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OSMANIA UNIVERSITY LIBRARY
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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, interatomic 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 nonmathematical, 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 LANOEV1NDEBYE FORMULA
10. Polar versus Non polar Molecules ...... 27
11. Rudimentary Proof of the LangeviiiDebye Formula . . 30
12. More Complete Derivation of the LangevhiDebyo Formula . 32
13. Derivation of a Generalized LaiigevinDebye 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 InfraRed 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. QUANTUMMECHANICAL 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. RussellSaunders 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 SecondOrder Zeeman Term . . .178
VII. QUANTUMMECHANICAL DERIVATION OF THE LANGE
VLNDEBYE FORMULA
44. First Stages of Calculation . . . . . . .181
45. Derivation of the LangevinDebye Formula with Special Models 183
46. General Derivation of the LangevinDebye Formula . . 186
47. Limit of Accuracy of the LangevinDebye 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 AtomCore 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 LuiigeviiiDebye 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 LangevinDebye formula % N (a + Q ^r Usually a
\ OrCL 1
arises from induced polarization or diamagnetic induction.
j8 = Bolir magneton 09174 x 10" 20 e.m.u. ( = 495 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  477 x 1()" 10 e.s.u.,w = 904 X 10~ 28 gm.
Jf = Hamiltonian function (to be distinguished from the magnetic field H).
K = 'molar polarizability ' 4:nLP/3NE.
L Avogadro number 6004 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 unitvolume. (B H \knM.}
p, permanent moment of the molecule. (On pp. 117 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 unitvolume ( 2706:: 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 unitvolume. (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 unitvolume (electric or magnetic). Xmnl~ ^X/^ sus "
ceptibihty per gramme znol.
Expressions in boldface 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 boldface type are
vectors. Between the four field vectors there exist the socalled 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 nonhomogeneous 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 186173. This electrical origin implied that
by probing down to subatomic 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 4770 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 nonhomogeneous media the
term 'number of molecules per c.c.' is to be understood to mean the
' numberdensity', 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 twentiethcentury 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) 060 092 117 118 120 126 140 146 160 X 10  8 cm.
a (Kinetic Theory) 112 135 182 183 191 190 230 231 278 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
EgradO , B^curlA;
C 8t (14)
1 da
e ~ grad<  , h = curl a.
c ot
The differential equations for determining the potentials are
=4 w [pdivP], nA==~~[i+ 1ccurlMl;
^ (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 wellknown 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(tKi/ 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 righthand 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. 5712.
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 Intelchangeability 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^curlMMxgracl^^ = 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 vectordifference 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/rgrad \dv
r 1 1 r r r r / i \ / i\i
2c inxgrad = \p vlrgrad 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(AC)C(AB).
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
Pgradi. (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 OlausiusMossotti 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 needleshaped cavity whose axis is parallel to E,
and D E+4?rP within a slabshaj)ed cavity whose surfaces are per
pendicular to E. The term 4?rP/3 in (34) is sometimes spoken of as the
* intermolecular 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 socalled LorenzLorentz 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 1484 4213 6924 9616 12304 14953 17627
l^^XlO 7 1953 1947 1959 1961 1961 1956 1956 1953
p n 2 2
7XlO 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 1415 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 intermolecular 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 wavelengths 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
nonpolar 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
'ClausiusMossotti' correction resulting from the difference between the
macroscopic and local fields, one has only to substitute 3(e l)/4?r(e f2)
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
socalled 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 intermolecular 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
'intramolecular' 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 socalled
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,Fe,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 secondorder equations
(41) are equivalent to the 677 firstorder 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 nonhomogeneous
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^je^^/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^
il
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 socalled 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 eWdq 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/eW 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 wellknown 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 socalled 'orgodic hypothesis', which supposes that all assemblies
having tho saino energy have the samo lifehistory, 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 FermiDirac statistics in the now quantum
mochanicfi do indeed show that only configurations colresponding 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 FermiDirao interference effects unimportant, and so wo use Boltzmann
statistics throughout except in 801.
II
CLASSICAL THEORY OF THE LANGEVINDEBYE 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 nonpolar. 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 nonpolar. 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 nonpolar if highly
symmetrical, as in e.g. CC1 4 , CH 4 (see 1920). Monatomic molecules
are always nonpolar, 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 LANGEVINDEBYE FORMULA II, 10
the electric susceptibility Xe == (* l)/4rr is given by the expression
=*(*+ )
*
which we shall call the LangevinDcbye 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 nonpolar 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 LANGEVINDEBYE FORMULA 29
and negative charges will then no longer coincide, as they did before
the field was applied if the molecule is nonpolar. 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 antiparallel 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 nonpolar 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 LANGEVINDEBYE 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 LangevinDebye formula. (In the electric case,
a formula such as (1) is commonly called just the Debye formula, but
we use the compound title LangevinDebye 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 LangevinDebye 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 LANGEVTNDEBYE 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 LANGEVINDEBYE 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 LaiigcvinDebye 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 LANGEVINDEBYE 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
LangevinDebye 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 rederive (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 LANGEVINDEBYE 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 LANUEV1NDEBYE 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 underdeterminant 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 LANGEVINDEBYE 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 LANUEVINDKBYE 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 ajZkT, 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 LangevinDebye Formula 8
We shall now deduce a very general form of the LangeviiiDebye
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 ANGEVINDEB 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 2component 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 righthanded 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 LANGEVINDEBYE 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^Pr' 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 timeaverage 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 LANGEVINDEBYE 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 LangevinDebye 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
LangevinDebye 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,
ITO, 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 LANGEVINDEBYE 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 LangevinDebye 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 LangevinDebye formula derived in Chapter II, but it will first
be necessary in 1418 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
wavelength. That n is really a function of frequency or wavelength
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. VV1I.
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 nonisotropic, 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
wavelengths 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 nonconservative 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 wavelengths. 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
wavelength 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 ultraviolet, 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 __ _
., 4rrNiJi 2 ,.
e == 3
as the LangcvinDebye 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 Infrared 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
infrared 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 internuclear
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 infrared 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 infrared, 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 infrared 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 infrared 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 = 0001040
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 = 0000871.
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 internuclear distance. For
a nonpolar diatomic molecule, for instance, e efl is zero. One method
of determination of e Ga is by measurement of the absolute intensities of
infrared 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 infrared
intensity measurements are probably those of Bourgiii 7 for HC1, who
finds that here e off 086 x 10~ 10 e.s.u. Introducing this value of e cjl and
the values m off  162 X ]0~ 24 , v vib = 882 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 17 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 onefifth the
charge 477 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 17x10 4 in HC1, for instance, the effective
charge would have to be about 87 x 10 ' 10 instead of 086 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 infrared (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 0949 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 086 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 infrared). 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
onethird the standard value 103 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 onehalf 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 infrared. 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 onesixth 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 infrared 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 infrared 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 infrared are
rather limited in number. Koch 14 measured the refractive indices of
2 , H 2 , CO, 00 2 , and CH 4 at 670tyx and at 8078/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 wavelength as long as 1319/4,
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
infrared, as was undoubtedly to be expected since nonpolar diatomic
molecules have no pure vibration spectra. The value 1000332 which
he found for the index of refraction n of CO at both 67/4 and 87/4
was slightly lower than that in the optical region (e.g. 1000335 at
0589/4) and agreed quite well with the value 1000327 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 465/4, so that Koch's measurements extended beyond
the vibrational singularity. The slight discrepancy between 1000332
and 1000327 may be merely experimental error or perhaps indicate
that there is a very small contribution 0000005 of the atomic polariza
tion to n 1 or 000001 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 09e, whereas Dennison 8 estimated
013e 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 wavelengths:
A 10 20 30 40 50 67 87 11 1319 ^
(n I)xl0 4 ^ 441 434 418 289 632 484 458 447 400
The behaviour is thus different from that given by an ordinary optical
dispersion formula which takes no cognizance of infrared resonance
points, and which predicts a steady and very gradual decrease of n from
its value 1000449 at optical wavelengths (NaD lines) to a value
1000441 at infinite wavelengths. 18 The anomalies shown by the table
at 40 and 50/4 are due to the influence of the vibration band at 43/4.
The abnormally low value 1000400 of n at the longest wavelength
18 B. Burmeister, Verh. d. D. Phys. Ges. 15, 689 (1913).
17 W. W. Coblentz, Investigations of Infrared 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 ultraviolet regions.
3595.3
50 DIELECTRIC CONSTANTS, REFRACTION, AND THE III, 15
1319jLt is undoubtedly because of another known vibration band at
149/i. Reference to curve III, p. 44, shows that the index of refraction
should be abnormally low on the short wavelength or highfrequency
side of a resonance point, but high on the long wavelength side. Hence
if measurements were available beyond 149/x, they would record a
value of n considerably larger than the value 100044 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
infrared as well as optical region. The characteristic feature of his
formula is that, besides the ordinary terms due to resonance with visible
or ultraviolet frequencies, it contains two terms of the form (4) in which
the resonance wavelengths are taken as respectively 4 31 /A and 14 9 I/A,
and in which the effective charges are taken to be 228e and 06 le,
where e = 477 X 10" 10 e.s.u. Pcimison 8 deduced an effective charge of
only 009e for the 149^ 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
149 band in CO 2 are too small by a factor no less than (061/009) 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 wavelengths. An ordinary dispersion formula such as
that of Cuthbertson, which includes no atomic polarization, yields
n ^_ 1^1000882, whereas Fuchs's formula with the two infrared
resonance points yields ri^l = 1000975. The value which Zahn 21
finds for the dielectric constant of C0 2 under standard conditions is
1000968, while Stuart 22 finds 1000987. 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 018 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 infrared
resonance points. It is well known that the OO a molecule has numerous other
infrared vibration bands besides those at 43/>t and 149/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 272/*.
In his dissertation (Upsala, 1924), not available to tho writer, Torston Wetter
blatt is reported to have explored tho dispersion in the vicinity of 272/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
infrared 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 149 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 43/A vibration of CO 2 ; (d) one of the various fundamental vibrations
in a polyatomic molecule usually has a longer wavelength 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 wavelengths 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 infrared
dispersion measurements available for methane (CH 4 ), viz. n = 1000419 at
6557fA and n 1000450 at 8678/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 77^, tho effective charge
must be roughly 020e, 26 again larger than the effective charge 0095e deduced by
Dennison 8 from the absorption measurements by Coblentz. An effective charge
020e for this vibration, will remove about onetenth of the discrepancy between
ganger's value 27 000096 for c 1 and the value 000086 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
infrared resonance points besides that at 77/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 1034 to 106 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 000087 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 100399 rather than 100416. 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 03 per cent, in the moment, a change clearly
within the experimental error, will diminish Zahn's 21 value 0000768 for 47rA/ r a to
a value 0000729 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 084 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 228, but Stuart suggests that a more
accurate value is 220. 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 003 X 10' 18 ,
K O == 259, while Stinger iinds p  1 15 00 1 X 1 0" 18 , /c 261 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 39 or 41 as compared to 16 with the old data. Both Hanger and Stuart
consider that even a discrepancy 39 is larger than the experimental error in K O .
which they consider to be about 15. They therefore make the traditional sugges
tion that the refractive extrapolation is in error because of infrared 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 0300 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 PaleGladstone 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 wavelength region 7500 8700 A, the temperature coefficient of n I
for air is exactly the samo as that 0000367 of the density, but that the former
coefficient increases in numerical magnitude to 0000387 when the wavelength
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
DaleGladstone law which they find at 2500 A arc perhaps because air has an
absorption band in the ultraviolet. 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 infrared. 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 wavelength were so very long
as to be in the short radio rather than far infrared 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
deorient 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 wavelengths 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 infrared
and ultraviolet.
18. The Dielectric Constants of Solutions
A pure polar liquid cannot in general be treated by the standard
LangevinDebye 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 ClausiusMossotti expression E+47rP/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 intermolecular 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 WeissGans 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. 12730.
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 nonpolar
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 ClausiusMossotti
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 nonpolar and the solute polar, the
LangevinDebye 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  = 264 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 nonpolar 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.
f0 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 ClausiusMossotti relation.
From the various polarizabilityconcentration 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 nonpolar substances the molar
40 P. Dobye, Polar Molecules, pp. 467.
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/(ef2)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 3869 and 4395 respectively
for O 2 and N 2 gas, while the corresponding values in the liquid state
are 3878 and 4390, 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 wavelengths, 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 nonpolar materials, (b) binary
mixtures of either polar or nonpolar materials in which the optical
refractivity rather than static dielectric constant is investigated, and
at sufficiently short wavelengths 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 nonpolar
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 gaslike 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 righthand 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 nonpolar 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 wavelengths, 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 meltingpoin t . hi the case of ice, for instance, the < lielcctric constant is near tho
meltingpoint 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 freezingpoint 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 46 at 2 0. when the wavelength is 8 kilo
metres. The corresponding molar polarizability nevertheless still has an excessively
high value 10, as 2 C. is too near the meltingpoint 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 LangevinDebye 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 infrared 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 nonpolar 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 DipoleMoment, 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 coworkers.
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
vacuumtube 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. 478 that the electric
moment can be directly calculated from absorption coefficients for the
pure rotation spectra in the far infrared 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 surfacelayer 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 SternGerlach
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
104 X 10 18 and 166 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 17 X 10" 18 and 020 X 10' 18 for SO a and CO 2
respectively.
66 Rideal, Surface Chemistry, pp. 2367, 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 BornLertes 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 questionmarks 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 SternGerlach
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. 1918) 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
nonpolar.
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 nonpolar. For this reason the finite
moments recorded for Br 2 and I 2 are hard to believe. The moment
040 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 030. 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 nonvanishing 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 boldface, 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 )
040'2
10? 78
Pure Br 2
Iodine (T 2 )
12? 88
Benzene
,,
10? 74
,,
O 74
Hexano
Hydrochloric acid (HC1)
103
106 78
Hydrobromic acid (HBr)
079
080 78
Hydriodic acid (HI)
038
041 78
Carbon monoxide (CO)
012 53
014 78
(012)
012 75
V79
,,
O10 45
013 78
(Oil)
Carbon dioxide (CO 2 )
014? 53
m
,,
006* 45
017 78 (0)
,,
000 5
018 78 (0)
Nitrous oxide (N 2 O)
Q 80
013 78
,,
GU
017 78
Carbon disulphide (CS 2 )
80,45
0. 06 * 58
Benzene
f .
Q60
028 78
008* 58
Hexane
Sulphur dioxide (SOo)
161 45
163 78
Water (H 2 O)
184 49
181 81
. .
181 82
Benzene
Hydrogen sulphide (H 2 S)
MO 53
092 78
093
094 78
Ammonia (NH 3 )
1.4888
147 78
,,
144 45
144 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
018 78
Methylchloride (CH 3 C1)
169 85 ?
189 86
186"
190 78
,,
189 M
Methylene chloride
159 6
155 74
Benzene
(CH 2 C1 2 )
,,
161 87
Chloroform (CHC1 3 )
095
11 ?V
105 57
Hoxane
10585
110 58
Benzene
ijr> 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
122 88
Benzene
,,
114"
121 78
115 90
j.22 56 ' 58
n
115
124 58
CC1 4
112 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
129 51
133 78
132 47
137 78
ft
123 86
Propyl other (C 3 H 7 ) 2 O
086 47
124 86
t
Acetone (CH 3 COCH 3 )
294 51
288 51
27(> M
CC1 4
297 8
272 89
Benzene (C 6 H 6 )
. .
033 ? 8
006
CC1 4
,,
o' 9 ' 1
Pure benzene
,,
010 58
CS 2
tt
008 M
Hoxatio
Fluorbenzene (C 6 H 5 F)
139 92
Benzene
Chlorobonzene (C 6 H 5 C1)
152"
155 58
Hexano
,,
152 58
CS 2
,,
155 58
Benzene
)t
156 93
tt
ft
,
m
157 74
tt
tt
161 57
t)
Hoxano
Bromobenzene (C 6 H & Br)
156"
Beiizono
>f
151 5t
>}
152 74
lodobeiizene (C 6 H 5 I)
;;
130 9S
"
tf
125"
ff
Nitrobenzene (C 6 H 5 NO 2 )
389 58
Hexane
,,
389 58
cs a
390 58
Bonzono
)f
,
384 M
ft
Hexane (C 6 H 14 )
. .
010 94
}J
,,
008 9 *
CC1 4
,,
0"
Pure Hexane
Kthyl alcohol (C 2 H 6 OH)
170 05
172 86
.74 96
Benzene
,,
. .
63 58
CC1 4
Methyl alcohol (CH 8 OH)
168 95
173 8
64 96
Benzeno
. 67 96
CC1 4
nPropyl alcohol (C 3 H 7 OH)
166 95
53 5
Benzene
isoPropyl alcohol
.7590
Benzene
isoArnyl 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 nonpolar, 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 infrared data of Koch, is 1000273 ; tho corresponding values of the
dielectric constant are 1000273 (Tangl, Ann. dcr Phyaik, 23, 559, 26, 59 (19078),
1000263 (Fritts, Phy*. Rev. 23, 345, 1924), 1000259 (Braummihl), 53 1000265 (Zahn). 45
For nitrogen, tho value of ri^ ranges from 1000580 to 1000589 according to the
observer, while Fritts finds e 1000555, and Zahii obtains e 1000581. For oxygen
n^ is very approximately 1000530 (Lowery 29 ), while = 1000507 (Fritts), 1000518
(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 infrared 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 boldface 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 infrared 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^OOSx 10~ 18 .
The agreement of the latter with Sanger's value 184 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 muchmooted 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) Xray 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 infrared 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.
Bandspectrum 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 nonpolar. 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. Stiiuhcomb 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 nonpolar 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
startingpoint 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
nonpolar 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 19, 28, 30, 08, 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 nonpolar 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 nonpolarity 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 COC 2 H 5 , CH 3 COC 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 COC 2 H 5 ,
C 3 H 7 COC 3 H 7 , (CH 3 ) 3 COC(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 271 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 27x 10~ 18 e.s.u. than
that of a free CO atom (OlxlQ 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 25 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, 16 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 nonpolar. 110 Errera 111 has investigated the
various isomers of acetylene dichloride and finds that the cis and
socalled asymmetrical forms have moments 1*89 and ll8xlO~ 18 re
spectively, while the trans form has no moment. More recently Miiller
and Sack 74 find 174 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 meltingpoint 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 typeform
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. 'Disubstituted' 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 nonpolarity see I)ebye, Polare
Molekdn, pp. 5960 (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
phydroquinone diethyl ether and phydroquinone diacetate have
electric moments l7x!0~ 18 and 22xlO~ 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 pC 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 Xray 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 Xray 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 coworkers. Some typical results for the dichloro
benzenes are shown in the following table:
oC 6 H 4 01 2 mC 6 H 4 C! 2 pC 6 H 4 C! 2
p, o}a 225 x 10 18 148 x 10 18 (04 X lQ 18 e.s.u.)
/d(Eq.!2) 205 153
^ calc (S.&H.) 213 142
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 pstates.
The value employed for / is 153 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 halogensubstituted 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.
oC!G 6 H 4 N0 2 mClC 6 H 4 N0 2 p01C 6 H 4 NO 2
/* 01)S 425X10 18 338XKH 8 255 X 10~ 18 e.s.u.
Mcaic 4 '78 326 211
/z ww 378 318 236
The values taken for / and /' in (12) are the moments 164 and
375 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 476, 330 and 225 x 10' 18 for the calculated
values for the three isomers of chloronitrobenzene, and 361, 410 and 420 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 043 X lO" 18 and 375 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 / = +043 X lO" 18 , I' = 375 X 1Q 18 (or else
1= _ 043 XlO 18 , /'^+375xlO 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
oCH 3 C G H 4 NO a mOH 3 C 6 H: 4 N0 2 pCH,0 6 H 4 NO a
/*obs(H6j) 364 X 10 18 431 X 10 18 e.s.u.
jLt calc 356 398 418
jLt w>a . 375 420 450
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 39 28 17 15 15  13 12 09 (04 +15
fi X 10 l8 (Cl 3 a) 3t 10 20 19 10
/tXl0 18 (Uaj 18 10 0S 04 15
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, nonpolar 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 pnitraniline (O 2 N C 6 H 4 NH 2 ), for
instance, use of (12) with the values of /, /' given in the table yields
15 (39) = 54 x 10~ 18 , whereas the observed value 64 is 71 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 socalled 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 threedimensional 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 128 X 10~ 8 cm., 134 and hence er is 6 11 X 10" 18 e.s.u., whereas the actual
moment is only 103X 10" 18 . 135 The reason the actual moment is so small is that
131 The moments found are of tho order 02 X 10' 18 or less, which may be considered
virtually zero. Symtrinitrobonzono seems to have a real electric moment of approxi
mately 08 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 103 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 internuclear distance r and the effective charge 086X 10' 10 found by Bourgin
from spectral intensities for the infrared 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 ' overpolarizes ' 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 = 128 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 HjCl, for
recent developments in the quantum mechanics seem to show that the valence
in HC1 is perhaps more nonpolar 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 = 09e deduced from infrarod dispersion (see 16) is correct, for then
e eft r = 5 X 10~ 18 , whereas \L 01 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 socalled
'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 bookreview 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 18624. Like many other investi
gators in this field, he defined the molecular refractivity by means of
the GladstoneDale 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^SOO, ^=130, ^ = 300, (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 132 360 212 516 364 820 258
&' obB 132 361 211 616 362 821 261
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)aiid4nn.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 GladstoneDale rather than the LorenzLorentz 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 272 421 350 392 386 461 429 603 632 682
jfc' ob 271 422 351 393 387 450 422 501 632 688
The compounds with three double bonds are of the socalled aromatic
type. The contributions of the various atoms are taken by Briihl to
have the following values slightly different from (14):
^=486, 4^=129, A^290, A" =20. (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 13 which must be assumed for hydrogen in applying
the rules is not the same as either the actual refractivity 25 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 31 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 26, 31 aro the values of (nl)Mjp corresponding to the
theoretical dielectric constant 1000225 for atomic hydrogen (see 48) and the measured 78
dielectric constant 1000273 of H 2 . The first resonance linos of both H and H 2 are so
far in the ultraviolet that the refractivitios in the optical region differ very little from
those for infinite wavelengths.
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 nonpolar. 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 nonpolar 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 DE curve rather than as the ratio D/E of its ordinate and abscissa.
If we assume that the local field has the ClausiusMossotti 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 righthand 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 = 430, N = 583 X 10 21 , p,  114 X 1Q 18 , and (18)
reduces at T = 293 to dD/dE ^ 430 025 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 430. 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
120 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 ClausiusMossotti 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 nonpolar, 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 nonpolar 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. Malsrh, 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 IXXI 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 wellknown 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. KamorliiighOnnos, Leiden Communications, 167 c (or Versl.
Amsterdam Akad. 32, 772, 1923).
IV,23 CLASSICAL THEOKY OF MAGNETIC SUSCEPTIBILITIES 91
nonmonatomic 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 twothirds 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 meltingpoint its atomic susceptibility has the constant value
73 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 interatomic
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. 3724 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 faW ^...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 kTdfajeWydpj. 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 faewr]^ (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 78. 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, nonferromagnetic 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 nonvanishing 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 245 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 'ultraviolet 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 nonconvergence 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 semitheoretical derivation of his celebrated
formula for this effect, but besides introducing the anomalous factor
two in the magnetic moment of what he called the atomcore (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 Jf .
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 bestknown 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 wellknown experiments of
Gerlach and Stern 3 on the deflexion of atoms in a nonhomogeneous
magnetic field. (The field must be nonhomogeneous 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 nongyroscopic rotating dipoles (the socalled 'dumbbell' 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 SternGorlach 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 154. 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  1034X 10' 18 e.s.u.
154 '; Whole quanta, 1921  0481 X 10' 18
457 Half quanta, 1925 ; 0332 xlO' 18
J , New mechanics, 1926 ! 1034X10' 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 dumbbell model would, for instance, furnish
a different C than 154 or 457. 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 nongyroscopic dumbboll
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 'dumbbell'
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 140X 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 nongyroscopic 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 15 times the classical
or highpressure 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 PaschenBack 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 PaschenBack
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 PaschenBack 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 SternGerlach 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
nonexistent. 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 nonexistence 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 nonpolar 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 001 to 01 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 firstorder Zeeman effect but only
a secondorder 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. MottSmith 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 04 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
halfquantum 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 nongyroscopic, as will bo soon in Eq.
(64) of 37.
21 L/M. MottSmith 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 MottSmith 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 = 457 already tabulated on p. 107 by assuming halfquantum 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 Jf 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> P2> /*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 righthanded 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
nonexistence 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 = 54l.
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
crossproduct 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 nonpolar, 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* = 09 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 = 137 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 quantummechanical
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 = 541, 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 Xrays. 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. 1501.
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 Stherer, ibid. 190, 700 ; Salceaunu,
ibid. 190, 737; 191, 486; ZadocKahn, ibid. 190, 672; general survey by Cotton in the
proceedings of the 1930 Solvay Congress.
VI
QUANTUMMECHANICAL FOUNDATIONS
32. The Schrodinger Wave Equation
Schrftdinger's equation 1
is today 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
noncom 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 singlevalued, 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 QUANTUMMECHANICAL 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.hWnh  0, Jt*.*KW 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 boldface 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.* . hW*. .) dv. (5)
124 QUANTUMMECHANICAL 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 righthand 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 nondegenerate 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 socalled 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 QUANTUMMECHANICAL 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 socalled
SturmLiouvillo type is given in CourantHilbert, 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 QUANTUMMECHANICAL 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 fg.*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 noncommutative, 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 fifijri, 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 QUANTUMMECHANICAL 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 (1213) 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 QUANTUMMECHANICAL 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^VAf0* , 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(fggf) is Hermitian if/, g are. The electrical and magnetic
moments are necessarily Hermitian, since
PSX*. ^
8 P. Jordan, Zeits.f. Physik, 37, 383 (1926); cf. also Podolsky, I.e.*
VI,33 QUANTUMMECHANICAL FOUNDATIONS 129
with analogous formulae for m y , m s and since therefore no noncommutative
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 selfadjoint 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 wellknown 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. Nondiagonal
elements are important only as intermediaries to the calculation of the
130 QUANTUMMECHANICAL 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 QUANTUMMECHANICAL 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 QUANTUMMECHANICAL 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 shorthand of matrix notation, the totality of homo
geneous linear equations (19) for determining the S(n',n) arc equivalent
to the singlematrix 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.
Nondegenerate Systems. A dynamical system in quantum mechanics
is termed degenerate if two or more energy levels coincide. If the
original system is nondegenerate we may develop the coefficients
VI,34 QUANTUMMECHANICAL 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^nOStn^nO^^SWfn'snJSfn^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
wellknown 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 secondorder 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 QUANTUMMECHANICAL 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 QUANTUMMECHANICAL 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 nondiagonal 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 QUANTUMMECHANICAL 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 higherorder 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 higherorder terms). To obtain the complete solution we
choose the sums (32) as new unperturbed wave functions 0^. We may
then proceed as in a nondegenerate 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 loworder 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 QUANTUMMECHANICAL 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 energylevels, 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 nondegenerate 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 QUANTUMMECHANICAL 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 QUANTUMMECHANICAL 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 righthand 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 farreaching 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 LangevinDebye 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 QUANTUMMECHANICAL 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 'quantumallowed' 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 onethird 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 unitvector 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 QUANTUMMECHANICAL 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 onethird 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 QUANTUMMECHANICAL 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 PaschenBack 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 submatrix (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 nonvanishing
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 QUANTUMMECHANICAL FOUNDATIONS 143
mean that the lefthand 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 mfifin 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 QUANTUMMECHANICAL FOUNDATIONS VI, 36
To obtain specific formulae for W (1) , W (2) ... we use (24) and (26). We
assume that the system is nondegenerate 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 righthand 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 QUANTUMMECHANICAL 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
secondorder term ,# (2) in (51) is because (2) contains a nonvanishing
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. 224, 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^ejs' 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 QUANTUMMECHANICAL 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 noncommutative
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 QUANTUMMECHANICAL 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 'endoverend' rotation of a diatomic
molecule by using a simplified, idealized model, sometimes called the
'rigid rotator' or c dumbbell 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 threedimensional wave equation (54) is, of course, like that
of the spherical pendulum. Despite the very simple form of the twodimensiona lequation ,
its characteristic functions cannot be expressed except in series, as there are no closed
L2
148 QUANTUMMECHANICAL 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 endoverend rotation may be expected to resemble that of
a dumbbell 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 dumbbell. Let 0, < be the usual polar
coordinates specifying the position of the axis of the dumbbell 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 sm0J
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 twodimensional 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 threedimensional dumbbell model.
23 E. Schrodinger, Ann. der Physik, 79, 748 (1926).
VI,37 QUANTUMMECHANICAL FOUNDATIONS 149
and we obtain the first two terms of (54) on assuming that dift/dr = 0, suggested
by the fact that the internuclear 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
quantummechanical 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 endovercrid 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 singlevaluedness,
&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 (lf.)" / '+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 dumbbell' is not necessarily
to be taken as representing accurately an actual molecule.
150 QUANTUMMECHANICAL FOUNDATIONS VI,37
normalized to unity. If we use the usual Ssymbol, 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 sidestep 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 QUANTUMMECHANICAL 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 bandspectroscopy. 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) + (jm+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 nonvanishing 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^ll)/STr*I. By
(62) and (63), Eq. (26) yields
(65)
The case j = requires special consideration. Here the summation
reduces to a single term/ =^41 = 1. There is no term/ = j 1 = 1
as states of negative j are nonexistent, 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 QUANTUMMECHANICAL FOUNDATIONS 153
which has been used to represent molecular rotations is that of the
socalled 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 dumbbell, 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 dumbbell 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 noncollinear 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 QUANTUMMECHANICAL 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 bandspectra.
Stark Effect. Eqs. (64) and (71) reveal an important distinction:
namely, that there is no firstorder 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 groundlevels, whereas measurements of dielectric con
stants test only the statistical average of the energies of the various
component rotational states of the groundlevel only. In other words,
Starkeffect measurements will isolate individual values of the quantum
numbers j, m, whereas susceptibility determinations will not. The
technique of Starkeffect observations in molecular spectra is, of course,
a difficult one. The secondorder 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 firstorder
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 SternGerlach effect 29 will
produce only very small deflexions in molecules. It must be mentioned
that besides the secondorder term (65) or (72) due to the permanent
dipoles, there is also another secondorder 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 firstorder effect in such molecules should, however, appear only when the energy
due to the external electric field is large compared to the socalled 'Atype doubling'.
See 70. This restriction does not appear in Eq. (71), as the model is too simple to take
account of the Adoubling phenomenon. If h&v(j) denote the width of the Adoublet,
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 semimechanical model or by means of Dirac's 'quantum
theory of the electron'.
In the semimechanical 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 EinsteinRichardsondeHaas 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 QUANTUMMECHANICAL FOUNDATIONS VI, 38
prior to 1926, by assuming that the atom contained a rather mysterious
'atomcore' (Atomrumpf) whose ratio of magnetic moment to angular
momentum has the value (73). This mystical 'atomcore' now turns
out in reality to be the spin.
Besides the arbitrary character of its postulates, the semimechanical
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 Xray doublets in heavy atoms. The interaction of the spin
with external magnetic fields, which is our particular concern, is handled
perfectly well by the semimechanical 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 semimechanical 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 secondorder
Schrodinger wave equation by four simultaneous firstorder 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 semimechanical 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 antiparallel. 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 semimechanical model except for terms too small to be observable.
VI, 38 QUANTUMMECHANICAL FOUNDATIONS 157
senting an electron of negative mass. If Dirac's quartet of wave func
tions were separable into two noncombining 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
quantummechanical 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 presentday quantum mechanics, 35 and very
likely will not be cleared up until the longawaited 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 firstorder 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 semimechanical 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 semimechanical 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 semimechanical 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 UhlenbeckGoudsmit 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 QUANTUMMECHANICAL FOUNDATIONS VI, 38
matrix elements of the spin, which in the semimechanical 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
semimechanical 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 purematrix theory is too
generally ignored.
We shall present the semimechanical 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 singlevaluedness, 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 UhlenbeckGoudsmit model
as 'semimechanical' 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 multiplevalued
wave functions, but this appears a little fortuitous. Pauli 37 later showed
that the difficulty of multiplevaluedness 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 QUANTUMMECHANICAL 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 multiplevalued 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 nonvanishing 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 righthand 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 QUANTUMMECHANICAL 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 . . /Zrridz, &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 QUANTUMMECHANICAL 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 SternGerlach 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 multipletstructure. 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 PaschenBack 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 fis compared to the new wif2;/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 noncommutativeness 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 QUANTUMMECHANICAL 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 socalled boundary
conditions on the matrices (zero probability of transition to nonexistent
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 boldface 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 QUANTUMMECHANICAL 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=\L8\, \L8+l\,..., L+S1, 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,..., J1, 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 RussellSaunders 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, oversimplified calculations.
44 For more complete discussion of tho various coupling possibilities in this socalled
'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 QUANTUMMKCHANICAL 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 antiparallel 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 nonvanishing 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 antiparallel 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 nondiagonal 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 QUANTUMMECHANICAL 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 RussellSaunders 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 QUANTUMMECHANICAL 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 socalled 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 QUANTUMMECHANICAL 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
RussellSaunders 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. RussellSaunders 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 RussellSaunders 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 RussellSaunders 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 RussellSaunders 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 nonvanishing 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+SL+l)x ] ( 88 )
, / x(J+L8+l)(J+M+l)(JM+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 QUANTUMMECHANICAL FOUNDATIONS VI, 40
PaschenBack 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 PaschenBack 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.
jj 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 toxtbooks (e.g. Born and Jordan, Elementare
Quantenmechanik, p. 150).
VI, 40 QUANTUMMECHANICAL 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 PaschenBack effect can be worked out for
jj coupling, but we ornit them, as this typo of coupling is much less common than
RussellSaunders coupling, especially in the case of the 'normal' or 'ground'
levels such as are involved in the study of magnetic susceptibilities. The jj
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 jj 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
jj 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 noncommutative algebra, and the derivations of (89)
and (90) by means of wave functions which have so far been published
all involve rather abstruse grouptheory 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 QUANTUMMECHANICAL FOUNDATIONS VI, 41
Eqs. (88) have seldom 61 * been explicitly given in the literature, but
are adaptations of general intensity formulae derived on semiempirical
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 semiempirical 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 QUANTUMMECHANICAL 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+^x7+^+^W+^^^^
/el   
172 QUANTUMMECHANICAL 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 RussellSaunders 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 QUANTUMMECHANICAL FOUNDATIONS 173
a, 6, c in (79), the problem is excessively complicated. 56 We shall there
fore henceforth assume that the atom has RussellSaunders 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 nonvanishing elements of the perturbative potential
are
; JM) .
' ' 47TWC_
; J+ 1 Jlf )  / /(J, M),
(98)
(Jl,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 jj and some limiting forms of coupling other than RussollJSaundors 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 QUANTUMMECHANICAL 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 firstorder term in (99) is the familiar Lande ^formula. 58 The
presence of second and higherorder terms is too commonly overlooked.
The secondorder 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 569). The secondorder 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 secondorder 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 energylevels 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 QUANTUMMECHANICAL 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 secondorder 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 089 Ml 102 100 096 104 096
9und 092 108 101 099 098 102 101
? uew 089 111 102 100 096 104 097
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 PaschenBack 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
energylevels 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 QUANTUMMECHANICAL 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 PaschonBack effect, the energylevels 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 nondiagonal 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 PaschenBack 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 QUANTUMMECHANICAL 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 firstorder terms in (99) by the
same quantummechanical refinement of the constant term of (95) as that already
discussed after (95). The firstorder 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 nondiagonal elements (88) and hence of the secondorder 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 thenondiagoiial 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 diagonalsum 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 socalled principle
of permanence of F and {/sums, 88 which was known on semiempirical grounds
before tho advent of quantum mechanics. The terms Fsum and psum 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. 0282. 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 QUANTUMMECHANICAL 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 cos0). 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
'dumbboll* 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 RussollSaunders
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 QUANTUMMECHANICAL 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 nonCoulomb central field if one uses in place
of n the socalled 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 firstorder displacement is 019 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 QUANTUMMECHANICAL 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
QUANTUMMECHANICAL DERIVATION OF THE
LANGEV1NDEBYE 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 4W (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 QUANTUMMECHANICAL 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^ ewjmiKi
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 intermolecular forces, for in the absence of all fields there
can be no preference between directions parallel and antiparallel to E.
In solids there are in reality important intermolecular 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 nonexistence 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 LANGEVINDEBYE FORMULA 183
45. Derivation of the LangevinDebye 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 'dumbbell', 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 firstorder 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 QUANTUMMECHANICAL DERIVATION OF VII, 45
analogue of the fact that with the dumbbell 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 dumbbell, 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 dumbbell, 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 dumbbell 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 fieldfree 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 LANGEVINDEBYE 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
LangcviriDebye 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 LangeviiiDebye formula for
the dumbbell 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 QUANTUMMECHANICAL 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 unitvolume 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 LangevinDebye formula
46. General Derivation of the LangevinDebye 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 
LangevinDebye 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 LangcvmDebye 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 QUANTUMMECHANICAL DERIVATION OF VII, 46
levels be small compared to kT. In Fig. 7 an interval such as bc must
be much less than kT. It is not necessary to demand that the energy
difference between two widely separated normal states, such as Orc 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 energydifferences of two widely separated
normal levels such as ac 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 LangeviiiDebye 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 LangevinDebye 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 energylevels 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 spacetime geometry. It is immaterial for the proof
how many frequencies are superposed, i.e. how complicated the motion.
The low and highfrequency elements will be found to contribute re
spectively the first and second terms of the LangevinDebye formula (10).
Thus the highfrequency elements of the moment matrix are responsible
for the induced polarization term, while only the lowfrequency elements
VII, 46 THE LANCEVINDEBYE 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 lowfrequency 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 nonpolar
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^ _ eir^kT /in
"M 1 " 2*^ Wnjmin'j'm 7 ) J ' ( '
190 QUANTUMMECHANICAL 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 lowfrequency part of p.
We shall therefore use * to denote the matrix formed from p by
dropping the highfrequency 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 { ciiumber', 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 LANGEVINDEBYE 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 highfrequency 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^WWI
(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 QUANTUMMECHANICAL DERIVATION OF VII, 46
please, as the separation of consecutive energylevels for the simple
dumbbell 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 LangevinDebye 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 LANGEVINDEBYE FORMULA 193
electric and magnetic fields, or intermolecular 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
LangevinDebye 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 intermolecular 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 _., , . . . ,., , xl ,
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
LangevinDebye 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
eirl/fcr (2 2)
hv(ri ; n)
,m.,j,/n) v '
Simplification of Lowfrequency Elements. From the rule for matrix
multiplication (Eq. (9), Chap. VI) it follows that
2
Here the righthand 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 highfrequency 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 QUANTUMMECHANICAL 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 LangevinDebye 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 ei^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 Highfrequency 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 'sumrules' 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 sumrules were first estab
lished on semiempirical 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 sumrule see Pauling and Goudsmit, The
Structure of Line Spectra, p. 137. The ordinary statement of the sumrule 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 sumrule 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 LANGEVINDEBYE 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 sumrules 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 sumrule 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
sumrules.
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 LangevinDebye formula. Combination of the simplifications
affected in the high and lowfrequency parts of (22) yields the complete
LangevinDebye formula (10).
lae of Kronig, Russell, Sommerfeld ami Honl (noto 52 , Chap. VI) for intensities in
multiplots. These formulae contain more information than the sumrules, 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 QUANTUMMECHANICAL 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 LangevinDebye 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 endoverend 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 dumbbell 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 firstorder 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 'lowfrequency 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 lowfrequency 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 LangevinDebye formula. Hence,
unlike the old quantum theory (31), a magnetic field cannot distort
VII, 46 THE LANGEVINDEBYE 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 higherorder 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 LangevinDebye 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 LangevinDebye 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 LangevinDebye 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 LangevinDebye 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 QUANTUMMECHANICAL 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 dumbbell 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 LangevinDebye 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 'dumbbell' 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
l32xl(H
( '
IT '
The factor lf(T) evidently enters in (31) as a correction to the
LangevinDebye 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= 265 x 10 40 gm. cm. 2 appropriate to HOI yields
/= 50/T, so that at room temperatures f(T) is only 001G. To make
the correction as great as 5 per cent, it would be necessary to reduce the
temperature to 100 K., or below the freezingpoint of HC1, which is
obviously not feasible. Gases which have low boilingpoints, and which
hence might be measured when the denominator of (31) is small, are
invariably nonpolar (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
dumbbell 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 LANGEVINDEBYE FORMULA 199
common slightly polar gases which have relatively low boilingpoints,
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
LangevinDebye 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 1034 x 10~ 18
to 1050X 10~ 18 , and correspondingly diminishes ^irNoL from 0*00104 to
000099, in somewhat better accord with the optical value 0000871
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 QUANTUMMECHANICAL 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 LangevinDobyo 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 HC1 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 nonrigidity 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 LANGEVINDEBYE 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 lowfrequency
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 infrared
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 highfrequency 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 LANGEVINDEBYE FORMULA VII, 47
our focal point for discussing infrared 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 LangevinDebye 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
LangevinDebye formula which arises from 'highfrequency' 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 firstorder 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 'Atype 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 lowfree iiioncy as well as no diagonal elements. There can bo no
lowfroquency 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 LangevinDebye 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 Zvalue 0, and hence no bothersome degeneracy
and no firstorder 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 nonhyclrogenic
atoms the effect of I on the energy is usually large, and there is no firstorder Stark
effect, except perhaps for a few excited states of very small quantum defect. The
kinematical meaning of this is that nonhydrogenic 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 semimajor 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
secondorder 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 borderline case of the Stark effect of nonhydrogoiiie
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 == 663 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 663 X 10~ 25 . The
corresponding numerical value of the 'molar polarizability' 47rL/3
introduced on p. 53 is K 168, 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 = 10000653 at C., 76 cm., while Basse's first calcu
lation gives 10000691 in excellent accord with the experimental value
10000693. This gratifying agreement, however, turns out to be rather
accidental, as in his second paper Hassc finds 1000079, using a pre
sumably more accurate unperturbed wave function J/T O . Still more
recently, a theoretical value 10000715 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 wellknown 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 timeaverage 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 hydrogenlike 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
= 2832X 10
Xmol
The normal state of atomic hydrogen has 7&=1,Z = 0, Z=l, and thus
its molar diamagnetic susceptibility is 237 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
079X10 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 nonhydrogenic 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 = 185 x 10~ 8 , in gratifying accord with Hector and
Wills' experimental value 188x1 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 ThomasFermi
theory the mean value of ] r 2 is e~ l JJJ pr 2 dxdydz, where the integra
tion is 3 rather than 3Zdimensional 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 346
per cent, of the charge at a distance greater than 2 06 A from the
nucleus. Better results than with the ThomasFermi charge distribu
tion are obtainable not only by the more refined Hartree selfcon
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 nonhydrogenic 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= 271 Zl fi volts, e 1 + 0000450 Z^, Xmol = 474X 10 Z^.
These may now be equated to the experimental values, 16 viz.
(He) W= 788, = 10000693, x mol = l'88x 10 6 ;
(H 2 ) W = 314, e  1000273, Xmo i = 394X 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 171(171) 159(110) 159(092)
H a 108 (108) 114 (078) 111 (063)
10 The value 10000093 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 1000273 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 51 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 1000225 of atomic H from tho Stark effect formulae of
Wentzel, Waller, and Epstein, and of its diamagnetism (3) from the quantummechanical
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 113 and 078. 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 = 0000050,
r* = (0528 X 10~ 8 ) 2 as compared to the new e 1 == 0000225, r* = 3(0528 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 secondorder Stark effect. 18
Application to Heavier Atoms. The most comprehensive and searching
application of Eqs. (1) and (3) to nonhydrogenic 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 PauliFermiDirac
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 subshell.
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 subshell, 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 nonCoulomb central fields, such as
Hartreo 14 has shown can be so chosen as to portray fairly well nonhydrogenic
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
firstorder Stark effect characteristic of a Coulomb field, also the part of the
secondorder effect which has hv(nl ;nZl) in the denominator in a nonCoulomb
central field, disappears on summing over a complete shell.
Consider now the effect of nondiagonal 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, wi 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 ~ 1000225 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 wavelength 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)
0391
0397
Ne (n = 2;
= 0,1)
445, 564
431, 650
Ar (n = 3;
= 0,1)
970, 1099
1111, 1240
Kr (n = 4;
= 0,1)
2128, 2292
2669, 2833
Xe (n = 5;
= 0,1)
3429, 3663
4226, 4460
Cu'(n = 3;
= 0, 1, 2)
144, 161, 195
149, 166, 200
Ag f (n = 4;
 0, 1, 2)
257, 275, 311
322, 339, 375
Au^(n = 5;
= 0, 1, 2)
460, 481, 524
599, 620, 663
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. 0506 114 172 072 088
/cobs. 0513 0995 4132 625 1016
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. 188 67 181 370 590
XmoiXl0 6 calc. (Pauling) 154 57 136 172 254
XmoiXl0 6 calc. (Slater) 164 56 185
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 (HeF), 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 wavemechanical
basis than those of Pauling.
51. Polarizability of the AtomCore 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 wellknown
RydbergRitz 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 hydrogenlike 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 atomcore. By the atomcore 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 semiempirical 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 atomcore, 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 atomcore 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 atomcores 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 168/16 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. 0056 00026 000045 0044 00021
_004n a 0005n 2 0002n' 2 003n' 2 0004W 2
Aobs. 0029 00025 0001 0020 00022
We have omitted S states as too penetrating, and have taken the mean
of the experimental values of A for par and orthohelium (or Li + ), as
the parortho 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,
0031, 0020 for the orthopar 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 stumblingblock. 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 atomcores.
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, orthopara means are 00274, 00278, 00207, whereas the table
gives the experimental Rydborg correction which best approximates experiment for all
values of n.
Li+
Na'
K 1 '
Rb^ Cs+
009
048
197
76?
080
103
427
165?
019
053
221
85?
He
Ne
A
Kr Xe
051
099
413
625 102
218 THE DIELECTRIC CONSTANTS AND DIAMAGNETIC VIII, 51
The questionmark 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 nonpenetrating
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 0096 0094 0093 0094 0068 0076 0011 004 022 13
Mg++ K = 0398 0434 0447 0460 0194 0195
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 isoelectric sequence.
Comparison of the value 028 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/(nfaf 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 067, 2, 4 for P, D, F terms (13, 3, 53
according to the old quantum theory). Some experimental values are : P terms, Li
063; D terms, Na 24, Mg+ 22, A1++ 26, Si+++ 24; F terms, Na 32, Mg+ 36,
A1++ 36, Si+++ 48.
If a frequency of motion (in the quantum sense) of the excited electron
nearly coincides with an absorption frequency of the atomcore, 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 ultraviolet, 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 D6 3 ^ and 3 3 D7 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 atomcore. 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 31 is about oneeighth 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 socalled '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
energycurve 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 interelectronic 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 isoelectronic 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 interatomic 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 wavelength, 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 wavelengths 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 048 (NaF),
053 (NaCl), 051 (NaBr), 033 (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 isoelectronic with neon which have been
obtained by the various methods.
IONIC REERACTIVITIES
F Ne Na+ Mg++ A1+++ S1++++
(a) (Fajans & Joos SG ) K = 250 (100) 050 028 017 01
(b) (B. & H. from Spang. 26 ) 251 (100) 046
(c) (Pauling 19 ) 265 (100) 046 024 014 008
(d) (B. & H., Spectro. 26 ) (100) 049 028 015 010
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 startingpoint. 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 LandoltBomstein'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) 13 115 65 195 146 395 605
(a) (Iken.) 40 139 104 204 169 348 313 493 457
(a) (Reich.) 219 325 502
(6) (Pascal & Pauling) 02 103 62 241 146 346 232 480 370
(c) (Stoner) 65 404 301
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 188, 67, and 181 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
59, 24, and 35 for halogen, alkali, and alkaline earth ions, and Cj is
080 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 080 x 10" 16 /283 or
(053 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 054, 071, 080, 092 x 10 8 cm. respectively for
Na + , F~, K+, Cl", which are in remarkably close agreement with the
ionic radii 063, 075, 079, 095 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 semiempirical 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 isoelectronic 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 ^
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d? &S +
N 00 cogo Op ^co g2c,
s
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^6<N MO(T   ^
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L 1 " 1 rl '^' ^O* bC 1 "" 1
J/2^130 O(7t^ OQCOCO bdCO
ft\ w co w ^
til isS? , i^^ t,S
1^66 66 ^o^ HH M co QrHc
+ 1 + CO + ^ + I
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pq co 10 ij ^ ^ O co 10 "^ 
CO ^ IO
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ou
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H9^ MCO f^ 5 ?^
rH CI 10 00 <N 00
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3595.3
IX
THE PARAMAGNETISM OF FREE ATOMS AND
RARE EARTH IONS
53. Adaptation of Proof of LangevinDebye Formula given in
46
The general proof of the LangevinDebye 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 'antiparallel' 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 lowfrequency 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 lowfrequency 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 nonparamagnetic
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
= (09m00013) 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  01241 +0004 X 10*3
7^ ~
= 000500 ^f +6064 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 unitvolume at 0C., 76 cm. in the gaseous state, the
numerical factors in the first and second forms of (4) would become
respectively 554XK) 6 , 2705X10 19 and 2258X1Q 7 , 2705xl0 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 1853 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 1/495 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 11235 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 (19249) he raised the estimate 1o 11256 per
gramme mol. (cf. Weiss and Ferrer, Annales de Physique, 12, 279 (1929) ) ; while si ill more
recently Cabrera suggests 11249iLO3 per mol. or 1855 J^O6 X 10~ 21 per atom as the most
probable value (Report of the Solvay Congress, 1930). Our numerical value 0917 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 5504 e.m.u. In the
literature the Bohr magneton seems to be very often taken as 497 rather than 495 Weiss
magnetons, or as 0921 instead of 0917 x 10" 20 e.m.u., due to use of older values of ejm
and h. We employ the spectroscopic value 1761 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 01 (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 nonintegral multiples 142 and 92
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 highfrequency elements basic
to the validity of (1) or (4). Thus, as already mentioned in Chapter VII,
the LangeviiiDebye 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 RussellSaunders 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 3942. 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 (bc 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 nondiagonal 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 nonhydrogenic 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 lowfrequency 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 highfrequency
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 righthand 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 PaschenBack 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 ML _ _ , J
~ ft [ ^ e ^IM L lkT T"
We now expand the exponents as power series in H, and retain as in
44 the nonvanishing 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 PaschenBack 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 PaschenBack 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 PaschenBack 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 highfrequency 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 lowfrequency 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 grfactor
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]Jl)Y V }
with the abbreviation
F(J) = [(8+L+ir J*$J*(8L)*\. (12)
The existence of this is a concomitant of the secondorder 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 secondorder
term in (99), is due to the component bc 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 ab is small compared to hc 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 firstorder 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 firstorder 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 lowfrequency part p of the magnetic moment
vector, then in a weak field the firstorder energy is W Q /iJF/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. 63048 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, interatomic 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 bc is usually
small compared to ab in Fig. 9. For instance, Laporte and vSommerfeld calculate
effective Bohr magneton numbers 292, 4'35, 497, 523, and 592 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 30, 447, 520, 548, 592. 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 subshell is 4/[2, the value of A is, in other
words, positive if 7; < 2J1, and negative if 77 > 2ZJ1, 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 wellknown '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 iionhydrogenic 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 'overall* 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 'overall' width when expressed in
wave numbers rather than ergs is
?ii) . 5.82(2L+1) _ 4 i
fl
c total Ac 2S
The factor 582 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 Psums
( 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 LJAV, 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 lowestlying 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, vapourpressure. 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 vapourpressures of from 0'5 to 30 mm., the
molar susceptibility of potassium vapour obeys Curie's law and is
038/T. The normal states of alkali atoms are 2 $ ones, and so by Eq. (6)
the theoretical molar susceptibility is LfPjlcT 0372/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 vapourpressure
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 vapourpressures (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 12 and 08 volt respectively) than that of K (about
06 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 wellknown 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 subshells 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 wellknown 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 546. 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 firstorder
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 277 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 SommerfeldKossel 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 224 for Pr+ + ' +, or 274 if the Weiss modification of (22) is used, in
which T is replaced by Tf 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
000
000
diam.
4/ 2 J^5/ 2
254
256
239
277
237
210
023?
4f a 3 //
358
362
360
347
347
341
229?
4/ 3 4 / 9 / 2
362
368
362
351
352
345
343
4 / 4 '%..
268
283
4/ 5 6 // 6 /2..
084
(l65J
154
132
153
163
157
4/ fi 7 ^ fl
000
J340)
361
312
\351 )
4/ 7 *jS'
794
794
82
81
781
786
746
4/ 8 7 F 8
97
97
9 6
90
94
98
4/ f (> // J5 / 3
106
!()()
105 100
109
100
4/ 10 r '/ 8
KM)
KM)
105 , 104
103
104
101
4/ 11 <1 / 1B y < ,
IM>
9 6
9 5 , 94
96
95
92
4/ 12 :< // 6
7 (>
76
72 75
*/ I3a F 7 / 2 i 45
45
44 46
44
45
4/ M ^S 00
00
diam.
diam.
12?
049 ?
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 151 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 interatomic 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 157 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 78 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, 347t, 332*;
Nd, 352t, 340*; Sin, l58t, 150*; Eu, 354t, 332H ; Od, 79I", 77"; Tb, 96t, ;
Ds, 103t, 102"; Ho, 104t, 102t; Er, 94t, 93t; Tu, 7Ot, 68t; Yb, 43t, 39t.
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(TfA) 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 interatomic 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, 376, 371; Nd, 379, 371; Gd, 79, 79; Tb, 97, ; Ds, 105, 105;
Ho, 106, 105; Kr f 96, 95; Tu, 75, 72; Yb, 48, 45. (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 'overall' 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
overall width, as for a 1 F term
[^max(^max+l)^ninWnin+l)] IX70= 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
Xray data. This data must be taken from atoms heavier than the
rare earths, as Xray 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 Xray 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
wellknown BowenMillikan result on the similarity of Xray and optical
multiplets. The uncertainty in the Xray 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 Xray data. This will be shown
particularly clearly when we study the temperature variation in Sm
( 60). Screening numbers in the ranges 3032 and 3538 definitely will
not fit the experimental magnetic measurements on Sm, even though
within Wentzel' s estimate of the Xray precision. Substitution of the
value er= 34 in (21) makes the overall 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
Xray 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 (19289).
34 G. Wentzel, Zeits.f. Physik, 33, 849 (1925).
36 D, Coster, in MullerPoulliets' Handbuch der Physik, ii. 2057.
248 THE PARAMAGNETISM OF FREE ATOMS IX, 59
pared to kT, inasmuch as kT is 0698T 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
01241
xT
) x mol (23)
01241
~~ yT
where x and y are respectively 1/21 and 1/55 of the ratio of the overall
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 351
and 165 respectively are obtained for Eu and Sm if a =34, while
340 and 155 result if or = 33. Either of these values is in as good
agreement with experiment as for the other members of the rareearth
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 BohrHund 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 17 instead of 3*51. As the susceptibility varies as the square
of this number, the secondorder 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 bc 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 interatomic 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 secondorder 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
084
084
20
091
092
74
107
09
091
85
1 09
12
096
112
116
20 
108
, ,
123
118
23
108
170
129
35
120
205
137
43
135
240
144
52
144
293
155
65
158
1 57
150
158
375
173
185
177
169
t .
400
178
191
175
500
200
245
197
543
209
225
214
206
600
220
238
217
800
258
278
.
1000
291
314
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^ 00628(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
107
1J5 j
70
201
214
100
238
253
200
309
320
,
293
340
351
334
3 53
400
305
375
3 57
3 7..
470
377
381)
3 61)
38{)
402
4 14
411
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 7lxlO 3 ((j=33) or 8lxlQ 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 03 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 secondorder 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 interatomic 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 interatomic 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
interatomic 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
(13 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, 19; Gd 2 (S0 4 ) 3 8H 2 O, 00 (or
possibly 026). In CeF 3 , however, pronounced deviations appear below
65 K. The corresponding experimental values of the effective Bohr
magneton number /i cff are 106, 251, 90, 78, in quite satisfactory agree
ment with the theoretical values 100, 254, 1MJ, and 79 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
interatomic 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 interatomic 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
75, 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^QTT 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 7374. 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 = 128007. 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
= 0270 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 133 mc/e which would be ob
tained if one forgot the terms proportional to otj in (24) contributed
by the secondorder 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 righthand 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 unitvolume 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
(79), and also primarily by the use of exceedingly low temperatures
(down to 13 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
watermolecules 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
088 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
084JV/x off , or about 95 per cent, of the full saturation allowance,
088Afyi lff . The theoretical value at this H and T is 0831J!Vfi eff , whereas
the Langevin value is 0859JV/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 extranuclear (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 PaschenBack 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 'hypermultiplet' 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.
extranuclear) 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 extranuclear
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 6870, 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 interelectronic 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 highfrequency 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 lowfrequency 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(Af 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 (Efl). 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
lowfrequency 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 'highfrequency* 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
SLpar 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 tierLondon 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 0993
Xmol
for the molar susceptibility, corresponding to an effective Bohr mag
neton number & = 283. As usual, L denotes the Avogadro number.
At 20 C., Eq. (4) gives a molar susceptibility 339 x 10 ~ 3 . The value
observed by Bauer and Piccard, 12 which seems to be usually accepted
as the most accurate, is 345 x 10~ 3 . The early determinations of Curie 13
yield 335 X 10~ 3 when recalibrated 14 on the basis of 072 x 10~ 6 rather
than 079 x 10~ 6 as the mass susceptibility of water. Still other observa
tions arc: 331 X 10~ 3 by Onnes and Oosterhuis, 15 333 X 10~ 3 by Sone, 16
348 xlO~ 3 by Wills and Hector, 17 and more recently 334 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 rhotype doubling*, of the order QQIJ 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 342 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
PasehcriBack effect. The triplet width for tho normal oxygen molecule is about
14 cm 1 , whereas the normal Zecman displacement is 467 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 PaschenBack 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 290720 K. At lower temperatures the validity
of this law was confirmed approximately for oxygen over the interval
143290K. 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 1365 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 1365K. 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. 393436 (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 highfrequency
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 004, then
Xtoi/X 2Y291 increases from at T = 291 K. to about 001 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 secondorder 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 highfrequency 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 22 per
cent, greater than Bauer and Placard's value, instead of 18 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.
1414 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 bandspectrum furnishes unambiguous
termassignments 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 1209 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 146 x 10~ 3 per gramme mol.
These measurements yield an effective Bohr magneton number 186
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. 2712. 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 1836. The discrepancy of about 1 per cent, with
the experimental value 186 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 1209 cm" 1 , rather than the value 1244 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.
00 K.
00
2384K.
1794
(l794)(W.,doH.,&C.)
500
1098
2506
1807
(1807) (St.)
1000
1489
2892
1833
1841 (St.)
1128
1546
535(W.,doH.,&C. 29 '*)
2902
1834
(1834) (Ah. & S.)
1355
1624
627 (St. 23 )
2921
1836
1852 (W.,doH.,&0.)
1572
678
679 (St.)
2960
1837
(1837) (Bit.)
1654
695
691 (W., deH., & C.)
3500
1864
1780
718
713 (St.)
5000
1908
1947
741
732 (Ah. & S. 29 )
10000
1955
2160
771
768 (Bit. 28 )
00
2000
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 boilingpoint 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 1128 and
2921.
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,
2902, 2506, and 2384 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 = 287 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 lowfrequency 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 highfrequency
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 lowfrequency 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 highfrequency 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, highfrequency 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 lowfrequency 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 highfrequency 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 134 x 10 4 , while (10) gives 127 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 highfrequency 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 127 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 +21 x 10~ 4 , a value less than onefifth 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)~ 2rrimv(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 highfrequency 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 nonmonatomic 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 centrosymmetric. 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 123.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. 190813.
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) 051 x 10~ 6 per mol. Wang 42 calculated the first term of (11) to
be 471xlO~ 6 . The computed molar susceptibility of H 2 is hence
420X 1C 6 . The experimental values 43 are 394 x 10~ 6 (Wills and
Hector) and 399 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 centrosym
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 dividingline 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
magnetoelectric 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 counterclockwise about the axis
of electric polarity. These left and righthanded 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 magnetoelectric 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 Atype 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 otypo doublet. Tho term litypo 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 (19289), Band Spectra and Molecular Structure,
Chap. II; J. H. Van Vlock, Phy. Rev. 33, 467 (1929); R. S. Mulliken, ibid. 33, 507
(1929). Tho Adoubling 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 637 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 lowfrequency
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 Adoublet components are respectively even and odd with
respect to the transformation (A), p. 203, and so have no firstorder
Stark effect, as there explained, although there is a firstorder Zeeman
one. A very powerful electric field, however, produces a PaschenBack
transformation on the doublet and gives a firstorder 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 interatomic 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 interatomic 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 interatomic 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 interatomic 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 interatomic 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 Xray 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 argonlike 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 "
00
00
00
00
00
00
rsc i *
30
155
257
173
I Ti i i i
30
155
218
173
\V >+'.
30
155
178
173
175
179
fTi 1 " 4 "
447
163
336
283
"1 y i H i
447
163
273
283
276285
rv^
520
077
3tiO
387
381386
Cr"i
520
077
297
387
368386
382
iMn 1 < M
520
077
247
387
400
jCr'^
548
00
425
490
480
\Mn * * '
548
00
380
490
505
f Mn f h
592
592
592
592
52596
585
\Fe +4 '
592
592
592
592
594
5460
Fo 1 ""^
548
G 70
654
490
533
5055
Co 1 '
520
664
656
387
4650
4452
Ni 1 '
447
559
5 56
283
323
2934
Cu 1 '
30
355
353
173
1820
1822
'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 323, 261, 201, 374, 316 instead of 360, 297,
247, 425, 380. 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 citod 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 lowestlying 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 interatomic forces. If,
then, these interatomic 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 interatomic 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
interatomic 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
interatomic 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 interatomic 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 higherorder terms in the Taylor's expan
sion than (2). We assume throughout that the external fields are never
strong enough to destroy appreciably the RussellSaunders 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(Ljl)
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 highfrequency
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 highfrequency elements one easily
finds that x^N^^T. Hence, supposing \A Cz 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 wellknown LenzEhrenfcst 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 higherorder 'saturation' terms in the
field strength one would obtain the socalled 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
nondegenerate 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 energylevels 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 highfrequency 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 nondegenerate states into which the
energylevels are separated by (2) or by an asymmetric interatomic
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 RussellSaundcrs 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 (192930), 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 higherorder 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 higherorder terms in the
Taylor's development are included (e.g. the fourthorder 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 nondegenerate state. The orbital
magnetic moment can be completely quenched only if the single or
nondegenerate 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 nonmagnetic 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 firstorder
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 firstorder 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 noncoinciding 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 Xray 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 interatomic 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 socalled 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 interatomic 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 nonpolar 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 interatomic 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 interatomic 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'LS (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 overall
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 16 x 10 2 to 08 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 01 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 nonmagnetic 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 interatomic 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 overall 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 fourthorder 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 secondorder 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 secondorder 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 dipolefields 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
155, 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 soelectronic
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 interatomic forces, there should pre
sumably be tremendous variations with the precise character of the
interatomic 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
44 to 52. 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 18 to 20, 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 WeissCurie law
(Eq. (6), p. 303 ) almost perfectly even down to 14K., with /x efl = 192
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 highfrequency 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 1745, 2760, and 38053855, respectively for the effective
Bohr magneton numbers, whereas the theoretical values predicted by
(1) arc 1732, 2828, and 3873. 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 28132848, 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 32 for Ni^ + in the
temperature interval 15125 C. } and 34 for 150500C., to be compared
with the values 283 and 556 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 gi 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 nonlinear with tho concave Hide of the curvature towards the T axis (Sci. Rep.
Tohoku Univ., vols. 34) 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 173 to that 353 appropriate to free ions. Jttrch
gives a magneton number 182 for CuCl a in solution from 040 and 20 from 4085 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 [^. (Tf 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. 185 and
186 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
0f actor 187 for pure iron, quite definitely less than 2. This is doubtless because the orbital
magnetic moment is not completely quenched. The 0factor 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 grfactor 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 592
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 socalled 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 18 and 20; Cr(NH 3 ) 6 T 3 and other chromic salts numbers from
34 to 38; and Ni(NH 3 ) 6 Br 2> &c., numbers ranging from 26 to 32. These
values arc nearly the same, though usually somewhat higher than those
173, 387, and 283 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 atomions,
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 highfrequency 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
highfrequency 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 interatomic forces not merely quench most of the orbital
paramagnetism if the constituent atomions 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 socalled 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, 1327, 1449, 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,
Leiden 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
583
MnSO 4
4H 2
31
. ,
31
516
FoSO 4 .
7H 2
45
509
CoSO 4 .
7H a O
79
321
NiSO 4 .
7H 2
CuS0 4 .
5H 2
02*
383
CrCl 3
0*
589
MnCl a
)0*
584
3*
555
FeCl a
22
500
CoCl a
4
321
NiCl a
A
Me
587
1
14
59
07*
522
505
295
191
325
361
561
20
33
~67
503
317
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 65135 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 interatomic 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 amonosulphates 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 interatomic 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 firstorder 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^ 078 095 118 144 204 297 337
O+ 1 ^ t . w = 174 252 290 351 425 455
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 (361) 2 /3k(T 325) 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
'ungaslike' 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 100800 K
they studied, barring oxidation effects at high temperatures and
Weiss Acorrections 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 wellknown 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 interatomic 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 = 98, 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 interatomic 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, 59, and 46 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 noncentral field. Both these potential causes of magnetic
anisotropy are secondorder 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 371 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+
00
00
diam.*
d 2 #3/3
m
155
173
[
Mo 41 "
. .
. .
163
283
(02)
W 4+
, .
163
283
18*
, .
2 I
t
u 4 '
163
283
24*
MO+++
077
387
36*
a f 3/2 
>
W l ' f
%
077
387
04*
d**D
Ru ++l '
00
490
(06)
r
Ru ++H
. .
592
592
21
03
18
. .
OB+ 4 *
t
592
592
diam.
I
Ir 4 +
592
592
19
(07)
f
Rh +f h
670
490
03
04
(04)
I
Os+ +
670
490 04
4 1
j r ++ +
670
490
03
05*
diam.
I
Pt 44 "
diam.
diam.
664
387
, .
d 8 *FA
Pd + '"
559
283
01
diam.
Pt if
559
283
diam.
diam.
^ 2 >6/ 2
355
173
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 Xray 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 12 and 25 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 interatomic 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. nonexchange) interatomic 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 lowestlying 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
moleculeions of zero resultant spin, possibly through electronsharing,
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 microcrystalline, 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
physicalchemical 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 interatomic 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 * highfrequency 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 socalled '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
TF( 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 twoelectron
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
'cnumbers' 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 parortho 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 antiparallel, 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 (223)
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 righthand
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 twoelectron 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 interatomic 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 nonvanishing 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 wellknown
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 quantTimmechanical
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 interatomic 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 facecentred 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 interatomic 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
microcrystal and that the macrocrystal is composed of a large number
of microcrystals, whose spins have random orientations and hence
compensate each other without an external alining field.
2.  J/kT  <: 1 . Here the interatomic 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 LangevinDebye 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 interatomic 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 antiparallel, and the normal states
are those of least 8'. The interatomic 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 pseudosaturation, which is the saturation
observed in the laboratory and which will be discussed more quanti
tatively on pp. 3346. This pseudosaturation, 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 wellknown 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 eW'+WrfWw, (17)
then the magnetic moment per unitvolume 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 energylevels 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
energylevels 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 eWW**'***' 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 )^nS f )log^n8 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
ZiT * 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, nonnegative solution is the nonmagnetic
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 firstdegree 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 01 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 221, 153, 403, and 025 (per gramme
mol.), and A the values 1047, 1063, 1340, and 1543 K. in the
intervals 10471101, 11011193, 11931668, above 1668 K. (Fe ft,
j8 2 , y, 8). Nickel behaves much better, as it conforms quite accurately to
a formula with C = 0325, A = 645 from 6891 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 socalled '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 pseudosaturation
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 microcrystals at H = to the state
of pseudosaturation. 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 162 from 6891173 K., and 191 above 1173 K., while
V(4 !!) = 173. For cobalt, j* efl =321, A ^1404 from 1443
1514 K., and /* eff :293, A = 1422 from 15141576K. > while the
theoretical jn efl is 283 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 halfintegral 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 microcrystals 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 microcrystals 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 FermiDirac 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 495 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 interatomic 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 HeitlerLondon 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 energylevels. 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
bodycentred cubic (z = 8) or facecentred (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 grouptheory 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
energylevels 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 pseudosaturation 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 bodycentred 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 higherorder 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 HeitlerLondon 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 interatomic
distance be not less than Q22h 2 /me 2 = 0*5. 10 7 cm. This is only a
rough approximation, but it is noteworthy that the actual interatomic
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^)(vs7) = ^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 highestorder 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 noninteracting
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 interatomic than intraatomic 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 thermomagnetic 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^oajr 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 = 30(flf = J), 40(flf=l), 441(3 = }), 50(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 = 22 (Ni), 61 (Fc 3 4 ), 68 (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 06 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 29 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
HeitlerLondon perturbation theory is a poor approximation at small
interatomic 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 interatomic 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
thermomagnetic effects of Weiss are related to the change of specific
heat at this point. Both the Curiepoint 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 pseudosaturation is
35 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 interatomic interactions, which we shall consider first,
and on the ground of the degeneracy phenomena in the FermiDirac
statistics. Actually both effects are doubtless to a certain extent super
posed, but they are too complicated to discuss when together.
Interatomic Interactions 'Exchange Demagnetization'. We have
shown at length in 73 that if the spacial separation due to interatomic
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 interatomic 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 energylevels 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(TT 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 HeitlerLondon 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 microcrystal
in the present discussion.
FermiDirac 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 FermiDime 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 FermiDirac 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
unitvolume 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
FermiDirac statistics of free electrons, Bloch 50 has shown that a more
accurate formula than (57) is
2 l
J
5S
= 220 X 10 103 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
leapfrog from one valley to another, thus giving an electric current.
Bloch shows that the 'overlapping' 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 leapfrog
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 overlapping 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 'overlapping' 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 leapfrog effect was so small that the FermiDirac 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 MaxwellBoltzmann 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 twodimensional 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 twodimensional 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 unitvolume 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 righthand 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 onethird
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 wellknown
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
FermiDirac Statistics. In actual solids the calculations should be
made with the FermiDirac 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 xy 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)dxhF'(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 twothirds 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 leapfrog 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 FermiDirac 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 twothirds 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.
68xlO
102xlO
157 x 10e
182 x 10 6
21 5 X lO 6
,. Calc.
06
42
167
313
457
moi Calc.
50
62
10
131
242
moi Obs.:
Honda & Owen 45
3521
12
1623
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 0372/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 wavelength
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. 1935 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 nonpolar part ^TrNoc of the static dielectric constant
(see Eq. (28), Chap. VII). This agreement seems trivial today, 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 wavelengths
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 'ThomasKuhn 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 ! = 225 X 10~ 4 (l + l228x 10 10 A(T 2 + 165 x 10 20 A/+...)
at C., 76 cm., provided the incident wavelength 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, Natwwissenschaften, 13, 627 (1925); Thomas and Reiche, Zeits.f. Physik,
34, 510 (1925); Kuhn, ibid. 33, 408 (1925).
8 See, for instance, A. H. Compton, Xrays 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, 024; Na, 004;
H, 044. 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 012,
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 antiStokes, 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 nonconservative 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 345, 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 nonpolar 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
MullerPouillot's Lehrbuch tier Physik, llth od., vol. ii, second half, Chaps. XXXV1XL.
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 ClausiusMossotti 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 nonpolar 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
(nonpolar); in these cases the nonvanishing term of highest order in
l/T (c 2 in polar, Cj in nonpolar molecules) thus has a preponderant
influence. The secondorder 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 CottonMouton or Voigt effect, 24
already discussed, like the Kerr effect, in 31 for the static case or
limit v Q = 0. Because of the secondorder 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 socalled
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 righthanded circularly polarized beams. Ele
mentary classical theory based on Larmor's theorem and a rather too
simple atomic model yields the socalled 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 (192930), 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' \nYvl MT /' V '
368 BRIEF SURVEY OF SOME RELATED XIII, 84
Faraday Effect in Atoms. A quite complete quantummechanical
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, nonexistent, while the nonexistence of the transition L = > // =
is a wellknown 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 050 to 070, but for H 2 it is 099, 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
infrared 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 (wf2) a ro7(w/[2) 2 n.
whose origin is closely related to that of the ordinary ClausiusMossotti 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 onethird 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 wavelengths 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 interatomic 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 interatomic 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 (192930), 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 WeissCurio formula # O/(Tf 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 secondorder effect. 42 Possibly exchange forces between para
magnetic atoms and also interatomic 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 interatomic
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 hyperfine
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. 35360, or the writer's
Quantum Principles and Line Spectra t pp. 1719.
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, 2669.
Beck, J., 344.
Becker, R., 311, 335.
Becquerel, H., 3679.
Becquerel, J., 292, 3702.
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, 3513.
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, 21719, 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., 2579, 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, 2425, 248, 2503, 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, 568, 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, 868, 110, 117, 118, 152,
181, 257, 279, 289, 322, 331; see also
LangevinDobyo formula.
Dokor, H., 243, 245.
Dennison, D. M., 4752, 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,
1568, 161, 162, 169, 170, 190, 194, 195,
211, 227, 229, 291, 31621, 336, 340,
342, 347, 34951, 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, 2503,
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., 2213.
Falkenhagen, H., 88.
Faraday, M., 5, 170, 279, 36671, 373.
Farquharson, J., 278.
Formi, E., 26, 51, 131, 181, 208, 211, 336,
339, 347, 34951, 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., 246, 172, 181, 284, 310,
330, 344 7, 350.
Franck, J., 365.
Frank, A., 242, 243, 248 50, 253, 256, 278.
Freed, S., 245, 247, 24952, 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., 502, 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, 3702.
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, 21719, 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, 7780.
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., 2435, 248, 2524.
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., 2213, 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, 3447.
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, 3614,
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, 3557.
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, 8991,
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, 902, 109, 206, 255,
277, 354, 367.
Loeuwen, J. H. van, 947, 99, 101, 102,
104, 105.
Logcndrc, A. M., 159, 160.
Lohrer, E., 112, 113, 209, 267.
LcnnardJoiiPH, 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, 1315, 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, 568, 87,
88, 366, 370.
j MottSmith, 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, 2668,
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, 1068, 147, 1569,
161, 162, 165, 170, 177, 183, 184, 206,
21012, 237, 241, 277, 278, 283, 317, 319,
338, 339, 349, 351, 353.
Pauling, L., 107, 108, 114, 115, 117, 163,
165, 194, 209, 21115, 2214, 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., 2669, 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, 21517, 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,
21821, 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, 31921, 338, 3469.
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, 32931, 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, 2668, 2857, 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, 3336, 339, 3435, 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., 2435, 2503, 308.
Williams, J. W., 59, 624, 6870, 72, 757,
7981.
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.
ZadocKahn, 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., 2435, 248, 2524.
Zoellor, H., 311.
Zwann, A., 175.
SUBJECT INDEX
Absorption intensities, 47, 50, 51, 362.
Additivity relations, 57, 60, 77, 825, 207,
2205.
Alkalis: diamagnetism and refractivity of
ions, 2225 ; 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,
15962, 16772 ; behaviour in molecules,
2624, 274, 27781; quantization of,
106, 150, 1645; quenching of orbital,
2734, 28797, 314, 324, 334; relation to
magnetic moment, 4, 155, 174, 255, 256,
300; true v. canonical, 234, 145, 354;
we also spin.
Anisotropy, magnetic (in crystals), 309
11.
Association, molecular, 17, 57, 58, 65, 294,
314; see also complexes.
Atomcore, polarization of, 21520, 222.
'Atomic polarization', 46, 51, 52, 68.
'Austasoh', sec exchange effect.
Axis of quantization, 108, 114, 115, 140,
196.
Azimuthal quantum number, 159, 166,
2034, 238, 354.
Band spectra: relation of dielectric con
stant to, 4553, 72, 280.
Benzeno molecule and derivatives, 7281.
Birefringence, see Kerr effect.
Bismuth, diarnagnetic susceptibility of, 92,
93.
Boltzmann distribution formula, 246, 96,
99, 1024, 181, 196, 353, 355.
Brillouin function, 2579, 325, 331, 332,
343, 371, 372.
Canonical: coordinates, 25; transforma
tion, 138.
Carbon bonds, 72, 76, 84.
C0 a molecule, 4952, 70 ; CS 2 , 70 ; CC1 4 , 71.
Centrifugal expansion, 200.
Characteristic: functions, 123; values, 123,
143.
Charge: * effective', 45, 4752, 81, 200;
electronic, 4; statistical density of, 129,
131, 354.
C1O 2 molecule, paramagnetism of, 275.
Chromium salts, 285, 301, 304, 3068.
Classical theory and statistics, 1104;
absence of magnetism in pure, 94102;
inadequacy of, 102; quantum analogies
with, 16972, 184.
ClausiusMossotti formula, 14, 16, 56, 87,
370.
Cobalt: pure metal, 335, 336; salts, 285,
2979, 302, 304, 309.
Commutation rules, 119, 1278, 161, 166,
258, 323.
Complexes, 293, 295, 301, 306, 314; com
plex atoms, 301, 304, 349.
Conduction electrons, 6, 1002, 336, 339,
346, 34960.
Contact transformation, 25, 138.
Continuity, equation of, 8, 10, 130.
Copper salts, 285, 297, 299, 300, 304.
Correspondence principle, 16972, 107,
109, 110, 357, 361, 362, 373.
Corresponding states, Weiss theorem of,
334.
Cosine law, 162, 165, 174, 177, 230 2, 236,
318.
Coupling: jj, 168; RufisellSaundors,
1638, 173, 178, 229, 288; spinorbit,
163, 237, 264, 267, 282, 295, 313, 317,
348; spinspin, 311, 163, 237, 283, 318,
335 ; typos (a)(b) in molecules, 263.
Cryornagnetic anomalies, 249, 3078, 333,
372.
Crystalline fields, 28797, 30911, 31415,
372, 373.
Crystals, 2, 30911, 323; liquid, 120;
mixed, 3703; single, 335.
Cubic symmetry, 291, 311, 324, 337, 338.
Curie's law, 89, 239, 249, 2534, 267, 269 ;
Weiss modification, 245, 253, 288, 300,
3039, 31415, 333, 372.
Curio point, 326, 3313, 3369, 3448.
Current density in quantum mechanics,
130.
Degenerate electron gas, 34952, 3579.
Degenerate systems, 1347, 112, 124, 182,
280; nearly degenerate, 137, 175, 179;
nondegenerate, 1324; invariance
theorems in, 13942; relation to sym
metry of field, 2907, 273, 274 ; required
with odd no. of electrons, 296, 297, 348.
Density, dependence on, see Clausius
Mossotti formula, dilution.
Diagonal: matrices, 127, 1423, 171, 172;
matrix elements, significance of, 129,
143; sum (invariarico of), 142, 177, 342.
Diamagnotism : of atoms and ions, 904,
99, 20615, 2205; of free electrons,
1002, 35360; of molecules, 2769;
pressure dependence, 110, 113; dia
magnetic correction to paramagnetic
susceptibility, 227, 243, 252, 359;
diamagnetic Zoeman term, 17880.
Diatomic molecules: electric moment, 27,
66; magnetic moment, 26272, 27681;
spectral notation, 262; dumbbell and
top models, 14755, 1835, 327, 115.
SUBJECT INDEX
381
Dielectric constant, 1 ; of atoms, 27, 2035,
20925; determination of molecular
moments by, 609; effect of magnetic
field on, 11321, 190; relations to mole
cular structure, 7082; relation to
refractive index, 4253, 60; saturation,
858; of solutions, 5660; see also
LangeviiiDebye formula.
Diffraction, electron arid Xray by single
molecules, 76.
Dilution, magnetic, 244, 255, 259, 268,
283, 294, 3045, 308, 309, 314, 321.
Dipole moment, determination of numeri
cal values, 60 82.
Dipole, models for rotating, see dumbboll,
top.
Directive effect, magnetoelectric, 182,
279 81.
Dispersion, see refractive index.
Doubling: Atype, 154, 203, 280; ptypo,
266, relativity, 157, 161, 203; see also
multiplets.
Dumbbell model, 106, 14752, 1835, 196,
198.
Effective: charge, 45, 4752, 200; mag
neton number, see magneton numbers.
Electric moment, see moment, electric.
Electrons: diamagnctism of free, 1002,
35360; Dirac theory of, 1568;
'equivalent', 165, 237, 241; feeble spin
paramagnotism in conductors, 34953,
359