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.

The Scherrer equation can be written as: where: 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 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.

[3] 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.

Other techniques, such as sieving, image analysis, or visible light scattering do directly measure particle size.

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.

This system, effectively a one dimensional perfect crystal, has a structure factor or scattering function S(q):[4]

To convert to an expression for crystal size in terms of the peak width in the scattering angle

used in X-ray powder diffraction, we note that the scattering vector

and hence the peaks, depends on the crystal lattice type, and the size and shape of the nanocrystallite.

The underlying mathematics becomes more involved than in this simple illustrative example.

However, for simple lattices and shapes, expressions have been obtained for the FWHM, for example by Patterson.

[2] Just as in 1D, the FWHM varies as the inverse of the characteristic size.

For example, for a spherical crystallite with a cubic lattice,[2] the factor of 5.56 simply becomes 6.96, when the size is the diameter D, i.e., the diameter of a spherical nanocrystal is related to the peak FWHM by

The finite size of a crystal is not the only possible reason for broadened peaks in X-ray diffraction.

Fluctuations of atoms about the ideal lattice positions that preserve the long-range order of the lattice only give rise to the Debye-Waller factor, which reduces peak heights but does not broaden them.

[5] However, fluctuations that cause the correlations between nearby atoms to decrease as their separation increases, does broaden peaks.

This can be studied and quantified using the same simple one-dimensional stack of planes as above.

To derive the model we start with the definition of the structure factor

Finally, the product of the peak height and the FWHM is constant and equals

Thus finite-size and this type of disorder both cause peak broadening, but there are qualitative differences.

For sufficiently large m the pair of planes are essentially uncorrelated, in the sense that the uncertainty in their relative positions is so large that it is comparable to the lattice spacing, a.

which is in effect an order-of-magnitude estimate for the size of domains of coherent crystalline lattices.