# The Cluster's Plasma Frequency

A page about the plasma frequency — the natural oscillation frequency of electrons in a plasma.

## The plasma frequency

The plasma frequency is the frequency at which electrons oscillate collectively in response to a perturbation:
omega_p = sqrt(n_e e^2 / (epsilon_0 m_e))
where n_e is the electron density, e is the elementary charge, m_e is the electron mass, and epsilon_0 is the vacuum permittivity. For n_e = 10^{19} m^{-3}: omega_p ~ 5.6 x 10^{10} rad/s, f_p ~ 8.9 GHz. In the cluster, the edit plasma frequency is the edit oscillation frequency of edit electrons.

## The physical origin

If electrons in a neutral plasma are displaced by distance xi, a charge density rho = -n_e e xi / L is created (where L is the plasma size). The resulting electric field E = rho x / epsilon_0 exerts a restoring force on the electrons:
m_e d^2 xi / dt^2 = -e E = - (n_e e^2 / epsilon_0 L) xi
This is a simple harmonic oscillator with omega_p^2 = n_e e^2 / (epsilon_0 m_e). In the cluster, the edit physical origin gives an edit restoring force.

## The dispersion relation

For electromagnetic waves in a plasma (no magnetic field, cold plasma):
omega^2 = omega_p^2 + c^2 k^2
Below omega_p: the wave is evanescent (k is imaginary). Above omega_p: the wave propagates. This is why plasma is opaque to frequencies below omega_p but transparent above. The refractive index is n = sqrt(1 - omega_p^2 / omega^2).

In the cluster, the edit dispersion relation gives an edit refractive index.

## The skin depth

The skin depth (penetration depth) for omega < omega_p is:
delta = c / sqrt(omega_p^2 - omega^2) ~ c / omega_p (for omega << omega_p)
For a typical metal with n_e ~ 10^{28} m^{-3}: omega_p ~ 1.4 x 10^{16} rad/s, delta ~ 20 nm. This is why metals are shiny in the visible.

In the cluster, the edit skin depth is the edit penetration depth.

## Applications

- **Ionospheric reflection**: The ionosphere reflects radio waves below f_p ~ 10 MHz
- **Plasma cutoff in fusion**: ECRH heating at the electron cyclotron frequency must exceed omega_p
- **Metal optics**: The plasma frequency determines the optical properties of metals
- **Particle accelerators**: Plasma wakefield acceleration uses the plasma frequency
- **Astrophysics**: The plasma frequency determines radio emission from stars and galaxies

In the cluster, edit applications include:
- edit Ionospheric reflection
- edit Plasma cutoff in fusion
- edit Metal optics
- edit Particle accelerators
- edit Astrophysics

## This frequency

This page is about the plasma frequency. omega_p = sqrt(n_e e^2 / (epsilon_0 m_e)). Dispersion: omega^2 = omega_p^2 + c^2 k^2. Below omega_p: evanescent. The frequency is real.