Each guide begins with a short answer and follows one original record through definitions, calculations, evidence checks, changed contexts, misconceptions and bounded reporting.
The worked numbers are stipulated for learning. Authoritative sources define the operational methods; the examples show how to keep their boundaries visible.
The worked records are original teaching examples. Each keeps the observation, method boundary, calculation and decision in separate fields so the reader can see exactly where a conclusion begins and where it stops.
A changed-context check alters one consequential condition while leaving the other inputs fixed. This makes transfer visible and prevents the original numerical answer from becoming a rule without a scope.
Keep a compact evidence receipt: object, sampling or measurement method, units, reference condition, transformation, result and nearest rival explanation. A future reader should be able to reproduce the narrow claim from that receipt.
The five reading routes move from definitions to mechanism, evidence, transfer and reporting. Readers may enter by task, but the examples stay connected so the explanation does not become a list of detached tips.
The practical standard is an answer that can be checked. Preserve raw inputs, intermediate calculations, exclusions and versioned sources; then state one stronger claim that the evidence still cannot support.
Before choosing a formula, name the reader job in one sentence. A process check, research comparison, operational report and teaching exercise may use the same observation differently.
Every central case includes a near-match with one familiar surface feature and a different mechanism underneath. The pair matters because a definition becomes useful only when a learner can reject the tempting substitute.
The evidence route treats missing information honestly. An unknown method, interval, calibration or boundary stays blank until a source supplies it; a plausible assumption may complete arithmetic while weakening the conclusion.
The transfer route changes scale, timing, composition or sampling without silently changing every other field. Predict the direction first, then calculate or inspect the result.
The reporting route keeps measurement and judgement separate. State what was observed, what rule produced the derived value, what uncertainty remains and which action follows only under the declared conditions.
The final reader check is practical: someone entering through the short answer should know the distinction, someone following the worked example should reproduce it, and someone opening the sources should see where the example ends.
A useful misconception test constructs two cases with the same headline number and different underlying records. If their interpretations differ, the omitted condition belongs in the result.
Where a standard leaves behaviour to a competent laboratory, observing authority, analyst or software workflow, the article says so directly. That boundary tells the reader which conclusion can travel.
The examples avoid hidden precision. Rounded display values remain linked to the unrounded calculation, and stipulated inputs are labelled as teaching data.
A complete explanation keeps three layers apart: the thing in the world, the method that turns it into a value, and the decision made from that value. Numerical agreement between layers is evidence to investigate rather than permission to merge them.
For each central example, sketch the boundary before calculating. Mark what enters, what is retained or excluded, what the instrument or algorithm can observe, and what must be inferred.
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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Read the short answer first: corrected intensity estimates a scaled magnitude squared, while amplitude is a nonnegative magnitude derived with the scale and uncertainty model. Preserve negative observations and use a validated weak-data treatment rather than clipping.
ROUTE 1 · CHAPTERS 1–4
Define the quantities
Name the quantities, reference conditions and reporting rules.
ROUTE 3 · CHAPTERS 9–12
Audit the evidence
Check instruments, algorithms, assumptions and rival explanations.
ROUTE 5 · CHAPTERS 17–20
Report and continue
Repair shortcuts, answer questions and follow verified routes.
Chapter contents
Define the quantities · Chapters 1–4
1. Name intensity and amplitude
Work the mechanism · Chapters 5–8
5. Work the strong-reflection square root
6. Work a scale-factor example
Audit the evidence · Chapters 9–12
9. Audit centric and acentric classes
10. Audit resolution-dependent expectations
Transfer the reasoning · Chapters 13–16
13. Transfer to refinement targets
14. Transfer to electron-density maps
Report and continue · Chapters 17–20
17. Repair amplitude equals square root of any number
18. Repair negative intensity equals no reflection
A corrected, scaled reflection intensity is an estimate related to the squared magnitude of a structure factor. A structure-factor amplitude is a nonnegative magnitude used in many crystallographic calculations. They are linked by a model and scale, not identical file columns.
Write the evidence in the order a reader can audit: the X-ray Crystallography quantity, its reference object, the observation or browser rule, the transformation, and the decision being made. Keeping those fields visible prevents a familiar label from hiding a changed basis.
On a teaching scale with I equal to |F| squared, I=400 corresponds to amplitude 20. Preserve the intensity uncertainty and scaling basis.
First-principles test: ask what physical, observational or algorithmic operation produces each value in this chapter. Then ask whether the second value could change while the first stays fixed. If it can, the two labels are not substitutes. For X-ray Crystallography, this counterfactual is often clearer than a definition learned in isolation.
Multiply all intensities by four under a scale change; amplitudes multiply by two.
Evidence check: another reader should be able to reproduce the narrow result from the stored definitions, inputs and intermediate steps, then identify the nearest stronger claim that the evidence does not support.
Real workflows include scale factors, uncertainties, merging and statistical estimation.
A structure factor is complex and has magnitude plus phase. Converting intensity to amplitude does not recover phase, because diffraction intensities measure magnitude-squared information.
Trace the mechanism before interpreting the result. For X-ray Crystallography, mark which value belongs to the source, which belongs to the measurement or layout algorithm, and which is a reported summary. A calculation is useful only when those roles remain attached.
Amplitude 20 can pair with many phases. Draw two complex vectors of length 20 at different angles.
Practical transfer: build two records that share the headline number but differ in the condition discussed here. If the correct interpretation changes, the omitted condition is decision-relevant and belongs in the report. This test turns a vocabulary distinction into a reusable X-ray Crystallography checking habit.
Keep amplitude fixed and rotate phase. Intensity remains 400.
Practical check: keep one counterexample beside the worked case. If the wording fails when that one condition changes, narrow the sentence until it describes only the evidence actually in hand.
Phase information comes from models or experimental strategies beyond this conversion.
Corrected intensities may be placed on an arbitrary but consistent dataset scale. A square root inherits the square root of that scale, so amplitudes from different datasets cannot be compared without scaling evidence.
Use a compact decision table with columns for condition, input, operation, result and limit. This makes the comparison reproducible and exposes where a missing field would force an assumption instead of an observation.
Dataset A reports I=400 and B reports I=1600 for equivalent reflections because B has fourfold intensity scale. After scale reconciliation their amplitudes can agree.
Misconception check: a plausible output does not prove the right reference, denominator, ancestor, spectrum or observation rule was used. Reconstruct one intermediate step and one changed-context prediction. If either cannot be stated, keep the result provisional rather than strengthening its wording.
Change only the global intensity scale factor.
Transfer check: repeat the reasoning with one consequential condition changed while holding the rest fixed. State what changes, what stays comparable and why the original label alone was insufficient.
Absolute electron-density scale and relative scaling require the crystallographic workflow.
CHAPTER 4 OF 20 · Define the quantities
4. Separate observation from latent true value
A measured intensity can be positive or negative after background subtraction because it is a noisy estimate of a nonnegative underlying signal. The observation and the latent physical intensity occupy different statistical roles.
Preserve the raw inputs beside the derived result. If a later reader changes one condition, the original X-ray Crystallography reasoning should still be recoverable without reverse-engineering a rounded headline or an unlabeled screenshot.
A weak reflection is reported Iobs=-3 with sigma 5. Keep that measurement instead of calling the physical scattering negative.
Evidence design: retain both a positive case and a near-miss. The positive case shows when the rule applies; the near-miss changes one consequential condition and shows when the conclusion fails. Together they teach the boundary of the X-ray Crystallography mechanism better than another unlabelled example.
Repeat the observation many times around true zero and expect positive and negative estimates.
Audit check: retain the source version, units, settings, exclusions and rounding rule. A future correction should be able to alter one field without rebuilding the entire explanation from memory.
Negative observations are information about noise and should not be silently deleted.
For a strong positive observation with small relative uncertainty, a square root gives useful intuition and first-order propagation gives sigmaF approximately sigmaI divided by 2F.
Ask what would have to stay fixed for the comparison to be fair. Record those controls explicitly, then name the first uncontrolled condition that could reverse the conclusion. This is the practical transfer step, not an optional footnote.
Let I=400 and sigmaI=20. F=20 and the first-order sigmaF is 0.5 on the teaching scale.
Communication test: replace any sentence that says two things are “the same” or “different” with a sentence naming the operation, reference condition and supported consequence. The extra words are useful when they let a future reader repeat the decision without access to the original author.
Keep sigmaI=20 and reduce I to 4. The linear approximation becomes far less reliable.
Decision check: end with an action or interpretation that follows from the stated conditions, followed by a boundary sentence explaining when a specialist rule, local procedure or additional measurement is needed.
The approximation is not a substitute for the workflow’s statistical treatment.
If corrected intensity follows I=k|F| squared, amplitude on the target scale is the square root of I/k for the simple positive case. The factor must be known and consistently applied.
Write the evidence in the order a reader can audit: the X-ray Crystallography quantity, its reference object, the observation or browser rule, the transformation, and the decision being made. Keeping those fields visible prevents a familiar label from hiding a changed basis.
With I=900 and k=2.25, calculate |F|=20.
First-principles test: ask what physical, observational or algorithmic operation produces each value in this chapter. Then ask whether the second value could change while the first stays fixed. If it can, the two labels are not substitutes. For X-ray Crystallography, this counterfactual is often clearer than a definition learned in isolation.
Hold I at 900 and change k to 1.0; the derived amplitude becomes 30.
Evidence check: another reader should be able to reproduce the narrow result from the stored definitions, inputs and intermediate steps, then identify the nearest stronger claim that the evidence does not support.
In practice, scale models can depend on batch, resolution, absorption and damage.
The square-root transformation is nonlinear. For strong reflections a differential approximation can work; near zero it becomes asymmetric and can imply impossible symmetric negative amplitudes.
Trace the mechanism before interpreting the result. For X-ray Crystallography, mark which value belongs to the source, which belongs to the measurement or layout algorithm, and which is a reported summary. A calculation is useful only when those roles remain attached.
Compare I=400±20 with I=4±20. The second case cannot be summarized responsibly as F=2±5 by simple propagation.
Practical transfer: build two records that share the headline number but differ in the condition discussed here. If the correct interpretation changes, the omitted condition is decision-relevant and belongs in the report. This test turns a vocabulary distinction into a reusable X-ray Crystallography checking habit.
Increase signal while keeping sigmaI fixed and observe the approximation improve.
Practical check: keep one counterexample beside the worked case. If the wording fails when that one condition changes, narrow the sentence until it describes only the evidence actually in hand.
Use the implemented statistical method for weak data.
French–Wilson treatment combines the intensity observation and uncertainty with a Wilson-distribution prior to estimate sensible amplitude information, including for negative observations. It is not equivalent to clipping I at zero and taking a square root.
Use a compact decision table with columns for condition, input, operation, result and limit. This makes the comparison reproducible and exposes where a missing field would force an assumption instead of an observation.
For I=-3, sigma=5, retain the observation and let the validated algorithm produce a posterior amplitude estimate rather than setting F=0.
Misconception check: a plausible output does not prove the right reference, denominator, ancestor, spectrum or observation rule was used. Reconstruct one intermediate step and one changed-context prediction. If either cannot be stated, keep the result provisional rather than strengthening its wording.
Keep I fixed and change sigma; the posterior should respond to evidence strength.
Transfer check: repeat the reasoning with one consequential condition changed while holding the rest fixed. State what changes, what stays comparable and why the original label alone was insufficient.
Exact results depend on centricity, resolution-dependent prior and implementation.
Wilson distributions differ for centric and acentric reflections. A French–Wilson conversion needs the correct symmetry class and related prior information.
Preserve the raw inputs beside the derived result. If a later reader changes one condition, the original X-ray Crystallography reasoning should still be recoverable without reverse-engineering a rounded headline or an unlabeled screenshot.
Process the same weak teaching observation under centric and acentric assumptions and expect different posterior summaries.
Evidence design: retain both a positive case and a near-miss. The positive case shows when the rule applies; the near-miss changes one consequential condition and shows when the conclusion fails. Together they teach the boundary of the X-ray Crystallography mechanism better than another unlabelled example.
Change only the reflection class.
Audit check: retain the source version, units, settings, exclusions and rounding rule. A future correction should be able to alter one field without rebuilding the entire explanation from memory.
Space-group assignment and systematic absences must be correct before conversion.
Expected intensity varies with resolution through scattering falloff and overall disorder. Statistical amplitude estimation uses local or resolution-binned expectations rather than one universal prior across the dataset.
Ask what would have to stay fixed for the comparison to be fair. Record those controls explicitly, then name the first uncontrolled condition that could reverse the conclusion. This is the practical transfer step, not an optional footnote.
Place two equal weak intensities in low- and high-resolution bins with different mean intensity.
Communication test: replace any sentence that says two things are “the same” or “different” with a sentence naming the operation, reference condition and supported consequence. The extra words are useful when they let a future reader repeat the decision without access to the original author.
Keep I/sigma fixed while changing the bin expectation.
Decision check: end with an action or interpretation that follows from the stated conditions, followed by a boundary sentence explaining when a specialist rule, local procedure or additional measurement is needed.
Anomalous shells or poor binning can distort the prior.
Multiple corrected observations may be scaled and merged as intensities with uncertainties before amplitude estimation. Averaging prematurely transformed square roots can give a different and biased result.
Write the evidence in the order a reader can audit: the X-ray Crystallography quantity, its reference object, the observation or browser rule, the transformation, and the decision being made. Keeping those fields visible prevents a familiar label from hiding a changed basis.
Observations I1=100 and I2=400 have mean intensity 250, whose square root is about 15.81; mean of square roots is 15.
First-principles test: ask what physical, observational or algorithmic operation produces each value in this chapter. Then ask whether the second value could change while the first stays fixed. If it can, the two labels are not substitutes. For X-ray Crystallography, this counterfactual is often clearer than a definition learned in isolation.
Keep observations but change their uncertainties and use weighted intensity merging.
Evidence check: another reader should be able to reproduce the narrow result from the stored definitions, inputs and intermediate steps, then identify the nearest stronger claim that the evidence does not support.
Merging model, outlier handling and symmetry must be versioned.
Rejecting every negative intensity biases weak data upward because only positive noise remains. Outlier rejection should target supported measurement problems rather than the sign alone.
Trace the mechanism before interpreting the result. For X-ray Crystallography, mark which value belongs to the source, which belongs to the measurement or layout algorithm, and which is a reported summary. A calculation is useful only when those roles remain attached.
A true-zero teaching set [-6,-2,1,3,4] becomes falsely positive if negative rows are removed.
Practical transfer: build two records that share the headline number but differ in the condition discussed here. If the correct interpretation changes, the omitted condition is decision-relevant and belongs in the report. This test turns a vocabulary distinction into a reusable X-ray Crystallography checking habit.
Restore all valid observations and compare the merged estimate.
Practical check: keep one counterexample beside the worked case. If the wording fails when that one condition changes, narrow the sentence until it describes only the evidence actually in hand.
Validated pipelines may reject observations for documented geometric or statistical reasons.
Some refinement programs work directly against intensities, while others use amplitudes. Converting data is therefore a model choice tied to the target, not a mandatory deletion of the original intensities.
Use a compact decision table with columns for condition, input, operation, result and limit. This makes the comparison reproducible and exposes where a missing field would force an assumption instead of an observation.
Store both merged I and derived F columns with provenance and note which target the refinement used.
Misconception check: a plausible output does not prove the right reference, denominator, ancestor, spectrum or observation rule was used. Reconstruct one intermediate step and one changed-context prediction. If either cannot be stated, keep the result provisional rather than strengthening its wording.
Switch to an intensity-based target without re-integrating images.
Transfer check: repeat the reasoning with one consequential condition changed while holding the rest fixed. State what changes, what stays comparable and why the original label alone was insufficient.
Program documentation and model assumptions govern the preferred input.
Map coefficients often use observed amplitudes and model phases or weighted combinations. The amplitude conversion does not itself create a map or solve phase uncertainty.
Preserve the raw inputs beside the derived result. If a later reader changes one condition, the original X-ray Crystallography reasoning should still be recoverable without reverse-engineering a rounded headline or an unlabeled screenshot.
Combine a derived Fobs with two different model phases and observe different Fourier contributions.
Evidence design: retain both a positive case and a near-miss. The positive case shows when the rule applies; the near-miss changes one consequential condition and shows when the conclusion fails. Together they teach the boundary of the X-ray Crystallography mechanism better than another unlabelled example.
Keep Fobs fixed and improve the phase model.
Audit check: retain the source version, units, settings, exclusions and rounding rule. A future correction should be able to alter one field without rebuilding the entire explanation from memory.
Map interpretation requires complete coefficient and weighting provenance.
Bijvoet mates may be kept separate to analyse anomalous differences. Scaling, uncertainty and correlation matter before amplitude differences are interpreted.
Ask what would have to stay fixed for the comparison to be fair. Record those controls explicitly, then name the first uncontrolled condition that could reverse the conclusion. This is the practical transfer step, not an optional footnote.
Two mates have close intensities with uncertainties larger than their difference. Preserve both observations rather than declaring a strong anomalous signal.
Communication test: replace any sentence that says two things are “the same” or “different” with a sentence naming the operation, reference condition and supported consequence. The extra words are useful when they let a future reader repeat the decision without access to the original author.
Increase redundancy while holding mean difference fixed.
Decision check: end with an action or interpretation that follows from the stated conditions, followed by a boundary sentence explaining when a specialist rule, local procedure or additional measurement is needed.
Anomalous-signal assessment needs dedicated statistics and model checks.
Isomorphous or time-resolved comparisons require scale reconciliation and control of systematic error. Equal amplitude labels from separately processed datasets do not guarantee directly comparable physical scale.
Write the evidence in the order a reader can audit: the X-ray Crystallography quantity, its reference object, the observation or browser rule, the transformation, and the decision being made. Keeping those fields visible prevents a familiar label from hiding a changed basis.
Dataset A and B each report F=30 after independent scaling. Compare scale models and common reflections before subtracting them.
First-principles test: ask what physical, observational or algorithmic operation produces each value in this chapter. Then ask whether the second value could change while the first stays fixed. If it can, the two labels are not substitutes. For X-ray Crystallography, this counterfactual is often clearer than a definition learned in isolation.
Change only one dataset’s global scale and observe the apparent difference.
Evidence check: another reader should be able to reproduce the narrow result from the stored definitions, inputs and intermediate steps, then identify the nearest stronger claim that the evidence does not support.
Structural changes require uncertainty-aware difference methods.
CHAPTER 17 OF 20 · Report and continue
17. Repair amplitude equals square root of any number
The square-root rule applies to a positive intensity on a matched scale under a simple model. Raw counts, uncorrected integrated values, negative observations and differently scaled datasets require earlier or statistical steps.
Trace the mechanism before interpreting the result. For X-ray Crystallography, mark which value belongs to the source, which belongs to the measurement or layout algorithm, and which is a reported summary. A calculation is useful only when those roles remain attached.
Refuse to take the square root of a raw detector sum from the prior specialist article.
Practical transfer: build two records that share the headline number but differ in the condition discussed here. If the correct interpretation changes, the omitted condition is decision-relevant and belongs in the report. This test turns a vocabulary distinction into a reusable X-ray Crystallography checking habit.
Apply detector and scaling corrections first, then identify the valid conversion stage.
Practical check: keep one counterexample beside the worked case. If the wording fails when that one condition changes, narrow the sentence until it describes only the evidence actually in hand.
Pipeline stage names and file columns must be checked, not guessed.
CHAPTER 18 OF 20 · Report and continue
18. Repair negative intensity equals no reflection
A negative observation can arise from noise around a weak or absent signal and still constrains the estimate. Setting every negative value to zero creates bias and discards uncertainty information.
Use a compact decision table with columns for condition, input, operation, result and limit. This makes the comparison reproducible and exposes where a missing field would force an assumption instead of an observation.
Retain I=-3±5 and compare with I=-30±5; both are negative but carry different evidence about model mismatch or processing.
Misconception check: a plausible output does not prove the right reference, denominator, ancestor, spectrum or observation rule was used. Reconstruct one intermediate step and one changed-context prediction. If either cannot be stated, keep the result provisional rather than strengthening its wording.
Increase sigma for the -30 observation and reconsider its extremeness.
Transfer check: repeat the reasoning with one consequential condition changed while holding the rest fixed. State what changes, what stays comparable and why the original label alone was insufficient.
Extreme negatives can also flag background or scaling problems needing investigation.
Store image/integration provenance, corrected observations, scale model, merge group, symmetry, centricity, resolution bin, I and sigmaI, rejection flags, conversion algorithm and version, F and sigmaF, and downstream target.
Preserve the raw inputs beside the derived result. If a later reader changes one condition, the original X-ray Crystallography reasoning should still be recoverable without reverse-engineering a rounded headline or an unlabeled screenshot.
Package the 400±20 strong case and -3±5 weak case with every stage.
Evidence design: retain both a positive case and a near-miss. The positive case shows when the rule applies; the near-miss changes one consequential condition and shows when the conclusion fails. Together they teach the boundary of the X-ray Crystallography mechanism better than another unlabelled example.
Delete sigmaI and ask whether the weak conversion remains reproducible.
Audit check: retain the source version, units, settings, exclusions and rounding rule. A future correction should be able to alter one field without rebuilding the entire explanation from memory.
A deposited amplitude column alone cannot reconstruct discarded intensity evidence.
A defensible statement names the corrected intensity scale, uncertainty and conversion method, and keeps amplitudes distinct from phases and raw detector counts.
Ask what would have to stay fixed for the comparison to be fair. Record those controls explicitly, then name the first uncontrolled condition that could reverse the conclusion. This is the practical transfer step, not an optional footnote.
Write: “Merged I and sigmaI were converted to nonnegative amplitude estimates with the versioned French–Wilson procedure; strong-reflection square roots are explanatory checks, not the weak-data algorithm.”
Communication test: replace any sentence that says two things are “the same” or “different” with a sentence naming the operation, reference condition and supported consequence. The extra words are useful when they let a future reader repeat the decision without access to the original author.
Change scale or prior class and identify which amplitudes must be regenerated.
Decision check: end with an action or interpretation that follows from the stated conditions, followed by a boundary sentence explaining when a specialist rule, local procedure or additional measurement is needed.
Structural conclusions require the validated complete crystallographic workflow.
Contents · Previous chapter · Return to X-ray Crystallography
Continue through the verified reading route
Raw counts and corrected intensity
Raw counts and corrected intensity. Use this for the detector, background, integration and correction stages before scaling.
Integrated intensity and peak height
Integrated intensity and peak height. Use this for profile area versus maximum sample.
Detector saturation and integrated strength
Detector saturation and integrated strength. Use this for overload and censored detector measurements.
Frequently asked questions
Why can an observed intensity be negative?
Background subtraction and measurement noise can give a negative estimate even though the latent physical intensity is nonnegative.
Why not set every negative intensity to zero?
Clipping discards uncertainty information and biases weak data upward; use the validated statistical treatment.
Does amplitude recovery solve the phase problem?
No. Amplitude is only the magnitude; diffraction intensity does not provide the structure-factor phase.
Authoritative sources and further reading
IUCr: An introduction to data reduction, scaling and intensity statistics. Explains scaled intensities, negative observations, Wilson statistics and French–Wilson treatment.
IUCr: On the treatment of negative intensity observations. The original French–Wilson method estimates sensible structure-factor moduli from positive and negative intensities.
IUCr: Intensity likelihood targets and structure-factor estimation. Explains limitations of square-root conversion and the Bayesian treatment of weak observations.
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