High-resolution study of $T_z=+1\to 0$ Gamow-Teller transitions in the ${}^{26}\mathrm{Mg}({}^3\mathrm{He},t$)${}^{26}\mathrm{Al}$ reaction

In order to study the $T_z=+1\to 0$ Gamow-Teller (GT) transitions in the mass $A=26$ system, a charge-exchange reaction ${}^{26}\mathrm{Mg}({}^3\mathrm{He},t){}^{26}\mathrm{Al}$ was performed at an incident energy of 140 MeV/nucleon and scattering angle at and near $0^{\circ }$, where $T_z$ is the $z$ component of isospin $T$ defined by $(N-Z)/2$. In this ($p,n$)-type reaction, it is expected that GT states with $T=0$, 1, and 2 are excited. An energy resolution of $\mathrm{\Delta }E=23$ keV allowed us to study many discrete states. Most of the prominent states showed $0^{\circ }$-peaked angular distributions, which suggested that they are the states excited by $\mathrm{\Delta }L=0$ GT transitions. Candidates of GT states were studied up to an excitation energy $E_x=18.5$ MeV. The reduced GT transition strengths, $B\left(\mathrm{GT}\right)$, were derived assuming the proportionality between cross sections and $B\left(\mathrm{GT}\right)$ values. Standard $B\left(\mathrm{GT}\right)$ values were obtained form the ${}^{26}\mathrm{Si}\beta$ decay, where the mirror symmetry of $T_z=\pm 1\to 0$ GT transitions was assumed. The GT strength, as a whole, is divided in two energy regions: the region of up to 8.5 MeV and the higher-energy region of $8.5\text{–}12.8$ MeV, where the strength in the latter region distributed like a resonance. The obtained GT strength distribution is compared with the results of random phase approximation calculations. The $T=2$ GT states are expected in the region $E_x\ge 13.5$ MeV. By comparing with the results of ($n,p$)-type ${}^{26}\mathrm{Mg}(d,{}^2\mathrm{He}){}^{26}\mathrm{Na}$ and ${}^{26}\mathrm{Mg}(t,{}^3\mathrm{He}){}^{26}\mathrm{Na}$ reactions, the isospin symmetry of $T=2$ GT states is discussed. Owing to the high-energy resolution, we could study the decay widths $\mathrm{\Gamma }$ for the states in the $E_x>9$ MeV region. The $T=2$ state at 13.592 MeV is not noticeably wider than the experimental energy resolution. The narrow width of the state is explained in terms of isospin selection rules.

Figure 1

Schematic view of the isospin analogous states (analog states) and analogous GT transitions in the $A=26$, $T_z=+2$, $+1$, 0, and $-1$ isobars, where only ${}^{26}\mathrm{Mg}$ is stable. The Coulomb displacement energies are removed so that the isospin symmetry of the states and transitions becomes clearer. The type of the reaction or decay is shown alongside the arrow indicating the transition.

Figure 2

The ${}^{26}\mathrm{Mg}({}^3\mathrm{He},t){}^{26}\mathrm{Al}$ energy spectrum on two scales, measured at $0^{\circ }$ and at an incoming energy of 140 MeV/nucleon. Events with scattering angles $\mathrm{\Theta }\le 0.5^{\circ }$ are included. Most of the prominent states are populated by $\mathrm{\Delta }L=0$ transitions (see Table 1 and Table 2), and we identify that they are GT states, except the isobaric analog state (IAS) at 0.228 MeV. States excited with $\mathrm{\Delta }L=0$ are indicated by their $E_x$ values, where the values up to 8 MeV are taken from Ref. [15]. The states excited with $\mathrm{\Delta }L\ge 1$ and discussed in the text are indicated by italic characters. All of them are weakly excited at $0^{\circ }$. (a) The full count and full range spectrum. Prominent peaks are observed mainly in the low-lying region below 5 MeV and also in the $9-12$ MeV (GTR) region. (b) The energy spectrum of the $E_x=7\text{–}14$ MeV region. The vertical scale is expanded by a factor of 5. Owing to the good energy resolution, many individual states and decay widths of them are observed.

Figure 3

(a) The $B\left(\mathrm{GT}\right)$ strength distribution as a function of excitation energy ($E_x$) in the final nucleus ${}^{26}\mathrm{Al}$. Compared to the peak height of the energy spectrum shown in Fig. 2, the strengths in the GT resonance (GTR) region of $8\text{–}12.5$ MeV appears to be larger. This is due to the broad peak widths of the states in this region. (b) Cumulative-sum strength of the experimental $B\left(\mathrm{GT}\right)$ values. The GT strength is divided into the low-lying ($E_x<7$ MeV) region and the GTR.

Figure 4

Strength distributions of the ${}^{26}\mathrm{Mg}\to {}^{26}\mathrm{Al}$ GT transitions obtained by HFB + spherical-QRPA calculations using the Skyrme interaction SGII. The vertical dotted line (seen at 2.3 MeV) is the calculated IAS position (seen at 0.23 MeV in the experiment). The IS pairing interaction can be included in the QRPA calculation. As the IS strength ratio $f$ increases, the GT strength moves to lower $E_x$. In addition, the lower-$E_x$ peak accumulates more GT strength.

Figure 5

Comparison of ${}^{26}\mathrm{Mg}({}^3\mathrm{He},t){}^{26}\mathrm{Al}$ and ${}^{26}\mathrm{Mg}(d,{}^2\mathrm{He}){}^{26}\mathrm{Na}$ spectra. (a) The $E_x=12.5-20$ MeV region of the $0^{\circ }$, ${}^{26}\mathrm{Mg}({}^3\mathrm{He},t){}^{26}\mathrm{Al}$ spectrum. States excited with $\mathrm{\Delta }L=0$ or probably with $\mathrm{\Delta }L=0$ [indicated by (0)] are shown by their $E_x$ values. (b) The ${}^{26}\mathrm{Mg}(d,{}^2\mathrm{He}){}^{26}\mathrm{Na}$ spectrum at $E_p=135$ MeV/nucleon [16] up to 6.5 MeV. In order to enhance the forward-peaked states excited with $\mathrm{\Delta }L=0$, the $3^{\circ }$ spectrum is subtracted from that of $0^{\circ }$. Corresponding isobaric analog GT states are observed in $T_z=0$ nucleus ${}^{26}\mathrm{Al}$ and $T_z=+2$ nucleus ${}^{26}\mathrm{Na}$.

Figure 6

Isospin selection rules in the proton decay from excited states of ${}^{26}\mathrm{Al}$. The $z$ component of isospin $T$, i.e., $T_z$ follows the (scalar) addition rule. On the other hand, vector nature should be taken into account in the addition of $T$ values. As a result, a $T=2$ state in ${}^{26}\mathrm{Al}$ cannot decay into a low-lying $T=1/2$ states in ${}^{25}\mathrm{Mg}$ and a proton. It can decay only into a $T=3/2$ state existing at $E_x=7.79$ MeV and higher and a proton. For details, see text.