Gamow-Teller properties of the double-$\beta$-decay partners ${}^{116}\mathrm{Cd}\left(\mathrm{Sn}\right)$ and ${}^{150}\mathrm{Nd}\left(\mathrm{Sm}\right)$

The two Gamow-Teller (GT) branches connecting the double-$\beta$ decay partners (${}^{116}\mathrm{Cd}$, ${}^{116}\mathrm{Sn}$) and (${}^{150}\mathrm{Nd}$, ${}^{150}\mathrm{Sm}$) with the intermediate nuclei ${}^{116}\mathrm{In}$ and ${}^{150}\mathrm{Pm}$ are studied within a microscopic approach based on a deformed proton-neutron quasiparticle random-phase approximation built on a Skyrme self-consistent mean field with pairing correlations and spin-isospin residual forces. The results are compared with the experimental GT strength distributions extracted from charge-exchange reactions. After combining the two branches, the nuclear matrix elements for the two-neutrino double-$\beta$ decay are evaluated and compared to experimental values derived from the measured half-lives.

Figure 1

Energy-deformation curves for ${}^{116}\mathrm{Cd}$ (a) and ${}^{116}\mathrm{Sn}$ (b) obtained from HF + BCS calculations with various Skyrme forces.

Figure 2

Same as in Fig. 1, but for ${}^{150}\mathrm{Nd}$ (a) and ${}^{150}\mathrm{Sm}$ (b).

Figure 3

GT strength distributions $B\left({\mathrm{GT}}^{-}\right)$ in ${}^{116}\mathrm{Cd}$ as a function of the excitation energy in the daughter nucleus (a) and accumulated strengths (b). SLy4-QRPA calculations for various deformations are compared to data (Sasano 2012: [6]) from $(p,n)$ reactions.

Figure 4

GT strength distributions $B\left({\mathrm{GT}}^{+}\right)$ in ${}^{116}\mathrm{Sn}$ as a function of the excitation energy in the daughter nucleus (a) and accumulated strengths (b). SLy4-QRPA calculations for various deformations are compared to data (Ref. [6]) from $(n,p)$ reactions, as well as from $(d{,}^2\mathrm{He}$) (Ref. [5]).

Figure 5

Same as in Fig. 3, but for $B\left({\mathrm{GT}}^{-}\right)$ in ${}^{150}\mathrm{Nd}$. Data (Ref. [7]) are from ${(}^3\mathrm{He},t$) CERs.

Figure 6

Same as in Fig. 4, but for $B\left({\mathrm{GT}}^{+}\right)$ in ${}^{150}\mathrm{Sm}$. Data (Ref. [7]) are from ($t{,}^3\mathrm{He}$) CERs.

Figure 7

GT strength distributions in ${}^{116}\mathrm{Cd}$ (a), ${}^{116}\mathrm{Sn}$ (b), ${}^{150}\mathrm{Nd}$ (c), and ${}^{150}\mathrm{Sm}$ (d). The results correspond to the prolate shapes with the Skyrme forces SLy4, SGII, and Sk3. Data are as in the previous figures.

Figure 8

Same as in Fig. 3, but in the low-energy range. Also shown is the GT strength extracted from electron capture of ${}^{116}\mathrm{In}$ (Ref. [10]).

Figure 9

Same as in Fig. 4, but in the low-energy range. Also shown (open circle) is the GT strength extracted from ${\beta }^{-}$ decay of ${}^{116}\mathrm{In}$ to which the data from Ref. [5] are normalized.

Figure 10

Same as in Fig. 5, but in the low-energy range.

Figure 11

Same as in Fig. 6, but in the low-energy range.

Figure 12

Nuclear matrix element for the $2\nu \beta \beta$ decay of ${}^{150}\mathrm{Nd}$ as a function of the coupling strengths ${\chi }_{\mathrm{ph}}^{\mathrm{GT}}$ (a) and ${\kappa }_{\mathrm{pp}}^{\mathrm{GT}}$ (b) for prolate shapes in both ${}^{150}\mathrm{Nd}$ and ${}^{150}\mathrm{Sm}$. The shaded area indicates the experimental range extracted from the measured half-life using bare $g_A=1.273$ (lower line) and quenched $g_A=1$ (upper line) factors.

Figure 13

Running sums of the $2\nu \beta \beta$ NME in ${}^{116}\mathrm{Cd}$ as a function of the intermediate excitation energy in ${}^{116}\mathrm{In}$ for various calculations using different quadrupole deformations for parent and daughter nuclei. Thin lines correspond to calculations with the energy denominator $D_1$ [Eq. (7)] while thick lines correspond to $D_2$ [Eq. (8)].

Figure 14

Same as in Fig. 13, but for the double-$\beta$ decay of ${}^{150}\mathrm{Nd}$.

Figure 15

Running sums of the $2\nu \beta \beta$ NME in ${}^{116}\mathrm{Cd}$ as a function of the intermediate excitation energy in ${}^{116}\mathrm{In}$ for the decay between prolate shapes. The results are obtained with the energy denominator $D_2$ for three different Skyrme forces.

Figure 16

Same as in Fig. 15, but for the decay of ${}^{150}\mathrm{Nd}$.