A measured rocking-curve width is not automatically the crystal mosaic spread. Beam divergence, wavelength bandpass, crystal size, strain, scan geometry, sampling and the fitted profile can contribute. The reader’s job is to preserve those terms before assigning a width to the crystal.
The central invented case uses an observed Gaussian FWHM of0.18° and a matched instrument contribution of0.10°. Under independent Gaussian broadening, the deconvolved contribution is sqrt(0.18²-0.10²), about0.150°. A Lorentzian assumption would give0.08°, showing why the model cannot be hidden.
This is an educational diffraction example. It does not prescribe a beamline setup, certify crystal quality, replace facility procedures or claim that two-component deconvolution describes every experiment.
A careful width audit also keeps the reflection, scan axis, angular step, exposure, background model, profile family and uncertainty together. When the instrument contribution is close to the observed width, deconvolution becomes especially sensitive: a small change in either fitted width can create a large relative change in the inferred crystal contribution. Multiple reflections and a matched reference measurement therefore provide stronger evidence than one visually narrow peak. The changed Gaussian and Lorentzian results are deliberately placed side by side so readers can see that a subtraction rule belongs to a model, not to the degree symbol printed beside a number. A residual plot, repeated scan and uncertainty interval can reveal whether the fitted width is stable enough for the intended comparison. They cannot make an omitted broadening mechanism disappear. Record the raw scan and the fitted model together so later reviewers can test a different profile without reconstructing missing observations.
This specialist guide belongs to the X-ray Crystallography root. It also connects to The Gold Standard of Education | Learning That Transfers into Life and the How X Works hub. The worked cases are original teaching examples; their numbers are stipulated so that the reasoning can be checked.
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Treat the measured rocking width as a combined observation and deconvolve a crystal contribution only under an explicit profile and instrument model.
ROUTE 5 · CHAPTERS 21–24
Report and continue
Repair shortcuts, state limits and follow verified routes.
Chapter contents
Define the quantities · Chapters 1–5
Work the mechanism · Chapters 6–10
6. Complete the Gaussian deconvolution
7. Check the impossible subtraction
8. Compare Lorentzian broadening
Audit the evidence · Chapters 11–15
11. Match the reflection and instrument calibration
Transfer the reasoning · Chapters 16–20
16. Transfer to data-collection strategy
17. Transfer to crystal comparison
18. Transfer to optics changes
Report and continue · Chapters 21–24
21. Repair observed width equals mosaicity
22. Repair narrower always means better
Start by naming the physical or logical quantity rather than the displayed number. An observed rocking-curve width contains crystal and instrument contributions; mosaic spread is a crystal contribution after the stated instrument model is accounted for.
Work the original teaching case in a visible order. The teaching scan reports an observed Gaussian FWHM of0.18° and a beam contribution of0.10°.
Make the reasoning independently checkable. Label every width by source and convention.
Test transfer by changing one condition while holding the others still. Hold the crystal fixed and change collimation.
Keep the claim boundary visible. The observed width is not automatically the crystal mosaic spread.
Before accepting the result, predict what should change and what should remain fixed. Apply this check: Label every width by source and convention. Then run the changed case: Hold the crystal fixed and change collimation. Record any mismatch without rewriting the original assumptions.
The evidence receipt for “Name the observed and latent widths” records source, input, transformation, changed-context prediction and limit on separate lines. Keep reflection, scan axis, width convention, instrument reference and profile model visible.
Keep observation, transformation and conclusion in separate evidence columns. A rocking curve records diffracted intensity while crystal angle is scanned through a reflection condition.
Work the original teaching case in a visible order. The fictional reflection is measured over a narrow omega scan.
Make the reasoning independently checkable. Preserve axis, step size, exposure and background treatment.
Test transfer by changing one condition while holding the others still. Repeat with a coarser angular step.
Keep the claim boundary visible. A plotted peak width depends on the measurement and fit protocol.
Build a compact audit receipt with input, unit, model, calculation, observation and limit. Use this independent check: Preserve axis, step size, exposure and background treatment. For transfer, Repeat with a coarser angular step. Preserve the first result beside the changed one.
The evidence receipt for “Define the rocking curve” records source, input, transformation, changed-context prediction and limit on separate lines. Keep reflection, scan axis, width convention, instrument reference and profile model visible.
Use the short answer only after the boundary and unit have been fixed. Mosaic spread represents a distribution of orientations in a mosaic-block model rather than a universal defect score.
Work the original teaching case in a visible order. The latent teaching parameter describes angular block orientation spread.
Make the reasoning independently checkable. Name the adopted crystal model.
Test transfer by changing one condition while holding the others still. Use a strained but nearly single-domain crystal.
Keep the claim boundary visible. Different imperfections can broaden features in different ways.
Ask a second reader to reproduce the decision from the stated evidence. They should be able to apply: Name the adopted crystal model. Next, Use a strained but nearly single-domain crystal. If the conclusion changes, name the changed condition explicitly.
The evidence receipt for “Define mosaic spread” records source, input, transformation, changed-context prediction and limit on separate lines. Keep reflection, scan axis, width convention, instrument reference and profile model visible.
Trace the mechanism in the order another reader would need to reproduce it. Incident rays occupy an angular distribution set by source and optics.
Work the original teaching case in a visible order. The instrument contribution is stipulated as0.10° Gaussian FWHM.
Make the reasoning independently checkable. Measure or justify the instrument resolution under matched settings.
Test transfer by changing one condition while holding the others still. Open a slit and increase divergence.
Keep the claim boundary visible. A nominal optic setting is not a measured resolution function.
Turn the claim into a falsifiable test. First, Measure or justify the instrument resolution under matched settings. Then change one consequential condition: Open a slit and increase divergence. State which observation would count against the explanation.
The evidence receipt for “Define beam divergence” records source, input, transformation, changed-context prediction and limit on separate lines. Keep reflection, scan axis, width convention, instrument reference and profile model visible.
CHAPTER 5 OF 24 · Define the quantities
5. Keep wavelength spread and geometry visible
Treat a plausible result as a prompt for a check, not as proof by itself. Rocking width can also depend on wavelength bandpass, crystal size, reflection geometry and scan direction.
Work the original teaching case in a visible order. The central case stipulates those other contributions negligible.
Make the reasoning independently checkable. List excluded terms before deconvolution.
Test transfer by changing one condition while holding the others still. Move to a reflection where spectral spread matters.
Keep the claim boundary visible. Two-term subtraction is invalid when material terms are omitted.
Before accepting the result, predict what should change and what should remain fixed. Apply this check: List excluded terms before deconvolution. Then run the changed case: Move to a reflection where spectral spread matters. Record any mismatch without rewriting the original assumptions.
The evidence receipt for “Keep wavelength spread and geometry visible” records source, input, transformation, changed-context prediction and limit on separate lines. Keep reflection, scan axis, width convention, instrument reference and profile model visible.
Start by naming the physical or logical quantity rather than the displayed number. Independent Gaussian broadening variances add, so matching FWHM values can be combined in quadrature.
Work the original teaching case in a visible order. sqrt(0.18²-0.10²)=about0.150°, the stipulated mosaic contribution.
Make the reasoning independently checkable. Square unrounded widths, subtract, take the square root, then round.
Test transfer by changing one condition while holding the others still. Use observed0.12° and instrument0.10° to obtain about0.066°.
Keep the claim boundary visible. Quadrature requires the Gaussian and independence assumptions.
Build a compact audit receipt with input, unit, model, calculation, observation and limit. Use this independent check: Square unrounded widths, subtract, take the square root, then round. For transfer, Use observed0.12° and instrument0.10° to obtain about0.066°. Preserve the first result beside the changed one.
The evidence receipt for “Complete the Gaussian deconvolution” records source, input, transformation, changed-context prediction and limit on separate lines. Keep reflection, scan axis, width convention, instrument reference and profile model visible.
Keep observation, transformation and conclusion in separate evidence columns. If the observed width is smaller than the asserted instrument width, the square-root argument becomes negative.
Work the original teaching case in a visible order. An observed0.08° with instrument0.10° flags inconsistent inputs or models.
Make the reasoning independently checkable. Stop rather than replacing the negative value with zero.
Test transfer by changing one condition while holding the others still. Repeat the instrument measurement under the same settings.
Keep the claim boundary visible. A failed deconvolution is evidence about the method, not a negative mosaicity.
Ask a second reader to reproduce the decision from the stated evidence. They should be able to apply: Stop rather than replacing the negative value with zero. Next, Repeat the instrument measurement under the same settings. If the conclusion changes, name the changed condition explicitly.
The evidence receipt for “Check the impossible subtraction” records source, input, transformation, changed-context prediction and limit on separate lines. Keep reflection, scan axis, width convention, instrument reference and profile model visible.
Use the short answer only after the boundary and unit have been fixed. For ideal Lorentzian components, FWHM values add rather than variances.
Work the original teaching case in a visible order. The same0.18° and0.10° would imply0.08° for a Lorentzian subtraction.
Make the reasoning independently checkable. Fit and inspect residuals before choosing the rule.
Test transfer by changing one condition while holding the others still. Use a pseudo-Voigt model.
Keep the claim boundary visible. One width pair cannot identify the profile family uniquely.
Turn the claim into a falsifiable test. First, Fit and inspect residuals before choosing the rule. Then change one consequential condition: Use a pseudo-Voigt model. State which observation would count against the explanation.
The evidence receipt for “Compare Lorentzian broadening” records source, input, transformation, changed-context prediction and limit on separate lines. Keep reflection, scan axis, width convention, instrument reference and profile model visible.
Trace the mechanism in the order another reader would need to reproduce it. For a Gaussian, FWHM and sigma differ by about2.35482.
Work the original teaching case in a visible order. Convert both observed and instrument widths consistently before variance arithmetic.
Make the reasoning independently checkable. Recover the same result after converting back.
Test transfer by changing one condition while holding the others still. Mix one FWHM with one sigma and expose the error.
Keep the claim boundary visible. Numbers with degree units can still be incomparable.
Before accepting the result, predict what should change and what should remain fixed. Apply this check: Recover the same result after converting back. Then run the changed case: Mix one FWHM with one sigma and expose the error. Record any mismatch without rewriting the original assumptions.
The evidence receipt for “Preserve FWHM versus standard deviation” records source, input, transformation, changed-context prediction and limit on separate lines. Keep reflection, scan axis, width convention, instrument reference and profile model visible.
CHAPTER 10 OF 24 · Work the mechanism
10. Separate scan-axis and detector-plane widths
Treat a plausible result as a prompt for a check, not as proof by itself. A rocking width in rotation angle and a spot width on the detector are related to different geometry projections.
Work the original teaching case in a visible order. The example uses the omega-scan FWHM only.
Make the reasoning independently checkable. Label axis and coordinate for each fitted width.
Test transfer by changing one condition while holding the others still. Compare an azimuthal spot arc.
Keep the claim boundary visible. A broad detector spot does not directly equal a rocking-curve width.
Build a compact audit receipt with input, unit, model, calculation, observation and limit. Use this independent check: Label axis and coordinate for each fitted width. For transfer, Compare an azimuthal spot arc. Preserve the first result beside the changed one.
The evidence receipt for “Separate scan-axis and detector-plane widths” records source, input, transformation, changed-context prediction and limit on separate lines. Keep reflection, scan axis, width convention, instrument reference and profile model visible.
CHAPTER 11 OF 24 · Audit the evidence
11. Match the reflection and instrument calibration
Start by naming the physical or logical quantity rather than the displayed number. Resolution can vary with angle, wavelength, optics and reflection orientation.
Work the original teaching case in a visible order. The instrument reference is measured near the teaching reflection.
Make the reasoning independently checkable. Use a standard and settings close to the sample measurement.
Test transfer by changing one condition while holding the others still. Move to a distant scattering angle.
Keep the claim boundary visible. A single calibration width need not apply everywhere.
Ask a second reader to reproduce the decision from the stated evidence. They should be able to apply: Use a standard and settings close to the sample measurement. Next, Move to a distant scattering angle. If the conclusion changes, name the changed condition explicitly.
The evidence receipt for “Match the reflection and instrument calibration” records source, input, transformation, changed-context prediction and limit on separate lines. Keep reflection, scan axis, width convention, instrument reference and profile model visible.
Keep observation, transformation and conclusion in separate evidence columns. Background slope and neighbouring reflections can alter a fitted width.
Work the original teaching case in a visible order. A weak shoulder is added to the fictional peak.
Make the reasoning independently checkable. Refit with a justified multi-peak model and inspect residuals.
Test transfer by changing one condition while holding the others still. Increase the neighbour intensity.
Keep the claim boundary visible. A wider one-peak fit can be a model failure.
Turn the claim into a falsifiable test. First, Refit with a justified multi-peak model and inspect residuals. Then change one consequential condition: Increase the neighbour intensity. State which observation would count against the explanation.
The evidence receipt for “Audit background and peak overlap” records source, input, transformation, changed-context prediction and limit on separate lines. Keep reflection, scan axis, width convention, instrument reference and profile model visible.
Use the short answer only after the boundary and unit have been fixed. Step size and scan range limit how well width and tails are observed.
Work the original teaching case in a visible order. A0.02° step gives only a few samples across the narrow changed case.
Make the reasoning independently checkable. Repeat with finer steps and compare fitted uncertainty.
Test transfer by changing one condition while holding the others still. Truncate one peak tail.
Keep the claim boundary visible. A fit can return a number when the scan cannot support it.
Before accepting the result, predict what should change and what should remain fixed. Apply this check: Repeat with finer steps and compare fitted uncertainty. Then run the changed case: Truncate one peak tail. Record any mismatch without rewriting the original assumptions.
The evidence receipt for “Audit angular sampling” records source, input, transformation, changed-context prediction and limit on separate lines. Keep reflection, scan axis, width convention, instrument reference and profile model visible.
Trace the mechanism in the order another reader would need to reproduce it. Deconvolution becomes unstable when observed and instrument widths are close.
Work the original teaching case in a visible order. The0.12° versus0.10° case is more sensitive than0.18° versus0.10°.
Make the reasoning independently checkable. Propagate uncertainty or resample the fitted parameters.
Test transfer by changing one condition while holding the others still. Increase instrument-width uncertainty.
Keep the claim boundary visible. A small deconvolved value may have a large relative uncertainty.
Build a compact audit receipt with input, unit, model, calculation, observation and limit. Use this independent check: Propagate uncertainty or resample the fitted parameters. For transfer, Increase instrument-width uncertainty. Preserve the first result beside the changed one.
The evidence receipt for “Carry width uncertainty” records source, input, transformation, changed-context prediction and limit on separate lines. Keep reflection, scan axis, width convention, instrument reference and profile model visible.
Treat a plausible result as a prompt for a check, not as proof by itself. Crystal and instrument effects can vary with direction and reflection geometry.
Work the original teaching case in a visible order. The audit measures several non-equivalent reflections.
Make the reasoning independently checkable. Look for consistent trends instead of selecting one peak.
Test transfer by changing one condition while holding the others still. Rotate the crystal orientation.
Keep the claim boundary visible. One reflection does not establish isotropic mosaic spread.
Ask a second reader to reproduce the decision from the stated evidence. They should be able to apply: Look for consistent trends instead of selecting one peak. Next, Rotate the crystal orientation. If the conclusion changes, name the changed condition explicitly.
The evidence receipt for “Use multiple reflections” records source, input, transformation, changed-context prediction and limit on separate lines. Keep reflection, scan axis, width convention, instrument reference and profile model visible.
Start by naming the physical or logical quantity rather than the displayed number. Rocking width informs rotation range and partial-reflection treatment in a stated experiment.
Work the original teaching case in a visible order. Compare a narrow and broad teaching crystal at the same image width.
Make the reasoning independently checkable. Keep dose, overlap and processing assumptions visible.
Test transfer by changing one condition while holding the others still. Change oscillation width per image.
Keep the claim boundary visible. This article does not prescribe beamline settings.
Turn the claim into a falsifiable test. First, Keep dose, overlap and processing assumptions visible. Then change one consequential condition: Change oscillation width per image. State which observation would count against the explanation.
The evidence receipt for “Transfer to data-collection strategy” records source, input, transformation, changed-context prediction and limit on separate lines. Keep reflection, scan axis, width convention, instrument reference and profile model visible.
Keep observation, transformation and conclusion in separate evidence columns. Comparing samples requires matched instrument resolution and fit protocols.
Work the original teaching case in a visible order. Two crystals measured on different beamlines have0.15° observed widths.
Make the reasoning independently checkable. Deconvolve each with its own resolution evidence.
Test transfer by changing one condition while holding the others still. Swap beamlines using a common reference crystal.
Keep the claim boundary visible. Equal observed widths can conceal different crystal contributions.
Before accepting the result, predict what should change and what should remain fixed. Apply this check: Deconvolve each with its own resolution evidence. Then run the changed case: Swap beamlines using a common reference crystal. Record any mismatch without rewriting the original assumptions.
The evidence receipt for “Transfer to crystal comparison” records source, input, transformation, changed-context prediction and limit on separate lines. Keep reflection, scan axis, width convention, instrument reference and profile model visible.
Use the short answer only after the boundary and unit have been fixed. Changing collimation can narrow beam divergence while reducing flux or changing other conditions.
Work the original teaching case in a visible order. The fictional beam width falls from0.10° to0.06°.
Make the reasoning independently checkable. Record flux, exposure and resolution together.
Test transfer by changing one condition while holding the others still. Lengthen exposure to restore counts.
Keep the claim boundary visible. A narrower curve after optics changes is not automatically improved crystal quality.
Build a compact audit receipt with input, unit, model, calculation, observation and limit. Use this independent check: Record flux, exposure and resolution together. For transfer, Lengthen exposure to restore counts. Preserve the first result beside the changed one.
The evidence receipt for “Transfer to optics changes” records source, input, transformation, changed-context prediction and limit on separate lines. Keep reflection, scan axis, width convention, instrument reference and profile model visible.
Trace the mechanism in the order another reader would need to reproduce it. Real profiles may combine Gaussian cores, Lorentzian tails and asymmetry.
Work the original teaching case in a visible order. The teaching residual plot shows systematic tail mismatch.
Make the reasoning independently checkable. Compare supported profile models without chasing the smallest residual alone.
Test transfer by changing one condition while holding the others still. Add an asymmetric source term.
Keep the claim boundary visible. A fitted component has meaning only within the model.
Ask a second reader to reproduce the decision from the stated evidence. They should be able to apply: Compare supported profile models without chasing the smallest residual alone. Next, Add an asymmetric source term. If the conclusion changes, name the changed condition explicitly.
The evidence receipt for “Transfer to non-Gaussian tails” records source, input, transformation, changed-context prediction and limit on separate lines. Keep reflection, scan axis, width convention, instrument reference and profile model visible.
Treat a plausible result as a prompt for a check, not as proof by itself. Record reflection, axis, wavelength, optics, step, exposure, background, profile model, instrument reference and uncertainty.
Work the original teaching case in a visible order. Another analyst reconstructs the0.150° teaching result.
Make the reasoning independently checkable. Remove the profile convention and identify the ambiguity.
Test transfer by changing one condition while holding the others still. Version the receipt after a slit change.
Keep the claim boundary visible. A complete receipt does not certify crystal quality for every purpose.
Turn the claim into a falsifiable test. First, Remove the profile convention and identify the ambiguity. Then change one consequential condition: Version the receipt after a slit change. State which observation would count against the explanation.
The evidence receipt for “Build the evidence receipt” records source, input, transformation, changed-context prediction and limit on separate lines. Keep reflection, scan axis, width convention, instrument reference and profile model visible.
Start by naming the physical or logical quantity rather than the displayed number. The instrument and other broadening terms must be addressed first.
Work the original teaching case in a visible order. The central observed0.18° becomes about0.150° only under stated Gaussian assumptions.
Make the reasoning independently checkable. State the model beside the reported value.
Test transfer by changing one condition while holding the others still. Set instrument width nearly zero.
Keep the claim boundary visible. Approximate equality in one setup is not a universal definition.
Before accepting the result, predict what should change and what should remain fixed. Apply this check: State the model beside the reported value. Then run the changed case: Set instrument width nearly zero. Record any mismatch without rewriting the original assumptions.
The evidence receipt for “Repair observed width equals mosaicity” records source, input, transformation, changed-context prediction and limit on separate lines. Keep reflection, scan axis, width convention, instrument reference and profile model visible.
Keep observation, transformation and conclusion in separate evidence columns. A narrow width can be useful, but crystal volume, diffraction strength, disorder, damage and the experimental aim also matter.
Work the original teaching case in a visible order. The teaching comparison changes only angular spread.
Make the reasoning independently checkable. List the decision’s other required evidence.
Test transfer by changing one condition while holding the others still. Reduce width while halving useful diffracting volume.
Keep the claim boundary visible. No single width ranks every crystal for every experiment.
Build a compact audit receipt with input, unit, model, calculation, observation and limit. Use this independent check: List the decision’s other required evidence. For transfer, Reduce width while halving useful diffracting volume. Preserve the first result beside the changed one.
The evidence receipt for “Repair narrower always means better” records source, input, transformation, changed-context prediction and limit on separate lines. Keep reflection, scan axis, width convention, instrument reference and profile model visible.
Use the short answer only after the boundary and unit have been fixed. The method can test whether a reported width has been deconvolved consistently under an explicit profile model.
Work the original teaching case in a visible order. It supports an evidence audit, not an operating instruction.
Make the reasoning independently checkable. Write the decision and the remaining unknowns.
Test transfer by changing one condition while holding the others still. Move from comparison to beamline setup.
Keep the claim boundary visible. Qualified crystallography and facility procedures govern real measurements.
Ask a second reader to reproduce the decision from the stated evidence. They should be able to apply: Write the decision and the remaining unknowns. Next, Move from comparison to beamline setup. If the conclusion changes, name the changed condition explicitly.
The evidence receipt for “State the supported decision” records source, input, transformation, changed-context prediction and limit on separate lines. Keep reflection, scan axis, width convention, instrument reference and profile model visible.
CHAPTER 24 OF 24 · Report and continue
24. Continue through verified crystallography routes
Trace the mechanism in the order another reader would need to reproduce it. This lesson sits beside detector saturation, completeness and systematic absence.
Work the original teaching case in a visible order. Follow the sibling that owns the next uncertainty.
Make the reasoning independently checkable. Keep width, intensity and reflection-count questions separate.
Test transfer by changing one condition while holding the others still. Move to detector saturation without reusing the width ledger.
Keep the claim boundary visible. This article owns mosaic-spread versus beam-divergence reasoning in rocking curves.
Turn the claim into a falsifiable test. First, Keep width, intensity and reflection-count questions separate. Then change one consequential condition: Move to detector saturation without reusing the width ledger. State which observation would count against the explanation.
The evidence receipt for “Continue through verified crystallography routes” records source, input, transformation, changed-context prediction and limit on separate lines. Keep reflection, scan axis, width convention, instrument reference and profile model visible.
Contents · Previous chapter · Return to X-ray Crystallography
Continue through the verified reading route
Detector saturation and integrated strength
Detector saturation and integrated strength. Use this when pixel linearity and intensity integration, rather than angular width, are at issue.
Reflection completeness and redundancy
Reflection completeness and redundancy. Use this when unique reciprocal-space coverage is confused with repeated observations.
Systematic absence and weak unobserved reflections
Systematic absence and weak unobserved reflections. Use this when symmetry extinction is confused with detection failure.
Frequently asked questions
Is the observed rocking width the mosaic spread?
Only when instrument and other contributions are negligible or have been handled under a justified model.
Why do Gaussian and Lorentzian subtraction differ?
Their convolution rules differ: Gaussian variances add, whereas ideal Lorentzian FWHM values add.
Can one reflection rank crystal quality?
No. Direction, instrument settings, intensity, disorder and the intended experiment also matter.
Authoritative sources and further reading
International Tables: single-crystal X-ray techniques. IUCr reference describing rocking width as a combination of mosaic spread, divergence, wavelength spread and geometry.
IUCr: integration of macromolecular diffraction data. Primary IUCr article explaining combined mosaic spread and beam-divergence estimation from reflection width.
IUCr: matching beam and detector properties. Primary IUCr analysis of crystal imperfections, rocking width and diffracted-beam divergence.
IUCr: protein crystal perfection. Primary IUCr source stating that mosaicity requires deconvolution of instrument contribution.
Return to the X-ray Crystallography subject guide, The Gold Standard of Education, or the How X Works hub.