Report 192 · Lab Science
Nearly every nanomaterials paper reports a “crystallite size” from X-ray diffraction, usually one number from one equation. That number is real, but it is not the average size of your particles. It is weighted so heavily toward the largest crystals that, in one published simulation, a powder whose most common particle was 10 nm came out at about 49 nm.
In spectroscopy you learn early that a line width is information, but it is information about everything that broadens the line. X-ray diffraction peaks work the same way. Small crystals make broad peaks, and Paul Scherrer turned that into an equation more than a century ago. The trouble starts when the output gets read as something it was never built to report.
The Scherrer equation is usually written L = κλ / (β cos θ). Here λ is the X-ray wavelength, θ is the Bragg angle of the peak, β is the peak's full width at half maximum, and κ is a constant. Make the crystals smaller and the peak gets wider, so L goes down.
Two details hide in that line. First, β has to be the width the sample causes, after the instrument's own broadening is removed. Valério and Morelhão write the correction as subtracting the instrumental width in quadrature from the measured width, one common approach. Skip it and every size comes out too small. Second, κ is not a universal number. The ceria-nanocube paper by the same group states that the constant “is determined by the NP shape, may differ from one diffraction peak to another in the case of non-spherical NPs.” Pick a constant for spheres and apply it to cubes or plates, and the answer shifts.
The equation sizes the regions that diffract coherently, the crystals. What you see in an electron microscope is particles. They are the same thing only when each particle is a single crystal, which is why the two University of São Paulo papers I read are careful to say “monocrystalline” nanoparticles in their titles. A particle built from several crystal domains will look bigger under the microscope than its Scherrer size.
Uvarov and Popov compared XRD sizes with transmission and scanning electron microscopy across 183 powder samples. Their abstract reports “a good correlation in size only for crystallites smaller than 50 – 60 nm.” Above that, the two methods drift apart.
Real powders are not one size. They have a distribution, often a long tail of bigger crystals. So which size does a single peak width report?
The usual answer in the literature is a volume-weighted average. Valério and colleagues argue that assumption is “limited to samples with narrow distributions.” Their reasoning is simple. A crystal's diffraction peak height grows steeply with its size, so the big crystals dominate the peak's shape. Working through the math, they find the Scherrer size behaves as the median of the size distribution weighted by size to the fourth power, which means the largest crystals carry enormous weight.
Their simulated example makes the point vividly. For spheres following a lognormal distribution from 1 to 90 nm whose most common size was 10 nm, the XRD-weighted median came out at 49.4 nm, and the small-angle scattering equivalent at 66.1 nm. In their words, “both values are much larger than the 10 nm value of the PSD mode.” Same powder, three very different “sizes,” and none of them wrong. They just answer different questions.
An important caveat: this is a simulation in a preprint, and the group's earlier paper says its result leads to “a different understanding” from the prevailing one. I'd treat the exact weighting as a strong, well-argued claim, not settled consensus. The direction of the effect, toward the bigger crystals, is the part that holds whichever weighting you use.
Strain. Small crystals are not the only cause of broad peaks. Lattice strain broadens them too, and it changes with angle differently than size broadening does. That is why careful work checks widths across several peaks before blaming size alone. In the ceria study, the authors treated their particles as strain free because “the peak widths show no systematic variation as a function of Q.” A single-peak Scherrer number can't make that check at all.
The upper limit. The equation also stops working for big crystals. Miranda and Sasaki, using dynamical diffraction theory, put that limit at “approximately 11.9% of the extinction length,” a value that depends on the material and the reflection. Past it, the simple relation between width and size no longer holds.
Two sources are preprints. The Valério and Morelhão papers are on arXiv and I found no journal record for either. The ceria paper's v2 PDF also carries a different title from its arXiv listing, so I cite the specific version I read.
Two sources are abstracts only. I read the Uvarov and Popov and the Miranda and Sasaki abstracts, not their full texts, which sit behind publisher paywalls. I quote only what the abstracts state.
This isn't my own method. My research is in microwave spectroscopy. I am explaining the published diffraction literature, not reporting XRD work of my own.
A Scherrer crystallite size is a measurement of crystal domains, read from a peak width that the instrument, the shape and the strain all help set. In a powder with a spread of sizes, it leans hard toward the biggest crystals, and it agrees with microscopy only for small crystals. Read it as “the size the diffraction peak reports,” ask how it was corrected, and never quote it as your average particle.
Scope note: this report explains what the Scherrer equation's output represents, using two openly available preprints read in full and two peer-reviewed abstracts. It does not evaluate any specific published crystallite-size result. No provider link: this beat carries no commercial stake.
U.S. Army combat veteran (Counter-IED / Electronic Warfare), peer-reviewed researcher in microwave spectroscopy, and founder & CEO of Shroombiosis. Consults on laboratory operations, AI, and supplement formulation.