THE LONDON, EDINBURGH, AND DUBLIN

PHILOSOPHICAL MAGAZINE
AND
JOURNAL OF SCIENCE.

VOL. XXVII—SIXTH SERIES.

JANUARY-JUNE 1914.

LIX. On the Effect of Electric and Magnetic Fields on Spectral Lines.

By N. BOHR,
Dr. phil. Copenhagen[1].

CONTENTS

IN a previous paper[2] the writer has shown that an explanation of some of the laws of line spectra may be obtained by applying Planck’s theory of black radiation to Rutherford’s theory of the structure of atoms. In the present paper these considerations will be further developed, and it will be shown that it seems possible on the theory to account for some of the characteristic features of the recent discovery by Stark[3] of the effect of an electric field on spectral lines, as well as of the effect of a magnetic field first discovered by Zeeman. It will also be shown that the theory seems to offer an explanation of the appearance of ordinary double spectral lines[4].

§ 1. The Emission of a Line Spectrum.

The theory put forward by the writer to explain the emission of a line-spectrum may be summarized as follows:—

The principal assumption of Planck’s theory is that the energy of a system of vibrating electrified particles cannot be transferred into radiation, and vice versa, in the continuous way assumed in the ordinary electrodynamics, but only in finite quanta of the amount

, where

is a universal constant and

the frequency of the radiation[5]. Applying this assumption to the emission of a line-spectrum, and assuming that a certain spectral line of frequency

corresponds to a radiation emitted during the transition of an elementary system from a state in which its energy is

to one in which it is

, we have

According to Balmer, Rydberg, and Ritz the frequency of the lines in the line-spectrum of an element can be expressed by the formula

where

and

are whole numbers and

,

, ... a series of functions of

, which can be expressed by

where

is a universal constant and

a function which for large values of

approaches the value unity. The complete spectrum is obtained by combining the numbers

and

as well as the functions

,

..., in every possible way.

On the above view this can be interpreted by assuming:

(1) That every line in the spectrum corresponds to a radiation emitted by a certain elementary system during its passage between two states in which the energy, omitting an arbitrary constant, is given by

and

respectively;

(2) That the system can pass between any two such states during emission of a homogeneous radiation.

The states in question will be denoted as “stationary states.”

The spectrum of hydrogen observed in ordinary vacuum-tubes[6] is represented by (2) and (3) by putting

Accordingly we shall assume that this spectrum is emitted by a system possessing a series of stationary states in which, corresponding to the

th state, the energy, omitting the arbitrary constant, is given by

According to Rutherford’s theory, the atom of an element consists of a central positive nucleus surrounded by electrons rotating in closed orbits. Concordant evidence, obtained in very different ways, indicates that the number of electrons in the neutral atom is equal to the number of the corresponding element in the periodic table[6].

On this theory the structure of the neutral hydrogen atom is of extreme simplicity; it consists of an electron rotating round a positive nucleus of opposite charge. In such a system we get on the ordinary mechanics the following equations for the frequency of revolution

and the major axis

of the relative orbit of the particles

where

and

are the charges,

and

the masses of the nucleus and the electron respectively, and where

is the amount of energy to be transferred to the system in order to remove the electron to an infinite distance from the nucleus. It may be noticed that the expressions are independent of the degree of eccentricity of the orbits.

In order to obtain a mechanical interpretation of the above-mentioned stationary states, let us now in (6) put

[7]. This gives

According to this view, a line of the hydrogen spectrum is emitted during the passage of the atom between two stationary states corresponding to different values for

. We must assume that the mechanism of emission cannot be described in detail on the basis of the ordinary electrodynamics. However, it is known that it is possible on the latter theory to account satisfactorily for the phenomena of radiation in the region of slow vibrations. If our point of view is sound, we should therefore expect to find in this region some connexion between the present theory and the ordinary ideas of electrodynamics.

From (7) we see that

vanishes for large values of

, and that at the same time the ratio

tends to unity. On the present theory the frequency of the radiation emitted by the transition from the

th to the

th stationary state is equal to

. When

is large, this approaches to

. On the ordinary electrodynamics we should expect the frequency of the radiation to be equal to the frequency of revolution, and consequently it is to be anticipated that for large values of

Introducing the values for

and

given by (5) and (7), we see that

disappears from this equation, and that the condition of identity is

From direct observations we have

. Introducing recent values for

,

, and

[8], we get for the expression on the right side of (9)

. The agreement is inside the limit of experimental errors in the determination of

,

, and

; and we may therefore conclude that the connexion sought between the present considerations and the ordinary electrodynamics actually exists.

From (7) and (9) we get

For

, corresponding to the normal state of the atom, we get

; a value of the same order of magnitude as the values for the diameters of atoms calculated on the kinetic theory of gases. For higher values of

, however,

is great compared with the values of ordinary atomic dimensions. As I pointed out in my former paper, this result may be connected with the non-appearance in vacuum-tubes of hydrogen lines corresponding to high numbers in Balmer’s formula and observed in the spectra of stars. Further, it will appear from the considerations of the next section that the large diameter of the orbits offers an explanation of the surprisingly great magnitude of the Stark effect.

From (10) it appears that the condition (8) holds, not only for large values of

but for all values of

. In addition, for a stationary orbit

is equal to the mean value of the total kinetic energy

of the particles; from (10) we therefore get

In using the expressions (6) we have assumed that the motion of the particles in the stationary states of the system can be determined by help of the ordinary mechanics. On this assumption it can be shown generally that the conditions (8) and (11) are equivalent. Consider a particle moving in a closed orbit in a stationary field. Let

be the frequency of revolution,

the mean value of the kinetic energy during a revolution, and

the mean value of the sum of the kinetic energy and the potential energy of the particle relative to the stationary field. Applying Hamilton’s principle, we get for a small variation of the orbit

If the new orbit is also one of dynamical equilibrium, we get

, where

is the total energy of the system, and it will be seen that the equivalence of (8) and (11) follows immediately from (12).

In these deductions we have made no assumptions about the degree of eccentricity of the orbits. If the orbits are circular (11) is equivalent to the simple condition that the angular momentum of the system in the stationary states is equal to an entire multiple of

[9].

In Planck’s vibrators the particles are held by quasi-elastic forces, and the mean value of the kinetic energy is equal to the mean value of the potential energy due to the displacements. Consequently (11) forms a complete analogy to Planck’s original relation

between the energy

of a monochromatic vibrator and its frequency

. This analogy offers another way of representing the present theory—a way more similar to that used in my former paper[10]. Considering, however, the widely different assumptions underlying the relation (11) and Planck’s relation, it may seem more adequate not to seek the basis of our considerations in the formal analogy in question, but directly in the principal condition (1) and in the laws of the line-spectra.

In dealing with the more complicated structure of the spectra of other elements, we must assume that the atoms of such elements possess several different series of stationary states. This complexity of the system of stationary states, compared with that of the hydrogen atom, might naturally be anticipated from the greater number of electrons in the heavier atoms, which render possible several different types of configurations of the particles.

According to (1), (2), and (3) the energy of the

th state in the

th series is, omitting the arbitrary constant, given by

The present theory is not sufficiently developed to account in detail for the expression (13). However, a simple interpretation may be obtained of the fact that in every series

approaches unity for large values of

.

Suppose that in the stationary states one of the electrons moves at a distance from the nucleus which is large compared with the distance of the other electrons. If the atom is neutral, the outer electron will be subject to very nearly the same forces as the electron in the hydrogen atom. Consequently, the expression (13) may be interpreted as indicating the presence of a number of series of stationary states of the atom in which the configuration of the inner electrons is very nearly the same for all states in one series, while the configuration of the outer electron changes from state to state in the series approximately in the same way as in the hydrogen atom.

It will appear that these considerations offer a possible simple explanation of the appearance of the Rydberg constant in the formula for the spectral series of every element. In this connexion, however, it may be noticed that on this point of view the Rydberg constant is not exactly the same for every element, since the expression (8) for

depends on the mass of the central nucleus. The correction due to the finite value of

is very small for elements of high atomic weight, but is comparatively large for hydrogen. It may therefore not be permissible to calculate the Rydberg constant directly from the hydrogen spectrum. Instead of the value 109675 generally assumed, the theoretical value for a heavy atom is 109735.

§ 2. The Effect of an Electric Field.

As mentioned above, J. Stark has recently discovered that the presence of an external electric held produces a characteristic effect on the line-spectrum of an element. The effect was observed for hydrogen and helium. By spectroscopic observation in a direction perpendicular to the held, each of the lines of the hydrogen spectrum was broken up into five homogeneous components situated very nearly symmetrically with regard to the original line. The three inner components were of feeble intensity and polarized with electric vector perpendicular to the field, while the two outer stronger components were polarized with electric vector parallel to the field. The distance between the components was found to be proportional to the electric force within the limits of experimental errors. With a field of 13,000 volt per cm. the observed difference in the wave-length of the two outer components was

and

for

and

respectively. For both systems of lines emitted by helium, Stark observed an effect on the lines of the Diffuse series which was of the same order of magnitude as that observed for the hydrogen lines, but of a different type. Thus the components were situated unsymmetrically with regard to the original line, and were also not polarized relative to the field. The effect of the field on the lines of the Principal series and the Sharp series was very small and hardly distinguishable.

On the theory of this paper the effect of an external field on the lines of a spectrum may be due to two different causes:—

(1) The field may influence the stationary states of the emitting system, and thereby the energy possessed by the system in these states.

(2) It may influence the mechanism of transition between the stationary states, and thereby the relation between the frequency of the radiation and the amount of energy emitted.

Considering an external electric field we shall not expect an effect of the second kind. Having assumed the atoms to be systems of particles governed by electrostatic forces, we may consider the presence of the field simply as a complication of the original system; but on the interpretation given in the former section of the general principle of Ritz of combination of spectral lines, we may expect that the relation (1) will hold for every system of electrified particles.

It appears that a necessary condition for the correctness of this view is that the frequencies of the components of spectral lines produced by the electric field can be expressed by a formula of the type (2). As we shall see, this seems to be consistent with Stark’s experiments.

Let us first consider the effect of an electric field on the hydrogen spectrum. In order to find the effect of the field on the energy of the atom in the different stationary states, we shall seek for its influence on the relation between the energy and the frequency of the system. In this calculation we shall make use of the ordinary mechanics, from analogy with the considerations of the former section.

For simplicity, let us suppose that the mass of the nucleus is infinitely great in comparison with that of the electron. Consider an electron originally moving in a circular orbit round the nucleus. Through the effect of an external electric field the orbit will be deformed. If the force is not accurately perpendicular to the plane of the orbit, this deformation will in course of time be considerable, oven if the external electric force is very small compared with the attraction between the particles. In this case, the orbit may at every moment be considered as an ellipse with the nucleus in the focus, and the effect of the field will consist in a gradual variation of the direction of the major-axis as well as of the eccentricity. During this variation, the length of the major-axis will approximately remain constant and equal to the diameter of the original circular orbit. A detailed investigation of the motion of the electron may be very complicated; but it can be simply shown that the problem only allows of two stationary orbits of the electron. In these, the eccentricity is equal to 1 and the major-axis parallel to the axis of the external field; the orbits simply consist of a straight line through the nucleus parallel to the axis of the field, one on each side of it. It can also be shown that orbits which are very near to these limiting cases will be very nearly stationary.

Neglecting quantities proportional to the square of the magnitude of the external electric force, we get for the rectilinear orbits in question

where

is the frequency of vibration and

the amplitude of the orbit.

is the external electric force, and the two signs correspond to orbits in which the direction of the major-axis from the nucleus is the same or opposite to that of the electric force respectively. For the total energy of the system we have

where

is an arbitrary constant. The mean value of the kinetic energy of the electron during the vibration is

Leaving aside for a moment the discussion of the possibility of such orbits, let us investigate what series of stationary states maybe expected from the expressions (14) and (15). In order to determine the stationary states we shall, as in the former section, seek a connexion with ordinary electrodynamics in the region of slow vibrations. Proceeding as on [page 4], suppose when

is large

where

and

denote the energy and the frequency in the

th state. By help of (14) and (15) we get

This gives

or

Introducing this in (14), (15), and (16) we get

and

It should be remembered that these deductions hold only for large values of

. For the mechanical interpretation of the calculations we need therefore only assume that the eccentricity is very nearly unity for the large orbits. On the other hand, it appears from (17), (18), and (19) that the principal terms in the expressions for

,

, and

are the same as those deduced in the former section directly from the Balmer formula. If we therefore suppose that these quantities in the presence of an electric field can be expressed by a series of terms involving ascending powers of

, we may regard the above deduction as a determination of the coefficient of the second term in this series, and may expect the validity of the expressions for every value of

. It may be considered, in support of this conclusion, that we obtain the same simple relation (11) between the frequency of revolution and the mean value of the kinetic energy as was found without the field, c. f. [page 5].

In the presence of an electric field we shall therefore assume the existence of two series of stationary states of the hydrogen atom, in which the energy is given by (19). In order to obtain the continuity necessary for a connexion with ordinary electrodynamics, we have assumed that the system can pass only between the different states in each series. On this assumption we get for the frequency of the radiation emitted by a transition between two states corresponding to

and

respectively:

This formula gives for every hydrogen line two components situated symmetrically with regard to the original line. Their difference in frequency is proportional to the electric force and equal to [11]

According to the deduction of (21) we may expect that for high values of

the radiation corresponds to vibrations parallel to the electric force. From analogy with the above considerations and in order to obtain agreement with Stark’s result we shall assume that this polarization holds also for small values of

.

Introducing in (21) the experimental values for

,

, and

, and putting

corresponding to an electric force of 13,000 volt per cm., we obtain for the distance between the components of

and

,

and

cm. respectively. We see that these values are of the same order of magnitude as the distance observed by Stark between the two components polarized parallel to the electric force, viz.

and

cm. The values calculated are somewhat higher than those observed; the difference, however, might possibly be due to the difficulties, mentioned in Stark’s paper, of the determination of the magnitude of the electric force in his preliminary experimental arrangement.

For the ratio between the displacements of

and