Editing Cyclotron (section)

=== 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}}