Radii in the $sd$ shell and the $s_{1/2}$ "halo" orbit: A game changer

TL;DR

This paper investigates the anomalously large proton radii in the $sd$ shell, especially the $s_{1/2}$ orbit, using microscopic shell model calculations to better understand shell formation and nuclear size deviations.

Contribution

It introduces a microscopic calculation approach to analyze the large $s_{1/2}$ orbit radii and their impact on nuclear radii trends, challenging existing shell formation models.

Findings

The $s_{1/2}$ orbit remains significantly larger than $d$ orbits up to $N,Z=14.

A sudden drop in the $s_{1/2}$ radius occurs beyond $N,Z=14$, but it remains larger than $d$ orbits.

The behavior of the $s_{1/2}$ orbit challenges current understanding of shell structure.

Abstract

Proton radii of nuclei in the $sd$ shell depart appreciably from the asymptotic law, $\rho_{\pi}=\rho_0A^{1/3}$. The departure exhibits systematic trends fairly well described by a single phenomenological term in the Duflo-Zuker formulation, which also happens to explain the sudden increase in slope in the isotope shifts of several chains at neutron number $N=28$. It was recently shown that this term is associated with the abnormally large size of the $s_{1/2}$ and $p$ orbits in the $sd$ and $pf$ shells respectively. Further to explore the problem, we propose to calculate microscopically radii in the former. Since the (square) radius is basically a one body operator, its evolution is dictated by single particle occupancies determined by shell model calculations. Assuming that the departure from the asymptotic form is entirely due to the $s_{1/2}$ orbit, the expectation value $\langle s_{1/2}|r^2|s_{1/2}\rangle$ is determined by demanding that its evolution be such as to describe well nuclear radii. It does, for an orbit that remains very large (about 1.6 fm bigger than its $d$ counterparts) up to $N,\,Z=14$ then drops abruptly but remains some 0.6 fm larger than the $d$ orbits. An unexpected behavior bound to challenge our understanding of shell formation.