*3.1. Superdeformation in 42Ca*

The potential of Coulomb excitation as a tool to study superdeformation has been demonstrated in the very first experiment using the AGATA *γ*-ray tracking array [19]. The superdeformed (SD) structure in 42Ca was populated following Coulomb excitation of a 42Ca beam on 208Pb and 197Au targets [3,18]. From the measured *γ*-ray intensities, magnitudes and relative signs of numerous *E*2 matrix elements coupling the low-lying states in 42Ca were determined. In particular, two key pieces of information were obtained for the first time, which confirm that the band built on the 0<sup>+</sup> <sup>2</sup> state in 42Ca has a SD character at low spin: the spectroscopic quadrupole moment of the 2<sup>+</sup> <sup>2</sup> state, which corresponds to *β*<sup>2</sup> = 0.48(16), as well as the enhanced *B*(*E*2; 2<sup>+</sup> <sup>2</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> <sup>2</sup> ) = <sup>15</sup>+<sup>6</sup> <sup>−</sup><sup>4</sup> W.u. value. As discussed in [12], even though the 2<sup>+</sup> <sup>2</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> <sup>2</sup> transition is too weak to be observed and, prior to the study of K. Hady ´nska-Kle¸k et al. [3,18] only an upper limit for the branching ratio was known, the corresponding matrix element has a strong influence on excitation cross sections of the observed states, and hence it could be determined from the intensities of other transitions measured in the Coulomb-excitation experiment.

The obtained transitional and diagonal *E*2 matrix elements were further interpreted in terms of quadrupole invariants of the 0<sup>+</sup> 1,2 and 2<sup>+</sup> 1,2 states, leading to the conclusion that the spherical ground state of 42Ca exhibits large fluctuations into the *β*2–*γ* plane, while the excited structure has a large quadrupole deformation of *β*<sup>2</sup> = 0.43(4) for the 0+ <sup>2</sup> state, comparable to those deduced from lifetime measurements for other SD bands in this mass region. The important increase of the *Q*2 quadrupole invariant for the 2<sup>+</sup> 1 state with respect to that for the ground state was attributed to the mixing of the 2<sup>+</sup> states. Additionally, the triaxiality parameter cos <sup>3</sup>*δ* obtained for the 0<sup>+</sup> <sup>2</sup> state, corresponding to *γ* = (13+<sup>5</sup> <sup>−</sup>6)◦, provided the first experimental evidence for the non-axial character of SD structures around *A* ≈ 40. The value obtained for the ground state, *γ* = 28(3)◦, was interpreted as resulting from its softness.

This experimental study triggered new Large-Scale Shell Model (LSSM) calculations for 42Ca [3,18]. They were performed using the SDPF.MIX interaction in the *sdpf* model space for neutrons and protons, with a virtual 28Si core, as in the earlier study [20] that successfully described properties of the deformed 4*p*–4*h* and 8*p*–8*h* structures in 40Ca. Up to six particle –hole excitations from the 2*s*1/2 and 1*d*3/2 orbitals into the *p f* orbitals were allowed, and the electric effective charges were 1.5*e* for protons and 0.5*e* for neutrons. The overall agreement of the calculations with the experimental level energies and decay patterns is remarkable, see Figure 2. The experimental values of the *Q*2 and *Q*<sup>3</sup> cos <sup>3</sup>*δ* invariants for the 0<sup>+</sup> 1,2 states were also well reproduced. The only notable systematic difference is the overestimation of *E*2 matrix elements in the SD band and underestimation of those in the yrast band as well as intra-band ones, which suggests that the mixing between the two bands is not fully reproduced by the calculations.

The LSSM results provide insight into the configurations of normal-deformed and SD states in 42Ca: the 0<sup>+</sup> <sup>2</sup> and 2<sup>+</sup> <sup>2</sup> states are predicted to be dominated by the 6*p*–4*h* excitation s, while the ground-state band has a predominantly two-particle configuration, with considerable 4*p*–2*h* and 6*p*–4*h* admixtures. Furthermore, they suggest that the experimentally known 2<sup>+</sup> <sup>3</sup> state is the band head of a *K* = 2 *γ* band related to the SD structure, with the configuration dominated by almost equal contributions of 6*p*–4*h* and 8*p*–6*h* excitations (≈40% each). This gives further support for the slightly triaxial shape of the SD band

in 42Ca, while the identification of higher-spin members of the predicted *K* = 2 *γ* band represents a challenge for future experimental studies.

**Figure 2.** Comparison of the experimental low-energy part of the 42Ca level scheme with that calculated using Large-Scale Shell Model (LSSM) [3,18]. States are labelled with their energies in keV, transitions with *E*2 transition probabilities in Weisskopf units and spectroscopic quadrupole moments for the 2<sup>+</sup> states, expressed in *e*fm2, are reported in brackets.
