Bragg Diffraction, Crystal Cross-Section

What is modelled

N identical parallel planes are spaced d apart. A ray of wavelength λ hits at glancing angle θ and each plane reflects a little of it. Waves from neighbouring planes differ in path by 2 d sinθ; they add up in phase only if that equals a whole number of wavelengths.

Equations

• Bragg law n λ = 2 d sinθ.
• Amplitude A = (1/N) Σj ei q zj, q = 4π sinθ / λ, intensity I = |A|2 (normalised so a perfect Bragg peak is 1).
• Perfect stack: I = sin2(Nφ) / (N2 sin2φ), φ = 2π d sinθ / λ, with N - 2 weak ripples between peaks.
• Peak width Δθ ≈ 2.783 λ / (2π N d cosθ).
• Disorder σ (random plane displacements) reduces the mean peak height to 1/N + (1 - 1/N) e-q2σ2 (Debye-Waller).

Checks run in the page

• Full intensity exactly at the Bragg angles of the first orders.
• The scanned peak width against the analytic width.
• Number of ripples between peaks equals N - 2.
• Mean peak height of many random disordered stacks against the Debye-Waller formula.

Honest simplifications

Kinematic (single-scattering) theory only: no absorption, polarisation, atomic basis or multiple scattering, and each plane is a point-like reflector. Real X-ray diffraction adds structure factors and dynamical effects.

• Drag the angle slowly through the first peak: the phasor chain straightens exactly at the Bragg angle.
• Cut the number of planes to 6: the peaks broaden.
• Raise the disorder: the peaks shrink and the background rises.