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 the determining 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 $(1.7~\mathrm{keV})^2$ in calcium to $(14~\mathrm{keV})^2$ in uranium.