The abnormal behavior of nuclear charge radii can be generally associated with the observed structure phenomena along the specific isotopic family. In this work, the relativistic mean-field theory with the effective Lagrangian NL3 force is employed to feature the systematic trend of changes of charge radii near $N=90$ regions. The pairing correlations are tackled by solving the state-dependent Bardeen-Cooper-Schrieffer equations with a density-dependent zero-range force. The neutron- and proton-pairs correlation around the Fermi surface has been incorporated into the charge radii formula. The potential energy surfaces of even-even rare-earth isotopes with the proton numbers $Z=60$, 62, 64, 66, and 68 are depicted by constraining the axially symmetric quadrupole and hexadecapole moments simultaneously. The calculated results suggest that the strong quadrupole-hexadecapole correlations can be significantly observed from the tilted potential energy surfaces of the corresponding nuclei. Especially, this phenomenon can be obviously predicted near the shape coexistence $N=90$ regions. Meanwhile, the quadrupole-hexadecapole coupling effect plays an indispensable role in reproducing the evolutionary trend of nuclear charge radii with high precision. The calculated results suggest that the modified charge radii formula can improve the local variations of charge radii, especially the odd-even staggering phenomena. Furthermore, this odd-even oscillation trend in charge radii can also be influenced by the pairing force type. This means that charge radii provide a robust indicator to identify the structure phenomena.

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• Nuclear forces
• Nuclear radii
• Nuclear structure & decays
• 150 ≤ A ≤ 189
• 90 ≤ A ≤ 149