Effect of anharmonicity on the hcp to bcc transition in beryllium at high-pressure and high-temperature conditions

High Pressure Phase Diagram of Beryllium from \emph{Ab Initio} Free Energy Calculations

“…The thermodynamic integration technique captures the full anharmonicity of the crystal, making this study the first attempt to calculate free energies of beryllium without relying on the quasiharmonic approximations or its extensions. We compare our resulting solid-solid phase boundary with a recent study based on the phonon quasiparticle method [19] and with other works based on the quasiharmonic approach, demonstrating that the QHA tends to underestimate the stability of hcp phase, lowering the hcp-bcc transition pressure as well as the hcp-bcc-liquid triple point. We also derive the melting line for pressures up to 1600 GPa, where we found a melting temperature of 10000 K, as well as the shock Hugoniot curve, which is found to be in good agreement with shock wave experiments.…”

“…where ω q s corresponds to the phonon eigenfrequency with vector q of branch s in the Brillouin zone. We do not consider contribution from electron F e (V, T ) here since it is negligible compared with other two terms [8,17,19]. At T = 0 K, the second term of Eq.…”

“…Robert et al [16] predicted a solid phase boundary with a lower transition temperature using QHA, with the hcp-bcc-liquid triple point located at 85 GPa and 3400 K. Another work based on the QHA by Luo et al [17] gave similar result using the LDA functional. A recent work by Xian et al [19] used a method based on phonon quasiparticles to calculate free energies and found that this transition occurs at higher temperatures than previous QHA estimations made by Robert et al, promoting the triple point to 165 GPa and 4200 K. Benedict et al [8] calculated the free energy using QHA and a global EOS model and led to much higher transition pressures. Regarding experiments at high temperature, Laziki et al [27] used x-ray diffraction in a laser-heated diamond anvil cell to study hcp phase of beryllium up to 205 GPa and 4000 K, finding no evidence of a bcc phase.…”

“…Over the past decades, several theoretical and experimental studies have been performed in order to understand the phase diagram and EOS of beryllium. Theoretical studies suggest that at 0 K and high pressure, Be transforms from the hcp to the bcc structure [8,[11][12][13][14][15][16][17][18][19][20]. The linear-muffin-tin-orbital (LMTO) method by McMahan [11] as well as ab initio pseudopotential simulations by Lam et al [12] predicted this transition to occur between 100-200 GPa.…”

“…Calculations predict a negative Clapeyron slope along the phase boundary, meaning that the transition temperature decreases with increasing pressure [8,[17][18][19]. The quasiharmonic approximation (QHA), a standard method to calculate free energies at high temperature, has been used to study the vibrational properties of Be at high temperature, but this approach does not fully consider the anharmonic effects [17][18][19]. Under the QHA, the free energy of solids at high temperatures is obtained from phonon frequencies.…”