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Odd-even staggering and shell effects of charge radii for nuclei with even $Z$ from $36$ to $38$ and from $52$ to $62$

A unified theoretical model reproducing charge radii of known atomic nuclei plays an essential role in making extrapolations for unknown nuclei. Recently developed new ansatz which phenomenologically takes into account the neutron-proton short-range correlations ($np$-SRCs) can describe the discontinuity properties and odd-even staggering (OES) effect of charge radii along isotopic chains remarkably well. In this work, we further review the modified root-mean-square (rms) charge radii formula in the framework of relativistic mean field (RMF) theory. The charge radii are calculated along various isotopic chains that include the nuclei featuring the $N=50$ and $82$ magic shells. Our results suggest that RMF with and without considering a correction term give an almost similar trend of nuclear size for some isotopic chains with open proton shell, especially the abrupt increases across the strong neutron closed shells and the OES behaviors. This reflects that the $np$-SRCs have almost no influence for some nuclei due to the strong coupling between different levels around Fermi surface. The weakening OES behavior of nuclear charge radii is observed generally at completely filled neutron shells and this may be proposed as a signature of magic indicator.