*7.2. Shell Model States*

The best view that one possesses of shell model states is of excited states in isotopic and isotonic sequences adjacent to closed shells. Examples are shown in Figures 17, 28 and 29, namely, the low-energy systematics of the odd-mass Tl, In and Sb isotopes, respectively. These views are the best because they are not dominated by pairing correlations with respect

to the unpaired nucleon: it is a single nucleon outside of a singly closed shell. Furthermore, the singly closed shell cores are dominated by spherical, seniority type excitations (when intruder states do not appear at low energy). Thus, one expects that the degrees of freedom of these closed-shell plus or minus one nucleon nuclei are dominated by independentparticle degrees of freedom. If there are any monopole energy shifts, i.e., changes in energy of shell model states across an isotopic sequence, they will be easy to see and easy to interpret. This was the universally held view until the observations on Coulomb excitation of 129Sb summarized in Figure 27, and the implications of these data when compared to the normal weak-coupling model case represented by 115In in Figure 26. The message from the data in Figures 27 and 26 is that, while a nucleon in a unique-parity configuration exhibits weak-coupling *E*2 strength, i.e., the summed strength in the odd-mass nucleus equals the singly closed-shell core strength, manifested in *B*(*E*2; 0<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>2</sup><sup>+</sup> <sup>1</sup> ), cf. Figure 26; when *j* mixing occurs, the *E*2 strength may exceed the weak-coupling value, cf. Figure 27. Thus, *B*(*E*2) data such as those in Figures 26 and 27 become a key focal point for exploring the emergence of collectivity in nuclei—*j* mixing must be quantified. In turn, the issue of *j* mixing is critical for assessing monople energy shifts in nuclei: any use of data in nuclei must first be assessed for *j* mixing before single-*j* energy shifts can be extracted.

The focus here on the role of *j*-mixing differs from the emphasis of the discussion of the increased *E*2 strength in 129Sb by Gray et al. [64], where the discussion in terms of shell model calculations identified that the collectivity of the neutron core was not increased by the addition of the extra proton, but rather the increased *E*2 strength arose primarily from the proton–neutron term, thus pointing to overall coherent contributions to the *E*2 strength. The role of *j* mixing is not immediately evident in this approach.

To explore the role of *j*-mixing explicitly, schematic particle–vibration model calculations were performed for 115In and 129Sb. A code developed by one of the authors (AES) was employed (see Refs. [171–173]). With 115In modeled as a 1*g*9/2 proton hole coupled to the ground and first-excited state of 116Sn, the sum rule was confirmed for the weakcoupling case; however, it was observed that a shortfall in the summed *E*2 strength occurs when the particle–vibration coupling becomes finite. Turning to 129Sb, the low excitation states were described by allowing the odd proton to occupy the *π*1*g*7/2 and *π*2*d*5/2 orbits, coupled to the 128Sn core. This is a minimal model to describe the low-excitation positiveparity states shown in Figure 27. The sum rule was confirmed for the weak-coupling limit, and a short-fall in *E*2 excitation strength from the ground state was again observed when the particle–vibration coupling became finite. However, when the proton was also allowed to occupy the *π*2*d*3/2 orbit as well as *π*1*g*7/2 and *π*2*d*5/2, the summed *E*2 strength in 129Sb exceeded that of the 128Sn core.

The above calculations demonstrate the role of *j*-mixing in the enhanced *E*2 strength observed in 129Sb compared to 128Sn, perhaps in a more transparent way than the largebasis shell model calculations. No tension between the shell model and these schematic particle-vibration model calculations is seen. The important concept is that *j* mixing means that the sphericity of the mean field has been broken. This is a fundamental point, based on symmetry, behind the emergence of nuclear collectivity. Some additional observations are made on *j* mixing in the discussion that follows in this Section.

Let us take the above points and consider the systematic features of the odd-mass copper isotopes, shown in Figure 48. A natural first look at *j* mixing is to assemble information for single-nucleon transfer reactions. As already noted, spectroscopic factors must be handled with caution. However, far more directly, fragmentation of *j* strength is often observed, as shown in Figure 49 for the copper isotopes. One makes the following observation: *j* is a quantum number characteristic of a spherical mean-field with spin–orbit coupling; if *j* strength is fragmented, the mean field is not spherical.

A particular feature of note in the odd-mass copper isotopes is the sudden change in the relative energies of the lowest states with spin–parity 1/2−, 3/2−, and 5/2− above *N* = 40, cf. Figure 48. These abrupt changes have been interpreted as a major illustration of monople energy shifts [91], based on the assumption that the observed states are the 2*p*1/2, 2*p*3/2, and 1 *f*5/2 shell model states. However, *E*2 transition strengths in 69,71,73Cu, compared to the closed shell cores, 68,70,72Ni, cf. Figure 33 reveal that *E*2 transition strength exceeds that in the cores, as seen also in 129Sb, cf. 128Sn (Figure 27). This collectivity is independent of intruder state structures, which are identified in Figure 48. However, the current status of monopole energy shifts requires detailed spectroscopy before it can be discussed quantitatively. Indeed, a recent paper [174] appears to give it dominant status in its role behind the appearance of intruder states at low energy. We would counsel greater caution, and consideration of the structures, interactions and energy dependencies of multiparticle-multihole intruder configurations as shown in Figure 40, in the pursuit of interpretations of the energies of intruder states.

**Figure 48.** The lowest energy states in the odd-mass Cu isotopes. The naïve interpretation is that these states are simply the manifestation of the expected *f pg* shell model states: 2*p*3/2, 2*p*1/2, 1 *f*5/2, 1 *f*7/2 and 1*g*9/2. The existence of rotational band patterns in 69,71,73Cu supports the 7/2<sup>−</sup> states as the Nilsson state 7/2−[303]. The situation with respect to the other states remains confused: transfer reaction data (shown in Figure 49) reveal fragmentation of single-particle *j* strength. The experimental situation in <sup>71</sup>−77Cu is very incomplete and any interpretation is premature, except—see comments in the text. Horizontal bars with vertical arrows indicate excitation energies above which states are omitted. The *B*(*E*2) data of Stefanescu et al. [175] for 67,69,71,73Cu can be compared with the Ni cores, cf. Figure 33: an investigation into collective enhancement in the odd-Cu isotopes relative to the corresponding Ni core nuclei appears to be in order.

**Figure 49.** The three lowest states in <sup>59</sup>−65Cu and states for which there is significant population in the one-proton transfer reaction (3He,d). The lengths of the coloured bars are directly proportional to the strength of population of the states in the transfer reaction. The ground-states can be naïvely interpreted as the shell model configuration 2*p*3/2; similarly, the first excited states would appear to be the configuration 2*p*1/2. However, the assumption that these are spherical shell model states breaks down for the second excited state where the one-proton transfer strength is severely fragmented. The colour coding indicates the values of the transferred nucleon — blue (- = 1 with *j* = - + 1/2), green (- = 1 with *j* = - − 1/2), red (- = 3 with *j* = - − 1/2). The direct interpretation is that the quantum number is not a good quantum number in these nuclei, i.e.,they are deformed. Further discussion of such issues must await a more advanced level of treatment. The data are taken from [176]. Reproduced from [8].

Whereas the issue of monopole shifts in subshell energies remains open, no systematic study has been made; the idea has only been applied selectively where there is unfortunately a lack of detailed spectroscopic information [91]. However, detailed information exists, for example in the odd-mass Sc (*Z* = 21) isotopes, as shown in Figures 50 and 51. The parabolic pattern in these figures points to a dominance of deformation-producing forces controlling intruder state energies. Intruder states are strongly deformed structures with both large correlations that originate in their multi-shell structure and in their pairing structure. It would be interesting to make a thorough study of such structures across all nuclei to clarify the role of monopole energy shifts as a factor underlying intruder states and shape coexistence.

Deformation in nuclei immediately adjacent to closed shells has become a recent focus in the odd-mass F (*Z* = 9) isotopes [177]. The data are consistent with deformed ground states. This appears to lie outside of any shell model expectations. Indeed, the surprise that the double-closed shell nucleus 28O does not have a bound ground state, but its neighbour, 29F does, may be because the double-closed shell of 28O does not favour ground-state deformation, but 29F can deform in its ground state. This would appear to be a simple

explanation of the surprise that 28O is unbound (but, to our knowledge, has never been pointed out).

**Figure 50.** Intruder states in K, Sc, V isotopes. These states are the heads of bands, which are consistent with *K* quantum numbers equal to the band head spins. *B*(*E*2) data for 45Sc are shown in Figure 51. The pattern matches the parabolic trend shown schematically in Figure 46 and supports the dominant role of a quadrupole interaction between protons and neutrons. Note that 45Sc is almost an "island of inversion", if one ignores the complete range of occurrence of the structure across the entire shell. The figure is adapted from [41].

**Figure 51.** Bands built on the intruder states (cf. Figure 50) in the odd-mass Sc and V isotopes. The numbers given for 45Sc are magnitudes of the intrinsic quadrupole moments, *<sup>Q</sup>*<sup>0</sup> (*e*· b) deduced from *B*(*E*2) measurements [178]. Adapted from [41].

At present, information on odd-mass nuclei adjacent to *N* = 50 and *N* = 82 remains very limited. Intruder states are observed in the *N* = 49 and *N* = 81 isotones as shown in Figures 52 and 53, respectively. These manifestations are not at the mid-shell points. Possibly, the proton structures, i.e., a subshell gap and/or proximity to a *j* = 1/2 subshell, at *Z* = 40 and *Z* = 64 have something to do with this. Excited 0<sup>+</sup> states for the *N* = 50 and *N* = 82 isotones are shown in Figure 54. At present, the reason for the dissimilarity between *N* = 50, 82 and *Z* = 50, 82 remains an open question. Whether or not there are low-lying excited 0<sup>+</sup> states in, e.g., 82Ge and 150Er, would be worth exploring. The situation at *N* = 48, i.e., in 80Ge, is of two contradicting reports [179,180] and a very recent result that casts further doubt on the existence of a low-energy excited 0<sup>+</sup> state in 80Ge [181]; unlike at *Z* = 48 (the Cd isotopes) where low energy deformed excited 0<sup>+</sup> states are well established (see, e.g., [41,116]).

**Figure 52.** Intruder states in the *N* = 49 isotones. The state in 79Zn is from Orlandi et al. [182] and Yang et al. [183]. The configuration involved may be a prolate deformed structure built on the 1/2+[431] Nilsson state. Note that these structures are nearly identical to the intruder state structures in the odd-In (*Z* = 49) isotopes, some details of which are noted in Section 4. Other data are taken from ENSDF [22]. For comments on the energy maximum at the mid-shell point, see Figure 54 caption.

**Figure 53.** Intruder states in the *N* = 81 isotones (shown in red with all observed decay branches). Horizontal bars with vertical arrows indicate excitation energies above which states are omitted. The mid-shell point is indicated. The configuration involved may be an oblate deformed structure built on the 7/2−[503] Nilsson state. The data are taken from ENSDF [22].

**Figure 54.** Excited 0<sup>+</sup> and 2<sup>+</sup> states in the *N* = 50 and *N* = 82 isotones which are candidate states for intruder configurations. They possess *ν*(2*p* − 2*h*) character as determined by two-neutron transfer studies [184–188]. Population of these states as percentages relative to the ground states are given as: blue for (t,p) and red for (p,t) reactions, respectively. These numbers are taken from [188–193]; other data are taken from ENSDF [22]. The mid-shell points are indicated. Possibly, the local high energy in 90Zr is due to a weak energy gap at *Z* = 40. See remarks on 82Ge in the text.

## **8. Survival of Seniority Structures Away from Closed Shells**

The picture of the intrusion of deformed structures into the domain of spherical structures is summarized in the foregoing, but what about the survival of seniority structures away from closed shells? This is an issue with only a few circumstantial focal points; it has never been subjected to systematic study, to our knowledge.

A leading illustration of the survival of seniority away from closed shells is shown in Figure 55 for the even-even *N* = 80 isotones. The dominance of a neutron 1*h*−<sup>2</sup> 11/2 broken pair is manifested at *J* = 10. Furthermore, as *Z* = 64 is approached, *J* = 10 states involving a proton 1*h*11/2 broken pair appear. Magnetic moment data strongly complement this observation. More specifically, the *g* factors of the 10<sup>+</sup> <sup>1</sup> states in 138Ce and 140Nd, *<sup>g</sup>* <sup>=</sup> <sup>−</sup>0.176(10) and *<sup>g</sup>* <sup>=</sup> <sup>−</sup>0.192(12), respectively, indicate their *<sup>ν</sup>*1*h*−<sup>2</sup> 11/2 structure. For 144Gd, however, *g*(10<sup>+</sup> <sup>1</sup> )=+1.276(14) [194] indicates the *<sup>π</sup>*1*h*<sup>2</sup> 11/2 configuration. But how far from closed shells does this broken-pair structure dominate *J* = 10 states, notably yrast states? A similar view is provided for the *J* = 6 state, due to the proton 1*g*<sup>2</sup> 7/2 broken pair in the tellurium isotopes in Figure 56. These and other issues are discussed in this Section.

**Figure 55.** Low-energy systematics of the positive-parity states in the *N* = 80 isotones. The high-spin states are discussed in the text. Vertical arrows indicate energies above which other positive-parity states are observed. First-excited 0<sup>+</sup> states are observed at (keV): 134Xe (1636), 136Ba (1579), 138Ce (1466), 140Nd (1413), 142Sm (1451), 144Gd (1887). The data are taken from ENSDF [22].

**Figure 56.** Yrast systematics in the even-mass Te (*Z* = 52) isotopes. Transition *B*(*E*2) values in W.u., where measured, are shown in blue between levels. Quadrupole moments of 2<sup>+</sup> <sup>1</sup> states, where measured, are shown below the isotope mass numbers. Additional details are given in the text. Data from [195–199] and ENSDF [22]. Figure is from [8].

High-*J* broken-pair states appear in localized regions across the entire mass surface. In spherical nuclei, they are manifested as seniority isomers; in deformed nuclei, they are manifested as *K* isomers. The topic of *K* isomerism is a time-honoured branch of nuclear structure study with comprehensive reviews [200–203]. The situation for transitional nuclei is poorly characterized. Two factors determine the excitation energies of highspin broken-pair states: pairing energy and rotational energy. Pairing contributions to broken-pair excitation energies are well understood and are well characterized. Rotational energy contributions to broken-pair excitation energies are epitomized by Figure 25. This aspect of nuclear structure is generally described as "rotational-alignment" effects: there is an enormous literature addressing this topic using the so-called cranked shell model. This model approximates the effects depicted in Figure 25 by "cranking" a deformed mean-field about a fixed axis at right angles to the symmetry axis of the deformed mean field. It has been extended to "tilting" the axis about which cranking occurs [204]. The cranked shell model has completely dominated the study of high-spin states in nuclei. Our concern here is with low-medium spin states in nuclei. Note that, at high spin, an axis of directional quantization approaches a semi-classical description in that the cone of uncertainty becomes narrow; thus, cranking about a fixed axis improves asymptotically with increasing total spin.

To move forward on the topic of the breaking of pairs away from closed shells, it is important to recognize that the prototype signature is properties of the 2<sup>+</sup> <sup>1</sup> states in nuclei, which manifestly involve breaking pairs. The leading question is: Which broken-pair configurations underlie a given 2<sup>+</sup> <sup>1</sup> state? While important insights can be gained through large-basis shell model calculations in the valence shell, the full answer must extend far beyond the valence shell, as manifested in the need for effective charges to describe *B*(*E*2; 2<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> <sup>1</sup> ) values (see Table 2). By looking at systematics of *B*(*E*2) values near closed shells, one expects to learn something about this fundamental aspect of the emergence of collectivity in nuclei. A natural first step is to look at even-even nuclei with one valence proton pair and one valence neutron pair, particles or holes, as will now be discussed.

It turns out that 132Te is one of the more accessible nuclei for a detailed study of what might be termed "prototype emergence of quadrupole collectivity in nuclei". The region around 132Sn is attractive for this purpose because 132Sn is an *N* > *Z* doubly magic core without low-excitation intruder states, and because detailed spectroscopic studies (including transfer reactions, *B*(*E*2), and *g*-factor measurements) show it to be a "good" doubly magic core. However the challenge, which makes performing detailed spectroscopy difficult, is that 132Te is accessible to radioactive beams, by beta-minus decay and as a fission fragment—but not at stable-beam accelerators.

The current knowledge of excitations in 132Te is shown in Figure 57. The extent of detailed information is best described as "inadequate". For example, a naïve broken-pair view would predict two low-lying 2<sup>+</sup> states, one due to a broken neutron (hole) pair, cf. 130Sn (*E*(2<sup>+</sup> <sup>1</sup> ) = 1221 keV), the other due to a broken proton (particle) pair, cf. 134Te (*E*(2<sup>+</sup> <sup>1</sup> ) = 1279 keV). Thus, (naïvely) there should be two excited 2<sup>+</sup> states in 132Te at 1221 and 1279 keV. The lowest-lying 2<sup>+</sup> states in 132Te are 2<sup>+</sup> <sup>1</sup> (974 keV), (2+) (1665 keV), (2+) (1778 keV), and then (2+) states at 2249 and 2364 keV, where the parentheses indicate that the spin–parity assignment is tentative. To pursue the naïve view, the broken-pair configurations can be viewed as mixing and repelling, so that one resulting state appears pushed down by 1221 − 974 = 247 keV and the other state is at 1270 + 247 = 1517 keV, cf. (above) 1665 keV. This raises many questions, such as: What is the structure of the states at 1778, 2249 and 2364 keV? What are the detailed properties of these states? Are they 2<sup>+</sup> states? What are their lifetimes, magnetic moments, and quadrupole moments? At present, all unanswered experimentally, except for some information on Coulomb excitation of the 1665 keV state. Note that the *J* = 0-coupled neutron-hole pair and the *J* = 0-coupled protonparticle pair also interact and cause an energy shift in the 0<sup>+</sup> configuration that dominates the ground-state of 132Te; but [(*π*2)*<sup>J</sup>* ⊗ (*ν*−2)*J*]<sup>0</sup> configurations can also be expected to contribute to the ground-state structure. Allowing for pair occupancies across the many shell-model subshells, there are many possibilities. In addition, note that there is an extensive literature that discusses the shell model configurations underlying the structure of 132Te [33,62,205–215].

To begin to answer some of the questions raised concerning the lowest few 2<sup>+</sup> states in 132Te, one can note that there is a single low-excitation proton configuration that forms a 2<sup>+</sup> state, namely *π*(1*g*7/2)<sup>2</sup> <sup>2</sup><sup>+</sup> , with *g* factor *g* = 0.82. In contrast, the neutron orbits 2*d*3/2, 3*s*1/2, and 1*h*11/2 are "almost degenerate", which means that low-excitation 2<sup>+</sup> states can be formed by the two-neutron-hole configurations *ν*(2*d*3/2)−<sup>2</sup> <sup>2</sup><sup>+</sup> , *<sup>ν</sup>*(1*h*11/2)−<sup>2</sup> <sup>2</sup><sup>+</sup> , and *ν*(2*d*−<sup>1</sup> 3/23*s*−<sup>1</sup> 1/2)2<sup>+</sup> , with *g* factors 0.54, −0.24, and −0.27, respectively.

**Figure 57.** Excited states in 132Te cf. 130Sn and 134Te excited states. Naïvely there should be an appearance of both sets of states, corresponding to the independent "breaking" of the proton particle pair or the neutron hole pair. However, these broken-pair configurations can be expected to interact; thus, the 2<sup>+</sup> <sup>1</sup> in 132Te is lower in energy than that in 130Sn and 134Te. See the text for details. The data are taken from ENSDF [22].

Table 7 shows the results of shell model calculations for the lowest five 2<sup>+</sup> states in 132Te. The calculations were performed with NUSHELLX [44] in the jj55 basis space and with the sn100 interactions; see Table 2 and [31,48,64,68] for additional details including the parameters of the effective *M*1 operator. Along with a comparison of the experimental and theoretical level energies, Table 7 lists the *g* factors and the decomposition of the wavefunctions into the dominant proton and neutron components coupled to 0<sup>+</sup> and 2+. Note that there is no relative phase information available in these structures. The mixing of the lowest two 2<sup>+</sup> states discussed above in relation to Figure 57 is qualitatively consistent with the shell model calculations. The considerable variation in the calculated *g* factors is an indication of the marked differences in the structures of these 2<sup>+</sup> states. As collectivity emerges, the *g* factors of all of the low-excitation states would be expected to approach the collective value, typically *g*coll ≈ 0.7*Z*/*A*. Such measurements are extremely challenging even for stable nuclides.

Given the complexity of the low-excitation states in 132Te due to the small energy spacing of the 2*d*3/2, 3*s*1/2, and 1*h*11/2 neutron hole orbits, one might consider nuclei like 136Te (approximately *<sup>π</sup>*1*g*7/2 ⊗ *<sup>ν</sup>*<sup>2</sup> *<sup>f</sup>*7/2) and 212Po (approximately *<sup>π</sup>*1*h*9/2 ⊗ *<sup>ν</sup>*2*g*9/2) as alternative "prototypes" to study the emergence of collectivity. Shell model calculations for these nuclei (see Table 2 for details of basis spaces and interactions) show that the configuration mixing in the lowest 2<sup>+</sup> states of these nuclei is already considerable.


**Table 7.** Shell model calculations for the five lowest 2<sup>+</sup> states in 132Te. The excitation energies and *g* factors are shown, along with the structure of the state. The structure indicates how the angular momentum is apportioned between protons and neutrons. It is not the wavefunction. The weights indicated sum to unity when all contributions are included.

There are limited simple and accessible cases to study in detail the proton plus neutron broken-pair structures of 2<sup>+</sup> states adjacent to a closed shell. Extending beyond this simplest case, the stable Te isotopes below 132Te provide the opportunity for detailed spectroscopy, including (n,n *γ*) studies [216], Coulomb excitation, and *g*-factor measurements [115], to track the emergence of collectivity as increasing numbers of neutron holes are added to the two protons outside the *Z* = 50 shell closure. The stable Xe isotopes, with four protons, are likewise accessible to detailed measurements [30,32,217–223]. In these iotopes, the cancellation of *E*2 strength for four protons in the 1*g*7/2 orbit (see Equation (1)) makes the observed *E*2 strengths in the Xe isotopes below 136Xe sensitive to the breakdown of the seniority structure and emerging collectivity.

Returning to the high-spin broken-pair states in this region, specifically the *J* = 10 broken-neutron-pair configurations and the *J* = 10 broken-proton-pair configurations, these do not mix strongly, as manifested in Figure 55, cf. 142Sm and 144Gd. This suggests that broken-pair high-*j*, high-spin configurations do not play a role in the emergence of collectivity. Figure 56 suggests survival of both the proton-broken-pair and the neutronbroken-pair structures, respectively for *J* = 6 and *J* = 10 in <sup>126</sup>−132Te. The *g* factor data, where available, support this suggestion. In the *N* = 82 case of 134Te, *g*(6<sup>+</sup> <sup>1</sup> ) = +0.847(25) [224], as expected for the *π*1*g*<sup>2</sup> 7/2 configuration. The *<sup>g</sup>* factors of the 2<sup>+</sup> <sup>1</sup> [48] and 4+ <sup>1</sup> [225] states in 134Te are consistent with *<sup>g</sup>*(6<sup>+</sup> <sup>1</sup> ), and hence the same configuration. In 132Te, with two neutron holes, *<sup>g</sup>*(2<sup>+</sup> <sup>1</sup> )=+0.46(5) [206,210,226,227] is closer to the collective *<sup>g</sup>* <sup>≈</sup> *<sup>Z</sup>*/*<sup>A</sup>* <sup>≈</sup> 0.39, whereas *<sup>g</sup>*(6<sup>+</sup> <sup>1</sup> )=+0.78(8) [228] remains consistent with that of the pure *π*1*g*<sup>2</sup> 7/2 configuration. Recent work at the Australian National University (ANU) tracks the persistence and eventual weakening of the proton-broken-pair structure in the 4<sup>+</sup> states of 124,126,128,130Te [115].

Indeed, discontinuities in yrast state energies persist throughout the open-shell, *Z* > 50, *N* < 82 region and as an example yrast *γ*-ray energies, *Eγ*, versus the spins of the initial states, *Ii*, are shown for the Ba isotopes in Figure 58. It is important to note that *K* isomerism can emerge in this region, as manifested in Figure 59, which shows the yrast sequences for the even-even *N* = 74 isotones. The band structures show that the deformation increases from 128Xe to 140Dy. An important issue in the emergence of collectivity in nuclei is: where and how is the validity of the *K* quantum number established?

In principle, measurements of the magnetic dipole and electric quadrupole moments along the sequence of isotones could help answer this question. Unfortunately, the data are limited. The *g* factors of the 8<sup>−</sup> isomers in 128Xe and 130Ba have been measured to be −0.036(9) [229] and −0.0054(35) [230], respectively. The quadrupole moment of the isomer in 130Ba has also been measured to be *Q* = +2.77(30)b [230], which corresponds to a deformation of <sup>2</sup> 0.2.

In 128Xe, the configuration of the isomer is assigned as *<sup>ν</sup>*(*h*11/2 ⊗ *<sup>g</sup>*7/2)8<sup>−</sup> . Evaluating the *g* factor of this configuration with the spin *g* factor, *gs*, quenched from the freeneutron value by the standard factor of 0.7 gives *g*(8−) = −0.046, consistent with

experiments. Empirical values for *g*(1*g*7/2) and *g*(1*h*11/2) from neighboring nuclei give *<sup>g</sup>*(8−) ≈ −0.07, somewhat larger than the experiment. For 130Ba, the isomer is assigned as 9/2−[514] ⊗ 7/2−[404]. The parentage of these Nilsson orbits is *νh*11/2 and *νg*7/2, i.e., as assigned to the isomer in 128Xe. Evaluating the *g* factor of the *K*-isomer with standard Nilsson wavefunctions at <sup>2</sup> = 0.2 and again quenching *gs* by the standard 0.7 factor, gives *g*(8−) = −0.003, in excellent agreement with the experiment. This result is not sensitive to the deformation. Thus, the moment data suggest that the validity of the *K* quantum number is established in 130Ba. It appears not to be established in 128Xe. Further insights could be gained from observation and characterization of bands built on the isomers.

**Figure 58.** Yrast *E<sup>γ</sup>* vs. *Ii* for <sup>120</sup>−128Ba (*Z* = 56). Note the discontinuity above spin 10. The interpretation is that this is due to dominance of a broken neutron pair, 1*h*<sup>2</sup> 11/2. See discussion in the text. The data are taken from ENSDF [22].

Finally, with respect to the data shown in Figure 59, note that hindrance of the *E*1 isomeric transitions appears to increase with decreasing deformation: this appears counterintuitive. *E*1 transitions are an observable for which systematic features often remain elusive. In the normal valence space, they are forbidden. When looking at *E*1 strength, probably this involves the net result of many small contributions to the matrix element. Nevertheless, there is a visible systematic trend in Figure 59, which lacks an explanation.

**Figure 59.** *K* isomers in *N* = 74 isotones. The figure is based on a more limited view presented in Królas et al. [231]. Hindrance factors are *<sup>f</sup>* <sup>−</sup><sup>7</sup> *<sup>ν</sup>* <sup>=</sup> *<sup>B</sup>*(*E*1) (in W.u.); the exponent is given by <sup>Δ</sup>*<sup>K</sup>* <sup>−</sup> *<sup>λ</sup>* where Δ*K* = 8 and *λ* = 1, i.e., decay between the *K<sup>π</sup>* = 8<sup>−</sup> isomer and 8<sup>+</sup> state of the *K* = 0 ground-state band occurs by *E*1 multipole radiation. See text for details. The data are taken from ENSDF [22].
