Nuclear Charge Radii of Silicon Isotopes

The nuclear charge radius of ${}^{32}\mathrm{Si}$ was determined using collinear laser spectroscopy. The experimental result was confronted with ab initio nuclear lattice effective field theory, valence-space in-medium similarity renormalization group, and mean field calculations, highlighting important achievements and challenges of modern many-body methods. The charge radius of ${}^{32}\mathrm{Si}$ completes the radii of the mirror pair ${}^{32}\mathrm{Ar}\text{−}{}^{32}\mathrm{Si}$, whose difference was correlated to the slope $L$ of the symmetry energy in the nuclear equation of state. Our result suggests $L\le 60\mathrm{MeV}$, which agrees with complementary observables.

Figure 1

Normalized resonance spectra of ${}^{28,29,30,32}\mathrm{Si}$. Inelastic collisions during the charge exchange process led to slightly asymmetric line shapes and were considered in the fit function [39]. The frequency is relative to the centroid of ${}^{28}\mathrm{Si}$.

Figure 2

Experimental and theoretical differential mean square charge radii of Si. Only the nuclear lattice calculation provided an uncertainty, which is plotted as a gray band. Together with the DFT calculations using the SVmin functional, the lattice results agree with the experimental results.

Figure 3

$\mathrm{\Delta }{R}_{\mathrm{ch}}({}^{32}\mathrm{Ar}-{}^{32}\mathrm{Si})$ as a function of $L$. The experimental $1\sigma$ constraint of $\mathrm{\Delta }{R}_{\mathrm{ch}}$ is indicated by the horizontal dashed lines. The solid circles are the results of Skyrme energy-density functionals, and the crosses are for the covariant density functional calculations. The overlapping area, highlighted in gray, shows our constraint for $L\le 60\mathrm{MeV}$. It is in good agreement with the result from the ${}^{54}\mathrm{Ni}\text{−}{}^{54}\mathrm{Fe}$ [81], the ${}^{36}\mathrm{Ca}\text{−}{}^{36}\mathrm{S}$, and the ${}^{38}\mathrm{Ca}\text{−}{}^{38}\mathrm{Ar}$ mirror pairs [80], with the findings from the electric dipole polarizability ${\alpha }_{\mathrm{D}}$ [85], the neutron star merger GW170817 [84] but smaller than the PREX II result [83]. Please note that those are only plotted as reference on the $L$ axis and are not correlated to $\mathrm{\Delta }{R}_{\mathrm{ch}}$. From our theoretical calculations on the lattice and from VS-IMSRG calculations with the EM1.8_2.0 and $\mathrm{\Delta }{\mathrm{N}}^2{\mathrm{LO}}_{\mathrm{GO}}$ interaction, we deduced $\mathrm{\Delta }{R}_{\mathrm{ch}}$ and related those with corresponding calculations for $L$ [20, 25, 73].