ADVANCED MATERIALS & PROCESSES | OCTOBER 2025 47 the magnitude of atomic displacement[17]. Thus, phonon calculations were performed using ABINIT with an energy cutoff of 60 Hartree and a 6×6×6 q-mesh, corresponding to a 432-atom supercell used in VASP. When the atomic displacement is between 0.06 and 0.12 Å, VASP predicts a similar imaginary frequency at (1/3, 1/3, 0) to that obtained by ABINIT, while for a displacement ≤ 0.05 Å or > 0.20 Å, such a vibrational instability is absent. Displacing atoms according to the eigenmode at (1/3, 1/3, 0) leads to the rhombohedral R phase, whose unit cell has 18 atoms and whose space group is identified to be P31m, instead of previously determined P3[3,4,33]. But these two rhombohedral structures are very similar, differing in energy only by 0.05 meV/atom with P3 slightly higher in energy. Figure 1c plots the transition path from Pm3m to P31m (without varying the lattice), which shows an energy drop of 1.79 meV/atom. The team notes that near Pm3m a small magnitude of in eigenmode slightly increases energy, and this explains why a small atomic displacement (≤ 0.05 Å) in VASP calculations leads to absence of imaginary phonon modes at (1/3, 1/3, 0). Full relaxation of the rhombohedral structure results in a slightly increased ΔE = 2.49 meV/atom (relative to B2) and that the lattice angle is slightly deviated from 90° (cubic) to 90° – α. The calculated α = 0.47° is in excellent agreement with the measured value of 0.45°, compared with α ≈1.0° in NiTi-Fe0.032 [13,34]. Figure 1d plots phonon distribution at 0 K of the R phase, which has no imaginary modes, indicating that the P31m structure is dynamically stable. Calculated lattice constants at 0 K are 3.187 Å for the B2 phase and 7.779 Å and 5.570 Å for the R phase, agreeing well with the corresponding experimental data of 3.149 Å[1,35], 7.658 Å, and 5.476 Å[35]. At 0 K, the volumes of these two phases differ only by 0.248% with R phase being slightly larger in volume. As temperature increases, the difference in volume is further reduced. Phonons at finite temperature. Next, the team ran the AIMD simulations at 50, 75, 100, and 125 K for 5 ps Fig. 1 — Phonon distributions at 0 K of cubic phase calculated using (a) VASP and (b) ABINIT; (c) energy change (ΔE) along transition path from cubic ( ) phase to rhombohedral ( ) phase; and (d) phonon distribution at 0 K of rhombohedral phase. Fig. 2 — Phonon distributions at 50 K (turquoise), 75 K (green), 100 K (black), and 125 K (red) of cubic phase. FEATURE (a) (b) (c) (d) 1 1
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