Why do ions appear fixed?

Ions are much heavier than electrons (m_i ≫ m_e), so on the electron oscillation period they barely move. The Langmuir mode is an electron sloshing mode against a neutralizing ion background.

What is ω_p physically?

It is the natural frequency at which electrons collectively try to restore charge neutrality after a displacement. Higher density means stronger restoring fields → higher ω_p.

Why does ω increase with k at finite temperature?

Thermal pressure adds a restoring contribution at finite k, giving the Bohm–Gross correction 3k²v_th² under the square root in the dispersion relation.

How is this related to Debye shielding?

Both use the same n_e and T_e. λ_D describes static spatial shielding of a charge; ω_p describes the fastest collective oscillation timescale of the electron fluid.

Langmuir Plasma Oscillations

Collective electron oscillations at ω_p = √(n_e e²/ε₀m_e): animated δn wave, Bohm–Gross dispersion ω(k), and ω_p vs density.

About this model

Langmuir oscillations are the simplest collective mode of an electron plasma: electrons oscillate against a stationary ion background on a timescale set by the plasma frequency ω_p = √(n_e e²/ε₀ m_e). In the electrostatic limit, a uniform displacement of electrons creates a restoring electric field that drives harmonic motion at ω_p. Propagating Langmuir waves add spatial structure with wavevector k and, when thermal motion is included, the Bohm–Gross dispersion ω² = ω_p² + 3 k² v_th² with v_th = √(k_B T_e/m_e). At long wavelength the frequency approaches ω_p; at shorter wavelengths thermal corrections raise ω above ω_p. This simulator animates a sinusoidal density perturbation δn/n ∝ sin(kx − ωt) with fixed ions, plots the ω(k) dispersion against the cold-plasma asymptote ω_p, and shows how ω_p scales as √n_e. It complements the Debye shielding page: λ_D sets spatial shielding while ω_p sets the fastest collective timescale. Landau damping and electromagnetic effects are omitted.

Who it's for: Undergraduate plasma physics or electromagnetism students after Debye shielding and before magnetized waves or kinetic theory.

Key terms

Langmuir oscillation
Plasma frequency
Bohm–Gross dispersion
Electron plasma wave
Phase velocity
Quasineutrality

How it works

Langmuir plasma oscillations: ω_p = √(n_e e²/ε₀m_e), animated electron density wave, and Bohm–Gross dispersion ω(k) phase diagram.

Frequently asked questions

Why do ions appear fixed?: Ions are much heavier than electrons (m_i ≫ m_e), so on the electron oscillation period they barely move. The Langmuir mode is an electron sloshing mode against a neutralizing ion background.
What is ω_p physically?: It is the natural frequency at which electrons collectively try to restore charge neutrality after a displacement. Higher density means stronger restoring fields → higher ω_p.
Why does ω increase with k at finite temperature?: Thermal pressure adds a restoring contribution at finite k, giving the Bohm–Gross correction 3k²v_th² under the square root in the dispersion relation.
How is this related to Debye shielding?: Both use the same n_e and T_e. λ_D describes static spatial shielding of a charge; ω_p describes the fastest collective oscillation timescale of the electron fluid.

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About this model

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Frequently asked questions

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Frequently asked questions