---
title: "Photoemission & Hemispherical Analyzer Interactive Lab | Sandbox Physics"
description: "Can you resolve two close peaks in a photoelectron spectrum without losing most of the counts? You can, if you give up only as much throughput as the separation needs. Lowering the pass energy, the energy at which electrons travel through the analyzer, and narrowing the slit sharpen the energy response but let fewer electrons through. Pick settings that just resolve the two peaks, then make up the counts with a longer exposure."
canonical_url: "https://sandboxphysics.com/labs/photoemission-analyzer/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/photoemission-analyzer/"
last_modified: "2026-10-25"
---

E50 · Electron imaging / spectroscopy

# Photoemission: Retarding Current & Energy Analysis

**Can you resolve two close peaks in a photoelectron spectrum without losing most of the counts?** You can, if you give up only as much throughput as the separation needs. Lowering the pass energy, the energy at which electrons travel through the analyzer, and narrowing the slit sharpen the energy response but let fewer electrons through. Pick settings that just resolve the two peaks, then make up the counts with a longer exposure.

Record retarding-current curves at several photon frequencies and intensities with a guarded radial collector. Find each edge from the saved shutter-open and shutter-closed counts, test the limits of the calibration, and then switch to the hemispherical energy analyzer.

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

Model assumptions

The Lab has two separate instruments. Retarding mode assumes radially collimated electrons in an ideal guarded spherical field, with Poisson counts taken with the shutter open and closed. When the whole source band escapes, only the combined offset of work function and contact potential can be identified; the source widths and the frequency scale need independent calibration. Hemispherical mode keeps its 150 mm central-field analyzer, its Gaussian-equivalent response and a fixed 4.5 eV analyzer work function. Neither model includes material-specific cross sections, space charge or a detailed lens solver.

Predict first, then record a current scan

## Intensity changes the flux, and frequency moves the edge.

How to read this experiment

A small spherical emitter and a concentric retarding grid select total kinetic energy. A shielded spherical-cap collector receives radial trajectories within its angular aperture. This is a separate apparatus from the hemispherical analyzer. Each ray starts on the finite emitter surface and is radially collimated about the common centre, with zero angular momentum. The prescribed fan weight resembles a cosine law but is not general Lambertian surface emission. The full spherical guard field is ideal; sizes are magnified. The energy and angular distributions are prescribed teaching sources.

Predict the effect of doubling intensity, then record a scan.

How to read this experiment

Space on the focused workbench acquires or stops; R resets. Sliders retain native arrow keys. Hiding the page or leaving the workbench pauses acquisition and preserves every observation.

No saved session.

Open- and closed-shutter counts

### Finding where the current crosses zero does not find the endpoint.

> Interactive data plot; use the surrounding model explanation and exportable records.

$V_R/\mathrm V,\quad I=e(C_{\rm open}-C_{\rm dark})/t$

How to read this experiment

Teal: observed current with the shutter open. Gray: background with the shutter closed. Amber: response fitted to the observations. The three-frequency run repeats the full open and closed scan at each frequency. An unfinished scan counts as missing data and is left out of the slope.

Central endpoint and the calibrated finite-source response

Record-fitted central endpoint

$\text{—}$

Diagnostic profile half-range

$\text{—}$

Latest open minus dark current

$\text{—}$

Record-fitted voltage / frequency slope

$\text{—}$

Intercept and identifiability

Intercept: work function plus contact

$\text{—}$

Work function with chosen contact reference

$\text{—}$

How to read this experiment

The fitted central endpoint accounts for a uniform photon-energy band and a uniform energy-loss band. The extreme-support stopping voltage is the central endpoint plus photon half-width. The score-plus-four profile range is a diagnostic without calibrated confidence coverage; response systematic errors are excluded. Broad, weak, unbracketed, escape-boundary and poor-fit scans do not enter the slope. The conservative escape check uses the independently declared contact operating range from minus 0.4 to plus 0.8 volts.

> Interactive data plot; use the surrounding model explanation and exportable records.

$\nu_{\rm reported}/\mathrm{PHz},\quad \widehat V_0/\mathrm V$

How to read this experiment

A common unknown contact barrier shifts the intercept. The current scans identify only the combined offset; an independent contact reference is required to interpret a work function. A common frequency-scale error biases the slope even when every curve fits well. The displayed reference value is not supplied to the regression.

$(h/e)_{\rm SI}=4.135667696\,\mathrm{V\,PHz^{-1}}$

Retained scans and latest raw observations

| Scan / group | Reported frequency | Rows / status | Inspect |
| --- | --- | --- | --- |

| Voltage / V | Exposure / s | Open counts | Closed counts |
| --- | --- | --- | --- |

How to read this experiment

Assumptions: independent Poisson equivalent electron counts, constant background measured with the shutter closed, no space charge, no secondary emission, no collector photoemission, no patch fields, and a guarded radial field. The endpoint fit cannot test those assumptions. The hemispherical mode retains its previous energy-analysis model and settings.

[Primary measurement: retarding curves and limitations](https://arxiv.org/html/1611.04310) [Ohio University: contact potential and background](https://people.ohio.edu/piccard/radnotes/penetrate.html) [SIMION · HDA](https://simion.com/info/hda.html)

Hemispherical analyzer data, shown separately

How to read this experiment

Energy-analysis diagnostics from the hemispherical mode. These calculated responses and synthetic detector images belong to that mode and are separate from the retarding-current records.

Physics tutorial

## Separate energy selection from an endpoint measurement.

**Background** This Lab has two different instruments. The hemispherical analyzer still sorts electrons by energy; the other instrument records the current reaching a guarded radial collector as a retarding voltage changes. Endpoints are inferred only from the recorded shutter-open and shutter-closed counts, and you compare several frequencies and intensities.

**Why it matters** Before drawing conclusions about the specimen, see how each instrument setting shapes the measured result.

Start with the essentials

- **Focus question**: Which setting changes the flux, which moves the edge, and which quantities cannot be separated?
- **One-sentence intuition**: The current curve has a finite source width and a background. Intensity changes only the flux. The slope of voltage against frequency needs several endpoints fitted from the recorded scans and a calibrated frequency scale. The shared contact barrier and the work function add up to one intercept that cannot be split.

### Core mathematical model

#### Kinetic energy against two vacuum levels

$$
\begin{gathered}E_{\mathrm{kin,s}}=h\nu-\phi_s-E_B\\E_{\mathrm{kin,a}}=h\nu-\phi_a-E_B\end{gathered}
$$

This assumes the sample is grounded in electrical contact with the analyzer, with no sample charging and no external bias.

#### Gaussian-equivalent energy width

$$
\begin{gathered}\sigma_a^2=E_p^2\left(\frac{s^2}{24R_0^2}+\frac{4\alpha^4}{45}\right)\\\Delta E_G=2\sqrt{2\ln2}\,\sigma_a\\\Delta E=\sqrt{\Delta E_G^2+\Delta E_\gamma^2}\end{gathered}
$$

Positions across the two equal slits and the entrance angle in the dispersive plane are each uniformly distributed, independently of one another. The angular half-width is in radians, and the average energy shift from the entrance angle is calibrated out. This Gaussian only matches the variance of the response; it is not the exact analyzer line shape or its base-width formula.

#### Noise and throughput

$$
\bar N=\dot N(E_p,s,\alpha,\theta)\,\tau,\quad \sigma_N=\sqrt{\bar N}
$$

The count rate follows an empirical teaching law, not the specification of a commercial instrument.

#### Retarding potential between concentric electrodes

$$
\begin{gathered}U(r)=e(V_R+V_c)g(r)\\g(r)=\frac{a^{-1}-r^{-1}}{a^{-1}-R^{-1}}\end{gathered}
$$

The emitter has a radius of two millimetres and the grid twenty millimetres. Electrons launched along the radius carry no angular momentum. The potential changes monotonically between the concentric electrodes, so an electron reaches the grid only if its initial kinetic energy is larger than the positive barrier. An equipotential collector cap sits at twenty-two millimetres. The field is an ideal guarded one; it is not a planar collector or a general model of emission from a spherical cathode.

#### Finite source width and transmitted fraction

$$
\begin{gathered}E=h\nu+\delta-\phi_s-\epsilon\\A_\alpha=\sin^2\alpha\\T=\Pr\!\left[E>\max(0,e(V_R+V_c))\right]\end{gathered}
$$

The photon-energy offset is spread uniformly over the calibrated half-width, and the energy loss is spread uniformly and independently from zero to its calibrated width. Electrons leave radially from launch sites spread over a prescribed cosine-weighted fan of directions. The angular size of the collector cap changes the accepted flux. Electrons whose energy on the vacuum side is zero or negative never escape, and the simulator applies this limit before any detector count.

#### Finding the central edge from the counts

$$
\begin{gathered}V_0=\frac{h\nu-\phi_s}{e}-V_c\\T=\Pr[\delta-\epsilon>e(V_R-V_0)]\end{gathered}
$$

The second line is used only when the whole source band is sure to escape for every contact value in the independently bounded range. Otherwise the plateau is clipped by the escape limit, and the scan is rejected. The shutter-open and shutter-closed Poisson counts give the background, the amplitude and a profiled central edge. The outermost cutoff is the central edge plus the photon half-width divided by the electron charge. The score-plus-four profile range is a diagnostic; its confidence coverage has not been calibrated.

#### Slope and the offset that cannot be split

$$
\begin{gathered}V_0=m\nu-B\\m=h/e,\qquad B=\phi_s/e+V_c\end{gathered}
$$

The fit needs at least three different reported frequencies from one preparation group. Planck’s constant, the source work function and the contact value are not inputs to the regression. An independently supplied contact reference lets you interpret the intercept, but the result is only as good as that reference. An unknown error in the frequency scale, common to all points, biases the slope while the current curves still fit well.

### Common difficulties

#### Work function and binding energy

**Typical misconception** In a properly grounded, calibrated analyzer, the sample work function shifts every binding-energy peak.

**Better mental model** The contact potential difference shifts the local vacuum level. Once the Fermi levels line up, binding energies are calibrated with the analyzer work function. The emission threshold still depends on the sample. An electron escapes if its kinetic energy above the sample vacuum level is positive, but it reaches the analyzer only if its kinetic energy above the analyzer vacuum level is also positive.

#### The intercept does not give a unique work function

**Typical misconception** A straight line of voltage against frequency measures both the emitter work function and the contact potential.

**Better mental model** As long as every source electron escapes, you can raise the work function by some amount, lower the contact voltage by the equivalent amount, and every current reading stays the same. Only an independent reference can tell the two apart. Near threshold the escape limit does carry extra information, but this restricted endpoint estimator deliberately leaves that range out and does not claim to recover both quantities independently.

### Run the experiment

1. 01

   #### Predict the resolution

   Predict whether the fastest preset can separate the doublet. Then compare Fast survey with Resolve doublet against the fixed reference of the intrinsic lines.

   **What to observe:** Lowering the pass energy or narrowing the slit makes the measured peaks narrower but lets fewer electrons through. When the sample and analyzer work functions differ, the kinetic energy just outside the sample is not the kinetic energy the analyzer measures.

2. 02

   #### Resolve the doublet

   Resolve the doublet, whose peaks are 0.25 eV apart, with a Gaussian-equivalent FWHM below 0.10 eV and at least 500 expected integrated counts. Electrons from the selected state must also be able to escape.

   **What to observe:** Check the numbers against the target, then compare the instrument view with the data it produces.

3. 03

   #### Change the sample work function

   Change only the sample work function. Compare surface kinetic energy with analyzer kinetic energy, then reduce photon energy until emission stops.

   **What to observe:** The analyzer disperses energies as an ideal central field, with a variance-matched Gaussian response. A full spectrum needs an energy scan, because one detector window covers only a narrow energy range. The analyzer work function is fixed at 4.5 eV, and electrical contact lines up the Fermi levels. The Gaussian line response and the empirical transmission are teaching assumptions; there are no material-specific cross sections, no space charge and no detailed lens solver.

4. 04

   #### Predict, then record the current

   With the Retarding collector, record three frequencies, then double only the photon flux and record one more scan. Look at the raw shutter-open and shutter-closed counts and at the fitted edges. Stopping partway and resuming keeps the same random sequence.

   **What to observe:** Flux changes how large the current is; frequency moves the edge. Because exposure is finite, the fitted values scatter. Earlier raw scans stay available in the table and the CSV file.

5. 05

   #### Try to break the inference

   Try Low signal, No emission, Broad source and Escape boundary, then the two calibration counterexamples, Frequency scale error and Wrong contact reference. Before comparing the two instruments, use Save exact session and then Restore saved session.

   **What to observe:** A scan that shows only dark current does not prove that zero electrons were emitted. Curves that are broad, do not bracket the edge, are clipped by the escape limit or disagree with each other do not give a point for the slope. A wrong contact reference spoils the inferred work function but leaves the slope unchanged.

Physical conclusion

The current curve has a finite source width and a background. Intensity changes only the flux. The slope of voltage against frequency needs several endpoints fitted from the recorded scans and a calibrated frequency scale. The shared contact barrier and the work function add up to one intercept that cannot be split.

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)