Cyclotron motion

In physics, cyclotron motion, also known as gyromotion, refers to the circular motion exhibited by charged particles in a uniform magnetic field.

The circular trajectory of a particle in cyclotron motion is characterized by an angular frequency referred to as the cyclotron frequency or gyrofrequency and a radius referred to as the cyclotron radius, gyroradius, or Larmor radius. For a particle with charge ${\displaystyle q}$ and mass ${\displaystyle m}$ initially moving with speed ${\displaystyle v_{\perp }}$ perpendicular to the direction of a uniform magnetic field ${\displaystyle B}$, the cyclotron radius is: $${\displaystyle r_{\rm {c}}={\frac {mv_{\perp }}{|q|B}}}$$ and the cyclotron frequency is: $${\displaystyle \omega _{\rm {c}}={\frac {qB}{m}}.}$$ An external oscillating field matching the cyclotron frequency, ${\displaystyle \omega =\omega _{c},}$ will accelerate the particles, a phenomenon known as cyclotron resonance. This resonance is the basis for many scientific and engineering uses of cyclotron motion.

In quantum mechanical systems, the energies of cyclotron orbits are quantized into discrete Landau levels, which contribute to Landau diamagnetism and lead to oscillatory electronic phenomena like the De Haas–Van Alphen and Shubnikov–de Haas effects. They are also responsible for the exact quantization of Hall resistance in the integer quantum Hall effect.