Editing Plasma oscillation (section)

=== 'Warm' electrons ===

When the effects of the [[electron]] thermal speed <math display="inline">v_{\mathrm{e,th}} = \sqrt{k_\mathrm{B} T_{\mathrm{e}} / m_\mathrm{e}}</math> 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<ref>*{{Citation |last=Andreev |first=A. A. |title=An Introduction to Hot Laser Plasma Physics |year=2000 |publisher= [[Nova Science Publishers, Inc.]] |location=Huntington, New York |isbn=978-1-56072-803-0}}</ref> wave:
<math display="block">
\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,
</math>
called the [[David Bohm|Bohm]]–[[Eugene P. Gross|Gross]] [[dispersion relation]]. If the spatial scale is large compared to the [[Debye length]], the [[oscillation]]s 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 <math>\sqrt{3} \cdot v_{\mathrm{e,th}}</math>. For such waves, however, the electron thermal speed is comparable to the [[phase velocity]], i.e., 
<math display="block">
v \sim v_{\mathrm{ph}} \ \stackrel{\mathrm{def}}{=}\  \frac{\omega}{k},
</math>
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 function|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 [[ion]]s' periodic potential must be taken into account. This is usually done by using the electrons' [[effective mass (solid-state physics)|effective mass]] in place of ''m''.