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.^[1][2] 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.