The Plasma Frequency

The natural oscillation frequency of electrons displaced from equilibrium, setting the plasma's electromagnetic response.

Electron oscillation

Displace the electrons of a plasma relative to the ions and the resulting electric field pulls them back, causing them to overshoot and oscillate. The natural frequency of this oscillation is the electron plasma frequency:

omega_pe = sqrt( n_e e^2 / (epsilon0 m_e) )

It depends only on the electron density (and fundamental constants), not on temperature. The plasma frequency is the fastest natural timescale of the plasma and the boundary between electromagnetic waves that propagate and those that are reflected.

Cutoff and reflection

Electromagnetic waves with frequency below omega_pe cannot propagate through the plasma; they are reflected. This is why the ionosphere reflects radio waves and why the plasma density sets the cutoff for heating and diagnostic frequencies. Wave heating schemes must operate above the relevant cutoffs, or exploit mode conversion, to deposit energy in the core.

Related frequencies

• The ion plasma frequency is lower by the square root of the mass ratio
• The electron cyclotron frequency omega_ce = eB/m_e sets gyration, distinct from oscillation
• The ratio of plasma to cyclotron frequency classifies the wave regimes

Numerical relevance

Explicit particle-in-cell codes must resolve the plasma period (time step times omega_pe less than about 0.2) for stability, another reason full kinetic simulation is expensive. Gyrokinetic and fluid models order out omega_pe to reach confinement timescales.

In fusion practice

The plasma and cyclotron frequencies determine which radio-frequency and microwave systems can heat and diagnose the plasma. Choosing electron-cyclotron and ion-cyclotron heating frequencies for a device like the Hyperion breeder starts from these characteristic frequencies at the design density and field.