Higher-Order Relativistic Contributions to the Zeeman Effect in Helium

A calculation of higher-order relativistic contributions to the Zeeman effect in the $2{}_{}{}^3{}_{}{}^{}P$, $2{}_{}{}^3{}_{}{}^{}S$, $3{}_{}{}^3{}_{}{}^{}P$, $4{}_{}{}^3{}_{}{}^{}P$, and $5{}_{}{}^3{}_{}{}^{}P$ states of helium is given. The contributions to the $g$ factors $g_S^{\prime }$, $g_L^{\prime }$, and $g_X$, have been calculated to order ${\alpha }^2$ and $\frac{m}{M}$ where $m$ is the electron mass and $M$ is the helium mass. Relativistic contributions are obtained from the matrix elements of a reduced Breit Hamiltonian with a series of numerical wave functions using a Hylleraas basis. The comparison of the theoretical values with the experimental determinations of the $g$ factors is given for the $2{}_{}{}^3{}_{}{}^{}P$ and $2{}_{}{}^3{}_{}{}^{}S$ states. The value of $g_S^{\prime }$ calculated for $2{}_{}{}^3{}_{}{}^{}P$ helium is within two standard deviations of the experimental error of the measurement of Lewis et al. A recent precise measurement of the $g$ factor in $2{}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}$ helium by Brossel et al. is three standard deviations of the experimental error from the value calculated here and from the result of Perl and Hughes. These higher-order contributions are necessary for the determination of the $2{}_{}{}^3{}_{}{}^{}P$ fine-structure intervals from the Zeeman effect, and hence to test the theory of fine structure for helium and to determine precisely the fine-structure constant $\alpha$ from the helium fine structure. The relativistic contributions to $g_S^{\prime }$ have been calculated for the helium isoelectronic series from Li II through Ne IX for the $2{}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{}$, $3{}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{}$, $4{}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{}$, and $5{}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{}$ states.