# Scherrer grain size analysis

The average grain size in a material may be estimated from the peak width. Sharp peaks indicate large grains, whereas broad peaks indicate small grains. This inverse relationship is, as always, due to the inverse nature of reciprocal-space. It can also be rationalized in terms of the Fourier transform: an infinitely large grain means that the lattice repeats forever without defect; this periodic oscillation can be described by a single peak in Fourier space. However, a finite grain requires additional Fourier information to encode this truncation to the repetition. A broad Fourier peak describes this 'localized periodicity' (c.f. wave packets).

Note that in scattering analysis, we often convert peak width into a distance, and call this the 'grain size' or 'crystallite size'. This is conceptually meant to refer to the average size of the region over which the repeating lattice is similarly-aligned. This average size thus might describe the average distance between one grain boundary and another. However, in some types of systems, the material may not have well-defined grain boundaries. For instance, in block-copolymers, the domains may 'meander', in which case the 'grain size' is instead a characteristic distance over which the lattice directionality is roughly preserved (an orientational correlation length). More generally still, some materials, the inverse peak width is perhaps better thought of as simply a correlation length: a distance over which the repeating lattice is well-correlated.

## Equation

The Scherrer equation is:

\begin{alignat}{2} \xi_{hkl} & = \frac{K\lambda}{ B_{hkl} \cos(\theta_{hkl}) } \\ & = \frac{2 \pi K}{ \Delta q_{hkl} } \end{alignat}

where the peak width (B) for a particular reflection (hkl) is inversely proportional to the crystallite size (ξ). Note that the x-ray wavelength (λ) and the scattering angle (2θ) are implicated. The constant K is of order unit.

### K constant

The constant K takes on different values depending on the the conversion; it particular it is affected by the grain shape, the grain size distribution, and how the peak width is defined. Typically it is given the value:

$K = 2 \sqrt{\ln(2)/\pi} \simeq 0.9394$

This value is appropriate for spherical crystals with cubic symmetry, where the peak width is defined using the FWHM. If one instead uses the integral breadth, then the constant would be 0.89. In general, the constant K varies between 0.62 and 2.08.

If the peak width is instead defined in terms of the Gaussian width (standard deviation), then $\scriptstyle \Delta q_{\mathrm{fwhm}} = 2\sqrt{2 \ln{2} } \sigma_{q}$, or:

$K = \frac{2\sqrt{\ln(2)/\pi}}{2 \sqrt{2 \ln(2)}} = \frac{1}{\sqrt{2 \pi}} \simeq 0.3989$

One must be careful in naively interpreting the peak width as the grain size. As noted above, it should more generally be thought of as a correlation distance. A scattering peak can be broadened by many things beyond merely a finite grain size. For instance, if the material includes a variety of grains with slightly different repeat spacings, each grain will have a slightly different peak position and thus the combined signal will have apparent peak broadening due to the distribution in lattice spacings. More specifically, strain can thus broaden peaks.

The following effects influence peak width:

• Instrumental resolution
• Crystallite size
• Crystallize size distribution
• Microstrain
• non-uniform lattice distortions
• faulting
• dislocations
• antiphase domain boundaries
• grain surface relaxation
• Solid solution inhomogeneity
• Temperature Factors

Note that differentiating these effects from one another is in general difficult. The instrumental resolution can be independently calculated or measured. The other effects can sometimes be disentangled by gathering appropriate data. For instance, detailed analysis of peak shape can be beneficial. Moreover, measuring the peak shape and peak width as a function of peak order, can be used to differentiate various effects.

## Resolution Limit

Note that the instrumental resolution contributes to the experimental peak width. Measured peaks are broadened due to the wavelength spread in the incident radiation, the divergence of the beam, the detector pixel size, etc. To obtain accurate grain sizes, this effect should be accounted for in the data analysis. Various publications describe how to quantitatively account for various instrumental effects. The simplest method is to correct the peak width values:

$\sigma_{\mathrm{corrected}} = \sqrt{ \sigma_{\mathrm{measured}}^2 - \sigma_{\mathrm{instrumental}}^2 }$

Importantly, if the peak are extremely sharp, and thus instrumental-limited, then the Scherrer analysis cannot be applied.