Temperature driven $alpha$ to $beta$ phase-transformation in Ti, Zr and Hf from first principles theory combined with lattice dynamics

Lattice dynamical methods used to predict phase transformations in crystals typically deal with harmonic phonon spectra and are therefore not applicable in important situations where one of the competing crystal structures is unstable in the harmonic approximation, such as the bcc structure involved in the hcp to bcc martensitic phase transformation in Ti, Zr and Hf. Here we present an expression for the free energy that does not suffer from such shortcomings, and we show by self consistent {\it ab initio} lattice dynamical calculations (SCAILD), that the critical temperature for the hcp to bcc phase transformation in Ti, Zr and Hf, can be effectively calculated from the free energy difference between the two phases. This opens up the possibility to study quantitatively, from first principles theory, temperature induced phase transitions.

💡 Research Summary

The paper addresses a long‑standing challenge in first‑principles modeling of temperature‑driven structural phase transitions: the inability of conventional harmonic phonon calculations to treat phases that are dynamically unstable in the harmonic approximation, such as the body‑centered cubic (bcc) structure that competes with the hexagonal close‑packed (hcp) phase in Ti, Zr and Hf. To overcome this limitation the authors employ the Self‑Consistent Ab‑Initio Lattice Dynamics (SCAILD) method, which simultaneously excites all commensurate phonon modes in a supercell, computes the Hellmann‑Feynman forces from density‑functional theory (DFT), and iteratively updates phonon frequencies until self‑consistency is achieved. Because all phonons are present at once, phonon‑phonon interactions are automatically incorporated, leading to temperature‑dependent renormalized phonon spectra that capture anharmonic effects to infinite order.

A central contribution of the work is a new free‑energy expression that can be evaluated within the SCAILD framework. The total free energy F(T,V) is split into three parts: the static ground‑state energy U₀(V), the phonon contribution F_ph(T,V), and the electronic contribution F_el(T,V). The temperature‑dependent part, F_ph+F_el, is obtained by averaging over a large ensemble (≈400) of atomic configurations generated during the SCAILD run. For each configuration the electronic free energy is calculated using a finite‑temperature Fermi‑Dirac smearing of the Kohn‑Sham occupations, and the phonon potential energy is taken directly from the DFT total energy of the displaced structure. The phonon entropy is evaluated using the standard Bose‑Einstein expression S_ph = k_B Σ_qs