Tim White Michael F. Toney Scherrer Equation: A Practical Guide

The Scherrer equation is a formula used to estimate the size of crystallites in a powdered sample from the broadening of its X-ray diffraction peaks. The “tim white michael f. toney scherrer equation” refers to a specific interpretation or application of this formula, often discussed in the context of materials science and thin-film analysis. It is the same fundamental Scherrer equation, but the names are sometimes attached to highlight a particular methodology or correction factor used when analyzing diffraction data.

Understanding the Scherrer Equation

The Scherrer equation relates the width of an X-ray diffraction peak to the average size of the crystallites in the sample. The basic form is:

τ = Kλ / (β cos θ)

Where:

• τ (tau) is the mean size of the crystallites.
• K is the shape factor (a dimensionless constant, typically around 0.9 for spherical crystallites).
• λ (lambda) is the X-ray wavelength.
• β (beta) is the full width at half maximum (FWHM) of the diffraction peak, measured in radians.
• θ (theta) is the Bragg angle.

The equation assumes the broadening comes only from the finite size of the crystallites. In reality, other factors like microstrain and instrument effects also broaden the peaks. The “Tim White Michael F. Toney” association often appears when researchers apply the equation to thin films or nanomaterials, where careful correction for these other broadening sources is critical.

The Shape Factor (K)

The shape factor K is not always 0.9. It depends on the shape of the crystallites. For cubic crystals, a value of 0.94 is sometimes used. For other shapes, K can range from about 0.8 to 1.3. The International Union of Crystallography provides general guidance on selecting the appropriate K value based on the expected crystal morphology. Using the wrong K value can lead to systematic errors in the calculated crystallite size.

Applying the Equation to Real Data

To use the Scherrer equation, you first collect an X-ray diffraction pattern. You then identify a peak, measure its FWHM, and correct for the instrument’s own broadening. This correction is often done by measuring a standard material (like silicon or lanthanum hexaboride) with large, strain-free crystallites. The instrument broadening is then subtracted from the measured peak width using a simple formula or a more sophisticated deconvolution method.

A common approach is to use the equation:

β_sample = √(β_measured² - β_instrument²)

This assumes the peaks are Gaussian in shape. If the peaks are Lorentzian, the subtraction is linear. Many modern analysis programs handle this automatically, but understanding the correction is important for reliable results.

A Simple Worked Example

Assume you have a diffraction peak from a sample of nanocrystalline cerium oxide. You measure the peak at a 2θ angle of 28.5 degrees (so θ = 14.25 degrees). The FWHM after instrument correction is 0.008 radians. You use a copper X-ray source with λ = 0.15418 nm. Assuming K = 0.9, the calculation is:

τ = (0.9 * 0.15418 nm) / (0.008 rad * cos(14.25°))

cos(14.25°) ≈ 0.969. So:

τ = 0.1388 nm / (0.008 * 0.969) = 0.1388 nm / 0.00775 ≈ 17.9 nm

The estimated crystallite size is about 18 nanometers. This is a typical size for a nanocrystalline powder.

Limitations and Common Pitfalls

The Scherrer equation has several limitations. It only gives a volume-weighted average crystallite size, not the full size distribution. It assumes the crystallites are strain-free. If the sample has microstrain, the peak broadening will be larger, and the equation will underestimate the crystallite size.

Another common issue is peak overlap. If two peaks are close together, the measured FWHM can be inaccurate. Software that fits peaks with mathematical functions (like pseudo-Voigt or Pearson VII) can help separate overlapping peaks. The equation also assumes the crystallites are roughly equiaxed. For plate-like or needle-like crystals, the Scherrer equation gives an average size that may not represent any real dimension.

When the Equation Does Not Apply

The Scherrer equation works best for crystallites between about 1 nm and 100 nm. Below 1 nm, the peaks become very broad and hard to distinguish from the background. Above 100 nm, the peak broadening from size is very small and can be lost in the instrument broadening. For larger crystallites, other methods like electron microscopy or light scattering are more appropriate.

Separating Size and Strain Broadening

A single diffraction peak cannot tell you whether the broadening comes from small crystallites or from microstrain. To separate these two effects, you need multiple peaks across a range of 2θ angles. The Williamson-Hall method plots β_cos(θ) against sin(θ) for several peaks.

The slope of this line gives the microstrain, and the y-intercept gives the crystallite size. If the data points form a horizontal line, strain is negligible, and the Scherrer equation alone is sufficient. If the line has a clear positive slope, strain is present, and the Scherrer equation will underestimate the true crystallite size.

Key Takeaways

• The Scherrer equation estimates crystallite size from X-ray diffraction peak broadening.
• The formula is τ = Kλ / (β cos θ), with K typically around 0.9.
• Instrument broadening must be corrected for accurate results.
• The equation assumes strain-free, roughly spherical crystallites.
• It is most reliable for crystallite sizes between 1 nm and 100 nm.