Abstract
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 $\ensuremath{\alpha}Z$, is found to be $[\frac{\ensuremath{\alpha}}{2\ensuremath{\pi}}\ensuremath{-}{\ensuremath{\alpha}}^{2}Z(\frac{5}{2}\ensuremath{-}\mathrm{ln}2)]{E}_{H}$, 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.
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