On shape coexistence and possible shape isomers of nuclei around 172Hg

his study explores the phenomenon of shape coexistence in nuclei around $^{172}$Hg, with a focus on the isotopes $^{170}$Pt, $^{172}$Hg, and $^{174}$Pb, as well as the $^{170}$Pt to $^{180}$Pt isotopic chain. Utilizing a macro-microscopic approach that incorporates the Lublin-Strasbourg Drop model combined with a Yukawa-Folded potential and pairing corrections, we analyze the potential energy surfaces (PESs) to understand the impact of pairing interaction.For$^{170}$Pt, the PES exhibited a prolate ground-state, with additional triaxial and oblate-shaped isomers. In $^{172}$Hg, the ground-state deformation transitions from triaxial to oblate with increasing pairing interaction, demonstrating its nearly $\gamma$-unstable nature. Three shape isomers (prolate, triaxial, and oblate) were observed, with increased pairing strength leading to the disappearance of the triaxial isomer.$^{174}$Pb exhibited a prolate ground-state that became increasingly spherical with stronger pairing. While shape isomers were present at lower pairing strengths, robust shape coexistence was not observed. For realistic pairing interaction, the ground-state shapes transitioned from prolate in $^{170}$Pt to a coexistence of $\gamma$-unstable and oblate shapes in $^{172}$Hg, ultimately approaching spherical symmetry in $^{174}$Pb. A comparison between Exact and Bardeen-Cooper-Schrieffer (BCS) pairing demonstrated that BCS pairing tends to smooth out shape coexistence and reduce the depth of the shape isomer, leading to less pronounced deformation features.The PESs for even-even $^{170-180}$Pt isotopes revealed significant shape evolution. $^{170}$Pt showed a prolate ground-state, whereas$^{172}$Pt exhibited both triaxial and prolate shape coexistence. In $^{174}$Pt, the ground-state was triaxial, coexisted with a prolate minimum. For $^{176}$Pt, a $\gamma$-unstable ground-state coexists with a prolate minimum. By $^{178}$Pt and $^{180}$Pt, a dominant prolate minimum emerged. These results highlight the role of shape coexistence and $\gamma$-instability in the evolution of nuclear structure, especially in the mid-shell region.These findings highlight the importance of pairing interactions in nuclear deformation and shape coexistence, providing insights into the structural evolution of mid-shell nuclei.