Mie-Grüneisen Equation of State Based on the Physical Mechanics Analysis of Three-Dimensional Lattice Thermal Vibration
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摘要: 根据晶体结构建立了三维定容热振动的微观物理力学模型，由力学方程推导出了点阵热振动能量与外力的微观热振动的力学关系。在微观热振动关系中引入宏观统计物理量，直接从物理力学模型得到了Mie-Grüneisen固体状态方程和Grüneisen系数的表达式。根据晶体结构中的原子排列规律，对简单立方、面心立方、体心立方、金刚石立方晶体和理想密排六方晶体的Grüneisen系数的表达式进行了证明。结果表明，这些对称性晶体结构的Grüneisen系数与冷压具有统一的微分关系，与晶体结构无关。
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关键词:
- 物理力学 /
- Mie-Grüneisen状态方程 /
- 原子热振动 /
- 点阵动力学
Abstract: According to crystal lattice structures, a three-dimensional lattice physical mechanics model in constant volume state was built up to study the atom thermal vibration.The equation of thermal energy and external force on the thermal vibrating lattice was deduced absolutely from the principle of mechanical vibration.By introducing macroscopic physical statics into microscopic atomic thermal vibration equation, Mie-Grüneisen equation of state for solids and the formula of Grüneisen parameter were deduced directly from the physical mechanical model.Finally, based on the arrangement of atoms in simple cubic, face-centered cubic, body-centered cubic, diamond cubic and close-packed hexagonal crystal, it was proved that the Grüneisen parameters of these symmetrical crystals can be expressed in a uniform formula, irrelevant to their actual arrangement of atoms.-
Key words:
- physical mechanics /
- EOS of Mie-Grüneisen /
- atom thermal motion /
- lattice dynamics
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图 2 原子在三维空间热振动物理力学模型
Figure 2. Thermal-vibration physical mechanical model of an atom in space
表 1 部分晶体结构参数
Table 1. Structural parameters of crystals
Crystal structure Number of atom
in the unit cellNumber of nearest
neighbor atom naCrystal lattice
constantsVolume of
an atom μr3Bearing area
of a bond ηr2$\frac{6 \mu}{n_{\mathrm{a}} \eta}$ Simple cubic 1 6 a=r μ=1 η=1 1 Body-centered cubic (bcc) 2 8 $a=2 r / \sqrt{3}$ $\mu=4 \sqrt{3} / 9$ $\eta=\sqrt{3} / 3$ 1 Face-centered cubic (fcc) 4 12 $a=\sqrt{2} r$ $\mu=\sqrt{2} / 2$ $\eta=\sqrt{2} / 4$ 1 Diamond cubic 8 4 $a=4 r / \sqrt{3}$ $\mu=8 \sqrt{3} / 9$ $\eta=4 \sqrt{3} / 3$ 1 Close-packed hexagonal (hcp) 2 12 $a=r, c=2 \sqrt{2} r / \sqrt{3}$ $\mu=\sqrt{2} / 2$ $\eta=\sqrt{2} / 4$ 1 
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