Probing symmetry energy slope with mirror nuclei: unmeasured charge radii of proton-rich nuclei from binding energies

Knowledge of the slope of nuclear symmetry energy at the saturation density L is crucial for guiding the extrapolation of the symmetry energy to both lower and higher densities, yet its value remains with significant uncertainty. A promising probe for constraining L is the difference in mirror pair root-mean-square (rms) charge radii Δ R ch . However, the Δ R ch data are scarce due to the difficulty in the measurements of rms charge radii R ch for the proton-rich nuclei, hindered by their short half-lives and low production yields. In this letter, we overcome this by introducing a novel method to deduce the unmeasured R ch of proton-rich nuclei based on the relation between binding energies B and R ch for mirror nuclei, B ( Z , N ) − B ( N , Z ) = − a ˜ c [ Z 2 / R ch ( Z , N ) − N 2 / R ch ( N , Z ) ] . The Coulomb coefficient a ˜ c of the relation is quasi-experimentally deduced to be 0.714(14) MeV/fm, with the aid of Bayesian neural networks which explicitly incorporate the experimental uncertainties and the model parametrization uncertainties. Using this method, we report the first determination of the R ch values for ten proton-rich nuclei 26 P, 27 S, 28 S, 41 Ti, 43 V, 45 Cr, 46 Cr, 47 Mn, 50 Fe and 53 Ni from their newly measured masses with the advanced B ρ -defined isochronous mass spectrometry at the experimental cooler storage ring (CSRe) in Lanzhou. In conjunction with the known R ch of their mirror partners, the Δ R ch values are deduced. Comparing the obtained Δ R ch values with the correlations between Δ R ch and L within a recently developed mean field code based on Skyrme energy density functionals (Sky3D), L is constrained to be 15 − 58 MeV at a 95% confidence level, via a least squares Monte Carlo analysis. Our result, well consistent with those from recent studies on mirror pairs, 36 Ca/ 36 S, 38 Ca/ 38 Ar, 54 Ni/ 54 Fe and 32 Ar/ 32 Si, strongly favors a soft symmetry energy.