Nucleon-pair states of even-even $N=82$ isotones

In this paper we study low-lying states of five $N=82$ isotones, ${}^{134}\mathrm{Te}, {}^{136}\mathrm{Xe}, {}^{138}\mathrm{Ba}, {}^{140}\mathrm{Ce}$ and ${}^{142}\mathrm{Nd}$, within the framework of the nucleon-pair approximation (NPA). For the low-lying yrast states of ${}^{136}\mathrm{Xe}$ and ${}^{138}\mathrm{Ba}$, we calculate the overlaps between the wave functions obtained in the full shell-model (SM) space and those obtained in the truncated NPA space, and find that most of these overlaps are very close to 1. Very interestingly and surprisingly, for most of these yrast states, the SM wave functions are found to be well approximated by one-dimensional, optimized pair basis states, which indicates a simple picture of “nucleon-pair states”. The positive-parity yrast states with spin $J>6$ in these nuclei, as well as the $8_2^{+}$ state, are found to be well described by breaking one or two $S$ pair(s) of the $6_1^{+}$ or $6_2^{+}$ state (low-lying, seniority-two, spin-maximum, and positive-parity); similarly, negative-parity yrast states with spin $J>9$ are well represented by breaking one or two $S$ pair(s) of the $9_1^{-}$ state (low-lying, seniority-two, spin-maximum, and negative-parity). It is shown that the low-lying negative-parity yrast states of ${}^{136}\mathrm{Xe}$ and ${}^{138}\mathrm{Ba}$ are reasonably described to be one-octupole-phonon excited states. The evolution of the $6_1^{+}$ and $6_2^{+}$ states for the five isotones are also systematically investigated.

Figure 1

Calculated energy levels for low-lying states of ${}^{134}\mathrm{Te}$, in comparison with experimental data [33].

Figure 2

Energy levels for ${}^{136}\mathrm{Xe}$ and ${}^{138}\mathrm{Ba}$ calculated in three configuration spaces, in comparison with experimental data [12, 33]. The first space, denoted by “SM,” corresponds to the full shell-model configuration space. The second, denoted by “NPA,” corresponds to space constructed by using collective pairs with positive parity and spin zero, two, and four, as well as two positive-parity spin-six pairs for positive-parity states, or by coupling the above positive-parity pairs to one negative-parity pair with spin three or nine for negative-parity states. The third, called “pair state,” is simply a one-dimensional, optimized nucleon-pair basis state constructed by collective pairs.

Figure 3

The overlap between the SM wave function and corresponding NPA wave function (solid black squares), the overlap between the NPA wave function and corresponding one-dimensional, optimized pair basis state (solid red circles), and the overlap between the SM wave function and the optimized pair basis state (solid blue up-triangles), for yrast states of ${}^{136}\mathrm{Xe}$ and ${}^{138}\mathrm{Ba}$. Panels (a) and (${\mathrm{a}}^{\prime }$) [(b) and (${\mathrm{b}}^{\prime }$), (c) and (${\mathrm{c}}^{\prime }$), (d) and (${\mathrm{d}}^{\prime }$)] correspond to the yrast states with positive parity and even spin (positive parity and odd spin, negative parity and odd spin, negative parity and even spin), respectively.

Figure 4

Energy levels of ${}^{140}\mathrm{Ce}$ and ${}^{142}\mathrm{Nd}$ calculated in the truncated NPA spaces, in comparison with experimental data from Refs. [12, 33].

Figure 5

(a) Ratio between the two structure coefficients $y\left(g_{7/2}{d}_{5/2}6\right)$ and $y\left(g_{7/2}{g}_{7/2}6\right)$ (denoted as $y_2/y_1$) for the $I_1$ and $I_2$ pairs, versus proton number $Z$. (b) $\mu \left(6_1^{+}\right)$ and $\mu \left(6_2^{+}\right)$ values versus $Z$. (c) $B\left(E3\right)$ values of the two transitions $9_1^{-}\to 6_1^{+}$ and $9_1^{-}\to 6_2^{+}$, versus $Z$. Experimental data are taken from Ref. [33].