E28 · Crystal diffraction

Dynamical Diffraction: When Intensity Returns

Tilt a synthetic crystal, change its thickness and select a reflection. Compare counted bright-field and dark-field images with a two-beam prediction, then hide the specified defect contrast while retaining a strong reflection.

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

Recorded dark-field share $\text{—}$

Model dark-field fraction $\text{—}$

Selected local thickness $\text{—}$

Dimensionless excitation departure $\text{—}$

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

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

Selected stored horizontal coordinate $\text{—}$

Experiment target $\text{—}$

Predict · acquire · explain

Does a thicker crystal always send more intensity into diffraction?

Central thickness

t_0

70\,\mathrm{nm}

Assumed extinction distance

\xi_g

100\,\mathrm{nm}

Tilt from exact Bragg orientation

\Delta\theta

0\,\mathrm{mrad}

Bend angle at right field edge

\theta_b

1\,\mathrm{mrad}

Thickness difference across field

\Delta t

40\,\mathrm{nm}

Specified defect: 0 absent, 1 present

d

1

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}

Target: keep the specified defect present but set its reflection projection to zero. At the selected stored pixel obtain at least 1000 total counts, a model diffracted fraction of at least 0.75 and an absolute dimensionless departure at most 0.25.

Nonabsorbing, local-column two-beam teaching model at fixed 200 keV with an assumed 4 inverse-nanometer reflection. Extinction distance is an adjustable material input. Tilt is departure from that reflection’s exact Bragg orientation. A linear bend and thickness wedge vary across a 400 nm field; 40 by 40 pixels sample local intensities without coherent image transfer. The regularized defect excitation shift is an assumed strain proxy scaled by the specified reflection–Burgers-vector projection; it is not a depth-dependent elastic dislocation solution. Zero projection removes only this modeled contribution. General edge or mixed dislocations can retain contrast even with zero projection. No absorption, multiple reflections, anomalous absorption, surface relaxation, defocus, inelastic background, damage or calibrated thickness inversion. BF and DF use independent seeded count channels at identical illumination and 80 percent detection efficiency. Counts above 40 use a normal approximation. Their expectation is complementary; individual noisy counts need not sum to a fixed incident count. Channel selection and cursor changes reuse stored data. The first-order kinematic comparator may exceed unit intensity outside its weak-scattering limit and is never clipped into a physical probability. Apparatus angles, thickness and image magnification are schematic.

Physics tutorial

Dynamical Diffraction: When Intensity Returns

BackgroundTilt a synthetic crystal, change its thickness and select a reflection. Compare counted bright-field and dark-field images with a two-beam prediction, then hide the specified defect contrast while retaining a strong reflection.

Why it mattersTarget: keep the specified defect present but set its reflection projection to zero. At the selected stored pixel obtain at least 1000 total counts, a model diffracted fraction of at least 0.75 and an absolute dimensionless departure at most 0.25.

Start with the essentials

Focus question: Does a thicker crystal always send more intensity into diffraction?
One-sentence intuition: A nonabsorbing crystal repeatedly exchanges intensity between transmitted and diffracted waves. Thickness fringes and bend contours can occur without compositional changes; absence of a modeled defect contribution does not prove absence of a defect.

Core mathematical model

Two-beam intensity

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

The lossless solution exchanges intensity instead of increasing monotonically with thickness.

Departure and conservation

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

The conservation statement concerns expected intensities before independent counting noise.

Weak-scattering comparator

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

Sinc means sine of the argument divided by the argument. Values above one signal a failed first-order approximation.

Common difficulties

Read the model boundary

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

Better mental modelA nonabsorbing crystal repeatedly exchanges intensity between transmitted and diffracted waves. Thickness fringes and bend contours can occur without compositional changes; absence of a modeled defect contribution does not prove absence of a defect.

Run the experiment

01
Predict the return
Compare half and one extinction distance at exact Bragg orientation.
What to observe: The thicker crystal returns intensity to the transmitted wave.
02
Separate thickness from bending
Use the contours preset, move the stored cursor and compare the known thickness reference.
What to observe: Both thickness and excitation error change the same diffraction intensity.
03
Hide contrast without removing the defect
Compare visible and hidden presets, then reach the target. Switch apparatus channels without reacquiring data.
What to observe: The defect remains a specimen input when its modeled contribution vanishes.