Relativistic Contributions to the Magnetic Moment ofS13Helium

Some relativistic contributions of order ${\alpha }^2\sim {\mathrm{}\left(137\right)\mathrm{}}^{-2\mathrm{}}$ 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 ${\alpha }^2$ 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 ${\alpha }^2$. The result derived was that the $g$ value for two interacting electrons bound in a ${}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}$ state is $2(1-\frac{1}{3}〈T〉-\frac{1}{6}〈\frac{e^2}{r_{12}}〉)$ 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 $-\frac{1}{3}〈T〉$ corresponds to the Breit-Margenau result for one electron and $-\frac{1}{6}〈\frac{e^2}{r_{12}}〉$ arises from the Breit interaction. For ${}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}$ helium the preceding $g$ value was evaluated numerically as 2[1-(38.7+2.3)×${10}^{-6\mathrm{}}$]. Comparison of theory and experiment tends to substantiate the nonradiative contribution $-\frac{1}{3}〈T〉$ 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.