Evolution of $\beta$-decay half-lives in stellar environments

$\beta$-decay properties of nuclei are investigated within the relativistic nuclear energy density functional framework by varying the temperature and density, conditions relevant to the final stages of stellar evolution. Both thermal and nuclear pairing effects are taken into account in the description of nuclear properties and in the finite-temperature proton-neutron relativistic quasiparticle random-phase approximation (FT-PNRQRPA) to calculate the relevant allowed and first-forbidden transitions in the $\beta$ decay. The temperature and density effects are studied on the $\beta$-decay half-lives at temperatures $T=0$–$1.5\mathrm{MeV}$ and at densities $\rho Y_e={10}^7\mathrm{g}/{\mathrm{cm}}^3$ and ${10}^9\mathrm{g}/{\mathrm{cm}}^3$. The relevant Gamow-Teller transitions are also investigated for Ti, Fe, Cd, and Sn isotopic chains at finite-temperatures. We find that the $\beta$-decay half-lives increase with increasing density $\rho Y_e$, whereas half-lives generally decrease with increasing temperature. It is shown that the temperature effects decrease the half-lives considerably in nuclei with longer half-lives at zero temperature, while only slight changes for nuclei with short half-lives are obtained. We also show the importance of including the de-excitation transitions in the calculation of the $\beta$-decay half-lives at finite-temperatures. Comparing the FT-PNQRPA results with the shell-model calculations for $pf$-shell nuclei, a reasonable agreement is obtained for the temperature dependence of $\beta$-decay rates. Finally, large-scale calculations of $\beta$-decay half-lives are performed at temperatures $T_9\left(\text{K}\right)=5$ and $T_9\left(\text{K}\right)=10$ and densities $\rho Y_e={10}^7$ and ${10}^9\mathrm{g}/{\mathrm{cm}}^3$ for even-even nuclei in the range $8\le Z\le 82$, relevant for astrophysical nucleosynthesis mechanisms.

Figure 1

Comparison between $\beta$-decay half-lives calculated using the FT-PNRQRPA with the ${\mathrm{D}3\mathrm{C}}^{*}$ interaction using $g^{\prime }=0.76$ (black full line) and the experimental data from Refs. [76, 77] (red dashed line) for Ti, Fe, Cd, and Sn isotopic chains. Additionaly, half-lives for the Sn chain obtained by setting $g^{\prime }=0.5$ are also shown (green full line).

Figure 2

[(a)–(d)] Total $\beta$-decay half-lives $T_{1/2}$ for Ti, Fe, Cd, and Sn isotopes as a function of temperature, including negative-energy transitions (de-excitations). The calculations are performed using the FT-PNRQRPA and the ${\mathrm{D}3\mathrm{C}}^{*}$ functional. Red triangles denote results of the FT-RTBA calculation for ${}^{132}\mathrm{Sn}$ from Ref. [38]. [(e)–(h)] Ratio between $\beta$-decay rate due to negative-energy transitions ${\lambda }_{\beta }^{+}$ and total $\beta$-decay rate ${\lambda }_{\beta }$. All calculations are performed at stellar density $\rho Y_e={10}^7\mathrm{g}/{\mathrm{cm}}^3$.

Figure 3

Temperature dependence of $\beta$-decay rates ${\lambda }_{\beta }$ for both allowed ($1^{+}$) and first-forbidden transitions ($0^{-},1^{-},2^{-}$) together with their total sum (Total) for ${}^{54}\mathrm{Ti}, {}^{62}\mathrm{Fe}, {}^{120}\mathrm{Cd}$, and ${}^{132}\mathrm{Sn}$. Calculations are performed at stellar density $\rho Y_e={10}^7\mathrm{g}/{\mathrm{cm}}^3$.

Figure 4

The temperature evolution of the Gamow-Teller strength, located within the $Q_{\beta }$ window, in ${}^{54}\mathrm{Ti}$ [(a)–(e)] and ${}^{62}\mathrm{Fe}$ [(g)–(k)] at temperatures in the range $T=0$–$1.5\mathrm{MeV}$ with respect to the excitation energy of parent nucleus. The neutron-proton chemical potential difference ${\lambda }_{np}={\lambda }_n-{\lambda }_p$ is labeled by the black dashed line and it separates the negative-energy transitions (determined by ${\mathrm{GT}}^{+}$ strength located at $E<{\lambda }_{np}$) from ${\mathrm{GT}}^{-}$ strength (located at $E>{\lambda }_{np})$. Within the panels (a)–(e) [and (g)–(k)], thick lines denote the Gamow-Teller strength B(GT) while thin lines represent the weighting prefactors ${(1-e^{-\beta (E-{\lambda }_{np})})}^{-1}$. In panels (f) and (l), the cumulative sum of half-lives is shown obtained by restricting the summation in Eq. (4) for ${}^{54}\mathrm{Ti}$ and ${}^{62}\mathrm{Fe}$, respectively.

Figure 5

$\beta$-decay rates ${\lambda }_{\beta }$ for selected nuclei in the temperature range $T=0$–$2\mathrm{MeV}$ for densities $\rho Y_e={10}^7\mathrm{g}/{\mathrm{cm}}^3$ (upper panels) and $\rho Y_e={10}^9\mathrm{g}/{\mathrm{cm}}^3$ (lower panels). The FT-PNRQRPA calculations based on the ${\mathrm{D}3\mathrm{C}}^{*}$ interaction (black dashed line) and DD-ME2 interaction (blue dash-dotted line) are shown together with the LSSM (red squares) [20] and shell-model calculations based on the pf-GXPF1J interaction (green triangles) [84, 85].

Figure 6

Left panel: the $\beta$-decay half-lives $T_{1/2}$ of nuclei at zero temperature. Large-scale calculations are performed for even-even nuclei in the range $8\le Z\le 82$, and the stellar density is fixed to $\rho Y_e={10}^7\mathrm{g}/{\mathrm{cm}}^3$. The even-even nuclei are shown side-by-side for demonstration purposes. Right panel: The same but the density is fixed to $\rho Y_e={10}^9\mathrm{g}/{\mathrm{cm}}^3$.

Figure 7

[(a), (b)] The percentage change of the $\beta$-decay half-lives $T_{1/2}$ at finite temperatures $T_9\left(\text{K}\right)=5, T_9\left(\text{K}\right)=10$ with respect to $T_9\left(K\right)=0.01$ (zero-temperature) results. Large-scale calculations are performed for even-even nuclei in the range $8\le Z\le 82$, and the stellar density is fixed to $\rho Y_e={10}^7\mathrm{g}/{\mathrm{cm}}^3$. The even-even nuclei are shown side by side for demonstration purposes. [(c), (d)] The same but the density is fixed to $\rho Y_e={10}^9\mathrm{g}/{\mathrm{cm}}^3$.