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Wikipedia / 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:

where:

is the mean size of the ordered (crystalline) domains, which may be smaller or equal to the grain size;
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;
is the X-ray wavelength;
is the line broadening at half the maximum intensity (FWHM), after subtracting the instrumental line broadening, in radians;
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):

where for N:

each sum is a simple geometric series, and the other series analogously gives:

which is further simplified by converting to trigonometric functions:

and finally:

Determination of the profile near the peak, and hence the peak width

, as the peak height is N. If we take the plus sign (peak is symmetric so either sign will do)

and

if N. Therefore, the size of the set of planes is related to the FWHM in q by

used in X-ray powder diffraction, and so

which is the Scherrer equation with K = 0.88.

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. Just as in 1D, the FWHM varies as the inverse of the characteristic size. For example, for a spherical crystallite with a cubic lattice, 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

Peak broadening due to disorder of the second kind

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. 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. The derivation follows that in chapter 9 of Guinier's textbook. This model was pioneered by and applied to a number of materials by Hosemann and collaborators over a number of years. They termed this disorder of the second kind, and referred to this imperfect crystalline ordering as paracrystalline ordering. Disorder of the first kind is the source of the Debye-Waller factor.

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

. So, then

is the probability density function lattice spacings apart. For the separation of neighbouring planes we assume for simplicity that the fluctuations around the mean neighbour spacing of a are Gaussian, i.e., that

s, etc. As the convolution of two Gaussians is just another Gaussian, we have that

is then just a sum of Fourier Transforms of Gaussians, and so

and so the structure factor of the infinite but disordered crystal is

. These peaks have heights

and obtain

which is a Lorentzian or Cauchy function limit.

. This, in principle, allows the two effects to be distinguished. Also, it means that the Scherrer equation is best applied to the first peak, as disorder of this type affects the first peak the least.

Coherence length

, for a pair of planes m planes apart. 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 equals a. This gives

, so the coherence length is approximately 1/FWHM for the first peak.