*2.3. Knockout Reactions at Intermediate Energies and a Case Study: The Tz* = −<sup>2</sup> *Nuclei* <sup>48</sup>*Fe and* <sup>56</sup>*Zn*

Spectroscopy of the most proton-rich systems (i.e., *Tz* ≤ −<sup>3</sup> <sup>2</sup> ) presents significant challenges for fusion reactions, since evaporation of at least three neutrons will be required to access the nuclei of interest. Indeed the majority of the recent in-beam *γ*-ray spectroscopic studies of *Tz* ≤ −<sup>3</sup> <sup>2</sup> nuclei have been performed with one- (or two-) neutron knockout reactions from relativistic fragmentation beams. The knockout reaction, being a direct process, will populate specific, usually low-lying, states, those bound states for which there is a large spectroscopic overlap between the ground-state configuration of the beam and the final state of the residue, with respect to neutron removal from a specific orbital. Whilst the range of final states can be rather limited, compared with fusion reactions, the reactions (and final spectra) can be easier to interpret, especially when combined with cross-section calculations based on a reaction model using shell-model spectroscopic factors. This, in turn, helps give confidence to the *J<sup>π</sup>* assignment of the observed states, when comparing with the analogue states in the mirror nucleus. Moreover, population of high-*J* states in proton-rich systems is possible in specific conditions, e.g., through knockout from isomeric states (e.g., [7]) or through two-neutron removal from a beam species with a *J* = 0 ground state (e.g., [5]).

The case studies discussed here are the very recent works related to the observation of excited states in *Tz* = −2 nuclei 56Zn [9] and 48Fe [8]. These studies have enabled the examination of *T* = 2, *Tz* = ±2, mirror pairs, providing stringent tests of the shell-model prescription for "distant" mirror pairs (large difference in *Tz*). In both of these examples, one-neutron knockout reactions were performed on odd-*A* relativistic fragmentation beams. For 56Zn [9], the experiment was performed at the RIBF facility (Radioactive Isotope Beam Factory), at the RIKEN Nishina Center, Japan. Fragmentation of a beam of 78Kr at 345MeV/u produced a secondary beam of 57Zn fragments, separated and identified using the BigRIPS spectrometer [35]. The Be reaction target was surrounded by the DALI2+ NaI *γ*-ray array [36] and the final knockout residues identified by the Zero Degree Spectrometer [35]. For 48Fe [8], the experiment was performed at NCSL (National Superconducting Cyclotron Laboratory, East Lansing, MI, USA). A primary beam of 58Ni at 160 MeV/u was used to create a 49Fe fragment beam, separated using the A1900 spectrometer [37]. The reaction target was surrounded by the GRETINA Ge *γ*-ray tracking array [38] and the final knockout residues identified by the S800 Spectrograph [39].

In both the above reactions, the ground-state of the beam species was *J<sup>π</sup>* = <sup>7</sup> 2 <sup>−</sup>, where the Fermi-level for the odd, unpaired, neutron was in the *f* <sup>7</sup> 2 shell. In both cases, excited states of *J<sup>π</sup>* = 2+, 4+, and 6<sup>+</sup> were observed (6<sup>+</sup> is the highest-*J* state that can be populated directly). The predicted spectroscopic factors for both reactions suggest that the yrast and yrare states of *J<sup>π</sup>* = 2+, 4+, and 6<sup>+</sup> are expected to be directly populated, with strong populations of 6<sup>+</sup> states, which matched the experimental observations [8,9]. It was not possible to identify decays from the yrare states in 56Zn, but the higher resolution of the *γ*-ray array in the 48Fe study enabled the yrare state decays to be tentatively identified.

In the 48Fe case, the experiment also used the "mirrored knockout" technique, which has proven to be especially powerful for the observation and assignment of analogue states in mirror pairs. In this approach, as well as using the 49Fe−1*<sup>n</sup>* reaction, the mirror partner to 48Fe, 48Ti, was studied through a 49V−1*<sup>p</sup>* reaction (this required a separate setting of the A1900 spectrometer). Since the two beam species, 49Fe and 49V, are also mirror nuclei, these reactions comprise a complete pair of "analogue" knockout reactions—i.e., reflected around the *N* = *Z* line. Isospin symmetry also implies that the spectroscopic factor for each specific knockout path (removal from a specific orbital to a specific final state) should be essentially identical in both mirror nuclei, and this should, in turn, lead to very similar distributions of knockout strength when the mirror nuclei are studied in the same experimental conditions. Since the scheme of 48Ti is known, this helps considerably in the assignment of their analogue states in 48Fe. The mirrored knockout approach was first demonstrated in [6] and has been employed in a number of other cases [10,14,40].

The spectra in Figure 3 show the resulting *γ*-ray spectra for this mirrored reaction: Figure 3a shows the 49V−1*<sup>p</sup>* →48Ti reaction and Figure 3b the mirrored 49Fe−1*<sup>n</sup>* →48Fe reaction. One can see very similar population distribution from the spectra. The spectra, as expected, are dominated by the decays from the 2<sup>+</sup> 1,2 states (labelled with blue squares), the 4<sup>+</sup> 1,2 states (green diamonds) and the 6<sup>+</sup> 1,2 states (red stars) [8,41]. The spectra also show the clear benefit of using a Ge *γ*-ray tracking array (i.e., GRETINA) for in-beam spectroscopy with relativistic beams. The position-sensitivity afforded by the pulse-shapeanalysis approach allowed for accurate Doppler reconstruction (e.g., [38]), reducing the otherwise huge impact of Doppler broadening at these high fragment velocities.

**Figure 3.** The *γ*-ray spectra observed in the case study [8]. The spectra are measured with the GRETINA array following the identification and selection of the relevant incoming and outgoing fragment beams. Panel (**a**) shows the 49V−1*<sup>p</sup>* <sup>→</sup>48Ti reaction and (**b**) the mirrored 49Fe−1*<sup>n</sup>* <sup>→</sup>48Fe reaction. The peaks are labelled by the *γ*-ray energy and the symbols refer to the angular momentum/parity, *Jπ*, of the states from which these decays proceed. Decays from the 2<sup>+</sup> 1,2 states are labelled with blue squares, the 4<sup>+</sup> 1,2 states with green diamonds and the 6<sup>+</sup> 1,2 states with red stars. The insert in (**b**) shows how the peak around 970 keV comprises three *γ* rays. Adapted from [41].

The use of knockout reactions, and the mirrored-knockout technique, has provided a wealth of data on MED in the upper *f* <sup>7</sup> 2 region which has, in turn, helped shed light on the role of isospin-non-conserving interactions in the shell-model analysis; see Section 4.1. The 56Zn case has also yielded information on how occupation of specific shell-model orbitals have a shape-driving effect; see Section 4.2.

## **3. Shell Model Approach for Energy Differences between Excited Analogue States**

Without a reliable model to describe MED and TED as a function of *J*, the experimental observations of the variation of MED and TED with *J* cannot be interpreted in any physically meaningful sense. The shell-model approach to modelling MED and TED has transformed this field of research, allowing interpretation in terms of detailed nuclear structure phenomena including particle alignments and changes in nuclear shape/radii. Indeed, the happy coincidence that occurred around 20 years ago was that exceptionally powerful large-scale shell-model calculations were becoming available (e.g., [42,43]) in exactly the region where major experimental advances in the spectroscopy of mirror nuclei were taking place—i.e., the lower part of the *p f* shell.

If perfect isospin symmetry between analogue states is assumed, and that the contributions to MED and TED are entirely related to electromagnetic effects, there are a number of effects that can contribute to MED/TED, and their variation with excitation energy/*J*, which can in principle be calculated in the shell-model approach. The key factor is the multipole effect of re-coupling the angular momentum of pairs of protons, resulting in a decrease in spatial overlap of the protons, with increasing coupled *J*, and hence a reduction in the Coulomb energy. This is straightforward to model in large-scale shell-model calculations through the application of Coulomb matrix elements, calculated in the usual harmonic oscillator (HO) basis, in addition to the nuclear effective interaction. Initial attempts to model MED, using just this approach, were only partially successful (e.g., [44,45]) and it was concluded from that analysis that additional ingredients (including of multipole origin) were missing in the model. Indeed, better agreement was obtained using "empirical" effective *f* <sup>7</sup> 2 Coulomb matrix elements, extracted from the *A* = 42 mirror nuclei (e.g., [46]) or sets of ad hoc Coulomb matrix elements derived from fits to the data in the centre of the *f* 7 shell [44].

2 It was clearly important to develop a consistent shell-model approach for prediction of MED and rooted correctly in the physics. The breakthrough came with the seminal work of Zuker et al. [16], in which multipole and monopole effects were treated together in the same shell-model prescription. The model was developed and tested using MED measured in the centre of the *f* <sup>7</sup> 2 shell, with shell-model calculations performed with the ANTOINE code [42,43] in the full *p f* space, using the mass-dependent effective interaction for the *p f*-shell, KB3G [47]. This model has formed the basis of the large-scale shell-model approach to MED and TED ever since; see, e.g., [3] for an earlier review. In this approach, the energy differences between analogue states within the shell model can be separated into four components, which can be calculated individually, so that the impact of each can be evaluated.

The first and last terms below are multipole terms. These can be calculated by determining the appropriate matrix elements of the interactions and calculating expectation values through first-order perturbation theory using a set of wave functions calculated in an isoscalar basis. The remaining two components are monopole terms, associated with bulk Coulomb effects and EM-induced shifts in single-particle energies. The four components are as follows.
