Elasticity behavior, phonon spectra, and the pressure-temperature phase   diagram of HfTi alloy: A density-functional theory study
  The pressure-induced phase transition, elasticity behavior, thermodynamic properties, and $P\mathtt{-}T$ phase diagram of $\alpha$, $\omega$, and $\beta$ equiatomic HfTi alloy are investigated using first-principles density-functional theory (DFT). The simulated pressure-induced phase transition of the alloy follows the sequence of $\alpha\mathtt{\rightarrow}\omega\mathtt{\rightarrow}\beta$, in agreement with the experimental results of Hf and Ti metals. Our calculated elastic constants show that the $\alpha$ and $\omega$ phases are mechanically stable at ambient pressure, while the $\beta$ phase is unstable, where a critical pressure of 18.5 GPa is predicted for its mechanical stability. All the elastic constants, bulk modulus, and shear modulus increase upon compression for the three phases of HfTi. The ductility of the alloy is shown to be well improved with respect to pure Hf and Ti metals. The Mulliken charge population analysis illustrates that the increase of the d-band occupancy will stabilize the $\beta$ phase under pressure. The phonon spectra and phonon density of states are studied using the supercell approach for the three phases, and the stable nature of $\alpha$ and $\omega$ phases at ambient pressure are observed, while the $\beta$ phase is only stable along the [110] direction. With the Gibbs free energy calculated from DFT-parametrized Debye model as a function of temperature and pressure, the phase transformation boundaries of the $\alpha$, $\omega$, and $\beta$ phases of HfTi are identified. 
Yong Lu, Ping Zhang
cond-mat.mtrl-sci