E38 · Electron imaging

DPC: Beam Deflection & Projected Fields

Pass a convergent probe through a smooth junction and compare its disk with an independently recorded vacuum reference. Recover an electric-equivalent projection from the measured center of mass, compare quadrant differences and test how clipping, counting noise and crystalline redistribution mimic a field.

Recovered electric-equivalent projection $\text{—}$

Known-model RMS error $\text{—}$

Minimum expected mask acceptance $\text{—}$

Selected frame · accepted counts $\text{—}$

Recorded vacuum centroid $\text{—}$

Known field-induced deflection $\text{—}$

Known blurred source projection $\text{—}$

Assumed force source $\text{—}$

Incident specimen scan dose $\text{—}$

Experiment target $\text{—}$

A centroid is a measurement, not a diagnosis

Calibrate the shift before naming the field

Subtract the measured vacuum referenceThe same noisy reference is shared by all scan points.Use quadrant difference for the readoutSmall-shift circular-disk calibration; both estimates remain visible.Use a magnetic junction · electric field absentSet a transverse induction giving the same deflection. One image cannot distinguish the force source.Convergence semi-angle

\alpha

1.5\,\mathrm{mrad}

Peak projection · electric equivalent

\mathcal E_0

200\,\mathrm{V}

Junction field width

w

7\,\mathrm{nm}

Probe current

I

2\,\mathrm{pA}

Dwell per probe and vacuum reference

\tau

80\,\mathrm{\mu s}

Detector zero offset

\delta_0

0.8\,\mathrm{mrad}

Analysis mask · inner radius

\beta_{\mathrm{in}}

0\,\mathrm{mrad}

Analysis mask · outer radius

\beta_{\mathrm{out}}

1.5\,\mathrm{mrad}

Maximum redistributed fraction

f

0

Inspect probe · horizontal

x_p

0\,\mathrm{nm}

Predict the sign of the disk shift for a positive electric projection. With the electric-only specimen, use a vacuum-corrected center of mass, full mask acceptance and at least 1000 incident electrons per probe. Keep model RMS error below 30 V and scan dose at most 50 electrons per square angstrom. Remove scattering redistribution, then check. Compare the magnetic preset to expose the ambiguity.

Fixed 200 kV, 48 nm field, 24 by 24 probes and a 48 by 48 detector spanning ±8 mrad. The specimen has an invented Gaussian projected junction field, uniform vertically. A rigid circular-disk shift uses the relativistic Lorentz impulse and an approximate Gaussian probe convolution. Electric and magnetic presets deliberately give identical force deflection; one scan cannot separate them. Every displayed projection is an electric equivalent, not a unique electric-field identification or a local three-dimensional field. The magnetic toggle sets the corresponding projected transverse induction, with electric field absent. An adjustable fraction transferred into a disk displaced by 3 mrad illustrates crystalline redistribution but is not dynamical diffraction. Actual acquired pixel counts and a separate vacuum exposure drive both center-of-mass and quadrant estimates; the reference is shared across the scan, so its noise is correlated. Quadrants use a small-shift calibration for an unmasked circular disk. Angular masks can invalidate that calibration. Missing counts remain invalid, signed estimates retain negative noise and model RMS compares with the known probe-blurred reference. Dose is incident specimen exposure per scan; vacuum exposure is separate. No damage, aberration, multiple scattering or field inversion solver is included.

Start with the essentials

Focus question: When does a shifted disk measure a field?
One-sentence intuition: A complete center of mass and a quadrant difference are different estimators. The electric-equivalent projection is interpretable as an electric field only with the magnetic contribution and scattering assumptions controlled.

Core mathematical model

A relativistic Lorentz impulse

p\,\Delta\boldsymbol\theta=-e\int\left(\frac{\boldsymbol E_\perp}{v}+\hat{\boldsymbol z}\times\boldsymbol B_\perp\right)\mathrm dz

The incident direction is positive specimen z. A positive projected electric field in x shifts negative x; a negative projected magnetic induction in y gives the same shift. The beam is fixed at 200 kV.

Compute a centroid from acquired pixels

\begin{gathered}\overline{\theta}_x=\frac{\sum_{m,n}M_{mn}D_{mn}\theta_{x,m}}{\sum_{m,n}M_{mn}D_{mn}}\\ \Delta\theta_x=\overline{\theta}_{x,\mathrm{spec}}-\overline{\theta}_{x,\mathrm{vac}}\end{gathered}

The independent vacuum frame uses the same angular mask. One shared reference makes its counting error common to every scan point. Clipping can bias both centroids differently.

Electric-equivalent projection

\mathcal E_{x,\mathrm{eq}}=-\frac{pv}{e}\Delta\theta_x=\int E_x\,\mathrm dz-v\int B_y\,\mathrm dz

The unit is volts because this is a field integrated over depth. It is not the local field in volts per meter. The magnetic preset reproduces the same counts with electric field absent.

Quadrants approximate a small displacement

D_x=\frac{N_R-N_L}{N_R+N_L},\qquad \Delta\theta_x\simeq\frac{\pi\alpha}{4}\left(D_{x,\mathrm{spec}}-D_{x,\mathrm{vac}}\right)

This slope belongs to a complete uniform circular disk at small displacement. It becomes nonlinear for large offsets; annular masking or lost disk support changes the calibration.

Counts and specimen dose

N_e=\frac{I\tau}{e},\qquad d=\frac{N_e}{(2\,\mathrm{nm})^2}

Each scan has 24 by 24 exposures at 2 nm spacing. A separate vacuum exposure adds acquisition time and data, but is not specimen dose. The displayed known-model error uses the probe-blurred projected field as its reference.

Common difficulties

Projection is not a local field

Typical misconceptionThe map gives a unique electric field at each atom.

Better mental modelThe map is an electric-equivalent depth integral after probe averaging. Magnetic force and scattering may contribute, and no atomic multislice calculation is performed.

A centered mask does not fix a shifted disk

Typical misconceptionA symmetric detector mask guarantees an unbiased centroid.

Better mental modelIf the shifted disk is cut differently from the vacuum disk, the symmetry of the mask does not restore the missing intensity.

Run the experiment

01
Predict the shift sign
Compare the uncalibrated state with Calibrated center of mass.
What to observe: Subtracting measured vacuum counts removes the detector offset but retains reference noise.
02
Lose angular support
Reduce the outer mask or add an inner exclusion.
What to observe: A centroid of a clipped disk no longer represents the full momentum average. Missing counts can invalidate both estimators.
03
Separate an estimator from an interpretation
Compare the quadrant and scattering-artefact presets, then the magnetic preset.
What to observe: Quadrants depart from their small-shift calibration; invented scattering gives false field contrast. Electric and magnetic sources can generate identical displacement data.
04
Complete the calibrated electric target
Meet the stated acceptance, counting, RMS and dose limits with vacuum correction, then check.
What to observe: More exposure reduces counting noise but does not remove systematic clipping or scattering bias.