Second-Order Hyperfine Structure in Hydrogenic Atoms

A calculation of the hyperfine splittings in the $1s$ and $2s$ levels of hydrogenic atoms is made. Second-order terms in the nuclear magnetic dipole moment and terms of comparable magnitude arising from nuclear structure effects are calculated. Also included is a term derived by the use of "uncrossed" and "crossed" photon diagrams; this term is necessary because the one-electron Dirac Hamiltonian which is used evidently does not properly take certain quantum electrodynamic effects into account. In order to compare with experiment the ratio of the hyperfine splitting in the $2s$ state to the splitting in the $1s$ state is taken and evaluated for the case of the hydrogen atom; this result is combined with a previous result and the final theoretical value of $\frac{1}{8}(1.00003445\mathrm{}\pm 0.00000002)$ is in agreement with the experimental value of $\frac{1}{8}(1.000034495\mathrm{}\pm 0.000000060)$. Complete agreement has not yet been reached with respect to the splittings themselves. The calculation consists in solving the separated radial equations arising from the Dirac Hamiltonian in which the nuclear magnetic moment and the finite size of the nucleus are considered as perturbations. An iteration scheme is devised which uses certain properties of the unperturbed solution; this method may well have applications elsewhere.