Exchange contribution to the variance of the excitation energy of the electron shell of the daughter atom in double β decay

The excitation of electron shell of a daughter atom in a neutrinoless ββ decay changes the shape of the total energy peak of β electrons at the endpoint of the energy spectrum. The main parameters of the modified distribution are the average energy and the variance of the electron shell excitation energy. An expression is derived for the variance taking into account exchange effects, and numerical estimates of the average excitation energy and variance are made based on the non-relativistic Roothaan–Hartree–Fock method and the relativistic Dirac–Hartree–Fock method implemented in the General Relativistic Atomic Structure Package (Grasp2018). The estimates are made for eleven isotopes, the two–neutrino double–β decay of which is observed experimentally. The results are determined by the first two negative moments of the electron radii in the parent atom. The values obtained can be used to parameterize the energy distribution of β electrons, taking into account the electron shell excitation of the daughter atom.