Sibasankar Panigrahy

Bragg

Powder X-ray diffraction of cubic crystals. Watch peaks broaden as crystallites shrink, shift with the lattice parameter, and recover size and strain from a Williamson–Hall plot.

About Bragg

X-ray diffraction is the quickest check of what a nanoparticle synthesis produced: which crystal phase, how large the crystallites are, and whether the lattice is strained. Bragg computes the full powder pattern from first principles, so you can see how each of these shows up in the data before you measure.

Python: pip install git+https://github.com/siba-p/nanomatkit, then nanomatkit xrd Au --size 8.

Method

Peak positions from Bragg's law for a cubic lattice,

λ=2dhklsin⁡θ,dhkl=ah2+k2+l2.\lambda = 2 d_{hkl} \sin\theta, \qquad d_{hkl} = \frac{a}{\sqrt{h^2+k^2+l^2}}.

Intensities from the structure factor with Cromer–Mann atomic form factors fj(s)f_j(s), s=sin⁡θ/λs = \sin\theta/\lambda:

Ihkl∝mhkl ∣Fhkl∣2 1+cos⁡22θsin⁡2θcos⁡θ e−2Bs2,Fhkl=∑jfj e2πi(hxj+kyj+lzj).I_{hkl} \propto m_{hkl}\, |F_{hkl}|^2\, \frac{1+\cos^2 2\theta}{\sin^2\theta\cos\theta}\, e^{-2Bs^2}, \qquad F_{hkl} = \sum_j f_j\, e^{2\pi i (h x_j + k y_j + l z_j)}.

Broadening: Scherrer size and Williamson–Hall strain terms, combined with the instrument width,

βcos⁡θ=KλD+4εsin⁡θ.\beta \cos\theta = \frac{K\lambda}{D} + 4\varepsilon \sin\theta.

Check: gold with Cu Kα gives (111) 38.19°, (200) 44.39°, (220) 64.58°, (311) 77.58° and relative intensities 100 : 49 : 31 : 35, against 100 : 52 : 32 : 36 in the reference pattern.

  1. B. D. Cullity & S. R. Stock, Elements of X-Ray Diffraction, 3rd ed. (2001).
  2. G. K. Williamson & W. H. Hall, Acta Metall. 1, 22 (1953).
  3. International Tables for Crystallography, Vol. C, Table 6.1.1.4 (form factors).