Background: The $\alpha$ cluster is a prominent feature, not only in light nuclei but also in heavy nuclei. To study the $\alpha$-particle formation in the mean-field calculation, the localization function has been extensively utilized. However, the localization function does not guarantee the proximity of four different nucleons which is required by the $\alpha$-particle formation. A simple indicator of the proximity is desired. Recently, experimental measurement of the quasifree $\alpha$-knockout reaction for Sn isotopes reveals the cross sections with a monotonic decrease with increasing neutron number. [Science 371, 260 (2021)]. This is interpreted as evidence of the surface $\alpha$ formation.

Purpose: We propose a simple and comprehensible quantity to assess the proximity of four nucleons with different spins and isospins. Using this, we examine the recent measurement of $\alpha$-knockout cross sections in Sn isotopes.

Methods: The local $\alpha$-removal strength is proposed to quantify the possibility to form an $\alpha$ particle at a specific location inside the nucleus. In addition, it provides the strength of ground and excited states in the residual nuclei after the removal of the $\alpha$ particle. To make the calculation feasible, we introduce several approximations, such as point-$\alpha$, mean-field, and no rearrangement approximations. We use the Hartree-Fock-plus-BCS method for the mean-field calculation for Sn isotopes. We also propose another measure, the local $\alpha$ probability, which should provide a better correlation with the $\alpha$-knockout cross sections.

Results: The calculation of the local $\alpha$-removal strength is extremely easy in the mean-field model with no rearrangement. For even-even Sn isotopes, the local $\alpha$-removal strengths to the ground state of residual nuclei are almost universal in the nuclear surface region. In contrast, the local $\alpha$ probability produces strong neutron number dependence consistent with the experiment.

Conclusions: The local $\alpha$-removal strength and the local $\alpha$ probability are easily calculable in the mean-field models. Recent experimental data for Sn isotopes may be explained by a simple model without explicit consideration of $\alpha$ correlation.

• Received 29 April 2023
• Accepted 29 June 2023

DOI:https://doi.org/10.1103/PhysRevC.108.014318