Large scale shell model calculations for the shape coexistence at N = 40 and N = 50 island of inversion

Shape coexistence---the competition between near-degenerate spherical and deformed configurations---emerges as a pervasive feature of neutron-rich nuclei in and around the islands of inversion. Using a large-scale shell model (LSSM) with a realistic effective interaction, we systematically compute low-lying spectra, average orbital occupations, and particle--hole (p--h) excitation probabilities to investigate the candidate shape-coexistence states for even--even \(N=40\) and \(N=50\) isotones. Where data exist, our LSSM reproduces the observed systematics, including the benchmark \(E(2_1^+)\) in \(⁷⁸\)Ni and the characteristic shape-coexisting structure near \(⁶⁸\)Ni. Along the \(N=40\) chain, \(E(0_2^+)\) closely tracks \(E(2_1^+)\) with a minimum at \(⁶⁴\)Cr, signaling maximal collectivity. Along the \(N=50\) chain, both \(E(2_1^+)\) and \(E(0_2^+)\) follow a parabolic evolution from \(⁷⁸\)Ni to \(⁷⁰\)Ca with a common minimum at \(⁷⁴\)Cr; microscopically, the \(0_2^+\) states are predominantly \(2p\!-\!2h\) (with modest \(3p\!-\!3h/4p\!-\!4h\) admixtures), while \(⁷⁴\)Cr is a controlled exception exhibiting comparable \(0p\!-\!0h\) and \(2p\!-\!2h\) components. Extending the survey to neutron-rich Ti, Cr, and Fe isotopes, the LSSM captures the pronounced variation of \(E(0_2^+)\)---peaking at \(⁶²\)Ti and reaching minima at \(N=50\) (\(⁷⁴\)Cr, \(⁷⁶\)Fe), suggesting a widespread shape coexistence and its evolution in these regions.