Lattice simulation of nucleon distribution and shell closure in the proton-rich nucleus $^{22}$Si

The proton-rich nucleus $^{22}$Si is studied using Nuclear Lattice Effective Field Theory with high-fidelity chiral forces. Our results indicate that $^{22}$Si is more tightly bound than $^{20}$Mg, thereby excluding the possibility of two-proton emission. The $Z = 14$ shell closure in $^{22}$Si is supported by the evolution of the $2^+$ state in the neighboring nuclei. We then focus on the charge radius and spatial distribution information of $^{22}$Si, considering the novel phenomena that may emerge due to the small two-proton separation energy and the shell closure. We present the distribution of the $14$ protons and $8$ neutrons obtained from our lattice simulation, revealing insights into the spatial arrangement of the nucleons. Moreover, the spatial localization of the outermost proton and neutron suggests that $^{22}$Si is a doubly magic nucleus. Furthermore, we develop the pinhole method based on the harmonic oscillator basis, which gives insight into the nuclear structure in terms of the shell model picture from lattice simulations. Our calculated occupation numbers support that $Z = 14$ and $N = 8$ are the shell closures and show that the $\pi 1s_{1/2}$ orbital component is minor in $^{22}$Si.