§ 2. Spectra emitted from systems containing only one electron.

In the former papers it was shown that the general assumptions led to the following formula for the spectrum emitted by an electron rotating round a positive nucleus , , , are the electric charges and the masses of the nucleus and the electron respectively. The frequency of rotation and the major axis of the relative orbit of the particles in the stationary states are given by The energy necessary to remove the electron to infinite distance from the nucleus is This expression is also equal to the mean value of the kinetic energy of the system. Since is equal to the total energy of the system we get from (4) and (5) If we compare (6) with the relation (1), we see that the connexion with ordinary mechanics in the region of slow vibration, mentioned in the former section, is satisfied.

Putting in (3) we get the ordinary series spectrum of hydrogen. Putting we get a spectrum which, on the theory, should be expected to be emitted by an electron rotating round a helium nucleus. The formula is found very closely to represent some series of lines observed by Fowler[9] and Evans[10]. These series correspond to and [11]. The theoretical value for the ratio between the second factor in (3) for this spectrum and for the hydrogen spectrum is 1.000409; the value calculated from Fowler’s measurements is 1.000408[12]. Some of the lines under consideration have been observed earlier in star spectra, and have been ascribed to hydrogen not only on account of the close numerical relation with the lines of the Balmer series, but also on account of the fact that the lines observed, together with the lines of the Balmer series, constitutes a spectrum which shows a marked analogy with the spectra of the alkali metals. This analogy, however, has been completely disturbed by Fowler’s and Evans’ observations, that the two new series contain twice as many lines as is to be expected on this analogy. In addition, Evans has succeeded in obtaining the lines in such pure helium that no trace of the ordinary hydrogen lines could be observed[13]. The great difference between the conditions for the production of the Balmer series and the series under consideration is also brought out very strikingly by some recent experiments of Rau[14] on the minimum voltage necessary for the production of spectral lines. While about 13 volts was sufficient to excite the lines of the Balmer series, about 80 volts was found necessary to excite the other series. These values agree closely with the values calculated from the assumption E for the energies necessary to remove the electron from the hydrogen atom and to remove both electrons from the helium atom, viz. 13.6 and 81.3 volts respectively. It has recently been argued[15] that the lines are not so sharp as should be expected from the atomic weight of helium on Lord Rayleigh’s theory of the width of spectral lines. This might, however, be explained by the fact that the systems emitting the spectrum, in contrast to those emitting the hydrogen spectrum, are supposed to carry an excess positive charge, and therefore must be expected to acquire great velocities in the electric field in the discharge-tube.

In paper IV. an attempt was made on the basis of the present theory to explain the characteristic effect of an electric field on the hydrogen spectrum recently discovered by Stark. This author observed that if luminous hydrogen is placed in an intense electric field, each of the lines of the Balmer series is split up into a number of homogeneous components. These components are situated symmetrically with regard to the original lines, and their distance apart is proportional to the intensity of the external electric field. By spectroscopic observation in a direction perpendicular to the field, the components are linearly polarized, some parallel and some perpendicular to the field. Further experiments have shown that the phenomenon is even more complex than was at first expected. By applying greater dispersion, the number of components observed has been greatly increased, and the numbers as well as the intensities of the components are found to vary in a complex manner from line to line[16]. Although the present development of the theory does not allow us to account in detail for the observations, it seems that the considerations in paper IV. offer a simple interpretation of several characteristic features of the phenomenon.

The calculation can be made considerably simpler than in the former paper by an application of Hamilton’s principle. 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 the revolution, and the mean value of the sum of the kinetic energy and the potential energy of the particle relative to the stationary field. We have then for a small arbitrary variation of the orbit This equation was used in paper IV. to prove the equivalence of the formulæ (2) and (6) for any system governed by ordinary mechanics. The equation (7) further shows that if the relations (2) and (6) hold for a system of orbits, they will hold also for any small variation of these orbits for which the value of is unaltered. If a hydrogen atom in one of its stationary states is placed in an external electric field and the electron rotates in a closed orbit, we shall therefore expect that is not altered by the introduction of the atom in the field, and that the only variation of the total energy of the system will be due to the variation of the mean value of the potential energy relative to the external field.

In the former paper it was pointed out that the orbit of the electron will be deformed by the external field. This deformation will in course of time be considerable even if the external electric force is very small compared with the force of attraction between the particles. The orbit of the electron may at any moment be considered as an ellipse with the nucleus in the focus, and the length of the major axis will approximately remain constant, but the effect of the field will consist in a gradual variation of the direction of the major axis as well as the excentricity of the orbit. A detailed investigation of the very complicated motion of the electron was not attempted, but it was simply pointed out that the problem allows of two stationary orbits of the electron, and that these may be taken as representing two possible stationary states. In these orbits the excentricity is equal to 1, and the major axis parallel to the external force; the orbits simply consisting of a straight line through the nucleus parallel to the axis of the field, one on each side of it. It can very simply be shown that the mean value of the potential energy relative to the field for these rectilinear orbits is equal to , where is the external electric force and the major axis of the orbit, 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. Using the formulæ (4) and (5) and neglecting the mass of the electron compared with that of the nucleus, we get, therefore, for the energy of the system in the two states respectively. This expression is the same as that deduced in paper IV. by an application of (6) to the expressions for the energy and frequency of the system. Applying the relation (1) and using the same arguments as in paper IV. p. 10, we are therefore led to expect that the hydrogen spectrum in an electric field will contain two components polarized parallel to the field and of a frequency given by

The table below contains Stark’s recent measurements of the frequency difference between the two strong outer components polarized parallel to the field for the five first lines in the Balmer series[17]. The first column gives the values for the numbers and . The second and fourth columns give the frequency difference corresponding to a field of 28500 and 74000 volts per cm. respectively. The third and fifth columns give the values of where should be a constant for all the lines and equal to unity.

28500 volts. per cm. 74000 volts. per cm.

2 3 0.46 0.83

2 4 1.04 0.79 2.86 0.83

2 5 2.06 0.89 5.41 0.90

2 6 3.16 0.90 7.81 0.85

2 7 4.47 0.90

Considering the difficulties of accurate measurement of the quantities involved, it will be seen that the agreement with regard to the variation of the frequency differences from line to line is very good. The fact that all the observed values are a little smaller than the calculated may be due to a slight over-estimate of the intensity of the fields used in the experiments (see Stark, loc. cit. pp. 38 and 118). Besides the two strong outer components polarized parallel to the field, Stark’s experiments have revealed a large number of inner weaker components polarized in the same way, and also a number of components polarized perpendicular to the field. This complexity of the phenomenon, however, cannot be considered as inconsistent with the theory. The above simple calculations deal only with the two extreme cases, and we may expect to find a number of stationary states corresponding to orbits of smaller excentricity. In a discussion of such non-periodic orbits, however, the general principles applied are no longer sufficient guidance.

Apart from the agreement with the calculations, Stark’s experiments seem to give strong support to the interpretation of the origin of the two outer components. It was found that the two outer components have not always equal intensities; when the spectrum is produced by positive rays, it was found that the component of highest frequency is the stronger if the rays travel against the electric field, while if it travels in the direction of the field the component of smallest frequency is the stronger (loc. cit. p. 40). This indicates that the components are produced independently of each other—a result to be expected if they correspond to quite different orbits of the electron. That the orbit of the electron in general need not be circular is also very strongly indicated by the observation that the hydrogen lines emitted from positive rays under certain conditions are partly polarized without the presence of a strong external field (loc. cit. p. 12). This polarization, as well as the observed intensity differences of the two components, would be explained if we can assume that for some reason, when the atom is in rapid motion, there is a greater probability for the orbit of the electron to lie behind the nucleus rather than in front of it.