Editing Cyclotron (section)

=== 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>