non-linear electron excitation of plasma waves

Electron Acceleration by Non-linear Plasma Wave Excitation

Consider an electron pulse (or “bunch”) of average density $\rho_B$ and average bunch velocity $\vec{v} _B$ in a surrounding plasma of average electron density $n_P$. One is interested in deriving the propagation equations for plasma waves with relativistic phase velocities. A simplifying assumption is the presence of relatively slow moving ions at a very small fraction of the speed of light c which is realistic for plasma ion temperatures of less than 10,000 K. One may also neglect in a first approximation the influence of the excited wake-field that affects the time-evolution of the electron pulse shape. Furthermore, one can consider the configuration of a cylindrical plasma in the absence of external magnetic fields; along the plasma containing tube $z$- axis one has a one-dimensional system for which Maxwell's equations can be written in the following simplified form for the electrical field $\vec{E}$, average electron velocity in plasma $v$, charge density $\rho = {\rho}_B + \delta n_P$, current density

\begin{displaymath}i = [(n_P +\delta n_P) v ~+ ~n_B v_B ]e\end{displaymath}

and perturbed electron density $+\delta n_P$:

\begin{displaymath}\partial E / \partial z = 4\pi \rho\end{displaymath}

and

\begin{displaymath}\partial E /\partial E t = - 4\pi i \end{displaymath}

.

The equation of motion of a plasma electron with momentum $p_e$ in the wake of a relativistic electron bunch of average velocity $\vec{v} _B$ can be then written as:


\begin{displaymath}\partial p_e / \partial t = e E. \end{displaymath}

Because the driving electron pulse has a relativistic average velocity one can expect solutions of the equations of motion to be of the form of travelling waves:


\begin{displaymath}E(z,t) = E (z~ - ~ v_B t)\end{displaymath}

.

Molecular dynamics experiments or computer simulations that include these equations provide results in the form of numerical data that are consistent with such travelling wave solutions.



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