the mother nucleus to the state ${\left| f
\right\rangle} $ of the daughter is related to the reduced transition probability $B_{ij} $ of the nuclear transition by

$$\label{GrindEQ__8_}
(ft)_{ij} \, =\, {D\mathord{\left/ {\vphantom {D B_{ij} .}} \right.
\kern-\nulldelimiterspace} B_{ij}.}$$

The D appearing in Eq. 8 is compound expression of physical constants,

$$\label{GrindEQ__9_}
D\, =\, \frac{2\pi ^{3} \hbar ^{7} \ln 2}{g_{V}^{2} m_{e}^{5} c^{4}
},$$

and the reduced transition probability of the nuclear transition is

$$\label{GrindEQ__10_}
B_{ij} \, =\, B(F)_{ij} \, +\, \left({g_{A} \mathord{\left/
{\vphantom {g_{A}  g_{V} }} \right. \kern-\nulldelimiterspace} g_{V}
} \right)^{2} B(GT)_{ij},$$

The value of D = 6146 $\pm$ 6 s [@Jokinen02] is adopted and the ratio of the axial-vector $(g_{A} )$ to the vector $(g_{V} )$ coupling constant is taken as -1.257. $B(F)_{ij} $ and $B(GT)_{ij} $ are reduced transition probabilities of the Fermi and GT transitions, respectively. These reduced transition probabilities of the nuclear transition are given by,

$$\label{GrindEQ__11_}
B(F)_{ij} \, =\, \frac{1}{2J_{i} +1} |<j||\sum _{k}t_{\pm }^{k} ||\,
i>|^{2},$$

$$\label{GrindEQ__12_} B(GT)_{ij} \, =\, \frac{1}{2J_{i} +1}
|<j||\sum _{k}t_{\pm }^{k} \vec{\sigma }^{k} ||\,  i>|^{2}.$$

Calculation of phase space integrals can be seen from Ref. [@Nabi99].

The number density of electrons associated with protons and nuclei is $\rho Y_{e} N_{A} ,$ 