Magnetic Microscopy
in Rock Magnetism
2026 IRM summer school
A bulk moment hides the grains that carry it
Bulk magnetometry measures:
\[\mathbf m = \int_V \mathbf M(\mathbf r)\,dV\]
Magnetic microscopy measures:
Stray magnetic field, flux, or field gradients outside magnetic grains
Local magnetization direction or magnetic-domain structure within, across, or at the surface of individual grains
The instrument measures field-, force-, phase-, or polarization-based contrast; magnetization (moment) is usually inferred through calibration or inversion.
The scale range from sub-mm to nm.
Every magnetic measurement begins with a forward problem
The sample contains magnetization, but most microscopes measure a field, force, phase, resonance, or polarization after geometry has filtered it.
\[\mathbf M(\mathbf r) \longrightarrow \mathbf B(\mathbf r,z) \longrightarrow s(\mathbf r) + \epsilon\]
Direct-ish contrast
XMCD, SEMPA, and Kerr microscopy show which way the magnetization points, but only for the part they can see—usually the sample surface or an average through its thickness.
Stray-field contrast
SQUID, Hall, MFM, NV, and Bitter imaging respond to the external field, its projection, or its gradients. Standoff suppresses short wavelengths.
A susceptibility bridge measures the response to a small AC field
Drive and balance
An AC current creates a small oscillating field. The sample responds with \(M(t)=\chi H(t)\) and changes the flux in one side of a balanced pickup bridge.
ReadoutLock-in detection separates the in-phase and out-of-phase voltage at the drive frequency.
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Measures reversible, low-field response.
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Frequency dependence can reveal relaxation time and thus grain size.
A VSM turns controlled motion into an induced voltage
Bulk moment along one axis
The specimen vibrates between pickup coils while an applied field sets its magnetic state.
ReadoutThe changing magnetic flux over the pick up coils produces a voltage, \(V=-d\Phi/dt\), proportional to the sample moment after calibration.
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Measure hysteresis, backfield curves, and FORCs.
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The result is the (calibrated) moment of the bulk specimen.
The JR-6 turns rotation into \(d\Phi/dt\)
Rotation encodes direction as phase
A defined-size specimen spins at constant \(\omega\) inside paired pickup coils, making the coil flux oscillate.
ReadoutFor a centered specimen, \(\Phi_{\rm link}(t)\!\approx\!K m_\perp\cos(\omega t-\alpha)\), so \(V=-d\Phi_{\rm link}/dt\). Fourier amplitude and phase give two components transverse to the spin axis.
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Reorient the specimen to recover the third component.
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Irregular shape or offset changes the coil coupling and waveform.
A SQUID converts flux into superconducting phase
\[I = I_c\sin\delta, \qquad \Phi_0 = \frac{h}{2e}\]
Quantum tunneling
Cooper pairs tunnel through Josephson junction barriers. Current depends on the phase difference of two superconducting condensates.
Flux-locked readout
Flux in the pickup loop changes SQUID voltage. Feedback cancels the change, and the feedback signal becomes a linear flux measurement.
Flux shifts phase; bias produces voltage
The weak links make flux observable
The two paths enclose flux, so identical junctions acquire different gauge-invariant phase drops.
\[
\begin{aligned}
\delta_1-\delta_2 &= 2\pi\frac{\Phi}{\Phi_0} \\[3pt]
I_{c,\mathrm{SQUID}} &= 2I_0\left|\cos\!\left(\pi\frac{\Phi}{\Phi_0}\right)\right| \\[3pt]
V &= \frac{\hbar}{2e}\frac{d\delta}{dt}
\end{aligned}
\]
Two current componentsThe bias splits through both arms: \(I_1=I_B/2-I_{\rm circ}\) and \(I_2=I_B/2+I_{\rm circ}\). The bias term points the same way; the circulating term points oppositely. Above \(I_c\), the phase runs and voltage appears.
A U-channel rock magnetometer couples specimen flux into a SQUID
Full vector moment
The sample moves through axial and transverse superconducting pickup coils. Their response functions separate \(M_x\), \(M_y\), and \(M_z\).
ReadoutPickup current couples flux into a DC SQUID. A flux-locked loop supplies the linear feedback voltage.
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Extremely sensitive remanence measurement.
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Translation gives a finite sensor-response curve.
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The output is a bulk vector moment.
An MPMS rejects background flux with a reverse-wound gradiometer
One axial moment component
The outer single turns are \(-1\) and \(-1\); the two central turns are \(+1\) and \(+1\). Their symmetric signed sum rejects uniform fields and linear gradients, but not a nearby dipole.
Two motion modesDC: fit SQUID signal versus a long translation. VSM: vibrate at the center and lock in the periodic signal.
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The U-channel instrument instead combines transverse saddle pairs with circular axial coils for vector remanence.
Pickup geometry and sample-shape context: Jackson, Sølheid & Marvin,
IRM Quarterly 11(4), 2001–2002. DC fitting and VSM lock-in operation:
Quantum Design MPMS 3 User’s Manual.
Scanning SQUID microscopy measures flux with exceptional sensitivity
How it works
A lithographic pickup loop rasters above the sample. Coupled Josephson junctions convert loop flux to a voltage held linear by a flux-locked loop.
CollectFlux on a rectangular grid plus height, background, and calibration scans; convert to \(B_z\) only with the loop response.
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Excellent for weak remanence.
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Loop diameter and standoff set resolution.
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Cryogenic sensor, not necessarily cryogenic sample.
Cooling through the Verwey transition creates monoclinic twins
A first-order structural transition
Near 120–125 K, magnetite changes from cubic \(Fd\bar{3}m\) to monoclinic \(Cc\). Charge/orbital order, strain, and magnetic anisotropy change together.
Cooling sequenceMonoclinic nuclei grow; differently oriented variants meet and form ferroelastic twin walls.
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Variants inherit parent cubic \(\langle100\rangle\) axes.
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Twin walls can redirect or pin magnetic domains.
Measurement workflow
Source
Magnetic stateremanent or induced moment
Spatial structuredomains and domain walls
Magnetic carriergrains, phases, and inclusions
→
Coupling
Fieldfield, flux, or field gradient
Mechanicalforce, torque, or deflection
Wave or spinphase, Zeeman shift, dichroism
→
Readout
Electricalvoltage or current
Opticalintensity or polarization
Mechanicalcantilever motion
Electronintensity or phase
→
Inference
Direct mapcontrast, field, or gradient
Calibration / inversionphysical units or source magnetization
Confidenceresolution, noise, and uncertainty
Instrument families differ in what they actually measure
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Family and examples
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Raw signal
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Best suited to
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Electron microscopyLorentz TEM · holography · Differential phase contrast
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Electron intensity, phase, or diffraction shift
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Nanometre-scale domains and walls
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X-ray and spin electronXMCD · ptychography · SEMPA
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Dichroism or electron-spin polarization
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Element-specific or vector contrast
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Flux and transportSQUID · Hall
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Pickup-loop flux or Hall voltage
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Quantitative stray-field maps
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Scanning probeMFM · single NV
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Force gradient or spin-resonance frequency
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Nanoscale maps near a surface
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Wide-field opticalQDM · Kerr · indicator film · Bitter
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Fluorescence, polarization, or particle density
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Fast context and in-situ changes
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Microscopy resolution length scale
Resolution, sensitivity, field of view, speed, perturbation, and sample preparation trade against one another.
atomic–nm
electrons / X-rays
10–100 nm
MFM / single NV
0.5–10 µm
QDM / Hall / Kerr / Bitter
10–200 µm
scanning SQUID
mm–cm
bulk moment
The Lorentz force
\[\mathbf F = q\,(\mathbf E + \mathbf v \times \mathbf B)\]
Electron optics
A transverse magnetic induction changes electron momentum. Lorentz TEM converts that deflection into intensity; DPC reads the diffraction-pattern shift.
Transport probes
In a Hall cross, the same sideways force creates a transverse voltage proportional to the local field component.
Lorentz TEM turns deflection into domain contrast
How it works
Electrons crossing in-plane induction are deflected. In Fresnel mode, a defocused image makes domain walls bright or dark; in Foucault mode an aperture selects a deflected beam.
CollectWide-field images at controlled defocus, often through field or temperature sequences.
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Excellent for dynamics and thin foils.
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Contrast is not a direct magnetization map.
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Rock preparation may alter stress or oxidation.
Lorentz TEM instrument overview
How it works
Electrons crossing in-plane induction are deflected. In Fresnel mode, a defocused image makes domain walls bright or dark; in Foucault mode an aperture selects a deflected beam.
CollectWide-field images at controlled defocus, often through field or temperature sequences.
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Excellent for dynamics and thin foils.
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Contrast is not a direct magnetization map.
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Rock preparation may alter stress or oxidation.
DPC and 4D-STEM retain the full diffraction pattern
How it works
A focused electron probe rasters over the foil. A segmented or pixelated detector measures the beam’s center-of-mass shift at every probe position.
CollectA diffraction pattern at every \((x,y)\) position—hence “4D”—then calibrate shifts into projected electric and magnetic fields.
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High data volume, flexible reconstruction.
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Structural diffraction can mimic field shifts.
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Scan distortion matters.
Hall effect: magnetic deflection becomes voltage
From force to voltage
A perpendicular field pushes moving carriers sideways. Edge charge builds a transverse electric field until the electric and magnetic forces balance.
Key readoutThe voltage between the two side contacts changes sign when either the field direction or dominant carrier sign reverses.
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Current supplies the moving charge carriers.
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\(B_\perp\) controls the transverse Lorentz force.
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Carrier density, charge, and sensor thickness set the response.
Teaching schematic uses the displayed current, field, and voltage sign convention.
A Hall cross gives a robust local field voltage
How it works
Bias current flows through a small cross. The Lorentz force separates charge, producing \(V_H \propto I B_\perp/(nqt)\).
CollectRaster \(V_H(x,y)\) while recording topography or height; subtract offset and calibrate field responsivity.
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Wide field range and simple readout.
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Works from ambient to cryogenic conditions.
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The active cross averages the local field.
Electron holography reconstructs magnetic phase
How it works
An electrostatic biprism overlaps an object wave with a vacuum reference wave. Fourier processing recovers phase; its spatial gradient gives the projected in-plane induction after electrostatic phase is removed.
CollectAn interferogram, vacuum reference, and usually reversed or flipped states to separate magnetic from mean-inner-potential phase.
Ptychographic tilt series can recover a 3D vector field
How it works
Overlapping probes recover phase at each tilt. Harrison et al. used pre-edge phase XMCD to measure a magnetization projection for each circular polarization.
CollectTwo perpendicular tilt series—73 and 71 projections—constrain \(M_x\), \(M_y\), and \(M_z\) inside the particle support.
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Curved vortex core and Bloch point.
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About 50 nm maximum spatial resolution.
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Limited angles create missing-wedge distortion.
Giant-magnetofossil vector tomography:
Harrison et al. (2025). Simplified reconstruction geometry; no paper figure is copied.
Zeeman splitting turns a spin into a field meter
\[
\begin{aligned}
\frac{H_{\mathrm{NV}}}{h}
&=D S_z^{2}+\gamma_e\,\mathbf B\!\cdot\!\mathbf S,\\[3pt]
f_{\pm}&\approx D\pm\gamma_e B_{\parallel},
\qquad \Delta f=2\gamma_e B_{\parallel}.
\end{aligned}
\]
Prepare
Green light preferentially returns the NV spin to the bright \(m_s=0\) state.
Measure
Microwaves sweep through spin resonances. Their field-dependent splitting gives the projection of \(\mathbf B\) along an NV axis.
A nitrogen–vacancy defect traps a spin inside diamond
From crystal defect to sensor
One nitrogen atom replaces carbon beside an empty lattice site. In the negative charge state, six defect electrons form an \(S=1\) ground state.
Zero-field splittingElectron–electron interactions place \(m_s=\pm 1\) about \(D=2.87\) GHz above \(m_s=0\), even before an external field is applied.
ODMR reveals the field-split spin transitions
Read an ODMR spectrum
Green light prepares the bright \(m_s=0\) state. A microwave sweep transfers population when \(hf\) matches an allowed spin transition, reducing red fluorescence.
Move the controlsA 0.9–1.4 mT bias field sets the baseline separation; the sample field shifts it locally. \(^{15}\)N gives a doublet and \(^{14}\)N a triplet.
A single NV trades speed for nanometre proximity
How it works
One NV center sits in an AFM tip. ODMR at each pixel gives the field projection along the known NV axis while the tip maintains close standoff.
CollectTopography plus one or more ODMR frequencies at each point, giving the quantitative projection \(B_{NV}=\mathbf B\!\cdot\!\hat{n}_{NV}\) rather than MFM’s tip-dependent force-gradient contrast.
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Quantitative, non-contact field projection.
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Ambient and nanoscale.
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Point spectroscopy makes large maps slow.
A wide-field QDM reads every pixel in parallel
How it works
A shallow NV ensemble lies beneath the diamond surface. A camera records fluorescence while microwaves sweep through ODMR resonances.
CollectAn image stack over microwave frequency, fit an ODMR spectrum per pixel, then transform four NV-axis projections into field components when justified.
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Micrometre maps over millimetres.
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Optical and magnetic images co-register.
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Resolution–sensitivity–NV-layer tradeoff.
MFM senses a field derivative, not the field itself
How it works
A magnetized tip oscillates above the surface. Magnetic force gradients shift cantilever phase or resonance frequency.
CollectTopography first, then phase or \(\Delta f\) at a defined lift height. The greyscale animation shows tip-dependent force-gradient contrast over a patterned magnetic film—not a calibrated \(B_z\) map.
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Fast access to domain-scale texture.
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Topographic and electrostatic cross-talk.
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The tip can move soft domain walls.
Bitter fluid makes surface domain walls visible
How it works
A thin colloid of superparamagnetic magnetite particles is placed on a polished magnetic surface. Particles concentrate where the stray-field gradient is strongest, commonly where domain walls meet the surface.
CollectAn optical micrograph after the particles settle; a saturated-state reference can help separate scratches and polishing relief from magnetic decoration.
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Simple, inexpensive, and wide-field.
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Shows wall locations—not magnetization direction.
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Surface preparation and fluid condition control contrast.
Kerr microscopy images domain motion in real time
How it works
Reflection from a magnetized surface slightly rotates polarization or changes ellipticity. An analyzer turns that small change into intensity contrast.
CollectReference-subtracted camera frames, often synchronized with a swept field to image nucleation and domain-wall motion.
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Fast, wide-field, and field-compatible.
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Surface-weighted and optically limited.
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Polish and reflectivity affect contrast.
Indicator films convert stray field into Faraday rotation
How it works
A magneto-optic film placed near the specimen rotates transmitted polarization in response to local \(B_z\). A reflective layer returns the light to the camera.
CollectWide-field intensity with background and calibration images; model the film response and physical gap.
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Large field of view and real-time response.
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Useful bridge from hand sample to microprobe.
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The indicator film is itself a spatial filter.
XMCD microscopy adds element specificity
How it works
Near an absorption edge, left- and right-circular X-rays interact differently with magnetic moments. PEEM images emitted electrons; STXM images transmitted photons.
CollectRegistered \(I^+\) and \(I^-\) images at selected energies and beam directions; compute asymmetry and, with rotations, vector constraints.
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Element- and orbital-sensitive contrast.
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Surface-sensitive PEEM; thin-section STXM.
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Synchrotron access and preparation required.
SEMPA maps surface spin polarization
How it works
A focused electron beam releases secondary electrons whose spin polarization follows the surface magnetization. A spin detector resolves two in-plane components.
CollectRaster intensity and spin-asymmetry channels, then rotate or add detector axes for vector reconstruction.
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Direct vector magnetization contrast.
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Nanometre-scale surface mapping.
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Requires ultrahigh vacuum and a very clean surface.
Correlative microscopy is stronger than any single map
Locate carriersQDM or SQUID on a thin section
Identify phasesreflected light, SEM/EDS, Raman
Resolve domainsMFM, electron, or XMCD method
Test stabilityfield, temperature, time, or demagnetization
Choose the observable before choosing the instrument
Locate weak remanence
StartQDM or scanning SQUID
Addoptical and chemical registration
Track domain changes
StartMFM or Kerr microscopy
Addheight, tip, and field controls
Resolve nanoscale induction
Startholography or DPC/Lorentz TEM
Addelectrostatic separation and simulation
Identify the magnetic element
StartXMCD or dichroic ptychography
Addspectroscopy and multiple projections
Quantify near-surface stray field
Startsingle NV, Hall, or SQUID
Addsensor response and height series