\[\frac{1}{d^2} = \frac{h^2 + k^2 + l^2}{a^2}\ (\text{立方}),\quad \frac{h^2 + k^2}{a^2} + \frac{l^2}{c^2}\ (\text{四方}),\quad \frac{h^2}{a^2} + \frac{k^2}{b^2} + \frac{l^2}{c^2}\ (\text{正交})\]\[\frac{1}{d^2} = \frac{h^2 + k^2 + l^2}{a^2}\ (\text{cubic}),\quad \frac{h^2 + k^2}{a^2} + \frac{l^2}{c^2}\ (\text{tetragonal}),\quad \frac{h^2}{a^2} + \frac{k^2}{b^2} + \frac{l^2}{c^2}\ (\text{orthorhombic})\]
六方晶系hexagonal
\[\frac{1}{d^2} = \frac43\cdot\frac{h^2 + hk + k^2}{a^2} + \frac{l^2}{c^2}\]含义Meaning
由米勒指数和晶胞参数计算一组平行晶面的间距，与布拉格定律联用即可指标化衍射图谱。
The spacing of a family of planes from Miller indices and cell parameters — combined with Bragg’s law to index diffraction patterns.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(h, k, l\) | 米勒指数Miller indices | 1 |
| \(a, b, c\) | 晶胞参数lattice parameters | Å |
推导Derivation
(hkl) 面在三轴上截距为 a/h、b/k、c/l。
The (hkl) plane cuts the axes at a/h, b/k, c/l.
原点到该平面的距离即 d。正交轴下平面方程 \(hx/a + ky/b + lz/c = 1\)，点到平面距离公式给出 \(1/d^2 = \sum(h/a)^2\)。
For orthogonal axes the plane is \(hx/a + ky/b + lz/c = 1\); the distance from the origin gives the result.
立方晶系中 \(h^2+k^2+l^2\) 只能取某些整数（7、15 等不可能），配合消光规则判断 P、I、F 点阵。
In cubic crystals h²+k²+l² skips 7, 15, …; with absences this identifies P, I, F lattices.
适用条件与局限Assumptions & limitations
单斜、三斜晶系公式更复杂（含夹角）；一般用倒易点阵度规张量计算。
Monoclinic and triclinic need angle terms (reciprocal metric tensor).