Plasma frequency

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The plasma frequency is the inertial frequency of particles in a quasi-neutral plasma. For example, for electrons it is the frequency at which displaced electrons will oscillate arnound the fixed ions. For electrons the frequency is given by:

\omega_{pe}=\sqrt{\frac{4\pi n_e e^2}{m_e}}

For ions:

\omega_{pi}=\sqrt{\frac{4\pi n_i Z^2 e^2}{m_i}}

Let us start out with ne0 = ni0 = n0 (a homogenous, quasineutral plasma with singly charged ions). Also lets take n_{e1}\ll n_0, so that we only need to look at the first order perturbation. We get from Poisson's equation :

\nabla\cdot\mathbf{E}_1=-4\pi e n_{e1}

From the continuity equation :

\frac{\partial n_{e1}}{\partial t} + n_{e0} \nabla \cdot \mathbf{v}=0

And from the Lorentz force :

m_e \frac{\partial \mathbf{v}_1}{\partial t} = - e \mathbf{E}_1

Operating on this last equation with \nabla\cdot, and plugging in from the Poisson and continuity equations:

m_e \frac{\partial}{\partial t}\left(-\frac{1}{n_{e0}} \frac{\partial n_{e1}}{\partial t} \right)= - e \left(-4\pi e n_{e1}\right)

Which becomes:

\frac{\partial^2 n_{e1}}{\partial t^2}= -\frac{n_{e0}}{m_e} e^2 4\pi n_{e1}

Or, using:

\omega_{pe}=\sqrt{\frac{4\pi n_e e^2}{m_e}}

we can write:

\frac{\partial^2 n_{e1}}{\partial t^2}= -\omega_{pe}^2 n_{e1}

Which gives the solution:

n_{e1}=A(x) e^{i\omega_{pe} t}

Where A(x) is an initial condition.

This page was recovered in October 2009 from the Plasmagicians page on Plasma_frequency dated 20:50, 13 June 2006.

Retrieved from "https://qed.princeton.edu/main/PlasmaWiki/Plasma_frequency"
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