Cyclotron motion
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In physics, cyclotron motion, also known as gyromotion, refers to the circular motion of a charged particle under the influence of a uniform magnetic field.
The cyclotron motion of a charged particle is characterized by the cyclotron frequency (also known as the gyrofrequency) and the cyclotron radius (also known as the gyroradius or Larmor radius).[1]
Parameters
[edit]The cyclotron frequency or gyrofrequency is the frequency of a charged particle moving perpendicular to the direction of a uniform magnetic field B (constant magnitude and direction).
SI units | CGS units |
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Derivation
[edit]If the charged particle is moving with velocity in a uniform magnetic field , then it will experience a Lorentz force given by The direction of the force is given by the cross product of the velocity and magnetic field. Thus, the Lorentz force will always act perpendicular to the direction of motion, causing the particle to gyrate, or move in a circle. The radius of this circle, , can be determined by equating the magnitude of the Lorentz force to the centripetal force as Rearranging, the gyroradius can be expressed as Thus, the gyroradius is directly 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 the period, can be calculated to be Since the period is the reciprocal of the frequency we have found and therefore
It is notable that the cyclotron frequency is independent of the radius and velocity and therefore independent of the particle's kinetic energy; all particles with the same charge-to-mass ratio rotate around magnetic field lines with the same frequency. This is only true in the non-relativistic limit, and underpins the principle of operation of the cyclotron.
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 approximately helical - in the direction parallel to the magnetic field, the motion is uniform, whereas in the plane perpendicular to the magnetic field the movement is, as previously circular. The sum of these two motions gives a trajectory in the shape of a helix.
Gaussian units
[edit]The above is for SI units. In some cases, the cyclotron frequency is given in Gaussian units.[2] In Gaussian units, the Lorentz force differs by a factor of 1/c, the speed of light, which leads to:
- .
For materials with little or no magnetism (i.e. ) , so we can use the easily measured magnetic field intensity H instead of B:[3]
- .
Note that converting this expression to SI units introduces a factor of the vacuum permeability.
Effective mass
[edit]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, so that:
- .
Relativistic case
[edit]For relativistic particles the classical equation needs to be interpreted in terms of particle momentum : where is the Lorentz factor. This equation is correct also in the non-relativistic case.
For calculations in accelerator and astroparticle physics, the formula for the gyroradius can be rearranged to give where m denotes metres, c is the speed of light, GeV is the unit of Giga-electronVolts, is the elementary charge, and T is the unit of tesla.
See also
[edit]- Ion cyclotron resonance
- Electron cyclotron resonance
- Beam rigidity
- Cyclotron
- Magnetosphere particle motion
- Gyrokinetics
References
[edit]- ^ Chen, Francis F. (1983). Introduction to Plasma Physics and Controlled Fusion, Vol. 1: Plasma Physics, 2nd ed. New York, NY USA: Plenum Press. p. 20. ISBN 978-0-306-41332-2.
- ^ Kittel, Charles. Introduction to Solid State Physics, 8th edition. pp. 153
- ^ Ashcroft and Mermin. Solid State Physics. pp12