The Theory of Spectra and Atomic Constitution: Three Essays

by Niels Bohr

III. DEVELOPMENT OF THE QUANTUM THEORY OF SPECTRA

We see that it is possible by making use of a few simple ideas to obtain a certain insight into the origin of the series spectra. But when we attempt to penetrate more deeply, difficulties arise. In fact, for systems which are not simply periodic it is not possible to obtain sufficient information about the motions of these systems in the stationary states from the numerical values of the energy alone; more determining factors are required for the fixation of the motion. We meet the same difficulties when we try to explain in detail the characteristic effect of external forces upon the spectrum of hydrogen. A foundation for further advances in this field has been made in recent years through a development of the quantum theory, which allows a fixation of the stationary states not only in the case of simple periodic systems, but also for certain classes of non-periodic systems. These are the conditionally periodic systems whose equations of motion can be solved by a "separation of the variables." If generalized coordinates are used the description of the motion of these systems can be reduced to the consideration of a number of generalized "components of motion." Each of these corresponds to the change of only one of the coordinates and may therefore in a certain sense be regarded as "independent." The method for the fixation of the stationary states consists in fixing the motion of each of these components by a condition, which can be considered as a direct generalization of condition (1) for a Planck oscillator, so that the stationary states are in general characterized by as many whole numbers as the number of the degrees of freedom which the system possesses. A considerable number of physicists have taken part in this development of the quantum theory, including Planck himself. I also wish to mention the important contribution made by Ehrenfest to this subject on the limitations of the applicability of the laws of mechanics to atomic processes. The decisive advance in the application of the quantum theory to spectra, however, is due to Sommerfeld and his followers. However, I shall not further discuss the systematic form in which these authors have presented their results. In a paper which appeared some time ago in the Transactions of the Copenhagen Academy, I have shown that the spectra, calculated with the aid of this method for the fixation of the stationary states, exhibit a correspondence with the spectra which should correspond to the motion of the system similar to that which we have already considered in the case of hydrogen. With the aid of this general correspondence I shall try in the remainder of this lecture to show how it is possible to present the theory of series spectra and the effects produced by external fields of force upon these spectra in a form which may be considered as the natural generalization of the foregoing considerations. This form appears to me to be especially suited for future work in the theory of spectra, since it allows of an immediate insight into problems for which the methods mentioned above fail on account of the complexity of the motions in the atom.

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

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

given in (12), we obtain for the energy, major axis and frequency of revolution in the

th state of the unperturbed atom the expressions

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

In order to fix the stationary states of systems in the presence of a given conservative external field of force, we shall have to investigate, on the basis of the principle of correspondence, how these forces affect the decomposition of the motion into harmonic oscillations. Owing to the external forces the form and position of the orbit will vary continuously. In the general case these changes will be so complicated that it will not be possible to decompose the perturbed motion into discrete harmonic oscillations. In such a case we must expect that the perturbed system will not possess any sharply separated stationary states. Although each emission of radiation must be assumed to be monochromatic and to proceed according to the general frequency condition we shall therefore expect the final effect to be a broadening of the sharp spectral lines of the unperturbed system. In certain cases, however, the perturbations will be of such a regular character that the perturbed system can be decomposed into harmonic oscillations, although the ensemble of these oscillations will naturally be of a more complicated kind than in the unperturbed system. This happens, for example, when the variations of the orbit with respect to time are periodic. In this case harmonic oscillations will appear in the motion of the system the frequencies of which are equal to whole multiples of the period of the orbital perturbations, and in the spectrum to be expected on the basis of the ordinary theory of radiation we would expect components corresponding to these frequencies. According to the principle of correspondence we are therefore immediately led to the conclusion, that to each stationary state in the unperturbed system there corresponds a number of stationary states in the perturbed system in such a manner, that for a transition between two of these states a radiation is emitted, whose frequency stands in the same relationship to the periodic course of the variations in the orbit, as the spectrum of a simple periodic system does to its motion in the stationary states.

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

where

is the intensity of the external electric field. From analogy with the fixation of the distinctive energy values of a Planck oscillator we must therefore expect that the energy difference between two different states, corresponding to the same stationary state of the unperturbed system, will simply be equal to a whole multiple of the product of

by the period

of the perturbations. We are therefore immediately led to the following expression for the energy of the stationary states of the perturbed system,

where

depends only upon the number

characterizing the stationary state of the unperturbed system, while

is a new whole number which in this case may be either positive or negative. As we shall see below, consideration of the relation between the energy and the motion of the system shows that

must be numerically less than

, if, as before, we place the quantity

equal to the energy

of the

th stationary state of the undisturbed atom. Substituting the values of

and

given by (17) in formula (19) we get

To find the effect of an electric field upon the lines of the hydrogen spectrum, we use the frequency condition (4) and obtain for the frequency

of the radiation emitted by a transition between two stationary states defined by the numbers

and

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

We shall now consider more closely the correspondence between the changes in the spectrum of hydrogen due to the presence of an electric field and the decomposition of the perturbed motion of the atom into its harmonic components. Instead of the simple decomposition into harmonic components corresponding to a simple Kepler motion, the displacement

of the electron in a given direction in space can be expressed in the present case by the formula

where

is the average frequency of revolution in the perturbed orbit and

is the period of the orbital perturbations, while

and

are constants. The summation is to be extended over all integral values for

and

.

If we now consider a transition between two stationary states characterized by certain numbers

and

, we find that in the region where these numbers are large compared with their differences

and

, the frequency of the spectral line which is emitted will be given approximately by the formula

We see, therefore, that we have obtained a relation between the spectrum and the motion of precisely the same character as in the simple case of the unperturbed hydrogen atom. We have here a similar correspondence between the harmonic component in the motion, corresponding to definite values for

and

in formula (22), and the transition between two stationary states for which

and

.

A number of interesting results can be obtained from this correspondence by considering the motion in more detail. Each harmonic component in expression (22) for which

is an even number corresponds to a linear oscillation parallel to the direction of the electric field, while each component for which

is odd corresponds to an elliptical oscillation perpendicular to this direction. The correspondence principle suggests at once that these facts are connected with the characteristic polarization observed in the Stark effect. We would anticipate that a transition for which

is even would give rise to a component with an electric vector parallel to the field, while a transition for which

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

The applications of the correspondence principle that have so far been described have been purely qualitative in character. It is possible however to obtain a quantitative estimate of the relative intensity of the various components of the Stark effect of hydrogen, by correlating the numerical values of the coefficients

in formula (22) with the probability of the corresponding transitions between the stationary states. This problem has been treated in detail by Kramers in a recently published dissertation. In this he gives a thorough discussion of the application of the correspondence principle to the question of the intensity of spectral lines.

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

where

is the intensity of the field and

the velocity of light.

Again we have a case where the perturbations are simply periodic and where the period of the perturbations is independent of the form and position of the orbit, and in the present case, even of the major axis. Similar considerations apply therefore as in the case of the Stark effect, and we must expect that the energy in the stationary states will again be given by formula (19), if we substitute for

the value given in expression (24). This result is also in complete agreement with that obtained by Sommerfeld and Debye. The method they used involved the solution of the equations of motion by the method of the separation of the variables. The appropriate coordinates are polar ones about an axis parallel to the field.

If we try, however, to calculate directly the effect of the field by means of the frequency condition (4), we immediately meet with an apparent disagreement which for some time was regarded as a grave difficulty for the theory. As both Sommerfeld and Debye have pointed out, lines are not observed corresponding to every transition between the stationary states included in the formula. We overcome this difficulty, however, as soon as we apply the principle of correspondence. If we consider the harmonic components of the motion we obtain a simple explanation both of the non-occurrence of certain transitions and of the observed polarization. In the magnetic field each elliptic harmonic component having the frequency

splits up into three harmonic components owing to the uniform rotation of the orbit. Of these one is rectilinear with frequency

oscillating parallel to the magnetic field, and two are circular with frequencies

and

oscillating in opposite directions in a plane perpendicular to the direction of the field. Consequently the motion represented by formula (22) contains no components for which

is numerically greater than

, in contrast to the Stark effect, where components corresponding to all values of

are present. Now formula (23) again applies for large values of

and

, and shows the asymptotic agreement between the frequency of the radiation and the frequency of a harmonic component in the motion. We arrive, therefore, at the conclusion that transitions for which

changes by more than unity cannot occur. The argument is similar to that by which transitions between two distinctive states of a Planck oscillator for which the values of

in (1) differ by more than unity are excluded. We must further conclude that the various possible transitions consist of two types. For the one type corresponding to the rectilinear component,

remains unchanged, and in the emitted radiation which possesses the same frequency

as the original hydrogen line, the electric vector will oscillate parallel with the field. For the second type, corresponding to the circular components,

will increase or decrease by unity, and the radiation viewed in the direction of the field will be circularly polarized and have frequencies

and

respectively. These results agree with those of the familiar Lorentz theory. The similarity in the two theories is remarkable, when we recall the fundamental difference between the ideas of the quantum theory and the ordinary theories of radiation.

Central perturbations. An illustration based on similar considerations which will throw light upon the spectra of other elements consists in finding the effect of a small perturbing field of force radially symmetrical with respect to the nucleus. In this case neither the form of the orbit nor the position of its plane will change with time, and the perturbing effect of the field will simply consist of a uniform rotation of the major axis of the orbit. The perturbations are periodic, so that we may assume that to each energy value of a stationary state of the unperturbed system there belongs a series of discrete energy values of the perturbed system, characterized by different values of a whole number

. The frequency

of the perturbations is equal to the frequency of rotation of the major axis. For a given law of force for the perturbing field we find that

depends both on the major axis and on the eccentricity. The change in the energy of the stationary states, therefore, will not be given by an expression as simple as the second term in formula (19), but will be a function of

, which is different for different fields. It is possible, however, to characterize by one and the same condition the motion in the stationary states of a hydrogen atom which is perturbed by any central field. In order to show this we must consider more closely the fixation of the motion of a perturbed hydrogen atom.

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

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

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

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

times the major axis. This condition was first derived by Sommerfeld from his general theory for the determination of the stationary states of a central motion. It is easily shown that this fixation of the value of the minor axis is equivalent to the statement that the parameter

of the elliptical orbit is given by an expression of exactly the same form as that which gives the major axis

in the unperturbed atom. The only difference from the expression for

in (17) is that

is replaced by

, so that the value of the parameter in the stationary states of the perturbed atom is given by

The frequency of the radiation emitted by a transition between two stationary states determined in this way for which

and

are large in proportion to their difference is given by an expression which is the same as that in equation (23), if in this case

is the frequency of revolution of the electron in the slowly rotating orbit and

represents the frequency of rotation of the major axis.

Before proceeding further, it might be of interest to note that this fixation of the stationary states of the hydrogen atom perturbed by external electric and magnetic forces does not coincide in certain respects with the theories of Sommerfeld, Epstein and Debye. According to the theory of conditionally periodic systems the stationary states for a system of three degrees of freedom will in general be determined by three conditions, and therefore in these theories each state is characterized by three whole numbers. This would mean that the stationary states of the perturbed hydrogen atom corresponding to a certain stationary state of the unperturbed hydrogen atom, fixed by one condition, should be subject to two further conditions and should therefore be characterized by two new whole numbers in addition to the number

. But the perturbations of the Keplerian motion are simply periodic and the energy of the perturbed atom will therefore be fixed completely by one additional condition. The introduction of a second condition will add nothing further to the explanation of the phenomenon, since with the appearance of new perturbing forces, even if these are too small noticeably to affect the observed Zeeman and Stark effects, the forms of motion characterized by such a condition may be entirely changed. This is completely analogous to the fact that the hydrogen spectrum as it is usually observed is not noticeably affected by small forces, even when they are large enough to produce a great change in the form and position of the orbit of the electron.

Relativity effect on hydrogen lines. Before leaving the hydrogen spectrum I shall consider briefly the effect of the variation of the mass of the electron with its velocity. In the preceding sections I have described how external fields of force split up the hydrogen lines into several components, but it should be noticed that these results are only accurate when the perturbations are large in comparison with the small deviations from a pure Keplerian motion due to the variation of the mass of the electron with its velocity. When the variation of the mass is taken into account the motion of the unperturbed atom will not be exactly periodic. Instead we obtain a motion of precisely the same kind as that occurring in the hydrogen atom perturbed by a small central field. According to the correspondence principle an intimate connection is to be expected between the frequency of revolution of the major axis of the orbit and the difference of the frequencies of the fine structure components, and the stationary states will be those orbits whose parameters are given by expression (25). If we now consider the effect of external forces upon the fine structure components of the hydrogen lines it is necessary to keep in mind that this fixation of the stationary states only applies to the unperturbed hydrogen atom, and that, as mentioned, the orbits in these states are in general already strongly influenced by the presence of external forces, which are small compared with those with which we are concerned in experiments on the Stark and Zeeman effects. In general the presence of such forces will lead to a great complexity of perturbations, and the atom will no longer possess a group of sharply defined stationary states. The fine structure components of a given hydrogen line will therefore become diffuse and merged together. There are, however, several important cases where this does not happen on account of the simple character of the perturbations. The simplest example is a hydrogen atom perturbed by a central force acting from the nucleus. In this case it is evident that the motion of the system will retain its centrally symmetrical character, and that the perturbed motion will differ from the unperturbed motion only in that the frequency of rotation of the major axis will be different for different values of this axis and of the parameter. This point is of importance in the theory of the spectra of elements of higher atomic number, since, as we shall see, the effect of the forces originating from the inner electrons may to a first approximation be compared with that of a perturbing central field. We cannot therefore expect these spectra to exhibit a separate effect due to the variation of the mass of the electron of the same kind as that found in the case of the hydrogen lines. This variation will not give rise to a splitting up into separate components but only to small displacements in the position of the various lines.

We obtain still another simple example in which the hydrogen atom possesses sharp stationary states, although the change of mass of the electron is considered, if we take an atom subject to a homogeneous magnetic field. The effect of such a field will consist in the superposition of a rotation of the entire system about an axis through the nucleus and parallel with the magnetic force. It follows immediately from this result according to the principle of correspondence that each fine structure component must be expected to split up into a normal Zeeman effect (Lorentz triplet). The problem may also be solved by means of the theory of conditionally periodic systems, since the equations of motion in the presence of a magnetic field, even when the change in the mass is considered, will allow of a separation of the variables using polar coordinates in space. This has been pointed out by Sommerfeld and Debye.

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

Theory of series spectra. Let us now turn our attention once more to the problem of the series spectra of elements of higher atomic number. The general appearance of the Rydberg constant in these spectra is to be explained by assuming that the atom is neutral and that one electron revolves in an orbit the dimensions of which are large in comparison with the distance of the inner electrons from the nucleus. In a certain sense, therefore, the motion of the outer electron may be compared with the motion of the electron of the hydrogen atom perturbed by external forces, and the appearance of the various series in the spectra of the other elements is from this point of view to be regarded as analogous to the splitting up of the hydrogen lines into components on account of such forces.

In his theory of the structure of series spectra of the type exhibited by the alkali metals, Sommerfeld has made the assumption that the orbit of the outer electron to a first approximation possesses the same character as that produced by a simple perturbing central field whose intensity diminishes rapidly with increasing distance from the nucleus. He fixed the motion of the external electron by means of his general theory for the fixation of the stationary states of a central motion. The application of this method depends on the possibility of separating the variables in the equations of motion. In this manner Sommerfeld was able to calculate a number of energy values which can be arranged in rows just like the empirical spectral terms shown in the diagram of the sodium spectrum ([p. 30]). The states grouped together by Sommerfeld in the separate rows are exactly those which were characterized by one and the same value of

in our investigation of the hydrogen atom perturbed by a central force. The states in the first row of the figure (row

) correspond to the value

, those of the second row (

) correspond to

, etc. The states corresponding to one and the same value of

are connected by dotted lines which are continued so that their vertical asymptotes correspond to the energy value of the stationary states of the hydrogen atom. The fact that for a constant

and increasing values of

the energy values approach the corresponding values for the unperturbed hydrogen atom is immediately evident from the theory since the outer electron, for large values of the parameter of its orbit, remains at a great distance from the inner system during the whole revolution. The orbit will become almost elliptical and the period of rotation of the major axis will be very large. It can be seen, therefore, that the effect of the inner system on the energy necessary to remove this electron from the atom must become less for increasing values of

.

These beautiful results suggest the possibility of finding laws of force for the perturbing central field which would account for the spectra observed. Although Sommerfeld in this way has in fact succeeded in deriving formulae for the spectral terms which vary with

for a constant

in agreement with Rydberg's formulae, it has not been possible to explain the simultaneous variation with both

and

in any actual case. This is not surprising, since it is to be anticipated that the effect of the inner electrons on the spectrum could not be accounted for in such a simple manner. Further consideration shows that it is necessary to consider not only the forces which originate from the inner electrons but also to consider the effect of the presence of the outer electron upon the motion of the inner electrons.

Before considering the series spectra of elements of low atomic number I shall point out how the occurrence or non-occurrence of certain transitions can be shown by the correspondence principle to furnish convincing evidence in favour of Sommerfeld's assumption about the orbit of the outer electron. For this purpose we must describe the motion of the outer electron in terms of its harmonic components. This is easily performed if we assume that the presence of the inner electrons simply produces a uniform rotation of the orbit of the outer electron in its plane. On account of this rotation, the frequency of which we will denote by

, two circular rotations with the periods

and

will appear in the motion of the perturbed electron, instead of each of the harmonic elliptical components with a period

in the unperturbed motion. The decomposition of the perturbed motion into harmonic components consequently will again be represented by a formula of the type (22), in which only such terms appear for which

is equal to

or

. Since the frequency of the emitted radiation in the regions where

and

are large is again given by the asymptotic formula (23), we at once deduce from the correspondence principle that the only transitions which can take place are those for which the values of

differ by unity. A glance at the figure for the sodium spectrum shows that this agrees exactly with the experimental results. This fact is all the more remarkable, since in Sommerfeld's theory the arrangement of the energy values of the stationary states in rows has no special relation to the possibility of transition between these states.

Correspondence principle and conservation of angular momentum. Besides these results the correspondence principle suggests that the radiation emitted by the perturbed atom must exhibit circular polarization. On account of the indeterminateness of the plane of the orbit, however, this polarization cannot be directly observed. The assumption of such a polarization is a matter of particular interest for the theory of radiation emission. On account of the general correspondence between the spectrum of an atom and the decomposition of its motion into harmonic components, we are led to compare the radiation emitted during the transition between two stationary states with the radiation which would be emitted by a harmonically oscillating electron on the basis of the classical electrodynamics. In particular the radiation emitted according to the classical theory by an electron revolving in a circular orbit possesses an angular momentum and the energy

and the angular momentum

of the radiation emitted during a certain time are connected by the relation

Here

represents the frequency of revolution of the electron, and according to the classical theory this is equal to the frequency

of the radiation. If we now assume that the total energy emitted is equal to

we obtain for the total angular momentum of the radiation

It is extremely interesting to note that this expression is equal to the change in the angular momentum which the atom suffers in a transition where

varies by unity. For in Sommerfeld's theory the general condition for the fixation of the stationary states of a central system, which in the special case of an approximately Keplerian motion is equivalent to the relation (25), asserts that the angular momentum of the system must be equal to a whole multiple of

, a condition which may be written in our notation

We see, therefore, that this condition has obtained direct support from a simple consideration of the conservation of angular momentum during the emission of the radiation. I wish to emphasize that this equation is to be regarded as a rational generalization of Planck's original statement about the distinctive states of a harmonic oscillator. It may be of interest to recall that the possible significance of the angular momentum in applications of the quantum theory to atomic processes was first pointed out by Nicholson on the basis of the fact that for a circular motion the angular momentum is simply proportional to the ratio of the kinetic energy to the frequency of revolution.

In a previous paper which I presented to the Copenhagen Academy I pointed out that these results confirm the conclusions obtained by the application of the correspondence principle to atomic systems possessing radial or axial symmetry. Rubinowicz has independently indicated the conclusions which may be obtained directly from a consideration of conservation of angular momentum during the radiation process. In this way he has obtained several of our results concerning the various types of possible transitions and the polarization of the emitted radiation. Even for systems possessing radial or axial symmetry, however, the conclusions which we can draw by means of the correspondence principle are of a more detailed character than can be obtained solely from a consideration of the conservation of angular momentum. For example, in the case of the hydrogen atom perturbed by a central force we can only conclude that

cannot change by more than unity, while the correspondence principle requires that

shall vary by unity for every possible transition and that its value cannot remain unchanged. Further, this principle enables us not only to exclude certain transitions as being impossible—and can from this point of view be considered as a "selection principle"—but it also enables us to draw conclusions about the relative probabilities of the various possible types of transitions from the values of the amplitudes of the harmonic components. In the present case, for example, the fact that the amplitudes of those circular components which rotate in the same sense as the electron are in general greater than the amplitudes of those which rotate in the opposite sense leads us to expect that lines corresponding to transitions for which

decreases by unity will in general possess greater intensity than lines during the emission of which

increases by unity. Simple considerations like this, however, apply only to spectral lines corresponding to transitions from one and the same stationary state. In other cases when we wish to estimate the relative intensities of two spectral lines it is clearly necessary to take into consideration the relative number of atoms which are present in each of the two stationary states from which the transitions start. While the intensity naturally cannot depend upon the number of atoms in the final state, it is to be noticed, however, that in estimating the probability of a transition between two stationary states it is necessary to consider the character of the motion in the final as well as in the initial state, since the values of the amplitudes of the components of oscillation of both states are to be regarded as decisive for the probability.

To show how this method can be applied I shall return for a moment to the problem which I mentioned in connection with Strutt's experiment on the resonance radiation of sodium vapour. This involved the discussion of the relative probability of the various possible transitions which can start from that state corresponding to the second term in the second row of the figure on [p. 30]. These were transitions to the first and second states in the first row and to the first state in the third row, and the results of experiment indicate, as we saw, that the probability is greatest for the second transitions. These transitions correspond to those harmonic components having frequencies

,

and

, and it is seen that only for the second transition do the amplitudes of the corresponding harmonic component differ from zero in the initial as well as in the final state. [In the next essay the reader will find that the values of quantum number

assigned in [Fig. 1] to the various stationary states must be altered. While this correction in no way influences the other conclusions in this essay it involves that the reasoning in this passage cannot be maintained.]

I have shown how the correspondence between the spectrum of an element and the motion of the atom enables us to understand the limitations in the direct application of the combination principle in the prediction of spectral lines. The same ideas give an immediate explanation of the interesting discovery made in recent years by Stark and his collaborators, that certain new series of combination line appear with considerable intensity when the radiating atoms are subject to a strong external electric field. This phenomenon is entirely analogous to the appearance of the so-called combination tones in acoustics. It is due to the fact that the perturbation of the motion will not only consist in an effect upon the components originally present, but in addition will give rise to new components. The frequencies of these new components may be

, where

is different from

. According to the correspondence principle we must therefore expect that the electric field will not only influence the lines appearing under ordinary circumstances, but that it will also render possible new types of transitions which give rise to the "new" combination lines observed. From an estimate of the amplitudes of the particular components in the initial and final states it has even been found possible to account for the varying facility with which the new lines are brought up by the external field.

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

The spectra of helium and lithium. We see that it has been possible to obtain a certain general insight into the origin of the series spectra of a type like that of sodium. The difficulties encountered in an attempt to give a detailed explanation of the spectrum of a particular element, however, become very serious, even when we consider the spectrum of helium whose neutral atom contains only two electrons. The spectrum of this element has a simple structure in that it consists of single lines or at any rate of double lines whose components are very close together. We find, however, that the lines fall into two groups each of which can be described by a formula of the type (14). These are usually called the (ortho) helium and parhelium spectra. While the latter consists of simple lines, the former possesses narrow doublets. The discovery that helium, as opposed to the alkali metals, possesses two complete spectra of the Rydberg type which do not exhibit any mutual combinations was so surprising that at times there has been a tendency to believe that helium consisted of two elements. This way out of the difficulty is no longer open, since there is no room for another element in this region of the periodic system, or more correctly expressed, for an element possessing a new spectrum. The existence of the two spectra can, however, be traced back to the fact that in the stationary states corresponding to the series spectra we have to do with a system possessing only one inner electron and in consequence the motion of the inner system, in the absence of the outer electron, will be simply periodic and therefore easily perturbed by external forces.

In order to illustrate this point we shall have to consider more carefully the stationary states connected with the origin of a series spectrum. We must assume that in these states one electron revolves in an orbit outside the nucleus and the other electrons. We might now suppose that in general a number of different groups of such states might exist, each group corresponding to a different stationary state of the inner system considered by itself. Further consideration shows, however, that under the usual conditions of excitation those groups have by far the greatest probability for which the motion of the inner electrons corresponds to the "normal" state of the inner system, i.e. to that stationary state having the least energy. Further the energy required to transfer the inner system from its normal state to another stationary state is in general very large compared with the energy which is necessary to transfer an electron from the normal state of the neutral atom to a stationary orbit of greater dimensions. Lastly the inner system is in general capable of a permanent existence only in its normal state. Now, the configuration of an atomic system in its stationary states and also in the normal state will, in general, be completely determined. We may therefore expect that the inner system under the influence of the forces arising from the presence of the outer electron can in the course of time suffer only small changes. For this reason we must assume that the influence of the inner system upon the motion of the external electron will, in general, be of the same character as the perturbations produced by a constant external field upon the motion of the electron in the hydrogen atom. We must therefore expect a spectrum consisting of an ensemble of spectral terms, which in general form a connected group, even though in the absence of external perturbing forces not every combination actually occurs. The case of the helium spectrum, however, is quite different since here the inner system contains only one electron the motion of which in the absence of the external electron is simple periodic provided the small changes due to the variation in the mass of the electron with its velocity are neglected. For this reason the form of the orbit in the stationary states of the inner system considered by itself will not be determined. In other words, the stability of the orbit is so slight, even if the variation in the mass is taken into account, that small external forces are in a position to change the eccentricity in the course of time to a finite extent. In this case, therefore, it is possible to have several groups of stationary states, for which the energy of the inner system is approximately the same while the form of the orbit of the inner electron and its position relative to the motion of the other electrons are so essentially different, that no transitions between the states of different groups can occur even in the presence of external forces. It can be seen that these conclusions summarize the experimental observations on the helium spectra.

These considerations suggest an investigation of the nature of the perturbations in the orbit of the inner electron of the helium atom, due to the presence of the external electron. A discussion of the helium spectrum from this point of view has recently been given by Landé. The results of this work are of great interest particularly in the demonstration of the large back effect on the outer electron due to the perturbations of the inner orbit which themselves arise from the presence of the outer electron. Nevertheless, it can scarcely be regarded as a satisfactory explanation of the helium spectrum. Apart from the serious objections which may be raised against his calculation of the perturbations, difficulties arise if we try to apply the correspondence principle to Landé's results in order to account for the occurrence of two distinct spectra showing no mutual combinations. To explain this fact it seems necessary to base the discussion on a more thorough investigation of the mutual perturbations of the outer and the inner orbits. As a result of these perturbations both electrons move in such an extremely complicated way that the stationary states cannot be fixed by the methods developed for conditionally periodic systems. Dr Kramers and I have in the last few years been engaged in such an investigation, and in an address on atomic problems at the meeting of the Dutch Congress of Natural and Medical Sciences held in Leiden, April 1919, I gave a short communication of our results. For various reasons we have up to the present time been prevented from publishing, but in the very near future we hope to give an account of these results and of the light which they seem to throw upon the helium spectrum.

The problem presented by the spectra of elements of higher atomic number is simpler, since the inner system is better defined in its normal state. On the other hand the difficulty of the mechanical problem of course increases with the number of the particles in the atom. We obtain an example of this in the case of lithium with three electrons. The differences between the spectral terms of the lithium spectrum and the corresponding spectral terms of hydrogen are very small for the variable term of the principal series (

) and for the diffuse series (

), on the other hand it is very considerable for the variable term of the sharp series (

). This is very different from what would be expected if it were possible to describe the effect of the inner electron by a central force varying in a simple manner with the distance. This must be because the parameter of the orbit of the outer electron in the stationary states corresponding to the terms of the sharp series is not much greater than the linear dimensions of the orbits of the inner electrons. According to the principle of correspondence the frequency of rotation of the major axis of the orbit of the outer electron is to be regarded as a measure of the deviation of the spectral terms from the corresponding hydrogen terms. In order to calculate this frequency it appears necessary to consider in detail the mutual effect of all three electrons, at all events for that part of the orbit where the outer electron is very close to the other two electrons. Even if we assumed that we were fully acquainted with the normal state of the inner system in the absence of the outer electron—which would be expected to be similar to the normal state of the neutral helium atom—the exact calculation of this mechanical problem would evidently form an exceedingly difficult task.

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

This effect of a magnetic field upon the doublets of the alkali spectrum is of interest in showing the intimate relation of the components and confirms the reality of the simple explanation of the general structure of the spectra of the alkali metals. If we may again here rely upon the correspondence principle we have unambiguous evidence that the effect of a magnetic field on the motion of the electrons simply consists in the superposition of a uniform rotation with a frequency given by equation (24) as in the case of the hydrogen atom. For if this were the case the correspondence principle would indicate under all conditions a normal Zeeman effect for each component of the doublets. I want to emphasize that the difference between the simple effect of a magnetic field, which the theory predicts for the fine structure of components of the hydrogen lines, and the observed effect on the alkali doublets is in no way to be considered as a contradiction. The fine structure components are not analogous to the individual doublet components, but each single fine structure component corresponds to the ensemble of components (doublet, triplet) which makes up one of the series lines in Rydberg's scheme. The occurrence in strong fields of the effect observed by Paschen and Back must therefore be regarded as a strong support for the theoretical prediction of the effect of a magnetic field on the fine structure components of the hydrogen lines.

It does not appear necessary to assume the "anomalous" effect of small fields on the doublet components to be due to a failure of the ordinary electrodynamical laws for the description of the motion of the outer electron, but rather to be connected with an effect of the magnetic field on that intimate interaction between the motion of the inner and outer electrons which is responsible for the occurrence of the doublets. Such a view is probably not very different from the "coupling theory" by which Voigt was able to account formally for the details of the anomalous Zeeman effect. We might even expect it to be possible to construct a theory of these effects which would exhibit a formal analogy with the Voigt theory similar to that between the quantum theory of the normal Zeeman effect and the theory originally developed by Lorentz. Time unfortunately does not permit me to enter further into this interesting problem, so I must refer you to the continuation of my paper in the Transactions of the Copenhagen Academy, which will contain a general discussion of the origin of series spectra and of the effects of electric and magnetic fields.