The ($t$,${}^3\mathrm{He}$) and (${}^3\mathrm{He}$, $t$) reactions as probes of Gamow-Teller strength

It is shown via a study on a ${}^{26}\mathrm{Mg}$ target that the ($t$,${}^3\mathrm{He}$) reaction at 115 MeV/nucleon reaction is an accurate probe for extracting Gamow-Teller transition strengths. To do so, the data are complemented by results from the ${}^{26}\mathrm{Mg}$(${}^3\mathrm{He}$, $t$) reaction at 140 MeV/nucleon that allows for a comparison of $T=2$ analog states excited via the mirror reactions. Extracted Gamow-Teller strengths from ${}^{26}\mathrm{Mg}$($t$,${}^3\mathrm{He}$) and ${}^{26}\mathrm{Mg}$(${}^3\mathrm{He}$, $t$) are compared with those from ${}^{26}\mathrm{Mg}$($d$,${}^2\mathrm{He}$) and ${}^{26}\mathrm{Mg}$($p,n$) studies, respectively. A good correspondence is found, indicating probe independence of the strength extraction. Furthermore, we test shell-model calculations using the new USD-05B interaction in the $\mathit{sd}$-model space and show that it reproduces the experimental Gamow-Teller strength distributions well. In anticipation of further ($t$,${}^3\mathrm{He}$) experiments on medium-heavy nuclei aimed at determining weak-interaction rates of relevance for stellar evolution, a second goal of this work is to improve the understanding of the ($t$,${}^3\mathrm{He}$) and (${}^3\mathrm{He}$, $t$) reaction mechanisms at intermediate energies because detailed studies are scarce. The distorted-wave Born approximation is employed, taking into account the composite structures of the ${}^3\mathrm{He}$ and triton particles. The reaction model provides the means to explain systematic uncertainties at the 10%–20% level in the extraction of Gamow-Teller strengths as being because of interference between Gamow-Teller $\Delta L=0,\Delta S=1$ and $\Delta L=2,\Delta S=1$ amplitudes that both contribute to transitions from $0^{+}$ to $1^{+}$ states.

Figure 1

Differential cross sections measured for the ${}^{26}\mathrm{Mg}$($t$,${}^3\mathrm{He}$) reaction at $E_t=115$ MeV/nucleon. (a) The spectrum for ${\theta }_{\mathrm{cm}}($${}^3\mathrm{He}$$)<1.1^{°}$. (b) The spectrum for $3.4^{°}<{\theta }_{\mathrm{cm}}($${}^3\mathrm{He}$$)<4.5^{°}$. Unlike transitions with higher angular momentum transfer, Gamow-Teller ($\Delta L=0$) transitions peak at forward angles. The energy region where Gamow-Teller transitions dominate the spectrum is indicated in (a), and the transitions to the $1^{+}$ states are indicated by down arrows. In contrast, dipole transitions ($\Delta L=1$) have a minimum at forward angles and peak near 3.5${}^{°}$. Such transitions are indicated by up arrows in (b).

Figure 2

Differential cross sections measured for the ${}^{26}\mathrm{Mg}$(${}^3\mathrm{He}$, $t$) reaction at $E_{{}^3\mathrm{He}}=140$ MeV/nucleon. (Top panel) Spectrum for ${\theta }_{\mathrm{cm}}(t)<0.3^{°}$. (Bottom panel) Spectrum for $2.0^{°}<{\theta }_{\mathrm{cm}}(t)<2.3^{°}$. As in Fig. 1, transitions with $\Delta L=0$ peak at forward angles. The broad bump in the spectrum, seen at larger angles (indicated in the bottom panel) is because of giant dipole resonances (IVSDR and IVGDR).

Figure 3

Energy spectra of the ${}^{26}\mathrm{Mg}$(${}^3\mathrm{He}$, $t$) and ${}^{26}\mathrm{Mg}$($t$,${}^3\mathrm{He}$) reactions. Peaks identified as the result of Gamow-Teller transitions are indicated. β denotes that the Gamow-Teller strength of the transition can be deduced from ${}^{26}\mathrm{Si}$ β decay under the assumption of isospin symmetry. The ${}^{26}\mathrm{Mg}$($t$,${}^3\mathrm{He}$) spectrum has been shifted by the Coulomb-energy difference so that the 1${}^{+}$ state at $E_x=0.08$ keV is aligned with its analog at $E_x=13.6$ MeV in the ${}^{26}\mathrm{Mg}$(${}^3\mathrm{He}$, $t$) spectrum.

Figure 4

Measured differential cross sections and comparison with DWBA calculations. (Left) Results for five states in the ${}^{26}\mathrm{Mg}$(${}^3\mathrm{He}$, $t$) spectrum with different $J^{\pi }$ and $T$ are shown. (Right) Results of MDA in regions A–E of Fig. 3 in the ${}^{26}\mathrm{Mg}$($t$,${}^3\mathrm{He}$) spectrum are shown.

Figure 5

(a) Measured Gamow-Teller strengths from ${}^{26}\mathrm{Mg}$(${}^3\mathrm{He}$, $t$) and comparison with the shell-model predictions using the USD-05B interaction, multiplied by the extracted quenching factor of 0.6. For $E_x>13$ MeV, Gamow-Teller strengths extracted from ${}^{26}\mathrm{Mg}$($t$,${}^3\mathrm{He}$) are included. The excitation energies in ${}^{26}\mathrm{Na}$ have been shifted as explained in Fig. 3 and the strengths have been divided by 6 to account for the difference in isospin Clebsch-Gordan coefficients. Regions where $T=0$, 1, or 2 are dominant are indicated. (b) Cumulative sums of Gamow-Teller strengths; below $E_x=13$ MeV, results from ${}^{26}\mathrm{Mg}$(${}^3\mathrm{He}$, $t$), ${}^{26}\mathrm{Mg}$($p,n$) are compared. Above $E_x=13$ MeV, results from ${}^{26}\mathrm{Mg}$($t$,${}^3\mathrm{He}$) and ${}^{26}\mathrm{Mg}$($d$,${}^2\mathrm{He}$) [10] experiments are compared. Note the changes in vertical scale for $E_x>13$ MeV. (c) Comparison between results from the ${}^{26}\mathrm{Mg}$(${}^3\mathrm{He}$, $t$) ($E_x<13$ MeV) and ${}^{26}\mathrm{Mg}$($t$,${}^3\mathrm{He}$) ($E_x>13$ MeV) experiments and the shell-model calculations using the USD and USD-05B interactions.

Figure 6

Results of a theoretical study into the effects of $\Delta L=2,\Delta S=1$ contributions mediated via the tensor-τ interaction that interfere with $\Delta L=0,\Delta S=1$ contributions to Gamow-Teller transitions. The calculations were performed for the ${}^{26}\mathrm{Mg}$(${}^3\mathrm{He}$, $t$) reaction at 140 MeV/nucleon and included all Gamow-Teller transitions to $T=0,T=1$, and $T=2$ $1^{+}$ states generated in the shell-model calculations described in the text up to an excitation energy of 20 MeV in ${}^{26}\mathrm{Al}$. Only transitions with $B(\mathrm{GT})>0.001$ are shown. In the top panel the expected relative error made in the extraction of the Gamow-Teller strength is shown as a function of the shell-model strength. The histograms in the bottom panel contain the relative errors for transitions with $0.001<B(\mathrm{GT})<0.01$ (left), $0.01<B(\mathrm{GT})<0.1$ (center), and $B(\mathrm{GT})>0.1$ (right). Standard deviations of the distributions are indicated.