**3. Hints of Correlations, beyond Pairing and Seniority, at Closed Shells**

The dominance of seniority, with intruding shape coexistence, in singly closed shell nuclei is not quite "the whole story". The following analysis of effective charges implied by the observed *B*(*E*2; 2<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> <sup>1</sup> ) in even-even nuclei adjacent to doubly closed shells demonstrates what can be encapsulated in the term "the effective charge problem".

Electric multipole transition rates in the shell model are usually evaluated using harmonic oscillator wavefunctions. For a single-particle transition *j<sup>β</sup>* → *jα*, the reduced matrix element *jα*||*T*(*E*2)||*jβ* can be evaluated from

$$\langle j\_a || T(E\lambda) || j\_\beta \rangle = \frac{e}{\sqrt{4\pi}} (-1)^{j\_\beta + \lambda - \frac{1}{2}} \frac{1 + (-1)^{l\_a + l\_\beta + \lambda}}{2} \hat{\lambda} j\_a \hat{j}\_\beta \left( \begin{array}{ccc} j\_a & j\_\beta & \lambda \\ \frac{1}{2} & -\frac{1}{2} & 0 \end{array} \right) b^\lambda \mathcal{R}\_{a\beta}^{(\lambda)},\tag{2}$$

where <sup>ˆ</sup>*<sup>j</sup>* <sup>≡</sup> (2*<sup>j</sup>* <sup>+</sup> <sup>1</sup>) and *<sup>R</sup>*˜(*λ*) *αβ* is the dimensionless radial integral that can be evaluated in closed form with harmonic oscillator wavefunctions. The oscillator length *b* is defined as

$$b = \sqrt{\frac{\hbar}{m\_N \omega}}\,\tag{3}$$

where ¯*h* is the reduced Planck constant, *mN* is the nucleon mass, and ¯*hω* can be evaluated as a function of the mass number *A* as

$$
\hbar\omega = (45A^{-1/3} - 25A^{-2/3}) \text{ MeV} \tag{4}
$$

which has been found to give satisfactory agreement with observed charge radii. In general,

$$B(E\lambda; I\_i \to I\_f) = |\langle I\_f || T(E2) || I\_i \rangle|^2 / (2I\_i + 1). \tag{5}$$

For transitions between the states of the pure *j* <sup>2</sup> configuration, the *B*(*E*2) values are related to the single-particle matrix element *j*||*T*(*E*2)||*j*, by

$$B(E2; l\_i \to j\_i - 2) = 4(2j\_i - 3) \left\{ \begin{array}{ccc} j & j\_i - 2 & j \\ j\_i & j & 2 \end{array} \right\}^2 |\langle j||T(E2)||j\rangle|^2. \tag{6}$$

It is instructive to begin with the textbook cases of 17O and 18O, which can be considered as adding one and two neutrons, respectively, to a 16O core. Identifying the first-excited state to ground, 1/2<sup>+</sup> <sup>1</sup> <sup>→</sup> 5/2<sup>+</sup> <sup>1</sup> , transition in17O as due to the neutron transition from the 2*s*1/2 to 1*d*5/2 orbits, the experimental value of *B*(*E*2) = 2.39(3) W.u. (Weisskopf units) requires an effective neutron charge of *en* = 0.534(3)*e*. This value is close to *en* = 0.5*e*, which is the default often adopted for shell model calculations. However, turning to 18O, and identifying the 2<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> <sup>1</sup> transition with *<sup>ν</sup>*(*d*5/2)<sup>2</sup> <sup>2</sup><sup>+</sup> <sup>→</sup> *<sup>ν</sup>*(*d*5/2)<sup>2</sup> <sup>0</sup><sup>+</sup> , requires *en* = 1.054(14)*e* to explain the observed transition strength of 3.32(9) W.u. One might hope that this discrepancy between 17O and 18O would be resolved by a shell model calculation in the full sd model space with one of the "universal" sd interactions, but it is not. Such shell model calculations describe 17O well. The same calculations, however, fall short of explaining the *B*(*E*2 : 2<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> <sup>1</sup> ) in <sup>18</sup> O by a factor of nearly 3. It is worth noting that the experimental *B*(*E*2) for 18O is based on about 20 independent measurements by four independent techniques, all in reasonable agreement. The conclusion must be that the effective charge handles 17O, but fails for 18O due to additional correlations.

Table 2 shows shell model calculations for the reduced transition rate, *B*(*E*2; 2<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> <sup>1</sup> ), in doubly magic nuclides plus or minus two like nucleons. The shell model calculations were performed with NUSHELLX [44] and generally use a contemporary set of interactions for the relevant basis space, and either the recommended effective charges for the selected basis space, or the default *ep* = 1.5*e* and *en* = 0.5*e*, for protons and neutrons, respectively. The effective charges required to bring the shell model calculations into agreement with experiment are shown. For those nuclides adjacent to 48Ca and 56Ni, calculations were run in a basis that treats these nuclei as doubly magic, as well as in the full fp shell, which allows for excitations from the 1 *f*7/2 shell across the *N*, *Z* = 28 shell gap into the 2*p*3/2, 1 *f*5/2, and 1*p*1/2 orbits. These calculations account for the neutron core excitation in 48Ca, including the *<sup>ν</sup>*(2*<sup>p</sup>* <sup>−</sup> <sup>2</sup>*h*) <sup>0</sup><sup>+</sup> state at 5.46 MeV, but cannot describe the *<sup>π</sup>*(2*<sup>p</sup>* <sup>−</sup> <sup>2</sup>*h*) <sup>0</sup><sup>+</sup> state at 4.28 MeV; see Figure 2, and cf. Figure 60.


**Table 2.** Effective charges, *ep* and *en*, in units of the elementary charge *e*, for *B*(*E*2; 2<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> <sup>1</sup> ) in doubly magic nuclides plus or minus two like nucleons. The experimental values are from the Evaluated Nuclear Structure Data File (ENSDF) [22], with the following exceptions: 46Ar [45], 130,134Sn [46,47], 134Te [48], 210Po [49].

*<sup>a</sup>* Model basis spaces:

p: *π* & *ν* (1*p*3/2, 1*p*1/2)

sd: *π* & *ν* (1*d*5/2, 2*s*1/2, 1*d*3/2)

f7: *π* & *ν* (1 *f*7/2)

fp: *π* & *ν* (1 *f*7/2, 2*p*3/2, 1 *f*5/2, 2*p*1/2)

sdpf: *π* (1*d*5/2, 1*d*3/2, 2*s*1/2); *ν* (1 *f*7/2, 2*p*3/2, 1 *f*5/2, 2*p*1/2)

ho: *π* (1 *f*7/2); *ν* (2*p*3/2, 1 *f*5/2, 2*p*1/2)

jj55: *π* & *ν* (1*g*7/2, 2*d*5/2, 2*d*3/2, 3*s*1/2, 1*h*11/2)

jj56: *π* (1*g*7/2, 2*d*5/2, 2*d*3/2, 3*s*1/2, 1*h*11/2); *ν* (1*h*9/2, 2 *f*7/2, 2 *f*5/2, 3*p*3/2, 3*p*1/2, 1*i*13/2)

jj67: *π* (1*h*9/2, 2 *f*7/2, 2 *f*5/2, 3*p*3/2, 3*p*1/2, 1*i*13/2); *ν* (1*i*11/2, 2*g*9/2, 2*g*7/2, 3*d*5/2, 3*d*3/2, 4*s*1/2, 1*j*15/2)

*<sup>b</sup>* The default charges are *ep* = 1.5 and *en* = 0.5, unless otherwise indicated.

*<sup>c</sup>* For usdb the recommended values *ep* = 1.36, *en* = 0.45 were used.

*<sup>d</sup>* For sdpgmu the recommended values *ep* = 1.35, *en* = 0.35 were used.

*<sup>e</sup>* This experimental value has been questioned; see text.

There is no overall pattern in the effective charges shown in Table 2. Most of the shell model *B*(*E*2) values are within a factor of 2 to 3 of the experiment; however, those for the calcium isotopes, 38Ca and 42Ca, are underestimated by an order of magnitude. The experimental *B*(*E*2) value for 46Ar is almost a factor of two smaller than theory. While a lifetime measurement [59] gave a *B*(*E*2) value consistent with theory, the weight of evidence from independent Coulomb excitation measurements [45,60,61] makes the adopted value in Table 2 firm and in tension with theory.

Good agreement in the fp-shell calculation is obtained for 50Ca and 54Fe. As noted above, in these cases, 48Ca and 56Ni are not doubly magic cores but part of the fp model space. It is puzzling that the calculation for 50Ti in the same model space is twice the experiment, but the restricted *f*7/2 model space agrees with experiment.

Moving to heavier nuclei, the effective charges in the 132Sn region are near the default values [62], although most recent calculations adopt *ep* ≈ 1.7*e* and *en* ≈ 0.8*e* [32,34,63,64]. The measured *B*(*E*2) for 130Sn [46,47] is lower than theory and the experimental systematics (see [65]); the experiment should be repeated.

In the 208Pb region, *en* approaches +*e*. The experimental result for 210Po is problematic. As shown below in this Section, an analysis of higher-excited states in 210Po corresponding nominally to the *π*1*h*<sup>2</sup> 9/2 configuration implies *ep* ≈ 1.5*e*. The experimental *B*(*E*2) in Table 2 for 210Po is deduced from a recent lifetime measurement by the Doppler shift attenuation method following the 208Pb(12C,10Be)210Po reaction, which gave *<sup>τ</sup>* = 2.6 ± 0.4 ps [49]. This new result is certainly an improvement on the previous measurement which used (d,d ) above the Coulomb barrier to excite a 210Po target [66]. However, it is difficult to measure such a short lifetime below the longer-lived 4+, 6<sup>+</sup> and 8<sup>+</sup> states that tend to also be populated in heavy ion reactions; Kocheva et al. [49] recommend additional experiments. Coulomb excitation of the radioactive beam (e.g., at ISOLDE where 210Po activity remains in used ion sources) would be a possibility, avoiding the problem of feeding from the longer-lived higher excited states.

In several cases in Table 2, a *j* <sup>2</sup> approximation is (at least at face value) a reasonable starting point. For the case of 14C, it is not: holes in 16O nominally occupy the 1*p*1/2 orbit which must couple with 1*p*3/2 to form a 2<sup>+</sup> state. In other cases, like 130Sn, the 2*d*3/2, 3*s*1/2, and 1*h*11/2 single-particle orbits are so close in energy that a single-*j* <sup>2</sup> approximation cannot be applicable.

In some respects, the comparison of effective charges from the 2<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> <sup>1</sup> transitions alone may be considered selective and not altogether fair. However, as discussed in this Section, it fits our purpose, which is to examine the emergence of collectivity in nuclei. To explore further the successes and limitations of the shell model approach, comparisons of *E*2 strengths and *g* factors are now made for a selection of the semimagic nuclides in Table 2 that can be approximated as a single-*j* <sup>2</sup> configuration adjacent to a doubly magic core. Later in this section and again in Section 8, we argue that the properties of 2<sup>+</sup> <sup>1</sup> states, especially their electromagnetic properties, play an important part in developing an understanding of the emergence of collectivity in nuclei.

Table 3 shows the effective charges required to explain *B*(*E*2) values between lowexcitation states associated with nominal *j* <sup>2</sup> configurations in doubly magic nuclides plus or minus two like nucleons. For most cases, only protons or neutrons are active in the basis space. For 50Ti and 54Fe, calculations were performed in the fp model space which allows neutron excitations across *N* = 28; thus, both protons and neutrons contribute to the transition rate. In these cases, the proton effective charge required by experiment was evaluated assuming that *en* = 0.5*e*. The uncertainty given is due to the uncertainty in the experimental *B*(*E*2) alone. Concerning the uncertainty in the assumed value of *en*, it can be noted that *ep* + *en* is near constant for 50Ti, so a decrease in *en* by say 0.1*e* leads to an increase in *ep* of approximately 0.1. For 54Fe, the value of *ep* is less sensitive to the assumed value of *en*.

As expected, the effective charge is generally reduced when the basis space is enlarged; the *j* <sup>2</sup> model is obviously an oversimplification. However, it is a better approximation for the nuclei adjacent to the *N* > *Z* doubly magic 132Sn and 208Pb. One reason is that, for nuclei adjacent to *N* = *Z* doubly closed shells, intruder configurations are present at

low energy and these place the active nucleons in a much larger Hilbert space than can be handled by the shell model.

From Table 3, one can conclude that the effective charge required to describe the *B*(*E*2; 2<sup>+</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> ) transition is often greater than that required to explain the transitions between the higher spins in the *j* multiplet (i.e., the *E*2 decays of the states with *J<sup>π</sup>* = 4+, +, ... (2*j*)+), particularly for the *j* model. One can also see that the effective charges exceed the bare nucleon values, even in the large basis shell model calculations. The effective proton charges are reduced significantly for 50Ti and 54Fe when the basis space is expanded to include the whole fp shell. The proton charge deduced for 50Ti even approaches unity, but this assumes that *en* = 0.5*e*.

**Table 3.** Effective charges for nominal *j* configurations in selected doubly magic nuclides plus or minus two like nucleons. The effective charges are evaluated assuming a pure *j* configuration and for the mixed configurations of the (large basis) shell model (SM) calculations in Table 2. The experimental transition rates, *B*(*E*2)exp, are from ENSDF [22] and from the references in Table 2.


*<sup>a</sup>* Evaluated in the fp basis with gx1a interactions and *en* = 0.5*e*. See text for details. *<sup>b</sup>* This experimental value has been questioned; see text.

There are broadly two scenarios to explain the effective charge. First, and universally applicable, is the coupling of the valence nucleons to collective excitations of the core, including the giant resonances, in such fashion that the concept of an effective charge as a renormalization procedure has some operational justification. Second, and specific to particular cases, is the coupling between the valence nucleons and low-excitation configurations outside the shell model basis. This later scenario means that the shell model configuration is wrong in a more fundamental way.

An examination of the magnetic moments, or rather the *g* factors (*g* = *μ*/*J*), can distinguish between these scenarios. To this end, Table 4 shows an evaluation of the *g* factors for those nominal *j* <sup>2</sup> configuration cases in Table 3 for which there are experimental data. It is useful to make use of the fact that *g*(*j <sup>n</sup>*) = *g*(*j*); that is, the *g* factor of any number of nucleons in a single-particle orbit is equal to the *g* factor of the single-particle orbit, independent of the number of nucleons (*n*) and the resultant spin.

The empirical *g* factor of the *j* <sup>2</sup> configuration was evaluated as the average of the *g* factors of the ground-states of the neighbouring nuclei with *A* ± 1 and odd-*Z* or odd-*N*, as appropriate. The shell model calculations in the sd and fp spaces use the default effective *M*1 operator for those basis spaces. For the jj55 space, the *M*1 operator is as in Refs. [48,64,65,67,68]. For 210Pb and 210Po (jj67), the effective *gs* was set to 70% of the free nucleon value and *gl* adjusted to reproduce the ground state *g* factors of 209Pb (*ν*2*g*9/2) and 209Po (*π*1*h*9/2). The values so obtained conform to expectations (*gl*(*π*) ≈ 1.1 and *gl*(*ν*) - 0). It is important to note that the renormalization of the *M*1 operator is due to processes quite distinct from those that give rise to the effective charge, namely meson exchange currents, and core polarization. Here, the core polarization involves particle– hole excitations between spin–orbit partners, which couple strongly to the *M*1 operator. It thus differs in a fundamental way from the core polarization associated with the *E*2 effective charge.

It is convenient to discuss the results in Table 4 beginning with the heavier nuclei, 210Pb and 210Po. For these nuclei adjacent to 208Pb, there is good agreement between the experimental *g* factors of the 6<sup>+</sup> <sup>1</sup> and 8<sup>+</sup> <sup>1</sup> states, and both the empirical *j* <sup>2</sup> estimate and the shell model. These can be considered text book examples. It is unfortunate that there are no data for the 2<sup>+</sup> <sup>1</sup> and 4<sup>+</sup> <sup>1</sup> states, which, as the following discussion in this Section suggests, might show additional collectivity.

Turning to 134Te, the *E*2 and *g* factor data for the *π*(1*g*7/2)<sup>2</sup> multiplet are complete, and there is reasonable agreement with both the *j* <sup>2</sup> model and the shell model calculations. A detailed analysis has been given in Ref. [48], wherein it is shown that there is additional quadrupole collectivity in the 2<sup>+</sup> <sup>1</sup> state of 134Te that is not accounted for by large-basis shell model calculations that assume an inert 132Sn core. It was demonstrated that coupling the valence *πg*<sup>2</sup> 7/2 configuration to a core vibration with the properties of the first-excited state in 132Sn can readily account for the observed 2<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> <sup>1</sup> transition strength in 134Te, and that the wavefunctions of the 2<sup>+</sup> <sup>1</sup> , 4<sup>+</sup> <sup>1</sup> and 6<sup>+</sup> <sup>1</sup> states of 134Te nevertheless remain dominated by the *πg*<sup>2</sup> 7/2 configuration. It can be concluded that 132Sn is a relatively inert shell-model core. The caveat, however, is that the shell model calculations still require relatively large effective charges.

In the fp shell, 50Ti shows quite good agreement with both the *j* <sup>2</sup> model and the shell model. For 54Fe, the experimental *g* factors show better agreement with the large-basis shell model than the *j* <sup>2</sup> model. The shell model calculations in the fp basis with the gx1a interactions do a reasonable job of describing the different behaviour of the *g* factors in 50Ti and 54Fe.

The isotopes with two neutrons outside the *N* = *Z* cores 16O and 40Ca show similar behaviour: *g*(2<sup>+</sup> <sup>1</sup> ) is reduced significantly in magnitude compared to both the *j* <sup>2</sup> model and the shell model calculation, whereas the higher excited states, 4<sup>+</sup> in 18O, and 6<sup>+</sup> in 42Ca, have *g* factors in agreement with both the *j* <sup>2</sup> model and the larger-basis shell model. In

these isotopes, both the *E*2 transition strengths and the *g* factors indicate that the 2<sup>+</sup> state must contain collective admixtures. Writing the 2<sup>+</sup> <sup>1</sup> wavefunction in the form

$$|2\_1^+\rangle = a|\text{SM}\rangle + \sqrt{1-a^2}|\text{coll}\rangle,\tag{7}$$

where SM denotes the part from the shell model basis space and "coll" denotes the collective part (from multiparticle-multihole excitations), implies that

$$\lg(2\_1^+) = \kappa^2 \lg \text{M} + (1 - \kappa^2) \lg \text{coll}.\tag{8}$$

Assuming that the collective *g* factor is *g*coll ≈ *Z*/*A* ≈ 0.5, and taking the shell model *<sup>g</sup>* factor from Table 4 implies that there is a collective contribution of *<sup>α</sup>*<sup>2</sup> = <sup>20</sup> ± 2% in the first excited state of 18O, and a huge 59 ± 5% collective contribution in the first-excited state of 42Ca. This mixing in 42Ca is in excellent agreement with a 50% collective contribution deduced from Coulomb excitation data and one-neutron transfer reaction data (see Figures 41 and 42 for full details). To explain the observed *g* factor in 42Ca, Ref. [69] requires that the basis space be expanded to include the sd as well as fp orbits for both protons and neutrons. This strongly collective structure of the 2<sup>+</sup> <sup>1</sup> state is in stark contrast with the near pure *<sup>ν</sup>*(*f*7/2)<sup>2</sup> structure of the 6<sup>+</sup> <sup>1</sup> state.

**Table 4.** *g* factors for nominal *j* <sup>2</sup> configurations in doubly magic nuclides plus or minus two like nucleons. Data are from [70] (with a correction for 54Fe *g*(2<sup>+</sup> <sup>1</sup> ) from [71]).


*<sup>a</sup> gs*(*ν*) = 0.7*g*free *<sup>s</sup>* (*ν*) = −2.678 and *gl*(*ν*) = −0.033 set to reproduce the ground-state moment of 209Pb. *<sup>b</sup> gs*(*π*) = 0.7*g*free *<sup>s</sup>* (*π*) = 3.910 and *gl*(*π*) = 1.16 set to reproduce the g.s. moment of 209Bi.

To sum up, for the nuclei with *N* = *Z* cores, the 2<sup>+</sup> <sup>1</sup> structure is apparently affected by mixing with low-excitation deformed multparticle-multihole states, whereas the higherspin states are closer to the naïve *j* <sup>2</sup> structure. For *N* > *Z* cores, the low-spin states are better approximated by the empirical *j* <sup>2</sup> model and quite well described by the shell model. However, in all cases, a substantial effective charge is required to explain the *E*2 strength, even when the *g* factor suggests a relatively pure shell model configuration.

Although a first assessment of the effective charges required to explain the *B*(*E*2; 2<sup>+</sup> 1 → 0+ <sup>1</sup> ) data adjacent to closed shells may appear to show no pattern, some features can be identified: (i) shape coexistence and mixing must be taken into account when the doubly magic core has *N* = *Z*, (ii) there are always non-zero corrections to the nucleonic charges. Defining *δep* and *δen*, where *e*eff *<sup>p</sup>* = (1 + *δep*)*e* and *e*eff *<sup>n</sup>* = *δen e*, the common assumption that *δep* ≈ *δen* ≈ 0.5 is seen to be valid in many cases. However, *δen* appears to increase in heavier nuclei.

The above data and discussion shows that, for *E*2 transition strengths, the bare electric charges, *ep* = +1*e* and *en* = 0, do not work for configurations confined to a valence shell. A correction to the effective charges *δep*(*n*) 0.5 is usually required, even when the low-lying core excitations are taken into account. Certainly, the use of effective charges has provided a means for exploring nuclear structure using the shell model applied to nuclei that do not have closed shells. However, such practice buries important aspects of the origin of quadrupole collectivity in nuclei; one cannot learn the whole story about the origin of nuclear collectivity using such theories. We suggest that the path forward is two-fold: first, to develop models that obviate the need for effective charges, and second, where the use of effective charges is unavoidable, to formulate appropriate strategies to understand and manage their use.

There are "standard" approaches to evaluate effective charge—often conceptually based on the particle-vibration model of Bohr and Mottelson for nuclei with a single valence nucleon. The vibration can be described microscopically by particle–hole excitations in a Random Phase Approximation (RPA)-type approach [72–76]. There is then some choice of and sensitivity to—the interaction used in the RPA calculation [76]. This procedure, based on single particle–hole excitations, will not account for the effects of mixing between the valence configurations and low-excitation multiparticle-multihole configurations, which will particularly affect the *E*2 effective charge. The procedure to generalize from one valence nucleon to many is less often discussed. The effective charge must vary to some extent with the number of valence nucleons, but, in practice, it is usually held constant.

Some further comments on the path forward are made in Section 10.

A wider view of what one means by the shell model as an independent-particle model is provided by quasi-elastic electron scattering knockout of protons from closed shell nuclei. A summary view is provided in Figure 15. Quasielastic electron-scattering knockout of protons is a probe of independent-particle behaviour in nuclei that is distinct from the more familiar one-nucleon transfer reaction spectroscopy such as (d, 3He). First, the interaction is purely electromagnetic; second, entrance and exit channel effects are limited to the outgoing (high-momentum) proton. Thus, confidence can be placed in the extracted spectroscopic factors for (*e*,*e p*) reactions and the revelation that the single-particle view is "incomplete". The important insight is that one is never dealing with independent particles in a quantum many-body system such as the atomic nucleus: correlations are ubiquitous. Indeed, there are severe warnings of this in the theoretical literature, e.g., [77,78]. These correlations go much deeper than pairing correlations. The subject of nucleon correlations in nuclei is broad. Reference to them in the narrative here is minimal because our focus is on systematics of low-energy phenomenology. For the interested reader, a useful entry point is Ref. [79]. For recent access to the topic, a useful source is Ref. [80].

**Figure 15.** Spectroscopic strengths from quasielastic electron scattering knockout of valence protons, *A*(*e*,*e p*). Adapted from [81] (taken from [82]). The conclusion is, relative to a mean-field view, that never more than 70% of independent-particle strength is manifested in valence nucleon structure, even at doubly closed shells, i.e., other degrees of freedom are contributing to these structures.

The dilemma presented by the data in Figure 15 is a direct confrontation of the shell model approach to nuclear structure, so it can be viewed as a restatement of the question that is used for the title of this review. The data raise two questions: (1) Where has the single-particle strength gone? (2) What has replaced the single-particle strength? We do not attempt to answer these questions. Note that we are in good (bad?) company with the Standard Model of particles and fields. The Standard Model has a plethora of parameters, and nobody knows where they come from. There is one difference in our favour: we believe that protons and neutrons underlie the low-energy degrees of freedom in nuclei, but to employ their bare parameters requires much larger model spaces. Let us note the subtle point regarding correlations: it is primarily the number of configurations involved, not the number of particles, that is relevant. Shell model computations are only tractable in (relatively) small Hilbert spaces: the accumulating evidence is that these spaces are too small. There is an exponential growth in matrix dimensions as the shell model space is increased. However, "symmetry guided" approaches are beginning to circumvent this limitation [83]. A few details are given in Section 10.

It is relevant to note here that the missing strength in (*e*,*e p*) knockout and the effective charge problem must be related at a fundamental level because the *T*(*Eλ*) matrix elements for mass *A* can be expanded in terms of one-body spectroscopic factors connecting *A* and *A* − 1. Whether the general missing strength in transfer reactions [84] is associated with short-range [85] or long-range [86] correlations is crucial for the question of emerging collectivity. Moreover, the role of this missing strength in the emergence of quadrupole collectivity in nuclei could possibly be illuminated by examining how the effective charges for higher multipolarities, particularly *E*4 and *E*6, compared to those for *E*2 transitions. The negative polarization charge required for the *E*6 transition in 53Fe remains a puzzle; see, e.g., [74,76]. Experimental verification of this sole example of an *E*6 transition is clearly important.

A useful tool that has been used to explore independent-particle degrees of freedom in nuclei has been one-nucleon transfer reactions. However, the so-called spectroscopic strengths extracted from such data must be treated with great caution. This was recognized long ago by Baranger [87], and even earlier by Macfarlane and French [88]. These issues have received renewed attention; see, e.g., [89,90] and references therein for a discussion of the problem. The key issue is: Which nuclei provide the best view of independent-particle degrees of freedom? The approach of looking at how degrees of freedom, which manifestly are not independent-particle degrees of freedom, "intrude" into nuclei where independentparticle degrees of freedom have the best chance of dominating (and are widely assumed to do so [91]) is explored here.

By now, it is recognized that structures, even highly deformed structures, "intrude" into the low-energy excitations of spherical nuclei [41]. However, there are subtleties in the mechanism by which such intruder states appear at low excitation energy. An example is shown in Figure 16 for low-energy excited states in 47,49Ca. The naïve interpretation of the low energies of the 3/2<sup>−</sup> state in 47Ca and the 7/2<sup>−</sup> state in 49Ca would be that the *N* = 28 shell gap has broken down; but, with an understanding of the manifestation of pairing correlations, the reality is that the *N* = 28 shell gap is strongly present. The persistence of the shell gap can be seen on the right side of Figure 16 where the difference between the observed excitation energies of the first-excited states in 47Ca and 49Ca (which correspond to excitation of a neutron across *N* = 28) and the shell gap energy of ≈ 5.1 MeV is very close to the pairing energy determined from the odd-even staggering in the neutron separation energy, *Sn*. However, one reads about "collapse of shells" and "dissolving of shells". This would be true if there were no correlations present; but correlations *are* present.

**Figure 16.** Intruder states in 47,49Ca. The low energy of the 3/2<sup>−</sup> state in 47Ca and the 7/2<sup>−</sup> state in 49Ca result from pairing correlations. The low *B*(*E*2) values associated with these states indicate little or no collective core excitation is involved. The left-hand side of the figure illustrates how a simple estimate of the pairing correlation energy can be made. This analysis shows that the energy gap for *N* = 28 at *Z* = 20 is 5.1 MeV, in line with a well-defined shell gap. The data are taken from ENSDF [22], AME2020 [92], and [90]. Reproduced from [8].

A classic example of intruder states that illustrate the role of deformation is shown in Figure 17 for the odd-mass thallium isotopes. The first hints of these deformed intruder states were recognized long ago [93]; the thallium isotopes were a prime motivational origin of the first review of shape coexistence [94]. The spectroscopic evidence resides in the hindrances of the isomeric transitions and in the band structures associated with the isomer (9/2− states). The key excitation is a proton across *Z* = 82 to leave a hole pair below *Z* = 82; this hole pair correlates with the valence neutron pairs. These correlations result in near-identical "parabolas" in Bi and Pb isotopes, scaled by the number of proton pairs (see Figure 17 in [41]) and the parabolas exhibit a near collinearity when plotted versus neutron number. The 9/2− intruder structure is the oblate Nilsson 9/2−[505] configuration. There are extensive band structures which are well-described by the Meyer-ter-Vehn model [95,96]. The cores are *<sup>A</sup>*−1Hg; the parameters are the same as for odd-Hg 1*i*13/2 bands and odd-Au 1*h*11/2 bands (viz. *β* = 0.15, *γ* = 37◦). However, these details raise serious questions about using simple shell model configurations when interpreting excited states even in nuclei with one nucleon coupled to a singly closed shell.

**Figure 17.** The lowest-energy intruder states in the odd-mass thallium isotopes. A naïve interpretation (from a spherical shell model perspective) would lead to the conclusion that because a 9/2− state is below an 11/2− state in excitation energy, spin–orbit coupling has "broken down" or "collapsed". In reality, the 9/2<sup>−</sup> states shown are dominated by proton 1*p* − 2*h* excitations and are deformed structures: the first collective excitation on these 9/2− states is shown. [41]. Further details are given in the text. Reproduced from [8].
