E28 · Crystal diffraction

Dynamical Diffraction: When Intensity Returns

Does a thicker crystal always send more intensity into diffraction? No. Even in a crystal that absorbs nothing, intensity swings back and forth between the transmitted and diffracted beams as the thickness grows. As thickness or local tilt varies across the specimen, this exchange draws thickness fringes and bend contours, with no change in composition at all.

Tilt a synthetic crystal, change its thickness and choose a reflection. Compare the counted bright-field and dark-field images with the two-beam prediction, then make the contrast of the specified defect disappear while keeping a strong reflection.

Selected stored BF + DF counts $\text{—}$

Recorded dark-field share $\text{—}$

Selected local thickness $\text{—}$

Model defect contribution to DF fraction $\text{—}$

More measurements and target

Model dark-field fraction $\text{—}$

Dimensionless excitation departure $\text{—}$

First-order kinematic intensity $\text{—}$

Selected stored horizontal coordinate $\text{—}$

Experiment target $\text{—}$

Simulated examples; each plot identifies its data source.

Central thickness

t_0

70\,\mathrm{nm}

Tilt from exact Bragg orientation

\Delta\theta

0\,\mathrm{mrad}

Bend angle at right field edge

\theta_b

1\,\mathrm{mrad}

Specified defect: 0 absent, 1 present

d

1

Advanced settings

Assumed extinction distance

\xi_g

100\,\mathrm{nm}

Thickness difference across field

\Delta t

40\,\mathrm{nm}

Reflection–Burgers-vector projection

\mathbf g\cdot\mathbf b

1

Mean incident electrons per pixel per channel

N_0

2000

Apparatus image: 0 BF, 1 DF

c

1

Inspect stored horizontal position

x

5\,\mathrm{nm}

Inspect stored vertical position

y

5\,\mathrm{nm}

Learning goal and model limits

Make a prediction, then test it

Does a thicker crystal always send more intensity into diffraction?

Target: keep the specified defect in the crystal but set its reflection projection to zero. At the selected stored pixel, record at least 1000 total counts, reach a model diffracted fraction of at least 0.75 and keep the magnitude of the dimensionless departure at most 0.25.

Model assumptions

This teaching model applies two-beam theory to each local column of a nonabsorbing crystal, at a fixed 200 keV and with an assumed reflection of 4 inverse nanometers. The extinction distance is an adjustable material input. Tilt is measured from the exact Bragg orientation of that reflection. A linear bend and a thickness wedge vary across a 400 nm field, and a grid of 40 by 40 pixels samples the local intensities; coherent image formation is not computed. The defect shifts the local excitation by an assumed, regularized strain term, scaled by the specified projection of the reflection on the Burgers vector. It is not a depth-dependent elastic solution for a dislocation. A zero projection removes only this modeled contribution; real edge or mixed dislocations can still show contrast at zero projection. Absorption, multiple reflections, anomalous absorption, surface relaxation, defocus, inelastic background, beam damage and calibrated thickness inversion are left out. Bright field and dark field are counted in independent seeded channels with the same illumination and 80 percent detection efficiency. Counts above 40 use a normal approximation. The expected fractions of the two channels add to one, but individual noisy counts need not add up to a fixed incident number. Switching channels or moving the cursor reuses the stored data. The first-order kinematic comparison can exceed unit intensity outside its weak-scattering range, and it is never clipped into a physical probability. Apparatus angles, thickness and image magnification are schematic.

Physics tutorial

Dynamical Diffraction: When Intensity Returns

BackgroundYou tilt a synthetic crystal, change its thickness and choose a reflection. You compare counted bright-field and dark-field images with the two-beam prediction, then make the contrast of the specified defect disappear while keeping a strong reflection.

Why it mattersThe goal is to keep the specified defect in the crystal while setting its reflection projection to zero. At the selected stored pixel you need at least 1000 total counts, a model diffracted fraction of at least 0.75 and a dimensionless departure no larger than 0.25 in magnitude.

Start with the essentials

Focus question: Does a thicker crystal always send more intensity into diffraction?
One-sentence intuition: In a crystal that absorbs nothing, intensity passes back and forth between the transmitted and diffracted waves as the thickness grows. Thickness fringes and bend contours can appear where the composition does not change at all. When the modeled defect contribution vanishes, that does not prove the defect is absent.

Core mathematical model

Diffracted intensity in the two-beam model

I_g=\frac{\sin^2\!\left(\pi t\sqrt{1+w^2}/\xi_g\right)}{1+w^2}

Without absorption, the diffracted intensity rises and falls with thickness instead of growing steadily.

Dimensionless departure and intensity conservation

w=s_g\xi_g,\qquad I_0+I_g=1

The two intensities add to one as expected values, before independent counting noise is added.

First-order weak-scattering approximation

I_g^{(1)}=\left(\frac{\pi t}{\xi_g}\right)^2\operatorname{sinc}^2(\pi t s_g)

The sinc of an argument is its sine divided by the argument. A value above one means the first-order approximation has broken down.

Common difficulties

What the scattering model can show

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

Better mental modelIn a crystal that absorbs nothing, intensity passes back and forth between the transmitted and diffracted waves as the thickness grows. Thickness fringes and bend contours can appear where the composition does not change at all. When the modeled defect contribution vanishes, that does not prove the defect is absent.

Run the experiment

01
Predict when intensity returns
At the exact Bragg orientation, compare a thickness of half an extinction distance with one full extinction distance.
What to observe: In the thicker crystal, the intensity has gone back to the transmitted wave.
02
Separate thickness from bending
Load Bend contours and a thickness wedge, move the cursor over the stored image and compare it with the known thickness reference.
What to observe: Thickness and excitation error both change the same diffracted intensity.
03
Hide contrast without removing the defect
Compare the visible and hidden defect presets, then reach the target. Switch the apparatus image between channels without acquiring new data.
What to observe: The defect is still in the specimen even when its modeled contribution vanishes.