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Bragg's Law Calculator

Find the Bragg diffraction angle for a crystal.

Result

14.87°
29.74°

How it is calculated

Bragg's law describes when X-rays reflected from parallel crystal planes reinforce each other: nλ = 2d · sin θ. n is the diffraction order (a whole number, 1 if left empty), λ is the X-ray wavelength and d is the spacing between lattice planes, both in nanometres (they must use the same unit). The tool solves θ = arcsin(nλ ÷ 2d) and also shows 2θ, the angle a diffractometer usually records. If nλ ÷ 2d is greater than 1 there is no angle that satisfies the law, so no result is given. Wavelength and spacing must be positive, and the order must be a whole number of at least 1 (0, negative or fractional orders give no result). Angles are in degrees.

Worked example

Copper K-alpha X-rays with λ = 0.154 nm, first order, and a plane spacing d = 0.3 nm. sin θ = (1 × 0.154) ÷ (2 × 0.3) = 0.154 ÷ 0.6 = 0.2567. θ = arcsin(0.2567) = 14.87°. The detector angle is 2θ = 29.74°. For second order the sine would be 0.5133 and the angle about 30.88°.

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Common mistakes

  • Mixing units, such as wavelength in ångströms and spacing in nanometres (1 Å = 0.1 nm).
  • Reporting θ as the angle between the incoming and outgoing beams. That angle is 2θ.
  • Measuring the angle from the surface normal rather than from the lattice plane itself.
  • Choosing an order so high that nλ ÷ 2d exceeds 1; there is simply no diffraction at that order.

About this calculator

This calculator finds the diffraction angle in crystallography using Bragg's law: nλ = 2d·sinθ, where n is the diffraction order (an integer), λ is the wavelength of the incident radiation, d is the spacing between crystal lattice planes, and θ is the angle of incidence measured from the plane (not the normal). Constructive interference — a detectable diffraction peak — occurs only when this condition is satisfied.

Crystallographers and materials scientists use Bragg's law in X-ray diffraction (XRD) to determine unknown crystal structures and lattice spacings by measuring the angles at which diffraction peaks appear, a technique that was foundational to solving structures like DNA and countless minerals and materials. Any one of the variables can be solved for, not just the angle.

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Comments (1)

  • Editorial team

    Tables often give plane spacings in angstroms; this tool wants nanometres, so divide by 10 first: 2.31 Å becomes 0.231 nm. For a cubic crystal with cell edge a, the spacing of the (hkl) planes is a divided by the square root of h² + k² + l², which gives you d to enter. If the tool says the values cannot be calculated, nλ ÷ 2d is above 1 or the order is not a whole number, so that reflection cannot be observed. Which crystal system are you working with?

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