Counter-propagating electron beams — Langmuir wave growth
Electric energy: 0.000
Growth rate: 0.000 ωₚ
Mode: —
4.0 vₜₕ
2.0
400
5%
Two counter-propagating electron beams (±v₀) in a neutralizing ion background. Electrostatic perturbation grows when wave phase velocity matches beam velocity. Instability criterion: ω²p/(ω−kv₀)² + ω²p/(ω+kv₀)² > 1. Max growth: γ_max ≈ (√3/2^(4/3)) ωₚ occurs near k_max ≈ ωₚ/v₀. Nonlinear: particle trapping forms cat-eye vortices in phase space.