Dispersion relation

From PlasmaWiki

The dispersion relation describes the relationship between the frequency $ω$ and the wavenumber $\mathbf{k}$ of a wave.

[edit] Zero order solution

In a homogenous unmagnetized cold plasma, the dispersion relation is given by:

$\omega^{2}-\omega_{p}^{2}=c^2 k^2$

Where $ω p$ is the plasma frequency. For a full derivation, see Electromagnetic waves (dispersion).

[edit] Solution from Vlasov Equation

One can derive a dispersion relation by using a perturbation expansion of the distribution function and linearizing the collisionless Vlasov equation. This gives the result:

$D(k,\omega)=1-\frac{4\pi e^2}{m_e} \int d^3v\,f_0\frac{1}{\left(\omega-kv_z\right)^2}=0$

[edit] Derivation

We set $f = f 0 + f 1$ , $f_1\ll f_0$ , and assume $E 0 = v 0 = B 0 = 0$ as well as $v\ll c$ . This reduces the collisionless Vlasov equation to:

$\frac{\partial f_1}{\partial t} + \mathbf{v}_1 \cdot \frac{\partial f_1}{\partial \mathbf{x}} - \frac{e}{m}\mathbf{E}\cdot \frac{\partial f_0}{\partial \mathbf{v}} =0$

Then we assume the form $f 1 ˜exp(i [k z - ω t])$ , and use the electrostatic approximation, $\mathbf{E}=-\nabla\phi$ . This gives: