Charge-exchange reactions on double-$\beta$ decaying nuclei populating $J^{\pi }=2^{-}$ states

The $\left({}^3\mathrm{He},t\right)$ charge-exchange reaction populating $J^{\pi }=2^{-}$ states has been examined at 420 MeV incident energy for a series of double-$\beta$ decaying nuclei, i.e., ${}^{76}\mathrm{Ge}$, ${}^{82}\mathrm{Se}$, ${}^{96}\mathrm{Zr}$, ${}^{100}\mathrm{Mo}$, ${}^{128}\mathrm{Te}$, ${}^{130}\mathrm{Te}$, and ${}^{136}\mathrm{Xe}$. The measurements were carried out at the Grand Raiden spectrometer of the Research Center for Nuclear Physics at the University Osaka with typical spectral resolution of 30–40 keV. It is found that the charge-exchange reaction leading to $2^{-}$ spin-dipole states is selective to the $\sigma \tau$ part of the interaction much similar to the observed selectivity to Gamow-Teller transitions. In the present case, the $\mathrm{\Delta }L=1$ peak cross sections at finite momentum transfers are used to extract the spin-isospin part of the low-lying transition strength near the Fermi surface (i.e., $E_x\le 5$ MeV). Relative strength values are confronted with various model calculations, i.e., the interacting shell model, the quasiparticle random-phase approximation, and the Fermi surface quasiparticle model. The impact on the nuclear matrix elements for the neutrinoless double-$\beta$ decay is discussed.

Figure 1

Low-energy excitation spectra obtained from (${}^3\mathrm{He},t$) charge-exchange reaction on seven $\beta \beta$ decaying nuclei. The spectra for the scattering angle intervals $[0\text{–}0.5^{\circ }]$ (in red) and $[2\text{–}2.5^{\circ }]$ (in blue) are overlaid to visualize the angular distribution effect of GT and spin-dipole transitions. Excitation energies are indicated for the most clearly visible $2^{-}$ transitions. The search for $2^{-}$ states was extended to 5 MeV, but no isolated $2^{-}$ states were observed even above 4 MeV in any of these nuclei. For a complete analysis of GT transitions, one may refer to Refs. [7, 8, 9, 10, 11, 17].

Figure 2

Cross-section angular distributions for the lowest-lying $2^{-}$ states in various $\beta \beta$ decaying nuclei. The solid lines are DW calculations for the sum of the $\mathrm{\Delta }L=1$ and $\mathrm{\Delta }L=3$ transitions as described in the text. The dotted lines show these transitions separately.

Figure 3

Comparison of the relative spin-dipole excitation strengths for the $A=76$, 96, 100, 128, 130, and 136 systems. The black bars indicate the experimental values from the (${}^3\mathrm{He},t$) charge-exchange reactions on the $\beta \beta$ decaying nuclei indicated on the lower axis. All theoretical values have been scaled in order to have the values for ${}^{128}\mathrm{Te}$ match the experimental one. This is for presentation purposes to show the general trend. Absolute numbers appear in Table 3.

Figure 4

Sketch of shell-model configurations near the Fermi surface showing $n\to p$ transitions for $\mathrm{\Delta }L=1$ in ($p,n$)-type charge-exchange reactions leading to $J^{\pi }=2^{-}$ states in the daughter nucleus. Full circles indicate occupied configurations, and open circles indicate vacant configurations.