Unblocking of stellar electron capture for neutron-rich $N=50$ nuclei at finite temperature

We have calculated electron-capture rates for neutron-rich $N=50$ nuclei (${}^{78}\mathrm{Ni}, {}^{82}\mathrm{Ge}, {}^{86}\mathrm{Kr}, {}^{88}\mathrm{Sr}$) within the thermal quasiparticle random-phase approximation approach at temperatures $T=0$, corresponding to capture on the ground state, and at $T=10\mathrm{GK}$ (0.86 MeV), which is a typical temperature at which the $N=50$ nuclei are abundant during a supernova collapse. In agreement with recent experiments, we find no Gamow-Teller (${\mathrm{GT}}_{+}$) strength at low excitation energies, $E<7\mathrm{MeV}$, caused by Pauli blocking induced by the $N=50$ shell gap. At the astrophysically relevant temperatures, this Pauli blocking of the ${\mathrm{GT}}_{+}$ strength is overcome by thermal excitations across the $Z=40$ proton and $N=50$ neutron shell gaps, leading to a sizable GT contribution to the electron capture. At the high densities, at which the $N=50$ nuclei are important for stellar electron capture, forbidden transitions contribute noticeably to the capture rate. Our results indicate that the neutron-rich $N=50$ nuclei do not serve as an obstacle of electron capture during supernova collapse.

Figure 1

Strength distributions of ${\mathrm{GT}}_{+}$ transitions in ${}^{78}\mathrm{Ni}, {}^{82}\mathrm{Ge}, {}^{86}\mathrm{Kr}$, and ${}^{88}\mathrm{Sr}$ at $T=0$ and $T=10\mathrm{GK}$ (0.86 MeV). The dashed vertical lines indicate the ground-state thresholds $Q=M_f-M_i$: $Q\left({}^{78}\mathrm{Ni}\right)=20.7\mathrm{MeV}, Q\left({}^{82}\mathrm{Ge}\right)=13.0\mathrm{MeV}, Q\left({}^{86}\mathrm{Kr}\right)=8.1\mathrm{MeV}$, and $Q\left({}^{88}\mathrm{Sr}\right)=5.8\mathrm{MeV}$ [54]. $A$ and $B$ label specific ${\mathrm{GT}}_{+}$ transitions: $A\equiv f_{7/2}^p\to f_{5/2}^n, B\equiv g_{9/2}^p\to g_{7/2}^n$. The notations $A\times 10$ and $B\times 10$ mean that the respective peaks are scaled by a factor of 10 for demonstration purposes.

Figure 2

Neutron ($f_{5/2}, g_{7/2}$) and proton ($f_{7/2}, g_{9/2}$) single-particle energies $E_j$ relative to the chemical potentials ${\lambda }_{n,p}$ for $T=10\mathrm{GK}$. The respective quasiparticle energies are given by ${\varepsilon }_j=|E_j-\lambda |$.

Figure 3

Electron-capture cross sections for ${}^{78}\mathrm{Ni}, {}^{82}\mathrm{Ge}, {}^{86}\mathrm{Kr}$, and ${}^{88}\mathrm{Sr}$ at $T=0,10,20\mathrm{GK}$. The blue (light gray) and brown (dark gray) curves represent results obtained with the ${\mathrm{SkM}}^{*}$ and ${\mathrm{SkO}}^{\prime }$ interactions, respectively. Like in Fig. 1, the dashed vertical lines indicate the ground-state thresholds $Q=M_f-M_i$.

Figure 4

Electron-capture rates calculated at $T=10\mathrm{GK}$ (0.86 MeV) as a function of density. The upper axis indicates the corresponding electron chemical potentials ${\mu }_e$. The blue (light gray) and brown (dark gray) lines are based on the Skyrme-TQRPA calculations with the ${\mathrm{SkM}}^{*}$ and ${\mathrm{SkO}}^{\prime }$ interactions, respectively. The total rates (full lines) include the contribution of allowed ($0^{+},1^{+}$) and first-forbidden ($0^{-},1^{-},2^{-}$) transitions. The dashed-dotted lines (labeled $1^{+}$) represent the unblocked ${\mathrm{GT}}_{+}$ contributions to the rates. The rates, indicated by dashed lines, have been calculated from the ground-state ${\mathrm{GT}}_{+}$ distribution. The shaded bands represent the results based on the experimental ${\mathrm{GT}}_{+}$ data [32, 33]. The labeled lines represent the rates calculated according to the parametrization (6).