Editing Cyclotron (section)

=== 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}}