*7.1. Intruder States*

In the lexicon of nuclear structure study, the term "intruder states" has become established. "Intruder" means a state that is observed where it is not expected or "does

not belong". These are states that appear to be shell model states which are observed at low energy on the "wrong side" of a shell model energy gap. Examples are presented in Figures 16 (47,49Ca) and 17 (Tl isotopes), and implicitly for the 9/2<sup>−</sup> intruder structure in 187Ir in Figure 22. However, intruder states are not simple shell model states because they have underlying correlations in their structure. Indeed, the reason they intrude is because of these correlations. Thus, in Figure 16, an example with important pairing correlations is shown, namely that a simple addition, deduced from one-neutron separation energies, "restores" the low energies of the intruder states to their uncorrelated energies, which reflect the 5 MeV shell gap. In Figure 17, an example with important pairing and deformation correlations is shown, notably that, in addition to the appearance at low energy (pairing correlations), there is a systematic "parabolic" trend in the excitation energies as a function of neutron number, with a minimum near the mid-shell point (*N* = 104), which is where the greatest number of neutrons are active. Much confusion exists in the literature regarding intruder state structures: it appears that they are often viewed as part of a shell model picture. Let us emphasize that the shell model is an independent-particle model based on a spherical mean field. Intruder states are usually strongly deformed and so they are of completely different character to "shell model" structures. For example, they can exhibit rotational bands which can expose their distinctly different character. In effect, the normal and deformed states largely exist in different basis spaces.

It is not implied that rotational bands cannot emerge from shell model calculations. If one could conduct shell model calculations in a sufficiently large space, intruders and their deformation should emerge, but, at present, such calculations are not generally feasible. Consequently, operationally, we have the "coexistence" of shell model descriptions and the Nilsson model plus rotations where nuclei with intruder states are concerned; and we observe actual structures characterized by different *E*2 properties, i.e., quadrupole moments and *B*(*E*2) values.

Having noted that rotational structures can begin to emerge in current shell model calculations, we also draw attention to the on-going challenge: the emergence of rotational bands in a finite many-nucleon system calculation is arguably the most profound challenge in a nuclear structure. From everything we understand by the term "a shell model calculation", it is fair to say that this must be a future reality. We look beyond the use of model interactions, such as employed in the Elliott model [160] (where the emergence of rotational bands is guaranteed), and we look to this question using the best effective interactions available. Note that the Elliott model is a single-shell description of states; intruder states demand a multi-shell view. The Elliott model bands are not the generally observed rotational bands in nuclei. It is noteworthy, however, that the Elliott model re-emerges as a submodel of the symplectic shell model, which is discussed in Section 10. By way of shape coexistence, especially in nuclei such as 40Ca, a convergence on a multi-shell view appears most promising (see, e.g., [158]). However, let us note that the key observables that characterize nuclear rotations are specific *E*2 properties: transition and diagonal *E*2 matrix elements with ratios given by rotor-model Clebsch–Gordan coefficients—these are observed to high precision in some nuclei. To be convincing, such a calculation must use the bare charges of the proton and neutron, *ep* = 1*e* and *en* = 0 and match the precision of these observed properties.

The most dramatic examples of intruder states are where they appear as the groundstate structures of nuclei. This only occurs in a few local mass regions (such as the 32Mg region, the 42Si region, and perhaps near 68Ni): the term "island of inversion" has become popular for the description of such an occurrence. There is a tendency to place an unphysical emphasis on this terminology: one reads about mapping the borders (or shores) of islands of inversion. However, the structures are not islands; they persist across the entire mass surface, albeit mostly as excited states. A leading example is shown in Figure 44, which depicts systematics in the even-mass *N* = 20 isotones. This is a celebrated historical example. The first clues came from mass measurements [161] and isotope shift measurements [162] in the sodium isotopes. This was shortly followed by a measurement

of *E*(2<sup>+</sup> <sup>1</sup> ) in 32Mg [163]. A suggested unified view of these observations, from earlier times, is reproduced in Figure 45. However, it took thirty years to establish the lowest spherical state in 32Mg [164], and to explore the structure as an excited state in the neighboring 34Si [165]. It appears that nobody has yet shown an interest in looking at the underlying structure in 36S, but it has been known for a long time in 38Ar. Note that the deformed band in 38Ar is nearly identical in energy spacing to the ground-state band in 32Mg. Our message is that: to refer to the ground-state structure of 32Mg as being part of an "island" is obscuring the discovery frontier of such structures, which must extend to higher excitation energies and broadly encompass nuclides in the region. This is a severe criticism of the misuse of language in science. A schematic view of the energies that contribute to intruder states is shown in Figure 46. A global view that recognizes the dominance of deformation in nuclear ground states is shown in Figure 47.

The challenge of the exploration of intruder states in nuclei is to arrive at the ability to predict their occurrence. With reference to Figure 46, there is a current interest in the so-called monopole energy contribution to the total energy that dictates the appearance of intruder states at low energy in various mass regions. This has received attention already a long time ago [166,167]; more recently, there has been attention from Heyde and collaborators [168], Zuker and collaborators [169], and a review by Otsuka et al. [91]. The theoretical formalism is not a critical concern; but identifying an empirical basis for fixing the relative magnitudes of the energy contributions shown in Figure 46 needs in-depth consideration. The problem is identifying manifestations of monopole energy effects that are free of correlations from pairing and from deformation. These correlations already feature in our chosen subject: they lie at the heart of emergent structures in nuclei, whether involving intruder states or not. Some mass regions of critical concern are addressed below in this Section and in Section 9.

**Figure 44.** Shape coexistence and intruder states in the *N* = 20 isotones. The 0<sup>+</sup> <sup>2</sup> state identifications are made in: 32Mg [164] and 34Si [165]. The intruder states can be understood in an exactly parallel manner to the situation in the Sn isotopes. Thus, here, the 2p-2h configurations involve neutron pairs interacting with protons. The excitation pattern reflects proton subshell structure (2*s*1/2, *ld*3/2) as these orbitals are filled: this is beyond the present level of discussion. Note that the deformed bands in 32Mg and 38Ar possess nearly identical energy spacing. Taken from [8].

**Figure 45.** (**a**) Excited state systematics in the even-mass *N* = 20 isotones. The low-lying 2<sup>+</sup> <sup>1</sup> state in 32Mg is interpreted as resulting from a ground state intruder configuration. The ground state of 32Mg should have an anomalously larger mean-square radius. The ground-state binding energy of 32Mg has been reported variously as anomalous and normal. (**b**) Two-neutron separation energy, *S*2*n*, and isotope shift, *<sup>δ</sup>r*2, systematics for the neutron-rich Na isotopes [161,162]. The discontinuity at *N* = 20 indicates an increased ground-state mean-square charge radius and increased binding energy. Reproduced from [170].

**Figure 46.** The different energy terms contributing to the energy of the lowest proton 2p-2h 0<sup>+</sup> intruder state for heavy nuclei. On the right-hand side, a schematic view of the excitation is given. On the left-hand side, the unperturbed energy, the pairing energy, the monopole energy shift, and the quadrupole energy gain are presented, albeit in a schematic way. Reprinted with permission from [41]. Copyright (2011) by the American Physical Society.

**Figure 47.** A schematic view of the intruder-state "parabolas", shown to dramatize the way that shells and subshells suppress the emergence of low-energy collectivity in nuclei. (**a**) The situation where deformed structures intrude to become the ground state at the middle of a singly closed shell, e.g., 32Mg. (**b**) The situation where the ground states for a sequence of singly closed shell nuclei remain spherical, but deformed structures form excited intruder bands, e.g., the Sn and Pb isotopes. (**c**) The situation where a subshell may suppress intrusion of a deformed structure from becoming the ground state or a low-lying excited band, e.g., *N* = 50, 82, cf. Figures 52–54. Reprinted with permission from [41]. Copyright (2011) by the American Physical Society.

The perspective that is presented in Figure 47 appears useful. This view "inverts" the parabolic energy perspective that can be applied to intruder states and, recognizing that most nuclei are deformed, and that shape coexistence probably occurs in all nuclei, expresses the occurrence of spherical shapes in nuclei as intruding to low energies only at and near closed shells. The competition between the controlling energies that lie behind this view devolve onto the configuration interaction problem that is foundational to the nuclear many-body problem.
