布拉格定律 Bragg's law — 公式推导、符号与局限

\[n\lambda = 2d\sin\theta\]

立方晶系晶面间距interplanar spacing for cubic crystals

\[\frac{1}{d_{hkl}^2} = \frac{h^2 + k^2 + l^2}{a^2}\ (\text{立方}),\qquad d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}\]\[\frac{1}{d_{hkl}^2} = \frac{h^2 + k^2 + l^2}{a^2}\ (\text{cubic}),\qquad d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}\]

含义Meaning

X 射线被晶体中一组间距为 d 的晶面反射时，只有满足光程差为波长整数倍的角度才发生相长干涉（衍射峰）。

X-rays reflected from planes of spacing d interfere constructively only when the path difference is a whole number of wavelengths.

符号说明Symbols

符号Symbol	含义 Meaning	单位 Unit
\(\lambda\)	X 射线波长（Cu Kα 1.5406 Å）X-ray wavelength (Cu Kα 1.5406 Å)	Å
\(d\)	晶面间距interplanar spacing	Å
\(\theta\)	布拉格角（入射线与晶面夹角）Bragg angle (to the plane)	°
\(n\)	衍射级数order	1
\(hkl\)	米勒指数Miller indices	1

推导Derivation

两束分别被相邻晶面反射的 X 射线，下一束多走的路程为 \(2d\sin\theta\)。
A ray reflected by the next plane travels an extra \(2d\sin\theta\).
相长干涉要求 \(2d\sin\theta = n\lambda\)。
Constructive interference needs \(2d\sin\theta = n\lambda\).
粉末衍射谱 2θ 位置 → d → 由晶系公式指标化得到晶胞参数；消光规律（如 fcc 要求 hkl 全奇或全偶）判断点阵类型。
Powder peaks at 2θ → d → indexing gives cell parameters; systematic absences (fcc: hkl all odd or all even) identify the lattice.

适用条件与局限Assumptions & limitations

只给出衍射位置；强度需由结构因子 \(F_{hkl} = \sum f_je^{2\pi i(hx_j + ky_j + lz_j)}\) 计算。
Gives positions only; intensities need the structure factor.
要求 \(\lambda \le 2d\)；晶粒很小时峰变宽（谢乐公式）。
Requires \(\lambda \le 2d\); small crystallites broaden peaks (Scherrer).

例题Worked example

NaCl（a = 5.640 Å）(200) 面：d = 2.820 Å，Cu Kα：sinθ = 1.5406/5.640 → 2θ = 31.7°。

NaCl (200): d = 2.820 Å, Cu Kα → 2θ = 31.7°.