**6. Collectivity in the Calcium Isotopes**

The calcium isotopes hold a unique position in the study of nuclear structure. With *Z* = 20 and a reach to either side of *N* = 20 and 28, they should be a perfect illustration of closed-shell behaviour in nuclei, except that they are not. Figures 1 and 2 open the focus of this contribution, with a perspective on 40Ca as an *N* = *Z* doubly closed-shell nucleus and on 48Ca as an *N* > *Z* doubly closed-shell nucleus: 48Ca conforms to expectations; 40Ca does not. Indeed, recently, the time-honored view that closed shells only occur at 2, 8, 20, 28, 50, 82, 126 has been questioned due to unusual systematic features in 52,54Ca: this is of high interest with respect to forthcoming prospects for new facilities which will provide access to very neutron-rich nuclei, and the calcium isotopes in particular. (The current "reach" into the neutron-rich calcium isotopes is two events in 39 h of beam time, assigned to 60Ca [137]).

A highly attractive feature of the calcium isotopes between *N* = 20 and 28 is that they should be dominated by a single *j* shell, the 1 *f*7/2 shell. Figure 35 shows data that support this view. The *j* = 7/2 seniority *v* = 2 states (*J* = 2, 4, 6) are highlighted in red; the *j* = 7/2 seniority *v* = 4 states (*J* = 2, 4, 5, 8) in 44Ca are highlighted in orange. Note that the *J* = 4, *v* = 4 configuration mixes with the *J* = 4, *v* = 2 configuration. Further note that the *J* = 5 state has not been observed. Figure 35 also shows that other states appear at low energy in 42,44,46Ca: these are discussed with reference to the following, Figures 36–40 and Table 5.

**Figure 35.** Seniority and shape coexistence view of <sup>42</sup>−46Ca. Data for 46Ca include recent results of Pore et al. [138] and Ash et al. [139]. The deformed band in 42Ca (shown in blue) is observed up to

spin 12 [140]. The deformed bands in 44Ca and 46Ca are indicated in purple. The seniority-two states are indicated in red (but see details below in this caption); the seniority-four states, unique to 44Ca, are indicated in orange (but, again, see details below in this caption). The distinction between the deformed bands is based on multi-nucleon transfer reactions: see Table 5. The seniority-2 and seniority-4 structures in 44Ca, the 4<sup>+</sup> states at 2.28 and 3.04 MeV, and the 2<sup>+</sup> states at 1.16, 2.66 and (probably) 3.30 MeV are mixed; see [141]. For the seniority structures associated with the *ν*1 *f*7/2 configuration, see Figure 3. Note that, while there is a high-spin study of 44Ca (Lach et al. [142]), the deformed band has not been characterized. For a complementary perspective of the calcium isotopes, see also details in Figures 36 and 41. The 3− states, which are not part of the present discussion, are shown in green. Horizontal bars with vertical arrows indicate excitation energies above which states are omitted.

**Figure 36.** Excited 0<sup>+</sup> states in <sup>40</sup>−48Ca. All known 0<sup>+</sup> states up to 10 MeV are shown. Assignments to particle–hole configurations are indicated where known and further details are given in Figure 35 and 37, and Table 5. Note the inset box which indicates when the *ν*1 *f*7/2 shell is half filled. Further note that the *ν*2*p* − 2*h* configurations are with respect to *N* = 28, and are identified by the neutron–pair–addition reaction (t,p).

**Figure 37.** Excited 0<sup>+</sup> states in 40Ca viewed from the perspective of multiparticle transfer reactions. The first excited 0<sup>+</sup> state at 3353 keV is usually labelled as "4p-4h" on the basis of its strong population in the 36Ar(6Li,d) reaction, but note the strong population of 0<sup>+</sup> states, particularly around 8.3 MeV. The second excited 0<sup>+</sup> state at 5212 keV is usually labelled as "8p-8h" on the basis of its strong population in the 32S(12C,*α*) reaction (the population of the 3353 keV state could involve partial filling of the hole states in 32S, and does not necessarily imply an 8p-8h admixture to the 3353 keV state). Figure 1 depicts the deformed and superdeformed bands built on the 3353 and 5212 keV states, respectively. Proton–pair configurations appear to dominate 0<sup>+</sup> states around 8 MeV, as supported by the 38Ar(3He,n)40Ca reaction. Based on the 42Ca(p,t)40Ca reaction, neutron–pair configurations do not dominate below 8.5 MeV. This leaves the 7301 keV excited 0<sup>+</sup> state as the leading structure of interest for an interpretation: possibly it is a "6p-6h" state (cf. Figure 40). Taken from [6].

**Figure 38.** Spectrum of deuterons following the reaction (6Li,d) on a 36Ar target. The most strongly populated excited states are members of the deformed band with *Ex* (*Jπ*): 3353 (0+), 3904 (2+), 5279 (4+), 6930 (6+), cf. Figure 1. Note that the peaks at 5.28 and 6.93 MeV are multiplets. Reprinted with permission from [143]. Copyright (1979) by the American Physical Society.

**Figure 39.** Spectrum of alphas following the reaction (12C, *α* ) on a 32S target. States in both the 4p-4h and the 8p-8h deformed bands are populated. The population of the 4p-4h band may involve a partial filling of the "eight holes" in the target. Reprinted from [144], Copyright (1972), with permission from Elsevier.

**Figure 40.** Estimate of the multiparticle-multihole basis state energies for 40Ca using a schematic *su*(3)particle ⊗ *su*(3)hole model with a *Q* · *Q* interaction of strength *C*, where *Q* = *Q*<sup>1</sup> + *Q*<sup>2</sup> and *Q*<sup>1</sup> and *Q*<sup>2</sup> act on the *Np* (*λ*1, *μ*1) and *Nh* (*λ*2, *μ*2), *p f* and *sd* irreps, respectively. The figure is from a collaboration between one of us (JLW) and the late David Rowe.

**Table 5.** Tabulation of multi-nucleon transfer reaction spectroscopic data that provide (limited) information on the multiparticle-multihole structures of excited 0<sup>+</sup> states in 42,44,46Ca. The terms "strong" and "weak" refer to strengths of population of states in these transfer reactions. Note that the inference of "particle" and "hole" structure depends on the transfer nucleons and the target configuration with respect to closed shells.


*<sup>a</sup>* <sup>4</sup>*<sup>p</sup>* − <sup>4</sup>*<sup>h</sup>* ⊗ *<sup>ν</sup>*<sup>1</sup> *<sup>f</sup> <sup>n</sup>* 7/2 , *<sup>b</sup> <sup>π</sup>*(2*<sup>p</sup>* − <sup>2</sup>*h*) , *<sup>c</sup> <sup>ν</sup>*(2*<sup>p</sup>* − <sup>2</sup>*h*).

Figure 36 shows the problem of the simple 1 *f*7/2 shell-based view of 42,44,46Ca: there are "too many 0<sup>+</sup> states" at low energy in the even-mass calcium isotopes. A singleshell-seniority view does not possess any excited 0<sup>+</sup> states; but the second excited state in 42,44,46Ca is a 0<sup>+</sup> state. Furthermore, the first-excited 0<sup>+</sup> states in 42,44,46Ca are not configurations due to a common origin. The evidence for this is presented in the following paragraphs.

The key to the structure of the calcium isotopes with *N* = 20 − 28 is manifested in multi-nucleon transfer reaction spectroscopy for 40Ca, summarized in Figure 37. The details are complex and counterintuitive. Indeed, the evidence "has to be seen to be believed"; Figures 38 and 39 show the evidence. It is important to recognize the role of the target nuclei in that they define "hole" structures with respect to which the transferred multinucleon "clusters" are "added". Added nucleons can completely fill the holes (ground-state population), or partially fill the holes, or not fill the holes at all. There is a dominance of transfer to states that involve the target holes remaining completely unfilled, i.e., a 4p-4h configuration in the (6Li,d) reaction and an 8p-8h configuration in the (12C,*α*) reaction. Note that the (6Li,d) reaction does not populate the states of the "8p-8h" band strongly, but band mixing is suggested. Further note that the (12C,*α*) reaction can populate the states of the "4p-4h" band by "partially filling" the holes.

A partial guide to the multiparticle-multihole structure of 42,44,46Ca is presented in Table 5. This view is only partial because of a lack of stable isotope targets. The view leaves open many questions, but overriding all questions is the clear view that the shell model, as a simple model view, catastrophically breaks down in these isotopes. A guide to a likely interpretation is provided by the schematic-model view presented in Figure 40. This treats particle "clusters" and hole "clusters" as distinct entities that interact. This is tractable using an *su*(3) algebra with a quadrupole–quadrupole, *Q* · *Q* interaction (see [150,151] for details). This schematic view suggests a viable "coupling scheme", which serves much like the Nilsson scheme serves in nuclei with deformed ground states, but here serving to handle multiparticle-multihole excitations at low energy in the calcium isotopes.

Figure 40 reveals the enormous energy shifts associated with interactions that produce nuclear deformation. In Figure 1, *B*(*E*2) values in association with the deformed bands in 40Ca are given: if the *<sup>B</sup>*(*E*2; 4<sup>+</sup> <sup>→</sup> <sup>2</sup>+) strength of 170 W.u. (in the 8p-8h band) is scaled to *<sup>A</sup>* <sup>=</sup> 172 (*A*4/3 dependence), it has a strength of 1200 W.u., cf. the 4<sup>+</sup> <sup>→</sup> <sup>2</sup><sup>+</sup> transition in the ground-state band of 172Yb, where the strength is 300 W.u. Note that the structural interpretations of the deformed bands in 40Ca are not model based; they are mandated by the transfer reaction data shown in Figures 38 and 39. Most importantly, Figure 40 and specifically the interaction strength of the quadrupole-quadrupole interaction, *C*, illustrates how configurations that are spread over ∼90 MeV in a spherical mean-field, i.e., 8*h*¯ *ω*, can appear almost degenerate in energy. Indeed, it is worthy of comment that Nature could be said to have "barely realized a spherical ground state for 40Ca" (as also for 16O). In the spirit of "islands of inversion", there is a veritable archipelago of islands of inversion, multiple inversions, within one-oscillator shell of excitation energy in 40Ca. Unfortunately, detailed spectroscopy of multiparticle-multihole states in nuclei is confined to this local mass region. A few more details are given before closing this Section.

Let us make some further comments on Figure 40. Just as one arrives at a shell model basis using an oscillator potential with spin–orbit coupling (and further, by deforming the oscillator potential, as a Nilsson model basis), so in Figure 40 one arrives at a multi-shell basis. The justification of invoking this basis is the observation of the 4p-4h and 8p-8h states in 40Ca at 3.5 and 5.2 MeV, respectively. The excitations of the 2p-2h and 6p-6h states in 40Ca are not characterized, but there are many 0<sup>+</sup> excited states known above 7 MeV, as presented in Figure 37. Note that a 5% change in the interaction strength corresponds to a 5 MeV shift in energy at *C* = 0.025 for the 8p-8h configuration. From this perspective, the very existence of a shell model description of nuclei is a "just-so" story, i.e., for a small change in this *su*(3)-model interaction, spherical states in nuclei would have only been encountered as rare, exotic excitations. An example of this is realized in 44Ti, as depicted in Figure 43. Details behind these schematic estimates are given in [150–152] (see also [153,154]).

A useful spectroscopic view of the persistence of multiparticle-multihole excitations in this mass region is provided by 42Ca. This nuclide is accessible to transfer reaction spectroscopy and to Coulomb excitation. A view which combines such spectroscopic data is presented in Figure 41. Figure 42 shows a simple view of the structure using "twostate mixing", applied to the lowest states with spins 0, 2, and 4; this description should be compared with shell model and collective model views summarized in Table 6. An important point to note is that, while these coexisting configurations mix, the mixing is sufficiently weak that the underlying dominant structures can be identified: they are a spherical (valence) neutron particle pair and a "6p-4h" structure resulting from the 40Ca core 4p-4h structure, cf. Figure 37.

The description presented in Figure 42 correctly reproduces the largest *E*2 transition strength, that between the 2<sup>+</sup> <sup>1</sup> and 0<sup>+</sup> <sup>2</sup> states, i.e., this strength is entirely due to mixing with zero contribution from intrinsic strength. The description fails for the *E*2 transition strength between the 2<sup>+</sup> <sup>2</sup> and 0<sup>+</sup> <sup>1</sup> states, and there is a serious failure for the diagonal matrix elements of the 2<sup>+</sup> <sup>1</sup> and 2<sup>+</sup> <sup>2</sup> states. The conclusion is that two-state mixing for spin 2 is inadequate: three (four)-state mixing is necessary. Experimentally, third and fourth 2<sup>+</sup> states are known at 3392 and 3654 keV (cf. ENSDF [22]); both are populated in the one-neutron addition reaction: spectroscopic characterization of these states is lacking.

**Table 6.** Comparison of *E*2 matrix elements in 42Ca with shell model (SM) and beyond meanfield (BMF) calculations. The differences between theory (th) and experiment (ex) are shown as [(*E*2ex − *E*2th)/*E*2ex] × 100%. Details of these calculations are given in [155]: the shell model calculations follow details similar to those employed in [156]. Comparison of the two-state mixing results, shown in Figure 42, suggests serious deficiencies in these two models, which are state of the art. Note that all three calculations obtain an incorrect sign for the 2<sup>+</sup> <sup>2</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> <sup>1</sup> E2 matrix element, and they all seriously fail for the diagonal matrix elements. Adapted from [155].


*<sup>a</sup>* Wrong sign.

**Figure 41.** Two key spectroscopic views of 42Ca and a comparison with 46Ca. On the left, the lowest positive-parity states are shown together with the *E*2 transition strengths in W.u. between these states and diagonal values for the *E*2 matrix elements in *e*b of the 2<sup>+</sup> <sup>1</sup> and 2<sup>+</sup> <sup>2</sup> states, as determined by Coulomb excitation [155]. In the centre of the figure, the population of these states by the 41Ca(d,p)42Ca reaction is shown (the spectrum is reproduced from [157], Copyright (1972), with permission from Elsevier.). This reaction should only populate the 0<sup>+</sup> ground state and one each for states of spin–parity 2+, 4+, and 6<sup>+</sup> corresponding to a seniority *v* = 2 multiplet in association with the expected 1 *f*7/2 orbital, which is the only shell model subshell for 20 < *N* < 28: these data provide evidence that there is mixing between these seniority configurations and other structure which is intruding to low energy. On the right, for comparison, the lowest positive-parity states in 46Ca together with known *E*2 transition strengths are shown. In addition to cited sources, data are taken from ENSDF [22].

**Figure 42.** Two-state mixing for the lowest pairs of states with spin–parity 0+, 2+, and 4<sup>+</sup> in 42Ca. The *E*2 matrix elements are shown for transitions and level quadrupole moments. The mixing amplitudes are fixed from the fragmentation of the one-neutron addition spectroscopy shown in Figure 41, viz. 0.807 in the ground state, 0.707 in the 2<sup>+</sup> <sup>1</sup> state and 0.807 in the 4<sup>+</sup> <sup>1</sup> for the 1 *f*7/2 components of these states. There are two fitted parameters: *Q*<sup>0</sup> = 10 *e* fm2 for the 1 *f*7/2 configurations and *Q*<sup>0</sup> = 40 *e* fm2 for the intruder configurations. The *Q*<sup>0</sup> value for intruder configurations is multiplied by a rotor model Clebsch–Gordan coefficient for the respective spin values, i.e., −1.195*Q*<sup>0</sup> for 21||*E*2||21 and 1.604*Q*<sup>0</sup> for 41||*E*2||21. Differences between theory (th) and experiment (ex) are shown in the lower part of the figure as [(*E*2ex − *E*2th)/*E*2ex] × 100%. Other theoretical views are tabulated in Table 6.

This conclusion that two-state mixing is inadequate is in line with recent experimental observations of *E*0 decays from the normal-deformed and superdeformed 0<sup>+</sup> states to the nominally spherical ground state in 40Ca where it is found that two-state mixing cannot explain the observed monopole decay strengths. Rather, three-state mixing is needed [158]. In this case large basis shell model calculations, which include multinucleon excitations of both protons and neutrons across the *Z* = *N* = 20 shell gap, are able to describe the *E*0 data. Of relevance for the present discussion is that these data confirm that the naïve spherical ground-state configuration of 40Ca is mixed with deformed intruder structures. This mixing contributes to the shortfall in single-particle strength displayed for valence proton knockout in Figure 15.

A rare view of a deformed nucleus, where a non-intruder spherical excited state has been identified, is 44Ti as depicted in Figure 43. The double-charge exchange reaction identifies the double-isobaric analog state of the 44Ca ground state, which is manifestly a spherical state as characterized by its seniority-dominated low-energy structure. This highlights the role of the many-body symmetrization in dictating deformation. Recall, the nucleon–nucleon interaction is short-ranged and attractive, and the total "space ⊗ spin ⊗ isospin" wave function is antisymmetric: thus, for maximum binding of nucleons in a nucleus, the space-part of the wave function must be as symmetric as possible (a spatially antisymmetric wave function results in "cancellations" in the many-body energy correlations). It also reveals, via the excitation energy of 9.3 MeV, why the identification of spherical states in nuclei with deformed ground states is extremely difficult and therefore essentially never discussed, but such states are present. Identification of deformed states in

nuclei with spherical ground states is usually achieved via the distinctive rotational bands associated with deformed structures in nuclei; spherical states do not exhibit such easily identified patterns.

**Figure 43.** Low-energy *T* = 0, 1 and 2 states of 44Ti and lowest-energy states of 44Ca which are states of isospin *T* = 2. *E*2 transition rates between states are given by *B*(*E*2) values (in W.u.). The key feature of the figure is the state at 9298 keV in 44Ti. This state is the double-isobaric analog state of the 44Ca ground state and so, manifestly, it is a "spherical" state. See text for details. Reproduced from [42]. Note: the 1884 keV state in 44Ca is incorrectly identified. As per Table 5 and Figures 35 and 36, it should be labelled as a *π*(2*p* − 2*h*) configuration.

The region of 40Ca could be regarded as the confrontational meeting point between shell model descriptions of nuclei and the true nature of the structure of nuclei. Multi-shell configurations manifestly dominate the low-energy structure. This is a proven spectroscopically based interpretation, i.e., it is not a model-inspired interpretation. Beyond this mass region, spectroscopic data that reveal the role of multiparticle-multihole configurations become sparse. Indeed, this region is a key meeting point, not only for shell-model based and multi-shell descriptions of nuclear structure, but also for incorporation of configurations from the continuum, as pointed out in [159].
