Editing Cyclotron (section)

==== 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" />