This study explores the phenomenon of shape coexistence in nuclei around 172\documentclass[12pt]{minimal}
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\begin{document}$$^{172}$$\end{document}Hg, with a focus on the isotopes 170\documentclass[12pt]{minimal}
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\begin{document}$$^{170}$$\end{document}Pt, 172\documentclass[12pt]{minimal}
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\begin{document}$$^{172}$$\end{document}Hg, and 174\documentclass[12pt]{minimal}
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\begin{document}$$^{174}$$\end{document}Pb, as well as the 170\documentclass[12pt]{minimal}
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\begin{document}$$^{170}$$\end{document}Pt to 180\documentclass[12pt]{minimal}
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\begin{document}$$^{180}$$\end{document}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\documentclass[12pt]{minimal}
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\begin{document}$$\hbox {For}\,$$\end{document} 170Pt, the PES exhibited a prolate ground state, with additional triaxial and oblate-shaped isomers. In 172\documentclass[12pt]{minimal}
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\begin{document}$$^{172}$$\end{document}Hg, the ground-state deformation transitions from triaxial to oblate with increasing pairing interaction, demonstrating its nearly γ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma$$\end{document}-unstable nature. Three shape isomers (prolate, triaxial, and oblate) were observed, with increased pairing strength leading to the disappearance of the triaxial isomer. 174\documentclass[12pt]{minimal}
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\begin{document}$$^{174}$$\end{document}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\documentclass[12pt]{minimal}
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\begin{document}$$^{170}$$\end{document}Pt to a coexistence of γ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma$$\end{document}-unstable and oblate shapes in 172\documentclass[12pt]{minimal}
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\begin{document}$$^{172}$$\end{document}Hg, ultimately approaching spherical symmetry in 174\documentclass[12pt]{minimal}
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\begin{document}$$^{174}$$\end{document}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\documentclass[12pt]{minimal}
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\begin{document}$$^{170-180}$$\end{document}Pt isotopes revealed significant shape evolution. 170\documentclass[12pt]{minimal}
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\begin{document}$$^{170}$$\end{document}Pt showed a prolate ground state, whereas 172\documentclass[12pt]{minimal}
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\begin{document}$$^{172}$$\end{document}Pt exhibited both triaxial and prolate shape coexistence. In 174\documentclass[12pt]{minimal}
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\begin{document}$$^{174}$$\end{document}Pt, the ground state was triaxial, coexisted with a prolate minimum. For 176\documentclass[12pt]{minimal}
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\begin{document}$$^{176}$$\end{document}Pt, a γ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma$$\end{document}-unstable ground state coexists with a prolate minimum. By 178\documentclass[12pt]{minimal}
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\begin{document}$$^{178}$$\end{document}Pt and 180\documentclass[12pt]{minimal}
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\begin{document}$$^{180}$$\end{document}Pt, a dominant prolate minimum emerged. These results highlight the role of shape coexistence and γ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma$$\end{document}-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.