[플라즈마 물리][Plasma Physics]CH1 Introduction - Plasma Frequency

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Plasma Frequency

Consider a hypothetical slab of plasma, where we assume that the ions have infinite mass(immobile) and the electrons can move freely through the ions.

1. Suppose the electron slab is displaced a distance $x$ to the right of the ion slab and then allowed to move freely.

2. An electric field will be set up, causing the electron slab to be pulled back toward the ions.

3. When the electrons exactly overlap the ions (when $x=0$), the net force is zero, but the electron slab overshoots.

4. The net result is harmonic oscillation. The frequency of the oscillation is called the electron plasma frequency.

Derivation for Plasma Frequency

From Gauss’s law,

$$\oint \vec{D} \cdot d\vec{a} = Q$$

we have

$$\epsilon_0 EA = neAx$$

or

$$E=\frac{nex}{\epsilon_0}$$

Since the force is given by

$$F=QE=(-neAx)(\frac{nex}{\epsilon_0})=nAxm_e \frac{d^2x}{dt^2}$$

we obtain the equation of motion

$$\begin{aligned} \frac{d^2x}{dt^2} = - \left( \frac{ne^2}{m_e \epsilon_0} \right) x\end{aligned}$$

or

$$\begin{aligned} \frac{d^2x}{dt^2} +\omega_{pe}^2x=0\end{aligned}$$

where

$$\begin{aligned} \boxed{\omega_{pe}=\sqrt{\frac{ne^2}{m_e \epsilon_0}}}\end{aligned}$$

This is electron plasma frequency.
$\bullet$ If there are no collisions, the disturbance will oscillate indefinitely.
$\bullet$ Debye length, thermal velocity, and plasma frequency are inter-related.

$$\frac{v_{th}}{\omega_{pe}}=\frac{\sqrt{\frac{K_BT}{m}}}{\sqrt{\frac{ne^2}{m\epsilon_0}}}=\sqrt{\frac{\epsilon_0 KT}{ne^2}}=\lambda_D$$

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