*4.2. Nuclear Radii and Neutron Skins*

In Section 3, it was demonstrated how occupation of low-*l* orbitals can contribute to MED and that this can be accounted for in the shell model through tracking of the total (proton plus neutron) occupation of low-*l* orbits: in the *f* <sup>7</sup> 2 region this would be the occupancy of the *p* <sup>3</sup> 2 , *p* <sup>1</sup> 2 orbitals. This provides the first indication that MED can yield real physical insight into changes in nuclear radii.

Recently, Bonnard et al. [19] have investigated the role of the occupation of low-*l* "halo" orbitals in driving radii and on their influence in the development of neutron skins. They have been able to show that the effect on the total radius of occupation of one of the low-*l* orbitals is strongly dependent on the extent of the occupation of that orbital. For example, in the *f* <sup>7</sup> 2 shell, the occupancy of the *p* orbits is generally expected to be low (the shell-model occupancies are 1). Moreover, the parameterisation of the *VCr* term (see Section 3) has been optimised for that region. However, in heavier nuclei, once the *f* 7 2 shell is full, the occupancies of the *p* orbits will increase significantly, and the work of Bonnard et al. [19] suggests that the radial-driving effect of the *p* orbit will be significantly smaller in this circumstance.

This has been investigated in the *A* = 56, *T* = 2 mirror nuclei following the spectroscopy of 56Zn [9], discussed as a case study in Section 2.3. Figure 6 shows the experimental MED compared with the shell-model calculations. These calculations have been performed with a modified KB3G interaction, KB3GR (Caurier, E.; Poves, A. *Unpublished work*) which has been optimised for this region. The calculation using the standard parameterisation for the radial term (*α* = 200 keV, see Equation (3)) is shown by the red dashed line. However, in this case, protons in 56Zn (and neutrons in its mirror, 56Fe) are already occupying the *p* <sup>3</sup> 2 orbital, and the results of Reference [19] therefore imply that the radial term *VCr* is likely to be overestimated. Therefore, in the analysis of the *A* = 56 mirrors, Fernández et al. [9] reduced the *α* parameter (see Equation (3)) for the *p* <sup>3</sup> 2 occupancies, from the standard value of 200 keV. The *α* parameter for the *p* <sup>1</sup> 2 , which remains largely unoccupied, was left unchanged. The results can be seen in Figure 6 where a smaller value of *α* = 50 keV is applied for the *p* <sup>3</sup> 2 orbital; see solid blue line. This gives a much better description, in qualitative agreement with the results of Bonnard et al. [19]. It is also noteworthy that the multipole contributions to the MED for this mirror pair turn out to be small, due to particle-hole symmetry; both nuclei have two particles and two holes with respect to 56Ni. This makes this mirror pair sensitive to the remaining significant monopole contribution, *VCr*, making this an ideal test case to examine radial effects.

**Figure 6.** Results from [9]. The experimental MED for the *A* = 56, *T* = 2 mirror nuclei compared with the results of shell-model calculations performed with the KB3GR interaction. The model uses the standard parameterisation, but with a varying value of the scaling parameter, *α* (Equation (3)), used in the determination of the radial contribution to the MED due to the occupation of the *p* <sup>3</sup> 2 orbital. See text for details.

As well as the *total* nuclear radius having an impact on the Coulomb energy, and hence MED, for a mirror pair, any *difference* between the neutron and proton radii (i.e., neutron skin) could also have an effect on MED if, as isospin symmetry would suggest, the neutron radius of one member of a mirror pair is equal to the proton radius of the other. This idea, also inspired by the study in Ref. [19], was pursued in the analysis of the *A* = 23 mirror nuclei by Boso et al. [11], work that was made possible by the spectroscopy of 23Mg, our remaining case study (see Section 2.1). The analysis was undertaken using a no-core shell-model approach based on the monopole-corrected interaction (MCI) [57], which contains all the necessary Coulomb and charge-symmetry breaking terms. The MCI matrix elements were computed using different size parameters for neutrons and protons (i.e., allowing for the possibility of different neutron and proton radii). Whilst the proton radius of 23Na is experimentally known, its neutron radius is not. The neutron radius (and hence neutron skin) was then determined following the method of Duflo and Zuker [48] by adjusting the neutron radius until the experimental ground state mirror displacement energy (MDE) is reproduced by the model. The method was then repeated state by state, in order to reproduce the MED, allowing for the variation of the skin thickness as a function of *J* for the excited states.

Full details can be found in Ref. [11] but the key results are shown in Figure 7. The neutron skin thickness parameter, *ζ*, is plotted using the blue circles. *ζ*, in the paramaterisation of Duflo and Zuker [48], is proportional to difference between the neutron and proton rms (root mean square) radii and is defined as *ζ* = Δ*rνπA*/(*Tzeg*/*A*), where the exponential factor is a correction term, applied for light nuclei [48]. These results show that the neutron skin, as derived from the MED, varies significantly from state to state. This, in turn, implies neutron skin sizes, and their variation with excitation energy/*J*, can influence the MED and, if so, it is an effect currently not included in the MED models. Another key observation is that the skin thickness, has a correlation with the difference between the neutron and proton occupancies of the *s* <sup>1</sup> 2 orbit. This difference is plotted as Δ*νπ* in Figure 7 (red squares). This analysis was repeated for a range of other odd-*A* mirror nuclei in the *sd* shell, and similar variations of neutron-skin thickness with *J* were suggested by that analysis; see Ref. [11] for the full results and discussion. Inclusion of effects of this kind in the calculation of MED is clearly an exciting future topic for investigation.

**Figure 7.** Data from Ref. [11]. Blue circles: the neutron skin thickness parameter, *ζ*, defined in Ref. [48], which is proportional to difference between the neutron and proton rms radii. This parameter has been extracted through fitting to the measured MED. Red squares: Δ*νπ*, the difference between the neutron and proton occupancies of the *s* <sup>1</sup> 2 orbit, for each state. See text and Ref. [11] for details.

## **5. Summary and Outlook**

In this short review, some of the latest experimental advances have been presented. The advent of the new radioactive beam facilities will allow some of these techniques to be applied to allow spectroscopy of the most exotic proton-rich systems, or to perform high precision tests of the predictions that come from the isospin formalism. The high-intensity intermediate-energy fragmentation beams available at the upcoming FRIB (Facility for Rare Isotope Beams, East Lansing, MI, USA) and FAIR (Facility for Antiproton and Ion Research, Darmstadt, Germany) facilities are expected to have particular impact. Techniques such as those described Section 2.3, can be applied to access nuclei with large proton excess and pursue spectroscopy of mirror nuclei in the upper half of the *f pg* region. The high-velocity beams also allow for a range of lifetime-measurement techniques to be applied, allowing for precision tests of the isospin-dependence of transition strengths. From a theoretical perspective, it will be especially important to develop a better understanding of the origin of the effective isovector isospin non-conserving (INC) interactions (see Section 4.1). Moreover, the link between mirror energy differences (MED) and radii/neutron skin is especially exciting and should be developed further in future shell-model work. As the study of energy splitting between isobaric multiplets develops in the future, the exciting developments in the shell-model, some of which have been discussed, will have crucial role to play.

**Funding:** This research was funded by the UKRI (UK Research and Innovation) Science and Technology Facilities Council under grant number ST/V001108/1.

**Data Availability Statement:** No new data are presented in this work. The data used can be found in the corresponding references.

**Conflicts of Interest:** The author declares no conflict of interest.

### **References**

