E41 · Continuous rotation diffraction

MicroED: Rotate, Integrate, Fit the Basis

If the crystal turns through only a limited angle, along which directions can its diffraction locate the atoms? Only along the directions the rotation actually sampled. Turning the crystal continuously sweeps diffraction spots across the detector, but a limited rotation covers only part of reciprocal space, the set of all possible diffraction spots. Fitted atom positions are reliable along the sampled directions and only weakly constrained along the rest.

Record diffraction frames while the crystal rotates, index the spots with the known cell, integrate the measured spots and fit the assigned two-atom basis.

Total recorded sensor counts $\text{—}$

Finite catalogue completeness $\text{—}$

Fitted second-atom fractional x $\text{—}$

Measured-fit relative RMS $\text{—}$

More measurements and target

Stored frame count $\text{—}$

Recorded total incident dose $\text{—}$

Selected recorded tilt $\text{—}$

Accepted indexed reflections $\text{—}$

Fitted second-atom fractional y $\text{—}$

Fitted second-atom fractional z $\text{—}$

Separated-candidate loss gap $\text{—}$

Experiment target $\text{—}$

Simulated examples; each plot identifies its data source.

Scattering: 0 kinematic, 1 local three-beam

0

Requested rotation half range

\theta_{\max}

45\,{}^\circ

Frame rotation width; endpoints round inward

\Delta\theta

2\,{}^\circ

Crystal and Gaussian-envelope thickness

t

40\,\mathrm{nm}

Total incident dose for next recording

D

10\,\frac{electrons}{nm^2}

Advanced settings

Assigned second-atom fractional x

x_2

0.26

Assigned second-atom fractional y

y_2

0.18

Assigned second-atom fractional z

z_2

0.34

Effective camera length

L

1\,\mathrm{m}

Counts: 0 expectation, 1 seeded statistics

1

Processing aperture radius in pixels

r

2

Selected stored frame position

f

50\,\%

Data color gain only

G

1

Learning goal and model limits

Predict · record · fit

Constrain a basis with measured spots

Target: accept at least 65 percent of the finite reflection catalogue, using an integration aperture radius of two pixels. The fit needs a relative RMS below 15 percent, and its loss must beat the best fit at a distant coordinate by more than 0.015.

Sliders plan the next recording; stored counts stay fixed until acquisition. Drag the foil to tilt and acquire in CBED, or browse recorded angles in MicroED.

Model assumptions

This teaching analogue compares weak scattering with a local three-beam approximation. It does not solve protein structures and is not a quantitative cryo-EM package. The orthorhombic cell is set to 0.5, 0.6 and 0.7 nm, the point scatterers have amplitudes 1 and 2, and atom one is fixed at the origin. Spots are indexed from the known cell and orientation, and the integrated intensities come only from the measured frame counts. Atom two is fitted on a restricted grid of fractional coordinates, with a free scale factor. Rocking profiles are Gaussian, spots are compact without streaks, and the scattering strength and form-factor envelope are assigned. The optional multiple-scattering branch couples the incident wave, each selected reflection and one auxiliary reflection at the Ewald crossing, then applies the assigned Gaussian rocking envelope. It illustrates thickness-dependent interference and bias in the unchanged kinematic basis fit; it is not a global dynamical crystal solution. Radiation damage, ice background, detector saturation and phase determination are left out, and no experimental resolution is claimed. Data lost at the sensor edges or in missing angles stay missing.

Start with the essentials

Focus question: If the crystal turns through only a limited angle, along which directions can its diffraction locate the atoms?
One-sentence intuition: How well you can fit the basis depends on how much of reciprocal space you actually measured.

Core mathematical model

Where the rotating reciprocal lattice meets the Ewald sphere

\begin{aligned}\mathbf q&=R_y(\theta)\mathbf g\\q_z+\frac{\lambda|\mathbf g|^2}{2}&=0\\(X,Y)&=L\lambda(q_x,q_y)\end{aligned}

A spot appears when its reciprocal-lattice point crosses the curved Ewald sphere, and its camera position uses the paraxial approximation. Spots at large angles fall off the finite sensor.

Structure factor of the two-atom basis

\begin{aligned}\phi_{hkl}&=2\pi(hx_2+ky_2+lz_2)\\F_{hkl}&=1+2e^{i\phi_{hkl}}\\|F_{hkl}|^2&=5+4\cos\phi_{hkl}\end{aligned}

Atom one is fixed at the origin. Knowing the amplitudes and limiting the coordinate box removes some ambiguities, but the inversion ambiguity and the phase limitations remain.

Integrating spot counts and fitting the basis

\begin{aligned}M_{hkl}&=\sum_{j,p\in A}C_{j,p}-bN_A\\\widehat I_{hkl}&=M_{hkl}v_{hkl}e^{0.05|\mathbf g|^2}\\(\widehat{\mathbf r}_2,\widehat S)&=\underset{\mathbf r_2,S\ge0}{\arg\min}\\&\quad\sum_{hkl}(\widehat I_{hkl}-S|F_{hkl}|^2)^2\end{aligned}

Spots are indexed with the known cell, the calibrated background is subtracted, and the counts are corrected for the rocking slope and the assigned envelope. One common scale is then fitted. Missing or weak reflections are left out.

Common difficulties

Limits of the model

Typical misconceptionIf I pass the task, I have solved a real structure from experiment.

Better mental modelThis teaching analogue compares weak scattering with a local three-beam approximation. It does not solve protein structures and is not a quantitative cryo-EM package. The orthorhombic cell is set to 0.5, 0.6 and 0.7 nm, the point scatterers have amplitudes 1 and 2, and atom one is fixed at the origin. Spots are indexed from the known cell and orientation, and the integrated intensities come only from the measured frame counts. Atom two is fitted on a restricted grid of fractional coordinates, with a free scale factor. Rocking profiles are Gaussian, spots are compact without streaks, and the scattering strength and form-factor envelope are assigned. The optional multiple-scattering branch couples the incident wave, each selected reflection and one auxiliary reflection at the Ewald crossing, then applies the assigned Gaussian rocking envelope. It illustrates thickness-dependent interference and bias in the unchanged kinematic basis fit; it is not a global dynamical crystal solution. Radiation damage, ice background, detector saturation and phase determination are left out, and no experimental resolution is claimed. Data lost at the sensor edges or in missing angles stay missing.

Run the experiment

01
Separate scattering bias from missing data
Choose the multiple-scattering preset. Keep its wide rotation range and high dose, then switch to kinematic scattering and acquire, integrate and fit again.
What to observe: The coupled waves redistribute reflection intensities. A broad angular range still leaves a systematic residual and a displaced best coordinate when a kinematic model is fitted to those counts. Turning off counting noise preserves that mismatch.
02
Plan the rotation range
Set the rotation range, frame width, dose and effective camera length, and choose a fictional basis.
What to observe: A longer camera length spreads the spots apart, but it can push some off the edge of the sensor.
03
Record while the crystal rotates
Acquire a rotation series and select stored frames.
What to observe: Each frame adds up the exposure over its own slice of rotation. A narrow rotation range leaves parts of reciprocal space unmeasured.
04
Index and integrate the spots
Use the known cell and angle calibration, choose an aperture radius, then run the integration.
What to observe: A small aperture misses the tails of each spot. Reflections that are weak, clipped, overlapping or only partly on the sensor edge are rejected.
05
Fit the second atom
Fit the second atom, then compare the measured and fitted intensities and look at the loss profile along each coordinate.
What to observe: The first atom, the cell and the scattering amplitudes were given in advance, so this is a restricted refinement, not a general solution of the phase problem.

Core mathematical model

Where the rotating reciprocal lattice meets the Ewald sphere

Structure factor of the two-atom basis

Integrating spot counts and fitting the basis

Common difficulties

Limits of the model

Run the experiment

Separate scattering bias from missing data

Plan the rotation range

Record while the crystal rotates

Index and integrate the spots

Fit the second atom