  Background: In the "island of inversion", ground states of neutron-rich
$sd$-shell nuclei exhibit strong admixtures of intruder configurations from the
$fp$ shell. The nucleus $^{30}$Mg, located at the boundary of the island of
inversion, serves as a cornerstone to track the structural evolution as one
approaches this region. Purpose: Spin-parity assignments for excited states in
$^{30}$Mg, especially negative-parity levels, have yet to be established. In
the present work, the nuclear structure of $^{30}$Mg was investigated by
in-beam $\gamma$-ray spectroscopy mainly focusing on firm spin-parity
determinations. Method: High-intensity rare-isotope beams of $^{31}$Mg,
$^{32}$Mg, $^{34}$Si, and $^{35}$P bombarded a Be target to induce nucleon
removal reactions populating states in $^{30}$Mg. $\gamma$ rays were detected
by the state-of-the-art $\gamma$-ray tracking array GRETINA. For the direct
one-neutron removal reaction, final-state exclusive cross sections and parallel
momentum distributions were deduced. Multi-nucleon removal reactions from
different projectiles were exploited to gain complementary information.
Results: With the aid of the parallel momentum distributions, an updated level
scheme with revised spin-parity assignments was constructed. Spectroscopic
factors associated with each state were also deduced. Conclusions: Results were
confronted with large-scale shell-model calculations using two different
effective interactions, showing excellent agreement with the present level
scheme. However, a marked difference in the spectroscopic factors indicates
that the full delineation of the transition into the island of inversion
remains a challenge for theoretical models.
