Calculation of stellar electron-capture cross sections on nuclei based on microscopic Skyrme functionals

A fully self-consistent microscopic framework for the evaluation of nuclear weak-interaction rates at finite temperature is introduced, based on Skyrme functionals. The single-nucleon basis and the corresponding thermal occupation factors of the initial nuclear state are determined in the finite-temperature Skyrme Hartree-Fock model and charge-exchange transitions to excited states are computed using the finite-temperature random-phase approximation (RPA). Effective interactions are implemented self-consistently: Both the finite-temperature single-nucleon Hartree-Fock equations and the matrix equations of RPA are based on the same Skyrme energy density functional. Using a representative set of Skyrme functionals, the model is applied in the calculation of stellar electron-capture cross sections for selected nuclei in the iron mass group and for neutron-rich Ge isotopes.

Figure 1

Occupation percentage of the proton orbitals $f_{7/2}$, $p_{3/2}$, $f_{5/2}$, $p_{1/2}$, and $g_{9/2}$ in ${}^{74}\mathrm{Ge}$, calculated in the finite-temperature Skyrme HF model with the SGII interaction, at zero temperature, $T=0.5$ MeV, and $T=1.3$ MeV.

Figure 2

Occupation percentage of the neutron orbitals $f_{7/2}$, $p_{3/2}$, $f_{5/2}$, $p_{1/2}$, and $g_{9/2}$ in ${}^{74}\mathrm{Ge}$, calculated in the finite-temperature Skyrme HF model with the SGII interaction, at zero temperature, $T=0.5$ MeV, and $T=1.3$ MeV.

Figure 3

Temperature dependence of GT${}^{-}$ and GT${}^{+}$ strength distributions in ${}^{74}\mathrm{Ge}$, calculated with the finite-temperature proton-neutron RPA model based on the Skyrme SGII interaction.

Figure 4

Electron-capture cross sections for the reaction ${}^{56}\mathrm{Fe}$$(e^{-},{\nu }_e)$${}^{56}\mathrm{Mn}$ at $T=0.5$ MeV as functions of the incident electron energy $E_e$. The Skyrme $\mathrm{HF}+\mathrm{RPA}$ results are compared with cross sections calculated from the SMMC GT strength distributions [9].

Figure 5

Same as Fig. 4, but for the reaction ${}^{48}\mathrm{Ti}$$(e^{-},{\nu }_e)$${}^{48}\mathrm{Sc}$.

Figure 6

Same as Fig. 4, but for the reaction ${}^{50}\mathrm{Cr}$$(e^{-},{\nu }_e)$${}^{50}\mathrm{V}$.

Figure 7

Electron-capture cross section for the ${}^{56}\mathrm{Fe}$ target nucleus at $T=0.5$ MeV, calculated with the $\mathrm{FTHF}+\mathrm{FTRPA}$ using the SLy5 Skyrme effective interaction. In addition to the total cross section, which includes multipole transitions $J^{\pi }=0^{\pm }$, $1^{\pm }$, and $2^{\pm }$, contributions from the individual channels are shown in the plot as functions of the incident electron energy $E_e$.

Figure 8

Electron-capture cross sections for the even-even Ni target nuclei ($A=56–62$) at $T=0.5$ MeV, calculated with the $\mathrm{FTHF}+\mathrm{FTRPA}$ using the SLy5 Skyrme effective interaction.

Figure 9

Electron-capture cross sections for the reaction ${}^{72}\mathrm{Ge}$$(e^{-},{\nu }_e)$${}^{72}\mathrm{Ga}$ at $T=0.5$ MeV as functions of the incident electron energy $E_e$. The Skyrme $\mathrm{HF}+\mathrm{RPA}$ results are compared with cross sections calculated with the hybrid SMMC/RPA model [10].

Figure 10

Same as Fig. 9, but for the temperature $T=1.3$ MeV.

Figure 11

Same as Fig. 9, but for ${}^{68}\mathrm{Ge}$ at $T=1.3$ MeV.

Figure 12

Same as Fig. 9, but for ${}^{76}\mathrm{Ge}$ at $T=1.3$ MeV.

Figure 13

Electron-capture cross sections for the ${}^{76}\mathrm{Ge}$ target nucleus at temperatures $T=0.5$, 1.3, and 2.0 MeV, calculated with the $\mathrm{FTSHF}+\mathrm{FTRPA}$ using the SLy5 Skyrme effective interaction.