Materials Science · Additive manufacturing and multiscale characterisation

Observation from atomic to component scales

Atomic-scale observations, diffraction, microscopy and component-scale tomography sample different features and volumes. Resolution, penetration and representative volume create trade-offs. A method that detects a pore may not resolve its sharpest tip.

Useful background: Processing routes and microstructural evolution; Measuring and checking material claims. Each worked case explains the quantities it uses.

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After reading, you can

Combine measurements that sample different structural scales.
Distinguish a voxel or pixel size from feature detectability, resolved shape and sampled material.
Calculate a bounded mixing response and compare geometric sampling volumes without claiming representativeness.
Use a declared location transform and error bounds to decide whether a targeted view contains a feature.
Connect registered geometry, local structure and an observed loading sequence to a limited mechanism inference.

No single image covers every question

A printed metal block contains a dark region in an X-ray volume image. A later electron image shows a sharp opening, and a tiny foil gives a diffraction pattern. Do these observations describe the same feature? Is the dark region definitely an empty pore, and did that pore start a crack? More detailed images help only when their signals, sampled material, locations and histories address those questions.

X-ray computed tomography, abbreviated XCT, combines many transmission projections to reconstruct a three-dimensional field related to X-ray attenuation: how strongly material weakens the beam along its path. A voxel is one small three-dimensional cell in that reconstruction. A pixel is a two-dimensional image cell. Their spacing describes a sampling grid, not a promise that adjacent small objects can be distinguished or that a pore of that diameter will be detected.

Microscopy supplies different contrast. A scanning electron microscope, or SEM, records signals from the surface and a nearby interaction region; a polished section exposes a plane inside the original object. In transmission electron microscopy, or TEM, electrons pass through a sufficiently thin specimen. Image contrast can depend on thickness, composition, crystal orientation and diffraction conditions. An atomic-scale image can reveal local periodic structure, but is not automatically a three-dimensional list of all atoms along the beam path.

Diffraction records interference from an illuminated crystalline region. Sets of reflections can constrain crystal structure and orientation; they do not by themselves locate every pore or show where a component first cracked. Chemistry and imaging can help distinguish possible assignments. Microscopy, diffraction and structure evidence provides the preparation on contrast, calibrated length and diffraction; here the task is to connect those different observations to the same material question.

A real pore need not produce a pore-labelled voxel

Use an intentionally reduced, fictional sampling model to isolate one effect. Each cubic voxel has edge s = 10.0 µm. Suppose its reported coefficient μ_v is exactly the volume average of the local linear attenuation coefficient μ. Take μ_s = 10.0 mm⁻¹ for one homogeneous solid and μ_p = 0 for an empty spherical pore. These are invented values. Ignore noise, blur, reconstruction artefacts and energy dependence; this is not a complete XCT forward-and-reconstruction model.

If fraction f of a voxel is pore and the rest is solid, its stipulated average is μ_v = (1 − f)μ_s + fμ_p = 10(1 − f) mm⁻¹. Define contrast reduction Δμ = μ_s − μ_v = 10f mm⁻¹. Averaging can therefore leave a pore-containing voxel close to the solid value. A real algorithm also depends on noise, contrast, shape and its decision rule.

For a sphere of diameter d = 4.00 µm entirely inside one voxel, pore volume is V_p = πd³/6 = 33.5103… µm³. The voxel volume is s³ = 1000 µm³, so f = 0.0335103… and μ_v ≈ 9.66490 mm⁻¹. The geometric size is 0.4 voxel edges across; that ratio alone does not decide whether it is detectable.

Move that same sphere so its centre lies exactly at the corner shared by eight voxels. Each receives one eighth of its volume. The affected fraction in each is f/8 = 0.00418879… and each reported value is about 9.95811 mm⁻¹. The pore amount and shape are unchanged, yet the largest single-voxel contrast is eight times smaller. This comparison is valid for the stipulated sphere locations and d ≤ s, not arbitrary pore shapes or scanner physics.

Now stipulate a simple flag: label a voxel as a pore candidate only when μ_v ≤ 9.80 mm⁻¹, equivalently Δμ ≥ 0.20 mm⁻¹. The centred 4 µm sphere flags one voxel; the corner-shared sphere flags none. A flag does not resolve the sphere’s shape, measure its tip curvature or identify its chemistry. Under this exact model the corresponding diameter boundaries are about 3.368 and 6.736 µm, respectively, while the geometry assumptions remain valid.

Experimental detection needs its own evidence. In a study of intentionally incorporated AM defects, changing acquisition and reconstruction conditions changed image noise and successful thresholding. The study also qualified its higher-resolution XCT reference sizes rather than treating them as perfectly known truth. A universal rule such as “three voxels always detects a pore” would discard those dependencies. The figure’s boundary belongs only to the invented averaging and flagging rule.

Original fictional volume-average model, not measured XCT performance. Cubic voxel edge is 10 micrometres; solid coefficient is 10 per millimetre and empty pore coefficient zero. The vertical axis is reduction from the solid coefficient, Δμ = 10 − μ_v, in inverse millimetres. The horizontal axis is spherical pore diameter, shown from zero to seven micrometres. The solid blue cubic curve with a circular marker is a sphere wholly inside one voxel: Δμ = 10πd³/(6s³). The dashed orange cubic curve with a square marker is the same sphere centred at an eight-voxel corner: each voxel has one eighth of that reduction. The dashed horizontal rule flags a candidate at Δμ at least 0.20 per millimetre, equivalent to μ_v at most 9.80. It is crossed at diameters about 3.368 and 6.736 micrometres, respectively. For the marked four-micrometre sphere, reductions are about 0.33510 and 0.04189 per millimetre, giving reported coefficients 9.66490 and 9.95811; the centred sphere flags but none of the shared-corner voxels does. Whole pore volume is unchanged. Sphere geometry requires d no greater than s here; the plotted curves use only the zero-to-seven part of that domain. Noise, blur, artefacts and real-scanner detection probability are excluded.

A larger image count is not a larger independent material sample

For a separate geometric account, take a fictional rectangular block 8.00 mm × 4.00 mm × 2.00 mm, total volume 64.0 mm³. A reconstruction covering this complete box on a 10.0 µm = 0.0100 mm grid contains 800 × 400 × 200 = 64 million voxels. Neighbouring reconstructed values can be correlated. They are not 64 million independently sampled parts, and observing the whole box does not remove its detection limits.

A chosen SEM section field is 0.400 mm × 0.300 mm, or 0.120 mm². Its area does not supply a sampled volume without an interaction-depth and signal model. A plane also cannot establish whether a feature continues, connects or changes shape out of that plane. Serial sections or suitable tomography can add that third-direction information; the actual removed material and section spacing must remain in the record.

Suppose a TEM lamella, the thin piece prepared for examination, measures 10.0 µm × 8.00 µm × 0.100 µm. Its geometric material volume is 8.00 µm³ = 8.00 × 10⁻⁹ mm³; the block-to-lamella volume ratio is eight billion. Preparing the lamella does not mean all of it was imaged at atomic resolution. A selected 20.0 nm × 20.0 nm atomic-scale field through that same 100 nm thickness has a geometric beam-path volume of 40000 nm³ = 4.00 × 10⁻¹⁴ mm³. This path volume is not a three-dimensionally resolved atomic volume.

A separately stipulated diffraction gauge box of 0.800 mm × 0.800 mm × 2.00 mm occupies 1.28 mm³, or 2.00% of the block. That geometric fraction does not mean 2% of every phase was measured with equal sensitivity. Absorption, orientation and the method’s weighting affect the signal; a set of diffraction peaks is not a spatial map of the rest of the block.

The volume ratios explain a sampling limit, not the probability that an unexamined region is good or bad. A deliberately targeted foil near an unusual feature can be valuable precisely because it is not a random sample. It can test a local structural explanation while remaining a poor estimator of whole-component phase fraction or defect frequency.

Invented geometric extents for one block. The entries count different objects: reconstruction volume, section area, gauge volume, prepared material and an illuminated path. They are not universal method resolutions or interchangeable representative volumes.
Observation or material region	Supplied extent	Geometric amount	Inference still limited
Whole-block XCT grid	8 × 4 × 2 mm; 10 µm grid	64.0 mm³; 64 million cells	Unresolved or weakly contrasted features
Selected SEM section field	0.400 × 0.300 mm	0.120 mm²	Depth and out-of-plane continuation not supplied
Diffraction gauge box	0.800 × 0.800 × 2.00 mm	1.28 mm³	Signal weighting and unilluminated material
Prepared TEM lamella	10.0 × 8.00 × 0.100 µm	8.00 µm³	Not all material atomically imaged
Selected atomic-scale field	20.0 × 20.0 nm through 100 nm	40000 nm³	Projected/path information, local selection

Connect the scales through the claim

Choose observations based on the proposed failure or transport mechanism. Register locations where useful and distinguish measured features from inferred ones. Combining images does not automatically make an unmeasured region representative.

Registration means putting observations into a common specimen coordinate system. A landmark, datum or fitted transform can connect two views. Preparation can remove material, rotate a section or distort it. Matching two bright shapes by eye is weaker than recording which physical region was retained and the uncertainty of the mapping. An experimental registration study found that changing sampling for its reference alignment changed the registered results even while the optical and XCT datasets themselves were unchanged.

Use an original one-dimensional targeting model. The other coordinates and section direction are stipulated to match; rotation and deformation are negligible. A target centre is estimated at x_CT = 1.250 mm in the tomography frame. A nominal translation b = 1.200 mm gives the microscopy coordinate x_M = x_CT − b = 0.050 mm = 50.0 µm. The subtraction converts frames; it does not improve the original position information.

Let the tomography centre error be bounded by ±12 µm and the residual mapping error by ±18 µm. These are supplied hard bounds for this illustration, not standard uncertainties or probability distributions. The total centre error can reach ±30 µm when signs align. No independence assumption permits replacing this worst-case sum by a root-sum-square bound.

The feature has stipulated half-width r = 20.0 µm along x. Its actual centre can lie from 20 to 80 µm, so the union of its possible extents runs from 0 to 100 µm. An 80 µm-wide field centred at 50 µm spans 10–90 µm. It contains the possible centres but need not contain the entire feature. Missing an edge of a targeted object is different from proving that edge does not exist.

Two separate changes can improve this one-dimensional containment. A 120 µm field, spanning −10 to 110 µm, contains the possible extent with 10 µm margin at each end. Alternatively, keep the original 80 µm field but independently reduce the mapping bound to ±5 µm. The centre bound becomes ±17 µm and the full possible extent becomes 13–87 µm, fitting with 3 µm margin. These are supplied improved conditions, not a prediction that changing magnification or a registration algorithm achieves them.

A general containment condition for a field centred on the nominal target is L/2 ≥ r + e_CT + e_map, with L its width and the two e values the stated absolute bounds. It is only an x-direction test. A good x margin does not establish the correct cutting depth, orientation, chemical identity or representativeness of the sampled material.

Original targeting model, all x coordinates in µm. The predicted centre is 50 µm; the other coordinates and orientation are fixed by stipulation. Bounds add without a statistical independence assumption.
Alternative	Field extent	Total centre bound	Union of possible feature extents	Containment margin
A: original field and mapping	10 to 90	±30	0 to 100	−10; entire feature not guaranteed
B: enlarge field only	−10 to 110	±30	0 to 100	10
C: improve mapping only	10 to 90	±17	13 to 87	3

Original one-dimensional targeting model, not a registration prescription or probability distribution. Nominal microscopy coordinate is 1.250 minus 1.200 mm = 50 micrometres. Other coordinates and orientation are fixed by stipulation. A capped violet line shows the union of all possible extents of a feature of half-width 20 micrometres. An outlined blue box shows the field of view; these are distinct shapes as well as colours. In A, absolute centre-error bounds 12 plus 18 micrometres give a possible feature extent 0 to 100 micrometres; the 80-micrometre field spans 10 to 90 and can miss an edge. In B, the same feature extent is contained by an enlarged field from −10 to 110, leaving 10-micrometre margin. In C, the mapping bound is separately improved from 18 to 5 micrometres. Together with the unchanged 12-micrometre tomography bound, the possible feature extent is 13 to 87; the original 10-to-90 field then has 3-micrometre margin. All three alternatives share the same linear x-coordinate scale. These are worst-case absolute bounds and add directly, without an independence or Gaussian assumption. Positive x containment does not establish the correct depth, phase or a representative sample.

Use the registered sequence to test a mechanism

Consider a fictional observation sequence on one selected region of the block. Before loading, XCT resolves a cavity. During a later held-load observation it resolves a new opening adjoining that cavity. A registered section subsequently confirms an opening at that location, while local microscopy and diffraction characterise the surrounding solid. Each observation contributes a different kind of evidence; none is simply “the highest-resolution answer”.

The before-and-after three-dimensional geometry makes a pore-linked opening a plausible explanation worth testing. It does not establish that the first resolved opening was the first microscopic crack to nucleate anywhere. Smaller openings may have existed below the detection capability; acquisition spans time; section preparation can also change a surface. Preserve the loading and imaging sequence, not just the final attractive image.

Local electron-image contrast alone cannot decide whether a nearby dark line is a crack, a thickness fringe or a diffraction effect. A section or tilt comparison, chemistry and an appropriately interpreted diffraction pattern can discriminate between proposed structures. Conversely, finding a second phase in a small nearby solid region does not establish that it caused the observed opening, or that it occurs throughout the component.

A real study combined interrupted tensile loading, XCT and fracture-surface SEM on intentionally defect-containing 17–4 stainless steel specimens. Image-based elastic finite-element calculations divide the imaged solid into small connected elements and solve an assumed elastic response. Comparing those calculations with the observed progression helped identify candidate initiation regions. The stress distribution came from a model, not direct stress pixels, and the result concerned those specimens and loading conditions. It illustrates the value of connecting geometry, sequence and a testable explanation rather than turning a post-fracture image into a universal AM failure mechanism.

Fictional registered observations, not a material test report. The examples separate a measurement from an additional inference.
Observation	Supported local conclusion	Not established by it alone
Pre-load XCT resolves a cavity	A cavity-compatible feature is present within that method’s capability	Its sharpest curvature or all smaller defects
Later XCT resolves an adjoining opening	Resolved geometry changed near that feature during the recorded history	The exact first nucleation event or a unique stress field
Registered section shows an opening	A corresponding exposed plane contains an opening	Its full 3D connectivity or freedom from preparation effects
Local TEM and diffraction describe adjacent solid	Structure/contrast constraints for the selected illuminated material	A component-wide phase fraction or causal failure attribution

Choose the missing link rather than another impressive image

If the claim concerns the shape of a small pore tip, a whole-part scan may locate the pore while a better resolved local observation is needed for curvature. If the claim concerns a connected internal path, one highly resolved section is insufficient without out-of-plane information. If the claim concerns phase identity, choose appropriate diffraction and chemistry rather than treating a grey level as a chemical label.

If the claim concerns the same feature across methods, improve the location record before comparing two potentially different objects. If it concerns the whole component, plan additional locations and appropriate detection evidence rather than multiplying a selected foil’s volume or image count into a population guarantee. Sampling, scale and extrapolation limits develops that wider inference.

Models can connect measurements at different scales, but comparison must use the same physical quantity, material state and boundary conditions. A calculated stress concentration, a measured opening and a local diffraction pattern are not interchangeable observables. The useful outcome is a supported chain from the recorded structure and history to a bounded explanation, with the remaining unmeasured links made explicit.

Worked examples

Work through the case

A high-resolution image covers a tiny region while tomography covers the whole component at lower resolution. Which is sufficient alone?

Identify the size and distribution of relevant defects.
Compare each method’s detection capability and sampling.
Use complementary measurements where needed.

Neither is universally sufficient; the claim determines the needed combination.

Move the pore without changing its size

Use the ideal 10 µm voxel model and μ_v ≤ 9.80 mm⁻¹ flag. Compare a 4.00 µm spherical pore centred in one voxel with the same pore centred at a shared corner of eight voxels. What does a flag establish?

The pore volume is π(4.00 µm)³/6 = 33.5103… µm³. A voxel is 1000 µm³. With the sphere inside one voxel, f = 0.0335103… and μ_v = 10(1 − f) ≈ 9.66490 mm⁻¹.
At an eight-voxel corner, each affected voxel has one eighth of that pore amount: f_i = 0.00418879…. Its value is 10(1 − f_i) ≈ 9.95811 mm⁻¹. All pore volume is still accounted for: eight contributions sum to the same 33.5103… µm³.
The centred voxel passes μ_v ≤ 9.80; none of the eight shared voxels passes. The averaging position changes the flag despite an unchanged pore.
This invented threshold does not resolve shape or provide a scanner detection probability. Even a flag could have another cause if the assumed solid coefficient or composition were wrong.

One centred voxel flags; no corner-shared voxel flags. Neither result by itself establishes the true shape, chemistry or absence of a pore.

Keep area, prepared volume and image path separate

Compare the 64.0 mm³ block, 10 × 8 × 0.100 µm TEM lamella and 20 × 20 nm atomic-scale field through its 100 nm thickness. Could the section’s 0.120 mm² area be inserted into the same volume ratio?

The lamella volume is 10 × 8 × 0.100 = 8.00 µm³. Because 1 mm³ = 10⁹ µm³, this is 8.00 × 10⁻⁹ mm³. The block-to-lamella volume ratio is 64/(8 × 10⁻⁹) = 8.00 × 10⁹.
The atomic-scale path volume is 20 × 20 × 100 = 40000 nm³. Since 1 mm³ = 10¹⁸ nm³, it is 4.00 × 10⁻¹⁴ mm³. The block-to-path ratio is 1.60 × 10¹⁵.
The SEM field is an area, not a specified volume. A depth/signal model is missing; dividing a block volume by that area would produce a length, not a dimensionless sampling ratio.
Neither geometric ratio is a probability or proof of representativeness. The illuminated path through the foil is not a three-dimensional atomic reconstruction.

The ratios are eight billion for the prepared lamella and 1.60 × 10¹⁵ for the selected atomic-image path. The section area cannot supply a volume ratio without additional depth information.

A well-centred field can still miss an edge

Apply the translation and bounded-error model to alternatives A, B and C. Does the original 80 µm field guarantee the entire 40 µm feature lies inside it?

Nominal microscopy position is 1.250 − 1.200 = 0.050 mm = 50 µm. Worst-case centre deviation is 12 + 18 = 30 µm. Including r = 20 µm gives a possible extent from 50 − 30 − 20 = 0 to 50 + 30 + 20 = 100 µm.
A spans 10–90 µm. Its half-width 40 is smaller than the needed 20 + 12 + 18 = 50 µm. The possible centre is inside, but an extreme edge can extend 10 µm beyond the field.
B has half-width 60 µm, leaving 60 − 50 = 10 µm margin. C instead needs 20 + 12 + 5 = 37 µm within half-width 40, leaving 3 µm margin.
The sum uses absolute bounds, not independent standard deviations. The conclusion is one-dimensional containment under stipulated alignment, not proof of cutting through the correct 3D feature.

A does not guarantee full containment. The separately supplied B and C alternatives do, with margins 10 and 3 µm in the x-direction only.

Try the reasoning

Does highest magnification imply largest representative volume?
Show the answer and reasoning
No.
Higher resolution often samples a smaller region.
Why report detection limits?
Show the answer and reasoning
Undetected does not mean absent.
The method may miss smaller or poorly contrasted features.
Keep the 4 µm sphere centred, but double the voxel edge from 10 to 20 µm with all other toy-model assumptions unchanged. Does it pass the same μ_v ≤ 9.80 flag?
Show the answer and reasoning
No. Its fraction is 33.5103…/8000 = 0.00418879…, giving μ_v ≈ 9.95811 mm⁻¹.
Voxel volume grows eightfold, so this centred contrast equals the earlier eight-way shared result. More sampled volume per cell is not automatically better small-feature detection.
Within the centred-sphere model, find the smallest diameter flagged by μ_v ≤ 9.80. State the geometric validity condition.
Show the answer and reasoning
About 3.368 µm; it must remain a sphere wholly within the 10 µm voxel.
The rule requires f ≥ (10 − 9.80)/10 = 0.0200. Set πd³/(6 × 1000) = 0.0200 to obtain d = (120/π)^(1/3) ≈ 3.36778 µm. This is the threshold of an invented exact averaging rule, not a universal imaging resolution.
A pore-free voxel of a different solid phase has coefficient 9.66490 mm⁻¹. Can its single value be distinguished from the centred 4 µm pore case by this scalar measurement alone?
Show the answer and reasoning
No, at the stated precision they give the same value.
The mixture calculation assumed solid μ_s = 10. A composition/phase change can supply another explanation for the contrast. Additional structure, chemistry or geometry evidence is needed; a dark voxel is not uniquely an empty-pore label.
The separately supplied TEM lamella thickness is halved to 0.0500 µm while its 10 × 8 µm lateral extent stays fixed. What happens to its volume and block-to-lamella ratio? Does this establish improved representative sampling?
Show the answer and reasoning
Volume becomes 4.00 µm³ and the block-to-lamella ratio becomes 1.60 × 10¹⁰. Representativeness is not established.
The material volume halves. The change may alter imaging conditions, but no image quality or sampling probability follows from the volume calculation alone.
Under the original ±12 and ±18 µm bounds, change the feature half-width to 40 µm. Is the formerly enlarged 120 µm field sufficient? What is the minimum centred width for full x containment, allowing equality?
Show the answer and reasoning
No. It needs at least 140 µm width.
Required half-width is 40 + 12 + 18 = 70 µm, exceeding 60. Doubling the feature size changes the containment demand even though the location estimate is unchanged.
An atomic-scale image from a second, unregistered foil shows a different phase. Does it demonstrate that the newly resolved opening in the original block began at that phase?
Show the answer and reasoning
No. The phase observation and the opening have not been tied to the same material region and history.
The second foil may be useful background evidence, but a causal local comparison requires location, state and sequence. Even correct registration would not by itself establish that phase as the unique cause.
No opening is resolved in an initial XCT scan; a later scan resolves one near a cavity. Can the later time be labelled the exact crack-nucleation time?
Show the answer and reasoning
No.
Earlier openings could have been below the detection capability, and each acquisition covers an interval. The record establishes the first resolved opening in those observations, not necessarily the first physical nucleation event.

What to carry forward

Match scale, resolution and sampled volume.
State detection and inference limits.
A small grid cell, a flag, a resolved shape and a chemical identity are different achievements.
A registered local observation can test a mechanism while remaining unrepresentative of unmeasured material.
Combine state, location and history before connecting component geometry with atomic-scale evidence.

References

Teaching resources and research supporting this lesson’s explanations and examples.