  The spin-orbit splitting between neutron 1$p$ orbitals at $^{33}$Si has been
deduced using the single-neutron-adding ($d$,$p$) reaction in inverse
kinematics with a beam of $^{32}$Si, a long-lived radioisotope. Reaction
products were analyzed by the newly implemented SOLARIS spectrometer at the
reaccelerated-beam facility at the National Superconducting Cyclotron
Laboratory. The measurements show reasonable agreement with shell-model
calculations that incorporate modern cross-shell interactions, but they
contradict the prediction of proton density depletion based on relativistic
mean-field theory. The evolution of the neutron 1$p$-shell orbitals is
systematically studied using the present and existing data in the isotonic
chains of $N=17$, 19, and 21. In each case, a smooth decrease in the separation
of the $1p_{3/2}$-$1p_{1/2}$ orbitals is seen as the respective $p$-orbitals
approach zero binding, suggesting that the finite nuclear potential strongly
influences the evolution of nuclear structure in this region.
