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Prompted by the simple proportional relation between critical temperature for pairing transition and pairing gap at zero temperature, we investigate the relation between critical temperature for shape transition and ground-state deformation by taking even-even Cm isotopes as examples. The finite-temperature axially deformed covariant density functional theory with BCS pairing correlation is used. Since the Cm isotopes are the newly proposed nuclei with octupole correlations, we studied in detail the free energy surface, the Nilsson single-particle (s.p.) levels, and the components of s.p. levels near the Fermi level in Cm. Through this study, the formation of octupole equilibrium is understood by the contribution coming from the octupole driving pairs with Ω[N,nz,ml] and Ω[N +1, nz±3,ml] for single-particle levels near the Fermi surfaces as it provides a good manifestation of the octupole correlation. Furthermore, the systematics of deformations, pairing gaps and the specific heat as functions of temperature for even-even Cm isotopes are discussed. Similar to the relation between the critical pairing transition temperature and the pairing gap at zero temperature Tc = 0.6∆(0), a proportional relation between the critical shape transition temperature and the deformation at zero temperature Tc = 6.6β(0) is found for both octupole shape transition and quadrupole shape transition for isotopes considered. PACS numbers: 21.10.-k, 21.60.Jz, 27.90.+b

Critical temperature for shape transition in hot nuclei within covariant density functional theory