Larmor Radius and Gyrofrequency

The size and rate of a charged particle's spiral around a magnetic field line, the basic scales of magnetization.

Gyration

A charged particle moving across a magnetic field spirals around the field line. Two quantities describe this gyration: the gyrofrequency (cyclotron frequency) at which it circles, and the Larmor radius (gyroradius) of the circle:

omega_c = q B / m
r_L = m v_perp / (q B) = v_perp / omega_c

The gyrofrequency depends only on the charge-to-mass ratio and field; the gyroradius also depends on the perpendicular speed, hence on temperature. Electrons gyrate far faster and in much smaller circles than ions of the same energy.

Magnetization

A plasma is magnetized when the gyroradius is small compared to the system size and the gyrofrequency is fast compared to collision and gradient timescales. Magnetization is what makes confinement possible: particles are tied to field lines transversely while moving freely along them.

Why these scales matter

• The ion gyroradius sets the scale of the turbulence that drives transport (gyro-Bohm)
• The cyclotron frequencies set the frequencies for resonant wave heating (ICRH, ECRH)
• The ratio rho_i / a (rho-star) is the key dimensionless parameter for confinement scaling

In simulation

Full particle codes must resolve the gyration in both time and space, which is expensive. Gyrokinetics removes the gyration analytically, keeping only its averaged effect through the gyroradius, which is why it can reach confinement scales. The finite-gyroradius Bessel-function operators in gyrokinetics come directly from averaging over the Larmor orbit.

High-field design

A strong field shrinks the gyroradius and raises the gyrofrequency, improving confinement (via gyro-Bohm scaling) and moving heating resonances to higher frequency. The Hyperion breeder uses a high field (16.84 T peak, 8 T on-axis) partly for this favorable small-gyroradius behavior.