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Plasma Frequency

Source lecture(s): pc368_lec03_frequency

Intuition

Pluck an electron cloud away from the ion background and it will oscillate. The natural frequency of this sloshing is the electron plasma frequency \(\omega_{pe}\). It sets the timescale for virtually every electrostatic process in a plasma, from Langmuir waves to Landau damping.

Formal Definition

For a cold electron–ion plasma with stationary ions:

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

where \(n_e\) is the electron number density.

Mathematical Formulation

Displace electrons by a small distance \(x\) relative to the fixed ion background. The charge imbalance creates an electric field \(E = n_e e x / \varepsilon_0\) (from Gauss’s law). Newton’s second law for the electron fluid gives:

\[m_e \ddot{x} = -e E = -\frac{n_e e^2}{\varepsilon_0} x\]

This is simple harmonic motion with frequency \(\omega = \sqrt{n_e e^2/(m_e \varepsilon_0)}\).

Derivation

  1. Linearize continuity, momentum, and Poisson equations about a uniform equilibrium with stationary ions.
  2. For isothermal electrons (\(T_e\) constant), momentum equation gives \(m_e \partial v_e / \partial t = -e E\).
  3. Combining with \(\partial n_1 / \partial t + n_0 \nabla \cdot v_e = 0\) and \(\nabla \cdot E = e n_1 / \varepsilon_0\), eliminate \(v_e\) and \(n_1\).
  4. The result is the wave equation \(\nabla^2 E = (\omega_{pe}^2 / c^2) E\), or more precisely, \(\omega^2 = \omega_{pe}^2 + c^2 k^2\) for electromagnetic disturbances.

Worked Example

Problem: What is \(\omega_{pe}\) in the solar corona (\(n_e \sim 10^{15}\,\text{m}^{-3}\))?

\(\omega_{pe} = \sqrt{10^{15} \times (1.6\times 10^{-19})^2 / (9.11\times 10^{-31} \times 8.85\times 10^{-12})} \approx 5.6\times 10^8\,\text{rad/s}\) (\(f_{pe} \approx 89\) MHz).

Common Mistakes

  • \(\omega_{pe}\) depends only on density. Confirm before using.
  • The ion plasma frequency matters for high-frequency electromagnetic waves.
  • Plasma oscillation is not a sound wave. It is an electrostatic Langmuir mode.

Quiz Questions

  1. Conceptual: If you suddenly removed all ions from a plasma, what would the electron frequency become?
  2. Computational: Compute \(\omega_{pe}\) for a dusty plasma with \(n_e = 10^{12}\,\text{m}^{-3}\).
  3. MCQ: For which of these parameters does \(\omega_{pe}\) increase?
  4. A) Lowering \(n_e\)
  5. B) Increasing \(m_e\)
  6. C) Increasing \(T_e\)
  7. D) Increasing \(n_e\)

Further Reading

  • T. J. M. Boyd & J. J. Sanderson, Plasma Dynamics via Atoms.