---
title: "Inverse Photoemission: Empty States Interactive Lab | Sandbox Physics"
description: "What can emitted photons tell us about empty electron states? Where they lie in energy."
canonical_url: "https://sandboxphysics.com/labs/inverse-photoemission/"
language: "en"
page_type: "lab"
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  zh-CN: "https://sandboxphysics.com/zh/labs/inverse-photoemission/"
last_modified: "2026-10-27"
---

E63 · Electron excitation and surface spectroscopy

# Inverse Photoemission: Empty States

**What can emitted photons tell us about empty electron states?** Where they lie in energy. An incoming electron drops into an empty state and emits a photon, and the detector accepts only one fixed photon energy. Scanning the electron energy then locates the unoccupied states, broadened by the instrument resolution.

Step the energy of the incoming electrons while a detector counts photons in one fixed energy band. Find the two peaks where electrons drop into empty states, then see how photon bandwidth, electron energy spread, exposure and energy calibration change them.

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

Model assumptions

This is isochromat inverse photoemission. The detector counts photons in a fixed band at 9.5 eV, modeled on published counters with an SrF2 window and approximated here by a Gaussian with adjustable FWHM. The electron current is fixed at 10 nA over one square millimetre. The scan has 220 electron-energy positions from 3 to 14 eV in 0.05 eV steps. Electron energy means the calibrated kinetic energy just outside the surface, not the gun voltage. The sample's true work function is 4.5 eV. The final-state energy equals the kinetic energy at the surface plus the work function minus the photon energy. The empty states are synthetic: two Gaussian peaks at 2 and 5 eV above the Fermi level. They are not gold (Au) data and not an absolute density of states. The numerical integral includes only empty final states. The Gaussian energy spread of the incoming electrons and the photon band both broaden the scan, and a wider photon band collects more counts. The assumed yield is two billionths per incident electron per unit spectral response, and the dark rate is five counts per second. Counts come from a fixed random seed and use a normal approximation above 40 expected counts. A set cumulative-dose rule shifts and weakens the empty-state response; it is not a calibrated damage model. Each centroid comes from one of two fixed kinetic-energy windows in the recorded scan, after the known dark counts are subtracted. Display calibration and smoothing reuse the same counts. The separate reference curves for filled and empty states are model inputs, not measured UPS spectra or reconstructed densities of states. The model leaves out momentum, spin, many-body matrix elements, charging and detector dead time.

Physics tutorial

## Inverse Photoemission: Empty States

**Background** An electron that enters a surface can drop into an empty state and give off a photon. Here you step the energy of the incoming electrons while a detector counts photons in one fixed band. You find two empty-state peaks in the counts, then test how photon bandwidth, electron energy spread, exposure and energy calibration affect them.

**Why it matters** The goal is to place each peak within 0.12 eV of its undamaged position. Each peak needs a signal-to-noise ratio of at least 10, the combined instrument FWHM must be at most 0.8 eV, and the set damage rule may change at most 15 percent of the response.

Start with the essentials

- **Focus question**: What can emitted photons tell us about empty electron states?
- **One-sentence intuition**: The electron brings in energy and the photon carries a fixed energy away; the difference tells you where the empty state sits.

### Core mathematical model

#### Energy of the empty final state

$$
\begin{aligned}\varepsilon_f&=E_{kin}+\Phi-h\nu\\E_{kin}&=\varepsilon_f+h\nu-\Phi\end{aligned}
$$

Kinetic energy is measured from the local vacuum level, and the final-state energy from the Fermi level. A calibration error shifts the whole energy axis.

#### Instrument resolution from two Gaussian widths

$$
\begin{aligned}w_{inst}&=\sqrt{w_e^2+w_\gamma^2}\\\sigma&=\frac{w_{inst}}{2\sqrt{2\ln2}}\end{aligned}
$$

The electron and photon widths are independent Gaussians, so they add in quadrature. The natural width of each state comes on top of this.

#### Expected photon counts at each electron energy

$$
\begin{aligned}\varepsilon_0&=E+\Phi-h\nu\\R(E)&=\int_0^\infty\rho_u(u)\\&\quad\times G_\sigma(u-\varepsilon_0)\,du\\\mu(E)&=b\tau+\frac{I\tau}{e}\eta A_\gamma R(E)\end{aligned}
$$

A wider photon band has a larger area, so it collects more counts. The recorded scan is the empty-state spectrum broadened and weighted by the instrument; it is not a reconstruction of the ideal input.

### Common difficulties

#### Reference curves are not data

**Typical misconception** The ideal input curve is the result inferred from the scan.

**Better mental model** The gold curves show the known input and serve only as an independent check. The teal result comes from the stored counts alone.

#### Longer exposure does not fix every error

**Typical misconception** If I collect enough counts, the answer will be right.

**Better mental model** More counts reduce counting noise. Errors from the continuum background, the energy calibration, the finite instrument width and the set damage remain.

### Run the experiment

1. 01

   #### Predict which energy is scanned

   Pick a stored electron-energy position and compare its photon count with the energy bookkeeping card.

   **What to observe:** The photon band the detector counts stays fixed; only the energy of the incoming electrons changes.

2. 02

   #### Recalibrate the same counts

   Change the assumed work function from 3.5 to 4.5 eV. Then change the smoothing and check that the raw scan does not change.

   **What to observe:** A wrong work function shifts both inferred peaks by the same energy.

3. 03

   #### Trade counts against resolution

   Compare a wide photon band with a broad spread of incoming electron energies. Look at the count rate and the combined FWHM.

   **What to observe:** More counts do not undo the broadening, and they cannot separate features that sit too close to resolve.

4. 04

   #### Recover the two peaks

   Set narrow electron and photon widths, the calibrated work function and enough dwell time. Compare the centroid energies with the separate reference, then check the target.

   **What to observe:** The target checks signal-to-noise, energy error, instrument width and the set amount of damage.

Physical conclusion

An isochromat scan records the empty states broadened by the instrument, and the result still carries calibration and counting uncertainty.

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)