xrd-peak-broadening.md

X-Ray Diffraction Peak Broadening Equations

1. Instrumental Broadening (βinst)

• Generally approximated using a standard reference material (like LaB₆)
• Instrumental Resolution Function (IRF):

βinst = U tan²θ + V tanθ + W

where U, V, and W are refinable parameters determined from standard measurements

2. Crystallite Size Broadening (βL)

• Scherrer Equation:

βL = Kλ/(L cosθ)

where:

• βL is the peak width due to size effects (in radians)
• K is the Scherrer constant (typically 0.9-1.0)
• λ is the X-ray wavelength
• L is the volume-weighted crystallite size
• θ is the Bragg angle

3. Strain Broadening (βε)

• Uniform Strain:

βε = 4ε tanθ

where ε is the strain parameter

• Wilson Formula for microstrain:

βε = 4(⟨ε²⟩)½ tanθ

where ⟨ε²⟩ is the mean square strain

4. Peak Shape Functions

Common profile functions used to model peak shapes:

Gaussian Profile:

G(x) = (1/(σ√(2π))) exp(-(x-x₀)²/(2σ²))
FWHM = 2√(2ln2)σ ≈ 2.355σ

Lorentzian Profile:

L(x) = (1/π) * (γ/2)/((x-x₀)² + (γ/2)²)
FWHM = γ

Pseudo-Voigt Profile (combination of Gaussian and Lorentzian):

pV(x) = ηL(x) + (1-η)G(x)

where η is the mixing parameter (0 ≤ η ≤ 1)

5. Total Peak Broadening

For Gaussian components:

β²total = β²inst + β²size + β²strain

For Lorentzian components:

βtotal = βinst + βsize + βstrain

Williamson-Hall Plot Equation:

βtotal cosθ = Kλ/L + 4ε sinθ

This equation allows separation of size and strain effects by plotting βcosθ vs 4sinθ:

• Slope = strain (ε)
• Y-intercept = Kλ/L (related to crystallite size)

6. Integral Breadth Methods

For more accurate analysis using integral breadth (β):

Warren-Averbach Method:

ln A(L) = ln As(L) + ln Ad(L)

where:

• A(L) is the Fourier coefficient
• As(L) is the size coefficient
• Ad(L) is the distortion coefficient

Double-Voigt Method:

βL = 1/⟨D⟩v
βG = 4ε tanθ

where:

• ⟨D⟩v is the volume-weighted crystallite size
• ε is the upper limit of strain distribution