§ 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.