XRD-Analysis · Scherrer equation with Caglioti correction · Cu KαHow well does a peak width tell the crystallite size?

Choose a true crystallite size, microstrain and instrument, and the five Cu peaks are drawn with the widths they would have; then the peak widths are analysed the way xrd_analysis does it, and the recovered size is compared with the true one. Everything is computed in the browser.

Repository Scherrer code

Simulated Cu pattern

One peak, close up

Peak heights scaled to 1; choose the peak below

Forward model: from the true sample to the observed peak

Peak positions from Bragg's law for fcc Cu, a = 3.6150 Å, Cu Kα₁ = 1.540562 Å and Kα₂ = 1.544390 Å (Bearden 1967), Kα₂/Kα₁ = 0.5; relative intensities of JCPDS 04-0836 as in the package.

size (Lorentzian): f_L = K_true λ / (D cos θ) · strain (Gaussian): f_G,ε = 4 ε tan θ · instrument (Gaussian): f_inst from Caglioti

f_G = √(f_G,ε² + f_inst²) → observed FWHM and η from the Thompson–Cox–Hastings formulas

The observed width and η are what a perfect, noise-free fit of the Kα₁ component would return; real fits add their own error.

Analysis: what xrd_analysis does with the observed width

quadratic: β = √(f_obs² − f_inst²) · Voigt: split f_obs into G and L parts with the inverse TCH mapping, subtract the instrument (G quadratically, L linearly), recombine with f = 0.5346 f_L + √(0.2166 f_L² + f_G²)

D = K λ / (β cos θ), λ = Kα₁; flags: f_obs/f_inst < 1.2 unreliable, D outside 2–200 nm warning

The Scherrer equation attributes all sample broadening to size, so microstrain makes it underestimate the size, more at high angle. The quadratic correction treats both widths as Gaussian; the size broadening is Lorentzian, so it under-corrects. With the Voigt correction and pure size broadening the size is recovered exactly; when microstrain adds a Gaussian part, splitting (TCH) and recombining (Olivero–Longbothum) use slightly different width formulas.

Checks

The instrument parameters are illustrative values for a laboratory diffractometer, not a calibration: the repository's config.yaml leaves U, V and W to be calibrated with a LaB₆ standard (NIST SRM 660c).

Sources

J. A. Bearden, Rev. Mod. Phys. 39, 78 (1967): wavelengths. J. I. Langford and A. J. C. Wilson, J. Appl. Cryst. 11, 102 (1978): Scherrer constants. P. Thompson, D. E. Cox and J. B. Hastings, J. Appl. Cryst. 20, 79 (1987): pseudo-Voigt width and η. G. Caglioti, A. Paoletti and F. P. Ricci, Nucl. Instrum. 3, 223 (1958): instrument width. J. J. Olivero and R. L. Longbothum, J. Quant. Spectrosc. Radiat. Transfer 17, 233 (1977): Voigt width.

Code: github.com/Shuruyue/XRD-Analysis. The physics block of this page is checked against the Python package by scripts/verify_web_page.py. Style shared with the ethanol–water page.