---
title: "DPC: Beam Deflection & Projected Fields Interactive Lab | Sandbox Physics"
description: "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."
canonical_url: "https://sandboxphysics.com/labs/dpc-field/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/dpc-field/"
last_modified: "2026-10-25"
---

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.

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

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.

Physics tutorial

## Measure momentum before assigning a field

**Background** An electron that crosses a region with a smooth sideways force picks up sideways momentum, and its diffraction disk shifts. If the disk shifts without changing shape, a pixelated detector can measure the shift from the centroid of the recorded intensity.

**Why it matters** The apparent field also depends on where the detector zero lies, on angles the detector misses, on counting noise and on intensity the crystal scatters elsewhere.

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 misconception** The map gives a unique electric field at each atom.

**Better mental model** The 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 misconception** A symmetric detector mask guarantees an unbiased centroid.

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

### Run the experiment

1. 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.

2. 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.

3. 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.

4. 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.

5. 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.

Physical conclusion

Set the zero with measured vacuum counts, keep the angles that matter and state what you assume produces the force. A field image alone cannot tell electric from magnetic contributions.

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)