On the Hyperfine Structure of Heavy Elements

The ordinary theory of hfs splitting is incomplete for two reasons in the case of heavy elements. (1) When the electron is close to the nucleus its velocity is high. Non-relativistic approximations to Dirac's equation become meaningless. (2) The probability of the electron being sufficiently close to the nucleus to interact with it at all may be appreciably different for different components of the same multiplet. General relativistic formulas (12), (12″) are derived for single electron spectra. Quantitative estimates are made for the specific case of the Tl lowest $p$ term. The nonrelativistic approximations with the same $\overline{r^{-3\mathrm{}}}$ for $p_{\frac{1}{2}}$ and $p_{\frac{3}{2}}$ are found to give values of $\frac{\left(\Delta \nu \right)p_{\frac{1}{2}}}{\left(\Delta \nu \right)p_{\frac{3}{2}}}$ which are too small by a factor of about 3.4=2×1.7. The factor 2 is attributable to relativistic corrections. The estimated factor 1.7 is due to the higher energy of the $p_{\frac{3}{2}}$ level which decreases the chance of the electron to be close enough to the nucleus to interact with it. Comparison with the observations of John Wulff on Tl I shows that even the corrected value of $\frac{\left(\Delta \nu \right)p_{\frac{1}{2}}}{\left(\Delta \nu \right)p_{\frac{3}{2}}}$ is too small by a factor of 2. A qualitatively similar disagreement exists for Bi I. The observed hfs of these elements is therefore not accounted for by the theory of a nuclear magnetic moment.