Electromagnetic electron wave

In an unmagnetized plasma, an electromagnetic electron wave is simply a light wave modified by the plasma. In a magnetized plasma, there are two modes perpendicular to the field, the O and X modes, and two modes parallel to the field, the R and L waves.

Waves in an unmagnetized plasma

Langmuir wave

The Langmuir wave is a purely longitudinal wave, that is, the wave vector is in the same direction as the electric field. It is an electrostatic wave; as such, it doesn't have an oscillating magnetic field.

A plasma consists of charged particles which react to electric fields, in contrast with dielectric matter. When electrons in a uniform, homogeneous plasma are perturbed from their equilibrium position, a charge separation occurs creating an electric field which acts as restoring force on the electrons. Since electrons have inertia the system behaves as a harmonic oscillator, where the electrons oscillate at a frequency ω_pe, called electron plasma frequency. These oscillations do not propagate—the group velocity is 0.

When the thermal motion of the electrons is taken into account a shift in frequency from the electron plasma frequency ω_pe occurs. Now the electron pressure gradient acts as the restoring force, creating a propagating wave analogous to a sound wave in non-ionized gases. Combining these two restoring forces (from the electric field and electron pressure gradient) a type of wave, named Langmuir wave, is excited. The dispersion relation is:

${\displaystyle {\omega }^2={\omega }_{\mathrm{p}\mathrm{e}}^2+3C_{\mathrm{e}}^2{k}^2}$

The first term on the right-hand side of the dispersion relation is the electron plasma oscillation related to the electric field force and the second term is related to the thermal motion of the electrons, where C_e is the electron thermal speed and k is the wave vector.

Electromagnetic wave

In an unmagnetized plasma, waves above the plasma frequency propagate through the plasma according to the dispersion relation:

${\displaystyle 1-\frac{{\omega }_{pe}^2}{{\omega }^2}-\frac{k^2{c}^2}{{\omega }^2}=0\to {\omega }^2=c^2{k}^2+{\omega }_{pe}^2}$

In an unmagnetized plasma for the high frequency or low electron density limit, i.e. for ${\displaystyle \omega \gg {\omega }_{\mathrm{p}\mathrm{e}}=(n_{\mathrm{e}}{e}^2/m_{\mathrm{e}}{ϵ}_0{)}^{1/2}}$ or ${\displaystyle n_{\mathrm{e}}\ll m_{\mathrm{e}}{\omega }^2{ϵ}_0/e^2}$ where ω_pe is the plasma frequency, the wave speed is the speed of light in vacuum. As the electron density increases, the phase velocity increases and the group velocity decreases until the cut-off frequency where the light frequency is equal to ω_pe. This density is known as the critical density for the angular frequency ω of that wave and is given by

${\displaystyle n_{\mathrm{c}}=\frac{{\epsilon }_0{m}_{\mathrm{e}}}{e^2}{\omega }^2}$ (SI units)

If the critical density is exceeded, the plasma is called over-dense.

In a magnetized plasma, except for the O wave, the cut-off relationships are more complex.