The Decay Q Value of Neutrinoless Double Beta Decay

An earlier publication "The implication of the atomic effects in neutrinoless double beta (0$\nu\beta\beta$) decay" written by Mei and Wei has motivated us to compare the decay Q value ($Q_{\beta\beta}$) derived from the decay of the parent nucleus to the daughter nucleus with the two ejected beta particles in the final state to the $Q_{\beta\beta}$ directly derived from the decay of the initial neutral atom to the final state of double-ionized daughter ion with the two ejected beta particles in the final state. We show that the results are the same, which is the mass-energy difference ($\Delta Mc^2$) subtracted by the total difference of the atomic electron binding energy ($\Delta E_{b}$) between the ground states of initial and final neutral atoms. We demonstrate that $\Delta Mc^2$ is the sum of $Q_{\beta\beta}$ and the atomic relaxation energy ($\Delta E_{b}$) of the atomic structure after the decay. Depending on the atomic relaxation time, the release of the atomic binding energy may not come together with the energy deposition of the two ejected beta particles.