Stellar electron-capture rates on nuclei based on a microscopic Skyrme functional

We compute electron-capture rates for ${}^{54,56}$Fe and Ge isotopes using a self-consistent microscopic approach. The single-nucleon basis and the occupation factors in the target nucleus are calculated in the finite-temperature Skyrme Hartree-Fock model, and the $J^{\pi }=0^{\pm }$, $1^{\pm }$, $2^{\pm }$ charge-exchange transitions are determined in the finite-temperature random-phase approximation (RPA). The scheme is self-consistent; i.e., both the Hartree-Fock and the RPA equations are based on the same Skyrme functional. Several interactions are used in order to provide a theoretical uncertainty on the electron-capture rates for different astrophysical conditions. Comparing electron-capture rates obtained either with different Skyrme sets or with different available models indicates that differences up to one to two orders of magnitude can arise.

Figure 1

Electron-capture cross sections for ${}^{54,56}$Fe for various temperatures as a function of the incident electron energy, for the Skyrme forces SLy4, SGII, SkM*, and BSk17. The FTSHF+RPA results are compared with cross sections obtained by the SMMC [30] and the FTRRPA [16] calculations.

Figure 2

Electron-capture cross sections for ${}^{76,78,80}$Ge for various temperatures as a function of the incident electron energy, for the Skyrme forces SLy4, SGII, SkM*, and BSk17. For comparison, the cross sections obtained by the SMMC+RPA [12] and the TQRPA [17] calculations are also shown.

Figure 3

Electron-capture cross sections for ${}^{76,78}$Ge for $T=1$ and 2 MeV, as a function of the incident electron energy, for the Skyrme forces SLy4, SGII, SkM*, and BSk17. For comparison, the cross sections obtained by the FTRRPA [16] calculations are also shown.

Figure 4

Occupation numbers in ${}^{76}$Ge as a function of temperature for the (a) $1f_{7/2}$ and the (b) $1g_{9/2}$ proton orbitals and for the (c) $1f_{5/2}$ and the (d) $1g_{7/2}$ neutron orbitals. The occupation numbers for the Skyrme forces SLy4, SGII, SkM*, and BSk17 are compared with those obtained by the SMMC and TQRPA models.

Figure 5

Electron-capture cross sections for Ge $A=70–80$ isotopes for various temperatures as a function of the incident electron energy, for the Skyrme forces SLy4, SGII, SkM*, and BSk17. The dotted lines correspond to $A=70$, while the solid lines correspond to $A=80$, and the other isotopes are between these boundaries.

Figure 6

Occupation numbers in ${}^{70,72,74}$Ge as a function of temperature, for the Skyrme forces SLy4 (left panels) and SGII (right panels), for the [(a) and (b)] $2p_{3/2}$ proton orbitals and for the [(c) and (d)] $2p_{1/2}$ neutron orbitals.

Figure 7

Electron-capture rates for ${}^{54,56}$Fe for different stellar conditions as a function of temperature, for the Skyrme forces SLy4, SGII, SkM*, and BSk17. The FTSHF+RPA results are compared with the rates obtained with the SMMC [9], the FTRRPA [16], and the TQRPA [17] calculations.

Figure 8

Same as in Fig. 7, but for ${}^{76,78,80}$Ge.

Figure 9

Electron-capture rates for Ge $A=70–80$ isotopes for different stellar conditions as a function of temperature, for the Skyrme forces SLy4, SGII, SkM*, and BSk17. The dotted lines correspond to $A=70$, while the solid lines correspond to $A=80$, and the other isotopes are between these boundaries.