Observation of a dominant $\boldsymbol{0f_{7/2}}$ neutron configuration in the $\boldsymbol{^{32}}$Si $\boldsymbol{J^{\pi}=5^-}$ isomeric state

Technical Summary: Observation of a dominant 0f7/2 neutron configuration in the 32Si Jπ = 5− isomeric state

Problem Statement
The nucleus $^{32}\text{Si}$ ($Z=14, N=18$) possesses a rare yrast spin-trap isomeric state with $J^\pi = 5^-$ at an excitation energy of $E_x = 5.505$ MeV. This state decays predominantly (>99%) via a highly hindered electric octupole (E3) transition to the first excited $2^+$ state, with a reduced transition probability of $B(E3) = 0.0841(10)$ W.u. Notably, the isomer bypasses a nearby $3^-$ level ($E_x = 5.288$ MeV) via an electric quadrupole (E2) transition. Previous shell-model calculations suggested this hindrance might arise from differences in neutron configurations between the pure single-neutron $5^-$ state and the mixed-configuration $3^-$ state. However, the extreme suppression of the $5^- \to 3^-$ transition strength ($< 0.053$ W.u.) compared to neighboring isotones like $^{34}\text{S}$ and $^{36}\text{Ar}$ (where strengths are $\approx 0.5 - 1$ W.u.) required further investigation into the single-particle nature of these levels and the specific role of proton and neutron participation in the transition.

Methodology
To investigate the single-neutron structure of the excited states in $^{32}\text{Si}$, the authors performed a $(d,p)$ transfer reaction study using a radioactive beam of $^{31}\text{Si}$.

• Beam Production: A primary $^{30}\text{Si}$ beam at 11.5 MeV/u was used at the ATLAS facility (Argonne National Laboratory) to produce a short-lived $^{31}\text{Si}$ beam via a $(d,p)$ reaction on a cryo-cooled deuterium gas cell. The resulting $^{31}\text{Si}^{14+}$ beam had an energy of 9.6 MeV/u.
• Reaction Setup: The secondary $^{31}\text{Si}$ beam impinged on a $^{12}\text{CD}_2$ target. Outgoing protons were detected in coincidence with $^{32}\text{Si}$ recoils using the HELIOS spectrometer, which operates within a 2.5 T solenoidal magnetic field.
• Analysis: Angular distributions were constructed for states in the excitation energy range $E_x \approx 3.5 - 8.5$ MeV. Data were analyzed using Distorted Wave Born Approximation (DWBA) calculations (via the code PTOLEMY) to extract orbital angular momentum transfer ($\ell$) values and spectroscopic factors ($C^2S$). The results were normalized relative to the $5^-$ isomeric state.

Key Results

1. Confirmation of Single-Particle Character: The $5^-$ isomeric state at 5.505 MeV was observed as the most abundant peak in the spectrum. Its angular distribution was dominated by an $\ell=3$ transfer, confirming a dominant $\nu 0f_{7/2}$ single-neutron character. The spectroscopic factor was normalized to 1.00, and contributions from $\ell=1$ (associated with the $1p_{3/2}$ orbital) were constrained to be less than 10%. This confirms the wave function is primarily a $(0d_{3/2})^1(0f_{7/2})^1$ neutron configuration.
2. Spectroscopic Factors for the $3^-$ State: The yrast $3^-$ state at 5.288 MeV was also populated via $\ell=3$ transfer but exhibited a reduced spectroscopic factor of $C^2S \approx 0.44$ relative to the $5^-$ state. This reduction suggests a mixed-configuration wave function, consistent with expectations for even-even systems where $3^-$ states are susceptible to octupole correlations.
3. Comparison with $^{34}\text{S}$: The ratio of the $3^-$ to $5^-$ spectroscopic factors in $^{32}\text{Si}$ ($\approx 0.44$) was found to be comparable to that observed in the neighboring nucleus $^{34}\text{S}$ ($\approx 0.25$). This indicates that the relative single-neutron $0f_{7/2}$ occupancies in the $3^-$ and $5^-$ states are similar in both nuclei.
4. Other States: A state at 5.772 MeV showed a summed $C^2S$ comparable to the $5^-$ state but with mixed $\ell=1$ and $\ell=3$ contributions, suggesting the presence of additional unresolved states in this energy region.

Significance and Conclusions
The primary conclusion of this work addresses the mechanism behind the highly hindered $5^- \to 3^-$ E2 transition in $^{32}\text{Si}$.

• Rejection of Neutron-Only Explanation: Since the relative single-neutron overlaps (spectroscopic factors) between the $3^-$ and $5^-$ states in $^{32}\text{Si}$ are similar to those in $^{34}\text{S}$, the authors argue that the difference in neutron wave function overlaps cannot be the primary cause of the order-of-magnitude reduction in the $B(E2)$ transition strength observed in $^{32}\text{Si}$ compared to $^{34}\text{S}$.
• Proton Participation Hypothesis: The paper proposes that the hindrance arises from a lack of participation by both protons and neutrons in the transition.
  • Neutrons: The neutron configurations in the $5^-$ and $3^-$ states of $^{32}\text{Si}$ are closely matched (dominantly $(0d_{3/2})^1(0f_{7/2})^1$), resulting in a small neutron matrix element ($M_n$).
  • Protons: The robust $Z=14$ sub-shell gap in $^{32}\text{Si}$ leads to a spherical proton distribution with weak deformation. The authors suggest that the proton configurations in the $5^-$ and $3^-$ states also do not differ significantly, leading to a small proton matrix element ($M_p$).
• Mechanism: The transition strength $B(E2)$ is proportional to $|e_p M_p + e_n M_n|^2$. The authors conclude that the extreme hindrance in $^{32}\text{Si}$ is due to the simultaneous smallness of both $M_p$ and $M_n$, likely exacerbated by potential destructive interference (opposite signs) between the proton and neutron contributions. This contrasts with $^{34}\text{S}$, where increased mixing and proton participation allow for a stronger transition.

The study successfully characterizes the single-neutron structure of the $^{32}\text{Si}$ isomer and refutes the hypothesis that neutron configuration mismatch alone explains the transition hindrance, pointing instead to a collective lack of nucleon participation driven by the specific shell structure of the $Z=14$ nucleus.