What is the trend for the SO-splitting? Approaching the island of inversion using $^{32}$Si$(d,p)^{33}$Si and $^{33}$P$(d,p)^{34}$P reactions

The Spin-orbital (SO) interaction plays a very important role in determining the magic numbers but is poorly constrained so far. We propose to study low-lying $l=1/l=3$ single-particle states to access the SO-splitting in $^{33}$Si and $^{34}$P using one neutron-adding transfer reactions. The SOLARIS magnet solenoid coupled with the HELIOS silicon array will be used to detect the protons. The goals of the experiment are to determine the $l=1$ and $l=3$ single-particle energies and SO-splitting in $^{33}$Si and $^{34}$P and compare with nuclei in the same isotonic chains. Special attention will be paid to determining the excitation energies of the $1/2^-$ and $5/2^-$ states in $^{33}$Si, which play an important role in determining the trend of the $p$-wave and $f$-wave SO-splitting. This information will determine if there is a sudden change of the SO-splitting in silicon isotopes, which links to the nucleus bubble effect, weak-binding effect, the SO interaction and the underlying mechanisms driving its evolution.