谢乐公式 Scherrer equation — 公式推导、符号与局限

\[\tau = \frac{K\lambda}{\beta\cos\theta}\]

威廉森–霍尔作图（分离尺寸与微应变）Williamson–Hall plot separating size and strain

\[\beta\cos\theta = \frac{K\lambda}{\tau} + 4\varepsilon\sin\theta\]

含义Meaning

由 X 射线衍射峰的展宽估算晶粒（相干散射畴）尺寸，常用于纳米材料（约 2–100 nm）。

Estimates crystallite (coherent domain) size from diffraction peak broadening — widely used for nanomaterials (~2–100 nm).

符号Symbol	含义 Meaning	单位 Unit
\(\tau\)	晶粒尺寸（垂直于晶面方向）crystallite size perpendicular to the planes	nm
\(K\)	形状因子 ≈ 0.9shape factor ≈ 0.9	1
\(\beta\)	扣除仪器展宽后的半高宽（弧度！）FWHM after instrument correction (radians!)	rad
\(\varepsilon\)	微应变microstrain	1

有限个晶面（N 层）衍射时，强度函数主峰宽度 ∝ 1/N，相当于狭缝衍射。
With only N planes, the interference peak width ∝ 1/N, like a finite grating.
把角宽度与 \(Nd = \tau\) 联系并用布拉格条件，得 \(\beta \approx \lambda/(\tau\cos\theta)\)，K 修正形状与宽度定义。
Relating angular width to τ = Nd via Bragg gives β ≈ λ/(τ cos θ).

给出的是平均相干畴尺寸，不等于颗粒尺寸（颗粒可含多个晶粒）；必须扣除仪器展宽与应变展宽。
Gives coherent-domain size, not particle size; subtract instrumental and strain broadening.
大于约 100–200 nm 时展宽太小，无法可靠测定。
Unreliable above ~100–200 nm.

Cu Kα，2θ = 38.2°（Au 111），β = 0.5° = 8.73×10⁻³ rad：τ = 0.9×0.15406/(8.73×10⁻³×cos19.1°) = 16.8 nm。

Au (111) at 2θ = 38.2°, β = 0.5°: τ ≈ 16.8 nm.