Effect of the Atomic Core on the Magnetic Hyperfine Structure

Abstract

Equations have been obtained for the magnetic field at the nucleus due to the current induced in the electron core by external valence electron. The magnetic field is written as $\ensuremath{-}4{\ensuremath{\mu}}_{0}[\frac{\mathbf{j}}{(j(j+1))}]{〈{r}^{\ensuremath{-}3}〉}_{p}(1+{R}_{m})$, where ${〈{r}^{\ensuremath{-}3}〉}_{p}$ is the average over the valence electron function, assumed in a $p$ state, ${R}_{m}$ is the correction due to the core, ${\ensuremath{\mu}}_{0}$ = Bohr magneton, $j$ = angular momentum. ${R}_{m}$ is of importance in obtaining the nuclear quadrupole moment $Q$ from the value of the magnetic moment ${\ensuremath{\mu}}_{I}$ and the ratio $\frac{b}{a}$ of the splittings due to $Q$ and ${\ensuremath{\mu}}_{I}$. The electric field gradient at the nucleus, $\ensuremath{-}\frac{2}{5}e{〈{r}^{\ensuremath{-}3}〉}_{p}(1+R)$, which determines $b$, contains a similar term $R$ for the distortion of the core by the valence electron. It is shown that ${R}_{m}$ approximately cancels the exchange terms of $R$, so that the correction factor for $Q$ is that predicted by the Thomas-Fermi model.