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THE LIBRARY 

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THE UNIVERSITY 
OF CALIFORNIA 

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The Theory of Spectra 

and 
Atomic Constitution 



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The Theory of Spectra 

and 

Atomic Constitution 

THREE ESSAYS 

BY 
NIELS BOHR 

Professor of Theoretical Physics in the University of Copenhagen 



CAMBRIDGE 

AT THE UNIVERSITY PRESS 
1922 



PRINTED IN GREAT BRITAIN 
AT THE CAMBRIDGE UNIVERSITY PRESS 



Collegt 
LibrMJT 



PREFACE 

THE three essays which here appear in English all deal with 
the application of the quantum theory to problems of atomic 
structure, and refer to the different stages in the development of 
this theory. 

The first essay " On the spectrum of hydrogen" is a translation of 
a Danish address given before the Physical Society of Copenhagen 
on the 20th of December 1913, and printed in Fysisk Tidsskrift, 
xn. p. 97, 1914. Although this address was delivered at a time 
when the formal development of the quantum theory was only at 
its beginning, the reader will find the general trend of thought 
very similar to that expressed in the later addresses, which 
form the other two essays. As emphasized at several points the 
theory does not attempt an "explanation" in the usual sense of 
this word, but only the establishment of a connection between facts 
which in the present state of science are unexplained, that is to 
say the usual physical conceptions do not offer sufficient basis for 
a detailed description. 

The second essay "On the series spectra of the elements" is a 
translation of a German address given before the Physical Society 
of Berlin on the 27th of April 1920, and printed in Zeitschrift fur 
Physik, vi. p. 423, 1920. This address falls into two main parts. 
The considerations in the first part are closely related to the con- 
tents of the first essay ; especially no use is made of the new 
formal conceptions established through the later development of 
the quantum theory. The second part contains a survey of the 
results reached by this development. An attempt is made to 
elucidate the problems by means of a general principle which postu- 
lates a formal correspondence between the fundamentally different 
conceptions of the classical electrodynamics and those of the 
quantum theory. The first germ of this correspondence principle 
may be found in the first essay in the deduction of the ex- 
pression for the constant of the hydrogen spectrum in terms of 
Planck's constant and of the quantities which in Rutherford's 



733173 



vi PREFACE 

atomic model are necessary for the description of the hydrogen 
atom. 

The third essay "The structure of the atom and the physical 
and chemical properties of the elements" is based on a Danish 
address, given before a joint meeting of the Physical and Chemical 
Societies of Copenhagen on the 18th of October 1921, and printed 
in Fysisk Tidsskrift, xix. p. 153, 1921. While the first two essays 
form verbal translations of the respective addresses, this essay 
differs from the Danish original in certain minor points. Besides 
the addition of a few new figures with explanatory text, certain 
passages dealing with problems discussed in the second essay are 
left out, and some remarks about recent contributions to the 
subject are inserted. Where such insertions have been introduced 
will clearly appear from the text. This essay is divided into 
four parts. The first two parts contain a survey of previous results 
concerning atomic problems and a short account of the theoretical 
ideas of the quantum theory. In the following parts it is shown 
how these ideas lead to a view of atomic constitution which seems 
to offer an explanation of the observed physical and chemical 
properties of the elements, and especially to bring the character- 
istic features of the periodic table into close connection with the 
interpretation of the optical and high frequency spectra of the 
elements. 

For the convenience of the reader all three essays are subdivided 
into smaller paragraphs, each with a headline. Conforming to the 
character of the essays there is, however, no question of anything 
like a full account or even a proportionate treatment of the subject 
stated in these headlines, the principal object being to emphasize 
certain general views in a freer form than is usual in scientific 
treatises or text books. For the same reason no detailed references 
to the literature are given, although an attempt is made to mention 
the main contributions to the development of the subject. As 
regards further information the reader in the case of the second 
essay is referred to a larger treatise "On the quantum theory of 
line spectra," two parts of which have appeared in the Transactions of 
the Copenhagen Academy (D. Kgl. Danske Vidensk. Selsk. Skrifter, 
8. Rsekke, iv. 1, 1 and II, 1918), where full references to the literature 
may be found. The proposed continuation of this treatise, mentioned 



PREFACE Vll 

at several places in the second essay, has for various reasons been 
delayed, but in the near future the work will be completed by the 
publication of a third part. It is my intention to deal more fully 
with the problems discussed in the third essay by a larger syste- 
matic account of the application of the quantum theory to atomic 
problems, which is under preparation. 

As mentioned both in the beginning and at the end of the 
third essay, the considerations which it contains are clearly still 
incomplete in character. This holds not only as regards the 
elaboration of details, but also as regards the development of the 
theoretical ideas. It may be useful once more to emphasize, 
that although the word "explanation" has been used more 
liberally than for instance in the first essay we are not concerned 
with a description of the phenomena, based on a well-defined 
physical picture. It may rather be said that hitherto every 
progress in the problem of atomic structure has tended to emphasize 
the well-known "mysteries" of the quantum theory more and more. 
I hope the exposition in these essays is sufficiently clear, never- 
theless, to give the reader an impression of the peculiar charm 
which the study of atomic physics possesses just on this account. 

I wish to express my best thanks to Dr A. D. Udden, University 
of Pennsylvania, who has undertaken the translation of the 
original addresses into English, and to Mr C. D. Ellis, Trinity 
College, Cambridge, who has looked through the manuscript and 
suggested many valuable improvements in the exposition of the 
subject. 

N. BOHR. 

COPENHAGEN, 
May 1922. 



CONTENTS 

ESSAY I 
ON THE SPECTRUM OF HYDROGEN 

PAGE 

Empirical Spectral Laws 1 

Laws of Temperature Radiation 4 

The Nuclear Theory of the Atom 7 

Quantum Theory of Spectra 10 

Hydrogen Spectrum 12 

The Pickering Lines 15 

Other Spectra 18 

ESSAY II 
ON THE SERIES SPECTRA OF THE ELEMENTS 

I. INTRODUCTION 20 

II. GENERAL PRINCIPLES OF THE QUANTUM THEORY OF SPECTRA . 23 

Hydrogen Spectrum 24 

The Correspondence Principle 27 

General Spectral Laws 29 

Absorption and Excitation of Radiation 32 

III. DEVELOPMENT OF THE QUANTUM THEORY OF SPECTRA . . 36 
Effect of External Forces on the Hydrogen Spectrum . . 37 

The Stark Effect 39 

The Zeeman Effect 42 

Central Perturbations .... ^ ... 44 

Relativity Effect on Hydrogen Lines 46 

Theory of Series Spectra 48 

Correspondence Principle and Conservation of Angular Mo- 
mentum 50 

The Spectra of Helium and Lithium 54 

Complex Structure of Series Lines 58 

IV. CONCLUSION . 59 



CONTENTS 

ESSAY III 

THE STRUCTURE OF THE ATOM AND THE PHYSICAL 
AND CHEMICAL PROPERTIES OF THE ELEMENTS 

PAGE 

I. PRELIMINARY 61 

The Nuclear Atom 61 

The Postulates of the Quantum Theory 62 

Hydrogen Atom 63 

Hydrogen Spectrum and X-ray Spectra 65 

The Fine Structure of the Hydrogen Lines .... 67 

Periodic Table 69 

Recent Atomic Models 74 

II. SERIES SPECTRA AND THE CAPTURE OF ELECTRONS BY ATOMS . 75 

Arc and Spark Spectra 76 

Series Diagram 78 

Correspondence Principle 81 

III. FORMATION OF ATOMS AND THE PERIODIC TABLE ... 85 

First Period. Hydrogen Helium 85 

Second Period. Lithium Neon 89 

Third Period. Sodium Argon 95 

Fourth Period. Potassium Krypton 100 

Fifth-Period. Rubidium Xenon 108 

Sixth Period. Caesium Niton 109 

Seventh Period Ill 

Survey of the Periodic Table 113 

IV. REORGANIZATION OF ATOMS AND X-RAY SPECTRA . . .116 
Absorption and Emission of X-rays and Correspondence Prin- 
ciple 117 

X-ray Spectra and Atomic Structure 119 

Classification of X-ray Spectra 121 

CONCLUSION 125 



ESSAY I* 

ON THE SPECTRUM OF HYDROGEN 

Empirical spectral laws. Hydrogen possesses not only the 
smallest atomic weight of all the elements, but it also occupies a 
peculiar position both with regard to its physical and its chemical 
properties. One of the points where this becomes particularly ap- 
parent is the hydrogen line spectrum. 

The spectrum of hydrogen observed in an ordinary Geissler tube 
consists of a series of lines, the strongest of which lies at the red 
end of the spectrum, while the others extend out into the ultra 
violet, the distance between the various lines, as well as their in- 
tensities, constantly decreasing. In the ultra violet the series con- 
verges to a limit. 

Balmer, as we know, discovered (1885) that it was possible to 
represent the wave lengths of these lines very accurately by the 
simple law 



where R is a constant and n is a whole number. The wave lengths 
of the five strongest hydrogen lines, corresponding to n = 3, 4, 5, 6, 
7, measured in air at ordinary pressure and temperature, and the 

values of these wave lengths multiplied by f j -- 3} are given in 
the following table: 



3 6563-04 91153-3 

4 4861-49 91152-9 

5 4340-66 91153-9 

6 4101-85 91152-2 

7 3970-25 91153-7 

The table shows that the product is nearly constant, while the devia- 
tions are not greater than might be ascribed to experimental errors. 
As you already know, Balmer's discovery of the law relating to 
the hydrogen spectrum led to the discovery of laws applying to 
the spectra of other elements. The most important work in this 
* Address delivered before the Physical Society in Copenhagen, Dec. 20, 1913. 

B. 1 



ON THE SPECTRUM OF HYDROGEN 



connection was done by Rydberg (1890) and Ritz (1908). Rydberg 
pointed out that the spectra of many elements contain series of 
lines whose wave lengths are given approximately by the formula 

\_ A R 

\ n (n -f- a)' 2 ' 

where A and a are constants having different values for the various 
series, while R is a universal constant equal to the constant in the 
spectrum of hydrogen. If the wave lengths are measured in vacuo 
Rydberg calculated the value of R to be 109675. In the spectra of 
many elements, as opposed to the simple spectrum of hydrogen, there 
are several series of lines whose wave lengths are to a close approxima- 
tion given by Rydberg's formula if different values are assigned to 
the constants A and a. Rydberg showed, however, in his earliest 
work, that certain relations existed between the constants in the 
various series of the spectrum of one and the same element. These 
relations were later very successfully generalized by Ritz through 
the establishment of the "combination principle." According to 
this principle, the wave lengths of the various lines in the spectrum 
of an element may be expressed by the formula 



(2) 



In this formula n^ and n z are whole numbers, and F l (n), F^ (n), ... is 
a series of functions of n, which may be written approximately 



where R is Rydberg's universal constant and a r is a constant which 
is different for the different functions. A particular spectral line \vill, 
according to this principle, correspond to each combination of n x 
and w 2 , as well as to the functions F lt F 2 , . . . . The establishment of 
this principle led therefore to the prediction of a great number of 
lines which were not included in the spectral formulae previously 
considered, and in a large number of cases the calculations were 
found to be in close agreement with the experimental observations. 
In the case of hydrogen Ritz assumed that formula (1) was a special 
case of the general formula 



ON THE SPECTRUM OF HYDROGEN 3 

and therefore predicted among other things a series of lines in the 
infra red given by the formula 

1 



In 1909 Paschen succeeded in observing the first two lines of this 
series corresponding to n = 4 and n = 5. 

The part played by hydrogen in the development of our 
knowledge of the spectral laws is not solely due to its ordinary 
simple spectrum, but it can also be traced in other less direct 
ways. At a time when Rydberg's laws were still in want of 
further confirmation Pickering (1897) found in the spectrum of a 
star a series of lines whose wave lengths showed a very simple re- 
lation to the ordinary hydrogen spectrum, since to a very close 
approximation they could be expressed by the formula 

1 



Rydberg considered these lines to represent a new series of lines 
in the spectrum of hydrogen, and predicted according to his theory 
the existence of still another series of hydrogen lines the wave 
lengths of which would be given by 

1 



By examining earlier observations it was actually found that a line 
had been observed in the spectrum of certain stars which coincided 
closely with the first line in this series (corresponding to n = 2) ; 
from analogy with other spectra it was also to be expected that this 
would be the strongest line. This was regarded as a great triumph 
for Rydberg's theory and tended to remove all doubt that the new 
spectrum was actually due to hydrogen. Rydberg's view has there- 
fore been generally accepted by physicists up to the present moment. 
Recently however the question has been reopened and Fowler 
(1912) has succeeded in observing the Pickering lines in ordinary 
laboratory experiments. We shall return to this question again 
later. 

The discovery of these beautiful and simple laws concerning the 
line spectra of the elements has naturally resulted in many attempts 
at a theoretical explanation. Such attempts are very alluring 

12 



4 ON THE SPECTRUM OF HYDROGEN 

because the simplicity of the spectral laws and the exceptional accu- 
racy with which they apply appear to promise that the correct expla- 
nation will be very simple and will give valuable information 
about the properties of matter. I should like to consider some of 
these theories somewhat more closely, several of which are extremely 
interesting and have been developed with the greatest keenness 
and ingenuity, but unfortunately space does not permit me to do 
so here. I shall have to limit myself to the statement that not 
one of the theories so far proposed appears to offer a satisfactory or 
even a plausible way of explaining the laws of the line spectra. 
Considering our deficient knowledge of the laws which determine 
the processes inside atoms it is scarcely possible to give an explana- 
tion of the kind attempted in these theories. The inadequacy of 
our ordinary theoretical conceptions has become especially apparent 
from the important results which have been obtained in recent years 
from the theoretical and experimental study of the laws of tem- 
perature radiation. You will therefore understand that I shall not 
attempt to propose an explanation of the spectral laws; on the 
contrary I shall try to indicate a way in which it appears possible 
to bring the spectral laws into close connection with other pro- 
perties of the elements, which appear to be equally inexplicable on 
the basis of the present state of the science. In these considerations 
I shall employ the results obtained from the study of temperature 
radiation as well as the view of atomic structure which has been 
reached by the study of the radioactive elements. 

Laws of temperature radiation. I shall commence by men- 
tioning the conclusions which have been drawn from experimental 
and theoretical work on temperature radiation. 

Let us consider an enclosure surrounded by bodies which are in 
temperature equilibrium. In this space there will be a certain 
amount of energy contained in the rays emitted by the surrounding 
substances and crossing each other in every direction. By making 
the assumption that the temperature equilibrium will not be dis- 
turbed by the mutual radiation of the various bodies Kirchhoff 
(1860) showed that the amount of energy per unit volume as well 
as the distribution of this energy among the various wave lengths 
is independent of the form and size of the space and of the nature 



ON THE SPECTRUM OF HYDROGEN 5 

of the surrounding bodies and depends only on the temperature. 
Kirchhoff's result has been confirmed by experiment, and the 
amount of energy and its distribution among the various wave 
lengths and the manner in which it depends on the tempe- 
rature are now fairly well known from a great amount of experi- 
mental work ; or, as it is usually expressed, we have a fairly 
accurate experimental knowledge of the "laws of temperature 
radiation." 

Kirchhoff's considerations were only capable of predicting the 
existence of a law of temperature radiation, and many physicists 
have subsequently attempted to find a more thorough explanation 
of the experimental results. You will perceive that the electro- 
magnetic theory of light together with the electron theory suggests 
a method of solving this problem. According to the electron theory 
of matter a body consists of a system of electrons. By making 
certain definite assumptions concerning the forces acting on the 
electrons it is possible to calculate their motion and consequently 
the energy radiated from the body per second in the form of 
electromagnetic oscillations of various wave lengths. In a similar 
manner the absorption of rays of a given wave length by a substance 
can be determined by calculating the effect of electromagnetic 
oscillations upon the motion of the electrons. Having investigated 
the emission and absorption of a body at all temperatures, and for 
rays of all wave lengths, it is possible, as Kirchhoff has shown, to 
determine immediately the laws of temperature radiation. Since 
the result is to be independent of the nature of the body we are 
justified in expecting an agreement with experiment, even though 
very special assumptions are made about the forces acting upon 
the electrons of the hypothetical substance. This naturally 
simplifies the problem considerably, but it is nevertheless suffi- 
ciently difficult and it is remarkable that it has been possible 
to make any advance at all in this direction. As is well known 
this has been done by Lorentz (1903). He calculated the 
emissive as well as the absorptive power of a metal for long 
wave lengths, using the same assumptions about the motions 
of the electrons in the metal that Drude (1900) employed in 
his calculation of the ratio of the electrical and thermal conduc- 
tivities. Subsequently, by calculating the ratio of the emissive 



6 ON THE SPECTRUM OF HYDROGEN 

to the absorptive power, Lorentz really obtained an expression 
for the law of temperature radiation which for long wave lengths 
agrees remarkably well with experimental facts. In spite of this 
beautiful and promising result, it has nevertheless become apparent 
that the electromagnetic theory is incapable of explaining the law 
of temperature radiation. For, it is possible to show, that, if the 
investigation is not confined to oscillations of long wave lengths, 
as in Lorentz's work, but is also extended to oscillations corre- 
sponding to small wave lengths, results are obtained which are 
contrary to experiment. This is especially evident from Jeans' 
investigations (1905) in which he employed a very interesting 
statistical method first proposed by Lord Rayleigh. 

We are therefore compelled to assume, that the classical electro- 
dynamics does not agree with reality, or expressed more carefully, 
that it can not be employed in calculating the absorption and 
emission of radiation by atoms. Fortunately, the law of temperature 
radiation has also successfully indicated the direction in which the 
necessary changes in the electrodynamics are to be sought. Even 
before the appearance of the papers by Lorentz and Jeans, Planck 
(1900) had derived theoretically a formula for the black body radia- 
tion which was in good agreement with the results of experiment. 
Planck did not limit himself exclusively to the classical electro- 
dynamics, but introduced the further assumption that a system of 
oscillating electrical particles (elementary resonators) will neither 
radiate nor absorb energy continuously, as required by the ordinary 
electrodynamics, but on the contrary will radiate and absorb dis- 
continuously. The energy contained within the system at any 
moment is always equal to a whole multiple of the so-called 
quantum of energy the magnitude of which is equal to hv, where 
h is Planck's constant and v is the frequency of oscillation of the 
system per second. In formal respects Planck's theory leaves much 
to be desired ; in certain calculations the ordinary electrodynamics 
is used, while in others assumptions distinctly at variance with it 
are introduced without any attempt being made to show that it 
is possible to give a consistent explanation of the procedure used. 
Planck's theory would hardly have acquired general recognition 
merely on the ground of its agreement with experiments on black 
body radiation, but, as you know, the theory has also contributed 



ON THE SPECTRUM OF HYDROGEN 7 

quite remarkably to the elucidation of many different physical 
phenomena, such as specific heats, photoelectric effect, X-rays and 
the absorption of heat rays by gases. These explanations involve 
more than the qualitative assumption of a discontinuous trans- 
formation of energy, for with the aid of Planck's constant h it 
seems to be possible, at least approximately, to account for a great 
number of phenomena about which nothing could be said previously. 
It is therefore hardly too early to express the opinion that, whatever 
the final explanation will be, the discovery of " energy quanta " 
must be considered as one of the most important results arrived at 
in physics, and must be taken into consideration in investigations 
of the properties of atoms and particularly in connection with any 
explanation of the spectral laws in which such phenomena as 
the emission and absorption of electromagnetic radiation are 
concerned. 

The nuclear theory of the atom. We shall now consider the 
second part of the foundation on which we shall build, namely the 
conclusions arrived at from experiments with the rays emitted by 
radioactive substances. I have previously here in the Physical 
Society had the opportunity of speaking of the scattering of a rays 
in passing through thin plates, and to mention how Rutherford 
(1911) has proposed a theory for the structure of the atom in 
order to explain the remarkable and unexpected results of these 
experiments. I shall, therefore, only remind you that the charac- 
teristic feature of Rutherford's theory is the assumption of the 
existence of a positively charged nucleus inside the atom. A number 
of electrons are supposed to revolve in closed orbits around the 
nucleus, the number of these electrons being sufficient to neutralize 
the positive charge of the nucleus. The dimensions of the nucleus 
are supposed to be very small in comparison with the dimensions 
of the orbits of the electrons, and almost the entire mass of the 
atom is supposed to be concentrated in the nucleus. 

According to Rutherford's calculation the positive charge of the 
nucleus corresponds to a number of electrons equal to about half 
the atomic weight. This number coincides approximately with the 
number of the particular element in the periodic system and it is 
therefore natural to assume that the number of electrons in the 



8 ON THE SPECTRUM OF HYDROGEN 

atom is exactly equal to this number. This hypothesis, which was 
first stated by van den Broek (1912), opens the possibility of 
obtaining a simple explanation of the periodic system. This as- 
sumption is strongly confirmed by experiments on the elements 
of small atomic weight. In the first place, it is evident that ac- 
cording to Rutherford's theory the a particle is the same as the 
nucleus of a helium atom. Since the a particle has a double positive 
charge it follows immediately that a neutral helium atom contains 
two electrons. Further the concordant results obtained from cal- 
culations based on experiments as different as the diffuse scatter- 
ing of X-rays and the decrease in velocity of a rays in passing 
through matter render the conclusion extremely likely that a 
hydrogen atom contains only a single electron. This agrees most 
beautifully with the fact that J. J. Thomson in his well-known 
experiments on rays of positive electricity has never observed a 
hydrogen atom with more than a single positive charge, while all 
other elements investigated may have several charges. 

Let us now assume that a hydrogen atom simply consists of an 
electron revolving around a nucleus of equal and opposite charge, 
and of a mass which is very large in comparison with that of the 
electron. It is evident that this assumption may explain the peculiar 
position already referred to which hydrogen occupies among the 
elements, but it appears at the outset completely hopeless to attempt 
to explain anything at all of the special properties of hydrogen, 
still less its line spectrum, on the basis of considerations relating 
to such a simple system. 

Let us assume for the sake of brevity that the mass of the nucleus 
is infinitely large in proportion to that of the electron, and that the 
velocity of the electron is very small in comparison with that of 
light. If we now temporarily disregard the energy radiation, which, 
according to the ordinary electrodynamics, will accompany the ac- 
celerated motion of the electron, the latter in accordance with 
Kepler's first law will describe an ellipse with the nucleus in one 
of the foci. Denoting the frequency of revolution by o>, and the 
major axis of the ellipse by 2a we find that 

2TF 3 P* 



ON THE SPECTRUM OF HYDROGEN 9 

where e is the charge of the electron and m its mass, while W is 
the work which must be added to the system in order to remove 
the electron to an infinite distance from the nucleus. 

These expressions are extremely simple and they show that the 
magnitude of the frequency of revolution as well as the length of 
the major axis depend only on W, and are independent of the 
excentricity of the orbit. By varying W we may obtain all possible 
values for &> and 2a. This condition shows, however, that it is not 
possible to employ the above formulae directly in calculating the 
orbit of the electron in a hydrogen atom. For this it will be necessary 
to assume that the orbit of the electron can not take on all values, 
and in any event, the line spectrum clearly indicates that the 
oscillations of the electron cannot vary continuously between wide 
limits. The impossibility of making any progress with a simple 
system like the one considered here might have been foretold from 
a consideration of the dimensions involved ; for with the aid of e 
and m alone it is impossible to obtain a quantity which can be 
interpreted as a diameter of an atom or as a frequency. 

If we attempt to account for the radiation of energy in the manner 
required by the ordinary electrodynamics it will only make matters 
worse. As a result of the radiation of energy W would continually 
increase, and the above expressions (4) show that at the same time 
the frequency of revolution of the system would increase, and the 
dimensions of the orbit decrease. This process would not stop until 
the particles had approached so closely to one another that they no 
longer attracted each other. The quantity of energy which would 
be radiated away before this happened would be very great. If we 
were to treat these particles as geometrical points this energy would 
be infinitely great, and with the dimensions of the electrons as 
calculated from their mass (about 10~ 13 cm.), and of the nucleus as 
calculated by Rutherford (about 10~ 12 cm.), this energy would be 
many times greater than the energy changes with which we are 
familiar in ordinary atomic processes. 

It can be seen that it is impossible to employ Rutherford's atomic 
model so long as we confine ourselves exclusively to the ordinary 
electrodynamics. But this is nothing more than might have been 
expected. As I have mentioned we may consider it to be an 
established fact that it is impossible to obtain a satisfactory 



JO ON THE SPECTRUM OF HYDROGEN 

explanation of the experiments on temperature radiation with the 
aid of electrodynamics, no matter what atomic model be em- 
ployed. The fact that the deficiencies of the atomic model we are 
considering stand out so plainly is therefore perhaps no serious 
drawback; even though the defects of other atomic models are 
much better concealed they must nevertheless be present and will 
be just as serious. 

Quantum theory of spectra. Let us now try to overcome these 
difficulties by applying Planck's theory to the problem. 

It is readily seen that there can be no question of a direct appli- 
cation of Planck's theory. This theory is concerned with the emission 
and absorption of energy in a system of electrical particles, which 
oscillate with a given frequency per second, dependent only on the 
nature of the system and independent of the amount of energy 
contained in the system. In a system consisting of an electron and 
a nucleus the period of oscillation corresponds to the period of 
revolution of the electron. But the formula (4) for <w shows that the 
frequency of revolution depends upon W, i.e. on the energy of the 
system. Still the fact that we can not immediately apply Planck's 
theory to our problem is not as serious as it might seem to be, for 
in assuming Planck's theory we have manifestly acknowledged the 
inadequacy of the ordinary electrodynamics and have definitely 
parted with the coherent group of ideas on which the latter theory 
is based. In fact in taking such a step we can not expect that all 
cases of disagreement between the theoretical conceptions hitherto 
employed and experiment will be removed by the use of Planck's 
assumption regarding the quantum of the energy momentarily 
present in an oscillating system. We stand here almost entirely on 
virgin ground, and upon introducing new assumptions we need only 
take care not to get into contradiction with experiment. Time will 
have to show to what extent this can be avoided ; but the safest 
way is, of course, to make as few assumptions as possible. 

With this in mind let us first examine the experiments on 
temperature radiation. The subject of direct observation is the 
distribution of radiant energy over oscillations of the various wave 
lengths. Even though we may assume that this energy comes from 
systems of oscillating particles, we know little or nothing about 



ON THE SPECTRUM OF HYDROGEN 11 

these systems. No one has ever seen a Planck's resonator, nor 
indeed even measured its frequency of oscillation ; we can observe 
only the period of oscillation of the radiation which is emitted. It 
is therefore very convenient that it is possible to show that to 
obtain the laws of temperature radiation it is not necessary to 
make any assumptions about the systems which emit the radiation 
except that the amount of energy emitted each time shall be equal 
to hv, where h is Planck's constant and v is the frequency of the 
radiation. Indeed, it is possible to derive Planck's law of radiation 
from this assumption alone, as shown by Debye, who employed a 
method which is a combination of that of Planck and of Jeans. 
Before considering any further the nature of the oscillating systems 
let us see whether it is possible to bring this assumption about the 
emission of radiation into agreement with the spectral laws. 

If the spectrum of some element contains a spectral line corre- 
sponding to the frequency v it will be assumed that one of the 
atoms of the element (or some other elementary system) can emit 
an amount of energy hv. Denoting, the energy of the atom before 
and after the emission of the radiation by E l and E a we have 



(5) 



During the emission of the radiation the system may be regarded 
as passing from one state to another ; in order to introduce a name 
for these states, we shall call them "stationary" states, simply 
indicating thereby that they form some kind of waiting places 
between which occurs the emission of the energy corresponding to 
the various spectral lines. As previously mentioned the spectrum 
of an element consists of a series of lines whose wave lengths may 
be expressed by the formula (2). By comparing this expression 

with the relation given above it is seen that since v = - , where c 

A. 

is the velocity of light each of the spectral lines may be regarded 
as being emitted by the transition of a system between two stationary 
states in which the energy apart from an additive arbitrary 
constant is given by chF r (n a ) and chF s (n 2 ) respectively. Using 
this interpretation the combination principle asserts that a series 
of stationary states exists for the given system, and that it can 



12 ON THE SPECTRUM OF HYDROGEN 

pass from one to any other of these states with the emission of 
a monochromatic radiation. We see, therefore, that with a simple 
extension of our first assumption it is possible to give a formal 
explanation of the most general law of line spectra. 

Hydrogen spectrum. This result encourages us to make an 
attempt to obtain a clear conception of the stationary states which 
have so far only been regarded as formal. With this end in view, 
we naturally turn to the spectrum of hydrogen. The formula 
applying to this spectrum is given by the expression 

i^R_R 

\ nf nf 

According to our assumption this spectrum is produced by tran- 
sitions between a series of stationary states of a system, concerning 
which we can for the present only say that the energy of the system 
in the nth state, apart from an additive constant, is given by 

-- . Let us now try to find a connection between this and the 
ri* J 

model of the hydrogen atom. We assume that in the calculation 
of the frequency of revolution of the electron in the stationary states 
of the atom it will be possible to employ the above formula for . 
It is quite natural to make this assumption ; since, in trying to 
form a reasonable conception of the stationary states, there is, for 
the present at least, no other means available besides the ordinary 
mechanics. 

Corresponding to the nth stationary state in formula (4) for to, 

let us by way of experiment put W = ^ . This gives us 



The radiation of light corresponding to a particular spectral line 
is according to our assumption emitted by a transition between 
two stationary states, corresponding to two different frequencies of 
revolution, and we are not justified in expecting any simple re- 
lation between these frequencies of revolution of the electron and 
the frequency of the emitted radiation. You understand, of course, 
that I am by no means trying to give what might ordinarily be 
described as an explanation; nothing has been said here about 



ON THE SPECTRUM OF HYDROGEN 13 

how or why the radiation is emitted. On one point, however, we 
may expect a connection with the ordinary conceptions ; namely. 
that it will be possible to calculate the emission of slow electro- 
magnetic oscillations on the basis of the classical electrodynamics. 
This assumption is very strongly supported by the result of 
Lorentz's calculations which have already been described. From 
the formula for a> it is seen that the frequency of revolution de- 

creases as n increases, and that the expression approaches the 

n+i 

value 1. 

According to what has been said above, the frequency of the 
radiation corresponding to the transition between the (n + l)th 
and the nth stationary state is given by 

v = Re I 



. 
* (n + 

If n is very large this expression is approximately equal to 

v = 2Rc(n 3 . 

In order to obtain a connection with the ordinary electrodynamics 
let us now place this frequency equal to the frequency of revolu- 
tion, that is 

w n = 2Rc/n 3 . 

Introducing this value of &> in (6) we see that n disappears from 
the equation, and further that the equation will be satisfied only if 



The constant R is very accurately known, and is, as I have said 
before, equal to 109675. By introducing the most recent values 
for e, m and h the expression on the right-hand side of the equa- 
tion becomes equal to 1'09 . 10*. The agreement is as good as 
could be expected, considering the uncertainty in the experimental 
determination of the constants e, m and h. The agreement between 
our calculations and the classical electrodynamics is, therefore, 
fully as good as we are justified in expecting. 

We can not expect to obtain a corresponding explanation of the 
frequency values of the other stationary states. Certain simple 
formal relations apply, however, to all the stationary ^tates. By 
introducing the expression, which has been found for R, we 
get for the wth state W n = %nha) n . This equation is entirely 



14 ON THE SPECTRUM OF HYDROGEN 

analogous to Planck's assumption concerning the energy of a 
resonator. W in our system is readily shown to be equal to the 
average value of the kinetic energy of the electron during a 
single revolution. The energy of a resonator was shown by Planck 
you may remember to be always equal to nhv. Further the average 
value of the kinetic energy of Planck's resonator is equal to its 
potential energy, so that the average value of the kinetic energy 
of the resonator, according to Planck, is equal to %nhco. This 
analogy suggests another manner of presenting the theory, and it 
was just in this way that I was originally led into these con- 
siderations. When we consider how differently the equation is 
employed here and in Planck's theory it appears to me misleading 
to use this analogy as a foundation, and in the account I have 
given I have tried to free myself as much as possible from it. 

Let us continue with the elucidation of the calculations, and in 
the expression for 2a introduce the value of If which corresponds 
to the nth stationary state. This gives us 

2a = w 2 . -^3 = n 2 . ^, t = w 2 . 1 1 . 10~ 8 . . . .(8) 
ckR 22 



It is seen that for small values of n, we obtain values for the 
major axis of the orbit of the electron which are of the same 
order of magnitude as the values of the diameters of the atoms 
calculated from the kinetic theory of gases. For large values of 
n, 2a becomes very large in proportion to the calculated dimensions 
of the atoms. This, however, does not necessarily disagree with 
experiment. Under ordinary circumstances a hydrogen atom will 
probably exist only in the state corresponding to n = 1. For this 
state W will have its greatest value and, consequently, the atom 
will have emitted the largest amount of energy possible ; this will 
therefore represent the most stable state of the atom from which 
the system can not be transferred except by adding energy to it 
from without. The large values for 2a corresponding to large n need 
not, therefore, be contrary to experiment ; indeed, we may in these 
large values seek an explanation of the fact, that in the laboratory 
it has hitherto not been possible to observe the hydrogen lines 
corresponding to large values of n in Balmer's formula, while they 
have been observed in the spectra of certain stars. In order that 
the large orbits of the electrons may not be disturbed by electrical 



ON THE SPECTRUM OF HYDROGEN 15 

forces from the neighbouring atoms the pressure will have to be 
very low, so low, indeed, that it is impossible to obtain sufficient 
light from a Geissler tube of ordinary dimensions. In the stars, 
however, we may assume that we have to do with hydrogen which 
is exceedingly attenuated and distributed throughout an enor- 
mously large region of space. 

The Pickering lines. You have probably noticed that we have 
not mentioned at all the spectrum found in certain stars which 
according to the opinion then current was assigned to hydrogen, 
and together with the ordinary hydrogen spectrum was considered 
by Rydberg to form a connected system of lines completely 
analogous to the spectra of other elements. You have probably 
also perceived that difficulties would arise in interpreting this 
spectrum by means of the assumptions which have been employed. 
If such an attempt were to be made it would be necessary to give 
up the simple considerations which lead to the expression (7) for 
the constant R. We shall see, however, that it appears possible to 
explain the occurrence of this spectrum in another way. Let us 
suppose that it is not due to hydrogen, but to some other simple 
system consisting of a single electron revolving about a nucleus 
with an electrical charge Ne. The expression for to becomes then 

2 W* 



Repeating the same calculations as before only in the inverse 
order we find, that this system will emit a line spectrum given by 
the expression 



I 

By comparing this formula with the formula for Pickering's and 
Rydberg's series, we see that the observed lines can be explained 
on the basis of the theory, if it be assumed that the spectrum is 
due to an electron revolving about a nucleus with a charge 20, or 
according to Rutherford's theory around the nucleus of a helium 
atom. The fact that the spectrum in question is not observed in 
an ordinary helium tube, but only in stars, may be accounted for 



16 ON THE SPECTRUM OF HYDROGEN 

by the high degree of ionization which is required for the produc- 
tion of this spectrum ; a neutral helium atom contains of course 
two electrons while the system under consideration contains 
only one. 

These conclusions appear to be supported by experiment. 
Fowler, as I have mentioned, has recently succeeded in observing 
Pickering's and Rydberg's lines in a laboratory experiment. By 
passing a very heavy current through a mixture of hydrogen and 
helium Fowler observed not only these lines but also a new series 
of lines. This new series was of the same general type, the wave 
length being given approximately by 



Fowler interpreted all the observed lines as the hydrogen spectrum 
sought for. With the observation of the latter series of lines, 
however, the basis of the analogy between the hypothetical 
hydrogen spectrum and the other spectra disappeared, and thereby 
also the foundation upon which Rydberg had founded his conclu- 
sions ; on the contrary it is seen, that the occurrence of the lines 
was exactly what was to be expected on our view. 

In the following table the first column contains the wave 
lengths measured by Fowler, while the second contains the limiting 
values of the experimental errors given by him ; in the third 
column we find the products of the wave lengths by the quantity 

( -) 10 10 ; the values employed for n^ and HZ are enclosed in 

\n 1 n% I 

parentheses in the last column. 

X . 108 Limit of error X . (^ - -^ . 10 10 

4685-98 0-01 22779'! (3:4) 

3203-30 0-05 22779'0 (3:5) 

2733-34 0-05 22777'8 (3:6) 

2511-31 0-05 22778-3 (3:7) 

2385-47 0-05 22777'9 (3:8) 

2306-20 0-10 22777'3 (3:9) 

2252-88 O'lO 22779'! (3:10) 

5410-5 1-0 22774 (4:7) 

4541-3 0-25 22777 (4:9) 

4200-3 0-5 22781 (4:11) 



ON THE SPECTRUM OF HYDROGEN 17 

The values of the products are seen to be very nearly equal, 
while the deviations are of the same order of magnitude as the 
limits of experimental error. The value of the product 



should for this spectrum, according to the formula (9), be exactly 
^ of the corresponding product for the hydrogen spectrum. From 
the tables on pages 1 and 16 we find for these products 91153 
and 22779, and dividing the former by the latter we get 4'0016. 
This value is very nearly equal to 4; the deviation is, however, 
much greater than can be accounted for in any way by the errors 
of the experiments. It has been easy, however, to find a theo- 
retical explanation of this point. In all the foregoing calculations 
we have assumed that the mass of the nucleus is infinitely great 
compared to that of the electron. This is of course not the 
case, even though it holds to a very close approximation; for a 
hydrogen atom the ratio of the mass of the nucleus to that of the 
electron will be about 1850 and for a helium atom four times as 
great. 

If we consider a system consisting of an electron revolving about. 
a nucleus with a charge Ne and a mass M, we find the following 
expression for the frequency of revolution of the system : 
a _2 W 3 (M+m) 
~7r 2 NWMm 

From this formula we find in a manner quite similar to that 
previously employed that the system will emit a line spectrum, 
the wave lengths of which are given by the formula 

1\ 



If with the aid of this formula we try to find the ratio of the 
product for the hydrogen spectrum to that of the hypothetical 
helium spectrum we get the value 4'00163 which is in complete 
agreement with the preceding value calculated from the experi- 
mental observations. 

I must further mention that Evans has made some experiments 
to determine whether the spectrum in question is due to hydrogen 
or helium. He succeeded in observing one of the lines in very 

B. 2 



18 ON THE SPECTRUM OF HYDROGEN 

pure helium ; there was, at any rate, not enough hydrogen present 
to enable the hydrogen lines to be observed. Since in any event 
Fowler does not seem to consider such evidence as conclusive it is 
to be hoped that these experiments will be continued. There is, 
however, also another possibility of deciding this question. As is 
evident from the formula (10), the helium spectrum under con- 
sideration should contain, besides the lines observed by Fowler, a 
series of lines lying close to the ordinary hydrogen lines. These 
lines may be obtained by putting % = 4>, n z = 6, 8, 10, etc. Even 
if these lines were present, it would be extremely difficult to 
observe them on account of their position with regard to the 
hydrogen lines, but should they be observed this would probably 
also settle the question of the origin of the spectrum, since no 
reason would seem to be left to assume the spectrum to be due to 
hydrogen. 

Other spectra. For the spectra of other elements the problem 
becomes more complicated, since the atoms contain a larger 
number of electrons. It has not yet been possible on the basis of 
this theory to explain any other spectra besides those which I 
have already mentioned. On the other hand it ought to be 
mentioned that the general laws applying to the spectra are very 
simply interpreted on the basis of our assumptions. So far as the 
combination principle is concerned its explanation is obvious. In 
the method we have employed our point of departure was largely 
determined by this particular principle. But a simple explanation 
can be also given of the other general law, namely, the occurrence 
of Rydberg's constant in all spectral formulae. Let us assume 
that the spectra under consideration, like the spectrum of hydrogen, 
are emitted by a neutral system, and that they are produced by 
the binding of an electron previously removed from the system. 
If such an electron revolves about the nucleus in an orbit which 
is large in proportion to that of the other electrons it will be 
subjected to forces much the same as the electron in a hydrogen 
atom, since the inner electrons individually will approximately 
neutralize the effect of a part of the positive charge of the nucleus. 
We may therefore assume that for this system there will exist a 
series of stationary states in which the motion of the outermost 



ON THE SPECTRUM OF HYDROGEN 19 

electron is approximately the same as in the stationary states of a 
hydrogen atom. I shall not discuss these matters any further, 
but shall only mention that they lead to the conclusion that 
Rydberg's constant is not exactly the same for all elements. 
The expression for this constant will in fact contain the factor 

-^17 , where M is the mass of the nucleus. The correction is 

M + m 

exceedingly small for elements of large atomic weight, but for 
hydrogen it is, from the point of view of spectrum analysis, very 
considerable. If the procedure employed leads to correct results, it 
is not therefore permissible to calculate Rydberg's constant directly 
from the hydrogen spectrum ; the value of the universal constant 
should according to the theory be 109735 and not 109675. 

I shall not tire you any further with more details ; I hope to 
return to these questions here in the Physical Society, and to 
show how, on the basis of the underlying ideas, it is possible 
to develop a theory for the structure of atoms and molecules. 
Before closing I only wish to say that I hope I have expressed 
myself sufficiently clearly so that you have appreciated the extent 
to which these considerations conflict with the admirably coherent 
group of conceptions which have been rightly termed the classical 
theory of electrodynamics. On the other hand, by emphasizing 
this conflict, I have tried to convey to you the impression that it 
may be also possible in the course of time to discover a certain 
coherence in the new ideas. 



3 2 



ESSAY II* 

ON THE SERIES SPECTRA OF THE ELEMENTS 

I. INTRODUCTION 

The subject on which I have the honour to speak here, at the 
kind invitation of the Council of your society, is very extensive and 
it would be impossible in a single address to give a comprehensive 
survey of even the most important results obtained in the theory 
of spectra. In what follows I shall try merely to emphasize some 
points of view which seem to me important when considering the 
present state of the theory of spectra and the possibilities of its 
development in the near future. I regret in this connection not to 
have time to describe the history of the development of spectral 
theories, although this would be of interest for our purpose. No 
difficulty, however, in understanding this lecture need be experienced 
on this account, since the points of view underlying previous 
attempts to explain the spectra differ fundamentally from those upon 
which the following considerations rest. This difference exists both 
in the development of our ideas about the structure of the atom 
and in the manner in which these ideas are used in explaining the 
spectra. 

We shall assume, according to Rutherford's theory, that an atom 
consists of a positively charged nucleus with a number of electrons 
revolving about it. Although the nucleus is assumed to be very 
small in proportion to the size of the whole atom, it will contain 
nearly the entire mass of the atom. I shall not state the reasons 
which led to the establishment of this nuclear theory of the atom, 
nor describe the very strong support which this theory has received 
from very different sources. I shall mention only that result 
which lends such charm and simplicity to the modern development 
of the atomic theory. I refer to the idea that the number of elec- 
trons in a neutral atom is exactly equal to the number, giving the 
position of the element in the periodic table, the so-called "atomic 
number." This assumption, which was first proposed by van den 
Broek, immediately suggests the possibility ultimately of deriving 
* Address delivered before the Physical Society in Berlin, April 27, 1920. 



ON THE SERIES SPECTRA OF THE ELEMENTS 21 

the explanation of the physical and chemical properties of the 
elements from their atomic numbers. If, however, an explanation 
of this kind is attempted on the basis of the classical laws of 
mechanics and electrodynamics, insurmountable difficulties are en- 
countered. These difficulties become especially apparent when we 
consider the spectra of the elements. In fact, the difficulties are 
here so obvious that it would be a waste of time to discuss them in 
detail. It is evident that systems like the nuclear atom, if based 
upon the usual mechanical and electrodynamical conceptions, 
would not even possess sufficient stability to give a spectrum con- 
sisting of sharp lines. 

In this lecture I shall use the ideas of the quantum theory. It 
will not be necessary, particularly here in Berlin, to consider in 
detail how Planck's fundamental work on temperature radiation 
has given rise to this theory, according to which the laws governing 
atomic processes exhibit a definite element of discontinuity. I shall 
mention only Planck's chief result about the properties of an ex- 
ceedingly simple kind of atomic system, the Planck "oscillator." 
This consists of an electrically charged particle which can execute 
harmonic oscillations about its position of equilibrium with a fre- 
quency independent of the amplitude. By studying the statistical 
equilibrium of a number of such systems in a field of radiation 
Planck was led to the conclusion that the emission and absorption 
of radiation take place in such a manner, that so far as a statistical 
equilibrium is concerned only certain distinctive states of the 
oscillator are to be taken into consideration. In these states the 
energy of the system is equal to a whole multiple of a so-called 
"energy quantum," which was found to be proportional to the fre- 
quency of the oscillator. The particular energy values are therefore 
given by the well-known formula 

E n = nha>, (1) 

where n is a whole number, a> the frequency of vibration of the 
oscillator, and h is Planck's constant. 

If we attempt to use this result to explain the spectra of the 
elements, however, we encounter difficulties, because the motion of 
the particles in the atom, in spite of its simple structure, is in general 
exceedingly complicated compared with the motion of a Planck 



22 ON THE SERIES SPECTRA OF THE ELEMENTS 

oscillator. The question then arises, how Planck's result ought to 
be generalized in order to make its application possible. Different 
points of view immediately suggest themselves. Thus we might 
regard this equation as a relation expressing certain characteristic 
properties of the distinctive motions of an atomic system and try 
to obtain the general form of these properties. On the other hand, 
we may also regard equation (1) as a statement about a property 
of the process of radiation and inquire into the general laws which 
control this process. 

In Planck's theory it is taken for granted that the frequency of 
the radiation emitted and absorbed by the oscillator is equal to its 
own frequency, an assumption which may be written 



(2) 



if in order to make a sharp distinction between the frequency of 
the emitted radiation and the frequency of the particles in the atoms, 
we here and in the following denote the former by v and the latter 
by o>. We see, therefore, that Planck's result may be interpreted to 
mean, that the oscillator can emit and absorb radiation only in 
"radiation quanta" of magnitude 



(3) 



It is well known that ideas of this kind led Einstein to a theory 
of the photoelectric effect. This is of great importance, since it 
represents the first instance in which the quantum theory was 
applied to a phenomenon of non-statistical character. I shall not 
here discuss the familiar difficulties to which the "hypothesis of 
light quanta" leads in connection with the phenomena of inter- 
ference, for the explanation of which the classical theory of radiation 
has shown itself to be so remarkably suited. Above all I shall not 
consider the problem of the nature of radiation, I shall only attempt 
to show how it has been possible in a purely formal manner to 
develop a spectral theory, the essential elements of which may be 
considered as a simultaneous rational development of the two wavs 
of interpreting Planck's result. 



ON THE SERIES SPECTRA OF THE ELEMENTS 23 



II. GENERAL PRINCIPLES OF THE QUANTUM THEORY 
OF SPECTRA 

In order to explain the appearance of line spectra we are com- 
pelled to assume that the emission of radiation by an atomic system 
takes place in such a manner that it is not possible to follow the 
emission in detail by means of the usual conceptions. Indeed, these 
do not even offer us the means of calculating the frequency of the 
emitted radiation. We shall see, however, that it is possible to give 
a very simple explanation of the general empirical laws for the 
frequencies of the spectral lines, if for each emission of radiation 
by the atom we assume the fundamental law to hold, that during 
the entire period of the emission the radiation possesses one and 
the same frequency v, connected with the total energy emitted by 
the frequency relation 

hv = E'-E" (4) 

Here E' and E" represent the energy of the system before and 
after the emission. 

If this law is assumed, the spectra do not give us information 
about the motion of the particles in the atom, as is supposed in the 
usual theory of radiation, but only a knowledge of the energy 
changes in the various processes which can occur in the atom. 
From this point of view the spectra show the existence of certain 
definite energy values corresponding to certain distinctive states 
of the atoms. These states will be called the stationary states of 
the atoms, since we shall assume that the atom can remain a finite 
time in each state, and can leave this state only by a process of 
transition to another stationary state. Notwithstanding the funda- 
mental departure from the ordinary mechanical and electrodynamical 
conceptions, we shall see, however, that it is possible to give a 
rational interpretation of the evidence provided by the spectra on 
the basis of these ideas. 

Although we must assume that the ordinary mechanics can not 
be used to describe the transitions between the stationary states, 
nevertheless, it has been found possible to develop a consistent 
theory on the assumption that the motion in these states can be 
described by the use of the ordinary mechanics. Moreover, although 
the process of radiation can not be described on the basis of the 



24 ON THE SERIES SPECTRA OF THE ELEMENTS 

ordinary theory of electrodynamics, according to which the nature 
of the radiation emitted by an atom is directly related to the har- 
monic components occurring in the motion of the system, there is 
found, nevertheless, to exist a far-reaching correspondence between 
the various types of possible transitions between the stationary 
states on the one hand and the various harmonic components of the 
motion on the other hand. This correspondence is of such a nature, 
that the present theory of spectra is in a certain sense to be regarded 
as a rational generalization of the ordinary theory of radiation. 

Hydrogen spectrum. In order that the principal points may 
stand out as clearly as possible I shall, before considering the more 
complicated types of series spectra, first consider the simplest spec- 
trum, namely, the series spectrum of hydrogen. This spectrum 
consists of a number of lines whose frequencies are given with great 
exactness by Balmer's formula 

K K 
~(n'J (n'r " 

where K is a constant, and ri and n" are whole numbers. If we put 
n" =* 2 and give to n' the values 3, 4, etc., we get the well-known 
Balmer series of hydrogen. If we put n" = 1 or n" = 3 we obtain 
respectively the ultra-violet and infra-red series. We shall assume 
the hydrogen atom simply to consist of a positively charged nucleus 
with a single electron revolving about it. For the sake of simplicity 
we shall suppose the mass of the nucleus to be infinite in comparison 
with the mass of the electron, and further we shall disregard the 
small variations in the motion due to the change in mass of the 
electron with its velocity. With these simplifications the electron 
will describe a closed elliptical orbit with the nucleus at one of the 
foci The frequency of revolution w and the major axis 2a of the 
orbit will be connected with the energy of the system by the fol- 
lowing equations: 



Here e is the charge of the electron and m its mass, while W is the 
work required to remove the electron to infinity. 

The simplicity of these formulae suggests the possibility of using 
them in an attempt to explain the spectrum of hydrogen. This, 



ON THE SERIES SPECTRA OF THE ELEMENTS 25 

however, is not possible so long as we use the classical theory of 
radiation. It would not even be possible to understand how hydrogen 
could emit a spectrum consisting of sharp lines; for since co varies 
with W, the frequency of the emitted radiation would vary con- 
tinuously during the emission. We can avoid these difficulties if 
we use the ideas of the quantum theory. If for each line we form 
the product hv by multiplying both sides of (5) by h, then, since 
the right-hand side of the resulting relation may be written as 
the difference of two simple expressions, we are led by comparison 
with formula (4) to the assumption that the separate lines of the 
spectrum will be emitted by transitions between two stationary 
states, forming members of an infinite series of states, in which the 
energy in the nth state apart from an arbitrary additive constant is 
determined by the expression 



The negative sign has been chosen because the energy of the atom 
will be most simply characterized by the work W required to remove 

the electron completely from the atom. If we now substitute 

for W in formula (6), we obtain the following expression for the fre- 
quency and the major axis in the nth stationary state: 



A comparison between the motions determined by these equations 
and the distinctive states of a Planck resonator may be shown to 
offer a theoretical determination of the constant K. Instead of 
doing this I shall show how the value of K can be found by a simple 
comparison of the spectrum emitted with the motion in the stationary 
states, a comparison which at the same time will lead us to the 
principle of correspondence. 

We have assumed that each hydrogen line is the result of a 
transition between two stationary states of the atom corresponding 
to different values of n. Equations (8) show that the frequency of 
revolution and the major axis of the orbit can be entirely different 
in the two states, since, as the energy decreases, the major axis of 
the orbit becomes smaller and the frequency of revolution increases. 



26 ON THE SERIES SPECTRA OF THE ELEMENTS 

In general, therefore, it will be impossible to obtain a relation be- 
tween the frequency of revolution of the electrons and the frequency 
of the radiation as in the ordinary theory of radiation. If, however, 
we consider the ratio of the frequencies of revolution in two stationary 
states corresponding to given values of ri and n", we see that this 
ratio approaches unity as n' and n" gradually increase, if at the 
same time the difference n n" remains unchanged. By consider- 
ing transitions corresponding to large values of n' and n" we may 
therefore hope to establish a certain connection with the ordinary 
theory. For the frequency of the radiation emitted by a transition, 
we get according to (5) 

K K > " n ' +n " 



If now the numbers n' and ri' are large in proportion to their differ- 
ence, we see that by equations (8) this expression may be written 
approximately, 



where &> represents the frequency of revolution in the one or the 
other of the two stationary states. Since n' n" is a whole number, 
we see that the first part of this expression, i.e. (n n") &>, is the 
same as the frequency of one of the harmonic components into 
which the elliptical motion may be decomposed. This involves the 
well-known result that for a system of particles having a periodic 
motion of frequency a>, the displacement f of the particles in a given 
direction in space may be represented as a function of the time by 
a trigonometric series of the form 



............... (11) 

where the summation is to be extended over all positive integral 
values of T. 

We see, therefore, that the frequency of the radiation emitted 
by a transition between two stationary states, for which the numbers 
n' and n" are large in proportion to their difference, will coincide 
with the frequency of one of the components of the radiation, which 
according to the ordinary ideas of radiation would be expected from 
the motion of the atom in these states, provided the last factor on 



ON THE SERIES SPECTRA OF THE ELEMENTS 27 

the right-hand side of equation (10) is equal to 1. This condition, 
which is identical to the condition 



(12) 



is in fact fulfilled, if we give to K its value as found from measure- 
ments on the hydrogen spectrum, and if for e, m and h we use the 
values obtained directly from experiment. This agreement clearly 
gives us a connection between the spectrum and the atomic model of 
hydrogen, which is as close as could reasonably be expected con- 
sidering the fundamental difference between the ideas of the quan- 
tum theory and of the ordinary theory of radiation. 

The correspondence principle. Let us now consider somewhat 
more closely this relation between the spectra one would expect on 
the basis of the quantum theory, and on the ordinary theory of 
radiation. The frequencies of the spectral lines calculated according 
to both methods agree completely in the region where the stationary 
states deviate only little from one another. We must not forget, 
however, that the mechanism of emission in both cases is different. 
The different frequencies corresponding to the various harmonic 
components of the motion are emitted simultaneously according to 
the ordinary theory of radiation and with a relative intensity de- 
pending directly upon the ratio of the amplitudes of these oscilla- 
tions. But according to the quantum theory the various spectral 
lines are emitted by entirely distinct processes, consisting of tran- 
sitions from one stationary state to various adjacent states, so that 
the radiation corresponding to the rth "harmonic" will be emitted 
by a transition for which n' n" = r. The relative intensity 
with which each particular line is emitted depends consequently 
upon the relative probability of the occurrence of the different 
transitions. 

This correspondence between the frequencies determined by the 
two methods must have a deeper significance and we are led to 
anticipate that it will also apply to the intensities. This is equiva- 
lent to the statement that, when the quantum numbers are large, 
the relative probability of a particular transition is connected in a 
simple manner with the amplitude of the corresponding harmonic 
component in the motion. 



28 ON THE SERIES SPECTRA OF THE ELEMENTS 

This peculiar relation suggests a general law for the occurrence 
of transitions between stationary states. Thus we shall assume that 
even when the quantum numbers are small the possibility of 
transition between two stationary states is connected with the 
presence of a certain harmonic component in the motion of the 
system. If the numbers n and n" are not large in proportion to 
their difference, the numerical value of the amplitudes of these 
components in the two stationary states may be entirely different. 
We must be prepared to find, therefore, that the exact connection 
between the probability of a transition and the amplitude of the 
corresponding harmonic component in the motion is in general 
complicated like the connection between the frequency of the radia- 
tion and that of the component. From this point of view, for 
example, the green line Hp of the hydrogen spectrum which cor- 
responds to a transition from the fourth to the second stationary 
state may be considered in a certain sense to be an "octave" of the 
red line H a , corresponding to a transition from the third to the 
second state, even though the frequency of the first line is by no 
means twice as great as that of the latter. In fact, the transition 
giving rise to Hp may be regarded as due to the presence of a har- 
monic oscillation in the motion of the atom, which is an octave 
higher than the oscillation giving rise to the emission of H a . 

Before considering other spectra, where numerous opportunities 
will be found to use this point of view, I shall briefly mention an 
interesting application to the Planck oscillator. If from (1) and (4) 
we calculate the frequency, which would correspond to a transition 
between two particular states of such an oscillator, we find 

v = (ri-n")a>, ........................ (13) 



where n' and n" are the numbers characterizing the states. It was 
an essential assumption in Planck's theory that the frequency of 
the radiation emitted and absorbed by the oscillator is always equal 
to to. We see that this assumption is equivalent to the assertion 
that transitions occur only between two successive stationary states 
in sharp contrast to the hydrogen atom. According to our view, 
however, this was exactly what might have been expected, for we 
must assume that the essential difference between the oscillator 
and the hydrogen atom is that the motion of the oscillator is simple 



ON THE SERIES SPECTRA OF THE ELEMENTS 29 

harmonic. We can see that it is possible to develop a formal theory 
of radiation, in which the spectrum of hydrogen and the simple 
spectrum of a Planck oscillator appear completely analogous. This 
theory can only be formulated by one and the same condition for 
a system as simple as the oscillator. In general this condition 
breaks up into two parts, one concerning the fixation of the stationary 
states, and the other relating to the frequency of the radiation 
emitted by a transition between these states. 

General spectral laws. Although the series spectra of the 
elements of higher atomic number have a more complicated struc- 
ture than the hydrogen spectrum, simple laws have been discovered 
showing a remarkable analogy to the Balmer formula. Rydberg 
and Ritz showed that the frequencies in the series spectra of many 
elements can be expressed by a formula of the type 

* =/*('')-/*("'), (14) 

where n' and n" are two whole numbers and f k > and f k are two 
functions belonging to a series of functions characteristic of the 
element. These functions vary in a simple manner with n and in 
particular converge to zero for increasing values of n. The various 
series of lines are obtained from this formula by allowing the first 
term fa (n") to remain constant, while a series of consecutive whole 
numbers are substituted for n' in the second terrn/f (n'). According 
to the Ritz combination principle the entire spectrum may then 
be obtained by forming every possible combination of two values 
among all the quantities f k (n). 

The fact that the frequency of each line of the spectrum may be 
written as the difference of two simple expressions depending upon 
whole numbers suggests at once that the terms on the right-hand 
side multiplied by h may be placed equal to the energy in the 
various stationary states of the atom. The existence in the spectra 
of the other elements of a number of separate functions of n compels 
us to assume the presence not of one but of a number of series of 
stationary states, the energy of the nth state of the &th series apart 
from an arbitrary additive constant being given by 

E k (n) = -hf k (n} (15) 

This complicated character of the ensemble of stationary states of 
atoms of higher atomic number is exactly what was to be expected 



30 



ON THE SERIES SPECTRA OF THE ELEMENTS 



from the relation between the spectra calculated on the quantum 
theory, and the decomposition of the motions of the atoms into 
harmonic oscillations. From this point of view we may regard the 
simple character of the stationary states of the hydrogen atom as 
intimately connected with the simple periodic character of this 
atom. Where the neutral atom contains more than one electron, we 
find much more complicated motions with correspondingly compli- 
cated harmonic components. We must therefore expect a more 
complicated ensemble of stationary states, if we are still to have a 
corresponding relation between the motions in the atom and the 
spectrum. In the course of the lecture we shall trace this corre- 
spondence in detail, and we shall be led to a simple explanation of 
the apparent capriciousness in the occurrence of lines predicted by 
the combination principle. 

The following figure gives a survey of the stationary states of 
the sodium atom deduced from the series terms. 




(k-l) 
(X-2) 
(k-3) 
(te-4) 



J a * 5 a 

Diagram of the series spectrum of sodium. 

The stationary states are represented by black dots whose distance 
from the vertical line a a is proportional to the numerical value 
of the energy in the states. The arrows in the figure indicate the 
transitions giving those lines of the sodium spectrum which appear 
under the usual conditions of excitation. The arrangement of the 
states in horizontal rows corresponds to the ordinary arrangement 
of the "spectral terms" in the spectroscopic tables. Thus, the states 
in the first row (8) correspond to the variable term in the "sharp 
series," the lines of which are emitted by transitions from these 
states to the first state in the second row. The states in the second 



ON THE SERIES SPECTRA OF THE ELEMENTS 31 

row (P) correspond to the variable term in the "principal series" 
which is emitted by transitions from these states to the first state 
in the S row. The D states correspond to the variable term in the 
" diffuse series," which like the sharp series is emitted by transitions 
to the first state in the P row, and finally the B states correspond 
to the variable term in the "Bergmann" series (fundamental series), 
in which transitions take place to the first state in the D row. The 
manner in which the various rows are arranged with reference to 
one another will be used to illustrate the more detailed theory 
which will be discussed later. The apparent capriciousness of the 
combination principle, which I mentioned, consists in the fact that 
under the usual conditions of excitation not all the lines belonging 
to possible combinations of the terms of the sodium spectrum appear, 
but only those indicated in the figure by arrows. 

The general question of the fixation of the stationary states of 
an atom containing several electrons presents difficulties of a pro- 
found character which are perhaps still far from completely solved. 
It is possible, however, to obtain an immediate insight into the 
stationary states involved in the emission of the series spectra by 
considering the empirical laws which have been discovered about 
the spectral terms. According to the well-known law discovered by 
Rydberg for the spectra of elements emitted under the usual con- 
ditions of excitation the functions fk(n) appearing in formula (14) 
can be written in the form 

/*<)-**<), (16) 

where <f> k (n) represents a function which converges to unity for 
large values of n. K is the same constant which appears in formula 
(5) for the spectrum of hydrogen. This result must evidently be 
explained by supposing the atom to be electrically neutral in these 
states and one electron to be moving round the nucleus in an orbit 
the dimensions of which are very large in proportion to the distance 
of the other electrons from the nucleus. We see, indeed, that in 
this case the electric force acting on the outer electron will to a 
first approximation be the same as that acting upon the electron 
in the hydrogen atom, and the approximation will be the better 
the larger the orbit. 



32 ON THE SERIES SPECTRA OF THE ELEMENTS 

On account of the limited time I shall not discuss how this 
explanation of the universal appearance of Rydberg's constant in 
the arc spectra is convincingly supported by the investigation of 
the " spark spectra." These are emitted by the elements under the 
influence of very strong electrical discharges, and come from ionized 
not neutral atoms. It is important, however, that I should indicate 
briefly how the fundamental ideas of the theory and the assumption 
that in the states corresponding to the spectra one electron moves 
in an orbit around the others, are both supported by investigations 
on selective absorption and the excitation of spectral lines by 
bombardment by electrons. 

Absorption and excitation of radiation. Just as we have 
assumed that each emission of radiation is due to a transition from 
a stationary state of higher to one of lower energy, so also we must 
assume absorption of radiation by the atom to be due to a transition 
in the opposite direction. For an element to absorb light corre- 
sponding to a given line in its series spectrum, it is therefore 
necessary for the atom of this element to be in that one of the two 
states connected with the line possessing the smaller energy value. 
If we now consider an element whose atoms in the gaseous state 
do not combine into molecules, it will be necessary to assume that 
under ordinary conditions nearlyall the atoms exist in that stationary 
state in which the value of the energy is a minimum. This state 
I shall call the normal state. We must therefore expect that the 
absorption spectrum of a monatomic gas will contain only those 
lines of the series spectrum, whose emission corresponds to tran- 
sitions to the normal state. This expectation is completely confirmed 
by the spectra of the alkali metals. The absorption spectrum of 
sodium vapour, for example, exhibits lines corresponding only to 
the principal series, which as mentioned in the description of the 
figure corresponds with transitions to the state of minimum energy. 
Further confirmation of this view of the process of absorption is 
given by experiments on resonance radiation. Wood first showed 
that sodium vapour subjected to light corresponding to the first 
line of the principal series the familiar yellow line acquires the 
ability of again emitting a radiation consisting only of the light of 
this line. We can explain this by supposing the sodium atom to 



ON THE SERIES SPECTRA OF THE ELEMENTS 33 

have been transferred from the normal state to the first state in 
the second row. The fact that the resonance radiation does not 
exhibit the same degree of polarization as the incident light is 
in perfect agreement with our assumption that the radiation from 
the excited vapour is not a resonance phenomenon in the sense of 
the ordinary theory of radiation, but on the contrary depends on a 
process which is not directly connected with the incident radiation. 
The phenomenon of the resonance radiation of the yellow sodium 
line is, however, not quite so simple as I have indicated, since, as 
you know, this line is really a doublet. This means that the variable 
terms of the principal series are not simple but are represented by 
two values slightly different from one another. According to our 
picture of the origin of the sodium spectrum this means that the 
P states in the second row in the figure as opposed to the S states 
in the first row are not simple, but that for each place in this row 
there are two stationary states. The energy values differ so little 
from one another that it is impossible to represent them in the 
figure as separate dots. The emission (and absorption) of the two 
components of the yellow line are, therefore, connected with two 
different processes. This was beautifully shown by some later re- 
searches of Wood and Dunoyer. They found that if sodium vapour 
is subjected to radiation from only one of the two components of 
the yellow line, the resonance radiation, at least at low pressures, 
consists only of this component. These experiments were later 
continued by Strutt, and were extended to the case where the 
exciting line corresponded to the second line in the principal series. 
Strutt found that the resonance radiation consisted apparently only 
to a small extent of light of the same frequency as the incident 
light, while the greater part consisted of the familiar yellow line. 
This result must appear very astonishing on the ordinary ideas of 
resonance, since, as Strutt pointed out, no rational connection exists 
between the frequencies of the first and second lines of the principal 
series. It is however easily explained from our point of view. From 
the figure it can be seen that when an atom has been transferred 
into the second state in the second row, in addition to the direct 
return to the normal state, there are still two other transitions 
which may give rise to radiation, namely the transitions to the 
second state in the first row and to the first state in the third row. 
B. 3 



34 ON THE SERIES SPECTRA OF THE ELEMENTS 

The experiments seem to indicate that the second of these three 
transitions is most probable, and I shall show later that there is 
some theoretical justification for this conclusion. By this transition, 
which results in the emission of an infra-red line which could not 
be observed with the experimental arrangement, the atom is taken 
to the second state of the first row, and from this state only 
one transition is possible, which again gives an infra-red line. This 
transition takes the atom to the first state in the second row, and 
the subsequent transition to the normal state then gives rise to the 
yellow line. Strutt discovered another equally surprising result, 
that this yellow resonance radiation seemed to consist of both 
components of the first line of the principal series, even when the 
incident light consisted of only one component of the second line 
of the principal series. This is in beautiful agreement with our 
picture of the phenomenon. We must remember that the states in 
the first row are simple, so when the atom has arrived in one of 
these it has lost every possibility of later giving any indication 
from which of the two states in the second row it originally came. 

Sodium vapour, in addition to the absorption corresponding to 
the lines of the principal series, exhibits a selective absorption in a 
continuous spectral region beginning at the limit of this series and 
extending into the ultra violet. This confirms in a striking manner 
our assumption that the absorption of the lines of the principal 
series of sodium results in final states of the atom in which one of 
the electrons revolves in larger and larger orbits. For we must 
assume that this continuous absorption corresponds to transitions 
from the normal state to states in which the electron is in a position 
to remove itself infinitely far from the nucleus. This phenomenon 
exhibits a complete analogy with the photoelectric effect from an 
illuminated metal plate in which, by using light of a suitable 
frequency, electrons of any velocity can be obtained. The frequency, 
however, must always lie above a certain limit connected according 
to Einstein's theory in a simple manner with the energy necessary 
to bring an electron out of the metal. 

This view of the origin of the emission and absorption spectra 
has been confirmed in a very interesting manner by experiments 
on the excitation of spectral lines and production of ionization by 
electron bombardment. The chief advance in this field is due to the 



ON THE SERIES SPECTRA OF THE ELEMENTS 35 

well-known experiments of Franck and Hertz. These investigators 
obtained their first important results from their experiments on 
mercury vapour, whose properties particularly facilitate such ex- 
periments. On account of the great importance of the results, these 
experiments have been extended to most gases and metals that can 
be obtained in a gaseous state. With the aid of the figure I shall 
briefly illustrate the results for the case of sodium vapour. It was 
found that the electrons upon colliding with the atoms were thrown 
back with undiminished velocity when their energy was less than 
that required to transfer the atom from the normal state to the 
next succeeding stationary state of higher energy value. In the 
case of sodium vapour this means from the first state in the first 
row to the first state in the second row. As soon, however, as the 
energy of the electron reaches this critical value, a new type of 
collision takes place, in which the electron loses all its kinetic 
energy, while at the same time the vapour is excited and emits a 
radiation corresponding to the yellow line. This is what would be 
expected, if by the collision the atom was transferred from the 
normal state to the first one in the second row. For some time it 
was uncertain to what extent this explanation was correct, since 
in the experiments on mercury vapour it was found that, together 
with the occurrence of non-elastic impacts, ions were always formed 
in the vapour. From our figure, however, we would expect ions 
to be produced only when the kinetic energy of the electrons is 
sufficiently great to bring the atom out of the normal state to the 
common limit of the states. Later experiments, especially by Davis 
and Goucher, have settled this point. It has been shown that ions 
can only be directly produced by collisions when the kinetic energy 
of the electrons corresponds to the limit of the series, and that the 
ionization found at first was an indirect effect arising from the 
photoelectric effect produced at the metal walls of the apparatus 
by the radiation arising from the return of the mercury atoms to 
the normal state. These experiments provide a direct and inde- 
pendent proof of the reality of the distinctive stationary states, 
whose existence we were led to infer from the series spectra. At 
the same time we get a striking impression of the insufficiency of 
the ordinary electrodynamical and mechanical conceptions for the 
description of atomic processes, not only as regards the emission 

32 



36 ON THE SERIES SPECTRA OF THE ELEMENTS 

of radiation but also in such phenomena as the collision of free 
electrons with atoms. 

III. DEVELOPMENT OF THE QUANTUM THEORY 
OF SPECTRA 

We see that it is possible by making use of a few simple ideas 
to obtain a certain insight into the origin of the series spectra. 
But when we attempt to penetrate more deeply, difficulties arise. 
In fact, for systems which are not simply periodic it is not possible 
to obtain sufficient information about the motions of these systems 
in the stationary states from the numerical values of the energy 
alone ; more determining factors are required for the fixation of 
the motion. We meet the same difficulties when we try to explain 
in detail the characteristic effect of external forces upon the spectrum 
of hydrogen. A foundation for further advances in this field has 
been made in recent years through a development of the quantum 
theory, which allows a fixation of the stationary states not only in 
the case of simple periodic systems, but also for certain classes of 
non-periodic systems. These are the conditionally periodic systems 
whose equations of motion can be solved by a " separation of the 
variables." If generalized coordinates are used the description of 
the motion of these systems can be reduced to the consideration 
of a number of generalized " components of motion." Each of these 
corresponds to the change of only one of the coordinates and may 
therefore in a certain sense be regarded as " independent." The 
method for the fixation of the stationary states consists in fixing 
the motion of each of these components by a condition, which can 
be considered as a direct generalization of condition (1) for a 
Planck oscillator, so that the stationary states are in general 
characterized by as many whole numbers as the number of the 
degrees of freedom which the system possesses. A considerable 
number of physicists have taken part in this development of the 
quantum theory, including Planck himself. I also wish to mention 
the important contribution made by Ehrenfest to this subject on 
the limitations of the applicability of the laws of mechanics to 
atomic processes. The decisive advance in the application of the 
quantum theory to spectra, however, is due to Sommerfeld and his 
followers. However, I shall not further discuss the systematic form 



ON THE SERIES SPECTRA OF THE ELEMENTS 37 

in which these authors have presented their results. In a paper which 
appeared some time ago in the Transactions of the Copenhagen 
Academy, I have shown that the spectra, calculated with the aid 
of this method for the fixation of the stationary states, exhibit a 
correspondence with the spectra which should correspond to the 
motion of the system similar to that which we have already con- 
sidered in the case of hydrogen. With the aid of this general 
correspondence I shall try in the remainder of this lecture to 
show how it is possible to present the theory of series spectra 
and the effects produced by external fields of force upon these 
spectra in a form which may be considered as the natural generali- 
zation of the foregoing considerations. This form appears to me 
to be especially suited for future work in the theory of spectra, 
since it allows of an immediate insight into problems for which 
the methods mentioned above fail on account of the complexity of 
the motions in the atom. 

Effect of external forces on the hydrogen spectrum. We 

shall now proceed to investigate the effect of small perturbing 
forces upon the spectrum of the simple system consisting of a single 
electron revolving about a nucleus. For the sake of simplicity we 
shall for the moment disregard the variation of the mass of the 
electron with its velocity. The consideration of the small changes 
in the motion due to this variation has been of great importance 
in the development of Sommerfeld's theory which originated in the 
explanation of the fine structure of the hydrogen lines. This fine 
structure is due to the fact, that taking into account the variation 
of mass with velocity the orbit of the electron deviates a little 
from a simple ellipse and is no longer exactly periodic. This devia- 
tion from a Keplerian motion is, however, very small compared 
with the perturbations due to the presence of external forces, such 
as occur in experiments on the Zeeman and Stark effects. In atoms 
of higher atomic number it is also negligible compared with the 
disturbing effect of the inner electrons on the motion of the outer 
electron. The neglect of the change in mass will therefore have no 
important influence upon the explanation of the Zeeman and Stark 
effects, or upon the explanation of the difference between the 
hydrogen spectrum and the spectra of other elements. 



38 ON THE SERIES SPECTRA OF THE ELEMENTS 

We shall therefore as before consider the motion of the un- 
perturbed hydrogen atom as simply periodic and inquire in the 
first place about the stationary states corresponding to this motion. 
The energy in these states will then be determined by expression (7) 
which was derived from the spectrum of hydrogen. The energy of 
the system being given, the major axis of the elliptical orbit of the 
electron and its frequency of revolution are also determined. Sub- 
stituting in formulae (7) and (8) the expression for K given in (12), 
we obtain for the energy, major axis and frequency of revolution 
in the nth state of the unperturbed atom the expressions 



We must further assume that in the stationary states of the 
unperturbed system the form of the orbit is so far undetermined 
that the excentricity can vary continuously. This is not only im- 
mediately indicated by the principle of correspondence, since the 
frequency of revolution is determined only by the energy and not 
by the excentricity, but also by the fact that the presence of any 
small external forces will in general, in the course of time, produce 
a finite change in the position as well as in the excentricity of the 
periodic orbit, while in the major axis it can produce only small 
changes proportional to the intensity of the perturbing forces. 

In order to fix the stationary states of systems in the presence 
of a given conservative external field of force, we shall have to 
investigate, on the basis of the principle of correspondence, how 
these forces affect the decomposition of the motion into harmonic 
oscillations. Owing to the external forces the form and position of 
the orbit will vary continuously. In the general case these changes 
will be so complicated that it will not be possible to decompose the 
perturbed motion into discrete harmonic oscillations. In such a 
case we must expect that the perturbed system will not possess 
any sharply separated stationary states. Although each emission 
of radiation must be assumed to be monochromatic and to proceed 
according to the general frequency condition we shall therefore 
expect the final effect to be a broadening of the sharp spectral lines 
of the unperturbed system. In certain cases, however, the perturba- 



ON THE SERIES SPECTRA OF THE ELEMENTS 39 

tions will be of such a regular character that the perturbed system 
can be decomposed into harmonic oscillations, although the ensemble 
of these oscillations will naturally be of a more complicated kind 
than in the unperturbed system. This happens, for example, when 
the variations of the orbit with respect to time are periodic. In 
this case harmonic oscillations will appear in the motion of the 
system the frequencies of which are equal to whole multiples of the 
period of the orbital perturbations, and in the spectrum to be 
expected on the basis of the ordinary theory of radiation we would 
expect components corresponding to these frequencies. According 
to the principle of correspondence we are therefore immediately 
led to the conclusion, that to each stationary state in the unper- 
turbed system there corresponds a number of stationary states in 
the perturbed system in such a manner, that for a transition 
between two of these states a radiation is emitted, whose frequency 
stands in the same relationship to the periodic course of the 
variations in the orbit, as the spectrum of a simple periodic system 
does to its motion in the stationary states. 

The Stark effect. An instructive example of the appearance of 
periodic perturbations is obtained when hydrogen is subjected to 
the effect of a homogeneous electric field. The excentricity and 
the position of the orbit vary continuously under the influence of 
the field. During these changes, however, it is found that the 
centre of the orbit remains in a plane perpendicular to the direc- 
tion of the electric force and that its motion in this plane is 
simply periodic. When the centre has returned to its starting 
point, the orbit will resume its original excentricity and position, 
and from this moment the entire cycle of orbits will be repeated. 
In this case the determination of the energy of the stationary 
states of the disturbed system is extremely simple, since it is found 
that the period of the disturbance does not depend upon the 
original configuration of the orbit, nor therefore upon the position 
of the plane in which the centre of the orbit moves, but only upon 
the major axis and the frequency of revolution. From a simple 
calculation it is found that the period <r is given by the following 
formula 



40 ON THE SERIES SPECTRA OF THE ELEMENTS 

where F is the intensity of the external electric field. From 
analogy with the fixation of the distinctive energy values of a 
Planck oscillator we must therefore expect that the energy difference 
between two different states, corresponding to the same stationary 
state of the unperturbed system, will simply be equal to a whole 
multiple of the product of h by the period a of the perturbations. 
We are therefore immediately led to the following expression for 
the energy of the stationary states of the perturbed system, 

(19) 



where E n depends only upon the number n characterizing the 
stationary state of the unperturbed system, while k is a new whole 
number which in this case may be either positive or negative. As 
we shall see below, consideration of the relation between the energy 
and the motion of the system shows that k must be numerically 
less than n, if, as before, we place the quantity E n equal to the 
energy W n of the nth stationary state of the undisturbed atom. 
Substituting the values of W n , ta n and a n given by (17) in formula 
(19) we get 

(20) 



To find the effect of an electric field upon the lines of the hydrogen 
spectrum, we use the frequency condition (4) and obtain for the 
frequency v of the radiation emitted by a transition between two 
stationary states defined by the numbers ri, k' and n", k" 

Sh.F , ,., , /// 



It is well known that this formula provides a complete explana- 
tion of the Stark effect of the hydrogen lines. It corresponds 
exactly with the one obtained by a different method by Epstein 
and Schwarzschild. They used the fact that the hydrogen atom in 
a homogeneous electric field is a conditionally periodic system 
permitting a separation of variables by the use of parabolic co- 
ordinates. The stationary states were fixed by applying quantum 
conditions to each of these variables. 

We shall now consider more closely the correspondence between 
the changes in the spectrum of hydrogen due to the presence of 



ON THE SERIES SPECTRA OF THE ELEMENTS 41 

an electric field and the decomposition of the perturbed motion 
of the atom into its harmonic components. Instead of the simple 
decomposition into harmonic components corresponding to a simple 
Kepler motion, the displacement f of the electron in a given 
direction in space can be expressed in the present case by the 
formula 

f = 2<7 T)(C cos 27r {t(T<o + Ka) + c Tilt } .......... (22) 



where a> is the average frequency of revolution in the perturbed 
orbit and er is the period of the orbital perturbations, while C T , K and 
c T>(e are constants. The summation is to be extended over all integral 
values for r and re. 

If we now consider a transition between two stationary states 
characterized by certain numbers n', k' and n", k", we find that in 
the region where these numbers are large compared with their 
differences n' n" and k' k", the frequency of the spectral line 
which is emitted will be given approximately by the formula 

v ~ (ri - n") a> + (k f - k") <r ................ (23) 

We see, therefore, that we have obtained a relation between the 
spectrum and the motion of precisely the same character as in the 
simple case of the unperturbed hydrogen atom. We have here a 
similar correspondence between the harmonic component in the 
motion, corresponding to definite values for r and in formula (22), 
and the transition between two stationary states for which n' n"= r 
and &'-&" = *. 

A number of interesting results can be obtained from this 
correspondence by considering the motion in more detail. Each 
harmonic component in expression (22) for which r + K is an even 
number corresponds to a linear oscillation parallel to the direction 
of the electric field, while each component for which r + K is odd 
corresponds to an elliptical oscillation perpendicular to this direc- 
tion. The correspondence principle suggests at once that these 
facts are connected with the characteristic polarization observed in 
the Stark effect. We would anticipate that a transition for which 
(n' n") + (k' k") is even would give rise to a component with an 
electric vector parallel to the field, while a transition for which 
(n' n") + (k' k") is odd would correspond to a component with an 



42 ON THE SERIES SPECTKA OF THE ELEMENTS 

electric vector perpendicular to the field. These results have been 
fully confirmed by experiment and correspond to the empirical rule 
of polarization, which Epstein proposed in his first paper on the 
Stark effect. 

The applications of the correspondence principle that have so 
far been described have been purely qualitative in character. It is 
possible however to obtain a quantitative estimate of the relative 
intensity of the various components of the Stark effect of hydrogen, 
by correlating the numerical values of the coefficients C T>K in formula 
(22) with the probability of the corresponding transitions between 
the stationary states. This problem has been treated in detail by 
Kramers in a recently published dissertation. In this he gives a 
thorough discussion of the application of the correspondence prin- 
ciple to the question of the intensity of spectral lines. 

The Zeeman effect. The problem of the effect of a homogeneous 
magnetic field upon the hydrogen lines may be treated in an 
entirely analogous manner. The effect on the motion of the hy- 
drogen atom consists simply of the superposition of a uniform ro- 
tation upon the motion of the electron in the unperturbed atom. 
The axis of rotation is parallel with the direction of the magnetic 
force, while the frequency of revolution is given by the formula 



.(24) 



where H is the intensity of the field and c the velocity of light. 

Again we have a case where the perturbations are simply 
periodic and where the period of the perturbations is independent 
of the form and position of the orbit, and in the present case, even 
of the major axis. Similar considerations apply therefore as in the 
case of the Stark effect, and we must expect that the energy in the 
stationary states will again be given by formula (19), if we sub- 
stitute for tr the value given in expression (24). This result is 
also in complete agreement with that obtained by Sommerfeld and 
Debye. The method they used involved the solution of the equations 
of motion by the method of the separation of the variables. The 
appropriate coordinates are polar ones about an axis parallel to 
the field. 

If we try, however, to calculate directly the effect of the field by 



ON THE SERIES SPECTRA OF THE ELEMENTS 43 

means of the frequency condition (4), we immediately meet with 
an apparent disagreement which for some time was regarded as a 
grave difficulty for the theory. As both Sommerfeld and Debye 
have pointed out, lines are not observed corresponding to every 
transition between the stationary states included in the formula. 
We overcome this difficulty, however, as soon as we apply the 
principle of correspondence. If we consider the harmonic com- 
ponents of the motion we obtain a simple explanation both of the 
non-occurrence of certain transitions and of the observed polariza- 
tion. In the magnetic field each elliptic harmonic component having 
the frequency TG> splits up into three harmonic components owing 
to the uniform rotation of the orbit. Of these one is rectilinear 
with frequency TO> oscillating parallel to the magnetic field, and 
two are circular with frequencies rco + ar and rw a- oscillating in 
opposite directions in a plane perpendicular to the direction of the 
field. Consequently the motion represented by formula (22) contains 
no components for which K is numerically greater than 1, in contrast 
to the Stark effect, where components corresponding to all values 
of K are present. Now formula (23) again applies for large values 
of n and k, and shows the asymptotic agreement between the 
frequency of the radiation and the frequency of a harmonic com- 
ponent in the motion. We arrive, therefore, at the conclusion that 
transitions for which k changes by more than unity can not occur. 
The argument is similar to that by which transitions between two 
distinctive states of a Planck oscillator for which the values of n 
in (1) differ by more than unity are excluded. We must further 
conclude that the various possible transitions consist of two types. 
For the one type corresponding to the rectilinear component, k 
remains unchanged, and in the emitted radiation which possesses 
the same frequency V Q as the original hydrogen line, the electric 
vector will oscillate parallel with the field. For the second type, 
corresponding to the circular components, k will increase or decrease 
by unity, and the radiation viewed in the direction of the field will 
be circularly polarized and have frequencies v + a and v cr re- 
spectively. These results agree with those of the familiar Lorentz 
theory. The similarity in the two theories is remarkable, when we 
recall the fundamental difference between the ideas of the quantum 
theory and the ordinary theories of radiation. 



44 ON THE SERIES SPECTRA OF THE ELEMENTS 

Central perturbations. An illustration based on similar con- 
siderations which will throw light upon the spectra of other ele- 
ments consists in finding the effect of a small perturbing field of 
force radially symmetrical with respect to the nucleus. In this case 
neither the form of the orbit nor the position of its plane will 
change with time, and the perturbing effect of the field will simply 
consist of a uniform rotation of the major axis of the orbit. The 
perturbations are periodic, so that we may assume that to each 
energy value of a stationary state of the unperturbed system there 
belongs a series of discrete energy values of the perturbed system, 
characterized by different values of a whole number k. The fre- 
quency a- of the perturbations is equal to the frequency of rotation 
of the major axis. For a given law of force for the perturbing 
field we find that a- depends both on the major axis and on the 
excentricity. The change in the energy of the stationary states, 
therefore, will not be given by an expression as simple as the 
second term in formula (19), but will be a function of k, which is 
different for different fields. It is possible, however, to characterize 
by one and the same condition the motion in the stationary states 
of a hydrogen atom which is perturbed by any central field. In 
order to show this we must consider more closely the fixation of 
the motion of a perturbed hydrogen atom. 

In the stationary states of the unperturbed hydrogen atom 
only the major axis of the orbit is to be regarded as fixed, 
while the excentricity may assume any value. Since the change 
in the energy of the atom due to the external field of force de- 
pends upon the form and position of its orbit, the fixation of the 
energy of the atom in the presence of such a field naturally 
involves a closer determination of the orbit of the perturbed 
system. 

Consider, for the sake of illustration, the change in the hydrogen 
spectrum due to the presence of homogeneous electric and mag- 
netic fields which was described by equation (19). It is found that 
this energy condition can be given a simple geometrical inter- 
pretation. In the case of an electric field the distance from the 
nucleus to the plane in which the centre of the orbit moves deter- 
mines the change in the energy of the system due to the presence 
of the field. In the stationary states this distance is simply equal 



ON THE SERIES SPECTRA OF THE ELEMENTS 45 

k 

to - times half the major axis of the orbit. In the case of a mag- 
netic field it is found that the quantity which determines the change 
of energy of the system is the area of the projection of the orbit 
upon a plane perpendicular to the magnetic force. In the various 

k 
stationary states this area is equal to - times the area of a circle 

whose radius is equal to half the major axis of the orbit. In the 
case of a perturbing central force the correspondence between 
the spectrum and the motion which is required by the quantum 
theory leads now to the simple condition that in the stationary 
states of the perturbed system the minor axis of the rotating orbit 

k 
is simply equal to - times the major axis. This condition was first 

derived by Sommerfeld from his general theory for the determina- 
tion of the stationary states of a central motion. It is easily shown 
that this fixation of the value of the minor axis is equivalent to 
the statement that the parameter 2p of the elliptical orbit is given 
by an expression of exactly the same form as that which gives the 
major axis 2a in the unperturbed atom. The only difference from 
the expression for 2a n in (17) is that n is replaced by k, so that 
the value of the parameter in the stationary states of the perturbed 
atom is given by 

*"$* < 25 > 

The frequency of the radiation emitted by a transition between 
two stationary states determined in this way for which n' and n" are 
large in proportion to their difference is given by an expression 
which is the same as that in equation (23), if in this case &> is the 
frequency of revolution of the electron in the slowly rotating orbit 
and <r represents the frequency of rotation of the major axis. 

Before proceeding further, it might be of interest to note that 
this fixation of the stationary states of the hydrogen atom perturbed 
by external electric and magnetic forces does not coincide in certain 
respects with the theories of Sommerfeld, Epstein and Debye. 
According to the theory of conditionally periodic systems the sta- 
tionary states for a system of three degrees of freedom will in general 
be determined by three conditions, and therefore in these theories 



46 ON THE SERIES SPECTRA OF THE ELEMENTS 

each state is characterized by three whole numbers. This would 
mean that the stationary states of the perturbed hydrogen atom 
corresponding to a certain stationary state of the unperturbed 
hydrogen atom, fixed by one condition, should be subject to two 
further conditions and should therefore be characterized by two 
new whole numbers in addition to the number n. But the per- 
turbations of the Keplerian motion are simply periodic and the 
energy of the perturbed atom will therefore be fixed completely 
by one additional condition. The introduction of a second condition 
will add nothing further to the explanation of the phenomenon, 
since with the appearance of new perturbing forces, even if 
these are too small noticeably to affect the observed Zeeman and 
Stark effects, the forms of motion characterized by such a condition 
may be entirely changed. This is completely analogous to the 
fact that the hydrogen spectrum as it is usually observed is not 
noticeably affected by small forces, even when they are large enough 
to produce a great change in the form and position of the orbit of 
the electron. 

Relativity effect on hydrogen lines. Before leaving the hydro- 
gen spectrum I shall consider briefly the effect of the variation of 
the mass of the electron with its velocity. In the preceding sections 
I have described how external fields of force split up the hydrogen 
lines into several components, but it should be noticed that these 
results are only accurate when the perturbations are large in com- 
parison with the small deviations from a pure Keplerian motion 
due to the variation of the mass of the electron with its velocity. 
When the variation of the mass is taken into account the motion 
of the unperturbed atom will not be exactly periodic. Instead we 
obtain a motion of precisely the same kind as that occurring in the 
hydrogen atom perturbed by a small central field. According to 
the correspondence principle an intimate connection is to be ex- 
pected between the frequency of revolution of the major axis of the 
orbit and the difference of the frequencies of the fine structure 
components, and the stationary states will be those orbits whose 
parameters are given by expression (25). If we now consider the 
effect of external forces upon the fine structure components of the 
hydrogen lines it is necessary to keep in mind that this fixation of 



ON THE SERIES SPECTRA OF THE ELEMENTS 47 

the stationary states only applies to the unperturbed hydrogen 
atom, and that, as mentioned, the orbits in these states are in 
general already strongly influenced by the presence of external 
forces, which are small compared with those with which we are 
concerned in experiments on the Stark and Zeeman effects. In 
general the presence of such forces will lead to a great complexity 
of perturbations, and the atom will no longer possess a group of 
sharply defined stationary states. The fine structure components 
of a given hydrogen line will therefore become diffuse and merged 
together. There are, however, several important cases where this 
does not happen on account of the simple character of the per- 
turbations. The simplest example is a hydrogen atom perturbed 
by a central force acting from the nucleus. In this case it is evident 
that the motion of the system will retain its centrally symmetrical 
character, and that the perturbed motion will differ from the un- 
perturbed motion only in that the frequency of rotation of the major 
axis will be different for different values of this axis and of the 
parameter. This point is of importance in the theory of the 
spectra of elements of higher atomic number, since, as we shall see, 
the effect of the forces originating from the inner electrons may 
to a first approximation be compared with that of a perturbing 
central field. We can not therefore expect these spectra to exhibit 
a separate effect due to the variation of the mass of the electron 
of the same kind as that found in the case of the hydrogen lines. 
This variation will not give rise to a splitting up into separate 
components but only to small displacements in the position of the 
various lines. 

We obtain still another simple example in which the hydrogen 
atom possesses sharp stationary states, although the change of mass 
of the electron is considered, if we take an atom subject to a homo- 
geneous magnetic field. The effect of such a field will consist in 
the superposition of a rotation of the entire system about an axis 
through the nucleus and parallel with the magnetic force. It follows 
immediately from this result according to the principle of corre- 
spondence that each fine structure component must be expected 
to split up into a normal Zeeman effect (Lorentz triplet). The 
problem may also be solved by means of the theory of conditionally 
periodic systems, since the equations of motion in the presence 



48 ON THE SERIES SPECTRA OF THE ELEMENTS 

of a magnetic field, even when the change in the mass is con- 
sidered, will allow of a separation of the variables using polar 
coordinates in space. This has been pointed out by Sommerfeld 
and Debye. 

A more complicated case arises when the atom is exposed to a 
homogeneous electric field which is not so strong that the effect 
due to the change in the mass may be neglected. In this case there 
is no system of coordinates by which the equations of motion can 
be solved by separation of the variables, and the problem, therefore, 
can not be treated by the theory of the stationary states of con- 
ditionally periodic systems. A closer investigation of the perturba- 
tions, however, shows them to be of such a character that the motion 
of the electrons may be decomposed into a number of separate har- 
monic components. These fall into two groups for which the direc- 
tion of oscillation is either parallel with or perpendicular to the 
field. According to the principle of correspondence, therefore, we 
must expect that also in this case in the presence of the field each 
hydrogen line will consist of a number of sharp, polarized compo- 
nents. In fact by means of the principles I have described, it is 
possible to give a unique fixation of the stationary states. The 
problem of the effect of a homogeneous electric field upon the fine 
structure components of the hydrogen lines has been treated in 
detail from this point of view by Kramers in a paper which will 
soon be published. In this paper it will be shown how it appears 
possible to predict in detail the manner in which the fine structure 
of the hydrogen lines gradually changes into the ordinary Stark 
effect as the electric intensity increases. 

Theory of series spectra. Let us now turn our attention once 
more to the problem of the series spectra of elements of higher 
atomic number. The general appearance of the Rydberg constant 
in these spectra is to be explained by assuming that the atom is 
neutral and that one 'electron revolves in an orbit the dimensions 
of which are large in comparison with the distance of the inner elec- 
trons from the nucleus. In a certain sense, therefore, the motion of 
the outer electron may be compared with *he motion of the electron 
of the hydrogen atom perturbed by external forces, and the appear- 
ance of the various series in the spectra of the other elements is 



ON THE SERIES SPECTRA OF THE ELEMENTS 49 

from this point of view to be regarded as analogous to the splitting 
up of the hydrogen lines into components on account of such forces. 
In his theory of the structure of series spectra of the type ex- 
hibited by the alkali metals, Sommerfeld has made the assumption 
that the orbit of the outer electron to a first approximation pos- 
sesses the same character as that produced by a simple perturbing 
central field whose intensity diminishes rapidly with increasing 
distance from the nucleus. He fixed the motion of the external 
electron by means of his general theory for the fixation of the 
stationary states of a central motion. The application of this 
method depends on the possibility of separating the variables in 
the equations of motion. In this manner Sommerfeld was able to 
calculate a number of energy values which can be arranged in rows 
just like the empirical spectral terms shown in the diagram of the 
sodium spectrum (p. 30). The states grouped together by Som- 
merfeld in the separate rows are exactly those which were charac- 
terized by one and the same value of k in our investigation of the 
hydrogen atom perturbed by a central force. The states in the 
first row of the figure (row S) correspond to the value k = I, those 
of the second row (P) correspond to k = 2, etc. The states corre- 
sponding to one and the same value of n are connected by dotted 
lines which are continued so that their vertical asymptotes corre- 
spond to the energy value of the stationary states of the hydrogen 
atom. The fact that for a constant n and increasing values of k 
the energy values approach the corresponding values for the unper- 
turbed hydrogen atom is immediately evident from the theory 
since the outer electron, for large values of the parameter of its 
orbit, remains at a great distance from the inner system during the 
whole revolution. The orbit will become almost elliptical and the 
period of rotation of the major axis will be very large. It can be 
seen, therefore, that the effect of the inner system on the energy 
necessary to remove this electron from the atom must become less 
for increasing values of k. 

These beautiful results suggest the possibility of finding laws of 
orce for the perturbing central field which would account for the 
spectra observed. Although Sommerfeld in this way has in fact 
succeeded in deriving formulae for the spectral terms which vary 
with n for a constant k in agreement with Rydberg's formulae, it 



50 ON THE SERIES SPECTRA OF THE ELEMENTS 

has not been possible to explain the simultaneous variation with 
both k and n in any actual case. This is not surprising, since it is 
to be anticipated that the effect of the inner electrons on the spec- 
trum could not be accounted for in such a simple manner. Further 
consideration shows that it is necessary to consider not only the 
forces which originate from the inner electrons but also to consider 
the effect of the presence of the outer electron upon the motion of 
the inner electrons. 

Before considering the series spectra of elements of low atomic 
number I shall point out how the occurrence or non-occurrence of 
certain transitions can be shown by the correspondence principle 
to furnish convincing evidence in favour of Sommerfeld's assump- 
tion about the orbit of the outer electron. For this purpose we 
must describe the motion of the outer electron in terms of its har- 
monic components. This is easily performed if we assume that the 
presence of the inner electrons simply produces a uniform rotation 
of the orbit of the outer electron in its plane. On account of this 
rotation, the frequency of which we will denote by cr, two circular 
rotations with the periods rto + tr and ro> a- will appear in the 
motion of the perturbed electron, instead of each of the harmonic 
elliptical components with a period r&> in the unperturbed motion. 
The decomposition of the perturbed motion into harmonic compo- 
nents consequently will again be represented by a formula of the 
type (22), in which only such terms appear for which K is equal 
to .+ 1 or 1. Since the frequency of the emitted radiation in the 
regions where n and k are large is again given by the asymptotic 
formula (23), we at once deduce from the correspondence principle 
that the only transitions which can take place are those for which 
the values of k differ by unity. A glance at the figure for the sodium 
spectrum shows that this agrees exactly with the experimental 
results. This fact is all the more remarkable, since in Sommerfeld's 
theory the arrangement of the energy values of the stationary 
states in rows has no special relation to the possibility of transition 
between these states. 

Correspondence principle and conservation of angular mo- 
mentum. Besides these results the correspondence principle sug- 
gests that the radiation emitted by the perturbed atom must 



ON THE SERIES SPECTRA OF THE ELEMENTS 51 

exhibit circular polarization. On account of the indeterminateness 
of the plane of the orbit, however, this polarization can not be 
directly observed. The assumption of such a polarization is a matter 
of particular interest for the theory of radiation emission. On 
account of the general correspondence between the spectrum of 
an atom and the decomposition of its motion into harmonic 
components, we are led to compare the radiation emitted during 
the transition between two stationary states with the radia- 
tion which would be emitted by a harmonically oscillating 
electron on the basis of the classical electrodynamics. In par- 
ticular the radiation emitted according to the classical theory 
by an electron revolving in a circular orbit possesses an angular 
momentum and the energy &.E and the angular momentum AP of 
the radiation emitted during a certain time are connected by the 
relation 

A^=27ro>.AP ...................... (26) 

Here CD represents the frequency of revolution of the electron, 
and according to the classical theory this is equal to the frequency 
v of the radiation. If we now assume that the total energy emitted 
is equal to hv we obtain for the total angular momentum of the 
radiation 



It is extremely interesting to note that this expression is equal 
to the change in the angular momentum which the atom suffers in 
a transition where k varies by unity. For in Sommerfeld's theory 
the general condition for the fixation of the stationary states of a 
central system, which in the special case of an approximately 
Keplerian motion is equivalent to the relation (25), asserts that 
the angular momentum of the system must be equal to a whole 

multiple of ~- , a condition which may be written in our notation 



We see, therefore, that this condition has obtained direct support 
from a simple consideration of the conservation of angular momen- 
tum during the emission of the radiation. I wish to emphasize 
that this equation is to be regarded as a rational generalization of 

42 



52 ON THE SERIES SPECTRA OF THE ELEMENTS 

Planck's original statement about the distinctive states of a har- 
monic oscillator. It may be of interest to recall that the possible 
significance of the angular momentum in applications of the 
quantum theory to atomic processes was first pointed out by 
Nicholson on the basis of the fact that for a circular motion the 
angular momentum is simply proportional to the ratio of the 
kinetic energy to the frequency of revolution. 

In a previous paper which I presented to the Copenhagen 
Academy I pointed out that these results confirm the conclusions 
obtained by the application of the correspondence principle to 
atomic systems possessing radial or axial symmetry. Rubinowicz 
has independently indicated the conclusions which may be obtained 
directly from a consideration of conservation of angular momentum 
during the radiation process. In this way he has obtained several 
of our results concerning the various types of possible transitions 
and the polarization of the emitted radiation. Even for systems 
possessing radial or axial symmetry, however, the conclusions which 
we can draw by means of the correspondence principle are of a 
more detailed character than can be obtained solely from a con- 
sideration of the conservation of angular momentum. For example, 
in the case of the hydrogen atom perturbed by a central force we 
can only conclude that k can not change by more than unity, while 
the correspondence principle requires that k shall vary by unity 
for every possible transition and that its value cannot remain un- 
changed. Further, this principle enables us not only to exclude 
certain transitions as being impossible and can from this point of 
view be considered as a "selection principle" but it also enables 
us to draw conclusions about the relative probabilities of the various 
possible types of transitions from the values of the amplitudes of 
the harmonic components. In the present case, for example, the 
fact that the amplitudes of those circular components which rotate 
in the same sense as the electron are in general greater than the 
amplitudes of those which rotate in the opposite sense leads us to 
expect that lines corresponding to transitions for which k decreases 
by unity will in general possess greater intensity than lines during 
the emission of which k increases by unity. Simple considerations 
like this, however, apply only to spectral lines corresponding to 
transitions from one and the same stationary state. In other 



ON THE SERIES SPECTRA OF THE ELEMENTS 53 

cases when we wish to estimate the relative intensities of two 
spectral lines it is clearly necessary to take into consideration the 
relative number of atoms which are present in each of the two 
stationary states from which the transitions start. While the in- 
tensity naturally can not depend upon the number of atoms in the 
final state, it is to be noticed, however, that in estimating the 
probability of a transition between two stationary states it is neces- 
sary to consider the character of the motion in the final as well as 
in the initial state, since the values of the amplitudes of the com- 
ponents of oscillation of both states are to be regarded as decisive 
for the probability. 

To show how this method can be applied I shall return for a 
moment to the problem which I mentioned in connection with 
Strutt's experiment on the resonance radiation of sodium vapour. 
This involved the discussion of the relative probability of the various 
possible transitions which can start from that state corresponding 
to the second term in the second row of the figure on p. 30. These 
were transitions to the first and second states in the first row and 
to the first state in the third row, and the results of experiment 
indicate, as we saw, that the probability is greatest for the second 
transitions. These transitions correspond to those harmonic com- 
ponents having frequencies 2<w + cr, w + a and <r, and it is seen 
that only for the second transition do the amplitudes of the corre- 
sponding harmonic component differ from zero in the initial 'as 
well as in the final state. [In the next essay the reader will find 
that the values of quantum number n assigned in fig. 1 to the 
various stationary states must be altered. While this correction 
in no way influences the other conclusions in this essay it involves 
that the reasoning in this passage can not be maintained.] 

I have shown how the correspondence between the spectrum of 
an element and the motion of the atom enables us to understand 
the limitations in the direct application of the combination principle 
in the prediction of spectral lines. The same ideas give an imme- 
diate explanation of the interesting discovery made in recent years 
by Stark and his collaborators, that certain new series of combina- 
tion lines appear with considerable intensity when the radiating 
atoms are subject to a strong external electric field. This phe- 
nomenon is entirely analogous to the appearance of the so-called 



54 ON THE SERIES SPECTRA OF THE ELEMENTS 

combination tones in acoustics. It is due to the fact that the 
perturbation of the motion will not only consist in an effect upon 
the components originally present, but in addition will give rise to 
new components. The frequencies of these new components may be 
To> + K(T, where K is different from + 1. According to the correspond- 
ence principle we must therefore expect that the electric field will 
not only influence the lines appearing under ordinary circumstances, 
but that it will also render possible new types of transitions which 
give rise to the "new" combination lines observed. From an esti- 
mate of the amplitudes of the particular components in the initial 
and final states it has even been found possible to account for the 
varying facility with which the new lines are brought up by the 
external field. 

The general problem of the effect of an electric field on the spectra 
of elements of higher atomic number differs essentially from the 
simple Stark effect of the hydrogen lines, since we are here con- 
cerned not with the perturbation of a purely periodic system, but 
with the effect of the field on a periodic motion already subject to 
a perturbation. The problem to a certain extent resembles the 
effect of a weak electric force on the fine structure components of 
the hydrogen atom. In much the same way the effect of an electric 
field upon the series spectra of the elements may be treated directly 
by investigating the perturbations of the external electron. A 
continuation of my paper in the Transactions of the Copenhagen 
Academy will soon appear in which I shall show how this method 
enables us to understand the interesting observations Stark and 
others have made in this field. 

The spectra of helium and lithium. We see that it has been 
possible to obtain a certain general insight into the origin of the 
series spectra of a type like that of sodium. The difficulties en- 
countered in an attempt to give a detailed explanation of the 
spectrum of a particular element, however, become very serious, 
even when we consider the spectrum of helium whose neutral atom 
contains only two electrons. The spectrum of this element has a 
simple structure in that it consists of single lines or at any rate of 
double lines whose components are very close together. We find, 
however, that the lines fall into two groups each of which can be 



ON THE SERIES SPECTRA OF THE ELEMENTS 55 

described by a formula of the type (14). These are usually called 
the (ortho) helium and parhelium spectra. While the latter con- 
sists of simple lines, the former possesses narrow doublets. The 
discovery that helium, as opposed to the alkali metals, possesses 
two complete spectra of the Rydberg type which do not exhibit any 
mutual combinations was so surprising that at times there has been 
a tendency to believe that helium consisted of two elements. This 
way out of the difficulty is no longer open, since there is no room 
for another element in this region of the periodic system, or more 
correctly expressed, for an element possessing a new spectrum. The 
existence of the two spectra can, however, be traced back to the fact 
that in the stationary states corresponding to the series spectra we 
have to do with a system possessing only one inner electron and in 
consequence the motion of the inner system, in the absence of the 
outer electron, will be simply periodic and therefore easily perturbed 
by external forces. 

In order to illustrate this point we shall have to consider more 
carefully the stationary states connected with the origin of a series 
spectrum. We must assume that in these states one electron De- 
volves in an orbit outside the nucleus and the other electrons. We 
might now suppose that in general a number of different groups of 
such states might exist, each group corresponding to a different 
stationary state of the inner system considered by itself. Further 
consideration shows, however, that under the usual conditions of 
excitation those groups have by far the greatest probability for which 
the motion of the inner electrons corresponds to the "normal" state 
of the inner system, i.e. to that stationary state having the least 
energy. Further the energy required to transfer the inner system 
from its normal state to another stationary state is in general very 
large compared with the energy which is necessary to transfer an 
electron from the normal state of the neutral atom to a stationary 
orbit of greater dimensions. Lastly the inner system is in general 
capable of a permanent existence only in its normal state, Now, 
the configuration of an atomic system in its stationary states and 
also in the normal state will, in general, be completely determined. 
We may therefore expect that the inner system under the influence 
of the forces arising from the presence of the outer electron can in 
the course of time suffer only small changes. For this reason we 



56 ON THE SERIES SPECTRA OF THE ELEMENTS 

must assume that the influence of the inner system upon the motion 
of the external electron will, in general, be of the same character 
as the perturbations produced by a constant external field upon 
the motion of the electron in the hydrogen atom. We must there- 
fore expect a spectrum consisting of an ensemble of spectral terms, 
which in general form a connected group, even though in the 
absence of external perturbing forces not every combination actually 
occurs. The case of the helium spectrum, however, is quite different 
since here the inner system contains only one electron the motion 
of which in the absence of the external electron is simple periodic 
provided the small changes due to the variation in the mass of the 
electron with its velocity are neglected. For this reason the form of 
the orbit in the stationary states of the inner system considered by 
itself will not be determined. In other words, the stability of the 
orbit is so slight, even if the variation in the mass is taken into 
account, that small external forces are in a position to change the 
excentricity in the course of time to a finite extent. In this case, 
therefore, it is possible to have several groups of stationary states, 
for which the energy of the inner system is approximately the same 
while the form of the orbit of the inner electron and its position 
relative to the motion of the other electrons are so essentially 
different, that no transitions between the states of different groups 
can occur even in the presence of external forces. It can be seen 
that these conclusions summarize the experimental observations 
on the helium spectra. 

These considerations suggest an investigation of the nature of 
the perturbations in the orbit of the inner electron of the helium 
atom, due to the presence of the external electron. A discussion 
of the helium spectrum from this point of view has recently been 
given by Land6. The results of this work are of great interest par- 
ticularly in the demonstration of the large back effect on the outer 
electron due to the perturbations of the inner orbit which themselves 
arise from the presence of the outer electron. Nevertheless, it can 
scarcely be regarded as a satisfactory explanation of the helium 
spectrum. Apart from the serious objections which may be raised 
against his calculation of the perturbations, difficulties arise if we 
try to apply the correspondence principle to Lande's results in 
order to account for the occurrence of two distinct spectra showing 



ON THE SERIES SPECTKA OF THE ELEMENTS 57 

no mutual combinations. To explain this fact it seems necessary 
to base the discussion on a more thorough investigation of the 
mutual perturbations of the outer and the inner orbits. As a 
result of these perturbations both electrons move in such an 
extremely complicated way that the stationary states can not be 
fixed by the methods developed for conditionally periodic systems. 
Dr Kramers and I have in the last few years been engaged in such 
an investigation, and in an address on atomic problems at the 
meeting of the Dutch Congress of Natural and Medical Sciences 
held in Leiden, April 1919, I gave a short communication of our 
results. For various reasons we have up to the present time been 
prevented from publishing, but in the very near future we hope to 
give an account of these results and of the light which they seem 
to throw upon the helium spectrum. 

The problem presented by the spectra of elements of higher 
atomic number is simpler, since the inner system is better defined 
in its normal state. On the other hand the difficulty of the mechani- 
cal problem of course increases with the number of the particles in 
the atom. We obtain an example of this in the case of lithium 
with three electrons. The differences between the spectral terms 
of the lithium spectrum and the corresponding spectral terms of 
hydrogen are very small for the variable term of the principal series 
(k = 2) and for the diffuse series (k = 3), on the other hand it is very 
considerable for the variable term of the sharp series (k= 1). This 
is very different from what would be expected if it were possible to 
describe the effect of the inner electron by a central force varying 
in a simple manner with the distance. This must be because the 
parameter of the orbit of the outer electron in the stationary states 
corresponding to the terms of the sharp series is not much greater 
than the linear dimensions of the orbits of the inner electrons. 
According to theprinciple of correspondence the frequency of rotation 
of the major axis of the orbit of the outer electron is to be regarded 
as a measure of the deviation of the spectral terms from the corre- 
sponding hydrogen terms. In order to calculate this frequency it 
appears necessary to consider in detail the mutual effect of all three 
electrons, at all events for that part of the orbit where the outer 
electron is very close to the other two electrons. Even if we assumed 
that we were fully acquainted with the normal state of the inner 



58 ON THE SERIES SPECTRA OF THE ELEMENTS 

system in the absence of the outer electron which would be 
expected to be similar to the normal state of the neutral helium 
atom the exact calculation of this mechanical problem would 
evidently form an exceedingly difficult task. 

Complex structure of series lines. For the spectra of elements 
of still higher atomic number the mechanical problem which has to 
be solved in order to describe the motion in the stationary states 
becomes still more difficult. This is indicated by the extraordinarily 
complicated structure of many of the observed spectra. The fact that 
the series spectra of the alkali metals, which possess the simplest 
structure, consist of double lines whose separation increases with 
the atomic number, indicates that here we have to do with systems 
in which the motion of the outer electron possesses in general a 
somewhat more complicated character than that of a simple central 
motion. This gives rise to a more complicated ensemble of stationary 
states. It would, however, appear that in the sodium atom the major 
axis and the parameter of the stationary states corresponding to 
each pair of spectral terms are given approximately by formulae 
(17) and (25). This is indicated not only by the similar part played 
by the two states in the experiments on the resonance radiation of 
sodium vapour, but is also shown in a very instructive manner by 
the peculiar effect of magnetic fields on the doublets. For small 
fields each component splits up into a large number of sharp lines 
instead of into the normal Lorentz triplet. With increasing field 
strength Paschen and Back found that this anomalous Zeeman effect 
changed into the normal Lorentz triplet of a single line by a gradual 
fusion of the components. 

This effect of a magnetic field upon the doublets of the alkali 
spectrum is of interest in showing the intimate relation of the com- 
ponents and confirms the reality of the simple explanation of the 
general structure of the spectra of the alkali metals. If we may 
again here rely upon the correspondence principle we have unam- 
biguous evidence that the effect of a magnetic field on the motion 
of the electrons simply consists in the superposition of a uniform 
rotation with a frequency given by equation (24) as in the case of 
the hydrogen atom. For if this were the case the correspondence 
principle would indicate under all conditions a normal Zeeman effect 



ON THE SERIES SPECTRA OF THE ELEMENTS 59 

for each component of the doublets. I want to emphasize that the 
difference between the simple effect of a magnetic field, which the 
theory predicts for the fine structure of components of the hydrogen 
lines, and the observed effect on the alkali doublets is in no way to 
be considered as a contradiction. The fine structure components 
are not analogous to the individual doublet components, but each 
single fine structure component corresponds to the ensemble of 
components (doublet, triplet) which makes up one of the series lines 
in Rydberg's scheme. The occurrence in strong fields of the effect 
observed by Paschen and Back must therefore be regarded as a 
strong support for the theoretical prediction of the effect of a mag- 
netic field on the fine structure components of the hydrogen lines. 
It does not appear necessary to assume the "anomalous" effect 
of small fields on the doublet components to be due to a failure of 
the ordinary electrodynamical laws for the description of the motion 
of the outer electron, but rather to be connected with an effect of 
the magnetic field on that intimate interaction between the motion 
of the inner and outer electrons which is responsible for the occur- 
rence of the doublets. Such a view is probably not very different 
from the "coupling theory" by which Voigt was able to account 
formally for the details of the anomalous Zeeman effect. We might 
even expect it to be possible to construct a theory of these effects 
which would exhibit a formal analogy with the Voigt theory similar 
to that between the quantum theory of the normal Zeeman effect and 
the theory originally developed by Lorentz. Time unfortunately 
does not permit me to enter further into this interesting problem, so 
I must refer you to the continuation of my paper in the Transactions 
of the Copenhagen Academy, which will contain a general discussion 
of the origin of series spectra and of the effects of electric and 
magnetic fields. 

IV. CONCLUSION 

In this lecture I have purposely not considered the question of 
the structure of atoms and molecules although this is of course most 
intimately connected with the kind of spectral theory I have de- 
veloped. We are encouraged to use results obtained from the spectra, 
since even the simple theory of the hydrogen spectrum gives a 
value for the major axis of the orbit of the electron in the normal 



60 ON THE SERIES SPECTRA OF THE ELEMENTS 

state (n = 1) of the same order of magnitude as that derived from 
the kinetic theory of gases. In my first paper on the subject I 
attempted to sketch a theory of the structure of atoms and of 
molecules of chemical compounds. This theory was based on a 
simple generalization of the results for the stationary states of the 
hydrogen atom. In several respects the theory was supported by 
experiment, especially in the general way in which the properties 
of the elements change with increasing atomic number, shown most 
clearly by Moseley's results. I should like, however, to use this 
occasion to state, that in view of the recent development of the 
quantum theory, many of the special assumptions will certainly have 
to be changed in detail This has become clear from various sides 
by the lack of agreement of the theory with experiment. It appears 
no longer possible to justify the assumption that in the normal 
states the electrons move in orbits of special geometrical simplicity, 
like "electronic rings." Considerations relating to the stability of 
atoms and molecules against external influences and concerning the 
possibility of the formation of an atom by successive addition of 
the individual electrons compel us to claim, first that the con- 
figurations of electrons are not only in mechanical equilibrium 
but also possess a certain stability in the sense required by 
ordinary mechanics, and secondly that the configurations employed 
must be of such a nature that transitions to these from other 
stationary states of the atom are possible. These requirements are 
not in general fulfilled by such simple configurations as electronic 
rings and they force us to look about for possibilities of more com- 
plicated motions. It will not be possible here to consider further 
these still open questions and I must content myself by referring 
to the discussion in my forthcoming paper. In closing, however, 
I should like to emphasize once more that in this lecture I have 
only intended to bring out certain general points of view lying at 
the basis of the spectral theory. In particular it was my intention 
to show that, in spite of the fundamental differences between these 
points of view and the ordinary conceptions of the phenomena of 
radiation, it still appears possible on the basis of the general corre- 
spondence between the spectrum and the motion in the atom to 
employ these conceptions in a certain sense as guides in the investi- 
gation of the spectra. 



ESSAY III* 

THE STRUCTURE OF THE ATOM AND THE PHYSICAL 

AND CHEMICAL PROPERTIES OF THE ELEMENTS 

I. PRELIMINARY 

In an address which I delivered to you about a year ago I 
described the main features of a theory of atomic structure which 
I shall attempt to develop this evening. In the meantime this 
theory has assumed more definite form, and in two recent letters 
to Nature I have given a somewhat further sketch of the de- 
velopment f. The results which I am about to present to you are 
of no final character; but I hope to be able to show you how this 
view renders a correlation of the various properties of the elements 
in such a way, that we avoid the difficulties which previously 
appeared to stand in the way of a simple and consistent explanation. 
Before proceeding, however, I must ask your forbearance if initially 
I deal with matters already known to you, but in order to intro- 
duce you to the subject it will first be necessary to give a brief 
description of the most important results which have been obtained 
in recent years in connection with the work on atomic structure. 

The nuclear atom. The conception of atomic structure which 
will form the basis of all the following remarks is the so-called 
nuclear atom according to which an atom is assumed to consist of 
a nucleus surrounded by a number of electrons whose distances 
from one another and from the nucleus are very large compared 
to the dimensions of the particles themselves. The nucleus 
possesses almost the entire mass of the atom and has a positive 
charge of such a magnitude that the number of electrons in a 
neutral atom is equal to the number of the element in the periodic 
system, the so-called atomic number. This idea of the atom, which 
is due principally to Rutherford's fundamental researches on radio- 
active substances, exhibits extremely simple features, but just this 
simplicity appears at first sight to present difficulties in explaining 
the properties of the elements. When we treat this question on 

* Address delivered before a joint meeting of the Physical and Chemical 
Societies in Copenhagen, October 18, 1921. 

t Xature, March 24, and October 13, 1921. 



62 THE STRUCTURE OF THE ATOM AND THE 

the basis of the ordinary mechanical and electrodynamical theories 
it is impossible to find a starting point for an explanation of the 
marked properties exhibited by the various elements, indeed not 
even of their permanency. On the one hand the particles of the 
atom apparently could not be at rest in a state of stable equilibrium, 
and on the other hand we should have to expect that every motion 
which might be present would give rise to the emission of electro- 
magnetic radiation which would not cease until all the energy of 
the system had been emitted and all the electrons had fallen into 
the nucleus. A method of escaping from these difficulties has now 
been found in the application of ideas belonging to the quantum 
theory, the basis of which was laid by Planck in his celebrated 
work on the law of temperature radiation. This represented a 
radical departure from previous conceptions since it was the first 
instance in which the assumption of a discontinuity was employed 
in the formulation of the general laws of nature. 

The postulates of the quantum theory. The quantum theory 
in the form in which it has been applied to the problems of atomic 
structure rests upon two postulates which have a direct bearing 
on the difficulties mentioned above. According to the first postu- 
late there are certain states in which the atom can exist without 
emitting radiation, although the particles are supposed to have an 
accelerated motion relative to one another. These stationary states 
are, in addition, supposed to possess a peculiar kind of stability, so 
that it is impossible either to add energy to or remove energy from 
the atom except by a process involving a transition of the atom 
into another of these states. According to the second postulate 
each emission of radiation from the atom resulting from such a 
transition always consists of a train of purely harmonic waves. 
The frequency of these waves does not depend directly upon the 
motion of the atom, but is determined by a frequency relation, 
according to which the frequency multiplied by the universal con- 
stant introduced by Planck is equal to the total energy emitted 
during the process. For a transition between two stationary states 
for which the values of the energy of the atom before and after the 
emission of radiation are E' and E" respectively, we have therefore 

hv = E'-E", (1) 



PHYSICAL AND CHEMICAL PROPERTIES OF THE ELEMENTS 63 

where h is Planck's constant and v is the frequency of the emitted 
radiation. Time does not permit me to give a systematic survey 
of the quantum theory, the recent development of which has gone 
hand in hand with its applications to atomic structure. I shall 
therefore immediately proceed to the consideration of those appli- 
cations of the theory which are of direct importance in connection 
with our subject. 

Hydrogen atom. We shall commence by considering the 
simplest atom conceivable, namely, an atom consisting of a nucleus 
and one electron. If the charge on the nucleus corresponds to that 
of a single electron and the system consequently is neutral we have 
a hydrogen atom. Those developments of the quantum theory which 
have made possible its application to atomic structure started with 
the interpretation of the well-known simple spectrum emitted by 
hydrogen. This spectrum consists of a series of lines, the frequencies 
of which are given by the extremely simple Balmer formula 



where n" and ri are integers. According to the quantum theory 
we shall now assume that the atom possesses a series of stationary 
states characterized by a series of integers, and it can be seen how 
the frequencies given by formula (2) may be derived from the 
frequency relation if it is assumed that a hydrogen line is con- 
nected with a radiation emitted during a transition between two 
of these states corresponding to the numbers n' and n", and if the 
energy in the nth state apart from an arbitrary additive constant 
is supposed to be given by the formula 



(3) 



The negative sign is used because the energy of the atom is 
measured most simply by the work required to remove the electron 
to infinite distance from the nucleus, and we shall assume that the 
numerical value of the expression on the right-hand side of formula 
(3) is just equal to this work. 

As regards the closer description of the stationary states we find 
that the electron will very nearly describe an ellipse with the 
nucleus in the focus. The major axis of this ellipse is connected 



64 THE STRUCTURE OF THE ATOM AND THE 

with the energy of the atom in a simple way, and corresponding to 
the energy values of the stationary states given by formula (3) 
there are a series of values for the major axis 2a of the orbit of the 
electron given by the formula 



where e is the numerical value of the charge of the electron and 
the nucleus. 

On the whole we may say that the spectrum of hydrogen shows 
us the formation of the hydrogen atom, since the stationary states 
may be regarded as different stages of a process by which the elec- 
tron under the emission of radiation is bound in orbits of smaller 
and smaller dimensions corresponding to states with decreasing 
values of n. It will be seen that this view has certain charac- 
teristic features in common with the binding process of an electron 
to the nucleus if this were to take place according to the ordinary 
electrodynamics, but that our view differs from it in just such a 
way that it is possible to account for the observed properties of 
hydrogen. In particular it is seen that the final result of the 
binding process leads to a quite definite stationary state of the 
atom, namely that state for which n = \. This state which corre- 
sponds to the minimum energy of the atom will be called the 
normal state of the atom. It may be stated here that the values of 
the energy of the atom and the major axis of the orbit of the 
electron which are found if we put n = 1 in formulae (3) and (4) 
are of the same order of magnitude as the values of the firmness 
of binding of electrons and of the dimensions of the atoms which 
have been obtained from experiments on the electrical and me- 
chanical properties of gases. A more accurate check of formulae 
(3) and (4) can however not be obtained from such a comparison, 
because in such experiments hydrogen is not present in the form 
of simple atoms but as molecules. 

The formal basis of the quantum theory consists not only of the 
frequency relation, but also of conditions which permit the deter- 
mination of the stationary states of atomic systems. The latter 
conditions, like that assumed for the frequency, may be regarded as 
natural generalizations of that assumption regarding the interaction 
between simple electrodynamic systems and a surrounding field of 



PHYSICAL AND CHEMICAL PROPERTIES OF THE ELEMENTS 65 

electromagnetic radiation which forms the basis of Planck's theory 
of temperature radiation. I shall not here go further into the 
nature of these conditions but only mention that by their means 
the stationary states are characterized by a number of integers, 
the so-called quantum numbers. For a purely periodic motion like 
that assumed in the case of the hydrogen atom only a single 
quantum number is necessary for the determination of the stationary 
states. This number determines the energy of the atom and also 
the major axis of the orbit of the electron, but not its excentricity. 
The energy in the various stationary states, if the small influence 
of the motion of the nucleus is neglected, is given by the following 
formula: 



where e and m are respectively the charge and the mass of the 
electron, and where for the sake of subsequent applications the 
charge on the nucleus has been designated by Ne. 

For the atom of hydrogen N=\, and a comparison with 
equation (3) leads to the following theoretical expression for K in 
formula (2), namely 



This agrees with the empirical value of the constant for the spectrum 
of hydrogen within the limit of accuracy with which the various 
quantities can be determined. 

Hydrogen spectrum and X-ray spectra. If in the above 
formula we put N=2 which corresponds to an atom consisting of 
an electron revolving around a nucleus with a double charge, we 
get values for the energies in the stationary states, which are four 
times larger than the energies in the corresponding states of the 
hydrogen atom, and we obtain the following formula for the 
spectrum which would be emitted by such an atom : 



This formula represents certain lines which have been known for 
some time and which had been attributed to hydrogen on account 
of the great similarity between formulae (2) and (7) since it had 



66 THE STRUCTURE OF THE ATOM AND THE 

never been anticipated that two different substances could exhibit 
properties so closely resembling each other. According to the theory 
we may, however, expect that the emission of the spectrum given by 
(7) corresponds to the first stage of the formation of the helium atom, 
i.e. to the binding of a first electron by the doubly charged nucleus 
of this atom. This interpretation has been found to agree with 
more recent information. For instance it has been possible to 
obtain this spectrum from pure helium. I have dwelt on this point 
in order to show how this intimate connection between the proper- 
ties of two elements, which at first sight might appear quite 
surprising, is to be regarded as an immediate expression of the 
characteristic simple structure of the nuclear atom. A short time 
after the elucidation of this question, new evidence of extraordinary 
interest was obtained of such a similarity between the properties of 
the elements. I refer to Moseley's fundamental researches on the 
X-ray spectra of the elements. Moseley found that these spectra 
varied in an extremely simple manner from one element to the 
next in the periodic system. It is well known that the lines of 
the X-ray spectra may be divided into groups corresponding to the 
different characteristic absorption regions for X-rays discovered by 
Barkla. As regards the K group which contains the most pene- 
trating X-rays, Moseley found that the strongest line for all ele- 
ments investigated could be represented by a formula which with 
a small simplification can be written 



K is the same constant as in the hj^drogen spectrum, and N the 
atomic number. The great significance of this discovery lies in 
the fact that it would seem firmly to establish the view that this 
atomic number is equal to the number of electrons in the atom. 
This assumption had already been used as a basis for work on 
atomic structure and was first stated by van den Broek. While 
the significance of this aspect of Moseley's discovery was at once 
clear to all, it has on the other hand been more difficult to under- 
stand the very great similarity between the spectrum of hydrogen 
and the X-ray spectra. This similarity is shown, not only by the 
lines of the K group, but also by groups of less penetrating X-rays. 



PHYSICAL AND CHEMICAL PROPERTIES OF THE ELEMENTS 67 

Thus Moseley found for all the elements he investigated that the 
frequencies of the strongest line in the L group may be represented 
by a formula which with a simplification similar to that employed 
in formula (8) can be written 



Here again we obtain an expression for the frequency which corre- 
sponds to a line in the spectrum which would be emitted by the 
binding of an electron to a nucleus, whose charge is Ne. 

The fine structure of the hydrogen lines. This similarity be- 
tween the structure of the X-ray spectra and the hydrogen spectrum 
was still further extended in a very interesting manner by Sommer- 
feld's important theory of the fine structure of the hydrogen lines. 
The calculation given above of the energy in the stationary states 
of the hydrogen system, where each state is characterized by a 
single quantum number, rests upon the assumption that the orbit 
of the electron in the atom is simply periodic. This is, however, 
only approximately true. It is found that if the change in the mass 
of the electron due to its velocity is taken into consideration the 
orbit of the electron no longer remains a simple ellipse, but its 
motion may be described as a central motion obtained by superposing 
a slow and uniform rotation upon a simple periodic motion in a 
very nearly elliptical orbit. For a central motion of this kind the 
stationary states are characterized by two quantum, numbers. In the 
case under consideration one of these may be so chosen that to a 
very close approximation it will determine the energy of the atom 
in the same manner as the quantum number previously used 
determined the energy in the case of a simple elliptical orbit. This 
quantum number which will always be denoted by n will therefore 
be called the "principal quantum number." Besides this condition, 
which to a very close approximation determines the major axis in the 
rotating and almost elliptical orbit, a second condition will be imposed 
upon the stationary states of a central orbit, namely that the angular 
momentum of the electron about the centre shall be equal to a whole 
multiple of Planck's constant divided by 2-n-. The whole number, which 
occurs as a factor in this expression, may be regarded as the second 
quantu m number and will be denoted by k. The latter condition fixes 

52 



68 THE STRUCTURE OF THE ATOM AND THE 

the excentricity of the rotating orbit which in the case of a simple 
periodic orbit was undetermined. It should be mentioned that the 
possible importance of the angular momentum in the quantum theory 
was pointed out by Nicholson before the application of this theory to 
the spectrum of hydrogen, and that a determination of the stationary 
states for the hydrogen atom similar to that employed by Sommer- 
feld was proposed almost simultaneously by Wilson, although the 
latter did not succeed in giving a physical application to his results. 
The simplest description of the form of the rotating nearly 
elliptical electronic orbit in the hydrogen atom is obtained by 
considering the chord which passes through the focus and is 
perpendicular to the major axis, the so-called "parameter." The 
length 2p of this parameter is given to a very close approximation 
by an expression of exactly the same form as the expression for the 
major axis, except that k takes the place of n. Using the same 
notation as before we have therefore 



For each of the stationary states which had previously been denoted 
by a given value of n, we obtain therefore a set of stationary states 
corresponding to values of k from 1 to n. Instead of the simple 
formula (5) Sommerfeld found a more complicated expression for 
the energy in the stationary states which depends on k as well as 
n. Taking the variation of the mass of the electron with velocity 
into account and neglecting terms of higher order of magnitude he 
obtained 



where c is the velocity of light. 

Corresponding to each of the energy values for the stationary 
states of the hydrogen atom given by the simple formula (5) we 
obtain n values differing only very little from one another, since 
the second term within the bracket is very small. With the aid of 
the general frequency relation (1) we therefore obtain a number of 
components with nearly coincident frequencies instead of each 
hydrogen line given by the simple formula (2). Sommerfeld has 
now shown that this calculation actually agrees with measurements 



PHYSICAL AND CHEMICAL PROPERTIES OF THE ELEMENTS 69 

of the fine structure. This agreement applies not only to the fine 
structure of the hydrogen lines which is very difficult to measure 
on account of the extreme proximity of the components, but it is 
also possible to account in detail for the fine structure of the helium 
lines given by formula (7) which has been very carefully in- 
vestigated by Paschen. Sommerfeld in connection with this theory 
also pointed out that formula (11) could be applied to the X-ray 
spectra. Thus he showed that in the K and L groups pairs of lines 
appeared the differences of whose frequencies could be determined 
by the expression (11) for the energy in the stationary states which 
correspond to the binding of a single electron by a nucleus of 
charge Ne. 

Periodic table. In spite of the great formal similarity between 
the X-ray spectra and the hydrogen spectrum indicated by these 
results a far-reaching difference must be assumed to exist between 
the processes which give rise to the appearance of these two types 
of spectra. While the emission of the hydrogen spectrum, like the 
emission of the ordinary optical spectra of other elements, may be 
assumed to be connected with the binding of an electron by an 
atom, observations on the appearance and absorption of X-ray 
spectra clearly indicate that these spectra are connected with a 
process which may be described as a reorganization of the electronic 
arrangement after a disturbance within the atom due to the effect 
of external agencies. We should therefore expect that the appear- 
ance of the X-ray spectra would depend not only upon the direct 
interaction between a single electron and the nucleus, but also on 
the manner in which the electrons are arranged in the completely 
formed atom. 

The peculiar manner in which the properties of the elements 
vary with the atomic number, as expressed in the periodic system, 
provides a guide of great value in considering this latter problem. 
A simple survey of this system is given in fig. 1. The number pre- 
ceding each element indicates the atomic number, and the .elements 
within the various vertical columns form the different "periods" of 
the system. The lines, which connect pairs of elements in successive 
columns, indicate homologous properties of such elements. Com- 
pared with usual representations of the periodic system, this method, 



70 



THE STRUCTURE OF THE ATOM AND THE 



proposed more than twenty years ago by Julius Thomsen, of indi- 
cating the periodic variations in the properties of the elements is 
more suited for comparison with theories of atomic constitution. 
The meaning of the frames round certain sequences of elements 
within the later periods of the table will be explained later. They 
refer to certain characteristic features of the theory of atomic 
constitution. 




118- 



Fig. 1. 

In an explanation of the periodic system it is natural to as- 
sume a division of the electrons in the atom into distinct groups 
in such a manner that the grouping of the elements in the system 
is attributed to the gradual formation of the groups of electrons 
in the atoms as the atomic number increases. Such a grouping 



PHYSICAL AND CHEMICAL PROPERTIES OF THE ELEMENTS 71 

of the electrons in the atom has formed a prominent part of all 
more detailed views of atomic structure ever since J. J. Thom- 
son's famous attempt to explain the periodic system on the basis 
of an investigation of the stability of various electronic configura- 
tions. Although Thomson's assumption regarding the distribution 
of the positive electricity in the atom is not consistent with more 
recent experimental evidence, nevertheless his work has exerted 
great influence upon the later development of the atomic theory on 
account of the many original ideas which it contained. 

With the aid of the information concerning the binding of 
electrons by the nucleus obtained from the theory of the hydrogen 
spectrum I attempted in the same paper in which this theory was 
set forth to sketch in broad outlines a picture of the structure of 
the nucleus atom. In this it was assumed that each electron in its 
normal state moved in a manner analogous to the motion in 
the last stages of the binding of a single electron by a nucleus. 
As in Thomson's theory, it was assumed that the electrons moved 
in circular orbits and that the electrons in each separate group 
during this motion occupied positions with reference to one another 
corresponding to the vertices of plane regular polygons. Such an 
arrangement is frequently described as a distribution of the electrons 
in "rings." By means of these assumptions it was possible to 
account for the orders of magnitude of the dimensions of the atoms 
as well as the firmness with which the electrons were bound by the 
atom, a measure of which may be obtained by means of experiments 
on the excitation of the various types of spectra. It was not 
possible, however, in this way to arrive at a detailed explanation 
of the characteristic properties of the elements even after it had 
become apparent from the results of Moseley and the work of 
Sommerfeld and others that this simple picture ought to be ex- 
tended to include orbits in the fully formed atom characterized by 
higher quantum numbers corresponding to previous stages in the 
formation of the hydrogen atom. This point has been especially 
emphasized by Vegard. 

The difficulty of arriving at a satisfactory picture of the atom is 
intimately connected with the difficulty of accounting for the pro- 
nounced "stability" which the properties of the elements demand. 
As I emphasized when considering the formation of the hydrogen 



72 THE STRUCTURE OF THE ATOM AND THE 

atom, the postulates of the quantum theory aim directly at this 
point, but the results obtained in this way for an atom containing 
a single electron do not permit of a direct elucidation of problems 
like that of the distribution in groups of the electrons in an atom 
containing several electrons. If we imagine that the electrons in 
the groups of the atom are orientated relatively to one another at any 
moment, like the vertices of regular polygons, and rotating in either 
circles or ellipses, the postulates do not give sufficient information to 
determine the difference in the stability of electronic configurations 
with different numbers of electrons in the groups. 

The peculiar character of stability of the atomic structure, de- 
manded by the properties of the elements, is brought out in an 
interesting way by Kossel in two important papers. In the first 
paper he shows that a more detailed explanation of the origin of 
the high frequency spectra can be obtained on the basis of the 
group structure of the atom. He assumes that a line in the X-ray 
spectrum is due to a process which may be described as follows: an 
electron is removed from the atom by some external action after 
which an electron in one of the other groups takes its place; this 
exchange of place may occur in as many ways as there are groups 
of more loosely bound electrons. This view of the origin of the 
characteristic X-rays afforded a simple explanation of the peculiar 
absorption phenomena observed. It has also led to the prediction 
of certain simple relations between the frequencies of the X-ray 
lines from one and the same element and has proved to be a suitable 
basis for the classification of the complete spectrum. However it has 
not been possible to develop a theory which reconciles in a satis- 
factory way Sommerfeld's work on the fine structure of the X-ray 
lines with Kossel's general scheme. As we shall see later the 
adoption of a new point of view when considering the stability of 
the atom renders it possible to bring the different results in a natural 
way in connection with one another. 

In his second paper Kossel investigates the possibilities for an 
explanation of the periodic system on the basis of the atomic theory. 
Without entering further into the problem of the causes of the 
division of the electrons into groups, or the reasons for the different 
stability of the various electronic configurations, he points out in 
connection with ideas which had already played a part in Thomson's 



PHYSICAL AND CHEMICAL PROPERTIES OF THE ELEMENTS 73 

theory, how the periodic system affords evidence of a periodic ap- 
pearance of especially stable configurations of electrons. These con- 
figurations appear in the neutral atoms of elements occupying the 
final position in each period in fig. 1 , and the stability in question is 
assumed in order to explain not only the inactive chemical properties 
of these elements but also the characteristic active properties of the 
immediately preceding or succeeding elements. If we consider for 
instance an inactive gas like argon, the atomic number of which is 18, 
we must assume that the 18 electrons in the atom are arranged in 
an exceedingly regular configuration possessing a very marked 
stability. The pronounced electronegative character of the preceding 
element, chlorine, may then be explained by supposing the neutral 
atom which contains only 17 electrons to possess a tendency to 
capture an additional electron. This gives rise to a negative chlorine 
ion with a configuration of 18 electrons similar to that occurring 
in the neutral argon atom. On the other hand the marked electro- 
positive character of potassium may be explained by supposing 
one of the 19 electrons in the neutral atom to be as it were super- 
fluous, and that this electron therefore is easily lost; the rest of the 
atom forming a positive ion of potassium having a constitution similar 
to that of the argon atom. In a corresponding manner it is possible 
to account for the electronegative and electropositive character of 
elements like sulphur and calcium, whose atomic numbers are 16 and 
20. In contrast to chlorine and potassium these elements are divalent, 
and the stable configuration of 18 electrons is formed by the addition 
of two electrons to the sulphur atom and by the loss of two electrons 
from the calcium atom. Developing these ideas Kossel has succeeded 
not only in giving interesting explanations of a large number of 
chemical facts, but has also been led to certain general conclusions 
about the grouping of the electrons in elements belonging to the 
first periods of the periodic system, which in certain respects are 
in conformity with the results to be discussed in the following 
paragraphs. Kossel's work was later continued in an interesting 
manner by Ladenburg with special reference to the grouping of the 
electrons in atoms of elements belonging to the later periods of the 
periodic table. It will be seen that Ladenburg's conclusions also 
exhibit points of similarity with the results which we shall discuss 
later. 



74 THE STRUCTURE OF THE ATOM AND THE 

Recent atomic models. Up to the present time it has not been 
possible to obtain a satisfactory account based upon a consistent ap- 
plication of the quantum theory to the nuclear atom of the ultimate 
causeof the pronounced stability of certain arrangements of electrons. 
Nevertheless it has been apparent for some time that the solution 
should be sought for by investigating the possibilities of a spatial 
distribution of the electronic orbits in the atom instead of limiting 
the investigation to configurations in which all electrons belonging 
to a particular group move in the same plane as was assumed for 
simplicity in my first papers on the structure of the atom. The 
necessity of assuming a spatial distribution of the configurations 
of electrons has been drawn attention to by various writers. Born 
and Land6, in connection with their investigations of the structure 
and properties of crystals, have pointed out that the assumption of 
spatial configurations appears necessary for an explanation of these 
properties. Lande" has pursued this question still further, and as 
will be mentioned later has proposed a number of different "spatial 
atomic models" in which the electrons in each separate group of 
the atom at each moment form configurations possessing regular 
polyhedral symmetry. These models constitute in certain respects 
a distinct advance, although they have not led to decisive results 
on questions of the stability of atomic structure. 

The importance of spatial electronic configurations has, in addition, 
been pointed out by Lewis and Langmuir in connection with their 
atomic models. Thus Lewis, who in several respects independently 
came to the same conclusions as Kossel, suggested that the number 
8 characterizing the first groups of the periodic system might in- 
dicate a constitution of the outer atomic groups where the electrons 
within each group formed a configuration like the corners of a cube. 
He emphasized how a configuration of this kind leads to instructive 
models of the molecular structure of chemical combinations. It is 
to be remarked, however, that such a "static" model of electronic 
configuration will not be possible if we assume the forces within 
the atom to be due exclusively to the electric charges of the 
particles. Langmuir, who has attempted to develop Lewis' con- 
ceptions still further and to account not only for the occurrence of 
the first octaves, but also for the longer periods of the periodic 
system, supposes therefore the structure of the atoms to be governed 



PHYSICAL AND CHEMICAL PROPERTIES OF THE ELEMENTS 75 

by forces whose nature is unknown to us. He conceives the atom 
to possess a "cellular structure," so that each electron is in advance 
assigned a place in a cell and these cells are arranged in shells in 
such a manner, that the various shells from the nucleus of the atom 
outward contain exactly the same number of places as the periods 
in the periodic system proceeding in the direction of increasing 
atomic number. Langmuir's work has attracted much attention 
among chemists, since it has to some extent thrown light on the 
conceptions with which empirical chemical science is concerned. 
On his theory the explanation of the properties of the various 
elements is based on a number of postulates about the structure of 
the atoms formulated for that purpose. Such a descriptive theory 
is sharply differentiated from one where an attempt is made to 
explain the specific properties of the elements with the aid of 
general laws applying to the interaction between the particles in 
each atom. The principal task of this lecture will consist in an 
attempt to show that an advance along these lines appears by no 
means hopeless, but on the contrary that with the aid of a con- 
sistent application of the postulates of the quantum theory it 
actually appears possible to obtain an insight into the structure 
and stability of the atom. 

II. SERIES SPECTRA AND THE CAPTURE OF ELECTRONS 
BY ATOMS 

We attack the problem of atomic constitution by asking the 
question : " How may an atom be formed by the successive capture 
and binding of the electrons one by one in the field of force sur- 
rounding the nucleus?" 

Before attempting to answer this question it will first be 
necessary to consider in more detail what the quantum theory 
teaches us about the general character of the binding process. We 
have already seen how the hydrogen spectrum gives us definite 
information about the course of this process of binding the electron 
by the nucleus. In considering the formation of the atoms of other 
elements we have also in their spectra sources for the elucidation 
of the formation processes, but the direct information obtained in 
this way is not so complete as in the case of the hydrogen atom. 
For an element of atomic number N the process of formation may 



76 TOT OTKircTCflKC or vat, ATOM AKB TOE 

be regarded af oceamag in JT *age, compano^ vita the soc- 



ItllMt DC SUMftUUtn wO CGfUBPpOHu V> Stfu Of tB9M? 

bat only for the first two element*, hydrogen and hefiam, do we 
p0M* detailed knowledge of tim spate For. 
of higher atomic number, where Kraal ipeefa* will be 
with the formation of the atom, we are at prevent acquainted with 
only two types, called the "arc" and "souk" spectra respectively, 
according to the experimental conditions of excitation. Although 
these spectra show a much more complicated structure than the 
hydrogen spectrum, given by formula (2) and the befinm spectrum 
given by formula (7), nevertheless in many cases it has been 
possible to find simple laws for the frequencies exhibiting a dose 
analogy with the laws expressed by these formulae. 

Arc and spark spectra. If for die sake of simplicity we dis- 
regard the complex structure shown by the lines of most spectra 
(occurrence of doublets, triplets etc.), the frequency of the lines of. 
many arc spectra can be represented to a dose approximation by 
the Rydberg formula 



where n' and n" are integral numbers, K the same constant as in 
the hydrogen spectrum, while o^ and a** are two constants be- 
longing to a set characteristic of the element A spectrum with a 
structure of this kind is, like the hydrogen spectrum, called a series 
spectrum, since the lines can be arranged into series in which the/ 
frequencies converge to definite limiting values. These series are 
for example represented by formula (12) i using two definite 
constants for a^ and a?, n" remains unaltered, while n' assumes a 
series of successive, gradually increasing integral values. 

Formula (12) applies only approximately, but it is always found 
that the frequencies of the spectral lines can be written, as in 
formulae (2) and (12), as a difference of two functions of integral 
numbers. Thus the latter formula applies accurately, if the 
quantities a k are not considered as constants, but as representatives 
of a set of series of numbers o* (n) characteristic of the element, 
whose values for increasing n within each series quickly approach 



PHYSICAL AND CHEMICAL PROPERTIES OF THE ELEMENTS 77 

constant limiting value. The fact that the frequencies of the 

jectra always appear as the difference of two terms, the so-called 

spectral terms," from the combinations of which the complete 

pectrum is formed, has been pointed out by Ritz, who with the 

stablishment of the combination principle has greatly advanced 

he study of the spectra. The quantum theory offers an immediate 

nterpretation of this principle, since, according to the frequency 

elation we are led to consider the lines as due to transitions 

>etween stationary states of the atom, just as in the hydrogen 

spectrum, only in the spectra of the other elements we have to do 

lot with a single series of stationary states, but with a set of such 

series. From formula (12) we thus obtain for an arc spectrum if 

we temporarily disregard the structure of the individual lines 

information about an ensemble of stationary states, for which the 

energy of the atom in the nth state of the &th series is given by 



very similar to the simple formula (3) for the energy in the station- 
ary states of the hydrogen atom. 

As regards the spark spectra, the structure of which has been 
cleared up mainly by Fowler's investigations, it has been possible 
in the case of many elements to express the frequencies approxi- 
mately by means of a formula of exactly the same type as (12), 
only with the difference that K, just as in the helium spectrum 
given by formula (7), is replaced by a constant, which is four times 
as large. For the spark spectra, therefore, the energy values in the 
corresponding stationary states of the atom will be given by an 
expression of the same type as (13), only with the difference that 
K is replaced by 4iK. 

This remarkable similarity between the structure of these types 
of spectra and the simple spectra given by (2) and (7) is explained 
simply by assuming the arc spectra to be connected with the last 
stage in the formation of the neutral atom consisting in the capture 
and binding of the Nth electron. On the other hand the spark 
spectra are connected with the last stage but one in the formation 
of the atom, namely the binding of the (N l)th electron. In these 
cases the field of force in which the electron moves will be much 



78 THE STRUCTURE OF THE ATOM AND THE 

the same as that surrounding the nucleus of a hydrogen or helium 
atom respectively, at least in the earlier stages of the binding 
process, where during the greater part of its revolution it moves 
at a distance from the nucleus which is large in proportion to the 
dimensions of the orbits of the electrons previously bound. From 
analogy with formula (3) giving the stationary states of the 
hydrogen atom, we shall therefore assume that the numerical value 
of the expression on the right-hand side of (13) will be equal to the 
work required to remove the last captured electron from the atom, 
the binding of which gives rise to the arc spectrum of the element. 

Series diagram. While the origin of the arc and spark spectra 
was to this extent immediately interpreted on the basis of the 
original simple theoryof the hydrogen spectrum,it was Sommerfeld's 
theory of the fine structure of the hydrogen lines which first gave 
us a clear insight into the characteristic difference between the 
hydrogen spectrum and the spark spectrum of helium on the one 
hand, and the arc and spark spectra of other elements on the other. 
When we consider the binding not of the first but of the subsequent 
electrons in the atom, the orbit of the electron under consideration 
at any rate in the latter stages of the binding process where the 
electron last bound comes into intimate interaction with those 
previously bound will no longer be to a near approximation a 
closed ellipse, but on the contrary will to a first approximation be a 
central orbit of the same type as in the hydrogen atom, when we 
take into account the change with velocity in the mass of the 
electron. This motion, as we have seen, may be resolved into a 
plane periodic motion upon which a uniform rotation is superposed 
in the plane of the orbit ; only the superposed rotation will in this 
case be comparatively much more rapid and the deviation of the 
periodic orbit from an ellipse much greater than in the case of the 
hydrogen atom. For an orbit of this type the stationary states, just 
as in the theory of the fine structure, will be determined by two 
quantum numbers which we shall denote by n and k, connected in 
a very simple manner with the kinematic properties of the orbit. 
For brevity I shall only mention that while the quantum number 
k is connected with the value of the constant angular momentum 
of the electron about the centre in the simple manner previously 



PHYSICAL AND CHEMICAL PROPERTIES OF THE ELEMENTS 79 



indicated, the determination of the principal quantum number n 
requires an investigation of the whole course of the orbit and for 
an arbitrary central orbit will not be related in a simple way to 
the dimensions of the rotating periodic orbit if this deviates essen- 
tially from a Keplerian ellipse. 




*- 



I 



I I 



Ot=D 
(k-2) 

ik=3> 
(k=4) 
Oc=5) 



Fig. 2. 



These results are represented in fig. 2 which is a repro- 
duction of an illustration I have used on a previous occasion 
(see Essay II, p. 30), and which gives a survey of the origin 
of the sodium spectrum. The black dots represent the sta- 
tionary states corresponding to the various series of spectral terms, 
shown on the right by the letters 8, P, D and B. These letters 
correspond to the usual notations employed in spectroscopic 
literature and indicate the nature of the series (sharp series, 
principal series, diffuse series, etc.) obtained by combinations of 
the corresponding spectral terms. The distances of the separate 
points from the vertical line at the right of the figure are pro- 
portional to the numerical value of the energy of the atom given 
by equation (13). The oblique, black arrows indicate finally the 
transitions between the stationary states giving rise to the 
appearance of the lines in the commonly observed sodium 
spectrum. The values of n and k attached to the various states 
indicate the quantum numbers, which, according to Sommerfeld's 
theory, from a preliminary consideration might be regarded as 
characterizing the orbit of the outer electron. For the sake of 
convenience the states which were regarded as corresponding to 
the same value of n are connected by means of dotted lines, and these 
are so drawn that their vertical asymptotes correspond to the 



80 THE STRUCTURE OF THE ATOM AND THE 

terms in the hydrogen spectrum which belong to the same value 
of the principal quantum number. The course of the curves illus- 
trates how the deviation from the hydrogen terms may be expected 
to decrease with increasing values of k, corresponding to states, 
where the minimum distance between the electron in its revolution 
and the nucleus constantly increases. 

It should be noted that even though the theory represents the 
principal features of the structure of the series spectra it has not 
yet been possible to give a detailed account of the spectrum of any 
element by a closer investigation of the electronic orbits which may 
occur in a simple field of force possessing central symmetry. As 
I have mentioned already the lines of most spectra show a complex 
structure. In the sodium spectrum for instance the lines of the 
principal series are doublets indicating that to each P-term not 
one stationary state, but two such states correspond with slightly 
different values of the energy. This difference is so little that 
it would not be recognizable in a diagram on the same scale as 
fig. 2. The appearance of these doublets is undoubtedly due to 
the small deviations from central symmetry of the field of force 
originating from the inner system in consequence of which the 
general type of motion of the external electron will possess a 
more complicated character than that of a simple central motion. 
As a result the stationary states must be characterized by more 
than two quantum numbers, in the same way that the occurrence 
of deviations of the orbit of the electron in the hydrogen atom from 
a simple periodic orbit requires that the stationary states of this 
atom shall be characterized by more than one quantum number. 
Now the rules of the quantum theory lead to the introduction of 
a third quantum number through the condition that the resultant 
angular momentum of the atom, multiplied by 2?r, is equal to an 
entire multiple of Planck's constant. This determines the orienta- 
tion of the orbit of the outer electron relative to the axis of the 
inner system. 

In this way Sommerfeld, Lande and others have shown that it 
is possible not only to account in a formal way for the complex 
structure of the lines of the series spectra, but also to obtain a 
promising interpretation of the complicated effect of external 
magnetic fields on this structure. We shall not enter here on these 



PHYSICAL AND CHEMICAL PROPERTIES OF THE ELEMENTS 81 

problems but shall confine ourselves to the problem of the fixation 
of the two quantum numbers n and k, which to a first approxi- 
mation describe the orbit of the outer electron in the stationary 
states, and whose determination is a matter of prime importance 
in the following discussion of the formation of the atom. In 
the determination of these numbers we at once encounter diffi- 
culties of a profound nature, which as we shall see are intimately 
connected with the question of the remarkable stability of atomic 
structure. I shall here only remark that the values of the quantum 
number n, given in the figure, undoubtedly can not be retained, 
neither for the S nor the P series. On the other hand, so far as 
the values employed for the quantum number k are concerned, it 
may be stated with certainty, that the interpretation of the pro- 
perties of the orbits, which they indicate, is correct. A starting 
point for the investigation of this question has been obtained from 
considerations of an entirely different kind from those previously 
mentioned, which have made it possible to establish a close con- 
nection between the motion in the atom and the appearance of 
spectral lines. 

Correspondence principle. So far as the principles of the 
quantum theory are concerned, the point which has been emphasized 
hitherto is the radical departure of these principles from our 
usual conceptions of mechanical and electrodynamical pheno- 
mena. As I have attempted to show in recent years, it appears 
possible, however, to adopt a point of view which suggests that the 
quantum theory may, nevertheless, be regarded as a rational 
generalization of our ordinary conceptions. As may be seen from 
the postulates of the quantum theory, and particularly the frequency 
relation, a direct connection between the spectra and the motion 
of the kind required by the classical dynamics is excluded, but at 
the same time the form of these postulates leads us to another 
relation of a remarkable nature. Let us consider an electrodynamic 
system and inquire into the nature of the radiation which would 
result from the motion of the system on the basis of the ordinary 
conceptions. We imagine the motion to be decomposed into purely 
harmonic oscillations, and the radiation is assumed to consist of 
the simultaneous emission of series of electromagnetic waves 
B. 6 



82 THE STRUCTURE OF THE ATOM AND THE 

possessing the same frequency as these harmonic components and 
intensities which depend upon the amplitudes of the components. 
An investigation of the formal basis of the quantum theory shows 
us now, that it is possible to trace the question of the origin of the 
radiation processes which accompany the various transitions back 
to an investigation of the various harmonic components, which 
appear in the motion of the atom. The possibility, that a parti- 
cular transition shall occur, may be regarded as being due to the 
presence of a definitely assignable "corresponding" component in 
the motion. This principle of correspondence at the same time 
throws light upon a question mentioned several times previously, 
namely the relation between the number of quantum numbers, 
which must be used to describe the stationary states of an atom, 
and the types to which the orbits of the electrons belong. The 
classification of these types can be based very simply on a decom- 
position of the motion into its harmonic components. Time does 
not permit me to consider this question any further, and I shall 
confine myself to a statement of some simple conclusions, which 
the correspondence principle permits us to draw concerning the 
occurrence of transitions between various pairs of stationary states. 
These conclusions are of decisive importance in the subsequent 
argument. 

The simplest example of such a conclusion is obtained by 
considering an atomic system, which contains a particle describing 
a purely periodic orbit, and where the stationary states are charac- 
terized by a single quantum number n. In this case the motion 
can according to Fourier's theorem be decomposed into a simple 
series of harmonic oscillations whose frequency may be written TO>, 
where r is a whole number, and <u is the frequency of revolution 
in the orbit. It can now be shown that a transition between two 
stationary states, for which the values of the quantum number are 
respectively equal to n' and n", will correspond to a harmonic 
component, for which r = n n". This throws at once light upon 
the remarkable difference which exists between the possibilities 
of transitions between the stationary states of a hydrogen atom 
on the one hand and of a simple system consisting of an electric 
particle capable of executing simple harmonic oscillations about a 
position of equilibrium on the other. For the latter system, which 



PHYSICAL AND CHEMICAL PROPERTIES OF THE ELEMENTS 83 

is frequently called a Planck oscillator, the energy in the stationary 
states is determined by the familiar formula E = nhca, and with the 
aid of the frequency relation we obtain therefore for the radiation 
which will be emitted during a transition between two stationary 
states v = (ri n") o>. Now, an important assumption, which is not 
only essential in Planck's theory of temperature radiation, but 
which also appears necessary to account for the molecular absorption 
in the infra-red region of radiation, states that a harmonic oscillator 
will only emit and absorb radiation, for which the frequency v is 
equal to the frequency of oscillation o> of the oscillator. We are 
therefore compelled to assume that in the case of the oscillator 
transitions can occur only between stationary states which are 
characterized by quantum numbers differing by only one unit, 
while in the hydrogen spectrum represented by formula (2) all 
possible transitions could take place between the stationary states 
given by formula (5). From the point of view of the principle of 
correspondence it is seen, however, that this apparent difficulty is 
explained by the occurrence in the motion of the hydrogen atom, 
as opposed to the motion of the oscillator, of harmonic components 
corresponding to values of r, which are different from 1; or using 
a terminology well known from acoustics, there appear overtones 
in the motion of the hydrogen atom. 

Another simple example of the application of the correspondence 
principle is afforded by a central motion, to the investigation of 
which the explanation of the series spectra in the first approxima- 
tion may be reduced. Referring once more to the figure of the 
sodium spectrum, we see that the black arrows, which correspond 
to the spectral lines appearing under the usual conditions of 
excitation, only connect pairs of points in consecutive rows. Now 
it is found that this remarkable limitation of the occurrence of 
combinations between spectral terms may quite naturally be 
explained by an investigation of the harmonic components into 
which a central motion can be resolved. It can readily be shown 
that such a motion can be decomposed into two series of harmonic 
components, whose frequencies can be expressed by TO) + a- and 
TCO cr respectively, where r is a whole number, o the frequency of 
revolution in the rotating periodic orbit and a the frequency of the 
superposed rotation. These components correspond with transitions 

62 



84 THE STRUCTURE OF THE ATOM AND THE 

where the principal number n decreases by r units, while the 
quantum number k decreases or increases, respectively, by one 
unit, corresponding exactly with the transitions indicated by the 
black arrows in the figure. This may be considered as a very 
important result, because we may say, that the quantum theory, 
which for the first time has offered a simple interpretation of the 
fundamental principle of combination of spectral lines has at the 
same time removed the mystery which has hitherto adhered 
to the application of this principle on account of the apparent 
capriciousness of the appearance of predicted combination lines. 
Especially attention may be drawn to the simple interpretation 
which the quantum theory offers of the appearance observed by 
Stark and his collaborators of certain new series of lines, which do 
not appear under ordinary circumstances, but which are excited 
when the emitting atoms are subject to intense external electric 
fields. In fact, on the correspondence principle this is immediately 
explained from an examination of the perturbations in the motion 
of the outer electron which give rise to the appearance in this 
motion besides the harmonic components already present in a 
simple central orbit of a number of constituent harmonic vibra- 
tions of new type and of amplitudes proportional to the intensity 
of the external forces. 

It may be of interest to note that an investigation of the 
limitation of the possibility of transitions between stationary 
states, based upon a simple consideration of conservation of angular 
momentum during the process of radiation, does not, contrary to 
what has previously been supposed (compare Essay II, p. 62), 
suffice to throw light on the remarkably simple structure of series 
spectra illustrated by the figure. As mentioned above we must 
assume that the "complexity" of the spectral terms, corresponding 
to given values of n and k, which we witness in the fine 
structure of the spectral lines, may be ascribed to states, cor- 
responding to different values of this angular momentum, in 
which the plane of the electronic orbit is orientated in a different 
manner, relative to the configuration of the previously bound 
electrons in the atom. Considerations of conservation of angular 
momentum can, in connection with the series spectra, therefore only 
contribute to an understanding of the limitation of the possibilities 



PHYSICAL AND CHEMICAL PROPERTIES OF THE ELEMENTS 85 

of combination observed in the peculiar laws applying to the 
number of components in the complex structure of the lines. So 
far as the last question is concerned, such considerations offer a 
direct support for the consequences of the correspondence principle. 

III. FORMATION OF ATOMS AND THE PERIODIC TABLE 

A correspondence has been shown to exist between the motion 
of the electron last captured and the occurrence of transitions 
between the stationary states corresponding to the various stages 
of the binding process. This fact gives a point of departure for a 
choice between the numerous possibilities which present themselves 
when considering the formation of the atoms by the successive 
capture and binding of the electrons. Among the processes which 
are conceivable and which according to the quantum theory might 
occur in the atom we shall reject those whose occurrence can not be 
regarded as consistent with a correspondence of the required nature. 

First Period. Hydrogen Helium. It will not be necessary to 
concern ourselves long with the question of the constitution of the 
hydrogen atom. From what has been said previously we may assume 
that the final result of the process of binding of the first electron in 
any atom will be a stationary state, where the energy of the atom 
is given by (5), if we put n = 1, or more precisely by formula (11), 
if we put n = 1 and k = 1. The orbit of the electron will be a circle 
whose radius will be given by formulae (10), if n and k are each 
put equal to 1. Such an orbit will be called a 1 -quantum orbit, 
and in general an orbit for which the principal quantum number 
has a given value n will be called an n-quantum orbit. Where it 
is necessary to differentiate between orbits corresponding to various 
values of the quantum number k, a central orbit, characterized by 
given values of the quantum numbers n and k, will be referred to 
as an n k orbit. 

In the question of the constitution of the helium atom we meet 
the much more complicated problem of the binding of the second 
electron. Information about this binding process may, however, be 
obtained from the arc spectrum of helium. This spectrum, as 
opposed to most other simple spectra, consists of two complete 
systems of lines with frequencies given by formulae of the type 



86 THE STRUCTURE OF THE ATOM AND THE 

(12). On this account helium was at first assumed to be a mixture 
of two different gases, "orthohelium" and "parhelium," but now we 
know that the two spectra simply mean that the binding of the second 
electron can occur in two different ways. A theoretical explanation of 
the main features of the helium spectrum has recentlybeen attempted 
in an interesting paper by Lande. He supposes the emission of the 
orthohelium spectrum to be due to transitions between stationary 
states where both electrons move in the same plane and revolve 
in the same sense. The parhelium spectrum, on the other hand, is 
ascribed by him to stationary states where the planes of the orbits 
form an angle with each other. Dr Kramers and I have made a 
closer investigation of the interaction between the two orbits in 
the different stationary states. The results of our investigation 
which was begun several years before the appearance of Lande's 
work have not yet been published. Without going into details 
I may say, that even though our results in several respects differ 
materially from those of Lande" (compare Essay II, p. 56), we agree 
with his general conclusions concerning the origin of the ortho- 
helium and parhelium spectra. 

The final result of the binding of the second electron is inti- 
mately related to the origin of the two helium spectra. Important 
information on this point has been obtained recently by Franck 
and his co-workers. As is well known he has thrown light upon 
many features of the structure of the atom and of the origin 
of spectra by observing the effect of bombarding atoms by 
electrons of various velocities. A short time ago these experiments 
showed that the impact of electrons could bring helium into a 
"metastable" state from which the atom cannot return to its 
normal state by means of a simple transition accompanied by the 
emission of radiation, but only by means of a process analogous to 
a chemical reaction involving interaction with atoms of other 
elements. This result is closely connected with the fact that the 
binding of the second electron can occur in two different ways, as 
is shown by the occurrence of two distinct spectra. Thus it is 
evident from Franck's experiments that the normal state of the 
atom is the last stage in the binding process involving the emission 
of the parhelium spectrum by which the electron last captured as 
well as the one first captured will be bound in a lj orbit. The 



PHYSICAL AND CHEMICAL PROPERTIES OF THE ELEMENTS 87 

metastable state, on the contrary, is the final stage of the process 
giving the orthohelium spectrum. In this case the second electron, 
as opposed to the first, will move in a 2! orbit. This corresponds to 
a firmness of binding which is about six times less than for the 
electron in the normal state of the atom. 

If we now consider somewhat more closely this apparently 
surprising result, it is found that a clear grasp of it may be obtained 
from the point of view of correspondence. It can be shown that 
the coherent class of motions to which the orthohelium orbits 
belong does not contain a l l orbit. If on the whole we would claim 
the existence of a state where the two electrons move in lj orbits 
in the same plane, and if in addition it is claimed that the motion 
should possess the periodic properties necessary for the definition 
of stationary states, then there seems that no possibility is afforded 
other than the assumption that the two electrons move around the 
nucleus in one and the same orbit, in such a manner that at each 
moment they are situated at the ends of a diameter. This extremely 
simple ring-configuration might be expected to correspond to 
the firmest possible binding of the electrons in the atom, and it 
was on this account proposed as a model for the helium atom in 
my first paper on atomic structure. If, however, we inquire about 
the possibility of a transition from one of the orthohelium states 
to a configuration of this type we meet conditions which are very 
different from those which apply to transitions between two of 
the orthohelium orbits. In fact, the occurrence of each of these 
transitions is due to the existence of well-defined corresponding 
constituent harmonic vibration in the central orbits which the outer 
electron describes in the class of motions to which the stationary 
states belong. The transition we have to discuss, on the other 
hand, is one by which the last captured electron is transferred from 
a state in which it is moving " outside" the other to a state in which 
it moves round the nucleus on equal terms with the other electron. 
Now it is impossible to find a series of simple intermediate forms 
for the motion of those two electrons in which the orbit of the last 
captured electron exhibits a sufficient similarity to a central motion 
that for this transition there could be a correspondence of the 
necessary kind. It is therefore evident, that where the two electrons 
move in the same plane, the electron captured last can not be 



88 THE STRUCTURE OF THE ATOM AND THE 

bound firmer than in a 2j orbit. If, on the other hand, we consider 
the binding process which accompanies the emission of theparhelium 
spectrum and where the electrons in the stationary states move in 
orbits whose planes form angles with one another we meet essen- 
tially different conditions. A corresponding intimate change in the 
interaction between the electron last captured and the one previously 
bound is not required here for the two electrons in the atom to 
become equivalent. We may therefore imagine the last stage of 
the binding process to take place in a manner similar to those 
stages corresponding to transitions between orbits characterized by 
greater values of n and k, 

In the normal state of the helium atom the two electrons must 
be assumed to move in equivalent 1 1 orbits. As a first approximation 
these may be described as two circular orbits, whose planes make 
an angle of 120 with one another, in agreement with the conditions 
which the angular momentum of an atom according to the quantum 
theory must satisfy. On account of the interaction between the 
two electrons these planes at the same time turn slowly around 
the fixed impulse axis of the atom. Starting from a distinctly 
different point of view Kemble has recently suggested a similar 
model for the helium atom. He has at the same time directed 
attention to a possible type of motion of very marked symmetry 
in which the electrons during their entire revolution assume 
symmetrical positions with reference to a fixed axis. Kemble has 
not, however, investigated this motion further. Previous to the 
appearance of this paper Kramers had commenced a closer investi- 
gation of precisely this type of motion in order to find out to what 
extent it was possible from such a calculation to account for the 
firmness with which the electrons are bound in the helium atom, 
that is to account for the ionization potential. Early measurements 
of this potential had given values corresponding approximately to 
that which would result from the ring-configuration already men- 
tioned. This requires 17/8 as much work to remove a single 
electron as is necessary to remove an electron from the hydrogen 
atom in its normal state. As the theoretical value for the latter 
amount of work which for the sake of simplicity will be repre- 
sented by W corresponds to an ionization potential of 13'53 volts, 
the ionization potential of helium would be expected to be 28 - 8 



PHYSICAL AND CHEMICAL PROPERTIES OF THE ELEMENTS 89 

volts. Recent and more accurate determinations, however, have 
given a value for the ionization potential of helium which is con- 
siderably lower and lies in the neighbourhood of 25 volts. This 
showed therefore the untenability of the ring-configuration quite 
independently of any other considerations. A careful investigation of 
the spatial atomic configuration requires elaborate calculation, and 
Kramers has not yet obtained final results. With the approximation 
to which they have been so far completed the calculations point to 
the possibility of an agreement with the experimental results. The 
final result may be awaited with great interest, since it offers in 
the simplest case imaginable a test of the principles by which we 
are attempting to determine stationary states of atoms containing 
more than one electron. 

Hydrogen and helium, as seen in the survey of the periodic 
system given in fig. 1, together form the first period in the system 
of elements, since helium is the first of the inactive gases. The great 
difference in the chemical properties of hydrogen and helium is 
closely related to the great difference in the nature of the binding 
of the electron. This is directly indicated by the spectra and 
ionization potentials. While helium possesses the highest known 
ionization potential of all the elements, the binding of the electron 
in the hydrogen atom is sufficiently loose to account for the tendency 
of hydrogen to form positive ions in aqueous solutions and chemical 
combinations. Further consideration of this particular question 
requires, however, a comparison between the nature and firmness 
of the electronic configurations of other atoms, and it can therefore 
not be discussed at the moment. 

Second Period. Lithium Neon. When considering the atomic 
structure of elements which contain more than two electrons in the 
neutral atom, we shall assume first of all that what has previously 
been said about the formation of the helium atom will in the main 
features also apply to the capture and binding of the first two 
electrons. These electrons may, therefore, in the normal state of 
the atom be regarded as moving in equivalent orbits characterized 
by the quantum symbol 1^ We obtain direct information about 
the binding of the third electron from the spectrum of lithium. 
This spectrum shows the existence of a number of series of 



90 THE STRUCTURE OF THE ATOM AND THE 

stationary states, where the firmness with which the last captured 
electron is bound is very nearly the same as in the stationary states 
of the hydrogen atom. These states correspond to orbits where k 
is greater than or equal to 2, and where the last captured electron 
moves entirely outside the region where the first two electrons 
move. But in addition this spectrum gives us information about a 
series of states corresponding to k = 1 in which the energy differs 
essentially from the corresponding stationary states of the hydrogen 
atom. In these states the last captured electron, even if it remains 
at a considerable distance from the nucleus during the greater part 
of its revolution, will at certain moments during the revolution 
approach to a distance from the nucleus which is of the same order 
of magnitude as the dimensions of the orbits of the previously 
bound electrons. On this account the electrons will be bound with 
a firmness which is considerably greater than that with which the 
electrons are bound in the stationary states of the hydrogen atom 
corresponding to the same value of n. 

Now as regards the lithium spectrum as well as the other alkali 
spectra we are so fortunate (see p. 32) as to possess definite evidence 
about the normal state of the atom from experiments on selective 
absorption. In fact these experiments tell us that the first member 
of the sequence of $-terms corresponds to this state. This term 
corresponds to a strength of binding which is only a little more than 
a third of that of the hydrogen atom. We must therefore conclude 
that the outer electron in the normal state of the lithium atom 
moves in a 2 X orbit, just as the outer electron in the metastable 
state of the helium atom. The reason why the binding of the 
outer electron can not proceed to an orbit characterized by a smaller 
value for the total quantum number may also be considered as 
analogous in the two cases. In fact, a transition by which the third 
electron in the lithium atom was ultimately bound in a l a orbit 
would lead to a state in the atom in which the electron would play 
an equivalent part with the two electrons previously bound. Such 
a process would be of a type entirely different from the transitions 
between the stationary states connected with the emission of the 
lithium spectrum, and would, contrary to these, not exhibit a 
correspondence with a harmonic component in the motion of the 
atom. 



PHYSICAL AND CHEMICAL PROPERTIES 6? THE ELEMENTS 91 

We obtain, therefore, a picture of the formation and structure of 
the lithium atom which offers a natural explanation of the great 
difference of the chemical properties of lithium from those of helium 
and hydrogen. This difference is at once explained by the fact that 
the firmness by which the last captured electron is bound in its 
2j orbit in the lithium atom is only about a third of that with which 
the electron in the hydrogen atom is held, and almost five times 
smaller than the firmness of the binding of the electrons in the 
helium atom. 

What has been said here applies not alone to the formation of 
the lithium atom, but may also be assumed to apply to the binding 
of the third electron in every atom, so that in contrast to the first 
two electrons which move in lj orbits this may be assumed to move 
in a 2j orbit. As regards the binding of the fourth, fifth and sixth 
electrons in the atom, we do not possess a similar guide as no simple 
series spectra are known of beryllium, boron and carbon. Although 
conclusions of the same degree of certainty can not be reached it 
seems possible, however, to arrive at results consistent with general 
physical and chemical evidence by proceeding by means of con- 
siderations of the same kind as those applied to the binding of the 
first three electrons. In fact, we shall assume that the fourth, fifth 
and sixth electrons will be bound in 2j orbits. The reason why the 
binding of a first electron in an orbit of this type will not prevent the 
capture of the others in two quanta orbits may be ascribed to the fact 
that 2j orbits are not circular but very excentric. For example, the 
3rd electron cannot keep the remaining electrons away from the inner 
system in the same way in which the first two electrons bound in 
the lithium atom prevent the third from being bound in a 
1-quantum orbit. Thus we shall expect that the 4th, 5th and 6th 
electrons in a similar way to the 3rd will at certain moments of 
their revolution enter into the region where the first two 
bound electrons move. We must not imagine, however, that these 
visits into the inner system take place at the same time, but 
that the four electrons visit the nucleus separately at equal 
intervals of time. In earlier work on atomic structure it was sup- 
posed that the electrons in the various groups in the atom moved 
in separate regions within the atom and that at each moment the 
electrons within each separate group were arranged in configurations 



92 THE STRUCTURE OF THE ATOM AND THE 

possessing symmetry like that of a regular polygon or polyhedron. 
Among other things this involved that the electrons in each group 
were supposed to be at the point of the orbit nearest the nucleus 
at the same time. A structure of this kind may be described as one 
where the motions of the electrons within the groups are coupled 
together in a manner which is largely independent of the interaction 
between the various groups. On the contrary, the characteristic 
feature of a structure like that I have suggested is the intimate 
coupling between the motions of the electrons in the various groups 
characterized by different quantum numbers, as well as the greater 
independence in the mode of binding within one and the same group 
of electrons the orbits of which are characterized by the same 
quantum number. In emphasizing this last feature I have two 
points in mind. Firstly the smaller effect of the presence of pre- 
viously bound electrons on the firmness of binding of succeeding 
electrons in the same group. Secondly the way in which the motions 
of the electrons within the group reflect the independence both of 
the processes by which the group can be formed and by which it 
can be reorganized by change of position of the different electrons 
in the atom after a disturbance by external forces. The last point 
will be considered more closely when we deal with the origin and 
nature of the X-ray spectra; for the present we shall continue the 
consideration of the structure of the atom to which we are led by 
the investigation of the processes connected with the successive 
capture of the electrons. 

The preceding considerations enable us to understand the fact 
that the two elements beryllium and boron immediately succeeding 
lithium can appear electropositively with 2 and 3 valencies respec- 
tively in combination with other substances. For like the third 
electron in the lithium atom, the last captured electrons in these 
elements will be much more lightly bound than the first two 
electrons. At the same time we understand why the electropositive 
character of these elements is less marked than in the case of 
lithium, since the electrons in the 2-quanta orbits will be much 
more firmly bound on account of the stronger field in which they 
are moving. New conditions arise, however, in the case of the 
next element, carbon, as this element in its typical chemical com- 
binations can not be supposed to occur as an ion, but rather as a 



PHYSICAL AND CHEMICAL PROPERTIES OF THE ELEMENTS 93 

neutral atom. This must be assumed to be due not only to the great 
firmness in the binding of the electrons but also to be an essential 
consequence of the symmetrical configuration of the electrons. 

With the binding of the 4th, 5th and 6th electrons in 2 l orbits, 
the spatial symmetry of the regular configuration of the orbits 
must be regarded as steadily increasing, until with the binding of 
the 6th electron the orbits of the four last bound electrons may be 
expected to form an exceptionally symmetrical configuration in 
which the normals to the planes of the orbits occupy positions 
relative to one another nearly the same as the lines from the centre 
to the vertices of a regular tetrahedron. Such a configuration 
of groups of 2-quanta orbits in the carbon atom seems capable 
of furnishing a suitable foundation for explaining the structure of 
organic compounds. I shall not discuss this question any further, 
for it would require a thorough study of the interaction between 
the motions of the electrons in the atoms forming the molecule. 
I might mention, however, that the types of molecular models to 
which we are led are very different from the molecular models 
which were suggested in my first papers. In these the chemical 
"valence bonds" were represented by "electron rings" of the same 
type as those which were assumed to compose the groups of 
electrons within the individual atoms. It is nevertheless possible 
to give a general explanation of the chemical properties of the 
elements without touching on those matters at all. This is largely 
due to the fact that the structures of combinations of atoms of the 
same element and of many organic compounds do not have the 
same significance for our purpose as those molecular structures in 
which the individual atoms occur as electrically charged ions. The 
latter kind of compounds, to which the greater number of simple 
inorganic compounds belong, is frequently called "heteropolar" and 
possesses a far more typical character than the first compounds 
which are called "homoeopolar," and whose properties to quite a 
different degree exhibit the individual peculiarities of the elements. 
My main purpose will therefore be to consider the fitness which 
the configurations of the electrons in the various atoms offer for 
the formation of ions. 

Before leaving the carbon atom I should mention, that a model 
of this atom in which the orbits of the four most lightly bound 



94 THE STRUCTURE OF THE ATOM AND THE 

electrons possess a pronounced tetrahedric symmetry had already 
been suggested by Landed In order to agree with the measurements 
of the size of the atoms he also assumed that these electrons moved 
in 2j orbits. There is, however, this difference between Lande's 
view and that given here, that while Lande deduced the character- 
istic properties of the carbon atom solely from an investigation of 
the simplest form of motion which four electrons can execute 
employing spatial symmetry, our view originates from a considera- 
tion of the stability of the whole atom. For our assumptions about 
the orbits of the electrons are based directly on an investigation of 
the interaction between these electrons and the first two bound 
electrons. The result is that our model of the carbon atom has 
dynamic properties which are essentially different from the proper- 
ties of Lande's model. 

In order to account for the properties of the elements in the second 
half of the second period it will first of all be necessary to show 
why the configuration of ten electrons occurring in the neutral atom 
of neon possesses such a remarkable degree of stability. Previously 
it has been assumed that the properties of this configuration were 
due to the interaction between eight electrons which moved in 
equivalent orbits outside the nucleus and an inner group of two 
electrons like that in the helium atom. It will be seen, however, 
that the solution must be sought in an entirely different direction. 
It can not be expected that the 7th electron will be bound in a 2 X orbit 
equivalent to the orbits of the four preceding electrons. The occur- 
rence of five such orbits would so definitely destroy the symmetry 
in the interaction of these electrons that it is inconceivable that a 
process resulting in the accession of a fifth electron to this group 
would be in agreement with the correspondence principle. On the 
contrary it will be necessary to assume that the four electrons in 
their exceptionally symmetrical orbital configuration will keep out 
later captured electrons with the result that these electrons will be 
bound in orbits of other types. 

The orbits which come into consideration for the 7th electron in 
the nitrogen atom and the 7th, 8th, 9th and 10th electrons in the 
atoms of the immediately following elements will be circular orbits 
of the type 2 2 . The diameters of these orbits are considerably larger 
than those of the lj orbits of the first two electrons ; on the other 



PHYSICAL AND CHEMICAL PROPERTIES OF THE ELEMENTS 95 

hand the outermost part of the excentric 2 X orbits will extend some 
distance beyond these circular 2 2 orbits. I shall not here discuss the 
capture and binding of these electrons. This requires a further in- 
vestigation of the interaction between the motions of the electrons 
in the two types of 2-quanta orbits. I shall simply mention, that 
in the atom of neon in which we will assume that there are four 
electrons in 2 2 orbits the planes of these orbits must be regarded not 
only as occupying a position relative to one another characterized 
by a high degree of spatial symmetry, but also as possessing a 
configuration harmonizing with the four elliptical 2! orbits. An 
interaction of this kind in which the orbital planes do not 
coincide can be attained only if the configurations in both sub- 
groups exhibit a systematic deviation from tetrahedral symmetry. 
This will have the result that the electron groups with 2-quanta 
orbits in the neon atom will have only a single axis of symmetry 
which must be supposed to coincide with the axis of symmetry of 
the innermost group of two electrons. 

Before leaving the description of the elements within the second 
period it may be pointed out that the above considerations offer a 
basis for interpreting that tendency of the neutral atoms of oxygen 
.and fluorine for capturing further electrons which is responsible for 
the marked electronegative character of these elements. In fact, 
this tendency may be ascribed to the fact that the orbits of 
the last captured electrons will find their place within the region, 
in which the previously captured electrons move in 2j orbits. This 
.suggests an explanation of the great difference between the pro- 
perties of the elements in the latter half of the second period of the 
periodic system and those of the elements in the first half, in whose 
atoms there is only a single type of 2-quanta orbits. 

Third Period. Sodium Argon. We shall now consider the 
structure of atoms of elements in the third period of the periodic 
system. This brings us immediately to the question of the binding 
of the llth electron in the atom. Here we meet conditions which 
in some respects are analogous to those connected with the binding 
of the 7th electron. The same type of argument that applied to 
the carbon atom shows that the symmetry of the configuration in 
the neon atom would be essentially, if not entirely, destroyed by 



96 THE STRUCTURE OF THE ATOM AND THE 

the addition of another electron in an orbit of the same type as 
that in which the last captured electrons were bound. Just as in 
the case of the 3rd and 7th electrons we may therefore expect to 
meet a new type of orbit for the llth electron in the atom, and the 
orbits which present themselves this time are the 3 X orbits. An 
electron in such an orbit will for the greater part of the time remain 
outside the orbits of the first ten electrons. But at certain moments 
during the revolution it will penetrate not only into the region of 
the 2-quanta orbits, but like the 2 X orbits it will penetrate to 
distances from the nucleus which are smaller than the radii of 
the 1-quantum orbits of the two electrons first bound. This fact, 
which has a most important bearing on the stability of the atom, 
leads to a peculiar result as regards the binding of the llth electron. 
In the sodium atom this electron will move in a field which so far 
as the outer part of the orbit is concerned deviates only very little 
from that surrounding the nucleus in the hydrogen atom, but the 
dimensions of this part of the orbit will, nevertheless, be essentially 
different from the dimensions of the corresponding part of a 3j 
orbit in the hydrogen atom. This arises from the fact, that even 
though the electron only enters the inner configuration of the first 
ten electrons for short intervals during its revolution, this part of 
the orbit will nevertheless exert an essential influence upon the 
determination of the principal quantum number. This is directly 
related to the fact that the motion of the electron in the first part 
of the orbit deviates only a little from the motion which each of 
the previously bound electrons in 2j orbits executes during a com- 
plete revolution. The uncertainty which has prevailed in the 
determination of the quantum numbers for the stationary states 
corresponding to a spectrum like that of sodium is connected with 
this. This question has been discussed by several physicists. From 
a comparison of the spectral terms of the various alkali metals, 
Roschdestwensky has drawn the conclusion that the normal state 
does not, as we might be inclined to expect a priori, correspond to 
a lj orbit as shown in fig. 2 on p. 79, but that this state corre- 
sponds to a 2 l orbit. Schrodinger has arrived at a similar result 
in an attempt to account for the great difference between the 
S terms and the terms in the P and D series of the alkali spectra. 
He assumes that the "outer" electron in the states corresponding 



PHYSICAL AND CHEMICAL PROPERTIES OF THE ELEMENTS 97 

to the S terms in contrast to those corresponding to the P and 
D terms penetrates partly into the region of the orbits of the 
inner electrons during the course of its revolution. These investi- 
gations contain without doubt important hints, but in reality the 
conditions must be very different for the different alkali spectra. 
Instead of a 2, orbit as in lithium we must thus assume for 
the spectrum of sodium not only that the first spectral term in 
the 8 series corresponds to a 3j orbit, but also, as a more detailed 
consideration shows, that the first term in the P series corresponds 
not to a 2 2 orbit as indicated in fig. 2, but to a 3 2 orbit. If the 
numbers in this figure were correct, it would require among other 
things that the P terms should be smaller than the hydrogen terms 



(k-J) 
(k-fl 
<k-3) 




J S S S i 

Fig. 3. 

corresponding to the same principal quantum number. This would 
mean that the average effect of the inner electrons could be described 
as a repulsion greater than would occur if their total electrical charge 
were united in the nucleus. This, however, can not be expected from 
our view of atomic structure. The fact that the last captured electron, 
at any rate for low values of &, revolves partly inside the orbits of the 
previously bound electrons will on the contrary involve that the 
presence of these electrons will give rise to a virtual repulsion 
which is considerably smaller than that which would be due to 
their combined charges. Instead of the curves drawn between 
points in fig. 2 which represent stationary states corresponding 
to the same value of the principal quantum number running from 
right to left, we obtain curves which run from left to right, as 
is indicated in fig. 3. The stationary states are labelled with 
B. 7 



98 THE STRUCTURE OF THE ATOM AND THE 

quantum numbers corresponding to the structure I have described. 
According to the view underlying fig. 2 the sodium spectrum 
might be described simply as a distorted hydrogen spectrum, 
whereas according to fig. 3 there is not only distortion but also 
complete disappearance of certain terms of low quantum numbers. 
It may be stated, that this view not only appears to offer an ex- 
planation of the magnitude of the terms, but that the complexity 
of the terms in the P and D series finds a natural explanation in 
the deviation of the configuration of the ten electrons first bound 
from a purely central symmetry. This lack of symmetry has its 
origin in the configuration of the two innermost electrons and 
"transmits" itself to the outer parts of the atomic structure, since 
the 2j orbits penetrate partly into the region of these electrons. 

This view of the sodium spectrum provides at the same time an 
immediate explanation of the pronounced electropositive properties 
of sodium, since the last bound electron in the sodium atom is still 
more loosely bound than the last captured electron in the lithium 
atom. In this connection it might be mentioned that the increase 
in atomic volume with increasing atomic number in the family of 
the alkali metals finds a simple explanation in the successively 
looser binding of the valency electrons. In his work on the X-ray 
spectra Sommerfeld at an earlier period regarded this increase in 
the atomic volumes as supporting the assumption that the principal 
quantum number of the orbit of the valency electrons increases by 
unity as we pass from one metal to the next in the family. His 
later investigations on the series spectra have led him, however, 
definitely to abandon this assumption. At first sight it might also 
appear to entail a far greater increase in the atomic volume than 
that actually observed. A simple explanation of this fact is how- 
ever afforded by realizing that the orbit of the electron will run 
partly inside the region of the inner orbit and that therefore the 
"effective" quantum number which corresponds to the outer almost 
elliptical loop will be much smaller than the principal quantum 
number, by which the whole central orbit is described. It may 
be mentioned that Vegard in his investigations on the X-ray spectra 
has also proposed the assumption of successively increasing quantum 
numbers for the electronic orbits in the various groups of the atom, 
reckoned from the nucleus outward. He has introduced assumptions 



IV. OF MO. LIBRARY 

PHYSICAL AND CHEMICAL PROPERTIES OF THE ELEMENTS 99 

about the relations between the numbers of electrons in the various 
groups of the atom and the lengths of the periods in the periodic 
system which exhibit certain formal similarities with the results 
presented here. But Vegard's considerations do not offer points of 
departure for a further consideration of the evolution and stability 
of the groups, and consequently no basis for a detailed interpretation 
of the properties of the elements. 

When we consider the elements following sodium in the third 
period of the periodic system we meet in the binding of the IZth, 
1.3th and l&th electrons conditions which are analogous to those 
we met in the binding of the 4th, 5th and 6th electrons. In the 
elements of the third periods, however, we possess a far more 
detailed knowledge of the series spectra. Too little is known 
about the beryllium spectrum to draw conclusions about the 
binding of the fourth electron, but we may infer directly from the 
well-known arc spectrum of magnesium that the 12th electron 
in the atom of this element is bound in a 3j orbit. As regards 
the binding of the 13th electron we meet in aluminium an 
absorption spectrum different in structure to that of the alkali 
metals. In fact here not the lines of the principal series but the 
lines of the sharp and diffuse series are absorption lines. Conse- 
quently it is the first member of the P terms and not of the 8 
terms which corresponds to the normal state of the aluminium 
atom, and we must assume that the 13th electron is bound in 
a 3 2 orbit. This, however, would hardly seem to be a general 
property of the binding of the 13th electron in atoms, but rather 
to arise from the special conditions for the binding of the last 
electron in an atom, where already there are two other electrons 
bound as loosely as the valency electron of aluminium. At the 
present state of the theory it seems best to assume that in the 
silicon atom the four last captured electrons will move in 3! 
orbits forming a configuration possessing symmetrical properties 
similar to the outer configuration of the four electrons in 2j orbits 
in carbon. Like what we assumed for the latter configuration we 
shall expect that the configuration of the 3! orbits occurring for the 
first time in silicon possesses such a completion, that the addition 
of a further electron in a 3j orbit to the atom of the following ele- 
ments is impossible, and that the 15th electron in the elements of 

72 



100 THE STRUCTURE OF THE ATOM AND THE 

higher atomic number will be bound in a new type of orbit. In this 
case, however, the orbits with which we meet will not be circular, 
as in the capture of the 7th electron, but will be rotating excentric 
orbits of the type 3 2 . This is very closely related to the fact, men- 
tioned above, that the non-circular orbits will correspond to a 
firmer binding than the circular orbits having the same value for 
the principal quantum number, since the electrons will at certain 
moments penetrate much farther into the interior of the atom. 
Even though a 3 2 orbit will not penetrate into the innermost con- 
figuration of l! orbits, it will penetrate to distances from the nucleus 
which are considerably less than the radii of the circular 2 2 orbits. 
In the case of the 16th, 17th and 18th electrons the conditions are 
similar to those for the 15th. So for argon we may expect a con- 
figuration in which the ten innermost electrons move in orbits of 
the same type as in the neon atom while the last eight electrons will 
form a configuration of four 3j orbits and four 3 2 orbits, whose 
symmetrical properties must be regarded as closely corresponding 
to the configuration of 2-quanta orbits in the neon atom. At the 
same time, as this picture suggests a qualitative explanation of the 
similarity of the chemical properties of the elements in the latter 
part of the second and third periods, it also opens up the possibility 
of a natural explanation of the conspicuous difference from a 
quantitative aspect. 

Fourth Period. Potassium Krypton. In the fourth period 
we meet at first elements which resemble chemically those at the 
beginning of the two previous periods. This is also what we should 
expect. We must thus assume that the 19th electron is bound in 
a new type of orbit, and a closer consideration shows that this will 
be a 4j orbit. The points which were emphasized in connection 
with the binding of the last electron in the sodium atom will be 
even more marked here on account of the larger quantum number 
by which the orbits of the inner electrons are characterized. In 
fact, in the potassium atom the 4j orbit of the 19th electron will, 
as far as inner loops are concerned, coincide closely with the shape 
of a 3j orbit. On this account, therefore, the dimensions of the 
outer part of the orbit will not only deviate greatly from the 
dimensions of a 4i orbit in the hydrogen atom, but will coincide 



PHYSICAL AND CHEMICAL PROPERTIES OF THE ELEMENTS 101 

closely with a hydrogen orbit of the type 2 t , the dimensions of 
which are about four times smaller than the 4 : hydrogen orbit. 
This result allows an immediate explanation of the main features of 
the chemical properties and the spectrum of potassium. Corre- 
sponding results apply to calcium, in the neutral atom of which 
there will be two valency electrons in equivalent 4 t orbits. 

After calcium the properties of the elements in the fourth period 
of the periodic system deviate, however, more and more from the 
corresponding elements in the previous periods, until in the family 
of the iron metals we meet elements whose properties are essenti- 
ally different. Proceeding to still higher atomic numbers we again 
meet different conditions. Thus we find in the latter part of the 
fourth period a series of elements whosechemical properties approach 
more and more to the properties of the elements at the end 
of the preceding periods, until finally with atomic number 36 we 
again meet one of the inactive gases, namely krypton. This is 
exactly what we should expect. The formation and stability of the 
atoms of the elements in the first three periods require that each 
of the first 18 electrons in the atom shall be bound in each suc- 
ceeding element in an orbit of the same principal quantum number 
as that possessed by the particular electron, when it first appeared. 
It is readily seen that this is no longer the case for the 19th 
electron. With increasing nuclear charge and the consequent 
decrease in the difference between the fields of force inside and 
outside the region of the orbits of the first 18 bound electrons, the 
dimensions of those parts of a 4 X orbit which fall outside will 
approach more and more to the dimensions of a 4-quantum orbit 
calculated on the assumption that the interaction between the 
electrons in the atom may be neglected. With increasing atomic 
number a point will therefore be reached where a 3 3 orbit will corre- 
spond to a firmer binding of the 19th electron than a 4 X orbit, and 
this occurs as early as at the beginning of the fourth period. This 
cannot only be anticipated from a simple calculation but is confirmed 
in a striking way from an examination of the series spectra. While 
the spectrum of potassium indicates that the 4! orbit corresponds 
to a binding which is more than twice as firm as in a 3 Z orbit 
corresponding to the first spectral term in the D series, the con- 
ditions are entirely different as soon as calcium is reached. We 



102 



THE STRUCTURE OF THE ATOM AND THE 



shall not consider the arc spectrum which is emitted during the 
capture of the 20th electron but the spark spectrum which corre- 
sponds to the capture and binding of the 19th electron. While the 
spark spectrum of magnesium exhibits great similarity with the 
sodium spectrum as regards the values of the spectral terms in the 
various series apart from the fact that the constant appearing in 
formula (12) is four times as large as the Rydberg constant we 
meet in the spark spectrum of calcium the remarkable condition 





4 


T ' ' 




j 


' [, - 


No. arc. 


* 


*--*f|-- 

1 1 


A 





6, \ ! 




4 


5" &^ 




5,- 


J "*^ ' 


* ~ 




i ! 


a A 


^ 




Afg ^park 


*| 








i i 




* 


'?0_ ^0...|^. 

--Q____jzQ.,.0 


C<z iiea.rk 










! i 



Fig. 4. 

that the first term of the D series is larger than the first term of 
the P series and is only a little smaller than the first term of the 
$ series, which may be regarded as corresponding to the binding 
of the 19th electron in the normal state of the calcium atom. 
These facts are shown in figure 4 which gives a survey of the 
stationary states corresponding to the arc spectra of sodium and 
potassium. As in figures 2 and 3 of the sodium spectrum, we 
have disregarded the complexity of the spectral terms, and the 
numbers characterizingthe stationary states are simply the quantum 



PHYSICAL AND CHEMICAL PROPERTIES OF THE ELEMENTS 103 

numbers n and k. For the sake of comparison the scale in which the 
energy of the different states is indicated is chosen four times as 
small for the spark spectra as for the arc spectra. Consequently 
the vertical lines indicated with various values of n correspond for 
the arc spectra to the spectral terms of hydrogen, for the spark 
spectra to the terms of the helium spectrum given by formula (7). 
Comparing the change in the relative firmness in the binding of 
the 19th electron in a 4 X and 3 3 orbit for potassium and calcium we 
see that we must be prepared already for the next element, 
scandium, to find that the 3 3 orbit will correspond to a stronger 
binding of this electron than a 4j orbit. On the other hand it 
follows from previous remarks that the binding will be much lighter 
than for the first 18 electrons which agrees that in chemical com- 
binations scandium appears electropositively with three valencies. 

If we proceed to the following elements, a still larger number of 
3 3 orbits will occur in the normal state of these atoms, since the 
number of such electron orbits will depend upon the firmness of 
their binding compared to the firmness with which an electron is 
bound in a 4j orbit, in which type of orbit at least the last captured 
electron in the atom may be assumed to move. We therefore meet 
conditions which are essentially different from those which we have 
considered in connection with the previous periods, so that here 
we have to do with the successive development of one of the inner 
groups of electrons in the atom, in this case with groups of electrons 
in 3-quanta orbits. Only when the development of this group has 
been completed may we expect to find once more a corresponding 
change in the properties of the elements with increasing atomic 
number such as we find in the preceding periods. The properties 
of the elements in the latter part of the fourth period show 
immediately that the group, when completed, will possess 18 
electrons. Thus in krypton, for example, we may expect besides 
the groups of 1, 2 and 3-quanta orbits a markedly symmetrical 
configuration of 8 electrons in 4-quanta orbits consisting of four 4j 
orbits and four 4 2 orbits. 

The question now arises : In which way will the gradual forma- 
tion of the group of electrons having 3-quanta orbits take place ? 
From analogy with the constitution of the groups of electrons with 
2-quanta orbits we might at first sight be inclined to suppose that 



104 THE STRUCTURE OF THE ATOM AND THE 

the complete group of 3-quanta orbits would consist of three sub- 
groups of four electrons each in orbits of the types 3 X , 3 4 and 3 S 
respectively, so that the total number of electrons would be 12 
instead of 18. Further consideration shows, however, that such an 
expectation would not be justified. The stability of the configura- 
tion of eight electrons with 2-quanta orbits occurring in neon must 
be ascribed not only to the symmetrical configuration of the elec- 
tronic orbits in the two subgroups of 2 a and 2 2 orbits respectively, 
but fully as much to the possibility ofbringing the orbits inside these 
subgroups into harmonic relation with one another. The situation 
is different, however, for the groups of electrons with 3-quanta 
orbits. Three subgroups of four orbits each can not in this case be 
expected to come into interaction with one another in a corre- 
spondingly simple manner. On the contrary we must assume that 
the presence of electrons in 3 3 orbits will diminish the harmony of 
the orbits within the first two 3-quanta subgroups, at any rate 
when a point is reached where the 19th electron is no longer, as 
was the case with scandium, bound considerably more lightly than 
the previously bound electrons in 3-quanta orbits, but has been 
drawn so far into the atom that it revolves within essentially 
the same region of the atom where these electrons move. We 
shall now assume that this decrease in the harmony will so to 
say "open" the previously "closed" configuration of electrons 
in orbits of these types. As regards the final result, the number 
18 indicates that after the group is finally formed there will 
be three subgroups containing six electrons each. Even if it has 
not at present been possible to follow in detail the various 
steps in the formation of the group this result is nevertheless 
confirmed in an interesting manner by the fact that it is possible 
to arrange three configurations having six electrons each in a simple 
manner relative to one another. The configuration of the subgroups 
does not exhibit a tetrahedral symmetry like the groups of 2-quanta 
orbits in carbon, but a symmetry which, so far as the relative 
orientation of the normals to the planes of the orbits is con- 
cerned, may be described as trigonal. 

In spite of the great difference in the properties of the elements 
of this period, compared with those of the preceding period, the 
completion of the group of 18 electrons in 3-quanta orbits in the 



PHYSICAL AND CHEMICAL PROPERTIES OF THE ELEMENTS 105 

fourth period may to a certain extent be said to have the same 
characteristic results as the completion of the group of 2-quanta 
orbits in the second period. As we have seen, this determined not 
only the properties of neon as an inactive gas, but in addition the 
electronegative properties of the preceding elements and the 
electropositive properties of the elements which follow. The fact 
that there is no inactive gas possessing an outer group of 18 
electrons is very easily accounted for by the much larger dimensions 
which a 3 3 orbit has in comparison with a 2 2 orbit revolving in the 
same field of force. On this account a complete 3-quanta group 
can not occur as the outermost group in a neutral atom, but only 
in positively charged ions. The characteristic decrease in valency 
which we meet in copper, shown by the appearance of the singly 
charged cuprous ions, indicates the same tendency towards the 
completion of a symmetrical configuration of electrons that we 
found in the marked electronegative character of an element like 
fluorine. Direct evidence that a complete group of 3-quanta orbits 
is present in the cuprous ion is given by the spectrum of copper 
which, in contrast to the extremely complicated spectra of the 
preceding elements resulting from the unsymmetrical character of 
the inner system, possesses a simple structure very much like that 
of the sodium spectrum. This may no doubt be ascribed to a 
simple symmetrical structure present in the cuprous ion similar to 
that in the sodium ion, although the great difference in the con- 
stitution of the outer group of electrons in these ions is shown 
both by the considerable difference in the values of the spectral 
terms and in the separation of the doublets in the P terms of the 
two spectra. The occurrence of the cupric compounds shows, how- 
ever, that the firmness of binding in the group of 3-quanta orbits 
in the copper atom is not as great as the firmness with which the 
electrons are bound in the group of 2-quanta orbits in the sodium 
atom. Zinc, which is always divalent, is the first element in which 
the groups of the electrons are so firmly bound that they can not 
be removed by ordinary chemical processes. 

The picture I have given of the formation and structure of the 
atoms of the elements in the fourth period gives an explanation of 
the chemical and spectral properties. In addition it is supported 
by evidence of a different nature to that which we have hitherto 



106 THE STRUCTURE OF THE ATOM AND THE 

used. It is a familiar fact, that the elements in the fourth period 
differ markedly from the elements in the preceding periods 
partly in their magnetic properties and partly in the characteristic 
colours of their compounds. Paramagnetism and colours do occur 
in elements belonging to the foregoing periods, but not in simple 
compounds where the atoms considered enter as ions. Many 
elements of the fourth period, on the contrary, exhibit paramag- 
netic properties and characteristic colours even in dissociated 
aqueous solutions. The importance of this has been emphasized 
by Ladenburg in his attempt to explain the properties of the 
elements in the long periods of the periodic system (see p. 73). 
Langmuir in order to account for the difference between the fourth 
period and the preceding periods simply assumed that the atom, 
in addition to thelayers of cells containing 8 electrons each, possesses 
an outer layer of cells with room for 18 electrons which is com- 
pletely filled for the first time in the case of krypton. Ladenburg, 
on the other hand, assumes that for some reason or other an 
intermediate layer is developed between the inner electronic 
configuration in the atom appearing already in argon, and the 
external group of valency electrons. This layer commences with 
scandium and is completed exactly at the end of the family of iron 
metals. In support of this assumption Ladenburg not only mentions 
the chemical properties of the elements in the fourth period, but 
also refers to the paramagnetism and colours which occur exactly 
in the elements, where this intermediate layer should be in 
development. It is seen that Ladenburg's ideas exhibit certain 
formal similarities with the interpretation I have given above of 
the appearance of the fourth period, and it is interesting to note that 
our view, based on a direct investigation of the conditions for the 
formation of the atoms, enables us to understand the relation 
emphasized by Ladenburg. 

Our ordinary electrodynamic conceptions are probably insufficient 
to form a basis for an explanation of atomic magnetism. This is 
hardly to be wondered at when we remember that they have not 
proved adequate to account for the phenomena of radiation which 
are connected with the intimate interaction between the electric 
and magnetic forces arising from the motion of the electrons. In 
whatever way these difficulties may be solved it seems simplest to 



PHYSICAL AND CHEMICAL PROPERTIES OF THE ELEMENTS 107 

assume that the occurrence of magnetism, such as we meet in the 
elements of the fourth period, results from a lack of symmetry in 
the internal structure of the atom, thus preventing the magnetic 
forces arising from the motion of the electrons from forming a 
system of closed lines of force running wholly within the atom. 
While it has been assumed that the ions of the elements in the 
previous periods, whether positively or negatively charged, contain 
configurations of marked symmetrical character, we must, however, 
be prepared to encounter a definite lack of symmetry in the 
electronic configurations in ions of those elements within the fourth 
period which contain a group of electrons in 3-quanta orbits in the 
transition stage between symmetrical configurations of 8 and 18 
electrons respectively. As pointed out by Kossel, the experimental 
results exhibit an extreme simplicity, the magnetic moment of the 
ions depending only on the number of electrons in the ion. Ferric 
ions, for example, exhibit the same atomic magnetism as manganous 
ions, while manganic ions exhibit the same atomic magnetism as 
chromous ions. It is in beautiful agreement with what we have 
assumed about the structure of the atoms of copper and zinc, that 
the magnetism disappears with those ions containing 28 electrons 
which, as I stated, must be assumed to contain a complete group 
of 3-quanta orbits. On the whole a consideration of the magnetic 
properties of the elements within the fourth period 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 element 
to element. It is to be hoped that a further investigation of the 
magnetic properties will give us a clue to the way in which the 
group of electrons in 3-quanta orbits is developed step by step. 

Also the colours of the ions directly support our view of atomic 
structure. According to the postulates of the quantum theory 
absorption as well as emission of radiation is regarded as taking 
place during transitions between stationary states. The occurrence 
of colours, that is to say the absorption of light in the visible region 
of the spectrum, is evidence of transitions involving energy changes 
of the same order of magnitude as those giving the usual optical 
spectra of the elements. In contrast to the ions of the elements of 
the preceding periods where all the electrons are assumed to be very 
firmly bound, the occurrence of such processes in the fourth period 



108 THE STRUCTURE OF THE ATOM AND THE 

is exactly what we should expect. For the development and com- 
pletion of the electronic groups with 3-quanta orbits will proceed, 
so to say, in competition with the binding of electrons in orbits of 
higher quanta, since the binding of electrons in 3-quanta orbits 
occurs when the electrons in these orbits are bound more firmly 
than electrons in 4j orbits. The development of the group will 
therefore proceed to the point where we may say there is equili- 
brium between the two kinds of orbits. This condition may be 
assumed to be intimately connected not only with the colour of the 
ions, but also with the tendency of the elements to form ions with 
different valencies. This is in contrast to the elements of the first 
periods where the charge of the ions in aqueous solutions is always 
the same for one and the same element. 

Fifth Period. Rubidium Xenon. The structure of the atoms 
in the remaining periods may be followed up in complete analogy 
with what has already been said. Thus we shall assume that the 
37th and 38th electrons in the elements of the fifth period are 
bound in 5j orbits. This is supported by the measurements of the 
arc spectrum of rubidium and the spark spectrum of strontium. 
The latter spectrum indicates at the same time that 4 3 orbits will 
soon appear, and therefore in this period, which like the 4th 
contains 18 elements, we must assume that we are witnessing a 
farther stage in the development of the electronic group of ^-quanta 
orbits. The first stage in the formation of this group may be said 
to have been attained in krypton with the appearance of a sym- 
metrical configuration of eight electrons consisting of two subgroups 
each of four electrons in 4j and 4 2 orbits. A second preliminary 
completion must be regarded as having been reached with the 
appearance of a symmetrical configuration of 18 electrons in the 
case of silver, consisting of three subgroups with six electrons each 
in orbits of the types 4 1( 4 2 and 4 3 . Everything that has been said 
about the successive formation of the group of electrons with 3- 
quanta orbits applies unchanged to this stage in the transformation 
of the group with 4-quanta orbits. For in no case have we made 
use of the absolute values of the quantum numbers nor of assump- 
tions concerning the form of the orbits but only of the number of 
possible types of orbits which might come into consideration. At 



PHYSICAL AND CHEMICAL PROPERTIES OF THE ELEMENTS 109 

the same time it may be of interest to mention that the properties 
of these elements compared with those of the foregoing period 
nevertheless show a difference corresponding exactly to what would 
be expected from the difference in the types of orbits. For instance, 
the divergencies from the characteristic valency conditions of the 
elements in the second and third periods appear later in the fifth 
period than for elements in the fourth period. While an element 
like titanium in the fourth period already shows a marked tendency 
to occur with various valencies, on the other hand an element like 
zirconium is still quadri-valent like carbon in the second period 
and silicon in the third. A simple investigation of the kinematic 
properties of the orbits of the electrons shows in fact that an 
electron in an excentric 4 3 orbit of an element in the fifth 
period will be considerably more loosely bound than an electron in 
a circular 3 3 orbit of the corresponding element in the fourth 
period, while electrons which are bound in excentric orbits of the 
types 5j and 4j respectively will correspond to a binding of about 
the same firmness. 

At the end of the fifth period we may assume that xenon, the 
atomic number of which is 54, has a structure which in addition to 
the two 1 -quantum, eight 2-quanta, eighteen 3-quanta and eight- 
teen 4-quanta orbits already mentioned contains a symmetrical 
configuration of eight electrons in 5-quanta orbits consisting of two 
subgroups with four electrons each in 5! and 5 2 orbits respectively. 

Sixth Period. Caesium Niton. If we now consider the atoms 
of elements of still higher atomic number, we must first of all 
assume that the 55th and 56th electrons in the atoms of caesium 
and barium are bound in Q l orbits. This is confirmed by the spectra 
of these elements. It is clear, however, that we must be prepared 
shortly to meet entirely new conditions. With increasing nuclear 
charge we shall have to expect not only that an electron in a 5, 
orbit will be bound more firmly than in a 6j orbit, but we must also 
expect that a moment will arrive when during the formation of the 
atom a 4 4 orbit will represent a firmer binding of the electron than 
an orbit of 5 or 6-quanta, in much the same way as in the elements 
of the fourth period a new stage in the development of the 3-quanta 
group was started when a point was reached where for the first 



110 THE STRUCTURE OF THE ATOM AND THE 

time the 19th electron was bound in a 3 3 orbit instead of in a 4^ 
orbit. We shall thus expect in the sixth period to meet with a new 
stage in the development of the group with 4-quanta orbits. Once 
this point has been reached we must be prepared to find with in- 
creasing atomic number a number of elements following one another, 
which as in the family of the iron metals have very nearly the same 
properties. The similarity will, however, be still more pronounced, 
since in this case we are concerned with the successive transforma- 
tion of a configuration of electrons which lies deeper in the interior 
of the atom. You will have already guessed that what I have in view 
is a simple explanation of the occurrence of the family of rare earths 
at the beginning of the sixth period. As in the case of the transforma- 
tion and completion of the group of 3-quanta orbits in the fourth 
period and the partial completion of groups of 4-quanta orbits in 
the fifth period, we may immediately deduce from the length of the 
sixth period the number of electrons, namely 32, which are finally 
contained in the 4-quanta group of orbits. Analogous to what 
applied to the group of 3-quanta orbits it is probable that, when 
the group is completed, it will contain eight electrons in each of the 
four subgroups. Even though it has not yet been possible to follow 
the development of the group step by step, we can even here give 
some theoretical evidence in favour of the occurrence of a sym- 
metrical configuration of exactly this number of electrons. I shall 
simply mention that it is not possible without coincidence of the 
planes of the orbits to arrive at an interaction between four sub- 
groups of six electrons each in a configuration of simple trigonal 
symmetry, which is equally simple as that shown by three subgroups. 
The difficulties which we meet make it probable that a harmonic 
interaction can be attained precisely by four groups each containing 
eight electrons the orbital configurations of which exhibit axial 
symmetry. 

Just as in the case of the family of the iron metals in the fourth 
period, the proposed explanation of the occurrence of the family of 
rare earths in the sixth period is supported in an interesting 
manner by an investigation of the magnetic properties of these 
elements. In spite of the great chemical similarity the members 
of this family exhibit very different magnetic properties, so that 
while some of them exhibit but very little magnetism others exhibit 



PHYSICAL AND CHEMICAL PROPERTIES OF THE ELEMENTS 111 

a greater magnetic moment per atom than any other element which 
has been investigated. It is also possible to give a simple interpre- 
tation of the peculiar colours exhibited by the compounds of these 
elements in much the same way as in the case of the family of iron 
metals in the fourth period. The idea that the appearance of the 
group of the rare earths is connected with the development of inner 
groups in the atom is not in itself new and has for instance been 
considered by Vegard in connection with his work on X-ray spectra. 
The new feature of the present considerations lies, however, in the 
emphasis laid on the peculiar way in which the relative strength of 
the binding for two orbits of the same principal quantum number 
but of different shapes varies with the nuclear charge and with the 
number of electrons previously bound. Due to this fact the presence 
of a group like that of the rare earths in the sixth period may be 
considered as a direct consequence of the theory and might actually 
have been predicted on a quantum theory, adapted to the explana- 
tion of the properties of the elements within the preceding periods 
in the way I have shown. 

Besides the final development of the group of ^-quanta orbits we 
observe in the sixth period in the family of the platinum metals the 
second stage in the development of the group of 5-quanta orbits. 
Also in the radioactive, chemically inactive gas niton, which com- 
pletes this period, we observe the first preliminary step in the 
development of a group of electrons with 6-quanta orbits. In the 
atom of this element, in addition to the groups of electrons of two 
1 -quantum, eight 2-quanta, eighteen 3-quanta, thirty-two 4-quanta 
and eighteen 5-quanta orbits respectively, there is also an outer 
symmetrical configuration of eight electrons in 6-quanta orbits, 
which we shall assume to consist of two subgroups with four electrons 
each in 6j and 6 2 orbits respectively. 

Seventh Period. In the seventh and last period of the periodic 
system we may expect the appearance of 7-quanta orbits in the 
normal state of the atom. Thus in the neutral atom of radium in 
addition to the electronic structure of niton there will be two 
electrons in l l orbits which will penetrate during their revolution 
not only into the region of the orbits of electrons possessing lower 
values for the principal quantum number, but even to distances 



112 THE STRUCTURE OF THE ATOM AND THE 

from the nucleus which are less than the radii of the orbits of the 
innermost 1-quantum orbits. The properties of the elements in the 
seventh period are very similar to the properties of the elements in the 
fifth period. Thus, in contrast to the conditions in the sixth period, 
there are no elements whose properties resemble one another like 
those of the rare earths. In exact analogy with what has already 
been said about the relations between the properties of the elements 
in the fourth and fifth periods this may be very simply explained by 
the fact that an excentric 5 4 orbit will correspond to a considerably 
looser binding of an electron in the atom of an element of the 
seventh period than the binding of an electron in a circular 4 4 orbit 
in the corresponding element of the sixth period, while there will be 
a much smaller difference in the firmness of the binding of these 
electrons in orbits of the types 7 a and 6 X respectively. 

It is well known that the seventh period is not complete, for no atom 
has been found having an atomic number greater than 92. This is 
probably connected with the fact that the last elements in the 
system are radioactive and that nuclei of atoms with a total charge 
greater than 92 will not be sufficiently stable to exist under con- 
ditions where the elements can be observed. It is tempting to 
sketch a picture of the atoms formed by the capture and binding 
of electrons around nuclei having higher charges, and thus to 
obtain some idea of the properties which the corresponding hypo- 
thetical elements might be expected to exhibit. I shall not develop 
this matter further, however, since the general results we should 
get will be evident to you from the views I have developed to 
explain the properties of the elements actually observed. A survey 
of these results is given in the following table, which gives a sym- 
bolical representation of the atomic structure of the inactive gases 
which complete the first six periods in the periodic system. In 
order to emphasize the progressive change the table includes the 
probable arrangement of electrons in the next atom which would 
possess properties like the inactive gases. 

The view of atomic constitution underlying this table, which 
involves configurations of electrons moving with large velocities 
between each other, so that the electrons in the "outer" groups 
penetrate into the region of the orbits of the electrons of the "inner" 
groups, is of course completely different from such statical models 



PHYSICAL AND CHEMICAL PROPERTIES OF THE ELEMENTS 113 



of the atom as are proposed by Langmuir. But quite apart from this 
it will be seen that the arrangement of the electronic groups in 
the atom, to which we have been lead by tracing the way in which 
each single electron has been bound, is essentially different from 
the arrangement of the groups in Langmuir's theory. In order to 
explain the properties of the elements of the sixth period Langmuir 
assumes for instance that, in addition to the inner layers of cells 
containing 2, 8, 8, 18 and 18 electrons respectively, which are 
employed to account for the properties of the elements in the 
earlier periods, the atom also possesses a layer of cells with room 
for 32 electrons which is just completed in the case of niton. 



Element 


*: 
^ 




f/umber of Electrons (n n^- orbits. 


1, 


2; 


2, 


4 


\ 


*> 


fj 


* 


* 


* 


$ 





4 


' 


^ 





4 


s> 


6* 


^ 


^ 


7, 


7, 


7, 




Htlium 
Nton 
Argon 
Krypton 
Xenon 
Niton 


2 

10 

n 

36 

S 

36 

us 


I 
I 

I 
2 
2 
2. 

2 


* 

? 
4< 
* 
* 

V- 


y 
* 


* 


4 









6 
<? 



8 

8 


4 
6 

8 


^ 

6 

J 


6 

<J 


<f 


- 




5 


6 














* 

v 
J 


6 
6 
6 


6 
6 
6 


6 

5 
6 


<5 
1 
g 


6 
g 

8 



In this connection it may be of interest to mention a recent 
paper by Bury, to which my attention was first drawn after the 
deliverance of this address, and which contains an interesting 
survey of the chemical properties of the elements based on similar 
conceptions of atomic structure as those applied by Lewis and 
Langmuir. From purely chemical considerations Bury arrives at 
conclusions which as regards the arrangement and completion of 
the groups in the main coincide with those of the present theory, 
the outlines of which were given in my letters to Nature mentioned 
in the introduction. 

Survey of the periodic table. The results given in this address 
are also illustrated by means of the representation of the periodic 
system given in fig. 1. In this figure the frames are meant to 
indicate such elements in which one of the " inner " groups is 
in a stage of development. Thus there will be found in the 



114 THE STRUCTURE OF THE ATOM AND THE 

fourth and fifth periods a single frame indicating the final com- 
pletion of the electronic group with 3-quanta orbits, and the 
last stage but one in the development of the group with 4-quanta 
orbits respectively. In the sixth period it has been necessary to 
introduce two frames, of which the inner one indicates the last 
stage of the evolution of the group with 4-quanta orbits, giving rise 
to the rare earths. This occurs at a place in the periodic system 
where the third stage in the development of an electronic group 
with 5-quanta orbits, indicated by the outer frame, has already 
begun. In this connection it will be seen that the inner frame 
encloses a smaller number of elements than is usually attributed 
to the family of the rare earths. At the end of this group an 
uncertainty exists, due to the fact that no element of atomic 
number 72 is known with certainty. However, as indicated in 
fig. 1, we must conclude from the theory that the group with 
4-quanta orbits is finally completed in lutetium (71). This element 
therefore ought to be the last in the sequence of consecutive 
elements with similar properties in the first half of the sixth period, 
and at the place 72 an element must be expected which in its 
chemical and physical properties is homologous with zirconium and 
thorium. This, which is already indited on Julius Thomsen's old 
table, has also been pointed out by Bury. [Quite recently Dauvillier 
has in an investigation of the X-ray spectrum excited in preparations 
containing rare earths, observed certain faint lines which he ascribes 
to an element of atomic number 72. This element is identified by 
him as the element celtium, belonging to the family of rare earths, 
the existence of which had previously been suspected by Urbain. 
Quite apart from the difficulties which this result, if correct, might 
entail for atomic theories, it would, since the rare earths according 
to chemical view possess three valencies, imply a rise in positive 
valency of two units when passing from the element 72 to the 
next element 73, tantalum. This would mean an exception from 
the otherwise general rule, that the valency never increases by 
more than one unit when passing from one element to the next in 
the periodic table ] In the case of the incomplete seventh period 
the full drawn frame indicates the third stage in the development 
of the electronic group with 6-quanta orbits, which must begin in 
actinium. The dotted frame indicates the last stage but one in 



PHYSICAL AND CHEMICAL PROPERTIES OF THE ELEMENTS 115 

the development of the group with 5-quanta orbits, which hitherto 
has not been observed, but which ought to begin shortly after 
uranium, if it has not already begun in this element. 

With reference to the homology of the elements the exceptional 
position of the elements enclosed by frames in fig. 1 is further 
emphasized by taking care that, in spite of the large similarity 
many elements exhibit, no connecting lines are drawn between 
two elements which occupy different positions in the system with 
respect to framing. In fact, the large chemical similarity between, 
for instance, aluminium and scandium, both of which are trivalent 
and pronounced electropositive elements, is directly or indirectly 
emphasized in the current representations of the periodic table. 
While this procedure is justified by the analogous structure of the 
trivalent ions of these elements, our more detailed ideas of atomic 
structure suggest, however, marked differences in the physical 
properties of aluminium and scandium, originating in the essenti- 
ally different character of the way in which the last three electrons 
in the neutral atom are bound. This fact gives probably a direct 
explanation of the marked difference existing between the spectra 
of aluminium and scandium. Even if the spectrum of scandium is 
not yet sufficiently cleared up, this difference seems to be of a much 
more fundamental character than for instance the difference between 
the arc spectra of sodium and copper, which apart from the large 
difference in the absolute values of the spectral terms possess a 
completely analogous structure, as previously mentioned in this 
essay. On the whole we must expect that the spectra of elements 
in the later periods lying inside a frame will show new features 
compared with the spectra of the elements in the first three periods. 
This expectation seems supported by recent work on the spectrum 
of manganese by Catalan, which appeared just before the printing 
of this essay. 

Before I leave the interpretation of the chemical properties by 
means of this atomic model I should like to remind you once again 
of the fundamental principles which we have used. The whole 
theory has evolved from an investigation of the way in which 
electrons can be captured by an atom. The formation of an atom 
was held to consist in the successive binding of electrons, this 
binding resulting in radiation according to the quantum theory. 



116 THE STRUCTURE OF THE ATOM AND THE 

According to the fundamental postulates of the theory this binding 
takes place in stages by transitions between stationary states 
accompanied by emission of radiation. For the problem of the 
stability of the atom the essential problem is at what stage such a 
process comes to an end. As regards this point the postulates give 
no direct information, but here the correspondence principle is 
brought in. Even though it has been possible to penetrate con- 
siderably further at many points than the time has permitted me 
to indicate to you, still it has not yet been possible to follow in 
detail all stages in the formation of the atoms. We cannot say, for 
instance, that the above table of the atomic constitution of the 
inert gases may in every detail be considered as the unambiguous 
result of applying the correspondence principle. On the other hand 
it appears that our considerations already place the empirical data 
in a light which scarcely permits of an essentially different interpreta- 
tion of the properties of the elements based upon the postulates of 
the quantum theory. This applies not only to the series spectra 
and the close relationship of these to the chemical properties of the 
elements, but also to the X-ray spectra, the consideration of which 
leads us into an investigation of interatomic processes of an entirely 
different character. As we have already mentioned, it is necessary 
to assume that the emission of the latter spectra is connected with 
processes which may be described as a reorganization of the com- 
pletely formed atom after a disturbance produced in the interior 
of the atom by the action of external forces. 

IV. REORGANIZATION OF ATOMS AND X-RAY SPECTRA 

As in the case of the series spectra it has also been possible to repre- 
sent the frequency of each line in the X-ray spectrum of an element 
as the difference of two of a set of spectral terms. We shall there- 
fore assume that each X-ray line is due to a transition between 
two stationary states of the atom. The values of the atomic energy 
corresponding to these states are frequently referred to as the 
"energy levels" of the X-ray spectra. The great difference between 
the origin of the X-ray and the series spectra is clearly seen, how- 
ever, in the difference of the laws applying to the absorption of 
radiation in the X-ray and the optical regions of the spectra. The 
absorption by non-excited atoms in the latter case is connected 



PHYSICAL AND CHEMICAL PROPERTIES OF THE ELEMENTS 117 

with those lines in the series spectrum which correspond to com- 
binations of the various spectral terms with the largest of these 
terms. As has been shown, especially by the investigations of 
Wagner and de Broglie, the absorption in the X-ray region, on 
the other hand, is connected not with the X-ray lines but with 
certain spectral regions commencing at the so-called "absorption 
edges." The frequencies of these edges agree very closely with the 
spectral terms used to account for the X-ray lines. We shall now 
see how the conception of atomic structure developed in the pre- 
ceding pages offers a simple interpretation of these facts. Let us 
consider the following question : What changes in the state of the 
atom can be produced by the absorption of radiation, and which 
processes of emission can be initiated by such changes ? 

Absorption and emission of X-rays and correspondence 
principle. The possibility of producing a change at all in the 
motion of an electron in the interior of an atom by means of radia- 
tion must in the first place be regarded as intimately connected 
with the character of the interaction between the electrons within 
the separate groups. In contrast to the forms of motion where at 
every moment the position of the electrons exhibits polygonal or 
polyhedral symmetry, the conception of this interaction evolved from 
a consideration of the possible formation of atoms by successive 
binding of electrons has such a character that the harmonic com- 
ponents in the motion of an electron are in general represented in 
the resulting electric moment of the atom. As a result of this it 
will be possible to release a single electron from the interaction 
with the other electrons in the same group by a process which 
possesses the necessary analogy with an absorption process on 
the ordinary electrodynamic view claimed by the correspondence 
principle. The points of view on which we based the interpreta- 
tion of the development and completion of the groups during the 
formation of an atom imply, on the other hand, that just as no 
additional electron can be taken up into a previously completed 
group in the atom by a change involving emission of radiation, 
similarly it will not be possible for a new electron to be added 
to such a group, when the state of the atom is changed by 
absorption of radiation. This means that an electron which belongs 



118 THE STRUCTURE OF THE ATOM AND THE 

to one of the inner groups of the atom, as a consequence of an 
absorption process besides the case where it leaves the atom 
completely can only go over either to an incompleted group, or 
to an orbit where the electron during the greater part of its revolu- 
tion moves at a distance from the nucleus large compared to the 
distance of the other electrons. On account of the peculiar conditions 
of stability which control the occurrence of incomplete groups in 
the interior of the atom, the energy which is necessary to bring 
about a transition to such a group will in general differ very little 
from that required to remove the particular electron completely 
from the atom. We must therefore assume that the energy levels 
corresponding to the absorption edges indicate to a first approxi- 
mation the amount of work that is required to remove an electron 
in one of the inner groups completely from the atom. The 
correspondence principle also provides a basis for understanding 
the experimental evidence about the appearance of the emission 
lines of the X-ray spectra due to transitions between the stationary 
states corresponding to these energy levels. Thus the nature of the 
interaction between the electrons in the groups of the atom implies 
that each electron in the atom is so to say prepared, independently 
of the other electrons in the same group, to seize any opportunity 
which is offered to become more firmly bound by being taken up 
into a group of electrons with orbits corresponding to smaller values 
of the principal quantum number. It is evident, however, that on 
the basis of our views of atomic structure, such an opportunity is 
always at hand as soon as an electron has been removed from one 
of these groups. 

At the same time that our view of the atom leads to a natural 
conception of the phenomena of emission and absorption of X-rays, 
agreeing closely with that by which Kossel has attempted to give 
a formal explanation of the experimental observations, it alsosuggests 
a simple explanation of those quantitative relations holding for the 
frequencies of the lines which have been discovered by Moseley and 
Sommerfeld. These researches brought to light a remarkable and 
far-reaching similarity between the Rontgen spectrum of a given 
element and the spectrum which would be expected to appear upon 
the binding of a single electron by the nucleus. This similarity we 
immediately understand if we recall that in the normal state of the 



PHYSICAL AND CHEMICAL PROPERTIES OF THE ELEMENTS 119 

atom there are electrons moving in orbits which, with certain 
limitations, correspond to all stages of such a binding process and 
that, when an electron is removed from its original place in the 
atom, processes may be started within the atom which will corre- 
spond to all transitions between these stages permitted by the 
correspondence principle. This brings us at once out of those 
difficulties which apparently arise, when one attempts to account 
for the origin of the X-ray spectra by means of an atomic structure, 
suited to explain the periodic system. This difficulty has been felt 
to such an extent that it has led Sommerfeld for example in his 
recent work to assume that the configurations of the electrons in 
the various atoms of one and the same element may be different 
even under usual conditions. Since, in contrast to our ideas, he 
supposed all electrons in the principal groups of the atom to move 
in equivalent orbits, he is compelled to assume that these groups 
are different in the different atoms, corresponding to different 
possible types of orbital shapes. Such an assumption, however, seems 
inconsistent with an interpretation of the definite character of the 
physical and chemical properties of the elements, and stands in marked 
contradiction with the points of view about the stability of the atoms 
which form the basis of the view of atomic structure here proposed. 

X-ray spectra and atomic structure. In this connection it is 
of interest to emphasize that the group distribution of the electrons 
in the atom, on which we have based both the explanation of the 
periodic system and the classification of the lines in the X-ray 
spectra, shows itself in an entirely different manner in these two 
phenomena. While the characteristic change of the chemical 
properties with atomic number is due to the gradual development 
and completion of the groups of the loosest bound electrons, the 
characteristic absence of almost every trace of a periodic change in 
the X-ray spectra is due to two causes. Firstly the electronic 
configuration of the completed groups is repeated unchanged for 
increasing atomic number, and secondly the gradual way in which 
the incompleted groups are developed implies that a type of orbit, 
from the moment when it for the first time appears in the normal 
state of the neutral atom, always will occur in this state and will 
correspond to a steadily increasing firmness of binding. The develop- 



120 THE STRUCTURE OF THE ATOM AND THE 

raent of the groups in the atom with increasing atomic number, 
which governs the chemical properties of the elements shows itself 
in the X-ray spectra mainly in the appearance of new lines. Swinne 
has already referred to a connection of this kind between the periodic 
system and the X-ray spectra in connection with Kossel's theory. 
We can only expect a closer connection between the X-ray pheno- 
mena and the chemical properties of the elements, when the con- 
ditions on the surface of the atom are concerned. In agreement 
with what has been brought to light by investigations on absorp- 
tion of X-rays in elements of lower atomic number, such as have 
been performed in recent years in the physical laboratory at Lund, 
we understand immediately that the position and eventual struc- 
ture of the absorption edges will to a certain degree depend upon 
the physical and chemical conditions under which the element 
investigated exists, while such a dependence does not appear in 
the characteristic emission lines. 

If we attempt to obtain a more detailed explanation of the 
experimental observations, we meet the question of the influence 
of the presence of the other electrons in the atom upon the firmness 
of the binding of an electron in a given type of orbit. This influence 
will, as we at once see, be least for the inner parts of the atom, 
where for each electron the attraction of the nucleus is large in 
proportion to the repulsion of the other electrons. It should also 
be recalled, that while the relative influence of the presence of the 
other electrons upon the firmness of the binding will decrease with 
increasing charge of the nucleus, the effect of the variation in the 
mass of the electron with the velocity upon the firmness of the 
binding will increase strongly. This may be seen from Sommerfeld's 
formula (11). While we obtain a fairly good agreement for the 
levels corresponding to the removal of one of the innermost electrons 
in the atom by using the simple formula (11), it is, however, already 
necessary to take the influence of the other electrons into considera- 
tion in making an approximate calculation of the levels corresponding 
to a removal of an electron from one of the outer groups in the 
atom. Just this circumstance offers us, however, a possibility of 
obtaining information about the configurations of the electrons in 
the interior of the atoms from the X-ray spectra. Numerous 
investigations have been directed at this question both by 



PHYSICAL AND CHEMICAL PROPERTIES OF THE ELEMENTS 121 

Sommerfeld and his pupils and by Debye, Vegard and others. It 
may also be remarked that de Broglie and Dauvillier in a recent 
paper have thought it possible to find support in the experimental 
material for certain assumptions about the numbers of electrons in 
the groups of the atom to which Dauvillier had been led by con- 
siderations about the periodic system similar to those proposed by 
Langmuir and Ladenburg. In calculations made in connection with 
these investigations it is assumed that the electrons in the various 
groups move in separate concentric regions of the atom, so that 
the effect of the presence of the electrons in inner groups upon the 
motion of the electrons in outer groups as a first approximation 
may be expected to consist in a simple screening of the nucleus. 
On our view, however, the conditions are essentially different, since 
for the calculation of the firmness of the binding of the electrons 
it is necessary to take into consideration that the electrons in the 
more lightly bound groups in general during a certain fraction of 
their revolution will penetrate into the region of the orbits of 
electrons in the more firmly bound groups. On account of this 
fact, many examples of which we saw in the series spectra, we can 
not expect to give an account of the firmness of the binding of the 
separate electrons, simply by means of a "screening correction" 
consisting in the subtraction of a constant quantity from the value 
for N in such formulae as (5) and (11). Furthermore in the calcu- 
lation of the work corresponding to the energy levels we must take 
account not only of the interaction between the electrons in the 
normal state of the atom, but also of the changes in the configu- 
ration and interaction of the remaining electrons, which establish 
themselves automatically without emission of radiation during the 
removal of the electron. Even though such calculations have not 
yet been made very accurately, a preliminary investigation has 
already shown that it is possible approximately to account for the 
experimental results. 

Classification of X-ray spectra. Independently of a definite 
view of atomic structure it has been possible by means of a formal 
application of Kossel's and Sommerfeld's theories to disentangle 
the large amount of experimental material on X-ray spectra. This 
material is drawn mainly from the accurate measurements of 



122 THE STRUCTURE OF THE ATOM AND THE 

Siegbahn and his collaborators. From this disentanglement of the 
experimental observations, in which besides Sommerfeld and his 
students especially Smekal and Coster have taken part, we have 
obtained a nearly complete classification of the energy levels corre- 
sponding to the X-ray spectra. These levels are formally referred 
to types of orbits characterized by two quantum numbers n and k, 
and certain definite rules for the possibilities of combination 
between the various levels have also been found. In this way a 
number of results of great interest for the further elucidation of 
the origin of the X-ray spectra have been attained. First it has 
not only been possible to find levels, which within certain limits 
correspond to all possible pairs of numbers for n and k, but it has 
been found that in general to each such pair more than one level 
must be assigned. This result, which at first may appear very 
surprising, upon further consideration can be given a simple 
interpretation. We must remember that the levels depend not 
only upon the constitution of the atom in the normal state, but 
also upon the configurations which appear after the removal 
of one of the inner electrons and which in contrast to the normal 
state do not possess a uniquely completed character. If we thus 
consider a process in which one of the electrons in a group 
(subgroup) is removed we must be prepared to find that after the 
process the orbits of the remaining electrons in this group may be 
orientated in more than one way in relation to one another, and 
still fulfil the conditions required of the stationary states by the 
quantum theory. Such a view of the "complexity" of the levels, as 
further consideration shows, just accounts for the manner in which 
the energy difference of the two levels varies with the atomic 
number. Without attempting to develop a more detailed picture 
of atomic structure, Smekal has already discussed the possibility 
of accounting for the multiplicity of levels. Besides referring to 
the possibility that the separate electrons in the principal groups 
do not move in equivalent orbits, Smekal suggests the introduction 
of three quantum numbers for the description of the various groups, 
but does not further indicate to what extent these quantum 
numbers shall be regarded as characterizing a complexity in the 
structure of the groups in the normal state itself or on the 
contrary characterizing the incompleted groups which appear 
when an electron is removed. 



PHYSICAL AND CHEMICAL PROPERTIES OF THE ELEMENTS 123 

It will be seen that the complexity of the X-ray levels exhibits a 
close analogy with the explanation of the complexity of the terms 
of the series spectra. There exists, however, this difference between 
the complex structure of the X-ray spectra and the complex 
structure of the lines in the series spectra, that in the X-ray 
spectra there occur not only combinations between spectral terms, 
for which k varies by unity, but also between terms corre- 
sponding to the same value of k. This may be assumed to be 
due to the fact, that in the X-ray spectra in contrast to the series 

Niton 



p 


















































6 


0, 


\ 



























I 


























h 


























& 






















0- A 


N 


2/ 






















6 




/, 






















a A. 


























f, 


M 


1 






















(, 


" 


* 






















a f. 




ia 
























L 


n 






















A 


K 


f*t 
























& 



Fig. 5. 

spectra we have to do with transitions between stationary states 
where, both before and after the transition, the electron concerned 
takes part in an intimate interaction with other electrons in orbits 
with the same principal quantum number. Even though this 
interaction may be assumed to be of such a nature that the 
harmonic components which would appear in the motion of an 
electron in the absence of the others will in general also appear 
in the resulting moment of the atom, we must expect that the 
interaction between the electrons will give rise to the appearance 
in this moment of new types of harmonic components. 



124 



THE STRUCTURE OF THE ATOM AND THE 



It may be of interest to insert here a few words about a new 
paper of Coster which appeared after this address was given, 
and in which he has succeeded in obtaining an extended and 
detailed connection between the X-ray spectra and the ideas 
of atomic structure given in this essay. The classification men- 
tioned above was based on measurements of the spectra of the 
heaviest elements, and the results in their complete form, which 
were principally due to independent work of Coster and Wentzel, 
may be represented by the diagram in fig. 5, which refers to 
elements in the neighbourhood of niton. The vertical arrows 



Xenon 



Krypton 



o- 
























(/, 












& 












% 












. t, 


1 










Va 

L 












'-'-*<," 


4s. 

h 




1 






13, 












b 


T 
?, 






T~ 


















o 






,1 




3 












ESi 




-V 


I 


. f 




, (0 












a I 


2*. 






!l 
ll 




f I 












a. L\ 


9 r 










K f, 












t> X 


ft. 











Fig. 6. 

represent the observed lines arising from combinations between 
the different energy levels which are represented by horizontal lines. 
In each group the levels are arranged in the same succession as 
their energy values, but their distances do not give a quantitative 
picture of the actual energy-differences, since this would require a 
much larger figure. The numbers n k attached to the different levels 
indicate the type of the corresponding orbit. The letters a and 6 
refer to the rules of combination which I mentioned. According 
to these rules the possibility of combination is limited (1) by the 
exclusion of combinations, for which k changes by more than one 
unit, (2) by the condition that only combinations between an a- 
and a 6-level can take place. The latter rule was given in this 



PHYSICAL AND CHEMICAL PROPERTIES OF THE ELEMENTS 125 

form by Coster; Wentzel formulated it in a somewhat different 
way by the formal introduction of a third quantum number. In 
his new paper Coster has established a similar classification for the 
lighter elements. For the elements in the neighbourhood of xenon 
and krypton he has obtained results illustrated by the diagrams 
given in fig. 6. Just as in fig. 5 the levels correspond exactly to 
those types of orbits which, as seen from the table on page 113, 
according to the theory will be present in the atoms of these ele- 
ments. In xenon several of the levels present in niton have dis- 
appeared, and in krypton still more levels have fallen away. Coster 
has also investigated in which elements these particular levels 
appear for the last time, when passing from higher to lower atomic 
number. His results concerning this point confirm in detail the 
predictions of the theory. Further he proves that the change in 
the firmness of binding of the electrons in the outer groups in 
the elements of the family of the rare earths shows a dependence 
on the atomic number which strongly supports the assumption that 
in these elements a completion of an inner group of 4-quanta 
orbits takes place. For details the reader is referred to Coster's 
paper in the Philosophical Magazine. Another important con- 
tribution to our systematic knowledge of the X-ray spectra is 
contained in a recent paper by Wentzel. He shows that various 
lines, which find no place in the classification hitherto considered, 
can be ascribed in a natural manner to processes of reorganization, 
initiated by the removal of more than one electron from the 
atom ; these lines are therefore in a certain sense analogous to 
the enhanced lines in the optical spectra. 

CONCLUSION 

Before bringing this address to a close I wish once more to 
emphasize the complete analogy in the application of the 
quantum theory to the stability of the atom, used in explaining 
two so different phenomena as the periodic system and X-ray 
spectra. This point is of the greatest importance in judging the 
reality of the theory, since the justification for employing con- 
siderations, relating to the formation of atoms by successive capture 
of electrons, as a guiding principle for the investigation of atomic 



126 THE STRUCTURE OF THE ATOM 

structure might appear doubtful if such considerations could not 
be brought into natural agreement with views on the reorgani- 
zation of the atom after a disturbance in the normal electronic 
arrangement. Even though a certain inner consistency in this 
view of atomic structure will be recognized, it is, however, hardly 
necessary for me to emphasize the incomplete character of the 
theory, not only as regards the elaboration of details, but also so 
far as the foundation of the general points of view is concerned. 
There seems, however, to be no other way of advance in atomic 
problems than that which hitherto has been followed, namely to let 
the work in these two directions go hand in hand. 



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