The mixing thermodynamics of the antifluorite-structured Mg₂Si${}_{1-x}$Ge${}_x$ is investigated using the first-principles calculations. We find that Mg₂Si and Mg₂Ge readily mix with each other leading to formation of a single-phase random solid solutions of Mg₂Si${}_{1-x}$Ge${}_x$ across the entire composition range from the temperature of about 50 K and above. At 0 K, Mg₂Si${}_{1-x}$Ge${}_x$ exhibits a weak energy preference toward local phase segregation into Mg₂Si and Mg₂Ge without forming any ordered patterns of Si and Ge atoms. Through a comparison with the mixing thermodynamics of Mg₂Sn with Mg₂Si or Mg₂Ge, a small lattice misfit between Mg₂Si and Mg₂Ge of less than 1 % is responsible for the formation of stable Mg₂Si${}_{1-x}$Ge${}_x$ random solid solutions at such a low temperature. Besides their thermodynamic stability, our prediction reveals that the random solid solutions of Mg₂Si${}_{1-x}$Ge${}_x$ are dynamically and mechanically stable. These findings justify the uses of structural models of Mg₂Si${}_{1-x}$Ge${}_x$, assuming a random distribution of Si and Ge atoms in the previous theoretical studies, and also provide an insight into the complete solubility of Mg₂Ge in Mg₂Si and vice versa at all temperature where the atomic diffusion is activated.

抗萤石结构的Mg ₂ Si的混合热力学${}_{\mathrm{1个}-X}$通用电器${}_X$使用第一性原理计算进行研究。我们发现Mg ₂ Si和Mg ₂ Ge容易相互混合，导致形成Mg ₂ Si的单相随机固溶体${}_{\mathrm{1个}-X}$通用电器${}_X$整个组合物的温度范围约为50K以上。在0 K下，Mg ₂ Si${}_{\mathrm{1个}-X}$通用电器${}_X$对局部相偏析成Mg ₂ Si和Mg ₂ Ge表现出较弱的能量偏好，而没有形成任何有序的Si和Ge原子。通过与Mg ₂ Sn与Mg ₂ Si或Mg ₂ Ge的混合热力学进行比较，Mg ₂ Si和Mg ₂ Ge之间小于1％的小晶格失配是形成稳定Mg ₂ Si的原因。${}_{\mathrm{1个}-X}$通用电器${}_X$在如此低的温度下形成无规固溶体。除了其热力学稳定性外，我们的预测还表明Mg ₂ Si的随机固溶体${}_{\mathrm{1个}-X}$通用电器${}_X$具有动态和机械稳定性。这些发现证明了使用Mg ₂ Si结构模型的合理性${}_{\mathrm{1个}-X}$通用电器${}_X$，假设在先前的理论研究中Si和Ge原子是随机分布的，并且还可以洞悉在激活原子扩散的所有温度下Mg ₂ Ge在Mg ₂ Si中的完全溶解性，反之亦然。