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{{Short description|Type of particle accelerator}}
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[[File:Cyclotron with glowing beam.jpg|300px|thumb|right|Lawrence's {{Convert|60|in|cm|0|adj=on}} cyclotron, {{Circa|1939}}, showing the beam of accelerated [[ion]]s (likely [[proton]]s or [[deuteron]]s) exiting the machine and ionizing the surrounding air causing a blue glow]]

A '''cyclotron''' is a type of [[particle accelerator]] invented by [[Ernest Lawrence]] in 1929–1930 at the [[University of California, Berkeley]],<ref>{{Cite web|url=http://www2.lbl.gov/Science-Articles/Archive/early-years.html|title=Ernest Lawrence's Cyclotron|website=www2.lbl.gov|access-date=2018-04-06}}</ref><ref>{{Cite web| url=https://www.nobelprize.org/nobel_prizes/physics/laureates/1939/lawrence-bio.html|title=Ernest Lawrence – Biographical| website=nobelprize.org|access-date=2018-04-06}}</ref> and patented in 1932.<ref name="Patent1948384">{{US patent|1948384}} Lawrence, Ernest O. ''Method and apparatus for the acceleration of ions'', filed: January 26, 1932, granted: February 20, 1934</ref><ref name="Lawrence">{{cite journal
 |last1 = Lawrence
 |first1 = Earnest O.
 |last2 = Livingston
 |first2 = M. Stanley
 |title = The Production of High Speed Light Ions Without the Use of High Voltages
 |journal = Physical Review
 |volume = 40
 |issue = 1
 |pages = 19–35
 |publisher = American Physical Society
 |date = April 1, 1932
 |doi = 10.1103/PhysRev.40.19
 | bibcode = 1932PhRv...40...19L
 |doi-access = free
 }}</ref> A cyclotron accelerates [[charged particle]]s outwards from the center of a flat cylindrical vacuum chamber along a spiral path.<ref name="Nave">{{cite web
 |last = Nave
 |first = C. R.
 |title = Cyclotron
 |publisher = Dept. of Physics and Astronomy, Georgia State University
 |date = 2012
 |url = http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/cyclot.html
 |access-date = October 26, 2014}}</ref><ref name=" Close">{{cite book
 |last1 = Close
 |first1 = F. E.
 |last2 = Close
 |first2 = Frank
 |last3 = Marten
 |first3 = Michael
 |title = The Particle Odyssey: A Journey to the Heart of Matter
 |publisher = Oxford University Press
 |date = 2004
 |pages = 84–87
 |url = https://books.google.com/books?id=PX87qqj5B2UC&pg=PA86
 |isbn = 978-0-19-860943-8
 |display-authors=etal|bibcode = 2002pojh.book.....C
 }}</ref> The particles are held to a spiral trajectory by a static [[magnetic field]] and accelerated by a rapidly varying [[electric field]]. Lawrence was awarded the 1939 [[Nobel Prize in Physics]] for this invention.<ref name=" Close"/><ref>{{Cite web|url=https://www.nobelprize.org/nobel_prizes/physics/laureates/1939/lawrence-facts.html|title=Ernest Lawrence – Facts|website=nobelprize.org|access-date=2018-04-06}}</ref>

The cyclotron was the first "cyclical" accelerator.<ref name="Serway" /> The primary accelerators before the development of the cyclotron were [[electrostatic accelerator]]s, such as the [[Cockcroft–Walton generator]] and the [[Van de Graaff generator]].  In these accelerators, particles would cross an accelerating [[electric field]] only once.  Thus, the energy gained by the particles was limited by the maximum [[electrical potential]] that could be achieved across the accelerating region. This potential was in turn limited by [[electrical breakdown|electrostatic breakdown]] to a few million volts. In a cyclotron, by contrast, the particles encounter the accelerating region many times by following a spiral path, so the output energy can be many times the energy gained in a single accelerating step.<ref name="Lawrence"/>

Cyclotrons were the most powerful particle accelerator technology until the 1950s, when they were surpassed by the [[synchrotron]].<ref name="Bryant">{{cite conference |url=https://cds.cern.ch/record/261062/files/p1_2.pdf |title=A Brief History and Review of Accelerators |last1=Bryant|first1=P.J. |date=September 1992 |publisher=[[CERN]] |book-title=Proceedings, Vol. 2  |page=12 |location=  Jyvaskyla, Finland|conference=CAS-CERN Accelerator School: 5th general accelerator physics course}}</ref> Nonetheless, they are still widely used to produce particle beams for [[nuclear medicine]] and basic research. As of 2020, close to 1,500 cyclotrons were in use worldwide for the production of [[radionuclide]]s for nuclear medicine.{{r|itnonline}} In addition, cyclotrons can be used for [[particle therapy]], where particle beams are directly applied to patients.<ref name="itnonline">{{cite web| date=March 10, 2020|title=MEDraysintell identifies close to 1,500 medical cyclotrons worldwide| url=https://www.itnonline.com/content/medraysintell-releases-unique-cyclotron-directory#:~:text=March%2010%2C%202020%20%E2%80%94%20MEDraysintell%20released,to%201%2C500%20medical%20cyclotrons%20worldwide.|work=ITN Imaging Technology News}}</ref>

== History ==

[[File:4-inch-cyclotron.jpg|thumb|upright=1.3|Lawrence's original {{convert|4.5|in|cm|adj=on}} cyclotron]]
[[File:Berkeley_60-inch_cyclotron.jpg|thumb|upright=1.3|Lawrence's {{convert|60|in|cm|adj=on}} cyclotron at [[Lawrence Radiation Laboratory]], [[University of California]], Berkeley, California, constructed in 1939. The magnet is on the left, with the vacuum chamber between its pole pieces, and the beamline which analyzed the particles is on the right.]]

In 1927, while a student at Kiel, German physicist [[Max Steenbeck]] was the first to formulate the concept of the cyclotron, but he was discouraged from pursuing the idea further.<ref>[http://ark.cdlib.org/ark:/13030/ft5s200764/ Lawrence and His Laboratory] - ''II — A Million Volts or Bust'' 81–82 in Heilbron, J. L., and Robert W. Seidel ''Lawrence and His Laboratory: A History of the Lawrence Berkeley Laboratory', Volume I.'' (Berkeley:  University of California Press, 2000)</ref> In late 1928 and early 1929, Hungarian physicist [[Leo Szilárd]] filed patent applications in Germany for the [[linear accelerator]], cyclotron, and [[betatron]].<ref name="Dannen-2001">{{cite journal |last1=Dannen |first1=Gene |title=Szilard's Inventions Patently Halted |journal=Physics Today |date=March 2001 |volume=54 |issue=3 |pages=102–104 |doi=10.1063/1.1366083 |bibcode=2001PhT....54c.102D |url=https://physicstoday.scitation.org/doi/10.1063/1.1366083?journalCode=pto&ver=pdfcov |access-date=31 January 2022}}</ref> In these applications, Szilárd became the first person to discuss the resonance condition (what is now called the cyclotron frequency) for a circular accelerating apparatus. However, neither Steenbeck's ideas nor Szilard's patent applications were ever published and therefore did not contribute to the development of the cyclotron.<ref>{{cite journal |last1=Telegdi |first1=Valentine L. |title=Szilard as Inventor: Accelerators and More |journal=Physics Today |date=October 2000 |volume=53 |issue=10 |pages=25–28 |doi=10.1063/1.1325189 |bibcode=2000PhT....53j..25T |url=https://physicstoday.scitation.org/doi/10.1063/1.1366083?journalCode=pto&ver=pdfcov |access-date=31 January 2022|doi-access=free }}</ref> Several months later, in the early summer of 1929, Ernest Lawrence independently conceived the cyclotron concept after reading a paper by [[Rolf Widerøe]] describing a drift tube accelerator.<ref>{{cite journal
 |last=Widerøe |first=R.
 |year=1928
 |title=Ueber Ein Neues Prinzip Zur Herstellung Hoher Spannungen
 |language=de
 |journal=[[Archiv für Elektronik und Übertragungstechnik]]
 |volume=21 |issue=4 |pages=387–406
 |doi=10.1007/BF01656341
|s2cid=109942448
 }}</ref><ref>{{cite web
 |date=8 December 2008
 |title=Breaking Through: A Century of Physics at Berkeley 1886–1968 2. The Cyclotron.
 |url=http://bancroft.berkeley.edu/Exhibits/physics/bigscience02.html
 |publisher=[[Bancroft Library]], [[UC Berkeley]]
 |archive-url=https://web.archive.org/web/20120527183442/http://bancroft.berkeley.edu/Exhibits/physics/bigscience02.html
 |archive-date=2012-05-27
 |url-status=dead
}}</ref><ref>{{cite conference |url=https://accelconf.web.cern.ch/c75/papers/j-01.pdf |title=The History of the Cyclotron |last1=Livingston |first1=M. Stanley |date=19–22 August 1975 |book-title=Proceedings of the 7th International Conference on Cyclotrons and their Applications |pages=635–638 |location=Zurich, Switzerland}}</ref> He published a paper in ''[[Science (journal)|Science]]'' in 1930 (the first published description of the cyclotron concept), after a student of his built a crude model in April of that year.<ref>{{cite journal |title= On the Production of High Speed Protons| author1=E. O. Lawrence |author2=N. E. Edlefsen| journal= Science | volume=72 |issue=1867 | pages= 376–377 |year=1930 |doi=10.1126/science.72.1867.372| pmid=17808988 | s2cid=56202243 }}</ref>
He patented the device in 1932.<ref name="Lawrence"/><ref name="Physics">{{cite book
 |last1=Alonso
 |first1=M.
 |last2=Finn
 |first2=E.
 |year=1992
 |title=Physics
 |publisher=[[Addison Wesley]]
 |isbn=978-0-201-56518-8
 |url-access=registration
 |url=https://archive.org/details/physics00alon
 }}</ref>

To construct the first such device, Lawrence used large electromagnets recycled from obsolete [[arc converter]]s provided by the [[Federal Telegraph Company]].<ref>{{cite journal |first=F. J. |last=Mann |title=Federal Telephone and Radio Corporation, A Historical Review: 1909–1946 |journal=Electrical Communication |volume=23 |issue=4 |date=December 1946 |pages=397–398}}</ref>
He was assisted by a graduate student, [[M. Stanley Livingston]]. Their first working cyclotron became operational in January 1931.  This machine had a diameter of {{convert|4.5|in|cm}}, and accelerated protons to an energy up to 80&nbsp;[[electronvolt|keV]].<ref name="aipexhibit">{{cite web
 |url=https://history.aip.org/exhibits/lawrence/first.htm
 |title=The First Cyclotrons
 |publisher=[[American Institute of Physics]]
 |access-date=7 June 2022
}}</ref>

At the Radiation Laboratory on the campus of the [[University of California, Berkeley]] (now the [[Lawrence Berkeley National Laboratory]]), Lawrence and his collaborators went on to construct a series of cyclotrons which were the most powerful accelerators in the world at the time; a {{convert|27|in|cm|abbr=on|0}} 4.8&nbsp;MeV machine (1932), a {{convert|37|in|cm|abbr=on|0}} 8&nbsp;MeV machine (1937), and a {{convert|60|in|cm|abbr=on|0}} 16&nbsp;MeV machine (1939). Lawrence received the 1939 [[Nobel Prize in Physics]] for the invention and development of the cyclotron and for results obtained with it.<ref>{{cite web |title=The Nobel Prize in Physics 1939 |publisher=Nobel Foundation |url=http://nobelprize.org/nobel_prizes/physics/laureates/1939/index.html |access-date=9 October 2008 |archive-url=https://web.archive.org/web/20081024052532/http://nobelprize.org/nobel_prizes/physics/laureates/1939/index.html |archive-date=24 October 2008 |url-status=live }}</ref>

The first European cyclotron was constructed in the [[Soviet Union]] in the physics department of the [[V.G. Khlopin Radium Institute]] in [[Leningrad]], headed by {{Interlanguage link multi|Vitaly Khlopin|2=ru|3=Хлопин, Виталий Григорьевич|preserve=1}}. This Leningrad instrument was first proposed in 1932 by [[George Gamow]] and {{Interlanguage link multi|Lev Mysovskii|2=ru|3=Мысовский, Лев Владимирович|preserve=1}} and was installed and became operative by 1937.<ref>{{cite journal |last=Emelyanov |first=V. S. |year=1971 |title=Nuclear Energy in the Soviet Union |journal=[[Bulletin of the Atomic Scientists]] |url=https://books.google.com/books?id=jQsAAAAAMBAJ&pg=PA38 |volume=27 |issue=9 |pages=39 |quote=State Institute of Radium, founded in 1922, now known as V. G. Khlopin Radium Institute| doi=10.1080/00963402.1971.11455411 |bibcode=1971BuAtS..27i..38E }}</ref><ref>{{cite web |url=http://www.khlopin.ru/english/memorial.php |title=History / Memorial |author=<!--Staff writer(s); no by-line.--> |publisher=V.G. Khlopin Radium Institute |access-date=February 25, 2012|archive-url=https://web.archive.org/web/20110426033005/http://www.khlopin.ru/english/memorial.php |archive-date=2011-04-26}}</ref><ref>{{cite web |url=http://www.khlopin.ru/english/hronology.php |title=History / Chronology |author=<!--Staff writer(s); no by-line.--> |access-date=February 25, 2012|archive-url=https://web.archive.org/web/20110426033000/http://www.khlopin.ru/english/hronology.php|archive-date=2011-04-26}}</ref>

Two cyclotrons were built in [[Nazi Germany]].<ref>{{Cite book |last=Ball |first=Philip |url=https://www.worldcat.org/oclc/855705703 |title=Serving the Reich: the Struggle for the Soul of Physics Under Hitler |date=2013 |publisher=The Bodley Head |isbn=978-1-84792-248-9 |location=London |pages=190 |oclc=855705703 |author-link=Philip Ball}}</ref> The first was constructed in 1937, in [[Otto Hahn]]'s laboratory at the [[Kaiser Wilhelm Society|Kaiser Wilhelm Institute]] in Berlin, and was also used by [[Rudolf Fleischmann]]. It was the first cyclotron with a [[Greinacher multiplier]] to increase the voltage to 2.8&nbsp;MV and 3&nbsp;mA current. A second cyclotron was built in [[Heidelberg]] under the supervision of [[Walther Bothe]] and [[Wolfgang Gentner]], with support from the [[Heereswaffenamt]], and became operative in 1943.<ref>{{cite web | title = Wolfgang Gentner 1906–1980 | language = de| url= http://www.physik.uni-frankfurt.de/paf/paf181.html | author=Ulrich Schmidt-Rohr 
|archive-url=https://web.archive.org/web/20070706153823/http://www.physik.uni-frankfurt.de/paf/paf181.html | archive-date=6 July 2007 }}</ref>

By the late 1930s it had become clear that there was a practical limit on the beam energy that could be achieved with the traditional cyclotron design, due to the effects of [[special relativity]].<ref>{{cite journal |last1=Bethe |first1=H. A. |last2=Rose |first2=M. E. |title=The Maximum Energy Obtainable from the Cyclotron |journal=Physical Review |date=15 December 1937 |volume=52 |issue=12 |pages=1254–1255 |doi=10.1103/PhysRev.52.1254.2|bibcode=1937PhRv...52.1254B }}</ref>  As particles reach relativistic speeds, their effective mass increases, which causes the resonant frequency for a given magnetic field to change.  To address this issue and reach higher beam energies using cyclotrons, two primary approaches were taken, [[synchrocyclotron]]s (which hold the magnetic field constant, but decrease the accelerating frequency) and isochronous cyclotrons (which hold the accelerating frequency constant, but alter the magnetic field).<ref name="craddock">{{cite conference|last=Craddock|first=M.K.|title=Eighty Years of Cyclotrons|book-title=Proceedings of Cyclotrons 2010|place=Lanzhou, China|date=September 10, 2010|url=http://accelconf.web.cern.ch/Cyclotrons2010/papers/mom1cio02.pdf | access-date=January 24, 2022}}</ref>

Lawrence's team built one of the first synchrocyclotrons in 1946. This {{convert|184|in|m|abbr=on}} machine eventually achieved a maximum beam energy of 350&nbsp;MeV for protons. However, synchrocyclotrons suffer from low beam intensities (<&nbsp;1&nbsp;μA), and must be operated in a "pulsed" mode, further decreasing the available total beam. As such, they were quickly overtaken in popularity by isochronous cyclotrons.{{r|craddock}}

The first isochronous cyclotron (other than classified prototypes) was built by F. Heyn and K.T. Khoe in Delft, the Netherlands, in 1956.<ref name="heyn">{{cite journal |last1=Heyn |first1=F. |last2=Khoe |first2=Kong Tat |title=Operation of a Radial Sector Fixed-Frequency Proton Cyclotron |journal=Review of Scientific Instruments |date=1958 |volume=29 |issue=7 |page=662|doi=10.1063/1.1716293 |bibcode=1958RScI...29..662H }}</ref> Early isochronous cyclotrons were limited to energies of ~50&nbsp;MeV per nucleon, but as manufacturing and design techniques gradually improved, the construction of "spiral-sector" cyclotrons allowed the acceleration and control of more powerful beams. Later developments included the use of more compact and power-efficient [[superconducting magnets]] and the separation of the magnets into discrete sectors, as opposed to a single large magnet.{{r|craddock}}

== Principle of operation ==

[[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 resonance|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 particle 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>

==Classifications==
[[File:1937-French-cyclotron.jpg|thumb|A French cyclotron, produced in [[Zürich]], Switzerland in 1937. The vacuum chamber containing the dees ''(at left)'' has been removed from the magnet ''(red, at right)''.]]

===Cyclotron types===
There are a number of basic types of cyclotron:<ref name="Chao">{{cite book
 | last1  = Chao
 | first1 = Alex 
 | title  = Handbook of Accelerator Physics and Engineering
 | publisher = World Scientific 
 | date   = 1999
 | pages  = 13–15
 | url    = https://books.google.com/books?id=Z3J4SjftF1YC&q=cyclotron&pg=PA13
 | doi    = 
 | id     = 
 | isbn   = 9789810235000
 }}</ref>

{{glossary}}
{{term|Classical cyclotron}}
{{defn|The earliest and simplest cyclotron. Classical cyclotrons have uniform magnetic fields and a constant accelerating frequency. They are limited to [[nonrelativistic]] particle velocities (the output energy small compared to the particle's [[rest energy]]), and have no active focusing to keep the beam aligned in the plane of acceleration.<ref name="seidel">{{cite report |last=Seidel|first=Mike |date= 2013|title=Cyclotrons for high-intensity beams |url=https://cds.cern.ch/record/1513944/files/CERN-2013-001-p17.pdf |publisher= [[CERN]] |access-date= June 12, 2022}}</ref>}}
{{term |Synchrocyclotron}}
{{defn |The synchrocyclotron extended the energy of the cyclotron into the relativistic regime by decreasing the frequency of the accelerating field as the orbit of the particles increased to keep it synchronized with the particle revolution frequency.  Because this requires pulsed operation, the integrated total beam current was low compared to the classical cyclotron. In terms of beam energy, these were the most powerful accelerators during the 1950s, before the development of the [[synchrotron]].{{r|craddock}}{{r|Bryant}}}}
{{term |Isochronous cyclotron (isocyclotron)}}
{{defn |These cyclotrons extend output energy into the relativistic regime by altering the magnetic field to compensate for the change in cyclotron frequency as the particles reached relativistic speed. They use specially shaped magnet pole pieces that are wider near the outer diameter of the cyclotron to create a nonuniform magnetic field stronger in peripheral regions.  Most modern cyclotrons are of this type. The pole pieces can also be shaped to cause the beam to keep the particles focused in the acceleration plane as they orbit.  This is known as "sector focusing" or "azimuthally-varying field focusing", and uses the principle of [[alternating-gradient focusing]].{{r|craddock}}}}
{{term | Separated sector cyclotron}}
{{defn| Separated sector cyclotrons are machines in which the magnet is in separate sections, separated by gaps without field{{r|craddock}}.}}
{{term | Superconducting cyclotron}}
{{defn | "Superconducting" in the cyclotron context refers to the type of magnet used to bend the particle orbits into a spiral.  Superconducting magnets can produce substantially higher fields in the same area than normal conducting magnets, allowing for more compact, powerful machines. The first superconducting cyclotron was the K500 at the [[Michigan State University]], which came online in 1981.<ref name="austin">{{cite book |last1=Austin |first1=Sam M. |title=Up from nothing : the Michigan State University Cyclotron Laboratory |date=2015 |publisher=Michigan State University |location=[East Lansing, Michigan] |isbn=978-0-99672-521-7}}</ref>}}
{{glossary end}}

===Beam types===

The particles for cyclotron beams are produced in [[ion source]]s of various types.

{{glossary}}
{{term|Proton beams}}
{{defn|The simplest type of cyclotron beam, proton beams are typically created by ionizing hydrogen gas.<ref name="clark">{{cite conference |url= https://accelconf.web.cern.ch/c81/papers/di-01.pdf|title= Ion Sources for Cyclotrons|last1=Clark |first1=David |date= September 1981 |pages= 231–240 |location= Caen, France |conference= 9th International Conference on Cyclotrons and their Applications }}</ref>}}
{{term|H− beams|H<sup>−</sup> beams}}
{{defn|Accelerating negative hydrogen ions simplifies extracting the beam from the machine.  At the radius corresponding to the desired beam energy, a metal foil is used to strip the electrons from the H<sup>−</sup> ions, transforming them into positively charged H<sup>+</sup> ions.  The change in polarity causes the beam to be deflected in the opposite direction by the magnetic field, allowing the beam to be transported out of the machine.<ref>{{cite journal |last1=Muramatsu |first1=M. |last2=Kitagawa |first2=A. |title=A review of ion sources for medical accelerators (invited) |journal=Review of Scientific Instruments |date=February 2012 |volume=83 |issue=2 |pages=02B909 |doi=10.1063/1.3671744 |pmid=22380341 |bibcode=2012RScI...83bB909M |doi-access=free}}</ref>}}
{{term|Heavy ion beams}}
{{defn|Beams of particles heavier than hydrogen are referred to as heavy ion beams, and can range from deuterium nuclei (one proton and one neutron) up to uranium nuclei.  The increase in energy required to accelerate heavier particles is balanced by stripping more electrons from the atom to increase the electric charge of the particles, thus increasing acceleration efficiency.{{r|clark}}}}
{{glossary end}}

===Target types===

To make use of the cyclotron beam, it must be directed to a target.<ref>{{cite conference |title= The Operation of Cyclotrons Used for Radiopharmaceutical Production |last1= Grey-Morgan |first1= T. |last2= Hubbard |first2= RE|date= November 1992 |publisher= World Scientific|pages=115–118 |location= Vancouver, Canada|conference= 13th International Conference on Cyclotrons and their Applications}}</ref>

{{glossary}}
{{term|Internal targets}}
{{defn|The simplest way to strike a target with a cyclotron beam is to insert it directly into the path of the beam in the cyclotron.  Internal targets have the disadvantage that they must be compact enough to fit within the cyclotron beam chamber, making them impractical for many medical and research uses.<ref name=Gelbart>{{cite conference |url=https://accelconf.web.cern.ch/c98/papers/a-21.pdf |title= Solid Targetry Systems: A Brief History|last1= Gelbart |first1=W.Z.|last2=Stevenson|first2=N. R. |date=June 1998  |pages= 90–93 |location= Caen, France|conference= 15th International Conference on Cyclotrons and their Applications }}</ref>}}
{{term|External targets}}
{{defn|While extracting a beam from a cyclotron to impinge on an external target is more complicated than using an internal target, it allows for greater control of the placement and focus of the beam, and much more flexibility in the types of targets to which the beam can be directed.{{r|Gelbart}}}}
{{glossary end}}

==Usage==
[[File:Cyclotron - University of Washington.jpg|thumb|upright=1.3|A modern cyclotron used for [[radiation therapy]]. The magnet is painted yellow.]]
=== Basic research ===

For several decades, cyclotrons were the best source of high-energy beams for [[nuclear physics]] experiments.  With the advent of strong focusing synchrotrons, cyclotrons were supplanted as the accelerators capable of producing the highest energies.<ref name="wilson" />{{r|Bryant}} However, due to their compactness, and therefore lower expense compared to high energy synchrotrons, cyclotrons are still used to create beams for research where the primary consideration is not achieving the maximum possible energy.{{r|austin}} Cyclotron based nuclear physics experiments are used to measure basic properties of isotopes (particularly short lived radioactive isotopes) including half life, mass, interaction cross sections, and decay schemes.<ref>{{cite web |title=About Rare-Isotope Research {{!}} TRIUMF : Canada's particle accelerator centre |url=https://www.triumf.ca/research-program/research-topics/rare-isotope-beam-science/rare-isotope#nuc |website=www.triumf.ca |access-date=27 January 2022}}</ref>

=== Medical uses ===
==== Radioisotope production ====

Cyclotron beams can be used to bombard other atoms to produce short-lived isotopes with a variety of medical uses, including [[medical imaging]] and [[radiotherapy]].<ref>{{cite web |title=Cyclotrons – What are They and Where Can you Find Them |url=https://www.iaea.org/newscenter/news/cyclotrons-what-are-they-and-where-can-you-find-them |website=www.iaea.org |publisher=International Atomic Energy Agency |access-date=27 January 2022 |language=en |date=27 January 2021}}</ref>  [[Positron]] and [[gamma radiation|gamma]] emitting isotopes, such as [[fluorine-18]], [[isotopes of carbon#carbon-11|carbon-11]], and [[technetium-99m]]<ref>{{cite news | last=Hume |first=M.  |date=21 February 2012 |title=In a breakthrough, Canadian researchers develop a new way to produce medical isotopes | url=https://www.theglobeandmail.com/news/british-columbia/in-a-breakthrough-canadian-researchers-develop-a-new-way-to-produce-medical-isotopes/article4092466/ | newspaper=[[The Globe and Mail]] |location=Vancouver}}</ref> are used for [[PET imaging|PET]] and [[Single-photon emission computed tomography|SPECT]] imaging. While cyclotron produced radioisotopes are widely used for diagnostic purposes, therapeutic uses are still largely in development.  Proposed isotopes include [[astatine]]-211, [[palladium]]-103, [[rhenium]]-186, and [[bromine]]-77, among others.<ref>{{cite book |title=Cyclotron produced radionuclides : principles and practice. |date=2008 |publisher=International Atomic Energy Agency |location=Vienna |isbn=978-92-0-100208-2}}</ref>

==== Beam therapy ====

The first suggestion that energetic protons could be an effective treatment method was made by [[Robert R. Wilson]] in a paper published in 1946<ref>{{Cite journal |last=Wilson |first=Robert R. |date=1946 |title=Radiological Use of Fast Protons |url=http://dx.doi.org/10.1148/47.5.487 |journal=Radiology |volume=47 |issue=5 |pages=487–491 |doi=10.1148/47.5.487 |pmid=20274616 |issn=0033-8419}}</ref> while he was involved in the design of the [[Harvard Cyclotron Laboratory]].<ref>{{Cite book |last=Wilson |first=Richard |url=https://books.google.com/books?id=4cnvAAAAMAAJ&q=A+Brief+History+of+the+Harvard+University+Cyclotrons |title=A Brief History of the Harvard University Cyclotrons |date=2004 |publisher=Harvard University Press |isbn=978-0-674-01460-2 |pages=9 |language=en}}</ref>

Beams from cyclotrons can be used in [[particle therapy]] to treat [[cancer]]. Ion beams from cyclotrons can be used, as in [[proton therapy]], to penetrate the body and kill tumors by [[radiation poisoning|radiation damage]], while minimizing damage to healthy tissue along their path. 

As of 2020, there were approximately 80 facilities worldwide for radiotherapy using beams of protons and heavy ions, consisting of a mixture of cyclotrons and synchrotrons. Cyclotrons are primarily used for proton beams, while synchrotrons are used to produce heavier ions.<ref>{{cite book |title=Regulatory control of the safety of ion radiotherapy facilities : a guide for best practice. |date=2020 |publisher=International Atomic Energy Agency |location=Vienna |isbn=9789201631190 |url=https://www-pub.iaea.org/MTCD/Publications/PDF/TE-1891web.pdf |access-date=27 January 2022}}</ref>

==Advantages and limitations==

[[File:M. Stanley Livingston (L) and Ernest O. Lawrence in front of 27-inch cyclotron at the old Radiation Laboratory at the... - NARA - 558593.tif|thumb|M. Stanley Livingston and [[Ernest O. Lawrence]] ''(right)'' in front of Lawrence's {{convert|27|in|cm|order=flip|abbr=on}} cyclotron at the Lawrence Radiation Laboratory. The curving metal frame is the magnet's core, the large cylindrical boxes contain the coils of wire that generate the magnetic field. The vacuum chamber containing the "dee" electrodes is in the center between the magnet's poles.]]

The most obvious advantage of a cyclotron over a [[linear accelerator]] is that because the same accelerating gap is used many times, it is both more space efficient and more cost efficient; particles can be brought to higher energies in less space, and with less equipment. The compactness of the cyclotron reduces other costs as well, such as foundations, radiation shielding, and the enclosing building. Cyclotrons have a single electrical driver, which saves both equipment and power costs. Furthermore, cyclotrons are able to produce a continuous beam of particles at the target, so the average power passed from a particle beam into a target is relatively high compared to the pulsed beam of a synchrotron.<ref name="peach">{{cite journal |last1=Peach |first1=K |last2=Wilson |first2=P |last3=Jones |first3=B |title=Accelerator science in medical physics |journal=The British Journal of Radiology |date=December 2011 |volume=84 |issue=special_issue_1 |pages=S4–S10 |doi=10.1259/bjr/16022594 |pmid=22374548 |pmc=3473892 }}</ref>

However, as discussed above, a constant frequency acceleration method is only possible when the accelerated particles are approximately obeying [[Newton's laws of motion]]. If the particles become fast enough that [[Special Relativity|relativistic]] effects become important, the beam becomes out of phase with the oscillating electric field, and cannot receive any additional acceleration. The classical cyclotron (constant field and frequency) is therefore only capable of accelerating particles up to a few percent of the speed of light. Synchro-, isochronous, and other types of cyclotrons can overcome this limitation, with the tradeoff of increased complexity and cost.{{r|peach}}

An additional limitation of cyclotrons is due to [[space charge]] effects – the mutual repulsion of the particles in the beam.  As the amount of particles (beam current) in a cyclotron beam is increased, the effects of [[Coulomb's law|electrostatic repulsion]] grow stronger until they disrupt the orbits of neighboring particles.  This puts a functional limit on the beam intensity, or the ''number'' of particles which can be accelerated at one time, as distinct from their energy.<ref>{{cite journal |last1=Reiser |first1=Martin |title=Space Charge Effects and Current Limitations in Cyclotrons |journal=IEEE Transactions on Nuclear Science |date=1966 |volume=13 |issue=4 |pages=171–177 |doi=10.1109/TNS.1966.4324198 |bibcode=1966ITNS...13..171R }}</ref>

==Notable examples==
{| class="wikitable sortable plainrowheaders"
|-
! scope="col" | {{Abbr|Name|Institution and Name}}
! scope="col" | Country
! scope="col" | Date
! scope="col" | {{Abbr|Energy|Maximum energy}}
! scope="col" | {{Abbr|Beam|Beam type}}
! scope="col" | Diameter
! scope="col" | In use?
! scope="col" | Comments
! scope="col" | {{Abbr|Ref|Reference}}
|-
! scope="row" | Lawrence 4.5-inch Cyclotron
| {{flagicon|USA}}United States
| 1931
| {{convert|0.08|MeV|keV|disp=out|sortable=on}}
| Protons
| {{convert|4.5|in|m|sortable=on}}
| {{No}}
| First working cyclotron
| {{r|aipexhibit}}
|-
! scope="row" | Lawrence 184-inch Cyclotron
| {{flagicon|USA}}United States
| 1946
| 380 MeV
| [[Alpha particle]]s, [[deuterium]], protons
| {{convert|184|in|m|sortable=on}}
| {{no}}
| First [[synchrocyclotron]] and largest single-magnet cyclotron ever constructed
| {{r|craddock}}
|-
! scope="row" | [[Delft University of Technology|TU Delft]] Isochronous Cyclotron
| {{flagicon|Netherlands}}Netherlands
| 1958
| 12 MeV
| Protons
| {{convert|0.36|m|m|disp=out|sortable=on}}
| {{no}}
| First isochronous cyclotron
| {{r|heyn}}
|-
! scope="row" | [[Lawrence Berkeley National Laboratory]] 88-inch Cyclotron
| {{flagicon|USA}}United States
| 1961
| 60 MeV
| Protons, Alpha Particles, Neutrons, Heavy Ions
| {{convert|88|in|m|sortable=on}}
| {{Yes}}
| Oldest continuously operated large cyclotron in existence; Lawrence's last cyclotron
| <ref>{{cite web|url=http://cyclotron.lbl.gov/|title=88-Inch Cyclotron, the oldest continuously operated large cyclotron in existence|work=cyclotron.lbl.gov}}</ref>
|-
! scope="row" | [[Paul Scherrer Institute|PSI]] Ring Cyclotron
| {{flagicon|CH}}Switzerland
| 1974
| 590 MeV
| Protons
| {{convert|15|m|m|disp=out|sortable=on}}
| {{Yes}}
| Highest beam power of any cyclotron
| <ref>{{cite conference
 |last=Grillenberger |first=J.
 |year=2021
 |title=The High Intensity Proton Accelerator Facility
 |url=https://scipost.org/SciPostPhysProc.5.002/pdf
 |book-title=SciPost Physics Proceedings issue 5, Review of Particle Physics at PSI
 |display-authors=etal}}</ref>
|-
! scope="row" | [[TRIUMF]] 520 MeV 
| {{flagicon|Canada}}Canada
| 1976
| 520 MeV
| H<sup>−</sup>
| {{convert|56|ft|m|sortable=on}}
| {{Yes}}
| Largest normal conducting cyclotron ever constructed
| <ref>{{cite web|url=http://www.guinnessworldrecords.com/records-7000/largest-cyclotron/|title=Largest cyclotron|work=guinnessworldrecords.com}}</ref>
|-
! scope="row" | [[Michigan State University]] K500 
| {{flagicon|USA}}United States
| 1982
| 500 MeV/u
| Heavy Ion
| {{convert|52|in|m|sortable=on}}
| {{Yes}}<ref name="MSUrefurb">{{cite web |last1=Koch |first1=Geoff |title=MSU to refurbish world's first superconducting cyclotron for chip testing |url=https://msutoday.msu.edu/news/2023/msu-to-refurbish-worlds-first-superconducting-cyclotron-for-chip-testing |website=MSUToday {{!}} Michigan State University |access-date=10 January 2024 |language=en}}</ref>
| First superconducting cyclotron
| <ref name="Blosser-2004">{{cite conference|first=H.|last=Blosser|title=30 Years of Superconducting Cyclotron Technology|book-title=Cyclotrons and their applications 2004. Proceedings of the seventeenth international conference|place=Tokyo, Japan|pages=531–534|date=2004|url=https://accelconf.web.cern.ch/c04/data/CYC2004_papers/22B1.pdf|access-date=January 24, 2022}}</ref><ref name="MSUrefurb"/>
|-
! scope="row" | [[RIKEN]] Superconducting Ring Cyclotron
| {{flagicon|JP}}Japan
| 2006
| 400 MeV/u
| Heavy Ion
| {{convert|18.4|m|m|disp=out|sortable=on}}
| {{Yes}}
| K-value of 2600 is highest ever achieved
| <ref>{{cite conference
 |last=Kamigaito
 |first=O.
 |year=2010
 |title=Status of RIBF accelerators RIKEN
 |url=http://accelconf.web.cern.ch/accelconf/Cyclotrons2010/papers/tum2cio01.pdf
 |book-title=Proceedings of the 19th International Conference on Cyclotrons and their Applications
 |display-authors=etal
 |access-date=2012-06-19
 |archive-date=2012-07-10
 |archive-url=https://web.archive.org/web/20120710142202/http://accelconf.web.cern.ch/accelconf/Cyclotrons2010/papers/tum2cio01.pdf
 |url-status=dead
 }}</ref>
|}

==Superconducting cyclotron examples==
A '''superconducting cyclotron''' uses [[superconducting magnets]] to achieve high magnetic field in a small diameter and with lower power requirements. These cyclotrons require a [[cryostat]] to house the magnet and cool it to -269°C or 4.2 K. Some of these cyclotrons are being built for medical therapy.<ref name="craddock"/>{{rp|6}}  
{| class="wikitable sortable plainrowheaders"
|-
! scope="col" | {{Abbr|Name|Institution and Name}}
! scope="col" | Country
! scope="col" | Date
! scope="col" | {{Abbr|Energy|Maximum energy}}
! scope="col" | {{Abbr|Beam|Beam type}}
! scope="col" | Diameter
! scope="col" | In use?
! scope="col" | {{Abbr|Ref|Reference}}
|-
! scope="row" | [[Michigan State University]] K500 
| {{flagicon|USA}}United States
| 1982
| 500 MeV/u
| Heavy Ion
| {{convert|52|in|cm|sortable=on}}
| {{Yes}}<ref name="MSUrefurb"/>
| <ref name="Blosser-2004"/><ref name="MSUrefurb"/> 
|-
! scope="row" | [[Texas_A%26M_University|Texas A&M University]] Cyclotron Institute K500 
| {{flagicon|USA}}United States
| 1987
| 70 MeV (protons), 15 MeV/u 
| Protons, Heavy Ions
| {{convert|1.15|m|in|sortable=on}}
| {{Yes}}
| <ref>{{cite web |title=K500 Superconducting Cyclotron
|url=https://cyclotron.tamu.edu/facilities/k500/|website=Cyclotron Institute|access-date=3 October 2024 |date=30 September 2019}}</ref>
|-
! scope="row" | {{interlanguage link|Laboratori nazionali del Sud|it}} K800 
| {{flagicon|Italy}}Italy
| 1994
| 80 MeV
| Protons, Heavy ions
| {{convert|0.9|m|in|sortable=on}}
| {{yes}}
| <ref>{{cite web |title=Laboratori Nazionali del Sud |url=https://www.lns.infn.it/en/accelerators/superconducting-cyclotron.html |access-date=2 October 2024}}</ref>
|-
! scope="row" | [[University Medical Center Groningen]] AGOR 
| {{flagicon|Netherlands}}Netherlands
| 1996
| 120-190 MeV (protons), 30-90 MeV/u (Heavy Ions)
| Protons, Light ions, Heavy ions
| {{convert|3.2|m|in|sortable=on}}
| {{yes}}
| <ref>{{cite web |last1=Berens |first1=Astrid |title=AGOR, the Netherlands |url=https://www.ionbeamcenters.eu/ion-beam-facilities/agor-the-netherlands/ |website=IonBeamCenters.eu |access-date=4 October 2024 |date=22 October 2019}}</ref>
|-
! scope="row" | [[Variable Energy Cyclotron Centre]] K500 
| {{flagicon|India}}India
| 2009
| 80 MeV/u (Light Ions), 5-10 MeV/u (Heavy Ions)<ref>{{cite journal |last1=Rana |first1=T. K. |last2=Kundu |first2=Samir |last3=Manna |first3=S. |last4=Banerjee |first4=K. |last5=Ghosh |first5=T. K. |last6=Mukherjee |first6=G. |last7=Karmakar |first7=P. |last8=Sen |first8=A. |last9=Pandey |first9=R. |last10=Pant |first10=P. |last11=Roy |first11=Pratap |last12=Shil |first12=R. |last13=Nayak |first13=S. S. |last14=Rani |first14=K. |last15=Atreya |first15=K. |last16=Paul |first16=D. |last17=Santra |first17=R. |last18=Sultana |first18=A. |last19=Pal |first19=S. |last20=Basu |first20=S. |last21=Pandit |first21=Deepak |last22=Mukhopadhyay |first22=S. |last23=Bhattacharya |first23=C. |last24=Debnath |first24=J. |last25=Bhunia |first25=U. |last26=Dey |first26=M. K. |title=Characterization of the first beam from the K500 superconducting cyclotron at VECC |journal=Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment |date=1 August 2024 |volume=1065 |pages=169530 |doi=10.1016/j.nima.2024.169530 |bibcode=2024NIMPA106569530R |url=https://www.sciencedirect.com/science/article/abs/pii/S016890022400456X |access-date=4 October 2024}}</ref> 
| Protons, dueterons, Alpha particles, Heavy ions
| {{convert|3|m|in|sortable=on}}
| {{Yes}}
| <ref>{{cite web |title=VECC's superconducting success |url=https://cerncourier.com/a/veccs-superconducting-success/ |website=CERN Courier |access-date=2 October 2024 |date=30 September 2009}}</ref>
|-
! scope="row" | Ionetix ION-12SC
| {{flagicon|USA}}United States
| 2016
| 12.5 MeV 
| Proton
| {{convert|88|cm|in|sortable=on}}
| {{Yes}}
| <ref>{{Cite journal |last1=Wu |first1=Xiaoyu |last2=Alt |first2=Daniel |last3=Blosser |first3=Gabe |last4=Horner |first4=Gary |last5=Neville |first5=Zachary |last6=Paquette |first6=Jay |last7=Usher |first7=Nathan |last8=Vincent |first8=John |date=2019 |title=Recent Progress in R for Ionetix Ion-12SC Superconducting Cyclotron for Production of Medical Isotopes |url=http://jacow.org/ipac2019/doi/JACoW-IPAC2019-THPMP052.html |access-date=2024-10-03 |journal=Proceedings of the 10th Int. Particle Accelerator Conf. |volume=IPAC2019 |pages=3 pages, 0.853 MB |language=en |doi=10.18429/JACOW-IPAC2019-THPMP052}}</ref>

|}

==Related technologies==
The spiraling of electrons in a cylindrical vacuum chamber within a transverse magnetic field is also employed in the [[magnetron]], a device for producing high frequency radio waves ([[microwaves]]). In the magnetron, electrons are bent into a circular path by a magnetic field, and their motion is used to excite [[microwave cavity|resonant cavities]], producing electromagnetic radiation.<ref>{{cite web |title=Magnetron Operation |url=http://hyperphysics.phy-astr.gsu.edu/hbase/Waves/magnetron.html |website=hyperphysics.phy-astr.gsu.edu |access-date=31 January 2022}}</ref>

A [[betatron]] uses the ''change'' in the magnetic field to accelerate electrons in a circular path. While static magnetic fields cannot provide acceleration, as the force always acts perpendicularly to the direction of particle motion, changing fields can be used to induce an [[electromotive force]] in the same manner as in a [[transformer]].  The betatron was developed in 1940,<ref>{{cite web |title=Betatron |url=https://physics.illinois.edu/people/history/betatron |access-date=31 January 2022 |website=physics.illinois.edu |publisher=Grainger College of Engineering, University of Illinois, Urbana-Champaign |language=en}}</ref> although the idea had been proposed substantially earlier.<ref name="Dannen-2001" />

A [[synchrotron]] is another type of particle accelerator that uses magnets to bend particles into a circular trajectory.  Unlike in a cyclotron, the particle path in a synchrotron has a fixed radius.  Particles in a synchrotron pass accelerating stations at increasing frequency as they get faster.  To compensate for this frequency increase, both the frequency of the applied accelerating electric field and the magnetic field must be increased in tandem, leading to the "synchro" portion of the name.<ref>{{cite web |title=Synchrotron |url=https://www.britannica.com/technology/synchrotron |website=Britannica Online |access-date=31 January 2022}}</ref>

==In fiction==
The [[United States Department of War]] famously asked for dailies of the ''Superman'' comic strip to be pulled in April 1945 for having Superman bombarded with the radiation from a cyclotron.<ref>{{cite book|title= Superheroes!:Capes cowls and the creation of comic book culture|author1=[[Laurence Maslon]]|author2=Michael Kantor|page=91}}</ref>

In the 1984 film ''[[Ghostbusters]],'' a miniature cyclotron forms part of the [[proton pack]] used for catching ghosts.<ref>{{cite book |last1=Aykroyd |first1=Dan |last2=Ramis |first2=Harold |editor1-last=Shay |editor1-first=Don |title=Making Ghostbusters : the screenplay |date=1985 |publisher=New York Zoetrope |location=New York, NY |isbn=0-918432-68-5}}</ref>

==See also==
{{Portal|Physics}}
<!-- Please keep entries in alphabetical order & add a short description [[WP:SEEALSO]] -->
* [[Cyclotron radiation]] – radiation produced by non-relativistic charged particles bent by a magnetic field
* [[Fast neutron therapy]] – a type of beam therapy that may use accelerator produced beams
* [[Microtron]] – an accelerator concept similar to the cyclotron which uses a linear accelerator type accelerating structure with a constant magnetic field.
* [[Radiation reaction force]] – a braking force on beams that are bent in a magnetic field
<!-- please keep entries in alphabetical order -->

==Notes==
{{Notelist}}

==References==
{{Reflist}}

==Further reading==
*{{cite journal
 |last1=Feder |first1=T.
 |s2cid=109712952
 |year=2004
 |title=Building a Cyclotron on a Shoestring
 |journal=[[Physics Today]]
 |volume=57 |issue=11 |pages=30–31
 |doi=10.1063/1.1839371
 |bibcode = 2004PhT....57k..30F
|doi-access=free
 }}
*{{cite magazine
 |last1=Jardin |first1=X.
 |date=12 Jan 2005
 |title=The Cyclotron Comes to the 'Hood
 |url=https://www.wired.com/politics/law/news/2005/12/69726
 |magazine=[[Wired (magazine)|Wired]]
}} About a neighborhood cyclotron in [[Anchorage, Alaska]].
*{{cite web
 |last=Niell
 |first=F. M.
 |year=2005
 |title=Resonance Mapping and the Cyclotron
 |url=http://www.niell.org/cyc2.html#tpaper
 |access-date=2005-05-27
 |archive-url=https://web.archive.org/web/20090505032339/http://www.niell.org/cyc2.html#tpaper
 |archive-date=2009-05-05
 |url-status=dead
 }} An experiment done by Fred M. Niell, III his senior year of high school (1994–95) with which he won the overall grand prize in the [[ISEF]].

==External links==
{{wiktionary}}
{{Commons category|Cyclotrons}}

===Current facilities===
* [https://user88.lbl.gov/ The 88-Inch Cyclotron] at [[Lawrence Berkeley National Laboratory]]
* [https://www.psi.ch/en/media/our-research/the-proton-accelerator-forty-years-of-top-flight-research PSI Proton Accelerator] – the highest beam current cyclotron in the world.
* The [https://www.nishina.riken.jp/facility/SRC_e.html Superconducting Ring Cyclotron] at the RIKEN Nishina Center for Accelerator Based Science – the highest energy cyclotron in the world
* [http://www.physics.rutgers.edu/cyclotron/ Rutgers Cyclotron] – Students at [[Rutgers University]] built a {{convert|12|in|cm|abbr=on|order=flip}} 1&nbsp;MeV cyclotron as an undergraduate project, which is now used for a senior-level undergraduate and a graduate lab course.
* [https://www.triumf.ca/research-program/research-facilities/main-cyclotron-beam-lines TRIUMF] – the largest single-magnet cyclotron in the world.

===Historic cyclotrons===
* [https://www2.lbl.gov/Science-Articles/Archive/early-years.html Ernest Lawrence's Cyclotron] A history of cyclotron development at the Berkeley Radiation Laboratory, now [[Lawrence Berkeley National Laboratory]]
* [https://nscl.msu.edu/ National Superconducting Cyclotron Laboratory] of the [[Michigan State University]] – Home of coupled K500 and K1200 superconducting cyclotrons; the K500, the first superconducting cyclotron, and the K1200, formerly the most powerful in the world.

{{Authority control}}

[[Category:1932 introductions]]
[[Category:Accelerator physics]]
[[Category:American inventions]]
[[Category:Nuclear medicine]]
[[Category:Particle accelerators]]