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

The finite-size correction to $\beta$ 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 $\beta$ 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 10–15% larger than the latter even though the densities differ significantly.

Figure 1

Charge density distribution of ${}^{120}\mathrm{Sn}$ as calculated by SLy4 EDF. The contribution from each single-particle state is given. The experimental data [32] are also plotted for comparison. A uniform density distribution (labeled “Uni.”) that extends to $R=1.2A^{1/3}$ fm is also shown.

Figure 2

(a) The radial wave functions of the neutron in the parent nucleus and the proton in the daughter nucleus participating in the allowed ground-state-to-ground-state ${\beta }^{-}$ decay of ${}^{100}\mathrm{Nb}$. (b) Normalized charge density in the daughter nucleus ${}^{100}\mathrm{Mo}$, normalized weak transition density calculated from the corresponding SPWFs, and the uniform density distribution for the $A=100$ nucleus, labeled by “Charge,” “Weak,” and “Uniform,” respectively. The convoluted densities ${\rho }_{\left(2\right)}\left(r\right)$ and ${\rho }_{\left(2\right)}^r\left(r\right)$ determining the Zemach moments ${\langle r\rangle }_{\left(2\right)}$ and ${\langle r\rangle }_{\left(2\right)}^r$, respectively, are plotted in panels (c) and (d). Assuming that the weak density is identical to the normalized charge density produces the curve labeled “Charge.” The curve labeled “Mix” uses the calculated charge and weak densities, while the curve labeled “Uniform” assumes that both densities have a common uniform density distribution.

Figure 3

The same as Fig. 2, but for the allowed ground-state-to-ground-state ${\beta }^{-}$ decay of ${}^{121}\mathrm{Sn}$.

Figure 4

(Left) The ratio (DFT/uniform) of the ground-state-to-ground-state ${}^{100}\mathrm{Nb}\to {}^{100}\mathrm{Mo}\beta$-decay spectrum. The DFT (uniform) finite-size correction was calculated using the values of ${\langle r\rangle }_{\left(2\right)}$ and ${\langle r\rangle }_{\left(2\right)}^r$ in Table 2 that are labeled “Mix (Uni).” Both spectra are normalized to unity. (Right) The FS correction ${\delta }_{\mathrm{FS}}$ for the DFT and the (equal) uniform-density approximation.