E38 · Electron imaging

DPC: Beam Deflection & Projected Fields

When does the shift of the diffraction disk measure the field in the specimen? Only when three things hold: the zero position is calibrated on a vacuum reference, the detector covers all the relevant angles, and scattering in the specimen doesn't reshape the disk. Even then, the shift measures only the total sideways push on the beam through the specimen. An electric or a magnetic field could give the same push, so one scan can't tell them apart.

Send a converging electron probe through a smooth junction and compare its diffraction disk with a separately recorded vacuum disk. Turn the measured center-of-mass shift into an electric-equivalent projected field, compare it with the quadrant-difference estimate, and see how clipping, counting noise and crystal scattering can imitate a field.

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

Known-model RMS error $\text{—}$

Minimum expected mask acceptance $\text{—}$

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

More measurements and target

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 $\text{—}$

Simulated examples; each plot identifies its data source.

More examples

Peak field equivalent

E_0

2\,\mathrm{V/nm}

Specimen thickness

t

100\,\mathrm{nm}

Mistilt from exact Bragg

\delta_g

0\,\mathrm{mrad}

Crystal coupling

\kappa/\kappa_0

0

Advanced settings

Subtract the measured vacuum referenceAll scan points share the same noisy reference.Use quadrant difference for the readoutThe calibration assumes a circular disk and a small shift. Both estimates stay visible.Use a magnetic junction · electric field absentSets a transverse magnetic induction that gives the same deflection. A single image cannot tell which force caused it.Convergence semi-angle

\alpha

1.5\,\mathrm{mrad}

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}

Inspect probe · horizontal

x_p

0\,\mathrm{nm}

Learning goal and model limits

What a centroid shift can tell you

Calibrate the shift before you interpret it as a field

Predict the shift direction, then double specimen thickness. Can a crystal move the centroid with no field? Use the electric-only specimen, a vacuum-corrected center of mass, full mask acceptance and at least 1000 incident electrons per probe position. Keep the RMS error against the known model below 30 V and the scan dose at or below 50 electrons per square angstrom. Set crystal coupling to zero, then check the target. Finally, load the magnetic preset: it gives the same shift, so the image alone cannot say which field caused it.

Model assumptions

The beam is fixed at 200 kV, the field of view is 48 nm, the scan has 24 by 24 probe positions, and the 48 by 48 detector spans ±8 mrad. The junction has an invented local field with a Gaussian transverse profile, uniform in depth and the vertical direction. Its local amplitude is in volts per nanometer; the selected thickness multiplies it to give the projected voltage. The assigned field does not model material breakdown. The disk shifts rigidly by the relativistic Lorentz impulse, and the probe blur is approximated by a Gaussian convolution. The electric and magnetic presets are set up to give the same deflection, and one scan cannot tell them apart. Every projection shown is an electric equivalent: it does not single out a unique electric field, and it is not the local three-dimensional field. The magnetic toggle sets the matching projected transverse magnetic induction and switches the electric field off. Crystal coupling instead solves lossless two-beam dynamical diffraction: the transmitted and one diffracted amplitude exchange intensity with thickness. The reference extinction distance is 80 nm at unit coupling. The disks are separated by 6 mrad; the aperture range keeps them non-overlapping. Mistilt and a prescribed smooth bend change the excitation error across the aperture and scan. Exact circular source-cell weights with midpoint excitation and 64 angular cells per diameter provide a finite quadrature. Missing detector support is not renormalized. The field impulse is then applied as a rigid shift to both branches, a separated-effects approximation rather than coupled atomistic electromagnetic propagation. Both the center-of-mass and quadrant estimates use the acquired pixel counts and a separate vacuum exposure. The whole scan shares that one reference, so its noise is correlated. The quadrant estimate uses a small-shift calibration for an unmasked circular disk, and angular masks can make that calibration invalid. Points with missing counts stay invalid, signed estimates keep their negative noise, and the model RMS error is measured against the known probe-blurred reference. Dose is the specimen exposure per scan; the vacuum exposure is counted separately. The model has no damage, aberrations, absorption, inelastic or many-beam scattering, or field-inversion solver. The depth glyph is schematic at 0.05 scene unit per nanometer; transverse scan coordinates retain 0.5 scene unit per nanometer. Known source arrows do not represent a recovered local field.

Start with the essentials

Focus question: When does the shift of the diffraction disk measure the field in the specimen?
One-sentence intuition: The full center of mass and the quadrant difference estimate the shift in different ways, and the electric-equivalent projection reads as an electric field only once magnetic contributions and scattering are accounted for.

Core mathematical model

Depth turns a local field into a projection

\mathcal E_0=E_0t,\qquad \Delta\theta_x=-\frac{eE_0t}{pv}

The local field is uniform through the chosen thickness. Without crystal coupling, doubling thickness doubles the expected impulse; it does not identify an unknown depth profile.

Lossless two-beam exchange

\begin{aligned}P_g(t)&=\frac{\kappa^2}{\kappa^2+s_g^2}\sin^2\!\left(\pi t\sqrt{\kappa^2+s_g^2}\right)\\P_0(t)&=1-P_g(t),\qquad \kappa=\frac{c}{80\,\mathrm{nm}}\end{aligned}

The coupling c and excitation error determine reversible elastic transfer. A nonzero disk centroid can therefore occur with no electromagnetic field.

Relativistic Lorentz impulse on the electron

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

The beam travels along positive z through the specimen. A positive projected electric field along x pushes the electron toward negative x, and a negative projected magnetic induction along y gives the same shift. The beam is fixed at 200 kV.

Centroid of the recorded 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 separate vacuum frame uses the same angular mask. Every scan point shares this one reference, so its counting error appears at every point. Clipping can bias the specimen and vacuum centroids by different amounts.

Electric-equivalent projection

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

Its unit is volts because it is a field integrated through the depth of the specimen, not the local field in volts per meter. The magnetic preset produces the same counts with no electric field at all.

Quadrant estimate for a small shift

\begin{aligned}D_x&=\frac{N_R-N_L}{N_R+N_L}\\\Delta\theta_x&\simeq\frac{\pi\alpha}{4}\Delta D_x\\\Delta D_x&=D_{x,\mathrm{spec}}-D_{x,\mathrm{vac}}\end{aligned}

This slope holds for a complete, uniform circular disk that moves only a little. Large offsets make the relation nonlinear, and an annular mask or a partly lost disk changes the calibration.

Electron count 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. The separate vacuum exposure adds acquisition time and data but no specimen dose. The error shown against the known model 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 shows an electric-equivalent field integrated through the depth and averaged over the probe. Magnetic forces and scattering can contribute, and the Lab does no atomic multislice calculation.

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
Separate thickness and lattice impulse
With crystal coupling off, drag the specimen from 100 to 50 nm and compare the recovered projection. Then choose No field · lattice impulse and vary thickness and mistilt.
What to observe: The field contribution scales with path length. Elastic two-beam exchange varies non-monotonically and can create an apparent field even when the true field is zero.
02
Predict the shift sign
Compare the uncalibrated state with Calibrated center of mass.
What to observe: Subtracting the measured vacuum counts removes the detector offset, but the noise in the reference stays.
03
Cut off part of the disk
Shrink the outer mask or exclude an inner region.
What to observe: The centroid of a clipped disk no longer gives the average momentum of the whole beam. Missing counts can make both estimates invalid.
04
Tell a measurement from its interpretation
Compare the quadrant and scattering-artefact presets, then the magnetic preset.
What to observe: The quadrant estimate drifts away from its small-shift calibration, and elastic dynamical diffraction creates false field contrast. Electric and magnetic sources can produce identical shift data.
05
Meet the calibrated electric target
Turn on vacuum correction, meet the stated limits on acceptance, counts, RMS error and dose, then check.
What to observe: More exposure lowers counting noise, but it cannot remove the systematic bias from clipping or scattering.