Gamow-Teller strengths and electron-capture rates for $pf$-shell nuclei of relevance for late stellar evolution

Background: Electron-capture reaction rates on medium-heavy nuclei are an important ingredient for modeling the late evolution of stars that become core-collapse or thermonuclear supernovae. The estimation of these rates requires the knowledge of Gamow-Teller strength distributions in the ${\beta }^{+}$ direction. Astrophysical models rely on electron-capture rate tables largely based on theoretical models, which must be validated and tested against experimental results.

Purpose: This paper presents a systematic evaluation of the ability of theoretical models to reproduce experimental Gamow-Teller transition strength distributions measured via ($n$,$p$)-type charge-exchange reactions at intermediate beam energies. The focus is on transitions from stable nuclei in the $pf$ shell ($45\le A\le 64$). In addition, the impact of deviations between experimental and theoretical Gamow-Teller strength distributions on derived stellar electron-capture rates is investigated.

Method: Data on Gamow-Teller transitions from 13 nuclei in the $pf$ shell measured via charge-exchange reactions and supplemented with results from $\beta$-decay experiments where available, were compiled and compared with strength distributions calculated in shell models (using the GXPF1a and KB3G effective interactions) and quasiparticle random-phase approximation (QRPA) using ground-state deformation parameters and masses from the finite-range droplet model. Electron-capture rates at relevant stellar temperatures and densities were derived for all distributions and compared.

Results: With few exceptions, shell-model calculations in the $pf$ model space with the KB3G and GXPF1a interactions qualitatively reproduce experimental Gamow-Teller strength distributions of 13 stable isotopes with $45\le A\le 64$. Results from QRPA calculations exhibit much larger deviations from the data and overestimate the total experimental Gamow-Teller strengths. For stellar densities in excess of ${10}^7$ g/cm${}^3$, ground-state electron-capture rates derived from the shell-model calculations using the KB3G (GXPF1a) interaction deviate on average less than 47$\%$ (31$\%$) from those derived from experimental data for which the location of daughter states at low excitation energies are well established. For electron-capture rates derived from Gamow-Teller strengths calculated in QRPA, the deviations are much larger, especially at low stellar densities.

Conclusions: Based on the limited set of test cases available for nuclei in the $pf$ shell, shell-models using the GXPF1a and KB3G interactions can be used to estimate electron-capture rates for astrophysical purposes with relatively good accuracy. Measures of the uncertainties in these rates can serve as input for sensitivity studies in stellar evolution models. Ground-state electron-capture rates based on the QRPA formalism discussed in the paper exhibit much larger deviations than those based on the shell-model calculations and should be used with caution, especially at low stellar densities.

Figure 1

GT transitions from ${}^{45}$Sc to ${}^{45}$Ca: (a) from ($n$,$p$) and $\beta$-decay data, (b) from calculations in QRPA (divided by a factor of 3), (c) from shell-model calculation using the KB3G interaction, and (d) from shell-model calculations using the GXPF1a interaction. In (e), running sums of $B\left(\mathrm{GT}\right)$ as a function of excitation energy are plotted. Here, and all similar graphs, the shell-model strengths have been scaled by ${\left(0.74\right)}^2$ (see text).

Figure 2

Electron-capture rates for ${}^{45}$Sc($e^{-},{\nu }_e$)${}^{45}$Ca based on experimental and theoretical GT strength distribution from Fig. 1, plotted as a function of temperature at $\rho Y_e={10}^7$ g/cm${}^3$ (a) and $\rho Y_e={10}^9$ g/cm${}^3$ (b).

Figure 3

GT transitions from ${}^{48}$Ti to ${}^{48}$Sc: (a) from ($n$,$p$), (b) from ($d$,${}^2$He) data, (c) from calculations in QRPA (divided by a factor of 6), (d) from shell-model calculations using the KB3G interaction, and (e) from shell-model calculations using the GXPF1a interaction. In (f), running sums of $B\left(\mathrm{GT}\right)$ as a function of excitation energy are plotted.

Figure 4

Electron-capture rates for ${}^{48}$Ti($e^{-},{\nu }_e$)${}^{48}$Sc, based on experimental and theoretical GT strength distributions shown in Fig. 3, plotted as a function of temperature at two densities, $\rho Y_e={10}^7$ g/cm${}^3$ (a) and $\rho Y_e={10}^9$ g/cm${}^3$ (b). Note that the rates based on the shell-model calculations with the KB3G and GXPF1a interactions overlap.

Figure 5

GT transitions from ${}^{50}$V to ${}^{50}$Ti: (a) from ($d$,${}^2$He) data (a known but unobserved $6^{+}$ state at 3.198 MeV is indicated), (b) from calculations in QRPA (divided by a factor of 3), (c) from shell-model calculations using the KB3G interaction, and (d) from shell-model calculations using the GXPF1a interaction. In (e), running sums of $B\left(\mathrm{GT}\right)$ as a function of excitation energy are plotted.

Figure 6

Electron-capture rates for ${}^{50}$V($e^{-},{\nu }_e$)${}^{50}$Ti based on experimental and theoretical strength shown in Fig. 5, plotted as a function of temperature at two densities, $\rho Y_e={10}^7$ g/cm${}^3$ (a) and $\rho Y_e={10}^9$ g/cm${}^3$ (b).

Figure 7

GT transitions from ${}^{51}$V to ${}^{51}$Ti: (a) from ($n$,$p$) data, (b) from ($d$,${}^2$He) data, (c) from calculations in QRPA (divided by a factor of 4), (d) from shell-model calculations using the KB3G interaction, and (e) from shell-model calculations using the GXPF1a interaction. In (f), running sums of $B\left(\mathrm{GT}\right)$ as a function of excitation energy are plotted.

Figure 8

Electron-capture rates for ${}^{51}$V($e^{-},{\nu }_e$)${}^{51}$Ti, based on experimental and theoretical GT strength distributions shown in Fig. 7, plotted as a function of temperature at two densities, $\rho Y_e={10}^7$g/cm${}^3$ (a) and $\rho Y_e={10}^9$ g/cm${}^3$ (b). Note that the rates based on the shell-model calculations with the KB3G and GXPF1a interactions overlap.

Figure 9

GT transitions from ${}^{54}$Fe to ${}^{54}$Mn: (a) from ($n$,$p$) data, (b) from calculations in QRPA (divided by a factor of 5), (c) from shell-model calculations using the KB3G interaction, and (d) from shell-model calculations using the GXPF1a interaction. In (e), running sums of $B\left(\mathrm{GT}\right)$ as a function of excitation energy are plotted.

Figure 10

Electron-capture rates for ${}^{54}$Fe($e^{-},{\nu }_e$)${}^{54}$Mn based on experimental and theoretical GT strengths shown in Fig. 9 plotted as a function of temperature at two densities, $\rho Y_e={10}^7$ g/cm${}^3$ (a) and $\rho Y_e={10}^9$ g/cm${}^3$ (b).

Figure 11

GT transitions from ${}^{55}$Mn to ${}^{55}$Cr: (a) from ($n$,$p$) and and $\beta$-decay data, (b) from calculations in QRPA, (c) from shell-model calculations using the KB3G interaction, and (d) from shell-model calculations using the GXPF1a interaction. In (e), running sums of $B\left(\mathrm{GT}\right)$ as a function of excitation energy are plotted.

Figure 12

Electron-capture rates for ${}^{55}$Mn($e^{-},{\nu }_e$)${}^{55}$Cr based on experimental and theoretical GT strengths shown in Fig. 11, plotted as a function of temperature at two densities, $\rho Y_e={10}^7$ g/cm${}^3$ (a) and $\rho Y_e={10}^9$ g/cm${}^3$ (b).

Figure 13

GT transitions from ${}^{56}$Fe to ${}^{56}$Mn: (a) from ($n$,$p$) data, (b) from calculations in QRPA (divided by a factor of 3), (c) from shell-model calculations using the KB3G interaction, and (d) from shell-model calculations using the GXPF1a interaction. In (e), running sums of $B\left(\mathrm{GT}\right)$ as a function of excitation energy are plotted.

Figure 14

Electron-capture rates for ${}^{56}$Fe($e^{-},{\nu }_e$)${}^{56}$Mn based on experimental and theoretical GT strengths plotted in Fig. 13, as a function of temperature at two densities, $\rho Y_e={10}^7$ g/cm${}^3$ (a) and $\rho Y_e={10}^9$ g/cm${}^3$ (b).

Figure 15

GT transitions from ${}^{58}$Ni to ${}^{58}$Co: (a) from ($n$,$p$) data, (b) from ($d$,${}^2$He) data, (c) from ($t$,${}^3$He) data, (d) from calculations in QRPA (divided by a factor of 6), (e) from shell-model calculations using the KB3G interaction, and (f) from shell-model calculations using the GXPF1a interaction. In (g), running sums of $B\left(\mathrm{GT}\right)$ as a function of excitation energy are plotted.

Figure 16

Electron-capture rates for ${}^{58}$Ni($e^{-},{\nu }_e$)${}^{58}$Co based on experimental and theoretical GT strengths from Fig. 15, plotted as a function of temperature at two densities, $\rho Y_e={10}^7$ g/cm${}^3$ (a) and $\rho Y_e={10}^9$ g/cm${}^3$ (b).

Figure 17

GT transitions from ${}^{59}$Co to ${}^{59}$Fe: (a) from ($n$,$p$) data, (b) from calculations in QRPA (without additional scaling), (c) from shell-model calculations using the KB3G interaction, and (d) from shell-model calculations using the GXPF1a interaction. In (e), running sums of $B\left(\mathrm{GT}\right)$ as a function of excitation energy are plotted.

Figure 18

Electron-capture rates for ${}^{59}$Co($e^{-},{\nu }_e$)${}^{59}$Fe based on experimental and theoretical GT strengths plotted in Fig. 17, as a function of temperature at two densities, $\rho Y_e={10}^7$ g/cm${}^3$ (a) and $\rho Y_e={10}^9$ g/cm${}^3$ (b).

Figure 19

GT transitions from ${}^{60}$Ni to ${}^{60}$Co: (a) from ($n$,$p$) data, (b) from $T=3$ states observed in ($p$,$n$) data, (c) from calculations in QRPA (divided by a factor of 2), (d) from shell-model calculations using the KB3G interaction, and (e) from shell-model calculations using the GXPF1a interaction. In (f), running sums of $B\left(\mathrm{GT}\right)$ as a function of excitation energy are plotted.

Figure 20

Electron-capture rates for ${}^{60}$Ni($e^{-},{\nu }_e$)${}^{60}$Co for experimental and theoretical GT strengths plotted in Fig. 19, as a function of temperature at two densities, $\rho Y_e={10}^7$ g/cm${}^3$ (a) and $\rho Y_e={10}^9$ g/cm${}^3$ (b).

Figure 21

GT transitions from ${}^{62}$Ni to ${}^{62}$Co: (a) from ($n$,$p$) data, (b) from $T=4$ states observed in ($p$,$n$) data, (c) from calculations in QRPA (divided by a factor of 2), (d) from shell-model calculations using the KB3G interaction, and (e) from shell-model calculations using the GXPF1a interaction. In (f), running sums of $B\left(\mathrm{GT}\right)$ as a function of excitation energy are plotted.

Figure 22

Electron-capture rates for ${}^{62}$Ni($e^{-},{\nu }_e$)${}^{62}$Co for experimental and theoretical GT strengths, plotted as a function of temperature at two densities, $\rho Y_e={10}^7$ g/cm${}^3$ (a) and $\rho Y_e={10}^9$ g/cm${}^3$ (b).

Figure 23

GT transitions from ${}^{64}$Ni to ${}^{64}$Co: (a) from ($n$,$p$) and $\beta$-decay data, (b) from ($d$,${}^2$He) and $\beta$-decay data, (c) from calculations in QRPA, (d) from shell-model calculations using the KB3G interaction, and (e) from shell-model calculations using the GXPF1a interaction. In (f), running sums of $B\left(\mathrm{GT}\right)$ as a function of excitation energy are plotted.

Figure 24

Electron-capture rates for ${}^{64}$Ni($e^{-},{\nu }_e$)${}^{64}$Co for experimental and theoretical GT strengths plotted in Fig. 23, as a function of temperature at two densities, $\rho Y_e={10}^7$ g/cm${}^3$ (a) and $\rho Y_e={10}^9$ g/cm${}^3$ (b).

Figure 25

GT transitions from ${}^{64}$Zn to ${}^{64}$Cu: (a) from ($d$,${}^2$He) and $\beta$-decay data, (b) from ($t$,${}^3$He) and $\beta$-decay data, (c) from calculations in QRPA, (d) from shell-model calculations using the KB3G interaction, and (e) from shell-model calculations using the GXPF1a interaction. In (f), running sums of $B\left(\mathrm{GT}\right)$ as a function of excitation energy are plotted.

Figure 26

Electron-capture rates for ${}^{64}$Zn($e^{-},{\nu }_e$)${}^{64}$Cu for experimental and theoretical GT strengths shown in Fig. 25, plotted as a function of temperature at two densities, $\rho Y_e={10}^7$ g/cm${}^3$ (a) and $\rho Y_e={10}^9$ g/cm${}^3$ (b).

Figure 27

Comparison of EC rates for 13 $pf$-shell nuclei calculated on the basis of theoretical (connected by lines) and experimental (indicated by markers) GT strength distributions. All rates are plotted relative to those calculated by in the shell-model with the KB3G interaction (for which the relative rate thus equals unity). The relative EC rates are shown for two combinations of stellar density and temperature: (a) $\rho Y_e={10}^7$ g/cm${}^3$ and $T=3\times {10}^9$ K and (b) $\rho Y_e={10}^9$ g/cm${}^3$ $T=10\times {10}^9$ K.