Line Broadening
https://xraytools.com/broadening
Powder peaks get wider when the crystallites get smaller — and also when the lattice spacings vary. Give the positions and widths you measured and the page takes the instrument out, applies Scherrer to each peak, and then separates size from strain the only way a single peak cannot: by how the broadening grows with angle.
- Before this
- A measured line has a width, which an ideal lattice would not give — that width is the whole subject. Peak fitting is where the numbers this page wants come from.
- You supply
- A wavelength, your instrument’s own width, and a list of peaks — each one a 2θ and a full width at half maximum in degrees. Three peaks are the minimum for a Williamson–Hall fit and they are worth more the more widely they are spread.
- Reading it
- Every answer comes in two versions, because taking the instrument’s width out of a measured width depends on the peak shape and a real peak is between the two limits: the answer is inside the bracket, not at either end. L is a coherently diffracting domain and not a particle size, K = 0.9 is a convention worth about ten per cent on its own, and the instrumental width has to be measured on a standard — the default here is a plausible diffractometer and is not yours.
Worked examples: size only, simulated at 15 nm · strain only, simulated at 0.30% · both at once, simulated at 25 nm and 0.15% · a varying instrument, read as a flat width · the same peaks, read with the resolution function
Earlier on the path: Powder Indexing Next on the path: Intensity Corrections On From a measured scan to sample information, step 3 of 4
Notation here: K — what each one means here
See also: Bragg Calculator · HKL Calculator · Peak Finding · Powder Indexing · Reduced Cell and Bravais Lattice
What each input changes
- The peaks
- Positions and widths. The spread of angles is what separates size from strain, so two peaks close together cannot do it however precisely they are measured.
- How the instrumental width is given
- One number, or a curve. A real diffractometer is at its sharpest at some middle angle and wider on both sides of it, which a single number cannot say — and subtracting a flat width from a pattern takes off too much at one end and too little at the other, which tilts the Williamson–Hall line rather than shifting it. The tilt is the microstrain, so that is the answer it damages first.
Teaching with this page
- Objective
- After this page a learner can separate crystallite size from microstrain, and say what neither of them measures.
- Start from
- this worked example
- Ask first
- The analysis returns a crystallite size of 24 nm for a powder whose grains are visibly micrometres across. Is something wrong?
- Watch for
- “Yes — the size must match what the particles measure”
- Then
- Intensity Corrections
Check yourself: The analysis returns a crystallite size of 24 nm for a powder whose grains are visibly micrometres across. Is something wrong?
No — a grain is not a crystallite Yes — the size must match what the particles measure
Line broadening measures the coherently diffracting domain: the region over which the lattice is unbroken. One particle can be a mosaic of many such domains, so a large particle made of small domains gives exactly this result, and it is not a contradiction. Nothing in a diffraction pattern can see a particle.
Input
These widths are simulated
Nothing was submitted, so the box holds a pattern this site generated rather than measured: the powder simulator next door, run at Cu Kα with 25 nm crystallites, 0.15% microstrain and a half-Lorentzian peak shape, on the same 0.08° instrumental width set above. So the answer is known, and the interesting thing is that neither column below gives it — the two bracket it, because the real peak shape is between the two limits each column assumes. Paste your own widths over them.
The peaks you entered, drawn
The 7 peaks as you entered them, each drawn at its own measured width. The heights are all the same because this page was given no intensities — a Williamson–Hall analysis does not need them — so the only thing changing across the picture is the width, which is the point. Under each one, in a lighter line, is the instrument’s own profile at the 0.080° you set above, the same at every angle: wherever a measured peak is barely wider than the line under it, that peak is telling you almost nothing about the sample. Drawn as a pseudo-Voigt with η = 0.5, which is a choice about ink and not about the analysis — the two columns below refuse to choose, which is why there are two of them. Hover any peak to read its angle, its measured width, the instrumental width under it and what is left for the sample — as a range, because the two shape limits give different answers and this page will not pick one. There is no hover on a touch screen: the table below carries the same numbers.
The same peaks, moved onto one centre so the widths can be compared — at full scale above they are all thin spikes, because a peak half a degree wide is a few pixels of a sixty-degree scan, and that is what a real pattern looks like. The axis here is distance from each peak’s own centre, not 2θ: the peaks are not all at one angle, they have been stacked. Across your list the width grows by a factor of 1.62, from the low-angle peak to the high-angle one, and the dashed curve is the instrument, which does not grow at all. Everything between those two facts is the sample.
The instrument’s width is not one number
A diffractometer’s own contribution to a line’s width is not one number. It is narrowest somewhere in the middle of the range and wider at both ends, because the terms that make it up — the divergence of the beam, the width of the receiving slit, the spread of the emission line — do not grow the same way with angle. The standard three-parameter description of that curve is Caglioti’s, and it is what U, V and W mean in every Rietveld program:
FWHM² = U tan² θ + V tan θ + W
The three are fitted to a standard — LaB6 or silicon, the same measurement the single constant comes from — so they are numbers your instrument has rather than numbers you choose. Note what the constant is in these terms: U = V = 0 with W the square of the width. It is the same model with two of its three parameters set to zero, not a rival to it. Two honest limits. Caglioti, Paoletti and Ricci derived this for the collimators of a neutron powder diffractometer; on a laboratory X-ray instrument it is an empirical three-parameter fit rather than physics, which is why nobody quotes a U without saying which instrument it came off. And in a Rietveld refinement it describes the Gaussian part of the peak, the Lorentzian part carrying two parameters of its own; this page uses it as the whole instrumental width in both deconvolution rules, which is the same simplification the constant makes and no larger. This analysis used the flat width above, the same at every angle. Switch the selector in the form to see what the curve does to the same peaks.
Each peak on its own
The instrument’s own width has to be taken out of each measured width, and how depends on the peak shape: widths subtract for Lorentzian peaks (β = βobs − βinst) and squares subtract for Gaussian ones (β² = βobs² − βinst²). A real laboratory peak is a mixture of the two, so both are given and the answer is between them. They differ most when a peak is only a little wider than the instrument, which is exactly when a size is least worth quoting.
| 2θ / ° | FWHM / ° | Lorentzian β / ° | Lorentzian L / nm | Gaussian β / ° | Gaussian L / nm |
|---|---|---|---|---|---|
| 20.00 | 0.4009 | 0.3209 | 25.2 | 0.3928 | 20.6 |
| 30.00 | 0.4262 | 0.3462 | 23.8 | 0.4186 | 19.7 |
| 40.00 | 0.4566 | 0.3766 | 22.5 | 0.4495 | 18.8 |
| 50.00 | 0.4926 | 0.4126 | 21.3 | 0.4861 | 18.0 |
| 60.00 | 0.5354 | 0.4554 | 20.2 | 0.5294 | 17.3 |
| 70.00 | 0.5864 | 0.5064 | 19.2 | 0.5809 | 16.7 |
| 80.00 | 0.6477 | 0.5677 | 18.3 | 0.6427 | 16.1 |
A dash means the peak is not wider than the instrument under that shape rule, so it says nothing about the sample and is left out of the fit below.
Each row applies the Scherrer equation to that peak alone, which assumes all of its width beyond the instrument comes from crystallite size. Any microstrain in the sample is counted as size as well, so these are lower bounds: the real crystallites are at least this large and are usually larger. The plot below is what separates the two.
Williamson–Hall, Lorentzian peaks
The classical plot: β cos θ against 4 sin θ, with β in radians. Size broadening does not depend on the abscissa at all, so it sits in the intercept as Kλ/L; strain broadening is proportional to it, so it is the slope, and the slope is ε directly.
| Crystallite size L | 29.6 | nm, from the intercept Kλ/L |
|---|---|---|
| Microstrain ε | 0.111 | %, from the slope |
| Peaks used | 7 | of 7 given |
| R² of the fit | 0.9975 | how much of the points’ spread the line accounts for |
Williamson–Hall, Gaussian peaks
Squares of both axes: β²cos²θ against (4 sin θ)², which is the form that follows when the two contributions add in quadrature. The intercept is (Kλ/L)² and the slope is ε², so both answers come out through a square root and a small negative value of either is not a small answer — it is no answer.
| Crystallite size L | 20.7 | nm, from the intercept Kλ/L |
|---|---|---|
| Microstrain ε | 0.213 | %, from the slope |
| Peaks used | 7 | of 7 given |
| R² of the fit | 0.9893 | how much of the points’ spread the line accounts for |
What the pair of them supports
Crystallite size. Between 20.7 nm (Gaussian) and 29.6 nm (Lorentzian).
Microstrain. Between 0.111% (Lorentzian) and 0.213% (Gaussian).
Both shape rules describe these peaks. Where they give a range, the answer is inside it and the width of the range is what it costs not to know the peak shape.
Four ways this goes wrong with the arithmetic right
| L is not a particle size | What diffraction measures is a volume-weighted mean column length along the scattering vector — the coherently diffracting domain. A particle made of several domains, or one with a fault running through it, gives an L smaller than anything an electron microscope would measure on the same powder, and the two disagreeing is the normal case rather than a discrepancy to resolve. |
|---|---|
| K is a convention, not a constant | 0.9 is the usual value for the full width at half maximum of roughly spherical crystallites. 0.89, 0.94 and 1.0 are all in print, for different shapes and for the integral breadth rather than the FWHM, so a size from this page is good to about ten per cent before anything else is considered. Quote the K you used. |
| The instrumental width has to be measured | The value here defaults to this site’s own 0.08°, which is a plausible well-aligned laboratory diffractometer and is not your instrument. It is measured on a standard with no size or strain broadening of its own — LaB6 or silicon — and it varies with angle, which this analysis is treating as constant. The panel above names the function that does not. |
| One peak cannot separate size from strain | Both broadenings grow with angle, so a single width is satisfied by a whole family of (L, ε) pairs. Only the difference in how they grow — 1/cos θ against tan θ — tells them apart, and reading that difference needs peaks spread as widely in 2θ as the pattern allows. Three clustered peaks are worth less than three spread ones. |
Where this comes from
- X-ray line broadening from filed aluminium and wolfram
G. K. Williamson and W. H. Hall, Acta Metall. 1953, 1, 22–31 · doi:10.1016/0001-6160(53)90006-6
The separation this page’s plot performs, in the paper that introduced it. The two contributions are told apart by their angular dependence alone — which is why several peaks are needed and why one is not enough, whatever its width. - The diffraction of X rays by distorted crystal aggregates - I
A. R. Stokes and A. J. C. Wilson, Proc. Phys. Soc. 1944, 56, 174–181 · doi:10.1088/0959-5309/56/3/303
Where the strain term 4ε tan θ comes from, and where L is defined as a mean column length rather than a particle size — the distinction the table of traps on this page spends the most words on. - Choice of collimators for a crystal spectrometer for neutron diffraction
G. Caglioti, A. Paoletti and F. P. Ricci, Nucl. Instrum. 1958, 3, 223–228 · doi:10.1016/0369-643X(58)90029-X
The resolution function this page offers instead of a single instrumental width: FWHM² = U tan² θ + V tan θ + W, which is what those three letters mean in every Rietveld program. Worth knowing what it was written about: neutron collimators, three decades before it became the standard empirical fit to a laboratory X-ray instrument’s measured widths.