X-Ray Diffraction Simulator: Bragg's Law & Crystal Structure Analysis

simulator intermediate ~10 min
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2θ = 29.7° — first-order Bragg peak for d = 3 Å, Cu Kα

X-rays of wavelength 1.54 Å (Cu Kα) striking crystal planes with 3 Å spacing produce constructive interference at 2θ = 29.7° — a sharp Bragg peak identifying the lattice parameter.

Formula

nλ = 2d sin θ (Bragg's law)
L = 0.9λ / (β cos θ) (Scherrer equation for crystallite size)
d_hkl = a / √(h² + k² + l²) (cubic lattice d-spacing)

Crystals as Diffraction Gratings

When Max von Laue proposed in 1912 that crystals could diffract X-rays, he revealed two things simultaneously: X-rays are electromagnetic waves with wavelengths comparable to atomic spacings, and crystals are periodic arrangements of atoms. The experiment, performed by Friedrich and Knipping, launched the era of crystallography and earned Laue the 1914 Nobel Prize.

Bragg's Elegant Law

William and Lawrence Bragg simplified Laue's formalism by treating diffraction as reflection from parallel crystal planes. Constructive interference occurs when the extra path length (2d sin θ) equals a whole number of wavelengths. This simple condition — nλ = 2d sin θ — connects the observable angle to the invisible lattice spacing, enabling direct measurement of atomic-scale distances from macroscopic diffraction patterns.

From Peaks to Structure

A powder diffraction pattern — intensity versus 2θ — encodes the crystal's symmetry and lattice parameters. Peak positions reveal d-spacings (via Bragg's law), peak intensities encode atomic positions and types, and peak widths indicate crystallite size and strain. The simulation generates diffraction patterns as you adjust lattice spacing and X-ray wavelength, showing how Bragg peaks shift and split.

Impact on Science and Technology

XRD has determined the structures of over a million crystals. Watson and Crick's DNA double helix relied on Rosalind Franklin's X-ray diffraction photographs. Semiconductor fabrication uses XRD for quality control. Pharmaceutical companies verify drug polymorphs — different crystal structures of the same molecule that can have dramatically different bioavailability. From geology to nanotechnology, XRD remains indispensable.

FAQ

What is X-ray diffraction?

X-ray diffraction (XRD) is a technique where X-rays scatter from the periodic arrangement of atoms in a crystal. Constructive interference occurs at specific angles determined by Bragg's law: nλ = 2d sin θ. The resulting diffraction pattern reveals crystal structure, lattice parameters, and atomic positions.

What is Bragg's law?

Bragg's law (nλ = 2d sin θ) states that constructive interference occurs when the path difference between X-rays reflected from adjacent crystal planes equals an integer number of wavelengths. Discovered by William Lawrence Bragg in 1912, it earned him and his father the 1915 Nobel Prize in Physics — at age 25, Bragg remains the youngest Nobel laureate.

What is the Scherrer equation?

The Scherrer equation (L = 0.9λ / β cos θ) relates diffraction peak breadth (β) to mean crystallite size (L). Smaller crystallites produce fewer scattering planes, reducing destructive interference of off-Bragg rays and broadening the peaks. It is widely used to estimate nanoparticle sizes from powder XRD data.

What materials can XRD analyze?

XRD works on any crystalline material — metals, minerals, ceramics, pharmaceuticals, proteins, and semiconductors. Powder XRD identifies phases in mixtures; single-crystal XRD determines complete 3D atomic structures. It cannot directly analyze amorphous (non-crystalline) materials, which produce broad halos instead of sharp peaks.

Sources

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