---
title: "Beam Diagnostics Bench Interactive Lab | Sandbox Physics"
description: "Why can a tiny spot hide a divergent beam? A screen measures position, so a small spot can still contain a wide range of particle angles."
canonical_url: "https://sandboxphysics.com/labs/beam-diagnostics/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/beam-diagnostics/"
last_modified: "2026-11-13"
---

# Beam Diagnostics Bench

**Why can a tiny spot hide a divergent beam?** A screen measures position, so a small spot can still contain a wide range of particle angles. Changing a quadrupole reveals several position–angle projections. Calibrated screen widths then recover the projected phase-space area, provided resolution, clipping and a changing beam do not hide it.

Read four-electrode signals, independent bunch charge and screen pixels. Scan a finite quadrupole to recover projected emittance and Twiss parameters.

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

### A spot is one projection

Drag the source, magnet or camera dial. Keep a scan to acquire new standards and pixels.

Entrance width scale $\beta/\mathrm{m}$ $1.5$ Position–angle correlation $\alpha$ $1.2$ Source steering $x_0/\mathrm{mm}$ $0$ Quadrupole scan span $f_k$ $1$

Optical blur and weak tails

Optical blur $\sigma_{\mathrm{opt}}/\mathrm{mm}$ $0.15$ Weak-tail fraction $f_h$ $0$

Acquiring independent standards and a screen scan…

Acquisition and model limits

Charge command $Q/\mathrm{pC}$ Acquisition seed

A linear hard-edge proton beamline with a rectangular absorbing bore. Constant beam moments, ideal linear-cut electrodes and uniform affine imaging are assumptions. Reference failures withhold affected inference; old raw records stay unchanged.

Model assumptions

Synthetic proton diagnostics with independently acquired electrical, charge, position, imaging and magnetic standards. The finite hard-edge quadrupole transports paraxial, fixed-momentum particles through a rectangular absorbing bore. Seven fit exposures, two held-out strengths and a repeat test a stationary second-moment model. Pixel-centre moments retain signed dark-subtracted weights; resolution comes from independent finite-width beams. Statistical standard uncertainty includes shared reference fits and pixel-noise covariance; deterministic pixel-bin bounds are reported separately in the area interval. Geometric RMS emittance has no extra pi factor; normalization assumes known protons and certified rigidity. The linear field omits fringe fields, aberrations, dispersion, coupling forces and collective dynamics; a conservative perveance screen limits the space-charge budget. Screen clipping, unqualified collection, calibration-domain failures and unresolved area withhold affected inference while retaining raw data. Finite checks cannot certify absence of arbitrarily weak invisible tails. The displayed electrode law assumes ideal linear-cut plates, the CT a calibrated finite gate, and the camera uniform affine imaging. Hardware noise and dimensions are teaching assumptions. Transverse size is magnified, markers represent the distribution and motion is slowed.

## From one spot to positions and angles

**Background** A profile screen measures where particles arrive. Their directions determine how quickly that spot changes farther down the beamline.

**Why it matters** A quadrupole changes which combination of entrance position and angle reaches the screen. Several projections supply information missing from one photograph.

- **Focus question**: How can equally small spots hide different divergence?
- **One-sentence intuition**: Projected phase-space area depends on position spread, angular spread and their correlation.

### Core mathematical model

#### Normalized electrode difference

$$
d_x=\frac{A_R-A_L}{A_R+A_L}
$$

Independent electrical and mechanical standards calibrate ideal linear-cut electrodes to a centroid. A separate charge ring supplies absolute bunch charge.

#### The screen is a transported projection

$$
\sigma_s^2=R_{11}^2\Sigma_{11}+2R_{11}R_{12}\Sigma_{12}+R_{12}^2\Sigma_{22}
$$

The matrix includes the finite quadrupole and drift. Each strength has a newly acquired image; held-out strengths are excluded from the fit.

#### Projected RMS emittance

$$
\varepsilon_{\mathrm{rms}}=\sqrt{\Sigma_{11}\Sigma_{22}-\Sigma_{12}^2}
$$

The moments are centred on the measured beam. Statistical standard uncertainty and deterministic pixel-bin bounds are separate; unresolved area is withheld.

### Common difficulties

#### Waist and area

**Typical misconception** A smaller bright spot always means a better beam.

**Better mental model** Particles can share a narrow waist and leave at very different angles. Emittance includes that directional spread.

#### Common reference errors

**Typical misconception** More photographs remove every calibration error.

**Better mental model** A shared scale or reference affects the entire scan. Repetition does not average that common error away.

### Run the experiment

1. 01

   #### Watch a scan

   Compare the default screen images and width curves.

   **What to observe:** One beam makes different spots as the field changes. Horizontal and vertical focusing act in opposite directions.

2. 02

   #### Compare equal spots

   Choose Same spot, different angles.

   **What to observe:** One projection can agree even when the full scan curves separate. The missing information is angular spread.

3. 03

   #### Blur the optics

   Drag the camera dial, then keep a scan.

   **What to observe:** Independent reference images are acquired again with the unknown images. Their measured widths support the resolution correction.

4. 04

   #### Steer off the screen

   Drag the source or choose the lost-beam demonstration.

   **What to observe:** Electrodes reveal the displacement. A mismatch between independent charge and screen light withholds full-beam area and preserves the pixels.

Physical conclusion

A waist gives a small spot; a qualified scan can measure projected emittance.

## Related experiments and articles

How physics instruments measure · Reading path

### How physics instruments measure

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **The directions hidden inside a small beam spot**](https://sandboxphysics.com/stories/instruments/beam-diagnostics/)

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)