---
title: "MicroED: Rotate, Integrate, Fit the Basis Interactive Lab | Sandbox Physics"
description: "If the crystal turns through only a limited angle, along which directions can its diffraction locate the atoms?"
canonical_url: "https://sandboxphysics.com/labs/microed-rotation/"
language: "en"
page_type: "lab"
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  zh-CN: "https://sandboxphysics.com/zh/labs/microed-rotation/"
last_modified: "2026-10-27"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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.

Physics tutorial

## MicroED: Rotate, Integrate, Fit the Basis

**Background** 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.

**Why it matters** A rotation series covers only part of reciprocal space, and that coverage limits what the fit can tell you about atom positions.

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 misconception** If I pass the task, I have solved a real structure from experiment.

**Better mental model** 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.

### Run the experiment

1. 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.

2. 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.

3. 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.

4. 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.

5. 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.

Physical conclusion

The fit uses only the recorded counts; the true structure used to simulate them never enters it.

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)