Bragg's Law Calculator
Find the Bragg diffraction angle for a crystal.
Result
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. 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°.
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. Enter the diffraction order, wavelength, and lattice spacing to find the angle, or solve for any of the other variables.
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