Higher-Order Relativistic Contributions to the Zeeman Effect in Helium

A calculation of higher-order relativistic contributions to the Zeeman effect in the 2³P, 2³S, 3³P, 4³P, and 5³P states of helium is given. The contributions to the g factors gS^', gL^', and gX, 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 2³P and 2³S states. The value of gS^' calculated for 2³P 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 2³S₁ 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 2³P 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 gS^' have been calculated for the helium isoelectronic series from Li II through Ne IX for the 2³P₁, 3³P₁, 4³P₁, and 5³P₁ states.