On the Serial Spectra of the Elements1

Niels Bohr.

The subject on which I have the honor to report today, in accordance with the friendly invitation of the presidium of the German Physical Society, is very extensive, and it would be quite impossible to give in a single lecture any at all complete survey of the extraordinarily important results achieved in the theory of spectra. In what follows I shall attempt to put forward certain starting points which seem to me important both for understanding the present state of the theory and for the further possibilities of its development in the near future. Unfortunately, time will not allow me to dwell in greater detail on the history of the development of spectral theories, although this would be of interest in connection with the rest; the absence of a historical survey will not hinder the understanding of the subsequent exposition, since the initial propositions on the basis of which, only a few years ago, attempts were made to explain spectra differ decisively from the propositions set forth below. This difference extends both to the development of our ideas concerning the structure of the atom and to the way in which these ideas are applied to the explanation of spectra. As regards the first point, we shall assume, according to Rutherford’s theory, an atom consisting of a positively charged nucleus around which a certain number of electrons revolve; the dimensions of the nucleus are exceedingly small in comparison with the size of the whole atom, yet it contains almost the entire atomic mass. Time does not permit me to present the foundations of this so-called nuclear theory of the atom, nor the very strong confirmations which it has received in investigations in the most varied fields. I wish to mention only one result which has given unusual simplicity and attractiveness to the modern theory of the atom: I mean the view that the number of electrons of a neutral atom is simply equal to the number determining the position of the corresponding element in the periodic system. This proposition, first expressed by van den Bröck, is the program to which

explanation of the physical and chemical properties of the elements—we must proceed from an atomic model based on this number, the so-called “atomic number.” An attempt to carry out such a program on the basis of the classical laws of mechanics and electrodynamics is beset with insurmountable difficulties. They are especially acute in the case of the spectra of the elements. The difficulties here are so obvious that it would be a mere waste of time to dwell on them. It is evident that systems of the above-mentioned type, according to the ordinary mechanical and electrodynamical conceptions, cannot possess sufficient stability and therefore cannot be the source of a spectrum consisting of sharp lines.

In what follows we shall base ourselves on the conceptions of the so-called quantum theory. There is no need, especially here in Berlin, to expound Planck’s fundamental works on heat radiation, which served as the impetus for the development of the quantum theory, according to which the laws determining the course of atomic processes contain an essential element of discontinuity. I shall mention here only Planck’s principal result concerning the properties of an extraordinarily simply constructed atomic system, the so-called Planck “oscillator.” This oscillator consists of a single electrically charged particle capable of executing harmonic oscillations about a position of equilibrium with a frequency independent of the amplitude. As a result of considering the statistical equilibrium of a group of such systems in a radiation field, Planck, as is well known, arrived at the conclusion that emission and absorption by such oscillators can take place only in such a way that, in considering statistical equilibrium, one has to take into account exclusively certain special states of the oscillator. In these states the energy of the system is an integral multiple of the so-called “energy quantum,” proportional to the oscillator’s number of oscillations. The special values of the energy may be expressed by the formula:

\[ E_n = n h \omega, \tag{1} \]

where \(n\) is an integer, \(\omega\) is the number of oscillations of the oscillator, and \(h\) is the universal, so-called Planck constant. An attempt to apply this result to the explanation of the spectra of the elements, however, encounters a difficulty consisting in the fact that the motion of a particle in an atom, despite the simplicity of the structure of the latter, is, generally speaking, extremely complicated in comparison with the motion of Planck’s oscillator. Hence the question arises: in what way should Planck’s conclusion be generalized in order to obtain an application in the present case? In this respect various points of view are possible. Thus, for example, equation (1) may be regarded as a condition determining the relations for special motions of an atomic system, and one may try to seek the general form of these conditions. On the other hand, one may look at the equation

(1), as a certain assertion about the properties of the radiation process, and, consequently, to seek general laws determining these processes. In Planck’s theory it is taken as something self-evident that the number of oscillations of the light absorbed and emitted by an oscillator is equal to the latter’s own frequency of oscillation. This assumption may be written as follows:

\[ \nu=\omega \tag{2} \]

where, here and below, in order to distinguish sharply, we denote by \(\nu\) the number of oscillations of the emitted light and by \(\omega\) the number of oscillations of a particle in the atom. We see from this that Planck’s result may be interpreted in the following way: an oscillator can emit and absorb light only in so-called “radiation quanta” of magnitude

\[ \Delta E=h\nu \tag{3} \]

As is known, this interpretation led Einstein to the theory of the photoelectric effect, which is of great importance as an application of the quantum theory to a phenomenon of non-statistical character. I shall not dwell on the well-known difficulties to which the so-called hypothesis of light quanta leads in interference phenomena, whose explanation is so simple in the classical theory of radiation. In general I do not intend to enter into a discussion of the riddle connected with the question of the nature of radiation, but propose only to try to show in what way it is possible, in a purely formal manner, to construct a theory of spectra whose fundamental elements would be the simultaneous rational development of both interpretations of Planck’s result.

For the interpretation of the phenomenon of the line spectra of elements on the basis of the view of atomic structure indicated above, we are forced to assume that the radiation of an atomic system occurs in such a way that the usual conceptions are not capable of interpreting the process in detail and do not give a method for determining the frequency of the oscillations of the radiation. We shall see, however, that there exists the possibility of a simple explanation of the general empirical laws of the frequencies of spectral lines, if it is assumed that for every radiation of an atomic system the following law of nature holds: throughout the entire process of radiation, the latter has one and the same frequency \(\nu\), determined by the following frequency condition

\[ h\nu=E'-E'', \tag{4} \]

where \(E'\) and \(E''\) are the energy of the system before and after the radiation. Taking this law as a basis, we conclude that spectra do not give us a picture of the motion of particles in the atom, as is assumed in the ordinary theory of radiation, and permit us to judge only the changes of energy in one or another possible process in the atom of the type indicated above. According to

According to this view, spectra testify to the existence of certain exceptional values of energy, corresponding to special states of the atom; below we shall call these states stationary states of the atom, since we assume that the atom can remain for a finite interval of time in each such state and, on leaving it, again passes into another stationary state. Despite the fundamental difference between such a view and the customary conceptions of mechanics and electrodynamics, we shall see the possibility of a rational combination of the above-mentioned theory of atomic structure with the facts of spectroscopy. It turned out precisely that, although we must renounce the application of mechanics in describing the transition from one stationary state to another, it is nevertheless possible to construct a coherent theory of these states, using ordinary mechanics to describe the motion in the stationary states themselves.

Further, the process of radiation connected with the transition from one stationary state to another cannot be traced in detail with the aid of ordinary electromagnetic conceptions. The properties of the radiation of the atom from the point of view of these conceptions are determined directly by the motion of the system and by the decomposition of these motions into harmonic components. Nevertheless, it turned out that there exists a deep correspondence (Korrespondenz) between the various types of possible transitions from one stationary state to another—on the one hand, and the various harmonic components of the decomposition—on the other. Thus the theory of spectra under consideration may be regarded, to a certain extent, as a generalization of the conceptions of the ordinary theory of radiation.

For the clearest exposition of the principal points, before passing to complex types of series spectra, I shall consider the simplest series spectrum of hydrogen. As is known, this spectrum consists of several lines, whose frequencies can, with great accuracy, be expressed by Balmer’s formula:

\[ \nu=\frac{K}{(n'')^2}-\frac{K}{(n')^2} \tag{5} \]

where \(K\) is a constant, \(n'\) and \(n''\) are integers. Putting \(n''=2\) and allowing \(n'\) to take the values \(3,4\ldots\), we obtain the well-known series of hydrogen lines of the visible spectrum; putting \(n''=1\), or \(n''=3\), we obtain in a similar way spectral series observed in the ultraviolet and infrared regions of the spectrum. Turning now to the structure of the hydrogen atom, we see that, according to Rutherford, it simply consists of one charged nucleus, around which a single electron moves. For simplicity we assume the mass of the nucleus to be infinitely large in comparison with the mass of the electron, and in addition we disregard the small changes in the motion caused by

with the change of the electron’s mass as a function of velocity, as required by the theory of relativity. Under these assumptions the electron, according to the usual mechanical conceptions, will describe a closed ellipse with the nucleus at one of the foci; the number of revolutions per unit time \(\omega\) and the major axis \(2a\) of this ellipse are connected with the energy of the system by the following simple formulas, which follow from Kepler’s laws:

\[ \omega=\sqrt{\frac{2W^{3}}{\pi^{2}e^{4}m}} \ ; \qquad 2a=\frac{e^{2}}{W} \tag{6} \]

where \(e\) is the charge, \(m\) the mass of the electron, and \(W\) the work required to remove the electron from the nucleus to infinity. The simplicity of these formulas prompts one to apply them to the explanation of the hydrogen spectrum, but this is impossible so long as our conceptions are based on the classical theory of radiation. Indeed, according to this theory it is quite impossible to understand how hydrogen emits a spectrum consisting of fine lines; \(\omega\) changes together with \(W\), and we would have to suppose that the frequency of the emitted light changes continuously during the emission. The situation changes if we consider the problem on the basis of the above-stated conceptions of quantum theory. Let us form, for each line, the product \(h\nu\), multiplying both sides of equation (5) by \(h\); we see that the right-hand side of the resulting relation can be written as the difference of two simple terms. Comparing with formula (4), we arrive at the conclusion that the individual spectral lines are emitted during transitions between two stationary states belonging to an infinite series; the energy of the \(n\)-th state is determined (if an arbitrary constant is discarded) by the expression:

\[ E_n=-\frac{K.h}{n^2} \tag{7} \]

The choice of the negative sign is determined by the fact that the energy of the atom is most simply characterized by the work required to remove the electron; above we denoted it by \(W\). Substituting expression (7) for \(W\) in formula (6), we obtain the number of revolutions of the electron and the major axis for the \(n\)-th stationary state:

\[ \omega_n=\frac{1}{n^3}\sqrt{\frac{2h^3K^3}{\pi^2e^4m}} \ ; \qquad 2a_n=\frac{n^2e^2}{h.K} \tag{8} \]

We could now investigate the relation of the motions described by these formulas to the special states of Planck’s oscillator. We shall not, however, dwell further on this question, whose rational treatment may lead to a theoretical determination of the constant \(K\); we shall only show how this determination can be obtained by a simple comparison of the emitted spectrum and the motions in the stationary states, and this comparison will lead us at once to the above-mentioned correspondence principle.

According to our assumptions, every hydrogen line is emitted in a transition between two states of the atom corresponding to different values of \(n\); the number of revolutions and the major axis of the ellipse may, in this case, be of the most varied kinds; as formulas (8) show, with the decrease of the energy of the atom during the process of emission the major axis of the electron’s orbit decreases and the number of revolutions increases. Thus, generally speaking, the possibility is excluded of obtaining a relation between the number of revolutions of the electron and the frequency of the radiation corresponding to the usual conceptions of radiation. Let us consider, however, the ratio of the numbers of revolutions \(\omega\) of two stationary states corresponding to the given quantities \(n'\) and \(n''\), if \(n'\) and \(n''\) increase gradually; we shall see that the ratio approaches unity, although the difference \(n' - n''\) remains unchanged. Thus there opens up the possibility of obtaining some basis for comparing our views with the usual conceptions of the theory of radiation in the case when transitions corresponding to large values of \(n'\) and \(n''\) are considered. For the frequency of the radiation in such a transition, on the basis of (5), we have:

\[ \nu=\frac{K}{(n'')^2}-\frac{K}{(n')^2}=(n'-n'')K\frac{n'+n''}{(n')^2(n'')^2} \tag{9} \]

If the numbers \(n'\) and \(n''\) are large in comparison with their difference, then this expression, on the basis of formulas (8), may approximately be rewritten as:

\[ \nu \sim (n'-n'')\omega \sqrt{\frac{2\pi^2 e^4 m}{Kh^3}} \tag{10} \]

where \(\omega\) is the number of revolutions in one of the two states. The number \(n' - n''\) is an integer; we therefore see that the first part of the expression, i.e. \(\omega(n'-n'')\), coincides with the frequency of one of the harmonic components into which the elliptical motion of the electron can be resolved. As is known, for every periodic motion with number of periods \(\omega\), the displacement \(\xi\) of a particle of the system in a definite direction of space can be represented, as a function of time, by a trigonometric series of the form:

\[ \xi=\sum C_{\tau}\cos 2\pi(\tau\omega t+C_t) \tag{11} \]

where the summation extends over all positive integral values of \(\tau\).

We thus see that the frequency of the light emitted in a transition between stationary states characterized by the numbers \(n'\) and \(n''\), large in comparison with their difference, coincides with one of the components of the radiation that one may expect for the chosen motion of the electron in a stationary state on the basis of the usual conceptions. This coincidence will occur in the case when the latter

the factor in formula (10) will be equal to unity. Such a condition is equivalent to the following:

\[ K=\frac{2\pi^2 e^4 m}{h^3} \tag{12} \]

It is in fact satisfied if for \(K\) one substitutes the value found on the basis of measurements in the hydrogen spectrum, and for \(e, m, h\) the quantities directly determined experimentally. Such an agreement establishes a connection between the spectrum and the model of the hydrogen atom. If one takes into account the fundamental difference between the conceptions of quantum theory and of the ordinary theory of radiation, then the connection found becomes all the more remarkable.

Let us now examine more closely the connection that has been found between the spectra expected according to quantum theory—on the one hand—and the ordinary theory of radiation—on the other, in the region where the stationary states differ very little from one another. As has already been shown, the frequencies of the spectral lines in this region, calculated by either method, coincide; we must not forget, however, that the mechanism of radiation in the two cases is completely different. According to the ordinary theory of radiation, the various components of the light corresponding to the various components of the motion are emitted simultaneously with a relative intensity determined by the ratio of the amplitudes of the oscillations. Matters stand quite differently in the interpretation of quantum theory. In this case the various spectral lines correspond to entirely different processes, consisting in transitions from one stationary state to various nearby states, in such a way that the radiation corresponding to the \(\tau\)-th overtone occurs during the transition determined by \(n' - n'' = \tau\). The relative intensity of the separate spectral lines emitted in this process depends on the relative probability of the various transitions. In asking about the deeper meaning of the correspondence found, we are naturally entitled to expect that the correspondence is not limited to the coincidence of the frequencies of the spectral lines calculated by the two methods, but extends also to the intensity. Such an expectation is equivalent to saying that the probability of a definite transition between two stationary states is connected in a known way with the amplitude of the corresponding harmonic component.

Further consideration leads us to the conclusion that this peculiar connection is a general law for the occurrence of transitions between stationary states; we must assume that the possibility of a transition between two given stationary states is connected with the presence of a definite harmonic component in the motion of the system. The fact that the magnitude of the indicated component may be quite different in the two states under consideration, when the numbers \(n'\) and \(n''\) are small compared with their difference, forces us to expect in advance that the connection between the probability of transition and the amplitude of the har-

of a harmonic component of the motion will, generally speaking, be complicated. The same may be said of the relation between the frequency of the radiation and the corresponding component. From this point of view we must, for example, regard the green hydrogen line \(H_\beta\), emitted in the transition from the fourth to the second state, in a certain sense as an “octave” of the red line \(H_\alpha\), corresponding to the transition from the second to the third state, although the frequency of oscillation of the first line is in no way equal to twice the frequency of the second line. We must regard the process causing the appearance of the line \(H_\beta\) as conditioned by the presence of a certain harmonic component in the motion of the atom, which is an octave of the component of that motion on which the possibility of the emission of the line \(H_\alpha\) depends.

Before passing to the consideration of other spectra, in which we shall find numerous applications of the point of view set forth, I wish to mention one interesting application of the foregoing to the theory of Planck’s oscillator. Let us compute from (1) and (4) the frequency of oscillation corresponding to a transition from one special state of such an oscillator to another; we shall find:

\[ \nu=(n'-n'')\omega \tag{13} \]

where \(n'\) and \(n''\) are the numbers determining the indicated states. The essential premise of Planck’s theory consists in the fact that the frequency of the light absorbed and emitted by the oscillator is always equal to \(\omega\); we see that this premise is equivalent to the assertion that in the given oscillator, in contrast to the hydrogen atom, only transitions between two neighboring stationary states are possible. From the point of view developed above, this could have been expected in advance, since the difference between the oscillator and the hydrogen atom, according to our assumption, consisted in the fact that the motion of the oscillator, in contrast to the motion of the electron in the atom, is purely harmonic. We thus see the formal possibility of constructing a theory of radiation in which the spectrum of hydrogen and the simple spectrum of Planck’s oscillator would figure on an entirely equal footing. It is obvious, however, that only for such a simple system as the oscillator can the theory be formulated by a single condition; in the general case this condition splits into two: one concerning the character of the motion in the stationary states, the other relating to the frequency of the light emitted in transitions between these states.

Passing now to the spectra of elements with higher atomic numbers, we see that they have a more intricate structure than the spectrum of hydrogen. As is known, however, for the spectra of many elements simple laws have been found that present a remarkable analogy with Balmer’s formula for the spectrum of hydrogen. According to Rydberg and Ritz, the frequencies of oscillation of the series spectra of these elements can be expressed by a formula of the type:

\[ \nu=f_{\kappa''}(n'')-f_{\kappa'}(n') \tag{14} \]

where \(n'\) and \(n''\) are two integers, \(f_{\varkappa'}\) and \(f_{\varkappa''}\) are two functions belonging to a series of definite functions characteristic of the given element, varying quite simply as functions of \(n\) and, in particular, approaching zero for large values of \(n\). Leaving the first term on the right-hand side of (14) constant and substituting for \(n'\) in the second term \(f_{\varkappa'}(n')\) various integral, successively increasing numbers, we shall obtain different series of lines. The complete spectrum can be obtained on the basis of the so-called Ritz combination principle, if in formula (14), in place of \(f_{\varkappa''}(n'')\) and \(f_{\varkappa'}(n')\), all combinations of two quantities from the totality of all possible values of \(f_k(n)\) are substituted.

The circumstance that the frequency of any line of the spectrum is the difference of two simple expressions depending on integers leads, on the basis of our interpretation of the process of emission, directly to the assumption that the terms on the right-hand side of equation (14), multiplied by \(h\), may be equated to the energies in different stationary states of the atom. In the spectra of the elements, in contrast to the hydrogen spectrum, there occurs not one but a whole series of functions of \(n\). This fact compels us to recognize, for atoms of the elements, the existence not of one but of several series of stationary states; the energy of the \(n\)-th state of the \(k\)-th series is represented (if an arbitrary constant is omitted) as follows:

\[ E_k(n)=-h\cdot f_k(n) \tag{15} \]

Such complexity of the totality of stationary states of elements with higher atomic numbers should also be expected on the basis of the connection between spectra calculated according to quantum theory and the decomposition of the motions of an atom into harmonic oscillations. As we have already seen, from this point of view the simplicity of the totality of stationary states of the hydrogen atom is most closely connected with the simple periodic character of the motion of this atom. In other elements, where the neutral atom contains several electrons, we find more complicated motions and have a more complicated decomposition into harmonic oscillations; in order to obtain the connection between the motions of the atom and the spectrum determined by the frequency condition, we must also expect a more complicated totality of stationary states. In what follows we shall see how such a correspondence can be traced in detail and how this path leads to a direct explanation of those obscure points which were associated with the application of the combination principle in connection with the seemingly arbitrary presence or absence of lines predicted by this principle.

In the drawing (see the following page) are presented those stationary states which can be determined in the indicated manner for the sodium atom. The states are denoted by black circles, and their distances from the vertical line \(aa\) are proportional to the numerical value of the energy corresponding to them. The arrows in the drawing denote transitions.

between states accompanied by the emission of those lines of the sodium spectrum which are obtained under ordinary excitation conditions.

The method of arranging states in horizontal rows in the drawing fully corresponds to the usual arrangement of the so-called “spectral terms” (Spektralterme) in spectroscopic tables. The states in the first horizontal row, denoted by the letter \(S\), correspond to the variable term in the expression of the so-called “sharp subordinate series” (scharfe Nebenserie), emitted in transitions from the named states to the first state in the second row. The states denoted by \(P\) correspond to the variable term of the so-called “principal series” (Prinzipalserie), associated with transitions from the states \(R\) to the first state of the row \(S\). The states \(D\) correspond to the variable term of the “diffuse subordinate series” (diffuse Nebenserie), emitted, similarly to the sharp subordinate series, in transitions to the first state of the second row. Finally, the states \(B\) correspond to the variable term of the so-called Bergmann series, associated with transitions to the first state of the third row. The way in which the various rows are mutually arranged will serve as an illustration of the more detailed theory with which we shall deal later. The already mentioned apparent arbitrariness connected with the application of the combination principle in the present case consists in the fact that, under ordinary excitation conditions, not all lines arise that correspond to all possible combinations of the terms of the sodium spectrum series, but only those which are indicated by arrows in the drawing.

The general question of determining the stationary states of an atom with several electrons is connected with considerable difficulties and, apparently, is still far from complete resolution. It is possible, however, to draw direct conclusions about the stationary states of such atoms that give rise to the emission of serial spectra, if one takes into account the empirical regularities of the spectral terms. According to Rydberg’s well-known law, the functions \(f_k(n)\) appearing in formula (14) can be written for serial spectra emitted under ordinary excitation conditions in the following form:

\[ f_k(n)=\frac{K}{n^2}\varphi_k(n) \tag{16} \]

where \(\varphi_k(n)\) is a function approaching 1 for large values of \(n\), and \(K\) is the constant in the formula of the hydrogen spectrum (5). Obviously, this result may be interpreted as follows: the atom in the correspo-

existing stationary states is neutral, and one of the electrons revolves about the nucleus in such an orbit whose dimensions are large in comparison with the distances of the other electrons from the nucleus. It is clear that in this case the electric force acting on the outer electron and due to the nucleus and the inner electrons will, in the first approximation, be the same as the force acting on the electron in the hydrogen atom; the approximation will be the better, the larger the dimensions of the orbit.

In view of the limited time, I shall not dwell in detail on how the explanation given for the occurrence of the constant of Rydberg in the so-called “arc spectra” of the serial radiation of the elements under ordinary conditions of excitation is convincingly confirmed by the study of the so-called “spark spectra,” i.e., spectra emitted by the elements under very strong electric discharges and caused not by neutral but by ionized atoms. I shall note, however, for what follows, that not only the basic ideas of the theory, but also the hypothesis that in the stationary states corresponding to spectra one electron revolves in a certain orbit about the others, is very interestingly confirmed by investigations of the selective absorption and excitation of spectral lines by the bombardment of atoms with electrons.

In accordance with our assumption that radiation is emitted in the transition from a stationary state with greater energy to a state with smaller energy, we must imagine that the absorption of radiation by an atom is connected with transitions in the opposite direction. For the possibility of the absorption by an element of light corresponding to one of the lines of the serial spectrum of the element, it is necessary that the atoms be in that one of the two states determining the given line which is connected with the smaller energy. For an element whose atoms in the gaseous phase are not bound into molecules, we must suppose that under ordinary conditions almost all atoms are in that stationary state to which the least possible value of the energy corresponds; we shall call it the normal state. This is fully confirmed for the spectra of the alkali metals; thus, for example, in the absorption spectrum of sodium vapor there are only the lines of the principal series; this series, as we said when describing the diagram, corresponds precisely to transitions to the state with the least energy. Further support for the representation just mentioned of the process of absorption is provided by experiments concerning the so-called resonance radiation. Sodium vapor, as Wood first showed, when illuminated by light of a frequency corresponding to the first member of the principal series (the well-known yellow sodium line), acquires the ability to emit light in which there is only that same line. According to our conceptions, this fact can be explained by the fact that the sodium atom, upon illumination, passes from the normal state into the first state

of the second row. The circumstance that in the experiments the resonance radiation is not polarized to the same degree as the incident light agrees perfectly with our views: the radiation of the illuminated vapor is not resonance radiation in the sense of the usual theory; it depends on a process whose course is not directly connected with the illumination. The phenomenon of resonance radiation of the yellow sodium line, however, is not so simple as we indicated above. As is known, the yellow line, like the other lines of the sodium spectrum, consists of two components situated close to one another. In the usual way of describing spectra by means of spectral terms, this circumstance is taken into account by the fact that the terms corresponding to the variable term of the principal series are not simple, but may be represented by two numbers differing slightly from one another. According to our conceptions of the origin of the sodium spectrum, this means that the stationary states corresponding to the second row of the drawing and denoted by \(P\), in contrast to the states \(S\) of the first row, are not simple; to each point of the row there correspond two states, whose energy values, however, differ so little that on the scale of the drawing they cannot be represented by two separate points. The radiation (and absorption) of each of the components of the yellow line is thus connected with two different processes, as is clearly seen from the later experiments of Wood and Dunoyer. These authors showed that, when sodium vapor is illuminated with the light of one of the two components of the yellow line, the resonance radiation (at least at low pressures), in turn, also gives only one (the same) component, corresponding to the return to the normal state from one or the other of the adjacent states into which the atom passed under illumination. These experiments were later continued by Strutt, and the experiments were also extended to the case of illumination by the second line of the principal series. Strutt found, first of all, that when illuminated by this line the resonance radiation corresponded only in small part to the same frequency of oscillations as the incident light, but for the most part consisted of the light of the yellow line. Such a result, quite unexpected from the point of view of the usual conceptions of resonance (all the more since, as Strutt notes, there is no rational relation between the frequencies of oscillation of the two lines of the principal series), is easily explained on the basis of our view. If the atom is brought into the second state of the second row, then, as is seen from the drawing, besides the return to the normal state two other transitions are possible that are connected with radiation: the transition to the second state of the first row and to the first state of the third row. The experiments compel one to make the plausible assumption that the second of the three named possible transitions, corresponding to the radiation of the infrared line, will be the most probable; with the given arrangement of the experiments this line could not be observed. Later-

we shall not give theoretical grounds speaking in favor of our supposition concerning the probability of emission of the infra-red line. If the atom has passed into the second state of the first row, there remains the only possible transition to the first state of the second row, again accompanied by the emission of an infra-red line. In the transition from the latter state to the normal one, a yellow line is emitted. Strutt further found, in excellent agreement with our scheme, that the yellow resonance light excited in the indicated way consists of both components of the first line of the principal series even in the case when the illumination of the sodium vapor was produced by light corresponding to only one component of the second line of the principal series. Such a result fully coincides with our suppositions, since an atom that, on the way to the normal state, has entered one of the states of the first row (where all states, as already indicated, are simple) has lost any possibility of giving any indication as to from which of the two possible states of the second place of the second row it arrived.

All these consequences may be derived from the scheme laid at the basis of the formal explanation of spectra. On the other hand, the fact of the existence in sodium vapor, besides absorption in the lines of the principal series, of a continuous selective absorption beginning at the series limit and extending farther into the ultra-violet region is a decisive confirmation of a further supposition. This supposition consists in the fact that, in absorption in the lines of the principal series of sodium, we are dealing with transitions in which, in the final state (Endzustand) of the atom, one of the electrons revolves around the nucleus and the remaining electrons in an orbit of ever greater and greater dimensions. We must imagine this absorption as corresponding to transitions from the normal state to such states in which the outer electron is capable of receding from the nucleus to infinity. Such a process is a complete analogy to the photoelectric effect occurring upon illumination of a metallic plate. As is known, we can obtain any velocity of the emitted electron by illuminating the metal with light of the corresponding frequency of oscillation. This frequency, however, cannot exceed a definite limit, depending on the nature of the metal and, according to Einstein’s theory, simply connected with the energy necessary for extracting the electron from the metal.

The general view set forth concerning the origin of emission and absorption spectra is confirmed by extraordinarily interesting experiments on the excitation of spectral lines and ionization by electron impacts. The beginning of decisive successes in this field was laid by the well-known experiments of Franck and Hertz. These investigators obtained the first significant results in experiments with mercury vapor, which possesses special properties that greatly facilitate such experiments. In view of the great importance of the results, the experiments were extended by the named and other physicists to the majority of gases and metals in the vaporous

form. With the aid of our drawing I illustrate the results for sodium vapor. Electrons, on colliding with atoms, as has been found, rebound with unchanged velocity in the case when this velocity corresponds to a smaller kinetic energy than that which is necessary for transferring the atom from its normal state to a neighboring one, i.e. in the case of sodium, from the first state of the first row to the first state of the second row. As soon, however, as the electron acquires a kinetic energy equal in magnitude to the energy difference just indicated, the collision becomes such that the electron loses all its kinetic energy and at the same time the vapor emits light corresponding to the yellow line, just as was to be expected if, upon impact, the atom passes from the normal state into the indicated state. For some time there was uncertainty as to the correctness of such an interpretation, since in experiments with mercury vapor, under the corresponding collisions, ions also arose simultaneously in the vapor. According to our scheme, we may expect the formation of ions only in the case when the kinetic energy is so great that it can transfer atoms from the normal state to the common boundary of the states in the various series. Further experiments, especially those of the American investigators Davis and Goucher, have shown, however, that in reality ions can be formed directly in collisions only in the case when the kinetic energy of the electrons reaches the value indicated above. The ionization observed in the experiments of Franck and Hertz is caused by the indirect influence of the photoelectric effect, excited by the illumination of the metallic parts of the apparatus with the light emitted by mercury atoms on returning to the normal state. In considering such experiments one can hardly escape the impression that we are dealing with a direct and independent proof of the reality of the special stationary states, to the supposition of whose existence the regularities of series spectra have led us. At the same time we obtain a decisive argument in favor of the insufficiency of our ordinary electrodynamical and mechanical conceptions of atomic processes, not only with regard to radiation, but also with regard to such phenomena as collisions between free electrons and atoms.

From all that has been said above it is clear that, on the basis of simple conceptions, we are able to compose a certain picture of the origin of the series spectra of the elements. In attempting, however, to penetrate more deeply into the problem of the detailed structure of these spectra, with the same methods as in the case of the spectrum of hydrogen, we encounter difficulties. For systems which are not purely periodic, it is impossible to obtain sufficient information about their motions in stationary

states solely on the basis of the energy values of these states, since several data (Bestimmungsstücke) are necessary for the characterization of such motions. We shall encounter such difficulties also for the hydrogen atom, if we wish to explain in detail the influence of external force fields on this atom. The basis for further progress in this area is provided by the development of quantum theory. In recent years a method has been developed for determining stationary states not only for simple periodic systems, but also for a certain class of non-periodic, so-called conditionally periodic systems, whose equations of motion can be solved by “separation of variables.” For systems of this type, as is known, generalized coordinates can be chosen in such a way that the description of the motions by the methods of general dynamics is reduced to the consideration of a certain number of generalized “components of motion,” each of which corresponds to the variation of only one coordinate during the motion and is in a certain respect “independent” of the others. The method mentioned for determining stationary states consists in the fact that each of the components of motion is connected with a condition which may be regarded as a direct generalization of condition (1) for Planck’s oscillator; in this way, in the general case, the stationary states are determined by as many integers as there are degrees of freedom in the system. Many physicists have taken part in this development of quantum theory, including Planck himself. I am pleased to mention here Ehrenfest’s works, which concern the limits of applicability of the laws of mechanics to atomic processes and shed new light on the principles underlying the indicated generalization of quantum theory. We owe decisive successes in the application of quantum theory to spectral questions to Sommerfeld and his collaborators. In what follows, however, I shall not dwell on the systematic form in which these authors presented their results. In a paper that recently appeared in the Proceedings of the Copenhagen Academy, I showed that the spectra calculated by means of the indicated methods for determining stationary states and the frequency condition (4) possess the same correspondence with the spectra obtained on the basis of the ordinary theory of radiation from the motion of the system as does the spectrum of hydrogen. On the basis of this general correspondence (allgemeine Korrespondenz), I shall try in the remainder of my lecture to set forth the point of view that is developed in greater detail in the paper mentioned. The theory of series spectra and of the action upon them of external force fields can be presented in such a form that it appears as a natural development of the preceding considerations. It seems to me that this point of view is especially suited to the consideration of future problems of the theory of spectra, for it makes it possible to approach even those problems for which Sommerfeld’s method is inapplicable because of the complexity of atomic motions.

We proceed at once to the study of the effects exerted by the presence of small perturbing forces on the spectrum of simple systems consisting of

of a single electron rotating around the nucleus. As before, for the sake of simplicity, we shall for the time being leave out of consideration the dependence of the electron’s mass on its velocity, required by that modification of the ordinary laws of mechanics which is connected with the theory of relativity. The small change of the motion caused by the variation of mass had, however, an essential significance for the development of Sommerfeld’s theory, which arose in connection with the explanation of the so-called fine structure of the hydrogen lines. This phenomenon consists in the fact that each hydrogen line, observed with instruments of very high resolving power, proves to consist of several components situated close to one another; the reason is that the motion of the hydrogen atom, if the change of mass is taken into account, differs somewhat from Keplerian motion, and the path of the electron is not exactly periodic. This deviation from Keplerian motion, however, is very small in comparison with those perturbations of the motion of the hydrogen atom which are caused by external forces in experiments on the Stark and Zeeman phenomena, and also in comparison with the perturbing influence of the presence of inner electrons on the motion of the outer one in the case of other elements. Thus, by neglecting the change of mass, we shall not introduce any essential change into the explanation of the influence of external forces, nor into the interpretation of that difference which exists between the spectrum of hydrogen and the spectra of other elements, of which we spoke above.

We shall therefore, as before, regard the unperturbed motion of the hydrogen atom as purely periodic, and first of all consider the question of the stationary states corresponding to this motion. The energy of these states is determined by expression (7), derived from the hydrogen spectrum. For a given energy of the system, as has already been mentioned, the major axis of the ellipse and the number of revolutions can be determined. Introducing into formulas (7) and (8) the expression for \(K\) from formula (12), we obtain for the energy, the major axis, and the frequency of revolutions in the \(n\)-th stationary state of the unperturbed atom the following expressions:

\[ E_n=-W_n=-\frac{1}{n^2}\frac{2\pi^2 e^4 m}{h^2};\qquad 2a_n=n^2\cdot\frac{h^2}{2\pi^2 e^2 m};\qquad \omega_n=\frac{1}{n^3}\frac{4\pi^2 e^2 m}{h^3} \tag{17} \]

As regards the shape of the orbit, we must further assume that in the stationary states of the system it remains indeterminate, i.e. the eccentricity may assume any continuously varying values. This follows directly from the correspondence principle, for the number of revolutions depends only on the energy, and not on the eccentricity. The same follows from the fact that the presence of arbitrarily small external forces causes, in the general case, a finite change in the position and eccentricity of the orbit, while the major axis undergoes only small changes proportional to the perturbing force.

In asking the question of the nearest determination of the stationary states of the system in the presence of a given constant external force field, we must investigate, by the correspondence principle, how the external forces affect the decomposition of the motion into harmonic oscillations. As has already been said, the influence of the external forces will show itself in the fact that the position and the form of the orbit will change continuously. In the general case the time course of these changes will be so complicated that we shall not be able to decompose the perturbed motion into harmonic components. In this case we must expect that the perturbed system has no sharply separated stationary states. Continuing to suppose that the radiation is always monochromatic and is governed by the frequency rule, we nevertheless cannot in this case expect a spectrum consisting of sharp lines; the external forces will cause a broadening of the spectral lines of the unperturbed system. In some cases, however, the perturbations may be so regular that the perturbed system admits a decomposition into harmonic oscillations, although the totality of these oscillations will, naturally, be of a more complicated type than in the unperturbed system. Such a change will occur, for example, in the case when the change of the orbit in time is periodic. In this case harmonic oscillations will appear in the motion of the system whose periods are multiples of the periods of the perturbations of the orbit, and in the spectrum, which may be expected according to the usual theory of radiation, components of the corresponding frequencies should appear. The correspondence principle therefore leads us directly to the assumption that to every stationary state of the unperturbed system there corresponds a certain number of states in the perturbed system, in such a way that, in the transition between each of two such states, light is emitted whose frequency is likewise connected with the periodic change of the orbit, just as the spectrum of a simple periodic system is connected with the motion in stationary states.

An instructive example of the appearance of perturbations of a periodic character may be furnished by the case of the hydrogen atom under the action of a uniform electric field. Under the influence of the field the position and eccentricity of the orbit change continuously. It turns out, however, that during these changes the center of the orbit remains in the plane perpendicular to the direction of the electric force, and its motion in this plane is purely periodic. When the center returns to its initial position, the orbit also assumes its original position and eccentricity, and from that moment the entire cycle of the orbit will be repeated with respect to its geometrical form and position. In this case the determination of the energy of the stationary states of the perturbed system is extremely simple, since it turns out that the period of the perturbation does not depend on the initial configuration of the orbit or on the position of the plane in which the center moves, but is determined by a large

axis and with the associated number of revolutions of the orbit. A simple calculation shows that the number of periods of the perturbation \(\sigma\) is expressed by the following formula:

\[ \sigma=\frac{3.eF}{8\pi^{2}ma\omega} \tag{18} \]

where \(F\) is the intensity of the external electric field. By analogy with the definition of the special energy values of Planck’s oscillator, one may expect that the energy difference between two different stationary states corresponding to one and the same stationary state of the unperturbed system is simply an integer multiple of the number of periods of the perturbation \(\sigma\), multiplied by \(h\). We thus arrive directly at the following expression for the energy of the stationary states of the perturbed system:

\[ E=E_n+kh\sigma \tag{19} \]

where \(E_n\) depends only on the number \(n\), determining the stationary states of the unperturbed system, and \(k\) is a new integer, which in the present case may be positive or negative. A closer consideration of the relation between the energy and the motion of the system, as we shall see below, leads to the conclusion that \(k\) must be numerically less than \(n\), if, as before, we equate the quantity \(E_n\) to the energy value \(W_n\) of the \(n\)-th stationary state of the unperturbed system. Substituting for \(W_n\), \(\omega_n\), and \(a_n\) the values (17) in formula (19), we find:

\[ E=-\frac{1}{n^2}\frac{2\pi^2 e^4 m}{h^2}+nk\frac{3h^2F}{8\pi^2em} \tag{20} \]

In asking about the influence of the electric field on the lines of the hydrogen spectrum, we obtain, on the basis of the frequency condition (4), for the number of oscillations of the light emitted in the transition from the state determined by the numbers \(n'\) and \(k'\) to the state \(n''\), \(k''\), the following expression:

\[ \nu=\frac{2\pi^2 e^4 m}{h^3}\left(\frac{1}{(n'')^2}-\frac{1}{(n')^2}\right)+\frac{3hF}{8\pi^2em}(n'k'-n''k'') \tag{21} \]

This formula exactly coincides with the formulae derived by Epstein and Schwarzschild, which, as is known, give a satisfactory explanation of the frequencies of the components of the Stark effect for the hydrogen lines. The derivation of these formulae is based on the fact that the hydrogen atom in a homogeneous electric field is a conditionally periodic system, whose equations of motion can be solved by separation of variables in parabolic coordinates; by the method indicated above, the stationary states can therefore be determined.

We shall now consider somewhat more closely the correspondence that exists between the changes of the hydrogen spectrum in an electric field (as observed in the Stark effect) and the decomposition of the perturbed motion of the atom into harmonic components. We find

above all, that instead of the simple expansion into harmonic components corresponding to Keplerian motion, the displacement \(\xi\) of the electron in a given direction in space is expressed in the present case as follows:

\[ \xi=\sum C_{\tau,\chi}\cos 2\pi\left\{t(\tau\omega+\chi\sigma)+c_{\tau,\chi}\right\} \tag{22} \]

where \(\omega\) is the mean number of revolutions in the perturbed orbit, \(\sigma\) is the number of periods of the perturbation of the orbit indicated above, \(C_{\tau,\chi}\) and \(c_{\tau,\chi}\) are constants, and the summation extends over all integral values of \(\tau\) and \(\chi\). Considering a transition between two stationary states characterized by certain numbers \(n'\), \(k'\) and \(n''\), \(k''\), we shall find, for those values of these numbers which are large in comparison with the differences \(n'-n''\) and \(k'-k''\), that the frequency of the emitted line is approximately expressed by the following formula:

\[ \nu \sim (n'-n'')\omega+(k'-k'')\sigma \tag{23} \]

Thus we have a connection between the spectrum and the motion of the atom of the same character as in the simple case, considered earlier, of the unperturbed hydrogen atom. Here there is the same correspondence between the harmonic components of the motion with definite values of \(\tau\) and \(\chi\) in formula (22) and the transition between two stationary states for which \(n'-n''=\tau\) and \(k'-k''=\chi\).

This correspondence will lead us, upon closer examination of the motion, to many interesting consequences. Thus, consideration of the motion shows that every harmonic component of expression (22) for which \(\tau+\chi\) is an even number corresponds to a rectilinear oscillation parallel to the direction of the electric field, while every component for which \(\tau+\chi\) is an odd number corresponds to an elliptical oscillation perpendicular to the electric field. Considering this fact in the light of the correspondence principle, we arrive at the idea of thus explaining the observed, characteristic polarization of the components of the Stark effect: radiation accompanying transitions for which the sum \((n'-n'')+(k'-k'')\) is an even number must give components for which the electric vector oscillates parallel to the electric field; conversely, for transitions with an odd value of \((n'-n'')+(k'-k'')\), there must arise a component with the electric vector oscillating perpendicular to the field. This supposition is fully confirmed by experiment and corresponds to the empirical rule of polarization established by Epstein in his first paper on the Stark effect. The applications of the correspondence principle of which we have so far spoken, relating to the question of the possibility of different types of transitions and to the polarization of the light emitted in such transitions, have been purely qualitative in character. It is possible, however, on the basis of this principle, by comparing the relative values of the amplitudes of the corresponding harmonic components of the motion, to give a quali-

quantitative estimate of the relative probability of the various possible transitions. This consideration was very instructively justified in the case of the Stark effect in hydrogen. By examining the numerical values of the coefficients \(C_{\tau k}\) in formula (22), one can fully elucidate the peculiar and seemingly arbitrary distribution of intensities among the various components into which each hydrogen line is split in an electric field. This question has been treated in detail by Kramers in his recently published dissertation, which contains a detailed discussion of the application of the correspondence principle to the question of the intensities of spectral lines.

Turning to the question of the influence of a uniform magnetic field on the lines of hydrogen, we may proceed in an entirely analogous way. As is known, the action of such a field on the motion of the hydrogen atom consists simply in the superposition of a uniform rotation upon the motion of the electron in the unperturbed atom. The axis of rotation is then parallel to the direction of the magnetic force, and the number of rotations is expressed by the formula:

\[ \sigma = \frac{eH}{4\pi mc} \tag{24} \]

where \(H\) is the field intensity, and \(c\) the velocity of light. We therefore have again the case in which the perturbations are of a purely periodic character and in which the number of periods of the perturbations does not depend on the form or position of the orbit, and in the present case does not even depend on its major axis. We may thus apply the same considerations as for the Stark effect, and are entitled to expect that the energy of the stationary states will again be expressed by formula (19), where for \(\sigma\) one must substitute the value (24). This result is in complete agreement with the expressions derived by Sommerfeld and Debye for the energy values of the stationary states of the hydrogen atom in a magnetic field. The derivation of these expressions is based on the fact that the equations of motion of the atom in a magnetic field can be solved by separation of variables if spatial polar coordinates are introduced with an axis parallel to the direction of the field. If, however, using directly the frequency condition (4), one attempts to calculate the influence of the field on the hydrogen lines from the energy values in the stationary states, one encounters apparent discrepancies which for some time constituted a serious difficulty for the theory. Sommerfeld and Debye pointed out that not every conceivable transition between two stationary states corresponds to a line observed in the Zeeman effect; in contrast to the state of affairs in the Stark effect, in the present case the theory would imply a considerably larger number of components than is observed experimentally. This difficulty, however, disappears as soon as we invoke the correspondence principle. Examining the decomposition of the motion into harmonic components, we find a direct explanation both of the impossibility of transitions corresponding to “superfluous” (überzählig)

components, and to the polarization of the observed components. Thus, for example, we simply find that every elliptic-harmonic component, with number of oscillations $\tau\omega$, appearing in the expansion of the unperturbed motion, splits in the presence of a magnetic field, under the influence of the above-mentioned uniform rotation of the orbit, into three harmonic components: one of them—rectilinear, with number of oscillations $\tau\omega$ and with direction parallel to the magnetic field; the other two—circular, with numbers of oscillations $\tau\omega+\sigma$ and $\tau\omega-\sigma$, oscillating in opposite directions in the plane perpendicular to the field. Consequently, the motion represented by formula (22) contains no components for which $\chi$ is greater than unity (in contrast to the Stark effect, where the motion contains components for all values of $\chi$). Comparing this result with formula (23), which expresses the “asymptotic” coincidence of the number of oscillations of the radiation and the number of oscillations of a harmonic component in the case of large values of $n$ and $k$, we come to the conclusion that transitions for which $k$ changes by more than one are impossible in this case. In a similar way, for the Planck oscillator, transitions between two special states for which $n$ in formula (1) differs by more than unity are excluded. Further, we must conclude that the possible transitions fall into two types. For transitions corresponding to the rectilinear components of the oscillation, $k$ in formula (19) does not change; the frequency $\nu_0$ of the original hydrogen line also does not change, and the electric vector oscillates parallel to the field. For the second type of transitions, corresponding to the circular components of the oscillation, $k$ decreases or increases by one, while the oscillation frequencies are respectively equal to $\nu_0+\sigma$ and $\nu_0-\sigma$; when observed parallel to the field, the oscillations will be circularly polarized. These results agree with Lorentz’s well-known theory of the normal Zeeman effect. Moreover, the character of our considerations presents the closest conceivable analogy with the indicated theory, provided only that one takes into account the fundamental difference between the representations of quantum theory and the ordinary theory of radiation.

An example of the application of similar considerations, shedding light on the structure of the spectra of other elements, may be provided by the action of a small perturbing force field, symmetric with respect to the center (nucleus), on the spectrum of hydrogen. In the present example, in contrast to the cases considered earlier, neither the form nor the position of the plane of the orbit changes with time, and the perturbing action of the field is expressed simply in the uniform rotation of the major axis of the orbit. In this case the perturbations are still periodic in character, and we may suppose that to each value of the energy of a stationary state of the unperturbed system there corresponds a series of discrete energy values of the perturbed system, characterizing states determined by integers $k$. In the case under consideration

the number of oscillations of the perturbation $\sigma$, equal to the number of revolutions of the major axis, depends, for a given force law, not only on the major axis of the orbit, but also on its eccentricity. The change of energy in stationary states caused by the presence of perturbing forces is not determined by so simple an expression as the second term of formula (19); the dependence of this change on $k$ will vary for different fields. We shall see, however, that it is possible to characterize the motion in the stationary states of the hydrogen atom in any central perturbing field by means of one and the same condition. To make this clear, we shall pause for a moment over the determination of the character of the motion of the perturbed hydrogen atom.

As we have already said, in the stationary states of the unperturbed hydrogen atom only the major axis of the orbit is completely determined; its eccentricity may take any values. On the other hand, the change of the atom’s energy under the action of an external force field depends on the form and position of the orbit; therefore, naturally, the determination of the atom’s energy in a force field is connected with a more detailed determination of the orbit in the stationary states of the perturbed system. In the cases considered above of changes of the hydrogen spectrum in a homogeneous electric and magnetic field, the energy condition (19) admits a simple geometrical interpretation.

In an electric field, the distance of the nucleus from the plane in which the center of the orbit moves and which determines the change of energy of the system produced by the field in the stationary states is equal to the semi-major axis of the orbit multiplied by $\frac{k}{n}$. In the case of a magnetic field, it can be shown that the quantity determining the change of energy of the system in the presence of the field, i.e. the area of the projection of the orbit onto a plane perpendicular to the magnetic force, in stationary states is equal to the area of a circle with radius equal to the semi-major axis of the orbit, i.e. $\pi \cdot a_n^2$, multiplied by $\frac{k}{n}$. In an analogous way it can be shown that the existence of the correspondence required by the quantum theory between the spectrum and the motion of the atom leads to a simple condition for the case of a hydrogen atom perturbed by a central force: in the stationary states of the perturbed system the minor axis of the rotating orbit is equal to the major axis $2a_n$, multiplied by $\frac{k}{n}$.

This condition was originally derived by Sommerfeld from his general theory of the determination of stationary states of central motion, which is a particularly simple example of the motion of a conditionally periodic system. It is easy to show that the indicated determination of the minor axis is equivalent to the fact that the parameter $2p$ of the elliptical orbit is determined by an expression of the same form as the major axis $2a_n$ in

in the unperturbed atom with \(n\) replaced by \(k\). Thus the parameter of the stationary states of the perturbed atom will have the value:

\[ 2p-k^{2}\frac{h^{2}}{2\pi^{2}e^{2}m} \tag{25} \]

With such a definition of the stationary states, we obtain for the frequency of the light emitted in transitions between states with \(n\) and \(k\) large in comparison with their difference an expression coinciding with (22), where \(\omega\) is the number of revolutions of the electron in the slowly rotating orbit, and \(\sigma\) is the number of revolutions of the major axis of the orbit.

Before going further, it is interesting to note that the above-stated definition of motion in the stationary states of the hydrogen atom in perturbing external force fields does not coincide in certain respects with the definitions of the same states in the theories of Sommerfeld, Epstein, and Debye. According to the essence of the theory of conditionally periodic systems, in the works of these authors the stationary states of a system with three degrees of freedom are determined by three conditions and, consequently, are characterized by three integers. In terms of the preceding exposition, this is equivalent to saying that the stationary states of the perturbed system, associated with one definite stationary state of the unperturbed atom, which in our case is specified by a single condition, are subject to two additional conditions and are characterized, besides \(n\), by two further new integers. Since the perturbations of the Keplerian motion in the cases considered are purely periodic, the energy of the perturbed system is determined by only one condition; the introduction of any additional condition, according to what was set forth above, contributes nothing new to the understanding of the phenomena. The appearance of new perturbing forces, even small ones, for a substantial change in the character of the Stark and Zeeman phenomena, may nevertheless completely change the forms of motion characterized by the indicated single condition. Here we have a complete analogy with the fact that the spectrum of hydrogen (as is usually observed) is not noticeably changed under the influence of small forces, even though the latter may still be sufficiently significant to cause large changes in the form and position of the electronic orbit.

For a proper illumination of the preceding considerations, as well as of those problems which we shall encounter, for the spectra of elements with high atomic numbers, it is not useless to say a few words about the influence of the change of the electron’s mass as a function of velocity on these problems. In accordance with what has already been said about the fine structure of the hydrogen lines, the preceding considerations have force only in the case when the influence of external forces is large in comparison with those deviations from purely Keplerian motion which are caused

dependence of the electron mass on velocity. If this dependence is taken into account, then the motion of the unperturbed atom ceases to be strictly periodic, and we obtain a motion of exactly the same character as in the case considered of a hydrogen atom perturbed by a small central field. By the correspondence principle, we must expect a close connection between the number of revolutions of the major axis of the orbit and the frequency differences of the fine-structure components; likewise, as in the case considered above, those orbits whose parameters are given by equation (25) correspond to stationary states. In considering the question of the influence of external fields on the fine structure of hydrogen lines, we must not forget that the definition given above applies only to the stationary states of an unperturbed atom; the orbits in these states, as we have already said, are strongly changed even in the presence of external forces small in magnitude compared with those with which one has to operate in experiments on the Stark and Zeeman phenomena. Owing to the complexity of the perturbations produced, the presence of such fields will lead to the atom no longer possessing a set of sharply defined stationary states, and in this connection the fine-structure components of a given hydrogen line will become diffuse and merge together. There are, however, several important cases in which the perturbations have a simple character. The simplest conceivable example is the case of perturbation of the hydrogen atom by a central force (the center being the atomic nucleus). It is clear that the motion of the system will retain its centrally symmetric character, and the difference between the perturbed motion and the unperturbed motion will reduce only to the fact that the number of revolutions of the major axis (for given values of the axis and the parameter) will become different. This point is essential for the theory of spectra of elements with high atomic number developed below, since the effect of the variability of the electron mass is of the same type as the change arising from the presence of forces caused by inner electrons; therefore, in the case of such spectra the effect of the variability of the electron mass will show itself differently than in the case of hydrogen lines. Taking into account the variability of the electron mass, we shall not be able to expect splitting into separate components; there will appear only a small displacement of the various lines of the series.

A simple example of the case in which the hydrogen atom will possess sharp stationary states even when the variability of the electron mass is taken into account is furnished by an atom in a homogeneous magnetic field. The action of such a field will consist in the fact that upon the motion of the unperturbed atom there will be superposed a rotation of the whole system about an axis passing through the nucleus parallel to the field. By the correspondence principle it follows immediately from this that each component of the fine structure must be split into a normal Lorentz triplet. This same problem can, however, also be solved by means of the theory of conditionally periodic systems, since the equations of motion in a magnetic field do—

allow separation of variables in spatial polar coordinates even in the case when the variability of the mass is taken into account, as has already been noted by Sommerfeld and Debye. A more complicated problem is presented by an atom in a homogeneous electric field not so insignificant that the change in mass can be neglected. In this case there exists no system of coordinates in which it would be possible to solve the equations of motion by separation of variables, and therefore the problem cannot be solved with the aid of the theory of stationary states of conditionally periodic systems. A closer investigation of the perturbations shows, however, that their character is such that in the present case too the motion of the electron can be decomposed into a series of discrete harmonic components of oscillation, falling into two groups with oscillations parallel and perpendicular to the field. According to the correspondence principle, we may expect in the present case as well a splitting of each hydrogen line into a certain number of sharp, polarized components. Using the same principle, we are able to determine unambiguously the stationary states of the system with the indicated motion. The problem of the action of an electric field on the components of the fine structure of hydrogen lines will be examined from this point of view in Kramers’s paper, which is to appear soon. It will show how it is possible to present in full detail the character of the transition of the fine structure of hydrogen lines into the ordinary Stark effect as the intensity of the electric force increases.

Let us now return again to the problem of the series spectra of elements with high atomic numbers. As we have already said, the generality of the Rydberg constant for these spectra leads to the conclusion that the atom in the stationary states under consideration is neutral and that there is one electron moving around the nucleus and the other electrons in an orbit whose dimensions are large in comparison with the distance of the inner electrons from the nucleus. The motion of the outer electron may therefore in certain respects be compared with the motion of the electron of a hydrogen atom perturbed by external forces; from this point of view, the appearance of different series in the spectra of other elements may be regarded as analogous to the splitting of hydrogen lines into components in the presence of external forces. In his theory of the structure of series spectra of the alkali-metal type, Sommerfeld made the assumption that the orbit of the outer electron lies in a certain plane and that the perturbing action of the inner electrons in the atom on the motion of the outer electron is, in first approximation, of the same character as a simple perturbing central field rapidly decreasing with distance from the nucleus. Proceeding from this assumption, Sommerfeld determined the motion of the outer electron in the stationary states of the atom, using his general theory of find-

of stationary states of central motion, based on the separability of the variables in the equations of motion. Sommerfeld showed how, in this way, it is possible to calculate the set of energy values of the stationary states of an atom, arranged, like the empirical spectral terms, in series, as is seen in our schematic drawing relating to the case of the sodium spectrum. The states assigned by Sommerfeld to separate series coincide with those which, in our analysis of the case of the hydrogen atom perturbed by a central field, we denoted by the same values of \(k\); the states of the first row of the diagram, \(S\), correspond to \(k=1\), those of the second, \(P\), to \(k=2\), and so on, as indicated in the diagram. States with one and the same value of \(n\) are joined by dotted curves, whose vertical asymptotes correspond to the stationary states of the hydrogen atom. The circumstance that, for constant \(n\) and increasing \(k\), the energy values become ever closer to the corresponding values of the unperturbed hydrogen atom follows directly from the theory. For large values of the orbital parameter, the outer electron remains throughout its revolution at a great distance from the inner system, and therefore the number of periods of rotation of the major axis of the almost Keplerian orbit of the outer electron decreases with increasing \(k\); at the same time there also decreases the influence of the inner system on the energy that must be imparted to the atom in order to remove the outer electron to infinity.

These admirable successes of the theory serve as an incentive to seek such force laws for the perturbing central field as would correspond to the observed spectra of the elements. Sommerfeld was in fact able, on the basis of simple assumptions about these laws, to calculate formulas for spectral terms which, for a given \(k\), vary with \(n\) as required by the empirical formulas discovered by Rydberg; with the aid of Sommerfeld’s assumptions, however, it proved impossible to explain those changes of the energy values with changes in \(k\) and \(n\) which are observed in the spectra. It is, of course, clear that one could not expect detailed agreement from such a simplified accounting of the action of the inner electrons on the spectrum. A more detailed analysis shows, for example, that in taking into account the influence of the inner electrons on the orbit of the outer one, it is necessary to consider not only the forces determined by the configuration of the inner system in the absence of the outer electron, but also to take into account the influence of the outer electron on the motion of the inner ones.

Before turning to questions connected with the attempt to explain the series spectra of elements with small atomic number, we shall show how Sommerfeld’s assumption concerning the character of the orbit of the outer electron is convincingly confirmed by considerations based on the correspondence principle with respect to the appear-

—or absence of lines possible according to the combination principle. For this purpose we shall take up the problem of resolving the motion of the outer electron into harmonic components; the problem is solved simply if one assumes that the presence of the inner electrons causes only a uniform rotation of the orbit of the outer electron in its plane. The presence of this rotation, with number of revolutions \(\sigma\), will show itself in the fact that, instead of some harmonic, elliptic component with frequency \(\tau\omega\), present in the motion of the unperturbed atom, there will appear two circular rotations with frequencies \(\tau\omega+\sigma\) and \(\tau\omega-\sigma\). The resolution of the perturbed motion into harmonic components will therefore again be represented by a formula of type (22), in which, however, there will be only terms for which \(\chi=\pm 1\). Further, the oscillation frequency of the radiation for large values of \(n\) and \(k\) will still be represented by the asymptotic formula (23); in accordance with the correspondence principle, we may therefore expect that only transitions between such two stationary states are possible for which \(k\) differs by unity. A glance at our diagram shows that these conclusions, in the case of sodium, agree with experiment: all the existing spectral series correspond exclusively to transitions between states plotted on two neighboring rows. This agreement is all the more remarkable because the distribution of the energy values of the stationary states among the rows has been made according to Sommerfeld’s theory, quite independently of the possibility of transitions between these states.

In addition to the conclusion concerning the possibility of only certain types of transitions, we may in the present case also expect, on the basis of the correspondence principle, circular polarization of the light emitted by the perturbed atom. In contrast to the Zeeman effect, the polarization in this case cannot be observed directly, owing to the indeterminacy of the planes of the orbits. This conclusion concerning polarization is, however, of fundamental interest in connection with the theory of the process of radiation. From the point of view of the general correspondence between the spectrum of an atom and the resolution of its motion into harmonic components, it is interesting to compare the light emitted in transitions between two stationary states with the light emitted by a harmonically oscillating electron according to the laws of classical electrodynamics. Considering, in particular, the radiation of an electron rotating in a circle, we shall find that it is associated with a definite angular momentum.

The energy \(\Delta E\) and the momentum \(\Delta P\) of the radiation over some interval of time are connected by the relation:

\[ \Delta E = 2\pi\omega\,\Delta P \tag{26} \]

where \(\omega\) is the number of revolutions of the electron, coinciding, according to classical electrodynamics, with the frequency of the radiation \(\nu\). Assuming that all the radi—

energy is equal to \(h\nu\), we find for the total momentum of the radiation the following expression

\[ \Delta P=\frac{h}{2\pi} \tag{27} \]

It is extremely interesting that this expression is in fact equal to the change in angular momentum experienced by the atom in transitions connected with a change of \(k\) by one unit. Indeed, the general condition for determining the stationary states of a central system in Sommerfeld’s theory, which in the particular case of approximately Keplerian motion coincides with formula (25), is equivalent to saying that the angular momentum of the system is an integral multiple of the quantity \(\frac{h}{2\pi}\), in our notation:

\[ P=k\cdot\frac{h}{2\pi} \tag{28} \]

We thus see that this condition is confirmed by simple considerations concerning the conservation of angular momentum during the process of radiation. As has already been said, from the point of view of the theory of stationary states of conditionally periodic systems, condition (28) is a rational generalization of Planck’s original postulate concerning the special states of the harmonic oscillator. In this connection it is interesting to recall that Nicholson was the first to point out the possible significance of angular momentum in applications of quantum theory to the atom, on the grounds that in circular motion the angular momentum is simply proportional to the ratio of kinetic energy to the frequency of revolution.

As the speaker showed in the already mentioned paper, published in the Proceedings of the Copenhagen Academy, similar considerations give an interesting confirmation of the consequences of the correspondence principle for the case of an atomic system with radial or axial symmetry. On the other hand, independently of us, Rubinowicz pointed out those consequences which can be derived from changes in angular momentum during radiation with regard to the possibility of various types of transitions and the polarization of radiation; Rubinowicz thus obtained many of the results indicated above. However, with respect to systems with radial and axial symmetry, more detailed conclusions can be obtained only by the simultaneous use of the correspondence principle and the principle of conservation of angular momentum during the process of radiation. Thus, for example, in the case of the hydrogen atom, perturbed by a central force, on the basis of the principle of conservation of angular momentum one can draw only the conclusion that in any transition \(k\) cannot change by more than one unit, whereas the correspondence principle definitely requires that in every possible transition \(k\) change by 1, and thus, for example, the case of an un-

of \(k\). In addition, the correspondence principle provides not only a means for excluding certain transitions as impossible (in this sense the correspondence principle is a “selection principle”), but also makes it possible to judge the relative probability of different types of transitions on the basis of studying the magnitudes of the amplitudes of the harmonic oscillations into which the motion may be resolved. In our case, for example, the circumstance that the amplitudes of the circular components with the direction of rotation coinciding with the rotation of the electron about the nucleus are greater than the amplitudes of the components with the opposite rotation leads to the conclusion that the lines corresponding to transitions in which \(k\) decreases by 1 must in general be brighter than those lines which are emitted in transitions with \(k\) increasing by 1. This conclusion, however, has force only for those spectral lines which are emitted in transitions from one and the same stationary state. In estimating the relative intensity of two spectral lines in other cases it is, of course, necessary to take into account the relative number of atoms which, under the given conditions of excitation, are in the initial states corresponding to the two transitions. Although the intensity, naturally, does not depend on the number of atoms in the final states, nevertheless, in estimating the probability of transition between two stationary states one must consider the character of the motion both in the initial and in the final orbit; the probability of transition is determined by the values of the amplitude of the corresponding component of the oscillation in both states. As an example, let us return for a moment to the problem already touched upon in connection with Strutt’s experiments on the resonance radiation of sodium vapor, namely to the question of the relative probability of different transitions of an atom from the second state of the second row of our diagram. Such transitions are possible to the first state of the first row, to the second state of the same row, and, finally, to the first state of the third row. As we have already said, the probability of the second of the named transitions is, according to experimental data, the greatest. We can now add that the correspondence principle apparently makes possible a theoretical interpretation of this result. The three indicated transitions must correspond to those harmonic components of the motion whose frequencies will be, in our notation, \(2\omega+\sigma\), \(\omega+\sigma\), and \(\sigma\). It is easy to see that only for the second transition is there a corresponding component of the oscillation in both the initial and the final state with an amplitude different from zero.

As we have already seen, the correspondence between the spectrum of an element and the motion of the atom explains the observed restrictions in the applications of the combination principle to predicting the presence of particular spectral lines. On the basis of the same correspondence, one can directly explain the fact discovered in recent years by Stark and his collaborators, consisting in the appearance in the spectra of many

elements of new series of combination lines of considerable intensity, not previously observed; these lines appear in the case when the luminous atoms are subjected to the action of strong electric fields. This phenomenon is quite analogous to the presence of the so-called combination tones in acoustics, and arises because the perturbation of the motion by the external field manifests itself not only in its action on the components of the oscillation that existed even without the field, but also in the appearance of new harmonic components that were absent in the unperturbed motion; the oscillation frequencies of these new components are determined by the expression \(\tau\omega+\varkappa\sigma\), where \(\varkappa\) is different from \(\pm 1\). By the correspondence principle we are therefore entitled to expect that the influence of the electric field will be expressed not only as an influence on lines existing under ordinary conditions, but also in the fact that the atoms will acquire the possibility of transitions of a new type, in which \(k\) either does not change at all or changes by an integer greater than unity; as a result, “new,” previously observed combination lines should appear. An estimate of the amplitudes of the oscillation components in the initial and final states of the atom corresponding to the new lines even makes it possible to take into account the degree of ease with which the indicated lines can be excited by electric fields.

The general question of the influence of an electric field on the spectra of elements with high atomic number is a problem substantially different from the Stark effect in hydrogen lines discussed above. In the present case we are dealing not with the perturbation of the motion of a purely periodic system, but with the perturbation of a periodic system already otherwise subjected to an extraneous action. The problem of a perturbation of this kind is similar to the problem of the influence of weak electric fields on the components of the fine structure of hydrogen lines. The action of an electric field on the serial spectra of elements may be determined, as follows from what has been said above, by studying the perturbations of the motion of the outer electron in the presence of the field. In the continuation of my work mentioned above, which is to appear shortly in the Proceedings of the Copenhagen Academy, it will be shown that in this way one can apparently obtain an interpretation of the interesting and substantial observations of Stark and others.

From the preceding it is clear how one may obtain a general conception of the origin of serial spectra of the type of the sodium spectrum. The difficulties that arise in attempting a detailed explanation of the spectrum of any element arise with full force already in the question of the spectrum of helium, the element following hydrogen and possessing, in the neutral state, only two electrons. As is known, the helium spectrum is simple, i.e. consists of simple lines, or of double lines with a very small separation between the components.

It was found, however, that the lines split into two groups, each of which is described by a formula of type (14); these groups are usually denoted as the spectrum of (ortho-) helium and that of para-helium; the latter spectrum consists of simple lines, the former of the above-mentioned doublets. This fact—that helium, in contrast to the alkali metals, has two complete serial spectra of the Rydberg type, with no mutual combinations—seemed so unexpected that for a time people were inclined to regard helium as a mixture of two elements. Such a conclusion is at present impossible, since in the corresponding region of the periodic system of the elements there is no place for a new element, or, more precisely, no place for an element with a new spectrum. The explanation for the presence of two spectra can, however, be reduced to the fact that, in the stationary states corresponding to serial spectra, we are dealing in the present case with a system possessing only one inner electron; consequently the motion of the inner system, in the absence of the outer electron, will be purely periodic and can therefore easily be perturbed by external forces.

To explain this point, we must pause briefly over the question of stationary states that are important for the occurrence of serial spectra. As has already been said, we must suppose that in these states one electron moves along a certain orbit far from the nucleus and the other electrons. One may think that, in the general case, several different groups of such states are possible, each of which corresponds to a different stationary state of the inner system considered in isolation. Closer examination shows that, under ordinary conditions of excitation, the greatest probability belongs to the group in which the motion of the inner electron corresponds to the “normal” state of the inner system, possessing 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 in comparison with the energy required to transfer the outer electron from the normal state of the neutral atom to a stationary orbit of large dimensions. Further, only in the normal state does the inner system, in general, possess a long duration of existence and therefore can withstand, without being destroyed, the transitions of the outer electron and the radiation connected with this process. The configuration of the atomic system in its stationary states, in particular in the normal state, is in general quite definite; therefore we may suppose that the inner system undergoes only small changes with the passage of time, caused by the presence of the outer electron. We may therefore suppose that the influence of the inner system on the motion of the outer electron is, in general, of the same character as perturbations

of the motion of the electron in the hydrogen atom, caused by a constant external field, and consequently one must expect the appearance of a spectrum corresponding to the totality of spectral terms which in general form a connected group, although even in the absence of external forces not every combination of two terms of this totality determines the appearance of a spectral line. In the question of the helium spectrum the situation is different, since the inner system, as we have already said, contains only one electron, whose motion, in the absence of the outer electron, is purely periodic, if only one neglects those small changes in the Keplerian motion which are caused by the change of the electron’s mass as a function of its velocity. The form of the orbit in the stationary states of the inner system is therefore in itself indeterminate, or, more precisely, the stability of the orbit is so small (even allowing for the variability of the mass) that already small external forces are capable of changing the eccentricity of the orbit by a finite amount in the course of time. Thus in the case of the helium atom there opens the possibility of the existence of several groups of stationary states, for which the energy of the inner system is approximately the same, although the form and position of the orbit of the inner electron are substantially different; by the same token the possibility of transitions between states within different groups is excluded even in the absence of external forces, in agreement with observations on the helium spectrum.

These considerations lead directly to the question of the type of perturbations of the orbit of the inner electron of the helium atom that arise owing to the presence of the outer electron. Landé has recently, on this basis, considered the question of the helium spectrum. Although the results of this work are in many respects very interesting (especially in the sense of establishing a strong reciprocal action upon the motion of the outer electron by virtue of those perturbations which are caused by its presence in the inner system), nevertheless these results cannot give a satisfactory explanation of the helium spectrum. Independently of those serious objections which may be made with regard to the calculations of the perturbations of the orbit, Landé’s considerations make it difficult to understand the occurrence of two separate spectra having no mutual combinations from the point of view of the correspondence principle. To clarify the state of affairs it is apparently necessary to make a thorough investigation of the mutual perturbations of the outer and inner orbits of the helium atom. These perturbations extremely complicate the character of the motion of both electrons of the helium atom, and the stationary states cannot be determined by the methods developed for conditionally periodic systems. The speaker has been engaged in the study of this question jointly with Dr. Kramers in recent years. In a report on the atomic problem at the congress of Dutch natural scientists and physicians in Leiden, in April 1919, I briefly communicated the results of our calculations of the perturbation problem, as well as of the investigation

certain special classes of motions that we have encountered in our work and that may be important for explaining the helium spectrum. External circumstances have so far prevented us from publishing our calculations; we hope to report them in the near future in connection with the problem of the helium spectrum.

The problem of the spectra of elements with higher atomic number is therefore simpler, since in this case the internal system in its normal state is more definite than in the helium atom. On the other hand, of course, the difficulties in the mechanical problem increase with the number of particles in the atom. An example of this may be furnished by the spectrum of lithium, whose atom in the neutral state contains three electrons. The deviations of the spectral terms of lithium from the corresponding terms of the hydrogen spectrum are very small for the running term of the principal series \((k = 2)\) and of the diffuse subordinate series \((k = 3)\), and very considerable for the running term of the sharp subordinate series \((k = 1)\). This result is quite different from what might have been expected if the action of the inner electrons could be replaced by a central force in simple dependence on the distance. The reason must be ascribed to the fact that, in the orbits of the stationary states of the sharp subordinate series, the parameter determined in first approximation by formula (25) only slightly exceeds the linear dimensions of the orbits of the inner electrons. In calculating the number of revolutions of the major axis of the orbit of the outer electron in stationary states—the number which, according to the correspondence principle, determines the deviations of the spectral terms from the corresponding terms of hydrogen—it is necessary to take into account in detail the interaction of the three electrons in their motions; such an accounting is especially essential for those intervals of the period of revolution of the outer electron when the latter is close to the other two electrons. Even if the normal state of the inner system in the absence of the outer electron were fully known (this state should be similar to the normal state of the helium atom), an exact solution of the mechanical problem arising here would present unusual difficulties.

In passing to the spectra of elements with still higher atomic number, the mechanical problem whose solution is necessary for describing the motion in stationary states becomes ever more difficult, as is already evident from the unusual complexity of many observed spectra. Even in spectra of the simplest structure, given by the alkali metals, the series lines are not simple but consist of doublets, the distance between whose components increases strongly with atomic number. This circumstance indicates that the motion of the outer electron is not a simple central motion, but has a more intricate character; consequently, we must reckon with a more complex aggregate of stationary.

states. The fact that in the sodium atom we are dealing with pairs of stationary states in which the major axis and the parameter of the outer electron are approximately determined by formulas (17) and (25) is indicated not only by the identical role of both states in the spectrum (as we saw in the analysis of experiments on the resonance radiation of sodium vapor), but is confirmed in a very instructive way in the study of the peculiar action of a magnetic field on doublets. This action consists in the fact that, for small field strengths, each component gives the so-called anomalous Zeeman effect, consisting in a splitting, unlike the Lorentz triplet, into a larger number of sharp components; as the field strength increases, as Paschen and Back first observed, both components of the doublet gradually merge into one simple serial line with the normal Zeeman effect.

This influence of a magnetic field on doublets of alkali spectra is of interest not only in connection with the preceding discussion, as showing the close connection of the components of a doublet and the reality of the simple explanation of the general structure of the spectra of the alkali metals, but it also has another significance: it shows quite definitely (if, of course, one may rely on the correspondence principle) that the action of an external magnetic field on the motion of the electrons of alkali atoms, in contrast to the Zeeman effect in hydrogen lines, cannot be reduced to the simple superposition of a uniform rotation, with the number of revolutions determined by formula (24), upon the possible stationary motion existing outside the field. Such a superposition, according to the correspondence principle, would always lead to the normal Zeeman effect for each component of the doublet. In this connection it must first of all be pointed out that the difference between the simple action of a magnetic field on the components of the fine structure of hydrogen lines, predicted by theory, and the observed action on alkali doublets is not a contradiction. The components of the fine structure do not correspond to the components of a doublet; each component of the fine structure corresponds, according to the theory, to the entire set of components of a doublet, triplet, etc., occupying the place of one spectral line in Rydberg’s scheme; the presence of the Paschen–Back effect in strong fields is therefore an essential support for the theoretical prediction concerning the character of the action of magnetic fields on the components of the fine structure of hydrogen lines. As for the “anomalous” action of weak fields on the components of a doublet, there is apparently no need to seek its cause in the inapplicability of the usual laws of electrodynamics to the motion of the outer electron in stationary states; rather, we are dealing with the action of the magnetic field on that delicate connection between the motions of the inner and outer electrons which apparently determines the appearance of doublets.

It is easy to see that such a representation is not entirely alien to the so-called “coupling theory” (Kopplungstheorie), by means of which Voigt was able, albeit in a purely formal way, to take account of the details of the anomalous Zeeman effect. One may even hope to construct a theory of this effect on the basis of quantum theory, formally similar to Voigt’s theory, despite the fundamental difference in the views of the two theories on the structure of the atom and on the process of radiation; a similar analogy is possessed by the theory, developed above, of the normal Zeeman effect and the original theory of Lorentz, based on classical electrodynamics. Unfortunately, time does not permit me to set out these interesting questions in greater detail, and I shall allow myself only to point to the already mentioned continuation of my article in the Proceedings of the Copenhagen Academy, in which there are discussed both questions of the origin of series spectra and the influence upon them of electric and magnetic fields, and also certain problems connected with the structure of atoms in connection with the study of spectra.

In the preceding exposition I deliberately did not dwell more closely on the question of the structure of atoms and molecules, although this question is naturally most intimately connected with the theory of the origin of spectra, such as that discussed above. The need to use, for this purpose, results obtained on the basis of the study of spectra arises already in connection with the simple theory of the hydrogen spectrum, since it turns out that the magnitude of the major axis of the electron orbit of the hydrogen atom in the normal state \((n=1)\) is of the same order as the values obtained for the sizes of atoms with the aid of the kinetic theory of gases. In his first paper on this subject the speaker already attempted to outline the foundations of a theory of the structure of atoms of the elements and molecules of chemical compounds. This theory was based on a simple generalization of the results for the stationary states of the hydrogen atom, obtained on the basis of the spectrum; its consequences were in many respects confirmed by experiment, especially with regard to the general type of variation of the properties of the elements with increasing atomic number, the best example of which may be the regularity in X-ray spectra discovered by Moseley. I take the opportunity, however, to note that many particular assumptions of this theory must be changed in detail; this is compelled by the modern development of quantum theory, both with regard to the elaboration of methods for determining stationary states and in the sense of the problem of the probability of various transitions between states; the discussion of the limits of applicability of mechanics and the insufficient agreement of theory with experiment also lead to this, as has been thoroughly pointed out from various sides. Thus, for example, it is no longer possible to justify the assumption, introduced for preliminary orientation, that in normal states the electrons move in geometrically especially simple orbits, po-

like “electron rings.” Considerations concerning the stability of atoms and molecules with respect to external influences, as well as the possibility of forming an atom by the successive addition of individual electrons, lead us, first, to the conclusion that the electron configurations under consideration are not only in mechanical equilibrium, but must also possess a certain stability in the sense required by ordinary mechanics; and, second, to the conclusion that the indicated configurations must be such that transitions to them from other stationary states of the atom are possible. Simple configurations such as electron rings do not satisfy these requirements, and we are forced to seek more complicated forms of motion. There is, however, no possibility of entering into a detailed discussion of these still open questions, and I must confine myself to referring once again to my article, which is soon to appear. In conclusion, I should like once more to note that in the present report I have sought only to set forth the starting points that underlie the theory of spectra. In particular, I wished to show that, despite the fundamental difference between these points of view and the usual conceptions of the process of radiation, it is nevertheless possible, on the basis of the general correspondence between spectra and motion in the atom, to make use of such conceptions as a guiding thread in the investigation of spectra.

Translated by S. Vavilov.

1. Lecture delivered on April 27, 1920, at a meeting of the German Physical Society in Berlin (Zeitschrift für Physik, 2, p. 423, 1920). ↩