The Nature of Optical and X-ray Doublets1

S. E. Frisch.

As is known, Millikan succeeded, with the aid of his vacuum spectrograph, in advancing far into the ultraviolet part of the spectrum. The source of light was a spark in vacuum (“hot-spark”). The elements investigated were chiefly those of the first row of the periodic system, which revealed in the region around 1000 Å a large number of spectral lines.

The question may be posed: what is the nature of the observed spectra, intermediate between optical and X-ray spectra?

Hydrogen, ionized helium, doubly ionized lithium, etc., being systems consisting of a nucleus and one electron, give, according to Bohr’s theory, spectra embraced by the generalized Balmer formula:

\[ \nu = RZ^2\left(\frac{1}{n'^2}-\frac{1}{n''^2}\right) \]

where \(Z\) is the atomic number. For \(n' = 1\) the indicated series is the analogue of the \(K\)-series in the X-ray region.

Ionized \(Be\), doubly ionized \(B\), etc., should give spectra like the spectrum of \(Li\), i.e. consisting of doublets analogous to the doublets of the alkali metals (only shifted into the ultraviolet region owing to the factor \(Z^2\)). Denoting that a spectrum belongs to a non-ionized atom by the index I, to an ionized atom by the index II, to a doubly ionized atom by the index III, etc., we obtain the following table of expected spectra:

In order to detect distant ultraviolet doublets, Millikan used a large diffraction grating of 4 inches, with a focus of 1 m, resolving 0.15 Å. It turned out that indeed \(Be_{\mathrm{II}}\), \(B_{\mathrm{III}}\), etc., have doublet lines. Since these doublets are analogues of X-ray \(L\)-doublets, it was natural to see whether the same laws apply to them as in the X-ray region.

As is known, the X-ray doublet \(L_2L_3\) is explained by the difference between the energies of an electron in the circular orbit \(2_2\) (\(L_3\)) and the elliptical orbit \(2_1\) (\(L_2\)), arising because of the dependence of mass on velocity, according to the principle of relativity. The theory leads to the formula:

\[ \nu = 0.365\,(Z-\sigma)^4 \ldots (*) \]

where \(\Delta \nu\) is the frequency difference between the components of the doublet, \(Z\) is the atomic number, and \(\sigma\) is the screening quantity (from experiments \(\sigma = 3.5\) for all elements).

In the following table, the second row gives the frequency differences between the components of the observed ultraviolet doublets; the third gives the values of \(\sqrt[4]{\Delta \nu/0.365}\), and the fourth gives the values of \(\sigma\) calculated by formula \((*)\).

\[ 2p_2 - 2p_1 \]

From the table it is clear that \(\sqrt[4]{\Delta \nu/0.365}\), in agreement with formula \((*)\), increases linearly with the atomic number \(Z\), and that \(\sigma\) is approximately equal to 2. This same law has proved applicable also to a whole series of other elements.

Thus, we obtain that the laws of relativistic X-ray doublets are applicable also in the optical region.

In the X-ray region, besides the relativistic \(L_2L_3\) doublets, there are also the so-called irregular doublets \(L_1L_2\), for which \(\Delta \sqrt{\nu}\) is approximately independent of \(Z\). These doublets are explained by two different screening quantities \(\sigma\) for the \(L_1\) and \(L_2\) orbits.

If the optical doublets turn out to be relativistic, then the optical analogues of the irregular doublets should be the differences between the \(s\)- and \(p\)-terms. The following table of Millikan shows that this is approximately justified:

\[ 2s - 2p_2 \]

Thus:

\[ \begin{aligned} L_3\ (2_2) &\text{ corresponds to } p_1,\\ L_2\ (2_1) &\text{ corresponds to } p_2,\\ L_1\ (2_1) &\text{ corresponds to } s. \end{aligned} \]

But the interpretation of optical doublets as relativistic presents a whole series of difficulties:

1) The orbits \(p_1\) and \(p_2\), differing greatly in form (circle \(2_2\) and ellipse \(2_1\)), ought to have different values of the screening quantity \(\sigma\), whereas in fact for them values of \(\sigma\) are obtained which almost coincide.

2) Since the orbits \(p_2\) and \(s\) are both ellipses \(2_1\) (they may differ only in their internal quantum numbers, i.e. in different orientation in space),

then for them the screening quantities \(s\) would have to be approximately the same, which in fact is not the case.

3) In the optical region there exist not only doublets, but also triplets. The relativistic theory, however, can assign to the terms \(p\) (total quantum number 2) only two different values, \(p_1\) and \(p_2\). Thus one would have to explain the occurrence of the \(p_1 p_2\) difference by one cause, and \(p_2 p_3\) by another, which is obviously inadmissible.

4) The entire classification of spectral series (selection principles), and especially the phenomena connected with the Zeeman effect, speak against the relativistic nature of optical doublets.

In conclusion: experiments lead to the identity of the laws obeyed by the X-ray relativistic and optical doublets. The interpretation of optical doublets as relativistic presents insurmountable difficulties.

It should be noted that the work of Landé1 has shown that, from the standpoint of the classification of X-ray spectra, the relativistic nature of the \(L_2L_3\) spectra is also called into question. Perhaps the following serves as especially weighty evidence: in the periodic system of the elements both levels \(L_2\) and \(L_3\) appear at once (in one, perhaps in two neighboring elements). If the orbit \(L_2\) were an ellipse \(2_1\), and \(L_3\) a circle \(2_2\), then they would belong to different shells, and their simultaneous appearance in the periodic system would be incomprehensible.

1. A. Landé. Das Wesen der relativistischen Röntgendubletts. Zs. f. Phys. 24, p. 58, 1924. ↩↩