Nuclear moments and nuclear structure near 132Sn

Several g-factor measurements have been performed recently on nuclei near the neutron-rich, double-magic nucleus 132Sn. The focus here is on 134Te, the N = 82 isotone which has two protons added to 132Sn. Comparisons are made with other nuclei that have two protons outside a double-magic core. The extent to which 132Sn is an inert core is discussed based on these comparisons.


Introduction
The recoil in vacuum (RIV) method has proved to be a powerful method to measure the g factors (gyromagnetic ratios) of the first 2 + states in neutron-rich isotopes near double magic 132 Sn. Following on from the pioneering work on 132 Te [1,2], RIV g-factor measurements have also been published recently for the isotopes 124 Sn, 126 Sn and 128 Sn [3]. The measurements were performed at the Holifield Radioactive Ion Beam Facility (HRIBF). The quality of the radioactive beams, which were produced by the ISOL method and accelerated through the 25UR tandem, together with the segmentation and angular coverage of the CLARION [4] and bare HYBALL [5] detector arrays contributed to the success of these measurements.
In execution, an RIV g-factor measurement on a radioactive beam is identical to a measurement of B(E2; 0 + 1 → 2 + 1 ) by safe Coulomb excitation. In simplified terms, the B(E2) is determined by the total intensity of the γ radiation emitted, whereas the g factor is determined by examining the spatial pattern of the γ radiation. As a consequence, a statistically precise B(E2) is measured along with the g factor. The measurements are complementary as the g factor probes single-particle aspects of the wavefunction whereas the B(E2) probes collectivity. This paper will focus on 134 Te [6], which has two valence protons added to 132 Sn. Comparisons will be made with other nuclides that have two valence protons added to a double magic core. Figure 1 shows the experimental [1,2,[6][7][8] and theoretical [9][10][11][12] g(2 + 1 ) systematics for the Te isotopes near N = 82. There is quite good agreement between theory and experiment for 130 Te and 132 Te, however the theories do not agree for 134 Te where the experimental g factor [6] falls between the predictions of the QRPA calculation [11] and the two shell model calculations [9,10]. Elsewhere we have shown that these three calculations in fact predict similar wavefunctions, dominated by the π(g 7/2 ) 2 configuration [6]. Once the differences in the M 1 operator are considered, the QRPA can be brought into agreement with the shell model. The Monte Carlo Shell Model (MCSM) calculation [12], however, used the same M 1 operator as the QRPA calculations. Thus the difference in the predicted g factors for these two models implies that there is a significant difference between their wavefunctions, which requires further investigation, particularly as the MCSM is two standard deviations from the experimental g(2 + 1 ) value. The g factors of the longer-lived 4 + 1 and 6 + 1 states in 134 Te have been measured [13,14], as have the moments of the ground states of 133 Sb and 135 I [15]. To the extent that these states can be associated with pure π(g 7/2 ) n configurations, their g factors should be the same. Shell model calculations, using OXBASH [16], have been performed for 134 Te and its neighbors. The interactions and model space were those of Brown et al. [10] (the two protons can occupy g 7/2 , d 5/2 , s 1/2 , h 11/2 , d 3/2 ), but the empirical M 1 operator of Jakob et al. [9] was used. The calculated g factors are compared to the experimental data in Table 1. It is evident that the shell model predicts only small differences between the g factors of the states considered. Experimentally, the g factor of the 6 + 1 state in 134 Te is the same, within uncertainties, as that of the ground-state of 133 Sb, which represents the single-proton case. The g(2 + 1 ) value is also consistent with the g factor of the single-proton g 7/2 state. Overall, the g-factor data point to rather pure π(g 7/2 ) 2 configurations for the yrast 2 + 1 , 4 + 1 and 6 + 1 states in 134 Te.

Comparison of nuclei with two protons added to a double-magic core
We have seen that the g factors of the 2 + 1 , 4 + 1 and 6 + 1 states in 134 Te are consistent with these states being predominantly due to the π(g 7/2 ) 2 configuration. Cases where two protons are added to a closed-shell, and both the 2 + 1 -state and the maximum-spin state with J max = 2j − 1 (usually an isomer) have known g factors are rare. Aside from 134 Te these are: 50 22 Ti 28 (two f 7/2 protons added to 48 Ca), 54 26 Fe 28 (two f 7/2 proton holes in 56 Ni), and 92 42 Mo 50 (two g 9/2 protons added to 90 Zr). Along with the discussion of the g factors, it is useful also to consider the E2 transition rates. For transitions between the states of the pure j 2 configuration, the B(E2) values are related to the single-particle matrix element ⟨j||T (E2)||j⟩, by The B(E2) data for 50 Ti, 54 Fe, 92 Mo, and 134 Te are compared with Eq. (1) in Figure 2.
In order to judge the degree to which the E2 properties of these nuclei are consistent with the predictions for a pure πj 2 configuration, we define the double ratio Thus R E2 is unity if the experimental ratio of B(E2) values is consistent with the pure πj 2 model, and it exceeds unity if there is additional quadrupole collectivity in the 2 + 1 state. Table 2 shows the R E2 and g(2 + 1 )/g(J π max ) ratios for the nuclei of interest. It is clear from Figure 2 and Table 2 that 54 Fe and 92 Mo show additional E2 collectivity in the 2 + state. These two cases also have g(2 + 1 )/g(J π max ) significantly less than unity (see Table 2). This behavior can be associated with added quadrupole collectivity in the 2 + 1 state, which has the effect of reducing g(2 + 1 ) toward Z/A ∼ 0.4. Both 56 Ni and 90 Zr are known to be soft cores. In the case of 54 Fe, for example, large-basis shell model calculations in the pf shell, which effectively include excitations of a 56 Ni core, can account for the observed B(E2) and g-factor data [17].
The B(E2) and g-factor ratios in Table 2 enable an assessment of the extent to which the core nuclei are inert. Elsewhere [6] we have concluded that there is additional quadrupole collectivity in the 2 + 1 state of 134 Te that is not accounted for by large-basis shell model calculations which assume an inert 132 Sn core. We have also shown that coupling the valence πg 2 7/2 configuration to a core vibration with the properties of the first-excited state in 132 Sn can readily account for the observed 2 + 1 → 0 + 1 transition strength in 134 Te [6]. It was found that the wavefunctions of the 2 + 1 , 4 + 1 and 6 + 1 states of 134 Te nevertheless remain dominated by the πg 2 7/2 configuration. Combining these insights with the discussion above, it can be concluded that 132 Sn is a relatively inert shell-model core, roughly comparable to 48 Ca.

Concluding comments
The g factor and B(E2) ↑ for the first-excited state of the neutron-rich N = 82 isotone 134 Te have been measured at HRIBF [6]. The precision achieved for these radioactive beam measurements is remarkable: a comparison of the precision of the B(E2) and g-factor ratios in Table 2 shows that the radioactive beam case of 134 Te matches the precision of the three stable-beam cases.
There is evidence of additional quadrupole collectivity in the 2 + 1 state of 134 Te due to coupling between the valence protons and excitations of the 132 Sn core. However, the electromagnetic properties of the low-excitation states of 134 Te are generally well described by the shell model, even in the approximation that the two protons are restricted to the g 7/2 orbit. The power of combined B(E2) and RIV g-factor measurements on radioactive beams has been demonstrated.