Modification of the Brink-Axel hypothesis for high-temperature nuclear weak interactions

We present shell-model calculations of electron capture strength distributions in $A=28$ nuclei and computations of the corresponding capture rates in supernova core conditions. We find that in these nuclei the Brink-Axel hypothesis for the distribution of Gamow-Teller strength fails at low and moderate initial excitation energy but may be a valid tool at high excitation. The redistribution of GT strength at high initial excitation may affect capture rates during collapse. If these trends which we have found in lighter nuclei also apply for the heavier nuclei which provide the principal channels for neutronization during stellar collapse, then there could be two implications for supernova core electron capture physics. First, a modified Brink-Axel hypothesis could be a valid approximation for use in collapse codes. Second, the electron capture strength may be moved down significantly in transition energy, which would likely have the effect of increasing the overall electron capture rate during stellar collapse.

Figure 1

${}^{28}\mathrm{Si}$ density of states per 1 MeV as functions of isospin. The inflection points for $T=0$ and $T=1$ near 30 MeV—which indicate a marked departure from the expected exponential growth—suggest we should be wary of results for states with energies above the mid-20s.

Figure 2

Average GT strength distribution in ${}^{28}\mathrm{Si}$ as a function of initial excitation energy $E_i$. Strength is binned in 0.5-MeV increments of transition energy. Also shown are fits of the distributions to the double Gaussian in Eq. (8). The dependence on initial excitation energy becomes small at high excitation. The distributions in each panel are averages over 2, 7, 42, 156, 435, 239, 296, and 329 initial states, respectively.

Figure 3

Total GT strength in ${}^{28}\mathrm{Si}$ as a function of excitation energy. Each point corresponds to an individual state computed from the shell model, giving a GT sum rule for that state. The black line shows the average total strength in 1-MeV bins.

Figure 4

Gamow-Teller strength distribution in ${}^{28}\mathrm{Si}$ at initial excitation energy $E_i=20–24$ MeV with initial isospin $T_i=0$ as a function of initial spin $J_i$. The strength distribution is not strongly dependent on $J_i$. We saw this trend in all nuclei we studied. The distributions in each panel are averages over 10, 10, and 12 initial states, respectively.

Figure 5

Gamow-Teller strength distribution in ${}^{28}\mathrm{Si}$ at initial excitation energy $E_i=20–24$ as a function of initial isospin $T_i$. The strength distribution has an apparent dependence on initial isospin. The distributions in each panel are averages over 108, 97, and 34 states, respectively.

Figure 6

Strength distribution for ${}^{28}\mathrm{Si}$ with initial isospin $T_i=2$ as a function of final-state isospin. Comparison with Fig. 5 shows that certain features of the total strength distribution are consequences of the distributions to specific final-state isospins. The distributions are averages over 34 initial states.

Figure 7

${}^{28}\mathrm{Si}$ single-particle state occupation for nuclear states with isospin $T=0$. The occupation numbers are the sum of protons and neutrons. The linear dependence on excitation energy of the $d$ orbital occupation numbers is understood to arise from the spin-orbit splitting and particle-hole repulsion energies of those orbitals. This dependence is consistent across all values of $T$, although the intercepts of the $d$ orbitals do shift as $T$ increases.

Figure 8

Total GT strength in ${}^{24}\mathrm{Mg}$ as a function of excitation energy. The black line shows the average total strength, computed from 1-MeV bins.

Figure 9

${}^{24}\mathrm{Mg}$ density of states. The inflection point is near 28 MeV, indicating that we cannot be confident of strength distributions for initial-state energies above the low 20s.

Figure 10

Computed average GT strength distribution in 0.5-MeV transition-energy bins and fits to a double Gaussian in ${}^{24}\mathrm{Mg}$ as a function of initial excitation energy $E_i$. As in the analysis of ${}^{28}\mathrm{Si}$, the fit curves are scaled to account for the choice of transition-energy bin width. Between the $E_i=12–16$-MeV and $20–24$-MeV bins, the strength varies slowly with excitation. The distributions in each panel are averages over 2, 8, 33, 127, 322, 689, 1265, and 1847 initial states, respectively.

Figure 11

${}^{28}\mathrm{Mg}$ density of states. The inflection point at $\sim 22$ MeV predicts that the region where the strength distribution is stable will be narrow.

Figure 12

${}^{28}\mathrm{Mg}$ strength distribution. The distribution stabilizes briefly between 8 and 16 MeV before model space restriction impacts the results. The distributions in each panel are averages over 2, 28, 102, 160, 199, 250, 338, and 385 initial states, respectively.

Figure 13

${}^{28}\mathrm{Na}$ density of states. Model space restrictions make the density of states very low, with the departure from exponential growth occurring at low excitation.

Figure 14

Strength distribution for ${}^{28}\mathrm{Na}$. There is no obvious energy regime where the strength is independent of initial excitation. The distributions in each panel are averages over 20, 60, 81, 91, 103, 125, 149, and 168 initial states, respectively.

Figure 15

${}^{28}\mathrm{Si}$ ground-state strength distribution. The solid line shows the distribution using our shell-model calculations, and the dotted line shows the strength from the FFN prescription. The large peak in the FFN distribution is the GT resonance used in those works.

Figure 16

Electron capture rates for ${}^{28}\mathrm{Si}$ as a function of electron Fermi energy and temperature. Solid lines show rates with all states up to 12 MeV considered individually and the rest of the statistical weight carried by a single high-energy average state, while dashed lines correspond to the rates when all states are assumed to have the same narrow GT resonance, in accordance with the FFN approach to the GT Brink-Axel hypothesis.

Figure 17

Electron capture rates for ${}^{28}\mathrm{Si}$ as a function of electron Fermi energy and temperature. Solid lines show rates with all states up to 12 MeV considered individually and the rest of the statistical weight carried by a single high-energy average state, while dashed lines correspond to the rates when all states are assumed to have the same bulk GT strength distribution as our shell-model calculation of the ground state.

Figure 18

Thermodynamically unweighted electron capture rates for high-energy average states in ${}^{28}\mathrm{Si}$. The solid lines are the rates for an HEA state with a cutoff energy of 12 MeV, the dashed lines show a cutoff of 15 MeV, and the dotted lines are for a cutoff of 20 MeV.

Figure 19

Electron capture rates for ${}^{28}\mathrm{Si}$ comparing two choices of cutoff energy. The solid lines correspond to a cutoff energy of 12 MeV, while the dashed lines are for a cutoff of 20 MeV. The dotted lines show the ratio of the cutoff = 12 rates to the cutoff = 20 rates. That the rates are nearly identical lends credence to the technique of using a high-energy average state.

Figure 20

Electron capture rates for ${}^{28}\mathrm{Si}$ showing the effect of quenching. Both sets of lines use a cutoff energy of 12 MeV. Solid lines show rates for unquenched strength, and dotted lines show the rates when a quenching factor of 0.6 is applied to computed strengths.