Second-Order Radiative Corrections to Hyperfine Structure

In this paper the effect of second-order radiative displacements of the energy levels of a bound electron on hyperfine structure separation is examined, and a correction to the Fermi formula for $s$ levels is obtained. The treatment is based on an approximate evaluation of a finite, completely renormalized, and exact expression for the second order energy. The correction, which is restricted to atoms of small $\alpha Z$, is found to be $[\frac{\alpha }{2\pi }-{\alpha }^{2}Z(\frac{5}{2}-\ln 2\left)\right]E_H$, where $E_H$ is the Fermi energy. The effect of this and other corrections to the Fermi formula on the presently accepted value of the fine structure constant is discussed.