Higher-Order Relativistic Contributions to the Combined Zeeman and Motional Stark Effects in Positronium

Abstract

A calculation of higher-order relativistic contributions to the combined Zeeman and motional Stark effects in positronium is presented. These contributions are necessary for the determination of the fine-structure interval in the ground state from the Zeeman effect and may be important in future experiments on the first excited states of positronium. The contributions to the ${g}_{J}$ factors have been calculated to order ${\ensuremath{\alpha}}^{2}$ for all the $S$ states and for the $2P$ states. The energy levels and the higher-order corrections to the motional Stark effect in the first excited state are also presented. Relativistic contributions are obtained from the matrix elements of a Hamiltonian containing the Breit interaction, a Pauli Hamiltonian, and the virtual annihilation interaction. For all $nS$ states the relativistic contributions to the Zeeman effect may be accounted for by the replacement of ${g}_{e}$ by ${g}_{e}(1\ensuremath{-}\frac{5{\ensuremath{\alpha}}^{2}}{24{n}^{2}}\ensuremath{-}\frac{T}{2m{c}^{2}})$, where $T$ is the kinetic energy of the atom and ${g}_{e}$ is the gyromagnetic ratio of the free electron.