Hong Life CalculatorsUnderstanding the Bragg's law and Scherrer calculator — concept and principle
X-ray diffraction (XRD) identifies crystalline materials from the angles at which they reflect X-rays. This calculator turns a peak position 2θ into the spacing d between lattice planes using Bragg's law, and estimates the average crystallite size from the peak width using the Scherrer equation. Materials scientists, chemists and geologists use it to index patterns, check phases and follow nanoparticle growth.
How it is calculated
- Bragg's law: n × λ = 2 × d × sin θ, so d = n × λ / (2 sin θ)
- Scherrer equation: τ = K × λ / (β × cos θ)
λ is the X-ray wavelength in ångström (default 1.5406 Å, Cu Kα₁). 2θ is the diffraction angle read from the pattern; the calculator halves it to get θ. n is the order of reflection, usually 1. For Scherrer, β is the full width at half maximum (FWHM) of the peak, entered in degrees 2θ and converted to radians, and K = 0.9 is the shape factor. The size τ is reported in nm (λ ÷ 10).
The principle
X-rays scattered from parallel planes of atoms interfere. Rays reflected from adjacent planes travel an extra distance 2d sin θ; when that equals a whole number of wavelengths, the waves add up and a strong peak appears. Measuring θ therefore reveals d. A perfect, infinitely large crystal gives a very sharp peak. In a small crystal there are fewer planes to cancel out slightly-off angles, so the peak broadens; Scherrer showed the width is inversely proportional to the crystal size.
Worked example
- Inputs: λ = 1.5406 Å, 2θ = 38.2°, n = 1, FWHM = 0.3°.
- θ = 19.1°, sin θ = 0.32722, cos θ = 0.94495.
- d = 1 × 1.5406 ÷ (2 × 0.32722) = 2.3541 Å, close to the (111) spacing of gold.
- β = 0.3 × π/180 = 0.0052360 rad.
- τ = 0.9 × 0.15406 nm ÷ (0.0052360 × 0.94495) = 28.02 nm.
| 2θ (°) | d-spacing (Å) |
|---|---|
| 9.55 | 9.253627229 |
| 19.1 | 4.642927937 |
| 28.6 | 3.113306885 |
| 38.2 | 2.354088827 |
| 47.8 | 1.903184749 |
| 57.3 | 1.606606261 |
| 76.4 | 1.245616984 |
How to read the result
Compare d-values with a reference database to identify phases. Peaks that shift to lower 2θ mean larger d, for example from lattice expansion. For Scherrer, narrower peaks mean larger crystallites; the method is most useful for sizes up to roughly 100–200 nm, above which the broadening is too small to separate from the instrument.
Common mistakes
- Entering θ instead of 2θ, or 2θ in a formula that needs θ.
- Not subtracting instrumental broadening measured on a standard sample.
- Using the Cu Kα wavelength for data taken with another anode such as Co or Mo.
Frequently asked questions
What K value should I use?
0.9 is common, but K depends on crystallite shape and the width definition; values between about 0.89 and 0.94 are typical for FWHM.
Why is Cu Kα 1.5406 Å?
It is the Kα₁ emission line of copper; Kα₂ at about 1.5444 Å is often stripped from the data.
Is crystallite size the same as particle size from microscopy?
Not necessarily. One particle can contain several crystallites, so XRD sizes are often smaller.
Sources: W. H. Bragg and W. L. Bragg, Proceedings of the Royal Society A 88, 428 (1913) · P. Scherrer, Göttinger Nachrichten, 98 (1918) · B. D. Cullity and S. R. Stock, Elements of X-Ray Diffraction, Pearson