*3.4. Evolution of Collectivity in Z* ≈ *50 Nuclei*

The tin nuclei, forming the longest chain of experimentally accessible isotopes between two doubly-magic nuclei, have traditionally been considered a prime example of the seniority scheme. While this description is supported by the almost constant energies of the 2<sup>+</sup> 1 states in the even–even Sn nuclei from 102Sn to 130Sn, the corresponding *B*(*E*2; 2<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> 1 ) values seem to deviate from the expected parabolic behaviour (see Figure 4). Extensive Coulomb-excitation studies of stable [55–57] and exotic [58–62] Sn nuclei yielded *B*(*E*2; 2<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> <sup>1</sup> ) values for 106–134Sn that were discussed in the context of Shell-Model calculations. In the Coulomb-excitation campaigns aiming at high-precision measurements of the *B*(*E*2; 2<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> <sup>1</sup> ) values in stable Sn isotopes, the experimental conditions minimised the role of multi-step excitation and the reorientation effect. The experiments at ORNL [55] were performed in strongly inverse kinematics, with a 12C target bombarded by 112,114,116,118,120,122,124Sn beams; a *nat*Ti target was also used for complementary *Qs*(2<sup>+</sup> <sup>1</sup> ) measurements.

In the IUAC campaign [56,57], a reaction partner with a much higher *Z* was used: a 58Ni beam impinged on 112,116,118,120,122,124Sn targets. However, due to the selection of events with the Ni beam particles scattered at forward angles, no excitation of higher-lying states was observed, although their possible weak influence on the 2<sup>+</sup> <sup>1</sup> excitation process was taken into account in the data analysis. The *B*(*E*2; 2<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> <sup>1</sup> ) values were obtained with relative uncertainties of 5% or less in all cases, and the results of the two campaigns agreed within 3*σ* for 120,122,124Sn and within 1*σ* for the other isotopes, demonstrating the level of accuracy and precision that can be achieved (see Figure 4).

Low-energy Coulomb-excitation experiments on neutron-deficient Sn isotopes were performed at ISOLDE [58,59] with 2.8-MeV/*A* 106,108,110Sn beams bombarding 58Ni targets. On the neutron-rich side, a campaign was performed at ORNL [60,61] to study 126,128,130,134Sn in very similar experimental conditions as those used for stable isotopes in [55]. In order to increase the excitation cross section for the 2<sup>+</sup> <sup>1</sup> state in 132Sn, located at 4.04-MeV excitation energy, targets of 48Ti and 206Pb were used in the ORNL [60] and HIE-ISOLDE [62] measurements, respectively.

While certain discrepancies with the values obtained using other methods exist (see e.g., [63] for a compilation of experimental data), the ensemble of experimental results points to an asymmetric shape of the *B*(*E*2; 2<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> <sup>1</sup> ) distribution as a function of *N*, with a plateau extending towards lighter nuclei. The reproduction of this plateau represented a challenge for model calculations. Recently, its appearance has been discussed [64,65] in the context of pseudo-SU(3) symmetry acting in the space of *gds* orbitals excluding 1*g*9/2. The calculations were performed using *V*low−*<sup>k</sup>* variants of the realistic N3LO interaction, with the monopole part of the interaction replaced by a Hamiltonian provided by the GEMO code [66], adding the single-particle energies for 101Sn. They successfully reproduced the evolution of the *B*(*E*2; 2<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> <sup>1</sup> ) values in 104–114Sn [64,65] (see Figure 4) and demonstrated that modifications of the pairing strength had a negligible effect on the calculated *B*(*E*2; 2<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> <sup>1</sup> ) values, in contrast to what was observed for the *<sup>B</sup>*(*E*2; 4<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>2</sup><sup>+</sup> 1 ) strengths [65].

**Figure 4.** Reduced transition probabilities *B*(*E*2; 2<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> <sup>1</sup> ) in the Sn isotopic chain determined from low-energy Coulomb-excitation measurements. The experimental results obtained at ORNL [55,60,61], IUAC [56,57], and ISOLDE [58,59,62] are compared with predictions from the Monte-Carlo Shell Model (MCSM) [63] and LSSM [64,65].

An alternative explanation was offered by the MCSM calculations [63] performed in the full *gds* model space complemented by the 1*h*11/2, 2 *f*7/2, and 3*p*3/2 orbitals for protons and neutrons. These calculations provide good reproduction of all measured *B*(*E*2; 2<sup>+</sup> 1 → 0+ <sup>1</sup> ) values in the Sn chain, including the local increase observed for 132Sn (see Figure 4), and link their enhancement for 108–114Sn to the development of quadrupole deformation driven by proton excitations from the 1*g*9/2 orbital. This scenario is consistent with the observed increase of the *Qs*(2<sup>+</sup> <sup>1</sup> ) values at mid shell [55], which was suggested to be due to the mixing with a deformed configuration, resulting in the presence of proton 2*p*–2*h* and 4*p*–4*h* components in the 2<sup>+</sup> <sup>1</sup> wave function [55]. Low-lying states of predominantly proton 2*p*–2*h* character have been identified in 114,116,118Sn via two-proton transfer reactions [67], and later also in 110,112Sn and 120,122,124Sn, although at higher excitation energies. The MCSM calculations [63] predicted indeed that the ground states of Sn nuclei involve a significant promotion of protons across the *Z* = 50 gap, with the largest 2*d*5/2 occupation predicted at *N* = 60. The occupation of proton orbitals above the *Z* = 50 gap becomes even larger for the 2<sup>+</sup> <sup>1</sup> states, and the corresponding T-plots indicate deformed shapes [63], in line with the measured non-zero quadrupole moments. Multi-step Coulomb-excitation studies aiming at the determination of deformation parameters of the deformed structures built on the 0<sup>+</sup> <sup>2</sup> states, as well as their mixing with the ground-state configurations, would be of much interest. One should note here that the quadrupole invariants for the 0<sup>+</sup> 1,2 states in 110Cd were measured in a recent Coulomb-excitation experiment [68].

The *B*(*E*2; 2<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> <sup>1</sup> ) and *<sup>B</sup>*(*E*2; 4<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>2</sup><sup>+</sup> <sup>1</sup> ) patterns in 100–110Cd nuclei closely resemble that of the *B*(*E*2; 2<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> <sup>1</sup> ) values in the corresponding Sn isotones. They were well reproduced by the calculation of [65], and found almost independent of the assumed pairing strength. This was linked [65] to their static quadrupole deformation, consistent with non-zero quadrupole moments measured for the 102,104Cd isotopes in a Coulombexcitation experiment at ISOLDE [69]. Interestingly, the obtained *Qs*(2<sup>+</sup> <sup>1</sup> ) values are positive, in contrast to those measured for stable Cd nuclei. Unfortunately, they are subject to large uncertainties, and the *Qs*(2<sup>+</sup> <sup>1</sup> ) value for 104Cd significantly changes if a previously measured lifetime of the 2<sup>+</sup> <sup>1</sup> state is used as an additional constraint in the Coulombexcitation data analysis.

Quadrupole deformation of light Cd isotopes was explored in an LSSM study [15] using a modified *v*3*sb* effective interaction [70] in the *π*(2*p*1/2, 1*g*9/2), *ν*(2*d*5/2, 3*s*1/2, 2*d*3/2, 1*g*7/2, 1*h*11/2) model space. The calculated *E*2 matrix elements provide a good reproduction

of the experimental *B*(*E*2; 2<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> <sup>1</sup> ) and *<sup>B</sup>*(*E*2; 4<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>2</sup><sup>+</sup> <sup>1</sup> ) values, and were analysed in terms of quadrupole invariants *Q*2 and *Q*<sup>3</sup> cos <sup>3</sup>*δ* pointing to a predominantly prolate character of 100–108Cd with both *β* and *γ* increasing with *N*. Very recently, Coulomb excitation of 106Cd was performed [71] at the NSCL ReA3 facility. Quadrupole moments of the 2+ <sup>1</sup> , 4<sup>+</sup> <sup>1</sup> , 6<sup>+</sup> <sup>1</sup> and 2<sup>+</sup> <sup>2</sup> states were obtained, as well as the *Q*2 and *Q*<sup>3</sup> cos <sup>3</sup>*δ* invariants for the ground state, which suggest its considerable triaxiality. This feature does not emerge from the LSSM calculations reported in [71], which also used a G-matrix-renormalized CD-Bonn nucleon –nucleon potential and the same model space as those of [15], but allowed at most two neutrons in the 1*h*11/2 orbital. While they well reproduced the experimental *Q*2 invariant for the ground state, the shapes that they predict for light Cd isotopes are decidedly prolate. The difference with respect to a more *γ*-soft behaviour suggested by [15] was attributed to the different 1*h*11/2 single-particle energies, as well as the adopted truncation. However, none of these calculations are able to explain the observed pattern of spectroscopic quadrupole moments in the light Cd nuclei, which will hopefully trigger future experimental and theoretical investigations aiming at understanding their quadrupole properties.
