---
title: "How a cyclotron keeps a proton on the accelerating beat"
description: "A magnetic field returns the proton to the gap. Relativistic turn times change when the next electrical push must arrive."
canonical_url: "https://sandboxphysics.com/stories/instruments/cyclotron-synchrocyclotron/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/instruments/cyclotron-synchrocyclotron/"
last_modified: "2026-11-12"
---

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

# How a cyclotron keeps a proton on the accelerating beat

A cyclotron sends a proton through an accelerating gap again and again. A magnetic field brings it back; an electric field gives it another push. Those pushes accumulate only when the voltage points the useful way at each crossing. As the proton gains momentum, relativity changes its return time and the original RF clock begins to miss the beat.

Sandbox Physics · 2026-11-12

## A useful push depends on the arrival time

Two hollow D-shaped electrodes face each other across a narrow gap. Inside either electrode, the electric field is approximately screened and the particle mainly bends. Across the gap, the voltage changes its kinetic energy. The voltage reverses between successive passages so that a proton travelling back the other way can receive another accelerating push.

The place reached in the electrical cycle when the particle arrives is its phase. Arrival near a useful voltage peak gives a strong push. Arrival after the field reverses can remove energy. A finite crossing also takes time: the voltage continues changing while the proton is in the gap, so the whole transit contributes to the work.

Sources: [1]

## A bigger orbit eventually takes longer

At low speeds, doubling the speed approximately doubles the bending radius in a uniform field. Distance and speed grow together, leaving the time per turn almost unchanged. The same pair of electrodes can therefore accelerate the proton repeatedly.

At higher energies, momentum keeps increasing as speed approaches the speed of light. Momentum still determines the bending radius. The path grows faster than the speed, and each turn takes longer. Small delays accumulate until the particle arrives outside the accelerating half-cycle.

$$
\omega_{\mathrm{rev}}=\frac{qB}{\gamma m}
$$

In a uniform field B, a particle with charge q and rest mass m revolves more slowly as its relativistic factor increases.

Sources: [2]

## Change the clock or strengthen the outer field

A synchrocyclotron gradually lowers the RF frequency. The voltage waits longer for each return, according to a programme designed around a reference particle. A group of protons gains energy during that sweep. Their differing starting conditions still affect how closely they follow it.

An isochronous cyclotron instead raises the average field at larger radii, bending energetic particles faster to maintain nearly equal turn times. Real machines also vary the field around the circumference to keep the beam vertically focused. A prescribed radial curve illustrates synchronism in the central plane; transverse stability needs the full magnet design.

Sources: [2]

## The last orbit must pass a physical exit

The outer turns crowd together. An extraction electrode must fit between them, with a thin septum separating the circulating beam from a deflecting channel. A proton entering on the useful side can leave towards an external beamline. One entering slightly differently may strike the septum. Turn separation and beam tails therefore affect extraction losses.

A collector belongs at the end of that path. Its signal comes from particles that reach its incoming face. Particles still circulating, or absorbed by a wall, contribute no collector event. Keeping the channel fixed lets different compensation programmes reveal their consequences through actual arrivals and losses.

Sources: [3]

## Independent standards connect signal with energy

Consider an ideal collector that fully stops each arriving proton and gives a proportional electronic response. The digitizer returns a code. Independently known particle energies determine the code-to-energy relation; another known energy, excluded from the fit, checks it. Unknown collector codes can then be converted into a distribution.

An uncertain energy standard shifts many converted events together. More particles reduce sampling fluctuations, but that shared reference error remains. Separately tagged incoming particles establish detection efficiency. In a physical instrument, the stopping material and its response must also be measured. This connects an accelerated orbit, the extracted population and the final energy readout.

Sources: [4]

## [Compare phase slip and extraction](https://sandboxphysics.com/labs/charged-particle/?instrument=relativistic-cyclotron)

Compare fixed RF with a frozen reference sweep in the relativistic cyclotron mode, then inspect collector events.

Finite-gap midplane dynamics feed a fixed ideal channel and an independently calibrated fully stopping linear response.

## Sources and further reading

1. [CERN, Accelerating: radiofrequency cavities](https://home.web.cern.ch/science/engineering/accelerating-radiofrequency-cavities/)

   Reads the public explanation of timed RF electric acceleration; no LHC operating parameters are adopted.

2. [M. Seidel, Cyclotrons and Fixed Field Alternating Gradient Accelerators, arXiv:2105.04477v3](https://arxiv.org/abs/2105.04477)

   Adopts printed pages 1–4 for frequency, rigidity and isochronicity, and 6–8 for phase slip and prescribed frequency/field compensation. Visually reads page 7, Fig. 3 and Eqs. 18–19. A given midplane profile is not a complete focusing magnet.

3. [M. Seidel, Injection and Extraction in Cyclotrons, CERN-2018-008-SP, 151–162](https://e-publishing.cern.ch/index.php/CYRSP/article/download/611/663/3742)

   Reads printed 151–153 and 157–159 on turn separation, deflection and septum extraction; visually inspects Fig. 7, page 158. PSI geometry, loss fractions and performance are not assigned to the teaching model.

4. [JCGM 100, Evaluation of measurement data, section 5](https://www.iso.org/sites/JCGM/GUM/JCGM100/C045315e-html/C045315e_FILES/MAIN_C045315e/05_e.html)

   Uses sensitivity coefficients in 5.1.3 and correlated-input propagation in 5.2 to explain energy-reference errors shared by multiple observations.

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

1 more group

### Other articles that use the same experiments

Each experiment from this article that another series also uses, followed by the articles there.

1. [Lab **Charged Particle**](https://sandboxphysics.com/labs/charged-particle/)
2. [Article **Lenard lets cathode rays out of the tube**](https://sandboxphysics.com/stories/particle-evidence/lenard-cathode-ray-window/)
3. [Article **Thomson finds a common particle in cathode rays**](https://sandboxphysics.com/stories/particle-evidence/thomson-electron-charge-mass/)
4. [Article **Wilson turns ionisation into a visible track**](https://sandboxphysics.com/stories/particle-evidence/wilson-cloud-chamber/)
5. [Article **The lead plate that revealed a positron**](https://sandboxphysics.com/stories/particle-evidence/anderson-positron-lead-plate/)
6. [Article **Glaser makes a particle trail start the boiling**](https://sandboxphysics.com/stories/particle-evidence/glaser-bubble-chamber/)
7. [Article **Finding the antiproton among thousands of pions**](https://sandboxphysics.com/stories/particle-evidence/chamberlain-antiproton/)
8. [Article **How a cyclotron gives the same ion another push**](https://sandboxphysics.com/stories/nuclear-particle/cyclotron-resonant-acceleration/)