---
title: "How defocusing magnets help a beam stay in a ring"
description: "Alternating magnets act on particles whose positions and directions have already changed. Position pickups show how small steering adjustments move the orbit."
canonical_url: "https://sandboxphysics.com/stories/instruments/synchrotron-storage-ring/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/instruments/synchrotron-storage-ring/"
last_modified: "2026-11-12"
---

[How physics instruments measure](https://sandboxphysics.com/stories/instruments/)

# How defocusing magnets help a beam stay in a ring

Alternating magnets can confine a beam because each magnet acts on the position and direction left by the previous one. Focusing and defocusing grow stronger with distance from the axis. They do not simply cancel. With suitable strengths and spacing, the sequence keeps particles oscillating inside the pipe as they travel around the ring.

Sandbox Physics · 2026-11-12

## The next magnet meets a different particle position

A bending magnet guides charged particles around a curve. Focusing magnets control their departures from the centre line. A quadrupole has four magnetic poles around the pipe. It can pull a horizontally displaced particle inward, yet push a vertical displacement outward. Rotate the poles through ninety degrees and those roles exchange.

Imagine particles moving parallel to the axis, slightly to one side. The first magnet turns them inward. During the following drift they move closer to the axis. A second magnet defocuses them, but now they are closer to its centre and receive a smaller outward turn. The combined action can leave an inward tendency. The distance between the magnets helps determine the result.

In the other direction, the first magnet defocuses. The particles reach the next magnet farther from the axis, where the focusing action is stronger. Following both position and angle explains why the same sequence can confine motion in both directions. The intervening flight changes the strength of the second turn.

A focusing magnet, an open section, a defocusing magnet and another open section form a FODO cell. Bending magnets can occupy the open sections. Repeated cells make a ring in which particles oscillate across a reference path. Suitable settings keep these excursions bounded. An unsuitable sequence lets them grow until the particles reach the pipe wall.

The particles in a bunch occupy different positions and travel at slightly different angles. Their width shrinks and expands around the ring. Stable focusing keeps that envelope within the aperture. A narrow spot can be a crossing point of many trajectories, followed by renewed expansion downstream.

Sources: [1] [6]

## Repeated passages reveal an oscillation

Insulated metal electrodes inside the pipe pick up a signal as a bunch passes. The nearer electrode usually receives the larger signal. Taking the right-minus-left difference and dividing by their sum reduces sensitivity to changes in bunch intensity. A second opposing pair measures the vertical position.

Those signals need a position scale. A thin wire carrying a high-frequency current can stand in for a beam on a test bench. Moving it through independently known positions measures how electrode response changes with displacement. Electronic gains and pedestals need their own checks. A very weak bunch or a displacement outside the calibrated region can leave the instrument without a reliable position reading.

At one pickup, a list of positions from successive turns reveals transverse oscillation. The number of oscillations per revolution is called the tune. Changing the focusing changes this rhythm. An error that repeatedly nudges particles at nearly the same point in their oscillation can build a much larger excursion.

One observation per turn also hides information, much like a wheel under a flashing light. Different rhythms can leave identical samples, and complete extra oscillations between observations disappear. More positions around the ring, or a small known change in focusing followed by another observation, can help distinguish the possibilities. The extra observations add spatial clues to the turn sequence.

Sources: [2] [5] [3] [4]

## Probe a small change before steering the ring

Particles with different momenta bend by different amounts in the same field. Their average paths can therefore differ even with unchanged focusing. Dispersion describes how that average position changes with momentum. Comparing independently measured momenta shows which locations around the ring are most sensitive to that difference.

A displaced magnet can move the average orbit around much of the ring. A small steering magnet changes the particle direction; the effect travels downstream and returns through the same lattice. Several pickups respond together. Improving the nearest reading alone may worsen positions elsewhere.

An operator can first move one corrector slightly in each direction and record every pickup. Repeating the experiment with the other correctors builds a table of measured responses. Combinations of small adjustments can then reduce several orbit deviations together. Nearly identical steering effects, or effects buried in noise, call for more independent information.

After the adjustment, the pickups observe another passage. Those new signals show whether the average orbit improved. A position pickup measures the bunch centre; a separate profile instrument is needed to measure its width. Alternating focusing confines the motion, and small measured probes help an operator choose the next useful adjustment.

Sources: [1] [3] [2]

## [Follow a bunch through alternating quadrupoles](https://sandboxphysics.com/labs/synchrotron-storage-ring/)

Compare a stable and unbalanced lattice, measure steering response and observe the corrected orbit.

A linear paraxial proton ring transports particles and second moments; calibrated electrodes reconstruct position and sampled tune aliases.

## Sources and further reading

1. [Wolfgang Hillert · Transverse Linear Beam Dynamics, CERN Accelerator School (2021)](https://arxiv.org/abs/2107.02614)

   PDF pp.21–27,35–36,43–45,50–52,58–59: linear transverse equations, thick quadrupoles/sector bends, periodic FODO, closed orbit and dispersion; no nonlinear or facility parameters adopted.

2. [Peter Forck · Beam Instrumentation and Diagnostics, CERN Accelerator School (2020)](https://arxiv.org/abs/2009.10411)

   PDF pp.16–18, Eqs.24–28: opposing electrode difference over sum, near-axis sensitivity and the two-dimensional finite-button image-current model and its relativistic-beam TEM boundary.

3. [Volker Ziemann · Imperfections and Corrections, CERN Accelerator School (2020)](https://arxiv.org/abs/2006.11016)

   PDF pp.9–10,14–17: measured orbit response, unrecoverable directions, periodic orbit constraints and tune aliases in turn-by-turn sampling.

4. [Alexahin, Gianfelice-Wendt & Marsh · Tune Evaluation From Phased BPM Turn-By-Turn Data (2012)](https://arxiv.org/abs/1202.1507)

   PDF pp.1–2: turn-by-turn frequency analysis and combining pickups; assumed optical phases remain a separate prior.

5. [D. Wang et al. · Wire Setup Calibration of Beam Position Monitors, PAC1995](https://accelconf.web.cern.ch/p95/ARTICLES/TPB/TPB04.PDF)

   PDF p.1 RF-excited wire fixture and p.3 polarity check; method only, no facility frequency, sensitivity or precision adopted.

6. [CERN · How an accelerator works](https://home.cern/science/accelerators/how-accelerator-works/)

   Main explanatory sections on RF cavities, vacuum pipe and magnet roles; conceptual context.

[How a vernier reads between millimetres](https://sandboxphysics.com/stories/instruments/length-metrology/) [Air helps hold up the object on a balance](https://sandboxphysics.com/stories/instruments/precision-balance/) [How Cavendish Measured the Attraction Between Spheres](https://sandboxphysics.com/stories/instruments/cavendish-balance/) [Which pressure does a liquid column measure?](https://sandboxphysics.com/stories/instruments/manometer/) [The cup absorbs heat too](https://sandboxphysics.com/stories/instruments/calorimeter/) [Rotational rheometers need both torque and speed](https://sandboxphysics.com/stories/instruments/rotational-rheometer/) [Why the air beyond a pipe changes its resonance](https://sandboxphysics.com/stories/instruments/resonance-tube/) [How a seismometer reads the ground through motion inside its case](https://sandboxphysics.com/stories/instruments/seismometer/) [Why does a meter change the circuit?](https://sandboxphysics.com/stories/instruments/electrical-meters/) [How a null detector reveals a resistance](https://sandboxphysics.com/stories/instruments/resistance-bridges/) [Why a capacitor can behave like an inductor](https://sandboxphysics.com/stories/instruments/lcr-impedance/) [Where does the slow wave on an oscilloscope come from?](https://sandboxphysics.com/stories/instruments/digital-oscilloscope/) [How a lock-in amplifier finds a signal buried in noise](https://sandboxphysics.com/stories/instruments/lock-in-amplifier/) [Where a small peak in a spectrum can come from](https://sandboxphysics.com/stories/instruments/spectrum-analyzer/) [How a network analyzer separates the cable from the device](https://sandboxphysics.com/stories/instruments/vector-network-analyzer/) [When a slower quartz crystal can measure added mass](https://sandboxphysics.com/stories/instruments/quartz-microbalance/) [How an echelle spectrograph gives overlapping colors a second direction](https://sandboxphysics.com/stories/instruments/echelle-spectrograph/) [How total brightness reveals many infrared colors](https://sandboxphysics.com/stories/instruments/ftir-spectrometer/) [Measuring a Solution’s Concentration with Light](https://sandboxphysics.com/stories/instruments/uv-vis-spectrophotometer/) [How an ellipsometer reads thin-film thickness from reflected light](https://sandboxphysics.com/stories/instruments/spectroscopic-ellipsometer/) [How a polarimeter reveals mixed polarization](https://sandboxphysics.com/stories/instruments/mueller-polarimeter/) [How individual photons reveal a fluorescence lifetime](https://sandboxphysics.com/stories/instruments/tcspc-fluorescence-lifetime/) [Brillouin Microscopy Listens to Sound with Light](https://sandboxphysics.com/stories/instruments/brillouin-microscope/) [How a white sphere compares a lamp’s total output](https://sandboxphysics.com/stories/instruments/integrating-sphere-radiometer/) [How powder rings reveal atomic spacing](https://sandboxphysics.com/stories/instruments/powder-xray-diffractometer/) [How Small-Angle X-ray Scattering Reads a Particle Halo](https://sandboxphysics.com/stories/instruments/small-angle-xray-scattering/) [Why X-ray fluorescence depends on sample thickness](https://sandboxphysics.com/stories/instruments/xray-fluorescence-spectrometer/) [Why an AFM makes a bump look wider](https://sandboxphysics.com/stories/instruments/atomic-force-microscope/) [How a small indentation separates hardness from elasticity](https://sandboxphysics.com/stories/instruments/nanoindenter/) [How Differential Scanning Calorimetry Reads a Melting Peak](https://sandboxphysics.com/stories/instruments/differential-scanning-calorimeter/) [How thermogravimetry separates mass loss from the gas around it](https://sandboxphysics.com/stories/instruments/thermogravimetric-analyzer/) [How a diamond anvil cell reads pressure in ruby light](https://sandboxphysics.com/stories/instruments/diamond-anvil-cell/) [How diamond fluorescence reveals a magnetic-field direction](https://sandboxphysics.com/stories/instruments/nv-center-magnetometer/) [Reading a magnetic field in the rhythm of atoms](https://sandboxphysics.com/stories/instruments/optically-pumped-magnetometer/) [Keeping a laser in time with atoms](https://sandboxphysics.com/stories/instruments/optical-lattice-clock/) [Why a Kibble balance needs two experiments](https://sandboxphysics.com/stories/instruments/kibble-balance/) [How a dilution refrigerator keeps carrying heat away](https://sandboxphysics.com/stories/instruments/dilution-refrigerator/) [Why a bigger vacuum pump may barely help the sample](https://sandboxphysics.com/stories/instruments/ultra-high-vacuum-system/) [How microwaves read a superconducting qubit](https://sandboxphysics.com/stories/instruments/superconducting-qubit-readout/) [How current leaves a voltage across a thin sheet](https://sandboxphysics.com/stories/instruments/four-probe-hall/) [How an ion flight becomes a mass spectrum](https://sandboxphysics.com/stories/instruments/time-of-flight-mass-spectrometer/) [How four straight rods select ion mass](https://sandboxphysics.com/stories/instruments/quadrupole-mass-filter/) [How a rapidly reversing field can hold ions](https://sandboxphysics.com/stories/instruments/paul-ion-trap-mass-spectrometer/) [How a cyclotron keeps a proton on the accelerating beat](https://sandboxphysics.com/stories/instruments/cyclotron-synchrocyclotron/) [Why the drift tubes in an RF linac get longer](https://sandboxphysics.com/stories/instruments/radio-frequency-linac/) [The directions hidden inside a small beam spot](https://sandboxphysics.com/stories/instruments/beam-diagnostics/) [Why a collider cannot keep every collision](https://sandboxphysics.com/stories/instruments/collider-trigger-daq/) [Getting synchrotron light onto a sample](https://sandboxphysics.com/stories/instruments/synchrotron-beamline/) [How a free-electron laser gathers electrons into a brighter beam](https://sandboxphysics.com/stories/instruments/xray-free-electron-laser/) [Following energy exchange with neutron flight times](https://sandboxphysics.com/stories/instruments/pulsed-neutron-time-of-flight/)

## Keep reading

How physics instruments measure · Reading path

### How physics instruments measure

1. [Article **How a vernier reads between millimetres**](https://sandboxphysics.com/stories/instruments/length-metrology/)
2. [Article **Air helps hold up the object on a balance**](https://sandboxphysics.com/stories/instruments/precision-balance/)
3. [Article **How Cavendish Measured the Attraction Between Spheres**](https://sandboxphysics.com/stories/instruments/cavendish-balance/)
4. [Article **Which pressure does a liquid column measure?**](https://sandboxphysics.com/stories/instruments/manometer/)
5. [Article **The cup absorbs heat too**](https://sandboxphysics.com/stories/instruments/calorimeter/)
6. [Article **Rotational rheometers need both torque and speed**](https://sandboxphysics.com/stories/instruments/rotational-rheometer/)
7. [Article **Why the air beyond a pipe changes its resonance**](https://sandboxphysics.com/stories/instruments/resonance-tube/)
8. [Article **How a seismometer reads the ground through motion inside its case**](https://sandboxphysics.com/stories/instruments/seismometer/)
9. [Article **Why does a meter change the circuit?**](https://sandboxphysics.com/stories/instruments/electrical-meters/)
10. [Article **How a null detector reveals a resistance**](https://sandboxphysics.com/stories/instruments/resistance-bridges/)
11. [Article **Why a capacitor can behave like an inductor**](https://sandboxphysics.com/stories/instruments/lcr-impedance/)
12. [Article **Where does the slow wave on an oscilloscope come from?**](https://sandboxphysics.com/stories/instruments/digital-oscilloscope/)
13. [Article **How a lock-in amplifier finds a signal buried in noise**](https://sandboxphysics.com/stories/instruments/lock-in-amplifier/)
14. [Article **Where a small peak in a spectrum can come from**](https://sandboxphysics.com/stories/instruments/spectrum-analyzer/)
15. [Article **How a network analyzer separates the cable from the device**](https://sandboxphysics.com/stories/instruments/vector-network-analyzer/)
16. [Article **When a slower quartz crystal can measure added mass**](https://sandboxphysics.com/stories/instruments/quartz-microbalance/)
17. [Article **How an echelle spectrograph gives overlapping colors a second direction**](https://sandboxphysics.com/stories/instruments/echelle-spectrograph/)
18. [Article **How total brightness reveals many infrared colors**](https://sandboxphysics.com/stories/instruments/ftir-spectrometer/)
19. [Article **Measuring a Solution’s Concentration with Light**](https://sandboxphysics.com/stories/instruments/uv-vis-spectrophotometer/)
20. [Article **How an ellipsometer reads thin-film thickness from reflected light**](https://sandboxphysics.com/stories/instruments/spectroscopic-ellipsometer/)
21. [Article **How a polarimeter reveals mixed polarization**](https://sandboxphysics.com/stories/instruments/mueller-polarimeter/)
22. [Article **How individual photons reveal a fluorescence lifetime**](https://sandboxphysics.com/stories/instruments/tcspc-fluorescence-lifetime/)
23. [Article **Brillouin Microscopy Listens to Sound with Light**](https://sandboxphysics.com/stories/instruments/brillouin-microscope/)
24. [Article **How a white sphere compares a lamp’s total output**](https://sandboxphysics.com/stories/instruments/integrating-sphere-radiometer/)
25. [Article **How powder rings reveal atomic spacing**](https://sandboxphysics.com/stories/instruments/powder-xray-diffractometer/)
26. [Article **How Small-Angle X-ray Scattering Reads a Particle Halo**](https://sandboxphysics.com/stories/instruments/small-angle-xray-scattering/)
27. [Article **Why X-ray fluorescence depends on sample thickness**](https://sandboxphysics.com/stories/instruments/xray-fluorescence-spectrometer/)
28. [Article **Why an AFM makes a bump look wider**](https://sandboxphysics.com/stories/instruments/atomic-force-microscope/)
29. [Article **How a small indentation separates hardness from elasticity**](https://sandboxphysics.com/stories/instruments/nanoindenter/)
30. [Article **How Differential Scanning Calorimetry Reads a Melting Peak**](https://sandboxphysics.com/stories/instruments/differential-scanning-calorimeter/)
31. [Article **How thermogravimetry separates mass loss from the gas around it**](https://sandboxphysics.com/stories/instruments/thermogravimetric-analyzer/)
32. [Article **How a diamond anvil cell reads pressure in ruby light**](https://sandboxphysics.com/stories/instruments/diamond-anvil-cell/)
33. [Article **How diamond fluorescence reveals a magnetic-field direction**](https://sandboxphysics.com/stories/instruments/nv-center-magnetometer/)
34. [Article **Reading a magnetic field in the rhythm of atoms**](https://sandboxphysics.com/stories/instruments/optically-pumped-magnetometer/)
35. [Article **Keeping a laser in time with atoms**](https://sandboxphysics.com/stories/instruments/optical-lattice-clock/)
36. [Article **Why a Kibble balance needs two experiments**](https://sandboxphysics.com/stories/instruments/kibble-balance/)
37. [Article **How a dilution refrigerator keeps carrying heat away**](https://sandboxphysics.com/stories/instruments/dilution-refrigerator/)
38. [Article **Why a bigger vacuum pump may barely help the sample**](https://sandboxphysics.com/stories/instruments/ultra-high-vacuum-system/)
39. [Article **How microwaves read a superconducting qubit**](https://sandboxphysics.com/stories/instruments/superconducting-qubit-readout/)
40. [Article **How current leaves a voltage across a thin sheet**](https://sandboxphysics.com/stories/instruments/four-probe-hall/)
41. [Article **How an ion flight becomes a mass spectrum**](https://sandboxphysics.com/stories/instruments/time-of-flight-mass-spectrometer/)
42. [Article **How four straight rods select ion mass**](https://sandboxphysics.com/stories/instruments/quadrupole-mass-filter/)
43. [Article **How a rapidly reversing field can hold ions**](https://sandboxphysics.com/stories/instruments/paul-ion-trap-mass-spectrometer/)
44. [Article **How a cyclotron keeps a proton on the accelerating beat**](https://sandboxphysics.com/stories/instruments/cyclotron-synchrocyclotron/)
45. [Article **Why the drift tubes in an RF linac get longer**](https://sandboxphysics.com/stories/instruments/radio-frequency-linac/)
46. [Article **The directions hidden inside a small beam spot**](https://sandboxphysics.com/stories/instruments/beam-diagnostics/)
47. [You are here **How defocusing magnets help a beam stay in a ring**](https://sandboxphysics.com/stories/instruments/synchrotron-storage-ring/)
48. [Article **Why a collider cannot keep every collision**](https://sandboxphysics.com/stories/instruments/collider-trigger-daq/)
49. [Article **Getting synchrotron light onto a sample**](https://sandboxphysics.com/stories/instruments/synchrotron-beamline/)
50. [Article **How a free-electron laser gathers electrons into a brighter beam**](https://sandboxphysics.com/stories/instruments/xray-free-electron-laser/)
51. [Article **Following energy exchange with neutron flight times**](https://sandboxphysics.com/stories/instruments/pulsed-neutron-time-of-flight/)