**4. Nuclei with Open Shells; Emergence of Collectivity**

Nuclear structure is dominated by open-shell nuclei. With excitation of nucleons across shell gaps, and the resulting correlations, "open-shell" configurations intrude to low energy, even to the ground state, in some closed-shell nuclei. Thus, one must understand open-shell nuclei from a microscopic perspective. There are excellent limiting cases for nuclear behavior in open-shell nuclei: these are the strongly deformed nuclei, but a detailed microscopic understanding is lacking. Some perspectives on the current situation are presented here. This is the main focal point of this paper.

The key criterion for this exploration is to identify signatures of shell model structure in open-shell nuclei. Doubly even nuclei obscure shell model structure because of the correlations of pairs of nucleons. Odd-mass nuclei manifestly provide a view, via the unpaired nucleon. However, correlations are still an issue because there can be mixing of the configurations with different *j* values within a given shell. However, spin–orbit coupling provides a way forward: each shell has a unique-parity orbital and configurations involving this orbital will be the least mixed of any structures observed.

The power of the systematics of unique-parity states is illustrated in Figures 18 and 19. These figures show the systematics of the positive-parity states in the odd-mass yttrium isotopes across two shells, Figure 18, and of the negative-parity states in the odd-mass *N* = 63 isotones across two shells, Figure 19. Noting that the "parent" *j* configurations are 1*g*9/2 and 1*h*11/2, i.e., they differ by one unit of spin, the patterns are similar to the point that they are close to identical. (We recognize that, in the yttrium isotopes, there is a "delayed" onset of collectivity in 91,93,95,97Y, an issue which does not concern us here). These patterns suggest that there is an underlying coupling scheme that is defined by just a few simple basic features. Since multi-*j* shell structures (as manifested in, e.g., the negative-parity states in the 28 < *Z* < 50 shell, involving the configurations 1 *f*5/2, 2*p*3/2, and 2*p*1/2) are dominated by mixing of these configurations, the unique-parity states may provide a basic guide, via recognized single-*j* shell dominated patterns, for a mixed *j*-shell description across all open-shell odd-mass nuclear structure. Thus, we point to patterns that are independent of specific open shells; and to the implication that "shell-specific" interactions may be unnecessarily complex and intricate.

At present, the best description of experimental data for odd-mass nuclei in regions where deformation is not large is: "incomplete". However, a small number of such nuclei have been sufficiently well studied that they can provide guidance to likely a more complete view of the structure of unique-parity states. The best experimental example of the structure we draw attention to is shown in Figure 20, i.e., the nucleus 125Xe. This is a pattern of organization related to the studies of a single-*j* particle coupled to a rigid triaxial rotor, by Juergen Meyer-ter-Vehn [95,96]; indeed, he suggested such a pattern in 187Ir, long before detailed spectroscopic information was available: an up-to-date view of 187Ir is shown in Figures 21 and 22, and these strongly support the view. Further note that a very similar pattern appeared in a weak coupling description [97].

**Figure 18.** Systematics of the positive-parity states across the yttrium isotopes. These states are the unique-parity states for protons in the 28 < *Z* < 50 open shell. Note the emergence of nearidentical excitation patterns at the extreme mass numbers (these are the Nilsson configurations Ω*π*[*Nnz*Λ] = 5/2+[422], which differ only in rotational energy parameters).

**Figure 19.** Systematics of negative-parity states across the *N* = 63 isotones. These states are the unique-parity states for neutrons in the 50 < *N* < 82 open shell. Note the emergence of nearidentical excitation patterns at the extreme mass numbers (these are the Nilsson configurations Ω*π*[*Nnz*Λ] = 5/2−[532], which differ only in rotational energy parameters and, slightly, in "staggering" or signature splitting).

**Figure 20.** Organization of the unique-parity states in 125Xe, associated with *j* = 11/2 into a "hyperband" pattern due to Meyer-ter-Vehn [95,96]. The inset shows the quantum numbers used to define this pattern [98]. The data are taken from [99], but the pattern was not recognized there. Note that "vertical" Δ*I* = 2 (i.e., *E*2) transitions are almost totally absent from the observed data. See the text for more details.

**Figure 22.** Organization of the unique-parity states in 187Ir associated with *j* = 9/2 intruder configuration into a "hyper-band" pattern due to Meyer-ter-Vehn [95,96]. The data are taken from [100], but the pattern was not recognized there.

The pattern shown in Figure 20 is an organization of experimental data to reflect the dominance of so-called "rotational-aligned" coupling, which occurs in odd-mass nuclei that are not strongly deformed. The leading rotationally aligned set of states is highlighted in red and extends from the lowest *I* = 11/2 state, diagonally upwards to the right. The spin 11/2 originates from the 1*h*11/2 spherical shell model state, which dominates all lowenergy negative-parity states in 125Xe. The lowest tier of states in this set has the spin sequence 11/2, 15/2, 19/2, 23/2, 27/2, ... ; the tier just above has the spin sequence 13/2, 17/2, 21/2, 25/2, ... However, as shown, this basic pattern is "repeated", within the set of states highlighted in red, with tiers possessing spin sequences 15/2, 19/2, 23/2, ... , 17/2, 21/2, ... Furthermore, with sets of states, highlighted in blue and green, the red pattern is repeated built on states of *I* = 9/2 and 7/2, respectively. The tiers of Δ*I* = 2 spin sequences, beyond the first two, result from axial asymmetry and the coupling of the *j* = 11/2 particle to an axially asymmetric rotor. The repeated sets of states, coded with different colours, identified as *I* = 9/2 and *I* = 7/2, arise from alignment of the *j* = 11/2 particle in the deformed quadrupole field of 125Xe (such as occurs in the Nilsson model) and rotations about the unfavored axis of the triaxial rotor. These multiple tiers have been considered in some nuclei, by some authors, as candidates for so-called "wobbling": such wobbling, however, requires strong *E*2 transitions between tiers of states, i.e., decays appearing as vertical arrows; in 125Xe, these transitions appear to be dominated by *M*1 transitions, which might be termed "magnetic" rotation; for further details, see Refs. [95,96]. There is controversy regarding *E*2 admixtures in Δ*I* = 1 transitions; see the general remarks in [101].

A perusal of the literature over recent decades suggests that the view of Meyer-ter-Vehn has been "forgotten". The question that arises from consideration of Figures 18–20 (and Figures 21 and 22) is: How small a deformation is meaningful in weakly deformed nuclei? We consider this question but do not reach a final answer. An important outcome of the Meyer-ter-Vehn model [95,96] has been a multi-*j* version of the model, which is usually described as the particle-triaxial-rotor model (PTRM) [102]. Indeed, it was applied to a description of 125Xe [103,104] before the more recent detailed data set [99]. Thus, the focus here is on a deeper look at the basics of these models, especially near their weak deformation limit.

A major factor in particle-rotor models, both axially symmetric and axially asymmetric, when the deformation is not large, is so-called "Coriolis" or "rotational" alignment. A milestone paper that pointed to this effect was by Frank Stephens and coworkers [105], based on observations in the odd-mass lanthanum isotopes; an up-to-date view of their perspective is shown in Figure 23. An up-to-date view of all known negative-parity states in the odd-mass lanthanum isotopes is shown in Figure 24. Except for 133La, the lowspin couplings are not yet observed. The coupling to low spin is addressed shortly in this Section. The pattern in Figure 23 is referred to as "rotation-aligned" coupling. A simple explanation is given in Figure 25. The essential mechanism is the competition between "rotation alignment" and "deformation alignment", where deformation alignment is embodied in the basic quantum mechanics of the Nilsson model. The quantum mechanics of rotation alignment is described by the *I · j* term of the particle-rotor model: Figure 25 is a semi-classical view of this term. A naïve view of the weak-coupling limit of this term is that it dominates the coupling, and the total spin, *I* and *j* become collinear. This already appears to happen in the odd-mass lanthanum isotopes for the *I* = *j* + 2 states, but this does not address the question for the other possible couplings of *j* (to the even-even core) to yield a resultant total spin *I*.

**Figure 23.** Energy pattern of the high-spin unique-parity states in the odd-mass lanthanum isotopes, compared to the ground-state bands of the (*A* − 1) even-mass barium isotopes. This figure is an up-to-date view of one first proposed by Stephens et al. [105], where the term "rotation-aligned coupling scheme" was introduced.

**Figure 24.** Systematics of unique-parity states in the odd-mass lanthanum isotopes for *N* ≤ 82. Note the sparse information on low-spin states. The data are taken from ENSDF [22].

**Figure 25.** Semi-classical view of the rotation-alignment or Coriolis term in the particle-rotor model. The two coupling schemes depict the two extremes of the Nilsson model for a single-*j* state in a spheroidally deformed mean-field, labelled by Ω = 1/2 and Ω = max. For the particle-rotor coupling, *I* <sup>=</sup> *R* <sup>+</sup> *j* and very different alignments of *I* and *j* are possible. Recognizing that, e.g., Figure 23 focuses on energy differences, one sees from these diagrams that *differences* in *I*, i.e., **<sup>Δ</sup>***I*, a vector quantity, result in very different values for **<sup>Δ</sup>***I · j* and hence for expectation values of this quantity. Reproduced from [8].

Coupling of *j* to an even-even core to yield low-spin states with unique parity is sparsely characterized in weakly deformed nuclei, as already noted. An extreme "weak coupling" example is shown in Figure 26. By weak coupling, one means that a set of states, resulting from coupling an odd-nucleon of spin *j* to the core 2<sup>+</sup> <sup>1</sup> excitation, *<sup>j</sup>* <sup>⊗</sup> <sup>2</sup><sup>+</sup> <sup>1</sup> , with spins <sup>|</sup>*<sup>j</sup>* <sup>−</sup> <sup>2</sup>| ≤ *<sup>J</sup>* ≤ |*<sup>j</sup>* <sup>+</sup> <sup>2</sup>|, appears as a closely spaced multiplet, at an excitation centred on the 2<sup>+</sup> 1 energy of the even-even core, connected by unfragmented *E*2 strength to the spin-*j* ground state, is observed. This simple view is approximately realized in Figure 26: Coulomb excitation strongly populates five states with *J* = 5/2, 7/2, 9/2, 11/2, and 13/2; it also weakly populates a 5/2<sup>+</sup> state at 941 keV and a 9/2<sup>+</sup> state at 1461 keV. These two states are due to a shape coexisting or intruder band (a Nilsson 1/2+[431] decoupled rotational band) details of which are not important to the present focus. It is sufficient to note that the weakly coupled multiplet is identifiable, with the provision that the spin 5/2 and 9/2 members of the multiplet are manifested with some configuration mixing due to near degeneracies with intruder band configurations. This would suggest that the weak coupling limit is a familiar pattern, and the quest is nearly complete, pending filling in some minor details. However, a recent result [64] shows that the situation is far from being the weak-coupling limit: this is illustrated in Figure 27. Even though the energies appear to approximate the weak-coupling pattern, significant collective *E*2 strength has been "acquired" by the addition of a single extra-core proton. More specifically, the odd-*A* nucleus 129Sb shows additional collectivity in Coulomb excitation from the ground state, above that of the 128Sn core. A shell model description with effective charges of *ep* = 1.7*e* and *en* = 0.8*e* set from the *B*(*E*2; 0<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>2</sup><sup>+</sup> <sup>1</sup> ) values of the semimagic neighbours 130Te for protons and 128Sn for neutrons, goes some way towards describing this additional collectivity. This simple

particle–core coupling situation therefore gives evidence of emerging collectivity over and above that implied by the significant effective charges associated with the individual proton and neutron contributions.

**Figure 26.** Example of near-weak coupling in 115In, observed by Coulomb excitation and shown in comparison to Coulomb excitation for the neighbouring even-even "core" nucleus, 116Sn. These states are due to the proton coupling 1*g*−<sup>1</sup> 9/2 <sup>⊗</sup> <sup>2</sup><sup>+</sup> <sup>1</sup> . There is some fragmentation of strength for specific spin-parities: this results from mixing with intruder states. The states at 941 keV, 5/2<sup>+</sup> and 1449 keV, 9/2<sup>+</sup> are members of a decoupled rotational band built on the 1/2+[431] Nilsson configuration [106]. This configuration has 1*g*7/2 parentage and results in a rotational band with decoupling parameter, *<sup>a</sup>* <sup>−</sup>2, which puts the 3/2<sup>+</sup> state below the 1/2<sup>+</sup> state. Note that the negative parity states are not shown; the lowest negative parity states in the odd-In isotopes are shown in Figure 28. The numbers in parentheses are *<sup>B</sup>*(*E*2) values for the excitation process, in units of *<sup>e</sup>*2fm4 <sup>×</sup> <sup>10</sup><sup>2</sup> (100 *<sup>e</sup>*2fm4 <sup>=</sup> 60 W.u. for *A* = 115.) The data are taken from [107]. Reproduced from [6].

**Figure 27.** *Cont*.

**Figure 27.** (**a**) Partial level scheme for 129Sb. Grey transitions result from excitation of the 1851-keV isomer present in the beam. (**b**): Fragmentation of the *<sup>E</sup>*2 strength in W.u. over the 2<sup>+</sup> <sup>⊗</sup> *<sup>π</sup>g*7/2 multiplet members and candidate *πd*5/2 state of 129Sb and enhancement of total strength as compared to the 128Sn core. The grey colored transition was not experimentally observed. The figure is reproduced from [64]. See also Ref. [64] for details of the large-basis shell model calculations SM1 and SM2, which employ the same basis space but alternative contemporary residual interactions.

The status of particle–core coupling presented above, and in additional calculations by Gray et al. [108], suggests that there is not a good understanding with respect to the *Z* = 50 closed shell and the odd-mass In and Sb isotopes. The issue extends across the entire mass surface due to a severe lack of critical data. The systematics of the low-lying states in the odd mass In and Sb isotopes are shown in Figures 28 and 29, respectively. The pattern of the In isotopes suggests that, for the negative-parity states, there may be important collective effects which would explain the energy minimum at mid shell. Weak deformation is supported by laser hyperfine spectroscopy studies [109] and is shown in Figure 30. Note that two views of deformation for the In isotopes are presented in Figure 30: a direct view via spectroscopic quadrupole moments—the lower sequence of data points centred on *β* ∼ 0.1, and an indirect view via isotope shifts—the upper sequence of data points. The latter view can be inferred to contain a dynamical contribution, but this aspect lies beyond the present discussion. The observed pattern for the Sb isotopes suggests a "crossing" of the 2*d*5/2 and 1*g*7/2 configurations. However, at present, the question of the collectivity associated with low-lying states in the odd-Sb isotopes suggests caution is needed in making the interpretation of the lowest 5/2<sup>+</sup> and 7/2<sup>+</sup> states as resulting from pure shell model configurations.

Skyrme Hartree–Fock calculations with the SKX interaction [110] correctly track the nominal 2*d*5/2 vs. 1*g*7/2 level ordering in the Sb isotopes, but the location of the 3*s*1/2 orbit does not track with the behaviour of the observed *J* = 1/2<sup>+</sup> state with its shift in energy across the observed 7/2<sup>+</sup> state. In the indium isotopes, the single-particle levels in the potential generated by SKX are more separated in energy and roughly track with the observed levels of the relevant spin–parity. It appears that the indium levels remain quite regular because the parent orbits are well separated in energy in the mean field and the observed states are less affected by residual interactions; however, one sees clear evidence from the quadrupole moments in Figure 30 that deformation develops at mid-shell. In contrast, the Sb isotopes have the 2*d*5/2 and 1*g*7/2 single-particle states quite close in the mean field calculation. Thus, the observed level ordering can be sensitive not only to changes in the mean field, but also to residual interactions and deformation effects.

**Figure 28.** Systematics of the negative-parity states in the odd-mass indium isotopes relative to the 9/2<sup>+</sup> ground states. Naïvely, these states could be interpreted as the shell model single proton–hole configurations 2*p*1/2, 2*p*3/2, and 1 *f*5/2; but *E*2 transition strengths would be desirable before such an interpretation is made. The "parabolic" energy trend suggests interactions with neutrons across the shell with a characteristic energy minimum near the mid-shell point (*N* = 66), as indicated. Note the severe deficiency of data for electromagnetic decay strengths: there is one half life, for the 589 keV state in 117In, and the *E*2/*M*1 mixing ratio for the decay of this state is ambiguous. Data for 107In are from [111] and for 131In are from [112]; other data are taken from ENSDF [22].

**Figure 29.** Systematics of selected states in the odd-mass antimony isotopes. The short blue lines show the energies of the 2<sup>+</sup> <sup>1</sup> states in the even-even *<sup>A</sup>*−1Sn isotopes with respect to which weak coupling in the *<sup>A</sup>*Sb isotopes can be assessed. The ground states of <sup>125</sup>−135Sb are not shown; they all have spin–parity 7/2+.

**Figure 30.** Deformations of the odd-In isotopes deduced from spectroscopic quadrupole moments and isotope shifts following laser hyperfine spectroscopy [109]. Reprinted from [109], Copyright (1987), with permission from Elsevier.

Mass regions where the issue of emergent collectivity needs detailed spectroscopic study are addressed in Section 5 through Section 9. In particular, Sections 5 and 6 focus on the Ni and Ca isotopes, respectively.

Let us emphasize that there is a substantial body of evidence for the role of triaxial shapes in nuclei that are of moderate deformation. This is supported by the observation of "too many low-energy states for axial symmetry" in unique-parity excitations, such as shown in Figures 20–22. It is also supported by the application of the Kumar–Cline sum rules [113,114] to shell model electromagnetic strengths, as summarized for calculations of the Bohr-model deformation parameters derived from the shape invariants for the tellurium isotopes in Figure 31. These features do not imply that <sup>128</sup>−134Te can be modeled as weakly deformed triaxial rotors in their low-lying states up to spin 6+. Scrutiny of the wave functions and predicted *g* factors, for example, indicates that the structures of the lowest few states are very different, despite their apparently similar shape parameters. These excitations are not rotations of a single intrinsic structure as is supposed in the triaxial rotor model. Although the magnetic moments indicate that the Te isotopes near the *N* = 82 shell closure cannot be accurately modelled as weakly deformed triaxial rotors, a triaxial rotor description may prove appropriate as the number of neutron holes increases. The fact that the excited-state shapes in Figure 31 are all triaxial with *γ* ≈ 30◦ may suggest that the pathway of emerging collectivity in this region progresses from near-spherical nuclei near 132Sn, to weakly-deformed triaxial rotors as an intermediate step, before finally reaching more strongly deformed prolate rotors near mid-shell. Further data and calculations across an extended range of Te and Xe isotopes would help to assess this conjecture.

**Figure 31.** Average Bohr-model deformation parameters for yrast states in 128,130,132,134Te, assuming an ellipsoidal deformed nucleus and determined from shell-model calculations using the Kumar– Cline sum rules. For clarity, the fluctuations are not plotted. They are similar in magnitude for all cases, and by happenstance, the "softness" or fluctuation associated with each point is comparable to the scatter in the plotted points. Reproduced from [115].

#### **5. Emergent Collectivity in the Nickel Isotopes**

Currently, there is a high interest in neutron-rich nuclei. This is because of unprecedented access to completely new mass regions, and soon to come facilities that may "reach" even further. In particular, the neutron-rich Ni isotopes and the adjacent open-shell isotopes have received much attention. The systematic features of the low-energy excited states in the even-mass Ni isotopes are shown in Figures 32 and 33. A naïve interpretation of <sup>58</sup>−66Ni (Figure 32) is that they are vibrational; however, the error of using only energies to make structural interpretations of weakly deformed nuclei has now been substantially demonstrated [116]. The structure of <sup>58</sup>−66Ni is addressed in detail in this Section, with attention to seniority and shape coexistence. An unequivocal interpretation of <sup>70</sup>−76Ni (Figure 33) is that these isotopes are dominated by seniority coupling. However, this is an incomplete view, as details in Figure 33 imply; the structure of <sup>70</sup>−76Ni is also addressed in detail in this Section.

**Figure 33.** Systematics of the lowest positive-parity excited states in <sup>68</sup>−78Ni. Data are taken from: [118,119] (68Ni); [120–122] (70Ni); [123] (72Ni); [124] (74Ni); [125] (76Ni). The *B*(*E*2) data are from ENSDF [22] for 68,70Ni, from [126] for 72Ni and from [127] for 74Ni. The dashed red lines are a conjecture regarding the possibility of deformed intruder bands based on the interpretation of 2<sup>+</sup> 2 and 4<sup>+</sup> <sup>2</sup> states being members of such bands, and would be consistent with the interpretation of such a structure in 70Ni, suggested by Chiara et al. [122]; they are interpreted as seniority-four states by Morales et al. [124].

Recently, a study of conversion electrons following (*p*, *p* ) excitation of 58,60,62Ni was made by Evitts et al. [128,129]. A notable result was the observation of strong *E*0 decay branches from second-excited 2<sup>+</sup> states to the first-excited 2<sup>+</sup> states. The details for 62Ni are shown in Figure 34. Previously, strong *E*0 decays had been established for a series of excited 0<sup>+</sup> states in 58,60,62Ni [130]. However, an unresolved puzzle was that, whereas this strength was associated with proton–pair excitations in 58,60Ni, this was not the situation in 62Ni. The paper of Evitts et al. [128,129] points to a possible resolution: Figure 34 suggests that the 0<sup>+</sup> <sup>2</sup> state at 2049 keV is the head of a strongly deformed band and the 2302-keV 2<sup>+</sup> state is the first-excited band member. The proton pair–excitation at 3524 keV is shown. Our interpretation of the 2049 keV 0<sup>+</sup> state in 62Ni is a "4p-4h" excitation of the 56Ni core. Such a structure would not be populated in (3He, n), (16O, 14C), (6Li, d) or (16O, 12C) reactions, which were the spectroscopic probes used to identify the proton–pair excitations in 58,60,62Ni [131–134].

**Figure 34.** Organization of the lowest excited states in 62Ni into seniority-dominated structures and a new strongly deformed band. At present, this view must be considered a conjecture. The proposed seniority-two structures are labelled by the shell model configurations with their associated spins and parities. The deformed band is discussed in the text. The 0<sup>+</sup> state at 3524 keV is assigned as a proton–pair excitation based on two-proton [134] and *α* [132] transfer reaction spectroscopic studies.

The seniority-dominated structure of 70,72,74,76Ni has an unusual complication. While it is simple in 76Ni, as established by direct observation of a cascade of four gamma rays from an isomer with half-life 590 ns [125], this isomerism has disappeared in the lighter even-mass nickel isotopes. The situation is now resolved at the level of the multiple decay branches from the candidate spin–parity 8<sup>+</sup> states characteristic of a (*j* = 9/2)<sup>2</sup> seniority, *v* = 2 multiplet; but an open question is the nature of the low-lying states that facilitate these "fast" decays. Two possibilities exist: the "extra" states are seniority four, *v* = 4 states or, the "extra" states are members of coexisting deformed bands. It is possible for *v* = 4 states to appear lower in energy than *v* = 2 states in the manner manifested in 72,74Ni [124]. It is also plausible that shape coexistence is occurring at low energy in these nuclei. In favor of the latter interpretation is that shape coexistence has been suggested to occur at low energy in 70Ni [121,122]. Furthermore, a near identical structure in the *N* = 50 isotones involving the proton 1*g*9/2 subshell exhibits robust seniority isomerism

with no involvement of *v* = 4 states producing fast decays (see, e.g., [135], although one has to note seniority breakdown at low spin inferred from lifetime measurements [136]).
