Langmuir Waves

Source lecture(s): pc368_lec09_wave1, pc368_lec10_wave2

Intuition

Langmuir waves are the natural electrostatic oscillations of electrons against a stationary ion background. They are the “sound” of a plasma’s electron fluid.

Formal Definition

Langmuir waves are electrostatic normal modes with dispersion relation

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

in the warm-fluid limit, where \(\\omega_{pe}\) is the plasma frequency and \(v_{te}\) is the electron thermal speed.

Mathematical Formulation

From the linearized Vlasov–Poisson system with a Maxwellian background, the dielectric function is

\[\\varepsilon(k,\\omega) = 1 + \\frac{1}{k^2\\lambda_D^2}\\Bigl[1 + \\zeta Z(\\zeta)\\Bigr] = 0\]

with \(\\zeta = \\omega/(k\\sqrt{2}v_{te})\). For \(k\\lambda_D \\ll 1\), this reduces to the fluid Langmuir dispersion above.

Derivation

Linearize electron continuity and momentum equations with \(p = n k_B T\).
Assume isothermal or adiabatic electrons (\(\\gamma_e = 1\) or \(5/3\)).
Combine with Poisson’s equation to eliminate density and potential.
The resulting wave equation gives \(\\omega^2 = \\omega_{pe}^2 + 3 k^2 v_{te}^2\).

Worked Example

For \(n_e = 10^{18}\\,\\text{m}^{-3}\) and \(T_e = 1\) eV: - \(f_{pe} \\approx 9\) GHz. - For \(k\\lambda_D = 0.1\), \(\\omega \\approx \\omega_{pe}\\sqrt{1 + 0.03} \\approx 1.015\\,\\omega_{pe}\).

Common Mistakes

Calling Langmuir waves “electromagnetic.” They are electrostatic; \(\\mathbf{E}\) is parallel to \(\\mathbf{k}\).
Forgetting the thermal correction \(3k^2 v_{te}^2\). At \(k\\lambda_D \\sim 1\), it dominates.

Quiz Questions

What is the phase velocity of a cold Langmuir wave?
Why does Landau damping vanish in the cold-fluid limit?