E10 · Electron acquisition / diffraction

Electron Diffraction & Reciprocal Space

Which reciprocal-lattice points become detector spots? The ones that lie close enough to the Ewald sphere. A thin crystal stretches each point into a short rod, so tilting the crystal changes which rods cross the sphere. Camera length only scales how far the spots sit from the center of the detector; it does not change which spots appear.

Tilt a thin crystal, record electron counts and measure the radius of a known diffraction spot or powder ring. Compare it with the geometric prediction, then halve the camera length and expose again. Low counts, overlapping peaks and saturated pixels limit what you can measure.

Known 100 window counts $0\,\mathrm{\mathrm{e}^{-}}$

Count-derived 100 radius $0\,\mathrm{mm}$

Indexed powder spacing $0\,\mathrm{nm}$

Reference-window separation $\text{—}$

More measurements and target

Relativistic wavelength $0\,\mathrm{pm}$

Reference 100 spacing $0\,\mathrm{nm}$

Predicted 100 angle $0\,\mathrm{mrad}$

Predicted 100 radius $0\,\mathrm{mm}$

100 rod displacement $0\,\mathrm{nm^{-1}}$

Predicted 100 shape $0\,\mathrm{\%}$

100 angular collection $\text{—}$

Pattern type $\text{—}$

Recorded incident expectation $0\,\mathrm{\mathrm{e}^{-}}$

Saved detector counts $0\,\mathrm{\mathrm{e}^{-}}$

Saturated detector pixels $0\,\mathrm{\mathrm{pixel}}$

Total-count shot-noise estimate $0\,\mathrm{\%}$

Experiment task $\text{—}$

Simulated examples; each plot identifies its data source.

More examples

Ideal random powderPowder averages orientations; its ring pattern is independent of the single-crystal tilt and azimuth.Beam energy

E_0

200\,\mathrm{keV}

Single-crystal slab thickness

t

5\,\mathrm{nm}

Tilt about laboratory transverse axis

\beta

0\,\mathrm{{}^\circ}

Calibrated camera length

L

600\,\mathrm{mm}

Advanced settings

Simple-cubic lattice parameter

a

0.4\,\mathrm{nm}

Azimuth about beam

\phi

0\,\mathrm{{}^\circ}

Collection semiangle

\alpha

25\,\mathrm{mrad}

Incident expectation for next exposure

N_0

200000\,\mathrm{\mathrm{e}^{-}}

Display gain on saved counts

G

4

Learning goal and model limits

Instrument settings

Electron Diffraction & Reciprocal Space

Record a well-separated 100 spot or the first powder ring with at least 100 counts in its window and no saturated pixels. The measured radius must lie within 5 percent of the prediction, and for a single crystal the shape factor must exceed 80 percent. These targets are set for teaching; they are not confidence limits. Then halve the camera length and expose again.

Model assumptions

This teaching model uses a monatomic simple-cubic slab with zero-order rods and single scattering. At most 12 percent of the beam is scattered, an assigned limit; the relative spot weights are not material cross sections. The ideal powder averages over orientations and ignores slab thickness. The simulated detector has unit efficiency, Poisson counting noise with an approximation at high means, an assigned 0.2 mm Gaussian blur and a cap of 65,535 counts per pixel. Read noise, beam damage, dynamical scattering, real-material structure factors, texture and camera calibration errors are left out. The radius is measured inside a window placed around the known 100 reference, and only the indexed powder ring is converted to a Bragg spacing. Predicted values are shown separately. The lens relay and the drawn ray paths are schematic, and all detector coordinates use one fixed field of view.

Physics tutorial

Which reciprocal-lattice points become detector spots?

BackgroundYou tilt a thin crystal, record electron counts and measure the radius of a known diffraction spot or powder ring. You compare that measured radius with the geometric prediction, then halve the camera length and expose again. Low counts, overlapping peaks and saturated pixels limit what you can measure.

Why it mattersA spot can look bright in the ideal pattern and still have too few counts in the real exposure to measure.

Start with the essentials

Focus question: Which reciprocal-lattice points become detector spots?
One-sentence intuition: Changing the camera length moves a spot closer to or farther from the center, but the scattering angles stay the same. Measure the radius from the recorded counts first, then compare it with the ideal reference shown separately. A tilted single crystal meets the Ewald sphere away from the exact Bragg condition, so its spots do not in general give a lattice spacing; only the powder ring, indexed in advance, is converted.

Core mathematical model

Elastic scattering condition in reciprocal space

\lvert\mathbf k_0+\mathbf g+\delta\mathbf n\rvert=\lvert\mathbf k_0\rvert,\quad \lvert\mathbf k_0\rvert=\lambda^{-1}

Spatial frequencies are counted in cycles per unit length, with no factor of two pi. Each rod runs along the slab normal, and the electron keeps its energy at the point where the rod meets the sphere.

Spot intensity from a slab of finite thickness

I_{hk}\propto e^{-0.16(h^2+k^2)}\operatorname{sinc}^{2}(\pi t\delta),\quad \operatorname{sinc}(u)=\frac{\sin u}{u}

The rod displacement is measured along the tilted slab normal, and its sign comes from this intersection. It is not the usual excitation error, whose sign is defined along the beam axis. The exponential prefactor is only illustrative.

Where a scattered beam lands on the detector

x_D=L\frac{q_x}{k_0+q_z},\quad y_D=L\frac{q_y}{k_0+q_z},\quad 2d\sin\theta_B=\lambda

Detector coordinates are in millimeters. The camera length already includes the magnification of the projector lenses. For an ideal powder, the scattering angle is twice the Bragg angle.

Radius measured from the counts

\bar{\mathbf r}=\frac{\sum_{j\in\mathcal W}n_j\mathbf r_j}{\sum_{j\in\mathcal W}n_j},\quad \bar R_{\mathrm{ring}}=\frac{\sum_{j\in\mathcal W}n_j\lvert\mathbf r_j\rvert}{\sum_{j\in\mathcal W}n_j}

For a spot, the radius is the length of the count-weighted centroid. For a powder ring, it is the count-weighted mean distance from the center. The analysis window is placed from the known reference, so the program does not find or index peaks on its own. The measured value comes from pixel positions, not from predicted positions.

Lattice spacing from the indexed powder ring

d_{100}=\frac{\lambda}{2\sin\!\left[\frac12\arctan(\bar R_{\mathrm{ring}}/L)\right]},\quad \frac{\sigma_N}{N}\approx N^{-1/2}

The exact angle conversion assumes the ring is the first simple-cubic shell and the camera length is correct. The shot noise on the total count is not a confidence interval for the spacing. If the counts fall below the minimum, peaks overlap or pixels saturate, no measurement is reported.

Common difficulties

How big the Ewald sphere is

Typical misconceptionThe Ewald sphere should look like a small ball around the reciprocal lattice.

Better mental modelAt 200 keV its radius is about 399 cycles per nm, far larger than the lattice spacing on screen. The separate Ewald plot uses different ranges on its transverse and axial axes so that the slight curvature is visible; it is a diagram, not a detector surface.

Predicted radius versus measured radius

Typical misconceptionThe radius readout is accurate even when the detector has no usable exposure.

Better mental modelThe predicted radius and the radius measured from counts are shown separately. A measured radius needs at least 100 counts in a window that lies inside the collection angle and is well separated from other peaks, with no saturated pixels. These thresholds are set for teaching; they are not confidence bounds.

Run the experiment

01
Record a usable exposure
Load Near zone axis and check whether the saved exposure meets the target. Then change the tilt or azimuth and acquire again. Read the predicted rocking curve, which uses the azimuth of the saved exposure.
What to observe: The detector shows a saved array of simulated counts. Moving the acquisition controls only sets up the next exposure; the recorded geometry and results stay the same until you acquire again.
02
Halve the camera length
Note the radius measured from the counts, load Half camera length and compare. Then switch to powder to convert the indexed ring into a lattice spacing.
What to observe: The radius drops to almost half, and the powder spacing stays close to the known lattice parameter. Detector blur, finite sampling and noise keep the values from matching exactly.
03
Find where the measurement fails
Load Too few counts, Pixel saturation and Overlapping peaks in turn. Raise the display gain and check the saved data again.
What to observe: A brighter display does not add counts, separate merged peaks or restore clipped pixels. The target check fails and no radius is reported.