# Overlap of electron shells in $\beta$ and double-$\beta$ decays

## TL;DR

This paper investigates how the overlap of electron shells affects decay rates in beta and double-beta decays, especially considering environmental influences on valence electrons and implications for neutrino mass measurements.

## Contribution

It provides estimates of shell overlap effects on decay suppression for various isotopes and discusses environmental impacts on valence versus core electrons.

## Key findings

- Valence electrons are significantly affected by the environment, altering decay rates.
- Core electron overlaps remain stable across different host materials.
- Shell overlap amplitudes influence the relationship between decay half-life and neutrino mass.

## Abstract

The $\beta$ and double-$\beta$ decay channels, which are not accompanied by excitation of the electron shells, are suppressed due to the nonorthogonality of the electron wave functions of the parent and daughter atoms. The effect is sensitive to the contribution of the outer electron shells. Since valence electrons participate in chemical bonding and collectivize in metals, the decay rates of the unstable nuclides are modified when they are embedded in a host material. Core electrons are less affected by the environment, and their overlap amplitudes are more stable. The suppression effect is estimated for $ \beta^- $ decay of $^{87}$Kr, electron capture in $^{163}$Ho, and $2\beta^-$ decays of $^{76}$Ge, $^{100}$Mo, $^{130}$Te, and $^{136}$Xe. The overlap amplitude of the electron shells enters the relationship between the half-life of neutrinoless $2\beta$ decay and the effective electron neutrino Majorana mass.