Atomic electron shell excitations in double-$\beta$ decay

The problem of the transition of electron shells of atoms to excited states in the process of neutrinoless double-$\beta $ decay is investigated. This subject is crucial for modeling the energy spectrum of $\beta $ -electrons, which is sensitive to the mass and Majorana nature of neutrinos. The dependence of the obtained results on the atomic number indicates an important role of the Feinberg–Migdal effect in the electron shell excitations. We report the overlap amplitudes of the electron shells of the parent atom and the daughter ion for eleven atoms, the two-neutrino double-$\beta $ decay of which was observed experimentally. In around one-fourth of the cases where the structure of the electron shells is inherited from the parent atom, there is a transition to the ground state or the excited state with the lowest energy. The de-excitation of the daughter ion in the latter scenario is accompanied by the emission of photons in the ultraviolet range, which can serve as an auxiliary signature of double-$\beta $ decay. The average excitation energy of the electron shells ranges between 300 and 800 eV, with the variance ranging from (1.7 keV)$^2$ in calcium to (14 keV)$^2$ in uranium.