Strength of $J^{\pi }=1^{+}$ Gamow-Teller and isovector spin monopole transitions in double-$\beta$-decay triplets

In this work we study systematically the energetics and intensity distributions of Gamow-Teller (GT${}^{\pm }$) and isovector spin monopole (IVSM${}^{\pm }$) transitions from the ground states of the pairs (${}^{76}\mathrm{Ge}$,${}^{76}\mathrm{Se}$), (${}^{82}\mathrm{Se}$,${}^{82}\mathrm{Kr}$), (${}^{100}\mathrm{Mo}$,${}^{100}\mathrm{Ru}$), (${}^{116}\mathrm{Cd}$,${}^{116}\mathrm{Sn}$), (${}^{128}\mathrm{Te}$,${}^{128}\mathrm{Xe}$), (${}^{130}\mathrm{Te}$,${}^{130}\mathrm{Xe}$), and (${}^{136}\mathrm{Xe}$,${}^{136}\mathrm{Ba}$), of double-beta-decay initial and final nuclei, to the $J^{\pi }=1^{+}$ excited states of the intermediate odd-odd nuclei ${}^{76}\mathrm{As}$, ${}^{82}\mathrm{Br}$, ${}^{100}\mathrm{Tc}$, ${}^{116}\mathrm{In}$, ${}^{128,130}\mathrm{I}$, and ${}^{136}\mathrm{Cs}$. We use a proton-neutron quasiparticle random-phase approximation (QRPA) theory framework with the Bonn-A two-body interaction in large no-core single-particle valence bases. Our studies indicate that the GT${}^{-}$ strength at the giant resonance (GTGR) region and the low-energy part of the IVSM${}^{-}$ strength overlap, which could be of significance for the analyses of the ($p,n$) type of reaction experiments at and around the GTGR.

Figure 1

Distribution of the GT${}^{-}$ [panels (a) and (b)] and IVSM${}^{-}$ [panels (c) and (d)] strength, (6) and (8), in the transition ${}^{76}\text{Ge}\to {}^{76}\text{As}$. The excitation energies are measured from the ground state of ${}^{76}\mathrm{Ge}$. The IVSM strength is given in units of fm${}^4$. The occupancies of the single-particle orbitals are based on either the WS + BCS model (left panels) or on experiment (right panels).

Figure 2

Distribution of the GT${}^{+}$ [panels (a) and (b)] and IVSM${}^{+}$ [panels (c) and (d)] strength, (7) and (9), in the transition ${}^{76}\text{Se}\to {}^{76}\text{As}$. The excitation energies are measured from the ground state of ${}^{76}\mathrm{Se}$. The IVSM strength is given in units of fm${}^4$. The occupancies of the single-particle orbitals are based on either the WS + BCS model (left panels) or on experiment (right panels).

Figure 3

Distribution of the GT${}^{-}$ [panel (a)], GT${}^{+}$ [panel (b)], IVSM${}^{-}$ [panel (c)], and IVSM${}^{+}$ [panel (d)] strength, (6), (7), (8), and (9), in the transitions ${}^{82}\text{Se}\to {}^{82}\text{Br}$ [panels (a) and (c)] and ${}^{82}\text{Kr}\to {}^{82}\text{Br}$ [panels (b) and (d)]. The excitation energies are measured from the ground state of ${}^{82}\mathrm{Se}$ for GT${}^{-}$ and IVSM${}^{-}$, and from the ground state of ${}^{82}\mathrm{Kr}$ for GT${}^{+}$ and IVSM${}^{+}$. The IVSM strength is given in units of fm${}^4$.

Figure 4

Distribution of the GT${}^{-}$ [panel (a)], GT${}^{+}$ [panel (b)], IVSM${}^{-}$ [panel (c)], and IVSM${}^{+}$ [panel (d)] strength, (6), (7), (8), and (9), in the transitions ${}^{100}\text{Mo}\to {}^{100}\text{Tc}$ [panels (a) and (c)] and ${}^{100}\text{Ru}\to {}^{100}\text{Tc}$ [panels (b) and (d)]. The excitation energies are measured from the ground state of ${}^{100}\mathrm{Mo}$ for GT${}^{-}$ and IVSM${}^{-}$, and from the ground state of ${}^{100}\mathrm{Ru}$ for GT${}^{+}$ and IVSM${}^{+}$. The IVSM strength is given in units of fm${}^4$.

Figure 5

Distribution of the GT${}^{-}$ [panel (a)], GT${}^{+}$ [panel (b)], IVSM${}^{-}$ [panel (c)], and IVSM${}^{+}$ [panel (d)] strength, (6), (7), (8), and (9), in the transitions ${}^{116}\text{Cd}\to {}^{116}\text{In}$ [panels (a) and (c)] and ${}^{116}\text{Sn}\to {}^{116}\text{In}$ [panels (b) and (d)]. The excitation energies are measured from the ground state of ${}^{116}\mathrm{Cd}$ for GT${}^{-}$ and IVSM${}^{-}$, and from the ground state of ${}^{116}\mathrm{Sn}$ for GT${}^{+}$ and IVSM${}^{+}$. The IVSM strength is given in units of fm${}^4$.

Figure 6

Distribution of the GT${}^{-}$ [panel (a)], GT${}^{+}$ [panel (b)], IVSM${}^{-}$ [panel (c)], and IVSM${}^{+}$ [panel (d)] strength, (6), (7), (8) and (9), in the transitions ${}^{128}\text{Te}\to {}^{128}\text{I}$ [panels (a) and (c)] and ${}^{128}\text{Xe}\to {}^{128}\text{I}$ [panels (b) and (d)]. The excitation energies are measured from the ground state of ${}^{128}\mathrm{Te}$ for GT${}^{-}$ and IVSM${}^{-}$, and from the ground state of ${}^{128}\mathrm{Xe}$ for GT${}^{+}$ and IVSM${}^{+}$. The IVSM strength is given in units of fm${}^4$.

Figure 7

Distribution of the GT${}^{-}$ [panel (a)], GT${}^{+}$ [panel (b)], IVSM${}^{-}$ [panel (c)], and IVSM${}^{+}$ [panel (d)] strength, (6), (7), (8) and (9), in the transitions ${}^{130}\text{Te}\to {}^{130}\text{I}$ [panels (a) and (c)] and ${}^{130}\text{Xe}\to {}^{130}\text{I}$ [panels (b) and (d)]. The excitation energies are measured from the ground state of ${}^{130}\mathrm{Te}$ for GT${}^{-}$ and IVSM${}^{-}$, and from the ground state of ${}^{130}\mathrm{Xe}$ for GT${}^{+}$ and IVSM${}^{+}$. The IVSM strength is given in units of fm${}^4$.

Figure 8

Distribution of the GT${}^{-}$ [panel (a)], GT${}^{+}$ [panel (b)], IVSM${}^{-}$ [panel (c)], and IVSM${}^{+}$ [panel (d)] strength, (6), (7), (8) and (9), in the transitions ${}^{136}\text{Xe}\to {}^{136}\text{Cs}$ [panels (a) and (c)] and ${}^{136}\text{Ba}\to {}^{136}\text{Cs}$ [panels (b) and (d)]. The excitation energies are measured from the ground state of ${}^{136}\mathrm{Xe}$ for GT${}^{-}$ and IVSM${}^{-}$, and from the ground state of ${}^{136}\mathrm{Ba}$ for GT${}^{+}$ and IVSM${}^{+}$. The IVSM strength is given in units of fm${}^4$.

Figure 9

Cumulative sums of the effective GT${}^{-}$ strength (6) [left panel] and IVSM${}^{-}$ strength (8) [right panel] for the transition ${}^{76}\text{Ge}\to {}^{76}\text{As}$ calculated by using the “WS + BCS” and the “exp” occupancies. The excitation energies are measured from the ground state of ${}^{76}\mathrm{Ge}$. The IVSM strength is normalized by dividing by $R^4$ where $R=5.08$ fm is the nuclear radius.

Figure 10

Left panel: The total effective GT${}^{-}$ (upper curve) and GT${}^{+}$ (lower curve) strengths, (6) and (7), as functions of the mass number. Right panel: The total effective IVSM${}^{-}$ (upper curve) and IVSM${}^{+}$ (lower curve) strengths, (8) and (9), as functions of the mass number in units of fm${}^4$. The small circles indicate the results obtained with the “exp” occupancies for the $A=76$ nuclei. For the IVSM the circles coincide in the scale of the figure (see also Tables 5 and 7).