Self-energy corrections to theK-electron binding in heavy and superheavy atoms

The self-energy corrections of order $\alpha$ to the $K$-electron binding energy in high-$Z$ atoms are studied numerically throughout the range $Z=70-160\mathrm{}$. Nuclear finite-size effects are included in the electron wave functions and in the electron propagator to avoid the Coulomb singularity at $Z\alpha =1\mathrm{}$. Denoting the self-energy correction by $\Delta E=\frac{\alpha {\left(Z\alpha \right)}^{4}F\left(Z\alpha \right)m{c}^2}{\pi }$, it is found that $F\left(Z\alpha \right)$ increases smoothly from a minimum near $Z=90\mathrm{}$ through $F\left(100\mathrm{}\alpha \right)=1.46\mathrm{}\pm 0.01$ to $F\left(160\mathrm{}\alpha \right)=3.34\mathrm{}\pm 0.16$. For comparison purposes, the self-energy corrections in a Coulomb field for $Z$ between 50 and 130 are evaluated, and the values of $F\left(Z\alpha \right)$ thus obtained are in close agreement with previously published values; the Coulomb field values of $F\left(Z\alpha \right)$ are very close to the finite-nucleus results for $Z<\mathrm{}90\mathrm{}$.