Bragg's Law
X-ray diffraction from crystal planes
Formula
n λ = 2 d sin θ
- n = order of diffraction (1, 2, 3...)
- λ = X-ray wavelength (m)
- d = spacing between crystal planes (m)
- θ = angle of incidence
Understanding Bragg's Law
Bragg's Law, formulated by William Henry Bragg and William Lawrence Bragg in 1913, describes the condition for constructive interference when X-rays are scattered by a crystalline material. When X-rays strike crystal planes at specific angles, they reflect and interfere constructively only when the path difference between rays equals an integer multiple of the wavelength.
The law is fundamental to X-ray crystallography, allowing scientists to determine atomic arrangements in crystals. Each set of parallel planes in a crystal has a characteristic spacing (d), and X-rays diffracted from adjacent planes travel different distances. Constructive interference produces detectable diffraction peaks only when the Bragg condition is satisfied.
The order of diffraction (n) represents how many wavelengths fit into the path difference. First-order diffraction (n=1) is typically strongest, while higher-order reflections (n=2, 3...) become progressively weaker.
Key Concepts
Constructive Interference
When X-rays from adjacent crystal planes arrive in phase (path difference = nλ), their amplitudes add to create strong reflection peaks.
Bragg Angle (θ)
The angle between incident X-ray beam and crystal planes, not the incident surface. Measured from the plane itself rather than the surface normal.
Lattice Spacing (d)
Distance between parallel crystal planes. Different crystal families (hkl planes) have different d-spacing values, creating unique diffraction patterns.
Worked Example
Given: λ = 1.54 Å (Cu Kα), θ = 15°, n = 1.
d = n λ / (2 sin θ)
d = (1 × 1.54 Å) / (2 × sin 15°)
d ≈ 1.54 / (2 × 0.2588) ≈ 2.98 Å
Answer: d ≈ 2.98 Å
This spacing is typical for metal crystalline structures and ceramics.
Applications in Science
Materials Science
Determine crystal structures, phase composition, grain size, and crystallinity in metals, ceramics, and semiconductors.
Structural Biology
Solve protein and DNA structures at atomic resolution using synchrotron X-ray sources and cryo-crystallography.
Geology & Mineralogy
Identify minerals and determine their crystallographic orientation in rocks and geological samples.
Pharmaceutical Industry
Analyze drug polymorphs, crystallinity of active ingredients, and quality control of pharmaceutical compounds.
Common X-ray Sources
| Source | Wavelength (Ã…) | Application |
|---|---|---|
| Cu Kα | 1.5418 | Most common for routine analysis |
| Mo Kα | 0.7107 | Protein crystallography |
| Co Kα | 1.7902 | Iron-containing samples |
| Synchrotron | Variable | High-resolution studies |
Practical Tips
- Always convert angles to radians when using calculators without degree mode.
- For powder diffraction, multiple grain orientations produce cone-shaped diffraction patterns captured as rings on detectors.
- Single crystals require careful alignment to obtain diffraction from specific planes.
- Sample preparation is critical—powder samples should be finely ground for accurate d-spacing measurements.
- Temperature affects lattice spacing; most standard measurements are at room temperature (25°C).