High-resolution study of Gamow-Teller transitions via the ${}^{54}$Fe(${}^3$He,$t$)${}^{54}$Co reaction

The Gamow-Teller transition strengths, $B$(GT), in $pf$-shell nuclei are of interest in nuclear physics as well as in nuclear astrophysics. A high-resolution (${}^3\mathrm{He},t$) charge-exchange (CE) reaction was performed on the $T_z=+1$ nucleus ${}^{54}$Fe at 0${}^{\circ }$ and at an intermediate incident energy of 140 MeV/nucleon for the study of precise GT transition strengths to the final $T_z=0$ nucleus ${}^{54}$Co. By applying dispersion matching techniques for a high-quality ${}^3$He beam at RCNP, an energy resolution of 21 keV and an angular resolution of 5 mr were realized. The bumplike structure of the GT resonance observed in low-resolution CE reactions at around the excitation energy ($E_x$) of 10 MeV was resolved in individual $L$ $=$ 0, GT states. Excitation strengths were obtained for these GT states. If the $R^2$ value that is defined by the ratio between GT and Fermi unit cross sections is known, the $B$(GT) values can be determined from the excitation strengths. For the derivation of the $R^2$ value, the “merged analysis” combining the GT strength distribution from the ${}^{54}$Fe(${}^3\mathrm{He},t$)${}^{54}$Co study and the half-life from a ${}^{54}$Ni $\beta$ decay was used, where $T=1$ isospin symmetry for $A=54$ isobars was assumed. The GT strengths were compared with a shell-model calculation using the GXPF1 interaction. The final GT states can have the isospin values $T$ $=$ 0, 1, and 2. The isospin $T$ of each GT state observed in the $8.3\le E_x\le 12.0$ MeV region of the ${}^{54}$Fe(${}^3\mathrm{He},t$)${}^{54}$Co spectrum was identified by comparing the excitation strength with that of corresponding $M1$ state observed in a ${}^{54}$Fe($p,p^{\prime }$)${}^{54}$Fe experiment. The $B$(GT) values of the states identified to have $T=2$, in particular, are of importance for the calculation of the electron capture rates at the core-collapse stage of presupernovae. The $B$(GT) strengths were further compared with $B\left(M1\right)$ strengths measured in the ${}^{54}$Fe($e,e^{\prime }$)${}^{54}$Fe reaction. In the $M1$ excitation using an electromagnetic probe, isoscalar (IS) and isovector (IV) orbital type operators are active in addition to the IV spin type operator that mediates the GT transitions. The contributions of the IS and IV orbital terms were studied by calculating the ratio of the strengths of analogous $M1$ and GT transitions.

Figure 1

The schematic view of the isospin structure in the $A=54$ system for the $T_z=+2,+1$, and 0 nuclei ${}^{54}$Mn, ${}^{54}$Fe, and ${}^{54}$Co. Coulomb displacement energies are removed so the isospin structure becomes clear. Analog states having $T=1$ and 2 are connected by dashed lines. The squared values of the CG coefficients for the transitions to $T$ $=$ 0, 1, and 2 GT and $M1$ states are shown for each excited state.

Figure 2

The “$0^{\circ }$” spectrum of the ${}^{54}$Fe(${}^3\mathrm{He},t$)${}^{54}$Co reaction up to $E_x$ $=$ 12.00 MeV. The energy resolution was 21 keV [full width at half maximum (FWHM)]. The events with scattering angles $0.0^{\circ }\le \Theta \le 0.5^{\circ }$ are included. All states below 4.90 MeV in Table are indicated by their excitation energies in MeV.

Figure 3

The enlarged ${}^{54}$Fe(${}^3\mathrm{He},t$)${}^{54}$Co spectrum for the energy region of $4.90\le E_x\le 8.60$ MeV. Major GT states in Table ($E_x\le 7.8$ MeV) and Table ($E_x\ge 7.8$ MeV) are indicated by their excitation energies in MeV.

Figure 4

The spectra of (a) ${}^{54}$Fe(${}^3\mathrm{He},t$)${}^{54}$Co and (b) ${}^{54}$Fe($p,p^{\prime }$)${}^{54}$Fe for the energy region of $8.15\le E_x\le 12.00$ MeV. Major GT and $M1$ states listed in Table are indicated by their excitation energies in MeV. The instrumental background appears below 8.27 MeV and under the observed peaks in the ${}^{54}$Fe($p,p^{\prime }$)${}^{54}$Fe spectrum. The identification of isospin value $T$ for GT and $M1$ states will be discussed in Sec. 5c by comparing both spectra.

Figure 5

The analogous GT transitions observed in the $T_z=+1\to 0$, ${}^{54}$Fe(${}^3\mathrm{He},t$)${}^{54}$Co reaction and the $T_z=-1\to 0$, ${}^{54}$Ni $\beta$ decay. Coulomb displacement energies are removed in this figure so the isospin symmetry structure becomes clear. Since the ground states of $T_z=\pm 1$ nuclei are IASs, the transition from the ground state of the $T_z=\pm 1$ nuclei to the $1^{+}$ excited state in $T_z=0$ are analogous. The GT states identified by the $\beta$-decay experiment from ${}^{54}$Ni nucleus ($E_x\le 4.544$ MeV) and the GT states to which $B$(GT) $\ge$ 0.090 identified from the present ${}^{54}$Fe(${}^3\mathrm{He},t$)${}^{54}$Co analysis ($E_x\ge 4.825$ MeV) are shown.

Figure 6

An example of the identification of $L=0$ and $L\ge 1$ states by comparing the spectra of the scattering angles of (a) $0.0^{\circ }\le \Theta \le 0.5^{\circ }$ and the scattering angles of (b) $1.0^{\circ }\le \Theta \le 1.5^{\circ }$ for the weakly excited states in the $5.0\le E_x\le 5.6$ MeV region. The vertical axes of these figures are normalized to the height of the state at 5.470 MeV, which was identified to be an $L=0$ state in the present analysis. Cross sections for three states at 5.115, 5.189, and 5.294 MeV are obviously larger in the spectrum (b). These states are identified to be $L\ge 1$ states.

Figure 7

Comparison of experimental and theoretical $B$(GT) distributions. (a) Measured $B$(GT) distribution from the ${}^{54}$Fe(${}^3\mathrm{He},t$)${}^{54}$Co reaction. (b) The $B$(GT) distribution from a shell-model (SM) calculation using the GXPF1 interaction. A quenching factor of (0.74)${}^2$ is included. (c) Cumulative sums of $B$(GT) values from the experiment (solid line) and the shell-model calculation (dotted line).

Figure 8

(a) Yields for GT states obtained from the ${}^{54}$Fe(${}^3\mathrm{He},t$)${}^{54}$Co experiment. The yields are extrapolated to $q=0$ to eliminate the dependence on the excitation energies $E_x$. The states identified as having isospins $T$ $=$ 0, 1, and 2 are shown by left hatched bars, blue open bars, and black solid bars, respectively. (b) Yields for $M1$ states obtained from the ${}^{54}$Fe($p,p^{\prime }$)${}^{54}$Fe experiment. The yields are extrapolated to $q=0$. The states identified as $T=1$ and $T=2$ states are shown by blue open bars and black solid bars, respectively. The states for which $T$ values are not identified are shown by right hatched bars. (c) The differences of the $E_x$ values of the GT states and the corresponding $M1$ states. (d) The $R_{\mathrm{GT}}$ values calculated from the yields between corresponding GT and $M1$ states. The method of extracting the $R_{\mathrm{GT}}$ values are explained in Sec. 5c. All identified isospin values $T$ are shown by open circles for $T=1$ states and black solid circles for $T=2$ states.

Figure 9

The $R_{\mathrm{ISO}}$ values as a function of $B$(GT) values.