New \(Q\)-\(\beta\)-Decay Theory Applied to the Calculations of the Rest Mass Energy \(m_0c^2\) of the Electron and of the \(Q\)-Value for \(\beta^+\)-Decay Transitions in Mirrors Nuclei \(A = 2Z - 1\)

Abstract

A new nuclear approach named \(Q\)-\(\beta\)-decay theory \((Q\beta T)\) is presented. This method is applied to express theoretically the \(\beta^+\)-\(Q\)-value for mirrors nuclei \(A = 2Z-1\). An important parameter named nuclear charge distribution coefficient (NCDC) noted \(\alpha (Z)\) is presented. In the framework of the liquid drop nuclear model where the nuclear charge is uniformly distributed in the nuclear volume, \(\alpha (Z) = 3/5\). It this work, it is demonstrated that the protons are not rigorously uniformly distributed within the nucleus. A slight correction is obtained with \(\alpha (Z) \approx \alpha_0 = 3/5 + 0.0557\). For \({}^{37}\)K and \({}^{65}\)As, we find for the electron rest mass energy \(m_0c^2=0.510\) 996 MeV agreeing excellently with the recommended value 0.510 998 950 MeV (CODATA, 2022). In addition, the \(Q\)-value calculated for \(A\) ranging between 11 and 99 agree very well with the recent Atomic Mass Evaluation -- AME2020 (Wang et al. [15]). New accurate $Q$-values are tabulated for nuclei mirrors with \(A\) ranging between 101 and 199. The present \(Q\)-\(\beta\)-DT make it possible to understand many nuclear properties and phenomena depending on the \(Q\)-value such as comparison between experimental and theoretical predictions of atomic masses, the understanding of the weak force and of the competitive processes between electron capture and \(\beta^+\)-decay.