Scherrer Equation Simulator — Crystallite Size from X-Ray Diffraction
Use the Scherrer equation D = K lambda / (beta cos theta) to compute, in real time, the crystallite size D, corrected FWHM beta_corr, uncorrected D and applicability flag (D less than 100 nm) from X-ray wavelength lambda, Bragg angle theta, measured FWHM beta_total and instrumental FWHM beta_inst. The tool visualises sample, instrument and total peak shapes and the D-beta hyperbola.
Parameters
X-ray wavelength lambda
A
Bragg angle theta
deg
Measured FWHM beta_total
deg
Instrumental FWHM beta_inst
deg
Defaults are lambda = 1.54 A (Cu K-alpha), theta = 30 deg, beta_total = 0.50 deg, beta_inst = 0.10 deg and K = 0.94 (spherical crystallite, FWHM basis). It is standard practice to calibrate beta_inst on a coarse standard (LaB6 or Si with crystallites larger than 1 um). Besides the Gaussian assumption beta_corr^2 = beta_total^2 - beta_inst^2, Lorentzian (linear subtraction) and Voigt-function analyses are also available.
Horizontal axis = corrected FWHM beta_corr (deg, log) / vertical axis = crystallite size D (nm, log) / blue = D inversely proportional to beta hyperbola / red dashed = D = 100 nm applicability boundary / yellow marker = current (beta_corr, D)
Theory & Key Formulas
The Scherrer equation estimates the crystallite size $D$ from the FWHM $\beta$ of an X-ray diffraction peak:
$$D = \frac{K\,\lambda}{\beta\,\cos\theta}$$
$K$ is the Scherrer constant (about 0.94 for spherical crystallites on a FWHM basis), $\lambda$ is the X-ray wavelength, $\beta$ is the corrected FWHM in radians and $\theta$ is the Bragg angle. The instrumental width $\beta_{\text{inst}}$ is removed from the measured width $\beta_{\text{total}}$ under a Gaussian assumption:
The equation is reliable for $D < 100$ nm (nano- to micro-crystalline regime); beyond this the size contribution falls below the instrumental width. To separate crystallite size from microstrain use the Williamson-Hall method:
$\lambda$ is the X-ray wavelength (A), $\theta$ and $\beta$ are angles (radians after conversion) and $D$ is the crystallite size (A; displayed in nm in this tool). 1 nm equals 10 A.
What is the Scherrer Equation Simulator?
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Do I really have to measure catalyst nanoparticles one by one in an electron microscope? I've heard XRD can do it, but how does a "peak width" turn into a particle size?
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Good observation. The link between XRD peak width and particle size is the Scherrer equation D = K lambda / (beta cos theta). When the coherent scattering domain (the crystallite) is small, a sharp Bragg line broadens into a peak because there are fewer crystal planes contributing to constructive interference. With the defaults of this tool (Cu K-alpha, theta = 30 deg, beta_total = 0.50 deg, beta_inst = 0.10 deg) the Results panel shows corrected FWHM about 0.490 deg, crystallite size D about 19.5 nm and Valid — right in the typical catalyst-particle range (5 to 50 nm).
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There's both a corrected FWHM and an uncorrected D in the panel. Why does the correction matter?
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The raw width beta_total combines sample broadening with instrumental broadening beta_inst from the X-ray tube, slits and monochromator. Assuming both are Gaussian, you remove the instrumental part with beta_corr squared equals beta_total squared minus beta_inst squared. Sweep beta_inst from 0.10 to 0.30 deg in this tool and the gap between corrected D and uncorrected D grows quickly. Skipping the correction systematically underestimates particle size and is one of the most common rookie mistakes in XRD analysis.
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What does "applicability" mean exactly? Doesn't the Scherrer equation work all the time?
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Far from it. When crystallites get too large (D above about 100 nm) their contribution to peak broadening drops below the instrumental width and the extracted size becomes unreliable. Shrink beta_total to 0.1 deg in this tool and D climbs past 100 nm and the badge flips to Out of range. Very small crystallites (D below 2 nm) violate the assumption of a well-ordered crystal. Also remember that microstrain (non-uniform strain) broadens peaks; to separate it from size you need the Williamson-Hall method (a cos theta versus sin theta plot).
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Is K = 0.94 fixed, or does the particle shape matter?
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The shape matters. K depends on the crystallite shape and the way "width" and "size" are defined. For spheres with FWHM and volume-averaged diameter, K is about 0.94. Cubes give K about 0.89, octahedra and columnar grains span K = 0.8 to 1.1. Many papers silently use K = 0.9 or K = 0.94, so any Scherrer-derived D carries roughly a 10 percent systematic uncertainty. The best validation is to cross-check with transmission electron microscopy. This tool keeps K = 0.94 fixed; if your shape demands a different value, rescale the output by the literature factor.
Frequently Asked Questions
The Scherrer equation D = K lambda / (beta cos theta) estimates the crystallite size D (coherent scattering domain) from the broadening of an X-ray diffraction peak. It was derived by Paul Scherrer in 1918. Here K is the Scherrer constant (about 0.94 for spherical crystallites), lambda is the X-ray wavelength, beta is the corrected full-width at half maximum (FWHM) in radians, and theta is the Bragg angle. With the defaults (Cu K-alpha at lambda = 1.54 A, theta = 30 deg, beta_total = 0.50 deg, beta_inst = 0.10 deg) this tool shows corrected FWHM beta_corr about 0.490 deg, crystallite size D about 19.5 nm, uncorrected D about 19.2 nm and Valid (D less than 100 nm).
The measured peak width beta_total combines the broadening from the sample crystallite size and the instrumental broadening beta_inst caused by X-ray tube divergence, monochromator and slits. Assuming both are Gaussian, the corrected width is beta_corr = sqrt(beta_total^2 - beta_inst^2). Skipping this correction systematically underestimates the crystallite size, with the largest error when beta_inst is close to beta_total. Sliding beta_inst from 0.10 to 0.40 deg in this tool clearly shows how the corrected D changes. In practice beta_inst is calibrated on a standard sample (LaB6 or Si with crystallites larger than 1 um).
The Scherrer equation is reliable for crystallite sizes D below about 100 nm (the nano- to micro-crystalline regime). For larger crystallites the size contribution to peak broadening drops below the instrumental width and the extracted size is no longer reliable. Microstrain (non-uniform strain) also broadens peaks, so to separate the two effects one applies the Williamson-Hall method (cos theta versus sin theta plot). When you reduce beta_total in this tool so that D exceeds 100 nm, the applicability badge switches from Valid to Out of range. Typical applications include catalyst nanoparticles (5 to 50 nm), Li-ion battery cathodes and magnetic nanoparticles.
The Scherrer constant K is a dimensionless factor that depends on the crystallite shape and the definition of width and size. For spherical crystallites with FWHM and volume-averaged diameter, K is about 0.94. Cubic crystallites give K about 0.89, while octahedral or columnar particles span K = 0.8 to 1.1. Many papers silently use K = 0.9 or K = 0.94, so the Scherrer-derived D carries roughly a 10 percent systematic uncertainty. This tool fixes K = 0.94; for non-spherical shapes the result should be rescaled with a literature value. Calibration against transmission electron microscopy (TEM) imaging is the recommended best practice.
Real-World Applications
Catalyst nanoparticle sizing: Automotive three-way catalysts (Pt/Pd/Rh on gamma-alumina), fuel-cell electrocatalysts (Pt/C) and PEMFC Pt-Ru alloys all depend strongly on particle size for their activity. Scherrer analysis of XRD peak widths combined with TEM imaging is the standard particle-sizing workflow in catalyst research. Sweeping beta_total from 0.5 to 2.0 deg in this tool walks D from about 19 nm down to 4.9 nm, covering the typical catalyst-particle window (5 to 50 nm). Catalyst ageing (particle coarsening) is routinely tracked as Bragg-peak sharpening in successive XRD scans.
Lithium-ion battery cathodes: LiCoO2, LiFePO4 and NMC (Ni-Mn-Co oxide) cathodes have crystallite sizes that strongly influence capacity, cycle life and rate capability. Engineers control Scherrer-derived crystallite sizes around 30 to 100 nm and combine that with nanocomposite or coating strategies for high-capacity, long-life cells. The applicability check in this tool flags when the Scherrer regime breaks down — a useful prompt to switch to Williamson-Hall analysis or direct TEM imaging.
Metal-oxide semiconductor gas sensors: SnO2, ZnO and WO3 thin films become more sensitive as their crystallite size drops below the Debye length (around 10 nm or less). Scherrer analysis quantifies the nanocrystalline regime during annealing-temperature and precursor optimisation. Changing K in this tool helps build intuition for how strongly the assumed crystallite shape biases the extracted size.
Magnetic nanoparticles and thin films: Magnetic recording media, MRI contrast agents, magnetic separation carriers and hyperthermia therapeutics all rely on size-dependent magnetic properties. Below about 10 nm a transition to superparamagnetism reshapes device behaviour. Scherrer sizing tunes the crystallite distribution and is paired with VSM or SQUID magnetometry to close the design loop. The corrected versus uncorrected D values in this tool make the importance of instrumental broadening immediately tangible.
Common Misconceptions
The most common misconception is that the Scherrer D is the particle size itself. In fact D is the volume-averaged coherent scattering domain. A particle that contains multiple crystallites (twin domains or polycrystalline grains) shows a Scherrer D smaller than its true particle size; a particle that connects to its neighbours epitaxially can show a D larger than its individual size. The Scherrer-derived D from this tool is an XRD-only number and must be cross-checked against TEM or SEM imaging before being called a "particle size".
The second is that instrumental broadening is small enough to ignore. Even on a modern high-resolution diffractometer beta_inst is roughly 0.05 to 0.15 deg, and as the crystallite grows (D above about 30 nm) the gap between beta_total and beta_inst shrinks, so the Gaussian-corrected beta_corr becomes very sensitive to small errors. Push beta_total close to beta_inst in this tool and the corrected D diverges almost immediately. A rigorous treatment calibrates beta_inst as a function of 2 theta on a standard (NIST SRM 660c LaB6, Si standard) and fits it with the Caglioti relation beta_inst squared (2 theta) = U tan squared theta + V tan theta + W.
The third is that a Gaussian assumption always works. Real peak profiles are usually mixtures of Gaussian and Lorentzian (pseudo-Voigt) or full Voigt functions. With significant Lorentzian content, beta_corr = beta_total - beta_inst (linear subtraction) is more appropriate, and a Gaussian model under-corrects. Rietveld refinement or full-pattern fitting (FullProf, TOPAS, GSAS-II) refines pseudo-Voigt parameters and is paired with Williamson-Hall analysis to separate crystallite size from microstrain. This tool covers the Gaussian-approximation entry point; publication-grade analysis requires dedicated software.