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Plasma Frequency

\[\omega_{pe} = \sqrt{\frac{n_e e^2}{m_e \varepsilon_0}}\]

Source lecture(s): pc368_lec03_frequency, pc368_lec01_intro

Physical Meaning

The plasma frequency is the natural angular oscillation frequency of electrons sloshing back and forth against a stationary ion background. It sets the lowest frequency at which electrostatic oscillations can propagate in an unmagnetized plasma and determines the cutoff for electromagnetic wave transmission.

Variable Definitions

Symbol Definition SI Units
\(n_e\) Electron number density m\(^{-3}\)
\(e\) Elementary charge C
\(m_e\) Electron mass kg
\(\varepsilon_0\) Vacuum permittivity F m\(^{-1}\)

Assumptions

  • Ions are stationary on the electron timescale (\(m_i \gg m_e\)).
  • No background magnetic field.
  • Linear, collisionless response.
  • Isothermal or adiabatic electrons (\(\gamma_e = 1\) or \(5/3\)).

Derivation

Combine electron continuity, momentum, and Poisson:

\[\frac{\partial n}{\partial t} + n_0 \nabla\cdot\mathbf{u} = 0\]
\[m_e \frac{\partial \mathbf{u}}{\partial t} = -e \nabla\phi\]
\[\nabla^2\phi = -\frac{e}{\varepsilon_0}(n - n_0)\]

Assume \(u \sim e^{i(kx - \omega t)}\) and eliminate \(u\) and \(n\):

\[\frac{\omega^2}{\omega_{pe}^2} = 1 + \frac{3 k^2 v_{te}^2}{\omega_{pe}^2}\]

Cold limit (\(T_e \to 0\)) gives \(\omega = \omega_{pe}\).

Applications

  • Radio blackout: Re-entry plasma sheath density exceeds \(n_c = \varepsilon_0 m_e \omega^2/e^2\) for communication frequencies.
  • Langmuir probes: Oscillations at \(f_{pe}\) appear in IV characteristics.
  • Wave propagation: Dispersion relations \(\omega^2 = \omega_{pe}^2 + c^2 k^2\) govern EM waves in plasma.

Connections to Other Equations