§ 3. Spectra emitted from systems containing more than one electron.

According to Rydberg and Ritz, the frequency of the lines in the ordinary spectrum of an element is given by where and are whole numbers and , , ...... are a series of functions of which can be expressed by where is a universal constant and a function which for large values of approaches unity. The complete spectrum is obtained by combining the numbers and , as well as the functions , ...... in every possible way.

On the present theory, this indicates that the system which emits the spectrum possesses a number of series of stationary states for which the energy in the th state in the th series is given by (see IV. p. 6) where is an arbitrary constant, the same for the whole system of stationary states. The first factor in the second term is equal to the expression (5) if .

In the present state of the theory it is not possible to account in detail for the formula (13), but it was pointed out in my previous papers that a simple interpretation can be given of the fact that in every series approaches unity for large values of . It was assumed that in the stationary states corresponding to such values of , one of the electrons in the atom moves at a distance from the nucleus 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, and the formula (13) indicates 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 the electron in the hydrogen atom. From the considerations in the former sections it will therefore appear that the frequency calculated from the relations (1) and (13) for the radiation emitted during the transition between successive stationary states within each series will approach that to be expected on ordinary electrodynamics in the region of slow vibrations[18].

From (13) it follows that for high values of the configuration of the inner electrons possesses the same energy in all the series of stationary states corresponding to the same spectrum (11). The different series of stationary states must therefore correspond to different types of orbits of the outer electron, involving different relations between energy and frequency. In order to fix our ideas, let us for a moment consider the helium atom. This atom contains only two electrons, and in the previous papers it was assumed that in the normal state of the atom the electrons rotate in a circular ring round a nucleus. Now the helium spectrum contains two complete systems of series given by formulæ of the type (11) and the measurements of Rau mentioned below indicate that the configuration of the inner electron in the two corresponding systems of stationary states possesses the same energy. A simple assumption is therefore that in one of the two systems the orbit of the electron is circular and in the other very flat. For high values of the inner electron in the two configurations will act on the outer electron very nearly as a ring of uniformly distributed charge with the nucleus in the centre or as a line charge extending from the nucleus, respectively. In both cases several different types of orbit for the outer electron present themselves, for instance, circular orbits perpendicular to the axis of the system or very flat orbits parallel to this axis. The different configurations of the inner electrons might be due to different ways of removing the electron from the neutral atom: thus, if it is removed by impact perpendicular to the plane of the ring, we might expect the orbit of the remaining electron to be circular, if it is removed by an impact in the plane of the ring we might expect the orbit to be flat. Such considerations may offer a simple explanation of the fact that in contrast with the helium spectrum the lithium spectrum contains only one system of series of the type (11). The neutral lithium atom contains three electrons, and according to the configuration proposed in paper II. the two electrons move in an inner ring and the other electron in an outer orbit; for such a configuration we should expect that the mode of removal of the outer electron would be of no influence on the configuration of the inner electrons. It is unnecessary to point out the hypothetical nature of these considerations, but the intention is only to show that it does not seem impossible to obtain simple interpretations of the spectra observed on the general principles of the theory. However, in a quantitative comparison with the measurements we meet with the difficulties mentioned in the first section of applying assumptions analogous with and to systems for which ordinary mechanics do not lead to periodic orbits.

The above interpretation of the formulæ (11) and (12) has recently obtained very strong support by Fowler’s work on series of enhanced lines on spark spectra[19]. Fowler showed that the frequency of the lines in these spectra, as of the lines in the ordinary spectra, can be represented by the formula (11). The only difference is that the Rydberg constant in (12) is replaced by a constant . It will be seen that this is just what we should expect on the present theory if the spectra are emitted by atoms which have lost two electrons and are regaining one of them. In this case, the outer electron will rotate round a system of double charge, and we must assume that in the stationary states it will have configurations approximately the same as an electron rotating round a helium nucleus. This view seems in conformity with the general evidence as to the conditions of the excitation of the ordinary spectra and the spectra of enhanced lines. From Fowler’s results, it will appear that the helium spectrum given by (3) for has exactly the same relation to the spectra of enhanced lines of other elements as the hydrogen spectrum has to the ordinary spectra. It may be expected that it will be possible to observe spectra of a new class corresponding to a loss of 3 electrons from the atom, and in which the Rydberg constant is replaced by . No definite evidence, however, has so far been obtained of the existence of such spectra[20].

Additional evidence of the essential validity of the interpretation of formula (13) seems also to be derived from the result of Stark’s experiments on the effect of electric fields on spectral lines. For other spectra, this effect is even more complex than for the hydrogen spectrum, in some cases not only are a great number of components observed, but the components are generally not symmetrical with regard to the original line, and their distance apart varies from line to line in the same series in a far more irregular way than for the hydrogen lines[21]. Without attempting to account in detail for any of the electrical effects observed, we shall see that a simple interpretation can be given of the general way in which the magnitude of the effect varies from series to series.

In the theory of the electrical effect on the hydrogen spectrum given in the former section, it was supposed that this effect was due to an alteration of the energy of the systems in the external field, and that this alteration was intimately connected with a considerable deformation of the orbit of the electron. The possibility of this deformation is due to the fact that without the external field every elliptical orbit of the electron in the hydrogen atom is stationary. This condition will only be strictly satisfied if the forces which act upon the electron vary exactly as the inverse square of the distance from the nucleus, but this will not be the case for the outer electron in an atom containing more than one electron. It was pointed out in paper IV. that the deviation of the function from unity gives us an estimate for the deviation of the forces from the inverse square, and that on the theory we can only expect a Stark effect of the same order of magnitude as for the hydrogen lines for those series in which differs very little from unity.

This conclusion was consistent with Stark’s original measurements of the electric effect on the different series in the helium spectrum, and it has since been found to be in complete agreement with the later measurements for a great number of other spectral series. An electric effect of the same order of magnitude as that for hydrogen lines has been observed only for the lines in the two diffuse series of the helium spectrum and the diffuse series of lithium. This corresponds to the observation that for these three series is very much nearer to unity than for any other series; even for the deviation of from 1 is less than one part in a thousand. The distance between the outer components for all three series is smaller than that observed for the hydrogen line corresponding to the same value of , but the ratio between this distance and that of the hydrogen lines approaches rapidly to unity as increases. This is just what would be expected on the above considerations. The series for which the effect, although much smaller, comes next in magnitude to the three series mentioned, is the principal single line series in the helium spectrum. This corresponds to the fact that the deviation of from unity, although several times greater than for the three first series, is much smaller for this series than for any other of the series examined by Stark. For all the other series the effect was very small, and in most cases even difficult of detection.

Quite apart from the question of the detailed theoretical interpretation of the formula (13), it seems that it may be possible to test the validity of this formula by direct measurements of the minimum voltages necessary to produce spectral lines. Such measurements have recently been made by Rau[22] for the lines in the ordinary helium spectrum. This author found that the different lines within each series appeared for slightly different voltages, higher voltages being necessary to produce the lines corresponding to higher values of , and he pointed out that the differences between the voltages observed were of the magnitude to be expected from the differences in the energies of the different stationary states calculated by (13). In addition Rau found that the lines corresponding to high values of n appeared for very nearly the same voltages for all the different series in both helium spectra. The absolute values for the voltages could not be determined very accurately with the experimental arrangement, but apparently nearly 30 volts was necessary to produce the lines corresponding to high values of . This agrees very closely with the value calculated on the present theory for the energy necessary to remove one electron from the helium atom, viz., 29.3 volts. On the other hand, the later value is considerably larger than the ionization potential in helium (20.5 volts) measured directly by Franck and Hertz[23]. This apparent disagreement, however, may possibly be explained by the assumption, that the ionization potential measured does not correspond to the removal of the electron from the atom but only to a transition from the normal state of the atom to some other stationary state where the one electron rotates outside the other, and that the ionization observed is produced by the radiation emitted when the electron falls back to its original position. This radiation would be of a sufficiently high frequency to ionize any impurity which may be present in the helium gas, and also to liberate electrons from the metal part of the apparatus. The frequency of the radiation would be , which is of the same order of magnitude as the characteristic frequency calculated from experiments on dispersion in helium, viz., [24].

Similar considerations may possibly apply also to the recent remarkable experiments of Franck and Hertz on ionization in mercury vapour[25]. These experiments show strikingly that an electron does not lose energy by collision with a mercury atom if its energy is smaller than a certain value corresponding to 4.9 volts, but as soon as the energy is equal to this value the electron has a great probability of losing all its energy by impact with the atom. It was further shown that the atom, as the result of such an impact, emits a radiation consisting only of the ultraviolet mercury line of wave-length 2536, and it was pointed out that if the frequency of this line is multiplied by Planck’s constant, we obtain a value which, within the limit of experimental error, is equal to the energy acquired by an electron by a fall through a potential difference of 4.9 volts. Franck and Hertz assume that 4.9 volts corresponds to the energy necessary to remove an electron from the mercury atom, but it seems that their experiments may possibly be consistent with the assumption that this voltage corresponds only to the transition from the normal state to some other stationary state of the neutral atom. On the present theory we should expect that the value for the energy necessary to remove an electron from the mercury atom could be calculated from the limit of the single line series of Paschen, 1850, 1403, 1269[26]. For since mercury vapour absorbs light of wave-length 1850[27], the lines of this series as well as the line 2536 must correspond to a transition from the normal state of the atom to other stationary states of the neutral atom (see I. p. 16). Such a calculation gives 10.5 volts for the ionization potential instead of 4.9 volts[28]. If the above considerations are correct it will be seen that Franck and Hertz’s measurements give very strong support to the theory considered in this paper. If, on the other hand, the ionization potential of mercury should prove to be as low as assumed by Franck and Hertz, it would constitute a serious difficulty for the above interpretation of the Rydberg constant, at any rate for the mercury spectrum, since this spectrum contains lines of greater frequency than the line 2536.

It will be remarked that it is assumed that all the spectra considered in this section are essentially connected with the displacement of a single electron. This assumption—which is in contrast to the assumptions used by Nicholson in his criticism of the present theory—does not only seem supported by the measurements of the energy necessary to produce the spectra, but it is also strongly advocated by general reasons if we base our considerations on the assumption of stationary states. Thus it may happen that the atom loses several electrons by a violent impact, but the probability that the electrons will be removed to exactly the same distance from the nucleus or will fall back into the atom again at exactly the same time would appear to be very small. For molecules, i. e. systems containing more than one nucleus, we have further to take into consideration that if the greater part of the electrons are removed there is nothing to keep the nuclei together, and that we must assume that the molecules in such cases will split up into single atoms (comp. III. p. 2).