Scherrer equation

The Scherrer equation converts diffraction-line broadening into an approximate crystallite size. Its usual form is D = Kλ/(β cos θ), where D is the volume-weighted mean dimension of coherently diffracting domains, K is a dimensionless shape factor, λ is the X-ray wavelength, β is the corrected peak breadth in radians, and θ is the Bragg angle.¹

The relation was formulated by Paul Scherrer in 1918 in work on colloidal particles examined with X-rays.² Broadening arises because finite domains contain only a limited number of lattice planes: smaller domains produce wider reciprocal-space features and therefore broader diffraction peaks. The equation is consequently a model-based estimate, not a direct image or a universal particle-size law.

The calculation requires a measured peak width, a known wavelength, and the peak position. Analysts commonly fit a selected reflection, determine its full width at half maximum or integral breadth, subtract or model the instrumental contribution, convert the remaining width to radians, and insert the Bragg angle into the equation.³

Instrumental broadening can be estimated with a strain-free reference material measured under comparable conditions. A simple correction may use β_sample = (β_observed² − β_instrument²)^1/2 for compatible Gaussian profiles; other line shapes require profile-based treatment. Results should be reported with the radiation, reflection, width definition, correction method, and value of K. Several reflections provide a useful check: consistent sizes support the approximation, while systematic differences may reveal anisotropy or strain.

The Scherrer equation measures coherent diffraction-domain size, which may be smaller than the visible particle size. A single nanoparticle can contain several domains separated by defects, whereas several particles can sometimes behave as one coherent domain. Consequently, transmission electron microscopy, scanning probe methods, or complementary scattering measurements may report a different size without either result being intrinsically wrong.

Peak broadening also comes from microstrain, faults, defects, compositional variation, unresolved overlapping reflections, and instrument optics. Applying the equation without separating these effects can underestimate or misassign the size. The shape factor K depends on crystallite shape, reflection, breadth convention, and profile assumptions; 0.9 is a convenient approximation rather than a constant of nature.³ For substantial size and strain broadening, whole-pattern methods such as Rietveld refinement or Warren–Averbach analysis are generally more informative.

The equation is most reliable when peaks are broadened mainly by finite size and when the crystallites are sufficiently small for the broadening to be measurable but not so disordered that distinct Bragg peaks disappear. It can be applied to powders, thin films, catalysts, minerals, ceramics, and nanoparticles, but the result is often a directional or reflection-specific dimension rather than a single geometric diameter.

Peak breadth is not interchangeable with peak separation: the Scherrer estimate uses the width of an individual reflection, while changes in peak position primarily indicate lattice spacing and therefore may reflect composition, temperature, or stress. Broadening can also be anisotropic, with different crystallographic directions producing different apparent sizes. Modern line-profile analysis therefore treats instrumental resolution, crystallite-shape effects, and strain together rather than forcing every broadened peak into one scalar estimate.

Crystallite

A coherently diffracting region of a crystalline material; it need not equal a whole particle or grain.

Peak breadth

The width of a diffraction peak, commonly expressed as full width at half maximum or integral breadth.

Microstrain

Small spatial variations in lattice spacing that broaden diffraction peaks.

Shape factor

The dimensionless constant K that incorporates crystallite shape, reflection, and the chosen breadth convention.