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:
τ = K λ β cos θ {\displaystyle \tau ={\frac {K\lambda }{\beta \cos \theta }}}
where:
τ {\displaystyle \tau } is the mean size of the ordered (crystalline) domains, which may be smaller or equal to the grain size;
K {\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;
λ {\displaystyle \lambda } is the X-ray wavelength;
β {\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 Δ ( 2 θ ) {\displaystyle \Delta \left(2\theta \right)} ;
θ {\displaystyle \theta } is the Bragg angle.
Applicability The Scherrer equation is limited to nano-scale crystallites, or more-strictly, the coherently scattering domain size, which can be smaller than the crystallite size (due to factors mentioned below). It is not applicable to grains larger than about 0.1 to 0.2 μm, which precludes those observed in most metallographic and ceramographic microstructures. The Scherrer equation provides a lower bound on the coherently scattering domain size, referred to here as the crystallite size for readability. The reason for this is that a variety of factors can contribute to the width of a diffraction peak besides instrumental effects and crystallite size; the most important of these are usually inhomogeneous strain and crystal lattice imperfections. The following sources of peak broadening are dislocations, stacking faults, twinning, microstresses, grain boundaries, sub-boundaries, coherency strain, chemical heterogeneities, and crystallite smallness. These and other imperfections may also result in peak shift, peak asymmetry, anisotropic peak broadening, or other peak shape effects. If all of these other contributions to the peak width, including instrumental broadening, were zero, then the peak width would be determined solely by the crystallite size and the Scherrer equation would apply. If the other contributions to the width are non-zero, then the crystallite size can be larger than that predicted by the Scherrer equation, with the "extra" peak width coming from the other factors. The concept of crystallinity can be used to collectively describe the effect of crystal size and imperfections on peak broadening. Although "particle size" is often used in reference to crystallite size, this term should not be used in association with the Scherrer method because particles are often agglomerations of many crystallites, and XRD gives no information on the particle size. Other techniques, such as sieving, image analysis, or visible light scattering do directly measure particle size. The crystallite size can be thought of as a lower limit of particle size.
Derivation for a simple stack of planes To see where the Scherrer equation comes from, it is useful to consider the simplest possible example: a set of N planes separated by the distance, a. The derivation for this simple, effectively one-dimensional case, is straightforward. First, the structure factor for this case is derived, and then an expression for the peak widths is determined.
Structure factor for a set of N equally spaced planes This system, effectively a one dimensional perfect crystal, has a structure factor or scattering function S(q):
S ( q ) = 1 N ∑ j , k = 1 N e − i q ( x j − x k ) {\displaystyle S(q)={\frac {1}{N}}\sum _{j,k=1}^{N}\mathrm {e} ^{-iq(x_{j}-x_{k})}}
where for N planes, x j = a j {\displaystyle x_{j}=aj} :
S ( q ) = 1 N ∑ k = 1 N e − i q a k × ∑ j = 1 N e i q a j {\displaystyle S(q)={\frac {1}{N}}\sum _{k=1}^{N}\mathrm {e} ^{-iqak}\times \sum _{j=1}^{N}\mathrm {e} ^{iqaj}}
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