# The Cluster's Bohm-Gross Dispersion

A page about the Bohm-Gross dispersion relation — the frequency of Langmuir waves in a warm plasma.

## The Bohm-Gross relation

The Bohm-Gross dispersion relation describes the frequency of longitudinal electron plasma waves (Langmuir waves) in a warm plasma:
omega^2 = omega_p^2 + 3 k^2 v_th^2 = omega_p^2 + 3 k^2 (k_B T_e / m_e)
where omega_p is the plasma frequency, k is the wave number, and v_th = sqrt(k_B T_e / m_e) is the thermal velocity. The 3 comes from the 3D velocity distribution. For 1D: the factor is 1. For 2D: the factor is 2. In the cluster, the edit Bohm-Gross relation gives an edit frequency.

## The derivation

The dispersion is derived from the fluid equations coupled with Poisson's equation:
nabla . E = rho / epsilon_0 (Poisson)
m_e n_e (dv_e/dt) = -n_e e E - grad p_e (momentum, with p_e = n_e k_B T_e)
dn_e/dt + nabla . (n_e v_e) = 0 (continuity)

Linearizing around n_0, v_0 = 0, p_0 = n_0 k_B T_e, and combining gives:
omega^2 = omega_p^2 + 3 k^2 v_th^2

In the cluster, the edit derivation combines edit fluid equations.

## The Landau correction

The fluid derivation gives the Bohm-Gross relation. Kinetic theory (Landau, Vlasov) gives a small correction:
omega^2 = omega_p^2 + 3 k^2 v_th^2 - 15 (k^2 v_th^2 / omega_p^2) omega_p^2 + i gamma
The damping rate is:
gamma = - sqrt(pi / 8) (omega_p / k v_th)^3 exp(-1/2k^2 lambda_D^2)
where lambda_D = v_th / omega_p is the Debye length. The kinetic correction is small for k lambda_D << 1 (long wavelength).

In the cluster, the edit Landau correction adds an edit damping.

## The Debye length

The Debye length is the screening length in a plasma:
lambda_D = sqrt(epsilon_0 k_B T_e / (n_e e^2)) = v_th / omega_p
For typical parameters (n_e = 10^{19} m^{-3}, T_e = 10^4 K): lambda_D ~ 2.1 x 10^{-4} m = 0.21 mm.
Plasma requires: N_D = (4 pi / 3) n_e lambda_D^3 >> 1 (many particles per Debye sphere).

In the cluster, the edit Debye length is the edit screening length.

## Applications

- **Langmuir waves**: The primary oscillation mode of a plasma
- **Plasma heating**: ECRH, ICRH rely on plasma wave dispersion
- **Fusion diagnostics**: Measuring omega_p gives the electron density
- **Space physics**: Langmuir waves in the magnetosphere and solar wind
- **Plasma oscillations in metals**: The conduction electron gas supports plasma oscillations

In the cluster, edit applications include:
- edit Langmuir waves
- edit Plasma heating
- edit Fusion diagnostics
- edit Space physics
- edit Plasma oscillations

## This relation

This page is about the Bohm-Gross dispersion. omega^2 = omega_p^2 + 3 k^2 v_th^2. Derived from fluid equations + Poisson. Landau damping adds the imaginary part. The relation is real.