Bragg Diffraction: X-Ray Crystallography

Beam & Crystal

Wavelength λ 1.50 Å

Plane spacing d 2.00 Å

Incidence angle θ 30.0°

Lattice planes N 6

Bragg Condition

nλ = 2d·sinθ

Path diff. 2d·sinθ

2.00 Å

Path diff. / λ

1.33

Reflected intensity I(θ)0%

How it works

A crystal is a stack of parallel atomic planes separated by distance d. When an X-ray beam of wavelength λ strikes the planes at grazing angle θ, each plane reflects a small fraction of the wave. The ray reflecting off a deeper plane travels an extra path of 2d·sinθ compared to the plane above it (Bragg, 1913):

Bragg's law:   nλ = 2d·sinθ   (n = 1, 2, 3, …)

Reflections from all N planes add constructively — a bright reflected beam — only when this extra path is a whole number of wavelengths. Off that condition, the N reflected waves fall out of phase and mostly cancel. Summing N equally-spaced wave contributions with phase step δ = (2π/λ)·2d·sinθ gives the exact interference intensity used here, the same finite-grating formula that governs diffraction gratings and multi-slit interference:

I(θ) = [ sin(Nδ/2) / (N·sin(δ/2)) ]²

More planes (larger N) make the peaks sharper and taller relative to their width — exactly why larger, more ordered crystals give sharper diffraction spots in real X-ray crystallography, and why powder/ crystal quality matters for resolving a structure.

λ, d, θ sliders — set the wavelength, the real inter-planar spacing being probed, and the incidence angle; the intensity readout updates in real time from the formula above.
N slider — number of contributing lattice planes; watch the Bragg peaks sharpen as N grows.
Snap to nearest Bragg peak — jumps θ to the closest angle satisfying nλ = 2d·sinθ for the current λ, d.
Auto-scan — sweeps θ from 0° to 89° and traces I(θ), the same curve a real θ–2θ diffractometer scan produces, with peaks exactly where Bragg's law is satisfied.

This is the physical basis of X-ray crystallography: measuring the angles and intensities at which crystals diffract X-rays lets scientists reconstruct the atomic-scale spacing and arrangement inside the crystal — from table salt to the double helix of DNA.

Incident / reflected ray Lattice atoms Detector (glows with I(θ))

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