---
title: "Photoelectron Structure: Direction and Depth Interactive Lab | Sandbox Physics"
description: "What can photoelectron diffraction and standing-wave scans tell you about where atoms sit?"
canonical_url: "https://sandboxphysics.com/labs/photoelectron-structure/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/photoelectron-structure/"
last_modified: "2026-10-27"
---

E64 · Structural photoemission

# Photoelectron Structure: Direction and Depth

**What can photoelectron diffraction and standing-wave scans tell you about where atoms sit?** An angle scan of photoelectron diffraction constrains the directions to nearby atoms, and a standing-wave scan constrains positions along one periodic direction. Neither tells you more than the measurement and its assumptions pin down. Counting noise, the range covered and fits that are not unique each need a separate check.

Take angle-resolved diffraction scans or standing-wave phase scans, fit the stored core-level counts, and see how much of the atomic structure those counts can really pin down.

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

Model assumptions

Both models are synthetic teaching models. The diffraction model has one surface emitter and one scatterer, with an assumed real scattering amplitude. The standing-wave model has an ideal, independently controlled phase at a fixed reflectivity. Element-specific phase shifts, multiple scattering, a realistic Bragg rocking curve and nondipole corrections are left out. The layer signal is weighted by the chance that electrons escape. The coherent position repeats every period, so it does not give a unique absolute depth. The apparatus, the atomic spacing and the layer thickness are drawn at different schematic magnifications.

Physics tutorial

## Photoelectron Structure: Direction and Depth

**Background** A photoelectron wave leaving an atom interferes with the part scattered by a neighbour, so the counts change with emission angle. In a standing-wave scan, the counts change with the phase of the standing wave instead. This Lab records both kinds of scan and fits the stored counts.

**Why it matters** A fit can match the data well and still leave the structure undetermined. Scan coverage, count statistics and the model assumptions decide what a scan can actually tell you.

Start with the essentials

- **Focus question**: What can photoelectron diffraction and standing-wave scans tell you about where atoms sit?
- **One-sentence intuition**: A good fit tells you only what the measurement and its assumptions are able to decide.

### Core mathematical model

#### Direct wave plus one scattered wave

$$
\begin{aligned}I(\theta)&=\left|1+\frac{f(\gamma)}{r}e^{ikr(1-\cos\gamma)}\right|^2\\\gamma&=\theta-\beta,\quad k=\sqrt{2m_eE_k}/\hbar\end{aligned}
$$

The assumed scattering amplitude is strongest in the forward direction. The direct and scattered paths differ by the neighbour distance times the angular path factor.

#### Coherent position and fraction, weighted by escape

$$
\begin{aligned}C&=\frac{\int_0^\infty\rho(z)e^{-z/\Lambda}\,e^{2\pi iz/d}\,dz}{\int_0^\infty\rho(z)e^{-z/\Lambda}\,dz}\\F&=|C|,\quad P=\frac{\arg C}{2\pi}\pmod 1\end{aligned}
$$

The depth distribution is a Gaussian cut off at the surface. The effective escape length includes the cosine of the takeoff angle. The coherent position does not have to equal the mean depth.

#### Core-level yield as the standing-wave phase changes

$$
\begin{aligned}Y(\varphi)/A&=1+R\\&\quad+2\sqrt R\,F\cos(\varphi-2\pi P)\end{aligned}
$$

The constant, cosine and sine coefficients are fitted to the stored counts, using the calibrated reflectivity. The overall amplitude is an unknown nuisance parameter.

#### Depth is known only up to one period

$$
z_{\mathrm{equiv}}=dP,\qquad z_{\mathrm{equiv}}+nd\quad(n\in\mathbb Z)
$$

Equivalent positions are one period apart. The unknown layer abundance can absorb any change in attenuation, so one scan cannot pick out the absolute depth.

### Common difficulties

#### A low residual does not fix the structure

**Typical misconception** A low residual proves there is only one possible atomic structure.

**Better mental model** The models here are synthetic teaching models: one surface emitter and one scatterer with an assumed real scattering amplitude, or an ideal standing wave with an independently controlled phase at fixed reflectivity. They leave out element-specific phase shifts, multiple scattering, a realistic Bragg rocking curve and nondipole corrections. The layer signal is weighted by escape. The coherent position repeats every period, so it cannot give a unique absolute depth. The apparatus, atomic spacing and layer thickness are drawn at different schematic magnifications.

#### The residual map is not a confidence interval

**Typical misconception** The residual map gives a confidence interval or the experimental resolution.

**Better mental model** The unweighted least-squares map only describes the fit. The angular acceptance is calibrated independently, and energy-dependent scale factors are fitted. The standing-wave fit shows noisy coherent fractions above one as they are, and it rejects rank-deficient or grossly unphysical fits. All counts are synthetic, and no uncertainty or external certification is claimed.

### Run the experiment

1. 01

   #### Trace the two electron paths

   Acquire an angular scan and move the selected point across the recorded angles.

   **What to observe:** The collector follows the signed angle from the outward normal. The direct wave and the singly scattered wave interfere.

2. 02

   #### Fit the stored angular counts

   Compare Completed angular scan with Narrow angle coverage and Too few counts.

   **What to observe:** The search grid shows the relative residual over distance and tilt. A sharp minimum holds only within the assumed one-neighbour model.

3. 03

   #### Measure a layer with a standing wave

   Acquire and fit a phase scan, then compare it with Broad layer distribution and No reflected wave.

   **What to observe:** When the depths spread across several phases, the coherent modulation weakens. With no reflection there is no phase information.

4. 04

   #### Test whether the absolute depth can be found

   Compare Completed phase scan with One period deeper.

   **What to observe:** The phase repeats while the absolute counts drop. Because the abundance is unknown, the absolute depth is not unique. The two examples use separate synthetic samples.

Physical conclusion

Check count statistics, scan coverage and non-uniqueness separately.

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)