Monopole effects, isomeric states, and cross-shell excitations in the $A=129$ hole nuclei near ${}^{132}\mathrm{Sn}$

We present results of large-scale shell-model calculations for the $A=129$ hole nuclei below ${}^{132}\mathrm{Sn}$. We discuss structures of ${}^{129}\mathrm{Sn}$, ${}^{129}\mathrm{In}$, and ${}^{129}\mathrm{Cd}$ with emphasis on the monopole effects and excitations across the neutron $N=82$ shell gap, and further predict low-lying levels for the more exotic ${}^{129}\mathrm{Ag}$. It is demonstrated that the monopole corrections in the Hamiltonian, which dynamically affect occupations of relevant orbits, can lead to interesting consequences for the shell evolution. It is found especially that the monopole terms, previously introduced to reproduce the cross-shell excitations of the $17/2^{+}$ and $21/2^{+}$ states in ${}^{131}\mathrm{In}$, shows more pronounced effects on the $A=129$ nuclei. In ${}^{129}\mathrm{In}$, the cross-shell excitations of $17/2^{+}$ and $21/2^{+}$ are pushed down significantly by the monopole terms, and in ${}^{129}\mathrm{Cd}$, the same monopole terms reverse the order of the single-hole states of $\nu d_{3/2}$ and $\nu h_{11/2}$, causing ${11/2}^{-}$ as the ground state for this nucleus. The structure of isomeric states in the $A=129$ nuclei is also discussed.

Figure 1

Calculated level spectrum of ${}^{129}\mathrm{Sn}$, ${}^{129}\mathrm{In}$, and ${}^{129}\mathrm{Cd}$ and comparison with experimental data [33]. Levels of blue color (also a star attached to the spin-parity notation) are the SP-like states. Levels of red color are cross-shell excitations. Theoretical results marked as Th. 1 (Th. 2) are those without (with) the Mod. 3 modification (see text).

Figure 2

Predicted cross-shell excited states in ${}^{129}\mathrm{Sn}$. Upper plot: Negative-parity states of the mixed configuration of $\nu h_{11/2}^{-2}{d}_{3/2}^{-2}{f}_{7/2}$ and $\nu h_{11/2}^{-4}{f}_{7/2}$. Lower plot: Positive-parity states of the mixed configuration by $\nu h_{11/2}^{-3}{d}_{3/2}^{-1}{f}_{7/2}$ and $\nu h_{11/2}^{-3}{s}_{1/2}^{-1}{f}_{7/2}$.

Figure 3

(a) The monopole effect induced by the $M1$ and $M2$ terms on the cross-shell excited states of 17/$2^{+}$ and 21/$2^{+}$ for ${}^{129}\mathrm{In}$. The results marked with M00 mean without any of these monopole corrections. Those marked with M1 mean with $M1$ only, and those with M1,2 mean with both $M1$ and $M2$. (b) Illustration of the monopole effect induced by $M2$ in the low-lying states of ${}^{129}\mathrm{Cd}$ and comparison with the SP states in ${}^{131}\mathrm{Sn}$. Note that, to show that the $11/2^{-}$ state is pushed down to be ground state by the $M2$ monopole effect, the $3/2^{+}$ levels are set to be the reference, and therefore, the two energy axes have a relative meaning only.

Figure 4

The monopole effect on the ground-state energy of the $A=129$ isobars. Part (a) marked with M00 means without any of these monopole corrections; part (b) marked with M1 means with $M1$ only, part (c) marked with M1,2 means both $M1$ and $M2$, and part (d) marked with M1,2+Mod3 means that the calculation includes $M1,M2$, and Mod. 3. Experimental data are taken from AME2012 in Ref. [38].