$\beta$-decay half-lives at finite temperatures for $N=82$ isotones

Using the finite-temperature quasiparticle random phase approximation (FTQRPA) on the basis of the finite-temperature Skyrme-Hartree-Fock$\hspace{0.5em}+\hspace{0.5em}$Bardeen-Cooper-Schrieffer (BCS) method, we study ${\beta }^{-}$-decay half-lives for even-even neutron magic nuclei with $N=82$ in a finite-temperature environment. We find that the ${\beta }^{-}$-decay half-life first decreases as the temperature increases for all the nuclei we study, although the thermal effect is found to be small at temperatures relevant to $r$-process nucleosynthesis. Our calculations indicate that the half-life begins to increase at high temperatures for open-shell nuclei. We discuss this behavior in connection to the pairing phase transition.

Figure 1

A schematic illustration for the excited states in QRPA. At zero temperature (left panel), excited states correspond to two-quasiparticle (2qp) states, while one-quasiparticle one-quasihole (1qp-1qh) states are also involved at finite temperatures (right panel). The occupation probability of one-quasiparticle states is given by $f_i(T)$ in Eq. (1).

Figure 2

A schematic illustration for the excitation scheme in FTQRPA. Transitions from the ground state correspond to a two-quasiparticle excitation, ${\alpha }^{†}{\alpha }^{†}$, while the transitions among two-quasiparticle states are described by the operator ${\alpha }^{†}\alpha$.

Figure 3

The GT strength functions for the ${}^{122}\mathrm{Zr}$, ${}^{126}\mathrm{Ru}$, and ${}^{130}\mathrm{Cd}$ nuclei at $T=0.3$ (left panels), $T=0.6$ (middle panels), and $T=0.8$ MeV (right panels) compared to $T=0.0$ MeV. These are plotted as a function of $E_m^{*}-E_i$, where $E_i$ and $E_m^{*}$ are the energy of the initial and final states, respectively. For ${}^{126}\mathrm{Ru}$ and ${}^{122}\mathrm{Zr}$, the strengths at $T=0.3$ MeV are scaled by a factor of 16. For ${}^{130}\mathrm{Cd}$, the strength at $T=0.6$ and $0.8$ MeV is scaled by one-third.

Figure 4

A schematic illustration for the ${\beta }^{-}$-decay scheme at finite temperatures. The transitions at zero temperature are indicated by the thick solid arrows; the additional transitions at finite temperatures are indicated by the thin solid arrows.

Figure 5

(Left panel): The $\beta$-decay half-lives for even-even Cd isotopes at zero temperature. The solid, dashed, and dotted lines show the results with the SLy5, SkM${}^{*}$, and the experimental data [42], respectively. (Right panel): The $\beta$-decay half-lives for even-even isotones with neutron number $N=82$ at zero temperature. The solid and dashed lines show the results of the QRPA with the SLy5 and SkM${}^{*}$ parameter sets, respectively. The dotted and dotted-dashed lines show the results of the FRDM$\hspace{0.5em}+\hspace{0.5em}$QRPA [7] and the shell-model calculations [3], respectively.

Figure 6

The $\beta$-decay half-life $T_{1/2}$ normalized to that at zero temperature, $T_{1/2}^0$. The solid and the dashed lines show the results with the SLy5 and SkM${}^{*}$ parameter sets, respectively.

Figure 7

The average proton pairing gap as a function of temperature for the ${}^{130}\mathrm{Cd}$, ${}^{126}\mathrm{Ru}$, and ${}^{122}\mathrm{Zr}$ nuclei. The top and bottom panels are results for the SLy5 and SkM${}^{*}$ parameter sets, respectively.