Screening Corrections to the Fermi Function for AllowedβDecay

Abstract

The influence of electron screening on the Fermi function $F(W,Z)$ for allowed $\ensuremath{\beta}$ decay is reconsidered. Numerical values for $F(W,Z)$ are obtained by integrating the radial Dirac equations for the decay electrons (positrons) in a self-consistent atomic potential. Relativistic Hartree-Fock-Slater equations are used to construct the atomic potential. The Fermi function is tabulated for various nuclei and energies, and comparisons are made with $F(W,Z)$ for a pure Coulomb field. Detailed comparisons are also made with the WKB screening formula of Rose as well as with the previous numerical results of Reitz and of B\uhring. It is found that the Rose formula provides an accurate approximation to $F(W,Z)$ over the dominant portion of the $\ensuremath{\beta}$ spectrum provided the endpoint energy is sufficiently high and provided further that an appropriate value of the potential at the nucleus ${V}_{0}$ is used. An empirical formula for ${V}_{0}$ from the Hartree-Fock-Slater calculations is provided. The numerical results of Reitz appear to be badly in error, especially for large electron momenta. The predictions of B\uhring for a screened-potential model are in good agreement with the present work. Corrections to the $\mathrm{ft}$ value of several superallowed Fermi transitions are recomputed and found to be negligibly different from the predictions made using the Rose approximation.