Bragg's law describes the condition for constructive interference when X-rays are reflected from parallel planes of atoms in a crystal: nλ = 2d sinθ. Here n is the diffraction order (positive integer), λ is the X-ray wavelength, d is the interplanar spacing (d-spacing), and θ is the Bragg angle (angle between the beam and the crystal plane).

What is d-spacing in crystallography?

D-spacing (d) is the distance between parallel planes of atoms in a crystal lattice. Different crystal planes have different d-spacings, labelled by Miller indices (hkl). For example, for a cubic crystal, d_hkl = a/√(h²+k²+l²), where a is the lattice parameter. D-spacings are typically 0.1–10 Å for common crystals.

What order of diffraction should I use?

First-order diffraction (n=1) gives the most intense peak and is most commonly measured. Higher orders (n=2,3...) are weaker but can be measured. For n=1, Bragg's law simplifies to λ = 2d sinθ. Most XRD analysis uses first-order reflections.

What wavelength do X-ray diffractometers use?

Most laboratory X-ray diffractometers use Cu Kα radiation with λ = 1.5406 Å (0.15406 nm). Co Kα (1.7902 Å) and Mo Kα (0.7107 Å) are also common. Synchrotron sources provide tunable wavelengths, often around 0.5–2.5 Å.

Science

Bragg Diffraction Calculator

Calculate interplanar spacing, X-ray wavelength, or diffraction angle using Bragg's law nλ = 2d sinθ. Leave any one variable blank to solve for it.

nλ = 2d sinθ — leave the variable you want to find blank.

Wavelength λ (Å)

d-spacing (Å)

Bragg angle θ (degrees)

Diffraction order n

Result

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Wavelength λ—

d-spacing—

Bragg angle θ—

2θ (diffractometer angle)—

Step-by-step—

Bragg's law

nλ = 2d sinθ n = diffraction order (1, 2, 3...) λ = X-ray wavelength (Å or nm) d = interplanar spacing (Å or nm) θ = Bragg angle (between beam and plane) Note: 2θ is the angle measured by diffractometers Common Cu Kα wavelength: λ = 1.5406 Å

Worked examples

Example 1 — NaCl d-spacing (2θ = 31.7°, Cu Kα):

λ = 1.5406 Å, θ = 31.7°/2 = 15.85°, n = 1 d = nλ/(2 sinθ) = 1.5406 / (2 × sin 15.85°) = 1.5406 / 0.5462 = 2.82 Å This is the (200) reflection plane of NaCl — a standard XRD reference peak

Example 2 — Find angle for silicon (d = 3.14 Å, Cu Kα):

sinθ = nλ/(2d) = 1.5406/(2 × 3.14) = 0.2453 θ = arcsin(0.2453) = 14.2°, 2θ = 28.4° (Si 111 peak)

Applications of Bragg diffraction

Crystal structure determination: XRD determined the double-helix structure of DNA in 1953 using Rosalind Franklin's X-ray patterns. Today it determines thousands of new crystal structures every year — from pharmaceuticals to battery materials.

Materials characterisation: Powder XRD identifies unknown materials, measures crystallite size (via peak broadening), and quantifies phase mixtures. Every materials science laboratory uses this routinely.

Protein crystallography: Over 200,000 protein structures in the Protein Data Bank were solved by XRD. Modern synchrotron sources provide X-ray beams bright enough to solve structures from micron-sized crystals.

D-spacings of common crystals

Material	d (Å)	2θ (Cu Kα)
Silicon (111)	3.14	28.4°
NaCl (200)	2.82	31.7°
Quartz (101)	3.34	26.6°
Aluminium (111)	2.34	38.5°
Iron α (110)	2.03	44.7°
Gold (111)	2.35	38.2°

Common questions

Bragg's law describes the condition for constructive interference when X-rays are reflected from parallel planes of atoms in a crystal: nλ = 2d sinθ. Here n is the diffraction order (positive integer), λ is the X-ray wavelength, d is the interplanar spacing (d-spacing), and θ is the Bragg angle (angle between the beam and the crystal plane).
D-spacing (d) is the distance between parallel planes of atoms in a crystal lattice. Different crystal planes have different d-spacings, labelled by Miller indices (hkl). For example, for a cubic crystal, d_hkl = a/√(h²+k²+l²), where a is the lattice parameter. D-spacings are typically 0.1–10 Å for common crystals.
First-order diffraction (n=1) gives the most intense peak and is most commonly measured. Higher orders (n=2,3...) are weaker but can be measured. For n=1, Bragg's law simplifies to λ = 2d sinθ. Most XRD analysis uses first-order reflections.
Most laboratory X-ray diffractometers use Cu Kα radiation with λ = 1.5406 Å (0.15406 nm). Co Kα (1.7902 Å) and Mo Kα (0.7107 Å) are also common. Synchrotron sources provide tunable wavelengths, often around 0.5–2.5 Å.