XIII. Report to Third Solvay Council

XIII. REPORT TO THIRD SOLVAY COUNCIL (April 1921*)

*

[See Introduction, sect. 17.1

1. THE THIRD SOLVAY COUNCIL: ATOMS A N D ELECTRONS (1921)

INSTITUT INTERNATIONAL DE PHYSIQUE SOLVAY

ATOMES ET ELECTRONS RAPPORTS ET DISCUSSIONS DU

CONSEIL D E PHYSIQUE TENU A BRUXELLES D U ley A U 6 AVRIL 1921 S O U 9 LES AUSPICES

D E L'INSTITUT 1NTERNATIONAL DE PHYSIQUE SOLVAY Publids par la Commission administrative de 1 Institut e t MM. l e s Secretaires du Conseil

PARIS GAUTHIER-VILLARS ET Cia, gD1TEURS LIBRAIRES DU BUREAU DES LONGITUDES, DE L'BCOLE POLYTECHNIQUE

55, Quai des Grands-Augustins, 55

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1923

TROISIEME CONSELL DE PHYSIQUE Le troisihme des Conseils de Physique, prevus par l’article 10 des Statuts de 1’Institut international de Physique fond6 par Ernest Solvay, le ~ e mai r 1912, s’est r h n i sur l’invitation de la Commission administrative ( l ) ,agissant de concert avec le ComitB scientifique international (”. I1 a tenu ses seances & Bruxelles, r 6 avril 1921. dans les locaux de I’Institut, du ~ e au Les participants h ce Conseil 6taient : Le President : le professeur H.-A. Lorentz, de Haarlem; Les Membres : les professeurs C.-G. Barkla, d’fidimbourg; W.-L. Bragg, de Manchester; M. et L. Brillouin, de Paris; M. de Broglie, de Paris; Mme Curie, de Paris; P. Ehrenfest, de Leyde; W.-J. de Haas, de Delft; H. Kamerlingh Onnes, de Leyde; M. Knudsen, de Copenhague; P. Langevin, de Paris; J. Larmor, de Cambridge; R.-A. Millikan, de Chicago; J. Perrin, de Paris; 0.-W. Richardson, de Londres; E. Rutherford, de Cambridge; M. Siegbahn, de Lund; Edm. van Aubel, de Gand; P. Weiss, de Strasbourg, et P. Zeeman, d’bmsterdam.

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(I) La Commission administrative se compose de : M. E. TASSEL,professeur honoraire B l’Universit8 libre de Bruxelles; M. P. HEGER,professeur honoraire A 1’Universitt libre de Bruxelles ; M. Ch. DE KEYSER,professeur ordinaire B 1’Universitb libre de Bruxelles. (9) Le Comitb scientifique international se compose de : M. H.-A. LORENTZ, professeur A 1’Universitb de Leyde, prbsident; Mme P. CURIE,professeur A la Facult6 des Sciences de Paris; M. W.-H. BRAGG,professeur A 1’Universitb de Londres; M. M. BRILLOUIN, professeur au College de France, B Paris; M. H. KAMERLINGH ONNES,professeur A l’Universit8 de Leyde; M. M. KNUDSEN, professeur & l’ficole Polytechnique de Copenhague; M. P. LANGEVIN, professeur au College de France, A Paris; M. E. RUTHERFORD, professeur B 1’Universitb de Cambridge; M. E. van AUBEL,professeur B l’Universit6 de Gand.

VI

TROISIEME COSSEIL DE PHYSIQUE.

Les SecrBtaires Btaient le professeur J.-E. Verschaffelt, de Haarlem, ancien membre de la Commission administrative de 1’Institut; le Dr M. de Broglie, de Paris; le professeur W.-L. Bragg, de Manchester, e t le Dr L. Brillouin, de Paris, membres du Conseil. Les collaborateurs de M. Ernest Solvay, qui assistaient au Conseil, Btaient le Dr Ed. Herzen et l’ingenieur Edm. Warnant, de Bruxelles. Le professeur A.-A. Michelson, de Chicago, de passage en Europe, assista aux reunions h titre d’invit6. Le professeur W.-H. Bragg, de Londres, membre du Comitb scientifique de l’Institut, ainsi que les professeurs N. Bohr, de Coperthague, A. Einstein, de Berlin, e t J.-H. Jeans, de Dorking, invites h participer comme membres aux reunions du Conseil, avaient B t B empeches de se rendre h Bruxelles. La mort d’A. Righi avait Bgalement l a k e une place vacante parmi les membres du Conseil. Le compte rendu ci-aprhs des Rapports present& aux reunions du Conseil et des discussions auxquelles ces Rapports ont donne lieu a 6 t h redig6 par les soins des Secretaires; 1’Bdition en a 6 t h confibe, comme precedemment, B la Librairie scientifique GauthierVillars e t Cie.

TABLE DES MATIkRES. . . . . . . . . . . . . . . . . . . . . . . . .. . .. . . . . . ... ... .... , .. , ... , . . . ... . .. . . ..... H.-A. LORENTZ. - Notes sur la thborie des electrons.. . . . . . . . . . . . . . .

Troisihme Conseil de Physique.. A la mbmoire d’Ernest Solvay.,

Pages.

v

VII

i

Discussion ................................................... 20 E. RUTHERFORD. - La structure de l’atome.. . . . . . . . . . . . . . . . . . . . . 36 66 Discussion .................................................. M, D E BROGLIE.- La relation hv = E dans lee phhomknes photoblectriques; production de la lumihre dans le choc des atomes par les 80 electrons e t production des rayons de Rontgen.. . 101 Discussion R.-A. MILLIKAN. - Sur l’absorption du rayonnement par quanta dans . . 120 les metaux ................................................ H. KAMERLINGH ONNES.- Le paramagnetisme aux basses temperatures consider6 a u point de vue de la constitution des aimants &mentaires e t de l’action que ceux-ci subissent de la part de leurs porteurs. 131 Discussion .................................................. I 58 I j8 P. WEISS.- Les actions mutuelles des moli!cules aimantkes. . . . . . . H. KAMERLINGH ONNES.- Les supraconducteurs e t le modkle de l’atome Rutherford-Bohr ...................................... Discussion .................................................. L. BRILLOUIN. - Sur la conductibiliti! des mbtaux.. . . . . . . . . . . W.-H. BRAGG. - L’intensith de la reflexion des rayons X par le diamant. W.-J. D E HRAS.- Le moment de la quantitk de mouvement dans un corps aimantb ................................................ 206 Discussion.. , . . . , . . . . . . . . . . . . . . . . . , . . . . . . , . , , , . , . . . . . . . 2 I 6 Note ajoutbe aprks l’exposi! et la discussion du Rapport.. . . . . . . . . . 222 N. BOHR.- L’application de la thkorie des quanta aux problkmes atomiques ........................................................ 228 P. EHRENFEST. - Le principe de correspondance.. . . . . . . . . . . . . . . . . . . 248 Discussion .................................................. 255 R.-A. MILLIKAN.- La disposition et le mouvement des blectrons dans les atomes., . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

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I X S T I T U T SOLVAY.

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2.

ON THE APPLICATION OF THE QUANTUM THEORY TO ATOMIC PROBLEMS* BY N . BOHR

REPORT TO THE THIRD SOLVAY CONGRESS APRIL 1921

N. BOHR: On the application of the quantum theory to atomic problems. Introductory remarks. According to the program for the Solvay conference the present report was originally conceived as consisting of two parts, of which the first should contain a survey of the general principles underlying the application of the quantum theory to atomic problems, while the second part should contain the special applications of these principles to the problem of the arrangement and the motion of the electrons in the atoms of the elements. Due to illness, however, which prohibited the author’s presence at the conference, only the present paper, which constitutes the former of the planned parts, was completed and presented at the meeting through the kind intervention of Mr. Ehrenfest. A brief indication of the contents of the planned latter part will be found at the conclusion of this paper.

5 1. THE GENERAL FEATURES OF THE QUANTUM THEORY The problem of the constitution of atoms to be dealt with in this report has in recent years acquired an aspect very different from that, in which this problem was considered a few years ago. In fact, as we have heard from the report of Sir Ernest Rutherford, we possess at present a detailed knowledge of the particles which form the constituents of the atom. Thus we may assume that an atom consists of a central positive nucleus, which possesses almost the whole mass of the atom and which is surrounded by a cluster of electrons, the number of which in the neutral atom is equal to the so-called atomic number, i.e., the number which indicates

*

[This is the manuscript on which the published French translation was based, except for the end, which was re-written at some later stage; the final form is given here as a re-translation from the French. As to the report which was handed to the participants at the meeting, and which we shall denote as the “Solvay report”, it does not differ essentially from the manuscript given here; it had another title: On theproblem of the arrangement and the motion ofthe electrons in the atom and lacked, of course, the Introductory remarks. Besides, it shows a number of minor deviations from the text reproduced here, which seems to be an improved version of it. At two places, which will be pointed out, Bohr has in fact re-introduced the text of the “Solvay report” as a basis for the French translation.]

THE CORRESPONDENCE PRINCIPLE

the position of the element within the periodic table. This picture of the atom exhibits a marked simplicity, and it might at first sight be expected that the problem with which we are presented in a closer investigation of the interatomic motions would be analogous to the problems with which we meet in celestial mechanics. In fact, since we are led to assume that the dimensions of the particles of the atom are very small compared with the dimensions of the whole system, the problem with which we meet is at first sight that of a system of mass points moving under the influence of mutual forces which vary as the inverse square of the distance just as the gravitational forces between celestial bodies. A closer consideration, however, of the physical and chemical evidence, teaches us soon, that such a comparison halts on essential points. In fact, if we consider for instance the orbits of the planets in the solar system, we may certainly assume that the motion of these bodies to a high degree of approximation may be accounted for by the general laws of gravitation, but the orbits of the various planets are not determined entirely by the masses of the sun and of the planets, but must be assumed also to depend essentially on the conditions which were prevalent during the formation of the solar system, or in other words on the previous history of this system. On the other hand, in order to account for the definite physical and chemical properties of the elements, we are forced to assume that the motion of the particles of the atom, at any rate in its normal state, will be completely determined by the values of the charges and masses of the constituting particles. This inherent stability of the atoms may perhaps be most clearly perceived if we consider the processes leading to the emission of the characteristic spectra of the elements. In these processes we may assume to witness an alteration in the state of the atom, produced by some external agencies by which energy is conveyed to the atom, and which is followed by a process of reorganisation by which the normal state is re-established and the energy re-emitted as electromagnetic radiation. When analysed, this radiation is found to consist of a number of harmonic trains of waves, each corresponding to a line in the spectrum. While, on the basis of the classical theory of electromagnetism, no possibility is offered to account either for the stability of the normal state of the atom or for the constitution of the radiation thus produced, a basis for a rational interpretation of the atomic properties under consideration is found in the so-called quantum theory, the germ of which was laid - as well known - in the theory of temperature radiation proposed by Planck about twenty years ago. In the form in which the quantum theory will be used in the following, the considerations will be based on the following fundamental postulate : An atomic system which emits a spectrum consisting of sharp lines possesses a number of separate distinguished states, the so-called stationary states, in which the system may exist at any rate for a time without emission of radiation, such an emission

THE CORRESPONDENCE PRINCIPLE

taking place only by a process of complete transition between two stationary states, whereby the emitted radiation always consists of a simple train of harmonic waves. In the theory, the frequency of the radiation emitted during a process of this kind is not directly determined by the motion of the particles within the atom in a way corresponding to the ideas of the classical theory of electromagnetism, but is simply related to the total amount of energy, emitted during the transition, the frequency v multiplied by Planck’s constant h being equal to the difference of the values E , and E2 of the energy of the atom in the two states involved in the process, so that we have hv

=

El-Ez.

(1)

This law, which in the following will be referred to as the general frequency relation, constitutes the formal basis of the quantum theory together with a set of relations, the state conditions which equally involve Planck’s constant, and which characterise certain properties of the motions in the stationary states of an atomic system by which they are distinguished among the continuous multitude of motions which might be possible according to the ordinary ideas of mechanics. Just as the frequency relation, these latter relations which will be referred to as state relations, and which will be discussed in detail in the following, may be considered as a natural generalisation of the assumptions regarding the interaction of a simple harmonic oscillator and an electromagnetic field of radiation, originally introduced in Planck’s theory of temperature radiation and applied with so great :success,especially by Einstein, in the theory of specific heat and the photoelectric effect. Before we enter into a detailed specification of the state relations and the application of the quantum theory to special atomic problems, it may be useful first to examine the general features of the theory more closely and especially to elucidate, on the one hand, the radical departure of the quantum theory from our ordinary ideas of mechanics and electrodynamics as well as, on the other hand, the formal analogy with these ideas which is nevertheless exhibited by the picture of atomic processes which may be developed on the basis of the formal postulates of the theory. 4 s will be seen, the analogy is of such a type that in a certain respect, we are entitled in the quantum theory to see an attempt of a natural generalisation of the classical theory of electromagnetism.

8 2.

THE PROPERTIES OF THE STATIONARY STATES

As regards the description of the motion in the stationary states, the first question which arises is to what extent the fundamental postulate of the quantum theory allows to base this description on the notions of the classical ideas of the

THE C O R R E S P O N D E N C E P R I N C I P L E

behaviour of systems of electrified particles*. In the first place, it is clear that these ideas cannot be applied without modifications, since such an application as mentioned would involve a continuous emission of electromagnetic radiation from the atom, which is at variance with the very existence of stationary states. If, however, we consider the effect on the motion of an atomic system, which in the electronic theory is directly connected with that part of the electromagnetic field which constitutes the radiation under emission, we find that in general these effects are small compared with the effect on the motion of the conservative part of the field, giving rise to forces analogous with those we meet in the simple theory of mechanics of mass points. Even if we are forced to assume modifications in the classical theory of electrodynamics involving radical departures as regards the mechanism of radiation, we are therefore by no means compelled to assume that the motion at any moment differs essentially from that which would follow from the classical electronic theory. On the contrary, we are naturally led to inquire whether it would not be possible to a close approximation to describe the motion of the particles in the stationary states of an atomic system as that of mass points moving under influence of their mutual repulsion and attraction due to their electric charges. If next we consider the problem of the “stability” of the stationary states, it is immediately seen that it is a necessary condition for this stability that, in general, the effect of the external angencies on the motion of the particles cannot be described by means of the ordinary laws of mechanics. In fact, as we shall see more closely, the properties expressed by the state relations which distinguish the stationary states among the possible mechanical motions of an atomic system, according to their nature, are not simply characterised by the velocities and configurations of the particles at a given moment but depend essentially on the periodicity properties of the orbits to which the momentary velocities and configurations belong. If, therefore, we consider an atom subject to varying external conditions, it is not enough-in order to find the alterations in the motion due to the variation in these conditions-simply to consider, as in ordinary mechanics, the effect of the forces acting on the particles at any given moment, but the ensuing motion must essentially depend on the variation of the character of the possible orbits corresponding to the given variation in the external conditions. As an example of this failure of mechanics in describing an atomic system under the effect of external agencies we may consider the remarkable effects of collisions between atoms and free electrons which we meet in such experiments

* [Here is a slight deviation between the manuscript and the published French version. The latter gives the text of the “Solvay report” : “the notions of the classical theory of electrons which are discussed in the report by Mr. H. A. Lorentz”.]

THE CORRESPONDENCE PRINCIPLE

as have first been carried out by Franck and Hertz, and in which we may see a source of great importance for the investigation of interatomic actions. Even if it is hardly necessary for the interpretation of these experiments, which have been mentioned in detail in the report of Mr. de Broglie, to assume that the velocities and positions of the particles undergo discontinuous alterations during the collision, the result of the experiments in question offers undoubtedly a most striking evidence of the essential non-mechanical laws which govern the motion of the particles in the atom. Thus we see that, under the conditions of these experiments, it is not possible to effect any alteration in the type of motion of the atom, unless this alteration consists in a transference of the atom from the original state to some other stationary state. Another illustrative example of the behaviour of atoms under varying external conditions we possess in the phenomena of absorption and dispersion of electromagnetic radiation by elements in the gaseous state. Thus, as regards the first phenomenon, we are, in order to account for the experimental facts on the basis of our fundamental postulates, forced to assume that true absorption can take place by a process which consists in a complete transition of the atom from the original state to some other stationary state of larger energy content. Further, the fact that the dispersion observed is in its character essentially connected with the lines in the absorption spectrum of the substance, -the frequencies of which, according to the interpretation of absorption just given, cannot be calculated from the motion of the particles in the atom, but are connected with the differences in the energy in the stationary states through the general frequency-relationproves that the interaction of the atom with the incident electromagnetic waves can by no means be described on the basis of the classical electronic theory, but must be intimately related to the unknown mechanism which underlies the various formal relations of the quantum theory. In the examples just mentioned, we have to do with effects on atoms of agencies which involve essential alterations in the external forces acting on the particles within short time intervals comparable with the periods characterising the mechanical motion of the particles in the atom. If, on the other hand, we consider a variation in the external conditions which takes place in a uniform way and so slowly that the comparative variations of the external forces within a time interval of the same order as the periods mentioned are very small compared with the total force to which the particles in the atom are subject, the problem stands differently. In this case we cannot beforehand, from a consideration of the fundamental postulate of the quantum theory, exclude the possibility that the alteration in the motion of the system due to such a slow transformation of the external conditions may be deduced by means of the laws of ordinary mechanics. The statement, that in most cases the use of mechanics is actually legitimate

THE CORRESPONDENCE PRINCIPLE

-or, more strictly expressed, that the deviations introduced from the actual stationary states which appear when the effect of a slow transformation is calculated by ordinary mechanics are the smaller the slower the transformation is performed-forms the content of the well-known principle, introduced in the quantum theory by Ehrenfest. This principle is often called the “adiabatic hypothesis’’, with reference to the fact that slow continuous alterations of external conditions, with regard to the part they play in the theory of thermodynamics, are sometimes termed “adiabatic”. In order sharply to distinguish between the important thermodynamic applications of the principle in question, on which we shall not enter here, and its direct content which refers to the limits of the applicability of mechanics in the quantum theory, we shall in the following call it the principle of “mechanical transformability” of stationary states. For the adequate representation of the quantum theory, the principle in question is of great importance, as it offers a rational means of obtaining information about the stationary states corresponding to general classes of mechanical motions from the knowledge of these states in case of typical examples of similar motions. The simplest possible illustration is obtained if we consider an atomic system of one degree of freedom consisting of an electrified particle performing oscillatory motions. Let the position of the particle be described by a generalised coordinate q and the momentary motion by means of the conjugated generalised momentum p, and let us form the integral

s

(2)

6E = o ~ J ,

(3)

J = pdq

taken over a complete oscillation of the system. Then it may be proved, that the quantity J thus defined possesses several important properties. In the first place, this quantity is found in a simple way to be connected with the total energy E of the system; thus, for two neighbouring motions which differ only very little from each other we have where tiE and 6 J represent the differences in E and J for the two motions respectively, while w is the frequency of the oscillation. Next, it is found that under a slow continuous transformation of the system-consisting in a gradual change of the forces acting on the particles during which the comparative alteration of these forces during the time of one period is very small, -the quantity J remains invariant if the effect of the transformation is calculated by means of ordinary mechanics. With reference to the principle of mechanical transformability of stationary states, we may therefore infer that the quantity J is suitable for a formulation of the laws which hold for the motion in the stationary states. Now, in case of a purely harmonic oscillator with constant frequency coo, we have ac-

THE CORRESPONDENCE PRINCIPLE

cording to Planck’s theory, that the stationary states are distinguished by the well-known relation E

=

(4)

nhw,

where n is a positive integer and h Planck’s constant. Next, it is seen from ( 3 ) that, in the case in question this equation is equivalent to the relation J

=

nh

(5)

and according to Ehrenfest’s principle we shall therefore assume that this relation is valid for the stationary states of all systems the motion of which is of the type under consideration. If we consider a system consisting of an electrified particle moving in space, the motion will in general be of a very intricate character, but, for important classes of systems, it is possible to obtain an immediate generalisation of the above result. Consider first the simple case of an atomic system consisting of a single particle which, in mutually perpendicular directions, can perform independent oscillatory motions, each of the same type as for a system of one degree of freedom. In this case, we must naturally expect that the stationary states are characterised by a set of conditions of the same type as ( 5 ) J , = n,h, J2 = n 2 h , J 3

=

n3h,

(6)

each corresponding to one of the possible independent motions. These relations are of course invariant for such transformations by which the dynamical independency of‘ the components of the motion in the three directions is preserved. They acquire, however, a greater interest from the circumstance that it can be shown that by certain general types of transformations, by which this independency is not preserved, the relations (6) may still be said to be invariant. In fact, it is possible at any stage of this transformation to describe the possible motions of the system by means of three quantities J , , J 2 , J 3 , which in the special case of independent motions coincide with integrals of the type (2) and the values of which remain invariant during the transformation if this is performed in an adiabatic way, so that in the stationary states they are always given by the formula (6). In general, these quantities may not be directly expressed by means of a set of generalised coordinates and momenta describing the motion in a similar way as in the simple case mentioned above, but there exists an important class of systems of this type, for which the expressions given for the Ss may be applied unaltered. This class of mechanical systems which are said to allow of a “separation of variables” is characterised thereby, that it is possible for a suitable choice of positional coordinates q l , q 2 ,q3 to obtain, that the value of each of the canonically conjugated momenta p , ,p 2 , p 3 during the motion depends on the

T H E CORRESPONDENCE PRINCIPLE

value of the conjugated coordinate only, in a way completely analogous to the case where the particle may be said to perform three motions independently of each other. In all cases where it has been possible to fix the stationary states by relations of the form (6), the motion is found to be “multiply periodic”, in such a way that the displacement of the particles in space may be resolved in a number of discrete elliptical harmonica1 vibrations, the frequencies of which are of the type z1 o1+ z 2 w 2+ z 3 w 3 , where zl, z2, z3 are positive or negative integers and the quantities ol, 0 2 0, , represent what might be called fundamental frequencies of the motion. In complete analogy to what holds for the case where the motion may be resolved in three independent motions, it is found that the difference in the energy 6E of two neighbouring motions of the system, for which the values of J 1 ,J 2 , J, , differ by 6Jl , 6 J 2 , 6 J 3 respectively, is given by

(7)

6E = 0 1 6 J 1 + 0 2 6 J z + W 3 6 J 3 .

While, in general, the energy will depend on the values of all three quantities J 1 ,J2,J , , and therefore all these quantities necessarily must be fixed in the stationary states, we meet in certain cases with systems, for which the energy depends on two, or one, linear combinations of the Ss with entire coefficients, due to the circumstance that, for all possible motions, there exist between the 0’s one or two linear relations

(8)

m l w 1 + m 2 w 2 + m 3 0 3= 0,

where the coefficients m are positive or negative integers. In these cases, where the system is said to be “degenerate”, the motion may be resolved into harmonic vibrations, the frequencies of which are built up by only two, or one, fundamental frequencies, and the stationary states will be completely determined by putting the mentioned two, or one, linear combinations of the Ss equal to entire multipla of Planck’s constant. Thus for a system consisting of a particle which, independently of the initial conditions, always will perform a purely periodic motion, the stationary states are, in complete analogy to a system of one degree of freedom, determined by a single condition which may be expressed by a relation of the form ( 5 ) : = [@l

6ql + p 2 6 q 2

fp3 6p3) =

nh,

(9)

where n is an integer, and where the integral has to be extended over one complete period of the motion, q l , q 2 , q3 representing some arbitrarily chosen set of generalised coordinates and p1,p 2 ,p 3 their corresponding conjugated momenta. The circumstance that the stationary states of a degenerate system are characterised by a smaller number of conditions is immediately connected with the

THE CORRESPONDENCE PRINCIPLE

circumstance that these systems possess a smaller degree of mechanical stability than a non-degenerate multiply-periodic system. In fact, in the latter case, the type of the motion will in the presence of a small constant external field of force only undergo small variations proportional to the intensity of the external forces, while, for a degenerate system, the presence of such forces may in the course of time, although the ecergy can of course only be affected by small quantities of this order, produce-so to say by accumulation4onsiderable alterations in the shape and position of the orbit of the particle. As regards the historical development of the theory of the stationary states of a multiply-periodic system, briefly exposed above, it may be remarked that the formulation of the state-relations has not been a direct result of the application of Ehrenfest’s principle, but is the result of a development in which a great number of physicists have taken part, also Planck himself. The essential progress as regards the applications of the theory to atomic problems, however, was due to Sommerfeld, while the consistency with the principle of mechanical transformability of the theory in its final form as presented above, which is mainly due to Epstein and Schwarzschild, has subsequently been proved by Ehrenfest and Burgers. Quite apart from the direct use of this principle, considered as a means of determining the stationary states of a given system, the existence of the transformability under consideration is of great interest in the quantum theory as it offers a means of connecting any two stationary states of a multiply-periodic atomic system by an imagined continuous transformation process, during which we do not for a moment leave the region of stationary states and thereby the legitimate field of the application of the ordinary laws of mechanics. In fact, in case of a multiply-periodic system, such a connection may be obtained by a suitably constructed “cyclic” transformation process, for the possibility of which the existence of the “degenerate” system plays an essential part. By this circumstance we obtain a means of an unambiguous definition of the difference between the energy of two stationary states of an atomic system which enters in the general frequency-relation, a determination which otherwise might give rise to a fundamental difficulty, as at the present state of the theory we do not possess any means of describing in detail the process of direct transition between two stationary states accompanied by an emission or absorption of radiation and cannot be sure beforehand that such a description will be possible at all by means of laws consistent with the application of the principle of conservation of energy.

5 3.

THE RADIATION PROBLEM IN THE QUANTUM THEORY

When now from the problem of the principles underlying the description and fixation of the stationary states we turn again our attention to the radiation

THE CORRESPONDENCE P R I N C I P L E

process, we see that the fundamental postulates of the quantum theory not only, as mentioned, involve that the process of emission or absorption of radiation cannot be described in detail on the basis of the classical theory of electrodynamics, but that the claim as to the constitution of the radiation accompanying the process even excludes any immediate connection between the motion of the particles in the atom and the radiation, analogous to that exhibited by the emission of periodical disturbances in an elastic medium from a vibrating mechanical system. In consequence of the postulates in question, we shall on the contrary assume that, instead of a direct connection between the motion in the atom and the emitted radiation, the emission of the various separate harmonic constituents of the spectrum must be ascribed to the occurrence of a number of independent interatomic processes, each of which gives rise to an emission of light corresponding to one of the spectral lines. Just this picture, however, presents us with an interpretation of a fundamental feature exhibited by the empirical laws governing the spectra of the elements and which has got its expression in the so-called “principle of combination” of spectral lines, which was originally established by Ritz on the basis of an examination of the laws holding for the frequencies of the series spectra, discovered by Balmer, Rydberg, and himself. According to this principle, which in the last years has found a wide field of application to various types of spectra, the frequency of the lines of a spectrum emitted by an atomic system may always be represented as the difference between the values of two among a set of numerical terms characteristic of the system. From the form of the general frequency relation, it will be directly seen that this circumstance is just what should be expected if each of the lines in the spectrum is emitted during the transition between two among a number of stationary states of the atomic system responsible for the spectrum, the energies in these, omitting an arbitrary constant, being numerically equal to the product of the mentioned spectral terms multiplied by Planck’s constant. For the simplest spectra, it has been found that the spectral terms may be arranged in a sequence, in such a way that their values may be represented by a simple function of a variable, each term corresponding to an integer value of the argument, just as it should be expected if the stationary states were determined in a way corresponding to that holding for a purely periodic system. In more complex cases, the spectral terms form a multitude which can be represented by the values of some simple functions of several variables for integer values of the arguments, as it should be expected if the multitude of stationary states were of the same type as that corresponding to a multiply-periodic system of several degrees of freedom. At this place, it might be of interest briefly to refer to a certain interesting conception of the frequency relation which has been advocated by several authors

THE CORRESPONDENCE PRINCIPLE

and the object of which has been to present this relation in a light analogous to that of the state relations. These considerations are based on the fact that under suitable conditions, standing electromagnetic waves are possible in the classical theory of electromagnetism, and that therefore in such cases the ether may be considered as analogous to a mechanical multiply-periodic system capable of stationary states. According to this conception, by which the 'formal basis of the quantum theory obtains a more unitary aspect, the process of emission or absorption of radiation by an atom becomes in a certain sense analogous with a process of interaction between two atomic systems, which before as well as after the process find themselves in stationary states'. This view, however, does not seem to offer a further guidance in the examination of the characteristic features of the radiation process as long as we do not possess a detailed picture of production and propagation of radiation. As regards the latter phenomena, Einstein has, as well known, several years ago, in connection with his considerations of the photoelectric effect, proposed the view that-quite apart from the problem of the mechanism of the emission and absorption of electromagnetic radiation from atomic systems-already the propagation through space of this radiation should take place in a way widely different from that corresponding to the classical electromagnetic theory. Thus, according to his theory of light-quanta, electromagnetic radiation from an atom should not spread as a system of spherical waves, but should be propagated in a definite direction as a concentrated entity, containing within a very small volume the energy hv. On one hand, such a conception seems to offer the only simple possibility of accounting for the phenomena of photo-electric action, if we adhere to an unrestricted application of the notions of conservation of energy and momentum; on the other hand, it presents apparently unsurmountable difficulties as regards the phenomena of interference of light, which constitutes our only means of analysing radiation in its harmonic constituents and determining the frequency and state of polarisation of these constituents. At present we do not possess any detailed picture of the mechanism of emission and absorption of electromagnetic radiation by atoms and the propagation of radiation through space. We shall see, however, how it is possible to trace a connection between the motion of an atomic system and the spectrum which, even if it must be essentially different from that which would follow from the classical electromagnetic theory, still preserves such features that it gives us hope of attaining a picture which includes the interpretation of the experimental evidence regarding atomic processes as well as the phenomena of interference of light waves, although the mechanism underlying this picture probably will For a more detailed discussion of this point of view, compare N. Bohr, Z. Phys. VI. p. 1. 1921.

THE CORRESPONDENCE PRINCIPLE

involve radical departure from fundamental notions on which the foundation of physical theories have hitherto been sought. In accordance with the fundamental postulate of the quantum theory, we shall take the view that an atomic system in a stationary state, although it can exist for a time without emission of radiation, in general possesses a certain probability of spontaneous transition to some other stationary states of less energy, in a manner corresponding to the assumptions on which Einstein has built his simple method of deducing the law of temperature radiation. Without possessing means of directly attacking the problem of the mechanism of the occurrence of the transitions, we shall now see how the closer examination of the relations of the quantum theory discloses a certain suggestive connection between the transitions and the motion of the system. For the multiply-periodic systems, for which the stationary states are fixed by the three conditions (6), the connection to which there is here alluded consists in the circumstance that it is possible to conjugate the appearance of a transition between two states where the numbers n l , n 2 , n3 have the values n; , n;, n j and n;l, n;l, nj,, respectively, with the appearance in the motion of the system of a harmonic vibration, the frequency of which in terms co2, co3 may be expressed by of the fundamental frequencies ol, (n; - n;’)o1

+ (n; - n y > o 2+ (n; - n 3 w 3

This peculiar connection reveals itself most clearly, if we consider such transitions, for which the motion in the stationary states, and consequently the values of the fundamental frequencies, only differ comparatively little from each other, as will be the case if we consider a transition for which the n’s are large numbers compared with their differences. In that case we get for the frequency of the radiation emitted during a transition, using (7) : 1

-

v = - {E(n;,n;, n;) - E(n;’, ny, ny)) h 1 - [(J;- J;’)Wl+ (s,- J ; l ) o 2 + ( J j - Y{)03] h

=

(n; - n;’)col + (n;- n;1)02+ (nj - nj,)w, *

Notwithstanding this intimate quantitative connection in the region of large n’s between the frequencies of the spectral lines emitted during the various transitions between the stationary states and the frequencies of the constituent harmonic vibrations into which the motion of the system may be resolved, it must be borne in mind, however, that there is so far no question of a gradual approach between the character of the radiation process on the quantum theory and the

THE CORRESPONDENCE PRINCIPLE

classical ideas of radiation. In fact, the calculation is entirely based on the postulate that radiation is always emitted as single trains of harmonic waves, and that accordingly the various trains of waves which coincide in frequency with the frequencies of the constituent harmonic components of the motion are not emitted simultaneously but by a number of independent processes, consisting in transitions between various sets of stationary states. Just by this circumstance, however, we are in the above result led to perceive a connection between the appearance of the various types of transition between stationary states and the various constituent vibrations of the motion, which hides itself at first sight in the region of states where the n’s are small numbers, and where a direct relation between the values of the frequencies of the spectral lines and the frequencies occurring in the motion of the atom is excluded already thereby that the values of the latter frequencies may be quite different in the different states. It is possible, however, to trace the correspondence in question also as regards other features connected with the radiation process. Thus the idea lies at hand that the direct connection between spectrum and motion, disclosed in the region of large n’s, not only holds as regards the frequencies of the spectral lines, but-in view of the internal consistency and wide field of successful application of the classical electronic theory-the assumption presents itself that the spectrum in this region also in a more complete way will reflect the properties of the motion. Now, in the classical theory, the polarisation and intensity of the different components of the radiation emitted by a system, the variation of the electric moment of which can be resolved into constituent harmonic vibrations, are simply related to the amplitude of these vibrations in the different directions in space. Corresponding to this circumstance we are led to conclude that, within the region of large n’s, the probability for the various possible spontaneous transitions between stationary states, and the polarisation of the radiation emitted, are connected with the character of the corresponding harmonic vibrations, in such a manner that the spectrum reflects the motion in the atom in exactly the same way as in the classical theory. This far-reaching correspondence can now be traced also in the region of small n’s, although it is here naturally excluded, just as in the case of the frequencies, to obtain a simple quantitative direct connection between the probabilities of the various transitions and the motion, because the amplitudes of the corresponding harmonic vibration as well as their frequencies in general will differ essentially in the two stationary states involved in the process. In fact, we are led to consider the possibility of the occurrence of a transition between two given stationary states as conditioned by the appearance in the motion of the corresponding harmonic vibration, and in consequence of this to expect that it is possible to draw definite conclusions about the polarisation of the emitted radiation in such cases where the state of polarisation of the radiation which ac-

THE CORRESPONDENCE P R I N C I P L E

cording to the classical electrodynamics should accompany the corresponding harmonic vibration is the same for all motions of the system. The examination of the atomic problems hitherto treated by means of the theory of multiply-periodic systems has given an unrestricted and convincing support for the above view, which may be referred to as the correspondence principle, and it may be specifically mentioned how the establishment of the correspondence in question offers an immediate interpretation of the apparent capriciousness involved in the application of the principle of combination of spectral lines, which consists in the circumstance that only a small part of the spectral lines which might be anticipated from an unrestricted application of this principle are actually observed”.

($4. APPLICATION

OF THE THEORY TO ATOMIC STRUCTURE**

As regards the application of the general considerations, exposed in the preceding, to the problem of the constitution of the atoms of the elements and their spectra, to be dealt with in details in the second part of this report, the theory of multiply-periodic systems in the form described above does not suffice. In fact, in case of atoms containing more than one electron, the multitude of possible mechanical motions will possess a character of a far more intricate type than that of the systemsjust mentioned. Thus for such atoms, motions which may be resolved in discrete harmonic vibrations will from a mechanical point of view only appear as exceptions, forming certain singular classes of distinguished solutions, while the general solution of the mechanical problem will not exhibit any such simple periodicity-properties. In consequence of this, it is not possible for the general motion to define quantities as the above mentioned S s , which possess the necessary invariant character claimed for a direct fixation of stationary states by means of Ehrenfest’s principle, and for such motions a correspondence with the spectrum of the kind discussed above is also beforehand excluded. The tendency, however, in this correspondence to perceive a general feature of the unknown mechanism underlying the quantum theory leads us-in view of the circumstance that the spectra of the elements exhibit sharp lines-to seek the stationary states involved in the emission of these spectra among motions belonging to the dis-

* See the second of the lectures mentioned in the note at the bottom of p. [380] ; a more detailed treatment of these problems is found there. ** [The manuscript here reproduced does not contain this paragraph (with the attached footnote), nor the sub-heading, all of which occur in the French translation. The text of the paragraph (except for an insignificant omission) does occur in the “Solvay report”, from which it has been reproduced; the word “capriciousness” replaces “capricity”, occurring in the original. The footnote and the subheading are later additions.]

THE CORRESPONDENCE PRINCIPLE

tinguished classes of multiply-periodic type, appearing as singular solutions of the mechanical problem. For these classes it is possible to define quantities which-although they do not possess the same general invariant character with respect to adiabatic transformations as the Ss which could be defined for a multiply-periodic system-still preserve a number of properties which make them suitable for the description of stationary states. Due to the fact, however, that the singular motions of multiply-periodic type do not form a single continuously connected class, but that in the atoms we have the choice between a large multitude of such classes for which an anticipation of stationary states is formally possible, such a formal procedure of selecting stationary states gives rise to a considerable amount of arbitrariness, which for a time has been a hindrance of fundamental character for the definite application of the quantum theory to the constitution of atoms of the elements. The tendency of considering the quantum theory not as a set of formal rules, but as a theory of radiation constituting a rational generalisation of the classical theory of electromagnetism leads, however, to a view as to the appearance of stationary states and transitions between them which allows to overcome this arbitrariness, and which tentatively may be formulated as follows. The stationary states of an atomic system which presents a spectrum consisting of sharp lines must be sought among such types of mechanically possible motions of the system for which the displacement of the particles allows a resolution in discrete harmonica1 vibrations. For a transition between two given stationary states accompanied by an emission or absorption of radiation to take place, it is a necessary condition that these states must belong to one and the same coherent class of such motions by which, so to say, the gap between the two states can be overbridged in a continuous way. The probability of a spontaneous transition from a given stationary state to one of the various other stationary states of less energy which belong to the same coherent class depends for each transition on the presence in the motion typical for this class of a certain harmonic vibration which may be said to “correspond” to the radiation which, according to the quantum theory, is emitted during the transition in question, in a way analogous to that holding for the transitions between the stationary states of a simple multiply-periodic system. The content of this statement will in the following be referred to as the “principle of correspondence”, and in the second part of this report we shall see that this principle seems not only suitable as a basis for the discussion of the spectra of the elements, but that it may also serve as a guidance in order to examine what types of stationary states we may expect to find in the normal state of actual atoms, formed by a process of binding of a number of electrons by a nucleus under emission of radiation. [The French translation follows quite closely (except for the two changes

THE CORRESPONDENCE PRINCIPLE

mentioned above) the manuscript here reproduced, until the sentence on p. [377] ending with “. . . is also beforehand excluded.” For the remainder of the report the following (here re-translated into English) has been substituted :] On the other hand,-if we admit that also in the complicated case the motion in the stationary states can be described by ordinary mechanics-a consideration of the necessary stability and “definition” of the stationary states demanded by the specific properties of the elements leads us to seek the stationary states in these ciises from among motions belonging to classes of the type of multiple periodicity that we have recognised and that present themselves as singular solutions of the mechanical problem. It will be possible to define for these classes quantities possessing a certain number of properties analogous to those of the Ss defined for the multiply-periodic systems, indicating that it may be possible to determine these states by means of relations of the same form as the relations (6), valid for the latter systems. But such a manner of proceeding does not lead directly to a determination of the motion of the electrons in the atom offering a convenient basis for a quantitative explanation of the physical and chemical properties of the elements, nor either to an interpretation of the general manner in which these quantities vary with the atomic number, a variation which finds, for many of the properties, its adequate expression in the well-known periodic table. This circumstance follows in the first place from the fact that the indicated procedure is arbitrary, because of the large number of types of electronic configurations for which the stationary states can be determined by the application of conditions analogous to (6). An indication for overcoming these difficulties is furnished, nevertheless, by a close examination of the spectra of the elements from which, according to the principal postulates of the quantum theory, one finds out what different types of transitions can take place in the atoms. Thus, the emission of the so-called series spectra must be considered as accompanying the binding of an additional electron by a system already consisting of a nucleus and a certain number of electrons bound by it, whereas the so-called characteristic X-ray spectra are emitted during a process which may be described as the reestablishment of the normal state of an atomic system from which one of the electronic constituents has been removed by some external agent. By considering the possible results of such processes from the point of view of the fundamental ideas on which the application to systems of multiple periodicity rests, the principle of transformability of the stationary states and the correspondence between the molion in such states and the radiation emitted during a transition between these states, it seems possible to limit the choice of the stationary states that may actually occur during the formation and reconstitution of atoms ;it seems possible, at the same time, to obtain a picture of atomic constitution suitable not only to provide a formal interpretation of the spectra of the elements but one which also

[3791

THE CORRESPONDENCE P R I N C I P L E

presents certain aspects capable of explaining the specific properties of the elements in accordance with the results expressed in the periodic table. But, for the details of these considerations which, as stated in the introduction, were to form the subject of the second part of this Report, I must refer the reader to a publication which will appear soon*.

* A brief account of the basis for the interpretation of the periodic table has been given in a letter to the editor of Nature published in the issue of 24 March, 1921. A French translation of a lecture given in Copenhagen in the month of October, 1923, containing a brief sketch of the principal viewpoints and essential results of these new ideas, as well as that of two other lectures made previously, will soon appear with the editors Herman et fils, Paris.

3.

SUPPLEMENTARY REPORT BY P. EHRENFEST AND DISCUSSION OF BOHR’S AND EHRENFEST’S REPORTS

LE YRINCIPE DE CORKESPONDANCE Pnn M. P. EHRENFEST

A.

1.. Les atomes de RUTHERFORD ne pouvaient pas rester tout ci

fait classiques, c’est-&-direse conformer pleinement ii la mkcanique e t & 1’6lectrodynamique classiques. D’aprks les idBes classiques, en effet, un atome d’hydrogkne par exemple devrait Bmettre un spectre continu, puisque 1’Blectron circulant autour du noyau devrait, par suite de son rayonnement ininterrompu, se rapprocher du noyau suivant une trajectoire en spirale.

2. BOHRsoumet les mouvements dans l’atome de RUTHERFORD i une censure de quanta. Dans cette censure il se laisse surtout guider d’une part par le fait de la discontinuit6 des series spectrales e t par le principe de combinaison de RITZ,qui se prksente dans l’btude de cette discontinuit&,d’autre part par 1’Bquation de PLANCK-EINSTEIN E

= hv.

3. Autant que possible il fait en sorte que son modkle d’atonie se conforme a u x rigles classiques (principe d’inertie, lois de Coulomb); l& OG cela n’est pas possible (rayonnement), il t h h e d’ktablir, entre les mouvements dans E‘atome et le rayonnement Bmis par lui, au moins une correspondance aussi Btendue que possible.

4. Pour trouver cette correspondance, BOHRse laisse guider par le principe heuristique suivant : I1 faut que lorsqu’on donne aux nombres de quanta d’un systkme quantit6 des valeurs de plus en plus BlevBes, le rayonnement Bmis tende asymptotiquement vers celui que le systkme Bmettrait suivant les rbgles classiques.

249

LE PRINCIPE DE CORHESPONDANCE.

B. 5. Dans tous les cas que l’on domine actuellement (cas oti (( les variables sont skparables )I), chaque mouvement dans l’atome jouit de la propriktk suivante : Les coordonnees 2, y, z de chaque klectron peuvent &tre reprksentees comme fonctions du temps a u moyen de skries trigonometriques multiples de la forme

I

z =

2

l’j/l2,,

Cp,,It

,

/’A

cos [ (PI w,

-+.. .+ p 1 s w x ) t +

*(/I,..

.pl1;

.Ill

k est Bgal ou infkrieur au nombre de degrks de liberti: de l’atome; pl, . . . , pk peuvent prendre, indkpendamment les uns des autres, toutes les valeurs entikres positives et negatives ; les frequences fondamentales wl, . . . , wk aussi bien que les amplitudes A,, . . p k , B, ,.., C, ,,..I,h de chaque K son de combinaison )) pi w

i t

. +p k w k * *

dkpendent encore de l’intensitk du mouvement considere. Ensuite, les (( moments angulaires )) 31,

(2)

32,

*. * I

3k,

qui correspondent aux coordonnkes angulaires (3)

(U1= W i t ,

w*=

W*t,

.,

.,)

w/<= w / r t ,

sont independants du temps.

6. Du point de vue de l’klectrodynamique classique on s’attendrait & ce que l’atome kmlt, en gknkral, & la fois tous les (( sons de combinaison )) ii nombres de vibrations

La perte continuelle d’knergie par rayonnement aurait en outre

2io

ATONES ET

BLECTROSS.

pour effet une variation continue de m1, . . . , ( , ) A , ce qui donnerait naissance B 1111 spectre d’bmission continu ( v o i v no 1).

7. On sait clue, d’aprbs la thborie de BOHII,il n’y a (contrairement B l’blectrodynamique classiclue) pas de rayonnetnent aussi

longtemps que le systeme execute un des mouvements (( stationnaires )), qni sont caractbrisCs, suivant BOHR,SOMMERFELD, E P s T E I N , Sc H WAR z s c H I LP , par 15)

= 11, it.

2 ZJ,

,

. ..

2%

3/,= I l / , I1

.

(nl, . ., i z / , sont des nombres entiers, indbpendants les uns des autres : B cliaclue coordonnbe angulaire w,.= (c), t correspond donc, dans le mouvement stationnaire considbrb, un nombre propre).

8. Ce n’est clue lors du passage tionnaire caractbrisb par les nombres autre dont les nombres sont n’:, . . ., u n rayonnement monochromatique dont est, comme on sait,

d’un mouvement stan‘,, ni, ., n:; B un n:. que 1’Clectron bmet le nombre de vibrations

..

0. A premikre vue, il ne semble pas exister de relation entre le nombre de vibrations (( quanteux )) (6) et une autre quelconque des vihrations de combinaison (( classiques 1) (4) de l’atome.

10. Guidb par les principes rnentionnbs sous (3) et ( 4 ) , BOHR

Ctablit cependant une pareille relation; c’est le t h d o r h e (de correspondance) formulb ci-dessous, qu’il complete ensuite par une hypothBse (de correspondance) des plus fructueuses.

C.

11. Pour formuler le theorbme trouvb par BOHR,nous prenons une transition ( n i , ,, ni,) -+ (n’i, , ., n’;#) et corzsidbrons dans les s h i es trigonombtriques ( I ) spbcialement la vibration de combinaison. pour laquelle

..

i7)

p , = n’, - n’;,

.

p2=

II;

-IL’~,

. . ,.

pi, =

?ti.- n%.

zi I

1.1< I'RINCIPE 1)E COilRBSPOSUAKCR.

S o u s I'appellerons la vibration de combinaison (( coinpktente )I de la transition ( n ; , .. . , n;;)+ ,(7( . . , , n : ' )Son . nomlwe vibratoire est done

E n vue de la formation d'une moyenne, nous devons maintenant envisager un certain groupe de niouvenients, cpi constituent un pont, d'interpolation (linkaire) entre les d e u s niouvemcnts stationnaires (n': . IZ;,) et (n', . n : );) ce sont les mouvements pour lesquels

..

. .

h pouvant prendre toutes les valeurs comprises entre

o et

(Dans ce cas les coefficients des seconds membres ne sont plus, en gknkral, des nombres entiers; ces mouvements interpoles ne sont donc pas des (( mouvements )) stationnaires. Provisoirement ils ne jouent que le r6le de grandeurs auxiliaires dans les calculs.) Comme les grandeurs q,, , ., (oh dkpendent, ainsi que nous l'avons dkjh dit, de l'intensitk du mouvement, le nonibre vibratoire (8) du son de combinaison compktent du passage (n',, ., ni) + (n",,. . ., n;) aura une valeur qui, pour les divers mouvements interpolks, dkpendra encore de h. Posant la valeur moyenne de cette (( hauteur de son 1) Bgale h I.

..

BOHRdkduit des propriktis fondarnentales de son inodkle d'atome le theorbme suivant :

E n toutes lettres, cela veut dire que le v (( quanteux )) (6) du rayonnement h i s lors du passage n' -+ n" est kgal h la moyenne du noinbre vibratoire du son de conibinaison competent, prise pour tous les mouvements (9') qui remplissent linkairement la lacune entre les deux mouvements stationnaires (n',, ., ni) et (d;, ., n;,.

..

..

a52

ATOMRS ET kLECTROSS.

12. Pour des vdeurs suflisamment grandes des n’, et n 6, mais telles que n: - n’i soit petit, N est dBj& presque independant de ?, I1~4-n”

m

e t se confond donc dBj& presque avec : conformement a u principe heuristique no 4*,le v quanteux et le nombre vibratoire du son de combinaison coinpktent coi’ncident donc ici asymptotiquement.

13. Grlce & des propriBtBs sphciales du systkme (par exemple dans le mouvement d’un electron dans u n champ & symBtrie axiale), il peut arriver que certains sons de combinaison manquent dans les shies de Fourier ( I ) , c’est-&-dire que pour taus les mouvements du syst8me leurs amplitudes sont exactement nulles. Dans ces conditions, le rayonnement des sons de combinaison correspondants n’est pas & prbvoir, meme du point de vue classique. Quelles sont, dans un tel cas, les conditions posBes par la thBorie des quanta ? 14. Se basant d’une part sur le principe heuristique 110 4, d’autre part sur I’idBe d’une (( correspondance )) entre les rayonnements Bmis et les sons de combinaison compktents, BOHRfait l’hypothkse suivante : Ce n’est pas seulement dans le cas limite d’un nombre de quanta tr8s grands, mais c’est d’une facon tout & Iait g6nBrale que l’existence ou la non-existence d’une transition spontanbe (ni, ., ni) -+ (n’i, ., n;,) correspondent & la prBsence ou l’absence du son de combinaison compBtent ( 7 ) dans les mouvements qui servent de pont d’interpolation dans cette transition [ooir Bquation (S)].

..

..

15. De cette faCon, des propriBtBs spkciales d’un s y s t t h e peuvent avoir pour conshquence l’ktablissement d’une (( selection )) parmi toutes les transitions imaginables, par exemple la disparition de raies spectrales qu’on pourrait s’attendre & observer d’aprks le principe de combinaison de Ritz ou la polarisation de certaines raies dans u n champ Blectrique ou magnBtique, lorsque les sons de combinaison correspondants dans les series de Fourier ( I ) ne fournissent par exemple pas de contribution a u mouvement

253

LC PRlSCIPE DE CORRESPONDAhCC.

parallele i l’axe des i (parallkle a u champ), mais bien a u inouvement diins u n plan parallble B x, y. BOHRa pu, de cette manibre, expliquer kgalement ce phknombne intkressant, que certaines raies, qui manquent dans les circonstances normales, apparaissent sous l’action d’un champ perturbateur. I1 montre notamment que les sons de combinaison qui Ctaient d’abord exclus en vertu d’une symktrie de l’atome se montrent lorsque cette symktrie est troublke par le champ extkrieur.

16. BOHRCtablit aussi u n certain parallklisme entre l’intensitk relative des divers sons de combinaison e t l’intensiti: relative avec laquelle la raie spectrale correspondante est i:mise par un ensemble d’atomes (c’est-&-direla frkquence statistique relative des transitions correspondantes), Mais ce n’est que pour la comparaison de raies d’espbces voisines (par exemple pour la dkcomposition d’une raie dans l’eff e t Stark) qu’il admet u n paral1i:lisme simple. Pour l’iiitensiti. relative de deux raies diffkrentes d’une m&meskrie, par exemple, il y a dCji d’autres facteurs qui interviennent aussi.

17. Soient (14’) et (11”) deux mouvements

stationnaires )) diffkrents d’un systbme. I1 e s t possible que dans les cas gi:nCraux oe pont d’interpblation de mouvements intermkdiaires, dont nous nous sommes servis en opposant i une certaine transition le rayonnement classique correspondant, n’existe pas du tout. Que faire alors? L’idke fondamentale de la (( correspondance )) impliyuet-elle qu’il est exclu que dans ce cas le passage 11’ -+ 11’’ s’effectue par u n rayonnernent spontank ? Cela serait une nouvelle source d e rbgles limitant la sklection des voies par lequelles u n atome peut se reconstituer apres une perturbation. (A ma connaissance, les publications d k j i existantes de BOHRne font pas encore connaitre avec certitude la position qu’il prend vis-i-vis de cette question.) ((

18. La signification la plus profonde des essais de BOHRsup la correspondance rkside bien en ceci, que. provisoirement ils

r 5 .i

17OMES E T i:IdECTlIOS3.

semblent nous rapprocher le plus de cette thkorie future, dont nous attendons qu’elle levera les difficultcis que nous rencontrons lorsque nous youlons traiter les phknomiines de rayonneinent h la Iois d’une maniiire classiqne e t en appliquant la inkthocle des quanta. C’est pourcpoi il n’est pas dksirable qu’en vile d’une npplicatioii au t a n t yue possible automatique, on c o d e clcijh daiis m e forrne rigide la condition de correspondance encore vari;tble et t Atonnante jusqu’ici.

DISCUSSION DU RAPPORT IIE M. BOHR ET DU RAPPORT COMPLkMENTAIRE DE M. EHHENFEST. 11 LORENTZ. - Pour exprimer les choses d’une facon un peu soiiiiiiaire, le passage de la cinquikme orbite stationnaire ?I la deiixibme et la lumibre produite dans ce passage doivent 6tre liks d’une manibre ou d’une autre h l’existence d u troisibme mouvement harmonique. De m6me le passage de la septibme orbite la deuxikme serait lii? a u cinquibme niouvement harmonique. C’est cc que M. Ehrenfest dit d’une fapon plus prkcise. Si les nombres de quanta des deux orbites ktaient grands en coinparaison de la diffkrence de ces nombres, alors la frkquence de la lumibre Bmise serait kgale k celle de l’harmonique en question. Dans le cas otI les nonihres de quanta sont plus petits, cette Bgalitk n’existe plus, inais il y a nkaninoins une certaine relation qui se niontre par excinple dans I’intensitB et dans l’ktat de polarisation entre l’une des composantes harmoniques des mouvements, qui existent dans l’atome, et la lumibre Bmise. C’est cela qui constitue le principe de correspondance. Cela s’accorde-t-il avec votre manibre d e voir ?

M E J ~ R E N W ES TOui. . Pour le cas oh les nombres de quanta sont petits, Bohr a donit6 uiie relation exacte entre’le nornbre de vibrations du rayoiine men t h i i s et la frkquence des vibrations harinoniqiies corrcspondnntes. Mais, pour ce qui regardr I’intenc c fapon plus prudente. sit6, Bohr se p ~ ~ ~ i i o nd’une M. BRAGG. - Conimeut est-ce qu’on dkfinit, pour la transition d’un &tat stationuairc h iin autre, la valeur moyenne de la grandeur to ? M. E H R E X F ~ ~S TC’est . la inoyenne la plus simple que vous puissiez iniaginer. Coiisidkrez une transition dans laquelle un des noinbres de quanta passe de la valeur n h la valeur n’, les autres nombres, s’ill-e n a, ne changeant pas. On peut alors se reprksenter

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les Btats de niouvement intermkdiaires entre l’ktat initial et 1’Ctat final. Ces ktats de mouvement ne sont pas e u x - m h e s stationnaires, mais ils sont tous possibles a u point de vue de la inkcanique classique. Pour un d’entre eux le nombre caractkristique g peut &re reprksenti: par y=nL)\(n’--n).

de cjorte que dans la transition X varie de o i3 I . Si 10 (I,) est la valcur de w pour le mouyement intermkdiaire caractkrisk par I,, on dkfinit la valeur moyenne dont il s’agit par la forniule

JI. LANGEVIR’. - Si, comme on le suppose d’ordinaire dans

l’exposk klementaire de la thkorie de Bohr, les klectrons parcourent des orbites stables circulaires autour du noyau, c’estB-dire si le dkveloppement de la loi du mouvement en. skrie de Fourier ne comporte que le terme fondamental, le principe de correspondance exige qu’un klectron ne puisse passer spontanbmerit d’une orbite circulaire a n quanta qu’h l’orbite immkdiatemerit infkrieure B n - I quanta. Chaque skrie ne comporterait ainsi qu’une seule raie, et la skrie de Balmer se rkduirait a sa premibre composante. Le fait qu’il en est autrement montre que les orbites ne sont pas circulaires.

M . EHRENFEST. - E n effet, si l’hydrogbne avait uniquement des orbites circulaires, un klectron ne pourrait passer d’une orbite qu’h la suivante. I1 ne se prksenterait que les raies

e t le spectre de l’hydrogkne serait trbs pauvre en raies.

M. LANGEVIN. - J e crois qu’il serait important de signaler ce fait dbs le dkbut de la theorie de Bohr, parce qu’il me parait tout fait essentiel. I1 me parait Cgalement important de remarquer que la loi d’action e q raison inverse d u carre de la distance, qui determine

1)ISCCSSION IIES RAPPORTS DE h1.V. B O l l R ET EHRI.:RFEST.

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1e inouveinent des Blectrons sup leurs orbites, dBterniine en indrne temps I’intensitB relative des raies d’une sCirie, puiequ’elle dBtermine l’importance des harmoniques dans les mouvements elliptiques. Pour une loi d’action proportionnelle 8 la distance, l’orbite elliptiyue, cornine l’orbite circulaire, ne comporterait que le terme 1)Briodique fondaiiiental e t chaque sCirie ne comporterait qu’une seule raie. L’existence de tous les termes de la sCirie dans le spectre tient 8 ce que les ellipses parcourues par les Blectrons sont des ellipses de KBpler.

11. LORENTZ. - Cette dernikre partie ajoutCe par 11. Bohr iiie parait t r b s remarquable, parce cpi’il ne se deiiiande pas coiiinient l’atonie est constitdb, inais coiiiiiient il a pu se former. Au sujet cle cette formation, il pose la question de telle manibre, qu’elle revient i~ demander s’il y a possibiliti: de transition entre deux Btats. I1 exclut ce passage, yuelquefois parcc que les deux Btats de iiiouveinent n’appartiennent pas la niZiiie (( classe )I, quelyuefois pour d’autres raisons. Ai-je bieii coinpris cela ? I1 est presque superflu d’insister sur la dificultk qu’on rencontre , quand o n cherchc ii se reprBsenter les mouveinents un peu en clktail. DCjh, dans le cas de I’hBlium, o h il y a deux Clectrons, le systbiiie est assez coiiipliqu8, car les forces agissant entre les deux blectrons soiit coiiiparables g celles qui sont exerckes par le iioyau. A cause de l’influence inuiuelle des Clectrons, leurs orbites lie saiiraient Ctrc planes. A la rigueur, o n ne del-rait pas quantifier les iiiouvenieiits dcs dcux Clectrons indBpendainment l’un de I’autre ; 0 1 1 dcvrait quantifier lc iiiou\-enient d u systbine tout cutier. I1 sciiiblc cependalit qu’on ait jusqu’ici quantifiB les mou\-einents des deux Blectrons en les prenant iiidividuelleiiient. I’eut-on avoir quelque id6e de l’influence que cela pourrait avoir siir le rksultat. ~

11. EIIRENFEST. -- Les inouveiiieiits ii’ont pas 6ti: calculBs exactcinent, iiiais vous avez raison, ils lie se font pas prBcisBinent d a m U l l pla11. 11. LANGEVIN. - J’ai cru coniprendre que hI. Bohr trouve 1)oiir le liiouveiiient d’un s y s t h e d’klectrons autour d’uii iioyau I \ b T I 1 1 T Z0I.VA.Y

‘7

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positif, plusieurs classes de solutions conformes ?I la dynainique clamique, e t telles que chaque classe comprend une infinitk de solutions formant un ensemble continu, c’est-&-dire qu’on peut passer d’une solution a une autre de la m&me classe par variatioii continue d’une ou plusieurs constantes arbitraires. Un tel passage ne :,erait pas possible entre deux solutions appartenant a des classes diff Brentes. Dans chaque classe, les conditions de q u a n t a font considkrer conime stables des solutions particulibres formant u n ensemble discontinu e t reprksentant les divers ktats permanents possibles pour l’atome. Le nouveau principe introduit par 31. Bohr afiiriiie q ue 1’a t o me ne p e u t pa s s er s p ont a n k men t d ’u n e con fi gura t io 11 stable q u ’ i une a u t r e de me“me classe en kmettant un ra? onnenient dont la frkquence Y est dBterminCe par la condition que la difTBrence des energies entre les deux configurations stables soit kgale B hv. Line application importante est kvidemment celle qui exclut les configurations trop symktriques, par exemple celle de l’atom? d’htilium avec deux Blectrons diamktralement opposBs sur la m&me orbite, comme incompatibles avec la genPse probable de l’atome par arrivke successive des klectrons depuis une distance infinie juscp’au voisinage d u noyau. La solution symktrique ne coniporterait, en effet, comme solutions de 7 d m e classe parcourues successivement pendant la genkse de l’atonie avec kmission de rayonnement, que des solutions symktriques exigeant une callture simultane‘e de plusieurs Blectroris depuis l’infini, ce qui apparait comme hautement improbable CL priori.

hl. LORENTZ. - Lorsqu’il s’agit de savoir si une structure esi poshible, Bohr se demande comment il peut la former. Ayant par exemple 17 Blectrons a u t o u r d u noyau, pour en mettre I S il se dit : nous pouvons placer les 17 premiers a peu prbs commc t a n t 8 t ; le dernier se trouve d’abord h grande distance e t n’a qu’une petite vitesse; c’est le premier ktat. L’Ctat final est imagink d e diverses fagons; la question est de savoir si le passage est possible. Bohr le dkcide Bvidemment en appliquant divers principes d’exclusion; un de ces principes c’est que, pour que la transition puissc se faire, les mouvements doivent 8tre de la mkme classe.

DISCLESION DES IL4PPORTS DE M b l . BOHR ET EHRENFLbT.

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hl. LANGEX I X . - Oui. L’autre principe, celui de correspondance, ajoutant que le passage d’une configuration ?I une autre de mime classe est kgalement exclu lorsque l’analyse harmonique de l’un ou de l’autre des mouvements extremes ou des inouvements intermkdiaires de la mdme classe, admis par la dynamique classiyue, ne comporte pas de terme correspondant, comme rang d’harmonique, a u x tliffkrences entre les norribres de quanta des configurations extrdmes. 11. RUTHERFORD. - Est-il nkcessaire, dans le modble lde Bohr, que les klectrons soient en mouvement autour d u noyau ? Les rksullats de Bohr sont-ils compatibles avec une autre loi que celle de l’inverse du c a r d ? 11. E H R E N F E S T . Nicholson a dkja indiquk des cas o h deux electrons circulent, d’une inanibre ceiitraleinent s)-niktrique par rapport a u noyau, sur deux cercles parallbles dont les plans ne passent pas par le noyau. hlais il semble que de pareils mouvements lie s’accordent pas avec les idkes de Bohr sur la synthkse des atomes par cap lures successives d’klectrons. Avec ces idkes s’accorde bien une inclinaison des diverses orbites les unes par rap port a u x autres. Toils lei calculs et toutes les considkrations actuelles de Bohr s’appuient sur l’hypothbse de forces agissant suivant la loi cle Coulomb, eiitre les divers klectrons et le no) au, moyennant uiie correction de relatilit$ pour la iiiassc. C’est tout ce que nous savons.

JI. L A > G E \ I ; \. J e voudrais demander aussi h )I. Ehrenfest cle nous donner quelques renseignenients sur les expkriences de AIR!. Franck e t Knipping. qui oiit permis d ’ a f h n e r l’existence de deux formes diff Crentes de l’atonie d’hkliuin, qui correspondraient deux classes de solutions a u sens de AI. Bohr, sans possibilitk dc passage spoiltank d’une forme h uiie autre. hl. BRAGG.- I’ourquoi, daiis les iiiolkcules coiiiposkes de plusieurs atornes, les klectrons sont-ils retenus par paires par les atomes ?

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ATOMES ET ELECTROXS.

M. EHRESFEST. - La thCorie n e doinine pas encore, pour le moment, le mouveinent des Clectrons d a n s les Cdifices d e plusieiirs atomes, pas in&me d a n s la niolCcule d’hydrogkne. D’un point de VUI: p u r e m e n t einpiriyue, il y a divers faits yui plaident en faveur d e la manikre d e voir d e Kossel, yu’en gCn6ral les atoiiies tendent i capturer, ou h a b a n d o n n e r a u t a n t d’dectrons qu’il est nkcessairc. pour yue les orbites des Clectrons extkrieurs .resseniblent B celle. dea gaz inertes. 11. ZEEMAS. - Quelle est l’explication d ’ u n spectre continu ? 11. L A X G E T I N. Le spectre continu d ’ h i s s i o n a u delg d e i t t t e s d e s6rie s’interprhte naturellement, coinine l’a niontri. hI. Bohr, par la rkunion d e l’atonie ionisi. et d’un irlectron T eiiaiit d e l’infini avec une Cnergie cinCtique iiiitiale q n i 1)eut Ctre ( p e l coriyue e t s’ajoute a u cluantuni d e la t e t e de sirrie pour augilientcr d e nianikre continue la frCquence iiiiiise a u nionieiit du retoiir h la configuration finale stable coininune B tous les ternlea d e la skrie. L’ensernble d e l’atoine ionisir e t d e l’klectrori en iiiou\ ciiiciit ii distance infinie constitue Line configuratioii stable ap1)artenant 11 la n i t h e classe q u e toutes celles c p i pr6c&dent 1’8niission d ’ i i i i t e r me yuelconcpe d e la s h i e spectrale e t clue la configurntioil finale commune B tous les terines. Ides coiifiguratioiis tle c e l t e c1a:ise stables a u sens d e la t h h r i e des q u a n t a i‘oriiiciit 1111 ciisciirll(. discontinu t a n t yue l’klectron p6riphi.riquc r c s ~ cit tlistancc fiiiic. d u n o y a u , puis u n ensemble continu qiiaiid l’irlectron est ii diat a n c e infinie a \ ec une \. itcsse susceptible d e T ariatioils coiilinucs. A t e t i t r e , le fond c o n t i n u est le prolongeineiit nature1 de In &lie.

?I. EHRENFEST. - J e ~ o u d r a i sia1)l)eler :i cc 1wo11w les 1)oasibilit6s suivantes, bien connues, d’ailleurs, de production d e raies spectrales diffuses : Effet S t a r k , irrkgulier sous l’action des champs d’ioni voisins (ce yu’on appelle (( l’irlargissement des raies spectrales pal effet d e pression e t l’irlargissement d a n s lei liquides e t le\ solides )I) ; 20 Bohr mentionne la possibiliti: clue d a n s des champs c o m pliquCs, par exemple d a n s les cas d e s6perposition d u c h a m p a to10

DISCUSSION DES RAPPORTS DE D I M . BOHR ET EHREN‘FEST.

>(it

iiiique et de champs irlectriyue et magnktique extkrieurs, il n’est plus possible de quantifier nettement le mouvement des klectrons.

M. D E BROGLIE.- Les nouvelles idires de M. Bohr expliquent-elles pourquoi le phknomene des raies de Fraunhofer, c’est &dire l’absorption d’un rayonnement accompagnk par exemple d u transport de l’klectron de la trajectoire I< sup une trajectoire L n’est pas possible?

M.ZEERIAN. - Y a-t-il dans la nouvelle thirorie une explication aussi belle de la dispersion anomale que dans l’ancienne. E t les bandes d’absorption clans le spectre, comment la thirorie de Bolir les expliyue-t-elle ? M. EHRENFEST. - I1 n’existe pas encore, pour le moment, de thirorie satisfaisante de la dispersion anomale basire sur le mod6le d’atome de Bohr. Une chose remarquable est la suivante : c’est que pour des champs qui varient avec une lenteur infinie (influence (( adiabaticlue ))), la perturbation de l’orbite de l’irlectron peut encore &tre calculire suivant les rkgles de la mircanique classique; pour les h a u t a frkquences, cela n’est certainement plus permis. M. LANGEVIN. - En rirflirchissant a u x deux principes de m’a seniblk yu’une liaison intime existe entre eux qui n’en fait en rbalitk qu’un seul principe, celui de correspondance. En eff et, l’application d u prineipe de correspondance, sous la forme que nous a rappelire M. Ehrenfest e t qu’a utilisire hi. Kramers dans sa These, fait intervenir l’intercalation entre les deux configurations extremes considirrkes coinme stables par la thCorie des quanta, tout une sirrie continue de configurations possibles a u sens de la dynamique classique. Cela n’a de sens que si lees configurations stables extremes appartiennent i la meme classe de mouvements, suivant la signification qu’il seinble nkcessaire d’attribuer h cette notion. L’knonck ineme d u principe de correspondance impliquerait l’existence de classes de mouvements comportant chacune un ensemble continu de solutions possibles a u point de vue de la dynamique classique e t ferait intervenir dans le calcul de la frhquence, de l’intensitk

M. Bohr dont nous avons parlir, il

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e t de la polarisation d u rayonnement Cmis lors d u passage de l’atome d’une configuration stable, au sens des quanta, & une autre, l’ensemble continu des solutions intermkdiaires de la n i h e classe. La condition que les configurations stables extremes appart iennent & une m&me classe serait ainsi contenue implicitement dans 1’CnoncC primitif d u principe de correspondance. Le nouveau principe fait siinplement remarquer yue plusieurs classes diflbrentes peuvent exister pour un meme atome.

11. BRAGG. - Pouyez-vous dire quelles sont approximati\ c ment, dans les cas d’atomes complexes, les dimensions des orbitcs des Clectrons, c’est-&-direquels sont les grands e t les petits axes des ellipses e t les rayons des orbites circulaircs ? 31. E H R E N F E ST . Bohr peut donner ces dimensions grossikrcnient.