Plasma oscillation

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 in the ultraviolet region.

The oscillations can be described as an instability in the dielectric function of a free electron gas.

The frequency depends only weakly on the wavelength of the oscillation.

The quasiparticle resulting from the quantization of these oscillations is the plasmon.

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 an electron or a group of 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

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.

taking the divergence on both sides and substituting the above relations:

This expression must be modified in the case of electron-positron plasmas, often encountered in astrophysics.

, depends only on physical constants and electron density

The numeric expression for angular plasma frequency is

is approximately 1023 cm−3, which brings the plasma frequency into the ultraviolet region.

This is why most metals reflect visible light and appear shiny.

are considered, the electron pressure acts as a restoring force, and the electric field and oscillations propagate with frequency and wavenumber related by the longitudinal Langmuir[4] wave:

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

For such waves, however, the electron thermal speed is comparable to the phase velocity, i.e.,

so the plasma waves can accelerate electrons that are moving with speed nearly equal to the phase velocity of the wave.

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. Plasma oscillations may give rise to the effect of the “negative mass”.

The system is subjected to the external sinusoidal force

and replace the entire system with a single effective mass

[5][6][7][8] The negative effective mass (density) becomes also possible based on the electro-mechanical coupling exploiting plasma oscillations of a free electron gas (see Figure 2).

[9][10] The negative mass appears as a result of vibration of a metallic particle with a frequency of

which is close the frequency of the plasma oscillations of the electron gas

The plasma oscillations are represented with the elastic spring

Thus, the metallic particle vibrated with the external frequency ω is described by the effective mass

Metamaterials exploiting the effect of the negative mass in the vicinity of the plasma frequency were reported.

A mechanical model giving rise to the negative effective mass effect
Figure 1. Core with mass is connected internally through the spring with to a shell with mass . The system is subjected to the sinusoidal force .
Figure 2. Free electrons gas is embedded into the ionic lattice ; is the plasma frequency (the left sketch). The equivalent mechanical scheme of the system (right sketch).