**2. Nuclei with Closed Shells: An Experimental Perspective**

Nuclei with closed shells, both singly and doubly closed, have been the base upon which the shell model has been built. However, such nuclei are neither manifestations of nor a sound basis for the shell model in its extreme independent-particle form. Such nuclei (i.e., closed shell) can usefully be classified into three types: doubly closed shell nuclei with equal numbers of protons (*Z*) and neutrons (*N*), i.e., *N* = *Z*; doubly closed shell nuclei with *N* > *Z*; and singly closed shell nuclei.

The distinction of doubly closed shell nuclei with *N* = *Z* is that they exhibit shape coexistence at low energy, even at the level of the first excited states in 16O and 40Ca, as shown in Figure 1. In doubly even nuclei with *N* > *Z*, shown in Figure 2, shape coexistence has not yet been observed. The simple explanation is that, for *N* = *Z*, spatial overlap of the

proton and neutron configurations is maximal, and it is proton–neutron correlations that are deformation producing.

**Figure 1.** Excited states in the *N* = *Z* doubly closed shell nuclei 16O and 40Ca. Collectivity associated with the 2<sup>+</sup> <sup>1</sup> and 3<sup>−</sup> <sup>1</sup> states is shown. Collectivity involving deformation is supported by large electricquadrupole transition rates, as indicated by the *B*(*E*2) values in Weisskopf units (W.u.). Inferred *K* quantum numbers for collective bands are indicated. The horizontal bars with upward pointing arrows indicate excitation energies above which states are omitted. Adapted from [6].

**Figure 2.** Excited states in the *N* > *Z* doubly closed shell nuclei. Collectivity associated with the 2+ <sup>1</sup> and 3<sup>−</sup> <sup>1</sup> states is indicated by *B*(*Eλ*) values. The lowest known pair excitations are labelled. The horizontal bars with upward pointing arrows indicate excitation energies above which states are omitted. Electromagnetic decay strengths for 132Sn are calculated from data appearing in [7]. Adapted from [8].

The distinction of singly closed shell nuclei is that they are dominated by the emergence of pairing correlations. Pairing correlations are concisely formulated using the concept of the seniority quantum number, *v*, i.e., the number of unpaired nucleons. This was first recognized by Maria Goeppert-Mayer [9,10]. The quantum mechanics of pairing correlations is concisely, even elegantly, described using quasispin, as introduced by Arthur Kerman [11]. The basic features of quasispin, as applied to a series of (*j* = 7/2)*<sup>n</sup>* configurations, where *n* denotes the occupation of the orbit, are shown in Figure 3; a view which complements that in Figure 3 is shown for a series of (*j* = 9/2)*<sup>n</sup>* configurations in Figure 4. The quasispin algebra is developed in detail in Chapter 6 of [6]. That Chapter includes a thorough treatment of the origins of the key ideas from Racah's seniority [12–14] through

Flowers' handling of *j* − *j* coupling [15], Helmers' unitary symplectic invariants [16], Lawson and Macfarlane's identification of the rank-1/2 quasispin su(2) tensorial character of one-body annihilation and creation operators [17], to Kerman's simple formulation [11]. Furthermore, it can be noted that there is a profound duality structure residing in these algebras [18], which shows how algebraic structure provides insight into the complexity of many-body quantum systems. A pedagogical treatment of the quasispin algebra is presented in Chapter 4 in [5]. That Chapter illustrates how P.W. Anderson's idea [19] provided the first conceptual recognition of quasispin as the essential algebraic structure underlying many-fermion systems with Cooper pairs [20].


**Figure 3.** A schematic view of basic features possessed by a seniority-dominated *j* = 7/2 shell with a many-proton or many-neutron structure. The excitation patterns and associated spins are shown relative to the seniority zero, *v* = 0 states across the filling of the shell, where the filling is designated by the particle number, *n*. The quasispin quantum numbers, *s* and *ν* are su(2) quantum numbers and their relationship to shell model quantum numbers is shown in the box. Adapted from [6].

**Figure 4.** A schematic view of basic features of a seniority-dominated *j* = 9/2 shell. The excitation patterns and associated spins *J* are shown relative to the uncorrelated, *v* = max. states across the filling of the shell, where the filling is designated by the particle number, *n*. For other details, see Figure 3. Taken from [6].

Experimentally, the seniority coupling scheme is realized essentially exactly when the low-energy structure of singly closed shell nuclei is dominated by a high-*j* orbital. This is shown in Figure 5 for *j* = 11/2 neutron subshell filling in the Sn isotopes and in Figure 6 for *j* = 11/2 proton subshell filling in the *N* = 82 isotones. The patterns are almost indistinguishable. The domination of seniority extends into patterns of electric quadrupole, *E*2 transition probabilities: this is shown in Figure 7 for *j* = 9/2 configurations in even-Cd and even-Pd nuclei with *N* = 50 and *N* = 82. The pattern of *E*2 matrix elements in nuclei dominated by seniority coupling shows a smoothly changing character which is well described by the following relationship for the reduced transition strength [6]:

$$B(E2; s\nu I\_i \to s\nu I\_f) \propto \langle s\nu 10 | s\nu \rangle^2 = \frac{\nu^2}{s(s+1)} = \frac{(n-\Omega)^2}{4s(s+1)},\tag{1}$$

where *Ji* and *Jf* are spins of initial and final states, *s*, *ν* are quasispin quantum numbers, details of which appear in Figure 3; *sν*10|*sν* is an su(2) Clebsch–Gordan coefficient and Ω = (2*j* + 1)/2, e.g., Ω = 6 for *j* = 11/2. This Clebsch–Gordan coefficient emerges from the quasispin su(2) algebra when applying the Wigner–Eckart theorem to the *E*2 operator: this operator is a rank-1 quasispin tensor. Details are beyond the present discussion and are given in [6]. (Note: *ν* (designated by the Greek letter nu) is distinct from the seniority quantum number, *v* (designated by the Latin letter vee).) This relationship is illustrated in Figures 8 and 9 for the *j* = 11/2 configurations in the even-mass Sn isotopes and *N* = 82 isotones, respectively. Indeed, these patterns are one of the best signatures of structure unique to singly closed shell nuclei. However, the clarity and interpretation of these structures are dictated by quantum mechanics that is beyond that of the independentparticle shell model in that correlations in the form of Cooper pairs have emerged. Pairing Hamiltonians can be derived as a simplification of the nucleon–nucleon residual interaction; however, the focus here is on the empirical simplicity of the seniority structures that persist toward mid-shell where the number of valence nucleons is large, in contrast with the connection between pairing correlations and the two-body residual interactions in a largebasis shell model calculation, which is not obvious. Stated in rhetorical terms: Could one ascertain the algebraic structure of Cooper pairs, in the guise of quasispin, and manifestly controlling structure in all singly closed shell nuclei, based on a shell model computational

program? Once the quasispin structure is recognized, its implications for the residual interactions required in the shell model can be explored so that the structure emerges from the calculations.

**Figure 5.** The seniority-dominated spectra versus the atomic mass number, *A*, in the neutron-rich tin isotopes, shown relative to the highest spin state in each multiplet (note the *J* = 27/2 state in the odd-mass isotopes is set at the same level as the *J* = 8 state in the even mass isotopes). These structures are dominated by neutrons filling the 1*h*11/2 orbital. Note: multiple *J* = 4 states are seen in 120,122,124Sn and multiple *J* = 19/2 states are seen in 125Sn. Reproduced from [8].

**Figure 6.** The seniority-dominated spectrum in the proton-rich *N* = 82 isotones, shown relative to the highest spin state in each multiplet (note the *J* = 27/2 state in the odd-mass isotones is set at the same level as the *J* = 8 state in the even-mass isotones). These structures are dominated by protons filling the 1*h*11/2 orbital. The structure of 146Gd and 147Tb involves two-state mixing, as depicted schematically. Reproduced from [8].

**Figure 7.** (**a**) Seniority isomers involving *j* = 9/2 structures. The inset shows the half lives of the states with spin 8, the corresponding 8<sup>+</sup> <sup>→</sup> <sup>6</sup><sup>+</sup> transition energies, and the deduced *<sup>B</sup>*(*E*2) values for these transitions. The constancy of the *B*(*E*2) values, independent of mass, is remarkable and shows the simple nature of seniority structures. The figure is adapted from one appearing in [21]. Data are from the Evaluated Nuclear Structure Data File (ENSDF) [22]. The 6+-state energy in 130Cd, which is uncertain in ENSDF, is from [23]. (**b**) Seniority isomers involving the proton (1*g*9/2)−<sup>4</sup> configurations in the palladium isotopes at the *N* = 50 and *N* = 82 shell closures. The inset shows the deduced *B*(*E*2) values. The 96Pd scheme is adapted from one appearing in [24] and the 128Pd scheme is from [25]. The tabulated half lives and *B*(*E*2) values are taken from ENSDF. There are more recent published values [26,27], but the conclusions do not change.

**Figure 8.** Illustration of Equation (1), expressed in square-root form, for the proton 1*h*11/2 configurations in the *<sup>N</sup>* <sup>=</sup> 82 isotones. The *<sup>B</sup>*(*E*2) data shown are for the 10<sup>+</sup> <sup>→</sup> <sup>8</sup><sup>+</sup> transitions in the even-mass nuclei and for the 27/2<sup>−</sup> → 23/2<sup>−</sup> transitions in the odd-mass nuclei, cf. Figure 6. The sign of the square root is allowed to change to match the matrix element changing from positive to negative as depicted. If the proton number is counted with reference to 146Gd as *n* = 0: with Ω = 6, according to Equation (1), the *B*(*E*2) value should vanish at 152Yb. Note that this is an effect emerging from the Wigner–Eckart theorem for su(2), applied to reduction of the *E*2 matrix elements with respect to their quasispin tensor structure. Redrawn from [28].

**Figure 9.** A pattern of *B*(*E*2) values, similar to that shown in Figure 8, for the even-mass and oddmass Sn isotopes. These data suggest that the half-filled shell, where the *B*(*E*2) value goes to zero, is at *A* ∼ 122, i.e., that the 1*h*11/2 orbital is not at the highest energy within the 50 < *N* < 82 shell: this is consistent with 129Sn (and likely 131Sn) exhibiting a ground-state spin–parity of 3/2<sup>+</sup>. Note: there is a scale factor of 0.514 applied between the even and odd-mass values, which accommodates the *v* = 2 and *v* = 3 seniorities involved via the Clebsch–Gordan coefficient in Equation (1). Reprinted with permission from [29]. Copyright (2008) by the American Physical Society.

In the remainder of this Section, some observations are made with respect to the mathematical structure on which quasispin is based, in order to place this shell model view into perspective.

The arrival at the concept of quasispin as a degree of freedom in nuclei requires the recognition of mathematical structures that are not obvious. A brief sketch of the essential details is given here in words. Full details are given by Rowe and Wood [6] and, at an introductory level, by Heyde and Wood [5]. Specifically, the quasispin algebra is recognized by expressing the Hamiltonian and the interaction using second quantization. The mathematics emerge by taking bilinear combinations of the elements (one-body fermionic creation and annihilation operators) of a Jordan algebra (anticommutator brackets of the creation and annihilation operators). These bilinear combinations obey a Lie algebra (commutator brackets). This is impossible to see until one works out the Lie bracket values of the bilinear combinations, which is done by expanding them using anticommutator bracket relations so as to express everything in terms of Jordan algebra elements in "normal order"; see Equation (4.93) in Ref. [5]. Normal order means annihilation operators all to the right and creation operators all to the left. Furthermore, the Lie bracket algebra for a Jordan algebra element (single creation or annihilation operator) with quasispin algebra elements (bilinear combinations of creation and annihilation operators) reveals that the creation and annihilation operators are rank-1/2 quasispin tensors. This is also impossible to see until one works out the Lie bracket values. Indeed, rank-1/2 tensors are unknown in spin-angular momentum theory; see p. 423 in Ref. [6] for additional details.

Spectroscopy of low-spin and medium-spin states is beginning to provide a comprehensive (near-complete) view of excited states in doubly even nuclei at and near closed shells. Consequently, seniority coupling has been shown to apply in nuclei where the structure is dominated by two medium-spin *j* shells. This is illustrated in Figures 10 and 11 for the *N* = 82 isotones with *Z* < 64. The *v* = 2 structures in 134Te, 136Xe, 138Ba, and 140Ce are labelled in Figure 10: these include the 1*g*7/2 structures, with *J* = 2, 4, and 6, and

the 1*g*7/2-2*d*5/2 structures, with *J* = 1, 2, 3, 4, 5, and 6. In 136Xe only, as expected, *v* = 4 structures are observed with the allowed spins, *J* = 2, 4, 5, and 8, cf. Figure 3. The comprehensive view of 136Xe is the result of an (*n*, *n γ*) study [30]. Note that this seniority-based organization of data is essentially complete; for example, there is no excited 7/2<sup>+</sup> state observed, as might be expected from a 1*g*7/2 <sup>⊗</sup> <sup>2</sup><sup>+</sup> <sup>1</sup> coupling—such a coupling is forbidden by the Pauli exclusion principle if the 2<sup>+</sup> <sup>1</sup> states are seniority-dominated structures. The *B*(*E*2; 2<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> <sup>1</sup> ) = *<sup>B</sup>*<sup>20</sup> values and the magnetic moments, *<sup>μ</sup>*(2<sup>+</sup> <sup>1</sup> ), are shown for reference and discussed further in Figure 12 as the *g* factors, where *g*(2<sup>+</sup> <sup>1</sup> ) = *<sup>μ</sup>*(2<sup>+</sup> <sup>1</sup> )/2.

**Figure 10.** A view of the systematics of the even-mass *N* = 82 isotones with 50 < *Z* < 64. The lowenergy structure of these isotones is dominated by occupancy of the *π*1*g*7/2 and *π*2*d*5/2 shell model configurations: the Fermi surface progressing from the *π*1*g*7/2 to the *π*2*d*5/2 orbit is schematically indicated by dashed lines between 140Ce and 142Nd. The seniority structures are identified. The 3<sup>−</sup> states are shown for reference. Horizontal bars with vertical arrows indicate excitations above which states are omitted from the figure.

**Figure 11.** A view of the systematics of the odd-mass *N* = 82 isotones with 50 < *Z* < 64. The low-energy structure of these isotones, as noted in Figure 10, is dominated by occupancy of the 1*g*7/2 and 2*d*5/2 shell model configurations: here, the completion of the filling of the 1*g*7/2 orbital at *Z* = 58 ( 140Ce) is manifest in the change in ground-state spins between 139La and 141Pr. In 135I and 137Cs only, as expected, *v* = 3 structures are observed with the allowed spins, *J* = 3/2, 5/2, 9/2, 11/2, and 15/2, cf. Figure 3. Note: the spin of 1010 keV state in 135I is not known but is consistent with 3/2<sup>+</sup>. A state with spin–parity 3/2<sup>+</sup> is predicted at about 1 MeV excitation energy in 137Cs. Horizontal bars with vertical arrows indicate excitations above which states are omitted from the figure. Additional data for 141Pr, 143Pm, and 145Eu are not shown because they are not part of the present focus. Taken from [8].

**Figure 12.** Shell model calculations of *B*(*E*2) and *g* factors in the *N* = 82 isotones with 50 < *Z* < 64. Reproduced from [31], with the permission of AIP Publishing.

The seniority structure of the *N* = 82 isotones and its breakdown is an issue for future detailed study. However, shell model calculations affirm the dominant seniority structures. The case of 136Xe has been studied comprehensively [30,32]. Table 1 shows experimental *B*(*E*2) values between low-excitation states in 136Xe in comparison to the (1*g*7/22*d*5/2) seniority model, as well as several shell model calculations that include all orbits in the 50 ≤ *Z* ≤ 82 major shell but use alternative interactions. The *B*(*E*2) data indeed demonstrate the pattern predicted by the seniority scheme. It should be noted that 136Xe represents the mid-shell for the *π*1*g*7/2 orbit, for which several *E*2 transitions are forbidden. In such cases, the observed transition strengths result from small components of the wavefunction, which can lead to considerable variations in the shell model predictions, despite the calculations agreeing on the dominant structure of the states. It was noted in [32] that the large-basis shell model calculations support the dominant configurations assigned in the (1*g*7/22*d*5/2) seniority model up to the 4<sup>+</sup> <sup>2</sup> state at 2.1 MeV excitation, although there is considerable configuration mixing. The (1*g*7/22*d*5/2) model accounts for all states up to about 2.8 MeV, with the exception of the 0<sup>+</sup> <sup>2</sup> state (more on the 0<sup>+</sup> <sup>2</sup> state below in this Section). However, above the 2.1-MeV 4<sup>+</sup> <sup>2</sup> state, where the level of density increases, the correspondence between the two-level and full basis is less clear.

The 0<sup>+</sup> <sup>2</sup> states are consistent with a multi-pair structure distributed over the 1*g*7/2 and 2*d*5/2 orbitals. For example, the jj55 model with sn100 interactions [33] has dominant configurations of *π*(2*d*5/2)<sup>2</sup> (76%) [134Te], *π*(1*g*7/2)2(2*d*5/2)<sup>2</sup> (45%) [136Xe], and *π*(1*g*7/2)<sup>6</sup> (51%) [138Ba], for the 0<sup>+</sup> <sup>2</sup> states.


**Table 1.** Electric quadrupole transition rates in 136Xe. The seniority model in the (1*g*7/22*d*5/2) space is described in [30] and in the text. The shell model calculations from [30,32,34] use alternative interactions in the model space 1*g*7/2, 2*d*5/2, 3*s*1/2, 2*d*3/2, 1*h*11/2, which covers the 50 ≤ *Z* ≤ 82 major shell.

(*a*) The seniority model uses proton effective charge *ep* = 1.81, set to reproduce the experimental *B*(*E*2; 2<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> <sup>1</sup> ). (*b*) The calculated 2<sup>+</sup> <sup>2</sup> state is identified with the experimental 2<sup>+</sup> <sup>3</sup> state and vice versa.

Figure 12 shows the experimental *g* factors of the 2<sup>+</sup> states and the *B*(*E*2; 2<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> 1 ) values of the even-even *N* = 82 isotones with 50 < *Z* < 64, and compares them with largebasis shell model calculations. In addition, the ground-state *g* factors of the interleaving odd-*A* isotones are shown, which indicate that the Fermi surface moves from the 1*g*7/2 orbit into the 2*d*5/2 orbit at *<sup>Z</sup>* = 59. The *<sup>B</sup>*(*E*2) trend is quite well described, but the *<sup>g</sup>*(2<sup>+</sup> 1 ) trend is not well described, particularly when the Fermi surface moves into the 2*d*5/2 orbit. In contrast, the odd-*A* isotopes are well described. Focusing on the range 51 ≤ *Z* ≤ 57, the *g* factor data in Figure 12, for both odd and even-*A* isotones, are near constant and thus consistent with a simple *π*1*g<sup>n</sup>* 7/2 structure in both the ground states (odd-*Z*) and 2<sup>+</sup> <sup>1</sup> states (even-*Z*). The lowered experimental *g*(2<sup>+</sup> <sup>1</sup> ) values for 140Ce, 142Nd and 144Sm have been attributed to increasing contributions from *ν*(1*h*−<sup>1</sup> 11/22 *f*7/2) excitations [36]. Nevertheless, the basic seniority structure appears to persist in these nuclei.

The complete pattern of excitations in odd-mass, singly closed shell nuclei is somewhat more complex than for even-mass singly closed shell nuclei. This is shown for *j* = 11/2 in the tin isotopes in Figure 13. Note that the states expected for seniority *v* = 3 range over 14 spin values for *j* = 11/2, viz. 2*J* = 3, 5, 7, 9, 9, 11, 13, 15, 15, 17, 19, 21, 23, and 27 (see, e.g., [37]). The experimental view is incomplete, but there is sufficient detail to conclude that the seniority scheme provides a reliable basis for understanding the low-energy excitations in these isotopes. This perspective is supported by a more global view of odd-mass nuclei shown in Figure 14, wherein patterns for seniority-three multiplets in selected nuclides and selected spin couplings are visible for *j* = 7/2, 9/2, and 11/2. This global behavior appears not to have been recognized. We conjecture that there may be a geometric interpretation of this pattern, similar to the geometrical interpretation of two-body interactions for a pair of identical nucleons in a moderate to high *j* orbit, as introduced by Schiffer and True [38]. An angle between the two spins can be defined, which gives a measure of the overlap of the two orbits for different resultant spins; see discussions in Refs. [3,8].

One can conclude that seniority likely provides a complete description of the lowestenergy excited states in singly closed shell nuclei—with one proviso: singly closed shell nuclei exhibit low-energy deformed structures that "coexist" with the low-excitation senioritydominated structures.

**Figure 13.** A view of the systematics of the seniority-three *νh<sup>n</sup>* 11/2 states in the neutron-rich odd-mass tin isotopes. There are some states missing, according to seniority-dominated coupling; the full set contains: 2*J* = 3, 5, 7, 9, 9, 11, 13, 15, 15, 17, 19, 21, 23, and 27 (see, e.g., [37]). Because of ambiguities in some parity assignments, other potential candidate states are omitted. Note there are "second" 19/2<sup>−</sup> states observed in 123,125,127Sn.

**Figure 14.** A global view of seniority-three multiplets in selected nuclides and selected spin couplings, for *j* = 7/2, 9/2, and 11/2. Energies are omitted to avoid cluttering the figure; energies are also relative, per isotope. To our knowledge, this universal behaviour has not been recognized. Note that the *j* = 7/2 multiplets (with the proviso made for 135I in Figure 11) are complete; the *j* = 9/2 and 11/2 multiplets contain more states than shown here, cf. Figures 4 and 13.

The manifestation of shape coexistence in singly closed shell nuclei was recognized already forty years ago [39] and was reviewed thirty years ago [40]. It is well established for *Z* = 20, 50, and 82 and for *N* = 20 and 28; there are hints to its presence for *Z* = 8 and 28, and for *N* = 8, 50, and 82. Details can be found in the most recent review [41], together with some details in the earlier review [40]. The emerging view is that shape coexistence likely occurs in all nuclei; including that spherical states occur in nuclei with deformed ground states [42]. A concise perspective of the occurrence of deformation in nuclei as compared to atoms can be encapsulated in: "The difference between atoms and nuclei is that atoms are a manifestation of many-fermion quantum mechanics with one type of fermion, which repel, whereas nuclei involve two types of nucleon, which attract. By deforming, the system can lower its energy via relaxing the constraints of the Pauli exclusion principle in such a manner that more spatially symmetric configurations become accessible, which leads to a lowering of the energy of the system". (It can be noted that the emerging view of baryons may signal correlated, even deformed structures, especially the recent realization [43] that the proton contains more (virtual) anti-down quarks than anti-up quarks: this is simply a manifestation of correlations that involve "particle–hole" excitations, i.e., quark–antiquark pairs, and the Pauli principle.)
