= 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 is
$$
\boxed{\omega_{pe}=\sqrt{\frac{n_e e^2}{\epsilon_0m_e}}},\qquad f_{pe}=\frac{\omega_{pe}}{2\pi}.
$$
Here $e$ is the magnitude of the electron charge and $m_e$ the electron mass. Angular frequency and cycles per second differ by $2\pi$.<NRL Plasma Formulary (2019)>[Fundamental Plasma Parameters, printed page 28: electron plasma frequency and practical units]{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> $E=n_eex/\epsilon_0$. Newton's equation for an electron then gives
$$
m_e\ddot x=-eE=-\frac{n_e e^2}{\epsilon_0}x,
$$
which is a <harmonic oscillator> with the frequency above. This argument describes a collective charge-separation mode, not an individual electron's orbital frequency.
For $n_e=10^6\,\mathrm m^{-3}=1\,\mathrm{cm}^{-3}$, $f_{pe}\simeq8.98\,\mathrm{kHz}$. 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.
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