E10 · Electron acquisition / diffraction

Electron Diffraction & Reciprocal Space

Tilt a thin simple-cubic crystal and follow its reciprocal rods into the elastic Ewald surface. Change beam energy, lattice spacing, angular collection and calibrated camera length. Compare a single-crystal spot pattern with ideal randomly oriented powder rings.

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

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

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

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

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

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

100 angular collection $\text{—}$

Pattern type $\text{—}$

Experiment task $\text{—}$

Instrument settings

Electron Diffraction & Reciprocal Space

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

E_0

200\,\mathrm{keV}

Simple-cubic lattice parameter

a

0.4\,\mathrm{nm}

Slab thickness

t

5\,\mathrm{nm}

Tilt about specimen axis

\beta

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

Azimuth about beam

\phi

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

Calibrated camera length

L

600\,\mathrm{mm}

Collection semiangle

\alpha

25\,\mathrm{mrad}

Use the single crystal. Bring the 100 shape intensity above 80 percent and keep that reflection inside the collection angle. Then halve camera length and explain the change in detector radius.

Single-scattering model of a monatomic simple-cubic slab; only zero-order reciprocal rods around the [001] direction, with thickness along the tilted slab normal. Sphere intersections and detector positions are geometric; Gaussian form-factor weights and detector glow are illustrative relative intensity. Powder ring positions follow Bragg geometry, while ring weights omit Lorentz, texture and detector corrections. No dynamical scattering, Kikuchi bands, higher Laue zones or real-material structure factors. The 3D Ewald view is a local patch at uniform scale, not a small sphere; ray fan angles are enlarged.

Start with the essentials

Focus question: Which reciprocal-lattice points become detector spots?
One-sentence intuition: At 200 keV its radius is about 399 cycles per nm, far larger than the displayed lattice spacing. The 3D view therefore shows a nearly flat local cap at uniform scale; the diagnostic plot explicitly uses different axis ranges.

Core mathematical model

Elastic reciprocal geometry

\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 use cycles per length, without a factor of two pi. The rod follows the slab normal; its intersection conserves electron energy.

Finite thickness selects the intersection

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. Its sign is defined by this intersection, not by a conventional beam-axis excitation-error sign. The prefactor is illustrative.

Calibrated detector position

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 mm. For ideal powder the scattering angle is twice the Bragg angle; the camera length includes projector magnification.

Common difficulties

Interpretation trap

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 displayed lattice spacing. The 3D view therefore shows a nearly flat local cap at uniform scale; the diagnostic plot explicitly uses different axis ranges.

Run the experiment

01
Locate the condition
Use Near zone axis. Tilt across the peak of the 100 rocking curve, then use Tilt away.
What to observe: A finite-thickness reciprocal rod can still intersect the sphere even when the lattice point does not lie exactly on it.
02
Measure reciprocal spacing
Keep the crystal fixed and halve camera length. Then change lattice parameter and beam energy.
What to observe: Camera length rescales the pattern without changing intrinsic intensity. Larger real-space spacing decreases reciprocal spacing and spot radius.
03
Compare spots and rings
Switch to Ideal random powder. Move the single-crystal tilt and azimuth controls, then close the collection angle.
What to observe: The powder pattern averages all grain orientations. Those single-crystal controls do not rotate the rings; angular collection removes outer rings.