Second-Order Corrections to the Fine Structure of Helium

Abstract

The fine-structure constant can be determined to high accuracy from precise measurements of the fine structure of the $2{}_{}{}^3{}_{}{}^{}P$ level in helium. One of the necessary calculations is to compute the contributions from the six Breit operators and the mass-polarization operator in second-order perturbation theory. The eighteen spin-dependent perturbations from intermediate ${}_{}{}^3{}_{}{}^{}P$ states are calculated by solving an inhomogeneous Schrödinger equation for the perturbation of the wave function by the variational method. The second-order contributions are then given by a single integral. These corrections are calculated using standard Hylleraas expansions with up to 165 terms for the perturbed wave functions, resulting in contributions to the two fine-structure intervals of the order of ${10}^{-4\mathrm{}}$ ${\mathrm{cm}}^{-1\mathrm{}}$, but only four of the results are sufficiently accurate.