Evolution of nuclear charge radii in copper and indium isotopes

Systematic trends in nuclear charge radii are of great interest due to universal shell effects and odd-even staggering (OES). The modified root mean square (rms) charge radius formula, which phenomenologically accounts for the formation of neutron-proton ($np$) correlations, is here applied for the first time to the study of odd-$Z$ copper and indium isotopes. Theoretical results obtained by the relativistic mean field (RMF) model with NL3, PK1 and NL3$^{*}$ parameter sets are compared with experimental data. Our results show that both OES and the abrupt changes across $N=50$ and $82$ shell closures are clearly reproduced in nuclear charge radii. The inverted parabolic-like behaviors of rms charge radii can also be described remarkably well between two neutron magic numbers, namely $N=28$ to $50$ for copper isotopes and $N=50$ to $82$ for indium isotopes. This implies that the $np$-correlations play an indispensable role in quantitatively determining the fine structures of nuclear charge radii along odd-$Z$ isotopic chains. Also, our conclusions have almost no dependence on the effective forces.