State-Dependent Mass Corrections to Hyperfine Structure in Hydrogenic Atoms

Abstract

The $\frac{{\alpha }^{2}m}{M}$ corrections to the ratio of the hyperfine structure of the $1s$ and $2s$ states of hydrogenic atoms have been evaluated using a Foldy-Wouthuysen reduction of the Dirac Hamiltonian for the electron plus additional nuclear motion terms. The covariant two-fermion Bethe-Salpeter equation gives the same result, as does an approximately covariant calculation based on the Breit equation, a simple nuclear model, and the correspondence principle. In units of ${10}^{-6\mathrm{}}$, the $\frac{{\alpha }^{2}m}{M}$ corrections to $R=\left(\frac{8{\nu }_2}{{\nu }_1}\right)-1\mathrm{}$ total -0.115, -0.029, and -0.101 for H, D, and T, respectively. The corresponding theoretical $R$ values are 34.45±0.02, 34.53±0.02, and 34.46±0.02. These agree with the available experimental values which are 34.495±0.060 and 34.2±0.6 for H and D, respectively.