**1. Introduction**

One of the challenging goals of the field of nuclear structure physics is to model atomic nuclei, including their properties and their reactions—rooted in the fundamental forces at play between protons and neutrons—with predictive power also for the shortestlived nuclear species located near the driplines of the chart. With this ultimate vision to extrapolate towards the most neutron-rich nuclei that may elude experimental study in the near future, much can be gleaned from nuclei in regions that display the effects of structural evolution away from the valley of stability and so offer a window into the driving forces of structural change and our understanding of it [1–3]. Specifically, the complex interplay between single-particle and collective degrees of freedom can provide exciting experimental challenges and demanding theoretical benchmarks.

The region of rapid structural change of interest in this review is the so-called "*N* = 40 island of inversion" [4,5], where the neutron-rich Fe and Cr nuclei around neutron number 40 become the most deformed in the region. In nuclear models, this is theorized to be caused by the strong quadrupole-quadrupole interaction producing a nuclear shape transition in which highly-correlated many-particle–many-hole configurations become energetically more favored than the normal-order (spherical) ones [4]. Such islands of inversion are characterized by rapid structural changes and shape coexistence [5,6], providing insight into nuclear structure physics far from stability [7]. Large-scale shell-model calculations with the LNPS (Lenzi-Nowacki-Poves-Sieja) effective interaction [4] in the full *f p* shell for protons and the *f*5/2, *p*3/2, *p*1/2, *g*9/2, and *d*5/2 orbitals for neutrons have confirmed the picture described above, with many successful predictions that preceded experimental results [3].

**Citation:** Gade, A Reaching into the *N* = 40 Island of Inversion with Nucleon Removal Reactions. *Physics* **2021**, *3*, 1226–1236. https://doi.org/ 10.3390/physics3040077

Received: 29 October 2021 Accepted: 2 December 2021 Published: 8 December 2021

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A recent prediction extends this island of inversion to *N* = 50 [5] and includes nuclei that will only be reached at next-generation rare-isotope beam facilities. This exciting prospect of extending the island towards the magic neutron number *N* = 50 is based on extrapolations of calculations using the LNPS shell-model effective interaction and its monopole drifts [4,5]. These prediction together with advances in experimentation continue to push the field forward on the journey to the *N* = 50 island of inversion.

The furthest experimental reach into the Fe, Cr, and Ti isotopes has been afforded by inverse-kinematics nucleon removal studies induced by fast rare-isotope projectile beams [8,9] to probe the nuclei of interest via in-beam *γ*-ray spectroscopy [10]. Often, such reactions provide the first glimpse of the excitation level scheme [11] and, in some cases, the direct character of such reactions is used to conclude on wave function overlaps within the shell-model framework [8,9]. This paper reviews the recent results for the very neutron-rich nuclei 66,68,70,72Fe, 64,66Cr, and 60,62Ti, all located near the center of the *N* = 40 island of inversion or already on the path to *N* = 50, obtained with such experimental approaches which have provided pioneering information the furthest away from the line of stability; see Figure 1.

**Figure 1.** Portion of the nuclear chart that shows the *N* = 40 island of inversion. Nuclei discussed in this review are highlighted with a red outline. This chart was generated with [12] (half-life color coding based on NuBase2020 and corresponding extrapolations).

#### **2. Experimental Approaches**

Experimental techniques aimed at tracking the changes in the structure of nuclei are multi-pronged. They include measurements of ground-state properties such as masses, radii, *β*-decay properties, and electromagnetic moments as well as the study of properties of bound and unbound excited states. One way of probing specific nuclear structure aspects in quantitative ways is the use of nuclear reactions that selectively probe a specific degree of freedom. Inelastic scattering of nuclei, including Coulomb excitation, has long been used to probe nuclear collectivity, characterized by the coherent motion of several protons and neutrons. The single-particle degree of freedom, on the other hand, is commonly associated with the single-particle composition of the many-body wave function in a shell-model picture. Such single-particle properties can be studied rather selectively by using *direct* reactions that add or remove one or a few nucleons from the nucleus of interest. Intriguing possibilities now arise in the above mentioned islands of inversion, where the telltale onset of collectivity and the underlying migration of single-particle levels can be tracked to provide a consistent picture.

At in-flight rare-isotope facilities, short-lived nuclei away from stability can be efficiently produced by fragmentation (or fission) of stable, primary beams impinging upon stable targets at high beam energy. The resulting secondary beams of rare isotopes are then available for experiments at velocities typically exceeding a *v*/*c* of 30%, where *c* is the speed of light. Well-established experimental techniques used for decades to study stable nuclei are not readily applicable in inverse kinematics and at the low beam rates encountered for the shortest-lived nuclear species. Instead, powerful new experimental approaches have been developed to enable in-beam nuclear spectroscopy studies of fast rare-isotope beams with intensities that are several orders of magnitude less than needed for typical low-energy techniques.

The intensities of rare-isotope beams are lower than stable-beam rates by several orders of magnitude. However, the experimental approach of in-beam *γ*-ray spectroscopy compensates for the reduced intensities by enabling thick reaction targets, due to the high beam velocity, and realizing measurements with luminosities comparable to stable-beam experiments but at beam rates of up to a factor of 104 less. Reactions such as nucleon removal are induced in thick reaction targets (several hundred mg/cm2 to g/cm2) and with the detection of *γ* rays for the identification of the reaction residue's final state [10]. Since the residue's *γ*-ray emission occurs *in flight*, the *γ*-ray detection systems have to be granular or position-sensitive to allow for an angle-dependent event-by-event reconstruction of the Doppler-shifted *γ*-ray energies into the rest frame of the emitter. The choice of the target material depends on the desired reaction; one- and two-nucleon knockout reactions [8] are often induced by light targets, for example 9Be or 12C, while quasi-free scattering of the (*p*, 2*p*) or (*p*, *pn*) type are nowadays performed with MINOS, an extended liquid hydrogen target that allows reaction vertex-reconstruction and tracking following the concept of a time projection chamber [13]. The projectile-like reaction residues exiting the target has to be identified with magnetic spectrographs or advanced detector systems to cleanly select the reaction channel of interest. In-beam *γ*-ray spectroscopy programs with fast beams are pursued at a number of fragmentation facilities around the world, while the work using nucleon removal reactions in the *N* = 40 region has been performed largely at NSCL [14] in the US and RIBF/RIKEN [15] in Japan with the GRETINA [16,17] and DALI2 [18] arrays for *γ*-ray spectroscopy, respectively. A sketch of the experimental scheme is shown in Figure 2.

**Figure 2.** Experimental scheme for inverse-kinematics nucleon knockout reactions at rare-isotope beam facilities that provide fast beams of rare isotopes via projectile fragmentation or fission with velocities, *v*, exceeding 30% of the speed of light, *c*.

For 9Be- or 12C-induced one-nucleon knockout reactions, the exit channel of interest is one where—in a single step—one proton or neutron is removed from the fast rareisotope beam and the projectile-like residue with one less nucleon survives in a bound final state. This channel is characterized by swift, surface-grazing collisions of the projectile and the target nuclei. From a large body of experiments performed at energies from 50 MeV/nucleon to more than 1 GeV/nucleon at (rare-isotope) facilities around the world, it has been established that, with large cross sections, the dominant single-hole states relative to the projectile ground state are populated in the projectile-like reaction residue, unambiguously demonstrating the unmatched sensitivity to the single-particle degree of freedom. The residue parallel momentum distributions encode in their shape and width the information of the orbital angular momentum and separation energy of the removed

nucleon [8]. The cross section of the selectively populated single-hole configurations scale with the respective spectroscopic factor or wave-function overlap in a shell-model picture. Statistical descriptions of these reactions will not capture these features. Comparisons of such one-nucleon removal data with nuclear structure calculations have been enabled by a direct reaction model [8,19,20] that uses the sudden (short interaction time) and the spectator-core approximation to many-body eikonal (forward-scattering) theory [19] with a detailed prescription provided in [21]. The single-particle nuclear structure information then enters the calculations through spectroscopic factors, or wave-function overlaps, that scale the calculated cross sections for the removal of one nucleon from the corresponding orbital. With that, the measured knockout cross sections can serve as formidable probes of shell-model interactions on the quest to identify the single-particle makeup of the projectile ground state and the residue final states [8,22].

It has been shown also that two-proton and two-neutron removal from neutron-rich and neutron-deficient projectiles, respectively, also proceed as direct reactions [23–25]. By combining eikonal reaction dynamics, that assumes a sudden single-step removal of two nucleons and shell-model calculations of the two-nucleon amplitudes (TNAs), the cross sections for two-nucleon knockout from the parent-nucleus ground state to each of the final states in the daughter nucleus can be calculated [26]. Also, it was shown that the shape of the parallel momentum distribution of the two-nucleon knockout residues depends strongly on the total angular momentum of the two removed nucleons, allowing spin values to be assigned to populated final states [27–29]. One step further, it was proposed and confirmed that since the two-nucleon overlaps contain components with different values of the total orbital angular momentum, information beyond the total angular momentum can be probed. This opens up the possibility to uniquely explore this composition and couplings within the wave functions of rare isotopes [30,31].

More recently, quasi-free (*p*, 2*p*) and (*p*, *pn*) reactions, extensively used in normal kinematics with stable targets, have been successfully adapted for inverse-kinematics studies of rare-isotope beams on proton targets [9]. Just as the heavy-ion-induced knockout reactions sketched above, the proton-induced knockout reaction selectively probes the single-particle structure of the nucleus of interest. Also, the shape of the momentum distribution of the knockout residue is connected to the momentum distribution of the knocked-out nucleon. Protons are a penetrating probe that interrogate the nuclear interior, and their rescattering inside the nucleus has to be understood and modeled [9]. In heavyion induced knockout, the orbital radii need to be modeled precisely due to their surface localization [21]. This experimental approach has been used recently at RIBF/RIKEN for measurements reviewed here. Various reaction models have been developed and their consistency remains a challenge for the future [32]. The different nucleon removal reactions were described and confronted with each other recently and extensive details on sensitivities and model dependencies can be found in reference [32].
