Nucleon-pair states of even-even Sn isotopes based on realistic effective interactions

In this paper we study yrast states of ${}^{128,126,124}\mathrm{Sn}$ and ${}^{104,106,108}\mathrm{Sn}$ by using the monopole-optimized realistic interactions in terms of both the shell model (SM) and the nucleon-pair approximation (NPA). For yrast states of ${}^{128,126}\mathrm{Sn}$ and ${}^{104,106}\mathrm{Sn}$, we calculate the overlaps between the wave functions obtained in the full SM space and those obtained in the truncated NPA space, and find that most of these overlaps are very close to 1. Very interestingly, for most of these states with positive parity and even spin or with negative parity and odd spin, the SM wave function is found to be well represented by one nucleon-pair basis state, viz., a simple picture of “nucleon-pair states” (nucleon-pair configuration without mixings) emerges. In ${}^{128,126}\mathrm{Sn}$, the positive-parity (or negative-parity) yrast states with spin $J>10$ (or $J>7$) are found to be well described by breaking one or two $S$ pairs in the ${10}_1^{+}$ (or $7_1^{-}$) state, i.e., the yrast state of seniority-two, spin-maximum, and positive-parity (or negative-parity), into non-$S$ pair(s). Similar regularity is also pointed out for ${}^{104,106}\mathrm{Sn}$. The evolution of $E2$ transition rates between low-lying states in ${}^{128,126,124}\mathrm{Sn}$ is discussed in terms of the seniority scheme.

Figure 1

Energy levels of yrast states for ${}^{128,126,124}\mathrm{Sn}$ obtained in three sets of calculations by using the monopole-optimized effective interactions based on the realistic CD-Bonn nucleon-nucleon potential, in comparison with experimental data. The first set (denoted as “SM”) is performed in the full shell-model configuration space. In the second set (denoted as “NPA”), the configuration space for positive parity is constructed by using collective positive-parity pairs with spin zero, two, four, six, eight, and ten; the configuration space for negative parity is constructed by coupling the above positive-parity pairs to one negative-parity pair with spin four, five, six, and seven, respectively. In the third set (denoted as “pair state”), the configuration space for a given spin and parity is constructed by the one-dimensional, optimized nucleon-pair basis state. In the NPA calculation for ${}^{124}\mathrm{Sn}$, up to two non-$S$ pairs are considered in the nucleon-pair truncated subspace. The experimental data is taken from Refs. [16, 17].

Figure 2

For the yrast states of ${}^{128,126}\mathrm{Sn}$, 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). (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] for ${}^{128}\mathrm{Sn}$ and ${}^{126}\mathrm{Sn}$, respectively.

Figure 3

The neutron-hole occupation number of the pseudo “$13/2^{+}$” and “$17/2^{-}$” shells for the $0_1^{+}, 2_1^{+}, 8_1^{+}, {10}_1^{+}, {12}_1^{+}, 5_1^{-}, 7_1^{-}, 9_1^{-}, {13}_1^{-}, {15}_1^{-}$ states versus mass number $A$, based on the SM calculation.

Figure 4

Same as Fig. 1 except for ${}^{104,106,108}\mathrm{Sn}$ and experimental data are taken from Ref. [41]. The building blocks of the NPA space include positive-parity pairs with spin zero, two, four, six, and negative-parity nucleon pairs with spin three, four, five, six, seven, eight, and nine. For ${}^{108}\mathrm{Sn}$, the maximum number for non-$S$ pairs of each pair basis is limited to be two.

Figure 5

Same as Fig. 2 except that the overlaps here are for the ${}^{104,106}\mathrm{Sn}$ nuclei.