Exchange contribution to the variance of the excitation energy of the electron shell of the daughter atom in double-$\beta$ decay

TL;DR

This paper derives an expression for the variance of electron shell excitation energy in double-beta decay, incorporating exchange effects, and provides numerical estimates for various isotopes to improve energy spectrum modeling.

Contribution

It introduces a new expression for the variance of excitation energy that accounts for exchange effects, with numerical estimates for multiple isotopes using advanced atomic structure methods.

Findings

Derived an expression for excitation energy variance including exchange effects.

Numerical estimates for eleven isotopes using non-relativistic and relativistic methods.

Results can parameterize beta electron energy distribution considering shell excitation.

Abstract

The excitation of electron shell of a daughter atom in a neutrinoless double $\beta$-decay causes change in shape of the total energy peak of $\beta$ electrons at the end of the energy spectrum. The main parameters of the modified distribution are the average energy and variance of the excitation energy of the electron shell. We derive an expression for the variance taking into account exchange effects and make numerical estimates of the average excitation energy and variance 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). Estimates are made for eleven isotopes, two-neutrino double-$\beta$ 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 for change in the peak shape can be used to parameterize the energy distribution of $\beta$ electrons, taking into account the excitation of the electron shell of the daughter atom.