Editing Plasma oscillation (section)

=== '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''
: <math>\omega_{\mathrm{pe}} = \sqrt{\frac{n_\mathrm{e} e^{2}}{m^*\varepsilon_0}}, \left[\mathrm{rad/s}\right]</math> ([[SI units]]),
:<math>\omega_{\mathrm{pe}} = \sqrt{\frac{4 \pi n_\mathrm{e} e^{2}}{m^*}}, \left[\mathrm{rad/s}\right]</math> ([[Centimetre gram second system of units|cgs units]]),

where <math>n_\mathrm{e}</math>  is the [[number density]] of electrons, <math>e</math> is the [[electric charge]], <math>m^*</math>  is the [[effective mass (solid-state physics)|effective mass]] of the electron, and <math>\varepsilon_0</math> 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, since electrons are much lighter than ions.

Proof using Maxwell equations.<ref name="Ashcroft">{{cite book | last1=Ashcroft|first1=Neil|author-link=Neil Ashcroft | last2=Mermin|first2=N. David|author-link2=N. David Mermin | title=[[Ashcroft and Mermin|Solid State Physics]] | publisher=Holt, Rinehart and Winston|location=New York | year=1976|isbn=978-0-03-083993-1 | page = 19}}</ref> Assuming charge density oscillations <math>\rho(\omega)=\rho_0 e^{-i\omega t}</math> the continuity equation:
<math display="block">\nabla \cdot \mathbf{j} = - \frac{\partial \rho}{\partial t} = i \omega \rho(\omega) </math>
the Gauss law
<math display="block">\nabla \cdot \mathbf{E}(\omega) = 4 \pi \rho(\omega)</math>
and the conductivity
<math display="block">\mathbf{j}(\omega) = \sigma(\omega) \mathbf{E}(\omega)</math>
taking the divergence on both sides and substituting the above relations:
<math display="block">i \omega \rho(\omega) =  4 \pi \sigma(\omega) \rho(\omega)</math>
which is always true only if
<math display="block">1+ \frac {4 \pi i \sigma(\omega)}{\omega} = 0</math>
But this is also the dielectric constant (see [[Drude Model]])
<math>\epsilon(\omega) = 1+ \frac {4 \pi i \sigma(\omega)}{\omega} </math>
and the condition of transparency (i.e. <math>\epsilon \ge 0</math> from a certain plasma frequency <math>\omega_{\rm p}</math> and above), the same condition here <math>\epsilon = 0</math> apply to make possible also the propagation of density waves in the charge density.

This expression must be modified in the case of electron-[[positron]] plasmas, often encountered in [[astrophysics]].<ref>{{cite book | last=Fu | first=Ying | title=Optical properties of nanostructures | year=2011 | publisher=Pan Stanford | pages=201}}</ref> Since the [[frequency]] is independent of the [[wavelength]], these [[oscillation]]s have an [[Infinity|infinite]] [[phase velocity]] and zero [[group velocity]].

Note that, when <math>m^*=m_\mathrm{e}</math>, the plasma frequency, <math>\omega_{\mathrm{pe}}</math>, depends only on [[physical constant]]s and electron density <math>n_\mathrm{e}</math>. The numeric expression for angular plasma frequency is
<math display="block">f_\text{pe} = \frac{\omega_\text{pe}}{2\pi}~\left[\text{Hz}\right]</math>

Metals are only transparent to light with a frequency higher than the metal's plasma frequency. For typical metals such as aluminium or silver, <math>n_\mathrm{e}</math> is approximately 10<sup>23</sup> cm<sup>−3</sup>, which brings the plasma frequency into the ultraviolet region. This is why most metals reflect visible light and appear shiny.