E40 · Crystal diffraction

Precession Diffraction: From Spots to Grains

Can swinging the electron beam around a cone make grain orientations easier to identify? Yes. Tilting the beam and swinging it around a cone averages over many diffraction conditions, so each pattern matches a library of reference patterns more reliably. Below the specimen the beam is scanned back, keeping every pattern in the same place on the detector. Points with too few counts or a weak match are better left without an orientation.

Precess a nanobeam about a pivot point on the specimen, cancel its motion below the foil and record a diffraction pattern at every scan position. Then index the stored patterns against a small, specified library and see where the confidence is too low.

Indexed selected orientation $\text{—}$

Indexed share of stored positions $\text{—}$

Selected similarity margin $\text{—}$

Forward reflection-share mismatch $\text{—}$

More measurements and target

Selected stored detector counts $\text{—}$

Selected counts outside central mask $\text{—}$

Mean similarity margin across raster $\text{—}$

Raster field width $\text{—}$

Stored count cube as float64 $\text{—}$

Experiment target $\text{—}$

Simulated examples; each plot identifies its data source.

More examples

Specified foil thickness

t

70\,\mathrm{nm}

Precession semi-angle

\alpha_p

0\,\mathrm{mrad}

Descan: 0 off, 1 on

d

1

Mean incident electrons per scan position

N_0

2000

Advanced settings

Mean incident tilt along horizontal axis

\theta_x

1\,\mathrm{mrad}

Raster step at fixed grid size

\Delta x

20\,\mathrm{nm}

Minimum similarity margin for indexing

q_{min}

0.15

Inspect stored scan column

i

5

Inspect stored scan row

j

5

Learning goal and model limits

Make a prediction, then test it

Can swinging the electron beam around a cone make grain orientations easier to identify?

Target: turn on descan, use a precession semi-angle of 5 to 15 mrad and a scan step of at most 30 nm. Index at least 90 percent of the stored positions with a mean similarity margin of at least 0.4. At the selected grain, keep the forward reflection-share mismatch at most 0.12. This mismatch is a model diagnostic; it does not prove that the indexing is accurate.

Model assumptions

This teaching model runs at a fixed 200 keV and couples eight reflections to the direct beam in a star pattern. The reciprocal net is rectangular, and three grains are rotated in the plane by 0, 35 and 75 degrees. The extinction distances are assumed and vary with reflection length; the model does not describe a real material or a full crystal structure. A finite Hermitian Hamiltonian couples each diffracted beam only to the direct beam. Unitary Strang steps keep the total elastic intensity constant without renormalization. Coupling between diffracted beams, absorption, higher-order Laue zones, continuous orientation spreads, beam convergence and mixing at grain boundaries are left out. Excitation errors use the paraxial Ewald-sphere expression. With precession on, the pattern is an equally weighted average over 256 directions at a fixed total fluence. Ideal descan holds the spots still; without it, the spots move across the finite detector and some intensity can fall outside the window. The angular response is Gaussian with a 0.65 mrad RMS width, and 48 by 48 detector pixels cover a 40 mrad square. A 12 by 12 scan stores seeded counts at 80 percent efficiency plus 0.002 dark counts per channel. Above 40 expected counts a normal approximation is used. Indexing masks the central 3 mrad and compares the counts with three ideal templates for a thin, untilted, descanned crystal. Confidence is a similarity margin, not a probability, and the method cannot find unknown orientations or handle arbitrary materials. Indexing needs at least 100 diffracted counts. Moving the cursor or changing the confidence threshold reuses the same data cube. At a fixed number of grid points, the step size changes the field of view; it does not make the probe resolution finer. The intensity-mismatch diagnostic compares reflection shares from the dynamical model and from a kinematic model at the same thickness. Both are forward calculations, labeled separately, and neither is reconstructed from measurements. Precession does not guarantee that dynamical scattering is removed. Beam paths and angles are enlarged, and the scene shows a few representative cone directions, not the real timing.

Physics tutorial

Precession Diffraction: From Spots to Grains

BackgroundYou precess a nanobeam about a pivot point on the specimen, cancel its motion below the foil and record a diffraction pattern at every scan position. Then you index the stored patterns against a small, specified library and look for places where the confidence is too low.

Why it mattersThe goal is to turn descan on, use a precession angle of 5–15 mrad and a scan step of at most 30 nm. At least 90 percent of the stored positions should be indexed, with a mean similarity margin of at least 0.4. At the selected grain, the forward reflection-share mismatch should stay at or below 0.12. That mismatch is a diagnostic of the model; meeting it does not certify accuracy.

Start with the essentials

Focus question: Can swinging the electron beam around a cone make grain orientations easier to identify?
One-sentence intuition: Precession adds up patterns from many excitation conditions, and descan puts them all on the same detector coordinates. A grain map counts as an inference only when its colors are computed from the recorded patterns and positions with low counts or low confidence are allowed to stay unindexed.

Core mathematical model

Elastic intensity is conserved

i\frac{d\boldsymbol\psi}{dz}=H\boldsymbol\psi,\quad H=H^\dagger

A finite Hermitian Hamiltonian with star coupling moves elastic amplitude between the beams without losing any. It leaves out coupling between diffracted beams.

Precession at fixed fluence

\bar I_g=\frac1{256}\sum_{m=0}^{255}|\psi_g(t;\phi_m)|^2

Incident directions are averaged, so enabling precession does not multiply the exposure by 256.

Confidence of a pattern match

q=\frac{c_{best}-c_{next}}{c_{best}}

The cosine similarities leave out the central beam. The margin compares the best match with the runner-up; it is not a posterior probability.

Common difficulties

Why the scattering model still matters

Typical misconceptionIf the image looks clean and the orientation is known with confidence, there is no need to check the scattering model.

Better mental modelPrecession adds up patterns from many excitation conditions, and descan puts them all on the same detector coordinates. A grain map counts as an inference only when its colors are computed from the recorded patterns and positions with low counts or low confidence are allowed to stay unindexed.

Run the experiment

01
Predict how the spots move
Turn precession on, then switch descan off and back on while keeping the incident counts the same.
What to observe: Without compensation the spots move. The pattern blurs, and spots can drift out of the detector window.
02
Index the stored patterns
Move the selected row and column in the stored scan. Raise the similarity threshold without changing the acquisition.
What to observe: Rejected positions turn gray. The specified grain reference stays separate from the indexed map.
03
Test the limits
Compare Thick foil retains dynamical mismatch with Too few diffracted counts, then reach the target.
What to observe: A grain can be indexed well and still show a dynamical intensity mismatch.