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Sandbox Physics

E53 · Electron imaging / spectroscopy

ARPES Band & Fermi-Surface Mapping

Follow the purple photon beam onto a gold crystal. Emitted electrons enter a fixed angle lens, bend between hemispherical electrodes and reach an energy–angle detector. Tilt the sample to select a momentum cut; compare the measured image with a converted band map.

Interactive modelARPES Band & Fermi-Surface Mapping
Photon incidence from normal0 ∘0\,\mathrm{{}^\circ}
Fermi-level electron kinetic energy0 eV0\,\mathrm{eV}
Central parallel momentum0 A˚−10\,\mathrm{\mathring{A}^{-1}}
Assumed perpendicular momentum0 A˚−10\,\mathrm{\mathring{A}^{-1}}
Local momentum FWHM0 A˚−10\,\mathrm{\mathring{A}^{-1}}
Thermal edge width0 eV0\,\mathrm{eV}
Voigt + edge width proxy0 eV0\,\mathrm{eV}
Dipole weight proxy00
Fermi crossings in slit00
Lower accepted momentum0 A˚−10\,\mathrm{\mathring{A}^{-1}}
Upper accepted momentum0 A˚−10\,\mathrm{\mathring{A}^{-1}}
Experiment task—\text{—}

Physics tutorial

How does an emission angle reveal an electronic band?

BackgroundMonochromatic photons strike the crystal; emitted electrons, rather than light, enter the analyzer. The lens preserves an angular coordinate, hemispherical electrodes disperse energies, and a two-dimensional electron detector records one energy–angle cut.

Why it mattersConnect the instrument setting to a measured result before interpreting the specimen.

Start with the essentials

Focus question
How does an emission angle reveal an electronic band?
One-sentence intuition
Energy and angle reconstruct an occupied spectral function. A dark band can reflect a matrix-element suppression; higher temperature smears the Fermi edge, and changing photon energy changes accessible momentum.

Core mathematical model

Energy and parallel momentum

EB=hν−ϕa−Ekin,k∥=2meEkinℏsin⁡θE_B=h\nu-\phi_a-E_{\mathrm{kin}},\quad k_\parallel=\frac{\sqrt{2m_eE_{\mathrm{kin}}}}{\hbar}\sin\theta

The model assumes equal 4.5 eV sample and grounded-analyzer work functions. The emission angle is measured from the sample surface normal. Each energy row uses its own kinetic energy; the photon incidence angle is a separate geometric quantity.

Occupied spectral weight

I(k,E)∝∣M∣2Γ/π(E−εk)2+Γ2f(E,T)∗RE∗RθI(\mathbf k,E)\propto |M|^2\frac{\Gamma/\pi}{(E-\varepsilon_{\mathbf k})^2+\Gamma^2}f(E,T)\ast R_E\ast R_\theta

Instrument blurring acts on the occupied intensity rather than shifting the underlying band.

Synthetic band and final state

εk=−2t[cos⁡(akx)+cos⁡(aky)]−μ+2tzcos⁡(ckz)\varepsilon_{\mathbf k}=-2t[\cos(ak_x)+\cos(ak_y)]-\mu+2t_z\cos(ck_z)

Hopping is 0.45 eV, in-plane spacing 3.8 angstrom and layer spacing 6 angstrom; perpendicular momentum requires the stated final-state assumption.

Common difficulties

Interpretation trap

Typical misconceptionAn absent bright band proves there are no electronic states at that momentum.

Better mental modelPhotoemission intensity includes spectral weight, occupation, dipole matrix elements and resolution. The fixed band reference separates visibility from dispersion; the model Fermi map is not a simultaneous full-field measurement.

Run the experiment

  1. 01

    Locate the light and the electrons

    Follow the purple beam to the illuminated spot. Trace the emitted electrons through the lens and around the hemispheres. Tilt the sample: the white normal rotates while the lens axis stays fixed. Then compare Resolved cut and Suppressed weight.

    What to observe: Energy and angle reconstruct an occupied spectral function. A dark band can reflect a matrix-element suppression; higher temperature smears the Fermi edge, and changing photon energy changes accessible momentum.
  2. 02

    Operate and check

    Capture a Fermi crossing, keep momentum FWHM below 0.025 inverse angstrom, thermal edge width below 0.03 eV, and matrix weight above 0.30. Then suppress the band by rotating polarization without moving it.

    What to observe: Use the numerical target, then compare the linked instrument and data views.
  3. 03

    Explain the limitation

    Rotate the crystal azimuth to change the cut. Compare a hot, blurred acquisition with the cold target. Add interlayer hopping and vary photon energy to explore the assumed perpendicular-momentum selection.

    What to observe: Synthetic square-lattice tight-binding band, Lorentzian spectral weight, Fermi occupation and Gaussian instrument blur. Polarization uses an assumed dipole-weight proxy. Perpendicular momentum uses a free-electron final state with assumed 10 eV inner potential. The full Fermi map is a model reference; a slit analyzer measures one cut at a time.