Scherrer equation

The Scherrer equation, in X-ray diffraction and crystallography, is a formula that relates the size of sub-micrometre crystallites in a solid to the broadening of a peak in a diffraction pattern. It is often referred to, incorrectly, as a formula for particle size measurement or analysis. It is named after Paul Scherrer. It is used in the determination of size of crystals in the form of powder.

The Scherrer equation can be written as:

${\displaystyle \tau ={\frac {K\lambda }{\beta \cos \theta }}}$

where:

• ${\displaystyle \tau }$ is the mean size of the ordered (crystalline) domains, which may be smaller or equal to the grain size, which may be smaller or equal to the particle size;
• ${\displaystyle K}$ is a dimensionless shape factor, with a value close to unity. The shape factor has a typical value of about 0.9, but varies with the actual shape of the crystallite;
• ${\displaystyle \lambda }$ is the X-ray wavelength;
• ${\displaystyle \beta }$ is the line broadening at half the maximum intensity (FWHM), after subtracting the instrumental line broadening, in radians. This quantity is also sometimes denoted as ${\displaystyle \Delta \left(2\theta \right)}$;
• ${\displaystyle \theta }$ is the Bragg angle.