Valence $1s-0d$ proton vacancy of the $^{32}$Si ground state

TL;DR

The work determines ground-state proton and neutron vacancies in the $1s-0d$ shell for $^{32}$Si by combining $^{32}$Si($^3$He,$d$) proton-adding data with neutron-adding information and applying DWBA with Macfarlane–French sum rules. It finds near-empty $1s_{1/2}$ and $0d_{3/2}$ proton orbitals and a gradual reduction of neutron vacancies toward the $N=20$ shell gap, with small changes across the $^{28-34}$Si isotopes. The results align with USDB shell-model calculations constrained to the $1s-0d$ space, supporting a robust $Z=14$ sub-shell and informing on the evolution of neutron spin-orbit partner energies near $N=20$. The study highlights the balance between single-particle occupancies and residual correlations in this region and discusses implications for core polarization and effective charges in nearby isotopes.

Abstract

The $^{32}$Si($^3$He,$d$)$^{33}$P reaction was studied in inverse kinematics at 6.3~MeV/$u$. States in $^{33}$P corresponding to the proton $1s-0d$ single-particle orbitals were identified up to $\sim$4.5 MeV in excitation energy. The ($^{3}$He,$d$) spectroscopic factors were determined from Distorted Wave Born Approximation calculations. When combined with complementary neutron-adding data, the $1s-0d$ proton vacancies in the $^{32}$Si ground state were extracted. In conjunction with a re-analysis of data from previous single-particle measurements, the trends in proton and neutron vacancy were explored across the $^{28,30,32,34}$Si isotopes. Both proton and neutron vacancy data show gradual changes in their occupancies. The proton $1s_{1/2}$ orbitals in $^{32}$Si and $^{34}$Si are both consistent with being empty. The ground-state nucleon distributions are described by shell-model calculations constrained to the $1s-0d$ model space.

Figures (5)

• Figure 1: The heavy-ion recoil particle identification (PID) plot from the $\Delta$E-E$_{res}$ energies of the recoil silicon telescope. A condition requiring the detection of a charged particle by a PSD within 200 ns has been applied. The element groups having $Z=14$ (red), 15 (green), and 16 (blue) are labeled and indicated by overlaid dashed lines. The inset histogram shows the relative (coincidence) time between a heavy-ion recoil and a corresponding light charged particle for only $Z=15$ recoils. Calibration of the relative time difference was done such that protons are centered around 10 ns and deuterons are centered around $\sim30$ ns, shaded in red.
• Figure 2: The excitation energy spectrum for states in $^{33}$P following the $^{32}$Si($^3$He,$d$) reaction at 6.3 MeV/$u$. States assigned to previously identified levels found in the literature are labeled by excitation and proton $\ell$ transfer. The unassigned level is labeled with its energy centroid in parenthesis.
• Figure 3: The measured ($^3$He,$d$) angular distributions for some of the observed states in $^{33}$P. The DWBA calculations for the expected $\ell$ transfer are also shown fit to the data. See text for details on the DWBA.
• Figure 4: Extracted proton vacancies for the even-$A$ Si isotopes from $A=28-34$ over the $1s-0d$ orbitals (Table \ref{['table:spectroscopic_values']}). The lighter shaded regions indicate the contributions from the $T_>$ states to the vacancy. The black rectangles indicate the error on the extracted proton vacancy. The thick black lines indicate the independent single-particle-model proton-vacancy expectations.
• Figure 5: A comparison of the empirical proton and neutron vacancies (data points) to those based on configuration-interaction shell-model calculations using the USDB $1s-0d$ effective interaction (lines) ref:Bro06. The normalization procedure applied to the data is given in the text.