Thermal quasiparticle random-phase approximation with Skyrme interactions and supernova neutral-current neutrino-nucleus reactions

The thermal quasiparticle random-phase approximation is combined with the Skyrme energy density functional method (Skyrme-TQRPA) to study the response of a hot nucleus to an external perturbation. For the sample nuclei ${}^{56}\mathrm{Fe}$ and ${}^{82}\mathrm{Ge}$, the Skyrme-TQRPA is applied to analyze thermal effects on the strength function of charge-neutral Gamow–Teller transitions, which dominate neutrino-nucleus reactions at $E_{\nu }\lesssim 20$ MeV. For the relevant supernova temperatures we calculate the cross sections for inelastic neutrino scattering. We also apply the method to examine the rate of neutrino-antineutrino pair emission by hot nuclei. The cross sections and rates are compared with those obtained earlier from the TQRPA calculations based on the phenomenological quasiparticle-phonon model Hamiltonian. For inelastic neutrino scattering on ${}^{56}\mathrm{Fe}$ we also compare the Skyrme-TQRPA results to those obtained earlier from a hybrid approach that combines shell-model and RPA calculations.

Figure 1

The ground-state ${\mathrm{GT}}_0$ strength distributions for ${}^{56}\mathrm{Fe}$ and ${}^{82}\mathrm{Ge}$. The Skyrme-QRPA calculations are performed with the ${\mathrm{SkM}}^{*}$, SLy4, and SGII energy density functionals. The ${\mathrm{GT}}_0$ strength calculated in Ref. [14] with the QPM Hamiltonian are also shown. Note that, for a clearer presentation, the low-energy peaks in ${}^{56}\mathrm{Fe}$ and ${}^{82}\mathrm{Ge}$ are scaled, i.e., the values displayed are obtained by multiplying the actual values by the indicated factor. The shell-model ${\mathrm{GT}}_0$ strength (scaled by a factor of 3.5) from Ref. [5] is shown for ${}^{56}\mathrm{Fe}$. The shell-model strength is split into the two isospin components: $\mathrm{\Delta }I=0$ (full line) and $\mathrm{\Delta }I=1$ (dashed line). The arrow indicates the experimental energy of the first $1^{+}$ state in ${}^{56}\mathrm{Fe}$ [$E_{\text{expt}}\left(1_1^{+}\right)=3.12$ MeV]. This energy corresponds to the threshold for inelastic neutrino scattering from the ground state of ${}^{56}\mathrm{Fe}$.

Figure 2

The ${\mathrm{GT}}_0$ strength function for ${}^{56}\mathrm{Fe}$ and ${}^{82}\mathrm{Ge}$ calculated at $T=0$ (dashed peaks) and $T=1.72$ MeV (solid peaks).

Figure 3

The energy centroid for the upward ${\mathrm{GT}}_0$ strength functions in ${}^{56}\mathrm{Fe}$ and ${}^{82}\mathrm{Ge}$ as a function of temperature. The arrows indicate the critical temperature $T_{\mathrm{cr}}$ when the pairing collapses.

Figure 4

Comparison of the running sums for the ${\mathrm{GT}}_0$ downward strength in ${}^{56}\mathrm{Fe}$ obtained by using and without using the Brink hypothesis (dashed and solid lines, respectively). Note that the values are scaled by a factor of ${10}^3$ ($T=0.86$ MeV) and ${10}^2$ ($T=1.72$ MeV).

Figure 5

Inelastic neutral-current neutrino scattering cross sections from the ground states of ${}^{56}\mathrm{Fe}$ and ${}^{82}\mathrm{Ge}$. The cross sections are calculated within the Skyrme-QRPA (${\mathrm{SkM}}^{*}$, SGII, Sly4) as well as the QPM-QRPA [14]. For ${}^{56}\mathrm{Fe}$, the results of shell-model calculations [5] are shown by the solid line.

Figure 6

Inelastic neutrino scattering cross sections for ${}^{56}\mathrm{Fe}$ and ${}^{82}\mathrm{Ge}$ at $T=0.86$ and 1.72 MeV, calculated within the TQRPA. For ${}^{56}\mathrm{Fe}$, the cross sections obtained within the hybrid model [6] are shown by the solid line.

Figure 7

Normalized spectra of outgoing neutrinos for ${}^{56}\mathrm{Fe}$ at $T=0.86,1.72$ MeV and two initial neutrino energies $E_{\nu }=5,15$ MeV. In all plots the energy of the incoming neutrino is indicated by an arrow. A comparison is made between the TQRPA results and those obtained in Ref. [6] within the hybrid model. The latter are shown by the solid lines.

Figure 8

Normalized spectra of $\nu \overline{\nu }$ pairs emitted by ${}^{56}\mathrm{Fe}$ (left panels) and ${}^{82}\mathrm{Ge}$ (right panels) at two different temperatures, $T=0.86$ MeV and $T=1.72$ MeV. For each temperature we show the spectra obtained by using and without using the Brink hypothesis (lower and upper panels on each plot, respectively).

Figure 9

The energy centroid for the spectrum of emitted neutrinos as a function of temperature.

Figure 10

Neutrino-antineutrino pair emission rates for ${}^{56}\mathrm{Fe}$ and ${}^{82}\mathrm{Ge}$ as functions of temperature.