On the Question of the Structure of Spectral Lines

(1. A. Sommerfeld. Sitzungsber. der K. Bayr. Akad. d. Wiss. 1915, p. 425, 1916, 459 p. 2. A. Sommerfeld. Zur Quantentheorie der Spectrallinien. Ann. d. Phys. 51, 1916. 3. F. Paschen. Bohrs Heliumlinien. Ann. d. Phys. 50, 1916).

In 1916 there appeared Paschen’s work devoted to the question of the finest structure of the helium lines, which almost all turned out to be complex, consisting of separate, regularly arranged components. This fact alone makes Paschen’s work deserving of attention; but it acquires still greater interest in connection with the works of Sommerfeld, who, proceeding from quite definite theoretical conceptions, explained such a structure of the lines of He and even quite accurately predicted the existence, position, and intensity of certain components that previously had not been noticed on photographs.

For an evaluation of Paschen’s work it is therefore necessary first to become acquainted with Sommerfeld’s work as well.

Sommerfeld starts from the model of the atom proposed by Bohr, considering in it, besides circular electron orbits, also elliptical orbits, and using the generalized quantum theory in the form in which it is given in Planck’s latest works,¹) which makes it possible to take into account the variable mass of the electron.

Let us first consider the simplest case: the hydrogen atom, consisting of a positive nucleus and an electron rotating about this nucleus. In order to pass from this model to the (ionized) He atom, it is necessary, as Sommerfeld showed, to introduce into the final formula only a definite numerical factor.

Using the above-mentioned assumptions, Sommerfeld calculates the total energy \(W\) of an electron moving along one of the possible orbits and finds it equal to:

\[ W = -\frac{Nh}{(n+n_1)^2}\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (1), \]

¹) M. Planck, Berliner Ber. p. 599, 1915; Verh. d. Deutsch. Phys. Ges. 17, p. 407, 1915. See also: A. Sommerfeld, Münchener Ber., p. 425, 1915.

where \(N\) is Balmer’s constant, \(n\) and \(n_1\) are integers, and \(h\) is Planck’s constant. Hence Balmer’s formula is also obtained in the following form:

\[ \nu = N\left(\frac{1}{(n+n_1)^2} - \frac{1}{(m+m_1)^2}\right)\ldots \tag{2}, \]

where \(m\) and \(m_1\) are integers corresponding to the electron’s orbit before, and \(n\) and \(n_1\) after, the emission, and \(\nu\) is the number of oscillations of the emitted ray.

In investigating this formula, let us take the simplest case, the so-called hydrogen Balmer series (the lines \(H\alpha, H\beta, H\gamma \ldots\) in the spectrum of hydrogen), which we obtain by putting \(n+n_1=2\), and \(m+m'=3,4,5\ldots\). As is found, the sums \(n+n'\) and \(m+m'\) give the magnitude of the major semiaxis of the elliptical orbit of the electron, while the quantities \(n\) and \(n'\), taken separately, determine the eccentricities of these orbits, making it possible to find the magnitude of the minor semiaxis.

We obtain the line \(H\alpha\) when the electron passes from one of the elliptical orbits to another, and correspondingly there are obtained 6 different ways of exciting this line. All these modes of excitation should give one and the same line only in the case if the mass of the electron does not depend on its velocity.

Taking this dependence into account, however, instead of expression (2) we obtain a more complicated expression, showing that the number of oscillations of the emitted light must vary somewhat according to which of the possible orbits, and to which one, the electron has jumped—in other words, in place of a single line \(H\alpha\) we should see a whole series of lines situated at quite definite distances from one another.

An experimental verification of the theory for the hydrogen lines is as yet impossible, since these lines are strongly broadened and the distances between the components, calculated theoretically, are very small. The verification was proposed and carried out by Paschen for the lines of \(He\), which, according to Bohr, consists of a nucleus with 2 charges and 2 electrons, and in the ionized state, having lost 1 electron, is in spectral relation entirely similar to the hydrogen atom, with the only difference that the distances between the components in the lines will be \(16\) times greater than for hydrogen. Paschen used a large Rowland spectrograph with a concave grating, which had previously served him for work together with Runge on the Zeeman effect. The dispersion in the spectra of the 3rd and 4th orders, which he used in photographing the lines, was respectively 0.86 and 0.62 Angström units per mm. The spectrum was studied when passed through a tube filled with helium, with direct current from 1000 accumulators and with a spark discharge.

The results quite satisfactorily confirmed Sommerfeld’s theory: the lines proved to be complex, and the mutual distances of the finest lines into which almost every spectral line was split coincided very accurately with the theoretical calculations.

These lines proved to be of different intensity, and, considering them, Sommerfeld gave a simple rule which proved generally correct for all the photographs. A line, evidently, is the more intense the more often the corresponding orbit of the electron occurs, the more probable it is. In the experiments it was found that the intensities observed in the photographs are explained...

that, if we accept that the smaller the eccentricity, the greater the probability, and that therefore the circular orbit has the maximum probability.

Further, some lines did not appear at all in the photographs. Simple calculations showed that these are precisely lines for which either \(m<n\) or \(m_1<n\), i.e., that, when radiating, not only can the sum \(m+m_1\) not increase, which is obvious, but individually \(m\) and \(m_1\) cannot increase either\(^1\).

After these preliminary remarks let us consider the results obtained by Paschen.

I. The so-called principal “hydrogen-like” series of He. Without correction for the variable mass of the electron, this series is given by the formula:

\[ \nu = 4N\left(\frac{1}{3^2} - \frac{1}{m^2}\right)\ . . . . . . . . . . \]

where

\[ m = 4,\ 5,\ 6\ . . . . . . . . . . \]

According to Sommerfeld, the first term of the formula must give a triplet repeated in all members of the series. The second term splits each of the lines of each triplet, respectively, into 4, 5, 6 components (i.e., each line of the first triplet gives 4 components, of the second—5, and so on).

1. For the first member of the series, \(4686\,\text{\AA}\), 9 components were found in the photographs, instead of the 12 required by the theory.

2. The second member of the series, \(\lambda = 3203\,\text{\AA}\) \(\left(\nu = 4N\left(\frac{1}{3^2} - \frac{1}{5^2}\right)\right)\), gives a triplet in which each line splits into 5 components.

3. The third member, \(\lambda = 2733\,\text{\AA}\) \(\left(\nu = 4N\left(\frac{1}{3^2} - \frac{1}{6^2}\right)\right)\), constitutes a triplet, each line of which splits further into 6 components. The distances between the lines of the triplet proved to correspond exactly to the computed ones, but it was not possible to resolve each line of the triplet into components.

II. The subsidiary series of helium. \(\left(\nu = 4N\left(\frac{1}{4^2} - \frac{1}{m^2}\right),\ \text{where } m = 5,6,7\ldots\right)\) Groups of four lines, each of which in turn splits into 5, 6, 7 lines; practically, however, because of its insignificance, this second splitting is impossible to detect. The groups \(6560,\ 5411,\ 4855,\ 4541,\ 4339,\ 4200\) and \(4100\,\text{\AA}\) were investigated. Of these, all except the first—which is very weak—proved to be complex; moreover, the first 3 components, standing close to one another in all these lines, proved to be merged, while the 4th component (very weak, but lying farther from the first three) was at first not noticed and was found in all the above-mentioned groups only after a theoretical calculation of its position, upon careful study of the photographs. Its distance from the first 3 components coincided exactly with the computed value.

In order to obtain an idea of the distances between the individual components found by Paschen (down to \(0.029\,\text{\AA}\)), let us recall that the dis—

\(^1\) The condition \(m_1>n_1\) is always observed under normal conditions; lines for which the second condition \(m>n\) is violated are noticed in some photographs.

the distance between the lines \(D_1\) and \(D_2\,Na\) is about \(6.5\,\text{\AA}\), i.e. 200 and more times greater than the smallest distance measured by Paschen.

Concluding this survey of the results obtained by Paschen, it is also necessary to note that, by measuring the distance of the 2 components into which the hydrogen line \(H_\alpha\) splits (apart from this line, only for \(H_\gamma\) was a doublet found, for which \(\Delta\lambda = 0.08\,\text{\AA}\)), it proved possible to calculate from Sommerfeld’s formula \(e\) and \(h\). This was done by Paschen; moreover, the values \(e=(4.776\pm0.07)\cdot10^{-10}\) and \(h=(6.533\pm0.16)\cdot10^{-27}\) were found to be in good agreement with Millikan’s latest data, \(e=4.77\cdot10^{-10}\), \(h=6.57\cdot10^{-27}\).

N. T. Fedorov.