Change of nuclear configurations in the neutrinoless double-$\beta$ decay of ${}^{130}\text{Te}\to {}^{130}\text{Xe}$ and ${}^{136}\text{Xe}\to {}^{136}\text{Ba}$

The change in the configuration of valence protons between the initial and final states in the neutrinoless double-$\beta$ decay of ${}^{130}\text{Te}\to {}^{130}\text{Xe}$ and of ${}^{136}\text{Xe}\to {}^{136}\text{Ba}$ has been determined by measuring the cross sections of the $\left(d,{}^3\text{He}\right)$ reaction with 101-MeV deuterons. Together with our recent determination of the relevant neutron configurations involved in the process, a quantitative comparison with the latest shell-model and interacting-boson-model calculations reveals significant discrepancies. These are the same calculations used to determine the nuclear matrix elements governing the rate of neutrinoless double-$\beta$ decay in these systems.

Figure 1

(a)–(h) Outgoing ${}^3\text{He}$ spectra following the $\left(d,{}^3\text{He}\right)$ reaction at an incident energy of 101 MeV on isotopes of ${}^{128,130}\text{Te},{}^{130,132,134,136}\text{Xe}$, and ${}^{136,138}\text{Ba}$ at ${\theta }_{\mathrm{lab}}=5.8^{\circ }$. The dominant peaks carrying proton strength corresponding to orbitals above $Z=50$ are labeled by their energy in keV and $\ell$ value.

Figure 2

Angular distributions for the outgoing ${}^3\text{He}$ ions following the ${}^{130}\text{Te}(d,{}^3\text{He}){}^{129}\text{Sb}$ reaction at 101 MeV. The curves are DWBA calculations normalized to fit the data. Examples of $\ell =0$ (blue triangles, dotted line), 2 (orange squares, dashed), 4 (gray circles, solid), and 5 (green diamonds, dot-dashed) transfer are shown. Those for $\ell =0$, 2, and 5 are scaled by factors of 0.1, 0.5, and 0.2, respectively. The three arrows mark the angles at which measurements were made for the other targets. The error bars represent the statistical uncertainty only.

Figure 3

Ground-state proton occupancies beyond $Z=50$ for ${}^{128,130}\text{Te},{}^{130,132,134,136}\text{Xe}$, and ${}^{136,138}\text{Ba}$ as derived from the experimentally determined cross sections. The uncertainties, discussed in the text, are estimated to be approximately $\pm 0.1$ nucleons for each orbital.

Figure 4

The bar charts to the left show the change in nucleon occupancies between the ground states for the $0\nu 2\beta$-decay of ${}^{130}\text{Te}\to {}^{130}\text{Xe}$ and ${}^{136}\text{Xe}\to {}^{136}\text{Ba}$. The experimental data are denoted EXP. The proton data are from the current work, while the neutron data for the ${}^{130}\text{Te}\to {}^{130}\text{Xe}$ system are from Ref. [22]. The experimental data are compared to four different calculations: SM1 [47]; SM2 [48] (both shell-model calculations); IBM [50] (interacting-boson model); and QRPA [49] (quasiparticle random-phase approximation). The plots to the right show a comparison of the theoretical calculations to the experimental data, for $2s_{1/2}$ (blue triangles, dotted line), $1d$ (orange squares, dashed), $0g_{7/2}$ (gray circles, solid), and $0h_{11/2}$ (green diamonds, dot-dashed) strength. The error bars reflect the uncertainty in the experimental data.