Editing Cyclotron (section)

=== 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>