Relativistic Treatment of the Shielding of the Electron and Proton Magnetic Dipole Moments in Atomic Hydrogen

Abstract

The Breit equation for two Dirac particles in an external magnetic field, with anomalous magnetic moments introduced phenomenologically as Pauli moments, is taken as the starting point for a Chraplyvy-Barker-Glover reduction. First-order perturbation theory employing the nonrelativistic wave function for the ${}_{}{}^2{}_{}{}^{}S$ ground state is then used to derive expressions for the diagonal matrix elements of the Hamiltonian which depend linearly upon the field. These expressions are symmetric in the masses and anomalous moments of the particles and agree with previous results to relative order ${\alpha }^2$. However, additional higher-order terms arise due to changes in the anomalous moments (in principle, obtainable from field theory), to reduced mass corrections, and to factorization of the moments as free-particle moments times shielding corrections. Contributions to the shielding expressions are interpreted as arising from (1) relativistic mass corrections, (2) spin-orbit coupling, and (3) spin-other-orbit coupling. In the case of atomic hydrogen the first two effects dominate the electron shielding and the latter dominates the proton shielding. Similar calculations and interpretations may be applied to magnetic shielding in larger atomic and molecular systems.