The Scherrer Equation: Math Proof & Limits

In nanotech and solid-state chemistry, measuring the dimensions of coherent crystalline domains is essential. The standard method used to estimate these sizes from powder diffraction data is the Scherrer equation. First proposed in 1918 by Swiss physicist Paul Scherrer, this formula remains highly popular due to its simplicity. This article provides a technical guide to the Scherrer equation, covering its mathematical basis, shape factor selections, and physical limits.

1. Derivation and Physical Principles

To understand peak broadening, we can model a crystal as a finite stack of $N$ parallel atomic planes spaced a distance $d$ apart. When monochromatic X-rays diffracted from these planes combine constructively, they produce a peak at the Bragg angle ($\theta_B$).

For an infinite crystal, destructive interference occurs at all angles slightly different from $\theta_B$, resulting in a delta-function peak. However, for a finite crystal containing only a few dozen planes, this destructive interference is incomplete. This incomplete cancellation allows X-rays to diffract over a small angular range around the Bragg angle, creating a broadened peak.

By analyzing the path difference between the top and bottom planes of the finite crystal stack, we can define the angles where the diffracted intensity drops to zero. This analysis leads to the classic Scherrer formulation:

Where:

• D: The mean crystallite size (in nanometers or Angstroms).
• K: A dimensionless shape factor. Typical values range from $0.89$ to $0.94$, with $0.90$ often assumed for spherical nanoparticles with cubic lattices.
• λ: The incident X-ray wavelength (typically $0.15406$ nm for copper targets).
• β: The peak FWHM in radians.
• θ: The Bragg angle (half the $2\theta$ coordinate of the peak centroid).

2. Selecting the Shape Factor (K)

The shape factor ($K$) in the Scherrer equation is a calibration constant that depends on:

• The shape of the crystallites (spherical, ellipsoidal, needle-like, or plate-like).
• The definition of crystallite size (e.g., volume-weighted or area-weighted average).
• The definition of peak width (FWHM vs. integral breadth).

The table below lists common shape factor values used in powder diffraction:

For general research where the crystallite shape is unknown, $K = 0.90$ is standard. If you need highly accurate size reporting, you must confirm crystallite shapes using transmission electron microscopy (TEM).

3. Subtracting Instrument Broadening

Peak broadening is caused by both microstructural features in the sample and the optical configuration of the diffractometer. The instrument contribution (β_inst) must be subtracted to determine the sample broadening (β_crystallite).

To subtract instrument broadening, you must measure a standard reference material (like LaB₆ or NIST Silicon) to establish the instrument's FWHM curve. The correction is then calculated using:

4. Limitations of the Scherrer Equation

While the Scherrer equation is widely used, researchers must keep its physical limitations in mind:

• Size Limit: Only works for crystallites smaller than 100 nm. Above this limit, peak broadening is too small to distinguish from instrument limits.
• Lattice Strain: Assumes the sample is free from lattice strain. If strain is present, it will also broaden peaks, causing the Scherrer equation to underestimate crystallite size. You must use the Williamson-Hall method to separate these contributions.
• Crystallites vs. Particles: Measures the size of coherent crystalline domains, not physical particles. A single physical particle can contain multiple boundaries, making it larger than the crystallite size.

5. Conclusion

The Scherrer equation remains a quick and reliable tool for estimating nanocrystalline sizes. By correcting FWHM measurements for instrument broadening and selecting an appropriate shape factor ($K$), materials scientists can estimate crystallite sizes directly from powder diffraction scans.