E16 · Reflection orientation mapping

EBSD: From Bands to Grains

Can you trust an EBSD grain map made with a bad calibration? No, and it may not look wrong. Each point's orientation comes from matching its diffraction pattern against candidate phases. The match needs enough signal, the right candidates and a calibrated pattern center, which tells the software where the screen sits relative to the beam spot. A bad calibration still paints a confident map, so report how well each point fits and how many were rejected.

Calibrate the pattern center and index the stored Kikuchi patterns. Then compare the beam-direction IPF, phase and grain maps with an independent check against the known specimen.

Indexed one-axis rotation $\text{—}$

Indexed teaching phase $\text{—}$

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

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

More measurements and target

Selected stored pattern counts $\text{—}$

Selected detector pixel counts $\text{—}$

Best candidate correlation $\text{—}$

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

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

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

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

Experiment target $\text{—}$

Simulated examples; each plot identifies its data source.

Exposure per scan pixel

\tau

0.5\,\mathrm{ms}

Stage tilt from horizontal

\alpha

70\,\mathrm{^\circ}

Assumed vertical pattern-center error / height

\Delta p_y

0.12

Stored detector binning factor

b

1

Advanced settings

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 grain map survive a bad calibration?

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.

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. The specimen field is 480 nm wide, scanned in 40 nm steps. The beam footprint is a set Gaussian with a standard deviation of 18 nm along the tilt direction and 7 nm across it; within the finite field, these weights mix the grain patterns before counts are drawn. Its FWHM is an assumed response, not a measured resolution. Across the 55 to 75 degree tilt range, collection stays fixed at 8 percent and band contrast at 0.7. Tilting rotates the crystal planes and changes the projection; the model does not solve how transport or yield changes with tilt.

Physics tutorial

EBSD: From Bands to Grains

BackgroundEBSD scans an electron beam across a tilted specimen and records a Kikuchi pattern at every point. Band positions encode the crystal orientation and band widths depend on the lattice spacing, so the patterns build maps of phases and grains.

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.

Start with the essentials

Focus question: Can you trust an EBSD grain map made with a bad calibration?
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.

Compare indexed neighbors

\begin{aligned}\delta&=\min(r,90^\circ-r)\\r&=|\phi_1-\phi_2|\bmod90^\circ\end{aligned}

For this one-axis cubic family only, the angle difference between two pixels of the same phase is reduced by the 90 degree symmetry. Angles are never compared across different phases.

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
Use short exposure and an overly strict threshold.
What to observe: A high rejection threshold does not make the missing pixels trustworthy.

Core mathematical model

Cone edges on the detector

Count first, index second

Compare indexed neighbors

Common difficulties

Map color is not a full orientation

Run the experiment

Predict before indexing

Calibrate and search

Read the maps

Test the tradeoff