Relativistic Contributions to the Magnetic Moment ofS13Helium

Abstract

Some relativistic contributions of order ${\ensuremath{\alpha}}^{2}\ensuremath{\sim}{(137)}^{\ensuremath{-}2}$ to the magnetic moment of helium in the lowest-energy triplet state $^{3}S_{1}$ have been calculated. These contributions arise from the effect of the electrostatic and Breit interactions in a relativistic wave equation. The purpose of the calculation was to isolate to order ${\ensuremath{\alpha}}^{2}$ the quantum-electrodynamic radiative contributions to the magnetic moment of a bound two-electron system for comparison with experiment. The method of calculation was to evaluate the sixteen-component form of the matrix element of magnetic interaction energy in terms of nonrelativistic wave functions in Pauli approximation and to use the angular and spin symmetry properties of the $^{3}S_{1}$ state. This procedure was possible because Russell-Saunders coupling in the Pauli approximation could be shown to hold rigorously to order ${\ensuremath{\alpha}}^{2}$. The result derived was that the $g$ value for two interacting electrons bound in a $^{3}S_{1}$ state is $2(1\ensuremath{-}\frac{1}{3}〈T〉\ensuremath{-}\frac{1}{6}〈\frac{{e}^{2}}{{r}_{12}}〉)$ where $〈T〉$ is the expectation value of total kinetic energy and $〈\frac{{e}^{2}}{{r}_{12}}〉$ of electrostatic interaction in the $^{3}S_{1}$ state, in units $m{c}^{2}$. The contribution $\ensuremath{-}\frac{1}{3}〈T〉$ corresponds to the Breit-Margenau result for one electron and $\ensuremath{-}\frac{1}{6}〈\frac{{e}^{2}}{{r}_{12}}〉$ arises from the Breit interaction. For $^{3}S_{1}$ helium the preceding $g$ value was evaluated numerically as 2[1-(38.7+2.3)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}6}$]. Comparison of theory and experiment tends to substantiate the nonradiative contribution $\ensuremath{-}\frac{1}{3}〈T〉$ and the additivity properties of radiative and nonradiative contributions to the magnetic moment of $^{3}S_{1}$ helium. The fourth-order radiative contribution is not contradicted. The Breit interaction contribution is too small to be noticed, with the present experimental error.