Gamow-Teller transitions to ${}^{45}\mathrm{Ca}$ via the ${}^{45}\mathrm{Sc}(t,{}^3\mathrm{He}+\gamma )$ reaction at $115 \text{MeV}/u$ and its application to stellar electron-capture rates

Background: Stellar electron-capture reactions on medium-heavy nuclei are important for many astrophysical phenomena, including core-collapse and thermonuclear supernovæ and neutron stars. Estimates of electron-capture rates rely on accurate estimates of Gamow-Teller strength distributions, which can be extracted from charge-exchange reactions at intermediate beam energies. Measured Gamow-Teller transition strength distributions for stable pf-shell nuclei are reasonably well reproduced by theoretical calculations in the shell model, except for lower mass nuclei where admixtures from the sd shell can become important.

Purpose: This paper presents a ${\beta }^{+}$ charge-exchange experiment on ${}^{45}\mathrm{Sc}$, one of the lightest pf-shell nuclei. The focus was on Gamow-Teller transitions to final states at low excitation energies, which are particularly important for accurate estimations of electron-capture rates at relatively low stellar densities. The experimental results are compared with various theoretical models.

Method: The double-differential cross section for the ${}^{45}\mathrm{Sc}(t,{}^3\mathrm{He}+\gamma )$ reaction was measured using the NSCL Coupled-Cyclotron Facility at $115\mathrm{MeV}/u$. Gamow-Teller contributions to the excitation-energy spectra were extracted by means of a multipole-decomposition analysis. $\gamma$ rays emitted due to the deexcitation of ${}^{45}\mathrm{Ca}$ were measured using GRETINA to allow for the extraction of Gamow-Teller strengths from very weak transitions at low excitation energies.

Results: Gamow-Teller transition strengths to ${}^{45}\mathrm{Ca}$ were extracted up to an excitation energy of 20 MeV, and that to the first excited state in ${}^{45}\mathrm{Ca}$ at 174 keV was extracted from the $\gamma$-ray measurement, which, even though weak, is important for the astrophysical applications and dominates under certain stellar conditions. Shell-model calculations performed in the pf shell-model space with the GXPF1A, KB3G, and FPD6 interactions did not reproduce the experimental Gamow-Teller strength distribution, and a calculation using the quasiparticle random phase approximation that is often used in astrophysical simulations also could not reproduce the experimental strength distribution.

Conclusions: Theoretical models aimed at describing Gamow-Teller transition strengths from nuclei in the lower pf shell for the purpose of estimating electron-capture rates for astrophysical simulations require further development. The likely cause for the relatively poor performance of the shell-model theory is the influence of intruder configurations from the sd shell. The combination of charge-exchange experiments at intermediate beam energy and high-resolution $\gamma$-ray detection provides a powerful technique to identify weak transitions to low-lying final states that are nearly impossible to identify without the coincidences. Identification of these weak low-lying transitions is important for providing accurate electron-capture rates for astrophysical simulations.

Figure 1

(Left) Double-differential cross section spectra for the ${}^{45}\mathrm{Sc}(t,{}^3\mathrm{He})$ reaction at various scattering angles. The error bars denote the statistical uncertainty only. The histograms also show the results from the multipole-decomposition analysis (MDA). (Right) Representative angular distributions at $E_{\mathrm{x}}=6.5$ and $15.3\mathrm{MeV}$ including the results from the MDA.

Figure 2

(a) $B\left(\mathrm{GT}\right)$ distribution extracted in the MDA of the ${}^{45}\mathrm{Sc}(t,{}^3\mathrm{He})$ data. The error bars denote the statistical and systematic uncertainties. The $B\left(\mathrm{GT}\right)$ distribution extracted from the $(n,p)$ data at 198 MeV [24] is also shown for comparison. (b) $B\left(\mathrm{GT}\right)$ distribution from the $(t,{}^3\mathrm{He})$ data is compared with the SM calculations with the GXPF1A, KB3G, and FPD6 interactions and with the QRPA calculation smeared with the experimental resolution. The results from the QRPA calculation are divided by a factor of 3. (c) Cumulative sum of the $B\left(\mathrm{GT}\right)$ distribution from the $(t,{}^3\mathrm{He})$ data and those from the theoretical calculations.

Figure 3

(a) $E_{\gamma }$ vs $E_{\mathrm{x}}\left({}^{45}\mathrm{Ca}\right)$. The $E_{\gamma }=E_{\mathrm{x}}$ line is shown, and the proton $\left(S_p\right)$ and neutron $\left(S_n\right)$ separation energies are also indicated. (b) A projection of (a) onto the $E_{\gamma }$ axis, gated on $E_{\mathrm{x}}\left({}^{45}\mathrm{Ca}\right)$ around $174\mathrm{keV}$ as indicated by the box in (a). The inset shows a schematic decay diagram of the 174-keV state. (c) $E_{\gamma }$ spectrum gated on $E_{\mathrm{x}}>8\mathrm{MeV}$. The selected region is indicated by the dashed rectangle in (a). The known $\gamma$-ray energies [47] are indicated, with the matched peaks shown by arrows.

Figure 4

(a) EC rates on ${}^{45}\mathrm{Ca}$ at $\rho Y_e={10}^7\mathrm{g}/{\mathrm{cm}}^3$ as a function of stellar temperatures. The shaded band denotes the EC rate based on the experimental GT strengths (including uncertainties), whereas the dotted and solid lines only represents the EC into the ground and 174-keV states, respectively. The total rates based on the SM (GXPF1A, KB3G, and FPD6) and QRPA calculations are also shown. (b) Same as the one on the top, but at $\rho Y_e={10}^9\mathrm{g}/{\mathrm{cm}}^3$.