Plasma frequency (${\omega }_{ps}=\sqrt{n_s{q}_s^2/({ϵ}_0{m}_s)}$)

The plasma frequency is the characteristic electrostatic oscillation frequency of a charged species displaced against a neutralizing background. For number density $n_s$, charge $q_s$ and mass $m_s$, the displayed formula uses SI units; in Gaussian units it is ${\omega }_{ps}=\sqrt{4\pi n_s{q}_s^2/m_s}$. The ion skin depth is $c/{\omega }_{pi}$ for the ion species. Different species have different plasma frequencies.

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.

With immobile ions, negligible collisions and electron density $n_e$, the plasma frequency is ${\omega }_p^2=n_e{e}^2/(m_e{ϵ}_0)$. Combining inertial electron current with vacuum displacement current gives ${\omega }^2={\omega }_p^2+c^2{k}^2$. Transverse fields are evanescent below ${\omega }_p$ and propagate above it.

A carrier response must appear once in an electromagnetic constitutive model. With harmonic convention $e^{-i\omega t}$, absolute permittivity $\epsilon$ describing the remaining polarization and an explicit conduction electric current density ${\mathbf{J}}_c=\sigma \mathbf{E}$, the Ampère-Maxwell equation containsThus an alternative description incorporates that carrier contribution into the complex permittivity ${\epsilon }_{\mathrm{eff}}=\epsilon +i\sigma /\omega$ and omits the explicit ${\mathbf{J}}_c$. Using this effective quantity and adding the same conduction current again doubles that response. For relative permittivity, the added term is $i\sigma /({\epsilon }_0\omega )$; changing the harmonic sign convention changes the displayed sign. Stating which charges are included in each term prevents a mathematical rearrangement from becoming an extra physical mechanism.

In an unmagnetized plasma, an electron displacement driven by $\mathbf{E}{e}^{-i\omega t}$ obeys $-m_e{\omega }^2{\mathbf{x}}_e=-e\mathbf{E}$. Its dipole moment is $-e{\mathbf{x}}_e$, giving electric polarization $\mathbf{P}=-n{e}^2\mathbf{E}/(m_e{\omega }^2)$. With $\mathbf{D}={\epsilon }_0\mathbf{E}+\mathbf{P}$, this yields the displayed relative permittivity and ${\omega }_p^2=n{e}^2/(m_e{\epsilon }_0)$. Ions contribute a further negative term smaller by $Z{m}_e/m_i$ for one ion species of charge $Ze$. Neglecting it requires heavy ions at electron-wave frequencies; collisions, background magnetization and pressure dispersion are excluded. The transverse electromagnetic wave relation is $c^2{k}^2={\omega }^2-{\omega }_p^2$.

For a downward transverse wave in an isotropic plasma, $E_z/E_x=\tan {\theta }_t$ fixes the transmitted direction. Tangential phase matching gives $\sin {\theta }_i=n_p\sin {\theta }_t$. Continuity of tangential electric and magnetic fields then separates incident and reflected amplitudes. A measured transmitted electric amplitude alone cannot be identified with the incident one; mean power for peak phasors includes a factor of one half, and interface-normal flux includes a cosine.

For vacuum incidence on a sharp planar, collisionless plasma interface, the transmitted normal wavenumber obeys $k_z^2=({\omega }^2/c^2)({\cos }^2\theta -{\omega }_p^2/{\omega }^2)$. Negative $k_z^2$ produces an evanescent field and total reflection in this lossless model.