*4.1. Isospin-Non-Conserving Interactions*

One of the key areas of study has been to map out the influence of the additional effective INC interactions (see Section 3.4). In the *f* <sup>7</sup> 2 shell, a large amount of data have become available which has enabled a more complete numerical evaluation of the influence of these INC effects. In Refs. [17,18], all available MED and TED data in the *f* <sup>7</sup> 2 shell (at that time) were gathered and modelled using a consistent shell-model approach. The shellmodel MED and TED were then fitted to the experimental data, allowing the magnitude of the *J*-dependent INC terms *V*(1) *<sup>B</sup>* and *<sup>V</sup>*(2) *<sup>B</sup>* to vary freely; the former (isovector) term was derived from the MED and the latter (isotensor) term from the TED. The key results are shown in Table 1. The results of two types of fit are presented: one where a single *T* = 1 *VB* matrix element is considered (at a coupling of *J* = 0) and the second where all four *T* = 1 matrix elements *J* = 0, 2, 4, 6 were allowed to be non-zero. Only *f* <sup>7</sup> 2 matrix elements were considered. See Refs. [17,18] for a full discussion of the analysis.

**Table 1.** Data collected from Refs. [17,18]. The isovector *<sup>V</sup>*(1) *<sup>B</sup>* (*J*) and isotensor *<sup>V</sup>*(2) *<sup>B</sup>* (*J*) INC matrix elements, for *f*7/2 pairs, extracted from fits across the whole *f*7/2 shell (see text for details). For the full fits, a monopole centroid has been subtracted as part of the fitting process to allow the *J*-dependence to be fully evaluated. The numbers in the parentheses are the errors on the fitted values.


Two key results emerged from the analysis. Firstly, for the purpose of MED and TED, it is having the correct *J*-dependence of these matrix elements that is crucial in the determination of the theoretical MED and TED; the extracted results do indeed have a strong *J*-dependence. Secondly, it was shown that a single matrix element at *J* = 0 gives, essentially, as good a fit as allowing all four matrix elements to vary. Hence a prescription in which a single *J* = 0 INC matrix element (*V*(1) *<sup>B</sup>* or *<sup>V</sup>*(2) *<sup>B</sup>* ) is included is now the commonly used approach for modelling MED and TED (e.g., [8–11,18,52]). The results in Table 1 suggest that an isovector *J* = 0 matrix element of the order of −100 keV is required for MED and an isotensor *J* = 0 matrix element of the order of +100 keV is required for TED. These conclusions are essentially consistent with the original study of Zuker et al. [16]. Indeed, the fits in Ref. [17] indicates that a positive isovector matrix element at *J* = 2 (as was originally extracted in [16]) has a similar effect as a negative matrix element at *J* = 0. Again, the key contributor is the *J*-dependence, not the absolute magnitude, of these matrix elements.

Figure 4 shows experimental and shell-model MED in the *f* <sup>7</sup> 2 shell. The solid blue lines contain the full shell-model calculation, performed exactly as described in Section 3, with a single isovector *J* = 0 matrix element of *VB* = −79 keV, the figure extracted from the fits [17] (see Table 1). The red dashed lines show the calculations without *VB* included. It is clear from data like these the crucial role that this effective INC interaction has, especially at low *J*, in the description of MED in the *f* <sup>7</sup> 2 region.

It is certainly of interest to understand the importance of this effect in other mass regions. In general, this is more challenging in regions where there are more orbitals in play and where the influence of the monopole contributions may be large. In the *sd* shell, inclusion of the INC *VB* term also appears to be necessary, with matrix elements of the same order as described above [11,52]. In the upper *f p* shell, it has been challenging to find a consistent picture for MED, and this remains an open question, e.g., [21]. However, a very recent analysis of the *A* = 58, *T* = 1 mirror nuclei has been performed [9], using the same modelling as that presented for the *A* = 56, *T* = 2 mirrors later in Section 4.2. In the *A* = 58 mirror nuclei, the *f* <sup>7</sup> 2 shell is expected to be almost fully filled and so the MED will be insensitive to the inclusion of *VB* in the *f* <sup>7</sup> 2 orbital, but will be sensitive to its inclusion in the other *p f* orbitals. The analysis indicated that a much better match to the experimental MED was obtained when a negative *J* = 0 matrix element for *VB* was included for *all* the *f p* orbitals.

The physical origin of the isovector INC interaction in the modelling of MED remains unclear, and the analysis presented in Reference [17] suggests that the matrix elements and their *J* dependence (see Table 1) cannot be reconciled easily with the properties of known nuclear charge-symmetry breaking interaction. This therefore points to other electromagnetic contributions missing in the model; see Ref. [17] for a discussion.

**Figure 4.** Experimental and shell-model MED in the *f* <sup>7</sup> 2 shell. The solid blue lines contain the full shell-model calculation, including a single isovector *J* = 0 INC matrix element of *VB* = −79 keV. See text for details. The red dashed lines show the calculations without *VB* included. Data for "Shell Model (no *VB*)" originally presented in Ref. [17]. Where error bars are not visible, they are smaller than the data markers.

Turning to TED, and the impact of the isotensor INC interaction *V*(2) *<sup>B</sup>* , the case study of 66Se (Section 2) and the *A* = 66, *T* = 1 isobaric triplet provides a practical example. The successful spectroscopy of 66Se [12] up to *J<sup>π</sup>* = 6<sup>+</sup> completed the *T* = 1 isobaric triplet and allowed for TED to be determined to *J<sup>π</sup>* = 6+. The experimental data are presented in Figure 5. The large negative TED observed are typical of all *T* = 1 triplets; see, e.g., [18]. Figure 5 also contains the result of the shell-model calculation performed following the prescription in Section 3 and using the JUN45 interaction [55]; see black line. The calculations were originally performed in Ref. [18], and updated for this review. The shell-model calculation does not contain calculations related to the two monopole components (*VCr* and *Vll*,*ls*) since these effectively cancel to zero due to the double-difference method of calculating the TED. Hence, only the two multipole interactions, *VCM* (Coulomb) and *VB* (INC), are relevant for this calculation. The shell-model results are plotted in Figure 5, for different strengths of the INC parameter *V*(2) *<sup>B</sup>* , for *J* = 0 couplings. The blue dotted line shows *VB* = 0 (i.e., just *VCM* contributes), the black solid line has *VB* = +100 keV and the red dashed line shows *VB* = +200 keV. The calculation with *VB* = +100 keV (black line) is consistent with the prescription in [16] and with the data in Table 1. The *VB* interaction was applied equally to all orbits in the *p* <sup>3</sup> 2 *f* 5 2 *p* 1 2 *g*9 2 valence space although, in this case, it is the contribution from the *f* <sup>5</sup> that dominates [18].

2 Two conclusions can be drawn from the comparison with the shell-model results when *VB* = +100 keV is applied. The first is that the agreement with experimental TED would fail badly without the inclusion of this additional effective isotensor INC term. Secondly, it can be shown from this analysis [18] that the two components, *VCM* (Coulomb) and *VB* (INC), have approximately the same magnitude when *VB* = +100 is applied. This is not that surprising since, as noted above, it is the *J*-dependence of the matrix elements that influences the TED, and the Coulomb matrix elements generally vary by around 100 keV from *J* = 0 to *J*max. The key point is that the contribution of the isotensor INC term, to the TED, is as large as that of the Coulomb two-body interaction.

**Figure 5.** Experimental and shell-model TED for the *A* = 66, *T* = 1 isobaric triplet. The black line shows the full shell-model calculation, including a single isotensor *J* = 0 INC matrix element of *VB* = +100 keV in all orbitals in the valence space. The other lines show the shell-model results using different strengths of the INC parameter, *VB*. See text for details. The calculations presented are based on the approach of Ref. [18]. The error bars on the data points are smaller than the data markers.

Lenzi et al. [18] performed a similar analysis for all *T* = 1 triplets between *A* = 22 and *A* = 66, using four different interactions, as appropriate to the valence space being used, and applying an isotensor *J* = 0 matrix element of *VB* = +100 keV in all orbitals. A remarkably consistent picture emerged, with observations very similar to that for *A* = 66; i.e., that the *VB* contribution is significant, and required, across the full range of triplets studied. An important point to note is that, for TED, we have seen that the monopole

terms of the shell-model prescription do not contribute significantly. Therefore, the TED is essentially only sensitive to multipole effects and thus represents an observable that can shed light on the nature of effective isospin-non conserving interactions. Since the size of the required *VB* interaction appears to be largely independent of mass region, orbital or shell-model interaction, it is natural to examine whether or not the true charge dependence of the nuclear interaction [1] could be the origin. It was shown by Ormand and Brown [56] that nucleon scattering data suggests that the *np* nuclear interaction is approximately 2–3% stronger than the *pp* and *nn* interactions. The analysis of Reference [18] indeed indicated that the scale, and sign, of the effective isotensor interaction *VB* interaction appear to be approximately consistent with that estimate for the charge-dependence of the NN interaction. This indeed highlights the power of using energy differences, coupled to a reliable shell-model calculation, to probe effective nucleon interactions.
