2018년 4월 18일 수요일

[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.

Suppose the electron slab is displaced a distance \(x\) to the right of the ion slab and then allowed to move freely.
An electric field will be set up, causing the electron slab to be pulled back toward the ions.
When the electrons exactly overlap the ions (when \(x=0\)), the net force is zero, but the electron slab overshoots.
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

\[\epsilon_0 EA = neAx\]

\[E=\frac{nex}{\epsilon_0}\]

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

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

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

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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2018년 4월 17일 화요일

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

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Plasma Parameter

\(\bullet\) The plasma parameter is defined as

\[\begin{aligned} \boxed{ N_D =n\frac{4}{3}\pi \lambda_D^3 }=1.38 \times 10^6 \frac{T^{\frac{3}{2}}}{n^{\frac{1}{2}}} [T](in^{\circ}K)\end{aligned}\]

which is the number of plasma particles in a Debye sphere.

\(\bullet\) For Debye shielding to occur, and for the description of a plasma to be statistically meaningful, the number of particles in a Debye sphere must be large; that is \(N_D>>1\)

\(\bullet\) Plasma parameter is a measure of the ratio of the mean plasma kinetic energy to potential energy.

\[\frac{K.E.}{P.E.} \simeq \frac{\frac{3}{2}K_BT}{\frac{e^2}{4\pi\epsilon_0 \lambda_D}} \simeq \frac{9}{2}N_D >> 1\]

\(\bullet\) Thus \(N_D>>1\) means that the potential energy of a particle due to its nearest neighbor is much smaller than its kinetic energy. If this were not the case, there would be a strong tendency for electrons and ions to bind together into atoms, thus destroying the plasma.

\(\bullet\) An ideal gas corresponds to zero potential energy between the particles. Since the plasma parameter is large, the plasma may be treated as an ideal gas of charged particles, that is, a gas that can have a charge density and electric field but in which no two discrete particles interact.

\(\bullet\) In deriving the Debye potential, we assumed that the electrostatic energy was small compared to the thermal energy. The largeness of the plasma parameter guarantees the validity of the Debye potential.