Plasma Frequency

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 a 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.

'Cold' electrons

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

\omega_{pe} = \sqrt{\frac{n_e e^{2}}{m^*\varepsilon_0}}, \left[rad/s\right] (SI units),
\omega_{pe} = \sqrt{\frac{4 \pi n_e e^{2}}{m^*}}, (cgs units),

where n_e is the number density of electrons, e is the electric charge, m* is the effective mass of the electron, and \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).[1] Since the frequency is independent of the wavelength, these oscillations have an infinite phase velocity and zero group velocity.

Note that, if m^* is electron mass (m^*=m_e), plasma frequency \omega_{pe} depends only on physical constants and concentration of electrons n_e. The numeric expression for plasma ordinary frequency



f_{pe} \approx 8980 \sqrt{n_e} Hz

with number density n_e in cm−3.

'Warm' electrons

When the effects of the electron thermal speed v_{e,th} = \sqrt{\frac{k_B T_{\mathrm{e}}}{m_e}} are taken into account, the electron pressure acts as a restoring force as well as the electric field and the oscillations propagate with frequency and wavenumber related by the longitudinal Langmuir[2] wave:

\omega^2 =\omega_{pe}^2 +\frac{3k_BT_{\mathrm{e}}}{m_e}k^2=\omega_{pe}^2 + 3 k^2 v_{\mathrm{e,th}}^2 , called the Bohm-Gross dispersion relation. If the spatial scale is large compared to the Debye length, the oscillations are only weakly modified by the pressure term, but at small scales the pressure term dominates and the waves become dispersionless with a speed of \sqrt{3} \cdot v_{e,th}. For such waves, however, the electron thermal speed is comparable to the phase velocity, i.e.,

v \sim v_{ph} \ \stackrel{\mathrm{def}}{=}\ \frac{\omega}{k}, so the plasma waves can accelerate electrons that are moving with speed nearly equal to the phase velocity of the wave. This process often leads to a form of collisionless damping, called Landau damping. Consequently, the large-k portion in the dispersion relation is difficult to observe and seldom of consequence.

In a bounded plasma, fringing electric fields can result in propagation of plasma oscillations, even when the electrons are cold.

In a metal or semiconductor, the effect of the ions' periodic potential must be taken into account. This is usually done by using the electrons' effective mass in place of m.

See also


Further reading

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