Electron plasma frequency

The electron plasma frequency is the characteristic angular frequency of small collective electron displacements against an effectively stationary neutralizing ion background. For electron number density $n_e$, its cold, unmagnetized value in SI units isHere $e$ is the magnitude of the electron charge and $m_e$ the electron mass. Angular frequency and cycles per second differ by $2\pi$.^[ref]

A displaced slab gives a simple derivation. Shift a uniform electron population by a small distance $x$ relative to fixed positive ions. The exposed charge produces a restoring electric field $E=n_{e}ex/{ϵ}_0$. Newton's equation for an electron then giveswhich is a harmonic oscillator with the frequency above. This argument describes a collective charge-separation mode, not an individual electron's orbital frequency.

For $n_e=10^6{\mathrm{m}}^{-3}=1{\mathrm{cm}}^{-3}$, $f_{pe}\simeq 8.98\mathrm{kHz}$. Increasing density by a factor of four doubles the frequency. An identified plasma oscillation can therefore diagnose density, while interpreting an arbitrary spectral peak this way would be unjustified.

Finite temperature changes the wave's dispersion relation; magnetic fields, collisions and ion motion introduce additional modes or damping. The same cold model also gives the transverse wave cutoff in a cold unmagnetized plasma, but an observed electromagnetic cutoff and a local electrostatic oscillation are distinct measurements.