E40 · Crystal diffraction

Precession Diffraction: From Spots to Grains

Precess a nanobeam around a specimen pivot, compensate its motion below the foil and record a diffraction pattern at every scan position. Index the stored patterns against a small declared library and inspect where confidence fails.

Selected stored detector counts $\text{—}$

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

Indexed selected orientation $\text{—}$

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

Selected similarity margin $\text{—}$

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

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

Raster field width $\text{—}$

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

Experiment target $\text{—}$

Predict · acquire · explain

Can averaging incident directions make grain orientations easier to identify?

Specified foil thickness

t

70\,\mathrm{nm}

Precession semi-angle

\alpha_p

0\,\mathrm{mrad}

Mean incident tilt along horizontal axis

\theta_x

1\,\mathrm{mrad}

Descan: 0 off, 1 on

d

1

Mean incident electrons per scan position

N_0

2000

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

Target: enable descan, use 5–15 mrad precession and a scan step at most 30 nm. Index at least 90 percent of stored positions with mean similarity margin at least 0.4. At the selected grain keep the forward reflection-share mismatch at most 0.12; this is a model diagnostic, not an accuracy certificate.

Fixed 200 keV, eight-reflection plus direct-beam star-coupled teaching model with a rectangular reciprocal net and three specified in-plane grain rotations: 0, 35 and 75 degrees. Assumed extinction distances vary with reflection length; no real material or full crystal structure is claimed. A Hermitian finite-beam Hamiltonian couples each diffracted beam only to the direct beam. Unitary Strang steps preserve total elastic intensity without normalization; diffracted-to-diffracted coupling, absorption, HOLZ, continuous orientation distributions, convergence and boundary mixing are omitted. Excitation errors use the paraxial Ewald-sphere expression. Nonzero precession averages 256 equally weighted directions at fixed total fluence. Ideal descan fixes spot positions; disabling it shifts spots over the finite detector and can lose intensity outside the window. Gaussian angular response is 0.65 mrad RMS; 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 matches counts against three ideal thin, untilted, de-scanned templates. Confidence is a similarity margin, not a probability; unknown orientations and arbitrary materials are not solved. At least 100 diffracted counts are needed for indexing. Cursor and confidence changes reuse the same cube. Step changes field of view at fixed grid count and does not confer finer probe resolution. The intensity-mismatch diagnostic compares separately labeled forward dynamical and same-thickness kinematic reflection shares, not reconstructed measurements. Precession is not guaranteed to remove dynamical scattering. Beam paths and angles are enlarged; the scene shows representative cone directions, not physical timing.

Physics tutorial

Precession Diffraction: From Spots to Grains

BackgroundPrecess a nanobeam around a specimen pivot, compensate its motion below the foil and record a diffraction pattern at every scan position. Index the stored patterns against a small declared library and inspect where confidence fails.

Why it mattersTarget: enable descan, use 5–15 mrad precession and a scan step at most 30 nm. Index at least 90 percent of stored positions with mean similarity margin at least 0.4. At the selected grain keep the forward reflection-share mismatch at most 0.12; this is a model diagnostic, not an accuracy certificate.

Start with the essentials

Focus question: Can averaging incident directions make grain orientations easier to identify?
One-sentence intuition: Precession integrates many excitation conditions; descan keeps them on a common detector coordinate system. Grain colors become an inference only when they are computed from recorded patterns, with low-count and low-confidence positions allowed to remain unindexed.

Core mathematical model

Elastic beam conservation

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

A finite Hermitian star Hamiltonian exchanges elastic amplitude. It omits diffracted-to-diffracted coupling.

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.

Stored-pattern confidence

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

Cosine similarities exclude the central beam. This margin is not a posterior probability.

Common difficulties

Read the model boundary

Typical misconceptionA clean image or confident orientation removes the need to check the scattering model.

Better mental modelPrecession integrates many excitation conditions; descan keeps them on a common detector coordinate system. Grain colors become an inference only when they are computed from recorded patterns, with low-count and low-confidence positions allowed to remain unindexed.

Run the experiment

01
Predict spot motion
Enable precession, then disable and restore descan at equal incident counts.
What to observe: Uncompensated spot motion blurs patterns and can leave the detector window.
02
Index the stored cube
Move the stored row and column. Raise the similarity threshold without changing acquisition.
What to observe: Rejected positions turn gray; the specified reference remains separate.
03
Test the limits
Compare the thick-foil and low-count presets, then reach the target.
What to observe: A well-indexed grain can still have dynamical intensity mismatch.