When electrons in a neutral plasma are displaced from their equilibrium positions, the restoring Coulomb force from the exposed positive ion background drives them back. The ions are effectively immobile on electron timescales (mi ≫ me), providing a fixed uniform neutralising background. The result is oscillation at a fixed frequency — the plasma frequency — regardless of wavelength (cold plasma).
E_k = i·ρ_k / (k·ε₀), k=0 mode set to zero (neutrality).v_new = v + (e/m)·E·dt, x_new = x + v_new·dt (positions advanced half-step staggered).ω_p = √(n·e² / (ε₀·mₑ))T_p = 2π / ω_p∂²φ/∂x² = −ρ/ε₀ → E = −∂φ/∂xBohm-Gross: ω² = ω_p² + 3v_th²k²
Irving Langmuir discovered plasma oscillations in 1928 while studying electric discharges. He called the ionised gas "plasma" after blood plasma, because it reminded him of a life-giving fluid carrying particles. Langmuir waves are the electrostatic analogue of sound waves in a neutral gas, but travel without collisions via collective Coulomb interactions. In laser-plasma wakefield accelerators, these waves are driven to enormous amplitudes (GV/m electric fields) to accelerate electrons to GeV energies in millimetre distances.
A plasma oscillation, also called a Langmuir wave, is a rapid oscillation of the electron density in a plasma caused by the restoring Coulomb force when electrons are displaced from their equilibrium positions relative to the fixed ion background. The electrons collectively oscillate at the plasma frequency ω_p.
The plasma frequency is ω_p = √(n·e² / (ε₀·mₑ)), where n is the electron number density, e is the electron charge, ε₀ is the permittivity of free space, and mₑ is the electron mass. Notably, it depends only on electron density — not on wave vector k — making cold Langmuir waves non-dispersive.
A particle-in-cell (PIC) simulation tracks macro-particles on a computational grid. Charge is deposited onto cells to compute the electric field via Poisson's equation, then the field is interpolated back to particle positions to compute forces. The leapfrog (Verlet) algorithm advances positions and velocities in a time-staggered manner for excellent long-term energy conservation.
When electrons are displaced, the exposed positive ion background creates an electric field that pulls them back — exactly like a spring. This restoring Coulomb force causes oscillation. The ions are assumed immobile on electron timescales (because m_i ≫ m_e), providing a fixed neutralising background that acts as the restoring spring.
For cold plasma: ω = ω_p (non-dispersive — all wavelengths oscillate at the same frequency). For warm plasma (Bohm-Gross): ω² = ω_p² + 3v_th²k², where v_th = √(k_B T/mₑ) is the thermal velocity. This gives a finite group velocity v_g = 3v_th²k/ω, allowing wave energy to propagate.
Landau damping is the collisionless attenuation of plasma waves, discovered theoretically by Lev Landau in 1946. Electrons moving slightly slower than the wave phase velocity are accelerated (taking energy from the wave), while those moving slightly faster are decelerated. If more electrons are slower than faster (as in a Maxwellian), the net effect is wave damping without any collisions.
The lower panel plots each electron's position (x) vs velocity (v_x) in phase space. For small-amplitude linear oscillations, electrons trace thin ellipses. For large amplitudes, nonlinear effects cause wave breaking and particle trapping — electrons circle in closed orbits within phase-space vortices (called phase-space holes or electron holes).
The charge density ρ(x) is computed on a grid. Poisson's equation ∂²φ/∂x² = −ρ/ε₀ is solved spectrally using FFT: in Fourier space, E_k = i·ρ_k / (k·ε₀), with k=0 set to zero (global neutrality). The inverse FFT gives E(x). This spectral solver is exact for periodic boundaries and runs in O(M log M) time.
For large perturbations, nonlinear effects dominate. The sinusoidal density wave steepens and can break. Electrons become trapped in the wave's electrostatic potential troughs, forming phase-space vortices. The oscillation frequency shifts downward (nonlinear frequency shift) and energy cascades from the fundamental mode to harmonics.
Langmuir wave physics underpins: plasma diagnostics via Langmuir probes (measuring n and T_e), laser-plasma wakefield particle accelerators (GeV electrons in mm), plasma heating in fusion reactors (electron Bernstein waves), ionospheric radar sounding (EISCAT), and plasma-based amplifiers. The plasma frequency sets a cut-off: EM waves below ω_p cannot propagate in plasma (which is why the ionosphere reflects AM radio).
Higher electron temperature broadens the velocity distribution, increasing the fraction of resonant electrons near the phase velocity. This enhances Landau damping, causing wave amplitude to decay faster. Temperature also adds thermal pressure corrections (Bohm-Gross term), making the waves dispersive and giving them a finite group velocity. In this simulation, raise the Temperature slider to observe dispersion and damping effects.
This simulation visualises plasma oscillations (Langmuir waves): when electrons in a neutral plasma are displaced from equilibrium, the exposed positive ion background — assumed immobile on electron timescales — exerts a restoring Coulomb force that pulls them back, overshoots, and rings at the characteristic plasma frequency ωp = √(n·e²/(ε₀·mₑ)). The shader renders this as a travelling electron-density wave whose frequency follows the warm-plasma Bohm-Gross dispersion relation ω² = ωp² + 3k²vth², so raising the temperature slider visibly adds dispersion and a Landau-damping-like envelope decay to the oscillation.
A glowing horizontal slab of plasma whose colour (orange = compression, blue = rarefaction) traces the longitudinal electron-density wave δn ∝ −cos(kx − ωt). Faint moving lines mark the oscillating restoring electric field, and drifting bright streaks represent electrons moving through the plasma column.
Drag Plasma density n to change ωp directly (higher n → faster oscillation), Temperature (v_th) to add Bohm-Gross dispersion and damping, Wavenumber k to change the spatial wavelength of the wave, and Amplitude to control how strongly the density is compressed and rarefied. Live readouts show ωp, the resulting ω, and vth; use ⏸ Pause / ↺ Reset or the P/R keys to control playback.
Irving Langmuir coined the term "plasma" in 1928 after studying oscillations in ionised gas, comparing it to blood plasma carrying particles. Because electromagnetic waves below the plasma frequency cannot propagate through a plasma, Earth's ionosphere reflects AM radio signals back down — the same physics that sets the cut-off you can explore here.
The plasma frequency ωp = √(n·e²/(ε₀·mₑ)) is the natural rate at which displaced electrons oscillate about the neutralising ion background. It depends only on the electron density n, not on wavelength, so in a cold plasma every Langmuir wave rings at the same frequency. It is also a cut-off: electromagnetic waves with frequency below ωp cannot propagate through the plasma and are reflected instead.
The Plasma density n slider scales ωp directly, since in the simulation's normalised units ωp = √n. Increasing n raises the plasma frequency and speeds up the visible oscillation of the density wave, exactly as higher electron density does in a real plasma.
For a warm plasma, thermal electron motion modifies the oscillation frequency according to ω² = ωp² + 3k²vth², where vth is the thermal velocity and k is the wavenumber. Raising the Temperature slider increases vth, which raises the displayed ω above ωp and adds a slow envelope decay to the wave, echoing how Landau damping attenuates real warm-plasma oscillations.
When electrons shift away from the fixed positive ions, the exposed ion charge creates a restoring electric field pulling the electrons back, much like a spring. Because the ions are far more massive than the electrons, they stay essentially fixed on the electron oscillation timescale, so this restoring force keeps pulling the electron cloud back and forth rather than letting it disperse.
This visualisation is a real-time WebGL shader that reproduces the correct physical relationships — the plasma frequency formula and the Bohm-Gross dispersion relation both drive the animation directly — rather than a full N-particle particle-in-cell code integrating individual electron trajectories. It is designed to build intuition for how density, wavenumber and temperature control the oscillation, using the same governing equations a PIC solver would obey.