Gamow-Teller strength for the analog transitions to the first $T=1/2,J^{\pi }=3/2^{-}$ states in ${}^{13}\mathrm{C}$ and ${}^{13}\mathrm{N}$ and the implications for type Ia supernovae

The Gamow-Teller strength for the transition from the ground state of ${}^{13}\mathrm{C}$ to the $T=1/2,J^{\pi }=3/2^{-}$ excited state at 3.51 MeV in ${}^{13}\mathrm{N}$ is extracted via the ${}^{13}\mathrm{C}$(${}^3\mathrm{He}$,$t$) reaction at 420 MeV. In contrast to results from earlier ($p,n$) studies on ${}^{13}\mathrm{C}$, a good agreement with shell-model calculations and the empirical unit cross-section systematics from other nuclei is found. The results are used to study the analog ${}^{13}\mathrm{N}$($e^{-},{\nu }_e$)${}^{13}\mathrm{C}$ reaction, which plays a role in the pre-explosion convective phase of type Ia supernovae. Although the differences between the results from the (${}^3\mathrm{He}$,$t$) and ($p,n$) data significantly affect the deduced electron-capture rate and the net heat deposition in the star resulting from this transition, the overall effect on the pre-explosive evolution is small.

Figure 1

Schematic of isospin analogous transitions in the $A=13$ and $T_z=\pm 1/2,\pm 3/2$ isobar system. The Coulomb displacement energies are set to zero so that the isospin symmetry of the system becomes clearer. For each state, spin parity $J^{\pi }$ and isospin $T$ are indicated. Excitation energies are given, where relevant for the present work, relative to the ground state of the nuclei. Transitions labeled by 1–4 are analogs and transitions with labels 5 and 6 are analogs. For transitions labeled with ${\beta }^{\pm }$, the log $\mathit{ft}$ values are known from β-decay experiments.

Figure 2

(a) Excitation energy spectrum of ${}^{13}\mathrm{N}$ taken with the (${}^3\mathrm{He}$,$t$) reaction at 420 MeV on a ${}^{\mathrm{nat}}$C target. The ground state and excited state at 3.51 MeV are indicated. (b) Differential cross sections for the ${}^{13}\mathrm{C}$(${}^3\mathrm{He}$,$t$)${}^{13}\mathrm{N}$($3.51\hspace{1em}\mathrm{MeV}$) reaction from the present data (with only statistical error bars) and from Ref. [18]. The latter have been scaled by a factor of 0.88. Also included is the result from the DWBA calculation, for which the scale was adjusted to fit the data. (c) Differential cross sections for the ${}^{13}\mathrm{C}$(${}^3\mathrm{He}$,$t$)${}^{13}\mathrm{N}$(g.s.) transition. The contribution from GT and Fermi components to this transition is described in the text. (d) Differential cross sections for the ${}^{13}\mathrm{C}$(${}^3\mathrm{He}$,$t$)${}^{13}\mathrm{N}$ (15.1-MeV) transition. Only data from Ref. [18] are available.

Figure 3

(a) Calculated electron-capture rates on ${}^{13}\mathrm{N}$ in the stellar environment as a function of temperature and $\rho Y_e$ [ranging from $\log {}_{10}(\rho Y_e)=8.7$ to 9.6 in steps of 0.1]. The $B$(GT) for the transition to the 3.68-MeV excited state in ${}^{13}\mathrm{C}$ was fixed to the shell-model value of 1.5. (b) Electron-capture rate at $\log {}_{10}(\rho Y_e)=8.8$ and $T_9=0.4$ as a function of the assumed $B$(GT) for the transition to the 3.68-MeV excited state in ${}^{13}\mathrm{C}$. The $B$(GT) values extracted from the ($p,n$) data (“1” refers to Refs. [13, 14, 15] and “2” to Ref. [17]) and (${}^3\mathrm{He}$,$t$) data and the shell-model calculation are indicated and associated with electron-capture rates. The dashed lines correspond to error margins in the $B$(GT) values deduced experimentally. (c) Same as (b) but at $\log {}_{10}(\rho Y_e)=9.5$.

Figure 4

(a) Calculated neutrino energy-loss rates from the electron-capture on ${}^{13}\mathrm{N}$ in the stellar environment as a function of temperature and $\rho Y_e$ [ranging from $\log {}_{10}(\rho Y_e)=8.7$ to 9.6 in steps of 0.1]. The $B$(GT) for the transition to the 3.68-MeV excited state in ${}^{13}\mathrm{C}$ was fixed to the shell-model value of 1.5. (b) Neutrino energy-loss rate at $\log {}_{10}(\rho Y_e)=8.8$ and $T_9=0.4$ as a function of the assumed $B$(GT) for the transition to the 3.68-MeV excited state in ${}^{13}\mathrm{C}$. The $B$(GT) values extracted from the ($p,n$) data (“1” refers to Refs. [13, 14, 15] and “2” to Ref. [17]) and (${}^3\mathrm{He}$,$t$) data and the shell-model calculation are indicated and associated with energy-loss rates. The dashed lines correspond to error margins in the $B$(GT) values deduced experimentally. (c) The same as (b) but at $\log {}_{10}(\rho Y_e)=9.5$.