---
title: "Synchrotron and Storage Ring Interactive Lab | Sandbox Physics"
description: "How can defocusing help confine a beam, and how do pickups guide an orbit correction? A magnet changes the position and angle seen by the next one."
canonical_url: "https://sandboxphysics.com/labs/synchrotron-storage-ring/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/synchrotron-storage-ring/"
last_modified: "2026-11-13"
---

# Synchrotron and Storage Ring

**How can defocusing help confine a beam, and how do pickups guide an orbit correction?** A magnet changes the position and angle seen by the next one. Suitable alternating strengths and spacing keep both transverse oscillations bounded. Turn-by-turn pickups and small measured steering steps then reveal how to correct the orbit.

Track a proton bunch through FODO magnets. Calibrate pickups, measure tune aliases and steering response, then observe a corrected orbit.

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

Model assumptions

A four-cell fixed-momentum linear paraxial ring transports a finite particle ensemble and second moments through hard-edge elements. Reference momentum is 100 MeV/c. Each scan platform independently reloads the source distribution after an ideal 20ms stabilized-current qualification; coherent motion has no damping. Pickups use a calibrated near-axis linear electrode response for a fixed bunch shape, with per-sample support; low-velocity proton electromagnetic frequency response is not solved. Uncertainty is conditional on the prescribed response and excludes omitted electromagnetic systematics. Envelope is a model prediction and electrodes measure the centroid. Turn sampling retains tune aliases. Uncertainty support is a finite-noise first-order approximation. Longitudinal RF, nonlinear multipoles, space charge and collective effects are omitted. Particle markers are enlarged and replay slowed. Radiation and beamlines belong to a separate instrument.

## Synchrotron and Storage Ring

**Background** Track a proton bunch through FODO magnets. Calibrate pickups, measure tune aliases and steering response, then observe a corrected orbit.

**Why it matters** Each quadrupole focuses one plane and defocuses the other. The next magnet acts after the particles have moved. Follow that change, then ask how electrode signals can guide a correction.

- **Focus question**: Which ring settings confine the bunch, and which steering changes do the recorded pickups support?
- **One-sentence intuition**: Position and angle must travel together. The orbit estimate comes from calibrated electrodes and measured scans; the propagated beam width is a separate model prediction.

### Core mathematical model

#### The two planes respond differently

$$
\begin{aligned}x''+(h^2+k)x&=h\delta\\y''-ky&=0,\quad\delta=(p-p_0)/p_0\end{aligned}
$$

The positive horizontal coordinate points outward and vertical points upward. Positive k focuses horizontally. Sector bends have curvature h. Only first-order transverse and momentum terms are retained.

#### Particle and beam matrix share transport

$$
z_{j+1}=M_jz_j+d_j\delta,\qquad\Sigma_{j+1}=M_j\Sigma_jM_j^{\mathsf T}
$$

The finite particle group and its covariance use the same thick-element map before losses. The homogeneous transverse matrix preserves the symplectic form. After clipping, surviving-particle moments are computed separately.

#### Calibrate position before averaging turns

$$
r_x=\frac{V_R-V_L}{V_R+V_L},\qquad x=K_xr_x+x_{\rm off}
$$

Independent voltage and wire-position standards determine the gains, pedestal, position scale and offset. Every electrode and position is checked against its own supported range. A low unresolved sum gives no position.

#### A sampled tune has aliases

$$
x_n=c+a\cos(2\pi qn)+b\sin(2\pi qn),\qquad q\leftrightarrow1-q
$$

The constant c separates the average orbit from undamped motion. Actual turn indices are retained. Once-per-turn real data cannot select an integer tune; every-other-turn data creates more aliases.

#### Measure the response and observe again

$$
D_i=\frac{\partial\bar x_i}{\partial\delta},\qquad R_{ij}=\frac{\partial\bar x_i}{\partial\theta_j},\qquad\Delta\bar x\simeq R\Delta\theta
$$

Independent momentum preparations measure dispersion. Positive and negative measured steering steps determine response columns, with a separate mixed-step check. A rank-supported solve is executed and tested with a fresh exposure. Shared calibration errors remain correlated.

### Common difficulties

#### A narrow spot is not a whole-ring condition

**Typical misconception** The smallest beam spot means the whole ring is stable.

**Better mental model** Position and angle evolve together. A crossing spot may expand downstream; stability concerns repeated transport around the complete lattice.

#### More turns do not erase a common zero

**Typical misconception** Averaging many turns removes all position error.

**Better mental model** Independent pickup noise can average down. The same calibration zero affects every turn and remains in the orbit and command uncertainty.

### Run the experiment

1. 01

   #### Predict the missing family

   Compare Stable FODO and Missing defocusing family. Inspect the quadrupole view and the physical loss marker.

   **What to observe:** Both coordinates stay bounded in the stable lattice. Removing one family lets an excursion reach the physical pipe. The raw observations already taken remain available.

2. 02

   #### Separate an orbit from oscillation

   Compare the baseline turn trace with Momentum scan. Select the independent negative and positive momentum platforms in the raw table.

   **What to observe:** A fitted constant estimates the orbit without adding damping. The slope uses measured momentum readbacks and a withheld preparation, not a hidden model dispersion.

3. 03

   #### Use a measured adjustment

   Choose Measure and correct, then compare before and newly observed residuals with the measured response map. Drag the first corrector control and acquire a fresh scan.

   **What to observe:** The controller uses the scanned response and actual actuator limits. A new exposure decides whether the proposed correction improved the observed orbit. Reset starts another experiment and keeps saved history.

4. 04

   #### Keep an unresolved observation

   Compare Every other turn and Weak pickup signal. Export JSON or CSV, import it and select the same record.

   **What to observe:** Missing turns preserve their indices and extra aliases. Weak signals retain codes while unsupported position or response stays unreported. Rebuilding does not replace the archived signals.

Physical conclusion

A four-cell fixed-momentum linear paraxial ring transports a finite particle ensemble and second moments through hard-edge elements. Reference momentum is 100 MeV/c. Each scan platform independently reloads the source distribution after an ideal 20ms stabilized-current qualification; coherent motion has no damping. Pickups use a calibrated near-axis linear electrode response for a fixed bunch shape, with per-sample support; low-velocity proton electromagnetic frequency response is not solved. Uncertainty is conditional on the prescribed response and excludes omitted electromagnetic systematics. Envelope is a model prediction and electrodes measure the centroid. Turn sampling retains tune aliases. Uncertainty support is a finite-noise first-order approximation. Longitudinal RF, nonlinear multipoles, space charge and collective effects are omitted. Particle markers are enlarged and replay slowed. Radiation and beamlines belong to a separate instrument.

## 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 **How defocusing magnets help a beam stay in a ring**](https://sandboxphysics.com/stories/instruments/synchrotron-storage-ring/)

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)