propyl other). 

79 Forro finds a value of e for CO smaller than n^ . This must bo experimental error, 
as it would yield an imaginary moment in method II. Similar remarks apply to the data 
of Ghosh, Mahanti, and Mukhorjee 80 on CS a . 

80 P. N. Ghosh, P. C. Mahanti, arid B. C. Mukhorjoe, Zcits. f. Physik, 58, 200 
(1929). 

81 Calculated from Zahn's 45 data by Stuart 51 . 

82 J. W. Williams, Phys. Zeits. 29, 204 (1928); but with the revisions we describe 
on p. 69. 

83 H. E. Watson, Proc. Roy. Soc. 117A, 43 (1927). 

84 C. P. Smyth and C. T. Zahn, J. Amer. Chem. Soc. 47, 2501 (1925). 

85 S. C. Sircar, Ind. J. Phys. 3, 197 (1928). 

88 Calculated from Pohrt's data 55 by Hojondahl 54 . 

87 P. C. Mahanti and R. N. Das Gupta, J. Ind. Chem. Soc. 6, 411 (1929). 

88 J. Rolinski, Phys. Zeits. 29, 658 (1928). 

89 O. Hassol and E. Naeshagon, Zeits. f. Phys. Chem. 4u, 217 (1929). 

90 L. Meyer, Zeits f. Phys. Chem. SB, 27 (1930). 

91 A. Parts, Zeits. f. Phys. Chem. 4u, 227 (1929). 

92 P. Walden and O. Werner, Zeits. f. Phys. Chem. 2, 10 (1929). 

93 Bergmann, Engel, and Sandor, Zeits. f. Phys. Chem. 10n, 106 (1930). 

94 L. Ebert and H. Hartel, Naturwissenschaften, 15, 669 (1927). 

95 J. B. Miles, Phys. Rev. 34, 964 (1929). 

96 J. D. Stranathan, Phys. Rev. 31, 653 (1928). 



Ill, 19 MEASUREMENT OF ELECTRIC MOMENTS 69 

polarization'. Especially convincing evidence is furnished by the figures 
in bold-face type, which are calculated with the dispersion data of 
Koch, 14 Statescu, 15 and Fuchs, 19 and thus, unlike the other values by 
method II, include the effect of the infra-red vibrations. Their data have 
been fully discussed in 15, and modify the results with method II 
sufficiently to remove all the discrepancy with method I in the case of 
CO and C0 2 . 

It is noteworthy that when different solvents have been tried in the 
solution methods, the values of the electric moment are, as shown in 
the table, virtually independent of the type of solvent which is em- 
ployed. This gives assurance that the moments obtained by using solu- 
tions have a real physical significance. When the same material has 
been measured both in the gaseous state and in solution, the moments 
obtained are seen to be the same within an experimental error no greater 
than the discrepancies among the different measurements for one kind 
of state. There is thus little evidence that molecules have a different 
'effective moment' in solutions than in the gaseous state, a suggestion 
which has sometimes been made. If it were necessary to assume such 
an 'effective moment', its value would presumably depend on the nature 
of the solvent, whereas actually the moments found for a given molecule 
are seen from the table to be virtually independent of the solvent, 
except perhaps in the case of I 2 mentioned above. The variations with 
the type of solvent are remarkably small, and clearly less than the 
experimental error. The determination of the moment of the water 
molecule by the solution method requires special mention. It is hard 
to achieve with precision, as an exact knowledge of the rather low 
solubility of water in benzene is necessary. Williams formerly used the 
value of the solubility given by Hill, Jr., which was in substantial 
agreement with earlier work by Groschuff and by Richards, Carver, 
and Schumb. 97 He thereby originally reported an electric moment 
l'7x!0~ 18 e.s.u. for the water molecule. 82 However, he informs the 
writer that when new, as yet unpublished, solubility determinations 
made by Cohen and Wcyling at Utrecht and also by Rosenbaum at 
Wisconsin are utilized, he obtains the higher value I'Sl^O-OSx 10~ 18 . 
The agreement of the latter with Sanger's value 1-84 x 10~ 18 measured 
in the vapour state is closer than one has any right to anticipate in 
view of experimental uncertainties, especially neglect of the atomic 
polarization in method II. 

97 Groschuff, Zeits. f. Elektrochem. 17, 348 (1911); Richards, Carver and Schumb, 
J. Amer. Chem. Soc. 41, 2019 (1919); Hill, Jr., ibid. 45, 1143 (1923). 



70 DIELECTRIC CONSTANTS, REFRACTION, AND THE III, 20 

20. Dielectric Constants and Molecular Structure 

By revealing the electric moments of molecules, measurements of the 
dielectric constants of chemical compounds often shed considerable light 
on the configurations in which the constituent atoms are grouped to 
form the molecule. Of course, the value of the electric moment alone 
does not enable one to determine the exact dimensions and geometry 
of the molecule, but it does disclose whether or not the atoms are 
arranged in a symmetrical way. The absence of an electric moment, 
of course, means a high degree of symmetry. The classic example of 
this is the once much-mooted subject of the model for the CO 2 molecule. 
Although early dielectric work on CO 2 seemed to demand an electric 
moment, the recent experiments of Zahn 45 and especially Stuart, 50 as 
well as use of method II with Fuchs's 19 dispersion formula explained 
in 15, shows quite definitely that the ( <O 2 molecule has no electric 
moment. Hence this molecule must be collinear, with the carbon atom 
at the centre and equidistant from the two oxygen atoms. The tri- 
angular and the unsymmetrical collinear models which have sometimes 
been proposed would clearly lead to an electric moment. A symmetrical 
collinear model for GO 2 is also demanded by other evidence than that 
on dielectrics. This other evidence has been nicely summarized by 
Wolf, 23 and includes (a) X-ray analysis of the structure of solid C0 2 , 98 
(6) the rotational specific heat of C0 2 , which has 24 the value approxi- 
mately E rather than 3JK/2 calories per mol, thus showing that there 
are only two moments of inertia appreciably different from zero, 
(c) absence of a third intense resonance point in the infra-red dispersion 
(p. 51), showing that one of the fundamental modes of vibration must 
be symmetrical. 

As sulphur and oxygen are in the same column of the periodic table, 
one would expect the CS 2 molecule to be symmetrical and collinear if 
this is true of C0 2 . Williams' s measurements 58 on CS 2 in solution, also 
more recently those of Ghosh, Mahanti, and Mukherjee, 80 of Zahn, 45 
and of Schwingel and Williams 60 on C\S 2 in the vapour state, do indeed 
yield a zero moment for the CS 2 molecule. In some of these papers 80 ' 60 
determinations are also made for N 2 O, and here also the moment proves 
to be zero, so that the nitrous oxide molecule is collinear. The per- 
ceptible, though small, moments reported for CS 2 and N 2 O respectively 
in earlier work of Zahn and Miles 45 and of Braunmiihl 53 doubtless arose 
through experimental error. 

98 J. do Smedt and W. H. Keesom, Proc. Amsterdam Acad. 27 , 839 ( 1 924) ; H. Mark, Zeits. 
/. Elektrochemie, 31, 623 (1925); H. Mark and E. Pohland, Zeita.f.Krist. 61, 293 (1925). 



fe 1)10' 



800 
700 



CH/TI 



-o CHCij 



III, 20 MEASUREMENT OF ELECTRIC MOMENTS 71 

The table shows that H 2 O and SO 2 have quite large moments. The 
polarity of water is also well known from other phenomena, such as 
association in the liquid state. The models of the H 2 O and S0 2 mole- 
cules must therefore be either triangular or unsymmetrical if collinear. 
The unsymmetrical collinear model encounters serious dynamical diffi- 
culties," at least in the case of H 2 0, and the triangular model for this 
molecule is the generally accepted one. 

Band-spectrum analysis, 100 also perhaps chemical evidence, 101 reveals 
that the correct model for the ammonia molecule is a pyramidal one, 
with the N atom at the vertex, and the H atoms at the corners of the 
base. Such a model would have a moment along the axis of the pyramid, 
in agreement with the polarity of 
NH 3 revealed by the observations on 
dielectric constants. 

The sequence OH 4 , CH 3 (!1, CH 2 C1 2 , 
GHC1 3 , CC1 4 is one of the standard 
illustrations of the valuable informa- 
tion oil molecular structure revealed 
by dielectric constants. Fig. 4 shows 
Stinger's observations of the dielectric 
constants of these materials as a 
function of temperature at constant 
density. The horizontal character of the curves for CH 4 and CC1 4 
shows clearly that these gases are non-polar. The methane and carbon 
tetrachloride molecules are thus highly symmetrical. The necessary 
symmetry can be secured by supposing that the four hydrogen or 
four chlorine atoms are at the corners of a regular tetrahedron, with 
the carbon atom at the centre. The valencies of the carbon atom thus 
have the tetrahcdral geometry so dear to the organic chemists. A pyra- 
midal model analogous to that for ammonia, which has sometimes been 
suggested 102 , is clearly out of the question, as it would be unsymmetrical. 
A coplanar model with the carbon atom at the centre of a square would 

99 See Debye, Polar Molecule*, p. 63 ff. 

100 G. A. Stiiu-hcomb and E. F. Darker, Phys. Rev. 33, 305 (1929); Barker, ibid. 33, 
084; R. M. Badger and C. W. Cartwright, ibid. 33, 692 (1929). 

101 A. Hant/suh and A. Werner, Her. d. D. Chcm. Ges. 23, 11 (1890); of. A. W. 
Stewart, {Stereochemistry, p. 197. This stereochemical evidence does not perhaps uniquely 
demand a pyramidal model, but at least shows that all three nitrogen valencies cannot 
be in the same plane. 

i 2 K. Weitwonborg, Phys. Zeits. 28, 829 (1927); Ber. d. D. Chcm. Ges. 59, 1526 
(1926); Naturwisscnschaften, 15, 662 (1927); also Ebert, Naturwissenschaften, 15, 669 
(1927) and ref. 104; Henri, Chem. Rev. 4, 189 (1927). 



Fio. 4. 



72 DIELECTRIC CONSTANTS, REFRACTION, AND THE III, 20 

be non-polar as well as the tetrahedron, but is very implausible from 
chemical and other grounds. 103 Also it would not explain the polarity 
of CH 2 C1 2 if one makes the natural assumption that alternate corners 
are filled by Cl and H atoms respectively. As one passes through the 
sequence CH 4 , CH 3 C1, CH 2 C1 2 , CHG1 3 , CC1 4 by replacing one H atom 
by one Cl at some corner of the tetrahedron, it is clear that with the 
tetrahedral model perfect symmetry is not secured except at the 
starting-point CH 4 or except when all the H atoms have been replaced 
by Cl in CC1 4 . This is in agreement with the finite electric moments 
found by Sanger for CH 3 C1, CH 2 C1 2 , and CHC1 3 . 

It may be mentioned that whereas molecules of the type Ca 4 are 
non-polar if a is an atom, polar molecules of the structure Ca 4 are 
known when a is a complicated radical rather than a simple atom. The 
molecules 

C(CH 2 0(0)CCH 3 ) 4 , C(COOCH 3 ) 4 , C(COOC 2 H 5 ) 4 , C(OCH 3 ) 4 , C(OC 2 H 5 ) 4 
are, for instance, revealed by their dielectric constant data 104 to be polar, 
having respectively the moments 1-9, 2-8, 3-0, 0-8, MX 10~ 18 . On the 
other hand, the moments of C(CH 2 Br) 4 , C(CH a Cl) 4> C(CH 2 T) 4 , C(N 2 O) 4 
are found to be zero, showing that Ca 4 can sometimes be non-polar even 
when a is not an atom. The existence of electric moments for any 
molecules of the type Ca 4 at first sight seems quite paradoxical in view 
of the non-polarity of methane. A pyramidal model for polar molecules 
of this form has been suggested 102 as a solution of the paradox, but it 
seems highly improbable that the carbon valences can have a tetra- 
hedral geometry in some instances and pyramidal in others. A much 
more plausible solution, proposed by Hojendahl 105 and by Williams, 106 
is that in molecules of the form C 4 the axis of electric moment of a need 
not coincide with its axis of valency if a is a complicated radical. Let 
us suppose that the angle between these two axes is 6, and that the 
axes of valency coincide with the axes of symmetry drawn from the 
centre of the carbon tetrahedron to its four vertices. The dipole axes 
are then free to rotate around the axes of valency subject only to the 
constraint that the angle between each corresponding pair of axes have 

103 Band spectrum evidence has been claimed to disqualify the tetrahedral model of 
methane (V. Guillemin, Ann. der Physik, 81, 173 (1926) ), but Dermison finds band spectra 
consistent with the tetrahodral model (Astrophys. J. 62, 84 (1925); also do Boer and 
van Arkel, Zeita. f. Physik, 41, 27 (1927)). 

104 L. Ebert, R. Eisenschitz, and H. v. Hartel, Naturwisaenachqften, 15, 668 (1927); 
Zeita. f. Phya. Chem. IB, 94 (1928). Cf. also J. W. Williams, Phys. Zeita. 29, 686 (1928). 

105 K. Hojendahl, Dissertation (Copenhagen, 1928), p. 60. 

106 J. W. Williams, Phya. Zeita. 29, 271, 683 (1928); J. Amer. Chem. Soc. 50, 2350 
(1928) ; Dipolmomente und ihre Bedeutung, p. 46. 



Ill, 20 MEASUREMENT OF ELECTRIC MOMENTS 73 

the given value 0. In other words, we have what is sometimes called 
a 'pliable* bond, which is a sort of socket in which the radical is free 
to turn. The dipole axes tend to set themselves in the position of 
minimum total energy subject to this constraint, and it is altogether 
probable, especially in view of the mutual interaction between the 
various dipoles, that in such a position the four a dipole axes are not 
arranged with sufficient symmetry to mutually compensate one another, 
and hence to yield zero resultant moment for the complete molecule. It 
is interesting to note that in all the polar molecules yet found of the 
type C 4 the radical a contains an oxygen atom, so perhaps the presence 
of the oxygen atom is responsible for the non -coincidence of the dipole 
and valency axes of a. 

Instances in which polar radicals compensate each other very com- 
pletely and which are thus the exact reverse of the examples cited in 
the previous paragraph are furnished by the group of ketones, of the 
form a CO a'. The following of these ketones, 

CH 3 -CO-C 2 H 5 , CH 3 -CO-C 3 H 7 , 

CH 3 CO C 4 H 9 , CH 3 CO C 6 H 13 , 

CH 3 CO C 9 H 19 , C 2 H 5 CO-C 2 H 5 , 

C 3 H 7 -CO-C 3 H 7 , (CH 3 ) 3 -CO-C(CH 3 ) 3 , 

have been found by Wolf 107 to have electric moments not differing by 
more than 2 or 3 per cent, from that 2-71 x 10~ 18 of acetone 

(CH 3 CO CH 3 ). 

The obvious inference is that the electric moment is due entirely to the 
CO radical, and that the dipole moments of the other radicals com- 
pensate each other completely. It may be noted, however, that here 
the CO radical has a very much larger moment 2-7x 10~ 18 e.s.u. than 
that of a free CO atom (O-lxlQ- 18 ), so that the CO radical in the 
ketones presumably borrows or loans electrons to or from the attached 
radicals in order to have a different structure from the free CO atom. 
According to Estermann, 108 benzophenol (C 6 H 5 CO C 6 H 5 ) has an 
electric moment 2-5 x 10~ 18 , about 10 per cent, lower than Wolf's values 
for the ketones. The difference is perhaps due to distortion of the CO 
structure by the polarization forces from the radicals, or may be experi- 
mental error, as Estermann measured the pure liquid. All the alcohols 
have approximately (within 20 per cent.) the same moment, 1-6 x 10~ 18 , 
which is probably due to the OH radical. 

107 K. L. Wolf, Zeits.f. Phys. Chem. 2u, 39 (1929). 

108 J. Estermann, Zeits.f. Phya. Chem. IB, 134 (1928). 



74 DIELECTRIC CONSTANTS, REFRACTION, AND THE III, 20 

It is interesting to note that Smyth and Stoops 109 find all nine isomers 
of heptane (C 7 H 16 ) to be non-polar. 110 Errera 111 has investigated the 
various isomers of acetylene dichloride and finds that the cis and 
so-called asymmetrical forms have moments 1*89 and l-l8xlO~ 18 re- 
spectively, while the trans form has no moment. More recently Miiller 
and Sack 74 find 1-74 x 10~ 18 for the cis form. This is in nice qualitative 
agreement with the structural formulae 

H H Cl H H 01 

C . .0, C- 0, 

Cl Cl Cl H Cl H 

(cis) (asymmetrical) (trans) 

which have been sometimes assumed by the organic chemists. If one 
measures the dielectric constants of these isomers in the solid state well 
below the melting-point and with a sufficiently high radio (heterodyne) 
frequency, the same polarization is found with all three isomers. 112 The 
reason is, of course, that under these conditions the relaxation time is 
too large to permit the dipoles to orient themselves in the field, so that 
there remains only the induced polarization which is independent, or 
very nearly so, of the symmetry of the atomic grouping in the molecule. 
The molecules obtained by substitutions in the benzene ring have had 
their geometry very thoroughly studied with the aid of dielectric con- 
stants, probably more than any other class of chemical compounds. 
The benzene molecule itself is non- polar, as one expects from the con- 
ventional coplanar hexagonal model which the chemists have for the 
benzene ring. 'Mono -substituted' benzene molecules, of the type-form 
C 6 H 5 a, which are formed from the benzene, molecule by substitution of 
an atom or radical a for one hydrogen atom are found to be invariably 
polar, as one might expect. 'Di-substituted' molecules, of the form 
C 6 H 4 a 2 , are found to be polar if the two 's are substituted in the ortho 
or meta configurations, but not if substituted in the para configuration 
as long as a is an atom or one of certain types of simple radicals. 113 

109 C. P. Smyth N. and W. Stoops, J. Amer. Chun. Hoc. 50, 1883 (1928). 

no f or structural diagrams and an account of tho rather complicated geometrical 
symmetries which must be assumed to explain this non-polarity see I)ebye, Polare 
Molekdn, pp. 59-60 (not in Engl. od.). 

111 J. Errera, Phya. Zcits. 27, 764 (1920); also the discussion by Estermann in Dipol- 
moment und cfiemische Struktur (Leipziger Vortrage, 1929), p. 36. 

112 J. Errera, Polarisation Dielectrique. (Paris, 1928), p. 106. 

113 For brevity we do not give, oven in our table, tho values of tho electric moments 
of all the numerous benzene derivatives. For those see various tables by other writers 
cited on p. 64 ; some new determinations when the substituents are hahdes have been 
made by Bergmann, Engel, and Sandor, Zeits.f. Phys. Chem. 10u, 106 (1930). 



Ill, 20 MEASUREMENT OF ELECTRIC MOMENTS 75 

The meaning of these various configurations is explained by the fol- 
lowing structural diagrams: 

a 

a 

a 
ortho meta para 

The results on the electric moments are exactly what one should expect, 
as the ortho and meta arrangements are unsymmetrical, while in the 
para arrangement the two atoms are diametrically opposite and there 
is perfect symmetry. On the other hand, the dielectric constant data 
reveal that molecules of the para form can be polar if the a's are 
certain complicated types of radicals. Williams, in fact, finds that 
p-hydroquinone diethyl ether and p-hydroquinone diacetate have 
electric moments l-7x!0~ 18 and 2-2xlO~ 18 e.s.u. respectively. 114 The 
ordinary structural formulae, viz. 



OC 2 H 5 H 3 CC(0)0<( 



which are given to represent the para configurations of these molecules, 
would at first thought lead one to expect a zero moment, as the two 
sides of the substituted benzene ring appear equal and opposite in 
character. The solution of the paradox is probably similar to that of 
the polarity of certain molecules of the type Ca 4 , viz. that the axis 
of polarity of a radical need not necessarily coincide with its axis of 
valency. Hence, in a polar molecule of the form p-C 6 H 4 a 2 , such as the 
two hydroquinone compounds mentioned above, the dipole axes of the 
radicals a need not necessarily fall in the plane of the benzene ring. 
Attempts have even been made to calculate from the observed moments 
the angles of inclination of the dipole axes to the benzene ring, and 
especially the angle between the dipole axes of the two like radicals 
present in the polar para compounds, but as yet the only sure con- 
clusion is that these two axes do not make an angle of 180 with each 
other, for otherwise there would be complete compensation of the 
moments and no polarity. The fact that some other angle than 180 
is the most stable appears somewhat startling from a dynamical stand- 
point, as dipoles tend to set themselves antiparallel. Very likely there 
is no angle of static equilibrium, but instead a continual internal pre- 

114 J. W. Williams, Phys. Zeits. 29, 683 (1928); A. Weissberger and J. W. Williams, 
Zeits.f. Phys. Chem. 3u, 367 (1929). 



76 DIELECTRIC CONSTANTS, REFRACTION, AND THE III, 20 

cession or oscillation of the molecule which makes the angle between 
the two dipoles a periodic function of the time. 

The question of the existence of such internal precessions has been the subject 
of considerable discussion in the literature. The assumption of a chemical bond 
which is merely a socket in which the radical can turn freely is usually termed 
the hypothesis of free rotation (freie Drehborkeit). The most extensive examina- 
tion appears to have been made in the case of ethyleno dichloride (( <1H 2 C CH 2 C1). 
As pointed out by Williams 115 and by Eucken and Meyer, 116 the observed moment 
is intermediate between the value zero, characteristic of the antiparailel alinement, 
and the value calculated under the assumption of free rotation. Hence it would 
appear that here the radicals C1H 2 arc not entirely free to turn and exhibit a 
preference for the antiparallel configuration, although also rotating (or perhaps 
vibrating) through other configurations. Also X-ray and electron diffraction data 117 
show that the molecule spends most of its time in certain particular configura- 
tions, perhaps favouring antiparallel alinement. If the potential energy resist- 
ing the free turning is comparable with kT, the dielectric constant should no longer 
be a linear function of l/T. This is shown by the general quantum -mechanical 
analysis of dielectric constants to be given in Chapter VII, in which tho Dobyo 
formula is obtained only if the separation of energy levels is very small or very 
large compared to kT. An analogous discussion of the departures from linearity 
on the basis of classical mechanics has been given by Meyer. 118 Until recently 
existing experimental data 119 did not seem adequate to test whether tho dielectric 
constant of ethylene dichloride, and other molecules where partial rotations are 
suspected, are really accurate linear functions of l/T. Some curvature is ap- 
parently exhibited in the measurements of Meyer, but not in those of Ghosh, 
Mahanti, and Gupta. Very recent measurements by Smyth and Walls 119 seem 
to show quite conclusively that there are pronounced departures from linearity 
in tho case of ethylorie dichloride. Sanger reports strict linearity, but perhaps 
this is because he did not use as low temperatures as Smyth. 

Unlike ethylene dichloride, acetylene dichloride (HC1C = CC1H) shows distinct 
isomers (cf . diagrams, p. 74), and this is direct experimental evidence that in the 
latter there is no appreciable internal rotation. To explain this, it has commonly 
been suggested that a carbon double bond is much more rigid as regards turning 
than a single bond. A theoretical basis for this rigidity has been given by Hiickel 120 
by means of quantum mechanics. 

Considerable work has been done on developing a quantitative 
'vector' theory of the electric moments of the disubstituted benzenes. 

* 15 J. W. Williams, Zeits. f. Phys. Chem. 138, 75 (1928). 
118 A. Euckeii arid L. Meyer, Phys. Zeits. 30, 397 (1929). 

117 P. Debye, Phys. Zeits. 31, 142 (1930) (report of experiments by Bowilogua and 
Ehrhardt; R. Wierl, ibid. 31, 366 (1930). Wierl's data seem to indicate two equilibrium 
positions, whereas the X-ray measurements apparently reveal only one. ganger 119 claims 
that none of these diffraction measurements are really precise enough to decide whether 
or not there is free rotation. 

118 L. Meyer, Zeits. f. Phys. Chem. SB, 27 (1930). 

119 Meyer, I.e.; Sanger, Phys. Zeits. 32, 21 (1931); Smyth and Walls, J. Amer. Chem. 
Soc. 53, 534 (1931); T. N. Ghosh, P. C. Mahanti, and Sen Gupta, Zeits. f. Physik, 54, 
711 (1929). 

E. Hiickel, Zeita.f. Physik, 60, 423 (1930). 



Ill, 20 MEASUREMENT OF ELECTRIC MOMENTS 77 

Let us suppose that the electric moment / due to the substitution of 
an atom or simple radical a in the benzene ring is directed from the 
centre of this ring to the position of the hydrogen atom replaced by a. 
An analogous assumption will be made about the moment /' due to 
substitution of another atom or simple radical a'. If, then, the two 
substituents a, a' be inserted simultaneously, thus forming a disub- 
stituted benzene, and if the dipole moments due to these two sub- 
stituents be supposed not to distort each other, then clearly the resultant 
moment of the molecule is, by the law of vector addition, 

^ = [/ a +J /a +2//'cosl*, (12) 

where <, the angle between the two constituent dipoles, is respectively 
60, 120, and 180 for the ortho, meta, and para positions respectively. 
Eq. (12) was first proposed by J. J. Thomson 121 in 1923, but it remained 
for other investigators to make a proper examination of the experi- 
mental validity of his suggestion (12), as unfortunately proper data 
were not available at the time of his paper. The first adequate experi- 
mental tests were made by Errera 122 and by Smyth and Morgan, 123 
while only slightly more recently a very great number of benzene com- 
pounds have been examined in the light of (12) by Hojendahl 54 and 
by Williams 58 and co-workers. Some typical results for the dichloro- 
benzenes are shown in the following table: 

o-C 6 H 4 01 2 m-C 6 H 4 C! 2 p-C 6 H 4 C! 2 

p, o}a 2-25 x 10- 18 1-48 x 10- 18 (0-4 X lQ- 18 e.s.u.) 

/d(Eq.!2) 2-05 1-53 

^ calc (S.&H.) 2-13 1-42 

The prefixes o-, m-, p-, of course, refer to the ortho, meta, and para 
states respectively. The first row gives the experimental moments 
obtained by Smyth and Morgan by the method II sol (see p. 61), while 
the second row gives the values computed from Eq. (12). As the two 
atoms which are substituted in the benzene ring are both chlorine 
atoms, / equals /' in Eq. (12), and the moments given by (12) reduce 
to the simple expressions V3/, /, and for the o-, m-, and p-states. 
The value employed for / is 1-53 X 10~ 18 , which is a mean of experi- 
mental values found by Williams and others for monochlorobenzene 
(see table, p. 67). The agreement between the calculated and observed 
values, while by no means perfect, shows that (12) has at least approxi- 
mate validity for the compounds in question. The agreement between 

121 J. J. Thomson, Phil. Mag. 46, 513 (1923). 

12 2 J. Errera, Comptes Rendus, 182, 1623 (1926); Phys. Zeits. 27, 764 (1926). 

123 Smyth and Morgan, J. Amer. Chem. Soc. 49, 1030 (1927). 



78 DIELECTRIC CONSTANTS, REFRACTION, AND THE III, 20 

the two sets of values would be perfect if the angle 6 in (12) were taken 
as 85 and 122 for the ortho and meta states instead of 60 and 120. 
These alterations in angle are in the direction one would expect, as two 
dipoles tend to set themselves antiparallel, but do seem excessively large 
as regards the ortho state. 

Apparently a more probable explanation of the departures of the 
angles from 60 and 120 is the induced polarization created by the 
forces between the different parts of the molecule. In other words, the 
field from one part polarizes the remainder of the molecule. This mutual 
induced polarization has been studied quantitatively by Smallwood and 
Hcrzfeld. 124 They endeavour to calculate quantitatively the resulting 
correction to Kq. (12) in the case of halogen-substituted benzenes under 
the assumption that the angles are 60, 120, &c. They find that the 
agreement with experiment is usually considerably improved. This is 
illustrated in the above table for dichlorobenzene, where the values given 
in the last row are inclusive of Smallwood and Herzfeld's correction for 
induced polarization. These are seen to agree with experiment much 
better than those without this correction. 

Numerous applications of (12) to other disubstituted benzenes could 
also be cited, notably those containing the NO 2 radical or other halogens 
than 01 as the substituents. As an example of a benzene derivative 
containing two unlike atoms we may consider Hojendahl's observations 
and calculations 54 on the chloronitrobenzenes, viz. 

o-C!G 6 H 4 N0 2 m-ClC 6 H 4 N0 2 p-01C 6 H 4 NO 2 

/* 01)S 4-25X10- 18 3-38XKH 8 2-55 X 10~ 18 e.s.u. 

Mcaic 4 '78 3-26 2-11 

/z ww 3-78 3-18 2-36 

The values taken for / and /' in (12) are the moments 1-64 and 
3-75 x 10~ 18 which Hojendahl found for monochlorobenzenc and nitro- 
benzene respectively. 125 In the last line we have listed the experimental 
values observed independently by Walden and Werner. 126 

124 H. M. Smallwood and K. F. Horzfeld, J. Amer. Chcm. fioc. 52, 2654 (1930). See 
also Borgmann, Engol, and Sandor, Z.c. 93 

125 The calculated values would be subject to slight revision if one used for / and I' the 
values of the moments of the C 6 H B C1 and C 6 H 5 NO 2 molecules as determined by other 
investigators than Hojendahl. These other do terminations are listed in the table on 
p. 67. The newer measurements are presumably more accurate, but the resulting changes 
are not largo enough to throw much additional light on the validity of (12). For instance, 
Sack, duo to different assumed /, /', gives 4-76, 3-30 and 2-25 x 10' 18 for the calculated 
values for the three isomers of chloronitrobenzene, and 3-61, 4-10 and 4-20 for those of 
nitrotoluene (Ergebniaae der exokten Naturwisaenschctften, viii. 345 (1929). 

126 P. Waldon and O. Werner, Zeits.f. Phys. Chem. 2s, 10 (1929). 



Ill, 20 MEASUREMENT OF ELECTRIC MOMENTS 79 

The case of nitrotoluene is particularly interesting, as it is necessary 
to suppose that here / and /' have opposite signs. Hojendahl finds the 
moments 0-43 X lO" 18 and 3-75 x 10" 18 for toluene and nitrobenzene 
respectively, but these measurements fix only the absolute values and 
not the signs of the moments for the toluene and N0 2 radicals, and to 
obtain any kind of agreement with his observations for the nitrotoluenes 
it is necessary to take / = +0-43 X lO" 18 , I' = 3-75 X 1Q- 18 (or else 
1= _ 0-43 XlO- 18 , /'^+3-75xlO- 18 , as only relative signs are of 
interest). The agreement with experiment is then quite good, as shown 
by the following table. The experimental values on the last line are 
those of Williams and Schwingel. 58 

o-CH 3 C G H 4 NO a m-OH 3 C 6 H: 4 N0 2 p-CH,0 6 H 4 NO a 
/*obs(H6j) 3-64 X 10- 18 4-31 X 10- 18 e.s.u. 

jLt calc 3-56 3-98 4-18 

jLt w>a . 3-75 4-20 4-50 

By way of summary we may give a table taken from Williams of the 
atoms whose effect on the moment in benzene substitutions has been 
found capable of approximate calculation by means of (12), at least in 
some cases. The table gives the values of the contributory moments 
/ which according to Williams 127 must bo used for each of them. For 
purposes of comparison the electric moments which are obtained when 
these various atoms or radicals arc substituted for a hydrogen atom in 
CH 4 and H 2 are also given when available: 

/K s?o 

N0 2 C^ OH Cl J3r I OCH 3 C< CH 3 NH 2 

^O N OII 

/XlO -3-9 -2-8 -1-7 -1-5 -1-5 - 1-3 -1-2 -09 -(-0-4 +1-5 

fi X 10 l8 (Cl 3 a) 3-t 1-0 2-0 1-9 10 

/tXl0 18 (Uaj 1-8 1-0 0-S 04 1-5 

Questions of sign have no significance in the last two lines, as only one 
constituent dipolc is involved in CK 3 or Hex. The agreement between 
the various lines in the table is surprisingly good, and shows that each 
substitution does often have approximately a characteristic dipole 
moment. The deviations in the hydrogen halides are to be expected, 
as here merely an atom, non-polar by itself, is substituted, and the 
moments in such cases must be due entirely to distortion of the elec- 
tronic distributions. 

The discussion in the preceding paragraphs has undoubtedly con- 
veyed an impression of excessive optimism regarding the universal 
approximate validity of Eq. (12). The numerical examples which we 

187 J. W. Williams, Dipolniomente und ihre Bedeutung, Tables X and XI. We have 
added the value of I for iodine given by Walden and Werner. 



80 DIELECTRIC CONSTANTS, REFRACTION, AND THE III, 20 

have given are some of the most favourable ones, and in some cases 
the agreement is very poor. For p-nitraniline (O 2 N C 6 H 4 NH 2 ), for 
instance, use of (12) with the values of /, /' given in the table yields 
1-5 (3-9) = 5-4 x 10~ 18 , whereas the observed value 64 is 7-1 x 1Q- 18 , 
showing that the mutual distortion between the two constituent radicals 
must be very great. A much more flagrant example of the inadequacy 
of (12) is furnished by the fact that the dipole moment of OH must be 
assTimed to have different signs in different cases. In both the cresols 
(CH 3 C1 6 H 4 OH) 128 and chlorophenols 129 (Cl C 6 H 4 OH) the dipole 
moment is found to be distinctly greater in the para than in the meta 
or the ortho configuration. Hence the OH radical must make contribu- 
tions of the opposite signs to those of CH 3 and of Cl. But reference to 
the table shows that CH 3 makes a positive contribution and Cl a 
negative one. Hence OH behaves negatively in one case and positively 
in the other. As regards the example we have given, the anomaly in 
sign might be blamed on CH 3 or on Cl, but comparison with a number of 
other examples shows that it is in all probability to be attributed to OH. 
Fogelberg and Williams 13 have recently found that similar anomalies 
also are unavoidable for NH 2 . When such anomalies in sign arise, 
Eq. (12) ceases to have much meaning. It is probable that the escape 
from the dilemma is that the electric moment of the OH radical does 
not fall in the plane of the benzene ring. This has already been men- 
tioned in connexion with the finite moment found for certain hydro - 
quinone compounds where (12) would demand zero. A radical of this 
type Hojendahl in his dissertation calls an 'inclined group', in distinc- 
tion to the ' positive' and 'negative' groups to which (12) is applicable. 
In the case of inclined groups, the geometric addition of the dipole 
moments of the various radicals to obtain the resultant moment of the 
entire molecule must be made in three rather than two dimensions, and 
then the usual so-called Vector models' of the polarity of the benzene 
substituents based on vector addition entirely in the plane of the 
benzene ring of course lose all meaning. In short, the approximate 
applicability of (12) to benzene substituents is for a limited class of 
compounds rather than a universal property, and doubtless this will be 
increasingly revealed by the continual extension of the experimental 
measurements to include more and more of these substituents. 

128 Smyth and Morgan, J. Amer. Chem. Soc. 49, 1036 (1927) (their calculations utilized 
earlier experimental measurements by Philip and Haynos, J. Chem. Soc. 87, 998 (1905)) ; 
cf. also Williams and Fogolborg, J. Amer. Chem. Soc. 52, 1356 (1930). 

129 J. W. Williams, Phys. Zeits. 29, 683 (1928). 

130 Fogolberg and Williams, Phys. Zeits. 32, 27 (1931). 



Ill, 20 MEASUREMENT OF ELECTRIC MOMENTS 81 

The data on the symmetrical trisubstituted benzenes are particularly 
interesting because of the light they shed on the structure of the benzene 
ring. It is found that mesitylene, 58 1,3,5 triethylbenzene, 58 and 1,3,5- 
tribromobenzene, 54 have zero moments within the experimental error. 131 
As implied by the suffixes 1,3,5, the substituent atoms or radicals 
replace every other hydrogen atom in the benzene ring. If the benzene 
ring is really a ring, i.e. six atoms evenly spaced in a plane, no electrical 
moment should result for these compounds. On the other hand, struc- 
tural formulae proposed by Korner, Baeyer, and Ladenburg 132 would 
require a three-dimensional instead of coplanar model, and would lead 
to a finite electric moment for all symmetrical trisubstituted benzenes, 
contrary to experiment. In particular, the model in which alternate 
hydrogen atoms are in different planes, and in which the familiar 
hexagon is thus replaced by two triangles in parallel planes, must be 
rejected. 

A study of the dielectric properties of the derivatives of diphenyl, 
which contains two benzene rings, has led to interesting information 
on the coupling between the two rings, but this would take us even 
farther afield into organic chemistry. 133 

A detailed knowledge of electronic motions and distributions would enable one 
to calculate directly by pure dynamics the moments of simple molecules and radi- 
cals, but so far the attempts made in this direction have boeii rather unsuccessful. 
The reader should particularly guard against the idea that the moment of a 
molecule such as HCl is anything like er Q , where e is the charge of an electron and 
r is the distance between the nuclei. The moment e / is, perhaps, what one would 
naively expect if one used the picture which has sometimes been given of HCl 
as having a proton at one end and a negatively charged chlorine ion at the 
other, so that the molecule would be merely H ' Cl . The value of /* for the HCl 
molecule is 1-28 X 10~ 8 cm., 134 and hence er is 6- 11 X 10" 18 e.s.u., whereas the actual 
moment is only 1-03X 10" 18 . 135 The reason the actual moment is so small is that 

131 The moments found are of tho order 0-2 X 10' 18 or less, which may be considered 
virtually zero. Sym-trinitrobonzono seems to have a real electric moment of approxi- 
mately 0-8 X 10' 18 e.s.u. 

132 Koriier, Gazz. Chem. Ital. 4, 444 (1874); Baoyor, Ann. der Chemie, 245, 103 (1888); 
Ladenburg, Ber. d. D. Chem. Ges. 2, 140 (1869). 

133 See J. W. Williams, 59 also Williams and Weissberger, J. Amer. Chem. Soc. 50, 2332 
(1928), Zeits. f. Phys. Chem. SB, 367 (1929); summary in Dobye, Polare Molekeln 
(German od. only), p. 66. 

134 From band spectra; Birgo, International Critical Tables, v. 414. 

135 C. T. Zahn, Phya. Rev. 24, 400 (1924). It may be noted that tho moment 1-03 X 10' 18 
e.s.u. found experimentally by Zahn for HCl is almost exactly tho product f eff r = 1 X 10' 18 
of the inter-nuclear distance r and the effective charge 0-86X 10' 10 found by Bourgin 
from spectral intensities for the infra-red vibration of HCl. This has sometimes been 
quoted as proof that Bourgin's value of the effective charge is very approximately correct, 
but there is no reason why the moment \i should bo identical with p e ff?V As a matter of 
fact tho two expressions are not even of comparable magnitude in CO if tho effective 

3595.3 G 



$2 DIELECTRIC CONSTANTS, HEFRACTION, AND THE III, 20 
the hydrogen atom does not lose all its charge to the chlorine atom. Another way 
of saying more or less the same thing is that a proton at distance r from a chlorine 
ion would attract some of the latter's negative charge and thus polarize the ion 
greatly. K. T. Compton and Debye 136 have both examined whether perchance 
the actual moment is approximately er Q oLE 9 where a is the polarizability of the 
chlorine ion as deduced from refractive data, and where E is the Coulomb field 
e/r$ which the H nucleus would exert at the geometric centre of the Cl~ ion. 
Actually the value of ex is so large that the term ocE is greater than er , and the 
molecule ' over-polarizes ' itself, an obvious absurdity. The absurdity has doubt- 
less arisen because the polarizability deduced from refractive measurements 
applies only to fields which are sensibly constant over the dimensions of an atom 
or ion, whereas a proton so close to the Cl~ ion as r Q = 1-28 X 10" 8 cm. gives rise 
to a highly divergent Coulomb field which is much larger on the near than on the 
far side of the chlorine ion. Furthermore, it is questionable whether it is a good 
approximation to consider HC1 as derived from H + + Cr~ rather than H-j-Cl, for 
recent developments in the quantum mechanics seem to show that the valence 
in HC1 is perhaps more non-polar than polar in nature. At any rate Compton's 
and Debye's calculations show that it is eminently reasonable that the actual 
moment be very much smaller than er . 

21. Optical Refractivities and Molecular Structure 

We have seen that a certain degree of success has attended the calcula- 
tion of the moments of complicated molecules by the vectorial addition 
of the dipole moments of the constituent radicals or groups. The optical 
refractivity of a chemical compound, or, what is essentially 137 equi- 
valent, the 'induced' part 4irNoc of its dielectric constant, can be 
calculated on the whole more accurately and more generally than can 
the dipole moment from the properties of the constituent atoms or 
radicals. The greater simplicity in the synthesis of the induced rather 
than permanent polarization of a molecule is to be expected, as each 
atom is capable of induced polarization, whereas permanent moments 
arise only from complicated interactions between atoms. In other 
words, induced polarization is to a considerable extent a purely atomic 
property. If this is true, the index of refraction n of a chemical com- 
pound should be capable of calculation from the indices n t of the 
constituents in the same way as for a mixture without chemical combina- 
tion, so that by considerations similar to those used in obtaining (9), 



where ^ is the number of atoms or radicals of type i contained in 

charge 4x 10~ 10 = 0-9e deduced from infra-rod dispersion (see 16) is correct, for then 
e eft r = 5 X 10~ 18 , whereas \L 0-1 x 10" 18 . Molecules with more than two atoms can have 
vibration spectra without having a permanent moment, so that in them there is no im- 
mediate connexion between the moment and effective charge. 

138 K. T. Compton, Science, 63, 53 (1926); P. Dobye, Polar Molecules, p. 62. 

187 Provided, as ia ordinarily the case, the atomic polarization is comparatively small. 



HI, 21 MEASUREMENT OF ELECTRIC MOMENTS 83 

a molecule of the compound, and where k and k t are the so-called 
'molar refractivities', 

l , _M i n\-\ 



of the compound and of a typical constituent, which have densities and 
molecular weights />, M and p i9 MI respectively. The scalar nature of 
the addition in (13) is to be contrasted with the vectorial addition of the 
dipole moments involved in an equation such as (12). The absence of 
vector properties simplifies the study of the refractivities of chemical 
compounds, but at the same time makes it much less illuminating on 
molecular structure than the study of dipole moments. 

The refractivities of various classes of molecules have been extensively 
analysed in the light of (13), especially in the nineteenth century. To 
quote from a book-review by C. P. Smyth : 138 'The polarization induced 
within the molecule, that is, the molecular refraction, was discussed so 
often a generation and more ago that it is now commonly regarded as 
an outworn subject and dismissed with a cursory treatment as a neces- 
sary preliminary to the discussion of the dipole polarization. 5 As a 
typical illustration of the early work with (13), we shall consider some 
calculations made by Landolt 139 in 1862-4. Like many other investi- 
gators in this field, he defined the molecular refractivity by means of 
the Gladstone-Dale formula k' = (nl)M/p instead of the Lorenz- 
Lorcntz one k ~ (n 2 l)M/p(n 2 +2). If n 1 is not too large, the 
difference between the two formulae is small except for a constant 
factor 2/3 which is of no interest in connexion with study of an additivity 
rule such as (13). The series expansion of (n 2 1)/(^ 2 +2) in n 1 is 
f[(w 1) \(n 1) 2 +...] Thus to the first approximation the two 
formulae are the same except for the constant factor, while the dif- 
ference in the second approximation is small because of the factor 1/6. 
Landolt found that by assuming the following values of k r for the 
constituent atoms 

fc^S-OO, ^=1-30, ^ = 3-00, (14) 

he could account nicely for the refractivities of certain compounds of 
these elements by using the additivity rule k r = 2 i\$i> This is shown 
in the following table : 

CH 4 C 4 H 10 C 2 H 4 O a C 6 H 12 2 C 4 H 8 O a C 10 H 20 O 2 C 3 H fl O 
fc' calc 13-2 36-0 21-2 51-6 36-4 82-0 25-8 

&' obB 13-2 36-1 21-1 61-6 36-2 82-1 26-1 

138 C. P. Smyth, Phya. Rev. 34, 166 (1929). 

139 l&ndolt.Pogg.Ann. 117,353(1862); 122,545(1864); 123, 596 (1864); also especially 
123,626(1864)aiid-4nn.derC%ew.4Supp.p. 1. Summary on p. 38ff.of Eisenlohr'sbook. 141 

02 



84 DIELECTRIC CONSTANTS, REFRACTION, AND THE III, 21 
In order to avoid confusion we have added primes to k to designate that 
the Gladstone-Dale rather than the Lorenz-Lorentz definition of the 
molar refractivity is used. It is to be emphasized that an equation 
such as (13) will not account for the refractivities of all compounds 
containing C, H, O if the values (14) for the constituent refractivities 
are used. Briihl 14 found in 1880 that the indices of refraction of certain 
other compounds of these elements could be approximately calculated 
by assuming that each double carbon bond contributed an amount K 
to the molecular refractivity. This makes k f = ] ^\+ rjK, where 17 is 
the number of double bonds. The calculated and observed values are 
then as follows: 

C 3 H 6 C B H 10 C 4 H 6 O a C 5 H 10 C 6 H 8 C 6 H 10 C 6 H e C 7 H 8 C 7 H 8 O C 9 H 12 O 
y 1111223333 

fc' ca1c 27-2 42-1 35-0 39-2 38-6 46-1 42-9 60-3 63-2 68-2 

jfc' ob- 27-1 42-2 35-1 39-3 38-7 45-0 42-2 50-1 63-2 68-8 

The compounds with three double bonds are of the so-called aromatic 
type. The contributions of the various atoms are taken by Briihl to 
have the following values slightly different from (14): 

^=4-86, 4^=1-29, A^-2-90, A" =2-0. (15) 
It must be mentioned that neither the atomic refractivities given in 
(14) or (15), nor the data in the tables, are the most recent or 
highly refined values, but the later work discriminates between many 
different types of bonds, and would take us too far afield into organic 
chemistry. 141 Oxygen, for instance, is attributed different refractive 
equivalents in different types of compounds. The introduction of num- 
berless such complications makes one feel that the highly developed 
elaborations and ramifications of (13) are to a certain extent numerical 
juggling. Complicated additivity rules are, of course, extremely useful 
in enabling one to calculate in advance the approximate value of the 
refractivity of a chemical compound whose dispersion has not been 
measured, but do not seem to have any very elemental physical signi- 
ficance. Also, it must be cautioned that such rules apply only to certain 
particular types of chemical compounds, especially the organic, and that 
their success is not at all universal. The artificiality of the additivity 
technique is nicely shown by the fact that the atomic refractivity of 
approximately 1-3 which must be assumed for hydrogen in applying 
the rules is not the same as either the actual refractivity 2-5 of the 

"o Bruhl, Ann. der Chemie, 200, 139 (1880). 

141 For a very comprehensive survey of additivity rules and the refraotivities of organic 
chemical compounds see F. Eisenlohr, * Spektrochemie organischer Verbindungen' 
(Chemie in Einzeldarstellungen, Band III). 



Ill, 21 MEASUKBMENT OF ELECTRIC MOMENTS 85 

hydrogen atom, calculated by quantum mechanics, or half the refrac- 
tivity 3-1 of the hydrogen molecule. 142 

The reader should not confuse the atomic refractivities which enter 
in the application of additivity rules to organic compounds with the 
'ionic refractivities' which enter in similar applications to solid salts 
and ionic solutions. The ionic refractivities have much more physical 
significance, as they are presumably identical with those of free ionized 
atoms, whereas we have seen, especially in the case of hydrogen, that 
there is no necessary immediate connexion between the atomic refrac- 
tivities utilized in the organic compounds and the true refractivities of 
free atoms. In order to permit use of the results of quantum theory, 
estimates and discussion of ionic refractivities will be deferred until 52. 

22. Saturation Effects in Electric Polarization 

Throughout the chapter we have supposed the field sufficiently small 
so that the moment is proportional to field strength. This is a condition 
usually fulfilled except in exceedingly strong fields. If e loc denote the 
effective local field to which the molecule is subjected, an accurate 143 
representation of the polarization as a function of field strength will 
involve a power series of the form 

P = Np E = ^Ke, uo +i?oc+ -I- (16) 

The ordinary approximation is obtained by retaining only the first term. 
Eq. (16) involves exclusively odd powers of e ]oc since in an isotropic 
medium the polarization changes sign with the field strength. If only 
the polarization due to orientation of the permanent dipolcs needed to 
be considered, the values of the coefficients a , a lv .. in the development 
(16) could be obtained by expanding the Langcvin function defined 
in (4), Chap. II, as a power series in the ratio 144 x = p,e lor lkT. Actually 
there are also terms resulting from the induced polarization, and 

142 Thoso molar rofractivios 2-6, 3-1 aro the values of (n-l)Mjp corresponding to the 
theoretical dielectric constant 1-000225 for atomic hydrogen (see 48) and the measured 78 
dielectric constant 1-000273 of H 2 . The first resonance linos of both H and H 2 are so 
far in the ultra-violet that the refractivitios in the optical region differ very little from 
those for infinite wave-lengths. 

143 Kven an expression such as (16) seems to the writer to involve some error, as the 
mean value of the cube of the effective field to which the atom is subjected is not 
necessarily the cube of the moan effective field, &c. In. other words, a different expression 
for c loc may bo necessary for oach power in the development. However, the error from 
this source is negligible, as all the terms but the first in the development (16) are extremely 
small, and so a small correction to them is insignificant. 

144 In Eq. (4), Chap. II, x was defined as the ratio pE/kT instead of ne\ oc lkT because 
we did not there bother to distinguish between the local and macroscopic fields. The 
distinction is, however, vital in considering saturation effects. 



88 DIELECTRIC CONSTANTS, REFRACTION, AND THE III, 22 

especially its interaction with the orientation effects. Even in the 
strongest field obtainable experimentally, it is sufficient to include only 
the first two terms of (16). The first term is, of course, the expression 
oi+p?/3kT which we have encountered so often, while further calcula- 
tion, which we omit, shows that the dependence of the second term on 
the temperature is of the form 

= ~ 



The constants <? , q l9 q 2 are expressible in terms of the dynamical charac- 
teristics of the model (i.e. matrix amplitudes in quantum mechanics), 
but are too complicated to be given here. The values of q Q , q lt q 2 were 
first calculated by Debye 145 in classical theory, using the conventional 
model ( 12) of harmonic oscillators mounted on a rigid rotating frame- 
work. Recently Nicssen, 146 using the quantum mechanics, has given 
general formulae for these ^-coefficients without the necessity of making 
any special assumptions concerning the nature of the electronic motions. 
It may be remarked that the coefficient q Q vanishes, in either quantum 
or classical theory, if one makes the unreal supposition that the elec- 
tronic motions can be represented by simple harmonic oscillators, and 
so q Q appears in Niessen's calculations but not in Debye's. If these 
oscillators be supposed isotropic as well as simple harmonic, the coeffi- 
cients q and q 2 also vanish. In an actual molecule all the q's are 
different from zero if it is polar, while only the terms q Q , q 1 remain if 
it is non-polar. The reason for this is that q Q and part of q l are due 
entirely to the induced polarization, while the remainder of q l and all 
of q 2 arise from the interaction between the induced and permanent 
polarity, provided there is any of the latter. It is clear that the tendency 
of the permanent di poles towards alinement parallel to the field destroys 
the random orientation, and so changes the induced polarizability if the 
molecule is optically anisotropic. Because even the induced moment 
tends to aline itself in the field, the term q l does not vanish even in 
a non-polar molecule. The 'interaction terms' q l9 q 2 are closely related 
to the Kerr effect, as the latter effect is the alteration in optical 
refractivity due primarily to orientation in a static electric field. The 
final term /z 4 /45FT 3 in (17) is due exclusively to the permanent 
moment, and is the same as obtained by series expansion of the Langevin 
function. 

In order to measure the effect of electrical saturation, it is usual to 

145 P. Debye, Handbuch der Madiologie, vol. vi, p. 754. 

146 K. F. Niessen, Phys. Rev. 34, 263 (1929). 



Ill, 22 MEASUREMENT OF ELECTRIC MOMENTS 87 

measure the change in the moment P or in the dielectric displacement 
D = E+^TrP when an already large field E is changed to E+dE with- 
out altering its direction. It is therefore convenient and customary to 
define the dielectric constant in a strong field as the slope dD/dE of 
the D-E curve rather than as the ratio D/E of its ordinate and abscissa. 
If we assume that the local field has the Clausius-Mossotti value 
J57+47rP/3, then , n 



where e is the ordinary dielectric constant for small field intensities, 
i.e. the value of dD/dE at the origin E = Q. To obtain (18) we sub- 
stitute e loc E-\-7rP/3 in the first right-hand term of (16), and the 
approximate value e loc = #(<r +2)/3 in the second term; we then solve 
the resulting equation for P and calculate d(E-\-4:TrP)/dE ) noting that 



The effect of the correction or 'saturation' term in (18) which is pro- 
portional to E 2 is very hard to measure, as it is exceedingly small, and 
appreciable only in such strong fields that it is difficult to eliminate 
error due to the effect of electrostriction on the size of the apparatus. 
As a numerical illustration let us, following Debye, 147 consider the case 
of ethyl ether; here <T O = 4-30, N = 5-83 X 10 21 , p, - 1-14 X 1Q- 18 , and (18) 
reduces at T = 293 to dD/dE ^ 4-30 0-25 x IQ~ 8 E 2 provided we 
neglect all but the last term of (17). In this formula the field strength 
must be expressed in electrostatic units, and hence in a field of 10,000 
volts/cm. = 33 e.s.u. the correction term is only of the order 10~ 6 as 
large as the main term 4-30. The reality of the saturation effect appears 
to have been first demonstrated by Herweg. 148 He even attempted to 
evaluate the constant a x in (18) from the amount of saturation observed 
experimentally, and hence determine the electric moment p,, assuming 
that the last term in (17) is predominant. He thus obtained a moment 
1-20 X 10~ 18 e.s.u. for ethyl ether, in exceedingly good agreement with 
the value l'14x 10~ 18 obtained by other, more standard, methods (see 
table, p. 66). It is hard to believe that this agreement is anything but 
accidental, for the derivation of moments from the amount of saturation 
is very difficult, not merely because the small saturation effects are very 
hard to measure with precision, but also because the first three terms 
of (17) are in reality not negligible, and further it is necessary to work 

"' P. Debye, Polar Molecules, p. 111. 

148 J. Herweg, Zeits.f. Physik, 3, 36 (1920); J. Herweg and W. Potzsch, ibid. 8, 1 
( 1922) ; An earlier attempt at measuring the saturation effect was made by S. Ratnowsky, 
Verh. d. D. Phys. Ges. 15, 497 (1913). 



88 MEASUREMENT OF ELECTRIC MOMENTS III, 22 

with pure liquids, where the Clausius-Mossotti expression for the local 
field is probably a poor approximation. Hence it appears undue optim- 
ism to expect quantitative rather than qualitative results from existing 
saturation experiments. The saturation effect has recently been 
measured for a number of materials by Kautsch, 149 and by Gunder- 
mann. 150 Kautsch finds that for ethyl ether, chloroform, and mono- 
chlorobenzene, all polar molecules, the saturation term in (IS) proves 
to be negative. In hexane and benzene, Kautsch and Gundermann 
respectively find this term too small to detect, and in carbon disulphide, 
which, like hexane, is non-polar, Kautsch finds it has a small positive 
value. This is in nice agreement with theory, as in highly polar mole- 
cules, the last term in (17), which is invariably negative, probably 
predominates, while in non-polar molecules only the first two terms of 
(17) remain, and these terms are probably small and usually positive. 

It has commonly been supposed that an elegant indirect way of 
observing saturation is furnished by the lowering of the dielectric con- 
stant of a liquid when a readily ionized salt is dissolved therein. Such 
a lowering is attributed by Debye and Sack 151 to the saturation of the 
molecules of the liquid by the intense fields arising from the dissolved 
ions. Clearly the effective susceptibility is not that appropriate to the 
applied field alone, but rather the much lower susceptibility appropriate 
to the resultant field obtained by compounding vectorially the applied 
field and the much larger ionic field. However, the experimental results 
have never been very consistent. The most recent experimental work, 
that by Wien, 152 seems to show that the dissolved ions raise rather than 
lower the dielectric constant. This, if true, is contrary to the earlier 
results of Sack, who found a lowering proportional to the 3/2 power of 
the valence of the dissolved salt, in accord with his saturation theory. 
If the results of Wien are accepted, the saturation effect is presumably 
still present, but masked by an increase of the susceptibility due to 
some other cause. 

149 F. Kautzsch, Phys. Zeits. 29, 105 (1928); measurements on water, glycerine, and 
othyl ether have been made by F. Malsr-h, Ann. der Physik, 84, 841 (1927). 

150 H. Gundermann, Ann. dcr Physik, 6, 545 (1930). 

161 P. Debyo, Polar Molecules* Chap. VI (includes references to other literature); 
H. Sack, Phys. Zcits. 27, 206 (1926); 28, 199 (1927). 

152 Unpublished work of Max Wien, communicated to the author by Professor 
Falkenhageri. 



IV 

THE CLASSICAL THEORY OF MAGNETIC 
SUSCEPTIBILITIES 

23. Conventional Derivation of the Langevin Formulae for Para- 
and Diamagnetism 

In Chapter II it was seen that if the molecule has a permanent electrical 
moment of magnitude p,, and if further the molecule is supposed rigid 
and hence incapable of induced polarization, the electric susceptibility 
is given by the expression -^ 2 



Let us now suppose that the molecule has a magnetic instead of electric 
moment, or, in other words, is a tiny permanent magnet. These mole- 
cular magnets will tend to aline themselves parallel to an applied 
magnetic field H, but are resisted by the 'temperature agitation' men- 
tioned in 10. By applying exactly the same physical reasoning and 
mathematical calculations as in Chapter II, except that the polarization 
is magnetic rather than electric, one concludes that the magnetic 
susceptibility is also given by an expression of the form (1), which 
is the Langevin formula for paramagnetism. According to Eq. (1) 
the paramagnetic susceptibility should be inversely proportional to the 
temperature, provided the density is kept constant. This relation is 
known as Curie's law, as it was discovered experimentally and enun- 
ciated by Curie 1 before it was obtained theoretically by Langevin. 2 Of 
course this law is not without numerous exceptions and refinements, 
which unfortunately tend to increase in number with improvements in 
experimental technique, but nevertheless Curie's law represents on the 
whole pretty well the gist of a large mass of experimental data for not 
merely gases, but many liquids and solids. 1 Of the two common para- 
magnetic gases, oxygen obeys Curie's law quite accurately right down 
to the temperature of liquif action, 3 whereas nitric oxide shows appre- 
ciable departures for reasons to be given in 67. From the magnitude 
of the temperature coefficient of the susceptibility it is possible to 
deduce the magnitude of the permanent magnetic moment of the mole- 
cule, in the same fashion as in the electrical case. A discussion of the 
numerical values so obtained experimentally for /z will be deferred until 

1 P. Curio, Ann. de Chim. ct Phys. (7) 5, 289 (1895); (Euvrea, Paris 1908, p. 232. 

2 P. Langovin, J. de Physique, (4) 4, 678 (1905); Ann. Chim. Phys. (8) 5, 70 (1905). 

3 P. Curio, 1. c. 1 and more recent other work to be cited and discussed in 66. 



90 CLASSICAL THEORY OF MAGNETIC SUSCEPTIBILITIES IV,23 
Chapters IX-XI as the theoretical estimates of p, with which they 
are to be compared are very intimately connected with the quantization 
of angular momentum. Just as in the electrical case, Eq. (1) holds only 
in fields inadequate for saturation effects, and in very strong fields the 
right side of (1) should be replaced by NnH^LdiHjkT), where L(x) is 
the complete Langevin function coth# (l/#), which has already been 
discussed and graphed on p. 32. The saturation effects are easier to 
detect experimentally in the magnetic than in the electric case, and for 
gadolinium sulphate, in particular, Woltjer and Onnes 4 have, by using 
very low temperatures and high fields (131 K. and 22,000 gauss), suc- 
ceeded in making the magnetic polarization reach over 80 per cent, of 
the saturation value iV/x corresponding to perfect alinement of all the 
molecular magnets parallel to the field. These measurements will be 
discussed more completely in 61 after we have developed the quantum 
theory of magnetism. 

Eq. (1) always makes x > an( l hence accounts only for para- 
magnetism. How can we explain the existence of diamagnetic media, 
which have susceptibilities x < ? In his celebrated paper Langevin 
answered this by showing that one indeed obtains diamagnetism if one 
considers the induced rather than permanent magnetic moment of the 
molecule. Such a result was also intimated, though less precisely, at 
considerably earlier dates by Weber and others. In Chapter I, especially 
Eq. (11), we -saw that the magnetic moment of an orbit is proportional 
to its angular momentum. Suppose now that the molecule is 'non- 
gyroscopic', i.e. has no electronic angular momentum in the absence of 
external fields (except the feeble electronic part of the angular momen- 
tum due to 'end over end' rotation of the molecule as a whole). If now 
a magnetic field is applied, the electronic motions are modified, and an 
'induced' angular momentum is created. The electronic orbits around 
the nuclei in many respects resemble a current undamped by resistance, 
and Lenz's well-known law states that currents induced by a magnetic 
field have such a sense that their magnetic fields tend to oppose the 
original field. The induced angular momentum thus has such a sign 
that the total microscopic magnetic field is less than the applied macro- 
scopic field //, and hence by Eq. (6), Chap. I, the magnetic induction 
B is less than H, making the material diamagnetic. These qualitative 
arguments may easily be made more precise if the molecule is mona- 
tomic, so that we can utilize Larmor's theorem (8). The more general 

* H. R. Woltjer and H. Kamorliiigh-Onnos, Leiden Communications, 167 c (or Versl. 
Amsterdam Akad. 32, 772, 1923). 



IV,23 CLASSICAL THEOKY OF MAGNETIC SUSCEPTIBILITIES 91 

non-monatomic case will be treated in 69. If p it z it fa be cylindrical 
coordinates of the electrons with the z axis coincident with the direction 
of //, then Larmor's theorem tells us that p t , Z L are the same functions 
Pt(0> 2 i(0 of the time as for a motion characteristic of H = 0, while the 
angular velocities fa about the axis are of the form fa(t)+He/2mc } the 
extra term Be/2mc representing thfc Larmor precession. The angular 
momentum ^mp^fa of the atom thus becomes (He/2c)^p^ if we 
suppose that J mtffa = when H = 0, or, in other words, that the 
atom has no permanent angular momentum. The summations, of 
course, extend over all the electrons in the atom. As the ratio of 
electronic magnetic moment to angular momentum is ej2mc (Eq. (11), 
Chap. I), the field thus creates a magnetic moment He 2 2 pf/4mc. 
The susceptibility is hence Ne 2 ]T/of /4rac, where the double bar in- 
dicates a statistical average over a very large number of molecules. If 
the orientations of an assembly of atoms are random, 5 then clearly the 
statistical mean of pf = x]-\-yl is two-thirds that of rf = ?+y?+zJ. If 
the molecules are all alike in size, there is no difference between the 
statistical mean rf over a large number of molecules, and the time- 
average 2rf for a single molecule. Thus we find 



which is Langevin's formula for the diamagnctic susceptibility in the 
form given by Pauli. 6 

Ono may also obtain Eq. (2) very simply by using Eqs. (48) and (49) of Chap. I. 
If ono assumes that there is no paramagiietism, so that Eq. (48) contains 110 
linear term in. H, then. it is a direct consequence of these equations that the 
susceptibility x - ~~ Nm t \H is 

< 2a > 



since the average of ^ 2 | y 2 is r 2 . The particles involved in the summation consist 
of the nuclei and electrons. The nuclei have such large masses compared to the 
electrons that their contributions to (2 a) may be neglected, inasmuch as (2 a) 
involves the masses in the denominator. In the case of electrons we can set 
e t = e, m t - m, and (2 a) thus becomes identical with (2). 

This second method of proof is short and does not explicitly uso Larmor's 
theorem. We have nevertheless first given the proof based on his theorem because 
such a proof is the usual one and gives more physical insight into the diamagnctic 
effect. Also the shorter proof is perhaps a little misleading because it gives the 

5 Tho random orientations will be slightly upset by the applied field, but the correction 
to tho diamagnetism on this account would only give a very small torm. 

6 W. Pauli, Jr., Zeite. f. Physik, 2, 201 (1920). Langevin's original paper gave the 
correct basic formula priori to tho spacial averaging, but in performing this average 
Langevin inadvertently took tho moan value of aJ 2 +2/ a as Jr 2 rather than $r 2 , thus giving 
an expression half as great as (2). 



02 CLASSICAL THEORY OF MAGNETIC SUSCEPTIBILITIES IV,23 
impression that (2) is valid even without Larmor's theorem. This, however, is not 
the case, as when his theorem is inapplicable, the linear terms in H in (48), Chap. I, 
cannot be disregarded. For this reason Eq. (2) cannot be applied to molecules, 
as will be discussed more fully in 69. 

Instead of the volume susceptibility x & ' s common practice to use 
a 'molar susceptibility' Xmoi similar to the molar electric suscepti- 
bilities mentioned in Chapter II. The molar susceptibility may be 
defined as the quotient obtained by dividing by the field strength the 
polarization of one gramme mol of the material rather than that of one 
c.c. Clearly x mol is given by an expression identical with (2) except that 
N is replaced by the Avogadro number L. Whenever the material is 
monatomic, the molar susceptibility is also sometimes termed the atomic 
susceptibility. 

There are two things particularly to be noted about Eq. (2): first, 
that it predicts that the diamagnetism per molecule be independent of 
the temperature provided the molecules always retain the same sizes, 
and, second, that the amount of diamagnetism should be propor- 
tional to 2 r 1> or approximately to the combined areas of the various 
orbits. The invariance of the diamagnetic susceptibility with respect 
to temperature was observed experimentally by Curie even before the 
Langevin theory. The molar susceptibilities of phosphorus, sulphur, 
and bromine, for instance, are independent of the temperature within 
the experimental error. For a great many elements, however, the 
independence of temperature is only approximate, 7 and in a few 
instances there arc very marked alterations in the diamagnetism at 
certain critical temperatures. Perhaps the worst offender is bismuth. 
Above its melting-point its atomic susceptibility has the constant value 
7-3 xlO- 6 , but at this point the susceptibility changes abruptly to 
about 200 X 10~ 6 and becomes even more highly diamagnetic as the 
temperature is lowered still further. De Haas and van Alphen have 
just found 8 a most remarkable periodic variation of the susceptibility 
of bismuth with field strength at the temperature of liquid hydrogen. 
These departures from the simple invariance of temperature and field 
strength predicted by the Langevin formula need not worry us too 

7 Stoner, Magnetism and Atomic Structure, p. 265, gives a comprehensive tablo of the 
sign of tho temperature coefficients and amounts of diamagnetism for tho different chemi- 
cal elements. Tho temperature variation of diamagnotism at low temperatures has 
recently boon measured for N 2 and H a by Bitter, Phys. Rev. 36, 1648 (1930). He finds 
a very large variation in the case of hydrogen, which is hard to understand from a 
theoretical standpoint, and ho suggests that his measurement on this gas ought to bo 
repeated. 

8 W. J. de Haas and P. M. van Alphen, Leiden Communications, 2 12 a (1931). 



IV,23 CLASSICAL THEORY OF MAGNETIC SUSCEPTIBILITIES 93 
much. In the first place small variations with temperature can be 
understood on the ground that the sizes of the orbits are not invariants, 
so that 2 rf changes somewhat with T. Of course the tremendous 
anomalies found in bismuth cannot be interpreted on the basis of any 
ordinary orbital contraction or expansion, but then bismuth has always 
been a black sheep because of its anomalous behaviour as regards 
electrical and other properties in the solid state. Ehrenfest and Raman 9 
have stressed that perhaps its large diamagnetism when solidified is due 
to the orbits extending around several atomic nuclei and hence having 
large diameters. The variations with field strength at low temperatures 
are presumably due to some sort of resonance between the radius of 
curvature of the electron's path in the magnetic field and inter-atomic 
distances. It is to be particularly emphasized that the large variations 
of diamagnetism with temperature are all found in the liquid or 
especially the solid state. The simple Langevin theory should be applic- 
able primarily to gases, and their molar diamagnetic susceptibilities are 
indeed invariant of the temperature, or at least very nearly so. 

From the absolute magnitude of the susceptibility it is possible to 
deduce an estimate of the sizes of the orbits. When numerical values 
are substituted for e, c, m, L, the formula for the molar susceptibility 
following from (2) is x ^ _ __ 2 -832 X 1C 10 2 r\. (3) 

Now the diamagnetic susceptibilities observed experimentally are 
usually of the order of magnitude 10~ 6 Z, where Z is the atomic number. 
As there are Z electrons that contribute to the sum in (3), the expression 
(3) becomes of this order of magnitude if the electronic orbits have on 
the average radii of the order 10~ 8 cm. This is in nice agreement with 
the estimates of molecular radii obtained by kinetic theory and other 
methods. It is to be clearly understood that the value of the orbital 
radius which we have deduced is only a crude average over all classes 
of orbits. The valence orbits may be somewhat larger than the estimate, 
while the innermost ones will usually be considerably smaller. Quanti- 
tative calculations of susceptibilities by means of (3) will be considered 
more fully in Chapter VIII. 

In case the molecule has a permanent magnetic moment, the dia- 
magnetic term (2) ought really to be added to the right side of (1) to 
obtain a complete expression for the susceptibility. It is always to be 
remembered that diamagnetic induction is a universal characteristic of 
all atoms and molecules, although the diamagnetic terms are usually 

9 P. Ehrenfest, Physica, 5, 388 (1925); Zeits.f. Physik, 58, 719 (1929); C. V. Raman, 
Nature, 123, 945; 124, 412 (1929). 



94 CLASSICAL THEORY OF MAGNETIC SUSCEPTIBILITIES IV,23 
overshadowed by the paramagnetic ones if the molecule has a per- 
manent moment. Thus in rough determinations of paramagnetism it 
is not necessary to add (2) to (1) unless the paramagnetism happens 
to be quite feeble, for strongly paramagnetic substances have molar 
susceptibilities of the order 10~ 4 or greater, whereas diamagnetic sus- 
ceptibilities are ordinarily of the order 10~ 6 to 10~ 5 . In accurate 
measurements or calculations, the correction for the diamagnetic part 
of the susceptibility should always be made if one wishes to deduce the 
purely paramagnetic susceptibility from the observed susceptibility. 
The diamagnetic susceptibility cannot, of course, be measured separ- 
ately, but can be estimated theoretically with accuracy sufficient for 
the correction, as it is a small one. When the diamagnetic term is added, 
Eq. (1) of the present chapter becomes of the same form as (1), Chap. II, 
if now No. denotes the right side of (2). The diamagnetic correction term 
thus resembles the second term of (1), Chap. II, arising from the induced 
polarization in the electrical case, inasmuch as both are independent of 
the temperature, and are due to distortion of the electronic motions by 
the applied field. The analogy is, however, a very incomplete one, for 
in the electrical case a is positive rather than negative and not generally 
small in magnitude compared to the term n z j3kT arising from the per- 
manent dipoles. Also somewhat different models have been used in the 
electric and magnetic cases, as in Chapter II we assumed the electrons 
had positions of static equilibrium, whereas we now picture them as 
circulating in orbits to endow the molecule with angular momentum. 
This inconsistency is an inherent classical one, and will be removed only 
in the quantum mechanical treatments given in later chapters. 

24. Absence of Magnetism with Pure Classical Statistics 

If we could stop at this point, we should feel exceedingly happy, for 
the simple Langevin theory has been shown to explain nicely many 
of the experimental phenomena, especially the difference between the 
temperature effects in para- and diamagnetism. However, in 1919 Miss 
van Leeuwcn 10 demonstrated the remarkable and rather disconcerting 

10 J. H. van Leeuwen, Dissertation, Leiden, 1919. A comprehensive summary is given 
in J. de Physique, (6) 2, 361 (1921). The work which wo quote is especially that given 
on pp. 372-4 of the summary. Besides the study of the magnetism for the general 
atomic dynamical system, Miss van Leouwen also examines special models in which the 
electrons are replaced by continuous currents. These models seem much less satisfactory 
than the general dynamical method which wo have reproduced. She mentions and dis- 
cusses at some length the fact that a susceptibility different from zero can bo obtained 
if in statistical mechanics there is imposed some auxiliary condition (Nebenbedingung) 
which restricts to a definite numerical value some other function of the dynamical 



IV,24 CLASSICAL THEORY OF MAGNETIC SUSCEPTIBILITIES 95 
fact that when classical Boltzmann statistics are applied completely to 
any dynamical system, the magnetic susceptibility is zero. We shall 
refer to this result as 'Miss van Leeu wen's theorem', but we must 
mention that other investigators 11 had previously predicted zero mag- 
netic susceptibilities under certain conditions, but it remained for Miss 
van Leeu wen to review critically the whole subject of susceptibilities in 
classical theory. There is no analogous theorem on null susceptibilities 
in the electrical case. One immediately wonders how Miss van Leeuwen's 
theorem can be reconciled with the fact that the simple Langevin theory 
predicts a susceptibility which can be either positive or negative, but 
not in general zero. The answer is that the conventional Langevin 
theory is open to the objection that it assumes a priori that the molecule 
has a definite 'permanent' magnetic moment which is the same for all 
molecules of similar chemical composition. As magnetic moment is 
proportional to angular momentum (if all the circulating particles are 
identical), we are thus supposing that the electronic angular momentum 
of the molecule has one definite value. Actually this cannot be the case 
with pure unadulterated classical statistics, as they always give a con- 
tinuous range of permissible values to all coordinates, and hence to the 
angular momentum. Thus the electronic angular momentum should 
have all values ranging from oo to +00. Similarly in the diamagnetic 
term, the radius of a given orbit can have a continuous range of values 
rather than the one particular size presupposed in the Langevin theory. 
In other words, it was not legitimate just before Eq. (2) to replace the 
double bars denoting the statistical average by the single bars denoting 
the time average for an individual atom. The relative prevalence of the 
different values of the angular momentum and radius should, of course, 
be determined by the Boltzmann probability factor e -W /T . In the con- 
ventional derivation given in 23, we have thus frozen ('ankylosed', as 
Jeans terms it) the electronic motions to one particular size and shape, 
rather than admitted the infinite number of possibilities allowed by 

variables of the assembly of molecules besides its total energy. There is, however, no 
known justification for the imposition of such an extra condition in assemblies such as 
are encountered in the theory of magnetism. 

11 W. Voigt, Ann. der Physik, 9, 115 (1902); J. J. Thomson, Phil. Mag. (6) 6, 673 
(1903). The mathematics of the theory of magnetic susceptibilities have been extensively 
developed by R. Gans in a number of papers: Gb'tt. Nachr., 1910, p. 197; 1911, p. 118, 
Verh. d. D. Phys. Ocs. 16, 780, 964 (1914); Ann. der Physik, 49, 149 (1916), summary 
by Wills in Theories of Magnetism (Bulletin 18 of the Nat. Res. Counc.) Gans's work 
contains stimulating features, but the typo of magneton which he assumes can scarcely 
be reconciled with modern knowledge of atomic structure. Many of Miss van Leeuwon's 
results were previously obtained in Bohr's dissertation (Copenhagen, 1911), but this 
unfortunately is probably rather inaccessible to most readers. 



96 CLASSICAL THEORY OF MAGNETIC SUSCEPTIBILITIES IV,24 
real classical statistics. One thus has to modify, or rather supplement, 
the classical statistics by an auxiliary condition (Nebenbedingung) that 
the angular momentum of the molecule be restricted to a particular 
value, and such a restriction appears highly artificial, to say the least. 
Of course the fact that the electronic motions do not contribute to the 
specific heat shows that real classical statistics cannot be applied to 
them, but it is nevertheless, from a logical standpoint, not at all satis- 
fying to apply, as the Langevin theory does, the classical distributions 
to the 'external' degrees of freedom specifying the rotation of the 
molecule as a whole, but not to the 'internal' or 'electronic' degrees 
of freedom. When one tries to be consistent and apply the classical 
Boltzmann distribution to all coordinates necessary to specify the con- 
figuration of the system (assuming this to be possible actually it is not 
in real atoms, as we shall see in 27), the paramagnetic and diamagnetic 
parts of the susceptibility exactly compensate each other. 

The proof of Miss van Leeu wen's theorem is very simple. The mag- 
netic moment of the molecule in any direction, say z, may be taken to 
be a linear function ^ = | ^ (4) 

of the generalized velocities </j,..., q f corresponding to any set of Lagran- 
gian positional coordinates adequate to specify the configuration of the 
molecule. The coefficients will in general be functions of the ^'s. These 
remarks are obvious in Cartesian coordinates, as here 

1 V / - - x 
W = 2c2, yi ~~ 

(Eq. (11), Chap. I), and the linearity in the velocities is preserved under 
any 'point' transformation to another set of generalized coordinates. 
The magnetic moment per unit volume in the direction z of the applied 
field is (Eq. (57), Chap. T) 

M 8 = CN J...J a k fa-W ^...dq f d Pl ...d P/ . (5) 

Let us consider any particular term a^q^ in the summation. By Hamil- 
ton's equations qj dJf/dpj, and hence it is clear that of the 2/ integra- 
tions, the one over pj can immediately be performed for this particular 
term, as the integrand is merely kTdfaje-Wydpj. If a and b denote 
the two limits of integration for pp the contribution of the term under 
consideration to (5) becomes 

-CNkT J...J fae-wr]^ (6) 

We now suppose that the energy becomes infinite when the momentum 



IV,24 CLASSICAL THEORY OF MAGNETIC SUSCEPTIBILITIES 97 
Pj assumes its extreme values a car b. The fulfilment of this condition 
is the essential requirement in Miss van Leeuwen's proof, and is 
obviously realized in a Cartesian system, as Cartesian momenta can 
range from oo to +00, and when the momentum is infinite the kinetic 
energy is, of course, also infinite. Thus we appear quite warranted in 
assuming that [a^e--^' 7 '] = regardless of the values of the remaining 
variables q v .., p v .., p^\, PJ+I-- - Hence the contribution (6) of a typi- 
cal term of the summation in (5) is zero, and as this demonstration is 
applicable to all terms, we see that (5) does indeed vanish. This null 
result holds quite irrespective of the presence of an applied magnetic 
field, as nothing in the proof requires that Jt be independent of H. 

25. Alternative proof of Miss van Leeuwen's Theorem 

It has occurred to the writer that the null suscej)tibility with pure 
classical theory can also be demonstrated by the following method as 
an alternative to Miss van Leeuwen's own one given above. 12 Let us, 
for simplicity, use a Cartesian system. The magnetic moment is 

N '" 



... 



dxdydzdp x dp v dp z 



where dxdydz means djc^dx 2 ...dy^dy 2 ...dz l dz 2 ..., with an analogous inter- 
pretation of dpjdpjjdp... We have here used p^ as an abbreviation for 
the expression 7n t x h which is not the same as the canonical momentum 
pt^mjCi+etAxJc, as already explained in 7-8. The p are essen- 
tially the velocity coordinates, as they differ from the velocities only 
by the constant mass factors m^ Now when expressed in terms of the 
canonical variables x, y, z, p x , p y , p z the Hamiltonian function Jt 
involves the magnetic field H as a parameter, whereas when expressed 
in terms of the x, y, z, p%, jpj), p it does not involve H explicitly, as it is 
exactly the same function of its arguments as for H = 0. This has been 
seen in 7, especially Eq. (51), and is associated with the fact that 
magnetic forces do no work, making the energy the same function of 
position and velocity as in the absence of the field. This expression 
,#* cannot be regarded as a true Hamiltonian function, as the variables 
x,..., #2 unlike x,..., p,,..., do not satisfy Hamilton's equations. For 

12 Since writing the present section, the author has learned that this alternative proof 
somewhat resembles one in Bohr's dissertation (Copenhagen, 1911) as Bohr also notes 
that absence of magnetism is a consequence of the fact that the functional determinant 
(8) is unity. 

3595.3 H 



98 CLASSICAL THEORY OF MAGNETIC SUSCEPTIBILITIES IV, 25 

clarity we attach an asterisk to J when it is expressed in terms of the 
2> rather than p. In view of the foregoing, 

^ = Jf*(x t y,z 9 p% 9 p*,p*) whereas J* = Jt(x,y,z,p x ,p y ,p z JI). 
Because of the independence of <#* of H, it is convenient to change 
the variables of integration in (7) from the p x , p y , p s to the p^ p$, p%. 
Because the transformation equations are of the form 

= 



the functional determinant 



of the transformation is unity. 13 Hence 

dp x dpydp z dxdydz = 
or in other words, the * weight' is the same in the p...x... as in the 
p x ...x... space. This is the crux of the whole proof, and enables us to 
calculate simply distributions in the p%...x... space. We could not have 
done this at the outset, as the theorems of statistical mechanics relate 
fundamentally to the 'phase space' of the canonical variables p x ...x... 
rather than to the space of the position and velocity ones. We now 
see that Eq. (7) retains its validity if wo write <#* in place of Jt and 
dp$dptylp% in place of dp x dp y dp s . Because of the kincmatical significance 
of the p Q as proportional to velocities, the limits of integration for the 
p are independent of H. From this and the fact that dJt*jdH = 0, it 
thus follows that by changing the variables to the p, we have made 
the right side of (7) completely independent of H. This means that the 
moment is the same as in the absence of the magnetic field, and hence 
is zero, since an isotropic, non-ferromagnetic body supports no out- 
standing moment when H 0. 

As a corollary of the above, it follows that the probability that the 
system be in a configuration corresponding to the element 



in which p^ falls between p^ and p^+dp^, &c., is 



The distribution of values of the coordinates and velocities is thus the 
same as in the absence of the magnetic field, since <&* is independent 
of //. (The distribution of the canonical variables p x ,..., #,..., on the 

13 Because the coordinates x, y, z are the same in the old arid now system of variables, 
this determinant is equal to the smaller determinant d(p x >P v >Pz)/d(Px>Pv>Pi)> wnicn is 
obviously unity. The identity of the large and small determinants is similar to that 
mentioned in footnote 7 of Chap. 1J. 



IV,25 CLASSICAL THEORY OF MAGNETIC SUSCEPTIBILITIES 99 
other hand, given by Eq. (55), Chap. I, involves the magnetic field 
through Jf.) The Maxwell distribution of translational velocities for 
free particles is, for instance, unmodified by a magnetic field. It is clear 
that the statistical mean of any function of the variables p%, p, p%, 
x, y, z which does not involve H explicitly is unaltered by application 
of a magnetic field. This may be regarded as a generalization of Miss 
van Leeuwen's theorem, and her null result on susceptibilities is merely 
the special case that the function is 2(^/2^)^^ 2/iA)- 

We have already mentioned that the reason that Langevin obtained 
a non-vanishing susceptibility is because he did not apply the Boltz- 
mann distribution to the internal or electronic degrees of freedom of 
the molecule. We may now amplify this point a little farther. Of the 
/ generalized coordinates in Eq. (5), three, say q v q 2 , # 3 , will, in the 
general poly atomic molecule, be what we may term 'external' coordinates, 
which specify the orientation of the molecule as a whole, as, for instance, 
the Eulerian angles in 12. The remaining coordinates g 4 , # 5 ,... will be 
internal coordinates. (We do not need to include coordinates specifying 
the translational motion of the molecule as a whole, as in Chapter I we 
agreed to consider the centre of gravity of the molecule to be at rest.) 
Now in the usual derivations of the Langevin formula, the Boltzmann 
distribution is applied to the canonical variables q l3 q 2 , q 3 , p^ p 2 , p 3 but 
not to g 4 . . . . , pi, . . . . Consequently the usual results ( 1 ) and (2) so obtained 
ought therefore for consistency to be integrated over these remaining 
variables. When we combine the paramagnetic and diamagnetic parts, 
the complete susceptibility should thus be given by the formula 14 



(') 



Here /i and the r i are to be regarded as functions of q 4 ,..., p&... rather 
than as molecular constants as in the ordinary Langevin treatment. 
Miss van Leeuwen's theorem tells us that when the integration is per- 
formed the para- and diamagnetic parts of (9) always cancel. Eq. (9) 
is not general enough to show the full sweep of Miss van Leeuwen's 
theorem, since the latter assures that the para- and diamagnetic effects 
always compensate to all powers of H, whereas (9) does not aim to 

14 In writing Eq. (9) wo use tho slightly modified form (2 a) of (2) which was given 
on p. 91. Eq. (2 a) includes the feeble contribution of the nuclei to the diamagnetism, 
without which tho cancellation of tho two parts of (9) would be very approximate rather 
than exact. 

H2 



100 CLASSICAL THEORY OF MAGNETIC SUSCEPTIBILITIES IV,25 

include saturation effects and gives only the part of the susceptibility 
which is independent of field strength. 

It is perhaps illuminating to verify explicitly for a very simple 
dynamical system that the two parts of (9) cancel, without appealing 
to the general proof. Let us suppose that we have a particle of charge 
e and mass m constrained to always remain at a distance I from a fixed 
centre. Let us further assume that the particle is subject to no other 
force except that of constraint, so that it will move in a circle with 
a constant angular velocity Q. The radius r of the orbit will have the 
constant value /, and our example is thus not illustrative of the most 
general case in which r is a statistical variable. The magnetic moment 
el 2 l/2c is, on the other hand, such a variable, as a molecule can acquire 
any amount of angular velocity 12. By the equipartition theorem, the 
statistical average of the kinetic energy irw/ 2 n 2 is kT, since the particle 
has two degrees of freedom. Now in this example the square of the 
magnetic moment differs from the kinetic energy only by a constant 
factor e 2 / 2 /2wc 2 . Hence the statistical mean square jrf the magnetic 
moment, such as results from the integration in (9), is /x 2 kTe 2 l 2 /2mc 2 . 
The proportionality of this expression to T cancels the T in the de- 
nominator of the Langevin formula, and we have indeed 



26. Absence of Diamagnetism from Free Electrons in Classical 
Theory 

In 25 we have shown that a magnetic field does not influence the 
Maxwellian distribution of translational velocities. This result is of 
particular interest when applied to free electrons, e.g. either stray 
electrons in a gas or conduction electrons in a solid. Of course when 
a magnetic field is applied, free electrons no longer move in rectilinear 
paths, but instead describe circular orbits about the direction of the 
field. One usually associates Maxwell's distribution with rectilinear 
motions, but it is not at all incompatible with the existence of such 
circular trajectories. Now since the Maxwellian velocity distribution is 
unaltered, the mean moment of the free electrons is uninfluenced by 
a magnetic field, and hence they cannot give either a paramagnetic or 
diamagnetic effect. Numerous attempts, 15 to be sure, have been made 
in the literature to show that free electrons behave diamagnetically, but 
if classical statistics are applied in their simplest and most direct manner 

15 J. J. Thomson, Rapports du Congres de Physique, Taris 1900, p. 140 ; E. Sfhrodingor, 
Wien.Ber. 121, 1305 (1912); J. N. Kroo, Dissertation, Gottingen, 1913, Ann.dcr Physik, 
42, 1354 (1913); H. A. Wilson, Proc. Roy. tioc. 9?A, 321 (1920). 



IV,26 CLASSICAL THEORY OF MAGNETIC SUSCEPTIBILITIES 101 
given above, their contribution to the magnetic moment is nil. This 
appears to have been first shown by Bohr, 16 and has also been observed 
by Lorentz and Miss van Leeuwen. 

This absence of a diamagnetic susceptibility from free electrons at 
first thought appears quite paradoxical. If each electron describes a 
circle about the field, it certainly possesses angular momentum about 
the centre of the orbit, and the sense of the rotation is such that the 
attendant magnetic moment is opposite to the field, apparently giving 
diamagnetism. However, the magnetic moment involved in the sus- 
ceptibility is not the magnetic moment of each electron with respect to 
the centre of its particular orbit, but instead the combined magnetic 
moment of all the electrons with reference to some one common point 
chosen as the origin for measuring angular momentum. In the schematic 
figure on p. 102, in which the magnetic field is supposed perpendicular 
to the plane of the paper, electron 2 clearly gives a diamagnetic moment 
with respect to point A, electron 3 with respect to point B, &c., but 
what we need is the combined moment of electrons 1, 2, 3..., with respect 
to some one point, say B. Now when electron 2 passes through the 
small element enclosed by the dotted square, its angular momentum 
relative to B is just equal and opposite to that of electron 1 when it 
passes through this element. Since actually electrons are distributed 
on the average continuously through space rather than with their orbits 
end to end as in the figure, it is clear that to every given electron 
passing through a given point in space with a velocity in a given direc- 
tion, there is another electron describing another circle and passing 
through the point with an equal velocity in an exactly opposite direction. 
In case the body containing the electrons is bounded in extent, the 
electrons near the boundary cannot describe complete circles but are 
reflected from the boundary (indicated by the heavy line). Instead, 
they describe cuspidal paths, such as are illustrated for electrons 
numbered 1 in the diagram. These boundary electrons are very vital, as 
without them there would be diamagnetism. An electron 1, for instance, 
is needed to compensate electron 2 at the point where their orbits touch. 
Fig. 5 is, of course, entirely too naive, but perhaps does afford some 
sort of a physical illustration of the general null result derived in 25. 
It may be noted that in 25 we did not need to use specifically the fact 
that the orbits are circles. This illustrates the beautiful freedom of the 

10 N. Bohr, Dissertation, Copenhagen, 1911. H. A. Lorciitz, Gottinger Vortrdge uber die 
kinetische Theorie der Materie widder Elcktrizitdt (Leipzig, 1914), p. 188; J. H. van Leeu- 
wen, Dissertation, Leiden, 1919, p. 49; J. de Phys. (6) 2, 361 (1921). 




102 CLASSICAL THEORY OF MAGNETIC SUSCEPTIBILITIES IV,26 
statistical method from the necessity of inquiry into the details of 
the motion of a dynamical system; only the Hamiltonian function is 
required. It might seem as if the characteristics of the bounding surface 
might make a difference in the proof. In Fig. 5, for instance, we assumed 
specular reflection at a cylindrical boundary of radius E. However, 

the boundaries for the spacial co- 
ordinates did not enter in the 
demonstration, and the medium 
could, in fact, be infinite in extent, 
or of a different shape or degree 
of smoothness than in Fig. 5. 
Also the electrons can suffer col- 
lisions. 17 The molecules move so 
slowly relative to the electrons 
that the former may be considered 
fixed scattering centres, and no 
harm is done in the j>roof if it is 

supposed that the potential energy 
FIG. fi. , ,7 , , u , 6J 

ot the electrons becomes very large 

at certain points, which consequently act as such scattering centres. 
A potential barrier is also required at the boundary to reflect the electrons. 
Of course in a true theory, quantum modifications must be taken into 
account, and it will be shown in 81 that in quantum mechanics there 
is a diamagnetic effect from free electrons, not to mention the spin 
paramagnetism ( 80). Thus the theorem on the absence of diamag- 
netism is valid only in classical theory. 

27. Inapplicability of Classical Statistics to any Real Atomic 
System 

Let us now revert to atoms and molecules rather than free electrons. 
Needless to say, the zero susceptibilities predicted in 24-5 are not 
the rule experimentally. In the theory the only escape from zero 
moment is not to apply the Max well -Boltzmann distribution to all 
coordinates and momenta, but instead to restrict their ranges of per- 
missible values. Such restrictions are effectively ' quantization', and we 
now anticipate the inevitable need of a quantum theory. Even in the 
electrical case the complete application of classical statistics to all 

17 Miss van Loouwon notes that the collisions might conceivably cause the free elec- 
trons to contribute to the susceptibility, provided some other function of the assembly's 
dynamical variables besides the total energy remains constant during the collisions. 
Of. note 10. 



IV,27 CLASSICAL THEORY OF MAGNETIC SUSCEPTIBILITIES 103 

degrees of freedom gives rather absurd results, though nothing as 
striking as a null susceptibility. For if we assume a statistical distribu- 
tion of electrons among orbits of various sizes, some molecules of a given 
chemical composition would be large, others small. Such a concept as 
the familiar atomic diameter in kinetic theory would be impossible, and 
also, more especially for our purposes, that of the permanent molecular 
moment. Instead we could speak only of the distributions of values for 
the diameters and moments, and we could employ only the statistical 
mean square moment (somewhat as in Eq. (22), Chap. II), which would 
in general vary with temperature and cease to be useful. 

All these things, along with the specific heat difficulty, force the con- 
clusion that classical statistics give only nonsense when applied to the 
internal or electronic degrees of freedom of the molecule. Another con- 
sideration which shows this even more urgently is the following. In the 
classical Boltzmann distribution formula it is ordinarily supposed that 
the numerical value of the energy or Hamiltonian function ,# can range 
from to oo, as the total integrated probability 6 f J...Je~^* T dq v ..dp f 
can then be made to converge to unity by proper normalization of the 
amplitude constant C. However, for real motions of the electrons, the 
energy approaches oo when an electron is close to the nucleus, and 
when it is removed to infinity. The energy thus ranges from oo 
to rather than from to oo. When M approaches oo the Boltzmann 
probability factor increases without limit. In a hydrogenic atom, for 
instance, the probability becomes infinite in the fashion e Xc ^ rkT as one 
approaches the nucleus, which we suppose located at the origin r = 0. 
It is thus infinitely more probable that the electron be infinitely close 
to the nucleus than anywhere else. The total integrated probability can 
clearly be finite only if we nonsensically take the amplitude C to be 
zero, and suppose the probability is infinitesimal of the electron being 
anywhere but right at the nucleus. In other words, we have a collision 
catastrophe, which is a little reminiscent of the 'ultra-violet catastrophe' 
in the classical theory of black body radiation, whereby the Rayleigh- 
Jeans formula u v ~ 8irv 2 kT/c* demands that the energy density u v 
increase without limit as we go to higher and higher frequencies. 
Modern physics shows a good deal of parallelism between matter and 
radiation, and they both have their catastrophes in classical statistics. 
The absurdities arising from non-convergence are frequently emphasized 
in connexion with the radiation problem, but, as the writer has men- 
tioned elsewhere, 18 do not seem commonly enough appreciated as 

18 Cf. J. H. Van Vlock, Quantum Principles and Line Spectra, p. 14. 



104 CLASSICAL THEORY OF MAGNETIC SUSCEPTIBILITIES IV,27 
regards the application of classical statistics to the Rutherford atom. 
It is because of this inherent limitation in classical theory that it has 
always been necessary in the classical theory of induced polarization to 
use not a real Rutherford atom but instead, as in Chapter II, an un- 
plausible model consisting of electrons oscillating harmonically about 
positions of static equilibrium. The limited range of molecular models 
which can be used in classical statistics makes Miss van Leeu wen's 
theorem rather academic, but nevertheless it is occasionally useful for 
other problems besides proving the absence of diamagnetism from 
classical free electrons. 

To summarize, the success of the Langevin and Debye theories shows 
that classical statistics give good results when applied to the external 
(i.e. rotational) degrees of freedom of the molecule. On the other hand, 
when one attempts to apply classical statistics to electronic motions 
within the atom, the less said the better. Hence, in the following 
chapters we must seek a quantum mechanics which constrains the 
electrons to move in certain discrete stationary states instead of giving 
a classical continuous distribution of orbits near the nucleus, but which 
when applied to rotational degrees of freedom gives nearly the same 
statistical results as classical theory except perhaps at very low tem- 
peratures. 



SUSCEPTIBILITIES IN THE OLD QUANTUM THEORY 
CONTRASTED WITH THE NEW 

28. Historical Survey 

To some readers it may seem like unburying the dead to devote a 
chapter to the old quantum theory. Every one knows that the original 
quantum theory developed by Bohr in 1913 has been refined and in 
a certain sense replaced by the new quantum mechanics of Heisenberg, 
Schrodinger, Born, and Dirac. However, there is perhaps no better field 
than that of electric and magnetic susceptibilities to illustrate the 
inadequacies of the old quantum theory and how they have been 
removed by the new mechanics. We shall merely summarize the results 
of applying the old theory, without giving any mathematical analysis. 
Also, we shall contrast descriptively with these results some of the out- 
standing features of the new quantum mechanics of susceptibilities, 
thereby giving a qualitative account of some of the new improvements 
which may perhaps satisfy some readers who do not wish to read the 
mathematics in the two following chapters. 

The old quantum theory was probably more successful as applied to 
magnetic than to electric susceptibilities. In the first place, inasmuch 
as it substituted discrete for continuous distributions, it clearly removed 
the difficulty found in the classical theory ( 27) of the overwhelming 
probability of orbits infinitesimally close to the nucleus. The smallest 
allowed quantum orbit had instead a finite radius. Also consistency no 
longer demanded zero susceptibility, as Miss van Leeu wen's theorem 
was no longer applicable. Furthermore, the old quantum theory (sup- 
plemented by the spin anomaly) was not very far from predicting 
quantitatively the magnetic moments p of atoms. To do this correctly 
it would have to yield the formula for the anomalous Zeeman effect. 
Lande 1 indeed gave a semi-theoretical derivation of his celebrated 
formula for this effect, but besides introducing the anomalous factor 
two in the magnetic moment of what he called the atom-core (but which 
we now know is electron spin), he found it necessary to give certain 
quantum numbers half -integral instead of integral values, and also even 
then to insert an extra term J in order to get agreement with experi- 
ment. 2 The old quantum theory was thus patched almost beyond 

1 A. Land6, Zeits.f. Physik, 15, 189 (1924). 

2 This was because the expression /(/+ 1) characteristic of the true quantum mechan- 
ics was interpreted as 7 ts i, where /t is a half quantum number J-f . 



106 SUSCEPTIBILITIES IN THE OLD QUANTUM THEORY V,28 
recognition, but Lande's work nevertheless distinctly showed that he 
was hot on the track of a true theory of the anomalous Zeeman effect, 
since supplied ( 42) by the new mechanics. There were other difficulties 
in the old quantum theory of magnetic susceptibilities, especially the 
troubles with weak and strong spacial quantization. These troubles will 
be discussed more specifically in connexion with dielectric constants, 
but the difficulties found in the electric case are usually also reflected 
in the magnetic one. 

One of the best-known and most characteristic features of the quan- 
tum theory is the phenomenon of spacial quantization. By this is meant 
the fact that according to the quantum conditions the molecule can 
only assume certain particular orientations in space. The particular 
condition responsible for the spacial quantization is usually the require- 
ment that the angular momentum of the molecule along some direction 
fixed in space be an integral or half -integral multiple M of A/277. Here 
M is called the 'equatorial', ' axial', or (even in the electric case!) the 
'magnetic' quantum number. A direct experimental confirmation of 
spacial quantization is furnished by the well-known experiments of 
Gerlach and Stern 3 on the deflexion of atoms in a non-homogeneous 
magnetic field. (The field must be non-homogeneous to give a transla- 
tional force on a magnetic dipole.) The discovery of a discrete rather 
than continuous set of deflexions in these experiments is conclusive 
evidence that the atoms can only orient themselves in particular direc- 
tions under the influence of an applied field. 

Because of the spacial quantization, one immediately expects the 
dielectric constant to be given by a different formula than in classical 
theory, where random orientations are assumed. This, indeed, proved 
to be so in the old quantum theory. Pauli, 4 treating polar molecules as 
rigid non-gyroscopic rotating dipoles (the so-called 'dumb-bell' model 
for molecules such as HC1), found that the electric susceptibility was 
still given by a formula of the form 



as in classical theory, with /x the moment of the molecule, and C a pure 

3 Gorlach and Stern, Zeits.f. Physik, 9, 349 (1922) and numerous subsequent papers 
by Gerlach and others. The ordinary Stern-Gorlach experiment is performed with mag- 
netic fields. The analogous experiment with electric fields is more difficult, but has 
recently boon performed by E. Wrede, Zeits.f. Physik, 44, 261 (1927) and by J. Ester- 
maim, Zeits.f. Phytt. Chem. 1, 161 (1928), Dipolmoment und chemisette Struktur (Leipziger 
Vortrage 1929), p. 17; Ergebnisse der exakten Naturwissenschaften, viii. 279. 

4 W. Pauli, Jr., Zeits.f. Physik, 6, 319 (1921). 



V,28 CONTRASTED WITH THE NEW 107 

number. However, the numerical factor C no longer had the value 1/3 
found in classical theory, but was instead 1-54. Later, progress in the 
analysis of band spectra made it increasingly apparent that the quan- 
tum numbers involved in the theory of the rotating dipole should be 
given half -integral instead of integral values to agree with experiment. 
Pauling 5 therefore revised Pauli's calculations by introducing half- 
quantum numbers. The result was still another value of C. These 
vicissitudes in C are listed in the following table, together with the 
corresponding values of the electrical moment p of the HC1 molecule 
deduced by applying the formulae to Zahn's 6 measurements on the 
temperature variation of the dielectric constant of HC1. The changes 
in C, of course, profoundly affect the value of /z, deduced from such 
experimental data. 

; Corresponding Value of 

| ] Electrical Moment /x of 

Value of Constant C. Form and Year of Theory, j HCl Molecule. 

I : Classical, 1912 | 1-034X 10' 18 e.s.u. 

1-54 '; Whole quanta, 1921 | 0-481 X 10' 18 

4-57 Half quanta, 1925 ; 0-332 xlO' 18 

J , New mechanics, 1926 ! 1-034X10' 18 



The last line gives the results obtained with the new quantum mechanics, 
which will be derived in detail in the following chapters. It is seen that 
this new dynamics restores the factor J characteristic of the classical 
Langevin formula. After quite a history, the computed electrical 
moment of the HCl molecule thus reverts to its original value. In 
general, the susceptibilities obtained with the new quantum mechanics 
are more closely akin to those of the classical theory than are those 
obtained with the old quantum theory. For this reason we were able 
in Chapter HE to discuss fairly completely the theoretical interpretation 
of experimental material on dielectric constants without deferring the 
discussion to the quantum chapters. It will be noted that in the old 
quantum theory, the discrepancy with the classical value of C persisted 
regardless of the temperature. Such a discrepancy did not seem 
plausible even before the discovery of the new mechanics, as the corre- 
spondence principle led us to expect usually an asymptotic connexion 
of the classical and quantum results at high temperatures. In the old 
quantum theory the value of the numerical factor C was not a universal 
constant, as it was very sensitive to the nature of the model employed; 

5 L. Pauling, Phys. Rev. 27, 568 (1926). 

6 C. T. Zahn, Phys. Rev. 24, 400 (1924). 



108 SUSCEPTIBILITIES IN THti OLD QUANTUM tHftORV V,28 
a gyroscopic 7 rather than a dumb-bell model would, for instance, furnish 
a different C than 1-54 or 4-57. On the other hand, we shall see that 
the new mechanics always yield C = % without the necessity of speci- 
fying the details of the model, and the generality of this value of C is 
one of the most satisfying features of the new theory. 

29. Weak and Strong Spacial Quantization 

A difficulty particularly characteristic of the old quantum theory is 
found in what is sometimes termed 'weak' and 'strong' quantization. 
Spacial quantization cannot be effective unless it has some axis of 
reference. In the calculations of Pauli and Pauling cited above, the 
direction of the electric field is taken as such an axis. If the electric 
field is the only external one, this choice for the axis of quantization 
has a good dynamical justification, for then the angular momentum 
about this particular axis, and no other, remains constant after applica- 
tion of the electric field. On the other hand, in the absence, of all 
external fields, the components of angular momentum in all directions 
remain constant, and there is no reason for choosing one direction in 
space rather than another for the axis of spacial quantization. The only 
escape from this ambiguity is to assume that in the absence of external 
fields the orientations of the atoms are random instead of quantized. 
Suppose now a field is gradually applied. As there are no impulsive 
forces to suddenly change the orientations of atoms, their spacial dis- 
tribution should presumably remain random for exceedingly weak fields. 
The constant C would then have 8 a value different from either of those 
calculated by Pauling or Pauli, and it is only when the field becomes 
strong enough for spacial quantization that their computations become 
applicable. If the spacial quantization is supposed achieved gradually, 
the term 'weak quantization' has sometimes been used to designate the 
case in which the quantization has only been acquired to a slight extent, 

7 By a gyroscopic molecule we mean one with an angular momentum about the axis 
of symmetry. In the 'symmetrical top' gyroscopic model which has often been used to 
represent the behaviour of symmetrical polyatomic molecules, Manneback showed that 
the constant C has the asymptotic value even in the old quantum theory, but that 
considerably higher temperatures are required than in the new mechanics in order to 
make this asymptotic value a valid approximation; Phys. Zeits. 28, 72 (1927). 

8 One's first guess by classical analogy would be that the constant C would be with 
random orientations (weak quantization) in the old quantum theory. Pauling showed 
that instead C would be zero under these circumstances if the non-gyroscopic dumb-boll 
model is used. The reason for this is that with such a model the classical polarization by 
orientation is due entirely to contributions from very slowly rotating molecules, and with 
a minimum of a half -quantum unit of rotational angular momentum, there are no mole- 
cules sufficiently slow to contribute. This difficulty is overcome in the new mechanics ; 
for details and references see 45. 



V,29 CONTRASTED WITH THE NEW 109 

and * strong quantization' the case in which the quantization is nearly 
perfect. 9 There should thus be a change of susceptibility with field 
strength due to the transition from weak to strong spacial quantization. 
This is not to be confused with the variation of susceptibility with field 
strength due to saturation effects, i.e. the effect of terms beyond the 
first power in the series development of the moment in terms of the field 
strength. The transition from weak to strong spacial quantization 
would involve the passage from one such series development to another 
one with totally different coefficients, and in either series only the 
coefficient of the linear term in E is ordinarily of interest. Saturation 
effects are found only in exceedingly large fields, whereas any changes 
in susceptibility attendant to passage from weak to strong quantization 
would be found in considerably smaller fields, at least at low pressures, 
for reasons given below. As far as the writer is aware, there is no 
experimental evidence for a variation of susceptibility with field strength 
in the peculiar fashion which would be characteristic of the change in 
quantization in the old quantum theory. 

It is apparent that at least in the old quantum theory one needed 
some sort of a quantitative estimate of how large a field is required for 
spacial quantization. Such an estimate was usually made by assuming 
that a quantum condition is 'completely' or 'strongly' fulfilled if the 
frequency with which it is associated by the correspondence principle 
is large compared to the frequencies of any disturbances which upset 
the regular motion of the molecule in a stationary state. Such dis- 
turbances were deemed due to the transitions of the molecule to other 
stationary states and to interruptions by collisions. In the 'dumb-bell' 
model of a dipole gas, the collision disturbances are the important ones, 
and the probability of transitions due to the absorption or emission of 
radiation is relatively slight. According to these ideas, the spacial 
quantization should be achieved when the frequency of precession about 
the field, which is the frequency associated with the 'equatorial quantum 
number', is so large that the atom can persist through several periods 
of precession without molestation by collision. As the mean free time 
between collisions is of the order 10~ 10 sec. under standard conditions, 
and as the Larmor precession frequency is 1-40X 10 6 # sec.^ 1 , a field of 
the order 10,000 gauss should on this view be required to establish 
spacial quantization at atmospheric pressure, while at low pressures, 

9 For papers on the old theory of weak and strong quantization see Ehrenfest and 
Breit, 1'roc. Atnsterdam Acad. 25, 2 (1922); Zeits.f. Physik, 9, 207 (1922); Ehrenfest and 
Tolman, Phys. Rev. 24, 287 (1924); Slater, ibid. 26, 419 (1925). 



110 SUSCEPTIBILITIES IN THE OLD QUANTUM THEORY V,29 

where collisions are less frequent, a considerably smaller field would be 
required. The sharpness of Zeeman patterns at comparatively high 
pressures indicates that these estimates are perhaps too high. Just how 
large an electric field is required for spacial quantization is not quite 
clear, due to the complication that non-gyroscopic molecules exhibit 
only a quadratic Stark effect (37). At any rate, either in the magnetic 
or electric case there should be certain critical pressures at which there 
is a pronounced pressure variation of the susceptibility per molecule due 
to the passage from 'weak' to 'strong' spacial quantization attendant 
on changes in the collision intervals. At one time it was thought that 
there was evidence for such a variation in the diamagnetic susceptibility 
of H 2 , N 2 , CO, and C0 2 . This result was named the 'Glaser effect' in 
honour of its discoverer. He found 10 the molar diamagnetic suscepti- 
bility for these gases to be approximately three times larger at low than 
at high pressures. Theoretical physicists 11 interpreted this as meaning 
that with spacial quantization the average value of # 2 +2/ 2 was greater 
than f r 2 , its mean with random orientation. When one particular axis 
is chosen as that of spacial quantization the different coordinate axes 
are not on a parity in the old quantum theory, and so there was no 
apparent reason why x 2 , y 2 , z 2 could not have mutually different values, 
thus making x 2 -{-y 2 different from the value fr 2 supposed in the classical 
equation (2), Chap. IV. On the other hand, it is hard to see how the 
effect could be as large as found by Glaser, because the average of 
x 2 +y 2 cannot possibly exceed r 2 , and so even with spacial quantization 
the molar susceptibility should not be more than 1-5 times the classical 
or high-pressure value, whereas Glaser found a value 3 times as great. 12 
The reader may well feel that such changes of susceptibility are very 
'unphysical', as they have no analogue in classical theory, contrary to 
the usual expectations from the correspondence principle. Now for- 
tunately it is found that the new quantum mechanics removes com- 
pletely this bugbear of weak and strong spacial quantization. It is 
a very remarkable fact that in the new mechanics the susceptibility 
is invariant of the choice of the axis of quantization, as we shall see in 
46. As random orientations are equivalent to a haphazard distribution 
of the axes of quantization, the susceptibility is the same with and 

10 A. Glaser, Phys. Zeite. 26, 212 (1925); Ann. der Physik, 75, 459 (1924); 78, 641 
(1925); 1, 814; 2, 233 (1929). 

11 Mathematical theories of the Glaser effect in the old quantum theory have been 
attempted by Dobyo, Phys. Rev. 25, 586 (1925) (abstract) and by Breit, J. Washington 
Acad. Sci. 15, 429 (1925). 

12 This difficulty is also noticed by Stoner, Magnetism and Atomic Structure, p. 276. 



V,29 CONTRASTED WITH THE NEW 111 

without spacial quantization. If the reader has felt that our presenta- 
tion of weak and strong quantization in the old quantum theory was 
somewhat mystifying (as indeed it had to be, as physicists themselves 
were hazy on the details of the passage from one type of quantization 
to another), he need now no longer feel alarmed, as the new mechanics 
gives no susceptibility effects without some analogue in classical theory. 

30. Spectroscopic Stability in the New Quantum Mechanics 

The theorem of the new quantum mechanics in virtue of which the 
question of weak versus strong spacial quantization becomes of no 
consequence for susceptibilities is termed the 'principle of spectroscopic 
stability' and will be proved in 35. The term is not a particularly 
happy one. It was originally introduced by Bohr 13 to designate the 
concept that a magnetic field should not influence the polarization of 
secondary radiation excited by temperature radiation or some other 
isotropic source. Later Born, Heisenberg, and Jordan 14 used the term 
to denote the invariance of a certain sum of matrix elements of the 
system of quantization, as this sum entered in the polarization problem 
studied by Bohr. Precisely this sum enters in the theory of suscepti- 
bilities, and for a mathematical formulation of the principle as a 'sum- 
rule' the reader will have to wait until 35, as we have not yet developed 
sufficient mathematical background. Tf a physical rather than mathe- 
matical definition of the principle of spectroscopic stability is desired, 
it can for our purposes be considered identical with the idea that the 
susceptibility is invariant of the type of quantization, or in the special 
case of spacial quantization, that summing over the various quantized 
orientations is equivalent, as far as results are concerned, to a classical 
integration over a random orientation of orbits. It is indeed remarkable 
that a discrete quantum summation gives exactly the same answers as 
a continuous integration. This was not at all true in the old quantum 
theory. In virtue of the principle, we can feel sure that in the new 
quantum mechanics the average of # 2 +?/ 2 over the various allowed 
spacial orientations is f r 2 , just as though the orbits could have random 
directions. Another example is that the average of the square of a 

13 N. Bohr, The Quantum Theory of Line Spectra, p. 85. It is clearly to be understood 
that when the excitation is by a directed beam of light rather than by primary radiation 
corning simultaneously from all directions, the secondary radiation may exhibit an out- 
standing polarization materially influenced by magnetic fields. The subject of spectro- 
scopic stability is intimately connected with the polarization of resonance radiation in 
magnetic fields ; for discussion and references see J. H. Van Vleck, Quantum Principles and 
Line Spectra, pp. 177 if. 

14 Born, Heisenberg, and Jordan, Zeits.f. Fhysik, 35, 590 (1926). 



112 SUSCEPTIBILITIES IN THE OLD QUANTUM THEORY V,30 

direction cosine is J, even though the angle can only take on particular 
values. 

The reader should not form the impression that the principle of 
spectroscopic stability applies only to spacial in distinction from other 
types of quantization. It assures equally well the invariance of the 
susceptibility of all questions concerning the choice of the system of 
quantization. Or, in more precise technical language, if the dynamical 
system is initially 'degenerate', the spectroscopic stability shows that 
the susceptibility is invariant of the manner in which the degeneracy 
is removed. As an example, the hydrogen atom should have the same 
dielectric constant in weak fields, in which polar coordinates are needed 
to separate the variables in the relativistic SchrOdinger equation, as in 
strong fields, in which parabolic coordinates are required even without 
the relativity corrections. Another example of spectroscopic stability 
is the invariance of magnetic susceptibilities of the Paschen-Back effect. 
In a very powerful magnetic field the orbital and spin angular momenta 
are quantized separately rather than only collectively relative to the 
axis of the field, and corresponding to this there is a complete re- 
organization of the Zeeman patterns, known as the Paschen-Back 
effect, but no change in the susceptibility. A big alteration in the 
Zeeman effect without any attendant change in the susceptibility may 
at first thought seem almost an impossibility, but it must be remem- 
bered that the position of any given Zeeman component of a spectral 
term involves only one value of the magnetic quantum number, whereas 
the calculation of a magnetic susceptibility always requires a summa- 
tion over all the stationary states corresponding to all possible values 
of the magnetic quantum number. The sum thus encountered is in- 
variant of the Paschen-Back reorganization, even though the individual 
energy -levels of which the sum is composed are altered. Similarly even 
in the new quantum mechanics the matter of spacial quantization still 
enters in the Stern-Gerlach effect, for this effect relates to the properties 
of individual Zeeman states, in contrast to the statistical nature of 
susceptibilities. 

What now becomes of the Glaser effect, which if real would contradict 
the principle of spectroscopic stability ? Glaser 's experiments were 
repeated by Lehrer 15 and by Hammar, 16 who found the molar dia- 
magnetic susceptibility invariant of the pressure, and the effect hence 
non-existent. In other words, the susceptibility per unit- volume, which 

15 E. Lehrer, Ann. der Fhysik, 81, 229 (1926). 

18 C. W. Hammar, Proc. Nat. Acad. Sci. 12, 597 (1926). 



V,30 CONTRASTED WITH THE NEW 113 

is proportional to the density, is a strictly linear function of the pressure. 
We must, however, remark that Glaser, despite the criticisms that 
have been made of his work, still claims that his effect is real. 17 The 
experiments of Lehrer and Hammar were performed at almost exactly 
the same time as the new mechanics was developed far enough to show 
the theoretical non-existence of the Glaser effect. It must have saved 
physicists a great deal of time and worry that the new results in theory 
and experiment came practically hand in hand. 

As emphasized by Ebert, 18 the fact that the same electric moments 
are obtained by the vapour and by the solution methods, also the fact 
that non-polar materials have the same molar polarizabilities in the 
gaseous and pure liquid states (see p. 59), is a nice confirmation of 
the spectroscopic stability characteristic of the new mechanics. In the 
old quantum theory, on the other hand, one might expect different 
results in the liquid and vapour states, because in the liquid the colli- 
sions are more frequent and the quantization hence 'weaker'. 

Before closing this section we must caution the reader not to attach 
too much physical reality to the spacial quantization discussed above, 
as the new mechanics does not endow orbits with as much geometrical 
reality as previously. Since in the new theory there is no detectable 
difference between weak and strong quantization as far as suscepti- 
bilities are concerned, the question of the mechanism by which spacial 
quantization is acquired loses much of its former interest. 

31. Effect of a Magnetic Field on the Dielectric Constant 

The influence of a magnetic field on the dielectric constant (or of an 
electric field on the magnetic susceptibility) was ludicrously large in 
the old quantum theory because of the spacial quantization. Ordinarily 
in studying dielectric constants, the quantization can be taken with 
respect to the axis of the applied electric field. Suppose, however, a 
powerful magnetic field is applied simultaneously, and at right angles 
to the electric one. If the former is made sufficiently large, it will make 
a stronger bid for the axis of spacial quantization than the latter. This 
axis then becomes perpendicular to the electric field, and under such 

17 Hammar 10 suggests that Glaser's anomalous results may bo duo to absorption of 
water as an impurity on the test body, while H. Buchnor (Ann. der Physik, 1, 40, 1929) 
attributes them to systematic variations in the temperature of this body. He shows that 
an undetected temperature alteration of the order 0-01 to 0-1 C. might explain the 
anomaly. These contentions are answered at length by Glasor in Ann. der Physik, 3, 
1119 (1929). Recently Bitter finds very convincing evidence in support of Hammar's 
claim that the anomaly is due to water (Phya. Rev. 35, 1672, 1930). 

18 L. Ebert, Naturwiasenschaften, 14, 919 (1926). 



114 SUSCEPTIBILITIES IN THE OLD QUANTUM THEORY V,31 
circumstances the old quantum theory yielded a dielectric constant 
radically different from that when the axis of quantization is parallel to 
the electric field. This has been shown by Pauling. 19 He demonstrated 
that a crossed magnetic field would make the constant C in Eq. (1) 
negative, an absurdity. Only a comparatively feeble magnetic field 
would be required. Jf the electric field were 100 volts/cm., the magnetic 
one would only have to be 1 gauss. The smallness of the necessary 
magnetic field relative to the electric one is a consequence of the fact 
that molecules such as HOI 20 have a first-order Zeeman effect but only 
a second-order Stark effect. An innocent little magnetic field of only a 
few gauss should thus in the old quantum theory change the sign of 
the temperature coefficient of the dielectric constant, and make the 
electric susceptibility negative in so far as the orientation rather than 
induced polarization is concerned. This is what one might term extreme 
spectroscopic instability. Needless to say, such a cataclysmic influence 
of a magnetic field on the dielectric constant is not found experi- 
mentally. Mott-Smith and Daily 21 showed that a field of 4,800 gauss 
did not alter the electric susceptibilities of NO or HC1 within the 
experimental error (8 per cent, in NO and 1 per cent, in HC1). Also 
a few months previously Weatherby and Wolf 22 found an analogous 
lack of effect of a magnetic field of 8,000 gauss on the electric suscepti- 
bilities of He, O 2 , and air (within 10 per cent, in He and 0-4 per cent, 
in O 2 and air). The results for HC1 and NO are perhaps a little more 
directly significant because they relate to polar substances, such as were 
assumed in Pauling's theory. 

As a matter of fact, oven the complication of a crossed magnetic field is not 
really required to yield the absurdity of a negative C in Eq. (1), provided one uses 
half-quantum numbers in the old theory. Pauling, to be sure, found the positive 

19 L. Pauling, Phijs. Rev. 29, 145 (1927). 

20 Tho magnetic moment of a diamagnotic molecule such as HC1 is developed solely 
in virtue of rotation of tho molecule as a whole, and so the corresponding Zeeman effect, 
though of tho first order, is only m/M times tho ordinary atomic Zeoman effect. Here 
mJM is the ratio of tho electronic to effective nuclear mass. Even though minute, this 
first order Zeoman effect is larger than the second order Stark effect. Tho first order 
Stark effect disappears as long as the molecule is non-gyroscopic, as will bo soon in Eq. 
(64) of 37. 

21 L/M. Mott-Smith and C. R. Daily, Phys. Rev. 28, 976 (1926). We give the percen- 
tage error in tho electric susceptibility rather than in the dielectric constant. The error 
in tho susceptibility is tho more significant because the dielectric constants of gases are 
nearly unity. Consequently a high precision in measuring the dielectric constant (1 part 
in 100,000 for NO, HC1; 1 in 500,000 for He, O 2 ) is necessary to determine the suscepti- 
bilities as accurately as mentioned above. Tho absence of the converse effect, the altera- 
tion of magnetic susceptibilities by an electric field, has been proved by Mott-Smith for 
HC1 and NO (Phya. Rev. 32, 817, 1928). 

22 B. B. Weatherby and A. Wolf, Phy*. Rev. 27, 769 (1926). 



V,31 CONTRASTED WITH THE NEW 115 

value C = 4-57 already tabulated on p. 107 by assuming half-quantum numbers 
and no magnetic field. However, he took the a priori probability of a rotational 
state J to be pj = 2.7+ 2. Here J denotes the integral rotational quantum num- 
ber of the new mechanics ; i.e. J+ J is the half integral effective rotational quantum 
number of the old theory. Modern experimental knowledge of band intensities and 
the Zeeman effect, as well as the theoretical dictums of the new mechanics, demand 
that instead the a priori probability should be pj = 2 J-f 1 . When the calculations 
are made with pj 2J-\- 1, the constant C turns out to be negative instead of 
+ 4- 57 when H = 0. It is for this reason that Pauling preferred to use pj = 2J + 2. 
With either choice of pj the application of a crossed magnetic field alters the sign 
of C. Thus if Pauling's theory is modified by taking pj =--= 2J+ 1, a magnetic field 
of a few gauss abruptly makes C positive. 

In the new quantum mechanics the choice of the axis of spacial 
quantization is no longer of importance, and so a magnetic field should 
be almost without effect on the dielectric constant, in agreement with 
the experiments reported above. We say 'almost 5 rather than 'com- 
pletely' without effect because a tremendously large magnetic field may 
still slightly distort the dielectric constant. This distortion is closely 
akin to a saturation effect, as it is proportional to the square of the 
magnetic field. It is thus negligible in any field of ordinary magnitude, 
and increases very gradually when the magnetic field is increased to 
a tremendous value, quite unlike the abrupt alterations reported in the 
old quantum theory. The order of magnitude of this distortion effect 
in the new quantum mechanics is precisely the same as in the classical 
theory. Consequently, if we are interested in seeing qualitatively about 
how large should be the influence of a magnetic field on the dielectric 
constant, we may make a classical calculation. This will be given in 
the next paragraphs. Such a classical digression may appear out of 
place in a chapter on old quantum theory, but it seems advisable to 
discuss magnetic distortion of the dielectric constant once for all. 

We shall use the same model and substantially the same notation as 
in 12, except of course for the addition of a magnetic moment. Let 
/*i> P-2> /*3 an d P* 9 /**> /** kc respectively the components of permanent 
electric and permanent magnetic moment along the principal axes x', 
y f , z' of the molecule. To facilitate printing, we here and henceforth 
use subscripts 1>2)3 in place of the previous x > tV ' >S '. The small induced 
diamagnetic moment will be neglected, and the various coefficients con- 
nected with the induced electric polarization will be taken as in 12. 
The angle between the applied electric and magnetic fields will be 
denoted by ft. The direction cosines of the angles which the principal 
axes of the molecule make with the field E will, as in 12, be denoted 
by A!, A 2 , A 3 , while those of the angles which these axes make with H 

12 



116 SUSCEPTIBILITIES IN THE OLD QUANTUM THEORY V,31 

will be denoted by A 1? A 2 , A 3 . Because of the potential energy of the 
permanent magnetic moment in the field H, we must add to the 
Hamiltonian function (9), Chap. II, the extra term 

-//(A 1 ^*+A 2/ 4+A 3/ **). (2) 

We must now develop the partition function (12), Chap. II, as a power 
series in the two variables E and //, instead of in the single variable E. 
This development will take the form 

Z = Z (l+A 2Q E*+AnH*+Ai Q E*+A 22 E*H*+AuH*+...), (3) 

where Z is the partition function in the absence of external fields. The 
factor Z Q is of no importance for us as it is independent of E and H. 
If we did not want the distortion or saturation effects it would have 
sufficed to stop with the second rather than fourth order terms. The 
omission of all odd powers of E and H from (3) requires some comment. 
It is obvious that there can be no terms proportional to E*H l when s+t 
is an odd integer, for on physical grounds the susceptibility, and hence 
Z, must be unaffected if we reverse simultaneously both E and //. The 
disappearance of such terms can also be verified by performing the 
integration over the Eulerian angles. Terms for which s and i are both 
odd are omitted on the ground that it is equally probable that the 
electrons rotate in either a left- or right-handed sense about the direc- 
tion of the electric moment. This idea of the equal probability of both 
senses of rotation is easily seen to imply that Z must be unaltered if 
the sign of // is reversed, and will be discussed more fully in 70 on the 
non-existence of a magneto -electric directive effect. 
The first two coefficients in (3) have the values 



(4) 



The first of these formulae has already been included in (17), Chap. II, 
and the derivation of the second is entirely analogous except that the 
polarization is magnetic rather than electric. In (4) and subsequent 
equations we use the abbreviations y for the 'polarization coefficients' : 

to. (4a) 

By adding the magnetic term (2) to the Hamiltonian function, and 
carrying the development of the exponent farther than in (15), Chap. II, 



V,31 CONTRASTED WITH THE NEW 117 

the coefficient A 22 in (3) is found to be 



A 22 = 

X [ft? Ai+/4 A 2 +/4A 3 ] 2 e- 2 </ 2 ^sin dOd^d^. . . (5) 
where Z x is a constant independent of E, introduced just before 
Eq. (14) in Chapter II, and where the p t are the components of com- 
bined permanent and induced moment, as given in Eq. (5), Chap. II. 
By (14), Chap. IT, and (3) the electric susceptibility is 



Xol = NUT " = X[<**+bH*+...(+c el E*+...)l (6) 



where a cl = 2kTA 2Q = + 



b = 2kT(A 22 -A 02 A 20 ), c ol - 
We are not at present interested in the purely electrical saturation term 
c el $ 2 , and we have hence not made the rather laborious calculation of 
^4 40 . 23 On use of (4) and evaluation of the integrals in (5) it proves that 
the explicit formula for b may be written 



_ 1] y[ (/i?2 _ /A * 

4t L J 

K(?'+)|- < 7 > 



In a similar fashion we find that the magnetic susceptibility is 

Xma = ^Ka g +^ 2 +"-(+W 7i N-...)], (8) 

where b is defined as previously and 

M*2 I M*2 





a mag 

Comparison of (6) and (8) shows that the correction for magnetic dis- 
tortion of the electric susceptibility has the same coefficient b as the 
corresponding term for electric distortion of the magnetic susceptibility. 
This identity of the coefficients would be true even with quantum- 
mechanical refinements. It may be noted that by (7) the coefficient b 
has opposite signs for parallel and crossed fields (given by Q = and 
D 7T/2 respectively) and vanishes at the critical angle 24 
Q = cos- 1 VJ = 54-l. 

28 The saturation coefficient c e i has been calculated with this model by Debye, in 
Handbuch dcr Radioloyie, vi, p. 779. 

24 In the old quantum theory Pauling found because of the spacial quantization 
a factor 3 cos 2 n 1 even in the main term a e \ of (6), and this is why his calculations gave 
such a tremendous anisotropy of the dielectric constant in a magnetic field. 



118 SUSCEPTIBILITIES IN THE OLD QUANTUM THEORY V,31 

Coefficients of the form a# (i =j) in (8) can be made zero by proper 
choice of axes as (4 a) shows that such coefficients are constructed of 
cross-product terms like the familiar 'products of inertia'; or, in other 
words, the axes #', y', z' can be made to coincide with the principal 
axes of Debye 's polarization ellipsoid. 25 There is only one common 
paramagnetic gas which is polar, viz. nitric oxide, and its electric 
moment is very small. Consequently the interplay between the per- 
manent magnetic moment and the induced electric one probably gives 
the most likelihood of experimental detection, and in the following 
numerical estimates we shall assume the molecule non-polar, making 
f jl>1 = p /2 = t ji 3 o. The distortion coefficient 6 is then inversely propor- 
tional to the square of the temperature and vanishes for all values of 
the angle 1 if the molecule is optically isotropic (a n = 22 a 33 , a^ = 0, 
i ^ j). Use of low temperatures would thus greatly favour the detection 
of the distortion, which is of course small. Most molecules have an 
electric polarizability a el of the order 10~ 23 , while a molecule with a Bohr 
magneton of permanent magnetic moment has JLL* = 0-9 x 10~ 20 . Hence 
if the molecular dissymetry is so large that ratios such as (ft* 2 p>* 2 )/p>* 2 > 
(! a 2 )/ 1 , &c., are comparable with unity, the coefficient b is of the 
order a ol ju* 2 /90& 2 T 2 or 10~ 33 /T 2 , as k = 1-37 X 10~ 16 . The percentage 
alteration in the electric susceptibility by a magnetic field is then of 
the order 1Q~ 10 H 2 /T 2 , while that in the magnetic susceptibility by an 
electric field is of the order 10~ 9 ^ 2 /T, where E is measured in electro- 
static units. These alterations are too small to have yet been detected, 
although it seems probable that with the recent technique of powerful 
magnetic fields, the magnetic distortion of dielectric constants will be 
observed before long. Also it is much easier than the converse effect 
of an electric field on magnetic susceptibilities, due primarily to the low 
rating of volts in electrostatic units. According to the above estimates, 
a magnetic field of 10,000 gauss would change the electric susceptibility 
by about one part in 10 7 at ordinary temperatures, while an electric 
field of 10,000 volts/cm. (= 33 e.s.u.) would alter the magnetic sus- 
ceptibility only by about one part in 10 8 . The purely electrical satura- 
tion effects for such a field strength are also of the order one part in 10 5 , 
as already mentioned in 22. A comparable distortion by a magnetic 
field certainly does not seem beyond the possibility of future detection, 
but on the other hand our estimates of this distortion may be a little 

23 Cf. Debye, I.e., pp. 760 ff. In Chap. II we took the x' y' t z' axes to be the principal 
axes of inertia, but as the kinetic part of the problem has been eliminated (cf. 12), we 
are now at liberty to take them as the principal axes of the Debye ellipsoid. 



V,31 CONTRASTED WITH THE NEW 119 

too high because the molecular dissymmetry is not as great as we 
assumed. 

We shall now mention qualitatively what modifications of the above 
results (6), (7), (8) are to be expected in the new quantum mechanics. 
The calculation of the distortion coefficient b has not yet been made in 
the new theory, but careful perusal of Niessen's 26 quantum-mechanical 
calculation of the somewhat related saturation coefficient c el reveals the 
general type of result to be expected. If we still were to assume that 
the electrons vibrate harmonically about positions of static equilibrium, 
formulae of exactly the type (6), (7), (8) would doubtless still remain 
valid, as the quantum and classical theories almost always give identical 
results for the harmonic oscillator. The quantum mechanics, however, 
frees us from the need of such an unreal assumption, and enables us to 
represent the electronic motions in their full dynamical generality. In 
the general case the dependence of the distortion coefficient b on the 
temperature would be of the form q Q +qJT+q 2 /T 2 +q 3 /T* rather than 
q 2 /T 2 +q 3 /T* as in (7). Here the expressions # , q v q 2 , </ 3 would be func- 
tions of the matrix elements and characteristic spectroscopic frequencies 
of the molecule, and would resemble Niessen's expressions, q Q) q l7 </ 2 , q z 
in their general type of structure, though numerically different as he 
calculated the expansion of c cl rather than b in 1/T. 27 The dependence 
of the coefficient b on angle through a factor of the form 3cos 2 } 1 
can be shown to remain valid only as long as we assume the molecule 
has the same moments rf, p,*, p* in its normal and excited states. 
Actually this will not be the case, so that we cannot expect b to vanish 
at Q = 54-1, although it may well be small at this angle. These diver- 
sities between the classical and quantum results in the dependence on 
temperature and on angle are not so much due to inherent differences 

2 K. F. Niessen, Phys. Itev. 34, 253 (1929). 

27 Tho calculation of the magnetic distortion is more difficult than that of the electric 
saturation effects in quantum mechanics because of the fact that the various components 
of magnetic momont do not commute with each other in matrix multiplication, or with 
the various components of electric moment. In a second paper (Zeits. /. Phymk, 58, 
63 (1929) Niosseii has purported to give a computation of the magnetic distortion of 
the part of the dielectric constant resulting from tho permanent dipolcs. He assumes 
that the same commutation (Vertaiischuny) relations apply to tho low frequency or 
'permanent' part of tho magnetic momont as to the total magnetic moment. This need 
not bo the caso, as those relations apply in general to complete matrices rather than to a 
portion thereof (Teilmatrixen). Niossen's calculations are correct from a formal mathe- 
matical standpoint once his initial assumption is granted, but it is hard to imagine any 
molecular model to which this is applicable. Instead it is quite conceivable that in a 
polyatomic molecule tho permanent magnetic moment consists entirely of diagonal 
elements when measured in the x\ y', z' system, in which caso tho peculiar quantum 
terms found by Niossen would be completely wanting. 



120 SUSCEPTIBILITIES IN THE OLD QUANTUM THEORY V,31 

of the two types of theories as to the more general model which can be 
utilized in the quantum theory. Because of the fact that the matrix 
calculations involve (in a sense 'scramble together') all the states of the 
molecule even though one is interested only in the normal state, it is 
not necessary to choose a molecule which has /A* different from zero 
in the normal state in order to obtain the paramagnetic distortion 
of electric susceptibilities or the converse effect. Instead one could 
employ a normally diamagnetic molecule, as such molecules are usually 
paramagnetic in excited states. For this reason nitric oxide should no 
longer be the sole polar molecule capable of a distortion of the dielectric 
constant by a magnetic field. The general order of magnitude of the 
distortion coefficient 6 is probably about the same as calculated above 
classically, and hence it still remains too small to be detected in experi- 
ments of the same precision as those of Wolf and Weatherby or Mott- 
Smith and Daily. 

Although we have especially emphasized the null results of these 
investigators, it must not be inferred that the dielectric constants of 
no substances whatsoever have yet been influenced by magnetic fields. 
The experiments of Friedel, Jezewski, and especially Kast 28 show that 
the dielectric constants of certain 'mesomorphic' substances (aiiisotropic 
liquids) are perceptibly altered by magnetic fields, to a greater extent 
than we would expect from the above calculations. Tn fact some of 
these substances thus distorted are merely diamagnetic rather than 
paramagnetic! This does not necessarily contradict the theory; for, as 
noted by Ornstein, 29 liquid crystals are probably built out of large com- 
plexes ('elementary crystals') rather than out of ordinary free molecules 
such as assumed in the theory intended primarily for gases. If each 
elementary crystal has a large moment and is oriented as a unit by 
external fields, it is clear that abnormally large effects may be expected. 
Particularly convincing evidence on this point is furnished by the 
scattering of X-rays. Investigations of Professor G. W. Stewart, which 
are to be published shortly, show that in mesomorphic liquids the inter- 
ference pattern is greatly changed by a magnetic field, whereas that in 
ordinary liquids or solids is not. This is only comprehensible if large 
groups of molecules, perhaps entire elementary crystals, are oriented 
en bloc by the field in the mesomorphic materials. 

Closely akin to the influence of a magnetic field on the static dielectric 

28 E. Friedel, Comptes Rendus, 180, 269 (1925) ; Jozowski, J. de Physique 5, 59 (1924) ; 
W. Kast, Ann. der Physik, 73, 145 (1924); also soo discussion by E. Bauer, Comptes 
Rendus, 182, 1541 (1926) and Errera, Polarisation dtilectrique (Paris, 1928), pp. 150-1. 

29 L. S. Ornstoin, Zeits.f. Physik, 35, 394 (1926) ; Ann. der Physik, 74, 445 (1924). 



V,31 CONTRASTED WITH THE NEW 121 

constant is the influence of such a field on the optical refractivity. The 
latter has been observed by Cotton and Mouton and others, 30 who find 
different refractive indices when the magnetic field is parallel and per- 
pendicular to the electric vector of the light wave. The difference in 
the two cases is found to be proportional to the square of the field, as 
one should expect by analogy with the result (6) for static fields. This 
' Cotton -Mouton effect' is the magnetic analogue of the Kerr effect, and 
is not to be confused with the familiar Faraday rotation of the plane 
of polarization ( 84) which is linear in H and hence much easier to 
measure. 

30 A. Cotton and H. Mouton, J. de Physique, 1, 5 (1911); Ann. Chim. Phys. 19, 153; 
20, 194 (1910) ; Raman and Krishrnan, Comptes Rendus, 184, 449 (1927) ; Proc. Roy. Soc. 
117A, 1 (1927); M. Rarnanadham, Indian J. of Physics, 4, 15, 109 (1929); Cotton and 
Dupouy, Comptes Rendus, 190, 630 (1930) ; Cotton and St-herer, ibid. 190, 700 ; Salceaunu, 
ibid. 190, 737; 191, 486; Zadoc-Kahn, ibid. 190, 672; general survey by Cotton in the 
proceedings of the 1930 Solvay Congress. 



VI 
QUANTUM-MECHANICAL FOUNDATIONS 

32. The Schrodinger Wave Equation 

Schrftdinger's equation 1 



is to-day so celebrated that we introduce it without further ado. Here 
J is the operator which is obtained by replacing each momentum p k 

-I r\ 

by in the classical Hamiltonian function <&(p v ...',q v ...). In case 



generalized coordinates are used, it is advisable to make the substitution 
of operators for momenta before rather than after transforming from 
Cartesian to generalized coordinates, as the direct formulation of the 
wave equation in generalized coordinates by extrapolation from classical 
theory encounters serious difficulties and ambiguities resulting from the 
non-com mutativeness of multiplication by q and d.../dq. 2 For an 
ordinary atomic or molecular system, subject to external electric and 
magnetic fields E, H directed along the z axis, the expression for the 
operator Jt in Cartesian coordinates is 



^~ 



# 2 <- / 2 , 2\1 . 77 



This can be seen by comparison with the classical Hamiltonian for a 
similar system given in Eq. (48), Chap. I. Here e t , m t are the charge 
and mass of a typical particle (nucleus or electron), and V is the ordinary 
internal electrostatic potential energy 2 e i e jl r ij (f- -E*!- (39), Chap. 1). 

i>i 

Unless otherwise stated, relativity corrections and, until 38, internal 
magnetic forces are always neglected. 

Solutions of (1) are required which are single-valued, vanish at infinity, 

1 For a full discussion of the properties of the Schrodinger wave equation, see, for 
instance, Schrodiiiger's original papers assembled in Abharullungen zur Wellenmechanik, 
or its English translation, Wave Mechanics ; also Condon and Morse, Quantum Mechanics ; 
Frenkel, Wellenmechanik ; Sommerfeld, Wellenmechanischer Ergrinzungsband ; Ruark and 
Uroy, Atoms, Molecules, and Quanta. 

2 For a good critical exposition of this difficulty see 13. Podolsky, Phys. Rev. 32, 812 
(1928). 



VT,32 QUANTUM-MECHANICAL FOUNDATIONS 123 

and are finite and twice differentiate over the whole of coordinate 
space, except perhaps for a few isolated points at which the solution 
becomes infinite in such a way that the integral J...J |0J 2 dv converges 
to a finite value even when these points are included. Solutions ^r n 
which meet these demands are called 'characteristic functions' (Eigen- 
funktionen) of (1), and constitute the c wave functions' of conservative 
systems in quantum mechanics. In the interest of simplicity we 
throughout the chapter consider only conservative systems, rather than 
the more general case in which t appears explicitly and the energy - 

7 o 

constant W n is replaced by the operator - - . Eq. (1) will in general 

ATTl Ct 

admit solutions which are characteristic functions only if the energy- 
constant W n has certain particular 'characteristic values', which furnish 
the quantized energies to be used in the Bohr frequency condition 
hv=W'-W". 

Two wave functions n , ft , representing states of different energy are 
orthogonal, i.e. possess the property that 

(dv^dxtdx^.dzj, (3) 

where the integration is to be extended throughout the coordinate space. 
Here and throughout the rest of the volume the asterisk * denotes the 
conjugate imaginary; thus i/r*> denotes the conjugate of ^,. To establish 
(3) we first observe that $ tl and $ satisfy respectively the equations 

V.h-Wnh - 0, Jt*.*K-W n .fii. = 0. (4) 

The second of these equations is equivalent to Jt. /r /t W^^ = since 
a complex expression and its conjugate vanish together; it is unneces- 
sary to attach an asterisk to W in passing to the conjugate as W is 
a real number. 

In order to distinguish clearly between operator functions and 
ordinary algebraic functions we shall insert a period between the 
operator and the subject of the operation except when the operator is 
written in full in terms of differentiations. Such a period is, of course, 
not to be confused with the dot involved in the scalar product of two 
vectors printed in bold-face type. 

To continue the proof of (3) we multiply the first equation of (4) by 
$*,, the second by ^ w , subtract the resulting expressions, and integrate 
over the coordinate space. This yields 



(W n -W n .) J... J ft. * n dv = J... J (ft.* . h-W*. .) dv. (5) 



124 QUANTUM-MECHANICAL FOUNDATIONS VI, 32 

The right side of (5) is most readily proved zero by specializing 3 M to 
be the operator (2), as this side then becomes 



and partial integration, as in Green's theorem, shows that an expression 
such as (6) vanishes, assuming that ift n , $*> vanish in the usual way at 
infinity. Thus the right-hand member of (5) equals zero, and hence 
the validity of Eq. (3) is demanded as long as the energies W n , W n . are 
different so that the factor (W n W n .) does not vanish in (5). Even 
wave functions belonging to states of coincident energy, as in a de- 
generate system, can be made orthogonal by taking proper linear com- 
binations (see Eq. (32) below). That it is possible to choose wave 
functions for degenerate systems in such a way that they are orthogonal 
is also obvious from the fact that degenerate systems are limiting cases 
of non-degenerate systems in which the difficulty of coincident energies 
is not encountered. Thus we may henceforth without loss of generality 
suppose that the wave functions belonging to different states are ortho- 
gonal regardless of whether or not these states all have different 
energies. 

Because Eq. (1) is linear, the ^'s all have arbitrary constant amplitude 
factors, which are, however, conveniently normalized by imposing the 
requirement that 



33. Construction of the Heisenberg Matrix Elements by Use of 
the Wave Functions 

Many readers will recall that before Schrodinger developed his wave 
equation, the quantum mechanics were first formulated in a matrix 
language by Born, Heisenberg, and Jordan. 4 The so-called Heisenberg 
matrix elements are readily constructed if we know all the characteristic 
functions of the given dynamical system. Suppose we desire these 
elements for an arbitrary matrix function /(<?!> -;Piv) of the co- 

3 More generally, it can be shown that the wave functions are orthogonal whenever 
the Hamiltonian operator is 'Hermitian' or 'self -adjoint'. See P. Jordan, Zeits. f. 
Physik, 40, 818 (1927). 

4 Heisenborg, Zeits. f. Physik, 33, 879 (1925); Born and Jordan, ibid. 34, 858 (1925); 
Born, Heisenberg, and Jordan, ibid. 35, 557 (1926); Born and Jordan, Elementare 
Quantenmechanik. 



VIJ33 QUANTUM-MECHANICAL FOUNDATIONS 125 

ordinates and momenta. The procedure is as follows. Construct from 

/ the operator /(ft,...; : - ,. 1 by substitution of - for each p k . 
\ 47Tl> d(i J ZTTI ufa 

If we let this operator operate on a typical wave function, we thereby 
generate a function f.$ n . It is to be clearly understood that while / is 
an operator, / . i/* n is an ordinary algebraic function of the coordinates 
q l9 ... . It can be shown that the complete set of wave functions corre- 
sponding to all possible stationary states constitutes a 'complete' 
(vollstdndig) orthogonal set, such that any arbitrary function may be 
expanded as a series in these functions. 5 Hence we may expand f.$ n 
as a series in the $ n > y so that 

;)^- (8) 



It will be proved, and this is the fundamental theorem of the present 
section, that the coefficients/^'; n) in this expansion are the Heisenberg 
matrix elements 6 (exclusive of the time factor). That is, f(n'\ri) is the 
element associated with a transition between a state characterized by 
a set of quantum numbers n' to one characterized by a set of quantum 
numbers n. Here the letters ri, n in general each signify more than one 
quantum number since a dynamical system with several degrees of 
freedom requires several quantum numbers to specify a stationary state. 
Proof. To show that the/(n';7i) defined by (8) are really the Heisen- 
berg matrix elements we must show that they possess all the charac- 
teristic properties of the latter. This means that we must show that 
they: (a) obey the matrix algebra, (/;) obey the quantum conditions, 
(c) make the energy a diagonal matrix, (d) are Hermitian if / is a func- 
tion only of the #'s but not the ^'s. The meaning of these terms will be 
explained when we shortly discuss the individual items (a) , (6), (c) , and (d). 
Born, Heisenberg, and Jordan 7 show that all the characteristic features 
of the matrix theory, including the validity of the canonical Eqs. (46), 
Chap. I, as matrix equations, follow uniquely from (a), (6), (c), and (d) if 
one impose the additional requirement that the time factor e 27rl ' v(n/; n * of 
a matrix element have its frequency determined by the Bohr frequency 
condition hv(n';n) = W n W n , where the IPs are the diagonal elements 
H(n\ri) of the energy matrix. As we use (8) to define merely the 
amplitude part of the Heisenberg matrix elements, it will clearly be 

5 The proof of the 'complete' property in case the wave equation is of the so-called 
Sturm-Liouvillo type is given in Courant-Hilbert, Methoden der Mathematischen Physik, 
pp. 278, 284, 291, 336, 337. 

8 This correlation between the wave and matrix theories was first established by 
Schrodinger, Ann. der Physik, 79, 734 (1926) and by Ekhart, Phys. Rev. 28, 711 (1926). 

7 Born, Heisenberg, and Jordan, Zeits.f. Physik, 35, 564 (1926). 



126 QUANTUM-MECHANICAL FOUNDATIONS VI, 33 

permissible to insert a time factor e 2lriv(n ' ;n)i of the above type. Hence 
it only remains to show that (a), (6), (c), and (d) follow from (8). 

(a) The proof of the addition law (f+g)(n'\ri) ~ f(ri \ri)+g(ri \ri) is 
trivial, as the coefficients in the expansion of (f+g) . i/t n in the fashion 
(8) are clearly the sums of the coefficients in the expansions of /. \jf n and 
g . if/ n . The proof of the matrix multiplication law 

nW;) (9) 



is only a trifle more difficult. We note that the function f-g.*ff tL may be 
expressed equally well as 

C/W /= 5; <#)('; , (10) 



or as 



n' 

The result (9) follows on comparison of coefficients in (10) and (11). 
The matrix multiplication is non-commutative, as in general 



(b) The quantum conditions on the coordinate and momentum 
matrices q k , p k are 

(12) 



where the elements of the unit matrix 1 are given by l(n;n)=l; 
l(n';n) = Q (n^n f ). 

Notation. If an equation is assigned a number, this number will 
throughout the rest of the volume be enclosed in angular rather than 
in round parentheses, e.g. (12) rather than (12), in case the equation is 
an equation between entire matrices rather than ordinary algebraic 
quantities. Not all expressions appearing in such a matrix equation 
need necessarily themselves be matrices, as some of the constants of 
proportionality may be ordinary numbers, like fifi-jri, for instance, in 
(12). An equality between matrix elements, as distinct from entire 
matrices, will not be given the distinctive numbering, as the occurrence 
of indices such as (n\ri), &c., indicates clearly that we are dealing with 
matrix elements. 

To prove the first relation of (12) from (8) we observe that 



VI, 33 QUANTUM-MECHANICAL FOUNDATIONS 127 

(o o \ 

-^^-i is thus equivalent to multiplication by 
v<lk vQk/ 

unity, and for the particular c&se f = p k q k q k p k > the right side of (8) 
reduces to the single term (hj27ri)i/J n , whence 

/(n; n) - &/2m*, /(n f ; n) - (n' ^ n), 

The remainder of the conditions given in (12-13) are obviously fulfilled 
since 



(c) By a diagonal matrix is meant one whose elements vanish except 
when n' = w. If the energy or Hamiltonian function Jt is to be a 
diagonal matrix, then the expansion (8) must reduce for the special 
case f=Jf to Jt . i/J n = Jt(n\ n)^ n , the right side thus consisting of but 
a single term. Comparison with (1) shows that this is merely the 
SchrOdinger wave equation, as W n is simply another notation for 
Jt(n;n). Thus his wave equation is equivalent to the requirement that 
the energy be a diagonal matrix. 

(d) A matrix /is termed Hermitian iif(n\ri) is the conjugate f*(ri\ri) 
of.f(n'-,n). Before discussing the Hermitian property it is convenient 
to derive a formula for the coefficients in the expansion (8). To do this 
we multiply Eq. (8) by some i/r*, say 0*-, and integrate over the entire 
coordinate space. In virtue of the orthogonality (3), only the particular 
term n' = n" remains on the right side after performing the integration, 
and this term becomes f(n" ; n) in virtue of the normalization (7). Hence 

/(n';n) = J...J **./.#. cfo. (14) 

If we use the expansion for /.0 n analogous to (8), multiply by ^*, 
integrate, and take the conjugate, we find that 

J*..d, (14a) 

as ^** = $ n . If the operator / is a function only of the generalized 

7 o 

coordinates q k and not of the momentum operators - , then /* is 

ATTi &q k 

identical with / since we may ordinarily suppose that i does not occur 
in any / explicitly, but only through the momentum operators. Also, 
with this restriction on /, the operator / degenerates into an ordinary 
algebraic function /, so that i/t n f. $* = 0*/,/. n . The expressions (14a) 
and (14) are then identical, demonstrating the Hermitian property for 
the particular f unction f~q k , or, more generally, for any function of 



128 QUANTUM-MECHANICAL FOUNDATIONS VI,33 

the q k 's alone. It will be noticed that the normalization (7) has been 
used in proving (d) though not in (a), (6), and (c). It is clear that the 
normalization must be involved somewhere, for otherwise the matrix 
elements defined by (8) would not be unique, as each wave function 
would have an arbitrary constant factor C n corresponding to the fact 
that C n *l* n is a solution of (1) if \jj n is one. 

In case p k is a Cartesian momentum w t # f -f ^A^/c, it is readily shown to be Hormi- 
tian, merely by using (14), and making a partial integration with respect to q 
as follows : 



If, however, p k be a canonical momentum in an arbitrary system of generalized 
coordinates, it need not necessarily be Hermitian. Tn such a system the volume 
element will take the form &dq l ...dq f instead of the Cartesian form dq l ...dq f used 
above, where A is the functional determinant of the transformation from the 
Cartesian to the generalized system. Partial integration with respect to q k will 

lead to an integrand - .^J^VA-f-0* , and the Hermitian property is 

secured only in the special case that d&/dq k 0. As a matter of fact, the genera- 
lized momenta can always be made Hermitian by taking the wave function to be 
0t ,_ ^k rather than j/r, and the wave equation to be that satisfied by ft rather 
than j/r. One then takes the generalized volume element as dq r ..dq f rather than 
A^.-.tfy. Jordan 8 has shown that this amounts to introducing a normalization 
in the definition ol the canonical momenta which are otherwise ambiguous as 
regards an arbitrary additive function of the coordinates. 

Even if we use a Cartesian system, so that the p's as well as <?'s are Hermitian, 
an arbitrary function f(p k , q k ) will still not in general be Hermitian. To see that 
this is true, we need only note that if/ and g bo any two Horrnitian matrices which 
do not commute in multiplication, such as /= q k , g = p k , their product will not 
be Hermitian. In fact the matrix law of multiplication (9) shows that if/, g be 
any two Honnitian matrices 

(fg)*(n"; n) = f*(n"; w')<7*(n'; n) = ff(n; n')/(n' ; n") = (gf)(n; n") 

n' ' 

so that the necessary and sufficient condition that their product be Hermitian is 
that those matrices satisfy the relation fg = gf. One sees, however, that the pro- 
duct fg+gf which involves what we shall call 'symmetrical' multiplication of/ 
and <7, is indeed Hormitian. Thus matrix functions constructed from a Hermitian 
set of coordinates and momenta, such as Cartesian ones, by repeated applications 
of addition and symmetrical multiplication will always be Hermitian. It is in 
general such symmetrical or 'Hermitianized' matrices which should bo used in 
quantum mechanics. The related operators are termed 'self -ad joint' or 'real'. 
The quantum conditions < 1 2>~< 1 3> can be made to appear Hermitian by multiplica- 
tion through by *, as i(fg-gf) is Hermitian if/, g are. The electrical and magnetic 
moments are necessarily Hermitian, since 



P-SX*. ^- 

8 P. Jordan, Zeits.f. Physik, 37, 383 (1926); cf. also Podolsky, I.e.* 



VI,33 QUANTUM-MECHANICAL FOUNDATIONS 129 

with analogous formulae for m y , m s and since therefore no non-commutative 
multiplications are involved in constructing these moments from the Cartesian 
p k '& and q k 'a. We suppose throughout the remainder of the volume that the 
Hamiltonian operator is always taken of a self-adjoint form, permitting us to set 



Eq. (14) is exceedingly useful, as it yields the Heisenberg matrix 
elements by a simple quadrature when the wave functions are known. 
We shall refer to it so frequently that it is convenient to give it a special 
name, and we shall therefore call it the 'fundamental quadrature'. If 
the reader is more fond of or familiar with the 'wave' than with the 
matrix formulation of quantum mechanics, he can take (14) to be 
definition of matrix elements without knowing anything more about 
them, and we have then proved the attributes (a, b, c, d) for these 
elements. Even if one tries to avoid explicit use of the matrix language 
and employ a purely wave picture, the wave functions inevitably appear 
in quadratures of the form (14), or equivalent expansions (8), especially 
in perturbation theory, so that the introduction of the matrix elements, 
even though not explicitly so called, is unavoidable. For our purposes 
it would really suffice to define matrix elements by means of (8) or (14) 
without bothering to show that they are the same as Heisenberg 's 
matrix elements, but the proof of the identity of Heisenberg's definition 
and the definition (8) or (14) in terms of wave functions, is so often 
omitted in texts on quantum mechanics, despite its simplicity and 
fundamental significance, that its incorporation in. the present chapter 
is, we hope, not too much out of the way. If, as in this chapter, the 
wave functions are used primarily in connexion with the expansion (8) 
or quadratures (14), these functions become primarily tools for cal- 
culating the matrix elements, and arc not given as much physical 
interpretation as they deserve, but this formal procedure seems better 
than going to the other extreme and constructing, as is sometimes done, 
hydrodynamical models of the molecule which localize and distribute 
the electronic charge with a definiteness contrary to the Heisenberg 
uncertainty principle. A diagonal Heisenberg matrix element f(n\ ri) 
has the physical significance of being the average value of / over all 
phases of the motion in a given stationary state. Only such an average, 
and not instantaneous values in a stationary state, are accessible to 
measurement. The well-known significance of |i/r n | 2 as proportional to 
the statistical charge density in a system with only one electron, can be 
obtained from the fundamental quadrature (14) by taking /to be unity 
in a small volume element dv and zero everywhere else. Non-diagonal 
elements are important only as intermediaries to the calculation of the 



130 QUANTUM-MECHANICAL FOUNDATIONS VI, 33 

diagonal elements of other functions, or of the same function under 
different conditions. It is not the purpose of the present volume to 
inquire further into the broad questions of interpretation in quantum 
mechanics, which would take us too far afield into transformation 
theory, hut instead to find the procedure for calculating energy -levels 
such as are involved in the study of electric and magnetic suscepti- 
bilities. 9 This requires primarily the development of two things: per- 
turbation theory and the theorem of spectroscopic stability. The 
aspects of the quantum mechanics which we present arc perhaps rather 
formal, but in the last analysis a theory is most 'physical' when it 
permits the calculation of a large number of experimentally observable 
quantities in terms of a few fundamental postulates. The triumph of 
the quantum mechanics is probably due more than in any one thing 
to its success and utility in making possible the formal numerical cal- 
culation of energy -levels and spectral intensities. 

In the hydrodyiiamical formulation of tho quantum theory, the expressions for 
tho charge and current densities for a system with a single electron are taken to 
be respectively 10 



p'= ecM>* and p'\ -^ - j-($*grad $-<J>grad3>*)- --A<M>*, (A) 
where A is the vector potential, and ^ is a normalized solution of the generalized 
wave equation obtained by replacing W by tho operator . * in ( 1 ). The hydro- 

< 2iTT1 <)t 

dynamical theory is not without its attractions. For instance, the charge and 
current thus defined satisfy the equation of continuity. However, difficulties are 
encountered in the generalization to systems with more than one electron, as with 
rj particles it is necessary to use a 877 -dimensional geometry, which has no direct 
physical significance. Also the spontaneous radiation in the hydroclyiiainieal 
theory, while in nice accord with the Bohr frequency condition, turns out to 
be proportional to the concentrations of electrons in both tho final and initial 
states rather than to that in the initial state alone. 11 

Eq. (10), Chap. I, shows that in any hydrodynarnical theory, the electric 
and magnetic moments of a stationary state containing only one electron are 
respectively 

J J J p'rdv and - J J J fr X p'v] dv. (B) 

Tho moments yielded by substitution of (A) in (H) are the same as those obtained 
by our own standpoint, in which we take tho average moment of a stationary 
state to be one of the diagonal elements of the appropriate Heiseriberg matrix, 
and which we shall later prove equivalent to defining tho moment by means of 

9 For tho postulational foundations of quantum mechanics, see Dirac's book, The 
Principles of Quantum Mechanics, in this scries. 

10 Cf., for instance, Srhrodinger, Ann. der Phyrik, 81, 137, 82, 265 (1927); Gordon, 
Zeite.f. Physik, 40, 117 (1927). 

11 For exposition of this difficulty soo, for instance, Sommerfeld, Weltenmechaniacher 
Erydnzungsband, p. 56; Condon and Morse, Quantum Mechanics, p. 90. 



VI,33 QUANTUM-MECHANICAL FOUNDATIONS 131 

Eq. (46) to be given subsequently. It Ls obvious in the electric case that (B) 
furnishes the same electric moment as that which we use, for if the atom is in 
a definite stationary state, then < = fa- **'*"* and with this restriction the first 
integral of (B) becomes identical with the fundamental quadrature (14) when in 
(14) we set / er. The proof of the identity of the two standpoints in the 
magnetic case is similar, 12 except that a partial integration of one term in the 
integral is necessary. In this case we take in ( 14) for the x component 

/ -_JLLM - + -A\ (~ ' () -4- G A\ 

J tone ^ \27Tt cte + c j ~ \27Tt ty + c V 

since in a magnetic field tnx=p x + -A A , &c. and since m f e(yzsy)/2c. 

c 

It may bo cautioned that in general the hydrodynarnical theory yields correctly 
only expressions which are linear in the charge or current. The reason for this is 
that it really gives only the average charge and current distributions of a station- 
ary state, since by the uncertainty principle the instantaneous distributions at 
a given point of space cannot be specified once the energy has a definite value. 
Unless one remembers this, the hydrodynamical theory can be quite misleading. 
For example, in the hydrogen atom one easily verifies that in the hydrodynamical 
theory the angular momentum is directed entirely along the z direction, if this is 
the 'axis of quantization' along which the angular momentum is given the 
quantized value mJiftTT. One can, however, verify by matrix methods (cf. Eqs. 
( 75)-( 7 )) that the squares of tho jr. and y components of angular momentum are then 
really not zero (except in >S f states). The explanation is, of course, that the square 
of tho mean and tho mean of the square are not the same. The moan of the first 
power, such as is yielded correctly by tho hydrodynamical theory, is indeed zero 
for tho x and y components, but tho mean square is not. As another example, the 
hydrodynamical theory yields zero current whenever tho wave functions are real 
except for tho time factor, provided there is no magnetic field H. This is seen 
by setting 3> ~i/t n e STT^/A^ ^ ^ A = in (A). This does not moan that tho 
electron is stationary, but only that it is as likely to bo moving in any given 
direction as in its opposite. 

34. Perturbation Theory 13 

Let us suppose that the Hamiltonian function consists of two parts: a 
main part Jf which is characteristic of the 'unperturbed problem', and 
a small 'perturbative potential' (titorungsfunktion) AJ (1) +A 2 J (2) +... . 
Here A is some small numerical parameter in which we suppose a series 
development can be effected. For our particular purposes the per- 
turbative potential will usually be the terms added to the Hamiltonian 
function by application of a constant external electric or magnetic field, 
and A will be proportional to the field strength. As usual in perturbation 

12 Tho identity of tho hydrodynamical with tho matrix viewpoint as regards magnetic 
moments has also boon noted by Bitter, Phys. Zcits.3Q, 497 (1029), and previously for 
tho special case of hydrogen atoms by Formi, Nature, 118, 876 (1926). 

13 The perturbation theory of quantum mechanics was first given by Born, Heisenberg, 
and Jordan, Zcits.f. Physik, 35, f>6. r > (1926) and by Schrodingor, Ann, dcr Physik, 80, 
437 (1926). 

K2 



132 QUANTUM-MECHANICAL FOUNDATIONS VI,34 

theory, we shall assume that the complete set of normalized charac- 
teristic functions and characteristic values t/r, W Q are known for the 
unperturbed problem. As the i/j for the unperturbed problem constitute 
a complete orthogonal set, the wave functions iff for the perturbed 
problem may be expanded in terms of the unperturbed ones, so that 

)fl, (15) 



We now substitute the expansion (15) in the complete wave equation 

(.#+X#W+A.A+...) .^-W n ^ = (16) 

which we wish to solve. We may utilize the fact that the ^ are solu- 
tions of the wave equation 

Jf.ft.-WX"Ki. = Q (17) 

for the unperturbed problem, and that by (8) 

;wm (18) 



n" 

where the <# (1) (n";n') are the matrix elements of the part Jt (l) of the 
perturbative potential calculated in the system of quantization of (i.e. 
with the wave functions of) the original unperturbed system. When we 
utilize (15), (17) and (18), Eq. (16) reduces to an expansion 



in terms of the unperturbed wave functions, with constant coefficients. 
Now if such an expansion is identically equal to zero, the coefficient of 
each \fj in this expansion must vanish separately. Hence 

2 [TFS(r^7i')+A^<%^^ - 0. 

(19) 

Here, as customary, (ri'\ri) means that 8(n'',n r )l, $(ri f ',ri) = 0, 
n" =n f . In the short-hand of matrix notation, the totality of homo- 
geneous linear equations (19) for determining the S(n'-,n) arc equivalent 
to the single-matrix equation (Jt Q +XJtW+X*Jf+...)S SW -=- 0, 
where S denotes the whole matrix whose elements are the S(ri\ri). 
Since there are an infinite number of states n or n', the simultaneous 
equations (19) for determining the S(n r ' 9 n) are infinite in number and 
so clearly can be solved only by successive approximations. 

Non-degenerate Systems. A dynamical system in quantum mechanics 
is termed degenerate if two or more energy -levels coincide. If the 
original system is non-degenerate we may develop the coefficients 



VI,34 QUANTUM-MECHANICAL FOUNDATIONS 133 

S(ri; n) and energy W as power series in A in the following fashion: 

S(n';n) = &(n';n)+XSM(ri;n)+\*SV\n';n)+..., (20) 

W n = H2+AH2 1 >+AB2+... . (21) 

The fact that $(ft';?i) = (ri',ri) is a consequence of the circumstance 
that $ n reduces to $> for A 0. We now substitute (20) and (21) in 
(19) and equate to zero the coefficients of successive powers of A. We 
shall carry the calculation only through terms of the second order in A. 
The equations obtained by equating to zero the first and second powers 
of A are respectively 

) - 0. (22) 



+| [^wCw^nO-Stn^nO^^SWfn'snJ-Sfn^n)^ - 0. (23) 
The solution of (22) is clearly 

"'^ (V), (24) 



where v(n\n") denotes a frequency of the unperturbed problem, which 
is, of course, given by the Bohr frequency condition 

hv(n\ n") = W*-W. = -hv(n'; n). (25) 

Thus v is really i>, but omission of the superscript simplifies the printing 
and is not likely to cause confusion in this particular case. The first 
relation of (24) is the expression in the new quantum mechanics of the 
well-known theorem, also true in the old quantum theory, 14 that the 
perturbed energy is to a first approximation the original energy W plus 
the perturbative potential averaged over an unperturbed orbit. We 
have already mentioned that diagonal matrix elements such as # (1) (w; n) 
are to be interpreted as average values. When we substitute (24) in 
(23) we obtain the second-order results: 



- ,~, y.^, . I 

n')Jr"M(ri ; n) J (l) (n"i n)J (l) (n ; n) 
hv(n"',n)hv(n f ]n) {tiv(n";n)}* 



The primes over the summation signs mean that the states n' = n and 
n" = n are to be excluded from the summation. 

14 For exposition of this theorem in the old theory, and references, see J. H. Van Vleok, 
Quantum Principles and Line Spectra, p. 203. 



134 QUANTUM-MECHANICAL FOUNDATIONS VI,34 

The equations (22) and (23) do not suffice to determine the diagonal 
elements S (l \n; n) or < 2 >(n; n) of the matrix S. These diagonal elements 
are, in fact, arbitrary unless one requires that the wave functions be 
normalized. Let us suppose that the wave functions for the unperturbed 
problem are normalized, i.e. 

l'| a <fo=l. (28) 



Let us seek to make the perturbed wave functions also normalized, so 

that they satisfy equation (7). If we substitute the expansion (15) in 

(7) and utilize (28), (3) (with /r's), the normalizing condition (7) becomes 

2S*(n';n)S(w';TO)=l. (29) 

n' 

On substituting the development (20), Eq. (29) becomes 



whence #(;) = 0, S(n;n) = -J 2 S>*(';n)/SW(n';n). 

n' 

Both the perturbed and unperturbed wave functions are orthogonal, as 
our proof of Eq. (3) by means of (4), (5), (6) is general. If one sub- 
stitutes (15) in (3) and utilizes at the same time the orthogonality 
property (3) applied to the unperturbed wave functions, one obtains 
2 8*(n' ; n")S(n' ; n) - 0, (n" + n) (30) 

n' 

a result which may also, of course, be verified explicitly to terms of 
the second order in A by use of (24) and (27). Eqs. (29) and (30) are 
equivalent to the single matrix equation 

8*8=^1, 

where S is the 'transposed' matrix 15 formed from S by interchanging 
initial and final indices, so that S(n'\n) =- S(n\ri). Since the product 
of 8* and 8 is a unit matrix, the matrix 8* is the reciprocal of the 
matrix 8, i.e. * __ $ .1 



A matrix possessing the property (31) is termed 'unitary'. Tt does not 
in general have the Hermitian property 8 = 8* as this would require 

Degenerate Systems. The preceding calculation fails in case the unper- 
turbed system is degenerate, as there will be states of coincident 
unperturbed energy, so that some of the denominators in equations 
such as (26) or (27) will be zero. To avoid confusion, we shall henceforth 
use a double index n, m rather than a single index n to specify a sta- 

15 In the literature S* is often called the matrix 'adjoint' to fl, and denoted by S}. 



VI, 34 QUANTUM-MECHANICAL FOUNDATIONS 135 

tionary state. The letter m will signify the totality of quantum numbers 
which are without effect on the unperturbed energy. Such quantum 
numbers are, of course, found only in degenerate systems. The letter 
n will designate the remaining quantum numbers. Thus two stationary 
states having the same n but different w's will possess the same original 
energy, so that frequencies of the type v(nm'\nm) will be zero. In 
a degenerate system an arbitrary linear combination 

#8.= I <S(';nM?,, (32) 

m'=l 

of the wave functions of all the states having the same energy is still 
a solution of the original wave equation, as all the */t% m having the same 
n satisfy the same unperturbed wave equation (<WWH)ifi ( * m 0. We 
suppose for concreteness that there are r states of coincident unper- 
turbed energy, which will be represented symbolically by giving the 
index m or m' the values 1, 2,..., r. The number r will in general be 
a function of n. It is to be especially noted that whereas the summation 
in (15) was over an infinite number of stationary states, that in (32) is 
over a finite, restricted number, as in any ordinary degenerate system 
only a finite number of states coincide in energy and so r is a finite 
number. Because of the arbitrariness (32) in the unperturbed wave 
functions we are not in general justified in supposing that 

S Q (nm' ; nm) = S(nm' ; nm) 

by analogy with Eq. (20) for degenerate systems. Instead 8 will possess 
terms which are non-diagonal in m (i.e. of the form m' ^ m). To deter- 
mine these terms we substitute (32) in (16), use (18), and equate to 
zero the coefficient of the first power of A, remembering that then the 
coefficient of each must vanish separately. This is tantamount to 
adapting (19) to the case (nm\nm r ) and yields 

f [J^\nm ff inm f )-^m tf ;m f )W 1 gl]S^nm f inm)^O f (m" =--l,...,r). (33) 

m'=l 

Although the original system of equations (19) was infinite, (33) is a 
finite set of r simultaneous homogeneous linear equations for deter- 
mining S(nl',nm), S(n2- 9 nm),.. fi 8 (nr m ,nm). The various equations 
belonging to a set are obtained by setting in turn m" = 1, 2,..., r. In 
other words, we have a finite number r rather than infinite number of 
simultaneous equations. Each value of n, i.e. each family of originally 
coincident levels, has its own characteristic set of such simultaneous 
equations. Now a set of homogeneous linear equations admits a non- 



136 QUANTUM-MECHANICAL FOUNDATIONS VI,34 

trivial solution only if the determinant of the coefficients is zero. This 
requirement gives the determinantal or 'secular' equation 



# (1) (nl;n2) 
,# (1 >(nl;n3) 



= (34) 



(1) (nl ; nr) ^ (1) (n2 ; w) o^ (l) (n3 ; nr) 

or in briefer notation 

\^\nm n \nm')-^(m n ,m')W^\ = (m',m" = l,2,...,r). (34 a) 
Eq. (34) or (34 a) is an algebraic equation of degree r and so has r roots 
for the unknown WJ&. The resulting values of W*+XW* are the first 
approximations to the perturbed energy values of the family of states 
in question. If these roots are all distinct, the perturbation has com- 
pletely removed the degeneracy, otherwise not. In case the roots are 
not all distinct, difficulty due to degeneracy may be encountered in 
higher order approximations, but discussion of this is beyond the scope 
of the present chapter, and the treatment of degeneracy which is 
removed only in higher-order terms is a fairly obvious extension of the 
method we have given for the first order. 16 

Having found the values of Wfi& we may substitute any one of them 
in (33) and then determine the $(nw'; wm) by solving these equations, 
which will be consistent with each other because (34) is satisfied. 
Eqs. (33), to be sure, determine only the ratios of the $(nw'; nm), but 
their absolute values may be found by invoking the aid of the normaliza- 
tion (29). 

Even after solving Eqs. (33), and thus finding the W and 
SP(nm' m ,nm), the complete solution of the wave equation has not been 
obtained, as in substituting (33) for (19) we have considered only the 
interaction between states of the same n but different m. Actually one 
must include also the effect of the matrix elements in the Hamiltonian 
function of the form J}M(rim'\nm), where n' ^n (also the effect of all 
of .# (2) and higher-order terms). To obtain the complete solution we 
choose the sums (32) as new unperturbed wave functions 0^. We may 
then proceed as in a non-degenerate system, and build up a power- 
series solution of the form (20), with of course the understanding that 
the P's rather than ^'s are to be used in equations such as (18) or (15). 
When we employ the 0''s rather than 0's, the difficulties characteristic 
of degenerate systems no longer appear, at least in low-order approxima- 

16 The procedure when the degeneracy is removed only in the higher orders is given 
by the writer in Phys. Rev. 33, 467 (1929), and more fully by Bom and Jordan, Elemen- 
tare Quantenmechanik, pp. 209 ff. 



VI,34 QUANTUM-MECHANICAL FOUNDATIONS 137 

tions, for use of the linear combinations (32) with coefficients S Q (nm'\ nm) 
determined by (33) makes diagonal in m the portion of the energy 
matrix for which n = n'\ in other words, 



The energy is affected only in the second approximation (26) by the 
portion of # (1) for which n r = n. Hence to a first approximation the 
energy is given by solution of (34), and the effect of the 'high frequency 
elements' n' ^ n is only secondary. 

Nearly Degenerate Systems. A case which commonly arises, and of 
which we shall give a specific example in Eq. (101), is that in which 
some of the unperturbed energy-levels, while not coincident, neverthe- 
less lie so close together that their separations are comparable in 
magnitude with the perturbative potential. A series of the usual type 
(20) for non-degenerate systems cannot then be used at the outset, as 
some of the denominators liv(nm\ nm r ) in (26) would be nearly zero. We 
here use the notation nm ; nm' to denote states of nearly the same energy. 
The procedure is quite similar to that in degenerate systems, and con- 
sists in finding a linear combination (32) of a finite number of unper- 
turbed wave functions which will dispose of the troublesome 'low 
frequency elements' n = n', m^m' in the Hamiltonian function. It is 
readily seen that the secular equation now becomes 
IW^^m'^m^+XH^nm^inm^-^m'im^W^l - 

(m',ra":=l,...,r) (35) 

instead of (34) or (34 a). Here the W^ are the unperturbed energy- 
levels, and W nm is the approximate energy inclusive of the perturba- 
tions, which cannot here be conveniently expressed as a power series 
in A. Such a development is useful only if there is little tendency 
towards degeneracy, or else complete degeneracy. (We must, however, 
mention that even in the intermediate case of near degeneracy, the 
secondary influence of the high frequency elements n' ^ n can still be 
handled by the series method.) 

35. Matrix Elements of a Perturbed System. Proof of Spectro- 
scopic Stability 

The matrix elements of any function /in the perturbed system are given 
by the fundamental quadrature (14) if we use in (14) the normalized 
functions appropriate to the perturbed system. If we substitute in (14) 
the expansions 

w = 2 S*(n"'m'"; nV)^. m ,,,, * nm = 

m"'*v> /tf n'm. 



138 QUANTUM-MECHANICAL FOUNDATIONS VI,35 

similar to (15) for the perturbed in terms of the unperturbed wave 
functions, then (14) yields 
f(n"m"; nm) = 2 S*(ri"m'"\ n*m")f>(n"'m'"', n'm')S(n'm' ; nm) 9 (36) 

n'"m"',n'm' 

where the f*(ri"m"'\ rim') are the matrix elements evaluated for the 
unperturbed system, given by 

f(n"'m'"; rim') = J...J ^. u ,,.f. #>,,, dv. 

In the brief matrix language (of. Eq. (31) ), Eq. (36) may be written 

j=8*f>S=S- l f>S. (37) 

The matrix 8 formed by the coefficients of the expansion of the per- 
turbed in terms of the unperturbed wave functions is called the trans- 
formation matrix. Thus we can evaluate the matrix elements of any 
function for the perturbed system if we know all the elements of the 
same function for the unperturbed state, and if in addition we know 
the transformation matrix. The formula (36) or (37) will be entirely 
accurate if we know the transformation matrix accurately, and approxi- 
mate if we know it only approximately by confining ourselves to low 
powers in the development (20). 

The transformation matrix S need not necessarily be used in con- 
nexion with the effect of a perturbation exerted upon a system. Another 
common use is in transforming from one system of quantization to 
another in a degenerate system not subject to perturbations that remove 
the degeneracy. We have already mentioned that in such a system any 
linear combination (32) of the wave functions of the states of identical 
energy is still a solution of the wave equation. When we pass from one 
set of wave functions t/rjj m to another set i/j% m by constructing arbitrarily 
such linear combinations, we make what is called a 'canonical' trans- 
formation. 17 Such a transformation amounts to changing the system of 
quantization, as the latter is not unique because of the ambiguity arising 
from the degeneracy. A familiar specific illustration is change in the 
direction of spacial quantization, which is arbitrary in the absence of 
external fields. Because we are now using instead of the infinite series 
(15) only the restricted sums (32) over the states of identical energy, 
the transformation matrix will now be diagonal in n, i.e. will have no 

17 Dirac notes (Quantum Mechanics, p. 82), that one must bo careful in dealing with 
transformations to note whether one is making a change of variables or a change in the 
representation, i.o*. in what wo call the system of quantization. Ho proposes the names 
'contact' and 'canonical' to designate the former and the latter typos of transformation. 
In the earlier literature both types of transformation were indiscriminately termed 
canonical transformations. 



VI, 35 QUANTUM-MECHANICAL FOUNDATIONS 139 

'high frequency' elements in which n' --/- n. Hence (36) may be written, 
on readjusting the prime notation, 

f(nm\rim')= J S*(nm''\nm)f\nm"\rim'")S(rim" r \rim r ). (38) 

m",m r " 

From (38) it is seen that 
2 f(nm\ rim')f*(nm', rim') ] {f(nm";rim'") X 

m,m' m,m' ,m" ,m'" ,mw ywv 



nm"\ nm)S(rim'"\ rim')8(niw\nm)8*(n'mv\ rim')}, 

(39) 

where we write w*v for m"" 5 &c. Now in the present case the normaliza- 
tion and orthogonality relations (29), (30) yield 

J 8*(nm"; nm)8(nm^f\ nm) ~ S(w"; m^). (40) 



The inversion of initial and final indices as compared with (29)-(30) is 
legitimate since 8* = S- 1 and since we have SS" 1 = 1 as well as 
$-1 S ~ }. There are, of course, equations similar to (40) in which n, m, 
m", w*v are replaced by ri, m' ', mv, m'" respectively. Thus (39) reduces to 
2 f(nm\rim')f*(nm\rim')= f*(nm"\n'm'")J**(nm"\n'm"'). (41) 

m,m' 7ii",m'" 

Now on the right-hand side we may replace m", m"' by m, m', for this 
is only a change in the notation for the variable of summation. Also 
the product of a complex number and its conjugate equals the square 
of its absolute magnitude. Hence (41) may be written 

2 |/(nm;n'm')| 2 - |/>m;nW)| 2 . (42) 

m t m' m,m' 

This rather formal identity of the sums in the two systems of quantiza- 
tion is the mathematical expression and formulation of the theorem of 
spectroscopic stability, whose far-reaching physical significance has 
already been discussed in 30. It doubtless seems to most readers a far 
cry from the abstract mathematical result .(42) to its superficially not 
at all related physical interpretation given in 30. To bridge the gap 
one must examine its application to spacial degeneracy, which will be 
discussed in the next few paragraphs, and also especially the specific 
use of the theorem in the proof of the Langevin-Debye formula, which 
will not be given until 46. Before proceeding to the discussion of spacial 
degeneracy we may note that the theorem (42) applies to all types of 
degeneracy, not merely to the particular type involved when the direc- 
tion of spacial quantization is ambiguous. Also, we may further note 
that the expression (42) is invariant even when n = ri , for there is 
nothing in the above demonstration which requires n^-ri. With n = ri 



140 QUANTUM-MECHANICAL FOUNDATIONS VI,35 

the summation in (42) extends over the various transitions within a 
multiple level rather than over those between two multiple levels. 

Application to Spatial Degeneracy. The most important application 
of (42) in calculating susceptibilities is to the case where the degeneracy 
arises from the absence of an external field, so that one direction in 
space is as good as another. Then the various values of the indices 
m and m' correspond to different values of the axial (often called 
'equatorial' or 'magnetic') quantum number belonging to a system of 
multiple levels whose components differ from each other only in that 
they represent different 'quantum-allowed' orientations relative to the 
axis of quantization. Ordinarily m then measures the component 
angular momentum of the entire molecule in the direction of this axis, 
in multiples of the quantum unit h/27r of angular momentum. The 
canonical transformation of the type considered above now simply 
involves a rotation of the coordinate axes, and means that the direction 
of spacial quantization is shifted from one direction in space to another. 
Clearly, if A is any vector, the double sum (42) has by symmetry the 
same value whether we take / equal to any one of the three components 
Aj., A y , A z provided we average (42) over all possible directions for the 
axis of quantization, for after the average there is no preference between 
the x, y, and z directions in the absence of external fields. But we have 
proved an expression of the form (42) invariant of the axis of quantiza- 
tion, and hence the average over all directions for this axis is unneces- 
sary. Thus (42) always has the same value with / equal to A x , A y , or 
A z . Hence it follows that 

2 \A t (nm',n'm')\* = % % \A(nm\rim')\*, (43) 

m,m' m,m' 

where 

\A(nm\ n'm')\* - \A r (nm; rim 1 ) | 2 + \A y (mn\ n'm')\*+A s (nm; rim')\\ (44) 
The expression (44), and hence (43), is clearly invariant of the choice 
of axis of quantization. There are, of course, equations analogous to 
(43) for the y and z components. Eq. (43) shows that summing over 
the axial quantum number has the same effect as a classical integration 
over random orientations, inasmuch as each Cartesian component con- 
tributes one-third of the total. Thus a quantum average over a discrete 
series of 'allowed' orientations is equivalent to a classical average over 
a continuous distribution of orientations. 

An example or two will perhaps make these results more concrete. 
If A be a unit-vector matrix, then A x may be regarded as the cosine of 
the angle between this vector and some fixed direction in space chosen 



VI,35 QUANTUM-MECHANICAL FOUNDATIONS 141 

as the x axis. In other words, A x is then a matrix representing a direc- 
tion cosine. Eq. (43) shows that the mean value of the square of a direction 
cosine is one-third when we average over the various allowed orienta- 
tions relative to the axis of quantization. This is the same mean value 
as classically. 

Another simple illustration of (42) is furnished by the theory of 
diamagnetism. It can be shown (see p. 91) that the diamagnetic 
susceptibility of an atom is proportional to x 2 -\-y 2 if the magnetic field 
is applied in the z direction. Now the average value of x 2 for the 
state n is 

\x(nm\rim')\ 2 . (45) 



Here we have utilized the matrix multiplication law (9). Hence n' is 
to be summed over all possible states, including n' = n. The 'a priori 
probability' g n is the number of different values of m belonging to the 
multiple state n. The multiplicity is, of course, due to the fact that 
the axial quantum number m may in general assume a variety of values 
for a given assignment of n. A diagonal matrix element x 2 (nm\nm) is 
the time average of x 2 for a component state having a particular value 
of m. Summation over m followed by division by g n is necessary in order 
to yield the mean taken over the various components. Now (45) is 
an expression of the form (43) summed over n', and there are, of course, 
similar expressions for the y and z components. Hence by (43) the 
average values of x 2 , y 2 , z 2 arc equal, and since r 2 = x 2 -{-y 2 -}-z 2 , we can 
take x 2 -{-y 2 = f r 2 , just as in classical theory. This has an important 
experimental application, as it shows that x 2 -\-y 2 has the same mean 
value as |r 2 , with or without spacial quantization, so that it is immaterial 
whether or not there are frequent collisions to upset the spacial 
quantization. Thus there should be no variation .of the diamagnetic 
susceptibility per molecule with pressure due to change in the collision 
frequency, and hence no basis for the G laser effect ( 29) on the ground 
of change in quantization. 

Application to the Intensities of Spectral Lines. Eq. (42) has an 
important application to the intensities of spectral lines. Let us suppose 
that the initial and final levels involved in the emission of a spectral 
line are both multiple, but that the spectral instruments do not have 
sufficient resolving power to reveal the multiplet structure. The ob- 
served intensity is then the sum of the multiplet components and is 
thus proportional to the sum of the squares of the matrix elements 
for the electric moment over all values of the subordinate indices m, m' 



142 QUANTUM-MECHANICAL FOUNDATIONS VI,35 

consistent with given n, ri. In other words, the intensity is proportional 
to an expression such as (42). If the multiplet structure is very narrow, 
it is very easily distorted (as in the Paschen-Back effect), and its pattern 
completely changed. By Eq. (42), however, the intensity in the entire 
pattern is the same as without the distortion. A magnetic field, for 
instance, should not affect the intensity of spectral lines unless we care 
to isolate the intensities of individual Zeeman components. 18 

Invariance of the Spur. If we take n' = n, m' = m in (38), sum over 
m, and then use (40), we have the very useful relation 
2 /(nm; nm) 2 f<*(nm\ nm). 

vi in 

The sum involved in this equation is called the 'diagonal sum' or 'spur' 
of the sub-matrix (Teilmatrix) formed from / by considering only the 
elements connecting the family of states of given n but variable m. 
The spur of any finite matrix is thus an invariant of a canonical trans- 
formation. The infinite matrices formed by varying n as well as m do 
not in general have bounded diagonal sums, as the sum is now an 
infinite one over both n, m. Hence the spur relation cannot be employed 
when the transformation matrix is not diagonal in n, unless perchance 
it involves only a finite number of states. 

The invariance of the spur requires that the sum of the roots of the 
secular equation equals the sum of the diagonal elements of the Hamil- 
tonian function calculated in the original system of representation, i.e. 
in the unperturbed system of quantization. This is true inasmuch as 
(33) takes the form W (1) = /S- 1 ^ (1 VS in matrix language, so that the 
roots of the secular equation are merely the diagonal elements of the 
energy matrix when transformed into diagonal form. Thus the sum of 
the roots of (34) is 2 Jt w (nm\ nm), and of (35) is 2 [W m +Mt(n'> nm)]. 

m m 

Without using the invariance of the spur, these values can also be 
verified by expanding the determinants in (34) or (35) so as to yield 
an explicit algebraic equation W r +a^W r - l +...+a r of degree r. The 
sum of the roots is, of course, a x , which is found to have the values 
given. 

Notation for Diagonal Matrices. A special symbolism will be con- 
venient for diagonal matrices, i.e. matrices whose only non-vanishing 
elements are the diagonal elements. A dot over the equality sign will 

18 If the multiplet width is at all different from zero, the case is, to be sure, that of 
near rather than complete degeneracy, and the transformation matrix will usually con- 
tain elements not diagonal in w, so that (42) is not rigorously applicable, but these 'high 
frequency elements' are usually small, since the corresponding denominators in (24) are 
relatively small, and so (42) is a good approximation. 



VI,35 QUANTUM-MECHANICAL FOUNDATIONS 143 

mean that the left-hand side is a diagonal matrix and that the right 
side gives the diagonal elements of this matrix. Thus p^ == m^/27r, for 
instance, is an abbreviated way of saying that p < f ) (nm',nm) = m l hl27r, 
2ty(nm;7i'm') = (ri =n,m' ^m). A symbol resembling an equality 
sign as much as possible has been desirable because the physicist likes 
to picture the diagonal elements of a physical quantity represented by 
a diagonal matrix as the values which it can assume in the stationary 
states. Thus one speaks of the axial component p^ of angular momen- 
tum as being mfifi-n in a stationary state. On the other hand, the 
equality sign without the modification of the dot over the equality "sign 
would not be mathematically correct as one cannot equate an entire 
matrix (the left side) to a diagonal element thereof (the right side). 
A bar is unnecessary to designate the time average of a function capable 
of representation by a dotted equality, as its matrix consists solely of 
diagonal elements and hence it is constant with respect to time. 

The diagonal elements of a diagonal matrix arc called its charac- 
teristic values. We tacitly consider throughout the volume only 
matrices in what is called the 'Heisenberg scheme of representation' in 
the parlance of the transformation theory of quantum mechanics. We 
do not need to occupy ourselves with the theorem of transformation 
theory 9 that any matrix can be brought into diagonal form if we are 
willing to sacrifice the diagonal form of the Hamiltoniaii function. 
When we say an expression is a diagonal matrix we mean that the 
diagonal form can be achieved without impairing the diagonal form of 
the energy. This restriction is necessary because we are dealing with 
conservative systems; otherwise every matrix would be potentially a 
diagonal matrix. 

36. Formulae for the Electric and Magnetic Moments of a 
Stationary State 

The average electric and magnetic moments in any given stationary 
state are obtainable from the formulae for the energy by a simple 
differentiation, viz. 

- "^ (46) 



The bar denotes the time average for a given stationary state, and is, 
of course, the same as a diagonal element of the Heisenberg matrix for 
the electric or magnetic moment. Thus if the series development of the 
energy in the field strength is W = W+EW w +E*W ( *>+... 9 then 

. (47) 



144 QUANTUM-MECHANICAL FOUNDATIONS VI, 36 

To obtain specific formulae for W (1) , W (2) ... we use (24) and (26). We 
assume that the system is non-degenerate or, if degenerate, that it has 
had the troublesome 'degenerate elements' n ri, m^m' eliminated 
from the Hamiltonian function by finding a new set of wave functions 
by a proper linear transformation (32). In case the only external field 
is the given electric or magnetic field, the degeneracy difficulty due to 
the arbitrariness of spacial orientation in the absence of the field is 
avoided by taking the direction of the axis of quantization as identical 
with the direction of the applied field. The matrices representing the 
components of electric and magnetic moment in this direction are 
readily shown to be diagonal in the axial quantum number m. 

We shall first derive formulae for W (1) and W (2) in the electric case. 
As usual, we suppose the applied field along the z axis. Here we may 
take the parameter A to be the field strength E, and comparison with 
Eq. (2) shows that 

#w = - 2 ei z ( = - p E , Jt* = 0. (48) 

Except for sign, the matrix elements Jt (l \nm\rim') involved in the 
perturbative potential EJt (l) are thus identical with those p^(nm\ rim') 
of the electric moment, provided the latter are calculated in the absence 
of the field, as indicated by the superscript . This proviso is necessary 
since the unperturbed wave functions are used in the definition (18) of 
the elements Jt (l \nm\rim'). These elements may be calculated by 
means of the fundamental quadrature (14) taking /= 2 e t z i> an( ^ the 
wave functions to be those of the unperturbed state. By (24), (26), 
and (48) 

a HO-.-* ss- l11 . (49) 



and hence by (47) and (49) 

'm')\ 2 /K ~. 

(50) 



rim' 

since 



m) = p^(nm\ rim')p%,*(nm\ rim') \p%,(nm\ rim') | a 
in virtue of the Hermitian property of the electric moment matrices, 
and since by (25) v(rim'\nm) = v(nm\n'm'). The presence of the 
second right-hand member of (50) means that the average electrical 
moment p E (nm\nm) of an atom or molecule in the presence of the field 
is not the same as the average p ( ^nm\ nm) for the same stationary state 
in the absence of the field. This is, naturally, because the presence of 
the electric field distorts the electronic (and nuclear) motions, and 



VI,36 QUANTUM-MECHANICAL FOUNDATIONS 145 

polarizes the atom so that there is an induced moment, given by the 
second term of (50). 

In the magnetic case we have A = //, and Eq. (2) of the present 
chapter or Eq. (48), Chap. I, shows that 



- Y ^L^( x *+y*)(nm',n'm'), (51) 

i 



where m^(nvn\n r m') denotes a matrix element of 

evaluated in the unperturbed state, i.e. an element of the magnetic 
moment in the absence of the magnetic field. The presence of the 
second-order term ,# (2) in (51) is because (2) contains a non-vanishing 
quadratic term in H, which has no analogue in the electric case. Pro- 
ceeding as before, we find from (24), (20), (46), and (51) that 
m n (nm\nm) 

\inj f (nm\ n'm')\ 2 TT ^ e?(#?+V?) / x 

l - // / v , -', ----- '-' // > ' V f (nm;nm). 
hv(nm:nm) *-< 4w,.c 2 

\ / i i 

(52) 

The last term of (52), which is not paralleled in (50), is a diamagnetic 
one, as can, for instance, be seen by comparison with the classical theory 
of magnetism previously given on p. 91. As we have mentioned on 
pp. 22-4, this third term is essentially a correction for the fact that in 
a magnetic field the 'canonical angular momentum' P Zl Xipyyip Xi 
is not the same as the true angular momentum m i (x i y i y i x i ). Hence 
in the field a matrix element m 7/ (?im; n'm') of the true magnetic moment 
is not the same as ]JT (e f /2m i c)f^ i (nm;?i'm / ). This distinction disappears 

when the field is absent, so that m*h(nm\n'm') = 2 (e /2m^c)P!f f (tim; w'm'). 

Proof of Eqs. (46). Having shown at some length how Eqs. (46) may 
be used to calculate the moment of a stationary state, it remains to 
give the proof of these equations. To do this, we note that Eqs. (49), 
Chap. I, viz. ^ 8jtf 

to^-e-js-' m "=- 8 ir <53) 

are valid in quantum mechanics provided p K , m w Jf are interpreted 
now as matrices, indicated by the angular parentheses around the equa- 
tion number, and provided in the differentiation the matrix elements 
of M are calculated for the system of representation appropriate to 
a particular field strength, say E (or // ), rather than the variable one 

3595.3 L 



146 QUANTUM-MECHANICAL FOUNDATIONS VI, 36 

EQ+&E (or // +A#). The reason that these equations are still valid 
is that the derivation of (49), Chap. I, from (48), Chap. I, involved no 
operations (such as the multiplication q k p k ) which are non-commutative 
in quantum mechanics, and so the various steps in going from (48) to 
(49) in Chapter I retain their validity in matrix as well as ordinary 
algebra. Let now the electric field be changed from E to 
The term added to the Hamiltonian function is then 
neglecting squares of A$. Further, if now we take A = &E, instead of 
A = E as previously, the change A IF in the quantized energy is to a first 
approximation in AJ by (21) and by (24) the average or diagonal 
value of the term added to the perturbative potential. Thus to this 
approximation we see, using (48), that 

p TU 

= A# - (nm- t nm) = ~-^Ep E (nm- i nm), 



and passing to the limit AJ57 we obtain the first relation of (46). 
The proof of the second relation is similar. 19 

Eqs. (46) and (53) are not to be confused, as W is the energy appro- 
priate to any given field strength, and is always a diagonal matrix, 
whereas Jf is the Hamiltonian function expressed in the system of 
quantization appropriate to one particular field strength E , and is not 
a diagonal matrix when E ^ E Q . Eq. (53) gives the matrix representing 
the instantaneous value of the moment, whereas (46) gives the time 
average. Because the distinction between (46) and (53) is a little subtle, 
some readers may prefer to take (46) to be the definition of the average 
moment, rather than falling back upon the definition of moment given 
in Eq. (11), Chap. I. This alternative is not without its advantages, 
and is sometimes used in the literature. However, if we regard (46) as 
a definition of the average moment, rather than as a consequence of 
(11), Chap. T, it is not at all obvious that the average moment per 
molecule, when multiplied by the number of molecules per c.c., is 
identical with the macroscopic polarization vector P given by the 
familiar macroscopic relation D = E+47rP. In other words, we have 

19 In performing the differentiation in <53> it is essential that the system of representa- 
tion, i.e. of quantization, for Jf bo held fast to that appropriate to a particular field 
strength E Q . Similarly Eqs. (49) of Chap. 1 are valid only in systems of coordinates 
obtained from Cartesian ones by transformations which are independent of the field 
strength E, but which can nevertheless involve the constant parameter E Q . Relations 
similar to (46) are readily proved valid in the classical or old quantum theory by essen- 
tially the same method as that which we use in the new. Then W nm denotes the energy 
express3d as a function of the angle and action variables w ky J k appropriate to the varia- 
ble field strength, and so W nm is independent of the w k corresponding to the fact that the 
energy is a diagonal matrix in (46) but not in <63>. 



VI, 36 QUANTUM-MECHANICAL FOUNDATIONS 147 

proved Eqs. (12), Chap. I, viz. 

P = #p, M = Mff, (12,1) 

from the definition (11), Chap. I, rather than from (46) of the present 
chapter. Even so, some readers may object that there is still a lack 
of rigour in our proof of the fundamental theorem (12, I) from (11), 
Chap. I, in the quantum mechanical case, as our proof of (12, I) in 3, 
which was inevitably rather long, was an entirely classical one, so that 
we are now reasoning only by analogy with classical theory. Apparently 
a completely rigorous justification of (12), Chap. I, would require a 
quantum theory of the electromagnetic field, 20 which is a very intricate 
subject beyond the scope of the present volume. However, one can 
always be almost certain that classical averages are replaced by diagonal 
matrix elements in quantum mechanics, and this is all we have used. 
As a matter of fact, considerable of the work in Chapter I can be 
repeated in the matrix language, at least when the fields are constant 
in time, taking E, H, D, B, &c., to be now matrices, and in this way 
one can see that the validity of the relations (12, I), with p, m defined 
by (11), Chap. I, is virtually unavoidable even in quantum mechanics. 

37. The Rotating Dipole in an Electric Field 21 

It is customary to treat the 'end-over-end' rotation of a diatomic 
molecule by using a simplified, idealized model, sometimes called the 
'rigid rotator' or c dumb-bell model'. The behaviour of this model in 
an electric field furnishes a simple illustration 22 of the perturbation 

20 A tentative form of such a theory has been given by Heisonberg and Pauli, Zeits. 
f. Physiky 56, 1, 59, 168 (1929), but is not without objections; (cf. Oppenheimer, Phys. 
Rev. 35, 461 (1930). 

21 This problem was treated more or less simultaneously by Mensing and Pauli, Phys. 
Zeite.27,609(1926); R. de L. Kronig, Proc. Nat. Acad. Sci. 12, 488(1926); C.Manneback, 
Phys. Zeits. 27, 563 (1926) ; and J. H. Van Vleck, Nature, 1 18, 226 (1 926) (abstract only). 
The wave equation for the rotating dipole in the absence of fields was first formulated 
and solved by Schrodinger, Ann. der Physik, 79, 520 (1926). The behaviour in fields so 
powerful as to prevent the use of the usual perturbation theory has been considered by 
Brouwer, Dissertation, Amsterdam, 1930; cf. also Lennard- Jones, Proc. Roy. Soc. 129 A, 
598 (1930). 

22 If the reader desires a still simpler example of perturbation theory, ho may consider 
the rotating dipolo in two dimensions subject to an electric field in the plane of motion. 
The unperturbed wave functions are then simple sines or cosines, and the development 
(15) of the perturbed wave function takes the form of a Fourier series, consisting ex- 
clusively of sine or cosine terms. The wave equation for the perturbed problem is of the 
form known as Mathieu's equation, and is similar to the wave equation for the two* 
dimensional simple pendulum, which has been discussed qualitatively by Condon, Phys. 
Rev. 31, 891 (1928). Our three-dimensional wave equation (54) is, of course, like that 
of the spherical pendulum. Despite the very simple form of the two-dimensiona lequation , 
its characteristic functions cannot be expressed except in series, as there are no closed 

L2 



148 QUANTUM-MECHANICAL FOUNDATIONS VI, 37 

theory given in 34. If we neglect the molecular vibrations, the nuclei 
of a diatomic molecule remain at a constant distance r from each other, 
and so the end-over-end rotation may be expected to resemble that of 
a dumb-bell of length r with masses M 19 M 2 at the two ends, equal to 
the masses of the nuclei. The moment of inertia is then 



If the molecule is polar, we may suppose that there is a constant dipole 
moment \L along the axis of the dumb-bell. Let 0, < be the usual polar 
coordinates specifying the position of the axis of the dumb-bell relative 
to a fixed direction in space, which we shall suppose to be the direction 
of an applied electric field E. The Schrodinger wave equation is then 



Eq. (54) is the specialization of the general wave equation (1) appro- 
priate to our particular model. 

The term p,E cos 6 is clearly the potential energy V of the dipole in the applied 
field E. The derivation of the first two terms of (54) is somewhat more compli- 
cated. The classical Hamiltonian function for the kinetic energy of a rotating 
dipole is 

< 55 > 



and if we replace p^,^ by hd.../27ridO, h&.../2nriV<l> the first term of (54) would be 
( h*/8iT*I)c) z ilf/VO*. The difficulty is the one mentioned at the beginning of 32, 
namely, that the transcription into the operator language is ambiguous because 
d.../?0 does not commute in multiplicationw ith/(0). The classical Hamiltonian can 
be written equally well as 



- p0sn^- , (56) 

21 ism 6 sm-0J 

since in any ordinary algebra PQ sin = sinflpg. Eq. (56) yields (54) on replacing 
momenta by operators in the fashion described above. That we should use (56) 
rather than (55) can be seen from a rule given by Schrodinger 23 for setting up his 
wave equation in generalized coordinates. Schrodinger first derived his rule by 
a variational method, but it is tantamount to throwing the Laplacian into 
generalized coordinates. In spherical coordinates the Laplacian operator is 

2 l 1L( * & \. l -*L('fi*L W _ l ^ 
V "' " + ~ 2 '" "'' 



formulae for those 'Mathiou functions'. The problem which we are treating maybe 
regarded as the generalization of the Mathieu problem to three dimensions. In the 
hypothetical two-dimensional problem the factor in the Langevin Debye formula is 
rather than , and the states for which j&Q make a negative rather than positive 
contribution to the susceptibility, whereas in 45 we shall see that they do not contribute 
at all with the three-dimensional dumb-bell model. 
23 E. Schrodinger, Ann. der Physik, 79, 748 (1926). 



VI,37 QUANTUM-MECHANICAL FOUNDATIONS 149 

and we obtain the first two terms of (54) on assuming that dift/dr = 0, suggested 
by the fact that the inter-nuclear distance r is constrained to the constant value 
r Q . This way of treating the constraint is more heuristic than rigorous, as (1) was 
intended for free particles. A better way of deriving (54) is to use the complete 
quantum-mechanical representation of molecular motions, taking into account 
the electronic and vibrational as well as rotational degrees of freedom. An 
elaborate theory has been developed by Born and Oppenheirner 21 for treating 
these different degrees of freedom by methods of successive approximations, 
beginning with the motions of largest energy, viz. the 'electronic;' motions rela- 
tive to fixed nuclei. It finally turns out that the end-over-crid motion is given 
approximately by (54), provided that the molecule is in what band spectroscopists 
call a S state, and provided oven then that we neglect * wobbles ' duo to nuclear 
vibrations, to departures of the instantaneous forces exerted on the nuclei by the 
electrons from the average of these forces, &c. Those wobbles are important in 
the precise spectroscopy of rotational fine structure, but are unimportant for us. 
By a S state we moan a state with no electronic angular momentum about the 
axis of figure (see 63 for further details). All common diatomic molecules except 
NO have states for their normal or 'ground levels'. 

Unperturbed System. The unperturbed system we can take to be that 
in the absence of the electric field. When we set E = the differential 
equation (54) becomes that of surface harmonics, and can be shown to 
have a solution having the necessary properties of single-valuedness, 
&c., outlined in 32 only if the constant 8?r 2 /TF/A 2 has the value j(j+ 1), 
where j is an integer, so that 25 



(57) 
The corresponding solutions are 



where P,-(*)- 2| (l-f.)" / '+j<*'- 1 >'- (59) 

The integer m can take on any integral value in the interval 



The functions (59) are called associated Legendre functions, and the 
ordinary Legendre polynomials are comprised as the special case m = 0. 
The radical is included as a constant factor in (58) to make the solutions 

24 Born and Oppoiiheimer, Ann. der Physik, 84, 457 (1927), or Condon and Morse, 
Quantum Mechanics, p. 153. 

25 We use small letters for the quantum numbers in the present section despite the 
fact that the latest usage in molecular spoctroscopy demands capitals. We do this for 
two reasons: first, because the recursion formulae, &c. would be rather awkward with 
capital subscripts and second, because the present 'rotating dumb-bell' is not necessarily 
to be taken as representing accurately an actual molecule. 



150 QUANTUM-MECHANICAL FOUNDATIONS VI,37 

normalized to unity. If we use the usual S-symbol, the normalization 
and orthogonality properties can be expressed in the single equation 

(60) 

Here the element da) = smdddd<l> of solid angle replaces the volume 
element dv in equations such as (3). It is not our purpose to show that 
(57) and (58) are the characteristic values and characteristic functions 
of (54) when E = 0, or that they fulfil (60). In fact, it is not our aim 
to discuss how accurate solutions are found in the rather limited number 
of cases in which the wave equation is exactly soluble, but rather how 
approximate solutions are obtained for a perturbed system if the unper- 
turbed one has been solved precisely. The necessary proofs connected 
with (57)-(59) will be found in all standard treatises on spherical har- 
monics, not to mention many recent texts on quantum mechanics, 26 
although it may be mentioned that often the treatments side-step the 
task of showing that (57) and (58) (or linear combinations thereof of 
the form (61)) are the only characteristic values and functions. How- 
ever, thorough investigation shows that the most general surface har- 
monic of degree j, i.e. the most general solution of (54) for E = 0, is 
obtained by taking an arbitrary linear combination 



of the 'tesseral harmonics' (58) over all values of ra consistent with 
given j. Eq. (61) is an illustration of the general theorem (32), and the 
arbitrariness (61) is thus to be expected since m is a 'degenerate' 
quantum number not appearing in the energy formula (57). The non- 
appearance of m in (57) expresses the fact that the spacial orientation 
of the axis of rotation is immaterial in the absence of external fields. 

The quantum numbers j, m have the following physical interpreta- 
tion. The square of the total angular momentum P of the molecule is 
P 2 =j(j-{-I)h?/4:7T 2 . The component angular momentum p^ about the 
axis of the polar coordinate system is p^ = mil/Sir. To prove the first 
of these statements we have only to use (57) and to note that the 
energy W has the value P 2 /27 in terms of angular momentum. To 
prove the second statement, take / in the fundamental quadrature 
(14) to be the operator hd.../27rid<l> corresponding to p$. As by (58), 
<tyj m l<ty = imiltfm, the integral (14) then differs from the normalizing 
relation (60) only by a factor m^/27r, and hence p^ is a diagonal matrix 
whose elements are 2ty(jw;j'w') = 8(jw; j'm')mh/27r. The energy, of 

2 * Cf., for instance, Sommerfeld, Wellenmechaniacher Ergdnzungsband, 2A. 



VI,37 QUANTUM-MECHANICAL FOUNDATIONS 151 

course, depends on the total angular momentum rather than on its 
component in a particular direction. The latter component merely deter- 
mines the spacial orientation. The energy formula (57) is a very familiar 
one in band-spectroscopy. As j(j+l) ^(j+|) 2 \ the energy is, apart 
from an unimportant additive constant, the same as though we used 
half -quantum numbers in the energy expression J 2 h 2 /8tr 2 l of the old 
quantum theory. 

Perturbed System. When E=Q, it has not been found possible to 
obtain accurate solutions of (54), and it is necessary to resort to the 
methods of perturbation theory, which yield the first few degrees of 
approximation very easily, if we take A = E. The first step is to cal- 
culate the elements of the perturbative potential XJfM = pE cos 0. In 
virtue of (14) these are given by 

Jfto(jm- 9 j'm') = -/*cos0(jm; j'm') = -/t J...J $* cos0$ m , <2o>. (62) 
Now the associated Legendre functions obey the recursion formula 
(2/+l)cos0Pf (cos0) = (j+m)P l (cwO) + (j-m+l)Pft l (co88), 

which by (58) is equivalent to the following relation between our 
normalized wave functions: 



COS t/iffint = / . _ . . _ , _ - _ . w ,_ -i , -i- / _. _.. . yjj i i j^. 



The integral (62) is thus reducible to two integrals of the form (60), 

and so one finds that 

^ . 

[j lw;jfw)= / . - -.- 

J -, (63) 



and that all other elements vanish. 

The fact that the non-vanishing elements are all diagonal in m shows 
that the degeneracy difficulty (i.e. appearance of elements diagonal in 
j but not in m) is avoided by taking the axis of quantization identical 
with that of the applied field. It has thus been allowable to use (58) 
for our initial wave functions instead of the more general linear com- 
binations (61). As by (62) and (63) # (1 > contains no diagonal elements 
in our problem, Eq. (24) shows that 

HW=0. (64) 

The summation in (26) reduces to but two terms, whose associated 



152 QUANTUM- MECHANICAL FOUNDATIONS VI,37 

frequencies are by (25) and (57) v(j;jl) = h(^2j^l-l)/STr*I. By 
(62) and (63), Eq. (26) yields 



(65) 

The case j = requires special consideration. Here the summation 
reduces to a single term/ =^4-1 = 1. There is no term/ = j 1 = 1 
as states of negative j are non-existent, and as one can also verify from 
(63). Thus m _rtcoBg(00;10)]_ 

^ ~ ~ ' ~ 



Spectroscopic Stability. From (63) and the rule (9) for matrix multi- 
plication it follows that the diagonal elements of cos 2 are 



/; _ i _ 
co(M( jm; jm) = ^ ^[OOB 0( jm; /m)] 2 - - (67) 

This gives the time average of cos 2 for one particular stationary state. 
To obtain the average of cos 2 over all the different allowed spacial 
orientations one must take the mean over the 2?'+l different allowed 
values of m ranging from j to +j. Now 

3 2 m*^(2j+I)j(j+l). (68) 

m = j 

This formula for the sum of squares of integers is one we shall have 
frequent occasion to use. It is readily proved inductively as follows. 
Assume it holds for a given/ Then it also holds for j+l as 



To complete the proof we need merely note that (68) is obviously 
correct for j = OTJ = 1. From (66) and (68) it follows that 

, +; 

(69) 



in agreement with the statements made in 30 and 35 that the square 
of a direction cosine has the same mean value 1/3 in quantum mechanics 
as in classical theory. 

The, Symmetrical Top. 21 Another, somewhat more general model 

27 The unperturbed wave equation for the symmetrical top was first solved by Roiche 



72, 1927), and by Kronig and Rabi (Phys. Rev. 29, 262 (1927)), although Donnison had 
previously obtained formula (70) by matrix methods (Phys. Eev. 28, 318 (1926)). The 
perturbed levels (72) in. an electric field wore obtained by R. de L. Kronig (Proc. Nat. 
Acad. Sci. 12, 608 (1926)), and more especially by C. Mannoback, I.e., and Debye and 



VI,37 QUANTUM-MECHANICAL FOUNDATIONS 153 

which has been used to represent molecular rotations is that of the 
so-called symmetrical top, which is a rigid body having two equal 
moments of inertia / and a third moment of inertia C. As compared 
to the rigid dumb-bell, the symmetrical top has an extra degree of 
freedom and moment of inertia, connected with rotation about the axis 
of symmetry and hence with the moment of inertia C. In the absence 
of external fields the wave equation for this model proves to be 
rigorously solvable, and has the energy 

A Q A 9.~l 

(70) 



where A is a quantum number specifying the angular momentum about 
the axis of symmetry. Perturbation calculations, which we omit, show 
that the change in energy produced by a field E is given by 

with 



(2) ._ 7T -- _ - 2 - A 2 ]] 

im ~ ' 3 ' < ; 



under the supposition, of course, that the dipole moment coincides in 
direction with the axis of symmetry of the top. 

The symmetrical top model has two applications to actual molecules. 
It can be shown 24 to represent (apart from an unimportant additive 
constant independent of j, m) the rotational energy of a diatomic mole- 
cule in II, A,O,... electronic states corresponding to A = 1, 2, 3,... . The 
S states which can be represented by the dumb-bell model are com- 
prised as the special case A = 0, where (70), (71), (72) reduce to (57), 
(64), (65). In applying the symmetrical top to diatomic molecules, the 
term A 2 / C of (70) must be dropped, as it is included in the internal or 
electronic energy of the molecule. This term would, in fact, be meaning- 
less, as the moment of inertia of a diatomic molecule about its axis of 
figure is virtually nil, being due entirely to the small electronic masses, 
and is not constant in time, as the electrons are continually moving. 
A second and simpler application of the symmetrical top model is to 
represent the rotational motion of a non-collinear polyatomic molecule 
with two equal moments of inertia, i.e. molecules such as NH 3 , &c. In 
this case the quantum number A is associated with a rotation of the 
whole molecule about the axis of symmetry, whereas in the previous 

Mannoback (Nature, 1 19, 83 (1927)), who considered Eqs. (71)-(72) explicitly in connexion 
with the Stark effect as well as implicitly in connexion with dielectric constants. 



154 QUANTUM-MECHANICAL FOUNDATIONS VI,37 

application it was only an 'electronic quantum number' which deter- 
mined the electronic level rather than the position within the band. 
A polyatomic molecule has two rotational quantum numbers j, A which 
can take on arbitrary integral values subject only to the restriction 
| A] <j, and hence it possesses exceedingly complicated band-spectra. 

Stark Effect. Eqs. (64) and (71) reveal an important distinction: 
namely, that there is no first-order Stark effect for an ordinary un- 
excited diatomic molecule in a 2 state, but that there is such an effect 
for a diatomic molecule not in a 2 state, or for a polyatomic molecule. 28 
Unfortunately, adequate measurements on the Stark effect, i.e. dis- 
placements of spectral frequencies in electric fields, are wanting in 
molecular spectra, but it would be exceedingly gratifying if such 
measurements could be made, as we would then be able to verify the 
theoretical predictions of equations such as (64), (65), (71), (72) on the 
energies of individual stationary states in electric fields, not necessarily 
the electronic ground-levels, whereas measurements of dielectric con- 
stants test only the statistical average of the energies of the various 
component rotational states of the ground-level only. In other words, 
Stark-effect measurements will isolate individual values of the quantum 
numbers j, m, whereas susceptibility determinations will not. The 
technique of Stark-effect observations in molecular spectra is, of course, 
a difficult one. The second-order Stark effect, which is the only type 
found in 2 levels, is so very small in any ordinary field strength that 
it would be hard to measure with any precision, and even the first-order 
effect in other levels or in polyatomic molecules is very minute except 
for the first few lines of a band, inasmuch as the rotational quantum 
number j appears in the denominator of (71). Hence, incidentally, 
experiments on the electric analogue of the Stern-Gerlach effect 29 will 
produce only very small deflexions in molecules. It must be mentioned 
that besides the second-order term (65) or (72) due to the permanent 
dipoles, there is also another second-order term due to the induced 
polarization, not included in our simple models. Experiments on the 
quadratic Stark effect would measure only the sum of the two terms. 
However, if the molecule should happen to be nearly isotropic optically, 

28 The first-order effect in such molecules should, however, appear only when the energy 
due to the external electric field is large compared to the so-called 'A-type doubling'. 
See 70. This restriction does not appear in Eq. (71), as the model is too simple to take 
account of the A-doubling phenomenon. If h&v(j) denote the width of the A-doublet, 
the true formula is (i^ 2 i'A 2 -f # 8 W (1)2 ) instead of (71), where W is defined by (71); 
cf. W. G. Penney, Phil. Mag., 11, 602 (1931). 

29 For references to such experiments see note 3, Chap. V. 



VI,37 QUANTUM.MECHANICAL FOUNDATIONS 155 

the induced portion would depend but little on m, and determinations 
of the relative in distinction from absolute displacements of the Stark 
effect components should then furnish a test of (65) or (72). 

38. The Electron Spin 

The writer begins this section with considerable trepidation, as the 
theory of the spin is neither particularly simple nor particularly rigorous. 
The concept that an electron has an internal degree of freedom about 
which it is free to spin has been extraordinarily fruitful in clarifying 
the analysis of spectra. This idea is due primarily to Uhlenbeck and 
Goudsmit, 30 although the spin has been proposed in other connexions 
at earlier dates by Compton, Kennard, 31 and others. The theory of the 
electron spin may be presented in two ways, viz. by means of what we 
shall call a semi-mechanical model or by means of Dirac's 'quantum 
theory of the electron'. 

In the semi-mechanical model, matrix expressions for the spin angular 
momentum are written down by analogy with the orbital angular mo- 
mentum matrices, with certain postulates regarding the occurrence of 
a half -quantum of spin per electron which will be explained below. It 
is further assumed that the ratio of spin magnetic moment M a to spin 
angular momentum P s has twice the classical value e/2mc for the 
ratio of orbital magnetic moment to orbital angular momentum, so that 

$--= 

The assumption (73) is made to explain the fact that in experiments on 
rotation by magnetization (the Einstein-Richardson-de-Haas effect) as 
well as on the converse magnetization by rotation (Barnett effect), the 
ratio of magnetic moment to angular momentum has approximately 
the value (73) instead of the classical orbital value. 32 The anomalous 
ratio (73) for the spin is also required by the anomalous Zeeman effect, 
as will be seen more fully in 42. Lande 33 found that his celebrated 
(/-formula could be explained, except for certain characteristic modi- 
fications resulting from the new quantum mechanics not understood 

30 Uhlenbeck and Goudsmit, Die Naturwissenachaften, 13, 953 (1925); Nature, 117, 
264(1926). 

81 A. H. Compton, J. Franklin Institute, 192, 145 (1921); E. H. Kennard, Phys. Rev. 
19, 420 (1922) (abstract). Kennard's note is often overlooked ; in it the spin was proposed 
explicitly in connexion with the gyromagnetic anomaly. 

38 For description of these gyromagnotic experiments, and references, see Stoner, 
Magnetism and Atomic Structure, p. 184. 

33 E. LandtS, Zeits.f. Physik, 15, 189 (1923) or Back and Land6, Zeemaneffekt und 
Midtiplettstruktur der Spektrallinien, pp. 43, 79. 



156 QUANTUM-MECHANICAL FOUNDATIONS VI, 38 

prior to 1926, by assuming that the atom contained a rather mysterious 
'atom-core' (Atomrumpf) whose ratio of magnetic moment to angular 
momentum has the value (73). This mystical 'atom-core' now turns 
out in reality to be the spin. 

Besides the arbitrary character of its postulates, the semi-mechanical 
model has the drawback that it is able to describe only to a first 
approximation (i.e. through terms of the order 1/c 2 ) the internal mag- 
netic forces of the atom. That is to say, it does not furnish an adequate 
dynamics of the interaction of the spins with each other and with 
orbital forces. Practically, this is not a serious handicap, as the terms 
of higher order 1/c 4 are entirely too small to be of any consequence in 
the optical region, although they are large enough to be observable in 
the case of X-ray doublets in heavy atoms. The interaction of the spin 
with external magnetic fields, which is our particular concern, is handled 
perfectly well by the semi-mechanical model. However, an approximate 
theory of internal magnetic forces is never as satisfying logically as an 
exact theory, and because these forces are only approximately described, 
the Hamiltonian function used in the semi-mechanical model does not 
behave properly under a Lorentz transformation, and so does not meet 
the requirements of the special theory of relativity. 

It is this need of relativity in variance which led Dirac to the discovery 
of his brilliant 'quantum theory of the electron'. 34 In the case of a 
system with one electron, he boldly replaced the single second-order 
Schrodinger wave equation by four simultaneous first-order wave equa- 
tions, involving the use of four wave functions. In a system with / 
electrons there would be 4/ wave functions, but the extension of Dirac's 
theory to many electron systems is at present in a rather unsettled state, 
and this is one reason we do not incorporate it in the present volume. 
Previously to Dirac, Pauli had shown that the existence of two wave 
functions per electron, and of two corresponding simultaneous second- 
order equations, was necessary in order to interpret in wave language 
the spin matrices of the semi-mechanical model. One wave function 
corresponds in a certain sense to the alinement of spin parallel to the 
axis of quantization, and the other to it anti-parallel. Four wave func- 
tions arc twice too many, and in order to vest them with a physical 
interpretation it seems necessary to interpret certain states as repre- 

* P. A. M. Dirac, Proc. Roy. tioc. 11?A, 610; 118A, 351 (1928); or The Principles of 
Quantum Mechanics, Chap. XIII. Tho explicit calculation of the susceptibility of an 
atom with one valence electron by means of Dirac's four simultaneous equations is given 
by Sommerfeld in the report of the 1930 Solvay Congress. The results are the same as 
with the semi-mechanical model except for terms too small to be observable. 



VI, 38 QUANTUM-MECHANICAL FOUNDATIONS 157 

senting an electron of negative mass. If Dirac's quartet of wave func- 
tions were separable into two non-combining pairs, i.e. into pairs such 
that integrals of the form (14) always vanish if the two wave functions 
belong to different pairs, the difficulty would not be so serious. Actually 
the two pairs of wave functions do 'combine', so that in the ordinary 
quantum-mechanical interpretation of wave functions there is a non- 
vanishing probability of the mass of the electron changing sign, an 
obvious absurdity. This difficulty is probably the most serious flaw in 
the logical framework of present-day quantum mechanics, 35 and very 
likely will not be cleared up until the long-awaited theory is evolved 
which explains the differences in mass of the electron and proton. How- 
ever, Dirac's theory is marvellously successful in explaining all spin 
phenomena. After setting up his four first-order equations, Dirac 
magically extracts all the properties of the spin, such as the anomalous 
ratio (73). His equations have the necessary relativity invariance, and 
give the internal magnetic interactions exactly rather than approxi- 
mately. They yield spin doublets of exactly the same width as Sommer- 
f eld's relativity doublets in the old quantum theory, thus yielding one 
of the most amazing fortuitous coincidences in the history of physics. 
The previous semi-mechanical model gave this coincidence only to terms 
of the order 1/c 2 inclusive. 

To many readers it will doubtless appear a step backwards that we 
shall dismiss Dirac's theory after this cursory qualitative discussion, and 
present the quantitative aspects of the spin entirely with the aid of the 
older semi-mechanical model. However, besides the difficulty of the 
physical interpretation of the superfluous pair of wave functions, Dirac's 
theory, with its four simultaneous equations, has necessarily a certain 
amount of mathematical complexity, and the semi-mechanical model 
is easier to visualize more 'anschaulich' as the Germans say. This pro- 
perty makes results on susceptibilities easier to remember and interpret, 
and perhaps less liable to computational errors if the semi -mechanical 
model is used. There is no loss of rigour, as it can be shown that Dirac's 
theory yields the same matrices for the spin energy in an external 
magnetic field as the previous Uhlenbeck-Goudsmit model. We can 
thus regard Dirac's theory as the most refined way of deriving the 

85 Dirac (Proc. Roy. Soc. 126A, 360 (1930)) has made the bold but interesting sugges- 
tion that the states with negative mass may bo nearly 'all full', as the Pauli exclusion 
principle allows each state to occur only once. What we interpret as ordinary electric 
neutrality is then really a maximum, infinite charge density of electrons with negative 
mass, and a proton is a vacancy or ' hole ' in the infinity of negative states. This idea, 
however, encounters many serious difficulties, and its ultimate significance is uncertain. 



158 QUANTUM-MECHANICAL FOUNDATIONS VI, 38 

matrix elements of the spin, which in the semi-mechanical model are 
taken as sheer postulates. Our omission of derivation of the spin matrix 
elements by Dirac's method is in accord with our policy of not attempt- 
ing to solve dynamical problems exactly, but only to show how the 
perturbed energy can be approximately found once the matrix elements 
of the perturbative potential are known. One reason that we use the 
semi-mechanical model is that while Dirac's quantum theory of the 
electron is discussed in most recent texts on quantum mechanics, 
Heisenberg and Jordan's very compact and elegant treatment of the 
anomalous Zeeman effect by means of the pure-matrix theory is too 
generally ignored. 

We shall present the semi-mechanical model in the pure matrix 
language, without giving the allied wave functions, as the latter do not 
help in setting up the appropriate secular equations (35). The first 
attempt at finding wave functions associated with the spin was made 
by Darwin. 36 In natural analogy with orbital motions he supposed that 
there was an azimuthal rotational coordinate </> 8 associated with pre- 
cession of the spin axis. The wave function would then contain a factor 
6*$; where m s is a quantum number specifying the axial component 
of spin angular momentum. Unfortunately this function then does not 
have the necessary property of single-valuedness, as for a single electron 
m a has the values J instead of being an integer, and e i ^ +27r) ^ e^. 
Because of this difficulty we speak of the Uhlenbeck-Goudsmit model 
as 'semi-mechanical' rather than 'mechanical'. As a matter of fact 
Darwin ingeniously found that spin matrix elements could be calculated 
by means of the fundamental quadrature (14) even with multiple-valued 
wave functions, but this appears a little fortuitous. Pauli 37 later showed 
that the difficulty of multiple-valuedness could be overcome by taking 
the arguments of the wave functions to be the axial component 8 Z of 
spin angular momentum instead of a rotational coordinate. The Dirac- 
Jordan transformation theory indeed permits us to use any set of 
coordinates and momenta as arguments of the wave function, which is 
a special case of a 'probability amplitude'. Now s z has only the two 
discrete characteristic values dzi(^/2^)> whereas <j> s assumes a con- 
tinuous range of values. A function whose argument only assumes two 
values is equivalent to a pair of functions, so Pauli' s scheme involves 
two wave functions per electron. For definition of the operators corre- 
sponding to spin angular momenta, which cannot be expressed as 

36 C. G. Darwin, Proc. Roy. Soc. 115A, 1 (1927). 

37 W. Pauli, Jr., Zeits.f. Phyaik, 43, 601 (1927). 



VI,38 QUANTUM-MECHANICAL FOUNDATIONS 159 

differentiations, and for modification of the fundamental quadrature 
(14) to include summation over the discrete spin characteristic values 
as well as integration over the continuous orbital coordinates, the reader 
is referred to Pauli's paper 37 and closely allied work by Darwin. 38 The 
treatment of the anomalous Zeeman effect either by the Pauli operators 
or by Darwin's multiple-valued wave functions is, of course, only super- 
ficially different from that with matrices (42). All methods inevitably 
lead to the same secular equation. 

39. Orbital and Spin Angular Momentum Matrices 

First let us consider the matrix elements of the orbital angular momen- 
tum of a single electron in a central field. Although we are now aiming 
to study the spin, these orbital elements will be useful for purposes of 
comparison. It is well known that in a central field the wave functions 
of a single electron, neglecting spin, are 



(74) 

where Pf ' is an associated Legendre function (59), and where R is a 
radial wave function, which we suppose normalized separately to unity, 

00 

so that J | R n i\ 2 r 2 dr = 1. The factors involving 9, <f> in (74) are the same 

as in (58), and the present calculations of angular momentum matrices 
are similar to those for the rigid rotator of 37, except that the notation 
/, m t rather than j 9 m is now used because the angular momentum is 
purely orbital and electronic. The quantum number I is the familiar 
azimuthal quantum number having the values 0, 1, 2,... for s, p, d,... 
electronic states. The maximum value of I is n 1, where n is the 
principal quantum number. The non-vanishing matrix elements of the 
x, y, and z components of orbital angular momentum are 

75) 

Here and throughout the balance of the volume we measure angular 
momentum in multiples of the quantum unit h/2Tr t as this saves con- 
tinually writing A/2?r on the right-hand side of equations such as (76). 
Also we give the x and y components in the x^4y combination, as this 
makes the formulae more compact and simpler. To prove the relations 
(75) we take in turn/ in the fundamental quadrature (14) to be one of 

88 C. G. Darwin, Proc. Roy. Soc. 116A, 227 (1927). 



160 QUANTUM-MECHANICAL FOUNDATIONS VI,39 

the following operators: 

. a 



iL._ y ll 11 (76) 

The operator on the second line is, for instance, that corresponding to 
the z component of angular momentum, since l s has the value xp y ~-yp x 
in terms of the components p x , p v , p z of linear momentum, and since 
the operator to be identified with p g is hd . . /Z-rridz, &c. The factor h/2ir 
can be dropped because of our choice of units. In (76) we have also 
stated the form which the operators take when transformed in an ele- 
mentary way to polar coordinates. The integrals are readily evaluated 
if we use the relations 39 



*'* [*^- cotfl ^l ^f'(cos0)e^ 

= -^ m f- ( ^^ Pf*'(cos0)^ 





G(p 

obeyed by the associated Legendre functions, as then the integrals are 
reducible to linear combinations of integrals of the type form (60). 
From (75) one, of course, finds that 

,; nlmj) = 2 fl^Mw/; nlm' l )\ 2 + {^ 



(77) 

in agreement with the value of the square of the angular momentum 
given in 37. 

Angular Momentum Matrices for Spins Subject to Individual Space 
Quantization. Let us hypothetically imagine that the spins are subject 
to no forces whatsoever from within the atom. If now an external 
magnetic field is applied, it will exert the only forces on the spin axes, 
which will hence be quantized individually relative to the direction of 
this field. By analogy with the orbital case, the matrix for each spin 

89 The second relation is trivial and the first relation with the upper sign is readily seen 
to be an identity when one makes the substitution (59). The first relation with the lower 
sign is perhaps most readily established inductively, as by differentiating the relation 
with respect to x = cos d and using (59), one can verify that if the relation is true for any 
given mi it holds also for mi one unit larger. To complete the proof one has only then to 
note that the formula is true for the easy case m t = j-{-l. Eqs. (75) follow when we 
express the formulae in terms of the normalized 0's instead of the P's, as on p. 151. 



VI, 39 QUANTUM-MECHANICAL FOUNDATIONS 161 

angular momentum can be assumed to be of the form (75), except that 
/, m l are replaced by quantum numbers s, m 8 determining the resultant 
spin angular momentum of an individual electron and the axial com- 
ponent thereof. It is further supposed that for each individual electron 
s = %, so that m s , which can range from s to -\-s, has only the two 
values \. This is demanded by the fact that two deflexions are found 
experimentally in the Stern-Gerlach effect for hydrogen atoms or alkali 
atoms, 40 which, of course, resemble hydrogen in having only one valence 
electron. All such atoms are normally in S states, so that the magnetic 
moment is entirely due to spin and must have two positions of quantiza- 
tion to give two deflexions. The result s = J is also demanded by the 
fact that the alkalis have a doublet multiplet-structure. The doublet 
structure requires two orientations of the spin, here relative to internal 
rather than external fields. 

By substituting s, m 8 for I, m t in (75) and further setting s = J-, 
MS i 2 we see tf ia ^ with individual quantization, the spin angular 
momentum matrices of any electron are 41 

(**-*)(- i; i) - (**+)(*; -J) - 1, 

* a (i;*) = -*c(-i;-l) = i. (78) 

We here, for brevity, write in only the indices m 8 , m 8 in which the 
matrices are not diagonal. It is to be clearly understood that experi- 
mentally it is impossible to achieve a magnetic field so extremely 
powerful that the 'internal forces' exerted on the spin can be neglected 
in comparison therewith, except for the valence electrons of light atoms 
in unusually strong fields producing a Paschen-Back effect. Hence the 
case represented by (78) is an idealized one, but it is nevertheless useful 

40 Phipps and Taylor, Phya. Rev. 29, 309 (1927); Wrodc, Zcits. /. Phyaik, 41, C69 
(1927), and references to earlier literature. The old quantum theory would give a third, 
undofloctod beam contrary to experiment, unless one ruled out HI as sometimes 
proposed. For in the old theory the lowest state of hydrogen had I 1, making 
mi = 1,0, -f- 1 fi-s compared to the new wi|-f-2;/j 2rn 8 i 1. 

41 Eqs. (78) are really the starting point of the Pauli operator theory. 37 From Eqs. 
(78) one can verify the celebrated ' Vertauschung ' relations 

aj, SyS x = w,, sfrWy -= iff,, 8^8^ = is y 

for the matrices representing the various components of spin angular momentum. 
Analogous formulae for I can be demonstrated from (75) or more elegantly, directly from 
the quantum conditions (12) (cf. Dirac, The Principles of Quantum Mechanics, p. 138). 
These relations measuring the non-commutativeness of the multiplication of various 
components of s or of 1 are very important for the establishment of general theorems 
involving angular momentum, but we shall not have particular occasion to use them. 
Angular momenta of different electrons, also the spin and orbital angular momenta of 
the same electron aro commutative, so that s r .? ~- .<? v .s* - 1jcl v l v ljc. Q(i^j) 



162 QUANTUM-MECHANICAL FOUNDATIONS VI, 39 

in exhibiting the simplest form of the spin matrices, and permits the 
most elementary formulation of the Pauli exclusion principle (viz. that 
no two electrons have the same n, I, m h m s ). The appearance of half- 
quantum numbers, of course, shows a fundamental difference as com- 
pared to orbital motions, but all the necessary so-called boundary 
conditions on the matrices (zero probability of transition to non-existent 
states) are fulfilled quite as well with half as with whole integers. In 
fact, in the early days of quantum mechanics, Born, Heisenberg, and 
Jordan 42 could not discriminate between whether half or whole quantum 
numbers should appear. (Their investigations demanded unit spacing 
of the values of m s from s to +s. This is possible if, and only if, s is 
an integer or half -integer.) 

40. Russell -Saunders Coupling, Spectroscopic Notation, &c. 

In the previous section we have neglected internal forces, but actually 
there are powerful forces of this character tending to couple together 
the various angular momentum vectors of the atom. The simplest 
assumption is that the energy of interaction between any two vectors 
is proportional to the cosine of the angle included between them or, 
what is equivalent, to their scalar product. The Hamiltonian function 
will then contain terms of the form 

a ik \ c s ki b iu \ c \ k , c lfc s<- s k9 (i, k - 1,..., 77), (79) 

where 77 is the number of electrons in the atom. Throughout the rest 
of the volume all expressions in bold-face type are to be construed as 
vector matrices, i.e. vectors of which each component is a matrix rather 
than an ordinary number. The proportionality constants a ik> b ik , c ik 
will in general be functions of all quantum numbers (such as, for 
instance, n, I) other than those quantizing the relative orientations of 
the vectors involved. The expression a^-s^ for instance, means the 
energy associated with the force which the orbital angular momentum 
of the ith exerts on the spin of the kth electron. Ordinarily it turns out 
that |^|;> K'fcl* &T^*J meaning that the orbital angular momentum 
of a given electron interacts more strongly with its own than with other 
spins. It can be shown that 43 __ 



42 Born, Hoisonborg, and Jordan, Zeit*. /. Physik, 35, 600 (1926). They have since 
beoii able to prove that the orbital Wj must be a whole integer by matrix methods 
without using the wave functions (74). See Elementare Atommechanik, p. 162. 

43 L. H. Thomas, Nature, 117, 514 (1926); Phil. Mag. 3, 1 (1927); J. Frenkol, Zeits. f. 
Physik, 37, 243 (1926). Formula (80), which is of course also yielded by Dirac's 'Quan- 



VI, 40 QUANTUM-MECHANICAL FOUNDATIONS 163 

where r is the distance of the electron from the nucleus, provided the 
given electron is not so highly perturbed by other electrons but that it 
may be regarded as moving in a Coulomb field. Although the validity 
of the cosine law (70) is only approximate, and can be justified theoreti- 
cally only with the aid of many simplifying assumptions, the departures 
from (79) need not cause us concern, as the general different types of 
quantization which we delineate by means of (79) are significant even 
when (79) is not strictly applicable. Actually, the coupling of the 
I vectors often departs widely from the cosine form given in the second 
term of (79). Also our arguments arc not affected by the fact, to be 
discussed in 76, that the constants c ik coupling the various spin vectors 
with each other are due primarily not to magnetic forces but to the 
Heisenberg exchange effect. 

Russell- launders Coupling. ** If the constants a, b, c in (79) are all 
of the same order of magnitude, the problem of the energy and nature 
of the atomic motion becomes one of extreme complexity. Very often, 
however, the factors a ik (including i = Ic) are small compared to the 
factors b ik , or to the factors c ik> or both. Tn other words, often the 
interaction between spin and orbital angular momenta is small com- 
pared to the interaction of orbital angular momenta among themselves, 
or else of the spins among themselves. Then the spins form a resultant 
S, and similarly the orbital angular momenta form a resultant L. If 
no external field is applied, the vectors S and L are compounded 
together to form a resultant J. The corresponding quantum number 
J can assume the range of values 

J=\L-8\, \L-8+l\,..., L+S-1, L+S. (81) 
The number J measures the resultant spin plus orbital angular momen- 
tum of the atom. Its projection along the axis of quantization is the 
'magnetic quantum number' M, which can have the values 

M^-J, -J+l,..., J-1, J (82) 

and which determines the component of total angular momentum in 
the direction of the axis of quantization, which we take as the z direc- 
tion. The present case of Russell-Saunders quantization is illustrated 
in (b) of Fig. 6. All such attempts at geometrical pictures should not, 

turn Theory of the Electron' with appropriate approximations, differs by a factor 2 from 
what one would oxpoct from elementary, over-simplified calculations. 

44 For more complete discussion of tho various coupling possibilities in this so-called 
'vector model of tho atom', and comparison with experimental spectroscopy, soo 
Pauling and Goudsmit, The Structure of Line Spectra. Russell and Saunclers first sug- 
gested their typo of coupling in Astrophysical Journal, 61, 38 (1925). 

M2 



164 QUANTUM-MKCHANICAL FOUNDATIONS VI, 40 

however, be taken too seriously, as in quantum mechanics angular 
momentum vectors are matrices rather than ordinary geometrical 
magnitudes. We shall frequently use rather loosely geometrical terms 
which must not be taken too literally, and which aim merely to indicate 
heuristically the analogues in classical mechanics. Thus we may say in 
a certain sense that the maximum and minimum values of J in (81) 
correspond respectively to 8 and L being mutually parallel and anti- 
parallel, and the maximum and minimum values of M in (82) to J being 
parallel and anti-parallel to the axis of quantization. However, the 





(c; 




Km. (i. 



inadequacy of this geometrical interpretation is shown by the fact that 
the matrices for both L and S always contain non-vanishing components 
perpendicular to J. 45 Also, the matrix elements for J% and J% never 
vanish for any state, indicating that there are always components of 
J perpendicular to the axis of quantization, so that the alinement is 
never perfectly parallel or anti-parallel relative to this axis. Whenever 
we use terms such as 'S and L form a quantized resultant J', or that 
'J measures the total angular momentum', this docs not mean that the 
resultant angular momentum is equal numerically to the inner quantum 



45 This is a consequence of tho fact that in TCq. (88) tho elements (JM \ . 
(JM ; J \M) never vanish simultaneously, except in the trivial ease that S or L is zero. 
These elements are seen by the correspondence principle ( 41) to arise from tho part of 
S or L which is perpendicular to J, inasmuch as non-diagonal elements in J correspond 
to classical trigonometric terms involving tho frequency o>j of procession about J (cf. 
Bq. 95). This frequency clearly appears only in the perpendicular in distinction from 
parallel components. (The first and second terms of (93) or (95) embody respectively 
the parallel and perpendicular components.) 



VI, 40 QUANTUM-MECHANICAL FOUNDATIONS 165 

number J. Instead, the square of the total angular momentum is a 
diagonal matrix of elements J(J+1) rather than J 2 , with analogous 
relations for 8 and L, so that 

U~L(L+l), S 2 ~ (+!), J 2 -J(J+1). (83) 
On the other hand, the projection of the total angular momentum J in 
the direction of the axis of quantization is a diagonal matrix of elements 
M rather than [M(M+ 1)]*. This may seem rather paradoxical, but the 
reason for the difference as compared with (83) is that we are dealing 
with a component of angular momentum in a single direction rather 
than with the sum of squares of three Cartesian components. 

The maximum values of L, 8 are, of course, ]T ^ and 2 s i J 7 ?* 
where 77 is the number of electrons. The minimum values are the 
smallest expressions of the form 2 i^ or 2 i' s '; obtainable with any 
choice of sign: e.g. L mill |Z X / 2 j for a system with only two electrons, 
while more generally $ min or J , depending on whether 77 is even or 
odd. 46 The quantum numbers *9, J, and M are half -integral or integral 
according as the number of electrons is odd or even, whereas L is always 
integral. For given 8 and L, the various values of J give the various 
components of a multiplet level. For example, the two upper levels 
involved in the D lines of sodium both have L = 1, 8 = 1, but one has 
e/~ 2> the other J \, and their difference in energy is such as to 
make the two D lines differ by 6 Angstroms. By (81) the number of 
multiplet components is 2$+ 1 if L ^ 8, or 2L+ 1 if L < 8. The multi- 
plicity of a spectral system is by definition 2*9+1. The full multiplicity 
is not developed in levels for which L < 8. For instance, if 8 ~- 2, 
L = 1, the multiplet level has only three components, but is still spoken 
of as a 'quintet level'. In Russell-Saunders coupling the multiplets 
conform approximately to a cosine law of the form 



so that W$=-.W w +lA[J(J+I)-L(L+l)-8(S+l)]. (84 a) 

Here o# denotes the part of the Hamiltonian function which is in- 
dependent of the coupling between L and 8 and which hence does not 
involve J. Eq. (84 a) has been obtained from (84) by using the vector 
addition relation J2 _ (L+S)2 _ L 2 +S 2 +2L- S (85) 

46 In case there are 'equivalent* electrons, i.e. electrons with identical n, I, not all 
combinations of L and S predicted by this rule are compatible with the Pauli exclusion 
principle. For instance, with two equivalent p electrons (/j 7 2 1) this rule gives 
L = 0, 1, 2 ; S = 0, 1, i.e. 1 S, *S, 1 P, 3 />, 1 D, 3 D, but out of those six terms only the throe 
1 *Si, 3 P, 1 D are in accord with the Pauli principle. For details on which states must be 
ruled out see Pauling and Goudsrnit, Z.c., 44 or Hund, Linienspectra, p. 118. 



166 QUANTUM-MECHANICAL FOUNDATIONS VI, 40 

together with (83). It can be shown that S and L 'commute* in multi- 
plication, so that in (85) it has been unnecessary to distinguish between 
S L and L S. The constant factor A is not the same as that a u given 
in (80) for a single electron, but can often be calculated in terms of the 
a ti for the individual electrons by methods developed by Goudsmit. 47 
A multiplet is called 'regular' or 'inverted' according as A > or A < 0. 
Spectroscopic Notation. In the now commonly accepted notation due 
primarily to Russell and Saunders, 48 a spectral term is indicated by 
a symbol such as i s *2s*2p3p6d P t . (86) 

The small letters give the azimuthal quantum numbers of the individual 
electrons, and the numbers preceding them their principal quantum 
numbers. The superscripts following each small letter give the number 
of electrons of each type. Thus in (86) there are two electrons having 
n = 1, 1 0, two having n = 2, 1 = 0, six having n ~ 2, 1 = 1, one having 
n = 3,lI, and one having n -- G, 1 = 2. Each electron, of course, has 
a spin 8=1, which it is unnecessary to record in the notation. The 
value of the capital letter gives the value of L, with the usual under- 
standing that 8, P, D, F, 0,... mean respectively L = 0, 1, 2, 3, 4,... . 
The subscript attached to the capital letter is the value of /, and thus 
fixes the multiplet component. The superscript preceding this letter is 
the multiplicity 2$+ 1, and has the values 1, 2, 3, 4... for singlet, doublet, 
triplet, quartet... terms. Very often the small letters specifying the 
quantum numbers of the individual electrons are omitted, in fact almost 
invariably those of electrons in closed shells. Otherwise the notation 
would be too cumbersome for a heavy atom such as uranium. Thus in 
the alkaline earths one gives only the quantum numbers of the two 
valence electrons. The normal state of magnesium, for instance, may 
be written as 3s 2 a $, and a singlet state in which one electron is excited 
to a 3jp level as 3s 3p IP, omitting ten electrons in closed shells which 
were written out explicitly in (86). Since L = I when the second valence 
electron remains in its normal s state, it is not really even necessary in 
these examples to list the individual quantum numbers of the valence 
electrons, so that the terms given in the preceding sentence are usually 
written as merely 3 1 8 9 3*P (also as 1 *8 9 2 1 P or even l 1 ^, IfP as, 
unfortunately, there is at present considerable diversity in usage in the 
choice of origin for the so-called ordinal number). When both valence 
electrons are excited, as in (86), this is sometimes indicated by attaching 

47 S. Goudsmit, Phys. Rev. 31, 946 (1928). 

48 For complete details on approved spectroscopic nomenclature see report of an 
informal committee on notation in Phys. Rev. 33, 900 (1929). 



VI, 40 QUANTUM-MECHANICAL FOUNDATIONS 167 

a prime instead of writing out the quantum numbers of the two 
individual electrons. 

It is to be emphasized that the above notation is intended only for 
Russell-Saunders coupling. Spectra possessing this type of coupling are 
sometimes termed 'normal multiplet structures', and fortunately they 
are characteristic of the simpler types of spectra, as, for instance, the 
alkalis, alkaline earths, and earths, except perhaps for complicated cases 
in which more than one electron is excited. Russell-Saunders coupling 
is also the rule in the iron and rare earth groups, so that it is the only 
type of quantization which we need consider for our magnetic work. 
Two commonplace illustrations of Russell-Saunders coupling are atoms 
with one valence electron, where L, 8 are identical with I, s; and inert 
gas atoms in their normal state, as completed shells (K, L, M shells, &c.) 
have zero resultant L, S, making them magnetically dead, so that they 
arc in a sense Russell-Saunders quantized to a null resultant. Excited 
inert gas atoms have exceedingly complicated spectra and do not con- 
form to any simple system of quantization. 

Angular Momentum Matrices for Russell-Saunders Coupling in Weak 
Fields (6, Fig. 6). Let us suppose that external fields are either absent or 
too weak to upset the tendency of L, S to form a quantized resultant J. 
Let us as usual take the z axis as the direction of spacial quantization. 
Then the non-vanishing elements for the z components of orbital and 
spin angular momentum can be shown to be 

\J(J+\)+8(8+\)--L(L+l)]M,\ 

(87) 

L(L+\)--8(8+\)]M, \ 



8 S (JM\ J+lM)~-= S S (J+IM; JM) =/, 
L,(JM; J+IM) = L S (J+IM; JM) = - 



where l\(J+L+S+2)(- J+8+L)(J+S-L+l)x ] ( 88 ) 

, / x(J+L-8+l)(J+M+l)(J-M+l)\ 

J N L 4(J+l) 2 (2J+l)(2J+3) J 

Elements of the form ( J IM ; JM), &c., are, of course, obtainable from 
(88) by lowering J one unit. From (87) and (88) it follows that S s + L z = M , 
as is to be expected from our previous remarks regarding the signi- 
ficance of the magnetic quantum number M. Formulae could also be 
worked out for the x and y components, but are not needed for our 
work. 49 

49 The x and y components differ from the z only as regards their factors depending 



168 QUANTUM-MECHANICAL FOUNDATIONS VI, 40 

Paschen-Back Effect, and Angular Momentum Matrices for Russell- 
Saunders Coupling in Strong Fields (c, Fig. 6). Suppose that a magnetic 
field is applied which is so strong as to produce a change in energy 
large compared to the coupling energy ^4L'S, but at the same time 
small compared to the terms c ik s t s k , b ik l t \ k in (79). Then the spin 
and orbital angular momenta will continue to have quantized resultants 
S, L respectively, but L and S will no longer form a quantized resultant 
J, as the field is by hypothesis strong enough to overpower the inter- 
actions between L and S, but not between the s or between the 1^ 
among themselves. Instead, L and S will be separately quantized with 
reference to the field, as illustrated in (c) of Fig. 6. The projections of 
L and S along the direction z of the applied field are diagonal matrices 
whose non- vanishing elements are respectively M fj and M s , where 
M L = ,...,+; M s = $,..., + $. The sum M L +M S is the magnetic 
quantum number M previously used. As one might expect, the formulae 
for the matrix elements of 8 X9 8y, S z and L X9 L y , L z are similar to (75) 
except that /, m t are replaced by L, M L or S, M K . Thus their non- 
vanishing elements are 

1 ; LM L ) - [L(L+ l)-M L (M L l)f, 



From (89) and (90) it, of course, follows in a fashion similar to (77) that 
L 2 , S 2 are diagonal matrices as given in (83), but J L+S is no longer 
quantized as in (83), and (87) and (88) are replaced by (89) and (90) 
since now L, S are not coupled together to give a constant resultant. 

The distortion in the spectroscopic multiplct structures and Zeeman 
pattern when a powerful magnetic field causes a passage from the 
quantization (b) to (c) in Fig. 6 is called the Paschen-Back effect. A still 
more powerful field might in principle overpower all the interactions 
in (79) and so give space quantization for individual electrons, studied 
in 39 and illustrated in (a), Fig. 6, but we have already mentioned 
that sufficiently powerful fields to do this cannot usually be built 
experimentally. 

j-j Coupling. If the terms of the form a (l l t - S t in <79> are large compared to the 
other terms c tt s t . s tt b^ . 1 A , <%!,- . S 4 (i k) then the spins no longer form a quan- 
tized resultant S, or the orbital angular momenta a quantized resultant L. Instead 

on M and independent of L, S. Such factors are easier to work out than those independent 
of M, and so arc commonly found in toxt-books (e.g. Born and Jordan, Elementare 
Quantenmechanik, p. 150). 



VI, 40 QUANTUM-MECHANICAL FOUNDATIONS 169 

the li and 8t of an individual electron form a quantized resultant associated with a 
quantum number j\ = l^ \ , so that each electron has, so to speak, its own private 
or individual inner quantum number. The total angular momentum of the whole 
atom is conserved, and this is expressed by the fact that the various j vectors 
form a quantized resultant J, whose projection in the direction of the axis of 
quantization is M. This is illustrated in (d) of Fig. 6. Angular momentum 
matrices, and various stages of the Paschen-Back effect can be worked out for 
j-j coupling, but we ornit them, as this typo of coupling is much less common than 
Russell-Saunders coupling, especially in the case of the 'normal' or 'ground' 
levels such as are involved in the study of magnetic susceptibilities. The j-j 
coupling is most likely to be realized in heavy atoms or in atoms which are 
multiply ionized. The reason for this is that the internal magnetic forces respon- 
sible for the constants a lt in <79> increase rapidly with the effective nuclear 
charge. In order for j-j coupling to occur it is usually necessary for there to be 
more than one ' uncompensated ' electron having 1*0 and so certain atoms may 
exhibit this type of coupling in excited states even though they do not in the 
normal level. For instance, high members of the 'primed' series of the alkaline 
earths, which represent excitation of both electrons, show some tendency towards 
j-j coupling. 

41. Classical Analogue of the Angular Momentum Matrices, 
and the Correspondence Principle 

We have stated that Eqs. (73) and (78) are the basic spin postulates, 
andso equations such as (87) and (88) or (89) and (90) should be derivable 
from (78) and the related orbital formula (75). Eqs. (89) and (90) are con- 
siderably easier to deduce than (87) and (88), although we shall not give 
the derivation of either. Formulae more or less equivalent to (89) and (90) 
were deduced in the early days of quantum mechanics by Born, Heisen- 
berg, and Jordan, and by Dirac with matrix and '^-number' methods 
respectively, as a consequence of the 'Vertauschung' relations satisfied 
by angular momentum matrices. 50 Their papers considered explicitly 
the compounding of orbital rather than spin angular momenta, but 
their results are readily adaptable to the spin because of the parallelism 
between (75) and (78). When there are several electrons, the proof of 
(89), which forms a part of what Dirac calls 'the elimination of nodes', 
is much more complicated by means of the Schrodinger wave functions 
than by use of non-commutative algebra, and the derivations of (89) 
and (90) by means of wave functions which have so far been published 
all involve rather abstruse group-theory considerations. 51 

80 Born, Hoisenberg, and Jordan, Zeits. f. Physik, 35, 603 (1926); Dirac, Proc. Hoy. 
Soc. 110A, 561 (1926). 

51 Weyl, Oruppentheorie und Quantenmechanik, p. 156; E. Wigiier, Zeits. f. Physik, 
43, 624, 45, 601 (1927); Neumann and Wignor, ibid. 47, 203, 49, 73 (1928). One must 
not confuse formula (89) for the resultant of several electrons with the easily proved 
similar formula (75) for one electron. 



170 QUANTUM-MECHANICAL FOUNDATIONS VI, 41 

Eqs. (88) have seldom 61 * been explicitly given in the literature, but 
are adaptations of general intensity formulae derived on semi-empirical 
grounds by Kronig, Russell, and Sommerfeld and Honl 52 just before 
the advent of quantum mechanics. The details of the adaptation are 
explained in an accompanying footnote, and were carried out by E. Hill 
in work unpublished except in abstract. 53 These semi-empirical formulae 
have been justified quantum mechanically by Dirac, 54 so that their use 
implies no loss of rigour. 

It is a general characteristic of the Heisenberg matrix elements that 
they merge asymptotically into the coefficients in classical Fourier 
expansions when the quantum numbers become very large. 55 It is easy 
to work out the classical amplitudes for various precessions of the 
angular momentum vectors, and it is illuminating to verify their 
asymptotic agreement with our previous quantum -mechanical formulae 
(87), (88), (89) and (90). This serves as an interesting check, but, of 
course, not a derivation of these formulae. 

First consider case (c), Fig. 6. Here L has a constant component M L 
along the z axis. Hence L%+L* equals LPM'b as in the old or 
* classical' quantum theory the square of the total angular momentum 

&u Formulae substantially equivalent to (88) have, however, been given by Rosenfeld 
in his paper on the Faraday effect, Zeits.f. Phyvik, 57, 835 (1929), especially his Kq. (75). 
Ho derives them by a group method due to Neumann and Wagner, ibid. 51, 844 (1930). 

52 R. do L. Kronig, Zeits.f. Physik, 31, 885 ; 33, 261 (1925) ; H. N. Russell, Proc. Nat. 
Acad. 11, 314, 322 (1925); Sommerfold and Honl, Sitz. Preuss. Akad. 9, 141 (1925). 

53 E. Hill and J. H. Van Vlock, Phys. Rev. 31, 715 (1927). To got the matrix elements 
or 'amplitudes' for L given in (87) and (88) from Kroiiig's formulae, one takes the Kronig 
formulae for Zeeman components relating to transitions of the form AL 0, given on 
p. 893 of his paper, and normalizes his constant B in such a way that L 2 = L(L-\-\}. 
This gives B = , as his elements sum to 4:BL(L+l). Kronig's K 9 E, J are the same 
as L-\-%, $+ J+% in our notation, and his elements are intensity ones, and hence pro- 
portional to the square of ours. The Kronig elements for A = 1 do not need to be 
considered for our purposes, as L has no component perpendicular to itself. The formulae 
for S given in (87) and (88) follow on interchanging L and S, with allowance for phase 
difference of 180 in the parts of L and S perpendicular to J, or more simply, on 
noting that L+S = J. 

64 P. A. M. Dirac, Proc. Roy. Soc. 1 1 IA, 281 (1926). Alternative methods of derivation 
have recently been given by Kramers, Proc. Amsterdam Acad. 33, 953 (1930) and by 
Cuttingor and Pauli, Zeits.f. Physik, 67, 754 (1931). Kramers employs the group theory 
of invariants, while Guttinger and Pauli use elementary matrix algebra. Their proof 
unfortunately appeared too late to include in the present volume. 

55 For exposition of Bohr's correspondence principle see N. Bohr, The Quantum 
Theory of Line Spectra, or for a more elementary discussion, Ruark and Urey, Atoms, 
Molecules, and Quanta, Chap. VI. That amplitudes (i.e. matrix elements) in the new 
mechanics do really merge asymptotically into classical Fourier coefficients has been 
shown by C. Eckart, Proc. Nat. Acad. 12, 684 (1926) and by J. H. Van Vleck, ibid. 14, 
178 (1927); tho related questions of convergence have boon covered by Jeffreys, Proc. 
Lon. Math. Soc. 23, 428 (1924) ; cf. also Eckart, Zeits.f. Physik, 48, 295 (1928). 



VI, 41 QUANTUM-MECHANICAL FOUNDATIONS 171 

is L 2 rather than L(L+l). If < be the angle between the x axis and 
the projection of L on the xy plane, then L x ~ (-L 2 Jfl^cos^, 
L y = (L 2 Mffiam(f>. But we may write < = 2rra} M t-{- Mj where M is 
a trivial epoch constant, and where OJ M is the frequency of precession 
of L about the z axis, i.e. the frequency associated with the quantum 
number M L . Thus 

Lx iL y = ( 2 Jf i)*e i(2irto * / ' l *\ L s ^ M L . (91 ) 

Now according to Bohr's correspondence principle, a quantum- 
mechanical matrix clement approaches asymptotically a coefficient 
in a classical multiple Fourier expansion 

2* '"TIT*"' 6 ' 1 l * * '" T 1 T 2'" \'*>) 

provided we select the particular overtone for which each r i (i= 1, 2,...) 
equals the change (i.e. difference between initial and final index) in the 
quantum number associated with the frequency co . Comparison of (89) 
and (91) shows that the requirements of the correspondence principle 
are indeed fulfilled, as L(L-\-\)M L (M L -}-\) is asymptotically the same 
as L 2 ~M^ if L, M L are very large. The fact that in (89) all the ampli- 
tudes for L x , L y vanish unless M' L M L I is an expression of the fact 
that the first part of (91) is a special case of the general series (92), in 
which all amplitudes for L x , L y vanish unless r = 1. The diagonality 
of (89) in all quantum numbers other than M L is because (91) involves 
a sole frequency a>, 1/5 i.e. is simply rather than multiply periodic. The 
proof of the asymptotic identity of (90) with classical theory is entirely 
analogous, as $, M$ simply replace L, M L . 

In case (b), Fig. 6, $ and L precess around J, and J in turn precesses 
around the axis z of quantization. Let D be the angle between the 
plane determined by the vectors J, $, L, and the plane determined by 
the vector J and the z axis. Then from the geometry 

8 S = 8 cos($, J)cos(J, z) -f 8 sin($, J)sin(J, z)cos}. (93) 

But Q = 27rco^+j, where <j*j is the frequency with which 8, L precess 
about J, and further 



<*A\ 
(94) 

Thus (93) becomes 

8 g = M(J*+S*L*) _|_yti[ 6 fer/a>y f e^e- 2 '^-^], (95) 

2iJ 
where 

/(j+L+^x-7+^+^W+^^^^ 

/el - -- --- 



172 QUANTUM-MECHANICAL FOUNDATIONS VI, 41 

and we see that the three non- vanishing amplitudes in (95) do indeed 
agree asymptotically with the corresponding elements of S a in (87) 
and (88). The proof for the elements of L z instead of 8 S is entirely 
similar. 

It is possible, in the following elementary way, to obtain the diagonal 
elements (87) from the constant term of (95) exactly rather than 
asymptotically, thus achieving what Fowler terms a 'refined' applica- 
tion of the correspondence principle. The constant term of (95) is the 
classical value of r/ 12 i C2 i 2\ 

- tj* ~ (%) 

Now with the Russell-Saunders coupling presupposed in case (b), Fig. 6, 
J c , J 2 , L 2 , and S 2 are all diagonal matrices, i.e. constant in time, and 
hence (96) is also diagonal. Furthermore, 

(l/e/ 2 )(J3I; JM) - l/e7 2 (JJf ; JM ) 

because J 2 is diagonal. Also J a = M . When we interpret (9(5) as a 
matrix expression, and substitute the values (83) of the elements, we 
have indeed the first equation of (87). The proof of the second equation 
of (87) is similar. 

42. The Anomalous Zeeman Effect in Atomic Spectra 

Once the matrix elements (87) and (88) or (89) and (90) are granted, the 
formulae for this effect are very easy to deduce by the perturbation 
theory of 34. Eq. (48), Chap. I, which corresponds to the wave Eq. (2) 
of the present chapter, gave the Hamiltonian function exclusive of spin. 
If we neglect the motion of the nucleus, we may take e^/w^ e/m, 
and then the part of this function which is linear in H may be written 
(Heh/4:7r?nc)L ffy where jkis the component of orbital angular momentum, 
measured in multiples of h/2ir, in the direction z of the applied field. 
To incorporate the spin, we add a term (Heh/27rmc)ti a due to the action 
of the external field H on the spin, not to mention new spin terms 
in the part <# of the Hamiltonian function which is independent of //. 
The factor e/27rwc rather than e/47rwc appears in the spin term propor- 
tional to H because of the anomalous ratio (73) of spin magnetic moment 
to angular momentum. Thus if we neglect the 'diamagnetic' part 
H 2 e 2 2 (#?+2/?)/8wc 2 , which is quadratic in //, the Hamiltonian func- 



tion is TI i 



For arbitrary couplings of the spin, i.e. arbitrary values of the constants 



VI, 42 QUANTUM-MECHANICAL FOUNDATIONS 173 

a, 6, c in (79), the problem is excessively complicated. 56 We shall there- 
fore henceforth assume that the atom has Russell-Saunders coupling. 
If we further assume the * cosine law 5 , then <# is of the form (84), but 
this extra restriction is unnecessary. As our unperturbed system let us 
take that of the atom in the absence of the external field. Then by (87), 
(88), and (97) the non-vanishing elements of the perturbative potential 
are 



; JM) -. 

' ' 47TWC_ 

; J+ 1 Jlf ) - / /(J, M), 



(98) 

(J-l,Jf), 

47TWC 

where /(J, Jlf) is defined as in (88). 

If there were no spin, i.e. if S 0, J /y, as is the case in singlet 
spectra, then/ would vanish and the perturbing potential would consist 
solely of diagonal elements HehM/^irmc. This can also readily be seen 
to have been the case if we had omitted to insert the anomalous factor 
2, so that we had L g +S s rather than 7^+2$,, in (97). The energy would 
then be given rigorously (neglecting the small diamagnetic effect to be 
discussed in 43) by the fornuila 



W=W+' M. 

4777/M/ 

There would then be only a normal Lorentz triplet v = v , 

since the selection principle for the magnetic quantum number allows 

only Alf = 0, 1 . 

Actually the perturbative potential is not a diagonal matrix except 
when 8 or L vanishes. If the applied field is small enough compared 
to the multiple! width to permit a series development (21) in the para- 
meter A 77, then Eqs. (24) and (26) show that the energy is 

(99) 

- 

where g - 1 + ( M)_ (1M) 



56 Tho ^/-factors and honno ilio onorgy to a first approximation in H have boon given 
for j-j and some limiting forms of coupling other than Russoll-JSaundors hy Goudsinit 
and Uhlonbeok, Zeils.f. Physik, 35, 618 (1926). 

57 Wo suppose the rcador at least a little familiar with the theory of the normal 
Xooinan effect, and the selection rules. Soo, for instance, liunrk and Uroy, Atoma t 
Molecules and Quanta, pp. 138, 143, 568. 



174 QUANTUM-MECHANICAL FOUNDATIONS VI, 42 

The frequencies appearing in the denominator of (99) are those separat- 
ing adjacent levels in a multiplet. If we assume the cosine law expressed 
inEq.(84a),then2/&v(J; Jl) = A^ (2 J+ l)A, but this specialization 
is unnecessary and it is better to substitute the experimental values of 
the multiplet intervals in (99), as the cosine law seldom holds precisely. 
In applying the Bohr frequency condition to get the spectroscopic 
frequencies predicted by (99), one must not forget that the selection 
principle allows 57 only AJ ~ 0, 1 ; AJf = 0, ^ 1. 

The first-order term in (99) is the familiar Lande ^-formula. 58 The 
presence of second- and higher-order terms is too commonly overlooked. 
The second-order term is, to be sure, ordinarily so small with the values 
of H used experimentally that it is confirmed by only a very limited 
amount of spectroscopic evidence, but is often quite important in the 
study of magnetic susceptibilities (vide 56-9). The second-order terms 
were calculated in the old quantum theory by Lande himself, 59 and the 
form (99) which they take in the new quantum mechanics was obtained 
by Hill and Van Vleck. 53 * 60 The physical significance of the second- 
order Zeeman term is that there is a component of magnetic moment 
perpendicular to the axis J of angular momentum, since the moment 
vector ( e/2mc)(L+2S) clearly is not in general parallel to L+S 
because of the factor 2. The ordinary first approximation involved in 
the Lande (/-formula utilizes only the component of magnetic moment 
along J, since by (24) the perturbed energy is to a first approximation 
the perturbative potential averaged over an unperturbed orbit, and 
such an average introduces only the component of moment parallel to 
J. This is most quickly seen by consulting the classical Eqs. (93), (94), 
(95), as the constant term in (95) involves 8 only through the projection 
$cos($, J) along J, and similarly for the contribution of L. 

Lande has shown that the second-order term in H is confirmed by 
a certain amount of direct spectroscopic evidence on the Zeeman effect, 
despite the smallness of this term in ordinary fields. The agreement 
with experiment is somewhat improved by using formula (99) instead 
of Lande 's analogous formula with the old theory. This is illustrated 
by the following table for the Mg triplet, 5184, 5173, 5167 A at 38,900 

58 For the abundant spectroscopic evidence confirming the (/-formula HOC, for instance, 
Back and Lande, Zeemaneffckt und Multiplcttstruktur. 

A. Lando, Zeits.f. Physik, 30, 329 (1924). 

60 Eq. (99) has also been given independently for the special case of triplets by A. 
/warm, Zeits.f. Physik, 61, 62 (1928). Extensive calculations of the energy-levels of 
triplets at intermediate field strengths by numerical solution of the secular equation, 
here a cubic, have been made by K. Darwin, Proc. Roy. Soc. 116A, 264 (1928). 



VI, 42 QUANTUM-MECHANICAL FOUNDATIONS 175 

gauss, which is the most comprehensive example quoted by Lande\ 
The entries in the table are the ratio q of the energy separation between 
components with the second-order terms to the separation without this 
term. Experimentally this ratio is determined from the dissymmetry 
between the two sides of the Zeeman pattern, as the departures of the 
ratio from unity gauge the distortion from a strictly linear or sym- 
metrical pattern: 

i5~ a 14 a ll~ a 10 W a ~ 1T 6 7T 5 7T 4 TT^ CT 3 ^3 ~ <*2 ^Z" 1 

q obB 0-89 Ml 1-02 1-00 0-96 1-04 0-96 

9und 0-92 1-08 1-01 0-99 0-98 1-02 1-01 

? uew 0-89 1-11 1-02 1-00 0-96 1-04 097 

The notation of the components is explained in Lande's paper. 59 Out 
of the fifteen ratios recorded in his table, we give only the seven which 
are changed by the new mechanics. The writer is indebted to E. Hill 
for his calculation (unpublished elsewhere) of the above g now from 
Eq. (99). 

Eq. (99) ceases to be a good approximation when the field is so 
powerful, or the multiplets so narrow, that there is an appreciable 
tendency towards a Paschen-Back effect, i.e. considerable progress in 
the passage from case (6) to case (c) in Fig. 6. We must then use the 
perturbation technique for nearly degenerate systems. By Eq. (35) the 
energy-levels are the roots of the secular equation 

XJtv>(JM',J'M)+8(J;J')(W$-W) -0, (101) 

where J, J' - Q, Q+l,..., L+8 with Q - \LS\ if \LS\ > |J/|, while 
Q = | M\ if \M | ^ | LS\. Each value of M furnishes a different secular 
equation. Eq. (98) shows that the determinant involved in (101) has 
zeros everywhere except along the principal diagonal and elements 
adjacent thereto. Even so, (101) cannot in general be solved explicitly, 
as it furnishes an algebraic equation of order L-\-SQ-}-l. Hence it is 
solvable in an elementary way for all allowable values of M only for 
the case of doublet spectra. Then (101) yields a quadratic equation 
for W, whose roots are 61 

r A HIT A.. / A_. \ S T\\ 

+const., (102) 

as can be seen by specializing (98) and (101) by setting 8 = \, J = L |. 
Here Av denotes the doublet separation in the absence of the field and 

61 The case M = di(^ + i) requires special consideration, as hore the secular equation 
is linear instead of quadratic, and has the solution W ^(L-}-l)h&v n - J r%h&v~\- const., 
the same as (102) only if one makes the proper choice of sign for the radical, which is 
arbitrary for other values of M. The physical significance of this is that only one of the 
doublet components can have M (L f-), as M cannot exceed J in magnitude. 



176 QUANTUM-MECHANICAL FOUNDATIONS VI, 42 

Av^ is the normal Lorentz displacement Av n = Hejknmc. Eq. (102) has, 
rather fortuitously, the same form as SommerfehTs adaptation 62 of 
Voigt's classical formula based on a model not in accord with modern 
knowledge of atomic structiire. 

The quantum mechanics of the anomalous Zeeman effect was first given by 
Heisonberg and Jordan. 63 One slight difference, however, may be noted between 
their procedure and that just given. They use as the unperturbed system of 
quantization that appropriate to a magnetic field powerful enough to give a 
Zoeman effect large compared to the multiplet structure, and hence separate 
spacial quantization of S and L, as shown in (c), Fig. 6. The energy duo to the 
external magnetic field is included in the unperturbed system, and the coupling 
forces between S and L treated as a perturbation, just the reverse of what we 
have done. Thus with the cosine law <84>, the perturbing potential is AL S. 
If the external field is really so strong that (o), Fig. 6, is a good approximation, 
one may replace AL S by its mean value, which is readily seen 64 to be AM L M a 
with the separate processions of S and L. Thus for very strong fields, which 
produce a complete Paschon-Back effect, the energy-levels are 



W -- W on + ~ (M L h2Af x ) + /lAfJM r / ,+ ... . (103) 

477 WC 

More accurate formulae than (103) can, of course, bo obtained by taking into 
account the non-diagonal terms arising in the Hamiltonian function for case (c) 
because L S is not identical with L S. Eq. (21) then takes the form of a power- 
series development in. a parameter of the order A/hkv n instead of the order 
hAv n /A as in (99). Here Av M is tho 'normal' Zeeman displacement He formic, and 
A gauges tho magnitude of the multiplot interval. Thus the neglected terms in 
(103) are of the order of magnitude A 2 /hkv n . The development (103) in A/hkv n 
is, of course, a poor approximation if A/h&v n is largo, i.e. if the field is not great 
enough for an almost complete Paschen-Back effect, and one can instead use the 
perturbation technique for nearly degenerate systems, and set up the proper 
secular equation. This Heisonborg and Jordan proceed to do. Their secular 
equation, being figured from (c) rather than (b) in Fig. 6 has a different super- 
ficial appearance from (101), but must yield tho same algebraic equation for W 
as (101), since the method based on the secular equation is always rigorous 
regardless of whether or not the assumed initial system of quantization is a close 
approximation to that appropriate to the actual field strength. A direct general 
proof has not yet been given that the two secular equations are the same, or what 
is partially equivalent, that our secular equation has asymptotically the roots 
(103) for very largo //, or that Heisenberg and Jordan's secular equation has 
asymptotically the roots (99) for small ? 65 . Such a proof would bo of interest 

62 Soniraorfold, Atombau, 4th od., p. 672. 

63 Hoisoriberg and Jordan, Zeit*. /. Phyxik, 37, 263 (1926); see Darwin, Z.c. 36 for the 
transcription into wave language. 

64 This can bo soon classically by taking the constant term of tho multiple Fourier 
series for Lj,S x -\-L y S y -\-rj z S z obtained by multiplying (91) with the corresponding 
formulae for the components of S; or quantum mechanically by the taking diagonal 
elements of this scalar product whon formed by multiplying together the matrices whose 
elements are given by (89) and (90). 

65 To show rigorously that tho two secular equations are tho same one would have to 



VI, 42 QUANTUM-MECHANICAL FOUNDATIONS 177 

only as mathematical manipulation, since the physical knowledge of the various 
appropriate quantizations assures the results must be the same with either method. 
In the particular case of doublets, it is, of course, directly verifiable that either 
secular equation yields the same quadratic equation (102), which agrees asymp- 
totically with (99) and ( 103) specialized to # = \, J L\. Heisenberg and Jor- 
dan treat weak fields separately, and deduce the first-order terms in (99) by the 
same quantum-mechanical refinement of the constant term of (95) as that already 
discussed after (95). The first-order terms in (99), and hence the celebrated Land6 
(/-factor, arc deducible in this elementary fashion involving only the quantum 
formulation of tho cosine law, whereas, as already mentioned, the derivation of 
tho non-diagonal elements (88) and hence of the second-order terms in (99), which 
are tho only thing of consequence not given in Hoisenberg and Jordan's eventful 
paper, are more difficult. Their use of case (c), Fig. 6, as compared to our use of 
(6) for the point of departure for deducing the rigorous secular equation has the 
advantage in that tho matrix elements (89) and (90) are easier to deduce from the 
basic spin postulate (78) than are thenon-diagoiial elements (88) of (87) and (88), but 
the disadvantage in that with ordinary field strengths case (c) is a poorer approxi- 
mation to the true state of quantization than (6), and so furnishes a less natural 
starting point even though use of (c) entails no loss of rigour or generality. Our 
use of (b) rather than (c) porhaps frees us a little more readily from the specializa- 
tions appropriate to the cosine law. We have, for instance, shown that tho Voigt 
formula (102) is valid without the assumption of the cosine law made by Heisen- 
berg arid Jordan, but this is not at all surprising when it is remembered that the 
doublet case inevitably loads to a quadratic secular equation. 

We have mentioned that, except when 8 -~ J, it is impossible because of 
algebraic difficulties, to trace readily tho transition of individual roots of the 
secular determinant from (99) to (103) as tho field is made very great. However, 
it is quite easy to verify that the sum of all tho roots of the secular determinant 
pass properly from one limit to tho other. As mentioned on p. 142, this sum is tho 
'spur 1 or diagonal-sum of the Hamiltoniaii matrix. Reference to the diagonal 
elements of tho determinant (101) shows that this sum must be linear in //, so 

that - (104) 



where cn i and a 2 are constants independent of H. Explicit formulae could be given 
for aj and a 2 , but are rather cumbersome inasmuch as it is necessary to differen- 
tiate between different cases depending on the relative magnitudes of >S Y , L, M. 
Now it is not hard to verify that Eqs. (99) and (103) yield identical values of oc^ 
or of a 2 . With (103) we of course sum over all values of M L , M 8 consistent with 
given M rather than over J. That (104) is linear in H is the so-called principle 
of permanence of F- and {/-sums, 88 which was known on semi-empirical grounds 
before tho advent of quantum mechanics. The terms F-sum and p-sum are used 
to designate the parts of (104) which are independent of H and proportional to 
//, i.e. the parts OC L and a 2 H respectively. 

prove that their roots when expanded in some parameter, say h&v n IA, are the same to all 
powers in H rather than merely to tho second power given in (99). Tho identity to tho 
first power has been shown by Darwin. 30 

66 For elaboration of these permanence principles, see Back and Lande, Zeemaneffekt 
und Multiplettstruktur, pp. 02-82. For explicit verification of the invariance of <x a 
see the writer's Quantum Principles and Line Spectra, p. 244. Pauli has shown (Zeits. 

3595.3 -M- 



178 QUANTUM-MECHANICAL FOUNDATIONS VI, 43 

43. The Diamagnetic Second -Order Zeeman Term 

Finally, we must not fail to note that besides the term in (99) propor- 
tional to 7/ 2 , there is another kind of quadratic term in II which we have 
so far neglected, viz. that arising from the term 2 //Vf(xf+?/?)/8m;C 2 
in Eq. (48), Chap. I, or Eq. (2) of the present chapter. If we call this 
term A 2 J (2) , and if we remember that diagonal matrix elements have 
the physical significance of being time averages, Eq. (26) shows that 
its effect on the energy is approximately 



provided degeneracy difficulties are not encountered. We have assumed 
the nucleus at rest at the origin, so the sum is only over the electrons, 
for which e t e, m t m. The expression (105) is very small unless 
one or more of the orbits is very large. Let us suppose for simplicity 
that there is only one electron not in a closed shell. Then (105) is very 
small compared to the quadratic term in (99) unless the valence electron 
has a very large principal quantum number n. The magnetic coupling 
is very small for such an orbit, as the factor A = a n in (84 a) can be 
shown to vary as n 3 . Thus under the conditions, viz. high field strength 
and large orbits, under which (105) might be capable of observation 
spectroscopically, there is certain to be separate spacial quantization 
of the orbit and spin, and (105) must then be added to the energy 
expression (103) rather than (99). Thus the two types of quadratic 
terms are never simultaneously of importance. When the spin is 
quantized separately, the time average (105) can be calculated as though 
the spin were entirely absent. By setting/ # 2 +?/ 2 in the fundamental 
quadrature (14), and taking the J/T'S to be of the central form (74), it 
follows 67 that for an electron moving without spin subject to a central 

/. J'hyaik, 16, 155 (1923)) that tho permanence of r/-sums permits calculation of the 
{/-factors for woak fields from tho strong field formula (103), assuming the linear term in 
H to bo proportional to M in woak fields. 

67 Because the wave functions (74) are products of the form R(r)S(9,^>), it follows from 
(74) that Hff cos~20) r*(l cos-0). Also tho value of cos 2 0has already been calculated 
in Eq. (67), as tho factor *S'(0, ^) is identical in form with the wave functions of tho 
'dumb-boll* used in 37. Tricidently, group theory considerations show that the depen- 
dence of Jijf+y? on MI, is of the form A^BM\ whenever there is Russoll-Saunders 
coupling and separate spacial quantization of S even though tho dynamical problem is 
that of many electrons rather than one (cf. H. A. Kramers, Proc. Amsterdam Acad. 32, 
1L79 (1929)). 



VI, 43 QUANTUM-MECHANICAL FOUNDATIONS J79 

The mean square radius r 2 depends on the nature of the central field. 
It can be shown 68 that if it is Coulomb, 



Unfortunately this value is never rigorously applicable to our Zeeman 
problem, for if the field is Coulomb the system is degenerate in the 
azimuthal quantum number I. Instead of using (106) one must then 
set up a secular equation, which has been given by Halpern and Sexl 69 
and which proves to be not explicitly solvable except for the uninterest- 
ing case of small values of n. If on the other hand the field is non- 
Coulomb, (106) is not accurate. It will perhaps be an approximately 
correct expression for a non-Coulomb central field if one uses in place 
of n the so-called effective quantum number n*, defined by n* = n A, 
where A is the 'quantum defect' in the Rydberg formula J?/(r& A) 2 for 
a spectral term. To avoid degeneracy difficulties, the expression (106) 
must be small compared to the departures from Coulomb character, 
which decrease rapidly with increasing n, as R/(n-~A)' 2 R/ri* ~ 2kR/n*. 
Fortunately the case of greatest interest, that of small values of I along 
with great values of n as in e.g. high numbers of the principal series of 
sodium, is accompanied by comparatively large values of A, at least in 
the alkalis. 

We shall not discuss this matter further, as suitable experimental 
evidence on the Zeeman effect of very large orbits, needed to test the 
theory, is wanting. The quadratic term (105) is of vital importance for 
the theory of diamagnetism, but the theory of this does not involve 
or test experimentally the mathematical problems connected with the 
degeneracy, as the susceptibility involves only the statistical average 
over all orientations, permitting us to replace x 2 +y* by |r 2 , regardless 

08 The result (107) is obtained by using a method developed by Waller for evaluating 
the mean, value of any power of r in Keplerian motion (Zcits.f. Physik, 38, 635 (1026)). 
The result is, of course, the same as evaluation of the fiuidamontal quadrature (14) with 
J r 2 , n - n', which is tedious if done by ordinary methods. 

09 O. Halpern and Th. Sexl, Ann. der Phyfdk, 3, 565 (1929). The analogous problem 
in the old quantum theory was considered by Burgers, Dissertation, p. 106, also Halpern, 
Zeite.f. Physik, 18, 352 (1923), and likewise was not soluble in closed foim. The order 
of magnitude of the quadratic effect arising from (105) has been estimated by E. Guth, 
Zeits. /. Physik, 58, 368 (1929). Ho finds it cannot give a displacement of more than 
0*08 A. for the 13th Balrner line of hydrogen, in a field of 30,000 gauss, whereas the 
normal first-order displacement is 0-19 A. If wo went to still higher lines the quadratic; 
term would soon become more important, as it varies as n*. All the writers consider 
primarily hydrogen, although the alkalis, where tho degeneracy difficulties are less 
bothersome, would appear likewise easier to tost experimentally. 

N2 



180 QUANTUM-MECHANICAL FOUNDATIONS VI, 43 

of the type of quantization. For the particular case when (106) is 
applicable, one can verify that the statistical average # 2 +2/ 2 is f r 2 not 
only by virtue of the general proof of spectroscopic stability, but also 
by direct evaluation of the summation over m l by means of Eq. (68), 
as already virtually done in Eq. (69). 



VII 

QUANTUM-MECHANICAL DERIVATION OF THE 
LANGEV1N-DEBYE FORMULA 

THROUGHOUT the present chapter we shall suppose that we are dealing 
with electric rather than magnetic polarization. The few modifications 
necessary to adapt the analysis to the calculation of magnetic instead 
of electric susceptibilities will be given in 53. 

44. First Stages of Calculation 

In 36 we showed that if an atom or molecule is in a given stationary 
state n, j, w, the time average of its electrical moment in the direction 
of the applied field E is given by the formula 



p E (njm',njm) = - |J = -W% m -*EW<$ M -..., (1) 

where the W$ m are the coefficients in the development 

W . = W- -4-W (l) E-\-WW E 2 4- (2} 

rr njm rr njm\ rr njm lj \ rr njtn^ i \*i 

of the energy in terms of the field strength E. We now use three 
indices to specify a stationary state, rather than two as in most of 
Chapter VI, inasmuch as later in the present chapter it will be necessary 
to distinguish between three kinds of quantum numbers. Eq. (1) gives 
only the moment for a single stationary state. The total polarization 
or moment P per unit volume is the statistical mean over all stationary 
states, weighted according to the Boltzmann factor er w "^ kT and 
multiplied by the number N of molecules per c.c Thus 

(3) 

Eq. (3) is, of course, the quantum analogue of (58), Chap. I. We assume 
here, and elsewhere unless otherwise stated, that the medium is suffi- 
ciently rarefied so that one may use the Boltzmann instead of the Fermi 
statistics. This assumption is fully warranted except for conduction 
electrons in solids. That the probability of a state is proportional to 
the factor er w ^ kT follows from exactly the same sort of statistical 
premises as in classical theory, 1 discussed on p. 25. The only difference 
is that there is now a discrete rather than continuous distribution of 
configurations. It is to be understood that we employ as many quantum 
numbers as degrees of freedom even though some of them are really 

1 See R. H. Fowler, Statistical Mechanics, Chap. II. 



182 QUANTUM-MECHANICAL DERIVATION OF VII, 44 

superfluous in degenerate systems, and we regard states of different 
quantum numbers as distinct even though they happen to have coinci- 
dent energies. We adopt this convention to avoid the necessity of 
introducing an 'a priori probability' or 'statistical weight'. If instead 
we treated a family of states of coincident energy as a single state, we 
would have to take the Boltzmann factor as y n je,~ w 'd kT rather than 
e - w W fcr , where the weight g nj is the number of states so coinciding. 
We can immediately substitute (1) and (2) in (3), and by expanding 

the exponentials as series e- Tr W*f -^ e - ] ^ kT ( I nlm E ...) in E, 

we can then develop the numerator and denominator as power series 
in E. We shall neglect saturation effects, and so consider only the 
portion of the susceptibility x = P/E which is independent of field 
strength. This means that we need develop the numerator of (3) only 
to the first power of E inclusive, and retain only the portion of the 
denominator which is independent of E. With this approximation 

2iP&>-^* r 

IA\ 
- " o ,,, ' V*) 

2^ e-wjmiKi 

njm 

In deducing Eq. (4) from (3) we have assumed, as is always done in 
calculations such as the present, that 

I p$.(njmi njm)e-^T = j _ Fpu.)^-'^ ^ 0, (5) 

n ; m 

or, in other words, that the medium does not possess a f permanent' or 
'residual' polarization per c.c. in the absence of the field E. This 
assumption clearly involves no loss of generality, as such residual effects 
are ordinarily found experimentally only in crystalline dielectrics, which 
are beyond the scope of the present volume. Also, from a theoretical 
standpoint, the expression (5) clearly vanishes on symmetry grounds if 
we neglect inter-molecular forces, for in the absence of all fields there 
can be no preference between directions parallel and anti-parallel to E. 
In solids there are in reality important inter-molecular fields, but if 
these directions are random, the sum (5) still vanishes by symmetry on 
averaging over a tremendous number of molecules. If there is some 
other applied external field, namely, say, a magnetic field, which remains 
even in the absence of the given electric field E, one might think that 
(5) could be different from zero due to alincment of molecules in this 
other field, but in 70 on the non-existence of a magneto -electric 
directive effect we shall show that at any rate a magnetic field cannot 
make (5) appreciably different from zero. 



VII, 45 THE LANGEVIN-DEBYE FORMULA 183 

45. Derivation of the Langevin-Debye Formula with Special 
Models 

Eq. (4) is the initial stage in the calculations of the susceptibility in 
quantum theory. Before one can proceed farther it is necessary to 
examine the structure of the formulae for W (l) and W < 2) . The completion 
of the calculation was first achieved simultaneously by Mensing and 
Pauli, Kronig, Manneback, and Van Vleck 2 for the special model of the 
'rigid rotating dipole' or 'dumb-bell', whose characteristics have been 
explained and energy -levels determined in 37. As two quantum mim- 
bers suffice for this model, we may omit the index n in Eq. (4), and 
take^" and m to be respectively the inner and axial or magnetic quantum 
numbers, just as in 37. Now in 37 we supposed that the electric field 
E is the only field to which the molecule is subjected, so that the energy 
in the absence of K is independent of spacial orientation, and hence 
W* m has a value W [ ] independent of m (of. Eq. (57), Chap. VI). Also 
we showed that this model had no first-order Stark effect, i.e. Wfy ~ 
(Eq. (64), Chap. VI). For a given value of j, there are 2jf+l possible 
values of m, viz. j, (j -1 ),..., j. Thus Eq. (4) reduces to 



J 

Now by Eqs. (65) and (68) of Chapter VI, 

2^-0 OVO). (7) 

ni;--j 

Iii other words, even the second -order energy vanishes on averaging 
over all values of the quantum number m consistent with a given j. An 
exception is the lowest rotational state j ~ 0, in which the summation 
involved in (7) reduces to the single term 

(7 a) 

(cf. Eq. (66), (Jhap. VI). Here p, and / denote respectively the mole- 
cule's dipole moment and moment of inertia. 

Eqs. (7), (7 a) bring to light the very remarkable fact that all the 
contribution to the susceptibility comes from the molecules in the lowest 
rotational state j -~ O. 3 This result is the very interesting quantum 

2 For references., Kee note 21 of Clinp. VI. 

3 This result cannot hold when higher powers of tho field strength, and tho resulting 
saturation effects aro considered, as in infinitely strong fiolds all molecules, regardless 
of j, alino themselves practically parallel to the field and contribute an amount, /u to 



184 QUANTUM-MECHANICAL DERIVATION OF VII, 45 

analogue of the fact that with the dumb-bell model in classical theory 
only the molecules whose total energy is less than pE contribute to the 
susceptibility. This property of the model in classical statistics was 
shown by Alexandrow and by Pauli. 4 In other words, classically the 
susceptibility arises entirely from molecules which possess so little 
energy that they would oscillate rather than rotate through complete 
circles in case their axis of rotation happened to be perpendicular to 
the axis of the field. Most molecules, of course, have some angular 
momentum about the axis of the field, so that their axes of rotation 
are not perpendicular to the field, and we mention the perpendicular 
case only because it admits a particularly simple interpretation ana- 
logous to the two types of motion for a simple pendulum. As the 
temperature is increased, the fraction of molecules which are located 
in the lazy' states that contribute to the susceptibility will steadily 
diminish, and hence we can see qualitatively why the susceptibility due 
to permanent dipoles decreases with increasing temperature. It must 
be cautioned that these theorems, both quantum and classical, that the 
susceptibility arises entirely from molecules of certain particularly low 
energies, are peculiar to the 'dumb -bell' model. In the general dynamical 
system to be considered in 46, or even in the symmetrical top model, 
which is almost as simple as the dumb-bell, the higher rotational states 
will make some contribiition to the susceptibility. Even then usually 
the bulk of the contribution conies from molecules with small rotational 
energies, since all rotational motions show at least a little resemblance 
to that of the dumb-bell, and since the susceptibility experimentally 
almost invariably decreases with increasing temperature. In the old 
quantum theory the susceptibility did not arise uniquely from the 
lowest rotational state even with the dumb-bell model, and this is per- 
haps one reason why the old theory gave such nonsensical results on 
dielectric constants (28). 

When we substitute (7) and (7 a), and utilize the familiar expression 
j(j+I)h 2 /$7T 2 I (Eq. (57), Chap. VI) for the field-free energy of the rigid 
rotator, the quantitative expression ((>) for the susceptibility becomes 



_ 



the polarization P. This is also evident from the classical analogy, as when E is arbitrarily 
large even the higher rotational states have original kinetic energies small compared to 
|u./, and so then correspond to classical motions which contribute to the .susceptibility. 
4 W. Aloxandrow, Phys. Zcita. 22, 258 (1921); W. Pauli, Jr., Zeits. f. Phynik, 6, 319 
(1921). 



VII, 45 THE LANGEVIN-DEBYE FORMULA 185 

If the temperature is sufficiently high so that most molecules have 
fairly large values of j, as is ordinarily the case, the summation in the 
denominator of (8) does not differ appreciably from the integral 



o h 

The formula (8) for the susceptibility, with this approximation, becomes 
X = NyPjSlcT, which is exactly the same as the part of the classical 
Langcviri-Debye formula which arises from the permanent dipoles. 
Characteristic deviations from the classical theory will, however, be 
found at low temperatures, where the summation cannot be replaced 
by an integral. In classical theory it is not legitimate near T to 
neglect saturation effects and consider only the first term in the expan- 
sion of the Langevin fimction (4), Chap. II, as the ratio x pH/kT 
becomes very large. At T = the classical theory will by (4), Chap. IT, 
give the full saturation polarization NJJL for infinitesimal fields, and 
hence an infinite susceptibility for such fields. On the other hand, the 
quantum theory expression (8) for the susceptibility has the finite value 
S7T 2 Nfju 2 l /3h 2 at T ~ 0, as here only the state j ~- gives a significant 
term in the denominator of (8). Thus even at the absolute zero the 
polarization is proportional to the field strength unless the latter is 
exceedingly large. 

After the asymptotic derivation of the Langeviii-Debye formula for 
the dumb-bell model, the next calculations of the susceptibility were 
made for the symmetrical top model (p. 153) independently by Kronig 
and by Mamieback. 5 Here also it is found that the susceptibility has 
asymptotically the Langevin- Debye value at high temperatures. We 
shall not give the details of the calculations, as the results are all a 
special case of the general derivation to be given in 46. We may, 
however, mention that it is unnecessary to replace the summation over 
the quantum number A used on p. 153 by an integration, as A drops 
out of the formulae on integrating over j. Hence the results apply for 
either use of the symmetrical top model mentioned on p. 153, viz. either 
to represent the rotational motion of a polyatomic molecule with two 
equal moments of inertia, in which A is a quantum number assuming 

8 R. de L. Kronig, Proc. Nat. Acad. 12, 608 (1926); 0. Manneback, Phys. Zcit.s. 28, 
72 ( 1 927). Thoso writers considered the application of the modol to polyatomic- molecules. 
The application to diatomic molecules was treated by the writer, the details being unpub- 
lished except in abstract (Nature, 118, 226 (1926)). Kronig and Manneback summed over 
A so that their publications do not show explicitly that each value of A contributes the 
same susceptibility. 



186 QUANTUM-MECHANICAL DERIVATION OF VII, 45 

a large and sensibly continuous range of values like j at ordinary 
temperatures, or to represent the motion of a diatomic molecule not in 
a 2 state, in which A is regarded as a fixed electronic quantum number. 
Induced Polarization. We have so far considered only the part of the 
susceptibility arising from the permanent dipoles. It is clear that the 
contribution of the induced polarization (p. 29) should also be con- 
sidered. If we adopt the common but rather cowardly artifice of 
attributing the induced polarization to a set of isotropic harmonic 
oscillators, the calculation is, as usual, particularly simple. The wave 
equation for a harmonically bound particle of charge e t and mass m t 
subject to an impressed field E along the z direction is 



If we make the substitutions z' z --e f E/a i9 W W + Jef/? 2 /^ this 
wave equation is of exactly the same form as in the absence of the 
field. Hence the characteristic values in the field differ from those in 
its absence by an amount lefEja^ so that 6 \V^=^ 0, W<= e < f/2a i , 
WW 0, (k 3,4,...). By Eq. (1) the polarization of one oscillator is 
EeJla t . To obtain the total polarization per unit-volume we must sum 
over all the oscillators in the molecule and multiply by the number of 
molecules per c.c. Weighting of the various states in accordance with 
the Boltzmann factor is unnecessary because the polarization of the 
oscillators has turned out not to involve the vibrational quantum 

numbers. If we set = 2 ej/a it and if we assume that the induced 

/ 

polarization from the oscillators can be superposed additively on 
the polarization previously calculated for the permanent dipoles, we 
have the full Langevin-Debye formula 



46. General Derivation of the Langevin-Debye Formula 7 

The models used in the preceding section to obtain Eq. (10) arc clearly 
too special, especially the ascription of the induced polarization to 

6 Ono can also verify as a nice easy oxainplo of perturbation theory that Eqs. (24) 
and (26), Chap. VT, when specialized to tho harmonic oscillator, give those same expres- 
sions for W (l) or ir (2) . This is done in Condon and Morse, Quantum Mechanics, p. 122. 
These writers also give and solve the wave Eq. (9) as ahove. The displacement ^E z la t 
in energy caused by the field E is, inciclently, exactly the total internal and external 
potential energy in the classical theory at the position r ~ CtEjat of static equilibrium 
between tho electric field and the restoring forco. 

7 This method was given by J. 11. Van Vleck, Phys. Rev. 29, 727 (1927). 




VIT, 46 THE LANGEVIN. DEBYE FORMULA 187 

harmonic oscillators, which involves the same kind of crude over- 
simplification as the first or preliminary classical treatment given in 
11. In the present section we shall therefore give a general derivation 
of the asymptotic validity of the Langevin- Debye formula at high 
temperatures, which frees us from the necessity of using special models. 
Thus in dealing with the polarization due to the permanent dipoles, 
we are no longer compelled to assume that the molecule has two equal 
moments of inertia, as in the symmetrical top model. Also, especially, 

the induced polarization can arise from real elec- 

tronic motions rather than from harmonic oscillators. - - ^i 

The vibrations of the nuclei arc, to be sure, very n n 

approximately simple harmonic, but have been 
shown in 15 to give only a very small portion of 
the total induced polarization, which is mainly elec- 
tronic. The general derivation is really simpler than 
the special ones, and is illuminating in that it shows 
generally under what conditions departures from the | 
Langevin-Debye formula should be expected, and 
hence what is the meaning of such departures when 
observed experimentally (cf. 47). IM(J * ' 

The only two assumptions which it is necessary to make in the general 
demonstration are that the atom or molecule has a permanent moment, 
and that the moment matrix involves only frequencies which are 'low' 
or 'high' compared to kT/k. What is meant by the latter terminology 
may be explained more fully as follows. We shall classify a state as 
'normal' if its Boltzmann distribution factor er u'l kT is appreciably 
different from zero, i.e. if its excess of energy over the very lowest state 
is either smaller than or comparable with kT. An 'excited' state is one 
which has such a small Boltzmann factor that its probability of being 
occupied is negligible, and whose energy thus exceeds the energies of 
the normal states by an amount large compared to kT. An energy -level 
diagram illustrating graphically the delineation into normal and excited 
states is given in Fig. 7. In order for the Langcvm-Debye formula to 
be valid, it is vital for the electrical moment to involve no 'medium 
frequency' elements, which involve energy changes of the same order 
of magnitude as kT. Thus here and throughout the remainder of the 
volume, the equipartition allowance kT of energy enters as the unit for 
determining whether an energy change is 'large' or 'small' for our pur- 
poses, or in other words, whether a frequency is 'high' or 'low'. It is 
essential that the spacing between consecutive normal states or energy- 



188 QUANTUM-MECHANICAL DERIVATION OF VII, 46 

levels be small compared to kT. In Fig. 7 an interval such as b-c must 
be much less than kT. It is not necessary to demand that the energy- 
difference between two widely separated normal states, such as Or-c in 
Fig. 7, be small compared to kT, as ordinarily there will be selection 
principles which require that the matrix elements connecting two nor- 
mal states be zero, or at least very small, unless the two states are 
adjacent, or nearly so (cf . the familiar selection rule Aj 0, 1 for the 
inner quantum number, as a special example). It is clear that it is 
impossible to require that the energy-differences of two widely separated 
normal levels such as a-c be small compared to kT, as the equipartition 
theorem demands that at high temperatures the average excess of rota- 
tional energy over the very lowest state be kT itself. At very low 
temperatures the 'unit' kT will become much smaller, and the separa- 
tion between adjacent normal states will become comparable with kT. 
Special calculations, depending on the model, must then be made, which 
yield a more complicated variation with temperature than that given 
by the Langeviii-Debye formula. An example of a case where such 
calculations are reqiiired will be encountered in 67 in connexion with 
the magnetic susceptibility of NO. Dielectric constants do not appear 
to have been measured at low enough temperatures to make the correc- 
tions to the Langevin-Debye formula appreciable, as will be discussed 
more fully in 47. 

The various normal levels usually result from giving the molecules 
different amounts of 'temperature' rotation about their centres of 
gravity, different orientations relative to the external fields, or from 
allowing the spin axis of the electron to assume different orientations 
relative to the rest of the system. Hence the frequencies v = (W l W 2 )/h 
associated with transitions between any two normal energy-levels W 19 
W 2 are connected by the correspondence principle with classical fre- 
quencies of rotation or precession. Thus another way of stating the 
fundamental assumptions is that the molecule possesses a permanent 
moment which has precession and rotation frequencies all small com- 
pared to kT/h. It is to be understood that such terms as precession 
are not to be taken too literally in quantum mechanics, as the atom 
has no ordinary space-time geometry. It is immaterial for the proof 
how many frequencies are superposed, i.e. how complicated the motion. 

The low- and high-frequency elements will be found to contribute re- 
spectively the first and second terms of the Langevin-Debye formula (10). 
Thus the high-frequency elements of the moment matrix are responsible 
for the induced polarization term, while only the low-frequency elements 



VII, 46 THE LANCEVIN-DEBYE FORMULA 189 

contribute to the permanent dipole term which is inversely proportional 
to temperature. The hypothesis of a permanent dipole moment means 
that the square of the low-frequency part of the moment matrix has 
the same value /x 2 for all the normal states. This is not at all a drastic 
assumption, as it is involved in all permanent dipole theories, and with- 
out it the expression //, in (10) would have no meaning. In non-polar 
molecules p, may be regarded as having the special value 0, and then 
the moment matrix will contain exclusively elements of the high- 
frequency type. 

Let p = ] e i r t be the vector moment matrix of the molecule. A typi- 
cal element (exclusive of the exponential time factor) of its component 
in the direction of the applied field may be denoted by p E (njm\rij'm'). 
Such an element is associated with a transition from a state specified 
by indices n, j, m to one by ri ', j', m' . We shall let the first of the three 
indices be identified with quantum numbers (e.g. 'electronic' and 
'vibrational') which have an effect on the energy large compared to 
IcT, so that one particular value of this index gives states of especially 
low energy. This value will be denoted by n, and yields the normal 
levels of the atom or molecule. The second index j or j' corresponds to 
quantum numbers (e.g. 'inner', 'rotational', 'spin') whose effect on the 
energy is comparable with or smaller than kT. We do not, however, 
include in the second index the 'axial' (also called 'equatorial' or 
'magnetic') quantum number which specifies the spacial orientation by 
quantizing angular momentum about some fixed direction in space. 
Instead, the third index m or in' is to be considered as representing the 
axial quantum number. Thus the various component levels of the 
normal state correspond to fixed n but different values of j and m, 
whereas the excited states have an index n' different from n. It is 
clearly to be understood that each index, except the third, in general 
symbolizes more than one quantum number. Hence we designate n, j, m 
as 'indices' rather than quantum numbers. Our proof is thus by no means 
confined to systems with three quantum numbers or degrees of freedom. 

If we substitute in Eq. (4) the formulae for W (l \ JF (2) given in Eq. (49) 
of Chapter VI, generalized to three rather than two indices (or what is 
equivalent, substitute in (1) the formula for the perturbed moment 
given in (50) of Chapter VI and expand as in 44), the formula for the 
susceptibility becomes 



v^ _ e-ir^kT /in 

"M 1 " 2*^ Wnjmin'j'm 7 ) J ' ( ' 



190 QUANTUM-MECHANICAL DERIVATION OF VII, 46 

with the abbreviation - 



In (11), as in 34, the prime means that the state ri = n,j' j, m' ~ m 
is to be excluded from the inner summation. 

Eq. (11) is a perfectly general expression for the susceptibility not 
requiring either of the two fundamental hypotheses of a permanent 
moment, and of the existence of solely 'low'- and 'high' -frequency 
elements in the moment matrix. It is perhaps well to restate these two 
hypotheses in equation form. The latter of them is that 

\hv(njm' 9 n'j'm') \ > kT (ri -- n), \ 

and that p E (njm-, nj'm') = unless \hv(njm\ nj'm') \ <<\ kT .} 

To exhibit most explicitly the significance of the other hypothesis of 
a permanent moment, it will be convenient to use a distinctive notation 
for the unperturbed matrix elements of the low-frequency part of p. 
We shall therefore use |* to denote the matrix formed from p by 
dropping the high-frequency elements ri ^n associated with transi- 
tions to excited states, so that 



It is unnecessary to use an index n or ri in JLA, as it is formed from those 
elements of p in which the first index has the same value n in both the 
initial and final states. The matrix /z is thus just a small square out of 
the larger matrix p, and is what Born and Jordan call a 'Toil matrix'. 
The assumption of a permanent moment means that the scalar magni- 
tude of the vector matrix p. is constant with respect to time, and the 
same for the vaiious normal states. The vector |x will usually not be 
constant in direction. Hence the individual Cartesian components JJL X , 
/V Pa w iU vary with time, having 'low '-frequency elements arising from 
the various precessions and rotations. These time variations, however, 
drop out of the scalar magnitude /z which is independent of j, m, and 

thUS I 2^O;/m'K 7 (jW;jV) = 3( j -m;j"m")^ j (15) 

q=x,u,s j'.m' 

with 8 as on p. 132. In other words, p. 2 is assumed to be a diagonal 
matrix whose elements are all equal, and in the terminology of Dirac 
it would be called a { c-iiumber', which is the square of the moment p, 
entering in Eq. (10). 
In terms of the distinctive notation introduced for the low -frequency 



VII, 46 THE LANGEVIN-DEBYE FORMULA 191 

elements, Eq. (II) becomes 



j.m 



^-v/ 



+213 V 

-^w 



hvin'j'm' ; 

^ J ' 

Here the first two and third lines represent respectively the contribu- 
tions of the low- and high-frequency elements. To bring out the fact 
that the third line is inherently positive, we have here introduced the 
positive or emission frequencies liv(n'j'm' ; njm) in place of the negative 
or absorption frequencies hv(njm\n'j'm'). In (16) we have written 
\l*>E(jm\jm)\* for \i*> E (jin\jm}\* 9 which is legitimate since the diagonal 
elements of Hermitian matrices are real. 

The terms in the summation in the second line of (16) may be grouped 
together in pairs of the form 



P = - 

12 iwtti;w w a W^WW-I 

(17) 

Now pu(J2 m '2>JLM'i) * s tne conjugate of ^(jtfn^jtfn^, and so has the same 
absolute magnitude. Also the denominator of the second term of (17) 
is the negative of that of the first term. We next make the substitution 
W% jtmit = JFS! JlMll +M w J2 w V 7 yi m i) in tuc second exponential of (17) and 
develop this exponential as a power series in the ratio 



Then (17) becomes 



If the fundamental hypothesis (13) is valid, w will be a very small 
quantity, and we may without serious error neglect terms of the order 
w* and beyond in the bracketed factor of (19). With this approximation, 
(19) is the same as 

PI* = j |,[l/0>i;wi)l'-'*' a -'* 7 '+ I/*B(JV.; ^iJI'e-"'^^]- ( 20 ) 

It may be objected by some readers that by giving the molecule suffi- 
cient quanta of rotation, the ratio (18) may be made as large as we 



192 QUANTUM-MECHANICAL DERIVATION OF VII, 46 

please, as the separation of consecutive energy-levels for the simple 
dumb-bell model by Eq. (57), Chap. VI, increases linearly with j. In 
other words, the spacing of the normal levels is not uniform as in Fig. 7, 
but becomes steadily greater as we go to higher normal states. For- 
tunately, however, this consideration gives no trouble, for the numerical 
magnitude of the exponent in the Boltzmann distribution factor er w \ kT 
increases much more rapidly than w. (One varies approximately as the 
square, the other as the first power of the rotational quantum number.) 
Hence terms for which w is comparable with unity will have such a 
small exponential factor or probability that they can be disregarded. 
As still further assurance that the higher powers of w can be discarded 
under the hypothesis (13), we shall give in 47 the quantitative correc- 
tion to the Langevin-Debye formula which results when the develop- 
ment is broken off after w* rather than w 2 . This correction proves to 
be very small if the fundamental hypothesis (13) is valid. 
When we utilize (20), Eq. (16) becomes 



B 
: kT 



i 



V 
^ 



Jiv(n'\n) 

,,,- 

where now the first sum includes the diagonal elements njm\ njm. 
The first two lines of (16) have been melted into a single line in (21), 
as the lirst line of (16) supplies just the diagonal elements wanting from 
its second after the simplification (20). 

In writing (21) we have introduced two simplifications. Firstly, in 
the second line we have replaced v(rij'm r \njrn) by a number v(n'\ri) 
independent of the indices j, m, j f , m' . This is clearly allowable as the 
separations between the various normal states are by hypothesis small 
compared to the interval between normal and excited states, and hence 
the 'high' frequencies v(ri\ri) (n* ^n) are affected but little by j 9 m, 
j', m'. Secondly, we have replaced W^ m by an expression W%j which 
is independent of m. This presupposes that the influence of orientation, 
i.e. of the axial quantum number m on the unperturbed energy W%j m , 
is small compared to kT, a condition which is certainly fulfilled with 
a high degree of precision in gases. In fact the ideal case ordinarily 
considered is that in which the molecule is subject to no external field 
except E, and then the unperturbed energy (i.e. the energy in the 
absence of E) is independent of orientation, so that the index m has 
absolutely no effect on W. To allow for the possibility of simultaneous 



VII, 46 THE LANGEVIN-DEBYE FORMULA 193 

electric and magnetic fields, or inter-molecular fields in liquids and 
solids, we admit the more general assumption that W% jm W Q ni is not 
identically zero but small compared to kT. Thus our derivation of the 
Langevin-Debye formula is applicable to solids provided, of course, the 
two fundamental hypotheses given on p. 187 are fulfilled, and provided 
the effect of orientation is small compared to kT. In other words, the 
1 turning over' of an atom or molecule against the inter-molecular field 
must require the expenditure of an amount of work considerably less 
than kT. This condition is really already embodied in our second 
fundamental hypothesis, as the requirement \W^ jm W^ jm '\<^kT is a 
special case of (13). 

It was proved in 35 that in virtue of the high degree of spectroscopic 
stability characteristic of the new quantum mechanics, an expression 

of the form _., , . . . ,., , x|l , 

2 \A z (njm',nj'm')\- 

m,m' 

is invariant of the direction of the axis of quantization, and equals 
2 |^(r^ra;w'j'm')| 2 , where \A(njm\rij'm')\*= 2 \A q (njm\rij'm'}\*. 

' 



m, 



This consequence of spectroscopic stability is vital, as it underlies the 
general occurrence of the factor J in the temperature term of the 
Langevin-Debye formula. Let us suppose that the field E is along 
the z axis. Then by taking A^p, (n = ri) and A=p Q (n^ri), one 
sees that (21) may be written 






2 

if 

e-irl/fcr (2 2) 



hv(ri ; n) 

,m.,j,/n) v ' 

Simplification of Low-frequency Elements. From the rule for matrix 
multiplication (Eq. (9), Chap. VI) it follows that 

2 



Here the right-hand side is a diagonal element of the matrix /x 2 , which 
is the square of the absolute magnitude of the vector matrix H formed 
from the complete moment p by deleting the high-frequency elements. 
The index m is not needed on the right side of (23) because the 
magnitude of a vector is independent of its spacial orientation. 

Now if the hypothesis of a permanent moment is valid, we may apply 

3595.3 Q 



194 QUANTUM-MECHANICAL DERIVATION OF VII, 46 

Eq. (15), which makes (23) independent of j, and the first line of (22) 
becomes T> 

^ V a-"***. (24) 

3kT Z ^ 

j,m 

Now we have already supposed that W^ jm can be replaced by W^j in 
the exponential factors, and so the sum in (24) is identical with the 
denominator of (12). Hence (24) is simply Np 2 /3kT, which is the 
'temperature part' of the Langevin-Debye formula (10). It is clear that 
if the hypothesis of a permanent moment were not valid, the first line 
of (22) would become instead 



_ ... __ _____ - (25) 

ZkT e-i^njlkr ' l ; 



=- 
Here ^ denotes the statistical mean square of the moment /z, i.e. the 

time average ju, 2 (j; j) of ^ for a given state j, m, with this average in 
turn averaged over all the normal states weighted in accordance with 
the Boltzmann factor. Eq. (25) represents a sort of generalized Langevin 
formula, somewhat analogous to the generalized classical expression 
which we derived in (22), Chap. IT. 

Simplification of High-frequency Elements. The important thing about 
the second line of (22) is that for given N it is independent of tem- 
perature and so may be denoted by a constant Not, as in Eq. (10). The 
demonstration is an easy consequence of the 'sum-rules' 8 for intensities, 
applied to absorption rather than emission, for it is the essence of these 
rules that an expression of the form 

'j'm')\* (26) 



is independent of the indices j and m. The sum-rules were first estab- 
lished on semi-empirical grounds, but the work of Born, Heisenberg, 
and Jordan, and of Dirac shows that they are required by the quantum 
mechanics, 9 provided j is associated with the one type of precession 

8 For references and description of the sum-rule see Pauling and Goudsmit, The 
Structure of Line Spectra, p. 137. The ordinary statement of the sum-rule for the inner 
quantum number is that the sum of the intensities of the multiplet components which 
have a common initial (or else common final) state j is proportional to its a priori 
probability gj. This statement, however, presupposes a summation over m inasmuch as 
the Zeeman components are assumed unresolved. The sum-rule for the magnetic quan- 
tum number shows that all the g f components of the state j contribute equally to the 
sum over m. Consequently when, as in (26), we do not sum over m, but only over m', 
the factor gj cancels out. 

Born, Heisenberg, and Jordan, Zeits. f. Physik, 35, 605 (1926). P. A. M. Dirac, 
Proc. Roy. Soc. Ill A, 281 (1926). The former deduce the formulae of Goudsmit and 
Kronig for the intensities of Zeeman components. Dirac proves the more difficult formu- 



VII, 46 THE LANGEVIN-DEBYE FORMULA 195 

ordinarily identified with the inner (or rotational) quantum number. 
Actually we have already stated that the index j may correspond to 
several quantum numbers, and hence represent the effect of several 
superposed precessions: e.g. simultaneous precessions resulting from 
interaction with electron spin and from molecular rotation. However, 
Dirac 9 notes that there is no difficulty in extending the proof of the 
intensity- or sum-rules to systems that are composed of any number 
of parts, and that so contain any number of precessions, provided the 
parts are coupled together by 'secular' forces which do not distort 
the motion but which instead give rise only to pure precession. This 
result is also obvious from the correspondence principle, inasmuch as 
the sum-rule is the quantum analogue of the fact that classically the 
intensity of radiation is not appreciably affected by precessions which 
do not sensibly alter the sizes and shapes of the orbits. This rules out 
centrifugal expansion and similar effects, but their effect is only sub- 
ordinate, as we shall see more fully in 47. The observed existence of 
a term in the dielectric constant which is independent of temperature, 
also especially the allied independence of the index of refraction of 
temperature 10 , discussed in 16, must be regarded as indirect but 
nevertheless very good experimental evidence for the validity of the 
sum-rules. 

From what has been said in the preceding paragraph we may replace 
(26) by an expression |#(n;fl/)| a independent of j and m, and so the 
second line of (22) reduces to 



n'(n'tti) 

The double sum in this equation is, as before, the same as the denomina- 
tor of (12). Thus (27) becomes an expression 

i^'Oi 2 , 2 

*<':> 

which is independent of T and which constitutes the 'constant' part 
of the Langevin-Debye formula. Combination of the simplifications 
affected in the high- and low-frequency parts of (22) yields the complete 
Langevin-Debye formula (10). 

lae of Kronig, Russell, Sommerfeld ami Honl (noto 52 , Chap. VI) for intensities in 
multiplots. These formulae contain more information than the sum-rules, but necessarily 
demand the validity of the latter. 

10 The proof that refractivities in the optical region are independent of temperature 
is similar to that of Eq. (28). Cf. Phys. Rev. 30, 41 (1927). 

02 



196 QUANTUM-MECHANICAL DERIVATION OF VII, 46 

Review. Now that the proof is over, it is perhaps well to caution 
against the misconception prevalent among many physicists that the 
reapportionment of molecules among the different stationary states due 
to alteration of the Boltzmann factors by the applied field is responsible 
for the term of the Langevin-Debye formula which is inversely propor- 
tional to T, and that the distortion of the motion within a stationary 
state gives rise to the term independent of T. Actually the distortion 
usually contributes to both terms, if we use the word * distortion' in the 
sense of any change in the motion produced by the field, such as, for 
instance, alteration in the end-over-end rotation. (Alteration in the 
internal structure of the molecule, such as stretching of the orbits, &c., 
as distinct from changes in the motion of the molecule as a whole, or 
perhaps its coupling to the spin, does, to be sure, give a term indepen- 
dent of T.) In fact, in the dumb-bell model ( 45) it was not necessary 
to consider the reapportionment at all, i.e. it was adequate to take 
W = W Q in the Boltzmann factors, as the first-order Stark effect W (l) 
vanished. (Note that in 44 we showed that to obtain the portion of 
the susceptibility independent of field strength, the energy is needed 
to the approximation W (2) in connexion with the moment factors 
(1) involved in (3), but need be carried only to the approximation W (l) 
in the Boltzmann factors. ) It may seem rather surprising that distortion 
without reapportionment can give rise to a term inversely proportional 
to T, but the situation is roughly the following. The 'low-frequency part' 
of the distortion of the moment by the applied field may bo enormous 
because of the 'low frequencies' v(njm;nj'm') in the denominator, but the 
different normal states would just compensate each other as regards 
this part, except for their differences in the value of the unperturbed 
Boltzmann factor, which after series expansion are found to have the 
effect of introducing k T in the denominator in place of hv (cf. Eq. (19) ). 
Only in the special case of the magnetic susceptibilities of atoms without 
spin or else with very wide multiplet structures does the l/T term arise 
entirely from W ( ^ without the aid of the low-frequency part of W ( 
(cf. Eq. (13), Chap. IX). The popular misconception has doubtless 
arisen as an incorrect generalization of this special case. The high- 
frequency part of W^, of course, gives the term independent of T. 

Case of a Simultaneous Magnetic Field. The expression (28), also of 
course the permanent moment //,, does not involve the index m, or the 
direction of the axis of quantization, and so the choice of this axis 
cannot influence either term of the Langevin-Debye formula. Hence, 
unlike the old quantum theory (31), a magnetic field cannot distort 



VII, 46 THE LANGEVIN-DEBYE FORMULA 197 

the dielectric constant merely by changing the direction of the axis of 
quantization. Instead a magnetic field will influence the dielectric con- 
stant only through higher-order terms, which we have neglected and 
which are analogous to saturation effects. Because of the spectroscopic 
stability characteristic of the true quantum mechanics, these small 
terms will be proportional to H 2 , and of the same order of magnitude 
as we calculated in 31 with classical theory. 

Special Case that E is the only External Field. In case other external 
fields, such as, for instance, the magnetic one just considered, are absent, 
the axis of quantization will coincide with the direction of E. The third 
index m will then have no influence on the energy in the absence of E, 
and consequently all frequencies of the form v(njm\njm') will vanish, 
as it is understood throughout that the frequencies v appearing in the 
denominators are to be calculated for the unperturbed system. Never- 
theless, there will be no trouble with zero denominators, 11 inasmuch 
as the matrices p K or p, E will contain no elements in which w^ra', 
since the component of moment in the direction of E clearly cannot 
involve the frequency of precession about the direction of A 7 , which is 
the frequency associated with the quantum number m. Thus the 
summation over m' may be replaced by the substitution m f m. 

47. Limit of Accuracy of the Langevin-Debye Formula 

It is clear that if the temperature is diminished sufficiently, the separa- 
tion of the normal levels cannot continue to remain small compared to 
kT. Hence at low temperatures there should be appreciable departures 
from the Langevin-Debye formula, which is, strictly speaking, in quan- 
tum mechanics only an asymptotic formula valid at high temperatures. 
This has already been emphasized in 4.5 in connexion with the 'dumb- 
bell' model. It was shown that for this model, Eq. (8) is a rigorous 
expression (neglecting saturation) for the susceptibility, valid right 
down to T = 0, whereas the Langevin-Debye formula is not. One could 
form an idea of the range of temperatures over which the latter formula 
is substantially valid with this model by determining the critical tem- 
perature at which it begins to depart appreciably from Eq. (8). We 
shall not do this, but instead shall give an approximate correction for 
the departures from the Langevin-Debye formula with a more general 

11 Conceivably there might bo degeneracies other than the apacial one. and which give 
rise to zero denominators. There is, however, 110 difficulty, as the degeneracy may be 
removed by applying a hypothetical infinitesmal field which removes the degeneracy, 
and the theorem of spectroscopie stability shows that the results are invariant of the way 
the degeneracy is removed. 



198 QUANTUM-MECHANICAL DERIVATION OF VII, 47 

model of a molecule, with three moments of inertia none of which are 
necessarily equal. This correction consists in determining the effect of 
the retention of terms to w* inclusive rather than to w 2 as previously 
in the series expansion made in the bracketed part of Eq. (19). A 
rigorous formula would involve the retention of all terms in this expan- 
sion, but such a formula would be intractable and umlluminating except 
for simplified models such as the dumb-bell or symmetrical top. The 
temperatures at which the terms in w 3 , w* begin to appreciably modify 
the susceptibility are evidently the temperatures at which we may begin 
to expect appreciable corrections to the Langevin-Debye formula. We 
shall not give the details 12 of the calculation of this second approxima- 
tion to the susceptibility by including the effect of w 3 , w 4 , but shall 
merely state the result, viz. 



(29) 
where 

(30) 



Here A, B, C denote the moments of inertia of the molecule about its 
three principal axes x', y', z', and p^, /^, ^ are the components of the 
permanent moment //, along these axes. For either the 'dumb-bell' or 
symmetrical top models, i.e. for either diatomic or symmetrical poly- 
atomic molecules, we may take ^ = /z, ^ = /^ = 0, A = B = 7, and 
(30) then reduces to 

l-32xl(H 

( ' 



IT ' 

The factor lf(T) evidently enters in (31) as a correction to the 
Langevin-Debye formula. To obtain a numerical estimate of its im- 
portance let us consider the particular case of HOI. Substitution in 
(31) of the value 7= 2-65 x 10- 40 gm. cm. 2 appropriate to HOI yields 
/= 5-0/T, so that at room temperatures f(T) is only 0-01G. To make 
the correction as great as 5 per cent, it would be necessary to reduce the 
temperature to 100 K., or below the freezing-point of HC1, which is 
obviously not feasible. Gases which have low boiling-points, and which 
hence might be measured when the denominator of (31) is small, are 
invariably non-polar (e.g. 2 , H 2 ) or nearly so. 00 and NO are two 

12 See J. H. Van Vleck, Phys. Rev. 30, 46 (1927). The correction (31) appropriate to 
diatomic and symmetrical molecules has also been deduced by Kronig and by Manne- 
back (Z.c. 5 , also ref. 21 of Chap. VI) by a different method than ours. In the case of the 
dumb-bell model their method consists in finding a somewhat bettor approximation to 
the denominator of (8) than merely replacing the sum by an integral as on p. 185. An 
analogous method is also used for the symmetrical top. 



VII, 47 THE LANGEVIN-DEBYE FORMULA 199 

common slightly polar gases which have relatively low boiling-points, 
but this feature is more than offset by their moments of inertia being 
over five times that of HC1, so that the values of f(T) obtainable 
experimentally for them are even smaller than for HC1. 

These numerical considerations show that the correction f(T) to the 
Langevin-Debye formula is ordinarily far too small to be of conse- 
quence, and experiments would have to be very sensitive to detect the 
departures from linearity which it occasions in the graph of x against 
I IT at constant density. The main value of the calculation of the 
second approximation involving /(T) is hence primarily to reassure us 
that in the case of dielectric constants the asymptotic agreement of 
classical and quantum theories is nearly completed at any temperatures 
ordinarily obtainable (barring possible 'internal rotations' in organic 
molecules discussed on p. 76). (In magnetism, because of the spin 
multiplets, 13 we shall later see that there may be important deviations 
from the classical Langevin law even at room temperature.) The cor- 
rection factor lf(T) docs modify slightly the experimental values of 
the electric moment determined by the method I which was described 
in 19. In this method p, is ordinarily calculated from the temperature 
coefficient of x * n the vicinity of a room temperature T , and hence 
with the correction /z 2 must be increased by a factor about l + 2/(2J,) 
to give this coefficient the same value as previously. Correspondingly 
the effect of this correction is to increase the contribution of permanent 
dipoles to the susceptibility by a factor [l+2/(T )][l-/(T )]~ 1+/(T ) 
and hence to diminish Not, as determined by method I, by an amount 
Afyi 2 /(T )/3fcT . In case most of the susceptibility arises from the per- 
manent polarity, this diminution in a may reduce slightly the dis- 
crepancy between the values of No. determined by methods I and II 
(respectively temperature variation and extrapolation of optical data), 
discussed at length in 14, 15, 19. In HC1, for instance, the correction 
raises the value of p with Zahn's temperature data from 1-034 x 10~ 18 
to 1-050X 10~ 18 , and correspondingly diminishes ^irNoL from 0*00104 to 
0-00099, in somewhat better accord with the optical value 0-000871 
furnished by method II. 

13 Spin multiplets comparable with kT may exist in the present electrical case, but 
do no harm. This is primarily because the electrical moment has no matrix components 
between states which differ solely as regards the alinoment of the spin relative to the 
rest of the molecule, thus avoiding the complication of 'medium frequency elements'. 
Although the multiplot structure thus does not influence the electric susceptibility, it 
docs modify the Stark effect, as the latter involves the stationary states individually 
rather than collectively in summations. 



200 QUANTUM-MECHANICAL DERIVATION OF VII, 47 

Correction for Centrifugal Expansion. Because of centrifugal force the moment \L 
is never rigorously * permanent', i.e. the same for all rotational states as supposed 
in 46, but increases slightly with the rotational quantum number and hence with 
T. Wo may calculate quantitatively the correction resulting from this effect by 
replacing the first term of the Langevin-Dobyo formula by the more general 
expression (25), which does not assume a permanent moment. To evaluate y? let 
us take a diatomic molecule whose nuclear separation, moment of inertia, and 
electrical moment are respectively r , J , and p, when the molecule is at rest, 
completely devoid of rotation or vibration. If now the molecule rotates with an 
angular velocity co, the centrifugal force Mra> z must equal the restoring force 
47T 2 v 2 M(r r ). Here v is the frequency of vibration, supposed simple harmonic. 
A small expansion of the molecule by an amount r r increases the electrical 
moment by ^(i" r ), where e ett is by definition the 'effective charge' (see p. 47). 
Conseq i ion tly we have approximately ___ 



since classically the mean of the rotational kinetic energy J/co 2 is kT, and quan- 
tum mechanically this value is valid at ordinary temperatures, as they are high 
enough to make rotational specific heats have substantially the classical equi- 
partition magnitude. The effect of centrifugal expansion is thus only a contribution 
jLt e eft r /37T 2 /v 2 to the constant a in Eq. ( 10). This contribution is, unlike the correc- 
tion (31) previously considered, as important at high as at very low temperatures, 
but is usually quite negligible. In HC1, for instance, if we use Bourgin's value 
( 15) of the effective charge, the centrifugal expansion gives rise to only 1 per cent. 
of a, and so may be neglected without appreciable error. 

Correction for Vibrational Distortion. By supposing a permanent moment we 
have assumed that the electrical moment is independent of the vibrational quan- 
tum number or else that all vibrational states but the lowest have negligible 
Boltzmann factors. The latter assumption is sufficiently warranted in ordinary 
stable diatomic molecules such as HC-1 at ordinary temperatures, but compli- 
cated polyatomic molecules may have some of their vibrational degrees of freedom 
less firmly bound, and so sometimes bo in higher vibrational states where the 
moment of inertia and hence the electrical moment is appreciably different from 
in the normal state. This effect has been considered in considerable detail by 
Zahn. 14 The correction can again be calculated approximately by replacing the 
first term of ( 10) by (25). For stable diatomic molecules such as HC1 it is readily seen 
to be of the same order of magnitude as that for rotational distortion and likewise 
independent of the temperature. In complicated polyatomic molecules where the 
vibrations are of large amplitude and not simple harmonic, this vibrational correc- 
tion may be more important and yield a complicated temperature dependence. 
We have already mentioned on p. 76 that in molecules possessing pliable bonds 
there will be departures from this formula if the radicals are only partially free to 
turn. This is closely akin to the effect considered by Zahn, the difference being that 
the non-rigidity is due to twisting rather than stretching. 

Correction for Saturation. By neglecting terms beyond E 2 in Eq. (2) we have 
disregarded all saturation effects. The effect of including terms through E* in (2) 

^* C. T. Zahn, Phys. Rev. 35, 1047, 1056 (1930). His Eq. (10) is essentially the same as 
(25). The departure from the Debyo formula which he finds experimentally in acetic 
acid is probably due to molecular association; ef. Phys. Rev. 37, 1516 (1931). 



VII, 47 THE LANGEVIN-DEBYE FORMULA 201 

has been calculated by Niessen 15 for the general quantum dynamical system, with 
the two same fundamental hypotheses as in 46. His results, especially the mode 
of temperature dependence have already been discussed in 22. The numerical 
estimates there given show that the saturation effects are very small at the largest 
field strengths E yet obtainable experimentally, but have nevertheless been 
detected by Herweg and others (22). We may mention that if one considers only 
the permanent dipole moment, or in other words retains only the low-frequency 
part of the total moment p, then Niesson has shown that it is possible to calculate 
asymptotically the effect of all powers of E in Eq. (2). lie finds that then the com- 
plete Langevin formula NL^E/JcT) (Eq. 4, Chap. II) for the polarization is valid, 
but it may be cautioned that this result applies only whoii the correction for 
saturation is larger than, or at least of greater interest than, corrections such 
as (30) for the finite magnitude of rotational energy intervals. In other words 
saturation is the main correction if fiE !>A'T' and kT'<^kT, where T r is the 
molecule's ' characteristic temperature ' h z /8ir*Ik. Existing experiments on electric 
saturation are not made at fields so large that fiE ^> kT', although made at 
temperatures for which T f <f, T. Under those conditions one can still use Niesseii's 
value of the correction for satiiration, as the corrections for finite intervals and for 
saturation are approximately additive if neither of them is great. 

Comparison with Experiment. As the quantum mechanics has re- 
stored the Langevin formula under ordinary conditions, the comparison 
with experiment, and deduction of numerical dipole values from the 
latter, proceeds as in classical theory, and so the material given in 
Chapter TIT is still applicable. The discussion of the effect of infra-red 
vibrations proceeds exactly as in 15. We have seen on pp. 30 and 186 
that the classical and quantum theories give exactly the same value 
2 e j/ tt i f r the polarizability a of a system of isotropic harmonic oscil- 
lators. The identity of results is also readily established 16 in the more 
general case of periodic rather than static impressed fields, and of one- 
dimensional harmonic oscillations along the figure axis instead of 
isotropic ones. Spectroscopic stability shows that the factor J arises 
from the spacial orientation, regardless of the type of spacial quan- 
tization or molecular rotation. 17 Thus (-4), Chap. Ill, which was 

15 K. F. Niesson, Phys. Rev. 34, 253 (1929). 

18 See Phys. Rev. 30, 44 (1927). 

17 Complete similarity with classical results on harmonic oscillators is an almost 
invariable characteristic of the new mechanics. On the other hand, the ^inharmonic 
correction' which results because the restoring forces on the nuclei are not strictly linear, 
has a different effect on the susceptibility than in classical theory. S. Bogiislawaki (Phys. 
Zfiitft. 15, 283, 1914) and K. Czukor (Vcrh. d. Dent. Phys. Get*. 17, 73 (1916) showed 
that classically this correction modified somewhat the nature of the temperature depen- 
dence of the dielectric constant, whereas in quantum mechanics it merely alters slightly 
the magnitude of a, inasmuch as (28) is a perfectly general expression for the contribu- 
tion of high-frequency elements. Such a divergence from classical results is, of course, 
only possible because the vibrational energy intervals, as distinct from the rotational, 
are usually large compared to kT in stable diatomic molecules. The vibrational intervals 



202 DERIVATION OF THE LANGEVIN-DEBYE FORMULA VII, 47 

our focal point for discussing infra-red vibrations, still retains its 
validity. 

cease to be largo compared to kT when there is an appreciable vibrational specific heat. 
The correction which then results to the Langevin-Debye formula is, however, very 
slight, as the nuclear vibrations are usually very nearly simple harmonic, and solution 
of Eq. (9) has shown that harmonic oscillators have exactly the same polarizability in 
all stationary states. The form of temperature dependence is altered only by super- 
position of the anharmonic correction and that for the excitation of higher vibrational 
states than the lowest. 



VIII 

THE DIELECTRIC CONSTANTS AND DIAMAGNETIC 
SUSCEPTIBILITIES OF ATOMS AND MONATOMIC IONS 

IT may seem strange that we mix electric and diamagnetic suscepti- 
bilities in the same chapter, but in the monatomic case it is convenient 
to discuss them together because of parallelism in the rigorous theory 
for hydrogen and in the adaptation to other atoms by the method of 
screening constants. 

48. The Dielectric Constant of Atomic Hydrogen and Helium 

As the electrons are on the time average symmetrically located with 
respect to any plane containing the nucleus, atoms and monatomic ions 
have no permanent moments, and so have only the term Not of the 
Langevin-Debye formula which arises from 'high-frequency' matrix 
elements (cf. Chap. VII). It has so far been possible to determine the 
numerical magnitudes of p and a for molecules only by the experimental 
methods of 19, but on the other hand it is easy to calculate by pure 
theory the absolute value of a for monatomic hydrogen. 

The rigorous proof that atoms have no permanent moments runs as follows. 1 
In the absence of external fields, the wave equation of any atom or molecule com- 
posed of rj particles is invariant with respect to the substitution 

x = -x l9 2/;--2/,, ;=- (t=l,...,q). (A) 

Hence its solutions t/ M may always be chosen to bo either oven or odd as regards 
the substitution (A). The conjugate i/j* is even or odd like ifi n , and i/j*^ n is then 
even. If we take / = S^r/, n' = n in the fundamental quadrature (14), Chap. VI, 
its integrand ifi n /tfe t r f is odd with respect to the substitution (A) and so the 
integral is zero when taken over the entire coordinate space. Horice the unperturbed 
electric moment of any atom or molecule has no diagonal elements. To be sure, 
Eq. (1) of this chapter and (71) of Chap. VI display first-order Stark effects, 
which would seem to imply the existence of such elements, but (1) neglects the 
relativity and spin precessions, 2 and the 'symmetrical top 1 model used to obtain 
(71) is too simple to include 'A-type doubling' (see note 28 of Chap. VI and 70). 
To prove that there is no permanent moment we must show that the moment 
matrix has 110 low-free iiioncy as well as no diagonal elements. There can bo no 
low-froquency elements if the separation of all levels, other than those differing 
only as regards spacial quantization or of quantization of spin relative to the rest 

1 E. Wigner. Zcits.j. Physik, 43, 646 (1927). 

2 For calculation of the Stark effect of atomic hydrogen to a first approximation inclu- 
sive of these precessions RCO K. Schlapp, Proc. Roy. Soc. 119A, 313 (1928); V. Rojansky, 
Phys. Rev. 33, 1 (1929). Because of a fortuitous degeneracy with respect to the azimuthal 
quantum number 7, the excited states of atomic hydrogen in the new quantum mechanics 
retain a portion of their linear Stark effect even when the relativity and spin corrections 
are included. This was not true in the old quantum theory. 



204 THE DIELECTRIC CONSTANTS AND DIAMAGNETIC VIII, 48 

of the atom, are separated by intervals large compared to kT. In other words this 
condition is that there be only one 'normal electronic' level. This is ordinarily 
the case in atoms, as electronic absorption frequencies are certainly large com- 
pared to kT, but not in molecules, as there is always a sequence of closely spaced 
rotational levels. Although we have thus shown the desired result that ordinary 
atoms have no ' dipole term ' proportional to 1 /T in the Langevin-Debye formula, 
such a term would bo found for excited hydrogen atoms . Such atoms are exceptional 
because of degeneracy with respect to the azimuthal quantum number I. The 
calculation of the 1 /T term for excited hydrogen is, of course, a purely academic 
affair, 3 which we omit, as susceptibilities involve only normal states, and the normal 
state of hydrogen has only the one Z-value 0, and hence no bothersome degeneracy 
and no first-order Stark effect even in Eq. ( 1 ). When there is near degeneracy, an 
experimentally obtainable field may greatly distort the symmetry properties of the 
wave function, and so the linear Stark effects predicted by (71) of Chap. VI and 
by (1) are a good approximation to reality in strong fields. In non-hyclrogenic 
atoms the effect of I on the energy is usually large, and there is no first-order Stark 
effect, except perhaps for a few excited states of very small quantum defect. The 
kinematical meaning of this is that non-hydrogenic atoms have fast orbital 
precessions owing to the departures of the field from Coulomb character, whereas 
hydrogen atoms have only the slow relativity precession which is easily stopped 
by an applied field resulting in the alinement of the semi-major axis in this field 
and a first -order Stark effect. 4 

To determine the moment of hydrogen in a definite stationary state 
one has only to calculate to terms in E 2 the characteristic values of 
its wave equation in an electric field in other words, to compute its 
second-order Stark effect. This has been done by Wentzel, Waller, and 
Epstein 5 with neglect of relativity corrections and spin, which permits 
a separation of variables and which does no harm for the normal state 
owing to the absence of fine structure (cf. end of fine print, p. 213). 
They find 

chRZ 2 



3 For this calculation see J. H. Van Vleok, Proc. Nat. Acad. 12, 665 (1926); and 
especially revision in footnote 31 on p. 37 of Phya. Rev. 30 (1927). 

4 For amplification and references on this distinction between hydrogeuic and non- 
hydrogenic atoms see Ruark and Urey, Atoms, Molecules, and Quanta, pp. 147, 343, or 
the writer's Quantum Principles and Line Spectra, pp. 62 and 131. A wealth of theoretical 
and experimental work on the border-line case of the Stark effect of non-hydrogoiiie 
atoms, especially neutral Ho, so highly excited that the field is nearly Coulomb, has been 
performed by J. S. Foster and by Miss Dewey, Proc. Roy. Soc. 117A, 137 (1927), Phys. 
Rev. 30, 770 (1927), and references; also Y. Fujioka, Sc. Rep. Phys. Chem. Res. Tokyo, 
10,99(1929). 

5 G. Wentzel, Zeits. f. Physik, 38, 527 (1926); I. Waller, ibid. 38, 635 (1926); P. S. 
Epstein, Phys. Rev. 28, 695 (1926). 



VIII, 48 SUSCEPTIBILITIES OF ATOMS AND MONATOMIC IONS 205 

where n v n 2 , n 3 are a set of parabolic quantum numbers, such that the 
principal quantum number n is w 1 +n 2 + 1^ 3 | + 1, while n 3 quantizes the 
angular momentum about the field and so is similar to the quantum 
number m l of 39. 

The normal state of hydrogen has n = n 2 n 3 = 0, Z = 1, so that 
by (I) its moment -dWjdE is 9/& 6 #/1287r 6 e 6 m 3 == 6-63 X 10~ 25 #, with 
the usual neglect of saturation effects resulting from higher powers of 
E not included in (1). Its polarizability is thus a 6-63 X 10~ 25 . The 
corresponding numerical value of the 'molar polarizability' 47rL/3 
introduced on p. 53 is K 1-68, and of the dielectric constant at 0, 
76 cm., is e 1 '000225. Unfortunately it has not yet been possible to 
make a direct experimental confirmation of these unambiguous theoreti- 
cal values, as it would be necessary to have a gas composed entirely of 
monatomic rather than molecular hydrogen. 

The dielectric constant of neutral helium has been calculated indepen- 
dently and simultaneously by Atanasoff 6 and by Hasse. 7 Both utilize 
the fact that the perturbed wave functions of the normal state are of 
the form i/r ^ 0(r l3 r 2 , r 12 ) + E{zJ(r 1 , r 2 , r 12 ) +z 2 /(r 2 , r v r 12 )} (neglecting 
E 2 , E 3 , &c.) and determine / by the Ritz method, while they take ^ 
from the work of Hylleraas. The dependence on the coordinates can be 
shown to be necessarily of this form by extension of Wigner's methods, 
though this is not demonstrated explicitly in either paper. 8 If one uses 
Birge's 'most probable values' of the atomic constants, Atanasoff's 
solution yields e = 1-0000653 at C., 76 cm., while Basse's first calcu- 
lation gives 1-0000691 in excellent accord with the experimental value 
1-0000693. This gratifying agreement, however, turns out to be rather 
accidental, as in his second paper Hassc finds 1-000079, using a pre- 
sumably more accurate unperturbed wave function J/T O . Still more 
recently, a theoretical value 1-0000715 is reported by Slater and Kirk- 
wood, 9 also by the Ritz method. The diversity in results seems to arise 
largely because the calculations are exceedingly sensitive to the choice 
of the unperturbed wave function. 10 

6 J. V. Atanasoff, Phys. Rev. 36, 1232 (1930). 

7 H. R. Hasse, Proc. Camb. PMl. Soc. 26, 542 (1930); 27, 66 (1931). 

8 The proof consists in showing that (1) involves a rotational group ('Darstellung') 
of tho type L 1 since is of tho typo L and J^ (1) is proportional to cos 0. Properties 
of the rotational groups are developed by Wigner, Zeits.f. Physik. 43, 640 (1927). 

9 Slater and Kirkwood, Phys. Rev. 37, 682 (1931). 

10 By a well-known theorem, the Ritz method always yields too high a value for the 
total energy, but one cannot tell whether it yields too high or too low a value of tho 
coefficient of E*, as the E* term is only a portion of the total energy. Hence it may 
yield too small or too largo a dielectric constant. A small alteration in can make 



206 THE DIELECTRIC CONSTANTS AND DIAMAGNETIC VIII, 49 

49. The Diamagnetism of Atoms, especially Hydrogen and 
Helium 

In considering diamagnetism, we may suppose the atoms in singlet S 
states, as otherwise there is an overwhelming paramagnetism. In such 
states the paramagnetic terms in the Hamiltonian function, which were 
given in Eq. <97>, Chap. VI, and which yield a perturbative potential 
proportional to the first power of //, disappear completely. This is so 
inasmuch as in atoms the squares of the orbital and spin angular 
momenta are respectively L(L-\-\) and S(S+l), and consequently in 
the 1 8 states, which have S = L 0, there cannot be even an instan- 
taneous magnetic moment in the absence of external fields. It may be 
cautioned that molecules have such a moment even in *S states, and 
for them the following formula (2) must be modified, as will be done 
in 69. In x $ atoms, there remains only the diamagnetic term in the 
perturbative potential, which is proportional to H 2 , and the magnetic 
moment is entirely an induced one coming from the Larmor precession. 
The resulting change in energy due to this term was seen in Eq. (105), 
Chap. VI, to be (e 2 /8mc 2 ) } (# 2 +2/ 2 )# 2 ; and furthermore, it was shown 
in 35 that on averaging over the different spacial orientations one 
may replace x 2 -\-y 2 by fr 2 because of spectroscopic stability. This, of 
course, assumes that the Boltzmann factor is sensibly the same for the 
different allowed spacial orientations, which it surely is in gases, and 
also in solids as long as the energy of orientation in the solid's inter- 
moleciilar field is small compared to kT. If we suppose that the atoms 
are all in the same stationary state except for spacial orientation, as is 
usually the case because the first excited states involve energy incre- 
ments large compared to kT, it is unnecessary to average over different 
electronic states weighted in accordance with the Boltzmann factor. 
The susceptibility L(dW/dH)/II per gramme mol is then 



where r 2 is the time-average value, i.e. the diagonal matrix clement for 
the state under consideration. Eq. (2) is exactly the same as the Pauli 
form of Langevin's formula in classical theory, already given in (2), 
Chap. IV. Thus again the new mechanics restores a classical formula. 
Eq. (2) is valid regardless of whether or not the atom is hydrogenic. 
For hydrogen-like atoms, we may, however, proceed farther and use 

a considerable error in the coefficient of K 2 t as only the unperturbed energy is stationary 
with respect to the parameters varied in obtaining the unperturbed wave function. 



VIII, 49 SUSCEPTIBILITIES OF ATOMS AND MONATOMIC IONS 207 

the formula for the mean value of r 2 given in (107), Chap. VI, and then 



= -2-832X 10 



Xmol 



The normal state of atomic hydrogen has 7&=1,Z = 0, Z=l, and thus 
its molar diamagnetic susceptibility is 2-37 x 10~ 6 . This value cannot, 
of course, be tested directly because of the difficulty of dissociating 
molecular hydrogen, also because monatomic hydrogen has a 2 /S f normal 
state and hence would be highly paramagnetic because of the spin. 
Instead we have only Pascal's 11 indirect value 293x 10~ 6 , obtained 
by applying the additivity method to diamagnetic organic compounds 
containing hydrogen. Exact agreement cannot be expected, as we have 
seen in 21, on refractivities, that the analysis of compounds by assumed 
additivity rules does not necessarily furnish true atomic properties. The 
error, however, is probably not so great as to permit the discrepancy 
by a factor about 3J which there was between his value and that 
0-79X10- 6 furnished by the old quantum theory (p. 210). Thus 
Pascal's result must be regarded as distinct evidence favouring the new 
mechanics in preference to the old. 

Direct Calculation of r 2 from the Wave Functions for Helium. Turning 
now to non-hydrogenic atoms, the theoretical calculation of diamagnetic 
susceptibilities is much easier than of the electric, as in the diamagnetic 
case it is only necessary to know the unperturbed wave function of the 
normal state. Once this is known, the requisite mean value needed for 
(2) is given by the simple quadrature 

l 1 2 '*, W 



the integration of course being over the coordinate space of all the 
electrons. On the other hand, to make calculations of electric suscepti- 
bilities such as were quoted on p. 205 one must know the effect of the 
perturbing electric field on the wave function. This is because the per- 
turbing potential was linear in the field rather than quadratic as in the 
magnetic case; and so to obtain the energy to the second power of 
the field strength, as needed for susceptibilities, it was necessary to find 
a second rather than a first approximation to the effect of the perturba- 
tion, which demands knowledge of the wave functions to the 1st rather 
than Oth approximation in E. 

11 A. Pascal, numerous references listed in Jahrb. d. Bad. und ElcMr. 17, 184 (1920); 
cf. also Weiss, J. de Physique, 1, 185 (1930). 



208 THE DIELECTRIC CONSTANTS AND DIAMAGNETIC VIII, 49 

The requisite quadrature (4) has been performed for neutral helium 
by Slater, 12 using a wave function which he shows to be a good ap- 
proximation to the three -body problem of the normal state of helium. 
He thus finds % mol = 1-85 x 10~ 8 , in gratifying accord with Hector and 
Wills' experimental value 1-88x1 0~ 6 . The discrepancy is less than 
the experimental error, as well as less than the amount of uncertainty 
in our knowledge of the helium wave functions. The quadrature for 
helium has also been evaluated independently by Stoner 13 with Har- 
tree's wave functions. He finds x m oi = 1'90 x 10~ 6 , likewise in exceed- 
ingly good agreement with experiment. 

Direct calculations of ]Tr 2 from the wave functions obtained with 
the Hartree method of the self -consistent field 14 have been made for the 
alkali ions and for Cl~ by Stoner. 13 His results will be given in 52. 
Similar calculations are at present wanting for other heavy atoms. 
Because it is easy, it is tempting to try calculating the mean value of 
r 2 and hence the susceptibility for the general heavy atom by means 
of the Thomas -Fermi 15 charge distribution />. In the Thomas-Fermi 
theory the mean value of ] r 2 is e~ l JJJ pr 2 dxdydz, where the integra- 
tion is 3 rather than 3Z-dimensional as in (4). One thus finds that 
Xmoi ~ 10~ 5 Z*, where Z is the atomic number, and where the factor 
10~ 5 has been estimated by a very crude numerical quadrature. 1511 This 
formula is not in accord with experiment, as according to Ikenmeyer 
( 52) observed susceptibilities in heavy atoms fit roughly the formula 
Xmoi = 0- 8 x 1 0~ 6 Z. The disagreement is not surprising, as the Thomas- 
Fermi field is primarily a good approximation to the distribution of the 
large number of inner electrons, rather than the few outer electrons 
that contribute the bulk of the susceptibility. The sensitiveness to 
errors in the outer distributions is illustrated by the fact that, according 
to Stoner, 33 per cent, of the susceptibility of Cl~ comes from the 3-46 
per cent, of the charge at a distance greater than 2 -06 A from the 
nucleus. Better results than with the Thomas-Fermi charge distribu- 
tion are obtainable not only by the more refined Hartree self-con- 
sistent field, but also by the method of shielding constants to be now 
given. 

12 J. C. vSlater, Fliys. Rev. 32, 340 (1928). 

13 E. C. Stoner, Froc. Leeds Phil. Roc. 1, 484 (1929). 

" D. R. Hartree, Proc. Cambr. Phil. Soc. 24, 89, 111, 426 (1928). 

18 L. H. Thomas, Proc. Cambr. Phil. Soc. 23, 542 (1927); E. Fermi, Zeits. f. Physik, 
48, 73; 49, 550(1928). 

i5 a This proportionality to Z* has also been noted by T. Takouchi, Phys. Math. Soc. 
Japan 12, 300 (1930). 



VIII, 50 SUSCEPTIBILITIES OF ATOMS AND MONATOMIC IONS 209 

50. Adaptation to Other Atoms by Screening Constants 

Because direct tests on monatomic hydrogen have so far been precluded 
by the difficulty of obtaining complete dissociation, the best existing 
way of testing the formulae for the susceptibilities of hydrogen atoms 
is to apply them to non-hydrogenic atoms by using screening constants. 
As a rough approximation one may assume the orbits are like those of 
hydrogen except that the effective nuclear charge is Z^e instead of the 
true charge Ze. The simplest illustrations are the helium atom and 
hydrogen molecule, even though the latter is not an atom, as each 
contains two electrons and so may be assumed to behave like two 
hydrogenic atoms in the normal state n I but with the effective 
charge Z^e. The formulae for the energy, dielectric constant, and 
diamagnetic susceptibilities are, then, respectively 
W= -27-1 Zl fi volts, e- 1 + 0-000450 Z^, Xmol = -4-74X 10- Z^. 
These may now be equated to the experimental values, 16 viz. 

(He) W= -78-8, = 1-0000693, x mol = l'88x 10- 6 ; 

(H 2 ) W = -31-4, e - 1-000273, Xmo i = 3-94X 10~ 6 . 
We then have three independent estimates of the effective charge Z efl , 
as follows: 17 _ f . _. _ 

Effective Charge Z M 

From: Energy. Dielectric Constant. Diamagnetism. 

Ho 1-71(1-71) 1-59(1-10) 1-59(0-92) 

H a 1-08 (1-08) 1-14 (0-78) 1-11 (0-63) 

10 The value 1-0000093 for Ho is that obtained by Herzfeld and Wolf, Ann. der Physik, 
76, 71 and 567 (1925), by extrapolation of the optical refraction, and is probably more 
accurate than direct determinations. The value 1-000273 for H, 2 is that obtained by 
Tangl; it is for C. rather than 20 C., contrary to the statement in the Laiidolt- 
Bornsteiii tables (5th ed., p. 1041), and so is in good agreement with dispersion data. 
Tho diamagnetic values are those of Wills and Hector, Phyft. Rev. 23, 209 (1926); also 
Hector, ibid. 24, 418. Their measurements are the only reliable ones at present available 
for He, while in tho case of H a they reassuringly agree within 2 per cent, with an indepen- 
dent determination by Sone, Phil. Mag. 39, 305 (1920). A much higher value 5-1 X 10 6 
for H 2 is reported by Lohrer, Ann. der Physik, 81, 229 (1926), but he himself states this 
may not bo accurate because of uncertainties in calibration. A value even higher than 
Lohror's is apparently indicated by tho graphs in a preliminary paper by C. W. Hammar, 
Proc. Nat. Acad. 12, 594 and 597 (1926), but here likewise it is not clear whether there 
has been an accurate calibration of absolute values. 

17 This table is taken from Proc. Nat. Acad. 12, 662 (1926). The calculations of tho 
theoretical dielectric constant 1-000225 of atomic H from tho Stark effect formulae of 
Wentzel, Waller, and Epstein, and of its diamagnetism (3) from the quantum-mechanical 
moan square radius wero given independently by the writer in this paper and by Pauling, 
Proc. Eoy. Soc. 1 14A, 181 (1927). In. reproducing the table slight revisions in the numerical 
values havo been made due to use of Birge's recent estimates of the most probable values 
of the atomic constants (Phys. Rev., Supp. July 1929). Wo havo not, however, made the 
corresponding small revisions due to these now atomic constants in quoting Pauling's 
numerical calculations on the following pages. 

3595.3 p 



210 THE DIELECTRIC CONSTANTS AND DIAMAGNETIC VIII, 60 

The results which would have been obtained with the old quantum 
theory are included in parentheses. In the new mechanics the values 
of Z Gff obtained by the different methods are seen to be roughly the 
same, and vastly more consistent and reasonable than with the old 
theory. Exact agreement between the various estimates cannot be 
expected even in the new, since screening constants are only crude 
representations of the interplay between electrons. Calculations of the 
dielectric constant of He by the Ritz method and of the diamagnetism 
of He directly from the wave functions have already been mentioned 
in 48 and 49, and, of course, represent a much higher degree of refine- 
ment. A similar improved calculation of the diamagnetism of H 2 by 
means of wave functions has been made by Wang and will be described 
in 69. 

To calculate the values in parentheses it has been necessary to know the old 
formulae analogous to (1) and (3). These can be shown to be 



(3a) 



where n n^Wi+lwJl an( i where n^ and V are axial and azimuthal quantum 
numbers one unit larger in numerical magnitude than in the new mechanics. Thus 
the normal state of hydrogen has \n' 3 \ = 1, or I' = 1, and S, P, D, F terms mean 
respectively V 1, 2, 3, 4 as compared to I 0, 1, 2, 3. In (1 a) and (3 a) we have 
tacitly supposed the field strong enough for spacial quantization in the electric 
case, but not in the magnetic. In the old quantum theory one cannot use the 
Pauli formula (2) unless one assumes random orientations, as otherwise x*-\-y z ^ $r* 
and one has all the ' Glaser effect ' difficulties discussed in 29. If instead one 
assumed spacial quantization in the magnetic field H, one would have x z -]-y z =r z 
for the normal state, the susceptibility would be increased by a factor |, and the 
values in parentheses in the last column would become 1-13 and 0-78. This would 
demand a powerful field, as diamagnetic effects are quadratic in H. Similarly 
other old values for use in the second column could be obtained if we supposed the 
electric field too weak to effect spacial quantization, or to overpower the relativity 
corrections, there being thus a double degeneracy difficulty. 

The divergence between (1) (3), and (la) (3 a) is, of course, most accentuated for 
small quantum numbers, and hence for the state n = 1 such as is involved in the 
table. For normal monatomic hydrogen (la) and (3 a) give e 1 = 0-000050, 
r* = (0-528 X 10~ 8 ) 2 as compared to the new e 1 == 0-000225, r* = 3(0-528 X 10~ 8 ) 2 . 
In other words the quadratic Stark effect and mean square radius for the normal 
state of hydrogen are respectively 4J and 3 times as large in the new mechanics as 
in the old. For larger values of n, the discrepancies are, naturally, much less pro- 
nounced, and so we shall not bother to include a comparison with the old theory 
in the discussion that follows of gases heavier than He. We may mention that for 
the excited states of hydrogen, the new formula (1) is favoured over the old (la) 



VIII, 50 SUSCEPTIBILITIES OF ATOMS AND MONATOMIC IONS 211 

by a certain amount of direct, though difficult experimental spectroscopic evidence 
on the second-order Stark effect. 18 

Application to Heavier Atoms. The most comprehensive and searching 
application of Eqs. (1) and (3) to non-hydrogenic atoms by using 
screening constants has been made by Pauling. 19 He considers primarily 
inert gases and ions with 'closed' electron shells, owing to the difficulty 
of obtaining reliable experimental data on other monatomic media. By 
the Pauli exclusion principle no two electrons have all their quantum 
numbers the same, and as the axial spin quantum number m 8 can have 
the two values 2 ( c f- 38), there are exactly two electrons for each 
set of orbital quantum numbers in a given complete shell. Thus one 
can find the total Stark effect for such a shell by multiplying Eq. (1) by 
2 and summing over all positive integral values of %, n 2 (including zero) 
and positive or negative of % 3 consistent with given n = n +n 2 + \n z \ + 1. 
Insertion of the Boltzmann distribution factor e~ w l kT would be ex- 
traneous, as we have here really a case of the Pauli-Fermi-Dirac 
statistics, although this name is rather formidable for the simple idea 
that orbits either occur twice or else practically not at all. - On per- 
forming these summations 20 and remembering that the moment is 
Pa = dW/dE (Eq. (46), Chap. VI) one thus finds for the 'molar 
polarizability' K = 4irL * p E /3E of an atom which has all shells com- 
pleted up to n = n f inclusive 



Similarly, on noting that for given I there are 21+1 possible values of 
m t and two of m 8 , one finds from (3) for the molar diamagnetic sus- 
ceptibility ~ t 7 

,0- 



n=lZ=0 

In these equations we have assumed that the true nuclear charge Z is 
screened by an amount cr, which may depend on n in (5) and on both 

18 Besides work quoted by Wontzel and Waller see also H. R. v. Traubenberg and 
B. Gebauer, Zeits. f. Physik, 54, 307, 56, 254, 62, 289; Naturwissenschaften, 17, 442 
(1929); C. Lanczos, ibid. 18, 329 (1930); M. Kiuti, Zeits.f. Physik, 57, 658 (1929). 

L. Pauling, Proc. Roy. Soc. 114A, 181 (1927). 

20 To perform this summation one notes that n x n a and n 8 can each assume integral 
values ranging from (n 1) to n 1, and that the weight of any given value of n^ n a 
is 2(n 1% n 2 |) and of w 3 is 2(n |n s |). These weights follow from counting the 
number of values of n lt n 2 , n a consistent with given n and with given w x n 2 or n s , and 
then multiplying by 2 on account of the spin. The resulting sums can be evaluated by 
means of Eq. (17a) of Chap. IX. Passage from (5) to (7) involves multiplication by 
(21+ l)/w 2 as there are 2n a electrons in a complete shell and 4J+2 in a closed sub-shell. 

P2 



212 THE DIELECTRIC CONSTANTS AND DIAMAGNETIC VIII, 50 

n and I in (6). There is an important distinction between (5) and (6). 
Eq. (5) supposes that if a shell of principal quantum number n occurs 
at all, it has its maximum allowance 2n 2 of electrons. Such a shell we 
shall call 'completed'. On the other hand, in (6) it has been assumed 
that if any state of given n and I is present, it has its full quota 4Z+2 
of electrons, but it has not been necessary to assume that all values of 
I are represented which are possible for a given n. In other words, for 
a given n in (6) we have summed over I up to some value l n , which is 
not necessarily as great as n 1. A full group of electrons with given 
n and I we shall call a 'closed' shell or sub-shell, as its resultant angular 
momentum is zero by the Pauli principle. A closed shell is with this 
usage not necessarily a complete shell, as it may be a subdivision of 
the latter. The outermost shells of inert gases beyond neon are closed 
rather than complete. Pauling assumes that (5) may be apportioned 
pro rata among the constituent closed shells composing a complete one, 
and makes use of the fact that practically all the electric polarizability 
comes from states with the maximum occurring value of n. He thus 
uses the formula 

'. 



where now we write simply n rather than n"\ for the maximum n. In 
our opinion Eq. (7) is not rigorous, since the principle of spectroscopic 
stability applies only to complete shells, as elaborated below. However, 
the use of (7) probably does not introduce serious error, as the method 
of screening constants itself is but an approximation. 

Eq. (1) is based on a separation of variables in parabolic coordinates peculiar 
to a Coulomb field, and does not apply to non-Coulomb central fields, such as 
Hartreo 14 has shown can be so chosen as to portray fairly well non-hydrogenic 
atoms. Although (1) fails completely for individual states in such fields, it does 
nevertheless yield a first approximation when one sums over all the states in a 
closed shell. This can be seen as follows. Consider first the part of the perturbative 
potential due to an electric field which is diagonal in n. Its effect on the energy 
is yielded by solution of a secular equation of finite order, embodying all the 
states having a given n. By the 'invariance of the spur' (p. 142) the sum of its 
roots is invariant of the system of quantization, and so there is no trouble arising 
from degeneracy, at least for diagonal elements in n, when we sum over a complete 
shell, but this does not apply to incomplete though closed shells. Incidentally, 
this sum is readily verified to be zero in the present problem, meaning that the 
first-order Stark effect characteristic of a Coulomb field, also the part of the 
second-order effect which has hv(nl ;nZl) in the denominator in a non-Coulomb 
central field, disappears on summing over a complete shell. 

Consider now the effect of non-diagonal elements in n having any given initial 



VIII, 50 SUSCEPTIBILITIES OF ATOMS AND MONATOMIC IONS 213 

and final values of n, say n f n'. Their effect on the energy of the state n t I, Wj is, by 
(26) of Chap. VI, given by an expression of the form 



l',m t 

Now if wo sum this expression over all values of I, mi consistent with n, and if we 
can neglect the dependence of the frequency in the denominator on I (as we can 
approximately unless there are very great departures from Coulomb character), 
the sum is invariant of the system of quantization, by Eq. (42) of Chap. VI. 
(Here I, w-i and V, m\ correspond to m and m' of (42), Chap. VI.) Thus the sums 
could bo equally well taken over a set of parabolic quantum numbers of given n or 
n' instead of over /, m t . Here again the in variance applies only to complete shells. 

It is evident that it is much less warranted to use (7) for incomplete than 
complete shells. Even with complete shells there is some lack of rigour as soon as 
one lets the screening constant a depend on I, as this implicitly assumes that (5) 
can bo apportioned pro rata to different Vs. The resulting error is, however, 
probably no greater than other unavoidable errors, such as, for instance, those 
resulting from the assumption that tho denominator of (8) does not depend on I. 

The normal state of hydrogen, discussed in 48, has no fine structure and so it 
is not even necessary to sum expressions such as (8) over I, m t (in distinction from 
I f 9 vrii) to establish in variance of the system of quantization. Thus Eq. (1) gives 
the Stark effect of the lowest state of hydrogen regardless of whether or not the 
field is able to overpower tho relativity correction. This has also been verified by 
Epstein 21 by making tho perturbation calculation for tho wave equation in polar 
coordinates. He finds the same result e ~ 1-000225 as ours. 

Pauling calculates the screening constants a nl in an interesting fashion 
by means of the old quantum theory, but with the substitution of 
1(1+1) for Z 2 . The various groups of electrons are assumed to influence 
each other like surface spherical distributions of electricity. Thus he 
makes approximate allowance for the large 'penetration' effects, but 
not for the smaller polarization terms to be given in 51. He neglects 
all powers above the first order in cr/Z, which seems legitimate in 
view of the approximate character of the calculations. The following 
table gives the values he thus calculates for the screening constants for 
the electric susceptibility of various closed configurations first reached 
at the listed atom or ion. The table also gives for comparison the 
'experimental' values of these screening constants which are deduced 
from observed values of the dielectric constant or polarizability of the 
atom or ion in question by means of Eq. (7). 22 

21 P. S. Epstein, Proc. Nat. Acad. 13, 432 (1927). 

22 The experimental values of the refractivitios for tho inert gases used by Pauling 
are those of C. and M. Cuthbortson, Proc. Roy. tioc. 84A, 13 (1911), extrapolated to 
infinite wave-length by Born and Heisenberg. 28 The values for Cu+, Ag+, Au+ are derived 
by tho additivity method from Heydweiller's data on salt solutions, with details as 
explained by Pauling, and are, of course, somewhat less certain than the direct measure- 
ments on the inert gases. 



214 THE DIELECTRIC CONSTANTS AND DIAMAGNETIC VIII, 50 



a^Ccdc. 



He (n= 1; 


= 0) 


0-391 


0-397 


Ne (n = 2; 


= 0,1) 


4-45, 5-64 


4-31, 6-50 


Ar (n = 3; 


= 0,1) 


9-70, 10-99 


11-11, 12-40 


Kr (n = 4; 


= 0,1) 


21-28, 22-92 


26-69, 28-33 


Xe (n = 5; 


= 0,1) 


34-29, 36-63 


42-26, 44-60 


Cu'(n = 3; 


= 0, 1, 2) 


14-4, 16-1, 19-5 


14-9, 16-6, 20-0 


Ag f (n = 4; 


- 0, 1, 2) 


25-7, 27-5, 31-1 


32-2, 33-9, 37-5 


Au^(n = 5; 


= 0, 1, 2) 


46-0, 48-1, 52-4 


59-9, 62-0, 66-3 



The Cu + , Ag + , Au + ions which are involved are, of course, in closed 1 S 
states. The values separated by commas represent the various sub- 
shells having the same n. As Eq. (7) involves a sum over I, the 'experi- 
mental method' is not able to isolate the individual cr^'s connected by 
commas, and in making the calculation from observed data for column 
2, Pauling assumes that their separations are as given by his theory in 
column 1 . Thus only a comparison of the means rather than differences 
of the values separated by commas serves as a real test of the theory. 
The agreement between the theoretical and experimental values is on 
the whole quite gratifying, especially for He and Ne. For He it is no 
greater than the experimental error. It must, however, be mentioned 
that a small error in a reflects itself in a much greater percentage error 
in K, as (7) involves Z a to the inverse fourth power. Thus, if one 
attempts to compute K for the rare gases from Eq. (7) with the ' cal- 
culated ' values for the screening constants given above, one obtains the 
following results, which show a rather wide discrepancy with observed 
values in the case of heavy atoms. 

He Ne Ar Kr Xe 

Kcalc. 0-506 1-14 1-72 0-72 0-88 

/cobs. 0-513 0-995 4-132 6-25 10-16 

Pauling also makes analogous theoretical calculations of the screening 
constant to be used in the formula (6) for diamagnetic susceptibilities. 
We shall not give his numerical results in detail, or enter upon certain 
rather elaborate distinctions between his procedures in the electric and 
diamagnetic cases. We must, however, not neglect to mention that his 
theory yields different screening constants for the same configuration 
for use in (6) than in (7), and it is to emphasize this distinction that 
we have used different notations (/ w) and a (c) in these two equations. 
He finds smaller screening constants in the diamagnetic than in the 
electric case, in agreement with experiment as regards sign of the 
difference. Thus Pauling's calculations represent considerable refine- 
ment in certain respects, though not in others. The agreement with 
experiment is of about the same order as in the electric case, with 



VIII, 50 SUSCEPTIBILITIES OF ATOMS AND MONATOMIC IONS 215 

greatest divergence for the heavy atoms, as shown by the following 
table for the inert gases: 23 

He Ne Ar Kr Xe 

XmoiXl0 6 obs. 1-88 6-7 18-1 37-0 59-0 

XmoiXl0 6 calc. (Pauling) 1-54 5-7 13-6 17-2 25-4 

XmoiXl0 6 calc. (Slater) 1-64 5-6 18-5 

On the last line we add the susceptibilities obtained by using a general 
system of screening constants recently proposed by Slater. 24 Zener 25 
calculated these screening constants by the Bitz method for the first 
period (He-F), while Slater extrapolated them to heavier atoms. They 
do not possess the refinement of being designed specifically for dia- 
magnetism, but probably possess a more immediate wave-mechanical 
basis than those of Pauling. 

51. Polarizability of the Atom-Core from Spectroscopic Quan- 
tum Defect 

Born and Heisenberg 26 first derived the following rather ingenious way 
of determining ionic polarizabilities spectroscopically. If one valence 
electron moves in a much more highly excited state than all other 
electrons, spectral energy -levels can be represented by the well-known 
Rydberg-Ritz formula 



where R= 27r 2 we 4 /c/& 3 , and Z = 1 in arc, 2 in spark, 3 in doubly 
enhanced spectra, &c. The quantum defect A, which measures the 
amount of departure from a hydrogen-like formula, owes its origin 
primarily to three causes: (a) penetration of the inner regions of the 
atom by the excited electron, (6) the Heisenberg exchange effect, 
(c) polarization of the atom-core. By the atom-core is meant the ion 
obtained by stripping the atom of its valence electron. The effect (a) is 
preponderant if the perihelion distance of the excited electron is small. 

23 The experimental values quoted for Ho, Ne, Ar are those of Hector and Wills, while 
those for Kr, Xe are only indirect determinations from Koenigsberger's work on salt 
solutions. The theoretical values are not the same as those which Pauling gives in his 
table VI, as the latter are based on semi-empirical shielding constants obtained by analogy 
with experimental refractive ones rather than from pure theory. It may be cautioned 
that while Pauling neglects the contribution of all but the outermost shell in the calcula- 
tion of rofractivities, he is obliged to include that of some of the inner shells in the 
calculation of the diamagnetism. He mentions, for instance, that the next to the outer- 
most shell of xenon contributes only 4 per cent, of the total refractivity, and hence can 
be approximately neglected in the optical case, whereas it contributes 20 per cent, of the 
diamagnetism. 

24 J. C. Slater, Phys. Rev. 36, 57 (1930). 

25 C. Zener, Phys. Rev. 36, 51 (1930). 

26 Born and Hoisenberg, Zeitsf. Physik, 23, 388 (1924) ; or Born, Atommechanik, p. 191. 



216 THE DIELECTRIC CONSTANTS AND DIAMAGNETIC VIII, 61 

If, however, its azimuthal quantum number exceeds a certain value, 
usually greater for heavy than light atoms, the orbit is nearly non- 
penetrating, and the effect (c) then may give rise to most of A. To 
calculate this polarization effect (c) we proceed as follows. If the radius 
r of the valence electron's orbit is large compared to the dimensions 
of the atom-core, this electron, will exert a sensibly homogeneous electric 
field e/r 2 on the rest of the atom, and so induce a dipole moment cxe/r 2 
in the core, where a is the latter 's specific polarizability. This dipole 
will in turn react on the valence electron with an attractive force 
F(r) = 2ae 2 /r 5 , since a dipole of strength //, gives a field 2ju,/r 3 at points 
along its axis. Tims the potential energy J F(r) dr due to polarization 
of the atom-core by the valence electron is 

F -_!^ fin) 

por ~ 2 r 4 ' ( ' 

If we regard this as a perturbative potential superposed on the ordinary 
Coulomb attraction, and if we neglect squares of a, then the change in 
quantized energy due to (10) is by (24), Chap. VI, 



r 3 _ i 

x ~' ( ' 



Here we have inserted the time average or diagonal element of 1/r 4 
obtained 27 by evaluating the integral (14), Chap. VI, with hydrogenic 
wave functions and with /= 1/r 4 . Now the departure of (9) from the 
Coulomb value is approximately 2&chRZl/n*, assuming that only 
the first term in the Taylor's development in A need be retained. 
Comparing this with (11) we see that 



(12) 

The parts of the quantum defect which are independent of n and pro- 
portional to 1/n 2 are known as respectively the Rydberg and Ritz 
corrections. In the second form of (12) we have substituted numerical 
values of the atomic constants and introduced the 'molar polarizability' 

K = 477/^/3. 

The simplest test of (12) is furnished by the spectra of neutral helium 
and of ionized lithium. Here the atom-cores are respectively He f and 

27 For details of the evaluation see I. Waller, Zeits.f. Phyaik, 38, 635 (1926). 



VIII, 51 SUSCEPTIBILITIES OF ATOMS AND MONATOMIC IONS 217 

Li ++ , which are exactly like the normal hydrogen atom except that the 
nuclear charge is 2e or 3e instead of e. Hence, by the theory of the 
dielectric constant of hydrogen given in 48, the values of the molar 
polarizability K should be respectively 1-68/16 and 1*68/81. Substitution 
of these K'S in (12) then yields the following quantum defects compared 
to the observed 28 for the P, Z>, and F series of He and Li+. 

He:JP He:D He: F Li+:P Li+ : D 

Acalc. 0-056- 0-0026- 0-00045- 0-044- 0-0021- 

_0-04n- a -0-005n- 2 -0-002n' 2 -0-03n' 2 -0-004W 2 

Aobs. 0-029 0-0025 0-001 0-020 0-0022 

We have omitted S states as too penetrating, and have taken the mean 
of the experimental values of A for par- and ortho-helium (or Li + ), as 
the par-ortho distinction is due to the Heisenberg exchange effect and 
taking this mean can be shown approximately equivalent to neglecting 
this effect. The agreement is at least qualitatively good. More pains- 
taking computations of Sugiura 29 based on systematic perturbation 
theory rather than the assumed polarization effect (11) yield A = 0*022, 
0-031, 0-020 for the ortho-par means of He 2P, He 3P, and Li+ 2P 
respectively. 

The calculations for He and Li + given in the preceding table were 
made by Waller and by Wentzel (for He only) soon after the advent of 
the new mechanics. Analogous calculations in the old quantum theory 
were unsuccessful, as helium always was its stumbling-block. Born and 
Heisenberg 26 showed that somewhat better results attended its applica- 
tion to the alkalis and analogous ions. The formula for A in the old 
theory was exactly like (12) except that the bracketed factor was 
(3/2Z' 5 ) (l/2rc 2 Z' 3 ) (with Z' as on p. 210) because of a different mean 
value of 1/r 4 . The following table compares values of the molar 
polarizability K obtained by the old and new mechanics from the 
spectroscopic quantum defects for the neutral alkalis and hence singly 
charged atom-cores. 

K calc. (new theory) 

K calc. (old, int. I') 

K calc. (old, half -int. I') 

K obs. 

28 The experimental values for Li+ are from Werner, Nature, 116, 574 (1925); 118, 
154 (1926). 

29 Y. Sugiura, Zeits.f. Physik, 44, 190 (1927); the direct experimental values for He 
2P, He 3JP, and Li+ 2P, ortho-para means are 0-0274, 0-0278, 0-0207, whereas the table 
gives the experimental Rydborg correction which best approximates experiment for all 
values of n. 



Li+ 


Na'- 


K 1 ' 


Rb^ Cs+ 


0-09 


0-48 


1-97 


7-6? 


0-80 


1-03 


4-27 


16-5? 


0-19 


0-53 


2-21 


8-5? 


He 


Ne 


A 


Kr Xe 


0-51 


0-99 


4-13 


6-25 10-2 



218 THE DIELECTRIC CONSTANTS AND DIAMAGNETIC VIII, 51 

The question-mark indicates that the spectroscopic data for Cs are some- 
what uncertain. Since the nuclear binding charge is larger, the polariza- 
bility of an alkali ion should always be lower than that of the preceding 
inert gas, which is directly revealed by ordinary measurements of the 
gas's dielectric constant. Reference to the table shows that this con- 
dition is met in the new mechanics but not in the old without the 
artificial introduction of half -quantum numbers another victory for 
the new theory. The reason that Born and Heisenberg found half- 
quantum numbers worked better is now apparent, as the denominators 
in the bracketed part of (12) are the same as 2(Z+|) 5 , 2ri*(l+%) 3 if in 
each case we keep only the two highest powers in I. This is a special case 
of Kramers' 30 theorem that half -quantum numbers in the old quantum 
theory are a better approximation to the new mechanics than are whole 
integers. The table shows an unusually large difference between the 
old and new theories for Li+, merely because the values are here cal- 
culated from P, rather than from F terms as for the others. The 
difference is naturally greater for I = 1 than I 3. 

A crucial test of (12) is found in the examination of whether the 
different spectroscopic values of A for the different non-penetrating 
terms of a given atom all yield the same value of K from (12). This test 
was extensively applied by Schrftdinger 31 in the old theory. He found 
it was not well fulfilled, and this is also true of the new. Constancy of 
K is unfortunately secured over only a limited range of terms or series. 
The following examples of Li + and Mg +f , in which we revise some of 
SchrOdinger's calculations in accordance with the new mechanics, are 
typical and by no means the worst. 

2P 3P 4P 5P 3D 41) CD 6D 4F 5F 

Li+ K -0-096 0-094 0-093 0-094 0-068 0-076 0-011 -0-04 -0-22 -1-3 
Mg++ K = 0-398 0-434 0-447 0-460 0-194 0-195 

The negative values for Li+ are, of course, nonsense. Eq. (12) does 
rather better in the dependence on Z than that on n, I, as in 52 we 
shall see that values of K deduced from (12) decrease in approximately 
the correct fashion as we go from left to right in an iso-electric sequence. 
Comparison of the value 0-28 for Mg++ given in the table on p. 222 with the 
values given above shows that Born and Heisenberg's and Schrodinger 's estimates 
of the polarizability of the Mg + ' ion by the spectroscopic method do not agree, 
even when one uses only the lattor's estimates from F terms. This discrepancy, 
which is not as bad as that between them for some other ions, exemplifies the 
rough character of the method, and arises in part because Schrodinger calculated 

ao H. A. Kramers, Zeits.f. Physik, 39, 828 (1927). 
81 E. Schrodinger, Ann. der Physik, 77, 43 (1925). 



VIII, 51 SUSCEPTIBILITIES OF ATOMS AND MONATOMIC IONS 219 

the polarizability by equating (12) to the entire experimental quantum defect for 
individual terms while Born and Heisenberg took a representation of the experi- 
mental data by the Ritz formula jRZj/(n-f-a-f 6/w 2 ) z and equated a to the corre- 
sponding part of (12). 32 Test of the theory by examination of the values of 6 is 
more difficult, as 6 is hard to determine accurately experimentally. According to 
(12) the ratio b/a should be respectively 0-67, 2, 4 for P, D, F terms (1-3, 3, 5-3 
according to the old quantum theory). Some experimental values are : P terms, Li 
0-63; D terms, Na 2-4, Mg+ 2-2, A1++ 2-6, Si+++ 2-4; F terms, Na 3-2, Mg+ 3-6, 
A1++ 3-6, Si+++ 4-8. 

If a frequency of motion (in the quantum sense) of the excited electron 
nearly coincides with an absorption frequency of the atom-core, clearly 
it is no longer a good approximation to regard the latter as subject to 
a static in distinction from periodic 
polarizing field. Near such coinci- 
dence we may surmise that A is 
abnormally large because of the 
resonance. Such resonance pheno- 
mena do not exist in alkali spectra, 
as ions homologous to inert gases 
have absorption frequencies too far 
in the ultra-violet, but are actually FlGt 8 - 

sometimes found in the spectra of atoms or ions with two valence 
electrons. An example is shown in Fig. 8, from SchrOdinger's paper, 
in which the observed quantum defect is plotted against the principal 
quantum number n for the 3 F terms of A1+. The similarity to anomalous 
dispersion curves is obvious. As the lines 3 3 D-6 3 ^ and 3 3 D-7 3 ^ of A1+ 
have wave numbers 43,000 and 47,700 cm." 1 respectively, it appears 
that A1++ might have an absorption line near 46,000 cm.- 1 , and the 
important line 1 2 -2 2 P of the A1++ spectrum does indeed have a wave 
number 53,918. (The latter line is written 3/Sf 3P if the principal quan- 
tum number is used as the ordinal number.) Exact agreement of the 
resonance point with an A1++ line cannot be expected, as the presence 
of the valence electron doubtless displaces somewhat the frequencies of 
the atom-core. Schrodinger shows that the observed variation of A can 
be nicely fitted by an 'undamped' dispersion curve. 




with v 46,000 cm.- 1 . The value 3-1 is about one-eighth that of 
the corresponding constant in the dispersion of sodium, which is very 
reasonable since A1++ should be harder to polarize than Na+. This 

32 Both Bom and Heisonborg and Schrudinger obtain the experimental values of the 
quantum defect from a comprehensive paper by A. Fowler, Proc. Roy. Soc., 103 A, 413 
(1923). 



220 THE DIELECTRIC CONSTANTS AND DIAMAGNETIC VIII, 51 

beautiful resonance phenomenon, discovered by SchrOdinger in the last 
days of the old quantum theory has been too much overlooked in 
the commotion of the new, although at the beginning of the latter it 
was considered quite noteworthy, though now perhaps commonplace, 
that the resonance frequencies in this phenomenon are those given by 
the Bohr frequency condition rather than the orbital frequencies of the 
old quantum theory. The need of more often using defects A with 
resonance points in spectroscopy has recently been emphasized by 
Langer. 33 We may also point out that this Schrodinger phenomenon is 
quite like the so-called 'perturbations' in band spectra, 34 wherein 
irregularities in the bands are found when rotational levels of two 
electronic states nearly coincide. If H 12 be the matrix element of the 
Hamiltonian function giving the interaction between two states other- 
wise of energy W v W z , then the secular equation (Eq. (35), Chap. VI) 
has the solution W = IW+W^^W^W^+IH^]*. When W, J-- W 2 
the effect of H 12 on W is thus of the second order, but at the point 
W W 2 it becomes of the first order. There is, then, a kink in the 
energy-curve at this point quite similar to that of a damped dispersion 
curve, or to the state of affairs in Fig. 8. This figure thus shows that 
there are 'perturbations' even in atomic spectra. The analysis of the 
secular problem connected with the inter-electronic interactions is, of 
course, a more complete way of handling the polarization phenomena 
than is the description by means of Eq. (10). This is especially true 
near resonance, as the dispersion analogy is only a more or less quali- 
tative one. The perturbation analysis for the F terms of A1+ is to be 
given in detail in future papers by Langer and by Whitelaw. They 
show that a 3p3d *F term (not previously classified as a vagrant) intrudes 
itself among the members of the F series, otherwise of the type 3snf 3 F. 
The interaction with this intruder distorts the other members and 
accounts for the irregularities in A and especially for the anomalies in 
multiplet widths. (This stray term is usually mislabelled 7 3 .F). Do not 
confuse with the 7 3 jP term of the revised notation used on p. 219.) 

52. Ionic Refractivities and Diamagnetic Susceptibilities 

The polarizabilities of ions which are iso-electronic with the inert gases 
may be obtained in the following four ways: 

(a) Application of the additivity method to salt solutions. 

(6) Application of the additivity method to salt crystals. 

** R. Langer, Phys. Rev. 35, 649 (1930). 

34 R. de L. Kronig, Zeits.f. Physik, 50, 347 (1928) (theory); J. Rosenthal and F. A. 
Jenkins, Proc. Nat. Acad., 15, 381, 896 (1929) (experiment). 



VIII, 52 SUSCEPTIBILITIES OF ATOMS AND MONATOMIC IONS 221 

(c) Use of the theoretical Eq. (7) with screening constants. 

(d) The spectroscopic method of 51. 

Method (a) has been extensively used by Heydweiller, 35 and by Eajans 
and Joos 36 with Heydweiller's measurements, while (6) has been used 
by Wasastjerna, 37 and by Born and Heisenberg 26 with Spangenberg's 38 
measurements. As already intimated, (c) has been employed by Paul- 
ing, 19 and (d) by Born and Heisenberg, 26 and by Schrodinger. 31 Each 
method has its limitations. Those of (c) and (d) have already been 
described. With (a) and (b) it is always necessary to assume some one 
ion as having a known refractivity, so that by subtracting this from 
that of the salt one obtains the refractivity of the other ion. Different 
values will, of course, be obtained depending on what values are 
assumed for the known ion. In the solid state (method b) the ions may 
well be considerably distorted by the inter-atomic forces, while in (a) 
there is always a more or less uncertain correction for the effect of 
hydration, or, in other words, for the distortion and saturation of the 
surrounding water molecules of coordination . This is especially evident in 
the light of the work of Sack and others ( 22) on saturation effects in ionic 
solutions, but these effects are not nearly as large for the optical region 
as for the static case considered in 22, as the bulk of the static satura- 
tion effect is due to polarization by orientation. The polarizabilities or, 
what is the same thing, the 'refractivities' which we shall give in the 
following tables are usually for infinite wave-length, but fortunately 
differ but little from those in the ordinary optical region (sodium D lines) 
as the absorption lines of inert gas configurations are well in the ultra- 
violet. Hence in methods (a) or (b) the refractivities should be deter- 
mined in the optical region, to rule out large orientation effects on the 
water molecules in (a) and the effect of atomic (i.e. crystalline) vibra- 
tions in (6). An extrapolation of the ionic refractivity thus obtained to 
infinite wave-lengths can then be made if desired. 

The approximate character of the additivity methods is shown by the 
fact that different refractivities for a given ion are obtained depending 
on the particular salt used. According to Born and Heisenberg, the 
values obtained for Na + from various salts are respectively 0-48 (NaF), 
0-53 (NaCl), 0-51 (NaBr), 0-33 (Nal), utilizing respectively assumed 
values for the contributions of the F~, Cl~, Br~, and I~ ions. 

35 Hoydwoiller, Phys. Zeits. 26, 526 (1925), and references to earlier work. 
38 Fajans and Joos, Zeits. f. Physik, 23, 1 (1924); K. Fajans, Zeits. /. Elektr. 34, 502 
(1928). 

37 Wasastjerna, Comm. Fenn. 1, 7 (1913); summary in Phys. Ber. 5, 226 (1924). 

38 Spangenberg, Zeits.f. Krist. 53, 499 (1923); 57, 517. 



222 THE DIELECTBIC CONSTANTS AND DIAMAGNETIC VIII, 52 

In the following table we compare by way of illustration some values 
for the refractivities of ions iso-electronic with neon which have been 
obtained by the various methods. 

IONIC REERACTIVITIES 

F- Ne Na+ Mg++ A1+++ S1++++ 

(a) (Fajans & Joos SG ) K = 2-50 (1-00) 0-50 0-28 0-17 0-1 

(b) (B. & H. from Spang. 26 ) 2-51 (1-00) 0-46 

(c) (Pauling 19 ) 2-65 (1-00) 0-46 0-24 0-14 0-08 

(d) (B. & H., Spectro. 26 ) (1-00) 0-49 0-28 0-15 0-10 

The value in parentheses is C. and M. Cuthbertson's direct measure- 
ment 22 of the refractivity of gaseous neon. The decrease in the re- 
fractivity as we go from left to right in the sequence is in good accord 
with the variation as (Z a)- 4 predicted theoretically by (7), since 
(a), (6), (d) agree quite well with (c). 

For the determination of ionic diamagnetic susceptibilities methods 
like (a), (6), (c) are available. Until recently the only calculations 
by the additivity method (a) were those of Joos 39 from old measure- 
ments of Koenigsberger and others. Recently, however, new determina- 
tions of the diamagnetic susceptibilities of alkali and alkaline earth 
halides in solution have been made by Ikenmeyer, 40 and of the halogen 
acids by Beicheneder. 41 The measurements with the acids have the 
advantage that the isolation of the individual ionic contributions by 
the additivity method is unique, inasmuch as the diamagnetism of H + 
is clearly zero. The method (6) based on measurement of solid salts has 
been extensively used by Pascal. 42 In the following table, which com- 
pares typical results with the various methods, we use Pauling's 19 
resolution of Pascal's data into the contribution of the individual ions. 
This differs from that originally proposed by Pascal, as a different con- 
tribution for the Na + ion is assumed as the starting-point. Still other 
resolutions have been proposed by Weiss. 11 As illustrative of method 
(c) we include purely theoretical calculations by Stoner, 13 in which the 
mean value of r 2 needed for Eq. (2) is calculated by Hartree's method 
of the self -consistent field. 43 

39 Joos, Zeits.f. Phynik, 19, 347 (1923) ; Joos and Fajans, ibid. 32, 835 (1925). 
4 K. Ikenmeyer, Ann. der Physik, 1, 169 (1929). 

41 Reicheneder, Ann. der Physik, 3, 58 (1929). 

42 Pascal, Comptes Rendus, 158, 37; 159, 429 (1914); 173, 144 (1921). For other 
measurements on solid salts see Landolt-Bomstein's tables and Pauling's comment on 
p. 203 of his paper. 

43 We for brevity omit from our tables the various estimates of refractivities and dia- 
magnetic susceptibilities of doubly charged alkaline earth ions. See the original paper 
for these. 



VIII, 62 SUSCEPTIBILITIES OF ATOMS AND MONATOMIC IONS 223 

MOLAB DIAMAGNETIC SUSCEPTIBILITIES X 10 fl 

Li+ F- Na+ 01- K+ Br~ Rb+ I- Cs+ 

(a) (Joos) 1-3 11-5 6-5 19-5 14-6 39-5 60-5 

(a) (Iken.) 4-0 13-9 104 20-4 16-9 34-8 31-3 49-3 45-7 

(a) (Reich.) 21-9 32-5 50-2 

(6) (Pascal & Pauling) 0-2 10-3 6-2 24-1 14-6 34-6 23-2 48-0 37-0 

(c) (Stoner) 6-5 40-4 30-1 

The molar susceptibility for an inert gas should, of course, be inter- 
mediate between those of the corresponding halogen and alkali ions, 
which should be respectively greater and smaller. In view of Hector 
and Will's 16 very careful values 1-88, 6-7, and 18-1 for He, Ne, and A, 
it would appear that Ikenmeyer's determinations for Li+ and Na+ are 
somewhat too high. Ikenmeyer finds that for a given column of the 
periodic table the molar susceptibility is almost exactly a linear func- 
tion 44 (Ci#+c 2 ) . 10~ 6 of the atomic number, where c 2 is respectively 
5-9, 2-4, and 3-5 for halogen, alkali, and alkaline earth ions, and Cj is 
0-80 for all three. In the light of Eq. (2) this means, since c 2 is small 
compared to c x Z except near the top of the periodic table, that the 
mean square radius per electron is approximately 0-80 x 10" 16 /2-83 or 
(0-53 X 10~ 8 ) 2 for all except the very lightest atoms. 45 This must mean 
that the effect of increasing nuclear charge and increasing mean quan- 
tum number nearly compensate as regards r 2 as we go down the periodic 
table. It is to be understood that this value r 2 is the mean over all 
classes of electrons. The outermost or valence electrons will have much 
larger values. Joos 39 notes that if one assumes that the outermost shell 
is responsible for practically all the diamagnetism, then his values of 
Xmoi yield atomic radii 0-54, 0-71, 0-80, 0-92 x 10- 8 cm. respectively for 
Na + , F~, K+, Cl", which are in remarkably close agreement with the 
ionic radii 0-63, 0-75, 0-79, 0-95 X 10~ 8 respectively deduced by Fajans 
and Herzfeld 46 from the grating energies of salt crystals. Only a rough 
agreement could be expected, as the discrepancies in the table show 
that there is considerable uncertainty in the diamagnctic measurements, 
while the estimate from grating forces is a purely classical one not 
utilizing the quantum 'exchange forces'. 

44 In our opinion these linear relations should not be taken too literally except perhaps 
for heavy atoms. In fact his analogous relation for the alkaline earths extrapolates into 
a paramagnetic susceptibility for Ba++, an absurdity. Ikenmeyer assumed the coefficient 
c to be the same for all ions in order to isolate their individual contributions in the salts, 
and this perhaps explains why his values for Na + and Li + are too high. 

45 H. Kulenkampff, Ann. der Physilc, 1 , 192 (1929) notes that this is exactly the radius 
of a one quantum hydrogen orbit in the old quantum theory. However, any agreement 
beyond that in order of magnitude is clearly fortuitous. 

46 K. Fajans and K. F. Herzfeld, Zeits.f. Phyaik, 2, 309 (1920). 



224 THE DIELECTRIC CONSTANTS AND DIAMAGNETIC VIII, 52 

One could go on with no end of numerical discussion on the best 
way of juggling the results in additivity methods, and we shall close 
by reproducing a semi-empirical table of ionic refractivities and dia- 
magnetic susceptibilities given by Pauling, which is probably as reliable 
as any. The values of the refractivities are obtained by assuming that 
the direct experimental values for the inert gases can be extrapolated 
to other members of an iso-electronic sequence by assuming a formula 
like Eq. (7) except that the effective charge is taken to be 



instead of Z a$. Here cr$ is determined from the experimental K for 
an inert gas of atomic number Z = Z Q> and af$ is a small, more or less 
empirical correction term, which is determined so as to best fit experi- 
ment in some cases (solution values for the Br~ and I~ ions) and by 
extrapolation in others. 47 The values for the diamagnetic susceptibility 
are calculated from Eq. (6), but with the assumption of an effective 
charge Z og?>+agp >(Z Z ). The values of o{ff> and ag?> are not deter- 
mined from experimental diamagnetic measurements, but by alteration 
of his theoretical affi described on p. 213. This is accomplished by 
analogy 48 with the alterations in the theoretical a$ necessary to secure 
agreement with the experimental refractivities, and by adaptation of 
the empirical ajf/. This analogy involves some rather bold extrapola- 
tions, and so the diamagnetic part of the table is probably not as 
dependable as the refractive part. 

47 The extrapolation is performed by assuming empirically that (in going from one 
row to another of the periodic table) the expression ojjf is proportional to the difference 
between the theoretical and experimental shielding constants for the inert gasos given 
in the table on p. 214. 

48 Pauling alters the calculated imperfection in shielding of an outer group by any 
given inner group by the same empirical factor as was required to give agreement with 
experiment on refractivities. For details see p. 201 of Pauling's paper. 



VIII, 52 SUSCEPTIBILITIES OF ATOMS AND MONATOMIC IONS 225 

- >o *7 t 

+ <M + CD .^. IO t- O> 

^OrH^oioPQocb M 6 1- 



O O 



H- ^ 

S -5 



^66 ^oc-i H6os ^5600 sic 



d? -&S + 

N 00 co-go Op ^co g2c, 



s 

c* 

o 



W 



co -L co + o f-H 

Hr-HlO o^OS ^0^0 + ** 

^6<N MO(T -- - ^ 



\) O CO 

^ 66 



I **> i- T i. T -r T 

L 1 " 1 rl '^' ^O* bC 1 "" 1 

J/2^130 O(7|t^ OQCOCO bdCO 

ft\ w co w -^ 



til -isS? , i^^ t,S 

1^66 66 ^o^ HH M co QrHc 



+ 1- + CO + ^ + I 

^D O bCCO 3 ^H Jj t 

pq co 10 ij ^ ^ O co 10 "^ - 

CO ^ IO 



s 

* 



ou 



i i iO 

j OJ PH , 10 , O 

H9^ MCO f^ 5 ?^ 

rH CI 10 00 <N 00 



" 6 c-i 



^? 



, o 

MO, O 

rH QO 



a 

I 



I I 



00 

6 o 



M fN IO 

<J t- oi 



-8 



f CO O T 00 O 

O C:| ^ OJ <N "- 1 



I I 



X 

I 



X 

I 



3595.3 



IX 

THE PARAMAGNETISM OF FREE ATOMS AND 
RARE EARTH IONS 

53. Adaptation of Proof of Langevin-Debye Formula given in 
46 

The general proof of the Langevin-Debye formula was given in Chapter 
VII explicitly only for electric polarization, but can be applied to the 
magnetic case merely by substituting everywhere the magnetic moment 
vector m (L+2S)^e/47mic for the electric moment vector p used 
in 44 and 46. There are, however, two points which require comment. 

In the first place, besides the paramagnetic part (L 3 -}-2S c )nek/^7Tmc 
( 42) of the Hamiltonian function, there is ever present the diamagnetic 
term proportional to 2 (# 2 +2/ 2 ) which has been discussed in 43 and 49, 
and which has no analogue in the electric case. Therefore, to all formulae 
for the susceptibility calculated by the methods of Chapter VII, we 
must add the expression for the diamagnetic susceptibility given in 
Eq. (2), Chap. VIII. 

The second point is the following. The magnetic moment in general 
consists of two parts, viz. the 'orbital' and 'spin' portions. In the 
different 'normal states' (cf. p. 187) these two parts may be inclined 
to each other at different angles. This will be the case if the normal 
states embrace a spin multiplet whose components are separated by 
intervals small compared to kT, as these different components corre- 
spond to different relative alinements of L and S ranging in atomic 
spectra from the 'anti-parallel' alinement J = \LS\ to the parallel 
one J = L+8 (cf. 40). Because of this flexibility in the coupling of 
L and 8 we cannot in general suppose that the resultant magnetic 
moment is 'permanent', i.e. the same for all normal states, and so we 
cannot always effect the simplification made in passing from Eq. (22) 
to (24) in Chapter VII. Instead we must use the more general expression 
(25), Chap. VII, for the contribution of the low-frequency elements, 
which does not require the hypothesis of a permanent moment. 

If, then, we make only the fundamental assumption that the moment 
matrix involves only elements whose frequencies are cither small or 
large compared to kT/h, the analysis in 46 shows that the formula for 
the susceptibility is ^r^ 

(1) 



IX, 53 _ THE PARAMAGNETISM OF FREE ATOMS 227 

where //, a is defined as in Eq. (25), Chap. VII, and is thus the time 
average of the square of the low-frequency part JJL of the magnetic 
moment vector, this average itself being averaged over the various 
normal states weighted in accordance with the Boltzmann factor 
e -wtykT t Tn e term JVa in (1) is the joint contribution of the high- 
frequency elements of the paramagnetic moment, and of the diamagnetic 
effect, similarly arranged. Consequently by (28), Chap. VII, and (2), 
Chap. VIII, 1 

(2) 
w 



- 
hv(n';n) (>rac 2 

u'-fn v ' ' 

In practice the diamagnetic correction given in the second term of 
(2) is relatively small if the material is really paramagnetic, as usually 
molar paramagnetic and diamagnetic susceptibilities are respectively of 
the orders 10- 4 ~10- 2 and 10- 6 -10- 5 . Consequently we shall henceforth 
omit writing the diamagnetic term except when we explicitly consider 
diamagnetism in 69 and 81. Of course allowance for diamagnetism 
ought to be made in the most refined calculations of paramagnetic 
moments from observed susceptibilities. Most of the experimental 
measurements of the susceptibilities of paramagnetic salts which we 
quote in the balance of the volume, also the 'effective Bohr magneton 
numbers' deduced therefrom, are corrected for the diamagnetism of the 
anion but not that of the cation. In other words, the quoted suscepti- 
bilities are the measured susceptibilities of the compound augmented 
by the absolute magnitude of the diamagnetism of the non-paramagnetic 
ingredient (anion), but not that of the paramagnetic ingredient itself 
(cation). One reason why the correction for the diamagnetism of the 
cation is usually omitted in the literature is that the diamagnetism of 
rare earth ions is rather hard to estimate quantitatively. 2 

It is convenient to introduce as a unit of magnetic moment the Bohr 
magneton 

= (0-9m00013) x 10- 20 erg. gauss- 1 . (3) 



1 We tacitly assume that the internal spin of the electron gives rise to no diamagnetic 
term. There is no term of this character in Dirac's 'quantum theory of the electron', 
and irrespective of the latter it appears quite obvious that any such term would necessarily 
be negligibly small since the orbital diamagnotic term is proportional to r a , and since the 
radius of the electron is negligible compared to that of an orbit. 

8 It must be cautioned that even though the diamagnotic correction is inconsequential 
for the given atom or ion itself, it can well be exceedingly important in solutions, since 
ordinary solvents are diamagnetic, or in salts of high 'magnetic dilution', where the 
diamagnetic atoms or ions greatly outnumber the paramagnetic. In these cases the total 
diamagnotism can clearly bo appreciable compared to the paramagiiotism. 

Q2 



228 THE PARAMAGNETISM OF FREE ATOMS IX, 53 

Then the formula for the molar susceptibility corresponding to (1) takes 
the numerical form 



S - 0-1241 +0-004 X 10*3 

7^ ~ 

= 0-00500 ^f +6-064 X 10 23 a, 

where p B denotes the low -frequency part of the magnetic moment 
measured in Bohr magnetons. If instead of the molar we used the 
susceptibility per unit-volume at 0C., 76 cm. in the gaseous state, the 
numerical factors in the first and second forms of (4) would become 
respectively 5-54XK)- 6 , 2-705X10 19 and 2-258X1Q- 7 , 2-705xl0 19 . 
Since the Weiss magneton is so commonly found in the literature and 
is in a sense also a convenient measure of moments, we have given in 
the second line of (4) the form which the formula takes if we measure 
/A in multiples of the Weiss magneton 1-853 X 10~ 21 e.m.u. 3 Historically 
it looked for a while 4 as if all atoms and molecules might turn out to 
have moments which are integral multiples of the Weiss magneton. 
Although many molecules arc still found to have moments which are 
integral multiples of the Weiss magneton within the experimental 
error, 5 this is probably purely fortuitous, for there are many reasons 
for believing the Weiss magneton phenomenon to be spurious. Til the 
first place, the Weiss magneton is 1/4-95 as large as the Bohr magneton, 

3 The Weiss magneton is often multiplied by the Avogadro number L to yield what 
may be termed a molar Weiss magneton, of magnitude 1123-5 e.m.u. We hero givo the 
value of the Weiss magneton which Weiss proposed in 1911, as this is the Weiss unit 
usually utilized in the literature. Later (1924-9) he raised the estimate 1o 1125-6 per 
gramme mol. (cf. Weiss and Ferrer, Annales de Physique, 12, 279 (1929) ) ; while si ill more 
recently Cabrera suggests 1124-9iLO-3 per mol. or 1-855 J^O-6 X 10~ 21 per atom as the most 
probable value (Report of the Solvay Congress, 1930). Our numerical value 0-917 X 10" 20 
of the Bohr magneton number embodies Dirge's estimate of the most probable values of 
the atomic constants, and yields a molar Bohr magneton number of 550-4 e.m.u. In the 
literature the Bohr magneton seems to be very often taken as 4-97 rather than 4-95 Weiss 
magnetons, or as 0-921 instead of 0-917 x 10" 20 e.m.u., due to use of older values of ejm 
and h. We employ the spectroscopic value 1-761 x 10 7 of ejm, generally conceded more 
reliable than the higher value sometimes furnished by the deflexion method. 

4 P. Weiss, J. de Physique, 5, 129 (1924); AnnaUs de Physique, 12, 279 (1929) and 
references to earlier work. 

5 In this connexion we must not fail to mention that Cabrera has collected some 160 
measurements of susceptibilities, primarily in the iron group, and finds that the vast 
majority of them seem to yield Weiss magneton numbers which do not deviate from 
integers by more than 0-1 (Report of the 1930 Solvay Congress). On the other hand, 
existing attempts to extract integral Weiss numbers from measurements on the rare 
earth group appear forced and artificial, quite irrespective of the clash of such attempts 
with quantum mechanics. 



TX,53 AND RARE EARTH IONS 229 

and so has no fundamental theoretical significance. Secondly, many of 
the apparent Weiss magneton numbers are fairly large (20 or so), so 
that the integral property is not very convincing unless the experiments 
are very precise. Finally, as noted by Stoner, 6 the Weiss magneton, if 
really fundamental, should manifest itself more clearly in gases than in 
solids or solutions, whereas the two common paramagnetic gases O 2 
and NO have moments which are non-integral multiples 14-2 and 9-2 
respectively of the Weiss magneton. Hence we shall not mention the 
Weiss magneton further. At the same time it must be cautioned that 
one must not expect magnetic moments to be integral multiples of the 
Bohr magneton either, due largely to the fact that in the new quantum 
mechanics, the absolute value of the angular momentum is ( J 2 + J) J ^/27T 
rather than an integral multiple J of h/27r. 

Tt will be desirable to discuss separately the limiting cases that the 
spin multiplets are very narrow or very wide compared to kT. These 
cases are particularly illuminating, and it is only to them that Eqs. (1) 
or (4) are applicable. It must be cautioned that not infrequently one 
encounters experimentally multiplet spacings comparable to kT, con- 
trary to the assumption of only low- and high-frequency elements basic 
to the validity of (1) or (4). Thus, as already mentioned in Chapter VII, 
the Langeviii-Debye formula is not as universal in the magnetic as in 
the electric case. 

54. Multiplet Intervals Small Compared to kT 

We shall throughout the balance of the chapter assume that the atom 
has Russell-Saunders coupling ( 40). As stated in 40, this supposition 
is usually fully warranted in the normal states involved in the calcula- 
tion of susceptibilities. Then the orbital and spin angular momentum 
vectors are constant in magnitude but not in direction, and the squares 
of their magnitudes have respectively the values L(L+l) and $($+1) 
(Eq. (83), Chap. VI). We throughout measure angular momentum in 
multiples of h/%7T, as in 39-42. In the absence of external fields the 
resultant angular momentum L+S will be constant both in direction 
and magnitude, and constitutes what Dirac calls a 'constant of the 
motion', whereas the vector L+2S which is important for magnetism 
will not be, due to the continual precession of L and S about J. This 
is illustrated in Fig. 9. In this precession the component (b-c in Fig. 9) 
of L+2S, which is perpendicular to J = L+S, will clearly not be con- 
stant in direction. However, the only periodic or non-diagonal elements 

E. C. Stoner, Magnetism and Atomic Structure, p. 15U. 



230 THE PARAMAGNETISM OF FREE ATOMS IX, 54 

in the moment vector m = j8(L+2S) are those associated with the 
precessions of L and S about J. Thus m is diagonal in all quantum 
numbers other than the inner quantum number J and the magnetic 
quantum number M . This, incidentally, is not true in molecules 
(Chap. X), and is also the underlying reason why the numerical para- 
magnetic susceptibilities of non-hydrogenic atoms can be calculated so 
much more definitely than the electric, as the electric moment vector 

p involves other quantum numbers, 
such as, e.g., the principal quantum 
numbers, in a complicated way. 

The preceding paragraph shows 
that when the multiplet intervals are 
small compared to kT (i.e. kine- 
matically when the precession of L 
and S about J is slow), the elements 
of the moment vector m are entirely 
of the low-frequency type. In other 
words, in the notation of Chapter VI 
we can dispense with the index n and 
take j to be identical with the inner 
quantum number J. Eq. (2) then 

shows that the part No. of the susceptibility vanishes if we neglect the 
second or diamagnetic term. Now in the absence of high-frequency 
elements there is no distinction between /* and the complete moment 
vector 0(L+2S), so that 




(5) 

Since the temperature factor e~ w lkT for the various multiplet com- 
ponents may be disregarded under the supposition of intervals small 
compared to kT, the statistical average of the product L'S may be 
taken as zero in very strong fields, where S and L are quantized 
separately relative to the axis of the field (case c of Fig. 6, 40), and 
hence have no correlation between their directions if we average over 
all orientations. Furthermore, the statistical average of this product 
is invariant of the field strength and hence vanishes in all fields if it 
does in very strong ones, for use of one of the spectroscopic stability 
relations given in 35 (viz. the 'invariance of the spur', p. 142) shows 
this average is invariant of the mode of quantization if the system is 
made degenerate with respect to J, M by neglecting the energy of 
interaction between L and S, and between H and both L and S. We 



IX, 54 AND RARE EARTH IONS 231 

can also verify directly that the statistical average of L . S vanishes in 
weak fields, since by the 'cosine law' (Eqs. <84>, (84a), Chap. VI) this 
average is proportional to the sum 7 



which is readily verified to be zero. The factor 2J-\-l appears since it 
is the a priori probability or number of M -components belonging to 
a given value of the inner quantum number J. 

The first two right-hand members of (5) are diagonal matrices having 
characteristic values 48(S-\-l)fi 2 and L(L-\-l)fi 2 invariant of /, so that 
double bars over them are unnecessary. We have just proved the third 
member is zero. Hence the expression (1) for the susceptibility becomes 



Eq. (6) may also be derived in the following elementary way for very 
strong fields adequate to produce a Paschen-Back effect. In such fields 
we may use the quantum numbers M s , M L explained in 40. The 
energy of a stationary state is then W w +Hfl(M L +2M s )+AM L -M 8 
(Eq. (103), Chap. VI), and its component of moment in the direction 
of the applied field is (M L +2M s )fl. The susceptibility is, of course, 
obtained by averaging over all states weighted with the Boltzmann 
factor (cf. Eq. (3), Chap. VII). If the multiplet widths are negligible 
we may neglect the term in the energy depending on A, and then the 
susceptibility is 

vo 2 I ( 



TJ 

, (7) 

" 



2 

Np M-L _ _ , J 

~ ft [ ^ e -^IM L lkT T" 

We now expand the exponents as power series in H, and retain as in 
44 the non-vanishing terms of lowest order in numerator and de- 
nominator. Then (7) becomes 



This agrees with (6) when we evaluate the sums by means of (68), 
Chap. VI. 

7 The sums can bo evaluated by using the formulae of Eq. (17a), 56. 



232 THE PARAMAGNETISM OF FREE ATOMS IX, 54 

The preceding elementary derivation of (6) was given only for fields 
adequate for a Paschen-Back effect, but the principle of spectroscopic 
stability, not to mention our first derivation of (6) by the methods of 
Chapter VIT, assures us that this expression for the susceptibility is 
invariant of the field strength (neglecting, as everywhere, saturation), 
and hence the formula will hold even when the field is not able to 
produce a Paschen-Back effect, or only a partial one. 

Since Eq. (6) is so obvious in strong fields, the writer has been fre- 
quently misquoted and misunderstood ever since he first proposed (6), 
even though he then explicitly said that (6) applied to all fields. Con- 
trary to statements often made, the validity of (6) does not by any 
means imply M L , M s quantization, and holds also with the J, M 
quantization, which is a better approximation at usual field strengths. 
The Paschen-Back effect will usually change the formulae for individual 
Zeeman components (cf. Eqs. (99) and (103), Chap. VI) but will not 
alter the expression for the susceptibility. To dispel all doubt we shall 
in 56 also prove Eq. (6) for weak fields in an elementary though 
tedious way that does not utilize spectroscopic stability. 

55. Multiplet Intervals Large Compared to kT 

When the separation of the multiplet components is large compared to 
kT, only the one component which has the lowest energy is a normal 
state, and the double bar is no longer needed in (1). The matrix ele- 
ments of m now are all of the high-frequency type unless A J = 0. Thus 
J may be identified with the index n used in Chapter VTT, while the 
index j of VII may be omitted. The low-frequency part of m is now 
only the part which is parallel to the resultant angular momentum J 
and so does not involve the now rapid precession of L, S about J. 
Hence 



where g is the Lancle gr-factor 



8(8+1)+ J(J+1)-L(L+}) 

' V ' 



Here we have utilized, as in <84>, <85> of Chapter VI, the cosine rela- 
tions 2L J = L 2 + J a -S 2 and the fact that J 2 = J(J+ 1), &c. The dot 



IX, 55 AND RARE EARTH IONS 233 

over the equality sign has the same meaning as on p. 142. Eq. (1) 
now becomes (1Q) 



The constant Na no longer has the value zero, as in (6) (except for the 
neglected diamagnetic part), but is instead 

N? f F(J+D _ F(J) I 
^J+\)\hv(J+l\J] hv(J]J-l)Y V } 



with the abbreviation 



F(J) = [(8+L+ir- J*$J*-(8-L)*\. (12) 



The existence of this is a concomitant of the second-order Zeenian 
term in (99), Chap. VI, not to be confused with the diamagnetic second- 
order Zeeman term of 43. The presence of a, or of the second-order 
term in (99), is due to the component b-c in Fig. 9, which now con- 
tributes only to the part of the susceptibility that is independent of 
temperature rather than as in 54 to the ordinary Curie part propor- 
tional to I /T. Usually the normal state is a minimum or maximum of 
J, depending on whether the multiplet is regular or inverted (p. 166), 
and then the second or first term respectively of (11) vanishes, as can 
be verified from the formula (12) for F. 

To prove formula (11) one might work out the expressions for 
L X +28 X9 L y +28 v analogous to those for L s +2S z in (88), Chap. VI, and 
add their squares together in the fashion (44), Ch. VI, to get the resultant 
amplitudes m<>(J',J')--= p /( 2 KM" 2 ^)^ J/ )l a ) needed for (2). 

'V \q=x,u,Z ' 

Another way, however, which is fundamentally very little different but 
is easier after one has once deduced the Zeeman formula to the second 
order in (99), Chap. VI, is as follows. The contribution of the second- 

order part H*\VM of (99) to the susceptibility is 2N TF< 2 Y(2J+1) 

j\t = J 

by (4), Chap. VII, inasmuch as under the present hypothesis of wide 
multiplets there is only one normal unperturbed value of J, and so the 
sum in (4), Chap. VII, reduces merely to one over M. The result (11) 
now follows on substitution of the expression for IF (2) obtained from 
(99) and (88), Chap. VI, and on evaluation of the sum over M . The 
terms proportional to M 2 are, as usual, summed by means of (68), 
Chap. VI, while the other terms are very simple to sum. 

Only the first term of (10) is ordinarily given in the literature. Due 
to the frequency factors in the denominator of (11), the second term 



234 THE PARAMAGNETISM OF FREE ATOMS IX, 56 

Noc will ordinarily be small compared to the first if the multiplet is 
really wide compared to kT, unless perchance the normal state involves 
an abnormally low value of J, so that a-b is small compared to h-c in 
Fig. 9, p. 230. The latter case does, however, occasionally arise, as will be 
shown and elaborated in connexion with the susceptibilities of Eu +++ 
and Sm+++ in 59. The first part of (10) is usually deduced in an 
elementary way analogous to the proof of (6) by means of (7), (8). If one 
uses only the first-order Zeeman formula W Q -\-MgpII for weak fields 
instead of the one for strong fields and supposes that only one value of 
J represents a normal state, then one sees that in place of (7) one has 



TT V-> nttfrilllL'i' ' > ' 



The rest of the procedure, which converts (13) into the first part of (10), 
is entirely similar to that explained after Eq. (7), the only essential 
difference being that gfi now replaces j8. 

We must not forget to mention that the first part of (10) was obtained 
by Sommerfeld 8 and others in the old quantum theory. In that theory 
the Lande (/-formula for the first-order Zeeman effect was taken as an 
empirical fact, and so the derivation by means of (13) could be used. 
There is, however, an important difference in the interpretation of (10) 
in the old and new mechanics. If 6 denote the angle between the 
magnetic field H and the low-frequency part p of the magnetic moment 
vector, then in a weak field the first-order energy is W Q |/i|JF/cos 0. If 
one uses this expression instead of the equivalent form Wo+MgpH and 
weights the various values of cos# in accordance with the Boltzmann 
factor, then one finds after the usual series expansion of the exponentials 
(e.g. as in 44 or as in passing from (7) to (8) ) that, except for satura- 
tion, the susceptibility is given by the expression 

X = *^P. (14) 

In the old quantum theory, the value of p 2 was considered to be g 2 f$ 2 J 2 
as the magnitude of the resultant angular momentum of the atom was 
Jh/27T. On using this value, and equating (14) to the first part of (10), 
one sees that the mean value of cos 2 was given by 



(old) 

3 o 

Here we have also included for comparison the analogous result with 

8 A Somraerfeld, Atombau, 4th ed., pp. 630-48 and references. 



IX, 65 AND RARE EARTH IONS 235 

the new mechanics, which one knows must be -J, but which one can 
also obtain by equating (10) and (14) with p? = g*P 2 J(J+l). The old 
value in (15) is one of the typical violations of the principle of spectro- 
scopic stability which were so common in the old theory. This difficulty 
did not affect the susceptibility as long as there was spacial quantiza- 
tion, for then Eq. (10) was valid in the old theory, and the only difference 
compared to the new mechanics was that the same value of the product 
/Aos 2 was apportioned in a different way between its two factors. If, 
however, inter-atomic collisions were frequent, due to high temperature 
or density, it would not be reasonable to assume spacial orientation in 
the old theory. Instead one would have to assume random orientations 
and use the second value in (15), which, when substituted in (14) with 
/* = ^J, gives x = Ng*fPJ*fikT instead of (10). Thus the susceptibility 
might vary with density or field strength, contrary to experiment, and 
again the new interpretation is superior. Also in the old theory the 
formula (6) would be valid only in fields powerful enough for a Paschen- 
Back effect. 

56. Multiple! Intervals Comparable to kT 

Let us now turn to the general or 'intermediate' case in which the 
effect of the inner quantum number J is comparable with kT. Then 
out of the total number of atoms a certain portion Nj will have any 
given value J of the inner quantum number. Their contribution to the 
susceptibility will be given by the expression (10) with N replaced by 
Nj, since (10) still applies as long as all atoms under consideration have 
the same */. We must now, however, add the contributions of the atoms 
with various different values of J. The number Nj is determined by 
the Boltzmaim temperature factor, and is hence proportional to 
(2J+l)e- w jl kT , inasmuch as for a given J there are 2J+1 component 
states having different values of M. Thus 



Here subscripts have been attached to g and a to show explicitly that 
they are functions of J. 

It is clear that (16) should reduce to (G) in the limiting case of very 
narrow multiplets. This may be verified as follows. If the dependence 
of W Q on J is negligible, it suffices to strike out entirely the exponential 
factors from the denominator and from the first term of the numerator, 
assuming for simplicity that the origin of the energy is at J = 0. With 



236 THE PARAMAGNETISM OF FREE ATOMS IX, 56 

the second term the procedure is not so simple, as by (11) the a's 
approach infinity when the multiplets become very narrow. One can, 
however, evaluate the second sum in the numerator in the limit 
(W J+l Wj)lkT = Q by using exactly the same type of expansion as 
in passing from (17) to (20) in Chapter VII. One thus finds that in this 
limit, (16) is the same as 



This expression can be verified with a bit of labour to be identically 
equal to (0) if one introduces the explicit expressions (9), (12) for g and 
F and uses the following formulae for the sums of series: 



~~ 



. 



(17a) 



4 Z*x(x+l) n+l 

in which x assumes consecutive integral values from 1 to n. These 
formulae are readily proved inductively in the same fashion as explained 
for (68), Chap. VI, which is essentially the second of these relations. 

Laporte and Sommorfelcl 9 use for narrow multiplets the expression obtained by 
omitting the F terms in (17). Their expression differs but little from the more 
rigorous and also more simple formula (6), the reason being that b-c is usually 
small compared to a-b in Fig. 9. For instance, Laporte and vSommerfeld calculate 
effective Bohr magneton numbers 2-92, 4'35, 4-97, 5-23, and 5-92 respectively for 
narrow 2 Z), *F, 4 -F, *D, fi *S f multiplets, which are of asymptotic interest in the iron 
group, whereas (6) gives 3-0, 4-47, 5-20, 5-48, 5-92. In quoting their results for 
narrow multiplets in the iron group in 72 we shall throughout give values modi- 
fied in accordance with (li). 

Before we can use formula (16) it is necessary to know quantitatively 
the multiplet intervals in order to evaluate the denominators in (11). 
These intervals should, of course, be taken if possible from direct 
spectroscopic data on the multiplets in question, but unfortunately such 
data are often not available for the type of ions encountered in para- 
magnetic solutions (58, 59). It is then necessary to resort to a 
theoretical expression for these intervals, and one uses the 'cosine law' 
(Eq. (84a), Chap. VI) 

Wj -- \AJ(J \- 1) + constant, (18) 

together with the proper value of. A. All the paramagnetic ions yet 
encountered seem to owe their magnetic moments to an incomplete 

9 O. Laporto and Soiurnerfnld, Zrit*. /. Phytik, 40, 333 (192(>); (). Laporto, ibid. 47, 
761 (1928). 



IX, 56 AND RARE EARTH IONS 237 

shell of 'equivalent electrons', i.e. electrons with similar n, I. Further- 
more, when all electrons not in closed groups are equivalent, the Hund 
theory of spectral terms 10 tells us that the normal or lowest lying' 
electronic state has the maximum multiplicity allowed these electrons 
by the Pauli exclusion principle, and also the maximum L consistent 
with this multiplicity. (Specific examples will be cited in 58.) With 
this specialization it will be shown in the next paragraph that an 
immediate application of Goudsmit's theory 11 for calculating A yields 

A- %. ' 

where a n is the A -factor which would result were only one electron of 
the group present. The plus or minus sign is to be usod according as the 
group is less than or more than half completed. 12 As the number of 
electrons in a completed sub-shell is 4/-[-2, the value of A is, in other 
words, positive if 7; < 2J-|-1, and negative if 77 > 2Z-J-1, where 77 is the 
number of electrons present of the type under consideration. The theory 
thus predicts that the multiplets of the normal state be respectively 
regular and 'inverted' for the first and second halves of the sequence 
formed by addition of consecutive electrons of the group. This rule is 
in good accord with experiment wherever spectroscopic measurements 
are available on multiple normal states. An abundance of such measure- 
ments is available for the iron group, though usually for no tas high 
a degree of ionization as directly involved in the study of pararnagnet- 
ism. The value of a n for a hydrogenic atom is 

( ' 

This is equivalent to the well-known 'relativity' or 'spin 1 doublet 
formula, and follows from Eq. (80), Chap. VI, after the mean value of 
1/r 3 involved therein is evaluated with hydrogenic wave functions. 13 It 
is assumed that a iion-hydrogenic atom can be approximately repre- 
sented by introducing a screening constant a into the hydrogen formulae. 
Instead of giving directly the value of A, it is often convenient to use 
the 'over-all* multiplet width AAv total = IF /inax -Tr, min , which by (18) 
is \A [( L+ 8)(L+ 8 + 1 ) - |L- tf | ( | L- tf \ + 1 )]! " The 'multiplets involved 
in the normal states which we need to deal with all have 8 ^ L. Hence 

J0 F. Hund, Linicnttpektri'Hi Chap. V. 

11 S. (jjoiulsniit, /'A//.?. Itcr. 31, 1)46 (1928). Eq. (19) is also derived by O. Laporto in 
Handbuch der A*trophi/nik, iii, p. 643. 

12 At tho point 17 *ll~\- 1 at which tho group is half complete, there is no trouble from 
ambiguity in sign, as here, tho 'lowest lying' state is an S one without multiplet structure. 

13 Seo Hoisenborg and Jordan, Zeits.f. Physik, 37, 263 (1926). 



238 THE PARAMAGNETISM OF FREE ATOMS IX, 66 

by (19), (20) the expression for the 'over-all' width when expressed in 
wave numbers rather than ergs is 

?ii)- . 5.82(2L+1) _ 4 -i 



fl 

c total- Ac 2S 

The factor 5-82 follows from substituting explicit numerical values for 
the constants e, h, m, c. It is perhaps well to reiterate the distinction 
between I and L. The former is an azimuthal quantum number measur- 
ing the angular momentum of a single electron, while L measures that 
of the whole incomplete group. 

Derivation ofEq. ( 19). By application of the principle of permanence of P-sums 
( 42) or iiivariance of the spur ( 35) to the passage from J, M to M St M L 
quantization (case (b) to (c) in Fig. 6, 40) one has, in virtue of Eqs. (84a), (103), 
Chap. VI, the relation 

v iA 



where S means a summation over all the states consistent with given S, L, M. 

By a similar application to the passage from M L9 M s to m lf m, quantization 
(case (c) to (a), Fig. 6), one has 2 AM L M a 2 (J n ?n,.mg , where 2 means a 

b i b * * b 

summation over all the states consistent with given M L , M K , n, I, even though 

involving different L or &', and where means a summation over all the individual 

f 

equivalent electrons, which wo suppose rj in number. In the first application the 
summation sign 2 can De omitted when Al = J L-J-AV, as there is then just one 

a 

term, and the relation becomes merely ALS = AM L M S . Similarly, in the second 
application, the summation 2 can he omitted when M L = L, M s = 8, S - - |iy, 

and when L has the maximum value L max which according to the exclusion princi- 
ple is consistent with this *S f . This is true because there is never more than one 
multiplet of the type S = JT/, L = J^ max . One then has AM L M S = Ja 11 Af L , as 
each m Si is necessarily J when M s = J^. Thus combining the special cases of the 
two applications one has ALS \i^L or A a n /2S. This is the desired result 
for the first half of the sequence. In the second half the exclusion principle 
demands # < JTJ so that the preceding method cannot be used. Instead, the value 
A = a n /2S follows as a special case of Goudsmit's general result that A values 
are reversed in sign in passing from the first to second half of the sequence. It is 
to be cautioned that we give in (19) only the formula for the lowest-lying multiplets 
rather than Goudsmit's necessarily more complicated relations and derivation for 
the general multiplet. 

57. Susceptibilities of Alkali Vapours. 

Unfortunately none of the preceding theory can easily be tested directly 
upon gases, as the only monatomic gases are the inert ones, which are 
diamagnetic. There are, to be sure, many materials whose vapours 
should be paramagnetic. However, the susceptibilities of vapours are 
very hard to measure, because of the difficulty of obtaining a sufficiently 



IX, 57 AND RABE EARTH IONS 239 

high, as well as accurately known, vapour-pressure. As far as the author 
is aware, the only quantitative determination of the susceptibility of 
a monatomic paramagnetic vapour is Gerlach's measurement 14 on 
potassium vapour. He finds that for temperatures between 600 and 
800 C. corresponding to vapour-pressures of from 0'5 to 30 mm., the 
molar susceptibility of potassium vapour obeys Curie's law and is 
0-38/T. The normal states of alkali atoms are 2 $ ones, and so by Eq. (6) 
the theoretical molar susceptibility is LfPjlcT 0-372/T. It is, of 
course, here not necessary to worry about the width of the multiplet, 
as a multiplet structure is wanting in 8 states. The agreement between 
the observed and theoretical Curie constants is extremely gratifying, 
as Gerlach's experiment is extremely difficult and he does not claim an 
accuracy of more than 10 per cent. Incidentally, this good agreement 
is a nice confirmation of the correctness of the assumed vapour-pressure 
curve, and shows that potassium vapour must be primarily monatomic 
at the temperatures employed. If diatomic, the vapour would be dia- 
magiietic, as the normal state of the K 2 molecule is of the type 1 2. 
Dissociation theory indicates that the fractional number of atoms asso- 
ciated into molecules at the temperatures employed by Gerlach should 
be 15 of the order 10~ 2 , and so the neglect of molecular association is 
not a dominant experimental error. Incidentally, magnetic measure- 
ments should be a sensitive means of determining the degree of mole- 
cular association when appreciable. Hence it would be of interest if 
Gerlach's experiments could be extended to conditions where there is 
more molecular association, viz. to still higher vapour-pressures (which 
means higher temperatures) in potassium, or to sodium or lithium. 
Compared to potassium, molecular association at a given p, T is 
favoured in Li and Na by their molecules having higher heats of dis- 
sociation (about 1-2 and 0-8 volt respectively) than that of K (about 
0-6 volt). 16 

58. Susceptibilities of the Rare Earths 

Because of the difficulty of obtaining paramagnetic materials in the 
vapour state, the best that can usually be done is to study the suscepti- 

14 W. Gorlaeh, Attl del Congrcsso hitcrnazionale del Fisici, 1, 119 (1927). 

15 Cf. R. W. Ditchhurn, Proc. Roy. Soc. 11?A, 486 (1928), or Gibson and Heitler's 
formula given on p. 175 of Ladciiburg and Thiclo, Zcits.f. Phys. Chem. 7, 161 (1930). 

16 Loomis and Nusbaum, Phys. Rev. 37, 1712 (1931); (Li,): Loomis, Ibid. 31, 323 
(1928); Kinsoy, Proc. Nat. Acad. 15, 37 (1929); Polanyi, Schay, and Ootuka, Zeits.j. 
Phys. Chcm. IB, 30 (1929), 7u, 407 (1930); Ladenburg and Thiele, I.e., all Na 2 . H. 
Ootuka, Zeits.f. Phys. Chcm. 7s, 422 (1930) ; Ditohburn, J.e. 15 ; A. Carrelli and P. Prings- 
hoim, Zcits.f. Physik, 44, (343 (1927); W. (). Crano and A. Christy, Phys. Rev. 36, 421 
(1930), all K 2 . 



240 THE PABAMAGNETISM OF FREE ATOMS IX, 58 

bilities of solutions containing paramagnetic salts, or of these salts in 
the solid state, preferably hydratcd. One hopes that in some of these 
cases the ions responsible for the paramagnetism may be so nearly free 
as far as orientation is concerned that the calculations for the gaseous 
phase may be utilized. We shall later see that disturbances from other 
atoms prevent realization of this hope in most instances. In the first 
place the number of paramagnetic ions is rather limited. In most atoms 
the only incomplete shells with an outstanding magnetic moment are 
those belonging to the valence electron, and in solution or in salts the 
atom is stripped of these valence electrons, leaving only ions with closed 
shells. For instance, Na or Cl atoms each contain an odd number of 
electrons, and are thus paramagnetic, but in NaCl or in solutions con- 
taining NaCl, the sodium and chlorine atoms respectively lose and 
capture one electron, yielding only closed configurations. The reason 
that solid pure alkali metals are not appreciably paramagnetic is rather 
complicated and will be considered in 80. 

There are, however, certain well-known places in the periodic table 
in which inner as well as outer groups of electrons are in process of 
formation. These cases are found in the iron, palladium, rare earth, 
and platinum groups, where respectively sub-shells of ten 3d, ten 4d, 
fourteen 4/, and ten 5d electrons have not been completely filled. 17 
Generally speaking, atoms in these groups lose all the valence electrons 
of their outer incomplete shells when in solution or in salt compounds, 
but not necessarily the electrons of the inner incomplete shells, so that 
paramagnetism may result. For instance, the Sm atom contains five 
4/, two 5^, and one 6s electrons, besides other electrons not in closed 
shells. This atom usually reacts trivalently, so that in, e.g., Sm 2 (S0 4 ) 3 , 
solid or dissolved, it loses the 5cZ and 6s electrons, but keeps the five 4/ 
ones. To quote Bohr, 17 'On the whole a consideration of the magnetic 
properties of the elements within the long periods gives us a vivid 
impression of how a wound in the otherwise symmetrical inner structure 
is first developed and then healed as we pass from one element to 
another.' 

The one case where one might expect the gaseous theory to apply 
better than usual is to salts or solutions containing the rare earths. 
The rare earths as a rule exhibit a trivalent behaviour, losing their two 

17 We assume the reader has at least a little familiarity with Bohr's well-known theory 
of the perodic table. See, for instance, N. Bohr, The Theory of Spectra and Atomic 
Constitution, Essay 3. The relation of magnetic properties to the structure of the periodic 
table has also been extensively discussed by Ladeiiburg, Zeits. f. Electrochem. 26, 263 
(1920); Zeita.f. Phys.Chem. 126, 133 (1927). 



IX, 58 AND RARK EARTH IONS 241 

5d and one 6s electrons as in the typical example of Sm cited above. 
Thus the paramagnetism of rare earth ions arises from the 4/ electrons, 
and such electrons are well in the interior of the atom, far more so 
than, e.g. the 3d electrons in the iron group, as the 4/ electrons are 
surrounded by eight electrons (viz. two 5s and six 5p) of higher principal 
quantum number, whereas the ions of the iron group of the type in- 
volved in paramagnetism have no electrons of principal qiiantum 
number greater than 3. Thus we must not become too surprised in 72 
at the complete failure of the ordinary gaseous theory in the iron group. 
The nearly identical chemical properties of the different rare earths, 
despite the fact that they have different numbers of 4/ electrons ranging 
from to 14, is direct evidence that these electrons are but little affected 
by the fields from other atoms, and hence by their neighbours in solu- 
tions or solid compounds. 

Before one can calculate the susceptibilities of rare earth ions it is 
necessary to know the values of the quantum numbers L, S, J to be 
used in the equations of 54-6. This information as to the 'lowest 
lying' spectral term is supplied by a theory of Hund. 10 When there 
are a number of equivalent electrons, the possible spectral terms are 
rather severely limited in number by the Pauli exclusion principle, and 
Hund supposes that out of the possible terms that of highest multi- 
plicity has the lowest energy. In case several values of L are possible 
when the multiplicity 2$+l has its maximum value, then the greatest 
L consistent with this S is assumed to give the least energy. All other 
states are assumed to have so much greater energy that they are not 
normal ones. For example, with two equivalent / electrons, the Pauli 
exclusion principle can be shown to admit the 1 S, 3 P, *D, 3 jP, 1 G, *H, l l 
and to rule out the 3 , *P, 3 A *F, 3 #, 1 H, 3 / states. Out of the former, 
the 3 // term alone is a normal state under Hund's assumptions. The 
normal states which he obtains with various numbers of equivalent 
/ electrons are shown in the table in the next paragraph. Generally 
speaking, Hund's assumptions are well confirmed spectroscopically, 
though in the particular case of the rare earths direct spectroscopic 
evidence is as yet wanting, so that here the only experimental con- 
firmation is by means of the magnetic theory itself. A first-order 
perturbation calculation by Slater 18 has confirmed theoretically Hund's 
assumptions in the case of the iron group, and similar confirming 
calculations could doubtless be made for the rare earths. 

It is, of course, necessary to know something about the effect of J as 
18 J. C. Slater, Phys. Rev. 34, 1293 (1929). 

3595.3 R 



242 THE PARAMAGNETISM OF FREE ATOMS IX, 58 

weU as of L, S as above, on the energy, so as to determine whether to 
use the theory of 54, 55, or 56. Hund 19 himself first calculated the 
susceptibilities under the assumption that the multiplets are very 
wide compared to kT. As discussed in 56 (Eqs. (18), (19), &c.), the 
component with minimum or maximum J is alone supposed to be 
a normal state depending on whether the multiplet is respectively 
'regular' or inverted, i.e. whether in the rare earth ions the number of 
4/ electrons is less or greater than 7. The results of Hund's calculations 
made by formula (10) without the term NOL are shown under the column 
headed 'Hund' in the following table and in Fig. 10. Various experi- 
mental values are shown for comparison. The entries throughout are 
what we shall call the 'effective Bohr magneton number', defined by 
the equation 



It is to be clearly understood that the p^ thus defined is a function of 
temperature if Curie's law x^const./T is not obeyed. Thus under 
Hund's assumptions /x eff should be independent of T 9 but not if instead 
the term NOL of (10) is important, or if it is necessary to use the 'inter- 
mediate formula' (16). The experimental values given in the table, also 
the theoretical values of Van Vleck and Frank (V. V. & F.), where 
different from Hund's, relate to room temperatures. The juxtaposition 
of theoretical and experimental values of /z off is merely one way of 
comparing the calculated and observed absolute susceptibilities, as /x eff 
is proportional to x*. A 'theoretical' or 'experimental' /x eff is obtained 
according to the kind of value of x which is used. Only when Curie's 
law is obeyed, is either the theoretical or experimental /z off defined by 
(22) to be identified with the permanent moment p, of the atom. 

Cassiopeium (Cp), of course, has a complete shell and so has no 
paramagnetism. Stefan Meyer's value 2-77 is for Pr++++ rather than 
Ce +++ , and its moderately good agreement with the values for Ce+ +f 
by other observers is in fair accord with the Sommerfeld-Kossel rule 
that ions with equal number of electrons often have very similar pro- 
perties. 20 The smaller and larger values for Sm and Eu in the column 
V. V. & F. correspond respectively to use of values 33 and 34 for the 
screening number a in (21), as will be elaborated in 59, 60. For the 

19 F. Hund, Zeits.f. Physik, 33, 855 (1925). 

20 Cabrera and Duporier 28 find a susceptibility for VrOg which gives an effective Bohr 
magneton number 2-24 for Pr+ + -' +, or 2-74 if the Weiss modification of (22) is used, in 
which T is replaced by T-f A. 



IX, 58 AND RARE EARTH IONS 243 

other ions it makes no appreciable difference which of these values 
is used. 



1 fon z] 



La 
Ce 
Pr 
Nd 
111 

Sm 

Ku 

Ucl 
Tfo 
l)s 
Ho 
Kr 
Tn 
Yh 



r<-^ 


EXPERIMENTAL EFF. BOHR MAGNETON 
NUMBER /* eff 27 






F. V. 






Z. & 










cfc 


Cabrera 


Meyer 29 


J. u 


Decker 


William* 


Term 


Hmid 


y\ 


Sulph. 


Sulph. 


Sulph. 


Sol. 


Oxide 


1 S 


0-00 


0-00 










diam. 


4/ 2 J^5/ 2 


2-54 


2-56 


2-39 


2-77 


2-37 


2-10 


0-23? 


4f a 3 // 


3-58 


3-62 


360 


3-47 


3-47 


3-41 


2-29? 


4/ 3 4 / 9 / 2 


3-62 


3-68 


3-62 


3-51 


3-52 


3-45 


343 


4 / 4 '%.. 


2-68 


2-83 












4/ 5 6 // 6 /2.. 


0-84 


(l-65J 


1-54 


1-32 


1-53 


1-63 


1-57 


4/ fi 7 ^ fl 


000 


J3-40) 


3-61 


3-12 












\3-51 ) 












4/ 7 *jS' 


7-94 


7-94 


8-2 


8-1 


7-81 


7-86 


7-46 


4/ 8 7 F 8 


9-7 


9-7 


9 6 


90 


9-4 


9-8 




4/ f| (> // J5 / 3 


10-6 


!()() 


10-5 10-0 




10-9 


10-0 


4/ 10 r '/ 8 


KM) 


KM) 


10-5 , 10-4 


10-3 


10-4 


10-1 


4/ 11 <1 / 1B y < , 


IM> 


9 6 


9 5 , 9-4 


96 


9-5 


9-2 


4/ 12 :< // 6 


7 (> 


76 


7-2 7-5 








*/ I3a F 7 / 2 i 4-5 


4-5 


4-4 46 


4-4 


4-5 




4/ M ^S 00 


00 


diam. 


diam. 


1-2? 


0-49 ? 



The experimental results call for some comment, especially in view 
of the different materials used. Cabrera, St. Meyer, and Zernicke all 
made their measurements primarily 28 on solid hydrated sulphates of 
the form M 2 (8O 4 ) 3 . 8H 2 O. Decker used solutions in which sulphates 
were dissolved, except that nitrates instead were dissolved for La, Ce, 

21 Throughout tho balance of the chapter we omit attaching three plus signs to the 
abbreviations for the chemical elements, and it is to bo uniformly understood that \ve 
are concerned with trivalcnt ions unless otherwise stated. 

22 B. Cabrera, Complcs Rcndus, 180, 668 (1925). 

23 St. Meyer, Mtys. Kelt*. 26, 51, 478 (1925). 

24 Zernicko and James, J. Amcr. Chcm. Soc. 48, 2827 (1926). 

25 11. Decker, Ann. dcr Physik, 79, 324 (1926). 

26 E. H. Williams, J'ltytt. Kev. 12, 158 (1918); 14, 348 (1919); 27, 484 (1926). 

27 In the calculation of the effective Bohr magneton number from experimental data 
there may bo slight differences in tho method of computation because different writers 
use different values of tho atomic constants in (22), and especially different estimates of 
tho diamngnotic corrections. Wo have not attempted to iron out these small inconsisten- 
cies, as we give in tho tablo the effective magneton numbers quoted by Hund from 
Meyer's and Cabrera's older data, and by Decker from his own work, while tho calcula- 
tion is our own for tho other data, including that given on p. 245. Zermcke and James 
make tho claim that tho proper diamagnotic correction raises Meyer's value for the molar 
susceptibility of Sin to 960 xlO' 8 , which gives a magneton, number 1-51 in improved 
agreement with other observations. 

28 Cabrera, however, used a ponta- rather than octohydratc for Ce. Meyer used oxides 
rather than sulphates for Tb and Tu. 

B2 



244 THE PARAMAGNETISM OF FREE ATOMS IX, 58 

and Pr. The good agreement of the values for solid sulphates with 
those for solutions and with the theoretical values for gases must mean 
that even in solids the ions are virtually free as far as the 4/ electrons 
are concerned. The similarity of the results with oxides, which have 
been measured not only by Williams but also by Cabrera and Duperier, 29 
to those for sulphates, is especially remarkable, as oxides are firmer 
chemical compounds than sulphates, and lack the 'magnetic dilution* 
of hydrated sulphates. The strikingly low value which Williams finds 
for Ce presumably means that he had largely 30 Ce0 2 rather than Ce 2 3 , 
as Ce ++++ should be diamagnetic like La+++, or else possibly that by 



10 



6 - 




LcTCe Pr" Nd 'ill Sm Eu Gd Tb Ds Ho Er Tu Yb Cp 
FIG. 10. 

exception the inter-atomic forces on the 4/ electron are here so large 
that the gaseous theory no longer holds. The rather wide discrepancies 
between some of the experimental values on the sulphates themselves 
are probably due to use of different samples or preparations of the rare 
earth salts, since the experimental error is due far more to difficulties of 
chemical purification than of magnetic technique. Zernicke and James 24 
suggest that close agreement between two different observers docs not 
necessarily imply that more weight should be attached to their results, 
but may simply arise because they used salts from the same original 
source of preparation. Cabrera and Duperier, 29 on the other hand, find 
remarkably consistent and reassuring values even with salts of different 

29 Cabrera and Duperior, Comptes Rendus, 188, 1640 (1929). 

30 Although trivalent valencies are the rule for the rare earths, there are exceptions 
near the beginning of the sequence, and the oxides CoO 2 , Ce 2 () 3 , Ce 4 O 7 are all known to 
exist. See Hevesy, Die seltenen Erdcn vom Standpunkte des Atombaues, pp. 52, 74. 



IX, 58 AND RAKE EARTH IONS 245 

preparation, which are given in detail below. Impurities doubtless 
explain the paramagnetic moments sometimes reported for Cp, also the 
abnormally low value found by Williams for Pr 2 3 , but not confirmed 
by Cabrera and Duperier. 29 

Additional Experimental Values. Unfortunately space compels us to omit in 
the table the older measurements by Du Bois, Urbain, and Wedekind. For a good 
summary of this older work, and references, see Zernicke and James, I.e. 2 * Roughly 
speaking, the older determinations agree fairly well with the newer work, but 
differ erratically in a few instances. Measurements in the Leiden laboratory on 
Gd, Er, Ds, Co will be cited on p. 254. The data of Freed 31 yield /A efl 1-57 for 
Sm at room temperatures. Zornicke and James measured anhydrous as well as 
hydrated Gd 2 (SO 4 ) 3 , finding the same effective magneton number 7-8 in both cases. 

It is particularly interesting and noteworthy that in addition to Cabrera's 
earlier work 22 on hydrated sulphates given in the table, Cabrera and Duperier 29 
have recently also measured the anhydrous sulphates and oxides, often from 
different sources of preparation. Their results for anhydrous sulphates and oxides 
yield in (22) the following effective Bohr magneton numbers: Pr, 3-47t, 3-32*; 
Nd, 3-52t, 3-40*; Sin, l-58t, 1-50*; Eu, 3-54t, 3-32H ; Od, 7-9I", 7-7"; Tb, 9-6t, ; 
Ds, 10-3t, 10-2"; Ho, 10-4t, 10-2t; Er, 9-4t, 9-3t; Tu, 7-Ot, 6-8t; Yb, 4-3t, 3-9t. 
The first and second values are respectively for anhydrous sulphates and oxides, 
and the t * II denote preparations obtained respectively from Auer v. Welsbach, 
Prandtl, and Urbain. Zernicke and James, and Williams used salts prepared 
respectively in the New Hampshire and Illinois laboratories, while Cabrera's, 
Meyer's, and Decker's measurements reported in the table are with salts prepared 
by Auer v. Wolsbach (except that Decker's, Ce, Pr, Nd were by Prandtl). 

]ii case the temperature variation is in accord with the Weiss generalization 
X = C7(T-fA) of the Curie law, it is probably preferable to deduce the effective 
Bohr magneton number from experiment by using a modified form of (22) in which 
T is replaced by 2 7 +A, provided A is really due to inter-atomic forces rather than 
primarily (as in 8m, Eu) to natural multiplet structures. If we apply this modifica- 
tion to Cabrera's measurements on the anhydrous sulphates and oxides, the only 
data refined enough to permit use of this modification, one obtains the following 
effective Bohr magneton numbers instead of those given in the preceding para- 
graph: Pr, 3-76, 3-71; Nd, 3-79, 3-71; Gd, 7-9, 7-9; Tb, 9-7, ; Ds, 10-5, 10-5; 
Ho, 10-6, 10-5; Kr f 9-6, 9-5; Tu, 7-5, 7-2; Yb, 4-8, 4-5. (In the cases of Pr, Nd, 
Tu, Yb, Cabrera and Duperier find that l/^ is not really a linear function of T as 
implied by the Weiss formula, but as a rough approximation we take A equal 
to the value of (d log x/dT)~^T at room temperature.) 

59. The Special Cases of Europeum and Samarium 

(The trivalent ions are to be understood throughout, Cf. note 21.) 
It is seen that Hund's calculations agree with experiment, within the 
discrepancies of the latter among themselves, except for Sm and Eu, 
where his theoretical values are much too low. Hund 19 himself, and 
also Laporte, 32 have suspected that this is perhaps because the multiplet 

31 S. Freed, J. Amer. Chem. Soc. 52, 2702 (1930). 
aa O. Laporto, Zeits.f. Physik, 47, 761 (1928). 



246 THE PARAMAGNRTISM OV FREE ATOMS IX, 59 

intervals in Sm and Eu are not really infinitely large compared to kT. 
This requires us to make a more careful examination of multiple! 
structures by means of (18). This formula shows that the various com- 
ponents are not evenly spaced, and instead crowd together for small 
values of J, as by (18) the interval between two consecutive components 
J, J+l equals A(J+l). Now if the multiplicity 2#+l is fairly large, 



I _ 



4- 
3- 




(i = kTat293*C*205cm H ) 

FIG. 11. 

and if L and 8 are nearly equal, so that J miu = \L S\ is small, the 
separation between the components with the two lowest values of J 
may well be small compared to kT, even though the 'over-all' multiplet 
width is considerably greater than kT. This is just the situation which 
arises par excellence in Eu, and to a lesser degree in Sm. In Eu the 
interval between the lowest multiplet components is only 1/21 of the 
over-all width, as for a 1 F term 

|[^max(^max+l)--^ninWnin+l)]- IX7-0= 21, J mln +l - 1. 

The corresponding value for Sm is 7/55. The multiplets for Eu and Sm 
are illustrated in Fig. 11 (Grotrian diagrams). The cases of Pr and Tb 



IX, 59 AND RAKE EAKTH IONS 247 

are also shown for comparison as illustrative of ions in which the lowest 
interval is an appreciable fraction (5/11 in Pr, 6/21 in Tb) of the total 
width and in which Hund's calculations are confirmed experimentally. 
To proceed further it is now necessary to evaluate quantitatively the 
multiplet intervals by determining the screening constant a in (21) from 
X-ray data. This data must be taken from atoms heavier than the 
rare earths, as X-ray emission lines terminating in the 4/ level are 
observed only after the 4/ group is complete, so that there is an initial 
concentration of electrons in higher levels. 33 Such a method is, of course, 
rather indirect, but a check on its accuracy is furnished by comparison 
with the iron group. In the latter, optical data are available on screening 
constants for incomplete shells as well as X-ray data for complete ones, 
and it is found that the screening constants for triply charged ions of 
the former never differ by more than one or two units from the standard 
values of the latter. This is naturally in a sense a special case of the 
well-known Bowen-Millikan result on the similarity of X-ray and optical 
multiplets. The uncertainty in the X-ray determinations of a is, how- 
ever, somewhat larger in the rare earth (4/) than in the iron (3d) group. 
Wentzel gives 34^4 as the value yielded by these determinations in 
the case of the 4/ shell, 34 while Coster 35 gives 33. In the table and 
elsewhere we have generally given the effective magneton numbers 
obtained both by taking or == 34 and a = 33. The screening number, as 
a matter of fact, is more accurately determined by the magnetic 
measurements themselves than by the X-ray data. This will be shown 
particularly clearly when we study the temperature variation in Sm 
( 60). Screening numbers in the ranges 30-32 and 35-38 definitely will 
not fit the experimental magnetic measurements on Sm, even though 
within Wentzel' s estimate of the X-ray precision. Substitution of the 
value er= 34 in (21) makes the over-all multiplet width 5,360 cm.- 1 for 
Eu and 7,320 cm.- 1 for Sm. The corresponding values of the intervals 
between the two lowest components are 5,360/21 = 255 cm." 1 and 
7,320(7/55) 932 cm." 1 , while somewhat higher values are obtained if 
we use or = 33. In either case these intervals are not very great com- 

33 Considerable work is in progress 011 the spectra of rare earth salts, whose analysis 
may ultimately yield the desired multiplet structures directly. Soo S. Freed and F. H. 
Spedding, Nature, 123, 525 (1929); Phys. Eev. 34, 945 (1929). A certain amount of 
X-ray data is available for some rare earths themselves, but this is for the neutral atom 
rather than trivalont ion and also the multiplet structures have not been adequately 
resolved and analysed; see E. Lindberg, Zeits.f. Physik, 50, 82; 56, 402 (1928-9). 

34 G. Wentzel, Zeits.f. Physik, 33, 849 (1925). 

36 D, Coster, in Muller-Poulliets' Handbuch der Physik, ii. 2057. 



248 THE PARAMAGNETISM OF FREE ATOMS IX, 59 

pared to kT, inasmuch as kT is 0-698T cm.- 1 when expressed in wave 
numbers, and is thus about 200 cm." 1 at room temperatures. It is 
therefore necessary to use the accurate 'intermediate' formula (16) 
instead of the asymptotic one (10). The explicit forms which (16) takes 
for TCu and Sm are respectively 



0-1241 
xT 

) x mol (23) 

0-1241 



~~ yT 

where x and y are respectively 1/21 and 1/55 of the ratio of the over-all 
multiplet width (in ergs) to kT. Ita--= 34, then x = 365/T 7 , y = 191/T, 
while with a = 33 the values are x = 418/T, y = 220/7 7 . We give the 
formulae in a form not requiring these specialized values of x and y to 
allow for the fact that in the future the multiplet widths will doubtless 
be known more accurately. 

When formula (23) is used, effective Bohr magneton numbers of 3-51 
and 1-65 respectively are obtained for Eu and Sm if a =34, while 
3-40 and 1-55 result if or = 33. Either of these values is in as good 
agreement with experiment as for the other members of the rare-earth 
series, 36 so that the agreement is now quite satisfactory for the entire 
series. Reference to the table in 58 shows that the confirmation in 
Eu and Sm is particularly close if one accepts the other experimental 
values in preference to St. Meyer's. It is altogether probable that his 
values for Sm and Eu are too low, 37 especially in view of the very 
careful recent work of Cabrera and Duperier. 

Calculations for the other rare earths can also be made by means of 

38 These theoretical values were given by Miss Frank and the writer, Phys. Rev. 34, 
1494, 1625 (1929), and clearly show that explanation of tho susceptibilities of Eu and 
Sm does not require modification of tho Bohr-Hund configurations for these ions. Such 
modifications, whereby tho numbers of 4/ electrons in Sm'* + + and Eu ++ f were taken to 
be different from 5 and 6 respectively, wore formerly debated because Hand's original 
calculations did not agree with experiment ; cf . Hevesy, Die seltenen Erden, p. 44 and 
references. 

37 Cf. especially Cabrera and Duperier's additional recent measurements given on 
p. 245, also Zernieke and Jamos' contention given at the end of note 27 . Their value for 
Sm should presumably be quite accurate, as they used the same preparation of Sin as 
employed for an atomic weight determination (Stewart and James, J. Amer. Chem. Soc. 
39, 2605, 1917). Meyer 23 intimates that tho usual values for Eu may be high on account 
of contamination by Gd, but this is disputed by Cabrera, 29 who states that in his own 
work the impurity is Sm, which would lower rather than raise the susceptibility. 



IX, 59 AND RARE EARTH IONS 249 

the accurate formula (16) instead of the asymptotic one (10), and the 
resulting values are given in the column of the table marked V. V. & F. 
The corrections other than those for Sm and Eu already studied are 
important only in the rare and magnetically unmeasured element 111. 
The absence of any revision in the bottom half of the table is because 
here the multiplets are inverted, so that the relatively narrow intervals 
separate only components which are excited rather than normal states. 
The diagram for Tb, for instance, is similar to that for Eu turned upside 
down (cf. Fig. 11). 

Jt is to be emphasized that in Eu and Sm it is very essential that 
one does not omit the commonly forgotten term / in (1C). Laporte 32 
showed that without this term the effective Bohr magneton number for 
Eu is only 1-7 instead of 3*51. As the susceptibility varies as the square 
of this number, the second-order Zeeman effect hence contributes about 
3/4 of the susceptibility of Eu. The kinematical origin of such an 
abnormal situation can readily be comprehended from Fig. 9, p. 230, 
for if J is very small compared to S and L, as in Eu, then sin( J, S) is 
substantially unity, and the component b-c in Fig. 9 is much larger 
than the ordinary component a -b. 

60. Temperature Variation in the Rare Earths The Gyro- 
magnetic Ratio 

The dependence on temperature for Sm and Eu is quite different from 
that given by Curie's law, as it is necessary to use the 'intermediate' 
formula (10) instead of (10). The variation with temperature of the 
effective Bohr magneton number for Sm computed by Miss Frank 38 
from (23) and (22) is exhibited in the table on p. 250. 

The experimental values of various observers are included for com- 
parison. The agreement with experiment is gratifying when it is 
remembered that the observations are made on solids rather than the 
theoretical ideal gas state. In particular, the ions in the oxides are 
doubtless far from free. The deviations from theory at low temperatures 
revealed by Freed 's measurements need not cause concern, as they are 
to be attributed to 'cryomagnetic anomalies', i.e. distortion by inter- 
atomic forces in the solid. We shall elaborate in Chapter XI the idea 
that the values for free ions should apply only as long as the work 
required to 'turn over' an ion against the inter-atomic forces is assumed 

38 A. Frank, Phys. Rc,v., in press. In Proc. Lon. Phys. Soc. 42, 388 (1930), W. Suek- 
srnith also has noticed that tho anomalies in iho temperature variation in Sm and Eu 
arc to be attributed to Van Vleck and Frank's second-order Zooman correction . 



250 THE PARAMAGNETISM OF FREE ATOMS IX, 60 

small compared to kT> a supposition clearly not warranted if the 
temperature is too low. 

BOHU MAGNETON NUMHEKS FOH SML' f '" 





/i eff Theory 


H ett Experiment 








William*** 


Freed? 1 


Cabrera cfc Dup. 


T 


(7 -= 33 


(7 -- 34 


(ofMe) 


(Injd. </.) 


(oxide) 


(anh. .?.) 


(>K 


0-84 


0-84 










20 


0-91 


0-92 










74 


1-07 


09 




0-91 






85 


1 -09 


12 




0-96 






112 


1-16 


20 | 


1-08 


, , 




123 


1-18 


23 


1-08 








170 


1-29 


35 




1-20 






205 


1-37 


43 




1-35 






240 


1-44 


52 




1-44 






293 


1-55 


65 


1-58 


1 -57 


1-50 


1-58 


375 


1-73 


1-85 


1-77 




1-69 


t . 


400 


1-78 


191 






1-75 




500 


2-00 


245 






1-97 




543 


209 


2-25 


2-14 




2-06 




600 


2-20 


238 






2-17 




800 


2-58 


278 




. 






1000 


2-91 


3-14 












A particularly interesting feature in Sm is that somewhat above room 
temperatures the susceptibility should reach a minimum and then 
increase slowly as the temperature is raised still further, in marked 
contrast to the usual Curie decrease with increasing temperature. This 
is shown in Fig. 12. 

From (23) Miss Frank finds that the temperature at which this 
minimum is located is approximately T min r^ 0-0628(Z o-) 4 , e.g. 386 K. 
with a = 34 or 444 with a 33. Precise experimental observation 
of this temperature would thus determine the screening constant a 
accurately, but such precision would be difficult as the curve is so flat 
near the minimum. The measurements of Cabrera and Duperier 40 do, 
however, definitely show that there is experimentally a minimum 
somewhere between 350 and 425 K., perhaps at 400. Furthermore, 
Williams finds a lower susceptibility at 375 than at either 293 or 
543. This is a striking confirmation of an unusual feature of the 
theory. 

39 The experimental values by Cabrera and Dupericr given in the tables for Sm and 
Ku at various temperatures arc obtained from a personal letter from Professor Cabrera 
giving more detail than rof. 29 . 

40 The writer is indebted to Professor Cabtvra for personal communication of this fact, 
not stated in the paper of Cabrera and Duporier. 



IX, 60 AND RARE EARTH IONS 251 

The theoretical effective Bohr magneton numbers for Eu at various 
temperatures are shown in the following table, along with the data of 
Cabrera and "Duperier. 



1900 - 



Cabrera & Dup (Oxide) 
O Cabrera &Dup(Anhsul) 
X Yreed(\\yd sul) 
A Williams (Oxide) 
n Other observers 




EiT* 

Cabrera & Dup (Oxide) 
O Cabrera & DupfAnh sul] 



100 200 300 400 SOO GOO 



Fic. 12. 




KKKKITIVK 



MUSNKTON NI>MI<UUS FOR Ku' 



l^ K Theory J ((\tbrrta <( Dup.f* 

T a 33 I a -- 34 | oxide j f/>j7*. s. 



0" K 


00 


00 






20 


1-07 


1-J5 j 




70 


201 


2-14 




100 


2-38 


2-53 






200 


309 


3-20 


, 




293 


3-40 


3-51 


3-34 


3 53 


400 


305 


3-75 


3 57 


3 7.-. 


470 


3-77 


3-81) 


3 61) 


3-8{) 




402 


4 14 




4-11 



The corresponding curves of susceptibility against temperature are 
shown in Fig. 12. Unfortunately no experimental data arc available for 
Eu at low temperatures, which would be particularly interesting, as 
the effective Bohr magneton number should approach zero at T ~ 0. 
This, however, does not mean that the paramagnetic susceptibility of 
Eu vanishes at the absolute zero. When the inner quantum number ./ 
vanishes, as in the lowest multiplet component of Eu, the first term of 



252 THE PARAMAGNETISM OF FREE ATOMS IX, 60 

(10) disappears, to be sure, but the second term Not is abnormally large. 
Thus in the vicinity of T = Eu should have a molar susceptibility 
about 7-lxlO- 3 ((j=33) or 8-lxlQ- 3 ((7=34) independent of tem- 
perature. 

Reference to Fig. 12 shows that use of a screening number 34 probably 
gives results in better accord with experiment than does 33 as regards 
the temperature at which the minimum is located in Sm, although the 
precise location of the experimental minimum is rather uncertain. Cer- 
tainly a = 34 fits the absolute values of the experimental susceptibility 
for Eu better than a = 33, if the anhydrous sulphate is considered a 
closer approach to gaseous behaviour than the oxide. In Sm the abso- 
lute values, in distinction from location of minimum, are perhaps in 
better accord with a 33 than with cr = 34. This need not be con- 
sidered a contradiction, as any departures from gaseous behaviour tend 
to lower the susceptibility, and hence make the apparent a less than 
the true cr. The screening constant a should be somewhat greater (per- 
haps 0-3 more) for Eu than for Sm, due to shielding by the additional 
4/ electron. As Sm has a lower paramagnetic susceptibility than the 
other rare earths, the omitted correction for the diamagnetism of the 
cation is more important than usual here, though still small. 

This correction would raise the experimental points in Fig. 1 2, but cannot be 
estimated precisely. It is perhaps about 30 X 10 6 per mol. This corresponds 
approximately to Pauling's estimate 38x 10 6 of the diamagnetism of La 1 ' + , 
which resembles Sm ' ++ except for the absence of the five 4/ electrons and a corre- 
sponding diminution of % by five units. The 4/ electrons contribute less to the 
diamagnetism than the surrounding 5s, 5/>, shells. The diamagnetic effect of 
the 4/ group is perhaps roughly counterbalanced by the fact that it screens the 
outer shells only imperfectly, thus contracting the 5 quantum orbits. If this is 
true, the diamagnetism of La and Sm is of the same order. The estimate 28 x 10~ 6 
furnished for Sm 1 ' ' by Slater's screening constants ( 50) is also in accord. The 
low magnitude 20 x 10" 6 found experimentally by Meyer for the susceptibility 
of HfO 2 suggests that even the estimate 30 X 10' 6 is excessive for Sm. (Hf 4+ 
resembles La 3 ' except for addition of the complete shell of 14 4/ electrons.) As the 
molar susceptibility ofO is 13 X 10" 6 , his determination would demand that 
Hf 4 1 be without appreciable magnetism, but possibly his low value is due to 
counterbalancing of the true diamagnetism by paramagnetic impurities. 

It is interesting to contrast the temperature coefficients ~x~ l dx/dT 
of the susceptibilities of Sm and Eu at room temperatures with the 
value 1/293 predicted by Curie's law. The theoretical value for Sm is 
1/1517 (a 33) or 1/2525 (cr 34), while the experimental determina- 
tions are 1/1GOO (Williams 26 ), 1/1700 (Zernicke and James 24 ), 1600 
(Freed 31 ). For Eu the corresponding theoretical values are 1/542 
(a = 33) and 1/525 (a = 34), while Cabrera and Duperier find 1/522 and 



IX, 60 AND RARE EARTH IONS 253 

1/500 for the oxide and anhydrous sulphate respectively. The agree- 
ment is as good as can be expected. The abnormally small value for 
Sm is because increasing temperature increases the concentration of 
ions in the states with larger values of the inner quantum number J 
and hence larger magnetic moments. This effect tends to offset the 
decrease with increasing temperature due to the factor T in the usual 
Curie denominator. In Eu the lowest state J = has an abnormally 
large second-order term Noi, so that promotion to higher values of J 
does not increase the susceptibility as much as in Sm. This is reflected 
in the temperature coefficient being nearer the ordinary Curie value in 
Eu than in Sm. 

The rare earths other than Eu, Sm, and 111 should conform very 
approximately to Curie's law and have a temperature coefficient about 
1/293 at room temperatures, as for these other ions the difference 
between the columns headed Hund and V. V. & F. in the table of 58 
is negligible. In Nd, for instance, Miss Frank's calculations show that 
the correction for the effect of the multiplet structure only changes this 
coefficient from 1/293 to 1/303, while for the remaining ions Hund's 
calculations apply still more accurately and the changes are smaller 
still. Some experimental results on the reciprocal of the temperature 
coefficient in the vicinity of 293 K. for ions in various salts are given 
in the following table. 

Co Pr Nd Gd Tb Ds Ho Er Tu Yb 



Anh. Sulph. Cabrera 41 . . 344 341 292 296 304 301 304 322 369 

Hyd. Sulph. Z. & J. 24 290 358 348 305 327 . . 320 252 . . 292 

n ., /Cabrera 41 (438) 366 344 306 316 312 307 308 330 390 

UXlrt \Williams 8a .. .. 337 305 .. 308 .. 306 .. 

The discrepancy between different observations on the same salt shows 
that the experimental error is considerable. The deviations from 293 
are, in most cases, relatively small, and, when real, are doubtless caused 
primarily by inter-atomic forces in the solid, and so shed no particular 
light on the ideal gas theory, but do reveal how much the orientations 
of the ions are constrained by inter-atomic forces. It is usually found 
that the temperature variation of the susceptibility can be fairly well 
represented at least over a range of a few hundred degrees and barring 
possible anomalies at very low temperatures by the Weiss generaliza- 
tion x = <7/(T+A) of the Curie formula. The value of A is approxi- 



41 Computed from Cabrera and Duperior's empirical formulae x k 
given in ref. 20 . An earlier paper (J. de Physique, 6, 252, 1925) gives other temperature 
coefficients differing by 5 per cent, or so in some cases. Zernicke and James report a value 
283 for anhydrous Gd 2 (SO 4 ) 8 . 



254 THK 1> ARAM AON RT18M OK FRHE ATOMS IX, 60 

mately -x( d xl<lT)~ l T, and is thus about 50 for, e.g. Nd. Tf the 
inter-atomic forces are adequate to produce this large A in Nd, they 
should produce deviations between theory and experiment in Sm at 
low temperatures, of about the order of magnitude found experimentally 
(p. 250). 

Because T occurs in the denominator, measurements at very low 
temperatures arc particularly desirable. At Leiden Ds 2 O 3 , 42 CeF 3 , 43 and 
Er 2 (S0 4 ) 3 8H 2 44 have been measured clown to the temperature of liquid 
hydrogen (c. 14 K.) and Gd 2 (S0 4 ) 3 8H 2 45 down to that of liquid helium 
(1-3 K.). Usually the law x = C/(T+A) is found to hold remarkably 
well down to the lowest temperature studied, with the following values 
of A: Ds 2 3 , 16; CeF 3 , 02; Er 2 (SO 4 ) 3 8H 2 O, 1-9; Gd 2 (S0 4 ) 3 8H 2 O, 0-0 (or 
possibly 0-26). In CeF 3 , however, pronounced deviations appear below 
65 K. The corresponding experimental values of the effective Bohr 
magneton number /i cff are 10-6, 2-51, 9-0, 7-8, in quite satisfactory agree- 
ment with the theoretical values 10-0, 2-54, 1MJ, and 7-9 respectively. 
These results yield temperature coefficients 1/309, 1/355, 1/295, 1/293 
respectively for Ds 2 O 3 , CeF 3 , Er 2 (SO 4 ) 3 8H 2 (), (ad 3 ) 3 (S() 4 ) 2 8H 2 O at room 
temperatures. The agreement with Zernicke and James's value 1/252 
for Er 2 (S0 4 ) 3 8H 2 O (see table) is poor, but the latter observers do not 
claim a high degree of precision, so that 1/295 is doubtless very close 
to the true value. The measurements at low temperatures are parti- 
cularly interesting because they reveal the order of magnitude of the 
inter-atomic forces tending to orient the 4/ orbits in the rare earths, 
which turn out to be surprisingly small. When the multiplet structure 
is so wide that the Langevin formula should hold for the ideal gas state, 
as is the case in the rare earths except for III, Sm, Eu, the theoretical 
considerations of Chapter VIT show that departures from the Langevin 
or Curie law should first be expected in a solid when the temperature 
becomes so low that the energy required to 'turn over' an atom against 
the inter-atomic field becomes comparable with kT. Thus A'A is a 

42 Onnes ami Oosterlmis, Leiden Communications, 129b, 132 c. 

43 W. J. do Haas and C. J. dorter, Leiden Communications, 210o, or Proc. Amsterdam 
Acad. 33, 949 (1930). 

44 W. J. cle Haas, K. C. Wiorsma, and W. H. Capcl, Leiden Communicationa, 201 b or 
Proc. Amsterdam Acad. 32, 739 (1929). 

45 H. R. Woltjor, Leiden Communications, 167 b; H. R. Woltjer and If. Kamorlingh 
Oimcs, ibid. 167c; earlier work, mostly at higher temperatures, by Oinios and Perrier 
and by Oimes and Oostorhuis, Leiden Communications, 122 a, 129b, 140d. Jackson and 
Onnes find that gadolinium ethyl sulphate obeys Curie's law x Vl^ down to T = 14 K., 
the lowest temperature they employed for this material (Leiden Communications, 168 a 
or Comptes Rendus, 177, 154, 1923) ; the value of C yields an effective magneton number 
7-5, nearly the same as for the hydrated sulphate. 



IX, 60 AND RARE EARTH IONS 255 

measure of the orientation energy in the crystal. In the case of Nd 
ions, for instance, this energy is of the order 30 cm.- 1 , as here A = 50, 
while kT/hc^Q-TT cm.- 1 . The fact that the unmodified Langevin 
formula holds so beautifully for hydrated gadolinium sulphate right 
down to the temperature of liquid helium is a consequence of the fact 
that the Gd ion is in a 8 $ state and has a paramagnetic moment only 
in virtue of the spin, so that questions of orbital dissymmetry do not 
arise. This point will be elaborated in 73-74. That Ds 2 3 and OeF 3 
have a larger A than Er 2 (SO 4 ) 3 8H 2 is probably because hydration 
causes 'magnetic dilution', and also because oxides and fluorides are 
firmer compounds than sulphates. 

The Gyromagnetic Effect. Besides the ordinary measurements on 
susceptibilities, the theory for the rare earths is confirmed by evidence 
of another sort, viz. the limited amount of data available on the gyro- 
magnetic effect 46 for those materials. To magnetize a body to the 
extent demanded by our formulae, it is necessary to supply the atoms 
with angular momentum. The atoms can secure this angular momentum 
only by stealing it from the body as a whole. Such a theft demands 
that the body acquire a mass rotation if it is at rest before the field is 
applied. (The angular momentum supplied by the field can be shown 
very generally to be only of the order of magnitude corresponding to 
the Larmor precession, 47 regardless of whether his theorem is applicable. 
Hence the field only supplies a small fraction A: of the necessary angular 
momentum, where & is of the order of the ratio of diamagnetic to para- 
magnetic susceptibility.) A free atom's angular momentum in the direc- 
tion of the field is Mhl"2ir 9 if measured now in ordinary rather than 
quantum units. The total angular momentum demanded per c.c. by 

the atoms is thus N ~ e~ w ^ T e-'/* r . As W - 



this expression may be simplified as \\\ passing from (13) to (10). The 
complication of a term analogous to Not in the susceptibility formulae does 
not enter in dealing with angular momentum, as the angular momentum 
is a constant of the motion not perturbed by the field except for a small, 

46 For an account of tho ordinary gyromagnetic experiments and references, seo 
Stoner, Magnetism and Atomic. Structure, Chap. VIII. This, however, is not recent enough 
to include Sucksmith's experiments on paramagnetic materials. Another summary, by 
Weiss, is given in tho report of tho 1930 Solvay Congress. 

47 This point has caused considerable uncertainty in. the literature, as it has sometimes 
been incorrectly conjectured that tho field might supply an appreciable fraction of the 
angular momentum, thus invalidating the gyromagnetic experiments. The writer hopes 
to discuss this subject more fully in a future paper. 



256 THE PARAMAGNETISM OF FREE ATOMS IX, 60 

essentially diamagnetic correction which we have neglected (viz. the 
difference between p yi and 2 m ip|& m an analysis such as that in 8), 
The ratio 6 of the angular momentum to the magnetic moment yll is 
thus found to be 

2mc 2 



after use of the formula (16) for x- It can be shown that the expression 
given by (24) is also the same as the ratio H/Q, in the converse (Barnett) 
experiment on magnetization by rotation, where H is the magnetic field 
which would produce the same magnetization as rotation of the solid 
with an angular velocity 1. 
If the multiplet is small compared to kT, (24) reduces to 

2mc[28(S+l)+L(L+l)] 
'" 



(cf . Eq. (6) ). If it is large compared to kT, and if the terms proportional 
to otj may be neglected, then 6 = Zmc/gje, where g tl is the ordinary 
Lande {/-factor. In dysprosium the theoretical value of g if is 1/33. 
Sucksmith's determination 48 of the gyromagnetic ratio for Ds 2 3 
yields g = 1-280-07. The agreement is especially gratifying when 
it is remembered that gyromagnetic experiments are vastly more 
difficult in paramagnetic than in the ferromagnetic bodies usually 
measured. 

In Sm and Eu it is, of course, necessary to use the 'intermediate' 
formula (24) without simplification, and the gyromagnetic ratio should 
vary with temperature. The numerical magnitudes of the values of 
(24) appropriate to various temperatures will be found in Miss Frank's 
paper. 38 At T = the ratio vanishes for Eu, while at room temperatures 
= 0-270 mc/e if a = 33 or 0*306 mc/e if a = 34. Measurements bravely 
undertaken by Sucksmith 49 on Eu at T = 293 have not yet achieved 
quantitative accuracy but definitely favour in a qualitative way these 
values as opposed to the higher value 1-33 mc/e which would be ob- 
tained if one forgot the terms proportional to otj in (24) contributed 
by the second-order Zeeman effect. 

Gyromagnetic measurements are at present wanting on rare earths 
other than Ds and Eu. 

W. Sucksmith, Proc. Roy. Soc. 128A, 276 (1930). 

49 W. Sucksmith, paper presented at 1930 meeting of British Association. 



IX, 61 AND RARE EARTH IONS 257 

61. Saturation Effects 

Hitherto we have supposed the field strength sufficiently small so that 
only the portion of the moment per c.c. which is linear in the field H 
need be retained. However, in the limiting cases of multiplets which 
are exceedingly narrow or wide compared to kT there is no difficulty 
in obtaining closed expressions for x even when the latter cannot be 
treated as independent of //. Namely, the right-hand sides of (7) and 
(13) can be summed without the necessity of expanding the exponents 
in scries as previously. The denominator of (13) is summed by making 
the substitution x = &uP u \ kT and using the elementary formula 
x~ J (l-\-x+x 2 +...+x ZJ ) = (x~ J x j !l )/(l x) for the sum of a geometric 
progression. Differentiation of this formula with respect to x yields 
a relation which sums the numerator of (13). One thus finds that 

(13) becomes / r.om 

M^NJtfB^f*), (25) 

with the abbreviation 



In Eq. (25) we have given the formula for the magnetic moment M n 
per unit-volume instead of the susceptibility. This difference is trivial, 
as M Jf = HX- It is convenient to have a name for the function (26), 
and so we shall call it a Brillouin function, as it was employed in the 
new quantum mechanics by Brillouin, 50 although also previously used 
111 the old theories by Debye 51 and others. One can also similarly 
evaluate Eq. (7) for narrow multiplets accurately, which becomes 
Mjf - 2NSpB H (2Spn/kT)+NLpB L (LpH/kT). This formula, however, 
is not especially useful, as temperatures low enough to permit experi- 
mental production of appreciable saturation effects do not warrant the 
assumption of multiplet intervals small compared to kT. 

When J becomes very great and jS is imagined to become small to 
keep flJ finite, the Brillouin function passes over asymptotically into 
the classical Langcvin function L(x) = coth^ (I/a:) in the following 
way, \imBj(y) L(y). The saturation moment predicted by (25) and 

J = oo 

(26) is NJgp, or N(J/J+ l)Wft where /* efl is the effective Bohr mag- 
neton number for weak fields, defined in the fashion (22) applied to (10). 

50 L. Brillouin, J. tie Physique, 8, 74 (1927). 

61 P. Debye in Marx, Handbuch der Radiologie, vi. 713; also Stoner, Magnetism and 
Atomic Structure, p. 116. Introduction of tho term 'Debye function' might load to con- 
fusion with his specific hoat function. 

3505.3 S 



258 THE PARAMAGNETISM OF FREE ATOMS IX, 61 

The Langevin expression M f = N^L^HjkT) gives a saturation moment 
Np, en , which is ( J+ 1//)* times greater than that given by the Brillouin 
for the same value of /x off , i.e. for the same initial slope of the magnetiza- 
tion curve. This difference is readily understandable, as the maximum 
(i.e. saturation value of the) z component of angular momentum is 
so that even here PJ Py never vanish, inasmuch as 
~ J(J+l)h*l4>n*. Thus cventhe < most p ara lier alinoment 
of magnetic moment is in a certain sense necessarily incomplete in 
quantum mechanics. On the other hand, in the electric case complete 



P s = 




alinement of the permanent electric moment vector of the molecule is 
possible, as in 47 we mentioned that the complete Langevin formula 
for electric polarization is obtained when saturation effects are con- 
sidered (assuming the rotational fine structure to be narrow compared 
to fcT). The difference between the magnetic and electric cases may 
seem strange, but Niessen 52 shows that it is due fundamentally to the 
fact that the various Cartesian components of angular momentum do 
not commute with each other in matrix multiplication, whereas the 
components 2 e t x i> 2 e i 2/i> 2 e i z i ^ electric moment do. Various typical 
Brillouin curves are contrasted with the Langevin one in Fig. 1 3. The 
ordinates and abscissae are taken as Jf y/ /iV/i off , Hjj, off /kT rather than 
M H , H, in order to make all curves have the same initial slope 1/3. 
The dotted line is drawn for later use and explanation in 77. 

52 K. F. Niesscn, Phys. Rev. 34, 253 (1929). 



IX, 61 AND RARE EARTH IONS 259 

To test Eq. (25) one has the celebrated Leiden measurements 45 on 
hydrated gadolinium sulphate. As far as the writer is aware, these are 
the only observations on true 'gaseous' saturation produced directly 
by the applied magnetic field, rather than through the agency of the 
molecular field as in ferromagnetic solids. The saturation is made 
appreciable by the use of a material with a comparatively large ju eir 
(7-9), and also primarily by the use of exceedingly low temperatures 
(down to 1-3 K.). At room temperatures the deviations due to satura- 
tion are too small to be detectable even with Kapitza's machinery 
(300,000 gauss)! Distortions from the theoretical gaseous behaviour 
arc minimized by using a material whose paramagnetic ion is in an 8 
state and which has a high 'magnetic dilution' in virtue of the eight 
water-molecules of hydration. The experimental results have usually 
been interpreted in terms of the classical Langcvin function, but the 
Brillouin one B^lfiHjkT) should, of course, be used instead. The value 
7/2 of J in Gd is sufficiently large that the difference between these 
two functions is not great. With the latter the saturation moment is 
0-88 Np on instead of Np, C ft(~ 7 'Q4N^). The experimental points arc in- 
dicated by crosses in Fig. 13. At the highest field strength and lowest 
temperature used by Woltjer and Onnes the magnetization reached 
0-84JV/x off , or about 95 per cent, of the full saturation allowance, 
0-88Afyi |lff . The theoretical value at this H and T is 0-831J!Vfi eff , whereas 
the Langevin value is 0-859JV/z off . Reference to the figure shows that 
the agreement of experiment with the theoretical curve is very grati- 
fying. As remarked by Giauque, 53 it is even better than with the 
classical Langevin formula. In fact, Woltjer and Onnes were puzzled 
with the perceptible, though small, deviations from the latter. 

62. Lack of Influence of Nuclear Spin 

The 'hyperfine' structure of series spectra makes it certain that nuclei 
possess internal spins, having an angular momentum of the same order 
of magnitude as that of electrons. 54 However, the narrowness of the 
hyperfine structure shows that the attendant magnetic moment is only 
of the order 10~ 3 j8, where j8 is the Bohr magneton 55 he/farmc. The direct 

63 W. b\ Giauqiie, J. Amcr. Chcm. Soc. 49, 1870 (1927). See this paper for numerical 
calculation of iho Brillouin function for Gd at various field strengths. 

64 For a good discussion of tho hyperfino structure and attendant evidence on the 
magnitude of nuclear spin and magnetic moment see Pauling and Goudsmit, The Struc- 
ture of Line Spectra, Chap. XI. 

65 This small value for tho magnetic moment of the nucleus is in part understandable 
because the ratio of charge to mass of a nucleus is of the order 10- 3 times the correspond- 
ing ratio for an electron. There is, however, tho difficulty emphasized by Kronig 

S2 



260 THE PARAMAGNETISM OF FREE ATOMS IX, 62 

effect of the nuclear spin on the susceptibility is clearly negligible, as 
it should be of the order lQ- B Nf3 2 /kT or lQ- 9 e.m.u. per gramme mol. 
at ordinary temperatures, whereas even diamagnetic susceptibilities 
are of the order 10~ 6 or greater. 

One might inquire whether the nuclear spin could indirectly modify 
the susceptibility by causing the ordinary extra-nuclear (i.e. orbital 
+ spin) angular momentum to be quantized in space in a different way. 
If / be the quantum number measuring the nuclear spin angular 
momentum, then in the absence of external fields the resultant of I and 
J has a quantized value F, just as L and S form a quantized resultant 
J in case (b) of Fig. 6, 40. Here J and F measure the total angular 
momentum of the atom respectively exclusive and inclusive of nuclear 
spin. Because the interaction of I and J, which yields the hyperfine 
structure, is small, an external field of ordinary magnitude could easily 
produce a Paschen-Back effect, so that I and J would have separate 
spacial quantization, analogous to case (c) for L, S in Fig. 6. If one 
did not have spectroscopic stability, as, for instance, in the old quantum 
theory, this would be an excellent opportunity to detect experimentally 
a dependence of susceptibility on field strength in virtue of the change 
in quantization. However, the analysis in Chapter VII and especially 
in 54 of the present chapter has made it sufficiently apparent that 
such an effect will not exist as long as the 'hyper-multiplet' width is 
small compared to kT. This condition is always met in practice (except 
possibly at the temperature of liquid helium) as the hyperfine structure 
Ai> hyp is of the order 1 cm.- 1 or less. Just as we showed that Eq. (6) 
applied in either weak or strong fields, one can prove that when the 
nuclear spin is considered the susceptibility is 

(27) 

regardless of whether the field distorts the hyperfine structure, provided 
only that one supposes that hkv hyl) /kT is small. As the nuclear (/-factor 

(NaturW'isncnschaften, 16, 335, 1928) that nuclei known from atomic weights to contain 
an odd number of electrons possess only this small magnetic moment. In ordinary (i.e. 
extra-nuclear) atomic dynamics there is always an odd multiplicity and hence a non- 
vanishing spin magnetic moment of the order of magnitude of a Bohr magneton 0, 
whenever there is an odd number of electrons. Consequently the mechanics within the 
nucleus must be still more complicated than tho ordinary quantum dynamics, pre- 
sumably in virtue of the close packing and very high velocities. If the spin magnetic 
moments of electrons within the nuclei did not very nearly compensate each other, our 
whole theory of susceptibilities would be upset, as it would be forced to involve nuclear 
properties (isotope effects, &c.) rather than just the configurations of the extra-nuclear 
electrons. 



IX, 62 AND RARE EARTH IONS 261 

ffmic i s f ^ e order 10~ 3 , the additive term in (27) duo to nuclear spin 
is negligible, and so the susceptibility can be calculated with disregard 
of the nuclear spin. In (27) we have supposed for concreteness that the 
ordinary (not hyper) multiplet is wide compared to kT, and that satura- 
tion effects do not need to be considered, but the extension to other 
cases occasions no difficulty. 



THE PARA- AND DIAMAGNETISM OF FREE MOLECULES 
63. Spectral Notation and Quantization in Diatomic Molecules 

Except in 68-70, we shall consider exclusively diatomic molecules in 
the present chapter. Probably the most important distinction between 
quantization in atoms and in diatomic molecules is that in the latter 
the resultant orbital electronic angular momentum is no longer constant 
in time and is thus incapable of quantization. Instead the combined 
field due to the two nuclear attracting centres has axial rather than 
central symmetry, and so only the parallel component of this angular 
momentum is conserved. This component can, however, be given a 
quantized value A. When we use the terms 'parallel' or ' perpendicular' 
in the present chapter we always mean relative to the axis of figure of 
the molecule, and we always measure angular momentum in multiples 
of the quantum unit h^n. A term is called of the 2, 11, A, type 
according as |A| -= 0, 1, 2, 3, &c. As in 40, small letters are used if it 
is desired to isolate the quantum numbers of individual electrons. Thus 
a notation l such as 

Isa 2 2p7T 5/5 <D 4> (1) 

which is the molecular analogue of (86), Chap. VT, means that there are 
two electrons having TI 1, Z A 0, one having n = 2, I = |A| =-= 1, 
and one having n = 5, I = 3, |A| == 2; furthermore, especially that the 
total orbital angular momentum A = ] ^ about the axis of figure is 3 
and that the multiplicity 2$+l is 3. Here S is the resultant spin, n is 
the usual principal quantum number, and I measures approximately an 
electron's total angular momentum, while A does the parallel component 
thereof. This significance of the quantum numbers Z, A is only an 
asymptotic one appropriate to small departures from central character, 
as the inter-electronic interactions destroy the constancy of an indi- 
vidual electron's angular momentum even to the parallel component. 
Besides the quantum numbers revealed by (1) there is the nuclear 
vibrational quantum number v, and the rotational quantum number J 
which orders the band structure and which determines the complete 
molecular angular momentum due jointly to nuclear and electronic 
orbits and to electron spin. 

1 For a fuller account of approved notation in molecular spectra soo R. S. Mulliken, 
Phys. Rev. 36, 611 (1930). The paper in which the writer first gave most of the theory 
in the present chapter (Phys. Rev. 31 , 587 (1928)) used an earlier notation in which Greek 
letters were less in vogue than now. 



X,63 THE PARA- AND DIAMAGNETISM OF FREE MOLECULES 263 
The matrix elements of the perpendicular components of orbital 
electronic angular momentum can readily be shown 2 to be exclusively 
of the type AA= 1. Essentially this point was already proved for 
a very special case in (75), Chap. VI, when allowances are made for 
differences in notation. 2 As molecular fields are far from central, the 
effect of the quantum number A on the energy is usually very large 
compared to kT ; or, interpreted kinematically, the electronic orbital 
angular momentum vector processes very rapidly about the axis of 
figure. This means that these perpendicular components contain ex- 
clusively high-frequency matrix elements, and so by the theory of 
Chapter VEI contribute to the susceptibility only a small additive term 
which is independent of temperature, and which we shall usually neglect 
except in 01). Hence the square of the low-frequency part of the 
orbital moment is proportional to A 2 instead of L(L-\-l) as in the 
atomic case when the multiplcta are narrow. 3 

We must now consider the spin. Hund 4 has emphasized that we 
must distinguish between two kinds of coupling of the spin axis relative 
to the rest of the molecule, which he designates as types (a) and (b). 
In (a) the energy of interaction between the spin and orbital angular 

2 The z axis in (7f>), Chap. VI, corresponds to the axis of symmetry and so m t to A 
(or better still to A). Kq. (75) was very special in that it assumed only one electron and 
central rather than merely axial symmetry. The substitution of axial for central sym- 
metry does not affect the factor involving^ in (74), Chap. VT, although it in general makes 
the factor involving 6 different. This change will alter the explicit form of the elements 
Awj - Jjl but does not modify the vanishing of other elements, our present concern. 
The generalization to more, than one electron is accomplished by noting that in tho 
coordinate system used in (52), (53), Chap. T, tho wave function involves y t only through 
an exponential factor; of., for instance, Kronig, Band Spectra ami Molecular Structure, 
p. 20. 

3 At first thought it may appear as though A(A-f 1) should appear in place of A 2 , as 
a similar combination appears so often in quantum mechanics. However, the square 
under consideration is one of a constant component rather than of the entire magnitude 
of a vector, and so proves to havo tho same voluo A 2 us in tho old quantum theory. (In 
tho hydrogen atom, for example, tho square of tho z component of orbital angular 
momentum is inf not wfon t -}-l) 9 if for simplicity we assume no coupling with tho spin.) 
Similarly, tho square of the parallel spin component is X 2 not S (E-fl). On the other 
hand, the squares of the angular momenta associated with tho quantum numbers J, K, 
*S' aro J (J -\- 1), &c., as these numbers measure resultants, rather than components. 

4 F. Hund, KcitH.f. Physik, 36, G57 (1926); 42, 93 (1927). We omit many other types 
of coupling which as a rule occur in excited rather than normal states. For these see 
Hund, I.e., also Mullikoii, Rev. Mod. Phys. 2, 00 (1930). For instance, the magnetic 
interaction between spin and orbit destroys the rigorous constancy of tho parallel com- 
ponents of spin and orbital angular momenta even for a stationary molecule, although 
not affecting that of tho sum measured by il. For this reason, the quantum numbers 
A, S sometimes cannot be used, but we assume that the distortion through this cause is 
negligible. This is warranted for tho molecular states with which we shall be con- 
cerned. 



264 THE PARA- AND DIAMAGNETISM OF X, 63 

momenta is large compared to that between the spin and the angular 
momentum due to rotation of the molecule as a whole, and consequently 
the spin axis is firmly quantized relative to the molecular axis. The 
parallel component of spin angular momentum can then be assigned 
a quantized value 2. The various values 2 = $,...,+$ yield the 
different components of a multiplet. 5 The notation 1 is employed for 
the sum A +2. Thus & gives the parallel component of spin and orbital 
angular momentum combined, and can be used in place of 2 to specify 
the multiplet component. In notation such as (1) the value of 1 is 
indicated by a subscript after the final Greek capital, but this subscript 
has a direct meaning only in case (a). In (1), for instance, O = 4, and 



i A ' 

.* * 




Case (a). 



Case (6). 



FIG. 14. 



so 2 = 1 A = 1. In (b) the magnetic coupling between spin and orbit 
is overpowered by the centrifugal forces caused by the molecular rota- 
tion. The spin axis then no longer makes a fixed angle with the axis of 
figure. Instead the angular momentum of the molecule exclusive of spin 
is first quantized to a resultant K, and then K and S are compounded 
vectorially to give the total angular momentum determined by J. This 
is shown in (/>), Fig. 14. Large rotational quantum numbers favour case 
(6), especially in light atoms where the multiplets arc narrow and the 
magnetic coupling is easily broken down. 

As in Chapter IX, it is convenient to consider the limiting cases of 
multiplets which are very small or very large compared to kT. 

64. Multiplet Intervals Small Compared to kT 

In this case the matrix elements of the spin will be entirely of the low- 
frequency type, for the only motion of the spin vector relative to the 
rest of the molecule is a precession about the axis of figure, whose 
frequency is correlated with the multiplet intervals. The square of the 
low-frequency part of the moment is therefore identical with the square 
of (2S+L par )j8. Furthermore, the statistical average of the product 

6 Do not confuse the quantum number S and X states. This double burden of the 
letter is approved usage. 



X, 64 FREE MOLECULES 265 

S ' L pttl , is zero, for with narrow multiplets we may neglect the Boltz- 
mann temperature factor, so that the components in which the sign 6 
of S is the same as or opposite to that of A have the same weight. 
Furthermore, we have S 2 =^ 8(8+1) and L2 )ar ~ A 2 . Thus the expres- 
sion given for the susceptibility in (1), Chap. IX, becomes, with neglect 
of the small term Not, 

X=^J[4(S+1)+AJ. (2) 

Formula (2) will apply regardless of whether the coupling is of type (a), 
type (&) r intermediate, provided only that the multiplets are small 
compared to kT. This is just another example of the 'spectroscopic 
stability' or invariance of the susceptibility of the mode of quantization 
so long as the magnitudes of frequencies relative to kTjh are unaltered. 

65. Multiple! Intervals Large Compared to kT 

Here the coupling will in general be of type (a), whereas in 64 it could 
be of either type. This is true because cases (a) and (6) arise when the 
multiplets are respectively very large and small compared to the spacing 
between the different rotational energy -levels, and because, further, this 
spacing is usually small 7 compared to kT. 

In the present case of wide multiplets the quantum number 
assumes in the normal state only the one value which gives the lowest 
energy. The matrix elements representing the perpendicular component 
of spin belong exclusively to the neglected 'high-frequency* category, 
for they represent transitions 8 AS = 1 to other multiplct components 
which must now be classed as excited states. The square of the low- 
frequency part of the moment is thus identical with the square of the 
parallel component of orbital and spin moment combined, and is thus 
(A+2V) 2 j82. By (1)j ohap IX the suscc ptibility is hence 



6 The introduction of tho quantum number S presupposes coupling of typo (a) and 
so ostensibly restricts our proof to this case. However, tho principle of spectroscopic 
stability or invariance of tho spur assures us that tho statistical average of the product 
S-Lpar will be invariant and hence zero with other types of coupling. For type (b) this 
ran. also be verified explicitly by averaging the diagonal matrix elements for this product 
in (6) given by Hill and Van Vleck, Phys. Rev. 32, 250 (1928); especially their Kq. (17). 

7 Cf. the observation on p. 192 that w is usually small compared to W jkT. 

8 The quantum number S plays the same role for spin as A for orbit. Analogy to the 
orbital case studied in note 2 thus shows that the perpendicular spin elements will bo 
exclusively of the form AS 1. 



266 THE PARA- AND DIAMAGNETISM OF X, 66 

66. The Oxygen Molecule 

The only two common paramagnetic gases are 2 and NO. Other im- 
portant diatomic molecules have 1 2 normal states and are diamagnetic. 
The case of the oxygen molecule is interesting because it is para- 
magnetic, despite containing an even number of electrons. 9 Its normal 
state is of the type 3 2, constituting something of an exception to the 
Hei tier-London valence rule that saturated valences yield singlet con- 
figurations. The 3 character of the normal state was first known from 
its magnetic behaviour (Eq. (4) below) but has subsequently been con- 
firmed spectroscopically in a careful analysis of oxygen bands by 
Mulliken. 10 

Molecular S states are practically devoid of multiplet structures, 
although experimentally 10 and theoretically 11 they do have a small line 
structure, of the order 1 cm.- 1 or less, if there is an outstanding spin, 
as in oxygen. Hence the susceptibility can certainly be calculated under 
the assumption that the multiplet structures are small compared to kT. 
With S = 1, A = Eq. (2) yields a value 

8 0-993 

Xmol 



for the molar susceptibility, corresponding to an effective Bohr mag- 
neton number & = 2-83. As usual, L denotes the Avogadro number. 
At 20 C., Eq. (4) gives a molar susceptibility 3-39 x 10 ~ 3 . The value 
observed by Bauer and Piccard, 12 which seems to be usually accepted 
as the most accurate, is 3-45 x 10~ 3 . The early determinations of Curie 13 
yield 3-35 X 10~ 3 when recalibrated 14 on the basis of 0-72 x 10~ 6 rather 
than 0-79 x 10~ 6 as the mass susceptibility of water. Still other observa- 
tions arc: 3-31 X 10~ 3 by Onnes and Oosterhuis, 15 3-33 X 10~ 3 by Sone, 16 
3-48 xlO~ 3 by Wills and Hector, 17 and more recently 3-34 x 10~ 3 by 

9 All odd molecules are necessarily paramagnetic . Paramagnetic even ones arc very 
rare except in the monatomic case of incomplete inner shells (cf. 58). Hy analogy with 
() 2 , one would expect S 2 to have a 3 S ground state and be paramagnetic*. This is con- 
firmed spectroscopically by Naude and Christy, P1iyn. Rev. 37, 490 (1031). 

10 R. S. Mulliken, Phys. Rev. 32, 880 (1928). 

11 II. A. Kramers, Zcite. f. Physik, 53, 422, 429. The fine structure in 2 states is 
somewhat greater for S > than for $ \ . In the latter case there is only the very minute 
4 rho-type doubling*, of the order Q-QIJ cm" 1 , duo to rotational distortion; cf. ,J. II. Van 
Vleck, Phys. Rev. 33, 497 (1929). 

12 Bauer and Piccard, J. de Physique, 1, 97 (1920). 

13 P. Curie, Ann. Chim. Phya. 5, 289 (1895); (Kuvre.s, p. 232. 

14 Cf. Stoner, Maynetiam and Atomic Structure, p. 126. 

15 H. Kamorlingh Onnes and E. Oostorhuis, Leiden Communications 134d. 

16 T. Sone, Phil. Mag. 39, 305 (1920). 

17 Wills and Hector, Phys. Rev. 23, 209 (1924). 



X,66 FREE MOLECULES 267 

Lehrer, 18 also 3-42 x 10~ 3 by Woltjcr, Coppoolse, and Wiersma. 19 ' 20 It is 
thus not improbable that Bauer and Piccard's value is slightly too high. 
Before the writer's theory with the new quantum mechanics, it was observed by 
Sommerfeld, 21 Stonor 22 , and others that the susceptibility of the oxygen molecule 
is the same as that of an atom in a 3 $ state. Such an atom likewise gives formula 
(4), since the atomic and molecular formulae (6), Chap. IX and (2), X, respectively 
are the same for 8 and U states, and these only. This can be seen by comparing 
(2) with (6), Chap. IX. The atomic formula for fl states is usually derived in an 
elementary way (cf. Eq. (7), Chap. IX) under the assumption that the spin is 
quantized relative to the applied field. Hence in the old quantum theory it was 
obvious that (4) would apply to an oxygen molecule only in the event that the 
applied field is able to break down the coupling of the spin relative to the rest of 
the molecule. (This coupling is of the type (6), Fig. 14, as the oxygen multiplet 
structure is very narrow.) In other words, the field must bo adequate for a 
Pasehcri-Back effect. The triplet width for tho normal oxygen molecule is about 
1-4 cm- 1 , whereas the normal Zecman displacement is 4-67 X 10~ 6 H cm- 1 , and so the 
magnetic field would have to be of about the order of 10 5 gauss to produce a com- 
plete Paschen-Back effect. Ordinary experimental fields thus become of just the 
transition range in which the susceptibility would presumably change with field 
strength in the old quantum theory because of the change in quantization. The 
principle of spectroscopic stability or our derivation of (4) by the general theory 
of Chapter VII shows that the susceptibility will, however, be invariant in the new 
mechanics. Derivations of (4) by the elementary method (7), Chapter IX, without 
appealing to this principle are obviously inadequate. 

Particularly significant is the mode of temperature variation. As the 
multiplet structure is almost negligible in width, Curie's law should be 
obeyed with considerable accuracy. This was first verified by Curie 13 
himself over the range 290-720 K. At lower temperatures the validity 
of this law was confirmed approximately for oxygen over the interval 
143-290K. by Onnes and Oosterhuis. 15 Recently more refined experi- 
ments have been made by Woltjer, Coppoolse, and Wiersma 19 down to 
157K. and by Stossel 23 down to 136-5 K. Both sets of observations 
agree in showing that Curie's law is at least very nearly valid. Stossel 
finds no perceptible departure from this law even at 136-5K. Woltjcr, 
Coppoolse, and Wiersma, on the other hand, contend that even after 

18 E. Lohror, Ami. der Physik, 81, 229 (1926). 

19 H. 11. Woltjer, C. W. Coppoolso and E. C. Wiersma, Loidoii Comm. 201 or Proc. 
Amsterdam Acad. 32, 1329 (1929). 

20 They do not give explicitly their absolute determination of the susceptibility, but 
this may bo obtained by calibrating all their observations by moans of their ' 1st series', 
instead of by comparison with Bauer and Piecard, and then extrapolating to zero 
pressure. 

21 A. Sommerfold, Atombau, 4th ed., pp. 630 ff. 

22 TC. C. Stoner, PMl. Mag. 3, 336 (1927). 

23 K. Stossel, Ann. der Physik, 10, pp. 393-436 (1931). Tho writer is indebted to 
Dr. Stossel for communication of his results in advance of publication. 



268 THE PARA- AND DIAMAGNETTSM OF X, 66 

their data are extrapolated to zero density to avoid interference effects 
between molecules, there is below 175K. a small deviation of 2 per 
cent, which is beyond the experimental error. This is not alarming. In 
the first place the experiment is a difficult one and no deviation is found 
by Stossel. Secondly, the theory has involved three distinct approxima- 
tions: (1) disregard of the additive term No. due to high-frequency 
elements and to diamagnetism (cf. Eq. (2), Chap. IX); (2) assumption 
that the spacing of rotational states is small compared to kT, or, in 
other words, that the ' characteristic temperature' h*/ Sir 2 Ik of the mole- 
cule is negligible compared to T\ and (3) assumption that the multiplet 
structure is likewise of negligible width. 

Of these (1) probably is to be most seriously considered, as the other corrections 
appear exceedingly small. If one assumes that JVa/^oi * s about 0-04, then 
Xtoi/X 2Y291 increases from at T = 291 K. to about 0-01 at T = 150 K., in 
accordance with the Leiden observations. 19 This makes the graph of this difference 
against T concave downwards, whereas experimentally it seems to bo upwards, 
but the curvature is a second-order effect which is very hard to measure. Such 
a value of NOL appears rather largo, some 200 times larger than we shall calculate 
in 69 for the effect of the high-frequency elements in H 2 . However, as the normal 
state of the O atom is of the type 3 P, the oxygen molecule is composed of 3 P 
atoms rather than 1 >S t like H 2 , and so the perpendicular orbital component 
responsible for these elements might conceivably be considerably greater than in 
H 2 . The agreement on absolute values is made somewhat worse by assuming this 
Not, as the calculated susceptibility at room temperature becomes about 2-2 per 
cent, greater than Bauer and Placard's value, instead of 1-8 per cent, lower, and 
deviates still more from the lower values found by most other observers. 

A wealth of Leiden data 24 exists on the susceptibility of liquid and 
solid oxygen, both pure and diluted in different amounts of N 2 . Great 
variations with density and abrupt discontinuities at certain critical 
temperatures are reported. Between these critical points a law of the 
Weiss form x = C/(T-{-&) is usually obeyed, where A increases rapidly 
with the concentration in numerical magnitude. (This Weiss generaliza- 
tion of Curie's law must also be used even for the gaseous state when 
at very high pressures.) Only the extrapolation to zero density, obtained 
by dissolving the oxygen in successively greater amounts of nitrogen, is 
of interest for the present theory, where forces between the molecules 
are disregarded. It is gratifying that within the rather large experi- 
mental error the extrapolations to zero density for the liquid conform 
to a Curie formula with the same constant C as for oxygen gas. We 
may note parenthetically that the existence of a A term in the Curie 
denominator which increases in magnitude with density is in at least 

24 A. Pcrrier and H. Kamerlingh Onnos, Leidoii Coinm. 139. Good summary on pp. 
141-4 of Stoner's Magnetism and Atomic Structure. 



X, 66 FREE MOLECULES 269 

qualitative accord with Heisenberg's theory of ferromagnetism to be 
developed in Chapter XII (cf. Eq. (39) of XII). 

67. The Nitric Oxide Molecule 

Nitric oxide gas furnishes the most striking confirmation of our entire 
theory, both because the NO band-spectrum furnishes unambiguous 
term-assignments and because the doublet width is comparable with 
kT and so furnishes a test for the finer points of the theory. The 
normal state of the NO molecule is known spectroscopically 25 to be 
a regular 2 11 doublet of width h&v approximately 26 120-9 cm 1 . The 
effective magneton number for very high temperatures is 2, as seen by 
taking S = i, A = 1 in Eq. (2). Eq. (3) shows that at very low tem- 
peratures the effective magneton number is zero, for the lower doublet 
component has S = 2, A = 1. The susceptibility observed by Bauer 
and Piccard 12 and by Sone 27 at 20 0. is 1-46 x 10~ 3 per gramme mol. 
These measurements yield an effective Bohr magneton number 1-86 
intermediate between the two asymptotic values just calculated. This 
is not surprising, for kT is about 200 cm.- 1 at room temperature, making 
hkvjkT about 0*0. Thus ordinary temperatures fall in the critical region 
in which the doublet width h&v is comparable with kT, and in which 
deviations from Curie's law should hence be expected. To verify the 
theory quantitatively, it is necessary to make calculations for the more 
complicated intermediate case, rather than the asymptotic ones pre- 
viously considered. Such calculations will be made on pp. 271-2. It 
will there be shown that the effective Bohr magneton number (defined 
by Eq. (22), Chap. IX) is the following function of temperature: 

173 . 



At 20 0. this yields 1-836. The discrepancy of about 1 per cent, with 
the experimental value 1-86 of Bauer and Piccard and of Sone is not 
excessive in view of experimental difficulties in absolute determinations, 
and of the fact that the theory itself involves certain small approxima- 

25 Cf., for instance, K. T. Birgo, Nature, Fob. 27 (1926) ; Jenkins, Barton, and Mulliken, 
Phys. Rev. 30, 150(1927). 

86 Wo take the doublet interval as 120-9 cm" 1 , rather than the value 124-4 quoted by 
Jenkins, Barton, and Mulliken, as for our purposes it is better to use energy differences 
which are inclusive rather than exclusive of the term 7* 2 Q 2 /87r 2 / in the rotational 
energy. (This term has been encountered in Eq. (70), Chap. VI, but we there neglected 
spin, so that A appeared rather than 11.) It makes little difference which value is used, as 
(5) shows that the corresponding change in the effective number of Bohr magnetons is 
only per cent. 

27 T. Sone", Tohoku Univ. Sci. Reports, 11, (3), 139 (1922). 



270 THE PARA- AND DIAMAGNETISM OF X, 67 

tions (viz. items (1) and (2), p. 268; also other approximations mentioned 
in notes 4 and 30). 
When the theoretical formula (5) was first developed by the writer, 

CALCULATED AND OB&KKVKD MAGNETON NUMB Kits von NO AS A 
FUNCTION OF TKMPIOKATURE. 



Temp. 


H c(t CaJc. 


/i,,, Obs. 


Temp. 


/x cf! Calc. 


He,, Obs. 


0-0 K. 


0-0 




238-4K. 


1-794 


(l-794)(W.,doH.,&C.) 


50-0 


1-098 




250-6 


1-807 


(1-807) (St.) 


100-0 


1-489 




289-2 


1-833 


1-841 (St.) 


112-8 


1-546 


535(W.,doH.,&C. 29 '*) 


290-2 


1-834 


(1-834) (Ah. & S.) 


135-5 


1-624 


627 (St. 23 ) 


292-1 


1-836 


1-852 (W.,doH.,&0.) 


157-2 


678 


679 (St.) 


296-0 


1-837 


(1-837) (Bit.) 


165-4 


695 


691 (W., deH., & C.) 


350-0 


1-864 




178-0 


718 


713 (St.) 


500-0 


1-908 




194-7 


741 


732 (Ah. & S. 29 ) 


1000-0 


1-955 




216-0 


771 


768 (Bit. 28 ) 


00 


2-000 





























Aeff 




.-H*~T 


&*&~ * J i c 


> *& 






OA.haroni & Scherrer 



Stossel 

X Wiersma, de Haas & Capel 
X Calibration points 



300 



350 



Kio. 15. 

there existed only the experimental data at room temperatures, so that 
it was then impossible to test the predicted dependence of magneton 
number on temperature or, in other words, the deviation from Curie's 
law. Subsequently this has been tested by Bitter, 28 by Aharoni and 
Scherrer, 29 by Stossel, 23 and by Wiersma, de Haas, and Capel, 29a each 

28 F. Bitter, Proc. Nat. Acad. 15, 632 (1929). 

29 Aharoni and Scherror, Zeits.f. Physik, 58, 749 (1929). 

29a Wiersma, de Haas, and Capel, Leiden Communications 212 b. 



X, 67 FREE MOLECULES 271 

to a lower temperature than the preceding. As the boiling-point of NO 
is 142K., it would be very difficult to go much below the lowest 
temperature 112*8 employed by Wiersma, de Haas, and Capel. The 
effective magneton numbers yielded by (5) for various temperatures, 
together with the experimental values reported by these different 
observers, are shown in the table opposite, and in Fig. 15. We, 
for brevity, include only two of Wiersma, de Haas, and Capel' s 
measurements at ten temperatures intermediate between 112-8 and 
292-1. 

The experimental measurements are all relative ones made on the 
ratios of the susceptibilities at different temperatures rather than on 
absolute magnitudes. In the table and figure, the data of Bitter, Aharoni 
and Scherrer, Stftssel, and Wiersma, de Haas, and Capel have been cali- 
brated so as to make them fit exactly the theoretical values at 296, 
290-2, 250-6, and 238-4 K. respectively. 

This quantitative verification of the deviations from Curie's law in 
NO must be regarded as a convincing proof of the correctness of the 
quantum theory of magnetic susceptibilities in gases. These deviations 
form a marked contrast to the validity of this law in O 2 . They are 
rather more striking than indicated in the figure, as the Curie constant 
varies as the square of the effective magneton number. It may be 
mentioned that although the effective magneton number vanishes at 
T = 0, the product p^y/T remains finite there, and in consequence the 
theoretical molar susceptibility approaches the finite limit 

v = 2-87 x 10~ 3 at T - 0. 

I 'roof of Eg. (5). As explained in. 63 we may neglect the perpendicular com- 
ponent of orbital magnetic moment because changes in. the electronic quantum 
number A give rise only to very high frequencies. On the other hand, we must 
not forget the perpendicular component of spin moment, because the effect on 
the energy of the spin quantum number X is comparable with kT (cf. p. 269). 
The same is also true of ft, since 1 S + A differs from H only by an additive 
constant -j- 1 in a II state. As wo have discarded transitions to excited orbital 
states, all elements diagonal in ft will bo of the low-frequency type, as they involve 
at most only changes in the molecular rotation.;. Elements involving transitions 
in S2 connect tho various spin components (hero only two in number), and so are 
of the 'medium frequency type'. We must now adapt the work of Chapter VII 
to admit a 'medium frequency ' quantum number ft in place of tho high-frequency 
one n. With this modification tho expression in magnetic notation corresponding 
to (22), Chap. VII, retains its validity if one adds a summation over the quantum 
number ft inasmuch as there is an appreciable Boltzmann factor for all values 
of ft. Also tho low-frequency moment will now involve ft as a parameter, so that 
the notation m(ftjm; ftj'm') is bettor than p(jm;j'm'). By (22), Chapter VII, the 



272 THE PARA- AND DIAMAGNETISM OF X, 67 

susceptibility is thus 30 

X = SkT BJ J,..h f (a/m;tV'')| 



+ ?? _ 

3 cw-.ffr*'<n'*n> M"';^) 
with the abbreviation (cf. Eq. (12), Chap. VII). 

^ e -H(nj)/*r 

The summations over H and O' each embrace only the two values J, which 
give the two doublet components of the 2 II state. The index j may be identified 
with the molecular rotational quantum number J 9 while w is the component of 
J along the field. 

The first and second linos of (6) arise respectively from the parallel and perpen- 
dicular components of moment. This follows from the fact that changes in 2 
are identified kinematically with processions about the axis of figure and so 
appear only in the perpendicular component. Hence we have 

j'm' > par 

since by the foregoing and the rules for matrix multiplication the sum in (8) is 
the square of the parallel component of combined spin and orbital moment. 
Similarly 

O'J'W'{Q' : n> 

where w porp is the perpendicular component of purely spin moment. The value 
of wjerp is that given in (9) since the square of the total spin magnetic moment 
is the sum of the squares of the perpendicular and parallel components, and since 
the square of the parallel spin component is 4S 2 j8 2 -- )3' 2 (as D -- i J-). These 
results would not bo true if any part of the spin were of the discarded high- 
frequency type such as all the perpendicular orbital component, but actually the 
motion of the spin axis is very closely that of a regular precession about the axis 
of figure without appreciable nutations. Now v(|; J) - ^(aJy) z Av, an( l a ^ so 
we may set W(,j) W(l f j)+h&v as v($jm;ljm) is approximately the doublet 
width Ay. Since furthermore, the expressions (8) and (9) have values independent 
of j 9 in, the sum over./, m is by (7) a common factor which can bo cancelled from 
numerator and denominator of (6). With these observations substitution of 
(7)-(9) in (6) yields (5). (The effective magneton number involved in (5) is of course 
defined as in (22), Chapter IX.) 

68. Polyatomic Molecules 

We saw that in diatomic molecules the perpendicular component of 
orbital moment was of the high-frequency type. When the molecule 

30 Eq. (6) assumes that to a sufficient approximation v(l'j'm'-, tljm) can bo replaced 
by v(Q'j 1), as an equivalent assumption was made in (22), Chap. VII. As Q, has less 
effect on. the energy than an electronic quantum number w, the resulting error is some- 
what larger than in the case of electronic frequencies. The ensuing error in the sucopti- 
bility is hard to estimate with precision, but is perhaps 1 per cent.; see Phya. Rev. 31, 
footnote, p. 611 (1928) for details. 



X, 68 FREE MOLECULES 273 

is polyatomic, i.e. contains more than two atoms, the entire orbital 
moment will be of this type unless the molecule has unusual symmetry. 
The first step in demonstrating this is to prove the following theorem: 
The existence of a mean magnetic moment for an atom or molecule in 
the absence of external fields implies the existence of at least a twofold 
degeneracy, i.e. at least two states of identical energy. The proof is as 
follows. If the degeneracy is completely removed, the wave functions 
are necessarily real in the absence of external magnetic fields. For when 
one supposes all the electrons are subject to only electrical forces (which 
can be external as well as internal to the atom), the wave equation docs 
not involve i V 1, as the potential energy is a real function, while 
the kinetic energy involves the imaginary momentum operators only 
in squares. It can also be shown that i occurs only in squares even 
when the magnetic coupling between spin and orbit is included. Hence, 
if it is possible to utilize what we shall call 'complex' wave functions 
of the form P(x l ,x 2 ,...)-}-iQ(x 1 ,x 2 ,...), where P and Q are different 
(i.e. linearly independent) real functions of the coordinates, then the real 
and imaginary parts must separately be solutions of the wave equation 
belonging to the same energy. The existence of two such linearly 
independent solutions would, of course, require at least a twofold 
degeneracy. Furthermore, whenever the wave functions are real, the 
average or diagonal part of the orbital angular momentum is zero, for 
if we take n' - n and take / to be the operator corresponding to any 

component of magnetic moment, say L z = i~ 1 lx- 2/7~i> the funda- 
mental quadrature (14) of Chapter VI vanishes, as it contains then as 

-i 

one factor either the expression ib* ' dy or 0* dx, which is 

J % J ' ^ c 

-- 00 

clearly zero when i/f* /t if I/J IL vanishes at infinity in the fashion proper 
to the characteristic functions for bound electrons. This argument no 
longer applies if $ is complex, as then /r* v^0 /r Hence the existence 
of some degeneracy is a necessary, though not a sufficient, condition 
for the existence of an average unperturbed magnetic moment. 

In diatomic molecules there is the twofold degeneracy associated 
with the fact that the sense of rotation about the axis of figure is 
immaterial, or, in other words, that the states +^ an d & give the 
same energy (neglecting a small rotational distortion effect to be men- 
tioned in 70). This is why diatomic molecules could have (barring this 
distortion) a constant orbital magnetic moment parallel to the axis of 

3595.3 T 



274 THE PARA- AND DIAMAGNETISM OF X, 68 

figure. In polyatomic molecules, however, there is no axis of symmetry 
about which the angular momentum is conserved, and this type of 
degeneracy is no longer encountered. In some cases the nuclei may be 
arranged with such a high degree of symmetry that some other de- 
generacy appears in its stead. The symmetry conditions necessary for 
this have been studied in detail by Bethe, 31 and his work will be further 
discussed in 73. He studied ostensibly the effect of different sym- 
metries in external fields, but his arguments are so general that they 
relate equally well to the fields arising from nuclei in polyatomic mole- 
cules. He shows that even if the fields are highly symmetrical, usually 
the states of lowest quantum number still do not admit complex wave 
functions. Hence we may assume that ordinarily in the normal states 
of polyatomic molecules the diagonal elements of the orbital magnetic 
moment are zero. (This is true even when the moment is referred to 
axes fixed in the molecule so that the frequencies of molecular rotation 
are avoided.) As all electronic quantum numbers in molecules usually 
have an effect on the energy which is large compared to M 7 , the non- 
diagonal elements of the orbital moment will be entirely of the high- 
frequency type. As there are no diagonal elements, this completes the 
proof that only such a type occurs. 

Molecules with a Resultant Spin. If the molecule has a spin quantum 
number $ different from zero, virtually all the paramagnetism will 
result from the spin, as we have seen that all the orbital moment is of 
the ineffective, high-frequency type. The multiplet structure which 
couples the spin relative to the rest of the molecule will usually be small 
compared to kT, for it becomes an effect of the second order, as in the 
states of diatomic molecules, rather than of the first order, as in the 
ordinary case of atoms and diatomic molecules not in states. This small 
size of the multiplet structure is a consequence of the fact that the 
average orbital moment vanishes. This will be shown in the analysis in 
73. 32 The spin is thus composed entirely of low-frequency elements, and 
is entirely free as far as the susceptibility is concerned. The latter is hence 



where No. is the small residual effect of the high-frequency orbital 

ai H. Bethe, Ann. der Physik, 3, 133 (192'.)); Zeits.f. Phyrik, 60, 218 (1929). 

32 In adapting this analysis to the present context, the 'spacial separation' ( 73) is to 
be considered due to the dissymmetry in the field from the molecule's own nuclei rather 
than to an external field. It is thus probably greater than for the external case considered 
explicitly in Chap. XI, so that the inequality (4) of XI is more apt to be satisfied. 



X, 68 FREE MOLECULES 275 

elements (Eq. (2), Chap. IX). All odd polyatomic molecules should 
presumably conform to this formula (unless some unusual case should 
arise where the orbit is less completely quenched, or else the spin is 
more firmly bound than we have anticipated). Experimental data for 
odd molecules are apparently available only for C1O 2 and NO 2 . 

The molar susceptibility observed by Taylor 33 for C10 2 in solution at 
20 C. is 1-34 x 10- 4 , while (10) gives 1-27 x 10~ 4 if we take 8 = \, a = 0. 
The discrepancy is scarcely greater than Taylor's estimate of the 
experimental error as 5 per cent. Furthermore, part of the difference 
might be due to the effect NOL of the high-frequency elements, which 
could conceivably be larger than in diatomic molecules. The appro- 
priate value of S cannot yet be deduced spectroscopically for polyatomic 
molecules, but we can note that our assignment S = 2 is consistent 
with the fact that C10 2 has an odd number of electrons. Very likely 
due to ionization or polymerization, the C1O 2 molecule loses its identity 
in solution, but the theoretical result 1-27 x 10~ 4 is still applicable if one 
ionic unit of spin quantum number i is formed for each molecule of 
C1O 2 which is dissolved. 

On the other hand, Sone 27 finds that at 20 C. the NO 2 molecule has 
a molar susceptibility +2-1 x 10~ 4 , a value less than one-fifth that given 
by (10) with S= \. Perhaps this low value is to be attributed to 
polymerization or some other spurious cause, for new measurements 
on NO 2 have just been completed by G. Havens, at the University of 
Wisconsin, and he finds a susceptibility which agrees with (10) (taking 
S - 1) within 5 per cent., which is less than the experimental error. 

Molecules without a Resultant Spin. When a polyatomic molecule has 
a spin quantum number zero, the commonest value for even molecules, 
there remains only the contribution of the orbital moment's high- 
frequency elements, represented by m(n\n') (ri ^n) in the notation 
of Chapter IX. The susceptibility is thus given by the expression (2), 
Chap. IX, viz.: 



h v (n'\n) 

n'/n ^ ' 

per gramme mol., and so should be very small and independent of tem- 
perature. The material is diamagnetic or paramagnetic depending on 
whether the first or second term of (11) is the greater. Nitrous oxide 
(N 2 0), for instance, is found to be diamagnetic, showing that here the 
first term has the greater magnitude. A more interesting and less 

a <> N. W. Taylor, J. Amer. Chem. Soc. 48, 854 (1926). 
T2 



276 THE PARA- AND DIAMAGNETISM OF X, 68 

common situation arises when the second term of (11) predominates. 
The substance should then have a feeble paramagnetism independent 
of temperature. Examples of materials containing complex ions which 
exhibit this behaviour will be cited on p. 302. These ions, of course, 
occur in solution or in solid salts, rather than as a gas, but they seem 
to have a distinct and fairly stable existence, so that they may be 
classed with gaseous polyatomic molecules for our purposes. 

Invariance of (11 ) of Origin. The reader has possibly wondered what 
point should be used as the origin for computing r, m in equations such 
as (11). This choice is immaterial, as (11) is invariant of the origin. To 
see this, let us change the origin of x, for instance, by an amount S#. 
The resulting change in the right side of (11) is 



~ 

x Ip^n'^p^n'-n^x^i^ 

hv(ri;n) 

"< n 5')^("5 w ' 

hv(n'\ri) I 

(12) 

inasmuch as (letting q denote ?/ or z) x(n\ri) = S#, x(n\ri) - (ri ^ n), 
Sjp^O, m q = (e/27nc)(p^p x q), &c., where p^ Py, P~ are the com- 
ponents of linear momentum. To simplify printing we have supposed 
that there is only one electron and have neglected the part of m fj due 
to spin; but removal of these restrictions occasions no difficulty. (As 
the radius of the electron is negligible, one can take 8s in the spin 
terms.) The frequency factors may be removed from the denominator 
by utilizing the relations 

p y (n',n')~ p u (n'\ri)~ 2-rrimv(n\ri)y(n\ri), &c. 

The expression (12) can be shown to vanish identically if we simplify 
the products by using repeatedly the quantum conditions and com- 
mutation rules given in (12), (13) of Chap. VI, applied to ('artesian 
coordinates. (See e.g. Eq. (4), Chap. XIII, for the explicit form of the 
first relation of (12), Chap. VI, in Cartesian coordinates.) 

69. The Diamagnetism of Molecules 

The fact that most gases are diamagnetic shows that ordinarily the 
first term of (11) is the greater in magnitude. It is therefore now con- 
venient to turn to consideration of the diamagnetism of molecules. Our 
discussion will include both diatomic and polyatomic molecules, for 



X, 69 FREE MOLECULES 277 

both are governed by formula (11). The only difference is that in the 
former the high-frequency matrix elements m(n; ri) (n r ^ n) arise only 
from the perpendicular component of orbital moment, while in the 
latter the components along all three of the principal axes of the mole- 
cule will contribute such elements. If a diatomic molecule is dia- 
magnetic, it is in a *X state, and the parallel component vanishes. 
Previously we neglected the feeble paramagnetic contribution of the 
perpendicular component, given by the second term of (11), but in 
dealing with small susceptibilities such as diamagnctic ones, 34 its inclu- 
sion is necessary. 

Without the second term, Eq. (11) would be the ordinary Langevin 
formula for the diamagnetism of atoms in the form given by Pauli 
(cf. Eq. (2), Chap. IV). Because of this term, however, Pauli' s formula 
does not apply to non-monatomic molecules. 35 This is closely connected 
with the fact that the validity of Larmor's theorem is confined to atoms 
(8). Because the additional term is inherently positive, Pauli 's formula 
will always be an upper limit to the diamagnetism, and estimates of the 
mean square orbital radii deduced from observed susceptibilities by means 
of this formula will tend to be somewhat too large except in atoms. 

One may inquire whether the second term of (11) can ever vanish, 
but this is possible only in atoms. The disappearance of this term would 
require that all the matrix elements P(n\ri) of the orbital angular 
momentum originating in the normal state n equal zero. As 

P(n;w) = TP(;n')P(n';n) = 2 |P(; ')!', (13) 

n n' 

this in. turn demands that the mean square angular momentum P 2 (n\ n) 
vanish for the normal state. The mean angular momentum P(n\ n) can, 
to be sure, vanish for diatomic molecules, but the mean square P 2 (n; n) 
cannot. By taking in (14), Chap. VI, /to be the operator corresponding 
to the square of the orbital angular momentum, it is not hard to show 36 
that P 2 (n', n) can vanish only if the wave function be invariant under 
a rotation of the coordinate system for the electrons without a corre- 
sponding rotation of the coordinate system for the nuclei. In case there 

34 Wo nevertheless neglect the paramagnetic term contributed by the molecular 
rotation, as we throughout disregard the contribution of the nuclei to the susceptibility. 
Duo to their large masses this term is in itself small compared oven to (11) ; see F. Bitter, 
1'hys. Zcits. 30, 497 (1929) for estimate. Furthermore, it is largely compensated by the 
diamagnetic contribution of the nuclei. This is true because the rotational quantum 
numbers arc so largo that we can almost use the classical theorem ( 24) on the cancelling 
of din- and paramagnotism, as far as the nuclei are concerned. 

35 Contrary to an. incorrect statement once made by the writer, Proc. Nat. Acad. 12, 
662 (1926). 3 For details see J. H. Van Vleck, Phys. Rev. 31, 600 (1928). 



278 THE PARA- AND DIAMAGNETISM OF X, 69 

is only one electron, this requirement may be more simply stated by 
saying that i/f must be a function of r alone. In any case, this demand 
can be satisfied only if the nuclear field is centro-symmetric. The 
latter, however, implies that there is only one attracting centre, i.e. an 
atom. In diatomic molecules, the physical interpretation of this non- 
disappearance of P 2 (n-,n) is that the two nuclei together exert a torque 
which causes fluctuations in the perpendicular component of orbital 
electronic angular momentum. The combined electronic and nuclear 
angular momentum is necessarily constant, but there are continual 
transfers back and forth between electrons and nuclei. 

As to the relative magnitude of the second term of (11), the fact that 
reasonable estimates of orbital radii can be obtained even in molecules 
by means of Pauli's formula shows that often this term must be 
fairly small compared to the first. Also Pascal's discovery 37 that the 
additivity method can be used to represent the diamagnetic suscepti- 
bilities of many organic compounds can only mean that atomic orbits 
are but little distorted by these molecular bonds, and that here the 
influence of the second part of (11), which is an interference effect 
between atoms, is subordinate. 38 

Quantitative calculation of the two parts of (11) has been attempted 
only for the hydrogen molecule. Even here, direct evaluation of the 
sum over the excited states ri would be excessively difficult, and it is 
necessary to adopt the artifice of replacing the variable denominator 
v(ri\ n) by a constant v v Then v l is a sort of mean absorption frequency 
which refractive data 39 lead one to take as 1-23.R, where R is the 
Rydberg constant. The elements of orbital magnetic moment differ from 
those of the angular momentum measured in multiples of h/27r, only 
by a constant factor /?. With the aid of the multiplication rule (13) the 
second term of (11) now becomes 



rac 
Miss Frank and the writer 40 showed that with Wang's 41 wave func- 

37 Pascal, various papers in Ann. Chim. Phys. 1908-13. 

88 In this connexion wo may cite particularly a paper by F. W. Gray and J. Farquharson, 
who examine critically the departures from additivity observed for various compounds, 
Phil. Mat/. 10, 191 (1930). Cf. also Gray and Dakors, ibid. 11, 81, 297 (1931). 

39 If, following Unsold (Ann. der Physik, 82, 380 (1927)), we replace l/r(n';n) by 
v(n';n)jv^; in Eq. (28), Chap. VII, the summation in the latter equation is readily per- 
formed in virtue of the quantum conditions. See Eq. (4), Chap. XIII. The evaluation 
of vi is then achieved by equating (28) to the observed electric susceptibility of H 2 . 

* J. H. Van Vleck and A. Frank, Proc. Nat. Acad. 15, 539 (1929). 

41 S. C. Wang, Phya. Eev. 31, 579 (1928). 



X, 69 FREE MOLECULES 279 

tions for the normal state of the hydrogen molecule, its mean square 
angular momentum P 2 (n\ri), which can be evaluated by the usual 
quadrature (14) of Chapter VI, is 0'394. This makes the expression 
(14) 0-51 x 10~ 6 per mol. Wang 42 calculated the first term of (11) to 
be 4-71xlO~ 6 . The computed molar susceptibility of H 2 is hence 
4-20X 1C- 6 . The experimental values 43 are 3-94 x 10~ 6 (Wills and 
Hector) and 3-99 Xl0~ 6 (Sonc). The agreement with these is quite 
satisfactory since the wave functions are not accurately known. 

Although the second term of (11) is only a little over 10 per cent, of 
the first in H 2 , it is quite probable that it is a somewhat larger fraction 
in other molecules where the nuclear field is less nearly centro-sym- 
metric. That H 2 departs less from atomic symmetry than most mole- 
cules seems to be evidenced by the fact that it alone among molecules 
has a normal Verdet constant in the Faraday effect ( 84). It is to 
be emphasized that there is no sharp dividing-line between diamagnetic 
molecules and feebly paramagnetic ones, mentioned in 68, where the 
second term of (11) predominates. One would expect this term to be 
particularly large for molecules formed out of atoms not of the 1 S type, 
and also molecules for which Mulliken's united atom, formed by 
collapsing the nuclei together, is not of the type 1 S ) for in these cases 
there is an overwhelming paramagnetism at large and small inter- 
nuclear distances respectively. 

70. Absence of Magneto -Electric Directive Effects 

It has often been conjectured 44 that especially in diatomic molecules 
with both electric and magnetic moments parallel to the molecular axis 
of figure, application of a magnetic field would produce electric as well 
as magnetic polarization, and that vice versa an electric field would 
magnetize the body. There would then be what may be termed a 
magneto-electric directive effect. The ground for this belief is the idea 
that when the molecules are oriented by an applied field of either nature, 
the electric and magnetic dipole axes would be alined together. 

Actually, experiments endeavouring to detect this effect have always 
yielded null results, even in liquids and solids. 45 ' 46 The only important 
diatomic gas for which such an effect might be expected is NO, for 

42 8. C. Wang, Proc. Nat. Acad. 13, 798 (1927). 

43 Wills and Hector, Phys. Rev. 23, 209 (1924); Hector, ibid. 24, 418 (1924); T. Sone, 
Phil. Mag. 39, 305 (1920). 

44 Debye and Huber, Physica, 5, 377 (1925) ; Debye, Zeite.f. Physik, 36, 300 (1926). 

45 Porrior and Borel, Archives des Sciences, 7, 289 and 375 (1925); Szivessy, Zetts.f. 
Physik, 34, 474 (1925); A. Perrier, Physica, 5, 380 (1925). 

4 A. Huber, Phys. Zeits. 27, 619 (1926). 



280 THE PARA- AND DIAMAGNETISM OF X, 70 

nitric oxide is the only common polar paramagnetic gas. The electric 
moment of the NO molecule is, to be sure, so small that it has not yet 
been measured quantitatively, but is undoubtedly different from zero, 
as the N and O atoms are not identical. The very sensitive experiments 
of Huber 46 show that even when extremely intense magnetization is 
produced by applying a magnetic field to liquid NO, there is no observ- 
able electric polarization. 

Despite the considerations advanced in the first paragraph, there is 
no difficulty in explaining theoretically why experiments invariably 
reveal no directive effect. The standard explanation is one first pro- 
posed specifically for NO by de Haas, 47 though previously suggested by 
Piccard 48 in connexion with experiments on certain solids. DC Haas 
suggests that there are two kinds of NO molecules, in which the electrons 
circulate respectively clockwise and counter-clockwise about the axis 
of electric polarity. These left- and right-handed molecules woidd pre- 
sumably be present in equal amounts. There is then on the average no 
correlation between the directions of electric and magnetic moments, 
and hence no magneto-electric directive effect. 

Hecent developments in the theory of band spectra, too complicated 
for us to give in much detail, show that this explanation by Piccard 
and de Haas is not quite correct. There are indeed two kinds of NO 
molecules, corresponding to the two components of what spectroscopists 
call a A-type doublet, 49 but each kind is in itself both left- and right- 
handed at once. There would thus be no directive effect even if we 
could isolate one of the kinds. The existence of such 'ambidextrous' 
molecules is a characteristic quantum effect which cannot very well be 
explained in terms of ordinary geometrical pictures. It arises because 
the molecular rotation removes the degeneracy associated with the 
identity of energies for the states Q, Q, in a stationary molecule. The 
correct wave function proves to be a linear combination of those corre- 
sponding to the states Q and LI. The parallel component of electronic 
angular momentum is thus of indeterminate sign, though of definite 
numerical magnitude |O|, when referred to an axis having invariably 

47 W. J. do Haas, Kon. Akad. Wet. Amsterdam, 35, 221. 

48 A. Piccard, Archives de Sciences, 6, 404 (1924). 

49 This was formerly called a o-typo doublet. Tho term li-typo doubling would bo tho 
most expressive, as with spin the signs of A and X must bo revorsod together to givo tho 
degeneracy in a stationary molecule. For theory of this doubling see B. de L. Kronig, 
Zcits. f. Ptiysik, 46, 814; 50, 347 (1928-9), Band Spectra and Molecular Structure, 
Chap. II; J. H. Van Vlock, Phy. Rev. 33, 467 (1929); R. S. Mulliken, ibid. 33, 507 
(1929). Tho A-doubling is superposed on tho much coarser true multiplot structure, 
such as e.g. the spin doublet in NO, and should not bo confused with tho latter. 



X, 70 FREE MOLECULES 281 

the N atom on one given end and the atom on the other. We talked 
in 63-7 as if this component were constant in both magnitude and 
sign, which we now see it is not, but this inaccuracy is admissible on 
two grounds. First, the frequency of oscillation in sign is measured by 
the width of a f A -type doublet', and is hence very small compared to 
kT, so that this component always remains of the very low-frequency 
category and hence as good as constant as far as the magnetic suscepti- 
bility is concerned. Secondly, it would be equally logical to say that 
the electric moment of the molecule fluctuates in sign relative to tho 
magnetic axis of the molecule, as the choice of axis is somewhat 
arbitrary. If one uses axes fixed in space the fluctuations in sign prove 
to be in the electric rather than magnetic moment. 50 This is because 
the two A-doublet components are respectively even and odd with 
respect to the transformation (A), p. 203, and so have no first-order 
Stark effect, as there explained, although there is a first-order Zeeman 
one. A very powerful electric field, however, produces a Paschen-Back 
transformation on the doublet and gives a first-order Stark effect. 51 The 
hypothesis of Piccard and de Haas is then correct. In any field strength 
there is on the average no correlation between the electric and magnetic 
axes, and hence no directive effect. 

50 As tho rotational frequencies enter in tho direction cosines connecting a coordinate 
system fixed in spaco with ono fixed in tho molecule, expressions which avorago to zero 
in ono system do not necessarily in tho other. Note that tho mean angular momentum 
relative to axes fixed in tho molecule definitely vanishes in a diatomic molecule only in 
virtue of removal of tho degeneracy by tho molecular rotation ; in a stationary ono it is 
really ambiguous because any linear combination of tho wave functions for the states 
i}, 12 could be used. On tho other hand, in a polyatomic molecule, the disappearance 
( 08) is duo to dissymmetry in. the nuclear field, and so found oven in stationary 
molecules. 

51 Cf. Penney, Phil. Mac/. 11, 602 (1931). 



XI 

THE PARAMAGNETISM OF SOLIDS, ESPECIALLY 
SALTS OF THE IRON GROUP 

71. Delineation of Various Cases 

We shall stress primarily only the new quantum developments rather 
than the innumerable classical theories of magnetization in solids. As 
the present and following chapters are a digression from our intent 
to study only rarefied media, and as the quantum theory of magnetism 
of solids has so far achieved success more in the bold qualitative outlines 
of the phenomena rather than quantitative detail, we shall not docu- 
ment the experimental measurements quite as completely as in the two 
preceding chapters. A whole volume would be required to digest the 
copious experimental work on the iron family alone. 

Different solids can exhibit susceptibilities of entirely different 
natures, and it may be well to outline in advance the various cases 
which can occur and in what materials they arc commonly found. 

(a) Instances where the inter-atomic forces are so small that the 
magnetism can be calculated by treating the atoms of the solid to be 
as free as in an ideal gas. The criterion for this is that the work required 
to orient an atom against the inter-atomic forces be small compared to 
kT. This case is exemplified remarkably well by rare earth salts, which 
have consequently been discussed at length in Chapter IX on free atoms 
and ions. As noted to the writer by Professor Bohr, the extraordinary 
f reeness of the 4/ orbits is revealed not only by the magnetism but also 
by the sharpness of the spectral lines from rare earth salts. This can 
only mean that the 4/ wave functions of the various rare earth atoms 
project out very little from the interiors of their respective atoms and 
so 'overlap' other atoms only very slightly even in the solid state. 

(/;) Solids or solutions in which inter-atomic forces quench the orbital 
angular momentum but leave the spin free. This is what probably 
occurs in most salts of the iron group, as we shall see in 72. 

(c) Solids in which there is such strong internal magnetic coupling, 
i.e. such wide multiplets, that irrespective of the Heisenberg exchange 
effect the inter-atomic forces of necessity quench the spin angular 
momentum when they do the orbital. It is hard to distinguish experi- 
mentally between this case (c) and (e), (f) below, but case (c) is possibly 
sometimes realized in some salts of the platinum and palladium groups 
( 75). 



XI, 71 THE PARAMAGNETI8M OF SOLIDS 283 

(d) Solids in which the Heisenberg exchange forces tend to aline the 
spins parallel and so create ferromagnetism. This is, of course, the case 
of iron, nickel, cobalt, also a few alloys which are ferromagnetic. 

(e) Solids in which these forces have the opposite sign from that in 
(d) and so tend to aline the spins antiparallel and destroy magnetism. 

(/) Materials in which the spin angular momenta compensate each 
other because of the restrictions imposed by the Pauli exclusion prin- 
ciple rather than because of the exchange effect. 

In cases (e) and (/) any orbital angular momentum is ordinarily 
quenched as in (6). Hence in (c), (e), and (/) the orbital and spin 
magnetic effects are both largely destroyed, so that these cases all give 
feeble paramagnetism, or even diamagnetism. One of these cases must 
be the commonest of all, as most elements (distinct as from salts) 
exhibit only a feeble paramagnetism, if any, in the solid state. 
Cases (e) and (/), which will be discussed in 80, are more probable 
than (c). 

We throughout use the term 'quenched' when the constancy of 
angular momentum is so completely destroyed by inter-atomic forces 
as to blot out most of the paramagnetism which would be found in the 
ideal gas state. The distinction between the various cases is, of course, 
usually not a hard and fast one. Besides ()-(/) there is also the trivial 
case of solids composed exclusively of atoms which are in 1 8 states when 
free and which are hence without appreciable magnetism. 

The Heisenberg 'exchange' or Austausch forces 1 play a very important 
role in the magnetism of solids, especially in ferromagnetism. As far 
as the present chapter is concerned, it will be sufficient to say that the 
exchange forces have the effect of introducing a very strong coupling 
between the spins of paramagnetic atoms or ions. Diamagnetic atoms 
or ions have no resultant spin and so do not give rise to any exchange 
forces tending to orient the spins of other atoms. The mathematical 
basis for these statements will be given in Chapter XTI. The important 
thing for present purposes is that the exchange forces become of sub- 
ordinate importance in media of considerable 'magnetic dilution', i.e. 
media in which the density of paramagnetic atoms or ions is low because 
the great majority of the atoms are diamagnetic. Such media are the 
primary concern of the present chapter, and so it seems best to defer 
until Chapter XII the detailed description of the nature and workings 
of the Heisenberg exchange effect. 

1 W. Hoisonberg, Zeits.f. Physik, 38, 411 (1926); 49, 619 (1928). 



284 THE PARAMAGNETISM OF SOLIDS, XI, 72 

72. Salts and Solutions Involving the Iron Group 

Pure solid elements of the iron group have high magnetic concentrations 
and large exchange effects, leading to the ferromagnetic phenomena to 
be discussed in the next chapter. On the other hand, most salts in- 
volving ions of the iron group are only paramagnetic, except possibly 
at extremely low temperatures. In these salts the magnetic dilution is 
usually sufficient to warrant neglect of the exchange forces. This is 
perhaps obvious only if the salt is in solution, or is highly hydrated 
in the solid state. However, it is found that in true salts (not oxides) 
the susceptibility is usually affected comparatively little (not over 10 
per cent, in many cases) by whether or not water molecules are present 
to increase the magnetic dilution. 

One is first tempted to try calculating the susceptibilities of salts of 
the iron group in the same fashion as for the rare earths, viz. under 
the assumption that the paramagnetic ions are perfectly free. The 
general nature of the procedure with the aid of the Hund spectroscopic 
theory has been fully explained in connexion with 58 and 59 on the 
rare earths, and so need not be repeated. The difference is that the 
incomplete inner group is now one of 3d rather than 4/ electrons. 2 
The comparison of theory with experiment is given in the table on 
p. 285, which corresponds to that in 58 for the rare earths. 

The values in the columns headed Ai> = and Ai> = oo are those obtained 
from the asymptotic formulae (6) and (10) of Chapter IX applicable 
respectively to multiplcts which are very narrow and very wide compared 
to kT. These limiting cases were first studied by Laporte and Sommer- 
feld. 3 The column 'actual Av' gives the values obtained at 293 K. 
by means of the accurate formula (16), Chap. IX, which must be used 
when the multiplet widths are comparable to kT. Such calculations 
were first made by Laporte, 4 and the reader is referred to his important 
paper for the details of the estimates of the screening constants <j used 
to determine the multiplet width by means of Eq. (21), Chap. IX. As 
a rule the values of a represent only a slight extrapolation from optical 
or X-ray data on other atoms and ions. The calculations in the column 

2 For details of the spectroscopic theory of the iron group RCO Hund, Linwnspeklrcn, 
33. Besides the 3d electrons and the closed groups already completed at argon, the 
neutral atoms of the iron group contain from one to two 4s electrons, but those 4a electrons 
are presumably the first to be lost when there is any ionization, so that all the ions 
involved in the table have only argon-like shells plus the 3d electrons. 

3 O. Laporte and A. Sommerfeld, Xeits.f. Physik, 40, 333 (1926). Somewhat similar 
calculations have also been made independently by Fowler, and the results briefly given 
in his Statistical Mechanics , p. 303. 

4 O. Laporto, Zeita.f. Physik, 47, 761 (1928). 



XI, 72 ESPECIALLY SALTS OF THE IRON GROUP 285 

EFFECTIVE BOHR MAGNETON NUMBERS FOR IONS OF THE IRON GROUP 5 



Term 



, 6/2 



3d* 



Theoretical /x r1t 


Experimental /x off 


Ion 


Av -= o 


Av oo 


Actual 

Av 


Spin 
only 


Solutions 


Salts 


K'...V 5t " 


0-0 


0-0 


0-0 


0-0 


0-0 


0-0 


rsc i * 


3-0 


1-55 


2-57 


1-73 






I Ti i i i 


3-0 


1-55 


2-18 


1-73 






\V >+-'-. 


30 


1-55 


1-78 


1-73 


1-75 


1-79 


fTi 1 " 4 " 


4-47 


1-63 


3-36 


2-83 






"1 y i- H -i- 


4-47 


1-63 


2-73 


2-83 


2-76-2-85 




rv-^ 


5-20 


0-77 


3-tiO 


3-87 


3-81-3-86 




Cr"i- 


5-20 


0-77 


2-97 


3-87 


3-68-3-86 


3-82 


iMn 1 < M 


5-20 


0-77 


2-47 


3-87 


4-00 




jCr'^ 


5-48 


0-0 


4-25 


4-90 


4-80 




\Mn * * ' 


5-48 


00 


3-80 


4-90 




5-05 


f Mn f h 


5-92 


5-92 


5-92 


5-92 


5-2-5-96 


5-85 


\Fe +4 -' 


5-92 


5-92 


5-92 


5-92 


5-94 


5-4-6-0 


Fo 1 ""^ 


5-48 


G 70 


6-54 


4-90 


5-33 


5-0-5-5 


Co 1 ' 


5-20 


6-64 


6-56 


3-87 


4-6-5-0 


4-4-5-2 


Ni 1 ' 


4-47 


5-59 


5 56 


2-83 


3-23 


2-9-3-4 


Cu 1 '- 


3-0 


3-55 


3-53 


1-73 


1-8-2-0 


1-8-2-2 



'actual Ai>' are inclusive of the term <*j in (10), Chap. IX, which 
Laporte neglected. This term is much less important in the iron group 
than in the rare earths Sm+++ and EU+++ ( 59), and has an appreciable 
influence only in V l+ , Cr ++f , Mn l+ ^ ( , Cr l+ , Mn +H , where Laporte's 
original values were 3-23, 2-61, 2-01, 3-74, 3-16 instead of 3-60, 2-97, 
2-47, 4-25, 3-80. The values given in the table for Sc++-V +++ inclusive 
are only slightly higher (about 5 per cent.) than Laporte's, while those 
in the bottom half are identical with his, as here the inversion of the 
multiplets makes the effect of a negligible. The rare earths other than 
Sm and Ku had such wide multiplet widths that the magneton number 
could be calculated without appreciable error under the supposition of 
multiplet widths extremely great compared to kT, but comparison of 
the columns 'Av = oo' and * actual Av' shows that this is very often not 
the case in the iron group. This is, of course, because here the atoms 
are lighter than in the rare earths, and the multiplets thus narrower. 

Because the multiplet widths are more precisely known, the present 
theoretical calculations for free ions of the iron group should be more 
accurate than for the rare earths, but comparison of the columns 'actual 
Ai/' and 'experiment' shows that the agreement with observations on 

5 The experimental magneton numbers quoted in the table are the same as those given 
by Stoner in a survey in Phil. May. 8, 250 (1929), except that wo have added Kroed's 
measurements on the vanadium ion to be c-itod in 74. A very complete documentation 
of experimental data is given by Cabrera in the report of the 1930 Solvay Congress. 



286 * THE PARAMAGNETISM OF SOLIDS, XI, 72 

salts and solutions is miserable, in marked contrast to the situation in 
the rare earths. One must therefore grope for some other explanation 
of the measured susceptibilities. As noted by Sommerfeld, 6 Bose, 7 and 
Stoner, 8 the latter are represented quite well if we use the formula 

(i) 

instead of the theoretical expressions based on the ordinary spectro- 
scopic theory for free ions. Here 8 is the spin quantum number for the 
appropriate spectral term listed in the table. For instance, 8 is 3/2 for 
Cr+++, as the multiplicity 28+1 of a *F term is 4. The magneton 
numbers calculated from (1) are given in the column marked 'spin only' 
and are seen to be in fairly good agreement with experiment. 

Mechanism for Leaving Only Spin Free. Our problem is now to 
obtain a theoretical justification for Eq. (1), which gives the same 
susceptibility as though we substituted 8 for D and F terms throughout 
the first column of the table, with the multiplicity unaltered. Two 
possibilities immediately suggest themselves. One is that the Hund 
theory of the assignment of spectral terms is in error, and that the ions 
in question are normally all in 8 states even when free. This proposal 
appears to have actually once been made by Sommerfeld, but is now 
abandoned by him. In our opinion it must be quite definitely rejected, 
as there is an abundance of experimental spectroscopic evidence for 
the correctness of the Hund theory in the iron group, not to mention 
Slater's perturbation calculation 9 which confirms the Hund predictions 
on the lowest-lying terms. In fact, S terms of the necessary multiplicity 
are not allowed by the Pauli exclusion principle, unless one supposes 
that there is some other incomplete group besides that of the equivalent 
d electrons. 

Another possibility, and one which we advocate, is that the assign- 
ment of spectral terms is correct, and that the theoretical calculations 
in the column ' actual Ai/' would be confirmed by experiment if measure- 
ments could be made on atoms or ions which are really free. The 
absence of such a confirmation is to be attributed to the fact that the 
existing observations are not on vapours or gases, but instead on solu- 
tions or salts, where there are inevitably large inter-atomic forces. If, 
then, these inter-atomic forces quench the magnetic effect of the orbital 

a A. Sommerfeld, Atombau, 4th od., p. 639, or Phys. Zeit*. 24, 360 (1923); ZritH. /. 
Physik, 19, 221 (1923). 7 D. M. Bose, ZeUs. f. Phyxik, 43, 864 (1927). 

8 E. C. Stouor, Phil Mag. 8, 250 (1929). 
J -C. Slater, Phys. Rev. 34, 1293 (1929). 



XI, 72 ESPECIALLY SALTS OF THE IRON GROUP 287 

angular momentum but leave the spin free, one will have precisely the 
expression (1) for the susceptibility. This point has been particularly 
emphasized by Stoner. The whole problem thus resolves itself into 
showing that from a theoretical standpoint it is reasonable to expect 
that the inter-atomic forces have a quenching effect of this type which 
blots out the orbital magnetic moment but not the spin. Stoner 8 showed 
that the necessary quenching could be obtained if one assumes that the 
inter-atomic forces are equivalent to extremely large random magnetic 
fields which are rather mysteriously supposed to act on only the orbital 
angular momentum. A somewhat similar assumption of random mag- 
netic fields was successfully used in Kapitza's theory 10 of the influence 
of magnetic fields on electrical conduction, although the true explana- 
tion of Kapitza's experiments is much more complicated. Stoner did 
not propose his calculation by means of random magnetic fields except 
in a preliminary, suggestive way. It should not be taken literally, as 
inter-atomic forces are, of course, primarily electrostatic rather than 
magnetic in nature, and the magnetic portion would be much too weak 
to do the requisite quenching. In the following section, however, we 
shall show that precisely the type of quenching that leads to Eq. (1) 
ensues if each atom or ion is subjected to sufficiently asymmetrical 
electrical forces. 10 * In our opinion such asymmetrical electrostatic fields 
are probable, and are the real explanation of the quenching phenomena 
first proposed and roughly described by Stoner. The reason why, on 
the other hand, the ordinary theory for free ions is applicable to the 
susceptibilities of rare earth salts is doubtless, as already stated, that 
their 4/ electrons are sequestered in the interior of the atom and so are 
not influenced nearly as much by neighbouring atoms as the 3d electrons 
involved in the iron group. 

73. Quenching of Orbital Magnetic Moment by Asymmetrical 
External Fields 

In a solid or liquid, the electrons of a given atom may to a first 
approximation be regarded as in an inhomogeneous external electric field 
which represents to this approximation the effect of other atoms on 
a given atom. This method has been extensively used by Bethe 11 and 
by Kramers, 12 and is admissible inasmuch as the other atoms are 

10 P. Kapitza, Proc. Roy. Soc. 123A, 342 (1929). 

10a During tho printing of the present volume this point has also been emphasized in 
a paper by Pauling, J. Artier. Chem. Soc., 53, 1367 (1931). 

11 H. Bethe, Ann. der Phyaik, 3, 133 (1929); Zeita.f. Physik, 60, 218 (1930). 
18 H. A. Kramers, Proc. Amsterdam Acad. 32, 1176 (1929). 



288 THE PAHAMAGNETISM OF SOLIDS, XI, 73 

approximately in the same configuration as in the absence of the given 
atom. Let us expand the potential energy of any electron of the 
given atom in this external field as a Taylor's series about the centre 
(nucleus) of the atom. The terms of successive order in the expansion 
usually involve the inter-atomic separation ('grating spacing') to suc- 
cessive negative powers so that the series converges rapidly. If there 
are similar equidistant atoms on either side of the given atom the linear 
terms disappear from the expansion, and the 'cross' products in the 
quadratic terms can, of course, be made to vanish by a proper rotation 
of axes. The terms of lowest non- vanishing order in the expansion of 
the external potential energy are then 



, (2) 

where the summation extends over all the electrons of the given atom. 
We do not necessarily claim that a simple quadratic form like (2) is 
a good quantitative representation of the interactions between atoms. 
However, only a rough qualitative portrayal, which is conveniently 
accomplished by means of (2), is needed to show that sufficient dis- 
symmetry will quench out the magnetic effect of the orbital angular 
momentum. 

Until further notice we shall neglect the spin and also any dissym- 
metries occasioned solely by higher-order terms in the Taylor's expan- 
sion than (2). We assume throughout that the external fields are never 
strong enough to destroy appreciably the Russell-Saunders quantization 
of the atom. In other words, electrostatic forces within the atom are 
supposed greater than forces from without the atom, so that the squares 
of the orbital and spin angular momentum are approximately L(L-j-l) 
and $($+1), even though we shall sec that the spacial quantization is 
greatly disturbed by the external fields. 

Case A = B C. If all three coefficients in (2) are equal, then the 
atom is as freely oriented in a solid as in a gas. The ideal magnetic 
theory for free atoms or ions should then apply. 

Case A = B,B^C. In case two of the three coefficients are equal, 
say A and B, the component of orbital angular momentum which is 
parallel to the z axis is conserved, and can be assumed to have a 
quantized value M L (in multiples of 7*/2?r) as the z axis now becomes 
one of symmetry. On the other hand, the matrix elements of the x and 
y components of angular momentum will be exclusively of the form 
&M L = 1 (cf. Eq. (89), Chap. VI). In other words, L z is a diagonal 
matrix, while L x , L u contain no diagonal elements. If the effect of M L 



XI, 73 ESPECIALLY SALTS OF THE IRON GROUP 289 

on the energy is large compared to kT, which means that 



then the contributions of L x , L y to the susceptibility when calculated 
by the methods of Chapter VII will be entirely of the high-frequency 
type, and hence will be relatively ineffective, as they then give terms 
having the high frequencies instead of k T in the denominator (cf. Eq. 
(28), Chap. VII). Thus only the contribution of the z component 
L Z = M L remains. If one assumes that the axes of symmetry of the 
micro -crystals can have a random spacial distribution, as in, e.g. a 
crystal powder, then on neglecting high-frequency elements one easily 
finds that x^N^^T. Hence, supposing \A C|z 2 > kT, one has 
X == or x ~- NL 2 /3kT according as the minimum in energy corresponds 
to a minimum or maximum in \M L \. In the former case the normal 
state has M L = 0; in the latter, M L = L. The present situation is 
similar to that in the well-known Lenz-Ehrenfcst 13 classical theory of 
magnetism, which was developed under the assumption that the atom 
has two positions of equal potential energy in a crystal, corresponding 
to the fact that in the present case A == B this energy is independent of 
the sign of M L . If one included higher-order 'saturation' terms in the 
field strength one would obtain the so-called Ehrenfest function as an 
expression for the paramagnetism. 14 Under the present supposition 
A ~ B, only two of the three components of orbital angular momentum 
are necessarily quenched. The third or z component is also quenched 
if one assumes that the level M L = is alone a normal state, but it is 
doubtful whether this could be the case universally enough to explain 
the widespread applicability of (1). In the following paragraphs, how- 
ever, we shall show that all three components are indeed quenched if 
instead we assume that all three coefficients A, B, C are unequal, so 
that there is complete dissymmetry. 

Case A, B, C All Unequal. Here the important fact is that the spacial 
degeneracy is completely removed. This is true inasmuch as (2) leads 
to the same secular problem as that of the asymmetrical top, which is 
non-degenerate iinless two or more of the coefficients in the quadratic 
form are equal. For this observation the writer is indebted to Professor 
Kramers. 15 Quite irrespective of the formal mathematical analogy to 

13 W. Lenz, Phys. Zeita. 21, 613 (1920); P. Ehrenfest, Proc. Amsterdam Acad. Dec. 
18, 1920, or Leiden Communications, Suppl. 44b. 

14 For elaboration oil the Ehrenfost function see p. 712 of Debye's article in vol. vi 
of the Handbuch, der jRadiologie. 

15 The Hamiltonian function involved in tho problem of the asymmetrical top is 
the quadratic form oPg,-f&PJ,-f cPj,, where P x , t P v ,, P z ,, are tho components of angular 

3595.3 n 



290 THE PAHAMAGNETISM OF SOLIDS, XI, 73 

the asymmetrical top it seems quite obvious that there is no remaining 
degeneracy when all three coefficients become unequal, as there was 
only the twofold degeneracy M L = \M L \ even when two coefficients 
were equal. In the present case there is no axis of symmetry about 
which the angular momentum is conserved, so that there is no longer 
this degeneracy associated with the equivalence of left- and right- 
handed rotations about such an axis. 

Now in 68 we showed that when the degeneracy is completely 
removed it becomes necessary to use real wave functions, and that 
hence the average magnetic moment is zero. Thus when the coefficients 
in (2) are all unequal, the orbital magnetic moment matrix contains no 
diagonal elements. If the sejmration of energy-levels occasioned by 
removal of the degeneracy is large compared to kT, the contribution of 
the orbital angular momentum to the susceptibility will be entirely 
of the high-frequency type (Eq. (2), Chap. IX) and hence relatively 
small. Hence, if the coefficients A, B, C in (2) are sufficiently large 
and unequal, the magnetic effect of the orbital angular momentum 
becomes practically entirely quenched. The residual effect of the high- 
frequency elements never disappears entirely but becomes negligible if 
the spacial separation is of the order 10 3 to 10* cm.- 1 or greater. Here 
and elsewhere we use the term 'spacial separation 5 for the difference 
in energy between the various non-degenerate states into which the 
energy-levels are separated by (2) or by an asymmetric inter-atomic 
field in general. If, on the other hand, the spacial separation is small 
compared to kT, the general theorem of Chapter VII shows that the 
orbital angular momentum will make its full contribution to the sus- 
ceptibility. 

Even though the potential (2) is probably not a close quantitative 

momentum referred to the principal axes, and whore a, b, c., are half tho reciprocals of the 
principal moments of inertia. A general theorem of group theory shows that tho resulting 
secular problem is similar to that of <2> inasmuch as x, y, z and P x , t P v ,, P z ,, transform 
similarly under a rotation of axes. The term ' secular' is here to be construed as moaning 
that we retain only the portion of <2> which is diagonal in L; otherwise the dynamical 
problem arising from <2> is more complicated than that of tho asymmetrical top. The 
retention of only tho diagonal elements is equivalent to our assumption that <2> does not 
destroy the Russell-Saundcrs coupling. Tho quantum numbers J, A in tho asymmetrical 
top correspond to L, M L in <2> ; this is a formal mathematical rather than physical 
correspondence; the asymmetrical top involves as a third quantum number a spacial 
quantum number physically somewhat similar to M&, but this does not enter in its 
secular problem. That tho asymmetry removes the degeneracy as regards tho sign of A 
has been shown independently by Kramers and Ittmann, Zeits.f. Phyaik, 53, 553; 58, 
217; 60, 663 (1929-30), and rather more explicitly by Wang, Phys. Rev. 34, 243 (1929) ; 
cf. especially his Eq. (12). The formal correspondence of the quantum numbers J, A and 
L y M L appears particularly clearly in tho work of Klein, Zeits.f. Phyaik, 58, 730 (1930). 



XI, 73 ESPECIALLY SALTS OF THE IRON GROUP 291 

approximation to the perturbing effect of other atoms, the preceding 
considerations nevertheless make it evident that sufficiently large and 
unsymmetrical external fields will quench the contribution of the orbital 
angular momentum to the susceptibility. In regular crystals it is usually 
necessary to retain higher-order terms than the second in the Taylor's 
expansion in order to reveal the exact degree of symmetry. An elaborate 
investigation of the effect of external fields of various types of crystalline 
symmetry has been made by Bethe, and readers interested in details 
are referred to his papers. 11 He finds that there can be at most only 
a rhomboidal symmetry if the spacial degeneracy is completely removed. 
Ordinary tetragonal, cubic, or hexagonal symmetries will not remove 
all the degeneracy. With these types of symmetries at least two of 
the coefficients in (2) are equal, but when higher-order terms in the 
Taylor's development are included (e.g. the fourth-order terms 
1) 2 (^H'Z/t+zJ) if there is cubic symmetry) then there is no longer an 
axis of symmetry about which angular momentum is conserved, and 
no component of the latter is what Dirac terms a ' constant of the 
motion'. Bethe shows 16 that nevertheless there is still a partial de- 
generacy, so that it is possible to use complex solutions and obtain an 
average magnetic moment, leading to an expression for the suscepti- 
bility of the form NC/ZkT, where in general 0< C< L(L+l)p*, so 
that there is only a partial quenching. For instance, he finds that an 
F term is split by an external field with cubic symmetry into two triply 
degenerate states and one single or non-degenerate state. The orbital 
magnetic moment can be completely quenched only if the single or 
non-degenerate state has so much less energy than the multiple ones 
that it alone is a normal state. 17 The situation is thus somewhat ana- 
logous to that when A B, B = C in (2), as there we found the 
quenching was complete only if the state M L = had much less energy 
than the pairs of states corresponding to each \M T \ >0. Thus suffi- 
ciently large external fields of rhomboidal or lower symmetry always 

18 H. Betho, ZcitH.f. Phynik\ 60, 218 (1030). 

17 Kvoii if the degeneracy is not completely removed, so that two or more wave func- 
tions, say P, Q, represent states of identical energy, it enii sometimes happen that there 
is no magnetic moment, as the most general linear combination of 1', Q may still yield 
a non-magnetic state. In. other words tho ability to use complex wave functions is a 
necessary but not sufficient condition for the existence of magnetic moments. For 
instance, Betho shows that a D term splits in a field with cubic symmetry into one triply 
and one doubly degenerate state, but tho doubly degenerate state is non -magnetic so that 
the orbital magnetic moment is quenched if tho doubly degenerate state has the lower 
energy. With tetragonal instead of cubic symmetry, however, the multiple levels are 
necessarily magnetic, according to Betho's analysis. 

U2 



292 THE PARAMAGNETISM OF SOLIDS, XI, 73 

quench completely, while with more symmetry the quenching is com- 
plete only if certain states are the lowest lying. 

Bethe shows that when an average magnetic moment persists despite 
the absence of an axis of symmetry (i.e. when there is more than 
rhomboidal but less than axial symmetry) that there can be a first-order 
Zeeman effect but that the selection rules are no longer the usual ones, 
and unusually large changes are permitted in the spacial quantum 
number, which no longer has the usual kinematical significance as pro- 
portional to a component of angular momentum. Bethe 's theory finds 
a direct confirmation in Becquerel's observation of abnormally large 
Zeeman effects in certain crystals containing rare earths. The external 
fields which Bethe utilizes to explain Becquerel's 18 observations do not 
necessarily contradict the calculation in Chapter IX of the suscepti- 
bilities of rare earth ions on the assumption that the latter are free, 
as these fields might produce a spacial separation small compared to 
kT } and hence not quench the orbital angular momentum as far as 
susceptibilities are concerned. Bethe shows that the appearance of the 
anomalous Zeeman lines is not contingent on the absolute value of 
the spacial separation, but only on the ratio of the fourth- to second- 
degree terms in the Taylor's expansion of the potential. This ratio must 
not be too small compared to unity. We have repeatedly emphasized 
that susceptibilities depend on second- as well as first-order Zeeman 
terms, and so are not necessarily altered when the usual spectroscopic 
Zeeman patterns are changed. A more serious difficulty is that Bethe 
needs fields with more than rhomboidal symmetry to explain Becquerel's 
results. If one assumes that crystals involving iron ions have fields of the 
same symmetry but of much greater magnitude than Becquerel's rare 
earth compounds so that the separation of non-coinciding levels becomes 
greater rather than smaller than TcT, the quenching of orbital angular 
momentum would not in general be as complete as needed for the 
validity of Eq. (1). Independently of Becquerel's work, Eq. (1) is found 
to be valid in crystals of the iron group in which it is known that there 
is more than rhomboidal symmetry. An example is NiO, which con- 
forms roughly to (1), at least at room temperature, but which has cubic 
symmetry like NaCl. Unfortunately X-ray analyses of crystal structure 
seem to be wanting for the sulphates of the iron group, for which the 
magnetic data are the most complete. When (1) is found to apply 
despite more than rhomboidal symmetry, the simplest 17 explanation is 
that for some reason a 'single' level lies below the multiple ones, a pos- 

18 J. Becquerel, Zeita.f. Physik, 58, 205 (1929). 



XI, 73 ESPECIALLY SALTS OF THE IRON GROUP 293 

sibility discussed at the end of the preceding paragraph. (Do not confuse 
this use of the terms 'single' and 'multiple' as regards the splitting due 
to inter-atomic fields with the 'singlets' and 'multiplets' introduced in 
connexion with spin fine structure.) The fact that most molecules are 
diamagnetic indeed seems to indicate that in complexes the non- 
magnetic states usually have the least energy. 

In solutions there is surely no difficulty in believing that the fields 
have the degree of dissymmetry required to quench the angular momen- 
tum, for in liquids the atoms are doubtless rather irregularly spaced. 
The coefficients A, B, Cm (2) might even be regarded as functions of 
the time which vary with the approach and recession of atoms from 
each other. Also linear as well as quadratic terms in x t > y t , Z L may be 
required. However, it is quite possible that in solutions the ions are 
not free, but attach themselves to water molecules, their so-called water 
molecules of coordination. In fact many chemists believe that an ion 
has the same definite number of water molecules attached to it in solu- 
tions as in the hydrated solid salt. A divalent ion containing iron, for 
example, would then really be of the form Fe++ . nH 2 O rather than Fe +f . 
The theory of polyatomic molecules ought, then, really to be invoked. 
This point has been particularly emphasized by Freed. 19 The magnetic 
theory for molecules has been discussed in Chapter X and shows a cer- 
tain amount of resemblance to the various cases just presented. For 
instance, a diatomic molecule is somewhat similar to the case A B, 
B ^ C, as there is an axis of symmetry and the orbital magnetic moment 
perpendicular to this axis is blotted out, while the parallel component 
is quenched only if one particular state (viz. the 2 state A 0) falls 
below the others in energy. A molecule with more than two atoms is 
similar to the case in which A, B, C are all unequal, since we showed 
in 68 that in general the orbital magnetic moment is largely quenched 
in unsymmetrical polyatomic molecules. This, we now see, is because 
all symmetry in the fields is lost in complicated molecules. Of course, 
the electrons in a molecule circulate freely from one atom to another, 
so that the representation of inter-atomic forces within the molecule as 
equivalent to a constant external field on each atom is but a poor 
quantitative approximation, but this does not affect the symmetry 
considerations. Furthermore, the individual atomic configurations 
probably preserve their identities rather more in complexes such as 
Fe ++ . nHgO than in true molecules, as these complexes are doubtless 
held together by weak polarization forces rather than true valence 

18 S. Freed, J. Amer. Chem. Soc. 49, 2456 (1927). 



294 THE PARAMAGNETISM OF SOLIDS, XI, 73 

forces. One might wonder whether it would not be possible to find 
a solvent in which the dissolved ions did not form such complexes, as 
then the ions would be less disturbed and it might be possible to test 
experimentally the theory for free ions. Unfortunately, however, the 
ability to attach molecules of coordination is probably the criterion 
for solubility, as well as for the dissociation of the dissolved salt 
into ions. 

It is conceivable that in certain solids the unit of crystalline structure 
is the molecule rather than the atom, in which case the considerations 
of the preceding paragraph or of Chapter X can be used. It is, of course, 
impossible to delineate sharply the point at which molecules or clusters 
begin to be formed in solids or liquids, and the representations by 
external fields and by isolated molecular entities are only asymptotic 
ones, but it seems quite safe to interpolate to intermediate or transition 
cases the result that sufficient dissymmetry quenches the orbital mag- 
netic moment. 

We have ostensibly assumed throughout that the atom of the iron 
group loses all of its s electrons and exists in a definite ionic form. 
However, the quenching considerations based on symmetry clearly 
apply equally well in the event of non-polar or coordination' bonds in 
which these electrons are merely shared or even traded, provided only 
that they are so grouped as to have zero resultant spin, thus leaving 
only the d electrons not in closed configurations. 

We have gone to considerable length to show that inter-atomic forces 
can quench out the magnetic effect of the orbital angular momentum. 
The question now arises as to why the spin is free and uiiquenched. 
There is no difficulty in understanding that forces from other atoms do 
not perturb the spin. The only inter-atomic forces which can have an 
appreciable orienting influence on the spin are the exchange forces 
between paramagnetic atoms or ions, cited on p. 283. It was there 
explained that the media studied in the present chapter are of sufficient 
'magnetic dilution' to make this influence unimportant. We must, 
however, show that the spin is free as regards the magnetic forces 
which arise from within the atom and which arc responsible for spin 
multiplets. (Magnetic forces between different atoms are, of course, 
negligible.) These internal magnetic forces will not quench the spin 
magnetic moment if the spin multiplet width in the ideal gas state is 
small compared to the spacial separation defined on p. 290. To show 
this let us take as an unperturbed system the atom without spin in 
a powerful external asymmetrical electronic field such as is embodied 



XI, 73 ESPECIALLY SALTS OF THE IKON GROUP 295 

in (2). Introduce as a perturbing potential the magnetic coupling 

A'L-S (3) 

between spin and orbit (cf. Eq. (84), Chap. VI. To avoid confusion 
with coefficients in (2) a prime is attached to A in (3. This case is, of 
course, unlike the ordinary ideal gas one, as the orbital magnetic moment 
has been already greatly distorted by the external potential (2) and no 
longer has matrix elements of the form (88) or (89), Chap. VI. In fact 
the quenching effect considered above has robbed L of all diagonal 
matrix elements. By (24), Chap. VI, this means that to a first ap- 
proximation in A', the perturbing potential (3) has no effect on the 
energy. When the second approximation (26), Chap. VI, is considered, 
however, (3) causes energy displacements of the order of magnitude 
^' 2 /^sop> as ^' 8 ar e of the order unity. Here 7n/ scp is an expression of 
the order of magnitude of the 'spatial separation', or of \A B\ %xf 9 
assuming \A B\^ \A <7|< \B C \- Thus the cffect of the internal 
magnetic coupling is of the order A'^/hv^, to be contrasted with the 
order A' for free ions. The theory of Chapter VII, especially the remarks 
on p. 193, show that the spin will make its full contribution (1) to the 
susceptibility, provided that the work required to 'turn over' the spin 
is small compared to kT. The only force interfering with the freedom 
of orientation of the spin is the coupling arising from (3). Thus the 
requirement is that 2 ^ < kT ^ (4) 

In the ions of the iron group with which we are concerned, the over-all 
multiplet width ranges from about 4 x 10 2 (Ti++ ) to 2 x 10 3 cm.- 1 (Cu++). 
The corresponding values of A' are roughly 1-6 x 10 2 to 0-8 x 10 3 cm.- 1 . 
Large departures from (1) (or more precisely from the Weiss formula 
(6) ) seem first to appear when T is reduced below about 70 K. (see 
p. 304), or in terms of wave numbers when kT is less than 50 cm.- 1 . 
Hence, for (4) to be satisfied down to this temperature, the spacial 
separation must bo of the order 10 3 to 10 4 cm.- 1 , or 0-1 to 1 volt. This 
requires fields of the order 10 7 to 10 8 volts/cm., as x, y, z are of the 
order 10~ 8 . Fields of this magnitude are perhaps not unreasonable, 
although they seem a bit high. In particular the higher estimate of 
1 volt is plausible only if the ions are intimately attached in complexes, 

20 The inequality (4) must bo satisfied in order for the, spin, to bo free ovon if the 
quenching of the orbital angular momentum ensues not because of complete dissym- 
metry, but because a non-magnetic component might have the least energy when there 
is more than rhomboidal symmetry. The inequality (4) is still needed, as the non- 
diagonal matrix elements of L, such as are involved in the second approximation, to the 
effect of <3>, never vanish regardless of the amount of symmetry or of quenching. 



296 THE PABAMAGNETISM OF SOLIDS, XI, 73 

so that the inter-atomic forces are large. It is perhaps well to compare 
(4) with the requirement that the spacial separation be large enough 
to quench the orbital angular momentum, which is 

^ SQp >kT. (5) 

At room temperatures in the first half of the iron period the multiplets 
are of the same order of magnitude as kT, so that fulfilment of (5) 
ensures that of (4). At lower temperatures and in the second half where 
multiplets are wider, (5) is a less severe condition than (4), as it only 
demands spacial separations of the order 10 s cm.- 1 . 

It is to be emphasized that our estimate by means of (4) and (5) of 
the magnitude of the spacial separation required to quench the orbital 
momentum and still leave the spin free is only a very crude one, and 
may well be in error by a factor, say 10 to 10 2 . If, for instance, one 
used in the left side of (4) the e over-all multiplet width 5 produced by 
(3) in free ions, rather than the proportionality factor A', one would find 
the spacial separation would have to be 1 to 10 volts, which seems 
rather too high. However, we believe the estimate by using A' comes 
closer to the truth, as what counts is the differences rather than absolute 
values of the energies of the various spacial spin quantizations produced 
by (3). The differences may well be considerably smaller than the 
absolute values. We must further caution that (4) and (5) cease to be 
good approximations when the multiplet structure for free ions is not 
small compared to (2), as then it is not allowable to treat (3) as a small 
perturbation superposed on (2). Hence we must not rely too exactly 
on (4) and (5) near the end of the group (Ni ++ , Cu H+ ) where the 
multiplets are wider. 

We must mention that even if the inequality (4) is not satisfied, the 
spin can sometimes be at least in part free, as some degeneracy may 
exist even in the secular problem connected with the superposition of 
(3) on (2). Kramers 21 shows that all states remain doubly degenerate 
when there is an odd number of electrons. This is because the inner 
quantum numbers for free ions are then half -integral, so that there is 
a 'Zweideutigkeit' in the representation of rotational transformations 
by group theory. 22 Even the persistence of this double degeneracy will 
leave the spin only partly free, except in doublet spectra, where it is 
very much freer than usual. Partial freedom of the spin when (4) is not 
satisfied would not explain an accurate validity of (1), but might mean 
that the departures from (1) are not very great even if the left side of 

21 H. A. Kramers, Proc. Amsterdam Acad. 33, 959 (1930), especially 2. 

22 Cf. pp. 153f. of Bethe's first paper. 



XI, 73 ESPECIALLY SALTS OF THE IRON GROUP 297 

(4) is somewhat greater than the right. The table on p. 304 shows that 
CuS0 4 .5H 2 (an odd ion, 2 D type) obeys Curie's law closer than 
NiS0 4 .7H 2 (even, *F). This is a beautiful manifestation of Kramers' 
degeneracy effect. 

Bethe estimates that the spacial separation in crystals is of the order 
250 cm." 1 if caused by fourth-order terms in the Taylor's expansion. 
He informs the writer that this estimate should be increased by a factor 
about 10 if the dissymmetries already appear in the second-order terms 
such as (2). There is also a further increase by a factor Z 2 in the second- 
order terms if the ions have a -fold rather than single ionization. The 
size of the second-order terms then agrees well with our estimate from 
(4), but Bethe supposes ionic crystals, and so for our purposes his 
estimates ought to be lowered if the units in solvents and hydrated 
salts are primarily neutral rather than ionic. One would conjecture 
that the dipole-fields from water molecules might be responsible for 
part of the quenching in solutions. On the other hand, Bethe does not 
consider clusters, which would raise the estimates. An obvious difficulty 
occurs should Eq. (1) be found valid in crystals of sufficient symmetry 
(e.g. cubic) so that the dissymmetry is occasioned only by the fourth- 
order terms, as then one has only a spacial separation of the order 
250Z 2 cm.~ 1 . 

Joos 23 has particularly emphasized in connexion with the problem 
of magnetism that the colours of solutions containing ions of the iron 
group must be associated with transitions between different energy- 
levels of complexes, as these ions when free have no normal absorption 
lines softer than 1300 A. If the characteristic colours are due to transi- 
tions between different levels which we have ascribed to spacial separa- 
tion, the latter must be about 2 or 3 volts, in good agreement with our 
upper estimate from (4). There is sometimes good evidence that similar 
associations are formed both in crystals and in solution, and this per- 
haps explains why the susceptibilities also are often so similar. The 
absorption spectra of cobalt compounds, for example, leads to the con- 
clusion that both in solids and solutions the cobalt ion is associated 
with four groups in the blue compounds and with six in the red. 24 

74. Further Discussion of Salts of the Iron Group 

In the present section we shall cite a number of experimental facts 
(a)-(h) which can for the most part be nicely correlated in a qualitative 

23 G. Joos, Ann. der Physik, 81, 1076 (1926); 85, 641 (1928). 

24 Quoted by Stoner from R. Hill and O. R. Howell, Phil. Mag. 48, 833 (1924). 



298 THE PARAMAGNETISM OF SOLIDS, XI, 74 

way with the theory just developed in 73, and which seem to preclude 
any other explanation of the susceptibilities of the iron salts. Probably 
the only alternative that warrants any consideration is that the orbital 
angular momentum is not fully quenched, but that the multiplet inter- 
vals are greatly distorted in solution or solids, and the susceptibility 
thus altered. However, a very serious objection to this alternative is 
that some experimental susceptibilities fall outside the limits for in- 
finitely narrow or wide multiplets in the latter half of the iron group 
(see table, 72). The only escape would be the highly improbable sup- 
position that in the solid or liquid state the multiplets become turned 
'upside down' as compared to the gaseous state, and thus become 
regular rather than inverted in the last half of the period. (To illustrate, 
the limiting magneton number for Cu ++ for Ai> = oo would then become 
1-55, the value for an infinitely wide regular 2 D multiplet; cf. So*-* in 
table.) 

Many of the points which we now give as favourable to Eq. (1) have 
already been mentioned by Stoner, 8 though not with exactly our 
quenching mechanism, and we have found the discussion in his paper 
very helpful in writing the present section. 

(a) Near Constancy of the Magneton Number in an I so-electronic 
Sequence. Reference to the table in 72 shows that as we pass down 
the sequence V +f , Cr +f+ , Mi^ ' ++ the magneton number should decrease 
in the ideal gas state (cf. column 'actual Av'). This is because the 
multiplet width is increasing. On the other hand, the experimental 
susceptibility for Mn 41 " is apparently higher than for Cr 1 *+. Hence 
all traces of the natural multiplet structure must bo pretty well 
quenched. 

(6) Small Variations with Concentration. If the anomalies in suscepti- 
bility were due to a fortuitous alteration (as distinct from quenching) 
of the multiplet structure by inter-atomic forces, there should pre- 
sumably be tremendous variations with the precise character of the 
inter-atomic fields, and hence with the concentration, nature of solvent, 
or nature of salt in solid compounds. There are some variations, but 
they are usually comparatively small, at least relative to the discrepancy 
between experiment and theory ('actual Av', p. 285) for free ions, as 
reference to the upper and lower limits to the experimental values given 
in the table of 72 will show. Even the same salt may behave differently 
according to its thermal treatment. The extensive investigations of 
Chatillon 25 yield magneton numbers for the cobaltous ion ranging from 

25 A. Chatillon, Annales de Physique, 9, 187 (1928). 



XI, 74 ESPECIALLY SALTS OF THE IRON GROUP 299 

4-4 to 5-2. Birch 26 has measured the magnetic moment of the cupric 
ion under no less than twenty different conditions (different solvents, 
concentrations, solid compounds, and temperature intervals) and finds 
the magneton number ranges from 1-8 to 2-0, so that the total variation 
is 10 per cent. 27 On the other hand, de Haas and Gorter 27a have 
recently found that solid CuS0 4 . 5H 2 O follows the Weiss-Curie law 
(Eq. (6), p. 303 ) almost perfectly even down to 14K., with /x efl = 1-92 
and with A only 0*70. 

Variations of the order of magnitude of those mentioned in the pre- 
ceding paragraph are understandable on the ground that the external 
fields such as (2) may not be adequate to quench the orbital angular 
momentum completely. Even if the inequality (5) is fulfilled, there is 
always some residual effect of the high-frequency elements which may 
vary from one material to another. Also it is quite likely that the left 
side of (4) is not entirely negligible compared to the right side, so that 
the spin is not completely free. In some cases Eq. (1) is a remarkably 
good quantitative approximation. For V++++, V +++ , V ++ in solution 
Freed 19 finds 1-745, 2-760, and 3-805-3-855, respectively for the effective 
Bohr magneton numbers, whereas the theoretical values predicted by 
(1) arc 1-732, 2-828, and 3-873. Freed even endeavours to distinguish 
between vanadium ions which have attached an oxygen atom to them 
and those which have not. Freed thinks his value given above for V j +++ 
is probably really for VO ++ rather than free V+ +++ , but that the other 
values arc for free ions, except for water molecules of coordination. 
The value quoted for V ++ ! was for the green variety. Freed suggests 
that the brown variety involves really V0 + instead of V+' 1 + , but the 
magneton number which he finds for it is 2-813-2-848, and thus in even 
closer accord with (1) than for the green variety. If the V 4 ++ ion really 
attaches itself to an O atom to form a diatomic ion, the latter must be 
in a 3 S state to explain the validity of (1). Perhaps the unusually good 
agreement for the vanadium ions is because (4) is better fulfilled in the 
first than second half of the iron period, as the multiplets are narrower. 
Also reference to the table shows that the discrepancy between (1) and 
the values 'actual Ai>' is, rather fortuitously, unusually small for the 
V ions, so that the susceptibility would be only slightly altered even if 
the quenching is only partial, provided, however, the ions are mona- 
tomic rather than diatomic. For this reason Freed' s data on vanadium 

20 Birch, J. de Physique, 9, 137 (1928). 

27 For further comparisons of the maguotoii numbers for tho same ion in. different salts 
see tho table to bo given in item (h). 
87a W. J. de Haas and (J. J. Gorier, Leiden Communications 210d. 



300 THE PARAMAGNETISM OF SOLIDS, XI, 74 

perhaps do not furnish quite as crucial a test of (1) as would equally 
careful measurements on other ions. 

(c) Experimental Values often Intermediate between (1) and Those for 
Free Ions. This should, of course, be the case if the quenching is only 
partial. Reference to the columns 'spin free', 'actual Av', and 'experi- 
ment* in the table of 72 shows that this is indeed usually the case, 
especially in the last half of the period, where the divergence between 
the values 'spin free' and 'actual Av' is greatest. Occasionally some of 
the experimental values lie outside these limits, but not very far. As 
the quenching condition (5) is better fulfilled at low temperatures, one 
might expect the beginning of a transition from (1) to the values for 
free ions to manifest itself as the temperature is raised. Stoner notes 
that a limited amount of experimental data seems to reveal just this 
situation. Theodorides 28 finds a magneton number 3-2 for Ni^ + in the 
temperature interval 15-125 C. } and 3-4 for 150-500C., to be compared 
with the values 2-83 and 5-56 given respectively by (1) and Laporte's 
theory for free ions. 

(d) Experiments on the Gyromagnetic Effect. This is one of the most 
satisfying arguments. Such experiments on rotation by magnetization, 
or the converse, yield a g-i actor approximately 29 2 when made as usual 
on Fe, Ni, Co, or alloys thereof. Now 2 is just the (/-factor characteristic 
of the spin alone, and so is indicative that the orbital magnetic moment 
is completely quenched. The usual experiments are, to be sure, made 
on the ferromagnetic pure metals and alloys rather than the para- 
magnetic salts now being studied, but since ferromagnetism is a spin 
phenomenon ( 77) there can be but little doubt but that the quenching 

88 Theodorides, Dissertation, Zurich, 1921; Comptes Rendutt, 171, 948 (1920); J. de 
Physique, 3, 1 (1922). Honda and lahiwara find that the graph of 1/x against T for 
CuCl a is non-linear with tho concave Hide of the curvature towards the T axis (Sci. Rep. 
Tohoku Univ., vols. 3-4) and Birch finds a similar ourvaturo for CuSO 4 between 10 and 
537 C. This is the type of curvature to bo expected if tho magneton number is beginning 
to change from the 'spin only' value 1-73 to that 3-53 appropriate to free ions. Jttrch 
gives a magneton number 1-82 for CuCl a in solution from 0-40 and 2-0 from 40-85 C. 
This difference gives, however, a rather exaggerated impression of tho amount of curva- 
ture, as the magneton number is taken by Birch to be proportional to [^. (T-f A)]* and 
there are nearly compensating changes in A, which he gives as respectively 10 and -f 65 
in the two intervals. If the magneton number is taken proportional to (^T)i i.e., without 
the Weiss modification in the Curio formula, it has nearly identical values, viz. 1-85 and 
1-86 in these two intervals. 

29 It is usually stated that gyromagnotic experiments yield a (/-factor which is 2 within 
the experimental error. However, the very careful recent work of S. J. Barnett and 
L. J. H. Barnett (J. Amer. Acad. 60, 127 (1925) ; Phys. Rev. 31, 1116 (1928) ) yields a 
0-f actor 1-87 for pure iron, quite definitely less than 2. This is doubtless because the orbital 
magnetic moment is not completely quenched. The 0-factor for a free Fe 4 ++ ion is 2, 
but that for Fo ++ is 3/2, and both types of ions are perhaps present. 



XI, 74 ESPECIALLY SALTS OF THE IRON GROUP 301 

on the orbital part is similar in both cases. Also particularly pleasing 
is the fact that gyromagnetic experiments made instead very recently 
by Sucksmith 30 on a paramagnetic salt containing Cr +++ still reveal 
a gr-factor 2, in marked contrast to the agreement of his gyromagnetic 
experiments on the rare earths with the theory for free ions (cf 60). 

(e) Confirmation in Fe f ++ and Mn++. These ions are in S states, so 
that there are no questions of multiplet structure, and Eq. (1) gives the 
same result as the theory for free ions. Hence the calculations should 
hold more closely than usual. Now it is noteworthy that Mn ++ does 
indeed have a remarkably constant magneton number. In a careful 
study of solutions of the chloride, sulphate, and nitrate, Cabrera 31 and 
his collaborators have found in each case a number close to 5-92 
independent of the concentration. On the other hand, a variation of 
nearly 10 per cent, in the magneton number of the Fe ++ j ion is observed 
when solutions of different concentrations or acid content are prepared. 
Perhaps this variation is due to chemical effects, such as the attaching 
of water molecules of coordination, or possibly, a suggestion made by 
Stoner, a partial formation of diamagnetic complex ions. 

(/) Complex Salts. There are many so-called complex salts in which 
the ion of the iron group is known to be present only as one of the 
constituents of a complicated radical. 32 Very often such salts have 
susceptibilities which can be approximately represented by (1) if $ is 
given the proper value. For instance, [Cu(NH 3 ) 4 (N0 3 ) 2 ] and also other 
similar complex cupric salts usually have magneton numbers somewhere 
between 1-8 and 2-0; Cr(NH 3 ) 6 T 3 and other chromic salts numbers from 
3-4 to 3-8; and Ni(NH 3 ) 6 Br 2> &c., numbers ranging from 2-6 to 3-2. These 
values arc nearly the same, though usually somewhat higher than those 
1-73, 3-87, and 2-83 given by (1) with 8 respectively 1/2, 3/2 and 1. 
These are just the values of S for isolated Cu++, Cr+++, and Ni+ 4 ions, 
and this suggests that the iron group ions often have a fairly indepen- 
dent existence even in the complex salts. Hence in (k) we shall draw 
freely on the temperature data for complex salts. Tn any case the 
obvious interpretation is that the complex radical has a nearly free 
resultant spin, and that any excess over (1) is due to an only partial 
quenching of the orbital angular momentum. 

There are, however, many complex salts, e.g. KMn0 4 , K 2 Cr 2 O 7 , many 

30 W. Sucksmith, paper communicated to tho British Association, 1930. 

31 B. Cabrera and A. Duperier, J. de Physique, 6, 121 (1925). 

32 For further discussion of complex salts see Stoner, Magnetism and Atomic Structure, 
pp. 325 if. ; Wolo and Baudisch, Nature, 116, 359, 606 (1925) ; S. Shaffer and N. W. Taylor, 
J. Amer. Chem. Soc. 48, 843 (1926); Rosenbohm. 33 



302 THE PARAMAGNETISM OF SOLIDS, XI, 74 

ferrous complex salts, and some sixty cobaltamines measured by Rosen- 
bohm, which are either diamagnetic or only feebly paramagnetic. The 
obvious explanation is that here the resultant spin of the complex ion 
is zero. This is allowable, as in each case the latter contains an even 
number of electrons. The spins of the individual constituent atom-ions, 
in so far as they have an isolated existence, need not be zero, but only 
the vector sum over the entire radical. 

(g) Feeble Paramagnetism of Certain Salts. Some of the complex salts 
cited in the preceding paragraph are feebly paramagnetic instead of 
diamagnetic. For instance, after allowing for the residual diamagnetic 
effects, Rosenbohm 33 finds that the molar susceptibility of the hexa- 
mines, pentamines, tetramines, and triamines of cobalt are respectively 
approximately 55xlO~ 6 , GOxlO- 6 , 73xlO~ 6 , 97xlO~ 6 . The ordinary 
paramagnetic susceptibilities which we have previously been treating 
are of the order 10~ 3 or greater. It seems reasonable to attribute this 
feeble paramagnetism to the residual effect of the high-frequency matrix 
elements, or, in other words, to the fact that the quenching of the 
orbital angular momentum is necessarily imperfect. This idea finds a 
beautiful confirmation in the fact that such feeble paramagnetism is 
usually found experimentally to be independent of the temperature, 34 
or nearly so, in accord with the general theorem of Chapter VII that 
high-frequency matrix elements give a contribution to the susceptibility 
which is independent of temperature. In this connexion the second 
term of Eq. (11), Chap. X, and remarks relating thereto should be 
consulted, as it was there found that the feeble paramagnetic term is 
unavoidable when the molecule or ion contains more than one atom. 

Ladenburg 35 has emphasized that certain ions which are in 1 S states 
when free sometimes seem to exhibit a small paramagnetism when in 
compounds which is usually independent of temperature. For instance, 
Sc 2 3 , Ti0 2 , V 2 5 , Ce0 2 all seem to exhibit paramagnetic susceptibilities 
(after allowance for diamagnetism) of the order 10~ 5 to 10~ 4 per gramme 
mol. independent of temperature, despite the fact that Sc 3 ' , Ti 4 , V 5 h , 
and Ce 4+ ions all have 1 S configurations. Such measurements are neces- 
sarily unprecise and difficult because of the smallness of the effect and 
uncertain estimate of the diamagnetism, but seem quite definite for 

33 Rosonbohm, Zeits.f. Phys. Chem. 93, 693 (1919). 

34 SeeT. Ishiwara, Sci. Kep. Tohoku Univ. 3, 303 (1914) ; P. Weiss and Mllo P. Collet, 
Comptes Rendus, 178,2147 (1924); 181, 1051 (1925); 182, 105 (1926); Freed arid Kaspor, 
J. Amer. Chem. Soc., 52, 4671 (1930). Freed explicitly mentions the agreement of his 
observations on temperature variation witli Eq. (11), Chap. X, or its equivalent. 

85 R. Ladenburg, Zcits.f. Phys. Chem. 126, 133 (1927). 



XI, 74 ESPECIALLY SALTS OF THE IRON GROUP 303 

V 6 + in view of the careful work of Weiss and Mile Collet. 34 Other con- 
vincing data have recently been given by Freed. 34 Such small residual 
paramagnetism finds its explanation along the lines of the preceding 
paragraph. Here the important thing to note is that in complicated 
molecules the inter-atomic forces not merely quench most of the orbital 
paramagnetism if the constituent atom-ions are not in *& states when 
free, but can actually create a small paramagnetism if they are all in 
1 S states and hence diamagnetic when isolated from each other. This 
is because the square of the orbital angular momentum never vanishes 
(and hence cannot be S(S+ 1) with S = 0) when there is more than one 
nuclear centre, as explained in 69. 

(h) Approximate Conformance of Temperature Variation to the Weiss 
Law. It is a remarkable and illuminating fact that the temperature 
variations of the susceptibilities of most salts involving the iron group 
are represented rather well down to a certain critical temperature by 
the so-called Weiss formula, 

(6) 

in which Curie's law is generalized by addition of the constant A to 
the denominator. In view of (1) the approximate value of the constant 
C is usually 4N/3 2 S (&-{-!)/ 3k. Whole pages could be devoted to recording 
the values of C and A reported by different investigators, not always 
in overly good accord with each other. 36 We shall content ourselves by 
giving in the following table some of the measurements made by Jackson 
and others at Leiden, 37 as the Leiden data usually extend to lower 
temperatures than elsewhere, and so furnish a more crucial test of (6), 
usually revealing the temperature below which (6) fails. 38 Instead of 

38 For an excellent survey of the different experimental determinations and references, 
see Stoiier, Magnetism and Atomic Structure, pp. 127, 132-7, 144-9, or Cabrera, I.e.* 

37 The measurements for the nickel and cobalt salts (except chlorides) given in the 
table are by Jackson, Leiden Communications 163, or Phil. Trans. Eoy. Soc. 224, 1 
(1923). The values for the ferrous salts are as quoted and calculated by Jackson from 
earlier Leiden work. The papers on the chlorides are cited in notes 42 and 44 below. The 
results on the ferric and manganous salts are taken from a variety of the Leiden, papers 
(Communications 129 b, 132e, 139 e, 168 b) mostly by Onnes and Oostorhuis. For Cu ++ 
see ref. 27 a. The determination for Cr 2 (SO 4 ) 3 K 2 SO 4 . 24H 2 O is by de Haas and Gorter, 
Lei-den Communications 208 c. 

38 In the early days of the old quantum theory several attempts were made to explain 
the deviations from Curie's law at low temperatures as found at Leiden, by quantizing 
the rotation of a free diatomic molecule. See, for instance, F. Reiche, Ann. der Physik, 
54, 401 (1917) ; S. Rotszajn, ibid. 57, 81 (1918). These attempts do not seem to have any 
physical significance, as an ionic crystal surely does not consist of such freely rotating 
molecules. 



A 


MeH 






24 


5-83 


MnSO 4 


4H 2 


31 




. , 




31 


5-16 


FoSO 4 . 


7H 2 


45 


5-09 


CoSO 4 . 


7H a O 


79 


3-21 


NiSO 4 . 


7H 2 






CuS0 4 . 


5H 2 


-0-2* 


3-83 


CrCl 3 




0* 


5-89 


MnCl a 




)0* 


5-84 






3* 


5-55 


FeCl a 




22 


5-00 


CoCl a 




4 


3-21 


NiCl a 





A 



Me 

5-87 


1 
14 
-59 
0-7* 


5-22 
5-05 
2-95 
1-91 


-32-5 




3-61 
5-61 


-20 
-33 
~67 


5-03 
3-17 



304 THE PARAMAGNETISM OF SOLIDS, XI, 74 

giving the Curie constant (7, we give in each case the effective Bohr 
jnagneton number defined by /i ejff = 



MnSO 4 . 
Fe a (S0 4 ) 3 
FeS0 4 . 
CoSO 4 . 
NiSO 4 . 



O a (S0 4 )K 2 S0 4 . 24H 2 . 
MnS0 4 (NH 4 ) 2 S0 4 .6H 2 0. 
Fe a (S0 4 ) 3 (NH 4 ) a S0 4 . 24H a O 0* 
FeS0 4 (NH 4 ) a S0 4 . 6H 2 O 
CoS0 4 (NH 4 ) 2 S0 4 . 6H 2 
NiS0 4 (NH 4 ) a S0 4 . 6H 2 

As a rule the values of A given in the table are found when substituted 
in (6) to yield a formula which represents the experimental data fairly 
well down to about 65-135 K. Below this critical region of temperatures 
Eq. (6) usually ceases to be valid, and the cryomagnetic anomalies 
discussed below begin to set in. In a few cases, however, Eq. (6) with 
the constants as given in the table is found to hold quite well down to 
the lowest temperature investigated (usually about 14 K.). These cases 
are indicated by asterisks in the table. 39 

As regards the theoretical interpretation of A, it is quite clear that 
it is not usually an atomic property, but is due primarily to distortions 
by inter-atomic forces. This is shown, for one thing, by the fact that 
A varies so much with the compound in which a paramagnetic ion of 
given valence occurs, in marked contrast to the comparative constancy 
of the magneton number /* ofl . Chlorides, for instance, usually yield a 
negative A, and sulphates a positive one. As a rule the values of A are 
lower in compounds of high 'magnetic dilution', such as the hydrated 
sulphates and amono-sulphates in the table. Heisenberg's theory, which 
we shall discuss in Chapter XII, shows that the exchange forces between 
paramagnetic atoms or ions have the effect of adding a constant A to 
the denominator of the usual Curie formulae, thus yielding an expres- 
sion of the desired form (6). We have, however, already mentioned 
that the exchange forces probably play only a subordinate effect in the 
ordinary salts, and if all of A is due to the exchange effect, then A 
should vanish when the magnetic dilution is high. Reference to the 
table, on the other hand, shows that A is still appreciably different 

39 In tho cases of MnSO 4 , MnSO 4 . 4H 2 O, FeSO 4 , 7H 2 O, and NiSO 4 (NH 4 ) a SO 4 . 6H a O, 
the deviations from (6) at low temperatures are, however, considerably loss pronounced 
than for the other salts not designated by asterisks in the table. 



XI, 74 ESPECIALLY SALTS OF THE IRON GROUP 305 

from zero even for some of the highly hydrated salts. One therefore is 
probably safe in attributing only a part of A to the exchange effect, 
and the balance to distortion effects involving the orbital angular 
momentum, probably because the inequalities (4) and (5) are not ful- 
filled with any great precision, although the requisite mathematical 
theory has not yet been developed to show that the temperature 
dependence is of the form (6). 40 

This view that A is often due mainly to the influence of orbital angular 
momentum finds support in the fact that A is usually smaller in the 
manganous and ferric than in most other salts. The Mn++ and Fe +++ 
ions are in S states and hence devoid of orbital angular momentum. 
Hence, only the part of A arising from the exchange effect should still 
remain, and this disappears at infinite magnetic dilution. This is in 
beautiful accord with the fact that A is zero for MnSO 4 (NH 4 ) 2 SO 4 . 6H 2 O 
and Fe 2 (S0 4 ) 3 (NH 4 ) 2 S0 4 .24H 2 within the experimental error. The 
latter of these (alum), with its 24 water molecules, of course represents 
an unusually high degree of magnetic dilution. Furthermore, for these 
two salts Curie's law holds right down to the temperature of liquid 
hydrogen, without the usual irregularities setting in at about 70K. 
We can, so to speak, say that in the salts of the iron group the orbital 
magnetic moment and all traces of gaseous multiplet structTire are 
pretty well exterminated, manifesting themselves only indirectly in A 
and in irregularities only at very low temperatures. In Mn++ and Fe+ ++ 
there are no multiplets to exterminate, and this is reflected in the closer 
applicability of Curie's law than for other ions, except possibly Cu++. 

The data on A which we have previously quoted have been for solids. 

40 Without a detailed calculation it can bo predicted that tho theoretical expression for 
the susceptibility can be developed in a series of the form X ^ + 7^4 + >^ "'~ when 

we consider corrections to (1) resulting from the fact that the left side of (4) is not negligible 
compared to tho right side, but at tho same time neglect any error resulting from the fact 
that tho condition (5) may not bo well fulfilled. This development is the same as (G) to 
terms of the order T' 2 if we take A = a 2 /(7. It is not clear without lengthy computa- 
tions of a 2 , 3 whether or not this development and that obtained by expansion of (6) 
differ appreciably in the terms of order T" 3 , T', &c., also how much tho development is 
spoiled because (5) is never ideally fulfilled. 

In this connexion wo may mention that a development of the susceptibility in descend- 
ing powers of T is likewise obtained in the theory of Cabrera and Palacios, An. Soc. eftp. 
Fia. Quvm. 24, 297 (1926). They also have tho idea that part of the susceptibility of tho 
free ion is suppressed by inter-atomic forces, but in our opinion tho numerical values of 
the coefficients in their series development are in error because they overlook the fact 
that tho second- as well as first-order Zeeman terms contribute to the Curio term of the 
susceptibility when the frequencies in the perturbation denominators are small compared 
to kTjh. 

3595.3 X 



306 THE PARAMAGNETISM OF SOLIDS, XI, 74 

The experimental data on A in solutions are rather hard to analyse, 
as there are complicated variations of A with acid content, with the 
temperature interval (indicating that here (6) is really not a good 
formula), &c. Generally speaking, A changes but slowly with con- 
centration, and does not vanish at infinite dilution. This is theoretically 
comprehensible if there are certain clusters or complexes which main- 
tain their existence at any dilution. The fact that the value of A 
depends somewhat on the nature of the negative radical of the dissolved 
salt indicates that the cluster apparently sometimes has a more com- 
plicated structure than Fe +1 .wH 2 0, &c. Cabrera and Duperier 41 find 
that A is about 23 to 28 for aqueous solutions of manganous salts. 
This is rather puzzling, as the Mn++ ion is in a *S state when free, and 
hence should presumably be affected but little by the surrounding 
molecules of hydration. In our opinion the data on hydrated solid salts 
furnish a more reliable and more easily analysed test of the theory than 
do the measurements on solutions. In particular, the determinations of 
A in solutions are often based on such restricted temperature intervals 
that they lack much significance (cf. end of note 28). 

It is to be understood that (6) is not claimed to be an entirely accurate 
representation of the temperature variation even at room temperatures 
and higher. Instead A must itself be regarded as a slowly varying func- 
tion of the temperature, in line with the transition effects mentioned 
at the end of item (c). One fact, however, stands out sharply, namely, 
that the large departures from Curie's law do not appear experimentally 
in the first half of the iron group which should appear theoretically in 
the ideal free or gaseous state because the multiplets are comparable 
to kT. Reference to the table on p. 285 shows that in the upper half 
of this table the effective magneton numbers for the free state calculated 
for ' Av oo ' or T = are quite different from those calculated for 
'actual Av' or ordinary temperatures. One can, for instance, calculate 
the following effective magneton numbers for free Cr+ ++ and Cr++ at 
various temperatures. 

T = 20 50 80 150 293 400 K. 

O m >eff^ 0-78 0-95 1-18 1-44 2-04 2-97 3-37 

O+ 1 ^ t . w = 1-74 2-52 2-90 3-51 4-25 4-55 

Analogous departures from Curie's law in the free state do not come 
in question in the second half of the iron period, as here the inversion 
of the multiplets makes the intervals between the lowest components 

41 B. Cabrera and A. Duporier, J. de Physique 6, 121 (1925). For further information 
on the values of A for solutions see Cabrera's article in the report of the 1930 Solvay 
Congress, also Stoner, Magnetism and Atomic Structure, p. 127. 



XI, 74 ESPECIALLY SALTS OF THE IRON GROUP 307 

large compared to IcT. Unfortunately there is a dearth of data, especially 
at low temperatures, on the temperature behaviour of the moments of 
the ions in the first half. In particular, no adequate temperature data 
are available for Cr ++ and Mn +++ , which resemble EU+++ inasmuch as 
the free magneton number drops to zero at T = 0. However, Woltjer 42 
finds that CrCl 3 obeys the formula x = Np 2 (3-61) 2 /3k(T 32-5) down to 
about 136K. The susceptibility thus increases more rapidly with de- 
creasing temperature than according to Curie's law, whereas we have 
seen above that the effective magneton number of the free Cr + f + ion 
instead diminishes with decreasing temperature, giving a departure from 
Curie's law in the opposite direction. This difference shows vividly how 
'un-gas-like' are conditions in salts of the iron group, in marked contrast 
to the rare earths. De Haas and Gorter find Cr 2 (SO 4 ) 3 K 2 SO 4 . 24H 2 
follows Curie's law almost perfectly down even to 14 K. Turning to 
measurements at somewhat higher temperatures, Honda and Ishiwara 43 
find that CrCl 3 , Cr 2 (SO 4 ) 3 , and Cr 2 O 3 .7H 2 O all approximately obey 
Curie's law throughout the entire temperature range about 100-800 K 
they studied, barring oxidation effects at high temperatures and 
Weiss A-corrections important only at low temperatures. This is in 
accord with the theory of 73, whereby the quenching effect effaces 
the multiplet structure, and yields Eq. (1), thus restoring Curie's law. 

Cryomagnetic Anomalies. We have already mentioned that (6) fails 
below a; certain critical temperature, usually about 70 K. This is at 
least in part understandable on the ground that the inequality (4) is 
less apt to be fulfilled at low temperatures, so that the coupling between 
spin and orbit becomes more important. Usually below the critical 
temperature the susceptibility increases less rapidly with decreasing 
temperature than predicted by (6), which is just what we should expect 
if the spin ceases to be free. 

At very low temperatures more anomalies than merely departure 
from the simple temperature variation (6) sometimes manifest them- 
selves. It is found that at the temperatures of liquid hydrogen the 
susceptibilities of CoCl 2 , CrCl 3 , FeCl 2 , and NiCl 2 , also Fe 2 (SO 4 ) 3 , are all 

42 H. R. Woltjor, Leiden Communications 173b. 

43 T. Ishiwara, tici. Rep. Tohoku Univ. 3, 303 (1014); Honda and Ishiwara, ibid. 4, 
215 (1915). They find that the susceptibility of Cr a O 3 decreases only very slowly as the 
temperature is raised, in marked contrast to CrCl 3 or O 2 O 3 . 7H 2 O. The meaning of this 
is not clear. The departure from Curio's law for Cr 2 O 3 arc, to be sure, in the same direc- 
tion as for the free Cr ' ' ' ion, but it is hard to believe that in the oxide the Cr ' f * ion 
is more free than elsewhere, although it is conceivable that there is different crystalline 
symmetry in the other salts. In general, the behaviour in the vicinity of 1000 K. is very 
irregular for the different chromic salts presumably because of chemical effects. 

X2 



308 THE PARAMAGNETISM OF SOLIDS, XI, 74 

dependent on the field strength H, in some cases increasing (NiCl 2 , 
CoCl 2 ), in others decreasing (CrCl 3 ) or even increasing and then de- 
creasing (FeCl 2 ) as the field strength is increased. 44 One has here a sort 
of incipient ferromagnetism, but not true ferromagnetism, in that there 
is no saturation or enormously high susceptibility. The explanation is 
probably that the exchange forces between magnetic ions, which 
Heisenberg shows can create ferromagnetism, may be vital at very low 
temperatures even though not at higher ones, as the importance of 
interaction energies is always gauged by comparing them with kT. This 
seems plausible since the chlorides have less magnetic dilution than 
many salts, so that exchange forces may be relatively more important 
than in the others. 

We must not give the impression that all compounds of atoms of the 
iron group with atoms not belonging to this group are not ferromagnetic 
except possibly at very low temperatures. Ishiwara, 45 for instance, finds 
thafc certain nitrides of Mn are ferromagnetic above room temperatures. 
The forces tending to create ferromagnetism are thus here stronger than 
in the chlorides; possibly this has something to do with the fact that the 
normal state of the N atom is a quartet state. Pyrrhotite (approxi- 
mately Fe 7 S 8 ), magnetite (Fe 3 4 ), haematite (Fe 2 3 ), and the Heusler 
alloys are well-known examples of ferromagnetic compounds consisting 
only in part of atoms of the iron group. Generally speaking, the sul- 
phides and oxides of this group, even when merely para- rather than 
ferromagnetic, often do not conform at all to Eq. (1) (cf. note 43). This 
is in marked contrast to the rare earths, where the oxides behave nearly 
as regularly as hydrated sulphates (58). Oxides have less magnetic 
dilution and more symmetric and simple crystal structures than salts 
composed of a variety of atoms, so that we need not be surprised that 
they often do not obey Eq. (1). 

It may be noted that Williams 46 finds that at room temperatures the 
pure rare earth metals, in distinction to the salts thereof, exhibit sus- 
ceptibilities dependent on field strength. This is probably the same 
sort of phenomenon as that of the chlorides of the iron group at low 
temperatures, as a true state of intense ferromagnetism has not been 
reached but yet x depends on H. That pure rare earths are thus less 
ferromagnetic than pure iron is doubtless because the deep sequestering 

44 Woltjor and Oimes, Leiden Communications 173; Woltjor and Wiersma, ibid. 
201 a. The dependence on field strength at low temperatures was first observed in ferric 
sulphate by Onnes and Oosterhuis, ibid. 129b. 

45 T. Ishiwara, Sci. Rep. Tohoku Univ. 5, 53 (1916). 

46 E. H. Williams, Phys. Rev. 29, 218 (1927). 



XI, 74 ESPECIALLY SALTS OF THE IRON GROUP 309 

of the 4/ orbits makes the inter-atomic exchange forces smaller than in 
the iron group. 

Crystalline Dissymmetries. The data which we have previously quoted 
have been for powders or other preparations in which the various 
crystalline axes cannot be isolated. In a few cases it has been found 
possible to use single crystals and so measure susceptibilities along the 
different crystal axes. Different salts behave quite differently. (We, of 
course, discuss primarily only paramagnetic crystals; ferromagnetic are 
much more complicated.) Three distinct cases which arise are: 

(I) Magnetically isotropic crystals. As an example, gadolinium ethyl 
sulphate 47 is found to obey Curie's law with the same constant for all 
three axes to within one part in a thousand. 

(II) Crystals in which the constant in (6) is the same for the 
different axes, but the constant A is different. As an example, Jackson 48 
finds that CoSO 4 (NH 4 ) 2 SO 4 . 6H 2 O has A = 9-8, 52, 15 respectively for 
various axes. Usually, though not invariably, the magnitude of A is 
least along the axes along which the atoms are spaced most sparsely, 
in accord with the general proposition enunciated by Onnes and 
Oosterhuis that A decreases when the 'magnetic dilution' becomes 
greater. This is in qualitative agreement with our interpretation of A 
as due to inter-atomic forces, although a quantitative theory for crystals 
is wanting. 

(Til ?) Crystals in which the Curie constant C lias different values for the princi- 
pal axes. Jackson and de Haas 49 report that in MnSO 4 (NH 4 ) 2 SO 4 . 6H a O the 
effective Bohr magneton numbers are 6*9, 5-9, and 4-6 for the three principal axes. 
As regards theory, the analysis of Chapter VII will yield Curie's law with different 
constants for the different axes if one abandons the isotropy relations given on 
p. 193, Chapter VII, and instead makes the supposition that the effect of the 
quantum number m 011 the energy is largo compared to kT. Then only the 
diagonal elements of the moment matrix contribute appreciably to the sus- 
ceptibility. This, however, would involve quenching most of the susceptibility 
of the free ion, and seems irroconcilible with tho fact that Jackson and Onnes find 
that the mean susceptibility for all three axes has the same value as for the free 
ion. An even greater difficulty is that it is hard to imagine appreciable forces 
causing magnetic dissymmetry when tho ions are in JS states, as are those of Mn ++ , 
Furthermore, according to the measurements of Rabi, 49a the magnetic anisotropy 
of MnSO4(NH 4 ) 2 SO 4 . 6H 2 O at room temperature does not exceed one per cent., 
which is convincing evidence that the constant C does not really depend much 
on the axis. It is therefore noteworthy that very recently K. S Krishnan. 49b 

47 Jaoksoii and Onnes, Leiden Communications 168 a (1923). 

48 L. C. Jackson, Leiden Communications 163; Phil. Trans. Roy. Soc. 224, 1 (1923); 
226, 107 (1926). 49 Jackson and de Haas, Leiden Communications 187. 

"I. I. Kabi, Phys. Rev. 29, 174 (1927). 

49b K. S. Krishiian, Zeite.f. Physik, 71, 137 (1931). 



310 THE PARAMAGNETISM OF SOLIDS, XI, 74 

claims to have found a computational error in the paper of Jackson and de Haas, 
and states that when proper corrections are made the dissymmetry proves to be 
of the usual type II. 50 

One would expect the magnetic dissymmetries in crystals to be due 
primarily to distortion effects involving the orbital angular momentum 
at least indirectly. The magnetic moment may well arise almost entirely 
from the spin magnetic moment and the anisotropy be in conjunction 
with the coupling of orbital and spin magnetic moment within a given 
atom. This coupling may have some disturbing effect because the left 
side of (4) is not negligible compared to the right, and need not be 
isotropic when the orbital angular momentum is exposed to an external 
field. The anisotropy of the crystal then makes itself felt directly on 
the orbital angular momentum, and hence indirectly on the spin. The 
beginnings of a theory based on this idea may be detected in interesting 
recent work by Powell, 51 though primarily in connexion with ferro- 
rather than paramagnetism. He subjects the spin to a Weiss molecular 
field having the same symmetry as the crystal. This may be regarded 
as a crude portrayal of the fact that the coupling (3) of spin and orbit 
will indirectly subject the spin to forces having the same type of sym- 
metry as the crystal if the orbit is itself first quenched by the crystalline 
fields associated with (2). Powell's model is, of course, merely a sub- 
stitute for the real dynamics connected with (2) and (3), but he shows 
that it can account quite nicely for certain crystalline dissymmetries 
in the magnetization curves of iron and nickel. 

It should be particularly noted that the exchange forces between 
atoms, which create effectively a coupling between spins, do not create 
any magnetic anisotropy in the crystal. This is one of the consequences 
of Heisenberg's theory of ferromagnetism (77) and is one of our main 
reasons for attributing the anisotropies to effects involving the orbital 
angular momentum. 

This idea that orbital distortions cause most of the magnetic aniso- 
tropy seems to be nicely confirmed by the almost perfect magnetic 
isotropy of crystals of gadolinium ethyl sulphate. The Gd f ' + ion is in 
a 8 S state and so has none of the complications coming from orbital 
angular momentum. Also, Rabi 49a finds manganouw salts much more 

50 Besides the measurements on cobalt ammonium sulphate quoted abovo, Jackson 
measured the three principal susceptibilities of NiSO 4 . 7H 2 O, while Foex had previously 
measured those of siderose. (Annalca <le Physique, 16, 174 (1921). In each case the dis- 
symmetry was found to bo of the type IT rather than III. 

51 F. C. Powell, Proc. Roy. Soc. 13(U, 167 (1930). Fowler and Powell, Proc. Camb. 
Phil. Soc. 27, 280 (1931). 



XI, 74 ESPECIALLY SALTS OF THE IRON GROUP 311 

isotropic magnetically than nickel, cobalt, or ferrous ones. This is 
what one would expect since the Mn ++ ion is in a 6 $ state. 

Even without coupling to the orbits, there can be some slight aniso- 
tropy in the part of the susceptibility coming from the spin because 
of the purely magnetic forces between the spins of different atoms. By 
a classical calculation, whose results no doubt hold in quantum 
mechanics, Becker 52 has shown that no anisotropy arises from this 
cause as long as the crystal is cubic. Also, Kramers 53 has shown that 
even in an 8 state a fine multiplet structure, interfering somewhat with 
the freedom of the spin, comes into existence as soon as the atom is 
subject to a non-central field. Both these potential causes of magnetic 
anisotropy are second-order effects, as magnetic forces between different 
atoms are small, and as the Kramers fine structure effect is very narrow. 
However, there are reasons for believing that crystalline dissymmetries 
in magnetization are themselves a secondary thing in origin. 

75. The Palladium, Platinum, and Uranium Groups 

Here respectively 4c, 5d, and Qd inner shells are in process of develop- 
ment. The structural situation is thus like that in the iron group in 
that the incomplete shell is one of d electrons, with a capacity of ten. 
The table on p. 312 sets forth the available experimental data, in 
comparison with the theory for free ions, and with the assumption that 
the spin only is effective, which was found so fruitful in the iron group. 
We include as ions illustrative of the various configurations only the 
smattering assortment with various valencies, for which experimental 
data are available. The salts employed by Cabrera, 54 by Bose, 55 and by 
Guthrie and Bourland 56 are the ordinary anhydrous chlorides, except 
that the values in parentheses are for oxides, and except for the values 
by Bose to which asterisks are attached. The latter are respectively for 
KTaF 6 , Th(C0 3 ) 2 , K 2 W(OH)C1 5 , U(C 2 O 4 ) 2 > K 3 MoCl 6 . 12H 2 O, K 3 W 2 C1 9 , 
2IrCl 3 . 3H 2 O. Thus the values designated by asterisks usually represent 
measurements on more complex salts of more magnetic dilution than 
the other data. The value for Mo 4+ (Mo0 2 ) is by Berkman and Zoellcr 57 

62 R. Becker, Zeits.f. Phyaik, 62, 253 (1030); related other work by G. S. Mahajani, 
Phil. Trans. Roy. Soc. 228, 63 (1920); N. S. Akulov, Zeits.f. Physik, 52, 389; 54, 582; 
57, 249 ; 59, 254 ; 64, 559, 817 (1930) ; L. W. McKeehan, Rev. Mod. Pl\ys. 2, 477 (1930) ; 
Nature, 126, 952 (1930). 53 H. A. Kramers, Zeits.f. Phyaik, 53, 422 (1929). 

54 B. Cabrera, Atti del Congrcsso (Conw) Internazionalc del Fisici, i. 95 (1927). 

55 D. M. Bose, ibid. p. 119, or Zeits.f. Phyaik, 48, 716 (1928). Besides the data given 
in the table, he finds a magneton number of 3-71 for Mo H + in Mo(SCN)e(NH 3 ) 4 . 4H 2 O. 

56 A. N. Guthrie and L. T. Bourland, Phys. Rev. 37, 303 (1931) 

87 S. Berkman and H. Zoellor, Zeits.f. Phys. Chem. 124, 318 (1927). 



312 



THE PARAMAGNETISM OF SOLIDS, 



XI, 75 





Jon 


Theoretical p ett 


Experimental /u efl 


















Outhrie 










Free 


Spin 






& 


Term. 


Pd. r. 


P*. (?r. 


Ur. Or. 


Ion 


only 


Cabrera 


Bose 


Bourl. 


l s 




Ta 61 - 


Th 4+ 


0-0 


0-0 




diam.* 




d 2 #3/3 






m 


1-55 


1-73 








[ 


Mo 41 " 


. . 


. . 


1-63 


2-83 


(0-2) 










W 4+ 


, . 


1-63 


2-83 




1-8* 


, . 


2 I 




t 


u 4 '- 


1-63 


2-83 




2-4* 






MO+++ 






0-77 


3-87 




3-6* 




a f 3/2 | 


> 


W l ' f 


% 


0-77 


3-87 




0-4* 




d**D 


Ru ++l '- 






0-0 


4-90 






(0-6) 


r 


Ru ++H 


. . 




5-92 


5-92 


2-1 


0-3 


1-8 




. . 


OB+ 4 *- 


t 


5-92 


5-92 




diam. 




I 




Ir 4 + 




5-92 


5-92 




1-9 


(0-7) 


f 


Rh +f h 






6-70 


4-90 


0-3 


0-4 


(0-4) 


I 




Os+ + 




6-70 


4-90 0-4 






4 1 




j r ++ + 




6-70 


4-90 


0-3 


0-5* 


diam. 


I 




Pt 44 " 










diam. 


diam. 










6-64 


3-87 




, . 




d 8 *FA 


Pd + '" 






5-59 


2-83 


0-1 


diam. 








Pt if 




5-59 


2-83 


diam. 


diam. 




^ 2 >6/ 2 









3-55 


1-73 









rather than Cabrera. In connexion with the measurements on oxides, 
we must remember that even in the iron group the oxides often had 
a different and more complicated behaviour than the ordinary salts. 
Consequently data on oxides can scarcely furnish a crucial test of the 
theory. 

The column headed spin only' gives the magneton numbers yielded 
by Eq. (1), while the theoretical values for free ions have been calculated 
under the assumption that the multiplets are infinitely wide compared 
to kT, permitting the use of Eq. (10), Chap. IX, with a= 0. This is 
legitimate since in the Pd, Pt, and U groups the multiplets are much 
wider than in the iron group. 58 

The agreement between the measurements of Cabrera and Bose is 

68 X-ray data show thai the screening constant for tho multiplets involved in the Pd 
group is about 24. This shows that the multiplets are here roughly about 6 times as wide 
as in the iron group, so that Mo~ M + and Ru + + + + should have about tho same magneton 
numbers at room temperatures as Cr+ l h and Cr ++ respectively at 50 K. Reference to 
the table on p. 306 shows that the theoretical values for Mo H + and Ru 41 thus become 
approximately 1-2 and 2-5 when the corrections for finite multiplet width are considered. 
This value for Mo 4 + + is not enough greater than that for Av = 00 to shed much light 
on tho glaring general discrepancy between theory and experiment, while in the case of 
R U f +++ experimental data are available only for the oxide. The corresponding change 
in the Pt and U groups is even less important, as the multiplets are still greater. The 
correction for finite multiplot width would be significant if adequate measurements were 
available for tho d* configuration, which is much tho most sensitive to this correction. 
In configurations other than d* and d* this correction becomes of subordinate importance. 



XI, 75 ESPECIALLY SALTS OF THE IRON GROUP 313 

none too good, 59 but the data are adequate to show that the magnetic 
behaviour of the Pd, Pt, and U groups is exceedingly complicated, and 
that here the susceptibilities do not conform except in isolated instances 
either to the theory for free ions so successful in the rare earths, or to 
the 'spin only* theory characteristic of the iron group. This is the exact 
opposite of the frequent conjectures made before the recent experiments 
that the heavier atoms with incomplete 3d configurations would be 
more amenable to theory than the iron group because the multiplet 
structures are wider and hence less easily distorted. 

An outstanding characteristic revealed by the table is the very low 
experimental susceptibilities, especially in the second half of the period. 
One naturally seeks to explain this on the ground that the 'internal 
magnetic' coupling (3) is stronger than the inter-atomic forces respon- 
sible for the spacial separation, so that an inner quantum number J 
can still be employed even in the presence of (2). One can then show 
that the spacial separation will tend to quench the total angular 
momentum rather than just the orbital part, for it is now necessary to 
treat the perturbations in the order (3) followed by (2) rather than (2) 
by (3), and so the argument used in 73 to demonstrate quenching of 
orbital angular momentum now demonstrates quenching of the total 
angular momentum. Even should (3) not be larger than (2), it is quite 
possible that the inequality (4) is not satisfied, so that the spin is not 
free and the susceptibility is hence low. On this view that the spacial 
separation should leave the multiplet structure intact in heavier atoms 
one would expect the theory for free ions to be more nearly applicable 
than the assumption that the spin only is free. This is supported by 
the fact that the susceptibilities of Ru 4+ , W + + + , and W 4 + conform much 
more closely to the values calculated for 'free ions' than for 'spin only', 
but the latter assumption seems to succeed better in MO+++ and U 4 +. 

A probably insuperable objection to the explanation attempted in the 
preceding paragraph is that abnormally low susceptibilities are observed 
for Ru 1 f+ , Os+++, and Tr 4 +, despite the fact that these ions are in 6 S 
states. This is in marked contrast to the close conformity to theory 
of the analogous Fe +++ , Mn' H ~ ions in the middle of the iron group. In 
8 states it is, of coiirse, no longer possible to impute the quenching of 
magnetic moment to ordinary (i.e. non-exchange) inter-atomic forces 

59 Perhaps a little of the discrepancy is duo to different assumptions concerning the 
corrections for diamagnetism which are omitted in some cases. The latter become of 
course, relatively more important when the effective magneton numbers are as small as 
they are for many of the salts in the table. However, the diamagnetic corrections are 
inadequate to shed any light on tho discrepancies between theory and experiment. 



314 THE PARAMAGNETISM OF SOLIDS, XI, 75 

which act upon the orbital angular momentum and hence indirectly on 
the spin if the multiplets are wide. Three possibilities seem to present 
themselves: (a) that the Hund theory of spectral terms is inapplicable 
and that the lowest-lying term for d 5 is here not an S one; (b) that the 
exchange forces between paramagnetic atoms are sufficiently large to 
quench the spin after the fashion to be explained in 80; (c) that the 
ions do not exist in monatomic form, but instead form complexes or 
molecule-ions of zero resultant spin, possibly through electron-sharing, 
or something of the kind, so that one might have, e.g. W 2 instead of 
W ions. Alternative (a) does not seem likely, although some slight 
anomalies in the position of spectral terms have been found in the Pd 
and Pt groups. 60 Cases (b) and (c) resemble each other in that both 
suppose that the exchange forces tend to create units of zero resultant 
spin, in one case micro-crystalline, the other molecular. If (b) is correct 
the anomalies should disappear if the 'magnetic dilution' is increased 
by adding more water molecules of hydration, or otherwise, while in 
(c) the anomalies should still persist when this is done, as the molecular 
units still remain. (This difference is really the definition of the distinc- 
tion between (b) and (c).) Measurements for RU+++, Os ++ '-, Tr 4+ in salts 
of different degrees of magnetic dilution would thus be of considerable 
interest, as in the anhydrous chlorides so far employed the density of 
paramagnetic atoms is probably sufficiently high so as not to preclude 
the explanation (b). However, (c) seems more probable, as the possi- 
bility of molecular units is cited by both Cabrera and Bose. Cabrera 54 
notes that Werner and Pfeiffer have remarked that there is considerable 
physical-chemical evidence that the halides of Fe, Co, Ni are true salts 
that can be dissociated electrolytically, but that in those of Ru, Rh, Pd 
and of Os, Tr, Pt the atomic clustering effects seem to be predominant, 
and it is questionable whether their halides are truly saline in nature. 

It will be noted that in the iron group as well as in the groups now 
under consideration the ideal theory for free ions seems to apply some- 
what better in the first than in the second half of the period. This 
seems reasonable, as the inter-atomic perturbing forces may well in- 
crease with the number of electrons in the incomplete group. 

One point at least stands out clearly. When none of the simple 
formulae are obeyed, Curie's law should not be followed, and Cabrera 
does indeed find complicated temperature variations for the salts of the 

60 Cf. Hund, Linienftpektren, diagram p. 166. Tho peculiarities are found in the arc 
spectra, and are presumably much less likely to occur in the doubly and triply enhanced 
spectra. 



XI, 75 ESPECIALLY SALTS OF THE IRON GROUP 315 

Pd and Pt groups which he has studied. In general, the graph of 1/x 
against T is curved rather than linear as given by (1) or (6). Usually 
the susceptibility decreases somewhat as the temperature is increased, 
but in one case (PdCl 2 ) it actually increases. The susceptibilities of 
RhCl 3 , IrCl 3 , OsCl 2 are nearly independent of temperature, suggesting 
immediately the predominance of * high-frequency elements'. On the 
other hand, Guthrie and Bourland find that RuCl 3 follows Curie's law 
with A = 37. 

Further experimental data on the different salts of the Pd and Pt 
groups are greatly to be desired. Without them further discussion 
would be too speculative. 



XII 

HEISENBERG'S THEORY OF FERROMAGNETISM. 
FURTHER TOPICS IN SOLIDS 

76. The Heisenberg Exchange Effect 

An outstanding characteristic feature of the new quantum mechanics 
is the so-called 'Austausch' or exchange effect, first discovered by 
Heisenberg. 1 It is concerned with the degeneracy associated with the 
possibility of two electrons trading places, and is best explained by 
considering first a system with only two electrons and with neglect of 
spin. First suppose that the electrons do not influence each other and 
that they are subject to fields derived from similar potential functions, 
so that the Schrodinger wave equation is 



T-F( a r 1 ,y 1 ,s 1 )-F( a - 1 , i , Zl )]V= 0. (1) 
A solution of this equation is 



Y! = V k (x v y lt zJYJ*,, y,, z 2 ), W - lV t +W m , (2) 

where v k , * m are solutions of the Schrodinger equation for a single 
electron subject to a potential F, as in the absence of interaction it is, 
of course, possible to consider each electron separately rather than 
together as in (1). We shall suppose the wave functions M^., x F) n are real, 
orthogonal, and normalized to unity. 2 The physical interpretation of 
solution (2) is that electron 1 is in the state k and electron 2 in the 
state ra (not necessarily states belonging to the same atom). This solu- 
tion is, however, not the only one belonging to the energy Tf^+W^. 
An alternative solution is clearly 

x Fii = ^fe2/ 3 ,2) x F m (.^2/i,^), W = W k -\-W m> (3) 

in which the electrons have traded places as compared to (2). More 
generally, any linear combination of (2) and (3) is a solution. The 
question now arises as to what is the proper combination to use when 
the degeneracy of interchange is removed by adding to the potential 
energy in (1) a potential energy F ]2 of interaction between the two 

1 W. Hoisonberg, Zeits.f. Phy^'ik, 38, 41 1 (1926). The same offect was also discovered 
almost simultaneously by Dirao, Proc. Roy. Hoc. 112A, 661 (1926). 

2 The restriction to real solutions involves no essential loss of generality for our pur- 
poses, and avoids the necessity of introducing complex coefficients in equations such as 
(4) or for distinguishing between J lz and J zl . The requirement of orthogonality is usually 
not met in the important case that fc, m relate to different atoms, but the resulting error 
is not great if the wave functions of the different atoms do not overlap too much (of. 
Heitler and London, Zeita.f. Phyaik, 44, 455 (1927) ). 



XII, 76 FURTHER TOPICS IN SOLIDS 317 

electrons, which we may suppose symmetrical in the coordinates x ly y^ 
z 1 and # 2 , 2/2> z v Most readers doubtless already know that the answer 
is the 'symmetric' and 'antisymmetric' combinations 

One way of proving (4) is to note that (4) diagonalizes the energy as 
far as the exchange degeneracy is concerned, since one can easily show 
that the fundamental quadrature (14), Chajx VI, vanishes if ^, *fi n are 
respectively symmetrical and antisymmetrical or vice versa, and if / is 
a symmetrical function, such as F 12 . Or one can set up the secular 
equation corresponding to the pair of wave functions (2) and (3). This is 

= (5) 

where W is the energy in the absence of the interaction term F 12 , 
and where 

A 'i2 =-- I- f M^Fia T r rftvfo, = f ... f x F n F J2 T u r/V^ 2 , (6) 

j j L * - jj 

The solutions of (5) are 

W - W + tf 12 + J 12 , W = W +K 12 -J 12 , (8) 

and correspond respectively to the solutions $(!;!) = 8(11} 1) and 
$(I;2) = 8(11] 2) for the simultaneous linear equations of type (33), 
Chap. VI, associated with the determinant (5). This agrees with (4). 
If we grant (4) instead of using perturbation theory the result (8), of 
course, follows directly from the fundamental quadrature (14) of 
Chapter VI on taking n' n, f. F J2 and using one of the wave func- 
tions (4). 

The Pauli exclusion principle demands that one use only antisym- 
metric wave functions. 3 The symmetry properties, however, are pro- 
foundly modified by inclusion of the spin. If we neglect the 'magnetic' 
coupling between the spin and orbital angular momenta, the wave 
functions are the product of the orbital and spin ones. Therefore, when 
the orbital wave function is symmetrical, the spin one must be anti- 
symmetrical and vice versa. Now it can be shown that in a two-electron 
problem the spin wave function is symmetrical when the spin quantum 
number $ is 1 and is antisymmetrical when it is O. 4 In other words, the 

8 The interpretation of the exclusion principle) in ten us of the symmetry of tho wave 
functions appears to have first been given by Heisonberg and by Dirac, Z.c. 1 

4 Cf, for instance, Dirac, The, Principles of Quantum Mechanics, p. 214, or Sominerfeld, 
Wellenmechanisclwr Erydnzunysband, p. 274. 



318 HEISENBERG'S THEORY OF FERROMAGNETISM XII, 76 

triplet and singlet spectra (e.g. ortho- and parhelium) are respectively 
antisymmetrical and symmetrical in the orbital part of the wave func- 
tion. Let Sj and s 2 be the spin angular momentum vectors of the two 
electrons, measured in multiples of the quantum unit h/27r as in previous 
chapters. The characteristic values of the matrix (s^Sa) 2 , the square 
of the magnitude of their resultant, are 8(8+1), with 8 equal to or 1. 
(By the characteristic values of a matrix are meant its diagonal elements 
after it is converted into a diagonal matrix by a proper canonical trans- 
formation. Cf. 35.) Now sf and s| are invariably diagonal matrices 
whose diagonal elements are all i(i + l) = ! as the spin quantum 
number for one electron is invariably J ; in other words, sj and s!j are 
'c-numbers' in Dirac's terminology. As (Sj+s 2 ) 2 8^+82+ 2s 1 -s 2 it 
now follows that the characteristic values of the scalar product s t s 2 
are 2 (0 2 . f ) = f and J(2 2 . J) = J corresponding respectively to 
8 = and 8 = 1, or to the symmetric and antisymmetric orbital solu- 
tions. The characteristic values of the potential energy F 12 of inter- 
action between the electrons are seen from (8) to be K 12 -{-J l2 and 
K 12 J 12 respectively for the symmetrical and antisymmetrical orbital 
solutions, as the remaining terms in the Hamiltoniaii function have 
the characteristic value W Q independent of the symmetry. Thus the 
matrix V 12 has the characteristic value ^ 12 +J 12 when S 1 -s 2 has the 
characteristic value f, and K 12 J 12 when the latter has +J. Tn 
other words, V 12 K 12 -\-^J 12 -{-2J n s 1 *s 2 has the characteristic values 
zero. Now a matrix whose characteristic values arc all zero is identically 
zero regardless of the system of representation, as any canonical trans- 
formation applied to a null matrix clearly still gives only a null matrix. 
Consequently we have the matrix equality 

r ia ^12~ 2*^12""" 2/i2 8 l 'S 2 , (9) 

which applies regardless of whether the matrices in question have been 
transformed to diagonal form. 

Eq. (9) shows that the two electrons behave as though there were 
a strong coupling between their two spins which apart from an additive 
constant is proportional to the scalar product of these spin angular 
momenta, or to the cosine of the angle between the two spin vectors. 
The latter is precisely the dependence of angle found in one term 5 of 
the mutual potential energy of two dipoles, so that the exchange effect 
has a partial semblance to a very powerful magnetic coupling between 
the spins. This is not at all the same as saying that actually there is 

5 The mutual potential energy of two dipoles is /^ ft 2 /' 3 --3(/i 1 r)(^ 2 r)/r r> . Tims only 
the first term is of the type form <10>. 



XII, 76 FURTHER TOPICS IN SOLIDS 319 

a real magnetic coupling of such magnitude, as the actual magnetic 
forces are so very weak that we have neglected them entirely in the 
present connexion. The semblance of large direct coupling between the 
spins is only because the exclusion principle requires one type of orbital 
solution when the spins are parallel and another when they are anti- 
parallel. Nevertheless, the interpretation (9), due to Dirac, of the 
exchange effect as formally equivalent to coupling between spins is 
exceedingly useful, as it enables us to picture and also to follow quanti- 
tatively the workings of the exchange effect by means of the vector 
model. The large par-ortho energy separations were shrouded in 
mystery before the new mechanics, as they require the constant of pro- 
portionality J 12 in the coupling (9) to be fairly large. This trouble now 
disappears, as J 12 is an exchange integral rather than a small magnetic 
factor. 

Let us now pass to systems with more than two electrons, say a 
crystal composed of N atoms each having Z electrons. The exchange 
degeneracy now becomes exceedingly complicated. It is, in fact, (NZ)\- 
fold rather than twofold as above, since in order to treat the inter- 
atomic forces such as interest us for magnetism in solids it is necessary to 
consider the permutations of electrons not necessarily in the same atom 
of the crystal. Even the problem of the Z!-fold exchange degeneracy 
for a single atom is complicated. Regardless of the number of electrons, 
the Pauli exclusion principle requires that the wave functions still be 
antisymmetric in any two electrons if both the spin and orbital co- 
ordinates be interchanged, but they will no longer in general be sym- 
metrical or antisymmetrical in the orbital and spin parts considered 
separately. (The latter characteristic is peculiar to systems with only 
two electrons.) Eq. (9) shows that this is equivalent to saying that the 
spins of two electrons taken at random in the crystal (or even in the 
same atom) will not in general be parallel or anti-parallel, a result which 
seems quite obvious. The proper linear combinations of the (NZ)\ 
original wave functions are usually deduced by rather involved group 
theory. We owe to Dirac 6 and Slater 7 the elucidation that this is not 

6 P. A. M. Dirac, Proc. Boy. Soc. 123A, 714 (1929), or The Principles of Quantum 
Mechanics, Chap. XI. 

7 Another method of avoiding group theory has boon given by Slater, Phys. Rev. 35, 
509 (1930). Slater's method could doubtless be used to obtain the moan values (22-3) 
which wo calculate in 78. In fact it is used by Bloch (ZeAts.f. Physik, 57, 545, 1930) 
and Pauli (Report of the 1930 Solvay Congress) to obtain the mean energy (22), or its 
equivalent, but they do not give the more difficult computation of the mean square 
energy (23). Dirac's and Slater's procedures resemble each other in that their strength 
arises from recognizing at the outset that the exclusion principle severely restricts the 



320 HEISENBERO'S THEORY OF FERROMAGNETISM XII, 76 

really necessary because the exclusion principle limits so severely the 
allowable 'characters' in the group theory. Dirac points out that the 
important results can instead all be obtained in an elementary way from 
the fact that Eq. (9) shows that any two electrons k, I in the crystal 
can be considered as having their spins coupled together by a potential 
of the form 9 T & .& /in\ 

^kl&k s /> \ 1U / 

where the coupling constant or exchange integral J kl will depend on the 
states assumed to be occupied by these two electrons, k and I, before 
allowing for the permutations. We here drop the first two right-hand 
terms of (9) as they do not depend on the orientations of the spin, and 
are of no interest for our problems in magnetism. These terms should, 
of course, be added when one requires absolute, as distinct from 
relative, energies. When there are more than two electrons, solution of 
the exchange degeneracy does not transform the matrix (10) into 
diagonal form, but only the expression 



which is the total exchange energy of the crystal except for the additive 
term J ]T J kt which we have dropped. The summation is over all the 
\NZ(NZ\) pairs of electrons in the crystal. The fact that individual 
terms in the sum (11) are not diagonalized does not impair the kine- 
matical representation (10) of the exchange effect, as wo have already 
mentioned that the validity of (9), which is basic to (10), (11), is 
invariant of the system of representation. We shall, for instance, show 
that use of (11) yields the mean values employed by Heisenberg in his 
theory of magnetism. 

It is clearly to be understood that (11) is only an approximation, in 
that it embodies only the Exchange' secular problem connected with 
the interaction between the various members of a family of (NZ)\ states 
having the same original energy, and neglects the interaction with the 
infinity of states with other unperturbed energies. An analogous ap- 
proximation in the two-electron problem was made in (4)-(9). In other 
words, we use (32) rather than (15) of Chapter VI, i.e. we seek to express 
the perturbed wave function as a linear combination of a finite number 
of unperturbed wave functions, whereas an infinity is required for a 
complete development. This means that by solving the secular problem 
connected with (11) the energy is obtained only to a first approximation 
in a parameter A proportional to the coupling forces between electrons. 

symmetry character. Slater's method is very powerful for computing purposes when 
spacial degeneracy in the orbital motion must bo considered, but does not give quite as 
much kinematical insight as Dirac's. 



XII, 76 FURTHER TOPICS IN SOLIDS 321 

This not only suffices to give all the essential qualitative features of 
the exchange effect, but is often a fair quantitative approximation in 
the case of inter-atomic forces, our primary interest. In the latter case 
the related integrals such as (6) and (7) are usually small since the wave 
functions of different atoms overlap but little. The most important 
thing for magnetism, however, is merely the fact that the exchange effect, 
though entirely orbital in nature, is, because of the exclusion principle, very 
sensitive to the way the spin is alined, and is formally equivalent to 'cosine 
coupling 9 between the spin magnets of the various atoms. 

A very vital point is that the alinement of the spin of a given atom 
having a non-vanishing spin is not influenced by the interaction with 
atoms which have closed shells of electrons and are thus in *S states. 
It is not correct to say that the exchange effects disappear entirely 
between a pair of atoms if at least one of them is in a 1 S state, as there 
is in any case the additive exchange term JJ 12 which we have dropped 
in going from (9) to (10). This term, however, does not involve the spin 
and so is not of significance for our magnetic work. The significant 
part of the exchange energy for us does, however, vanish if one of the 
atoms is in an $ state. To prove this 8 consider the interaction of a 
given electron k of one atom with a closed shell of r similar electrons 
(I = 1,..., r) in another atom. According to (10) the part of the exchange 
energy depending on alinement of the spin is 2/ w s fc -2s / . 'Ehis 

vanishes, as ]? s / is zero f r a closed shell. In other words, for our 
purposes (viz. neglecting terms which have no alining effect on the 
spin), the exchange forces can be considered as existing only between 
the paramagnetic atoms or ions of a solid. These forces will thus be 
subordinate if the material has a high 'magnetic dilution', i.e. consists 
primarily of diamagnetic rather than paramagnetic atoms. Hence, 
exchange effects have played only a subordinate role in the preceding 
chapter. 

8 The proof here given that the expression < 10> vanishes on being summed over a closed 
shell is a bit incomplete in that it takes no cognizance of the fact that in actual atoms the 
orbital spacial degeneracy is superposed 011 the exchange degeneracy. The extension of 
Dirac's procedure to include the former degeneracy will bo given in a future paper by 
the writer, where Dirac's and Slater's methods will be compared in detail. It will there 
be shown that full generality can be achieved by allowing the coefficients J 12 to be matrix 
functions of the orbital angular momentum vectors. A wave function for a closed shell 
can bo constructed by superposition and linear combination of wave functions based on 
m M mi quantization (case (a), fig. 6, 40), and the vital point is that S s* S i on summing 
over the two values m, | of m, possible for given mi. Another proof that closed 
shells do not influence spins of other electrons has boon given by Slater, 7 using considera- 
tions closely related to these. 

3595.3 Y 



322 HEISENBERG'S THEORY OF FERROMAGNETISM XII, 77 

77. Heisenberg's Theory of Ferromagnetism 9 

The explanation of ferromagnetism has long been a conundrum. The 
early work of Ewing, subsequently amplified by Honda and others, 10 
showed that many of the phenomena of hysteresis and of magnetization 
in crystals could be described by assuming a large potential energy 
between adjacent molecular magnets. Also Weiss, in his well-known 
theory, 11 showed that many properties of ferromagnetic media, especi- 
ally the thermal ones, could be imputed to a local field of the form 
H+qM. The portion qM proportional to the intensity of magnetization 
M is called the ' molecular field'. The great difficulty, however, has been 
that tremendously large values must be assumed for the constant q, of 
the order 10 4 , quite different from the value 4?r/3 calculated under 
ordinary electromagnetic assumptions ( 5). The magnetic forces be- 
tween molecules are clearly too feeble to account for such enormous 
values of q, or for the large amount of interaction between molecular 
magnets in the Ewing theory. Classical electrostatic forces lead to 
interactions of the right order of magnitude, but do not give the desired 
linearity of the Weiss molecular field in M or, what is more or less 
equivalent, the right dependence of the Ewing interaction energy on 
the angle between the elementary magnets. 12 

This dilemma has been beautifully solved by the quantTim-mechanical 
exchange forces described in 76. These forces are electrostatic, but 
because of the constraints imposed by the Pauli exclusion principle are 
formally equivalent to a tremendously large coupling between spins. 
In fact, reference to Eq. (10) or (11) shows that this coupling is pro- 
portional to the cosine of the angle between two spins, just as in the 
classical theories of Weiss and Ewing. Even without the following 
further analysis the empirical successes of these theories are thus already 
qualitatively understandable. 

A crystal is nothing but a large molecule. Hence, if we neglect the 
usually subordinate, purely magnetic coupling between spin and orbital 
angular momenta, the total spin of the entire crystal is conserved, like 

9 W. Heisonberg, Zeits.f. Physik, 49, 619 (1928). 

10 J. A. Ewing, Proc. Eoy. Soc. 48, 342 (1890) ; Proc. Roy. Soc. Edinb. 47, 141 (1927) ; 
K. Honda and J. Okubo, Sci. Rep. Tohoku Univ. 5, 153; 6, 183; 13, 6 (1919). Sum- 
maries by Terry in. Theories of Magnetism (Bull. Nat. Research Coun. No. 18), p. 144 
or by McKeohan, Rev. Mod. Phys. 2, 477 (1930). 

11 P. Weiss, J. de Physique, 6, 667 (1907); 1, 166 (1930). A good survey of the Weiss 
theory is given in Theories of Magnetism, p. 114, or Stoner's Magnetism and Atomic 
Structure, p. 75. 

12 For further discussion of this and related points and of the magnitude required for 
q see p. 703 of Dobyo's article in Handbuch der Radiologie, vol. vi. 



XII, 77 FURTHER TOPICS IN SOLIDS 323 

that of an ordinary free molecule, and its square has the characteristic 
values $'($'+!), where 8' is a whole or half -integer according as the 
number of electrons in the entire crystal is even or odd. Also, if we 
continue to neglect the magnetic forces in comparison with the larger 
electrostatic ones, the energy of the crystal will not depend on the 
orientation (as distinct from the absolute value of) its resultant spin. 
The truth of these propositions can be seen by invoking the formal 
similarity of a crystal to the arbitrary polyatomic molecule. Or they 
can be established more fundamentally by proving that the square 
S' 2 (2 s t -) 2 of the total spin of all the electrons of the crystal, also 
any Cartesian component thereof, say S' e , commute in matrix multi- 
plication with the part (11) of the energy which involves the spin. This 
can be seen from the commutation rules given in note 41 of Chapter VI. 
Since S' 2 , S' s commute with {11} and with each other, it follows that 
S' 2 , S' s can be assigned their characteristic values 8'(8'+l) and M' H in 
a stationary state. Instead of 8' s we could equally well choose S' x or S' y 
for this spacial quantization, and this implies that the energy is inde- 
pendent of the orientation of 8' relative to the crystal, a result already 
quoted in 74. 

At this point it is perhaps well to say a word on notation. We employ 
primes, as in S', M' K , &c., to distinguish 'crystalline quantum numbers' 
and other expressions which relate to the entire crystal, regarded as 
one big molecule. Quantum numbers written in capital letters without 
primes, such as S, M& refer to a complete single atom, while those 
written in small letters, such as s, m s , are, as usual, for a single electron 
within the atom. 

If a magnetic field is applied along the z direction, the z component 
of the crystal's spin assumes a quantized value M'g. Let us suppose 
that the crystal is composed of n identical atoms each having a given 
spin 8. The maximum value of 8' is then n8. The number of states 
&(Jtf#) of the crystal having a given M'g is best obtained by imagining 
a field so strong as to break down inter-atomic coupling and give each 
atom individual spacial quantization of spin and orbit. That ordinary 
laboratory fields are not adequate to do this is immaterial since we are 
merely counting the number of terms. Each atom, then, has a spacial 
spin quantum number M St and ft is clearly the number of different 
combinations of the M s consistent with the condition M# = 2 -M#- I n 
case 8 = J, the expression for ft takes the simple form 



Y2 



324 HEISENBERG'S THEORY OF FERROMAGNETJSM XII, 77 

as here \n-\- M' s atoms must have M s = +|, and \n M' s must have 
M s = - J; hence Q is merely the number of permutations of n things 
between two classes. For arbitrary 8 it is readily found that 

l(M'#) coefficient of X M 'S in (a t +x s *- l +...+x- s ) n . (13) 

The number of states co of the crystal having a given resultant spin 

>S/is13 a>(S') = Q(8')-a(8'+I) t (14) 

since M' s ~ $',...,+$', so that &(M' L ) contains all states having 
S 1 ^ \M' 8 \. In ferromagnetism we are interested in quite large values 
of 8'. In this region !(') J>Q($'+1), so that approximately 

(S') = (') (W) 

The physical significance of (15) is that the great bulk of the states of 
a given M' s have S f = \M' K \, provided \M' S \ is fairly large. 

Heisenberg's calculations appear to assume that the atoms are in 8 
states, but this is not really the case, as it is only necessary to suppose 
that the orbital angular momentum is quenched after the fashion 
explained in 73. He also assumes that a given atom has an appreciable 
exchange coupling only with adjacent atoms, and possesses z such 
neighbours equidistant from it. Thus z = 2 for a linear chain, 4 for 
a quadratic surface grating, 6 for a simple cubic grating, 8 for body- 
centred cubic, and 12 for face-centred cubic. Let us further suppose 
that the valence electrons, or electrons not in closed shells, are in similar 
states. The part of the Hamiltonian function which involves inter- 
atomic spin coupling, and which we shall denote by <&', is then 

J'--2e/ J s rS,. (16) 

neighbours 

Here J is the exchange integral (7) between two valence electrons of 
adjacent atoms, and the summation extends over all neighbouring 
pairs of atoms. The result (16) can be seen from (9) or (11), since 
2 Sj. Sj = 2 s fc " 2 s / ^ ^r S^- if k and I refer to different atoms i and 
j and if we sum over the valence electrons of both atoms. Closed shells 
contribute nothing to (11) or (16), as explained at the end of 76, while 
exchange effects between electrons of the same atom merely give an 
additive constant to the energy as far as we are concerned, since we 
may suppose the inter-atomic forces not large enough to destroy the 
quantization 8 of the spin of each individual atom. 
The fundamental problem of Heisenberg's theory of ferromagnetism 

13 The number o> here given does not include the apacial degeneracy factor 2S'-\-l 
which results because different orientations yield identical energies in the absence of 
external fields. 



XII, 77 FURTHER TOPICS IN SOLIDS 325 

is to calculate the characteristic values of (16) and hence the energy 
states belonging to various resultant spins of the crystal. Before 
explaining the mathematical details of how this is done, or rather 
circumvented, it will perhaps be illuminating to consider qualitatively 
three limiting cases. 

1. J/&T>1. Here the exchange coupling is so exceedingly great 
that the state S' nS of maximum crystalline spin has much less 
energy than all other states of less 8' and hence is the only normal 
state. By regarding the whole crystal as a single molecule of spin nS, 
its susceptibility is seen to be (2nSplH)B uS (2n8pH/kT), where B tt # is 
the Brillouin function defined in 61. As the number n of atoms is very 
great, virtually any field is sufficient to make nfiH/kT^l, and so 
B nS =l 9 thus giving the full saturation magnetization 2nSf3. The 
crystal is then, so to speak, infinitely ferromagnetic. In fact it would 
possess a magnetic moment even without an external field. This diffi- 
culty is, of course, avoided by supposing that our crystal is really a 
micro-crystal and that the macro-crystal is composed of a large number 
of micro-crystals, whose spins have random orientations and hence 
compensate each other without an external alining field. 

2. | J/kT | <: 1 . Here the inter-atomic exchange coupling is negligible, 
and the susceptibility will be x 4N8(8-}-l)^ 2 /3kT } disregarding the 
here negligible saturation effects. This is the case which arises in para- 
magnetic iron salts (Chap. XI). We may here remark that the derivation 
of the Langevin-Debye formula given in Chapter VII can still be applied 
if the unit of structure is the (micro-) crystal instead of individual 
atom. We showed in Eq. (6), 54, that the orbit and spin made the 
same contribution to the susceptibility as though both were entirely 
free, provided only their interaction energy is small compared to kT. 
Similarly, one can show that the susceptibility is the same as that coming 
from the individual atomic spins, considered separately, provided only 
the inter-atomic exchange couplings are small compared to kT. 

3. JjkT^ l. Here J is negative and the energy will be lowest 
when as many spins as possible are anti-parallel, and the normal states 
are those of least 8'. The inter-atomic coupling thus here erases practi- 
cally all the paramagnetism, as will be discussed more fully in 80. 

Even ferromagnetic bodies conform to case 1 only asymptotically at 
T = 0. In such bodies the state 8' = n8 of maximum spin for the 
crystal does, to be sure, represent the least energy, 14 as when the spins 

14 Another proof that the state of maximum spin is an extremum in energy has been 
given by Teller, Zeits.f. Phyaik, 62, 102 (1930). 



326 HEISENBERG'S THEORY OF FERROMAGNETISM XII, 77 

are all parallel, each term in the summation (11) has its minimum 
characteristic value (assuming J > 0). However, whereas there is only 
one state with the maximum spin nS, there are by (13)-(15) n I states 
of spin S' nS1, approximately n z /2 states of spin nS 2, &c. This 
increasing number of states as S' is diminished from its maximum is 
important for two reasons. First of all, strength of numbers will 
partially offset the smaller Boltzmann factors for spins less than the 
maximum. In other words, the probability of the crystal being in some 
state having a given S'<nS may be appreciable even though the 
probability of its being in one particular designated state of this 8' may 
be negligible. Secondly, all the states with a given S' < nS do not have 
the same energy, and a few favoured ones may have quite low energies, 
even though they can never lie as deep as the state S' = nS. Whereas 
the infinitely ferromagnetic case 1 is thus too much of an idealization, 
it may nevertheless well be that most of the crystals have very large 
resultant spins. This seems to be the characteristic of ferromagnetic 
materials. A field of ordinary magnitude is then not able to produce 
the true saturation magnetization 2nSf$, as this would require that the 
field be able to convert the crystal into the state /S" = nS, and only 
enormous fields can have an appreciable effect on the distribution of 
$' as distinct from M ' s . However, it is possible at the same time to 
have ///Sf'j8/A;T>l, though 8'<nS, HflkT^l, so that the field is 
able to aline the spin of the crystal in its direction. There is then 
what we may term a state of pseudo-saturation, which is the saturation 
observed in the laboratory and which will be discussed more quanti- 
tatively on pp. 334-6. This pseudo-saturation, of course, approaches 
asymptotically the true saturation at T = 0. On the other hand, if the 
temperature is raised sufficiently, case 2 above will become a better 
approximation than case 1, in agreement with the well-known experi- 
mental fact that ferromagnetism is obliterated if the temperature is 
raised above a certain critical point, called the Curie point. 

We must now seek to make these ideas more quantitative. If Z be 
the partition function 

Z = 2 e~ w l kT = 2 e-W'+WrfWw, (17) 

then the magnetic moment per unit-volume in the direction of the field 

(is) 

n on. 

These relations are readily seen to be the equivalent of (3), Chap. VII, 
or the quantum analogue of (59), Chap. I, applied to the magnetic 



XII, 77 FURTHER TOPICS IN SOLIDS 327 

rather than electric moment. In the second formpf (17) we have utilized 
the fact that the magnetic moment is assumed to come entirely from 
the spin, so that the crystal, regarded as one big molecule, possesses 
a Zeeman term 2M' s flH (cf. Eq. (103), Chap. VI). As usual N denotes 
the number of atoms per c.c., so that N/n is the 'density of micro- 
crystals'; Eq. (18) involves N/n rather than N as in previous chapters 
since our unit of structure is now the (micro-) crystal rather than the 
atom. 

The precise determination of the unperturbed energy-levels W be- 
longing to a given value of the spin /S", i.e. the determination of the 
characteristic values of (16), is virtually impossible, as it involves 
solution of a secular problem of degree co (cf. Eq. (14)). Heisenberg 
therefore makes the approximation that the discrete succession of w 
energy-levels belonging to a given S' can be replaced by a continuum. 
He further supposes that this continuum is distributed according to the 
Gaussian error law about the average energy for the spin S', which we 
shall denote by W$>. Thus he takes the number of states of the crystal 
of given spin which have energies between W s >-}-x and W s >-{-x-\-dx in 
the absence of the field H to be 

x. (19) 

The mean energy W s > and the constant A^. which determines the Gaus- 
sian 'spread' will be calculated later as functions of S'. The partition 
function (17) now becomes 



f e-WW-**'***' dx 2 e- 2M *0*H. (20) 






As M#, S' are large numbers, the summations over M'g, S' may be 
replaced by integrations. The integration over M' s is immediate, and 
here to a sufficient approximation we may take 



If we perform also the integration over x, Eq. (20) is transformed into 

Z =~~^G f ai(S')fi (MrpH " HV)/ * T A l' /afciri *& (21) 

2pH J 



The Special Case $ = J. Following Heisenberg, it is best to consider 
first the case $ = | of atoms with only one electron each not in a closed 



328 HEISENBERG'S THEORY OF FERROMAGNETISM XII, 7 

shell. We shall show in 78 that here 15 

W ff =- 8'*, (22) 

n 

Il^ 1 ). (23) 



The prevalent states have sufficiently large values of %nS', so that 
the Stirling approximation x\ ~x* can be used for the factorials in (12), 
and (15) can be employed instead of the exact formula (14). By (22), 
(23), and (12), Eq. (21) now reduces to 



(24) 
where 



~ kT nkT ~ 

+nlogn~ (^n+S f )lo^n+S f )-^n-S f )log^n-8 f ). (25) 
As the integral (24) cannot be evaluated in closed form, it is necessary 
to have recourse to the method of steepest descents. This method 
hinges on the fact that the integrand of (24) has a sharp maximum at 
the value of 8' which maximizes f(S') and which we shall denote by 
$ f . It can be shown that the equation 



for determining 8* has either (a) one real root, a maximum of 
or (b) three real roots which give respectively a maximum, a minimum, 
and a subordinate maximum. The correct root to use then is, of course, 
the dominant maximum. If we stop with quadratic terms in the 
Taylor's expansion of / about 8' - 8*, Eq. (24) becomes 



where y8'8*, 2a = d 2 f/dy z \ u ^ Q . The limits for y are not really 
i oo, but this is immaterial owing to the sharp peak of the integrand at 
y = 0. The expression (18) for the magnetic moment is now, by (25) 
and (27), 

M==2S^~, (28) 

7i 

15 In comparing results such as (22), (23) with Heisenberg's paper, it should be noticed 
that our n is the same as 2,n in his notation, and that he retains the additive term in the 
exchange energy which is independent of spin and which we dropped in passing from 
<9> to <10>. 



XII, 77 FURTHER TOPICS IN SOLIDS 329 

since in performing the differentiation involved in (17) the implicit 
dependence of /($*) on H through S* (which is a function of H) may 
be disregarded in virtue of (26), so that dfjdH = 2jB/S t /4T. Also the 
dependence of the factor I/ Ha* in (27) on H can be considered negligible 
compared to that of e>, as/>l. With these observations Eq. (28) is 
obtained immediately. The physical significance of (28) is that S* is 
so great that the spin S' ~ S* alines itself parallel to the field. 

It is convenient to use dimensionless measures of the moment, and 
also of the importance of the exchange integral J, by introducing special 
notations f, y respectively for the ratio of the moment to its true 
saturation value at T == 0, and for the ratio of zJ to kT. Thus 

r^_M_ __t _ZJ 

^ 2NpS n& y ~ kT' ( ' 

In the present case 8 = J, but we shall later use (29) also in the general 
case of any S. 

On working out the explicit form of (26) by means of (25) and using 
the new notation (29) it is found that 

(30) 

This is Heisenberg's final result, which will be discussed after the next 
paragraph. 

The General Case of Arbitrary 8. Unfortunately it has not yet been 
possible 16 to calculate the mean square deviation A^ of the energy for 
given S' from its mean value W# except for the special case S = \ of 
only one valence electron per atom. For S>\, one can therefore not 
yet calculate the susceptibility even under the assumption of the Gaus- 
sian distribution of characteristic values. Instead one must make the 

16 An attempt to extend the calculation of A^ to the case of atoms with arbitrary S 
has been made by Heisenborg in his article in Probleme der modernen Physik, p. 114 
(Sommerfeld Festschrift, edited by Debyo). Unfortunately, this article is marred by an 
error, and when appropriate corrections are made, the group theory method there used 
fails to give a result on A 5 ' except for S . The calculation of Ws r , on the other hand, is 
entirely correct. (Details of the error are as follows. It consists in overlooking the fact 
that in the case (2) considered by Heisenborg at the top of p. 119 his expression b it T T ' 
will have different values when T 9 T' define transpositions from the common element 
to elements located in a common second atom or to elements in entirely different 
atoms. This is allied to the fact that even when jf, I are in different atoms from i, the 
mean value of (#fty)(*rsj) is different when j, I are in the same atom or in different 
ones, as a constraint is imposed by the resultant of the spins for a single atom being S. 
Analogous difficulties are encountered in case 3. The number of different kinds of b's 
is thus greater than Heisenberg recognized, and this complication makes them incapable 
of calculation in the manner sought.) 



330 HEISENBERG'S THEORY OF FERROMAGNETISM XII, 77 
still cruder supposition that all crystalline states of the same spin have 
the same energy W#. Then W#> is the mean energy appropriate to the 
spin 8 f , which can be calculated by very elementary means for any S. 
This mean will be so shown in 78 to have very approximately the 
value given in Eq. (22), regardless of the magnitude of S. As the pre- 
ceding detailed analysis for the case 8 = J has made it clear that the 
prevalent states aline themselves parallel to the field, it will not be 
necessary to carry through the niceties of the method of steepest 
descents, and we can take the partition function to be 

WfiH ytf' 2 

Z = ] a>(S')e * 2 '- + n~. (31) 

Because of the sharp maximum of the summand at some particular S', 
denoted as before by 8* t it will suffice to retain only terms through the 
first order in the Taylor's expansion about $*, so that (31) becomes 

y#t 3 rt c.,r 2,S"j8/7 

Z-iT * 2^( s '^ ' < 32 ) 

As we may use the approximation (15), the expression (32) is, in virtue 
of (13), merely 



_ _ i 

Z^z n ( e v+ e ~N +...+e * +e- ?/ )"> with y= t 

to j. 

(33) 

This rather elegant observation is due to Heisenberg. 17 In the present 
method we assume ttd hoc that there is an overwhelming probability 
that the crystal be in the state \M'^\ = 8' = 8*. Hence the moment 
given by (18) must be the same as 208* (N/n) or 2pSN (cf. Eq. (29) ). 
In view of (33) this means that 



l=BM, with y-= +2y^~/7+6C, (34) 

where 

A Sfe*+(fl l)e<^^ 

~ " " 



17 W. Heisonberg, I.e. 9 , also especially Sommerfeld Festschrift, p. 122. It will bo noted 
that quite apart from the neglect of & s > wo have used a different method of calculation 
for the case of arbitrary S than for the case & %. Our procedure for S = , based on 
steepest descents, is taken largely from Fowler and Kapitza, Proc. Roy. Soc. 124 A, 1 
(1929) while that for any S follows more nearly the original papers of Heisenberg. The 
latter method avoids the necessity of using Stirling's theorem, but does not give such a 
good justification for the assumption of a sharp maximum without some further study, 
such as was given in Heisonberg's original paper. 9 Wo purposely use for variety one 
method in one case and one in the other. 



XII, 77 FURTHER TOPICS IN SOLIDS 331 

The expression B s (y) is what we called the Brillouin function in 61. 
When S = |, Eq. (34), of course, reduces to (30), provided we neglect 
the part of y in (30) which is quadratic in y. This part resulted from 
the Gaussian spread of energy -levels, now omitted. 

Discussion of Eqs. (30) and (34). Either of the pairs of simultaneous 
equations (30) or (34) bears an obvious relation to the classical Weiss ones 

(35) 

obtained by taking the argument of the Langevin function 
L(y) = cothy~ \\y 

to be proportional to the applied plus molecular field H-\-qM. In fact, 
(34) is identical in form with the Weiss expression, except for the 
difference, characteristic of quantum theory, that the Brillouin function 
occurs in place of the Langevin one. 18 We explained in 61 how the 
Brillouin function merged asymptotically for very large quantum num- 
bers (or fictitiously small h) into the Langevin. A classical analysis 
starting with (16) would thus yield the Weiss result 19 (35), provided 
one overlooks the distribution of energies for crystalline states of identi- 
cal angular momentum. The major role of quantum mechanics is thus 
to provide a real mechanism, viz. the exchange effect, for (16). The 
term in (30) which is cubic in or M is a refinement resulting from the 
Gaussian spread, and finds 110 analogue in the Weiss theory, but is 
unimportant, at least from a qualitative standpoint. Like other writers, 
we henceforth for simplicity omit this term in discussing the workings 
of Eqs. (30) or (34). 

The exposition of how Eqs. (30) or (34) yield the ferromagnetic 
phenomena of the Curie point proceeds largely as for the Weiss expres- 
sion (35), and so need be given only briefly here. 20 

The simultaneous equations (30) or (34) will have a real positive 
solution for even if the applied field // is zero, provided the propor- 
tionality constant b, which measures the effect of the apparent molecular 

18 Even, before the advent of the now quantum mechanics the substitution of the 
Brillouin for tho Langevin function in. the Weiss theory was proposed and studied by 
Debye, in Handbuch der Radioloyie, vi. 718. Tho justification of Eqs. of the form (34) 
by means of the exchange mechanism was first given by Heisenberg in the Sommerfeld 
Festschrift, p. 122. 

19 In seeming contradiction, Tsing found that classically there was no forromagnetism 
regardless of tho crystalline arrangement (Kelts. /. Physik, 31, 253 (1925)). This, how- 
ever, was because Ising arbitrarily took the coupling between elementary magnets to be 
proportional to /i^/i^ rather than to tho complete scalar product ^i>i 2 . 

20 For elaboration of the details of tho Weiss theory seo the references cited in note 11. 



332 HEISENBERG'S THEORY OF FERROMAGNETISM XII, 77 

field, is sufficiently large. This is most easily seen graphically, as the 
solutions of (34) for H = are the intersection of = B$(y) with a 
straight line = yjb passing through the origin in a -t/ diagram., There 
will be a real positive solution provided the slope tan ^ 1/6 of this line 
is positive, but less than the initial slope tan 6 (dB s /dy) Q of the 
Brillouin curve. This is illustrated in Fig. 13, p. 258, for the case 8 = J. 
The critical temperature at which 6 = < is the Curie point. Below this 
temperature, (f> is less than 6, and there is a solution of the type > 
when H = 0, so that spontaneous magnetization is possible. Above this, 
</> exceeds 6, and the only real, non-negative solution is the non-magnetic 
one 0, showing that here the strength of numbers of the crystalline 
states with S' ~ more than offsets the fact that some of the states 
with larger spins have lower energies. In other words, the prevalent 
states have large spins below and small spins above the Curie point, 
and correspondingly there is ferromagnetism below and only para- 
magnetism above. 

To derive a quantitative expression for the Curie point, and for the 
paramagnetism at high temperatures, we expand the Brillouin function 
as a power series in y. Eqs. (34) then become 

2B8H , 2JzS* ,_ x 

=-kT-+-kT*' (36) 

The cubic term could be omitted for present purposes, but is retained 
for later use in 79. As explained above, the first-degree portions of 
the two equations of (36) must become identical f or H = at the Curie 
point T c . Hence 9 f 

(37) 



If instead we used (30), we would similarly find 



t 1- /!-- (38) 



The observed Curie temperatures are roughly of the order 10 3 K., and 
in view of (37) or (38) this means that the atoms in ferromagnetic bodies 
must be close enough together so that the exchange integral J is of the 
order 10 3 fc or about 0-1 volt, which does not seem unreasonable. 

Behaviour above Curie Point. Above the Curie point it is adequate 
to retain only the linear portion of (36). The elimination of y between 
the two parts of (36) can then immediately be performed, and it is 
thus found on using (37) that the expression for the susceptibility 



XII, 77 FURTHER TOPICS IN SOLIDS 333 

x = ZNpSyH may be written 

If instead one uses the Gaussian refinement (30), the corresponding 
formula proves to be the same as (39), specialized to S = \, only if z is 
large and if in addition the temperature is high. 

Eq. (39) is in good agreement with experiment, inasmuch as the 
susceptibilities of ferromagnetic bodies are usually found to exhibit a 
temperature dependence of the form x=Cy(T+A) above the Curie 
point. Also the constant A is often found to be roiighly equal to the 
negative of the Curie temperature T c , but there is sometimes the com- 
plication, not envisaged by (39), that A changes at the various poly- 
morphic transitions. For instance, in iron, one of the worst offenders, 
C has respectively the values 2-21, 1-53, 4-03, and 0-25 (per gramme 
mol.), and A the values 1047, 1063, 1340, and 1543 K. in the 
intervals 1047-1101, 1101-1193, 1193-1668, above 1668 K. (Fe ft, 
j8 2 , y, 8). Nickel behaves much better, as it conforms quite accurately to 
a formula with C = 0-325, A = 645 from 689-1 173 K.; above 1173, 
however, departures seem to set in. 21 Typical experimental values of 
T c for Fe and Ni are respectively 1042 K. (Terry) and 637 K. (quoted 
by Weiss). Different determinations vary by as much as 20, and 
Weiss states that on the whole T c seems to be about 20 less than 
A in Ni. 

We have seen in 74 that a formula of the form x = CI(T+&) often 
represents quite well the temperature behaviour of purely paramagnetic 
bodies, such as iron salts, over considerable temperature ranges. When 
A is negative this implies, according to (39), that the paramagnetic 
body is really a ferromagnetic one, but with so low a Curie point that 
the ferromagnetic properties are not exhibited at ordinary temperatures. 
Some of the cryomagnetic anomalies cited on p. 307 lend some support 
to this view, but on the whole not as many traces of ferromagnetism 
are actually found at low temperatures as one would then expect. More 
often A is positive, and on Heisenberg's theory this simply means that 
the Curie temperatiire is negative. According to (34), (37) this will be 
the case if the molecular field (i.e. the part of y proportional to ) is 
negative. The question of the sign of this temperature will be discussed 
on p. 336, and we shall there see that one would expect negative tem- 

21 The experimental data above the Curie point are discussed by Stouer in Magnetism 
and Atomic Structure, p. 149 and ref . 23 ; also in the reports of Weiss and Cabrera for the 
1930 Solvay Congress. 



334 HEISENBERG'S THEORY OF FERROMAGNETISM XII,77 

peratures to be the more common. However, these considerations on 
paramagnetic atoms may be a bit irrevelant when the atoms are not 
in S states, as we have seen in 74 on salts of the iron group that the 
primary cause of A is probably here not the exchange effect but the 
incomplete quenching of the orbital angular momentum. On the other 
hand, the rough equality of A and T a in real ferromagnetic media 




above the Curie point must mean that in the latter, A is really due to 
the Heisenberg exchange effect. 

Behaviour below Curie Point. In this region can be determined as 
a function of the 'reduced temperature' T/T C by numerical solution 
of (34). The resulting curves for various values of 8 are shown in 
Fig. 16. Owing to the use of , T/T C instead of M, T as variables, the 
curves are uniquely determined, and do not involve / and N as para- 
meters. This absence of parameters constitutes the so-called 'law of 
corresponding states', a famous result of the Weiss theory (which 
behaves like S = oo), and is seen to hold in quantum mechanics only 
for materials with the same atomic spins, as the curves do depend on 
S. As explained on p. 326, the spontaneous magnetization represented 
as a function of temperature in Fig. 16, is not the experimental residual 
magnetization in the absence of any applied field, but rather the experi- 



XII, 77 FURTHER TOPICS IN SOLIDS 335 

mental saturation value, which we dubbed the pseudo-saturation 
magnetization. No adequate theory has yet been developed for the 
magnetization curves in weak fields or, in other words, for the transi- 
tion from the compensation of the micro-crystals at H = to the state 
of pseudo-saturation. 22 According to the preceding theory this transi- 
tion should take place with infinite rapidity, and there should be no 
hysteresis; this defect perhaps arises because the magnetic forces be- 
tween particles have been neglected in comparison with the exchange 
ones. The former forces are, to be sure, usually subordinate, but might 
be important during unstable equilibria involved in sensitive readjust- 
ment processes such as are involved in the study of hysteresis and 
retentivity. 

Magneton Numbers. The experimental saturation values for nickel 
and iron are represented respectively by dots and crosses in Fig. 16. 
The experimental points for nickel are seen to fit the quantum curve 
for 8 - J better than the original Weiss curve. 23 Also, the Curie 
constant for nickel above the Curie point agrees approximately with 
that obtained by taking 8 i in (39), as the experimental magneton 
number is 1-62 from 689-1173 K., and 1-91 above 1173 K., while 
V(4 !!) = 1-73. For cobalt, j* efl =3-21, -A ^1404 from 1443- 
1514 K., and /* eff -:2-93, A = 1422 from 1514-1576K. > while the 
theoretical jn efl is 2-83 for S~l. The existence of an apparent spin 
quantum number | for Ni, and possibly 1 for Co, is a bit hard to 
understand theoretically, as neutral nickel and cobalt atoms contain 
respectively even and odd numbers of electrons, and so should have 
respectively integral and half-integral 8. Perhaps the explanation of 
this dilemma lies in the fact that metals are composed of ions and con- 

82 The greatest progress along this lino appears to have been made in. the classical 
theories of a. S. Mahajani, Phil. Trans. 228, 63 (11)29); N. S. Akulov, Zcite. f. PkytiTc, 
54, 582; 57, 249, 254; 64, 559, 817; 67, 794; 69, 78, 822; and B. Becker, ibid. 62, 253 
(1930). Discussion of those papers is beyond the scope of our chapter, primarily on 
quantum developments. We may, however, mention that the question as to whether 
a demagnetized crystal really consists of spontaneously magnetized micro-crystals has 
been the subject of considerable controversy. Besides the papers of Akulov on this 
subject see also interesting articles by Frenkel and Dorfinaim, Nature, 126, 274 (1930) ; 
McKeehan, ibid. 126, 952 (1930). A very interesting study of single crystals has 
recently been given by Webster, Proc. Lon. Phys. Soc., 42, 431 (1930). Ho deduces 
convincing evidence that the micro-crystals are really spontaneously magnetized. The 
magnetization curves for single crystals are much more amenable to theory than those 
for other ferromagnetic bodies. 

23 A particularly careful comparison of theory and experiment for nickel has been 
given by F. Tyler, Phil. Mag. 9, 1026 (1930). An excellent resume of the status of the 
theory of ferromagnetism in comparison with experiment is given by Stoner in Phil. Mag. 
10, 27 (1930). 



336 HEISENBERG'S THEORY OF FERROMAGNETISM XII, 77 

duction electrons rather than of ordinary neutral atoms. 24 If the 
conduction electrons are sufficiently free, the magnetic effect of their 
spins virtually disappears because of the Fermi-Dirac interference 
effects to be considered in 80. If, then, an odd number (most likely 
one) of electrons are detached from each neutral atom of Ni or Co to 
b'ecome conduction electrons, the residual ions might have the desired 
values of S. A further difficulty is that the saturation moment of Ni 
at T = is found 25 to be about 40 per cent, lower than the value NfS 
which would be expected on the basis of S = i. The value for Co is 
probably about 10 or 20 per cent, lower than that 2Nf$ corresponding 
to S 1. Possibly these irregularities are because interaction with the 
conduction electrons hinders complete alinement of the spins of the ions. 

The variations in the Curie constant of iron at the polymorphic 
transition points are clearly too complicated to explain by any simple 
theory. There is thus, all told, no adequate existing theory of the 
numerical magneton values in ferromagnetic media, although the values 
of Ni and Co are, as we have seen, not entirely unreasonable. 

Question of Sign of T c . To permit a ferromagnetic solution of Eqs. 
(30) or (34) it is essential that the Curie temperature T (J be positive. 26 
Otherwise the condition < <f> < 6 is incapable of realization, as negative 
or imaginary temperatures have, of course, no physical meaning. The 
absence of a positive T c means either that the molecular field is 
negative, and so opposes ferromagnetism, or else is positive but too 
small to have an appreciable effect. Eq. (37) shows, if one makes the 
crude assumption of identical energies for all crystalline states of 
identical spin, that the necessary and sufficient condition for ferro- 
magnetism is that the exchange integral J be positive. Heisenberg 
estimates that J can be positive only if the principal quantum number 

24 The nood of considering ionic magnetic units oven in metals has been emphasized 
by Stoner, Proc. Leeds Phil. *Soc. 1, 55 (1926). Stoner is able to account more or less 
quantitatively for the various moments of nickel above the Curio point on the assumption 
that Ni, Ni + and Ni^ 1 atoms are present in the ratio 3: 1 : 1, but this supposition seems 
rather arbitrary. Another interpretation of the magneton numbers in nickel has very 
recently been given by A. Wolf, Zeits. f. Physik, 70, 519 (1931). He suggests that two 
states of the neutral nickel atom alternate in the crystal structure. 

25 The experimental saturation magnetization for Ni extrapolated to T = yields 
a moment almost exactly 3 Weiss magnetons per atom, whereas the theoretical value for 
S is 1 Bohr magneton or 4-95 Weiss magnetons. 

26 Ferromagnetism clearly cannot exist without a critical Curie temperature at which 
it disappears (though not necessarily discontinuously). For at very high temperatures 
the inter-atomic interactions become of negligible consequence and the ordinary formu- 
lae for paramagnetism in gases become applicable. Hence by raising T sufficiently, one 
can always reach a point at which the ferromagnetism is effaced from a body. 



XII, 77 FURTHER TOPICS IN SOLIDS 337 

n is 3 or greater, in agreement with the fact that ferromagnetism is not 
found in the first two Mendeleef periods, but the fact that the normal 
state of O 2 has a spin quantum number 1 rather than seems to require 
that J can sometimes exceed zero even when n = 2. This exception is 
not surprising as Heisenberg's estimates of the sign of the exchange 
integrals are only very crude ones. The whole Heitler-London theory 
of valence 27 is based to a large extent on the idea that stable chemical 
compounds as a rule (barring, e.g. 2 ) have as small spins as possible, 
implying that commonly J< 0. 

Even when J > 0, there may still fail to be a real Curie temperature 
if one assumes a Gaussian distribution of energy-levels. Reference to 
(38) shows that with the latter distribution a real T c is achieved only 
if 2^8. Ferromagnetism should then exist only if each atom has at 
least eight neighbours. Heisenberg cites in nice confirmation of this 
theory that the common ferromagnetic crystals Fe, Ni, Co are either 
body-centred cubic (z = 8) or face-centred (z = 12). However, as men- 
tioned to the writer by Dr. Wiersma, there are known ferromagnetic 
alloys and compounds in which the atoms of the iron group which are 
responsible for the ferromagnetism have themselves the simple cubic 
arrangement, and so have z = 6 if we consider only neighbours with 
outstanding spins. Such exceptions are not discomforting, as they can 
conceivably be explained on three different grounds: (a) the Gaussian 
distribution may be but a poor approximation to the actual distribution 
of energy values; (6) ferromagnetism may be due to free rather than 
bound electrons; (c) the behaviour may be substantially modified by 
the presence of an orbital magnetic moment. 

Alternative (a) seems the most likely. The accuracy of the Gaussian 
hypothesis can be tested by calculating (W W^) 3 or (WWg)* with 
the methods of 78 (or their group-theory equivalent) and examining 
whether these fluctuations agree with those obtained with the Gaussian 
distribution after the constant determining the spread has been obtained 
from ( W W s >) 2 . This has been done in unpublished work of Peierls, by 
the group method. His calculation shows that actually the Gaussian 
assumption is a poor approximation, but does not give enough further 
information to permit any real improvement in the theory. The only 
service of the calculation with the Gaussian distribution is to show 
qualitatively that there are other criteria for ferromagnetism besides 
a positive exchange integral. 

" W. Heitler and F. London, Zeita. f. Physik, 44, 455 (1927); Heitler, ibid. 46, 47; 
47, 835; London, ibid. 46, 455 (1928). 

3595.3 z 



333 HEISENBERG'S THEORY OF FERROMAGNETISM XII, 77 
Improved Method of Calculation near T = 0. Although a rigorous 
solution of the a)(S') -dimensional secular problem connected with (16) 
is impossibly difficult, a very ingenious approximate method of solution 
has been developed by Slater. 28 Unfortunately, space will not permit 
us to give details. The essence of the method is that by throwing away 
relatively few of the elements in the secular equation it can be thrown 
into a form which can be solved rigorously. In this way closed expres- 
sions for the different energy values belonging to a given S' can be 
obtained. It is, of course, far preferable thus to obtain the individual 
energy-levels than a rough distribution curve for all the levels regarded 
as a continuum. Unfortunately, Slater's solution is only a good ap- 
proximation when S' is nearly equal to its maximum value nS. Hence 
it is useful for ferromagnetism primarily at very low temperatures, 
where the pseudo-saturation is nearly the true saturation, or the pre- 
valent spins nearly the maximum spins. It does not thus aid in studying 
the phenomena of the Curie point. Slater did not consider his solution 
of the secular equation in connexion with the problem of ferromagnet- 
ism, and the application to the latter has been made in an important 
paper by Bloch. 29 The results, which relate entirely to atoms having 
S = \ y are as follows. A linear chain, the quadratic surface grating, and 
hexagonal surface grating, should all not exhibit ferromagnetism. These 
results are the same as with Heisenberg's Gaussian calculation, as the 
number of neighbours is less than eight. For simple cubic and body- 
centred cubic crystals, Bloch finds that there can be ferromagnetism. 
This is an advance over the Gaussian method, as we saw that the latter 
did not allow ferromagnetic simple cubic crystals. The refinement of 
the method is evidenced by the fact that it predicts ferromagnetism for 
simple cubic but not for hexagonal surface gratings, despite the fact 
that there is the same number, six, of neighbours in each; thus the 
arrangement as well as number of neighbours is important. Bloch cal- 
culates that in a simple cubic crystal the moment should approach 
its saturation value 1 in the following fashion: 

(T\ 3 / 2 ^ \ J 

-) , where = ^,= 0.0587, 8 o = r (40) 

For the body-centred cubic he finds the same form of temperature 
dependence but a different value of the coefficient a. Eqs. (34), on the 
other hand, give an asymptotic solution of the form = l/S~ 1 e- 3 W 5f+1)21 

28 J. C. Slater, Phys. Rev. 35, T>09 (1930). 

29 F. Bloch, Zeits.f. Physik, 61, 206 (1930). Summary by Pauli in Report of tho 1930 
Solvay Congress. 



XII, 77 FURTHER TOPICS IN SOLIDS 339 

in the vicinity of T = 0, regardless of the form of crystalline arrange- 
ment, provided only a positive molecular field is granted. As (40) is 
surely more accurate than (34) near T = 0, the true theoretical curves 
should be drawn somewhat differently than in Fig. 16 in the immediate 
vicinity of T/T C = 0. They should still remain horizontal at T 0, but 
exhibit a more pronounced downward curvature at this point. There 
are not adequate data available at extremely low temperatures to test 
experimentally this modification. (Data 30 at ordinary low temperatures 
favour an empirical formula of the form = 1 o^T 2 OL 2 T* ..., but con- 
ceivably higher-order terms in T/T C , neglected in obtaining (40), might 
make the theoretical formula sufficiently resemble the empirical one 
over a limited temperature range somewhat higher than covered by 
Eq. (40).) 

Possibility of Ferromagnetism from Free Electrons. A variant of 
Heisenberg's theory has been given in another paper by Bloch. 31 Here 
he assumed that the electrons whose exchange effects give the ferro- 
magnetism are the free rather than bound electrons. This corresponds 
to using the wave functions characteristic of the Sommerfeld 32 theory 
of conduction instead of the Heitler-London theory of valence. The 
type of crystal structure, of course, then does not enter. Instead Bloch 
shows that free electrons will give ferromagnetism if, and only if, the 
tendency towards ferromagnetism is able to conquer the tendency of 
the Fermi statistics or the Pauli exclusion principle to suppress the spin 
paramagnetism of free electrons, an effect to be discussed in 80. 
Assuming that there is one free valence electron per atom, and that 
the temperature is below the critical degeneracy temperature of the 
Sommerfeld theory, Bloch shows that this requires that the inter-atomic 
distance be not less than Q-22h 2 /me 2 = 0*5. 10- 7 cm. This is only a 
rough approximation, but it is noteworthy that the actual inter-atomic 
distances are considerably less than this critical value in the alkalis, 
thus giving insight into why the latter are only feebly paramagnetic. 
Stoner 33 further points out that if ferromagnetism were due to free 
electrons, the Curie point would have to be higher than the critical 
temperature of the Sommerfeld conduction theory, and this would yield 
absurdly high Curie points. 

Summary. Although it is not yet possible to formulate quantitatively 
the exact criteria for a positive molecular field, all the foregoing con- 

3 Weiss, J. de Physique, 10, 354 (1929). 
i F. Bloch, Zeits.f. Physik, 57, 545 (1929). 
3 * A. Sommerfold, Zeits.f. Physik, 47, 1 (1928). 
33 E. C. Stoner, Proc. Leeds Phil. Soc. 2, 50 (1930). 
Z2 



340 HEISENBERG'S THEORY OF FERROMAGNETISM XII, 77 
siderations make it clear that ferromagnetism is possible only if the 
exchange integrals are positive, and even then only under very special 
conditions of crystalline arrangement and spacing. This agrees with 
the fact that relatively few materials are ferromagnetic. 

78. Proof of Formulae (22) and (23) for the Mean and Mean 
Square Energy 

These formulae were first derived by Heisenberg with the rather in- 
volved machinery of group theory, but we shall show that Dirac's 
kinematical interpretation of the exchange effect as equivalent to a 
coupling (9) between spins frees us from the need of using this. 
The square of the resultant crystalline spin vector is 

( I S,) 2 = I S?+ I 8 t 'S t ~S'(8'+ 1). (41) 

whole crystal i/j 

Now the square Sf of the spin of any atom Hs a number S(S+1), and, 
further, if we average over all the different states belonging to a given 
spin, the mean value (i.e. mean diagonal element) of S^ S ; - will by 
symmetry be independent of the indices i, j, provided i ^=j. There are 
n terms in the single and n(n~ 1) in the double sum in (41), as the 
crystal contains n atoms. Therefore 34 



= . 



(42) 



The sum in (16) contains by definition \nz terms, and so {16} becomes 

(43) 



In ferromagnetism we are interested in states which have large spin, 
so that S' is of the order n. On the other hand, S is of the order unity, 
as one atom contains only a limited number of electrons. Also, it is 
only necessary to retain the terms of highest order in n, and with this 
observation (43) yields (22) immediately. 

We shall now calculate the mean square deviation Af,>, but with the 
specialization S = s = J, which was not needed for (43). 35 When one 

84 Tho bars throughout 78 are to be construed as denoting averages over all the states 
belonging to a given &', with all such states weighted equally. Such averages should not 
be confused with the usual statistical averages wherein the various states are weighted 
with the Boltzmann factor. 

85 Besides the usual case of one valence electron per atom, the present calculation is 
applicable to atoms which are one electron short of a closed configuration, as here S = J 
even though S is not identical with the spin s of an individual electron. Nevertheless it 
has seemed better in the ensuing equations to denote the spin of quantum number by 
8 rather than S in order clearly to distinguish that the formulae, except Eq. (44), do not 
apply with an arbitrary S substituted for s. 



XII, 78 FURTHER TOPICS IN SOLIDS 341 

squares (16) one encounters three distinct types of terms having 
respectively two, three, and four unlike indices. It is very important 
to note that these three types hence have different mean values. When 
one counts the number of times these different kinds of products occur, 
one sees that the mean square of (16) is 

W* = 4J 2 p(v^F+nz(z- l)(8,-8,)(8^iS+ 

+ (" 2 f 2 -^Hf)(V^KV^)]. (44) 

Here and henceforth it is to be understood that no two subscripts are 
identical, so as to free us from the necessity of writing out explicitly 
i ^j t i ^l, &c. On p. 318 it was shown that the scalar product s { Sj 
of two electron spins has the characteristic values J, f , and so always 
satisfies the algebraic equation (s t s^ J)(s^ s^+f ) = 0, quite indepen- 
dently of whether or not one uses a system of representation in which 
this product is a diagonal matrix. Hence 



. 

, , = 



MK\ 

(45) 



In the second form of this relation use has been made of (42) with 
S = s i. The algebraic equation satisfied by S t Sj would be of 
higher order than the second if we did not take $ = , as S^S^ 
has 28+ 1 characteristic values, and this explains our making such a 
restriction. 

The first kind of term in (44) is now evaluated. The method for the 
second is similar. We observe that the square (Sj+Sj+sJ 2 of the 
resultant of three electron spins? has the characteristic values $ 8 ($ 3 +l) 
with $ 3 = |, \. As sf = |, therefore x = s t Sj+s^ s^+s^ s k has the 
characteristic values J, +| and so always satisfies the algebraic equa- 
tion x 2 = ^g. On squaring the left side of this equation one has three 
terms of the form (i, j) 2 and six of the form (i,j)(i, k). Hence, with the 
aid of (45), 

(v^)(v-s7) = ^-3(ir^r 2 j = ?' ( ^^. (46) 

To find the third term of (44) we note that in virtue of formula (41) 
for the resultant crystalline spin, we have 

(2 vs,)* =['('+ 1)->] 2 , 

as now ^ S| = \n. Here the summation extends over the entire crystal. 
On reckoning the number of times the different kinds of terms occur 



342 HEISENBERG'S THEORY OF FERROMAGNETISM XII, 78 

in squaring out the left side, we find that 

Substitution of (45) and (46) in (47) now yields 



The mean square deviation is 

A|^^-Tf|, (49) 

One can now substitute (45), (46), and (48) in (44) and then in turn 
(43) and (44) in (49). One then finds the Heisenberg result (23) if one 
keeps only the terms of highest order in $' or n. These terms are those 
which increase as the first power of n or $' when n and $' become very 
great. To be sure, the highest-order terms in J' 2 and TF|, both increase 
as n 2 , but they cancel each other in (49). To the approximation under 
consideration one may ignore the distinction between n I and n in the 
denominators of (45) and (46), but in squaring (43) and in (48) one must 
use the somewhat better approximations (n l)~ 2 /^(n+2)/n 3 and 
[n(n l)(n 2)(n 3)]- l ~(n+6)/n 5 respectively, as the correspond- 
ing terms in (49) are larger. In this fashion one finally arrives 
at (23). 

One assumption which may well have bothered the reader is that we have sup- 
posed that all the scalar product terms in <I6> have the same mean value as given 
in (42), despite the fact that <16> involves only neighbouring pairs, whereas we 
obtained (42) by averaging over all pairs of atoms in the crystal, whether adjacent 
or not. The legitimacy of this procedure is not immediately obvious, as the 
presence of the coupling potential <16> might conceivably make the mean of 
s f Sj for interacting pairs (i,j neighbours) different than for the non-interacting 
pairs not involved in <16>. Superficially this might seom the more likely since in 
the actual system of quantization the expression < 16> is a diagonal matrix when the 
sum is taken over interacting pairs rather than over any arbitrary \nz pairs. The 
following argument somewhat resembling one of Dirac's 36 removes this objection. 
The average value in question is proportional to the 'spur' or diagonal sum of the 
oj(S') -dimensional matrix representing S, S^ for given 8'. The invariance of the 
spur ( 35) shows that this average is invariant of the mode of quantization, and 
by a proper transformation one can make a sum analogous to <16> a diagonal 
matrix for any \nz pairs, rather than for the actual interacting pairs (neighbours). 
In virtue of this invariance such transformations will not affect the average of 
any S, S jt Therefore the mean value of S 4 S^ does not depend on whether it is 

36 P. A. M. Dirac, Proc. Roy. Soc. 123A, 730 (1929), or The Principles of Quantum 
Mechanics, p. 211. 



XII, 78 FURTHER TOPICS IN SOLIDS 343 

included in < 16>, and so all pairs are on a par as regards their mean values, regard- 
less of whether or not they are neighbours. 37 One can similarly justify the 
assumption made in using (46), (46), (48) in connexion with (49) that these various 
mean products are the same regardless of whether i, j, k, I are neighbours. 

79. Magneto -caloric and Magnetostrictive Effects 

Since a body in a magnetic field has a different energy than in its 
absence, and since the amount of magnetization changes with tem- 
perature, the application of such a field will produce a change in the 
specific heat. If measured per gramme mol., this change, which we shall 
call the 'magnetic specific heat', is given by 

_ ***], (50, 

where W and Z are respectively the mean energy per molecule and the 
partition function in the presence of the field, and W 6 , Z are the 
corresponding expressions in its absence. 

The third law of thermodynamics requires that the entropy 6 remain 
finite at T = 0. As the specific heat at constant volume is c v = Td<S/dT, 
this means that the specific heat must approach the limit zero at T = 0. 
Contrary to this law, the magnetic specific heat in the Langevin theory 
has the non- vanishing value Lie at T= 0, as is seen by using in (50) 
the Langevin partition function 



kT' 

o 

On the other hand c tf e{J is indeed zero at T = if one employs the 
Brillouin function demanded by quantum mechanics, for the Brillouin 
partition function is 38 (x~ J x J ^ l )l(l x), where x e- ff P n l kT , and this 
makes (50) vanish at T = 0. The compliance of the magnetic specific 
heat in quantum mechanics with the third law still remains true even if 
a Weiss molecular field is included to represent the effect of the Heisen- 
berg exchange coupling, or if the Bloch modification of the Heisenberg 
theory appropriate to low temperatures is introduced. We omit explicit 
proofs, as all such results are merely special cases of the quite obvious 
result that any quantum distribution gives zero specific heat at T = 0. 

37 This is not to be construed as meaning that all pairs have the same mean value in 
problems where constraints are imposed on certain groups of electrons, such as e.g. 
Heiseiiberg's calculation cited in note 16. For instance, in atoms with &>, the mean 
of S 4 8j is different for inter-atomic than intra-atomic electron pairs because we con- 
strain 8 to one particular value rather than average over all values of S consistent with 
given S'. 

88 This partition function is the same as the denominator evaluated on p. 257. 



344 HEISENBERG'S THEORY OF FERROMAGNETISM XII, 79 

In fact, quantum partition functions are discrete summations %e- w l kT , 
so that near T = the expression log(Z/Z) is very approximately of 
the form (Wj^W^/kT, where W : W\ is the change in energy pro- 
duced in the lowest state by the field. This form makes (50) vanish, 
whereas the classical distribution functions involving integration rather 
than summation do not have this property. The physical significance 
is that near T = the molecules are not raised out of their quantum 
state of very lowest energy (including the spacial orientation of least 
energy) by increasing the temperature infinitesimally from the absolute 
zero. Instead kT must be made comparable with the excitation energy 
of the next state before c v becomes appreciable. 

The magnetic specific heat predicted by (50) is observed qualitatively 
in some ferromagnetic materials, usually with an adiabatic experimental 
method, whereby application of the field produces a change in tem- 
perature. We shall not give details, which have been discussed by 
Weiss, 39 and which he shows are nicely explained by his molecular fields. 
As we have seen that Heisenberg's theory gives results substantially 
equivalent to the Weiss theory (except for the difference between the 
Langevin and Brillouin functions), there is no difficulty in understanding 
in a general way these Weiss thermo-magnetic effects. 

Even in the absence of a magnetic field, there should be a discon- 
tinuity in specific heat as one passes through the Curie point, a result 
first noted by Weiss. 39 In terms of the Heisenberg theory, this is 
because the states of high resultant spin are probable for the crystal 
below the Curie point, while those of very low spin gain the upper hand 
above the latter. In other words, the prevalent spin $t is large for 
T< T c , but is very small for T > T c , as the crystal loses its spon- 
taneous magnetization above the Curie point. To calculate this dis- 
continuity in cjj it will suffice, as a first approximation, to assume that 
all states of the same crystalline spin possess the same energy. An 
equivalent assumption was made in the original work of Fowler and 
Kapitza. 40 Also we may neglect the statistical distribution of various 
values of S', as we saw in 77 that the probability had a steep maximum 
at 8' = S*. Thus for T< T c , we may take nW* 1 equal to the expres- 
sion W s . given in (22), with 8' = t, while for T > T c we take W= 0. 

39 Weiss and Beck, J. de Physique, 7, 249 (1908) ; Weiss, Piccarcl, and Carrard, Arch, 
des Sci. Phys. et. Nat. 42, 379 (1916); 43, 113 and 199 (1917); P. Weiss, J. de Physique, 
2, 161 (1921) ; Weiss and Ferrer, Annales de Physique, 15, 153 (1926) ; Stoner, Magnetism 
and Atomic Structure, p. 289. 

R. H. Fowler and P. Kapitza, Proc. Boy. Soc. 124A, 1 (1929). 

41 The factor n appears here because (22) relates to a crystal of n atoms. 



XII, 79 FURTHER TOPICS IN SOLIDS 345 

Here, of course, we have disregarded additive terms in the energy which 
do not involve the spin interaction, and so are continuous at the Curie 
point. The discontinuity in specific heat at the Curie point is thus 

"if... vVrsn ^ -r rrvnuC f&i\ 

(51) 



" rr _0 
""' - n a^-o-ajr u ~ ffi 

by (22) and (29). The value of d^ z /dT is found by differentiation of the 
first relation of (36) with respect to T after setting y = 2zJS 2 /kT. 
Since approaches zero when T does T c , this gives 

__i ( S i ) _L - ( 2zJ \ z d ^ r(Mn 4 ~ i ] 

3 "8" T c ~ \kf c ) df TssT [ 72CT J 
and so, after use of (37), Eq. (51) becomes 42 



As 7y&>^2cal., the numerical values given by (52) for some particular 
values of S are as follows in calories/gramme mol. : 

AcJ = 3-0(flf = J), 4-0(flf=l), 4-41(3 = }), 5-0(3 =00). (53) 
The value for S = oo is, as we should expect, the same as in the classical 
theory of Weiss. Some experimental values are 43 

Acg = 2-2 (Ni), 6-1 (Fc 3 4 ), 6-8 (Fe). (54) 

42 Fowler and Kapitza 40 derived (52) for the case & = |. The general expression (52) 
appears to have been first obtained in unpublished work of Landau, quoted by Dorfmann. 43 
Stoner notes that, in the case of *S = , inclusion of a Gaussian distribution makes the 
theoretical value lower than (52) and so makes for poorer agreement with experiment 
(Phil. Mag. 10, 27 (1930)). The resulting disagreement is not, however, as bad as repre- 
sented by Stoner as his formula contains an extraneous factor because of an algebraic 
error. With the Gaussian distribution the theoretical value for A*-? \ is 



We have already mentioned that this distribution is not an especially good approxima- 
tion, though doubtless bettor than neglecting the spread entirely, and so wo need not be 
worried at quantitative discrepancies with experiment. 

43 These experimental values are from Weiss, Piccard, and Carrard, Z.c., 39 except that 
the determination for Ni is by Mine Lapp, Annales de Physique, 12, 442 (1929). The 
agreement on Ni is ameliorated if, following Weiss, one writes (51) in the form 



and uses merely empirical values of the Curie point T, the Curie constant C, and the 
saturation magnetization 2Lf}Si/n. The discrepancy is then only a few per cent. Accurate 
agreement with the theoretical form (52) of (51) is clearly out of the question since the 
saturation magnetization is only 0-6 as largo as that to be expected from S = . 

By measuring changes in the Thomson heat, Dorfmann, Jaanus, and Kikoin (Zeits.f. 
Physik, 54, 277, 289 (1929)) report a discontinuity of 2-9 cal. mol. in the 'specific heat of 
electricity* of nickel at the Curio point. Tho agreement with (53) for S = $ is closer than 
the probable accuracy of (53), as (53) neglects entirely the 'spreading' of energies for 
given S'. Also, especially there is the further difficulty that the discontinuity in the 



346 HEISENBEBG'S THEORY OF FERROMAGNETISM XII, 70 

The agreement between (53) and (54) as regards order of magnitude is 
gratifying. As the calculation is only a crude one, we need not worry 
over the want of quantitative agreement, or the fact, sometimes urged 
against the theory, that the experimental discontinuities are somewhat 
gradual rather than perfectly sharp at T = T c . As also seen on p. 335, 
nickel seems to accord much better to S = | than any other value of S, 
but too much weight should not be attached to this fact as precision is 
wanting in both theory and experiment. 

The transition from large to small crystalline spins naturally implies 
a change of volume at the Curie point. Fowler and Kapitza 40 calculate 
that the order of magnitude of this change should be about 1 per cent., 
in accord with experiment. The reader is referred to their interesting 
paper for details. The sign of the observed change is such as to require 
dJ/dV > 0, or that the exchange integral increase with the volume. 
This seems at first thought a little mystifying, as one would expect \J\ 
to be greatest at small volumes, but is in accord with a theoretical 
prediction of Slater 28 that states of low crystalline spin have the least 
energy if the atoms are close enough together. This is more or less 
equivalent to saying that J would become negative if the volume were 
sufficiently diminished, although this statement is a little misleading, 
as Slater's whole argument is based on the fact that the Heisenberg or 
Heitler-London perturbation theory is a poor approximation at small 
inter-atomic distances, so that one should use instead a method de- 
veloped in Bloch's theory of conduction. In support of his view that 
sufficient concentration precludes ferromagnetism, Slater cites 44 the 
fact that in ferromagnetic bodies the ratio of the orbital radius of the 
3d electrons to the inter-atomic distance in ferromagnetic bodies seems 
to be smaller than the usual ratio of the radius of the valence orbits to 
this distance in most materials. Fowler and Kapitza emphasize that 
the sign and small magnitude of the volume change at the Curie point 
show quite conclusively that the forces between the electrons which are 
responsible for ferromagnetism cannot be the 'cement' which holds the 
solid together. Instead, forces between other groups of electrons, pre- 
sumably the outer or true valence electrons, as distinct from the 
ferromagnetic' electrons in inner incomplete shells, must be invoked. 

electrical specific boat has the opposite sign from that which one would expect on 
ordinary elementary views. Hence the theoretical significance of Dorfmann's interesting 
measurements is at present a little obscure. In particular, they cannot bo regarded as 
forcing the conclusion that the electrons responsible for ferromagnetisin are conduction 
electrons. 

** J. C. Slater, Phys. Rev. 36, 57 (1930). 



XII, 79 FURTHER TOPICS IN SOLIDS 347 

The same conclusion is reached independently on different grounds in 
Slater's work on cohesion. 

The ordinary phenomenon of magnetostriction, i.e. the change in size 
observed on actually magnetizing ferromagnetic bodies, must be related 
to the change in volume at the Curie point in the same way as the 
thermo-magnetic effects of Weiss are related to the change of specific 
heat at this point. Both the Curie-point phenomena are primary effects 
resulting from the obliteration of the spontaneous molecular fields (or 
rather their quantum equivalents), whereas the others are secondary 
ones resulting from the superposition of the external on the molecular 
fields. Fowler and Kapitza show that in view of the observed magnitude 
and sign of the volume change at the Curie point, the observed magneto- 
strictive effects are of the right sign and order of magnitude (viz. the 
relative change in length calculated for iron on pseudo-saturation is 
3-5 x 10- 5 , as compared to 2x 10~ 5 observed by Webster). 

80. Feeble Paramagnetism 

Numerous solids exhibit a feeble paramagnetism, which is comparable 
with diamagnetism in order of magnitude, and which is often indepen- 
dent of temperature, even though spectroscopic theory shows that the 
same materials would be strongly paramagnetic and conform approxi- 
mately to Curie's law if present in the gaseous state. The researches 
of Honda and of Owen 45 show that a great many pure solid elements, 
e.g. the alkalis and earths, are of this category. 

There is no difficulty in understanding theoretically the existence of 
such feeble paramagnetism. There are two possible explanations, viz. 
on the ground of inter-atomic interactions, which we shall consider first, 
and on the ground of the degeneracy phenomena in the Fermi-Dirac 
statistics. Actually both effects are doubtless to a certain extent super- 
posed, but they are too complicated to discuss when together. 

Inter-atomic Interactions 'Exchange Demagnetization'. We have 
shown at length in 73 that if the spacial separation due to inter-atomic 
interaction is large compared to kT, the orbital magnetic moment is 
largely quenched, leaving only a small residual effect, due to 'high- 
frequency matrix elements', which has the desired independence of 
temperature. Unless the atoms happen to be in singlet states, it is also 
necessary to have some mechanism for quenching the spin. One pos- 
sibility is the existence of such an intense magnetic coupling between 
spin and orbit within the atom that the inequality (4), Chap. XI, is 

K. Honda, Ann. der Physik, 32, 1027 (1910); M. Owen, ibid. 37, 657 (1912). 



348 HEISENBERG'S THEORY OF FERROMAGNETISM XII, 80 

reversed. The inter-atomic forces are then unable to loosen sufficiently 
the coupling between spin and orbit, and the spin magnetic moment 
can hence be quenched along with the orbital (cf. p. 313). This is likely 
only in very heavy atoms, where the multiplet structures are wide, 
whereas many feebly paramagnetic solids are light (e.g. aluminium) or 
even exist in S states devoid of multiplet structures (e.g. the alkalis). 46 
A further difficulty is that Kramers' theorem (p. 206) shows that the 
quenching of the spin is necessarily only a partial one if the atom 
contains only an odd number of electrons. 

It is probable that the spin is more commonly quenched through the 
operation of the Heisenberg exchange effect. If the exchange integral 
J is negative, this effect favours the states of low crystalline spin, and 
so tends to efface any paramagnetism which would be present in the 
gaseous state. We then have 'exchange demagnetization', the exact 
opposite of ferromagnetism. Except for the change in sign, the discus- 
sion proceeds entirely as in Heisenberg 's theory of ferromagnetism. If 
we neglect the refinement of the 'spread' of energy-levels belonging to 
a given S', the susceptibility will be given by the expression (39), which 
can always be used since now always T > T c as T c < 0. The molar 
susceptibility will thus be given by the expression 

4L8(8+1)P_ 8(8+1) 
Xmol ~ M(T-T C ) - U * JO T+A"' 

, A ,77 %sJS(S+l) 
where A = T c = * r ; 

OK 

(cf . (37) ). This susceptibility will be of the order of magnitude 10~ 5 -1 0~* 
observed experimentally in feebly paramagnetic media, provided one 
assumes that T c is of the order 10 4 . The temperature dependence 
is then very subordinate, as A>T. Such a value of T c or A requires 
that the exchange integral J be about - 1 volt, a larger numerical value 
than in Heisenberg's theory of magnetism, where T c ~ 10 3 , J /*w 0- 1 volt. 
This difference does not seem too unreasonable, as the exchange 
integrals between the true valence orbits, especially the highly eccentric 
s ones, may well be larger than between the nearly circular d orbits 
responsible for ferromagnetism. Also one has Slater's 28 ' 44 somewhat 
allied suggestion that the ratio of the orbital radius to the interatomic 
distance is less than usual in ferromagnetic media. 
Two points may be mentioned as favouring the above. One is that 

48 The alkali atoms have 2 S ground states, and hence have strong spin paramagnotism 
(given by Eq. (59) 80) when free. The normal levels of earth atoms, such as Al, are of 
the type 2 P and hence possess both spin and orbital moments. 



XII, 80 FURTHER TOPICS IN SOLIDS 349 

the relative abundance of feebly paramagnetic materials as compared 
to ferromagnetic is in agreement with the Heitler-London idea that the 
common bonds have negative J, also with Slater's proposition that low 
crystalline spins predominate at high densities not to mention the fact 
that the Gaussian calculation, &c., shows that even with positive J the 
exchange effects may still not lead to f erromagnetism without the proper 
crystalline arrangement, number of neighbours, &c. The second point 
is that many feebly paramagnetic media, notably, of course, pure ele- 
ments, are composed solely of naturally paramagnetic ions, so that 
practically all neighbours have exchange couplings between their spins. 
This is in marked contrast to the salts of the iron group, where we saw 
that the large magnetic dilution made the exchange effects subordinate 
and the spin hence free. We must, however, caution that there also 
exist feebly paramagnetic complex ions, whose feeble paramagnetism 
persists irrespective of the extent to which these ions are diluted in 
other media. Such ions involve the theory for polyatomic molecules, 
as discussed on pp. 272 and 293, and so are not to be confused with 
the present discussion of solids composed of simple atoms. We can, 
however, say that in the theory of these ions the complex ion is a unit 
of structure corresponding in a certain sense to the whole micro-crystal 
in the present discussion. 

Fermi-Dirac Statistics. Pauli has shown 47 that it is possible to explain 
the quenching of the spin in solids in quite a different way without 
invoking the exchange forces between electrons if one assumes that all 
the electrons not in closed shells participate to a certain extent in con- 
duction or, in other words, are at least partially free, so that they can 
be considered as wandering in, say, a cubical box of volume F = Z 3 . 
Each electron has then three translational quantum numbers n l9 n& n B 
besides a fourth quantum number m 8 which gives the component of 
spin along some axis. The Pauli exclusion principle states that no two 
electrons have all four quantum numbers the same. At the absolute 
zero the totality of electrons, regarded as one big system, will be in the 
state of lowest energy. In the absence of external fields the dependence 
of the energy on m 3 can be disregarded, and if there are n conduction 
electrons this means that at T = H = there are two electrons for each 
of the n/2 combinations of the quantum numbers n v n 2 , n 3 which yield 
the least energy. Each combination of these numbers, i.e. each transla- 
tional state, we shall call a cell. Because of the exclusion principle, two 
electrons in the same cell differ as to the sign of m s and compensate 
W. Pauli, Jr., Zeits.f. Physik, 41, 81 (1927). 



350 HEISENBERG'S THEORY OF FERROMAGNETISM XII, 80 

each other magnetically. Any paramagnetic alinement of the spins in 
a magnetic field, whereby more electrons have m 8 = \ than m s ~ }, 
can be secured only by 'boosting' some of the electrons out of the n/2 
originally occupied cells, and giving them higher values of the quantum 
numbers n v i& 2 > % We now see qualitatively why much of the para- 
magnetism is suppressed in the Fermi-Dime statistics based on the 
exclusion principle. 

The readjustment in the distribution among the cells in a magnetic 
field does, however, take place to a certain slight extent even at low 
temperatures, leaving a small residual paramagnetism. The latter may 
be calculated at the absolute zero without delving into any of the 
intricacies of the Fermi-Dirac statistics, 48 such as, for instance, what is 
meant by temperature, which can no longer be defined as proportional 
to the mean translational kinetic energy. We shall follow a simple and 
elegant method due to Frenkel. 49 At the absolute zero the distribution 
is, of course, that of minimum energy. When a magnetic field is applied, 
this is no longer that in which the electrons are paired in the n/2 
originally lowest cells. Instead we may suppose that the n/2k cells 
of least original energy have their full quota of two magnetically paired 
electrons, but that the next succeeding 2k cells each have a single 
electron with m a = . As k<^n, we may to a sufficient approximation 
regard these 2k cells as equally spaced in unperturbed energy, with the 
same spacing AIT as that in the vicinity of the highest originally occu- 
pied cells, which we shall call the critical spacing. The change from 
the original complete pairing to the new distribution involves an increase 
of amount k 2 &W in the 'unperturbed' part of the energy, as taking an 

electron from cell \n x to \n-\-x changes this energy by an amount 

k 

2x&W, and to a sufficient approximation J # = & 2 /2. However, this 

x=l 

change in the distribution diminishes by an amount 2kf$H the part of 
the energy due to the magnetic field, as 'turning over' an electron from 
m a = i to m a = \ gives an alteration gpH 2pH. The value of 
k appropriate to the absolute zero is that which minimizes the total 
energy k* A W2fikH+ constant, and is hence k = fiH/kW. As 2k elec- 
trons now have spins alined along the field, the susceptibility per 

unit-volume is O T,Q 902 

Y = f=_f_ (55) 

X VH V&W ( } 

48 For exposition of these statistics see, for instance, Fowler's Statistical Mechanics 
Chap. XXI. 

* e J. Frenkel, Zeite.f. Physik, 49, 31 (1928). 



XII, 80 FURTHER TOPICS IN SOLIDS 351 

The value of the critical spacing &W will depend on whether we 
consider the conduction electrons as absolutely free, or consider the 
binding effect of the atoms through which they migrate. We shall first 
treat the case that they are absolutely free, as in Sommerfeld's theory 
of conduction. Here the unperturbed translational wave function of 
an electron is *//~Aam(7Tn l x/l)ain(7Tn 2 y/l)8m('TrnyS/l) as it vanishes at 
the walls of the box and satisfies the appropriate wave equation 
> = provided 

*> ( W3 >0). (56) 



There is one cell at each corner of unit cubes in the n v n 2 , n z space, 
and so the number of cells with energies inferior to some given value 
J^inax is approximately the volume (47r/3)(2mZ 2 fF max /^ 2 ) 3 / 2 of one octant 
of a sphere of radius (SmZW^x/ft 2 )*. The critical energy, or energy of 
the highest cell occupied at T = H = 0, is (7i 2 /8raZ 2 )(3w/7r) 2 / 3 , as it defines 
an octant of volume n/2. The spacing ATT of the cells near the critical 
upper limit is 2dW/dn = (& 2 /6mZ 2 )(9/7r 2 tt)*. Substitution of this value 
of &W in (55) yields lam^/.u/^l 

x nr(v)(i) (57) 

where n/V is the number of conduction electrons per unit- volume. This 
is Pauli's celebrated formula, which marked the beginning of the quan- 
tum theory of electrons in metals. It will be compared with experiment 
at the end of 81. By considering a second approximation in the 
Fermi-Dirac statistics of free electrons, Bloch 50 has shown that a more 
accurate formula than (57) is 



2 l 

-J 



5S 

= 2-20 X 10- 1-03 X 10 7 ~ . (58) 



As n/V is of the order 10 22 , the second term is negligible compared to 
the first at ordinary temperatures, so that (57) can be regarded as an 
adequate approximation or, in other words, it is legitimate to treat the 
electron gas as completely degenerate. The independence of tempera- 
ture predicted by (57) or (58) is approximately confirmed in the 
measurements of McLennan, Ruedy, and Cohen, 51 which extend down 
to -190C. 

50 F. Bloch, Zeits.f. PhynJc, 53, 216 (1929). 

61 J. C. McLennan, R. Ruody, and E. Cohon, Proc. Roy. Soc. 116A, 468 (1927). 



352 HEISENBERG'S THEORY OF FERROMAGNETISM XII, 80 

As an alternative to the standard Sommerfeld theory based on free 
electrons, Bloch 52 has developed a theory of conduction based on bound 
electrons. Such electrons can pass from one atom to another because 
of the remarkable fact that in quantum mechanics there is a finite 
probability of a particle traversing a peak of potential energy greater 
than its own energy. In Bloch's theory an electron is in a Valley' of 
potential energy when bound in an atom, and is continually playing 
leap-frog from one valley to another, thus giving an electric current. 
Bloch shows that the 'over-lapping' of the wave functions of adjacent 
atoms removes the degeneracy associated with the fact that electron- 
levels are identical in identical free atoms. An electron playing leap-frog 
thus has a variety of closely spaced energy states even though it has 
only one normal state when in a perfectly isolated atom. If the total 
splitting of these closely spaced levels (i.e. the 'critical energy' or total 
energy spread W c of the n/2 different levels occupied at T = 0) is large 
compared to kT, the 'degeneration' will be practically complete, and 
the susceptibility will be given by the expression (55), as the various 
steps used in the derivation of (55) retain their validity. The constant 
A If will, of course, have a different value than that calculated for free 
electrons in the preceding paragraph. Thus, if the over-lapping of the 
wave functions of adjacent atoms is sufficient, Bloch's theory also will 
give a feeble paramagnetism independent of temperature. This is 
encouraging, as Bloch's conduction mechanism probably comes closer 
to reality than that by free electrons in many cases. It must, however, 
be cautioned that it is not at all certain whether the 'over-lapping' in 
his theory is in many cases adequate to make the splitting or diffusion 53 
W in the ground state large compared to kT. If inadequate, Eq. (55) 
no longer applies. In the limit W c <^kT, the susceptibility is given by 
the same formula, P2 



as in the Boltzmann statistics, and the solid is strongly paramagnetic. 
In the Sommerfeld theory for free electrons, Eq. (59) also, of course, 
applies to the analogous limiting case (h 2 J8ml 2 )(^n/Tr) 2 ^^kT 9 which is 

B2 F. Bloch, Zeits.f. Physik, 52, 655 (1928). 

53 This diffusion W c in Bloch's theory is not to be confused with the 'spacial separa- 
tion' introduced in 73, arid will usually bo much smaller than the latter or than the 
corresponding diffusion or critical energy (fc 2 /8ra)(3/7rF)l for free electrons. The 'spacial 
separation' is associated with the removal of the spacial degeneracy for a single atom, 
and gives a splitting into 2L+ 1 components (neglecting spin). The Bloch diffusion effect 
involves a further division of each of those 2L+1 components, removing the degeneracy 
associated with tho fact that these components are otherwise the same for all atoms. 



XIT, so FURTHER TOPICS IN SOLIDS 353 

usually realized only at extremely high temperatures. We have seen 
that the susceptibilities of rare earth and iron salts can be treated by 
the Boltzmann statistics. We now see that this implied that in these 
salts the Bloch leap-frog effect was so small that the Fermi-Dirac inter- 
ference effects were negligible. Another way of saying the same thing 
is that the electrons could be regarded as firmly bound to an individual 
atom, for Bloch 's theory is a sort of intermediary between that for 
isolated atoms and that for free electrons. This is in accord with the 
fact that these salts are much poorer conductors and have higher ioniza- 
tion potentials than, for instance, the feebly magnetic alkali metals. 

81 . The Diamagnetism of Free Electrons in Quantum Mechanics 

Landau 54 has discovered the very remarkable fact that the orbital 
motions of free electrons give a diamagnetic contribution in quantum 
mechanics, whereas we saw in 26 that classically they were without 
such an effect. This difference is a little hard to explain intuitively, 
but arises from the fact that the boundary electrons have different 
quantized velocities than those which do not touch the walls of the 
vessel, and so the magnetic moments of these two types of electrons 
do not compensate each other as in classical theory. (Classically, both 
types have the Maxwell-Boltzmann distribution of velocities.) 

The calculation is most easily made by using cylindrical coordinates 
p, z, <, with the applied magnetic field along the z direction. The z 
component of motion can then be disregarded for our immediate pur- 
poses, as there is no force on the electron in this direction. We have 
then to deal with the two-dimensional wave equation 



o, (60) 
p dp p irmc <> 8mc 2 v / 

as can be seen, for instance, by introducing cylindrical coordinates into 
(2), Chap. VI, and then ignoring the z degree of freedom. As the solu- 
tions of (60) are clearly of the form i/t = e in ^f(p), the term in (60) which 
is proportional to d$/d<t> has merely the effect of displacing W by an 
amount he^H/^rrmc. Without this term, Eq. (60) is identical in form 
with that of a two-dimensional oscillator of frequency v = 
The characteristic values of the latter are well known 55 to be 



** L. Landau, Zeits.f. Physik, 64, 629 (1930) ; also given by Pauli in the report of the 
1930 Solvay Congress. 
55 Soe, for instance, Condon and Morse, Quantum Mechanics, p. 78. 

3595.3 A a 



354 HEISENBERG'S THEORY OF FERROMAGNETISM XII, 81 

where n v n 2 are integers. Hence, as ]8 = &e/47rwc, 

Tf = (n 1 +|n l |+2n 1 +l)j5JI. (61) 

Thus a free electron, even when not enclosed by bounding walls, has 
a discrete rather than continuous spectrum in a magnetic field. The 
magnetic moment corresponding to the stationary state (61) is 

dW 

-^, = -(%+KI+2rvfl)j8. (62) 

Classically the azimuthal quantum number % has the geometrical 
significance rr 

w 1 = 7 ^_ (r-(P), (63) 

where r is the radius of the circle described by the electron under the 
influence of the field, and d is the distance of its centre from the origin 
p 0. To prove (63) we have only to note that 56 

p+ m(xij~yx)He(x*+y*)/2c 
and that classically we may take 

Pj njt'l"2rr 9 x = x -\~r cos(Hel/mc), y y +r&in(Het/mc), 

as in the field the electron moves with an angular velocity 57 He/me, in 
a circle about some point X Q , y Q . Of course the geometry of (63) is not 
to be taken too literally in wave mechanics, but will clearly have at 
least an asymptotic meaning for large quantum numbers, by means of 
which the most probable position of the statistical charge density can 
be approximately located. 

Tn point of fact we must consider not ideally free electrons, as above, 
but rather those enclosed by some vessel, as we saw on p. 101 that 
reflection at the boundary played a very vital role. It is most con- 
venient to take the vessel as a cylinder of radius R, with axis parallel 
to the field. In order to avoid the complication of distortion of charac- 
teristic values by the wall, the only case readily treated is that in which 
the classical radius of curvature r = *J(2mc 2 W/e*H 2 ) is much smaller 
than E for the great bulk of electrons. (Whether the susceptibility 
would be the same in the case r ^ E is at present uncertain, although 
the concept of spectroscopic stability suggests that perhaps it would 
be.) Since in Boltzmann statistics the prevalent energies are of the 

56 That this is the proper definition of the canonical generalized momentum associated 
with the coordinate pj can be seen from the theory given in 8. 

57 Note that this angular velocity, which is readily deduced by elementary mechanics, 
is twice that corresponding to the Lannor frequency. This difference arises because 
Larmor's theorem neglects second order terms in //. 



XII, 81 FURTHER TOPICS IN SOLIDS 355 

order kT, this condition r <^ R becomes 

an inequality clearly satisfied for ordinary values of T, H, and R. Out 
of all the electrons within the vessel a fraction of the order r/R have 
'boundary' orbits which classically hit the wall. This shows that in 
quantum mechanics the overwhelming majority of common stationary 
states will not have their characteristic values appreciably distorted 
from (61) by the influence of the boundary, although there may be 
a very slight distortion because some of the charge wanders outside 
the classical limits. 

Boltzmann Statistics. If we use Boltzmann statistics, the partition 
function is 

~l 00 



ko ^ < 64 > 

HI 

Here the choice of the limits used in the summation over % requires 
some discussion. Positive values of n v have been omitted because if 
R^>r, only a negligible number of electrons have r^d or, in other 
words, centres near the origin p ~ 0. It is very vital that, following 
Landau, we have taken the lower limit for the summation as 7reHR 2 /hc 
rather than oo. This value is obtained on the ground that the centre 
of the orbit cannot be distant more than approximately R from the 
origin, and so we can exclude all values of n which by (63) would give 
d > R. The sum over n 2 in (64) is readily evaluated, as it is merely 
a geometric series, and one finds Z = 
The moment per unit-volume is 



(65) 



and is hence 



where L(y) is the usual Langevin function, but the presence of the 
minus sign means that there is dia- instead of paramagnetism. In the 
limit h = j8 = the right-hand side of (66) reduces to zero, in agreement 
with the fact that classically there is no diamagnetism for free electrons. 
This asymptotic agreement with classical theory can also be verified by 
replacing the sum by an integral in (64), as then Z becomes independent 
of H. For the usual case fiH^kT, Landau's formula (66) reduces to 
X = MjH = nfPfiVkT, and so the orbital diamagnetism is one-third 

A a 2 



356 HEISENBERG'S THEORY OF FERROMAGNETISM XII, 81 

as great as the spin paramagnetism (59), making the total resultant 
susceptibility 2nfi*l$VkT. 

The moment (66) is not the same as that furnished by the more immediate 



formula ^ (-6>W/c>H)e- w l kT 



(67) 
{ ' 



if we use the same Jimits of summation as in (64) and same energy values as in 
(61). Eq. (67) then yields a preposterously large diamagnotism (viz. the first term 
alone in Eq. 66), as each of the states (61) has a strongly diamagnotic moment 
(62). This difference as compared with (66) arises because in (65) part of dZ/ftH 
comes from the fact that in (64) the limits of summation, or number of states, 
as well as the energy -levels W are functions of H. (In our previous work Z involved 
H only through W and as a matter of fact we derived (65) only for this special 
case, cf. p. 25). To obtain a proper expression for the moment by means of (67) 
it is very essential to include the 'boundary' electrons which are reflected at the 
walls of the cylinder (cf. Fig. 5, 26). These electrons are few in number, but 
have enormously greater moments than do the inner orbits so much greater in fact 
that they completely neutralize the latter in classical theory. If (66) is correct, 
the term in (65) resulting from the dependence of the limits on H must be the 
same as that which results from inclusion of the boundary electrons in (67). 
Another way of saying the same thing is that Landau's use of (65) without 
boundary electrons to obtain (66) implies that those electrons make a negligible 
contribution to (65) despite the fact that they make a large one to (67). The 
states n t < ireHR 2 /hc are those representing boundary electrons, 58 so that when 
they are included, the partition function contains a constant rathor than variable 
number of states, and so involves H only through W. There would thus bo no 
doubt as to the applicability of (65) were the boundary electrons included therein. 
It is obvious that Z itself would not be appreciably affected by including the 
boundary states, since, though great in number, they have such high energies that 
only a negligible fraction, of the order r/R, of the total number of electrons are 
located therein. It is, however, not quite so obvious (although justifiable on closer 
examination) that the boundary states make a negligible contribution to dZ/dH, 
since they have abnormally large moments 0W/8H. It is therefore reassuring 
to show that the moment (66) can also be calculated from (67). We shall follow 
a variant of a method due to Teller. 59 As R ^> r, the magnetic moment of a bound- 

58 This is most easily seen by regarding the wall as equivalent to a fictitious central 
field which is zero for p ^ R but which increases to a very large value when p slightly 
exceeds R. The dynamical problem is then one in central fields, where n x , n a are respec- 
tively the azimuthal and radial quantum numbers. All positive and negative values of 
n t are allowable in this problem, and the range of values of n t not absorbed by the inner 
electrons must be due to the boundary electrons. Values of w t less than 7re//JR a //ic 
rapidly take the electron into the region where this fictitious field is large, and so give 
large energies. 

59 E. Teller, Kelts. f. riiysik, 67, 311 (1931). Our procedure differs from his in the use 
of a cylindrical rather than infinite plane boundary. Still another method, which is quite 
simple, has been given by Darwin (Beport of the 1930 Solvay Congress or Proc. Camb. 
Phil. Soc., 27, 86). Instead of using bounding walls, or the equivalent sudden repulsive 
field cited in note 58, he introduces a radial linear restoring force ap which, of course, 
becomes large only gradually. The wave equation for this system in a magnetic field is 



XII, 81 FURTHER TOPICS IN SOLIDS 357 

ary electron is approximately 60 ( e/2c)2iTa) ni It 2 where co ni is the frequency with 
which this electron creeps around the complete circumference of the wall. 
Further by the correspondence principle we have w ni = dW/hdn lt a well-known 
kinematical result in the old quantum theory. 61 As the radius R of the cylinder 
is large compared with the radius of an ordinary orbit, ha) n JW is very small. 
Hence with given n a the states belonging to different consecutive values of n x lie 
very close together, and the summation over n^ for the boundary electrons may 
be replaced by an integration. Hence the expression (67) becomes 

- ml/IP Ihr 



_ 

n hc 

V " 

(68) 

Here the first term in the numerator arises from the overwhelming number of 
inner electrons, and tho second from the boundary ones, whoso energies T^(n 1 ,w 2 ) 
would be dif ucult to determine explicitly. Tho summation over n from ireHlt*/hc 
to 1 has already been performed for the former, and we have omitted tho 
contribution of the latter to the denominator, which is clearly negligible on account 
of tho high values of W n . The integration of the second term over /i x can immedi- 
ately bo performed. Furthermore W B has tho value oo at n L -^ GO and the 
value (61) at n t ireHltf/hc, as at tho latter limit the orbits just begin to touch 
the walls of the cylinder and so do riot have their energies appreciably distorted 
from (61). The second term in the numerator thus becomes kT/H times tho 
denominator, and so the boundary electrons contribute tho second term of the 
Langoviri function in (66). It is easily seen that now the expression (68) assumes 
the desired value (66). t 

Fermi-Dirac Statistics. In actual solids the calculations should be 
made with the Fermi-Dirac rather than Boltzmann statistics. Here 
also the orbital diamagnetism proves to be one -third as great as the 
spin paramagnctism. Landau has shown that this is true regardless of 
whether or not the degeneracy is complete, but we shall give an ele- 
mentary proof in which complete degeneracy is assumed, so that all 
orbital states may be supposed occupied by two electrons up to a certain 
critical energy W c , and vacant thereafter. A similar assumption was 
made on p. 349 and is amply warranted at ordinary temperatures, for 

readily integrated, as the harmonic form is preserved, and in the limit a = Darwin 
finds the same expression for tho susceptibility as Landau's. 

60 Tho boundary electrons havo inordinately large moments because they encircle the 
origin when they make a circuit of tho wall (Cf . Fig. fi, 26). Hoiico in thoir case we may 

replace inp*<j> by mR z <j> 2irmR z ti) n . On tho other hand, the inner electrons usually do 

not encircle the origin, so that they have 0; their resultant moment (62) is duo 
entirely to the fact that p is different at different points of the orbit, a factor of subordi- 
nate importance in tho case of the boundary electrons. 

fll Cf. for instance, J. H. Van Vleek, Quantum Principles and Line Spectra, p. 298. 
This relation will bo a good approximation with the new mechanics, as the quantum 
number n x is large for the boundary electrons. 



358 HEISENBERG'S THEORY OF FERROMAGNETISM XIT, 81 

the density of electrons in ordinary conductors is sufficient to make 
W c /k of the order 10 4 C. Unlike the case of the Boltzmann statistics, 
the component of motion in the direction z of the field H cannot be 
entirely forgotten, as the exclusion principle can be applied only when 
we consider all components. If the cylinder has a length I, the charac- 
teristic values of the energy associated with the z component are 



f. Eq. (56)), and consequently there are V8mZ 2 ^ nax /F values 
of % for which this part of the energy does not exceed any given limit 
W max . As there are two possibilities for the spin quantum number, and 
as a state is occupied if the z component does not require more energy 
than W c W, the weight of a given state n v n 2 of motion in the x, y 
plane is 2f(W), where /== ^*mP(W c - W )/h* if W < W c , and /= if 
W > W c . As previously, we use W to denote only the x-y part of the 
energy. In Eq. (68) we must now replace the Boltzmann exponential 
factor by this/(PF). The integration of the boundary term over n can 
be performed in the same fashion as previously explained in the Boltz- 
mann case. The expression for the moment thus becomes 

n 



with 2n c +l = We/fill. It is adequate to replace the sum by an integral 
in the denominator, but the numerator vanishes in classical theory, and 
here it is necessary to use the more accurate approximation formula 62 

?/ a v *+* 

2 F(x)= J F(x)dx-hF'(x)\ v \. Eq. (69) thus gives M=-np/4Vn . 
x Ul v^l 

By filling twice the n/2 lowest orbital states, one finds 



This is, as we should expect, the same value of W c as we calculated on 
p. 351 in the absence of the field. It is thus finally found that (69) 
becomes x= 4mj8%- a (w7r a /9F) 1 ' 8 , so that the resultant susceptibility 
inclusive of both spin and orbit has two-thirds as large a value as (57). 
It is to be emphasized that all these results apply only to electrons 
which are absolutely free. As soon as an electron becomes tightly bound 

62 Cf. Rungo and Willers in Encyl. der Math. Wiss. ii. 2, 92. Wo apply this approxima- 
tion formula to our function F despite the fact that this F has a discontinuity in its first 
derivative at n a = n since we must take F = for n a > n . The justification for so doing 
lies in the fact that this discontinuity disappears as soon as one makes any allowance, 
however small, for the effect of temperature. In other words, if T is greater than 0, but 
small compared to W jk the distribution function diminishes exceedingly rapidly but not 
discontinuously in the vicinity of W = W o . 



XII, 81 FURTHER TOPICS IN SOLIDS 359 

to the atom, its diamagnetism will be given by the ordinary atomic 
formula e 2 r 2 /6rac 2 (cf. Eq. (2), Chap. VIII), even though occasionally 
it plays leap-frog from one atom to another. The freak case of bismuth, 
in which the electron seems to migrate around frequently from one 
atom to another, and so has an abnormally large radius and dia- 
magnetism, has already been cited in 23. In general one has no 
adequate theory for the intermediate case of feebly bound electrons. 

It is tempting to calculate the susceptibilities for the alkali metals 
under the assumption that the valence electrons are completely free 
and conform to the Fermi-Dirac statistics. It is essential to include 
a correction for the diamagnetism of the residual positive ions (Na+, 
&c.), which can be estimated by any of the methods given in 52. This 
ionic diamagnetism we denote by x+ in the following table, while x~ e 
denotes two-thirds the expression (57), multiplied by the volume of 
a gramme mol. of the metal. We use Pauling's estimates of % + for Li > , 
Na+, and K+, but Ikenmeyer's for Rb+ and Cs - as the method of 
screening constants is probably more reliable for light atoms and the 
additivity method for heavy (see 52). 

CALCULATED AND OBSEKVED MOLAR SUSCEPTIBILITIES OP ALKALI METALS 





Li 


Na 


K 


Rb 


Cs 


_. Calc. 


6-8xlO- 


10-2xlO- 


15-7 x 10-e 


18-2 x 10- 6 


21 -5 X lO- 6 


,. Calc. 


-0-6 


4-2 


16-7 


31-3 


45-7 


moi Calc. 


5-0 


6-2 


1-0 


13-1 


24-2 


moi Obs.: 












Honda & Owen 45 


3-5-21 


12 


16-23 


6 


13 


Sucksmith 83 




14 


20 


6 


3 


McLomiaii, R. & 




14 


19 


17 


28 


C. 61 












Lane 64 


1 t 


15 


21 


18 


29 


Bitter 65 


26 











The discrepancy between the different observations shows that experi- 
mental as well as theoretical precision is difficult. It is possible that 
the susceptibility varies considerably with the physical treatment 
accorded the specimen, as Bitter 65 finds that stretching increases the 
susceptibility of copper almost 50 per cent. Hence no quantitative 
agreement with simple theory can be expected. The experimental 
values are invariably greater than the calculated, and this fact is 
probably to be explained on the ground that the electrons are not 
entirely free. Very tightly bound electrons have the strong spin para- 

63 W. Sucksmith, Phil. Mag. 2, 21 (1926). 

" C. Lane, Phil. Mag. 8, 354 (1929); Phys. Rev. 35, 977 (1930). 

65 F. Bitter, Phys. Rev. 36, 978 (1930). 



360 HEISENBERG'S THEORY OF FERROMAGNETISM XII, 81 

magnetism (59), which is 0-372/T per gramme mol., and so even feeble 
binding might make the free value (57) too low. Also the exchange 
effects discussed on p. 348 may be important. At any rate, there is no 
difficulty in understanding qualitatively the marked contrast between 
the feeble paramagnetism of the solid alkali metals, and the strongly 
paramagnetic behaviour of their vapours, evidenced by Gerlach's con- 
firmation of (59) in potassium vapour ( 57). 



XIII 

BRIEF SURVEY OF SOME RELATED OPTICAL PHENOMENA 
82. The Kramers Dispersion Formula 

This formula for the index of refraction n is l 



The frequency of the incident light is denoted by V Q , and its wave-length 
we suppose large compared to the atomic or molecular radius. The 
index I or I' denotes the totality of quantum numbers necessary to 
specify a stationary state, and the expressions p Q E (l\l') are the unper- 
turbed matrix elements of the component of the atom's or molecule's 
electric moment in the direction of the electric vector E of the primary 
beam. 

Eq. (1) was first obtained as an extrapolation from classical dynamics 
by means of the correspondence principle, but has since been deduced 
more rigorously with quantum mechanics. 2 It is hence the formula for 
dispersion. Classical theories of dispersion based on naive harmonic 
oscillators owe their measure of success to the fact that each term in 
(1) has the same 'Sellmeier' form of dependence on the frequency V Q as 
a conventional oscillator of appropriately chosen charge e t and mass m i9 
viz. et/m^S^l'Wp^lil'W/h. 

Some features of (1) on which we may comment briefly arc the 
following: 

(a) Presence of Negative Terms. Any term with v(l f >l)<0 has a 
negative value of e?/m^ for the corresponding fictitious or 'virtual' 
oscillator. Such a term is said to give 'negative dispersion'. 3 This can 

1 H. A. Kramers, Nature, 113, 673; 114, 310 (1924); H. A. Kramers and W. Hoisen- 
borg, Zeits.f. Physik, 31 , 681 (1925). Except for the negative terms the formula was first 
proposed by Ladenburg, Zeita.f. Physik, 4, 451 (1921). 

a See Born, Heisenberg, and Jordan, Zeita.f. Physik, 35, 570 (1926), or Born and 
Jordan, Elementare Quantenmechanik, p. 240; P. A. M. Dirac, Proc. Roy. Soc. 114A, 710 
(1927); E. Schrodingor, Ann. der Physik, 81, 109 (1926); Sommerfeld, Atombau, Wcllen- 
mechanisher JKrganzungsband, p. 193. 

8 The negative terms are difficult to detect experimentally because of the difficulty 
of obtaining a sufficient concentration of atoms in excited states, but seem to be definitely 
established in neon. See Ladenburg, Carst, and Kopfermaim, Zeits.f. Physik, 48, 15, 
26, 51, 192 (1927); Kopfermaim and Ladenburg, ibid. 65, 167; Ladenburg and Levy, 
ibid. 65, 189 (1929). 



362 BRIEF SURVEY OF SOME RELATED XITT, 82 

exist only when there is an appreciable concentration of atoms or mole- 
cules in excited states, as v(l';l) is necessarily positive if / is the lowest 
state. For this reason the negative dispersion is hard to detect experi- 
mentally. 

(b) Behaviour in Limit h Q. In this limit, as well as in that of very 
large quantum numbers, Eq. (1), of course, merges asymptotically 4 into 
the classical dispersion formula for the corresponding dynamical system, 
which is in general a 'multiply periodic one', not a harmonic oscillator. 
This requirement was, in fact, the clue to the initial discovery of (I). 

(c) Isotropic Media. If there are no fields other than that of the 
incident light, and if the dispersion is by a gas or even an isotropic 
liquid or solid, a spacial averaging may be performed by means of the 
principle of spectroscopic stability as on p. 1J)3. 

(d) Invariance of Temperature. Eq. (1) is a general expression not 
yet requiring the hypothesis of 46, that the quantum numbers can be 
divided into three categories n, j, m such that the effect of the index 
n on the energy is large, while that of j, and of the magnetic quantum 
number m, is small compared to TcT. If we make this hypothesis and 
also that (c) of isotropy, the procedure on pp. 193-5 reduces (1) to 



provided further that the incident light is far enough from resonance 
to permit assuming that the denominators in (1) are insensitive to the 
indices j 9 j'. At constant density the expression (3) is independent of 
temperature. This is in accord with experiment (see 16). Eq. (3), 
of course, involves the resultant amplitudes p Q (n',ri) in place of com- 
ponents thereof as in (1). The expression 87T 3 \p(n',n')\ 2 /3h 2 is the same 
as the Einstein absorption probability coefficient 5 B n _ >n * for the transi- 
tion n->n', with unresolved fine structure^', f. 

(e) Behaviour in Limit i/ 0. When v = 0, Eq. (3) is the same as 
the induced or non-polar part ^TrNoc of the static dielectric constant 
(see Eq. (28), Chap. VII). This agreement seems trivial to-day, but was 
not secured in the last days of the old quantum theory in which refined 
applications of the correspondence principle were used to obtain the 
dispersion formula (1) for periodic fields, but in which straight classical 

4 For proof seo Kramers and Heiseriberg Z.c. 1 , or J. H. Van Vleck, Phys, Rev. 24, 347 
(1924). 

5 For discussion and references on the Einstein A and B coefficients and their relation 
to the dispersion formula see Born and Jordan's Elementare Quantenmechanik, p. 240, 
or the writer's Quantum Principles and Line Spectra, pp. 120, 161. 



XIII, 82 OPTICAL PHENOMENA 363 

dynamics were used to calculate the orbits to be quantized in the 
analogous static case of the Stark effect. 

(/) Behaviour in Limit V Q = oo. For very short incident wave-lengths 
Eq. (1) reduces to the classical Thomson formula for the dispersion by 
free electrons. 6 This means that, as we would expect, impressed forces 
of very high frequency are resisted more by the electron's own inertial 
reaction than by the forces binding the electron to the rest of the mole- 
cule. We shall give only the very simple proof appropriate to a one- 
electron system, although the theorem is much more general. Here the 
matrix elements of p K are the same as those of ez if we suppose E 
directed along the z axis. If we use Cartesian coordinates in the quan- 
tum condition p k q k ~q k p k = h/27Ti (Eq. (12), Chap. VI) we may take 
<lk = *>Pic = z> so that p k (l\ V) = 2<rrimv(l\ l')z(l; I 1 ). As v(l\ I') = v(V ; Z), 
we then have (p k q k )(l\l) (q k p k )(l;l). The diagonal elements of this 
condition thus yield the 'Thomas-Kuhn relation' 7 



Eqs. (4) and (2) show that (1) will reduce to the Thomson formula 8 



provided we can neglect v(l\l')* in comparison with i/g in the denomina- 
tors, as will be the case if the incident frequency is large compared to 
the atom's absorption frequencies. To prove (5) classically, we observe 
that the solution of the differential equation mz = eE for a free elec- 
tron in a periodic field E = J57 cos 27rv Q l isz = eE/47T 2 mv$, plus arbitrary 
terms At+ C not of interest to us. Hence n 2 1 = InP/E = ^irNez/E 
has the value (5). 

(g) Explicit Values of (3). The various terms in (3) have been evaluated 
numerically in certain cases. Podolsky 9 and later Reiche 10 showed that 
in normal atomic hydrogen (3) becomes 

ft 2 -! = 2-25 X 10~ 4 (l + l-228x 10- 10 A(T 2 + 1-65 x 10- 20 A/+...) 
at C., 76 cm., provided the incident wave-length A is large compared 
to that 4/3 R= 1216 A of the softest absorption line, thus permitting 

6 This asymptotic connexion was first proved by Kramers, Physica, 5, 369 (1925), 
although in the old quantum theory Reiche and Thomas, also Kuhn, 7 proposed summa- 
tion rules equivalent to (4) in order to secure this connexion. 

7 Thomas, Natw-wissenschaften, 13, 627 (1925); Thomas and Reiche, Zeits.f. Physik, 
34, 510 (1925); Kuhn, ibid. 33, 408 (1925). 

8 See, for instance, A. H. Compton, X-rays and Electrons, p. 205. 

9 13. Todolsky, Proc. Nat. Acad. 14, 253 (1928). 

10 F. Reiche, Zeits.f. Physik, 53, 168 (1929). 



364 BRIEF SURVEY OF SOME RELATED XIII, 82 

use of a series development in descending powers of A . Explicit calcula- 
tions of (3) are usually difficult because in general the summation 
symbolizes an integration over the continuous spectrum beyond the 
'series limit* besides the usual summation over the discrete one. Podol- 
sky dodged this integration by an ingenious method due to Epstein, 11 
while Reiche showed great computational skill by performing it ex- 
plicitly. The various terms of (3) have also been estimated for some of 
the alkalis by various workers. 12 They confirm the experimental result 
that the first line of the principal series far overshadows succeeding 
lines in intensity. It is also calculated that the part of the dispersion 
due to the continuous spectrum is less important than in atomic 
hydrogen; the computed ratios of the continuous part to the total at 
very high incident frequencies being respectively: Li, 0-24; Na, 0-04; 
H, 0-44. This difference is primarily because the normal states of alkali 
valence electrons have principal quantum numbers greater than unity, 
for the corresponding value for the Balmer series of hydrogen is 0-12, 
against 0'44 for the Lyman series. 13 

(h) Quadrupole Effects. Eqs. (1) or (3) embody only the dipole part 
of the radiation. Both theoretically 14 and experimentally 15 the quadru- 
pole part sometimes gives an appreciable effect, though usually very 
small. 

(i) Raman Scattering. Eqs. (1) or (3) give the dispersive effect of the 
Rayleigh scattering, or resonance radiation which is emitted on return 
of the atom to its original state after excitation. There is also the now 
famous Raman scattering, first predicted by Smekal and by Kramers 
and Heisenberg, 16 in which the scattered light differs from the incident 
by an atomic (or molecular) frequency v(Z';Z), and which arises from 
fluorescent radiation, either Stokes or anti-Stokes, whereby the atom 
reverts after excitation to a different state than the initial. As the 
Raman radiation has a different frequency from the primary, it does 

11 P. S. Epstein, Proc. Nat. Acad. 12, 629 (1926). 

12 Hargreaves, Proc. Cambr. Phil. Soc. 25, 75 (1929); B. Trumpy, Zeits.f. Physik, 57, 
787 (1929) and earlier papers (Li); Y. Sugiura, Phil. Mag. 4, 495 (1927) (Na). 

13 Y. Sugiura, J. de Physique, 8, 113 (1927). 

14 A. Rubinowicz, Phys. Zeits. 29, 817 (1928) ; Zeits.f. Physik, 53, 267 (1929) ; Bartleii, 
Phys. Rev. 34, 1247 (1929); A. F. Stevenson, Proc. Roy. Soc. 128A, 591 (1930). 

i* W. Prokofjew, Zeits. f. Physik, 57, 387 (1929). 

18 A. Sraokal, Naturwissewchajten, 11, 873 (1923); 16, 612 (1928); Kramers and 
Heisenberg, I.e. 1 The experimental literature on the Raman effect is too copious for us 
to cite, but we may mention that probably the most careful measurements on gases, as 
distinct from liquids, are those of Rasetti, Proc. Nat. Acad. 15, 234, 515 (1929); Phys. 
Rev. 34, 367 (1929); Dickinson, Dillon, and Rasetti, ibid. 34, 582 (1929); and of Wood 
and Dieke, Phys. Rev. 35, 1355; 36, 1421 (1930). 



XIII, 82 OPTICAL PHENOMENA 365 

not react coherently with the latter to give a dispersive effect or altera- 
tion of the primary velocity of propagation, and can be observed only 
by analysing spectroscopically the scattered radiation. The intensity 
of a Raman line of frequency v +v(Z;0 can be shown proportional to 
the expression 



Here the Pq(l',l') are the matrix elements of the various Cartesian com- 
ponents of the unperturbed electrical moment; while the p Q ^\l') are 
those of the component along the incident electric vector E. As (6) 
involves the products p Q (l; l*)p Q (l*;V) rather than \p g (l; l')\ 2 , a necessary 
condition for a Raman line is that it involve a displacement of energy- 
levels obtainable by superposition of two consecutive allowed transi- 
tions. It need not be a possible emission or absorption line. The 
necessary condition just given is not also a sufficient one, as the various 
product terms in (6) may have such phases as to 'interfere destructively', 
i.e. cancel each other when the summation is performed, even though 
they do not vanish individually. For instance, it can be shown 17 that 
the only Raman displacements for the rotational quantum number J 
in molecular spectra are AJ 0, 2; the displacement AJ^^l is 
impossible, even though there be Q (AJ = 0) as well as P, E branches 
(AJ 1) in the absorption or emission spectra. For a harmonic 
oscillator the cancellation of the various terms in (6) (individually of 
the form kn = 0, 2) is so great that there is no Raman effect. The 
purely nuclear motions in diatomic molecules are to a first approxima- 
tion simple harmonic. Hence the observed Raman displacements in the 
vibrational quantum number v in such molecules (usually Av 1) 
must owe their origin to 'intermediate states' I" which represent 'elec- 
tronic' rather than just vibrational excitation. When there are electron 
transitions, the vibrational selection rules are governed by the Franck- 
Condon principle, and are more complicated than for the harmonic 
oscillator. This principle sometimes allows large transitions in v in 
electronic absorption bands, but the interference effects are such 18 that 
the only appreciable Raman lines are those for which At; = 1 ('funda- 
mental') or Av = 2 (first harmonic). The latter should be much fainter 
than the former, and neither of them nearly as intense as the Rayleigh 
line Av = 0. 

17 E. C. Kemble and E. Hill, Proc. Nat. Acad. 15, 387 (1929) ; this article contains an 
excellent survey of the theory of the Raman effect. 

18 C. Manneback, Naturwissenschaften, 17, 364 (1929); Zeits.f. Physik, 62, 224; 65, 
574 (1930) ; J. H. Van Vleck, Proc. Nat. Acad. 15, 754 (1929). 



366 BRIEF SURVEY OF SOME RELATED XIII, 83 

83. The Kerr Effect 

When a static electric field E' is applied (besides, of course, the periodic 
field E of the incident light), the medium no longer has isotropic 
refractive properties. Instead, it becomes birefringent and the index of 
refraction n has a different value n when E' is applied perpendicular 
to E than that n n when it is applied parallel thereto. The existence of 
this difference, i.e. influence of an electric field on dispersion, is known 
as the Kerr effect, and has been investigated in quantum mechanics 
by Kronig 19 and by Born. 20 The derivation of the Kramers dispersion 
formula (1), though not particularly difficult, would require us to 
develop the quantum mechanics of non-conservative systems. This is 
our main reason for omitting all mathematical analysis in the present 
chapter. We must, however, mention that once Eq. (1) is granted, the 
calculation of the Kerr effect is a straightforward, though rather tedious, 
piece of static perturbation theory. One simply uses in (1) not the 
amplitudes and frequencies for a free molecule, but rather those appro- 
priate to a molecule in a constant electric field E' . These can be found 
by means of Eq. (37) and other relations of 34-5, treating E' as a per- 
turbation parameter. The system perturbed by E' becomes in turn the 
unperturbed system for calculating the polarization and attendant 
dispersion (1) produced by the periodic field E. If one makes the usual 
hypothesis, that the molecule's frequencies are all either small or large 
compared to kT/h, it is finally found that 

(7) 

where c , c lt r, 2 are complicated sums of matrix elements involving also 
the incident frequency V Q . Calculation of explicit numerical values for 
c > Ci, c 2 would be very tiring, if not difficult, and so the quantum 
mechanics of the Kerr effect has as yet yielded little more than classical 
theory. 21 The constant c 2 vanishes if the molecule is non-polar or mon- 
atomic. The constant c x also vanishes for atoms in states devoid of 

19 R. de L. Kronig, Zeits.f. Physik, 45, 458 (1927); 47, 702 (1928). 

20 Born and Jordan, Elemcntare Quantenmechanik, p. 259. 

21 Wo do not attempt to include any of tho classical theory of tho Korr and Faraday 
effects, or tho experimental work. A good survey of this is given by Laden burg in the 
Muller-Pouillot's Lehrbuch tier Physik, llth od., vol. ii, second half, Chaps. XXXV1-XL. 
Much of the experimental work, especially in tho case of the Kerr effect (except Stuart*s 
recent data 23 ), is for liquids rather than gases, and then there is the complication of 
possible association. Also tho Clausius-Mossotti corrections, which we have omitted, 
then become important. 



XIIT, 83 OPTICAL PHENOMENA 367 

angular momentum. 22 The Kerr effect should hence usually be larger 
and vary more rapidly with temperature in polar molecules. It is indeed 
found experimentally that n\n\ varies less rapidly with temperature 
than l/T in non-polar molecules (also in the polar ones bromo- and 
chlorobenzol). On the other hand, it varies more rapidly than l/T in 
the polar materials chloroform and ethyl ether, showing clearly the 
effect of the term c 2 . It is particularly striking that recent experiments 
of Stuart 23 on the Kerr effect in gases show that after reduction to 
constant density riftn\ is very nearly proportional to l/T 2 in ethyl 
chloride and methyl bromide (polar), and to l/T in carbon disulphide 
(non-polar); in these cases the non-vanishing term of highest order in 
l/T (c 2 in polar, Cj in non-polar molecules) thus has a preponderant 
influence. The second-order dependence on the field strength E' de- 
manded by (7) as well as by all earlier theories is, of course, confirmed 
experimentally. 

84. The Faraday Effect 

If the applied field is magnetic rather than electric, one has formulae 
analogous to (7) with W replaced by H and with c 2 = unless the 
molecule is paramagnetic. This is the Cotton-Mouton or Voigt effect, 24 
already discussed, like the Kerr effect, in 31 for the static case or 
limit v Q = 0. Because of the second-order dependence on //, it is hard 
to measure, and the experimental data are rather meagre. 

Far more important is the fact that a magnetic (but not an electric) 
field applied parallel to the direction of propagation of the incident light 
produces a rotation of the plane of polarization. This is the so-called 
Faraday effect, which is of the first rather than the second order in H. 
The rotation in a length x is thus given by a formula of the form 
= liTJx. The factor of proportionality is called the Verdet constant, 
and is the same as Trv (n + n_)/Hc, where n_, n + are the refractive 
indices for left- and right-handed circularly polarized beams. Ele- 
mentary classical theory based on Larmor's theorem and a rather too 
simple atomic model yields the so-called Becquerel formula 25 

e = CHxv Q ^ with = ,. (8) 

dv Q 2mc 2 

22 Wo are unablo to agroo with the statement on. p. 267 of Born and Jordan's Elenien- 
tarc Quantenmechanik that c vanishes for all typos of atoms. 

23 H. A. Stuart, Zeits.f. Physik, 55, 358; 59, 13; 63, 533 (1929-30), especially p. 538 
of 63. 

24 A survey of existing experimental work will appear in Professor Cotton's paper in 
the report of the 1930 Solvay Congress. 

25 H. Becquorel, Comptes Rendus, 125, 679 (1897). 



/(, L)\p(n;n')\* L(L+l)\ 
v(n' \nY-vl MT /' V ' 



368 BRIEF SURVEY OF SOME RELATED XIII, 84 

Faraday Effect in Atoms. A quite complete quantum-mechanical 
treatment in the monatomic case has been given by Rosenfeld. 26 His 
calculations use many of the same general sorts of perturbation devices 
as in our preceding chapters, especially frequent use of the principle of 
spectroscopic stability and the measuring of multiplet intervals relative 
to kT. The results reduce to simple forms only in limiting cases, which 
we denote by (a), (6), (c). 

(a) Multiplet width small compared to kT and incident light well 
outside the multiplet. By the latter condition we mean that v v(n f ; n) 
is large in magnitude compared to the size of the multiplet, so that 
there is no especially small denominator or large 'resonance effect* for 
one particular multiplet component. Here Rosenfeld finds that 

0- 

The terms of (9) which are respectively independent of and inversely 
proportional to the temperature are usually, following Ladenburg, 27 
called the diamagnetic and paramagnetic parts of the Faraday rotation, 
but this is not to be construed as meaning that they have opposite 
signs, for this is not necessarily the case. The factor / in the second 
term has the value 3/4(^+1), 3/4Zr, or 3/4:(L 2 +L) according as the 
change L'L in the azimuthal quantum number in the transition 
n->ri is 1, 1, or 0. As n 2 l~2(nl), comparison with (3) show^ 
that the diamagnetic part of the rotation is given exactly by Becquerers 
formula (8). In this particular case the spin anomaly has thus com- 
pletely cancelled out, a result previously found in the old quantum 
theory by Darwin. 28 The need of adding a paramagnetic term to that 
given by Becqucrel's formula was stressed by Ladenburg. 27 As it con- 
tains the factor L(L-\-l), this term disappears if the atom is in an S 
state, regardless of whether there is a spin paramagnetism. 29 

(b) Incident light very close to resonance with one particular multiplet 
component. 30 Here we may omit the refractive effect of all lines but 

8 L. Rosenfeld, Zeits. f. Physik, 57, 835 (1930). 

27 R. Ladenburg, Zeits. /. Physik, 34, 898 (1925). The possible existence of a para- 
magnetic term appears also to have been intimated in previous work of Drude, Becquerel, 
and Dorfmann. 28 C. G. Darwin, Proc. Roy. Soc. 112A, 314 (1926). 

29 The explicit expressions for / contain a factor L in the denominator in the case of 
tho transitions L>Tj 1 and L >L, but this occasions no difficulty even in S states 
(L = 0), as the amplitudes p(w;n') vanish for these transitions if L = 0. States of nega- 
tive L are, in fact, non-existent, while the non-existence of the transition L = -> // = 
is a well-known selection rule. 

80 We, however, suppose throughout that the incident light is well outside the 
Zeetnan pattern, i.e. that v(n';) 1> is large in magnitude compared to Hej&irmc. 



XIII, 84 OPTICAL PHENOMENA 369 

this particular component. The expression for the rotation proves 
to be 31 



r AvK?"; nj) 

- 



SkT 

Because it contains a second rather than a first power of v(rif',nj) 2 vf } 
in the denominator, the diamagnetic part gives greater resonance than 
the paramagnetic, and so predominates except at very low temperatures. 
The paramagnetic part is present whenever the atom has a magnetic 
moment, and, unlike the previous case (a), remains even in an S state 
if the latter has a spin moment. Since in the present case one term in 
the summation in (1) has much the greatest resonance, Eq. (1) shows 
that vtflnjdvQ is proportional to v^l(v(n'j'\nj)^v^f. Hence the form of 
dependence on frequency for the diamagnetic part is such as to ensure 
the validity of Becquerel's formula (8), but in general with an ano- 
malous value of G', viz. e/ 1 /2mc 2 . The expressions f lf / 2 , YJ in (10) are 
functions of the quantum numbers which are too complicated for us 
to give explicitly, but we may mention that / x can be computed in an 
elementary manner 32 in which it is only necessary to consider the per- 
turbing effect of the magnetic field on the frequencies and not on the 
amplitudes. In all other cases, e.g. (a) and (c) where there is less 
resonance to one particular component, it is vital to consider also the 
perturbations in amplitudes; neglect of this fact has led to many 
erroneous articles in the literature. The anomalous factor / x proves to 
be just the ratio of the mean Zeeman displacement for the various 
transverse components, weighted according to their intensity, to the 
normal Lorentz value He/47rmc. The most extensive experimental 
measurements for the present case (b) appear to be those of Kuhn. 33 
From the observed rotation he is even able to deduce the Einstein 
probability coefficients. 

(c) Multiplet large compared to kT, incident light outside the multi- 

31 The existence of the part of (10) involving the factor y^ also the third term of (11), 
which is of similar form as regards dependence on v , appears usually to be overlooked in 
the literature. This sort of term in, roughly speaking, the parallel of the part Nat of the 
susceptibility which is independent of temperature (cf., for instance, Eq. (1) of Chap. 
IX). Except in exceptional cases it will hence be of subordinate importance compared 
to the term of 'Curio form' which is inversely proportional to temperature. 

38 For typical explicit calculations see the following reference to Kuhn. 

33 W. Kuhn, Math. Phys. Comm. Dan. Acad. vii. 12, 11 (1926). 

3595.3 B b 



370 BRIEF SURVEY OF SOME RELATED XIII, 84 

plet. Here the dependence of on T and v is of the form 







and Becquerel's formula is not in general valid even with an anomalous 
value of G. 

Faraday Effect in Molecules. Here an adequate analysis is wanting, 
although the beginnings of a theory for diatomic molecules have been 
made by Kronig. 34 About all one can say is that the dependence on 
v and T is of the general form (11), assuming one is not close to 
resonance with any one line. The second term vanishes in a non- 
paramagnetic state. It is a curious fact that the rotation for many 
molecules is represented quite well by a formula of the Becquerel form 
(8) but with an anomalous value of (7. The anomaly in C usually ranges 
from 0-50 to 0-70, but for H 2 it is 0-99, so that the unmodified Becquerel 
formula applies almost perfectly to hydrogen. 35 Eq. (11) gives a more 
complicated dependence on v than (H) even with an anomalous value 
of C. The ability to represent many molecules approximately by the 
latter probably means that the second and third terms of (11) are 
usually small compared to the first, and that a group of absorption 
lines having nearly equal values of v(n'\ri) have a predominant effect 
on the rotation. Oftentimes the dispersion and rotation arc measured 
in the visible, while the lowest absorption lines are in the ultra violet; 
then Eq. (3) is not greatly different from a dispersion formula with only 
one assumed molecular frequency. 

Experimental Confirmation of Paramagnetic TermSaturation Effects. 
At very low temperatures the paramagnetic part, if present, should 
preponderate. This is confirmed especially well in the measurements of 
Becquerel and de Haas 36 on mixed crystals (tysonite and xenotime) 

34 R. do L. Kronig, Zeits.f. Phy.tik, 45, 508 (1027). 

35 For an excellent compilation of the experimental values of the constant C see 
Darwin arid Watson, Proc. Hoy. Soc. 114A, 474 (1927). Those writers find that the 
behaviour of oxygen is anomalous, and its rotation cannot oven bo represented by a 
formula of the general form ( 11), at least if one assumes that there is only one important 
absorption frequency v (n'; n). The experimental work is also well surveyed in Laden- 
burg's article already cited. 21 In this article it is emphasized that in media in which the 
infra-red vibration bands are known to contribute appreciably to the dispersion, a 
formula of the form (8), even with an anomalous C, is found to be applicable only if one 
replaces dnfdv by dn'/c/v, whore n' is the part of the refractive index not arising from 
these bands. It is also then necessary to insert a correction factor (w-f-2) a ro7(w/-[-2) 2 n. 
whose origin is closely related to that of the ordinary Clausius-Mossotti correction in the 
case of static dielectric constants ( 5). See pp. 2150, 2163 of Ladenburg, J.c. ai 

86 J. Becquerel, Le Radium, 5, 16 (1908) ; Becquerel and Onnes, Leiden Communications 



XIII, 84 OPTICAL PHENOMENA 371 

containing rare earth atoms among the ingredients. We saw in Chapter 
JX that from a magnetic standpoint the rare earth ions behave as if 
free even when in solid compounds. Because of a factor v(n'-,n) 2 vf } 
rather than [v(n'- 9 n) 2 v%] 2 in the denominator, the paramagnetic rota- 
tion, unlike the diamagnetic, should change sign on passing through an 
absorption band, and this is verified experimentally. 37 The theoretical 
proportionality to l/T is found to hold only approximately. In the 
case of tysonite the deviations from this law are not great (about 10 per 
cent.) down to 20 K., but the rotation at temperatures of liquid helium 
is about one-third less than one would expect if it were inversely pro- 
portional to the temperature. In xenotime the measurements show 
quite definitely that the rotation does not involve H and T only through 
the ratio H/T. 

At very low temperatures one encounters the complication that the 
Faraday rotation is no longer linear in H, and instead saturation effects 
begin to enter, as ^HjlcT is no longer small compared to unity. This 
saturation is indeed found by Becquerel and de Haas 38 at liquid helium 
temperatures. From the curvature of the saturation curves information 
can be deduced concerning the apparent Bohr magneton numbers. 
Unlike the case of susceptibilities, such information cannot be deduced 
from the initial slope, as the numerical values of the amplitudes and 
hence the absolute magnitudes of the right sides of Eqs. (9) and (10) 
are unknown. It can bo shown tha*t when saturation effects arc con- 
sidered, the paramagnetic parts of the rotations in cases (a) and (b, c) 
become, as we would expect, proportional to B L (pLH/kT) and 
Bj(g,jJ^H/kT) respectively instead of being linear in H/T as in Eqs. (9), 
(10), and (11). Here B(y) denotes the 'Brillouin function' defined in 
61. The elements responsible for the rotation in the mixed crystals 
tysonite and xenotime are cerium and gadolinium respectively, at 
least at the wave-lengths used by Becquerel and de Haas. In view of the 
Hund theory of the rare earths ( 58) one should expect the saturation 

103 ; Bocquerel, Dimes, and do Haas, ibid. no. 177 ; Becquerel and do Haas, ibid, no 193, 
or Zeits.f. Physik, 52, 568; 57, 11 (1929); also further references cited in note 40. 

37 J. Bocquerel, Phil. Mag. 16, 153 (1908). The change in sign is found at ordinary 
temperatures, but with very low values of T he finds that the absorption band soems to 
separate into two components such that the rotation is positive on both sides of one 
component, and negative on both sides of the other. The phenomena at low temperatures 
thus seem to be more complicated than contemplated by the usual simple theory. As the 
diamagnetic part of the rotation is necessarily positive, at least in atoms, the observation 
of a negative rotation in certain cases, notably TiCl 4 , must mean that in these instances 
the influence of the paramagnetic part is quite appreciable. 

38 Becquorel and de Haas, Zeits.f. Physik, 52 , 678 ; 57 , 1 1 ( 1 929), or Leiden Communica- 
tions 193, 204. 

Bb2 



372 BRIEF SURVEY OF St)ME RELATED XIII, 84 

curves for tysonite and xenotime to be proportional to B^l&flH/lkT) 
and J5 7/2 (7/tf//&T) respectively. Actually, at the temperatures of 
liquid helium they are found to be proportional to B^H/kT) and 
JB J/2 (7/tf//&T) respectively. As noted by Becquerel and de Haas, 
Schiitz, 39 and especially Kramers, 40 one obtains the empirical curve for 
tysonite if one assumes that the inter-atomic fields are so powerful as 
to quench the orbital angular momentum and leave only the spin free. 
The theory of 73 shows that sufficiently large unsymmetrical fields 
will do this, but in our opinion any explanation by this mechanism is 
purely spurious. Because of the larger multiplet widths, even greater 
fields would be required than in the iron group (cf . especially Eq. (4), 
Chap. XI), whereas the close conformity of the susceptibilities of the 
rare earths to the gaseous theory undeniably evidences that the /shells 
in rare earth ions are remarkably free. Instead, the clue to the satura- 
tion curves in tysonite is, we -believe, to be found in the investigation 
of the distorting effects of fields, probably not of axial symmetry, which 
are larger than, or at least comparable with, kT at the temperature of 
liquid helium, but still very small compared to the multiplet structure. 
This only requires fields whose effect is of the order 10 cm.- 1 , and there 
is then no contradiction with susceptibility measurements at higher 
temperatures. 41 An explanation on this basis is now being attempted 
by Kramers in place of his original theory. 

As regards xenotime, Kramers 40 has shown that one obtains the 
empirical curve if one assumes that the spin & == \ of the gadolinium 
atom is subject to an inter-atomic axial field sufficiently large to make 
-3f#= 2 tne only normal state at the temperature of liquid helium. 
There is the obvious difficulty that such a field, if purely electrostatic, 

39 Schiitz, Kelts, f. Physik, 54, 731 (1929). 

40 H. A. Kramers, Proc. Amsterdam Acad. 32, 1176 ; 33, 959 (1929-30), also Kramers 
and Becquorel, ibid. 32, 1190 (1929); Beequerel, do Haas, and Kramers, ibid. 32, I20f> 
(1929). 

41 Susceptibility measurements just completed at Loideii (Communication 210c) by 
fie Haas and dorter show that in the case of CeF 3 the Weiss-Curio formula # O/(T-f A) 
holds quite well down to about 70 K, with A 62, and with a value of C which yields 
a magneton number 2 52 in excellent accord (1 %) with the Hund theory ( 58). The 
value of A and the temperature 70 at which the cryomagnetic anomalies begin to appear 
are, as we should expect, considerably higher than for the hydrated sulphates usually 
measured. As the composition of tysonite is (La, Co, Nd,-f Tr)F, it is more analogous to 
the fluoride of cerium than to the sulphate, and so the susceptibility measurements 
demand, rather than preclude, an external energy of about 50 cnr l . In short the ineasure- 
ments of the saturation rotation in tysonite are made at such low temperatures that 
one would hence expect great cryornagnotic anomalies, and so one cannot infer any 
contradiction between these measurements and the ordinary Hund theory of sus- 
ceptibilities. 



XIII, 84 OPTICAL PHENOMENA 373 

would have to be very large, as the energy coupling spins to electric 
fields is a second-order effect. 42 Possibly exchange forces between para- 
magnetic atoms and also inter-atomic magnetic forces have some 
influence; in this event the ultimate theory will be quite involved. At 
any rate the rotation measurements show indisputably that the spin 
of the gadolinium atom is much less free in xenotime than in hydrated 
gadolinium sulphate or gadolinium ethyl sulphate, where the suscepti- 
bility measurements confirm the gaseous theory remarkably well even 
at very low temperatures. 

Further Faraday measurements on other materials at very low" 
temperatures are in progress at Leiden. The saturation curves are 
apparently obtained more easily in this manner than by direct deter- 
minations of susceptibilities. Careful analysis of these curves should 
ultimately yield valuable quantitative information on the inter-atomic 
fields in rare earth crystals. 

Polarization of Resonance Radiation. As the Faraday rotation is 
linear in H, it is appreciable only at large field strengths. On the other 
hand, a small field, 100 gauss or so, often has a tremendous effect on 
the polarization of resonance radiation. An elaborate theory of the 
polarization of scattered radiation was developed by Heisenberg and 
others 43 in the old quantum theory by means of the correspondence 
principle, and the new quantum mechanics has justified the method. 
The large influence of a small magnetic field (also of the hyper-fine 
structure 44 ) on the polarization of scattered radiation does not contra- 
dict the principle of spectroscopic stability, though we have seen the 
latter demands that only exceedingly large magnetic fields appreciably 
distort the dielectric constant or index of refraction. This difference is 
because the principle in question imposes restraints on the total inten- 
sity but not on the polarization of the secondary radiation, except when 
there is excitation by isotropic rather than directed primary radiation. 

42 An additional difficulty is that S states aro without any paramagnetic rotary powar 
if 0110 neglects the influence of the inner quantum number J on the energy. However, 
the excited states of the Gd ion may well have such large multiplet widths that one is 
not justified in replacing v(n'j'; nj) by v(n'; n) and then the paramagnetic part can persist 
even in $ states. It still seems a bit suprising that the groat rotation found in Gd is thus 
duo entirely to distortion of the orbital motion by the spin. 

43 W. Hoisenborg, Zeits. f. Physik, 31, 617 (1925); for further references and dis- 
cussion see Ruark and Urey, Atoms, Molecules, and Quanta, pp. 353-60, or the writer's 
Quantum Principles and Line Spectra t pp. 171-9. 

44 A. Eliett, Phys. Rev. 35, 588 (1930). 



INDEX OF AUTHORS 



Abraham, M., 7, 9. 
Aharoni, J., 270, 271. 
Akulov, N. S., 311,335. 
Alombort, d', 7. 
Alexandrow, VY., 184. 
Allgeier, J., 62. 
Alphen, P. M. van, 92. 
Ampere, A. M., 6. 
Anderson, Annie I., 65. 
Arkel, A. E. van, 72. 
Aschkinass, K., 50. 
Atanasoff, J. V., 205. 

Back, E., 155, 174, 177; we alxo Paschen- 
Back effect. 

Badeker, K., 62. 

Badger, R. M., 48, 71. 

Baeyer, A., 81. 

Bahr, E. v., 48. 

Barker, E. F., 71. 

Barnott, S. J. and L. J. H., 155, 256, 300. 

Bartlett, J. H., 364. 

Barton, H. A., 269. 

Baudisch, O., 301. 

Bauer, E., 120, 266-9. 

Beck, J., 344. 

Becker, R., 311, 335. 

Becquerel, H., 367-9. 

Becquerel, J., 292, 370-2. 

Bergmann, E., 68, 74, 78. 

Berkman, S., 311. 

Bethe, H., 274, 287, 291, 292, 296, 297. 

Bewilogua, 76. 

Birch, J., 299, 300. 

Birge, R. T., 81, 209, 228, 269. 

Bitter, F., 92, 113, 131, 270, 271, 277, 359. 

Bloch, F., 319, 338, 339, 343, 346, 351-3. 

Bluh, O., 57. 

Bodareu, E., 16. 

Boer, J. H. de, 72. 

Boguslawski, S., 201. 

Bohr, N., 95, 97, 101, 105, 111, 123, 125, 
170, 171, 220, 227, 240, 248, 282. 

Bois, H. du, 245. 

Boltzmann, L. ; see Boltzmann distribu- 
tion. 

Borel, 279. 

Born, M., 34, 37, 38, 64, 105, 111, 124, 125, 
131, 136, 149, 162, 168, 169, 190, 194, 
213, 216, 217-19, 221, 222, 361, 362, 
366, 367. 

Bose, D. M., 286, 311, 312, 314. 

Bourgin, D. G., 47, 48, 52, 81, 200. 

Bourland, L. T., 311, 312, 315. 

Bowen, I. S., 247. 

Boyce, J. C., 62. 

Braurnnuhl, H. v., 61, 67, 70. 

Breit, G., 109, 110. 

Brillouin, L., 257-9, 325, 331, 332, 343, 
344, 371. 

Brouwer, F., 147. 



Briihl, J. W., 84. 
Buchner, H., 113. 
Burgers, J. M., 179. 
Burmeister, B., 48, 49. 

Cabrera, B., 228, 242-5, 248, 250-3, 285, 

301, 303, 305, 306, 311, 312, 314, 333. 
Capel, W. H., 254, 270, 271. 
Carrard, A., 344, 345. 
Carrelli, A., 239. 
Carst, A., 361. 
Cartwright, C. W., 71. 
Carver, E. K., 69. 
Chatillon, A., 298. 
Cheney, E. W., 54. 
Christy, A., 266. 
Clark, R. J., 64. 

Clausius, R., 14, 16, 56-8, 87, 88, 366, 370. 
Coblentz, W. W., 49, 50, 52. 
Cohen, E., 69. 
Cohen, Elizabeth, 351, 359. 
Collet, P., 302, 303. 
Compton, A. H., 155, 363. 
Compton, K. T., 82. 
Condon, E. U., 122, 130, 147, 149, 186, 353, 

365. 

Coppoolse, C. W., 267. 
Coster, D., 247. 
Cotton, A., 121, 367. 
Courant, R., 125. 
Crane, W. O., 239. 
Curie, P., 89, 92, 266, 269, 303, 332; see 

also Curie's law, Curio point. 
Cuthbertson, C. and M., 47, 49, 50, 53, 68, 

213, 222. 

Czemy, M., 48. 

| Czukor, K., 201. 

Daily, C. R., 114, 120. 

Dakern, J., 278. 

Dale, T. P., 54, 83, 84. 

Darwin, C. G., 25, 26, 158, 159, 176, 177, 
356, 357, 368, 370. 

Darwin, K., 175. 

Debyo, P., 28, 30, 55, 58, 60, 61, 64, 71, 
74, 76, 81, 82, 86-8, 110, 117, 118, 152, 
181, 257, 279, 289, 322, 331; see also 
Langevin-Dobyo formula. 

Dokor, H., 243, 245. 

Dennison, D. M., 47-52, 72, 152. 

Dewey, Jane, 204. 

Dickinson, R. G., 364. 

Dioko, G. H., 364. 

Dillon, R. T., 364. 

Dirac, P. A. M., 23, 26, 105, 130, 138, 
156-8, 161, 162, 169, 170, 190, 194, 195, 
211, 227, 229, 291, 316-21, 336, 340, 
342, 347, 349-51, 353, 357, 369, 361. 

Ditchburn, R. W., 239. 

Dorfmann, J., 335, 345, 346, 368. 

Drude, P., 42, 368. 



376 



INDEX OF AUTHORS 



Duperier, A., 242, 244, 245, 248, 250-3, 

301, 306. 
Dupouy, G., 121. 

Earnshaw, 41. 

Ebort, L., 45, 58, 59, 65, 68, 71, 72, 113. 

Eckart, C. H., 125, 170. 

Ehrenfest, P., 93, 109, 289. 

Ehrhardt, 76. 

Einstein, A., 155, 362, 369. 

Eisenlohr, F., 83, 84. 

Eisenschitz, K., 72. 

Ellett, A., 373. 

Engel, L., 68, 74, 78. 

Epstein, P. S., 204, 209, 213, 364. 

Errora, J., 56, 60, 74, 77, 120. 

Estermann, J., 62, 64, 73, 74, 106. 

Euckon, A., 51, 76. 

Ewing, J. A., 322. 

Fajans, K., 221-3. 

Falkenhagen, H., 88. 

Faraday, M., 5, 170, 279, 366-71, 373. 

Farquharson, J., 278. 

Formi, E., 26, 51, 131, 181, 208, 211, 336, 

339, 347, 349-51, 353, 357, 359. 
Foex, G., 310. 
Fogelberg, J. N., 80. 
Forror, R., 228, 344. 
Forro, Magdalena, 67, 68. 
Foster, J. S., 204. 
Fowler, A., 219. 
Fowler, R. H., 24-6, 172, 181, 284, 310, 

330, 344 7, 350. 
Franck, J., 365. 

Frank, A., 242, 243, 248 50, 253, 256, 278. 
Freed, S., 245, 247, 249-52, 285, 293, 299, 

302, 303. 

Frenkel, J., 7, 9, 122, 162, 335, 350. 

Friedel, E., 120. 

Fritts, E. C., 67. 

Frivold, (). E., 63. 

Frnmkin, A., 63. 

Fuehs, O., 50-2, 61, 69, 70. 

Fujioka, Y., 204. 

Gans, R., 32, 56, 57, 95. 

Gebauer, R., 211. 

Gerlaoh, W., 63, 64, 106, 112, 154, 161, 

239, 360. 
Gorolcl, E., 59. 
Ghosh, P. N., 68, 70, 76. 
Giauque, W. F., 259. 
Gibbs, W., 25. 
Gibson, G. E., 239. 
Gladstone, J. H., 54, 83, 84. 
Glaser, A., 110, 112, 1 13, 141, 210. 
Gordon, W., 130. 

Gorter, C. J., 254, 299, 303, 307, 372. 
Goudsmit, S., 155, 157, 158, 163, 165, 166, 

173, 194, 237, 238, 259. 
Gray, F. W., 278. 
Groschuff, E., 69. 
Grutzmacher, M., 59. 
Guillemin, V., 72. 



Gundermann, H., 88. 
Gupta, R. N. Das, 68. 
Gupta, D. N. Sen, 76. 
Guth, E., 179. 

Guthrie, A. N., 311, 312, 315. 
Guttinger, P., 170. 

Haas, W. J. do, 92, 155, 254, 270, 271, 280, 
281, 299, 303, 307, 309, 310, 370-2. 

Halpern, O., 179. 

Hammar, C. W., 112, 113, 209. 

HantzBch, A., 71. 

Hargreavos, J., 364. 

Hartol, H. v., 68, 72. 

Hartree, D. R., 208, 212, 222. 

Hasse, H. R., 205. 

Hassol, O., 63, 68. 

Havens, G., 275. 

Haynes, D., 80. 

Hector, L., 208, 209, 215, 223, 266, 279. 

Heiseiiberg, W., 105, 1 11, 124, 125, 129 31, 
143, 147, 158, 162, 163, 169, 170, 176, 
177, 194, 215, 217-19, 222, 237, 269, 283, 
304, 308, 310, 316 ff., 322 IT., 361, 362, 
364, 373. 

Heitler, W., 239, 266, 316, 337, 339, 346, 
349. 

Henri, V., 71. 

Horweg, J., 87, 201. 

Herzfold, K. F., 78, 209, 223. 

Housler, F., 308. 

Hevesy, G. v., 244, 248. 

Hoydwoiller, A., 213, 221. 

Hilbort, D., 125. 

Hill, A. E. Jr., 69. 

Hill, E., 170, 174, 175, 265, 365. 

Hill, R., 297. 

Hojendahl, K., 59, 61, 64, 68, 72, 77-80. 

Honda, K., 300, 307, 322, 347, 359. 

Honl, H., 170, 195. 

Howell, O. R., 297. 

Hubbarcl, J. 0., 59. 

Hubor, A., 279, 280. 

Hiickel, E., 76. 

Hund, F., 165, 237, 241 3, 245, 247, 248, 
253, 263, 284, 286, 314, 371, 372. 

Hylleraas, E., 205. 

Ikenmeyor, K., 208, 222, 223, 359. 
Jshiwara, T., 300, 302, 307, 308. 
Ising, E., 331. 
Isnardi, H., 56. 
Ittrnann, G. P., 290. 

Jaanus, R., 345. 

Jackson, L. C., 254, 303, 309, 310. 

James, C., 243-5, 248, 252-4. 

Joans, J. H., 5, 8, 25, 35, 95, 103. 

Jeffreys, H., 170. 

Jenkins, F. A., 220, 269. 

Jezewski, J., 120. 

Jona, M., 62. 

Joos, G., 221-3, 297. 

Jordan, P., Ill, 124, 125, 128, 131, 136, 

158, 162, 168, 169, 176, 177, 190, 194, 

237, 361, 362, 366, 367. 



INDEX OF AUTHORS 



377 



Kapitza, P., 250, 287, 330, 344-7. 

Rasper, C., 302. 

Kast, W., 120. 

Kautzseh, F., 88. 

Keesom, W. H., 59, 70. 

Kernble, E. 0., 365. 

Kennard, E. H., 155. 

Kerr, J. ; see Korr effect. 

Kikoin, I., 345. 

Kinsoy, E. L., 239. 

Kirkwootl, J. G., 205. 

Kiuti, M., 211. 

Klein, O., 290. 

Kliofoth, W., 63. 

Koch, J., 48, 49, 67, 69. 

Koenigsborger, J., 215, 222. 

Kopformarm, H., 361. 

Kornor, 81. 

Kossel, W., 242. 

Kramers, H. A., 55, 170, 178, 218, 266, 

287, 289, 290, 296, 297, 311, 348, 361-4, 

366, 372. 

Krchma, J., 59, 62. 
Krishnan, K. S., 63, 121, 309. 
Kronig, R. do L., 147, 152, 170, 183, 185, 

194, 195, 198, 220, 259, 263, 280, 366, 

370. 

Kroo, J. N., 100. 
Kuhn, W., 363, 369. 
Kulenkampff, H., 223. 

Ladonburg, A., 81. 

Ladenburg, R., 81, 239, 240, 302, 361, 366, 

368, 370. 
Lanczos, C., 211. 
Landau, L., 345, 353, 355-7. 
Lande, A., 105, 106, 155, 174, 175, 177, 

232, 234, 256. 
Landolt, H., 83. 
Lane, C., 359. 
Larigo, L., 61. 
Langor, R., 220. 
Langevin, P., 30, 32, 33, 42, 45, 85, 89-91, 

96, 99, 181 ff., 206, 226, 254, 257, 258, 

277, 331, 343, 355; see also Langovm- 

Dobye formula, &c. 
Laporto, ()., 236, 237, 245, 249, 284, 285, 

300. 

Lapp, E., 345. 
Lannor, J., 22, 23, 24, 90-2, 109, 206, 255, 

277, 354, 367. 
Loeuwen, J. H. van, 94-7, 99, 101, 102, 

104, 105. 

Logcndrc, A. M., 159, 160. 
Lohrer, E., 112, 113, 209, 267. 
Lcnnard-JoiiPH, J. E., 147. 
Lenz, W., 90, 289. 
Lortes, P., 64. 
Lovy, S., 361. 
Lindberg, E., 247. 
London, F., 266, 316, 337, 339, 346, 

349. 

Loomis, F. W., 239. 
Lorontz, H. A., 2, 6, 7, 13-15, 83, 84, 101 

156, 173, 176, 369. 



Lorenz, L., 15, 83, 84. 
Lowery, H., 53, 67, 68. 

McKeehan, L. W., 311, 322, 335. 

McLennan, J. C., 351, 359. 

Magri, L., 15. 

Mahajani, G. S., 311, 335. 

Mahanti, P. C., 68, 70, 76. 

Malsch, F., 88. 

1 Manneback, C., 55, 108, 147, 152, 153, 183, 
; 185, 198, 365. 

Mark, H., 70. 
1 Mason, M., 7, 12. 

Mathiou, 147, 148. 

Maxwell, J. C., 1, 2, 99, 100, 102, 353. 

Meggers, W. F., 54. 

Mensing, L., 147, 183. 

Meyer, L., 68, 76. 

Meyer, S., 242, 243, 245, 248, 252. 
, Miles, J. B., 59, 61, 68, 70. 
i Millikan, R. A., 247. 

Morgan, S. O., 62, 77, 80. 

Morse, P. M., 122, 130, 149, 186, 353. 

Mossotti, O. F., 5, 6, 14, 16, 30, 56-8, 87, 

88, 366, 370. 

j Mott-Smith, L. M., 114, 120. 
j Mouton, H., 121, 367. 
1 Mukhorjee, B. C., 68, 70. 

Muller, H., 65, 74. 

Mulhkon, R. S., 262, 263, 266, 269 279, 
280. 

1 Naeshagen, E., 68. 

Nando, S. M., 266. 
! Neumann, J. v., 169, 170. 

Niessen, K. F., 86, 119, 201, 258. 

Nusbaum, R. E., 239. 
l 

1 Occhialini, A., 16. 
i Ogg, E. F., 62. 

Okubo, J., 322. 

, Onnes, H. Kamerlingh, 90, 254, 259, 266-8, 
303, 308, 309, 370, 371. 

Oostorhuis, E., 254, 266, 267, 303, 308, 309. 
1 Ootuka, H., 239. 

Oppcnheimer, J. R., 147, 149. 
: Ornstoin, L. S., 120. 

Owen, M., 347, 359. 

Palacios, J., 305. 

Parts, A., 68. 

Pascal, A., 207, 222, 223, 278. 

Pauli, W., Jr., 26, 91, 106-8, 147, 156-9, 

161, 162, 165, 170, 177, 183, 184, 206, 

210-12, 237, 241, 277, 278, 283, 317, 319, 

338, 339, 349, 351, 353. 
Pauling, L., 107, 108, 114, 115, 117, 163, 

165, 194, 209, 211-15, 221-4, 252, 259, 

287, 359. 
Peierls, R., 337. 
Penney, W. G., 154, 281. 
Perrior, A., 254, 268, 279. 
Peters, C. G., 54. 
I Pfeiffor, P., 314. 
I Philip, J. C., 80. 



378 



INDEX OF AUTHORS 



Phillips, B., 51. 

Phipps, T. E., 161. 

Piccard, A., 266-9, 280, 281, 344, 345. 

Plancherel, M., 25. 

Podolsky, B., 122, 363, 364. 

Pohland, E., 70. 

Pohrt, G., 61, 62, 68. 

Polanyi, M., 239. 

Potzsch, W., 87, 

Powell, F. C., 310. 

Prandtl, L., 245. 

Pringsheim, P., 239. 

Prokofjow, W., 364. 

Rabi, 1. 1., 152, 309, 310. 

Rademacher, H., 152. 

Raman, C. V., 51, 63, 93, 121, 364, 365. 

Ramanadham, M., 121. 

Rasotti, F., 364. 

Ratnowsky, S., 87. 

Rayleigh, Lord, 103, 364, 365. 

Reiche, F., 152, 303, 363, 364. 

Reicheneder, K., 222, 223. 

Richards, T. W., 69. 

Richardson, O. W., 155. 

Rideal, E. K., 63. 

Ritz, W., 205, 210, 215, 216, 219. 

Rojansky, V., 203. 

Rolinski, J., 68. 

Rosonbaum, C. K., 69. 

Rosenbohm, E., 301, 302. 

Rosenfold, L., 170, 368. 

Rosenthal, A., 25. 

Rosenthal, J., 220. 

Rotszajn, S., 303. 

Ruark, A. E., 122, 170, 174, 204, 373. 

Rubons, H., 50. 

Rubinowicz, A., 364. 

Ruody, R., 351, 359. 

Hunge, C., 358. 

Russell, H. N., 170, 195; see also Russell- 

Saunders coupling. 
Rutherford, Lord, 30, 41, 104. 
Rydberg, J. R., 179, 215-17, 278. 

Sack, H., 62, 65, 74, 78, 88, 221. 

Salceanu, C., 121. 

Sandor, S., 68, 74, 78. 

ganger, R., 45, 52, 53, 61, 64, 69, 71, 72, 76. 

Schaefor, C., 51. 

Schaffer, S., 301. 

Schay, G., 239. 

Scherer, M., 121. 

Scherrer, P., 270, 271. 

Schlapp, R., 203. 

Schrodinger, E., 4, 100, 105, 112, 122, 124, 

125, 127, 130, 131, 147, 148, 156, 169, 

218-21, 316, 361. 
Schumb, W. C., 69. 
Schiitt, F., 59. 
Schutz, W., 372. 
Schwingol, C. H., 62, 70, 79. 
Sellmeier, W., 42, 361. 
Sexl, T., 179. 
Shaefer, S., 301. 



Sircar, S. C., 68. 

Slater, J. C., 109, 205, 208, 215, 241, 252, 

286, 319-21, 338, 346-9. 
Smallwood, H. M., 78. 
Smekal, A., 364. 
Smedt, J. de, 70. 

Smyth, C. P., 62, 64, 68, 74, 76, 77, 80, 83. 
Sommerfeld, A., 122, 130, 150, 156, 157, 

170, 176, 195, 234, 236, 242, 267, 284, 

286, 317, 329-31, 339, 351, 352, 361. 
Sone, T., 209, 266, 269, 275, 279. 
Spangenborg, K., 221, 222. 
Spedding, F. H., 247. 
Stark, J. ; see Stark effect. 
Statescu, J., 48, 69. 
Steiger, O., 45, 53, 61. 
Stern, O., 63, 64, 106, 112, 154, 161. 
Stevenson, A. F., 364. 
Stewart, A. W., 71. 
Stewart, G. W., 120. 
Stewart, O. S., 248. 
Stinchcornb, G. A., 71. 
Stoner, E. C., 92, 110, 155, 208, 222, 223, 

229, 255, 257, 266-8, 285-7, 297, 298, 

300, 301, 303, 306, 322, 333, 335, 336, 

339, 344, 345. 
Stoops, W. N., 62, 74. 
Stossel, R., 267, 268, 270, 271. 
Stranathan, J. D., 68, 76. 
Stuart, H. A., 45, 50, 51, 53, 61, 68, 70, 

366, 367. 

Sucksmith, W., 249, 255, 256, 301, 359. 
Sugiura, Y., 217, 364. 
Swann, W. F. G., 7. 
Szarvassi, A., 25. 
Szivessy, G., 279. 

Takouchi, T., 208. 

Tangl, K., 16, 67, 209. 

Taylor, J. B., 161. 

Taylor, N. W., 275, 301. 

Toller, E., 325, 356. 

Terry, E. M., 322, 333. 

Thoodoridos, P., 300. 

Thiele, E., 239. 

Thomas, L. H., 162, 208. 

Thomas, W., 363. 

Thomson, J. J., 77, 95, 100, 345, 363. 

Tolman, R. C., 48, 109. 

Traubenberg, H. R. v., 211. 

Trumpy, B., 364. 

Tumrners, J. H., 55. 

Tyler, F., 335. 

Uhlenbeck, G. E., 155, 157, 158, 173. 

Unsold, A., 278. 

Urbain, G., 245. 

Uroy, H. C., 122, 170, 174, 204, 373. 

Van Vleck, J.H., 24, 37, 103, 111, 133, 147, 
170, 174, 183, 186, 198, 204, 242, 243, 
248, 249, 253, 265, 277, 278, 280, 357, 
365. 

Verdet, E., 279, 367. 

Voigt, W., 95, 176, 177, 367. 



INDEX OF AUTHORS 



379 



Waibel, E., 16. 

Walden, P., 68, 78, 79. 

Waller, I., 179, 204, 209, 211, 216, 217. 

Walls, W. S., 76. 

Wang, S. C., 210, 278, 279, 290. 

Wasastjorna, J. A., 221. 

Watson, H. E., 68. 

Watson, W. H., 370. 

Woatherby, B. 13,, 114, 120. 

Weaver, W., 7, 12. 

Weber, W., 6, 90. 

Webster, A. G., 34. 

Webster, W. L., 335, 347. 

Wodokind, E., 245. 

Weiss, P., 56, 207, 222, 228, 229, 242, 245, 
253, 268, 295, 299, 300, 302, 303, 307, 
310, 322, 331, 333-6, 339, 343-5, 347, 
372. 

Weissborgor, A., 75, 81. 

Woissenberg, K., 71. 

Wolo, L. A., 301. 

Welsbach, A. v., 245. 

Wentzel, G., 204, 209, 211, 217, 247. 

Werner, A., 71, 314. 

Werner, O., 68, 78, 79. 

Werner, S., 217. 

Werner, W., 59. 

Wettorblatt, T., 51. 

Weyl, H., 169. 

Weyling, 69. 



Whitelaw, N., 220. 

Whittaker, E. T., 20. 

Wien, Max, 88. 

Wiorl, R., 76. 

Wiersma, E. C., 254, 267, 270, 271, 308, 

337. 

Wigner, E., 169, 170, 203, 205. 
Willers, F. A., 358. 
Wills, A. P., 32, 95, 208, 209, 215, 223, 266, 

279. 

Williams, E. H., 243-5, 250-3, 308. 
Williams, J. W., 59, 62-4, 68-70, 72, 75-7, 

79-81. 

Wilson, H. A., 100. 
Wintsch, F., 60. 
Wolf, A., 114, 120, 336. 
Wolf, K. L., 16, 51, 70, 73, 209. 
Woltjer, H. R., 90, 254, 259, 267, 307, 

308. 

Wood, R. W., 364. 
Wredo, E., 106, 161. 

Zadoc-Kahn, J., 121. 

Zahn, C. T., 46, 48, 50, 52, 53, 59, 61, 67, 

68, 70, 76, 81, 107, 199, 200. 
Zeeman, P. ; see Zocman effect. 
Zoner, C., 215. 

Zornicke, J., 243-5, 248, 252-4. 
Zoellor, H., 311. 
Zwann, A., 175. 



SUBJECT INDEX 



Absorption intensities, 47, 50, 51, 362. 

Additivity relations, 57, 60, 77, 82-5, 207, 
220-5. 

Alkalis: diamagnetism and refractivity of 
ions, 222-5 ; magnetic susceptibility of 
solid, 347, 348, 359, 360; of vapour, 
238, 239 ; quantum defect, 217 ; Zeomaii 
effect, 179. 

Aluminum spectrum: resonance pheno- 
menon, 219. 

Ammonia molecule, 52, 71. 

Anglo and action variables, 37, 146. 

Angular momentum: characteristic values 
of, 165 ; commutation rules for, 161 ; 
Fourier series for, 171 ; matrices for, 
159-62, 167-72 ; behaviour in molecules, 
262-4, 274, 277-81; quantization of, 
106, 150, 164-5; quenching of orbital, 
273-4, 287-97, 314, 324, 334; relation to 
magnetic moment, 4, 155, 174, 255, 256, 
300; true v. canonical, 23-4, 145, 354; 
we also spin. 

Anisotropy, magnetic (in crystals), 309- 
11. 

Association, molecular, 17, 57, 58, 65, 294, 
314; see also complexes. 

Atom-core, polarization of, 215-20, 222. 

'Atomic polarization', 46, 51, 52, 68. 

'Austasoh', sec exchange effect. 

Axis of quantization, 108, 114, 115, 140, 
196. 

Azimuthal quantum number, 159, 166, 
203-4, 238, 354. 

Band spectra: relation of dielectric con- 
stant to, 45-53, 72, 280. 

Benzeno molecule and derivatives, 72-81. 

Birefringence, see Kerr effect. 

Bismuth, diarnagnetic susceptibility of, 92, 
93. 

Boltzmann distribution formula, 24-6, 96, 
99, 102-4, 181, 196, 353, 355. 

Brillouin function, 257-9, 325, 331, 332, 
343, 371, 372. 

Canonical: coordinates, 25; transforma- 
tion, 138. 

Carbon bonds, 72, 76, 84. 
C0 a molecule, 49-52, 70 ; CS 2 , 70 ; CC1 4 , 71. 
Centrifugal expansion, 200. 
Characteristic: functions, 123; values, 123, 

143. 
Charge: * effective', 45, 47-52, 81, 200; 

electronic, 4; statistical density of, 129, 

131, 354. 

C1O 2 molecule, paramagnetism of, 275. 
Chromium salts, 285, 301, 304, 306-8. 
Classical theory and statistics, 1-104; 

absence of magnetism in pure, 94-102; 

inadequacy of, 102; quantum analogies 

with, 169-72, 184. 



Clausius-Mossotti formula, 14, 16, 56, 87, 

370. 
Cobalt: pure metal, 335, 336; salts, 285, 

297-9, 302, 304, 309. 
Commutation rules, 119, 127-8, 161, 166, 

258, 323. 

Complexes, 293, 295, 301, 306, 314; com- 
plex atoms, 301, 304, 349. 
Conduction electrons, 6, 100-2, 336, 339, 

346, 349-60. 

Contact transformation, 25, 138. 
Continuity, equation of, 8, 10, 130. 
Copper salts, 285, 297, 299, 300, 304. 
Correspondence principle, 169-72, 107, 

109, 110, 357, 361, 362, 373. 
Corresponding states, Weiss theorem of, 

334. 
Cosine law, 162, 165, 174, 177, 230 2, 236, 

318. 
Coupling: j-j, 168; Rufisell-Saundors, 

163-8, 173, 178, 229, 288; spin-orbit, 

163, 237, 264, 267, 282, 295, 313, 317, 

348; spin-spin, 311, 163, 237, 283, 318, 

335 ; typos (a)-(b) in molecules, 263. 
Cryornagnetic anomalies, 249, 307-8, 333, 

372. 
Crystalline fields, 287-97, 309-11, 314-15, 

372, 373. 
Crystals, 2, 309-11, 323; liquid, 120; 

mixed, 370-3; single, 335. 
Cubic symmetry, 291, 311, 324, 337, 338. 
Curie's law, 89, 239, 249, 253-4, 267, 269 ; 

Weiss modification, 245, 253, 288, 300, 

303-9, 314-15, 333, 372. 
Curio point, 326, 331-3, 336-9, 344-8. 
Current density in quantum mechanics, 

130. 

Degenerate electron gas, 349-52, 357-9. 

Degenerate systems, 134-7, 112, 124, 182, 
280; nearly degenerate, 137, 175, 179; 
non-degenerate, 132-4; invariance 
theorems in, 139-42; relation to sym- 
metry of field, 290-7, 273, 274 ; required 
with odd no. of electrons, 296, 297, 348. 

Density, dependence on, see Clausius- 
Mossotti formula, dilution. 

Diagonal: matrices, 127, 142-3, 171, 172; 
matrix elements, significance of, 129, 
143; sum (invariarico of), 142, 177, 342. 

Diamagnotism : of atoms and ions, 90-4, 
99, 206-15, 220-5; of free electrons, 
100-2, 353-60; of molecules, 276-9; 
pressure dependence, 110, 113; dia- 
magnetic correction to paramagnetic 
susceptibility, 227, 243, 252, 359; 
diamagnetic Zoeman term, 178-80. 

Diatomic molecules: electric moment, 27, 
66; magnetic moment, 262-72, 276-81; 
spectral notation, 262; dumb-bell and 
top models, 147-55, 183-5, 32-7, 115. 



SUBJECT INDEX 



381 



Dielectric constant, 1 ; of atoms, 27, 203-5, 
209-25; determination of molecular 
moments by, 60-9; effect of magnetic 
field on, 113-21, 190; relations to mole- 
cular structure, 70-82; relation to 
refractive index, 42-53, 60; saturation, 
85-8; of solutions, 56-60; see also 
Langeviii-Debye formula. 

Diffraction, electron arid X-ray by single 
molecules, 76. 

Dilution, magnetic, 244, 255, 259, 268, 
283, 294, 304-5, 308, 309, 314, 321. 

Dipole moment, determination of numeri- 
cal values, 60 82. 

Dipole, models for rotating, see dumb-boll, 
top. 

Directive effect, magneto-electric, 182, 
279 81. 

Dispersion, see refractive index. 

Doubling: A-type, 154, 203, 280; p-typo, 
266, relativity, 157, 161, 203; see also 
multiplets. 

Dumb-bell model, 106, 147-52, 183-5, 196, 
198. 

Effective: charge, 45, 47-52, 200; mag- 
neton number, see magneton numbers. 

Electric moment, see moment, electric. 

Electrons: diamagnctism of free, 1002, 
353-60; Dirac theory of, 156-8; 
'equivalent', 165, 237, 241; feeble spin 
paramagnotism in conductors, 349-53, 
359