Part 4, Chapter 2

Langmuir Waves

Electron plasma oscillations and the Bohm-Gross dispersion relation

2.1 Electron Plasma Oscillations

Langmuir waves are the most fundamental electrostatic oscillation in a plasma. When electrons are displaced from their equilibrium positions relative to the stationary ion background, the resulting charge separation creates a restoring electric field. The electrons overshoot, setting up oscillations at the plasma frequency.

Consider a uniform, unmagnetized plasma with equilibrium electron density $n_0$. If we displace a slab of electrons by a small distance $\delta x$, Gauss's law gives the restoring electric field:

$$E = \frac{n_0 e \,\delta x}{\epsilon_0}$$

Newton's second law for an electron in this field yields the equation of motion:

$$m_e \frac{d^2 (\delta x)}{dt^2} = -eE = -\frac{n_0 e^2}{\epsilon_0}\,\delta x$$

This is simple harmonic motion at the electron plasma frequency:

$$\omega_{pe} = \sqrt{\frac{n_e e^2}{m_e \epsilon_0}} \approx 56.4 \sqrt{n_e}\;\text{rad/s}$$

where $n_e$ is in m^-3. For a typical tokamak with $n_e \sim 10^{20}\;\text{m}^{-3}$, we get $\omega_{pe} \approx 5.6 \times 10^{11}\;\text{rad/s}$, corresponding to microwave frequencies.

2.2 Fluid Derivation: Bohm-Gross Relation

To include thermal effects, we use the electron fluid equations. The three equations governing electrostatic perturbations in an unmagnetized plasma are:

Continuity equation:

$$\frac{\partial n_e}{\partial t} + \nabla \cdot (n_e \mathbf{v}_e) = 0$$

Momentum equation (with pressure):

$$m_e n_e \left(\frac{\partial \mathbf{v}_e}{\partial t} + \mathbf{v}_e \cdot \nabla \mathbf{v}_e\right) = -e n_e \mathbf{E} - \nabla p_e$$

Poisson's equation (ions form a uniform background):

$$\nabla \cdot \mathbf{E} = -\frac{e(n_e - n_0)}{\epsilon_0}$$

We linearize by writing $n_e = n_0 + n_1$,$\mathbf{v}_e = \mathbf{v}_1$,$\mathbf{E} = \mathbf{E}_1$, and assume plane-wave perturbations $\propto e^{i(kx - \omega t)}$. For an adiabatic equation of state with $\gamma = 3$ (one-dimensional compression), the pressure perturbation is $p_1 = 3 T_e n_1$.

The linearized equations become:

$$-i\omega n_1 + i k n_0 v_1 = 0$$$$-i\omega m_e n_0 v_1 = -e n_0 E_1 - 3 i k T_e n_1$$$$i k E_1 = \frac{e n_1}{\epsilon_0}$$

Eliminating $n_1$, $v_1$, and $E_1$ yields the Bohm-Gross dispersion relation:

$$\boxed{\omega^2 = \omega_{pe}^2 + 3 k^2 v_{te}^2}$$

where $v_{te} = \sqrt{T_e/m_e}$ is the electron thermal speed. The thermal correction $3k^2 v_{te}^2$ gives Langmuir waves a finite group velocity $v_g = 3k v_{te}^2/\omega$, allowing energy to propagate. In the cold-plasma limit ($T_e \to 0$), the wave reduces to a pure oscillation at $\omega_{pe}$ with zero group velocity.

2.3 Landau Damping

The fluid derivation misses a crucial kinetic effect: Landau damping. Even in a collisionless plasma, Langmuir waves are damped through wave-particle interaction. Electrons with velocities slightly below the phase velocity $v_\phi = \omega/k$ are accelerated by the wave, extracting energy from it.

A rigorous kinetic treatment using the Vlasov equation gives the Landau damping rate for $k\lambda_D \ll 1$:

$$\gamma_L = -\sqrt{\frac{\pi}{8}}\,\omega_{pe}\,(k\lambda_D)^{-3}\,\exp\!\left(-\frac{1}{2k^2\lambda_D^2}\right)$$

where $\lambda_D = v_{te}/\omega_{pe}$ is the Debye length. Key features of Landau damping:

Damping is exponentially small for long wavelengths ($k\lambda_D \ll 1$)
Damping becomes strong when $k\lambda_D \gtrsim 0.3$
The mechanism is collisionless -- it arises from the resonance $v = \omega/k$
For a Maxwellian distribution, there are always more slow particles than fast ones near $v_\phi$, so net energy flows from wave to particles

Landau damping was predicted theoretically by Lev Landau in 1946 and confirmed experimentally by Malmberg and Wharton in 1964. It remains one of the most important examples of collisionless dissipation in plasma physics.

2.4 Computational Exploration

The following code plots the Bohm-Gross dispersion relation and the Landau damping rate as a function of wavenumber. Observe how the dispersion curve departs from the cold-plasma limit at short wavelengths, and how damping becomes significant for $k\lambda_D \gtrsim 0.3$.

Langmuir Waves Simulation

Python

script.py52 lines

import numpy as np
import matplotlib.pyplot as plt

# Parameters
omega_pe = 1.0        # normalize to plasma frequency
v_te = 0.1            # electron thermal speed (units of c or arbitrary)
lambda_D = v_te / omega_pe  # Debye length

k = np.linspace(0.01, 5.0 / lambda_D, 500)
k_norm = k * lambda_D  # normalized wavenumber

# Bohm-Gross dispersion relation
omega_BG = np.sqrt(omega_pe**2 + 3 * k**2 * v_te**2)

# Cold plasma limit
omega_cold = np.full_like(k, omega_pe)

# Landau damping rate (valid for k*lambda_D < ~0.5)
with np.errstate(over='ignore', divide='ignore'):
    gamma_L = -np.sqrt(np.pi / 8) * omega_pe * (k_norm)**(-3) * np.exp(-1.0 / (2.0 * k_norm**2))
    gamma_L = np.where(np.isfinite(gamma_L), gamma_L, 0.0)

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 5))

# Left: Dispersion relation
ax1.plot(k_norm, omega_BG / omega_pe, 'orange', linewidth=2, label='Bohm-Gross')
ax1.plot(k_norm, omega_cold / omega_pe, '--', color='gray', linewidth=1.5, label=r'Cold plasma ($\omega = \omega_{pe}$)')
ax1.plot(k_norm, np.sqrt(3) * k_norm, ':', color='cyan', linewidth=1.5, label=r'$\omega = \sqrt{3}\,k v_{te}$')
ax1.set_xlabel(r'$k\lambda_D$', fontsize=13)
ax1.set_ylabel(r'$\omega / \omega_{pe}$', fontsize=13)
ax1.set_title('Bohm-Gross Dispersion', fontsize=14)
ax1.set_xlim(0, 5)
ax1.set_ylim(0, 10)
ax1.legend(fontsize=10)
ax1.grid(True, alpha=0.3)

# Right: Landau damping rate
ax2.plot(k_norm, np.abs(gamma_L) / omega_pe, 'red', linewidth=2)
ax2.set_xlabel(r'$k\lambda_D$', fontsize=13)
ax2.set_ylabel(r'$|\gamma_L| / \omega_{pe}$', fontsize=13)
ax2.set_title('Landau Damping Rate', fontsize=14)
ax2.set_xlim(0, 1.0)
ax2.set_ylim(0, 0.5)
ax2.axvline(x=0.3, color='yellow', linestyle='--', alpha=0.5, label=r'$k\lambda_D = 0.3$')
ax2.legend(fontsize=10)
ax2.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('output.png', dpi=100, bbox_inches='tight', facecolor='#0a0a1a')
plt.close()
print("Langmuir wave dispersion and Landau damping plotted.")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

2.5 Applications and Physical Significance

Langmuir waves play a central role across plasma physics:

Plasma diagnostics: The plasma frequency sets the cutoff for electromagnetic wave propagation. Microwave interferometry exploits this to measure electron density in fusion experiments.
Laser-plasma interaction: Stimulated Raman scattering involves the decay of a laser photon into a scattered photon plus a Langmuir wave, a major concern in inertial confinement fusion.
Space physics: Type III solar radio bursts are produced when electron beams excite Langmuir waves in the solar wind, which then convert to electromagnetic radiation.
Particle accelerators: Plasma wakefield acceleration uses laser- or beam-driven Langmuir waves to produce accelerating gradients exceeding 100 GV/m, orders of magnitude beyond conventional accelerators.

← Previous Next →