Langmuir waves

Plasma oscillations, also known as "Langmuir waves" (after Irving Langmuir), are rapid oscillations of the electron density in conducting media such as plasmas or metals. The oscillations can be described as an instability in the dielectric function of a free electron gas. The frequency only depends weakly on the wavelength of the oscillation. The quasiparticle resulting from the quantization of these oscillations is the plasmon.

Langmuir waves were discovered by American physicists Irving Langmuir and Lewi Tonks in the 1920s. They are parallel in form to Jeans instability waves, which are caused by gravitational instabilities in a static medium.

Consider an electrically neutral plasma in equilibrium, consisting of a gas of positively charged ions and negatively charged electrons. If one displaces by a tiny amount all of the electrons with respect to the ions, the Coulomb force pulls the electrons back, acting as a restoring force.

If the thermal motion of the electrons is ignored, it is possible to show that the charge density oscillates at the plasma frequency

where ${\displaystyle n_{\mathrm {e} }}$ is the number density of electrons, e is the electric charge, m^* is the effective mass of the electron, and ${\displaystyle \varepsilon _{0}}$ is the permittivity of free space. Note that the above formula is derived under the approximation that the ion mass is infinite. This is generally a good approximation, as the electrons are so much lighter than ions. (One must modify this expression in the case of electron-positron plasmas, often encountered in astrophysics). Since the frequency is independent of the wavelength, these oscillations have an infinite phase velocity and zero group velocity.

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