Editing Cyclotron (section)

== Principle of operation ==
A cyclotron is essentially a [[linear particle accelerator]] wrapped in a circle. A uniform magnetic field perpendicular to the plane of particle motion causes the particles to orbit. During each orbit the particles are accelerated by electric fields.<ref name="Klaus-2000">{{Cite book |last=Wille |first=Klaus |title=The physics of particle accelerators: an introduction |date=2000 |publisher=Oxford University Press |isbn=978-0-19-850550-1 |location=Oxford ; New York}}</ref>{{rp|13}}
[[File:Cyclotron diagram.png|thumb|center|upright=2.8|Diagram of a cyclotron. The magnet's pole pieces are shown smaller than in reality; they must actually be at least as wide as the accelerating electrodes ("dees") to create a uniform field.]]

=== Cyclotron principle ===

[[File:Cyclotron patent.png|right|thumb|250px|Diagram of cyclotron operation from Lawrence's 1934 patent. The hollow, open-faced D-shaped [[electrode]]s (left), known as dees, are enclosed in a flat [[vacuum chamber]] which is installed in a narrow gap between the two [[magnet#Modelling magnets|poles]] of a large magnet (right).]]
[[File:Lawrence 27 inch cyclotron dees 1935.jpg|thumb|250px|Vacuum chamber of Lawrence {{convert|27|in|cm|order=flip|abbr=on}} 1932 cyclotron with cover removed, showing the dees. The 13,000&nbsp;V RF accelerating potential at about 27&nbsp;MHz is applied to the dees by the two feedlines visible at top right. The beam emerges from the dees and strikes the target in the chamber at bottom.]]
In a particle accelerator, charged particles are accelerated by applying an electric field across a gap.  The force on a particle crossing this gap is given by the [[Lorentz force|Lorentz force law]]:

<math display="block">\mathbf{F} = q   [\mathbf{E} + (\mathbf{v} \times \mathbf{B})]</math>

where {{mvar|q}} is the [[electric charge|charge]] on the particle, {{math|'''E'''}} is the [[electric field]], {{math|'''v'''}} is the particle [[velocity]], and {{math|'''B'''}} is the [[magnetic flux density]].  It is not possible to accelerate particles using only a static magnetic field, as the magnetic force always acts perpendicularly to the direction of motion, and therefore can only change the direction of the particle, not the speed.<ref name="conte">{{cite book |last1=Conte |first1=Mario |last2=MacKay |first2=William |title=An introduction to the physics of particle accelerators |date=2008 |publisher=World Scientific |location=Hackensack, N.J. |isbn=9789812779601 |pages=1 |edition=2nd}}</ref>

In practice, the magnitude of an unchanging electric field which can be applied across a gap is limited by the need to avoid [[Electrical breakdown|electrostatic breakdown]].<ref name="edwards">{{cite book |last1=Edwards |first1=D. A. |last2=Syphers |first2=M.J. |title=An introduction to the physics of high energy accelerators |date=1993 |publisher=Wiley |location=New York |isbn=9780471551638}}</ref>{{rp|21}} As such, modern particle accelerators use alternating ([[radio frequency]]) electric fields for acceleration. Since an alternating field across a gap only provides an acceleration in the forward direction for a portion of its cycle, particles in RF accelerators travel in bunches, rather than a continuous stream. In a [[linear particle accelerator]], in order for a bunch to "see" a forward voltage every time it crosses a gap, the gaps must be placed further and further apart, in order to compensate for the increasing [[speed]] of the particle.<ref name="wilson">{{cite book |last1=Wilson |first1=E. J. N. |title=An introduction to particle accelerators |date=2001 |publisher=Oxford University Press |location=Oxford |isbn=9780198508298 |pages=6–9}}</ref>

A cyclotron, by contrast, uses a magnetic field to bend the particle trajectories into a spiral, thus allowing the same gap to be used many times to accelerate a single bunch.  As the bunch spirals outward, the increasing distance between transits of the gap is exactly balanced by the increase in speed, so a bunch will reach the gap at the same point in the RF cycle every time.{{r|wilson}}

The frequency at which a particle will orbit in a perpendicular magnetic field is known as the [[Cyclotron motion|cyclotron frequency]], and depends, in the non-relativistic case, solely on the charge and mass of the particle, and the strength of the magnetic field:

<math display="block">f = \frac{qB}{2\pi m}</math>

where {{mvar|f}} is the (linear) frequency, {{mvar|q}} is the charge of the particle, {{mvar|B}} is the magnitude of the magnetic field that is perpendicular to the plane in which the particle is travelling, and {{mvar|m}} is the particle mass. The property that the frequency is independent of particle velocity is what allows a single, fixed gap to be used to accelerate a particle travelling in a spiral.{{r|wilson}}

=== Particle energy ===

Each time a particle crosses the accelerating gap in a cyclotron, it is given an accelerating force by the electric field across the gap, and the total particle energy gain can be calculated by multiplying the increase per crossing by the number of times the particle crosses the gap.<ref name="seidel" />

However, given the typically high number of revolutions, it is usually simpler to estimate the energy by combining the equation for [[frequency]] in [[circular motion]]:

<math display="block">f = \frac{v}{2 \pi r}</math>

with the cyclotron frequency equation to yield:

<math display="block">v = \frac{q B r}{m}</math>

The kinetic energy for particles with speed {{mvar|v}} is therefore given by:

<math display="block">E = \frac{1}{2}m v^2 = \frac{q^2 B^2 r^2}{2 m}</math>

where {{mvar|r}} is the radius at which the energy is to be determined. The limit on the beam energy which can be produced by a given cyclotron thus depends on the maximum radius which can be reached by the magnetic field and the accelerating structures, and on the maximum strength of the magnetic field which can be achieved.<ref name="Serway">{{cite book
 | last1  = Serway
 | first1 = Raymond A.
 | last2  = Jewett
 | first2 = John W.
 | title  = Principles of Physics: A Calculus-Based Text, Vol. 2
 | publisher = Cengage Learning 
 | edition = 5
 | date   = 2012
 | pages  = 753
 | url    = https://books.google.com/books?id=0d4KAAAAQBAJ&dq=cyclotron&pg=PA753
 | isbn   = 9781133712749
 }}</ref>

==== K-factor ====

In the nonrelativistic approximation, the maximum kinetic energy per atomic mass for a given cyclotron is given by:

<math display="block">\frac{T}{A} = \frac{(e B r_{\max})^2}{2 m_a}\left(\frac{Q}{A}\right)^2 = K \left(\frac{Q}{A}\right)^2</math>

where <math>e</math> is the elementary charge, <math>B</math> is the strength of the magnet, <math>r_{\max}</math> is the maximum radius of the beam, <math>m_a</math> is an [[atomic mass unit]], <math>Q</math> is the charge of the beam particles, and <math>A</math> is the atomic mass of the beam particles.  The value of ''K''

<math display="block">K = \frac{(e B r_{\max})^2}{2 m_a}</math>

is known as the "K-factor", and is used to characterize the maximum kinetic beam energy of protons (quoted in MeV).  It represents the theoretical maximum energy of protons (with ''Q'' and ''A'' equal to 1) accelerated in a given machine.<ref>{{cite web |last1=Barletta |first1=William |title=Cyclotrons: Old but Still New |url=https://uspas.fnal.gov/materials/12MSU/UTcyclotrons.pdf |website=U.S. Particle Accelerator School |publisher=Fermi National Accelerator Laboratory |access-date=27 January 2022}}</ref>

=== Particle trajectory ===
[[File:Spiral-fermat-1.svg|thumb|250px|The trajectory followed by a particle in the cyclotron approximated with a [[Fermat's spiral]]]]

While the trajectory followed by a particle in the cyclotron is conventionally referred to as a "spiral", it is more accurately described as a series of arcs of constant radius. The particles' speed, and therefore orbital radius, only increases at the accelerating gaps. Away from those regions, the particle will orbit (to a first approximation) at a fixed radius.<ref name="Chautard">{{cite journal |last1=Chautard |first1=F |title=Beam dynamics for cyclotrons |journal=CERN Particle Accelerator School |date=2006 |pages=209–229 |doi=10.5170/CERN-2006-012.209 |url=https://cds.cern.ch/record/1005052/files/p209.pdf |access-date=4 July 2022}}</ref>

Assuming a uniform energy gain per orbit (which is only valid in the non-relativistic case), the average orbit may be approximated by a simple spiral. If the energy gain per turn is given by {{math|&Delta;{{var|E}}}}, the particle energy after {{mvar|n}} turns will be:
<math display="block">E(n) = n \Delta E</math>
Combining this with the non-relativistic equation for the kinetic energy of a particle in a cyclotron gives:
<math display="block">r(n) = {\sqrt{2 m \Delta E} \over q B} \sqrt{n}</math>
This is the equation of a [[Fermat's spiral|Fermat spiral]].

=== Stability and focusing ===

As a particle bunch travels around a cyclotron, two effects tend to make its particles spread out.  The first is simply the particles injected from the ion source having some initial spread of positions and velocities. This spread tends to get amplified over time, making the particles move away from the bunch center.  The second is the mutual repulsion of the beam particles due to their electrostatic charges.<ref>{{cite conference |url= https://accelconf.web.cern.ch/HB2012/papers/tuo1a03.pdf|title= Space Charge Effects in Isochronous FFAGs and Cyclotrons|access-date=2022-07-19 |last1= Planche |first1= T. |last2= Rao |first2=Y-N |last3=Baartman|first3=R. |date= September 17, 2012 |publisher=CERN |book-title= Proceedings of the 52nd ICFA Advanced Beam Dynamics Workshop on High-Intensity and High-Brightness Hadron Beams |pages= 231–234|location= Beijing, China |conference= HB2012 |id=}}</ref> Keeping the particles focused for acceleration requires confining the particles to the plane of acceleration (in-plane or "vertical"{{efn|name=horz-vert|The terms "horizontal" and "vertical" do not refer to the physical orientation of the cyclotron, but are relative to the plane of acceleration. Vertical is perpendicular to the plane of acceleration, and horizontal is parallel to it.}} focusing), preventing them from moving inward or outward from their correct orbit ("horizontal"{{efn|name=horz-vert}} focusing), and keeping them synchronized with the accelerating RF field cycle (longitudinal focusing).<ref name="Chautard" />

==== Transverse stability and focusing ====

The in-plane or "vertical"{{efn|name=horz-vert}} focusing is typically achieved by varying the magnetic field around the orbit, i.e. with [[azimuth]]. A cyclotron using this focusing method is thus called an azimuthally-varying field (AVF) cyclotron.<ref name="sylee014">{{cite book
 |last=Lee |first=S.-Y.
 |year=1999
 |title=Accelerator physics
 |url=https://books.google.com/books?id=VTc8Sdld5S8C&pg=PA14
 |page=14
 |publisher=[[World Scientific]]
 |isbn=978-981-02-3709-7
}}</ref> The variation in field strength is provided by shaping the steel poles of the magnet into sectors<ref name="Chautard" /> which can have a shape reminiscent of a spiral and also have a larger area towards the outer edge of the cyclotron to improve the vertical focus of the particle beam.<ref>{{cite journal |last1=Zaremba |first1=Simon |last2=Kleeven |first2=Wiel |title=Cyclotrons: Magnetic Design and Beam Dynamics |journal=CERN Yellow Reports: School Proceedings |date=22 June 2017 |volume=1 |pages=177 |doi=10.23730/CYRSP-2017-001.177 |url=https://e-publishing.cern.ch/index.php/CYRSP/article/view/99/222 |access-date=30 March 2024}}</ref> This solution for focusing the particle beam was proposed by [[Llewellyn Thomas|L. H. Thomas]] in 1938<ref name="sylee014"/> and almost all modern cyclotrons use azimuthally-varying fields.<ref>{{cite book |editor1-last=Cherry |editor1-first=Pam |editor2-last=Duxbury |editor2-first=Angela |title=Practical radiotherapy : physics and equipment |date=2020 |publisher=John WIley & Sons |location=Newark |isbn=9781119512721 |page=178 |edition=Third}}</ref> 

The "horizontal"{{efn|name=horz-vert}} focusing happens as a natural result of cyclotron motion.  Since for identical particles travelling perpendicularly to a constant magnetic field the trajectory curvature radius is only a function of their speed, all particles with the same speed will travel in circular orbits of the same radius, and a particle with a slightly incorrect trajectory will simply travel in a circle with a slightly offset center.  Relative to a particle with a centered orbit, such a particle will appear to undergo a horizontal oscillation relative to the centered particle. This oscillation is stable for particles with a small deviation from the reference energy.<ref name="Chautard" />

==== Longitudinal stability ====

The instantaneous level of synchronization between a particle and the RF field is expressed by phase difference between the RF field and the particle. In the first harmonic mode (i.e. particles make one revolution per RF cycle) it is the difference between the instantaneous phase of the RF field and the instantaneous azimuth of the particle. Fastest acceleration is achieved when the phase difference equals 90° ([[Modular arithmetic|modulo]] 360°).{{r|Chautard|at=ch.2.1.3}} Poor synchronization, i.e. phase difference far from this value, leads to the particle being accelerated slowly or even decelerated (outside of the 0–180° range).

As the time taken by a particle to complete an orbit depends only on particle's type, magnetic field (which may vary with the radius), and [[Lorentz factor]] (see {{slink||Relativistic considerations}}), cyclotrons have no longitudinal focusing mechanism which would keep the particles synchronized to the RF field. The phase difference, that the particle had at the moment of its injection into the cyclotron, is preserved throughout the acceleration process, but errors from imperfect match between the RF field frequency and the cyclotron frequency at a given radius accumulate on top of it.{{r|Chautard|at=ch.2.1.3}} Failure of the particle to be injected with phase difference within about ±20° from the optimum may make its acceleration too slow and its stay in the cyclotron too long. As a consequence, half-way through the process the phase difference escapes the 0–180° range, the acceleration turns into deceleration, and the particle fails to reach the target energy. Grouping of the particles into correctly synchronized bunches before their injection into the cyclotron thus greatly increases the injection efficiency.{{r|Chautard|at=ch.7}}

=== Relativistic considerations ===

In the non-relativistic approximation, the cyclotron frequency does not depend upon the particle's speed or the radius of the particle's orbit. As the beam spirals outward, the rotation frequency stays constant, and the beam continues to accelerate as it travels a greater distance in the same time period. In contrast to this approximation, as particles approach the [[speed of light]], the cyclotron frequency decreases due to the change in [[Mass in special relativity|relativistic mass]]. This change is proportional to the particle's [[Lorentz factor]].{{r|conte|pages=6–9}}

The relativistic mass can be written as:

<math display="block">m = \frac{m_0}{\sqrt{1-\left(\frac{v}{c}\right)^2}} = \frac{m_0}{\sqrt{1-\beta^2}} = \gamma {m_0},</math>

where:
* <math>m_0</math> is the particle [[rest mass]],
* <math>\beta = \frac{v}{c}</math> is the relative velocity, and
* <math>\gamma=\frac{1}{\sqrt{1-\beta^2}}=\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^2}}</math> is the [[Lorentz factor]].{{r|conte|pages=6–9}}

Substituting this into the equations for cyclotron frequency and angular frequency gives:

<math display="block">\begin{align}
f & = \frac{q B}{2\pi \gamma m_0} \\[6pt]
\omega & = \frac{q B}{\gamma m_0}
\end{align}</math>

The [[gyroradius]] for a particle moving in a static magnetic field is then given by:{{r|conte|pages=6–9}}
<math display="block">r = \frac{\gamma \beta m_0 c}{q B} = \frac{\gamma m_0 v}{q B} = \frac{m_0}{q B \sqrt{v^{-2} - c^{-2}}}</math>

Expressing the speed in this equation in terms of frequency and radius
<math display="block">v = 2\pi f r</math>
yields the connection between the magnetic field strength, frequency, and radius:
<math display="block">\left(\frac{1}{2\pi f}\right)^2 = \left(\frac{m_0}{q B}\right)^2 + \left(\frac{r}{c}\right)^2</math>

=== 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>