pgm: A Python package for free energy calculations within the phonon gas model

The quasi-harmonic approximation (QHA) is a powerful method that uses the volume dependence of non-interacting phonons to compute the free energy of materials at high pressures (P) and temperatures (T). However, anharmonicity, electronic excitations in metals, or both, introduce an intrinsic T-dependence on phonon frequencies, rendering the QHA inadequate. Here we present a Python code, pgm, to compute the free energy and thermodynamic property within the phonon gas model (PGM) that uses T-dependent phonon quasiparticle frequencies. In this case, the vibrational contribution to the Helmholtz free energy is obtained by integrating the vibrational entropy, which can be readily calculated for a system of phonon quasiparticles. Other thermodynamic properties are then obtained from standard thermodynamic relations. We demonstrate the successful applications of pgm to two cases of geophysical significance: cubic CaSiO3-perovskite (cCaPv), a strongly anharmonic insulator and the third most abundant phase of the Earth's lower mantle, and NiAs-type (B8) FeO, a partially covalent-metallic system. This is the oxide endmember of a recently discovered iron-rich FenO alloy phase likely to exist in the Earth's inner core.准谐近似（quasi-harmonic approximation, QHA）是一种极具应用价值的方法，其通过非相互作用声子（non-interacting phonons）的体积依赖性，计算材料在高压（P）与高温（T）下的自由能（free energy）。然而，非简谐性（anharmonicity）、金属中的电子激发（electronic excitations），或二者兼具，会使声子频率出现固有的温度依赖性，导致准谐近似不再适用。本文提出一款名为pgm的Python代码，可基于含温度依赖的声子准粒子频率（T-dependent phonon quasiparticle frequencies），在声子气体模型（phonon gas model, PGM）框架内计算自由能与热力学性质（thermodynamic property）。在此框架下，亥姆霍兹自由能（Helmholtz free energy）的振动贡献项可通过积分振动熵（vibrational entropy）求得，而振动熵可便捷地针对声子准粒子体系进行计算。其余热力学性质均可通过标准热力学关系（standard thermodynamic relations）推导得到。我们将pgm成功应用于两类具有重要地球物理意义的体系：一是作为强非简谐绝缘体的立方相CaSiO3钙钛矿（cubic CaSiO3-perovskite, cCaPv）——它是地球下地幔（Earth's lower mantle）中丰度排名第三的物相；二是NiAs型（NiAs-type, B8）FeO，这是一种部分共价-金属混合体系。后者是近期发现的富铁FenO合金相的氧化物端元组分，该合金相大概率存在于地球内核（Earth's inner core）之中。