Plasma oscillation

Plasma oscillations, also known as Langmuir waves (after ), are rapid oscillations of the in conducting media such as  or s in the  region. The oscillations can be described as an instability in the. The frequency only depends weakly on the wavelength of the oscillation. The resulting from the  of these oscillations is the.

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

Mechanism
Consider an electrically neutral plasma in equilibrium, consisting of a gas of positively charged s and negatively charged. If one displaces by a tiny amount an electron or a group of electrons with respect to the ions, the 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
 * $$\omega_{\mathrm{pe}} = \sqrt{\frac{n_\mathrm{e} e^{2}}{m^*\varepsilon_0}}, \left[\mathrm{rad/s}\right]$$ ,
 * $$\omega_{\mathrm{pe}} = \sqrt{\frac{4 \pi n_\mathrm{e} e^{2}}{m^*}}, [rad/s]$$ ,

where $$n_\mathrm{e}$$  is the of electrons, $$e$$ is the, $$m^*$$  is the  of the electron, and $$\varepsilon_0$$ is the. Note that the above is derived under the  that the ion mass is infinite. This is generally a good approximation, as the electrons are so much lighter than ions. (This expression must be modified in the case of electron- plasmas, often encountered in ). Since the is independent of the, these s have an   and zero.

Note that, when $$m^*=m_\mathrm{e}$$, the plasma frequency, $$\omega_{\mathrm{pe}}$$, depends only on s and electron density $$n_\mathrm{e}$$. The numeric expression for angular plasma frequency is


 * $$f_\text{pe} = \frac{\omega_\text{pe}}{2\pi}~\left[\text{Hz}\right]$$

Metals are only transparent to light with a frequency higher than the metal's plasma frequency. For typical metals such as aluminium or silver, $$n_\mathrm{e}$$ 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.

'Warm' electrons
When the effects of the thermal speed $$v_{\mathrm{e,th}} = \sqrt{\frac{k_\mathrm{B} T_{\mathrm{e}}}{m_\mathrm{e}}}$$ 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  related by the longitudinal Langmuir wave:



\omega^2 =\omega_{\mathrm{pe}}^2 +\frac{3k_\mathrm{B}T_{\mathrm{e}}}{m_\mathrm{e}}k^2=\omega_{\mathrm{pe}}^2 + 3 k^2 v_{\mathrm{e,th}}^2 $$, called the -. If the spatial scale is large compared to the, the s are only weakly modified by the term, but at small scales the pressure term dominates and the waves become dispersionless with a speed of $$\sqrt{3} \cdot v_{\mathrm{e,th}}$$. For such waves, however, the electron thermal speed is comparable to the, i.e.,

v \sim v_{\mathrm{ph}} \ \stackrel{\mathrm{def}}{=}\  \frac{\omega}{k}, $$ so the plasma waves can 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. Consequently, the large-k portion in the is difficult to observe and seldom of consequence.

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

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