Editing Cyclotron (section)

=== Approaches to relativistic cyclotrons ===

{| class="wikitable floatright" style="text-align: center"
|+ Characteristic properties of cyclotrons and other circular accelerators<ref>{{cite web
| title = Cyclotrons – II & FFA
| series = CERN Accelerator School – Introductory Course
| author = Mike Seidel
| publication-date = 2019-09-19
| publication-place = High Tatras
| website = [[CERN]]
| url = https://indico.cern.ch/event/808940/contributions/3553715/attachments/1909807/3157187/CAS_Cyclotrons_II.pdf
| page = 36
}}</ref>
! rowspan=2 |
! rowspan=2 scope=col      | Relativistic
! colspan=2 scope=colgroup | Accelerating field
! colspan=2 scope=colgroup | Bending magnetic<br>field strength
! rowspan=2 scope=col      | Orbit<br>radius<br>variation
|-
! scope=col | Origin
! scope=col | Frequency<br>vs time{{efn|name=op-mode|Only accelerators with time-independent frequency and bending field strength can operate in continuous mode, i.e. output a bunch of particles in each cycle of the accelerating field. If any of these quantities sweeps during the acceleration, the operation mode must be pulsed, i.e. the machine will output a bunch of particles only at the end of each sweep.}}
! scope=col | vs time{{efn|name=op-mode}}
! scope=col | vs radius
|-
| colspan=7 style="text-align: left" | {{small|Cyclotrons}}
|-
! scope=row | Classical cyclotron
| No
| [[Electrostatic field|Electrostatic]]
| Constant
| Constant
| Constant
| Large
|-
! scope=row | Isochronous<br>cyclotron
| Yes
| Electrostatic
| Constant
| Constant
| Increasing
| Large
|-
! scope=row | [[Synchrocyclotron]]
| Yes
| Electrostatic
| Decreasing
| Constant
| Constant{{efn|Moderate variation of the field strength with radius does not matter in synchrocyclotrons, because the frequency variation compensates for it automatically.{{Citation needed|date=July 2022}}}}
| Large
|-
| colspan=7 style="text-align: left" | {{small|Other circular accelerators}}
|-
! scope=row | [[Fixed-field alternating gradient accelerator|FFA]]
| Yes
| Electrostatic
| DD{{efn|name=DD|Design-dependent}}
| Constant
| DD{{efn|name=DD}}
| Small
|-
! scope=row | [[Synchrotron]]
| Yes
| Electrostatic
| Increasing,<br>finite [[limit at infinity|limit]]
| Increasing
| N/A{{efn|name=NA|Not applicable, because the particle orbit radius is constant.}}
| None
|-
! scope=row | [[Betatron]]
| Yes
| [[Faraday's law of induction|Induction]]
| Increasing,<br>finite limit
| Increasing
| N/A{{efn|name=NA}}
| None
|}

==== Synchrocyclotron ====
{{main|Synchrocyclotron}}
Since <math>\gamma</math> increases as the particle reaches relativistic velocities, acceleration of relativistic particles requires modification of the cyclotron to ensure the particle crosses the gap at the same point in each RF cycle. If the frequency of the accelerating electric field is varied while the magnetic field is held constant, this leads to the ''synchrocyclotron''.{{r|wilson}} 

In this type of cyclotron, the accelerating frequency is varied as a function of particle orbit radius such that: 

<math display="block">f(r) = \frac{1}{2\pi \sqrt{\left(\frac{m_0}{q B}\right)^2 + \left(\frac{r}{c}\right)^2}}</math>

The decrease in accelerating frequency is tuned to match the increase in gamma for a constant magnetic field.{{r|wilson}}

==== Isochronous cyclotron ====
[[File:Lorentz factor.svg|thumb|250px|In isochronous cyclotrons, the magnetic field strength {{mvar|B}} as a function of the radius {{mvar|r}} has the same shape as the Lorentz factor {{mvar|γ}} as a function of the speed {{mvar|v}}.]]

If instead the magnetic field is varied with radius while the frequency of the accelerating field is held constant, this leads to the ''isochronous cyclotron''.{{r|wilson}}

<math display="block">B(r) = \frac{m_0}{q \sqrt{\left(\frac{1}{2\pi f}\right)^2 - \left(\frac{r}{c}\right)^2}}</math>

Keeping the frequency constant allows isochronous cyclotrons to operate in a continuous mode, which makes them capable of producing much greater beam current than synchrocyclotrons. On the other hand, as precise matching of the orbital frequency to the accelerating field frequency is the responsibility of the magnetic field variation with radius, the variation must be precisely tuned.

==== Fixed-field alternating gradient accelerator (FFA) ====
{{main|Fixed-field alternating gradient accelerator}}

An approach which combines static magnetic fields (as in the synchrocyclotron) and alternating gradient focusing (as in a [[synchrotron]]) is the fixed-field alternating gradient accelerator (FFA).  In an isochronous cyclotron, the magnetic field is shaped by using precisely machined steel magnet poles.  This variation provides a focusing effect as the particles cross the edges of the poles.  In an FFA, separate magnets with alternating directions are used to focus the beam using the principle of [[strong focusing]].  The field of the focusing and bending magnets in an FFA is not varied over time, so the beam chamber must still be wide enough to accommodate a changing beam radius within the field of the focusing magnets as the beam accelerates.<ref>{{cite journal | author=Daniel Clery | date=4 January 2010 | title=The Next Big Beam? | journal=[[Science (journal)|Science]] | volume=327 |pages=142–143 | doi=10.1126/science.327.5962.142 | pmid=20056871 | bibcode = 2010Sci...327..142C | issue=5962 }}</ref>