发表机构
Center for Basic Research on Materials, National Institute for Materials Science（物质材料研究创新中心，国立材料研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本报告介绍phonopy中通过自由能最小化计算六方晶体轴向热膨胀的方法，以钛的α和ω相为例，发现α-Ti的c轴低温负膨胀且最难精确计算。
AI 中文摘要
在六方晶体中，$a$轴和$c$轴随温度升高而变化的速率不同，热膨胀由两个系数描述，每个轴各一个。每个温度下的平衡晶格参数通过使亥姆霍兹自由能对晶格参数最小化来确定，而热膨胀系数则是这些参数对温度的对数导数。本报告描述了在phonopy中实现这一最小化过程的步骤。自由能使用谐波力常数在有限的一组$a$和$c$值下计算，并拟合出关于$a$和$c$的自由能曲面。在一种变体中，谐波力常数被来自自洽谐波近似的温度相关力常数所取代，其中力由在每个这些值处拟合的机器学习势计算。钛的$\alpha$相和$\omega$相作为工作示例。两种计算均给出$\alpha$-Ti的$c$轴在低温下的负热膨胀，而$\omega$-Ti则没有，且两者在$\alpha$-Ti的$c$轴上差异最大。轴向系数比体积系数更难计算，因为两个轴在自由能中是耦合的。在钛的两个相中，使$a$伸长的热效应同时使$c$缩短。因此，$c$轴的系数是两个符号相反、部分抵消的项之和，任一项的微小误差都会导致系数产生较大的相对误差。本报告衡量了该过程的每种设置如何改变热膨胀系数，其中$\alpha$-Ti的$c$轴是最敏感的情况。
英文摘要
In a hexagonal crystal the $a$ and $c$ axes change their lengths at different rates with increasing temperature, and the thermal expansion is described by two coefficients, one for each axis. The equilibrium lattice parameters at each temperature are found by minimizing the Helmholtz free energy over the lattice parameters, and the coefficients are their logarithmic derivatives with respect to temperature. This report describes a procedure for this minimization as implemented in phonopy. The free energy is computed with harmonic force constants at a finite set of values of $a$ and $c$, and a free-energy surface over $a$ and $c$ is fitted to these values. In a variant, the harmonic force constants are replaced by temperature-dependent force constants from a self-consistent harmonic approximation, with the forces computed by a machine-learning potential fitted at each of these values. The $α$ and $ω$ phases of titanium are the worked example. Both calculations give negative thermal expansion of the $c$ axis of $α$-Ti at low temperature and none in $ω$-Ti, and they differ most for the $c$ axis of $α$-Ti. The axial coefficients are harder to compute than the volumetric one, because the two axes are coupled in the free energy. In both phases of titanium the thermal effect that lengthens $a$ also shortens $c$. The coefficient of the $c$ axis is then the sum of two terms of opposite sign that partly cancel, and a small error in either term gives a large relative error in the coefficient. The report measures how much each setting of the procedure changes the thermal expansion coefficients, with the $c$ axis of $α$-Ti as the most sensitive case.
