Review of Gamow-Teller and Fermi Transition Strength Functions
We studied the temperature effect in isospin-singlet pairings in Gamow-Teller excitations. We use theories of a hole-particle in the mean field shell model studied decay transition using the one-particle-one-hole model for the $β$-decay of odd-even isotopes and the two-particle-hole models for the $β$-decay of even-even and/or odd-odd isotopes. Our reference isotopes for the one-particle-one-hole model are \ce{^{15}O}, \ce{^{15}N}, \ce{^{17}F}, and \ce{^{41}Sc}, whereas for the two-particle-hole model we use \ce{^{16}N} (for $β^-$-decay) and \ce{^{56}Ni} and \ce{^{40}Sc} (for $β^+$/EC). The calculations involve evaluating the matrix elements of Gamow -Teller and Fermi transitions, then calculate the reduced transition probabilities of Gamow-Teller and Fermi, from which we evaluate the half-lives and the strength function $ft$. The results are compared with the available experimental data. For one-particle-one-hole model we found there is a deviation from experimental values which indicates that the model is not valid for beta decay for the even-even nuclei in the ground state due to the residual nucleon-nucleon interaction. As for a two-particle-hole model, we calculated the transition amplitude, from which we calculated the strength of the transition $\log ft$ values. We found an excellent agreement between experimental and theoretical results. By drawing the relationship between temperature versus $\log ft$ values, we found the general trend is that the strength function values slowly decrease as temperatures increases. There are fluctuations $\log ft$ due to the strongly dependent of $\log ft$ on the shell configuration of the valence nucleons.