Pressure-induced structural phase transition of vanadium: A revisit from the perspective of ensemble theory

Abstract

For realistic crystals, the free energy strictly formulated in ensemble theory can hardly be obtained because of the difficulty in solving the high-dimension integral of the partition function, the dilemma of which makes it even a doubt if the rigorous ensemble theory is applicable to phase transitions of condensed matters. In the present work, the partition function of crystal vanadium under compression up to 320 GPa at room temperature is solved by an approach developed very recently, and the derived equation of state is in a good agreement with all the experimental measurements, especially the latest one covering the widest pressure range up to 300 GPa. Furthermore, the derived Gibbs free energy proves the very argument to understand most of the experiments reported in the past decade on the pressure-induced phase transition, and, especially, a novel phase transition sequence concerning three different phases observed very recently and the measured angles of two phases agree with our theoretical results excellently.

Thermal contributions to the structural phase transitions of crystalline aluminum under ultrahigh pressures

“…As formulated in statistical mechanics, the FE can be readily obtained without empirical parameters as long as the partition function (PF) is solved while unfortunately the exact solution to the PF of condensed matters is a long-standing problem because of the complexity of the high dimensional configurational integral [13,40,41], so that state-of-the-art numerical algorithm for PF can hardly afford a system consisting of more than several hundred particles even using empirical interatomic force field and the determined transition pressures for Al were far from the experimental results [42]. Very recently, we put forward a direct integral approach (DIA) to the PF of condensed state systems with ultrahigh efficiency and precision [43][44][45][46], and has been successfully applied to investigate the phase transitions of vanadium [47], the EOS of copper [43] and the optimum growth condition for 2-D materials [44] combined with density functional theory (DFT). Compared with quasi-harmonic phonon model, DIA was examined to be applicable to much wider realm with much higher precision [46].…”

“…( 5) and ( 6) depends on the geometric symmetry of the investigated structures. For the FCC and BCC structures, as the same procedures applied to copper [43] and vanadium [47] in our previous works, Eq. ( 5) is used to calculate the configurational integrals because the effective lengths along the three axes, L x,y,z , of any atom in a conventional cubic supercell of FCC or BCC are of a same value, while, in terms of the HCP structure, it is obvious that the values of L x,y,z are not equivalent to each other due to the asymmetric geometry along the three axes, and thus, Eq.…”