\[\varphi = \frac{Z\cdot\tfrac43\pi r^3}{V_\text{晶胞}}\]\[\varphi = \frac{Z\cdot\tfrac43\pi r^3}{V_\text{cell}}\]
常见结构common structures
\[\text{sc: } \frac{\pi}{6} = 52.4\%\qquad \text{bcc: } \frac{\sqrt3\pi}{8} = 68.0\%\qquad \text{fcc/hcp: } \frac{\pi}{3\sqrt2} = 74.0\%\qquad \text{金刚石: } \frac{\sqrt3\pi}{16} = 34.0\%\]\[\text{sc: } \frac{\pi}{6} = 52.4\%\qquad \text{bcc: } \frac{\sqrt3\pi}{8} = 68.0\%\qquad \text{fcc/hcp: } \frac{\pi}{3\sqrt2} = 74.0\%\qquad \text{diamond :} \frac{\sqrt3\pi}{16} = 34.0\%\]含义Meaning
晶胞中原子所占体积分数。等径球最密堆积（fcc、hcp）为 74.05%（开普勒猜想，2014 年完成形式化证明）。
Fraction of the unit-cell volume occupied by atoms; equal spheres pack at most 74.05 % (fcc/hcp — the Kepler conjecture, formally verified 2014).
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(Z\) | 晶胞中原子数atoms per unit cell | 1 |
| \(r\) | 原子半径atomic radius | m |
| \(a\) | 晶胞参数lattice parameter | m |
推导Derivation
简单立方：Z = 1，a = 2r → \(\varphi = \tfrac43\pi r^3/(2r)^3 = \pi/6\)。
Simple cubic: Z = 1, a = 2r → π/6.
体心立方：Z = 2，体对角线 \(\sqrt3 a = 4r\) → \(\varphi = 2\cdot\tfrac43\pi r^3/(4r/\sqrt3)^3 = \sqrt3\pi/8\)。
bcc: Z = 2, \(\sqrt3 a = 4r\) → √3π/8.
面心立方：Z = 4，面对角线 \(\sqrt2 a = 4r\) → \(\pi/(3\sqrt2)\)。
fcc: Z = 4, \(\sqrt2 a = 4r\) → π/(3√2).
适用条件与局限Assumptions & limitations
假设原子为等径刚球；离子晶体中阴阳离子半径不同，需分别计算。
Equal hard spheres; ionic crystals need separate radii.