Nuclear Zemach moments and finite-size corrections to allowed $\beta$ decay

The finite-size correction to $\ensuremath{\beta}$ decay plays an important role in determining the expected antineutrino spectra from reactors at a level that is important for the reactor-neutrino anomaly. Here we express the leading-order finite-size correction to allowed $\ensuremath{\beta}$ decay in terms of Zemach moments. We calculate the Zemach moments within a Hartree-Fock model using a Skyrme-like energy density functional. We find that the Zemach moments are increased relative to predictions based on the simple assumption of identical uniform nuclear charge and weak transition densities. However, for allowed ground-state to ground-state transitions in medium and heavy nuclei, the detailed nuclear-structure calculations do not change the finite-size corrections significantly from the simple model predictions, and are only 10\char21{}15% larger than the latter even though the densities differ significantly.