Nuclear Magnetic Moments from Hyperfine Structure Data

The nuclear magnetic moments determined from the hyperfine structure of the ${}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}$ and ${}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}$ states are systematically smaller than those determined by methods of magnetic resonance. The Breit-Crawford-Schawlow correction, which takes into account the finite dimensions of the nucleus, together with the Bohr-Weisskopf correction which takes into account the spatial distribution of nuclear magnetism, succeed in explaining at least the order of magnitude in the preceding difference. However, in the two corrections, certain factors are determined graphically, while others are to a certain extent erroneous, owing to incomplete solving of the Darwin-Gordon differential system. All these difficulties are removed in the present paper and the final result is a completely analytical expression for the total correction. The numerical calculations made for ${}_{80}{}^{}{}_{}{}^{}\mathrm{Hg}_{}^{199}{}_{}{}^{}$ starting from the ground state ${}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}$ (Hg ii) fall in good agreement with the value of the nuclear moment determined by magnetic resonance.