Speaker
Description
We investigate the atomic exchange effect between the final atom's bound electrons and those emitted in the allowed $\beta$ decay and $2\nu\beta\beta$ decay of the initial nucleus. The electron wave functions are obtained with the Dirac-Hartree-Fock-Slater self-consistent method, and we ensure the orthogonality between the continuum and bound electron states of the final atom by modifying the last iteration of the self-consistent method. We show that orthogonality plays an essential role in calculating the exchange correction. We argue that our findings can solve the mismatch between the predictions and experimental measurements in the low-energy region of the $\beta$ spectrum. First, we calculate the exchange effect for four low-energy $\beta$ transitions recently investigated in the literature. Additionally, we present the observables for the $2\nu\beta\beta$ decay of $^{100}$Mo, including exchange and radiative corrections. Next, we compute the exchange correction for the $\beta$ emitters with $Z$ from $1$ to $102$. From the systematic study, we found that for ultra-low energy, i.e., $5$ eV, the $Z$ dependence of total exchange effect is affected by $s_{1/2}$ and $p_{1/2}$ orbitals closure. Finally, we provide an analytical expression of the exchange correction for each atomic number for easy implementation in experimental investigations.
