2D companion to the 3D scene: the same finite-size diffraction physics, drawn as a flat stack of N reflecting planes instead of a rendered 3D lattice. A finite stack of N crystal planes spaced d apart scatters X-rays like an N-slit grating. The kinematic interference (Laue) function for the reflected intensity is:
I(θ) ∝ [ sin(Nδ/2) / sin(δ/2) ]²
δ(θ) = (4π d / λ) · sin θ
δ = 2π at the Bragg angle θ₀, where λ = 2d·sinθ₀ (Bragg's law, first order). As N grows, the peak sharpens toward a delta function; for a small N (a nanocrystal) it stays measurably broad — the physical basis of finite-size line broadening.
A real diffractometer adds its own Gaussian instrumental broadening βinst on top. This sim convolves the theoretical peak with that instrumental width, then removes it again the way a standards lab does — via Warren's quadratic correction:
β_corrected = √(β_observed² − β_inst²)
D_Scherrer = K·λ / (β_corrected · cos θ₀) (β in radians)
K ≈ 0.9 is the fixed Scherrer shape constant used here (spherical crystallites, cubic lattice). Comparing D_Scherrer against the true D = N·d shows exactly why ISO/IEC and OECD nanomaterial guidance require reference-material calibration and interlaboratory comparison before trusting a single XRD linewidth as a size number — the correction is only as good as the instrumental-broadening measurement feeding it.
- Crystallite size N — number of reflecting planes stacked along the diffracting direction; sets the true size D = N·d and directly narrows/broadens the peak.
- Lattice spacing d — interplanar spacing of the reflecting family of planes; also shifts the Bragg angle θ₀.
- Wavelength λ — the X-ray source line (Cu Kα ≈ 0.154 nm by default); changes θ₀ via Bragg's law.
- Instrumental broadening βinst — the diffractometer's own peak width; the sim subtracts it in quadrature before estimating D, exactly as a calibrated measurement should.