§ 1. General assumptions.
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According to the theory proposed by Sir Ernest Rutherford, in order to account for the phenomena of scattering of -rays, the atom consists of a central positively charged nucleus surrounded by a cluster of electrons. The nucleus is the seat of the essential part of the mass of the atom, and has linear dimensions exceedingly small compared with the distances apart of the electrons in the surrounding cluster. From the results of experiments on scattering of alpha rays, Rutherford concluded that the charge on the nucleus corresponds to a number of electrons per atom approximately equal to half the atomic weight. Concordant evidence from a large number of very different phenomena has led to the more definite assumption that the number of electrons per atom is exactly equal to the atomic number, i.e., the number of the corresponding element in the periodic table. This view was first proposed by van den Broek[3]. While the nucleus theory has been of great utility in explaining many important properties of the atom[4], on the other hand it is evident that it is impossible by its aid to explain many other fundamental properties if we base our considerations on the ordinary electrodynamical theory; but this can hardly be considered as a valid objection at the present time. It does not seem that there is any escape from the conclusion that it is impossible to account for the phenomena of temperature radiation on ordinary electrodynamics, and that the modification to be introduced in this theory must be essentially equivalent with the assumptions first used by Planck in the deduction of his radiation formula[5]. These assumptions are known as the Quantum theory. In my previous paper it was attempted to apply the main principles of this theory by introducing the following general assumptions:—
A. An atomic system possesses a number of states in which no emission of energy radiation takes place, even if the particles are in motion relative to each other, and such an emission is to be expected on ordinary electrodynamics. The states are denoted as the “stationary” states of the system under consideration.
B. Any emission or absorption of energy radiation will correspond to the transition between two stationary states. The radiation emitted during such a transition is homogeneous and the frequency is determined by the relation where is Planck’s constant and , and , are the energies of the system in the two stationary states.
C. That the dynamical equilibrium of the systems in the stationary states is governed by the ordinary laws of mechanics, while these laws do not hold for the transition from one state to another.
D. That the various possible stationary states of a system consisting of an electron rotating round a positive nucleus are determined by the relation where is the mean value of the kinetic energy of the system, the frequency of rotation, and a whole number.
It will be seen that these assumptions are closely analogous to those originally used by Planck about the emission of radiation in quanta, and about the relation between the frequency of an atomic resonator (of constant frequency) and its energy. It can be shown that, for any system containing one electron rotating in a closed orbit, the assumption C and the relation (2) will secure a connexion between the frequency calculated by (1) and that to be expected from ordinary electrodynamics, in the limit where the difference between the frequency of the rotation of the electron in successive stationary states is very small compared with the absolute value of the frequency (see IV. p. 5). On the nucleus theory this occurs in the region of very slow vibrations. If the orbit of the electron is circular, the assumption D is equivalent to the condition that the angular momentum of the system in the stationary states is an integral multiple of . The possible importance of the angular momentum in the discussion of atomic systems in relation to Planck’s theory was first pointed out by J. W. Nicholson[6].
In paper I. it was shown that the above assumptions lead to an interpretation of the Balmer formula for the hydrogen spectrum, and to a determination of the Rydberg constant which was in close agreement with the measurements. In these considerations it is not necessary to make any assumption about the degree of excentricity of the orbit of the electron, and we shall see in the next section that it cannot be assumed that the orbit is always circular.
So far we have considered systems which contain only one electron, but the general validity of the assumptions A and B seems strongly supported by the fact that they offer a simple interpretation of the general principle of combination of spectral lines (see IV. p. 2). This principle was originally discovered by Ritz to hold for the ordinary series spectra of the elements. It has recently acquired increased interest by Fowler’s work on the series spectra of enhanced lines emitted from many elements when subject to a powerful electric discharge. Fowler showed that the principle of combination holds for these spectra although the laws governing the numerical relation between the lines at an important point (see section 3) differed from those of the ordinary series spectra. There is also, as we shall see in section 4, some indication that the principle holds for the high frequency spectra revealed by interference in crystals. In this connexion it may also be remarked that the assumption A recently has obtained direct support by experiments of A. Einstein and J. W. de Haas[7], who have succeeded in detecting and measuring a rotational mechanical effect produced when an iron bar is magnetized. Their results agree very closely with those to be expected on the assumption that the magnetism of iron is due to rotating electrons, and as pointed out by Einstein and Haas, these experiments therefore indicate very strongly that electrons can rotate in atoms without emission of energy radiation.
When we try to apply assumptions, analogous with C and D, to systems containing more than one electron, we meet with difficulties, since in this case the application of ordinary mechanics in general does not lead to periodic orbits. An exception to this, however, occurs if the electrons are arranged in rings and rotate in circular orbits, and from simple considerations of analogy the following assumption was proposed (see I. p. 24).
E. In any atomic or molecular system consisting of positive nuclei and electrons in which the nuclei are at rest relative to each other, and the electrons move in circular orbits, the angular momentum of each electron round the centre of its orbit will be equal to in the “normal” state of the system, i.e. the state in which the total energy is a minimum.
It was shown that in a number of different cases this assumption led to results in approximate agreement with experimental facts. In general, no stable configuration in which the electrons rotate in circular orbits can exist if the problem of stability is discussed on ordinary mechanics. This is no objection, however, since it is assumed already that the mechanics do not hold for the transition between two stationary states. Simple considerations led to the following condition of stability.
F. A configuration satisfying the condition E is stable if the total energy of the system is less than in any neighbouring configuration satisfying the same condition of angular momentum of the electrons.
As already mentioned, the foundation for the hypothesis E was sought in analogy with the simple system consisting of one electron and one nucleus. Additional support, however, was obtained from a closer consideration of the formation of the systems. It was shown how simple processes could be imagined by which the confluence of different rings of electrons could be effected without any change in the angular momentum of the electrons, if the angular momentum of each electron before the process was the same. Such considerations led to a theory of formation of molecules.
It must be emphasized that only in the case of circular orbits has the angular momentum any connexion with the principles of the Quantum theory. If, therefore, the application of ordinary mechanics to the stationary states of the system does not lead to strictly circular orbits, the assumption E cannot be applied. This case occurs if we consider configurations in which the electrons are arranged in different rings which do not rotate with the same frequency. Such configurations, however, are apparently necessary in order to explain many characteristic properties of the atoms. In my previous papers an attempt was made in certain cases to overcome this difficulty by assuming, that if a very small alteration of the forces would make circular orbits possible on ordinary mechanics, the configuration and energy of the actual system would only differ very little from that calculated for the altered system. It will be seen that this assumption is most intimately connected with the hypothesis F on the stability of the configurations. Such considerations were used to explain the general appearance of the Rydberg constant in the spectra of the elements, and were also applied in discussing possible configurations of the electrons in the atoms suggested by the observed chemical properties. These calculations have been criticised by Nicholson[8], who has attempted to show that the configurations chosen for the electrons in the atoms are inconsistent with the main principles of the theory, and has also attempted to prove the impossibility of accounting for other spectra by help of assumptions similar to those used in the interpretation of the hydrogen spectrum.
Although I am quite ready to admit that these points involve great and unsolved difficulties, I am unable to agree with Nicholson’s conclusions. In the first place, his calculations rest upon a particular application to non-circular orbits of the principle of constancy of angular momentum for each electron, which it does not seem possible to justify either on the Quantum theory or on the ordinary mechanics, and which has no direct connexion with the assumptions used in my papers. It has not been proved that the configurations proposed are inconsistent with the assumption C. But even if it were possible to prove that the unrestricted use of ordinary mechanics to the stationary states is inconsistent with the configurations of the electrons, apparently necessary to explain the observed properties of the elements, this would not constitute a serious objection to the deductions in my papers. It must be remarked that all the applications of ordinary mechanics are essentially connected with the assumption of periodic orbits. As far as the applications are concerned, the first part of the assumption C might just as well have been given the following more cautious form:—
“The relation between the frequency and energy of the particles in the stationary states can be determined by means of the ordinary laws of mechanics if these laws lead to periodic orbits.”
The possible necessity for an alteration of this kind in assumption C may perhaps not seem unlikely when it is remembered that the laws of mechanics are only known to hold for certain mean values of the motion of the electrons. In this connexion it should also be remarked that when considering periodic orbits only mean values are essential (comp. I. p. 7). The preliminary and tentative character of the formulation of the general assumptions cannot be too strongly emphasized, and admittedly they are made to suit certain simple applications. For example, it has been already shown in paper IV. that the assumption B needs modification in order to account for the effect of a magnetic field on spectral lines. In the following sections some of the recent experimental evidence on line spectra and characteristic Röntgen rays will be considered, and I shall endeavour to show that it seems to give strong support to the main principles of the theory.