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Sandbox Physics

E41 · Continuous rotation diffraction

MicroED: Rotate, Integrate, Fit the Basis

Collect continuous-rotation diffraction frames, index a known cell, integrate measured spots and fit an assigned two-point basis.

Interactive modelMicroED: Rotate, Integrate, Fit the Basis
Stored frame count—\text{—}
Total recorded sensor counts—\text{—}
Recorded total incident dose—\text{—}
Selected recorded tilt—\text{—}
Accepted indexed reflections—\text{—}
Finite catalogue completeness—\text{—}
Fitted second-atom fractional x—\text{—}
Fitted second-atom fractional y—\text{—}
Fitted second-atom fractional z—\text{—}
Measured-fit relative RMS—\text{—}
Separated-candidate loss gap—\text{—}
Experiment target—\text{—}

Physics tutorial

MicroED: Rotate, Integrate, Fit the Basis

BackgroundCollect continuous-rotation diffraction frames, index a known cell, integrate measured spots and fit an assigned two-point basis.

Why it mattersWhich atomic coordinates can your recorded rotation wedge constrain?

Start with the essentials

Focus question
Which atomic coordinates can your recorded rotation wedge constrain?
One-sentence intuition
A good fitted basis depends on measured reciprocal coverage.

Core mathematical model

Rotating reciprocal lattice

q=Ry(θ)gqz+λ∣g∣22=0(X,Y)=Lλ(qx,qy)\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}

Curved-Ewald crossing and paraxial camera coordinates. A finite sensor clips large-angle spots.

Assigned two-point basis

ϕhkl=2π(hx2+ky2+lz2)Fhkl=1+2eiϕhkl∣Fhkl∣2=5+4cos⁡ϕhkl\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. Known amplitudes and a restricted coordinate box remove some ambiguities, but inversion and phase limitations remain.

Count integration and basis fitting

Mhkl=∑j,p∈ACj,p−bNAI^hkl=Mhklvhkle0.05∣g∣2(r^2,S^)=arg⁡min⁡r2,S≥0∑hkl(I^hkl−S∣Fhkl∣2)2\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}

Known-cell indexing, calibrated background subtraction, rocking-slope and assigned envelope corrections; fit a common scale. Missing or weak reflections are excluded.

Common difficulties

Scientific boundary

Typical misconceptionA passed task is an experimentally solved structure.

Better mental modelWeak-scattering teaching analogue, not a protein structure solution or quantitative cryo-EM package. Assigned orthorhombic cell of 0.5, 0.6 and 0.7 nm; point scattering amplitudes 1 and 2; atom one fixed at the origin. Known cell and orientation index spots; measured frame counts alone supply integrated intensities. A restricted fractional-coordinate grid fits atom two with a free scale. Gaussian rocking profiles, compact nonstreaked spots, assigned scattering strength and form-factor envelope; no multiple scattering, radiation damage, ice background, detector saturation, phase determination or experimental resolution claim. Sensor edges and missing angles remain missing.

Run the experiment

  1. 01

    Plan the wedge

    Set rotation range, frame width, dose and effective camera length; choose a fictional basis.

    What to observe: Longer camera length separates spots but can lose them at the sensor edge.
  2. 02

    Record continuous exposures

    Acquire a rotation series and select stored frames.

    What to observe: Each frame integrates its angular exposure. Narrow wedges omit reciprocal-space observations.
  3. 03

    Index and integrate measured spots

    Use the known cell and angular calibration; choose an aperture radius and commit integration.

    What to observe: A small aperture loses spot tails; weak, clipped, overlapping and edge-partial reflections are rejected.
  4. 04

    Fit the assigned basis

    Fit the second atom, inspect measured versus fitted intensities and the coordinate-loss profile.

    What to observe: The first atom, cell and scattering amplitudes were supplied; this is restricted refinement, not general phase solving.