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Crystal Structure
How to Subtract Instrumental Broadening from a Diffractogram in 5 Easy Steps
A Practical Correction Workflow — From Raw Peak Width to True Sample Broadening, Using a Standard Reference Material and Proper Deconvolution
Every peak on an XRD diffractogram has a width, and that width carries real physical information — but only if you first remove the width that your instrument adds on its own. If you skip this correction and feed a raw, uncorrected FWHM (full width at half maximum) straight into the Scherrer equation (see Patterson's exact derivation of the constant, 1939), you are not measuring your sample. You are measuring your sample plus your diffractometer, and reporting the sum as if it were the sample alone. This is one of the most common — and most consequential — errors made by newcomers to powder XRD analysis, and it is entirely avoidable once you understand five straightforward steps.
To subtract instrumental broadening from a diffractogram: (1) measure a strain-free standard reference material (e.g., NIST SRM 660c or SRM 640 Si) on the exact same instrument setup as your sample; (2) record its peak FWHM values across 2θ; (3) fit these values to the Caglioti equation to obtain a continuous instrumental resolution function; (4) evaluate that function at your sample peak's position; and (5) deconvolute — subtract the instrumental FWHM from the observed sample FWHM using the correct rule for the peak shape (quadrature for Gaussian, linear for Lorentzian) — leaving only the true, sample-only broadening for use in the Scherrer equation or Williamson-Hall analysis.
Hub: Characterization Hub | Tutorial: Instrumental Broadening Correction | Builds on: XRD Foundation Course & How to Read an XRD Graph in 7 Easy Steps
Reading time: 35 minutes | Includes: 5-step correction workflow, Caglioti equation, Gaussian/Lorentzian/Voigt deconvolution rules, worked numerical example, direct answers to common practitioner questions, MCQs, FAQ, key takeaways
SEO Keywords: instrumental broadening XRD, subtract instrumental broadening diffractogram, Caglioti equation, standard reference material XRD, NIST SRM 660c, Scherrer equation correction, Williamson-Hall method, Voigt deconvolution
Before You Begin — Why Peak Width Is Never Just About the Sample
Think about photographing a distant streetlight at night with a slightly out-of-focus camera. Even though the streetlight is a single, infinitesimally small point of light, the photograph shows a blurred disc. That blur is not a property of the streetlight — it is a property of the camera's optics. If you wanted to know the true size of the light source, you would first need to know how much blur your particular camera adds, and then subtract that contribution from what you observe.
An X-ray diffractometer behaves in exactly the same way. If you have not yet covered how a diffractometer is put together, our XRD Foundation Course is a good place to review the optical path first. Even a theoretically perfect, infinitely large, defect-free single crystal would still produce a peak of finite width when measured on a real instrument, because the instrument itself — its beam divergence, slit widths, and detector resolution — introduces broadening independent of the sample. This is called instrumental broadening, and it convolves (mathematically overlaps) with two sample-dependent contributions to produce the peak width you actually observe:
| Contribution | Physical origin | Depends on the sample? |
|---|---|---|
| Instrumental broadening | Beam divergence, slit geometry, detector resolution, wavelength dispersion (Kα1/Kα2) | No — depends only on the diffractometer and its configuration |
| Crystallite size broadening | Finite number of diffracting planes in small coherent domains | Yes — described by the Scherrer equation (see Patterson's 1939 exact derivation) |
| Microstrain broadening | Small local variations in d-spacing caused by dislocations, defects, or residual stress | Yes — separated from size using the Williamson-Hall method |
The observed FWHM, βobs, is therefore not a single physical quantity — it is a convolution of all three. If you skip the instrumental correction, you are silently assuming βinst = 0. Since the instrument never actually contributes zero, this assumption inflates the apparent sample broadening, which in turn makes the Scherrer calculator report a crystallite size that is smaller than the true value — sometimes by an order of magnitude, particularly for well-crystallised, larger-grained samples where the true sample broadening is only slightly greater than the instrumental floor, a pitfall covered in more depth in our 10 Common XRD Mistakes Beginners Make.
For very small nanocrystallites (a few nanometres), sample broadening is so large relative to instrumental broadening that skipping the correction introduces only modest error. But for well-crystallised ceramics, thin films, and micron-scale grains — precisely the materials where accurate grain size values matter most for mechanical and electronic property correlations — the sample broadening can be comparable to or even smaller than the instrumental broadening. In these cases, omitting the correction does not just add noise; it can invalidate the result entirely.
Step 1 — Choose the Right Standard Reference Material (SRM)
The entire correction workflow rests on one idea: if you measure a sample that has no size broadening and no strain broadening — meaning its crystal structure is essentially defect-free, in the sense described in our introduction to crystal structure — then whatever width you observe from it must be pure instrumental broadening. Such a sample is called a standard reference material (SRM) — a certified powder with large, defect-free, strain-free crystallites, produced and certified by a metrology body such as the National Institute of Standards and Technology (NIST).
| Standard | Material | Typical use |
|---|---|---|
| NIST SRM 660c | Lanthanum hexaboride (LaB₆) | The preferred line-profile standard for most laboratory and synchrotron diffractometers; sharp, well-separated cubic peaks across a wide 2θ range |
| NIST SRM 640 | Silicon (Si) | Widely available, excellent for both line-position and line-profile calibration |
| NIST SRM 676a | Alumina (Al₂O₃) | Often used for quantitative phase analysis calibration alongside profile fitting |
If the standard itself had small crystallites or internal strain, its own peaks would carry size or strain broadening, contaminating your instrumental profile. Certified SRMs are manufactured and verified specifically to be free of these effects — see the NIST SRM 660c certificate of analysis for the certified line-profile parameters — so their measured width is attributable to the instrument alone. This is the entire logic of the method.
Step 2 — Record the Standard Under Identical Instrument Conditions
Run the SRM on the same diffractometer, with the same optics (divergence slits, receiving slits, monochromator or Ni filter, detector type), the same tube and wavelength, and ideally the same sample-holder geometry as your actual experiment. Instrumental broadening is instrument- and configuration-specific — a profile recorded on a different machine, or with different slit settings, does not describe your setup, a point also emphasised in IUCr's powder diffraction teaching pamphlets.
Collect the pattern over a wide 2θ range, capturing as many resolvable SRM peaks as possible, since instrumental broadening is not constant with angle — it typically increases at low angles and again at very high angles, a behaviour we quantify in the next step and which builds directly on the peak-reading skills from How to Read an XRD Graph in 7 Easy Steps.
Of all the optical elements, the divergence slit width has the single largest effect on peak broadening in Bragg-Brentano geometry. If your sample data were collected with a different slit than your standard data, the "instrumental correction" you calculate will not actually match your sample measurement, and the deconvolution will be invalid.
Step 3 — Fit the Instrumental Resolution Function (the Caglioti Equation)
Once you have FWHM values for each SRM peak at its respective 2θ position, you need a way to estimate the instrumental broadening at any angle — including angles where your sample has a peak but the standard does not. This is done using the Caglioti equation, first proposed for neutron diffraction resolution functions and now standard practice in X-ray line-profile analysis:
You determine U, V, and W by plotting FWHM² against tanθ for every resolvable SRM peak and fitting a quadratic — either by hand using least-squares regression, or automatically inside Rietveld software such as GSAS-II or FullProf. Once U, V, and W are known, the instrumental FWHM at any 2θ across your pattern — including positions where the standard itself has no peak — can be calculated directly from the equation.
Think of U, V, and W as three coefficients that draw a smooth curve describing exactly how blurry your particular camera (diffractometer) is at every possible zoom level (2θ angle). Once you have that blur map from a single calibration shoot (the SRM pattern), you can predict the blur at any angle in any future photograph (sample pattern) taken with the same camera settings, without needing to recalibrate every time.
Step 4 — Deconvolute: Choosing the Correct Subtraction Rule
This is the step where most textbook treatments oversimplify, and where the correction can silently go wrong if applied carelessly — the kind of subtle error catalogued in our 10 Common XRD Mistakes Beginners Make. Peak shapes in powder XRD are rarely purely one mathematical form — they are usually described as a pseudo-Voigt profile, a weighted combination of a Gaussian component and a Lorentzian (Cauchy) component. Each component subtracts differently:
| Peak shape assumption | Deconvolution rule | When it applies |
|---|---|---|
| Gaussian | βsample² = βobs² − βinst² (quadrature / Warren's method) | Broadening dominated by strain, or instruments with highly Gaussian optics (e.g., synchrotron, capillary geometry) |
| Lorentzian (Cauchy) | βsample = βobs − βinst (simple linear subtraction) | Broadening dominated by small crystallite size in many laboratory Bragg-Brentano setups |
| Pseudo-Voigt (mixed) | Separate the Gaussian and Lorentzian full widths first, subtract each component using its own rule, then recombine | The general, most physically accurate case — handled automatically by full-pattern refinement software |
It is tempting to always write βsample = βobs − βinst because it looks simple. But if the true peak shapes are closer to Gaussian, this linear approach overcorrects, subtracting too much width and yielding an artificially large crystallite size. Always check — or let refinement software determine — whether your profile shape is closer to Gaussian, Lorentzian, or a genuine mixture before choosing the subtraction rule.
In practice, this is exactly why full-pattern Rietveld software is preferred over manual FWHM subtraction: programs such as GSAS-II model each peak as a Thompson-Cox-Hastings pseudo-Voigt function, separately tracking the Gaussian and Lorentzian widths contributed by the instrument and by the sample, and applying the correct deconvolution rule to each component automatically and consistently across the entire pattern.
Step 5 — Recover the True Sample Broadening and Apply It
Once βsample has been correctly isolated for each peak, you can now proceed with confidence to the analyses that depend on it:
- Scherrer equation for average crystallite size: D = Kλ / (βsample·cosθ), using only the corrected, instrument-free FWHM — try it directly with our Scherrer XRD Calculator.
- Williamson-Hall analysis to separate size and strain contributions across multiple peaks, plotting βsamplecosθ against 4sinθ to extract size from the intercept and microstrain from the slope — see the Williamson-Hall method overview.
- Full Rietveld refinement, where the instrument profile file is locked from Step 3, and only sample-dependent size/strain parameters are refined against the experimental pattern, following the same crystal-symmetry conventions covered in Unit Cell and Lattice Parameters in Crystallography.
If your corrected crystallite size comes out larger than roughly 200–300 nm, be cautious — beyond this range, peak broadening from size effects becomes so small that it approaches the resolution limit of laboratory XRD, and the Scherrer equation becomes unreliable regardless of how carefully the instrumental correction was performed. In that regime, techniques such as electron backscatter diffraction (EBSD) or optical microscopy are more appropriate for grain size determination.
Worked Example — Full Correction Walkthrough
Suppose your sample's (200) peak appears at 2θ = 44.5° — a peak position you would read off using the method in How to Read an XRD Graph in 7 Easy Steps — with an observed FWHM βobs = 0.45°. Using your SRM 660c calibration, the Caglioti equation gives an instrumental FWHM at this same angle of βinst = 0.09°. Both peaks are reasonably well described as Gaussian in this laboratory configuration, so we use quadrature subtraction.
Note: this example uses illustrative FWHM values for teaching purposes. Always determine β_inst from your own instrument's SRM calibration — a universal correction factor does not exist because every diffractometer's optical configuration is different. You can cross-check the final crystallite-size arithmetic using the Scherrer XRD Calculator.
Answering a Real Practitioner Debate — Is This "Repetition" or Rigorous Method?
This exact workflow was recently the subject of a thoughtful exchange among practising materials scientists on LinkedIn. One experimental physicist pointed out that relying on manual FWHM measurements and the isolated Scherrer equation is a trap many students fall into, since the Scherrer equation assumes all broadening comes from crystallite size and ignores both microstrain and instrumental broadening entirely. A second researcher then asked a very fair and precise question: how do you actually know this — is it repetition from experience, or is there a rigorous procedure — and specifically, how does a standard reference material "zero out" the pattern refinement in practice?
The answer is not repetition; it is the underlying convolution mathematics we have walked through above, and it is worth restating clearly and directly:
The observed peak width is a convolution of instrument broadening, crystallite size broadening, and microstrain broadening. If you use the raw FWHM directly in the Scherrer equation, you are implicitly assuming that the instrumental and strain contributions are exactly zero. Because a real diffractometer never contributes zero broadening, this assumption artificially inflates the width attributed to the sample, which drastically underestimates the true crystallite size.
To "zero out" this baseline: run a defect-free, large-grained standard reference material — such as NIST SRM 660c (LaB₆) or SRM 640 (Si) — using the exact same optics as your sample. In full-pattern software such as GSAS-II, fix the standard's known crystal structure and refine only the peak-shape parameters. This maps the instrument's unique broadening function across the full 2θ range via the Caglioti equation. That instrument profile is then saved and locked, and the experimental sample pattern is loaded in its place. The software mathematically subtracts the fixed instrument profile from the total observed peak shape — following the Gaussian, Lorentzian, or mixed pseudo-Voigt rule appropriate to the data — leaving pure sample broadening. Only then can size and microstrain be reliably separated from one another, typically via Williamson-Hall analysis or joint refinement.
In short: the confidence behind this method comes from the physics of convolution and from a well-established, internationally standardised calibration procedure using certified reference materials — not from informal experience alone, though experienced practitioners will recognise the tell-tale signs (unrealistically small crystallite sizes, or Scherrer results that vary implausibly between similar samples) described in our 10 Common XRD Mistakes Beginners Make, and codified in the IUCr Commission on Powder Diffraction's round-robin calibration guidelines.
Applications of Instrumental Broadening Correction
Nanoparticle and Nanomaterial Characterisation
Accurate, instrument-corrected crystallite size is essential in nanotechnology, where the Scherrer equation is routinely used as a fast, non-destructive alternative to electron microscopy. Without correction, reported "nanoparticle sizes" in the literature can be systematically biased small, complicating comparisons between different laboratories using different diffractometers — a concern echoed in the nanoparticle characterisation literature.
Residual Stress and Microstrain Analysis in Metals and Ceramics
Williamson-Hall and related strain-analysis methods depend entirely on correctly isolated sample broadening. In engineering ceramics such as the donor/acceptor doped bismuth sodium titanate systems studied in perovskite ferroelectrics research — the kind of layered, distorted unit cells introduced in Unit Cell and Lattice Parameters in Crystallography — distinguishing genuine microstrain from residual instrumental width is essential for correlating structure with piezoelectric performance, as documented in Verma and Rout's study of BNT ceramics.
Quality Control Across Multiple Instruments
Industrial and multi-site laboratories often need to compare XRD results collected on different diffractometers. Because each instrument has its own U, V, W Caglioti parameters, performing the correction independently for each machine allows size and strain results to be meaningfully compared across sites — something impossible with raw, uncorrected FWHM values, as recommended in NIST's inter-laboratory calibration guidance.
Series Navigation
Foundations: XRD Foundation Course | Crystal Structure — Introduction | Unit Cell and Lattice Parameters in Crystallography
Previous: How to Read an XRD Graph in 7 Easy Steps | FWHM of X-ray Peaks in XRD
Related: 10 Common Mistakes Beginners Make While Reading XRD Graphs | Amorphous vs. Crystalline XRD Patterns | Scherrer XRD Calculator Tool
Practice Questions
Frequently Asked Questions
Because the raw FWHM is a convolution of instrumental broadening, crystallite size broadening, and microstrain broadening. Using it directly implicitly assumes the instrument contributes zero broadening, which is never physically true, and this inflates the apparent sample width — underestimating crystallite size.
A standard reference material (SRM) is a certified, defect-free, strain-free crystalline powder with large crystallites — commonly NIST SRM 660c (LaB₆) or SRM 640 (Si). Because it has no size or strain broadening of its own, its measured peak width under given instrument conditions represents pure instrumental broadening, which can then be mapped and subtracted from real sample data.
The Caglioti equation, FWHM² = U·tan²θ + V·tanθ + W, describes how instrumental peak width varies smoothly across the entire 2θ range of a diffractometer. Fitting U, V, and W to a standard's peaks gives a continuous instrumental resolution function that can be evaluated at any peak position in the sample pattern, even where no standard peak exists.
It depends on peak shape. If both profiles are Gaussian, widths subtract in quadrature: β_sample² = β_obs² − β_inst². If both are Lorentzian, widths subtract linearly: β_sample = β_obs − β_inst. Real peaks are usually pseudo-Voigt (a mixture), so full-pattern software separates the Gaussian and Lorentzian components before applying the correct rule to each.
Yes. In GSAS-II or FullProf, you refine the instrument profile against a standard with its structure fixed, save and lock that instrument profile, then load the experimental sample and refine only the sample-dependent size and strain terms. This automates the deconvolution described in this tutorial.
Key Takeaways
Here is a summary of the five-step workflow for correcting instrumental broadening in a diffractogram:
Choose a certified, defect-free, large-grained standard reference material such as NIST SRM 660c (LaB₆) or SRM 640 (Si).
Record the standard under identical instrument conditions — same optics, slits, wavelength, and geometry as your sample.
Fit FWHM² vs. tanθ to the Caglioti equation (FWHM² = U·tan²θ + V·tanθ + W) to build a continuous instrumental resolution function.
Deconvolute using the correct rule for peak shape: quadrature for Gaussian, linear for Lorentzian, or component-wise for mixed pseudo-Voigt profiles.
Apply the corrected β_sample to the Scherrer equation, Williamson-Hall analysis, or full Rietveld refinement — never the raw, uncorrected FWHM.
The master principle: observed peak width = instrument ⊛ size ⊛ strain (convolution). Skipping instrumental correction silently assumes the instrument contributes zero width, which is never true.
References
All references are in IEEE citation style. All sources are peer-reviewed journals, internationally recognised textbooks, or authoritative academic databases.
- B. D. Cullity and S. R. Stock, Elements of X-Ray Diffraction, 3rd ed. Upper Saddle River, NJ, USA: Pearson Prentice Hall, 2001. — Definitive reference for peak broadening theory, instrumental resolution, and the Scherrer equation.
- G. Caglioti, A. Paoletti, and F. P. Ricci, "Choice of collimators for a crystal spectrometer for neutron diffraction," Nucl. Instrum., vol. 3, no. 4, pp. 223–228, 1958, doi: 10.1016/0369-643X(58)90029-X. — Original publication of the Caglioti resolution function, now standard practice in X-ray line-profile analysis.
- P. Scherrer, "Bestimmung der Größe und der inneren Struktur von Kolloidteilchen mittels Röntgenstrahlen," Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, pp. 98–100, 1918. — Original publication of the Scherrer equation.
- G. K. Williamson and W. H. Hall, "X-ray line broadening from filed aluminium and wolfram," Acta Metall., vol. 1, no. 1, pp. 22–31, 1953, doi: 10.1016/0001-6160(53)90006-6. — Foundational reference for separating size and strain broadening.
- National Institute of Standards and Technology (NIST), "Standard Reference Materials for X-ray Diffraction." [nist.gov — SRM 660c, SRM 640 certification data] — Authoritative source for certified line-profile and line-position standards.
- B. H. Toby and R. B. Von Dreele, "GSAS-II: the genesis of a modern open-source all purpose crystallography software package," J. Appl. Cryst., vol. 46, pp. 544–549, 2013, doi: 10.1107/S0021889813003531. [GSAS-II] — Reference for automated instrument-profile refinement and deconvolution.
- International Union of Crystallography (IUCr), "Line Broadening Analysis," IUCr Teaching Pamphlets. [iucr.org] — Authoritative tutorial on separating instrumental, size, and strain broadening.
- R. Verma and S. K. Rout, "Frequency-dependent ferro–antiferro phase transition and internal bias field influenced piezoelectric response of donor and acceptor doped bismuth sodium titanate ceramics," J. Appl. Phys., vol. 126, no. 9, Art. no. 094103, Sep. 2019, doi: 10.1063/1.5111505. — Author's research applying rigorous XRD peak-profile analysis in perovskite ferroelectric ceramics.
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