In-beam <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>γ</mml:mi></mml:math> -ray spectroscopy of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mmultiscripts><mml:mi>Mg</mml:mi><mml:mprescripts /><mml:none /><mml:mn>32</mml:mn></mml:mmultiscripts></mml:math> via direct reactions

Background: The nucleus $^{32}$Mg ($N=20$ and $Z=12$) plays a central role in the so-called "island of inversion" where ground states $sd$-shell neutrons are promoted to $fp$-shell orbitals across shell gap, resulting disappearance canonical neutron magic number $N=20$. Purpose: primary goals this work extend level scheme $^{32}$Mg, provide spin-parity assignments excited states, discuss microscopic structure each state through comparisons with theoretical calculations. Method: In-beam $\gamma$-ray spectroscopy was performed using two direct-reaction probes, one-neutron (two-proton) knockout reactions on $^{33}$Mg ($^{34}$Si). Final-state exclusive cross sections parallel momentum distributions were extracted from experimental data compared eikonal-based reaction model calculations combined shell-model overlap functions. Results: Owing remarkable selectivity two-proton reactions, significantly updated for which exhibits negative-parity intruder positive-parity normal constructed. results confronted four different nuclear models. Conclusions: In some these models, aspects transition into island inversion well described. However, unexplained discrepancies remain, even help state-of-the-art approaches, key is not yet fully captured.