E47 · Magnetic electron imaging

Lorentz TEM: Where the Walls Appear

Why does a magnetic domain wall switch from bright to dark when the defocus is reversed? Because defocus turns the curvature of the electron phase into brightness, and reversing the defocus flips the sign of that conversion. At exact focus a pure-phase film shows no contrast at all. Electric phase, for example from changes in thickness, can produce similar Fresnel contrast, so a line that flips is not by itself proof of a magnetic wall.

Defocus the image of a magnetic foil, flip the sign of the defocus, and use a Foucault aperture to pass only half of the deflection angles. Compare the counted images with the known induction, then check whether an electric phase can mimic a magnetic signal.

Mean recorded counts per pixel $\text{—}$

Peak model magnetic deflection $\text{—}$

Model relative RMS contrast $\text{—}$

Opposite-defocus model correlation $\text{—}$

More measurements and target

Model aperture throughput $\text{—}$

Assumed field-induced displacement $\text{—}$

Experiment target $\text{—}$

Simulated examples; each plot identifies its data source.

More examples

Peak prescribed induction

B_0

0.8\,\mathrm{T}

Applied field · assumed texture shift

B_{\mathrm{app}}

0\,\mathrm{mT}

Signed defocus · Fresnel only

\Delta f

0\,\mathrm{\mu m}

Image mode: 0 Fresnel, 1 Foucault

m

0

Advanced settings

Foil thickness

t

60\,\mathrm{nm}

Assumed mean electric potential

V_0

0\,\mathrm{V}

Detector Gaussian RMS blur

\sigma_d

5\,\mathrm{nm}

Accepted angular sign

s

1

Foucault angular cutoff

\theta_c

0\,\mathrm{\mu rad}

Mean incident electrons per pixel

N_0

2000

Learning goal and model limits

Make a prediction, then test it

Why does a magnetic domain wall switch from bright to dark when the defocus is reversed?

Target: use positive Fresnel defocus with the electric potential set to zero. Record at least 1000 counts per pixel on average and reach a model RMS contrast of at least 0.12. The contrasts at opposite defocus must have a correlation of at most -0.85.

Model assumptions

The beam is fixed at 200 keV and travels along positive z, with x to the right and y up. The field of view is periodic, 800 nm across, sampled on 128 by 128 points at 6.25 nm pitch. The in-plane induction is a prescribed smooth sinusoid, not a micromagnetic solution, and the foil has uniform thickness. The optional electric phase comes from a specified potential with a smooth modulation. Absorption, crystal diffraction, stray-field calculation, out-of-plane induction, aberrations, damage and calibrated material identification are not included. The detector records 80 percent of electrons. Fixed seeds generate independent Poisson records, with a normal approximation above 40 counts. Counts stay linear; display colors are mapped separately. In the apparatus view, thickness, electron paths, angular separation and magnification are schematic. Fresnel images come from full paraxial propagation of the complex wave, followed by a Gaussian detector blur. Positive defocus means propagation downstream. Foucault mode places a coherent half-plane aperture at focus: its cutoff is in microradians, it blocks the direct zero-order beam, and defocus has no effect. The usual immersion objective lens is off, and a separate Lorentz lens keeps the specimen field-free. The applied field only shifts the magnetic texture rigidly and reversibly, by an assumed 2 nm per mT, with no hysteresis or domain nucleation. The opposite-defocus image is a noiseless forward reference, not a second recorded image. At finite defocus the bright-dark reversal is only approximate, and a correlation coefficient measures how close it is. The Lab does not reconstruct height or magnetization.

Physics tutorial

Lorentz TEM: Where the Walls Appear

BackgroundA thin magnetic foil barely changes the electron intensity. It mainly shifts the phase of the wave and deflects the electrons slightly. Lorentz TEM makes the domains and their walls visible by defocusing the image or by blocking part of the deflected beam.

Why it mattersThe goal: use positive Fresnel defocus with the electric potential at zero. Record at least 1000 counts per pixel on average, reach a model RMS contrast of at least 0.12, and get an opposite-defocus contrast correlation of -0.85 or lower.

Start with the essentials

Focus question: Why does a magnetic domain wall switch from bright to dark when the defocus is reversed?
One-sentence intuition: In focus, a foil that only shifts the phase can look completely uniform; defocus turns the curvature of that phase into visible intensity.

Core mathematical model

Phase gradient and deflection

\begin{aligned}\theta_x&=\frac{\lambda}{2\pi}\frac{\partial\phi_m}{\partial x}\\&=\frac{e\lambda}{h}\int B_y\,dz\end{aligned}

The sign is for a negatively charged electron moving along positive z.

Fresnel transfer function

\begin{aligned}H&=\exp(-i\chi)\\\chi&=\pi\lambda\Delta f(q_x^2+q_y^2)\end{aligned}

Before the aperture, propagation keeps the total expected intensity unchanged.

Small-defocus limit

I(\Delta f)\simeq1-\frac{\lambda\Delta f}{2\pi}\nabla_\perp^2\phi

This approximation explains why the contrast reverses with the sign of defocus; the image itself is computed with full complex-wave propagation.

Common difficulties

The reference map is not data

Typical misconceptionThe known reference map is the field the microscope recorded or recovered.

Better mental modelThe reference map is the prescribed specimen input. The images are counted and processed without it; the reference is used only to score the error.

Run the experiment

01
Predict the focused foil
Start from the In focus preset, then set the Signed defocus first positive, then negative.
What to observe: Defocus turns the phase into visible contrast. Flipping the sign roughly swaps bright and dark, but at finite defocus the swap is only approximate.
02
Select half of the angles
Set Image mode to Foucault, flip the Accepted angular sign and raise the Foucault angular cutoff.
What to observe: The domain contrast changes, and fewer electrons reach the detector.
03
Test another possible cause
Load the Electric phase confound preset, move the Applied field slider, then reach the target.
What to observe: A wall-like line of intensity is not enough, by itself, to show that magnetism caused it.