Higher-order relativistic contributions to the Zeeman effect in helium

A calculation of higher-order relativistic contributions to the Zeeman effect in the $2^{3}P,2^{3}S,3^{3}P,4^{3}P, and5^{3}P$ states of helium is given. The contributions to the g factors g_S′, g_L′, and $g_X$, have been calculated to order α² and 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′ 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 α from the helium fine structure. The relativistic contributions to g_S′ have been calculated for the helium isoelectronic series from Li II through Ne IX for the $2^{3}P_{1},3^{3}P_{1},4^{3}P_{1}, and5^{3}P_{1}$ states.