The Scherrer Equation Explained for X-Ray Diffraction Crystallite Size

The Scherrer equation estimates the size of coherently diffracting domains from the broadening of an X-ray diffraction peak:

[ D = \frac{K\lambda}{\beta\cos\theta} ]

Here, (D) is the estimated crystallite size, (K) is a shape factor, (\lambda) is the X-ray wavelength, (\beta) is the sample-related peak broadening in radians, and (\theta) is the Bragg angle. The angle (\theta) is half the measured (2\theta) position. The basic relationship is simple: smaller crystallites produce broader diffraction peaks.

The equation estimates a coherently diffracting domain, not necessarily the size of a particle or grain seen in a microscope. A particle can contain several crystallites, so its physical size may be larger than the Scherrer estimate.

What the terms mean

• (D), crystallite size: The estimated length of the coherently diffracting domain in the direction associated with the selected reflection.
• (K), shape factor: A dimensionless factor that depends on crystallite shape, the definition of size, and how peak width is measured. A value near 0.9 is often used as an approximation, but it is not universal.
• (\lambda), wavelength: The wavelength of the X-rays used. For copper Kα radiation, a commonly used value is about 1.5406 Å.
• (\beta), peak broadening: The contribution to the peak width attributed to crystallite size, after accounting for instrumental broadening. It must be expressed in radians on the (2\theta) scale.
• (\theta), Bragg angle: Half the peak position recorded as (2\theta).

Scherrer’s work on estimating particle dimensions from X-ray diffraction was published in 1918 P. Scherrer, Göttinger Nachrichten 2, 98–100 (1918). The equation and its terminology are also described by the International Union of Crystallography IUCr, Scherrer equation.

The cosine term matters when comparing peaks at different angles. For the same crystallite size, the relationship between peak width and size depends on (\theta), so raw widths from different reflections should not be compared as if they were equivalent.

One worked example

Suppose a diffraction peak is measured at (2\theta = 40.0^\circ). Then (\theta = 20.0^\circ), and (\cos\theta \approx 0.940). Use copper Kα radiation, with (\lambda = 1.5406) Å, and assume (K = 0.9). If the corrected sample broadening is (\beta = 0.0080) radians, the calculation is:

[ D = \frac{0.9 \times 1.5406}{0.0080 \times 0.940} \approx 184\ \text{Å} \approx 18.4\ \text{nm} ]

The result is an estimate based on the selected reflection and the assumptions made for (K) and peak broadening. It should not be read as an exact particle diameter.

Correcting the peak width

A measured peak is broadened by the instrument as well as by the sample. The Scherrer equation requires the sample-related broadening, so the instrument contribution must be estimated, often by measuring a suitable standard under the same conditions.

The correction depends on the peak-shape model. For Gaussian profiles, widths are commonly corrected using a quadrature relationship. For Lorentzian profiles, a different correction applies; real peaks can also have mixed shapes. The correction method should match the way the peak widths are defined and fitted. Using the uncorrected measured width generally makes (\beta) too large and the calculated (D) too small.

Keep the units consistent: convert the corrected width from degrees to radians before putting it into the equation. A width reported on a (2\theta) scan is used as the (2\theta) width; do not halve (\beta) when calculating (\theta). Only the peak position is halved to obtain the Bragg angle.

What the result can and cannot tell you

The Scherrer equation attributes peak broadening to finite crystallite size. Microstrain, which comes from variations in lattice spacing, can also broaden peaks. A single peak cannot generally separate size broadening from strain broadening. When both effects may matter, analysis across several reflections can help assess how the broadening varies with angle. NIST provides an overview of crystallite-size and strain analysis through the NIST Center for Neutron Research.

The estimated size can also depend on the reflection chosen. Crystallites are not always spherical, and different reflections can probe different directions in an anisotropic domain. As a result, estimates from different peaks may vary. The IUCr describes crystallite size as a direction-dependent quantity IUCr, Crystallite size.

For a useful report, state the wavelength, the value of (K), the reflection used, the peak-width definition, and the method used to account for instrumental broadening. If you calculate sizes from several reflections, report how you summarized them and whether the values varied. These details make the result easier to interpret and reproduce.

The Scherrer equation is a practical way to estimate crystallite size from broadened diffraction peaks. Its result is most useful when instrumental broadening has been handled carefully and other sources of broadening, especially microstrain, are considered.

Key Takeaways

• The Scherrer equation is (D = K\lambda/(\beta\cos\theta)).
• It estimates the size of a coherently diffracting domain, which may be smaller than the particle or grain observed by microscopy.
• Use (\beta) in radians on the (2\theta) scale, corrected for instrumental broadening.
• In the worked example, (2\theta = 40.0^\circ), (\lambda = 1.5406) Å, (\beta = 0.0080) rad, and (K = 0.9) give an estimate of about 18.4 nm.
• Peak broadening can come from microstrain as well as crystallite size, so one peak alone may not distinguish the two effects.