Inphysics,cyclotron motion, also known asgyromotion, refers to thecircular motion exhibited bycharged particles in a uniformmagnetic field.

The circulartrajectory of a particle in cyclotron motion is characterized by an angular frequency referred to as thecyclotron frequency orgyrofrequency and a radius referred to as thecyclotron radius,gyroradius, orLarmor 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:$${\displaystyle r_{\mathrm{c}}=\frac{m{v}_{\perp }}{|q|B}}$$ and the cyclotron frequency is:$${\displaystyle {\omega }_{\mathrm{c}}=\frac{|q|B}{m}.}$$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.

Inquantum mechanical systems, the energies of cyclotron orbits are quantized into discreteLandau levels, which contribute toLandau diamagnetism and lead to oscillatory electronic phenomena like theDe Haas–Van Alphen andShubnikov–de Haas effects. They are also responsible for the exact quantization of Hall resistance in theinteger quantum Hall effect.

If a particle with electric charge${\displaystyle q}$ and mass${\displaystyle m}$ is moving with velocity${\displaystyle \overrightarrow{v}}$ in a uniform magnetic field${\displaystyle \overrightarrow{B}}$, then it will experience aLorentz force given by$${\displaystyle \overrightarrow{F}=q(\overrightarrow{v}\times \overrightarrow{B}).}$$The direction of the force is given by thecross product of the velocity and magnetic field. Thus, the Lorentz force will always act perpendicular to the direction of motion, causing the particle togyrate, or move in a circle. The radius of this circle,${\displaystyle r_{\mathrm{c}}}$, can be determined for non-relativistic particles by equating the magnitude of the Lorentz force to thecentripetal force as$${\displaystyle \frac{m{v}_{\perp }^2}{r_{\mathrm{c}}}=|q|v_{\perp }B.}$$Rearranging, the cyclotron radius can be expressed as$${\displaystyle r_{\mathrm{c}}=\frac{m{v}_{\perp }}{|q|B}.}$$Thus, the cyclotron radius isdirectly proportional to the particle mass and perpendicular velocity, while it is inversely proportional to the particle electric charge and the magnetic field strength. The time it takes the particle to complete one revolution, called theperiod, can be calculated to be$${\displaystyle T_{\mathrm{c}}=\frac{2\pi r_{\mathrm{c}}}{v_{\perp }}.}$$The period is thereciprocal of the cyclotron frequency:$${\displaystyle f_{\mathrm{c}}=\frac{1}{T_{\mathrm{c}}}=\frac{|q|B}{2\pi m}}$$or^[1]: 20$${\displaystyle {\omega }_{\mathrm{c}}=\frac{|q|B}{m}.}$$The cyclotron frequency is independent of the radius and velocity and therefore independent of the particle's kinetic energy; in the non-relativistic limit all particles with the same charge-to-mass ratio rotate around magnetic field lines with the same frequency.

The cyclotron frequency is also useful in non-uniform magnetic fields, in which (assuming slow variation of magnitude of the magnetic field) the movement is approximatelyhelical. That is, in the direction parallel to the magnetic field, the motion is uniform, whereas in the plane perpendicular to the magnetic field the movement is circular. The sum of these two motions gives a trajectory in the shape of a helix.^{[2]: 14–16}

An oscillating field matching the cyclotron frequency of particles creates a cyclotron resonance. For ions in a uniform magnetic field in a vacuum chamber, an oscillating electric field at the cyclotron resonance frequency creates a particle accelerator called acyclotron.^[3]: 13An oscillating radiofrequency field matching the cyclotron frequency is used to heat plasma.^[1]: 381

The above is forSI units. In some cases, the cyclotron frequency is given inGaussian units.^[4] In Gaussian units, the Lorentz force differs by a factor of 1/c, the speed of light, which leads to:

${\displaystyle {\omega }_{\mathrm{c}}=\frac{v}{r}=\frac{qB}{mc}}$.

For materials with little or no magnetism (i.e.${\displaystyle \mu \approx 1}$)${\displaystyle H\approx B}$, so we can use the easily measuredmagnetic field intensityH instead ofB:^[5]

${\displaystyle {\omega }_{\mathrm{c}}=\frac{qH}{mc}}$.

Note that converting this expression to SI units introduces a factor of thevacuum permeability.

For some materials, the motion of electrons follows loops that depend on the applied magnetic field, but not exactly the same way. For these materials, we define a cyclotron effective mass,${\displaystyle m^{\ast }}$ so that:

${\displaystyle {\omega }_{\mathrm{c}}=\frac{qB}{m^{\ast }}}$.

For relativistic particles the classical equation needs to be interpreted in terms of particle momentum${\displaystyle p=\gamma mv}$:$${\displaystyle r_{\mathrm{c}}=\frac{p_{\perp }}{|q|B}=\frac{\gamma m{v}_{\perp }}{|q|B}}$$where${\displaystyle \gamma }$ is theLorentz factor. This equation is correct also in the non-relativistic case.

For calculations inaccelerator andastroparticle physics, the formula for the cyclotron radius can be rearranged to give$${\displaystyle r_{\mathrm{c}}=3.3\mathrm{m}\times \frac{(\gamma m{c}^2/\mathrm{G}\mathrm{e}\mathrm{V})\cdot (v_{\perp }/c)}{(|q|/e)\cdot (B/\mathrm{T})},}$$wherem denotesmetres,c is the speed of light,GeV is the unit ofGiga-electronVolts,${\displaystyle e}$ is theelementary charge, andT is the unit oftesla.

Inquantum mechanics, the energies of cyclotron orbits of charged particles in a uniformmagnetic field are quantized to discrete values, known asLandau levels after the Soviet physicistLev Landau. These levels aredegenerate, with the number of electrons per level directly proportional to the strength of the applied magnetic field.^[6]

Landau quantization contributes towardsmagnetic susceptibility of metals, known asLandau diamagnetism. Under strong magnetic fields, Landau quantization leads to oscillations in electronic properties of materials as a function of the applied magnetic field known as theDe Haas–Van Alphen andShubnikov–de Haas effects.

Landau quantization is a key ingredient in explanation of theinteger quantum Hall effect.^[7]

• Ion cyclotron resonance
• Electron cyclotron resonance
• Beam rigidity
• Cyclotron
• Magnetosphere particle motion
• Gyrokinetics

• Calculate Cyclotron frequency with Wolfram Alpha

• Plasma phenomena
• Accelerator physics
• Kinematic properties
• Radii
• Condensed matter physics
• Electric and magnetic fields in matter
• Scientific techniques

• CS1 German-language sources (de)
• Articles with short description
• Short description is different from Wikidata