E17 · Thin-foil orientation mapping

TKD: Through the Nanocrystal

Can a thinner specimen resolve smaller grains? Yes. When the beam passes through a thin foil, it has little room to spread, so the diffraction pattern comes from a smaller region than in ordinary reflection EBSD. The price is a weaker pattern from a thinner foil, and the geometry still has to be calibrated.

Compare a reflection geometry with an off-axis transmission screen. Then trade off foil thickness, counts and binning to recover the orientation and phase of nanometre-sized grains.

Indexed one-axis rotation $\text{—}$

Indexed teaching phase $\text{—}$

Accepted fraction of all pixels $\text{—}$

Assumed footprint FWHM along tilt $\text{—}$

More measurements and target

Selected stored pattern counts $\text{—}$

Selected detector pixel counts $\text{—}$

Best candidate correlation $\text{—}$

Best versus runner-up score gap $\text{—}$

Independent check of phase and angle inputs $\text{—}$

Accepted-pixel mean angular error $\text{—}$

Incident dose per scan pixel $\text{—}$

Experiment target $\text{—}$

Simulated examples; each plot identifies its data source.

Geometry: 0 transmission, 1 reflection reference

m

0

Normal foil thickness

t

180\,\mathrm{nm}

Exposure per scan pixel

\tau

0.5\,\mathrm{ms}

Foil tilt in transmission only

\alpha

20\,\mathrm{^\circ}

Advanced settings

Assumed vertical pattern-center error / height

\Delta p_y

0

Stored detector binning factor

b

2

Minimum candidate correlation

c_{min}

0.65

Selected scan pixel index

k

26

Selected native detector pixel

j

1176

Pattern display gain only

G

1

Learning goal and model limits

Predict, then index the patterns and check them

Can a thinner specimen resolve smaller grains?

Target: accept at least 90 percent of the pixels, and get at least 90 percent of all pixels right in phase and within 2 degrees in angle. The accepted pixels must have a mean angle error of at most 2 degrees. The assumed footprint FWHM must be at most 12 nm.

Model assumptions

The electron counts are synthetic, generated from a fixed random seed; they are not experimental data. The beam is fixed at 20 keV and 20 pA. The scan has 12 by 12 pixels, and the detector natively has 48 by 48. Counts are drawn from a Poisson distribution when the mean is below 40 and from a normal approximation above that. Two fictional simple-cubic phases, with lattice constants 0.36 and 0.54 nm, produce smoothed bands from 13 plane normals, all with the same sign and the same weight. The cone geometry follows Bragg's law; band intensities, the flat background and the total collection efficiency are teaching assumptions. Dynamical diffraction, structure factors, excess-deficiency contrast, elastic strain, damage and calibrated material identification are not modeled. Indexing searches only the two phases and one known family of rotations about a single axis, from 0 to 60 degrees in 2 degree steps. It does not recover a general orientation with three angles. IPF colors show the incident-beam direction after reduction by cubic symmetry, and a single IPF does not fully specify an orientation. Connected grains and disorientation within one phase are computed from indexed pixels only. Unindexed pixels stay visible and count as failures in the check. The known inputs are kept separate from the maps derived from counts. Binning adds up the stored native counts; changing the pattern-center error, threshold, selected pixel or display gain does not acquire new data. Hardware and beam paths are schematic, and the nanometre specimen and millimetre detector are each magnified by their own factor. Detector rays ignore the nanometre scan shifts, which are tiny next to the millimetre screen distances. Both geometries use the same 120 nm field and 10 nm scan step, and the crystal does not change through the foil depth. The reflection reference uses a fixed footprint like that of a bulk specimen, with the stage fixed at 70 degrees; it does not solve the reflection response of a thin foil. In transmission the foil tilts from 0 to 30 degrees, the Gaussian footprint depends on thickness by a stated rule, and the efficiencies of pattern formation and attenuation are prescribed. The drawn foil is compressed fivefold in depth relative to its lateral scale. The primary beam enters the upper face, and diffracted rays leave the lower face toward an off-axis screen downstream. These rays are schematic connections, not trajectories from a transport solver. The spatial kernel leaves out the mean sideways walk-off of the beam and any change of grains with depth. Surface-specific layer identification, depth reconstruction, inelastic energy spectra and external resolution calibration are not included.

Physics tutorial

TKD: Through the Nanocrystal

BackgroundIn transmission Kikuchi diffraction the beam passes through a thin foil, and diffracted electrons leave the far side toward an off-axis screen. A thinner foil narrows the beam footprint, which helps with nanometre grains, but it can also give too few counts.

Why it mattersThe task asks for a map you can trust: at least 90 percent of pixels indexed, the correct phase and a rotation within 2 degrees in at least 90 percent of all pixels, and a mean angular error within 2 degrees for the accepted pixels. In TKD the assumed footprint FWHM must also stay at or below 12 nm.

Start with the essentials

Focus question: Can a thinner specimen resolve smaller grains?
One-sentence intuition: An indexing result depends on calibrated geometry, enough counts and a suitable family of candidates.

Core mathematical model

Cone edges on the detector

\begin{aligned}d_{hkl}&=a/\sqrt{h^2+k^2+l^2}\\|\widehat{\mathbf s}\cdot\widehat{\mathbf n}_{hkl}|&=\sin\theta_B=\lambda/(2d_{hkl})\end{aligned}

Ray directions that meet this condition lie on two Bragg cones. The trace of the lattice plane midway between them is the band center line; band intensity is prescribed.

Count first, index second

\begin{aligned}\mu_j&=\frac{I\tau}{e}\,\eta\,\frac{B_j}{\sum_k B_k}\\\widehat c&=\underset{c}{\operatorname{arg\,max}}\;\frac{(\mathbf N-\overline N)\cdot(\mathbf T_c-\overline T_c)}{\|\mathbf N-\overline N\|\,\|\mathbf T_c-\overline T_c\|}\end{aligned}

Each scan point stores one detector pattern. Indexing compares those counts with every entry in the stated dictionary of orientations and phases and keeps the highest normalized correlation.

Assumed foil response

\begin{aligned}p&=\frac{t}{(1\,{\rm nm})\cos\alpha}\\\sigma&=(1\,{\rm nm})\sqrt{4+(0.065p)^2}\\f&=1-e^{-p/15}\\\eta&=0.08f e^{-p/150}\end{aligned}

Tilting the foil turns its normal thickness into a longer path along the beam. A narrower footprint can still deliver too few counts. The coefficients are teaching assumptions.

Common difficulties

Map color is not a full orientation

Typical misconceptionA beam IPF map alone proves every orientation and phase.

Better mental modelThe search covers only the stated one-axis family and the fictional phases. Treat dark pixels and narrow margins between candidates as limits of the result, and look at the stored patterns.

Run the experiment

01
Predict before indexing
Select a stored pixel and inspect its detector pattern. Compare it with the separately labeled input.
What to observe: Band positions encode orientation; band widths also depend on the lattice spacing.
02
Calibrate and search
Try the wrong-center preset, then set the center error to zero. Sweep binning without reacquiring.
What to observe: A wrong geometry can reject a clean pattern, and coarse binning wipes out the difference in band widths.
03
Read the maps
Compare beam IPF, phases, connected grains and neighbor disorientation. Select a boundary pixel.
What to observe: Unindexed pixels stay dark, and grains are segmented from the recovered data only.
04
Test the tradeoff
Compare the same nanograin specimen in reflection and transmission. Sweep foil thickness and exposure.
What to observe: A thin foil narrows the assumed footprint, but low counts or a thick foil can still make indexing fail.

Core mathematical model

Cone edges on the detector

Count first, index second

Assumed foil response

Common difficulties

Map color is not a full orientation

Run the experiment

Predict before indexing

Calibrate and search

Read the maps

Test the tradeoff