The Gold Standard of X-ray Crystallography | Separating Resolution Cutoff from Detector Angular Coverage

“Resolution” may refer to a chosen high-resolution cutoff, the smallest observed d-spacing, an estimate of meaningful signal, a detector-edge geometry or the resolving ability of an instrument. Begin by identifying the object and rule. A bare value such as 2.5 Å is incomplete without that context.

Detector angular coverage asks where diffracted rays can land on active detector area for the stated placement. It is directional. The radial distance from the beam centre, panel shape, inactive borders, module gaps, beamstop shadow, offset and tilt can all make the maximum reachable angle vary with azimuth.

A reported cutoff asks which reflections were included in a particular processing, refinement or table. It may be deliberately lower than a geometric edge because of signal, overlap, radiation damage, scaling behaviour or an analysis policy. It is an evidence decision, not a measurement of the detector’s physical size.

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A detector coordinate gives the angle between the direct-beam direction and the diffracted ray, commonly written 2θ in this geometry. Bragg’s law uses θ: nλ=2d sinθ. Substituting the full scattering angle where θ belongs can produce a large and plausible-looking error.

For first-order diffraction n=1, solve d=λ/(2 sinθ). A larger scattering angle corresponds to a smaller d-spacing at fixed wavelength. “Higher resolution” in crystallography therefore commonly means a smaller d value, which reverses the ordinary intuition that a larger number means more detail.

Preserve units. Detector offsets and distance may be in millimetres, wavelength in ångströms and d in ångströms. The arctangent uses a dimensionless ratio of lengths, while trigonometric functions require a consistent angle convention in the calculator or code.

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For a flat detector perpendicular to the incident beam, let L be the sample-to-detector distance and r the distance on the detector from the beam centre to an active point. In the simple central geometry, tan(2θ)=r/L. This relation changes when the detector is tilted or when coordinates are not measured in the calibrated detector frame.

Mark the beam centre rather than assuming it lies at the panel centre. A detector shifted to reach higher angles on one side intentionally creates asymmetric coverage. A central radius derived from panel width can be wrong when the direct beam is offset, cropped or shadowed.

Also mark active rather than physical extent. A housing edge, dead module, gap or masked pixel does not collect a usable reflection. The geometric question is whether the ray intersects calibrated active area under the acquisition conditions, not whether it intersects the rectangle drawn in a brochure.

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A radial edge calculation says which scattering magnitude can reach some pixel. Completeness asks what fraction of expected unique reflections in a defined region were measured under the chosen crystal orientations and acceptance rules. A detector can reach a high angle in one direction while leaving other directions unobserved.

The earlier specialist article on completeness versus redundancy counts expected unique reflections and repeated observations. This article asks a prior geometric question: which directions can the detector intercept, and how does that envelope compare with the selected resolution shell? The two articles meet when missing-reflection maps are interpreted.

Never report “the detector covers 2.2 Å” as a complete data claim. A careful sentence is “the stated central geometry places the radial active edge at a d-spacing of about 2.2 Å along that direction.” Orientation, shadowing, symmetry, signal and processing determine what the dataset actually contains.

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Use L=200 mm and r=100 mm. The ratio r/L is 0.5, so 2θmax=arctan(0.5)=26.565°. Halving gives θmax=13.2825°. With λ=1.000 Å and n=1, dmin=1.000/(2 sin13.2825°), approximately 2.176 Å.

The result is a geometric edge value for the stipulated idealised direction. Keep more digits during the calculation than the inputs justify, then report a sensible rounded result such as 2.18 Å. Printing 2.176342 Å would imply information that the 200 mm and 100 mm inputs did not provide.

Check direction before trusting arithmetic. If r increases at fixed L, 2θ increases and d decreases. If L increases at fixed r, 2θ decreases and d increases. A result that moves in the opposite direction usually signals an angle, ratio or calculator-mode error.

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At d=2.50 Å and λ=1.000 Å, sinθ=1/(2×2.50)=0.2. Thus θ≈11.537° and 2θ≈23.074°. The corresponding detector radius in the simple geometry is r=L tan(2θ)=200 tan23.074°, about 85.1 mm.

Because 85.1 mm lies inside the 100 mm active edge along the studied direction, the radial geometry can intercept that scattering angle. The 2.50 Å analysis cutoff is more conservative than the calculated 2.18 Å edge. This comparison is valid only for the calibrated geometry and wavelength used.

The 14.9 mm radial margin is not a completeness margin. A gap may cross the ring, another azimuth may have a nearer edge, and a given crystal orientation may not place every expected reciprocal-lattice point on a recorded trajectory. Use a coverage simulation or observed-reflection map for those questions.

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A rectangular detector has different radii to an edge midpoint and a corner. The corner gives the largest radial distance, but it occupies only a direction and may contain module gaps. Quoting the corner d-spacing as if it applied in every azimuth overstates coverage.

Suppose the nearest active boundary is 80 mm horizontally and 120 mm vertically from the beam centre. Along the horizontal direction, 2θmax=arctan(80/200)=21.801°, giving d≈2.65 Å at λ=1 Å. Along the vertical direction, arctan(120/200)=30.964° gives d≈1.94 Å.

A circular 2.50 Å shell requires about 85.1 mm radius in this setup. It fits vertically but extends beyond the horizontal active edge. The detector has partial angular reach for that shell, even before crystal rotation and symmetry are considered. A single “maximum resolution” cannot express this anisotropy.

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IUCr detector-calibration work notes that detector position, beam centre, tilt, pixel dimensions, sample-to-detector distance and wavelength all matter. Mechanical readings alone may not determine the diffraction geometry at the precision needed. A standard and calibrated model connect pixel positions with scattering angles.

For a sensitivity check, change L from 200 mm to 202 mm while keeping r=100 mm and λ=1 Å. Then 2θ=arctan(100/202)=26.329°, θ=13.1645° and d≈2.196 Å. The 2 mm distance change moves the geometric edge by roughly 0.020 Å in this teaching case.

This is not a general uncertainty calculation. Real calibration parameters can be correlated, detector tilt changes different regions unequally and wavelength uncertainty also contributes. Preserve the fitted covariance or suitable validation residuals when a quantitative uncertainty statement is needed.

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Low-angle reflections may be hidden by the beamstop, while high-angle rays may cross panel gaps or leave the active area. Masked regions should be represented in the predicted coverage, not treated as random missing measurements after collection. Their geometry often creates structured holes.

A beamstop does not set the high-resolution edge, yet it affects low-resolution completeness and downstream scaling. Likewise, a narrow panel gap may remove only a small fraction of pixels but intersect a recurring trajectory for some orientations. Pixel percentage and reflection percentage are not interchangeable.

Audit masks by projecting expected reflection positions for the acquisition plan. Compare predicted losses with observed missing-reflection patterns. When the pattern matches geometry, collecting more exposure time at the same orientations may repeat the hole rather than fill it.

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A reciprocal-lattice point diffracts when it satisfies the Ewald-sphere condition for a crystal orientation. Rotating the crystal moves reciprocal-lattice points through diffracting positions; the detector records only rays that then intersect active area. Detector reach and rotation range therefore work together.

IUCr data-collection guidance uses the Ewald sphere to explain this geometry and notes that detector placement may trade maximum resolution, spot separation and completeness. A wider rotation range can fill some directional gaps, but it cannot make a ray land on inactive detector area.

Record orientation wedges, excluded ranges and any collision or shadow constraints. “We rotated 180°” is not a complete coverage statement without symmetry, wavelength, detector position and the actual valid frames. Use predicted completeness by shell and direction as the final bridge.

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Reflections may be geometrically reachable yet too weak or overlapped for reliable integration. Conversely, a strong reflection at the detector edge does not show that every expected reflection in the shell is measurable. Geometry defines opportunity; signal and processing determine observed evidence.

A resolution cutoff might be chosen using intensity statistics, correlation measures, model behaviour or a stated laboratory policy. This article does not prescribe a universal threshold. It requires the threshold, shell definitions and decision rationale to be reported rather than retrofitted to the detector edge.

Compare three maps: predicted geometric reach, integrated observations and accepted reflections after filtering. Their differences locate the cause of loss. Collapsing them into one number makes it impossible to tell whether a remedy needs geometry, exposure, orientation, indexing, integration or a revised claim.

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Change the central case to L=300 mm while keeping the 100 mm edge and λ=1 Å. Then 2θmax=arctan(1/3)=18.435°, θ=9.2175° and dmin≈3.12 Å. The farther placement provides less high-angle reach on the same detector.

Greater distance can improve spot separation or angular sampling per pixel in some contexts, so the smaller angular field is not automatically a worse experiment. IUCr guidance explicitly discusses matching distance to diffraction, overlap and completeness needs. The correct placement depends on the specimen and question.

If the analysis still reports to 2.50 Å, the simple central geometry cannot intercept that shell at the 100 mm edge. The discrepancy should trigger a check of detector offsets, a larger active radius, a different wavelength, multiple placements or an incorrect metadata assumption.

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Return to L=200 mm and r=100 mm, but use λ=0.800 Å. The angular edge is unchanged because geometry has not moved. Bragg’s law now gives dmin≈0.800/(2 sin13.2825°)=1.741 Å. A shorter wavelength maps the same angle to a smaller d-spacing.

This changed case proves why an angular coverage value cannot be converted to resolution without wavelength. It also shows why copying a detector-edge d value between experiments is unsafe when wavelength differs, even if the images have the same dimensions.

Shorter wavelength does not guarantee better useful data. Scattering strength, absorption, anomalous signal, instrument response, background and the scientific question also matter. The worked calculation transfers geometry only; it does not rank complete experimental strategies.

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Shifting a detector away from the direct-beam centre can extend high-angle reach on one side. IUCr data-collection guidance notes that offsets can collect higher diffraction angles, sometimes with additional rotation needed for complete data. The gain is directional rather than free circular coverage.

Suppose the active panel spans x from −40 to +160 mm relative to the beam centre at L=200 mm. The positive edge reaches 2θ≈38.66° and d≈1.51 Å at λ=1 Å, while the negative edge reaches only 11.31° and d≈5.08 Å. One headline edge would conceal the asymmetry.

Report the offset coordinate system and coverage map. If multiple placements are combined, identify which frames supply which regions and how scaling was checked. A merged dataset may be broad, but each placement still has its own geometry and calibration evidence.

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After geometry is modelled, compute expected and observed unique reflections in resolution shells. A shell can be nearly complete even when the rectangular detector does not contain a full circle because crystal symmetry and rotation distribute reflections in ways that the simple ring picture does not capture.

Conversely, a generous radial edge can coexist with poor completeness if orientation wedges are missing or reflections fall into systematic masks. This is why the companion article separates completeness from redundancy: repeated observations in accessible regions cannot manufacture unique reflections in unobserved regions.

Use directional completeness or reciprocal-space visualisation when a shell average hides anisotropy. Report whether a cutoff is spherical, ellipsoidal or otherwise filtered. A single percentage without the expected-set definition cannot diagnose whether detector angular coverage is the limiting cause.

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Replace “The detector resolution is 2.18 Å” with “Under the stated central flat-detector geometry and λ=1.000 Å, the 100 mm active radial edge corresponds to d≈2.18 Å.” The revised sentence names the model, wavelength, direction and geometric nature of the result.

Replace “The dataset is complete to 2.5 Å because the ring fits” with “The studied radial geometry reaches the 2.5 Å scattering angle; completeness must be evaluated from the expected and observed reflection set across orientations, masks and symmetry.” This keeps the calculation useful without making it carry a different claim.

Avoid treating the processing cutoff as a detector specification. Record who or what chose it, the statistic and shell used, and whether later refinement or deposition uses the same boundary. If several cutoffs appear in a workflow, name each rather than averaging them into a fictional consensus.

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Does moving a flat detector closer always improve the experiment? It increases angular reach for a fixed active radius, but can worsen spot overlap or other constraints. Does a corner define the resolution everywhere? No. It is a direction-specific maximum and may include inactive regions.

Can the detector reach a shell that is excluded from processing? Yes. Geometry can be more permissive than an evidence-based analysis cutoff. Can the analysis include a shell outside the modelled active reach? That inconsistency requires checking wavelength, beam centre, distance, offsets, multiple placements and metadata.

Is resolution the same as coordinate precision? No. The root guide already separates detail supported by diffraction from parameter uncertainty and displayed digits. The present article addresses yet another boundary: geometric interception of scattering angles versus a chosen reciprocal-space cutoff.

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Use the IUCr data-collection strategies guide for the Ewald construction, detector-distance trade-offs, offsets and completeness considerations. Use the IUCr article on area-detector calibration with multiple placements for the relation tan(2θ)=h/L and the calibration parameters needed to map pixels to angles.

Continue with Separating Reflection Completeness from Measurement Redundancy, Separating Systematic Absence from an Unobserved Weak Reflection and Separating Friedel-Pair Differences from Ordinary Symmetry Equivalence.

Final transfer test: with L=250 mm, r=75 mm and λ=1.200 Å, 2θmax=arctan(0.3)≈16.699°, θ≈8.3495° and dmin≈4.13 Å. Explain why this is an edge geometry, then list the additional evidence required before claiming a complete 4.2 Å dataset.

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The simple relation tan(2θ)=r/L assumes a detector normal to the direct beam and a coordinate measured from the calibrated beam centre. Tilt changes the path length and angular mapping across the panel. One scalar distance and one radius no longer describe every pixel.

Use the calibrated detector transformation supplied by the processing system and preserve its convention. Check predicted powder rings or a suitable standard across the full panel, not only near the centre. Systematic residuals that change with azimuth can reveal an incorrect tilt, beam centre or distortion model.

A tilted panel may extend angular reach in one region while reducing it elsewhere. Report the resulting map or directional extrema. Converting the farthest physical corner with the untilted formula creates a precise number for the wrong geometry and can misclassify the analysis cutoff.

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Several placements can cover a wider angular range than one image. Treat each placement as a separate calibrated geometry, project its valid active area into reciprocal-space coverage, then form the union. The union operation adds regions; it does not add duplicate observations as new unique coverage.

Where placements overlap, use common reflections to check scaling and consistency. A mismatch can reveal calibration, exposure, absorption, damage or processing differences. Merging by file order without this check can turn a geometric extension into an intensity discontinuity.

Record which placement supplies the limiting high-angle region and which supplies low-angle or shadowed regions. If one placement fails, the final dataset may retain most reflections but lose a structured shell sector. A global completeness percentage can hide that dependence.

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A missing wedge is directional: a region of reciprocal space is unobserved because the available orientations or geometry do not reach it. A spherical high-resolution cutoff excludes all reflections beyond a chosen d boundary. The shapes and remedies are different.

More exposure at the same orientations can improve counts in observed regions but cannot fill a geometrically inaccessible wedge. Extending the rotation range, changing orientation, moving the detector or using another strategy may address the geometry, subject to practical and safety constraints.

Visualize missing reflections by direction and shell. If the outer shell has 90% completeness but the missing 10% clusters in one wedge, report the anisotropy. A random-looking 10% loss and a coherent wedge can have the same percentage yet different consequences for maps and interpretation.

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Before any d-edge calculation, verify wavelength, detector model, pixel size, beam centre, distance, tilt, panel masks, binning, image orientation and units. Metadata copied from a template or stale calibration can produce internally consistent but physically wrong coordinates.

Check one observed ring or known reflection against the predicted position. Then test a second angle and a different azimuth. One match can be forced by an incorrect combination of distance and wavelength; multiple placements or known standards help separate parameters, as the IUCr calibration article explains.

Preserve raw headers and the corrected geometry with provenance. A later reprocessing should reveal what changed and why. Do not silently overwrite original metadata, because the discrepancy itself may explain a difference between historical and current resolution reports.

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A fictional setup has L=180 mm, active radial edge r=90 mm and λ=1.10 Å. The angle is 2θ=arctan(0.5)=26.565°, so θ=13.2825° and the geometric edge is d≈2.39 Å. Processing proposes a 2.70 Å cutoff.

At d=2.70 Å, θ=arcsin(1.10/(2×2.70))≈11.75°, so 2θ≈23.50° and r≈180 tan23.50°≈78.3 mm. That ring lies inside the radial edge in the idealised direction. The geometry is compatible with the proposed cutoff there.

The decision is still incomplete. Ask for the calibrated azimuthal active boundary, masks, orientation plan, predicted and observed unique reflections by shell, signal evidence and cutoff rule. The transfer answer is therefore two-part: the radial calculation passes, while complete dataset support remains to be demonstrated.

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A calibrated detector model has uncertainty in distance, beam centre, tilt, pixel scale and wavelength. Varying only the nominal distance gives a useful sensitivity check, but it does not describe their joint effect. Some parameters move the edge differently by azimuth, so the result can be an uncertainty band on a coverage map rather than one symmetric error bar.

For the simple normal-detector model, compute the edge at the bounds of the supported distance and wavelength ranges, then repeat for relevant beam-centre or active-radius uncertainty. If the proposed analysis cutoff lies comfortably inside every credible edge, geometry is unlikely to be the limiting boundary in that direction. If it overlaps the band, report the ambiguity and seek stronger calibration evidence.

Do not convert a fit residual directly into a resolution uncertainty without checking the model and parameter correlations. Validate predicted positions at several angles and directions. The final statement should name the calibrated geometry, uncertainty sources, directional reach and separate evidence used to choose the dataset cutoff.

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A circular radial calculation is a first screen, not a coverage map for a rectangular or segmented detector. For the proposed d cutoff, convert d to 2θ at the experiment wavelength, project that angle around the calibrated beam centre and test the locus against active pixels, masks and panel boundaries at each azimuth.

If the locus leaves active area in one direction, determine whether crystal rotation and symmetry can still supply the corresponding unique reflections elsewhere. The pixel-space gap and the reciprocal-space missing set are related through orientation; they are not identical shapes. Prediction software should preserve that mapping rather than label every off-panel point as an absent observation.

Report the smallest supported directional reach, any local extensions and the resulting expected-set completeness. A single corner value can be useful for planning, but it cannot certify a shell. The final claim joins calibrated geometry with orientation, symmetry and observed reflection evidence.

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An active pixel near a detector border may geometrically intercept a diffracted ray yet lack the surrounding profile region needed for reliable integration. Edge truncation, point-spread behaviour, module boundaries and spot size can shrink the usable area relative to the nominal sensitive rectangle.

Define an integration-safe mask for the processing method and project that mask into angular coverage. The largest active-pixel angle and the largest reliably integrated-reflection angle can differ. Report which one supports the claimed cutoff rather than switching definitions after examining the result.

Inspect reflections that approach the boundary at several azimuths and intensities. A gradual rise in partial profiles or rejected spots can reveal the usable edge. This is still separate from the evidence-based dataset cutoff, which can remain stricter for other reasons.

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If sweeps collected at different wavelengths or energies are combined, the same detector angle maps to a different d-spacing in each sweep. Preserve wavelength per image or sweep and calculate geometric reach under the corresponding value. One copied wavelength can make the merged reciprocal-space boundary internally inconsistent.

Use common indexed reflections and calibration metadata to check that orientation and scale transformations are coherent before forming the union. For anomalous experiments, wavelength choice may also change the scientific signal, so a geometric advantage is not the only consideration.

Report the merged cutoff as an analysis choice and show which sweep contributes each limiting region. A nominal detector placement shared across sweeps does not imply identical reciprocal-space coverage when wavelength differs.

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For each placement, store wavelength, calibrated beam centre, distance, tilt or panel transforms, active and integration masks, crystal orientation range, excluded frames and the predicted coverage version. Pair these with observed and accepted unique-reflection counts by shell and direction.

Add the selected processing cutoff and its evidence rule as a separate field. This receipt lets a reviewer ask whether loss came from geometric reach, orientation, masking, weak signal or filtering. It also prevents a later reprocessing choice from being misreported as an instrument limit.

The receipt need not contain every raw image, but it should point to immutable acquisition and calibration records. Recomputing the edge from remembered nominal settings is inadequate when the conclusion depends on millimetres, tilt, masks or multiple placements.

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A collection plan, integration run, scaling report, refined model and data deposition can each display a resolution value. Build a small provenance table that names the stage, numerical boundary, selection rule and data version. Differences may be legitimate, but unexplained differences invite the wrong geometric story.

Check that shell statistics use the same accepted-reflection set described by the cutoff and that any anisotropic treatment is visible. If a later step truncates the data further, preserve the earlier geometric reach as planning evidence rather than relabelling the final cutoff as the detector limit.

Before publication, read every sentence containing “resolution” and replace ambiguous uses with detector angular reach, geometric d edge, observed-reflection limit, processing cutoff or model-detail claim. Precise labels make the experimental reasoning auditable without demanding one universal number.

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A thick sensor can register an oblique photon at a position displaced from the entry point, and detection efficiency can vary with energy and angle. These effects do not change the ideal ray angle, but they can affect the calibrated mapping, spot profile and usable evidence near high-angle edges.

Use the detector’s supported calibration and correction model, then validate with appropriate standards or known reflections. Do not invent a universal parallax correction from sensor thickness alone; material, energy, geometry and readout design matter. Preserve the model version with the processed coordinates.

If edge reflections move or broaden systematically with azimuth, compare the pattern with tilt, distortion, parallax and integration-mask hypotheses. The goal is not to assign every discrepancy to the detector, but to keep physical interception, measured position and accepted intensity as distinct stages.

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A reflection can land on an active pixel yet exceed the detector or processing dynamic range. That observation is geometrically covered but may not yield a reliable integrated intensity. Saturation is therefore another reason why detector reach does not equal usable dataset support.

Mark overloaded pixels and affected reflection profiles, including blooming or neighbourhood effects under the actual detector model. A shorter exposure can recover strong low-angle reflections while a longer exposure supports weak high-angle data; merging requires scale and damage checks rather than selecting the largest value.

Report overload counts by shell and the strategy used to replace or exclude affected observations. Do not describe a saturated reflection as missing because it was outside angular coverage. Correct cause labels point to different remedies and make completeness statistics interpretable.

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If the proposed shell lies outside active high-angle area, a closer distance, offset, larger detector, shorter wavelength or additional placement may change reach. If the shell is geometrically present but weak, moving the detector may not solve signal, damage or background limitations. Diagnose before changing hardware geometry.

Write the desired d shell, convert it to 2θ at the chosen wavelength, then project it onto candidate calibrated placements with masks and collision constraints. Compare predicted completeness and overlap, not only the farthest edge. A plan that reaches one high-angle corner can still leave a directional hole.

After collection, compare predicted and observed coverage. Preserve discrepancies for calibration and strategy review. This closed loop turns detector geometry from a one-time calculation into evidence that improves the next collection without rewriting the meaning of the final analysis cutoff.

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