PlasmaWaveExcitation 2009-06-08 00:20:25 2009-06-08 00:27:17 Experiment electron pulse plasma electron plasma ion temperature plasma wave excitation excited wake-field average electron velocity in plasma current density in plasma relativistic phase velocities \subsection{Electron acceleration by non-linear plasma wave excitation \subsection{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 slowly 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 $$i = [(n_P +\delta n_P) v ~+ ~n_B v_B ]e$$ and perturbed electron density $+\delta n_P$: $$\partial E / \partial z = 4\pi \rho$$ and $$ \partial E /\partial E t = - 4\pi i $$. 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: $$ \partial p_e / \partial t = e E. $$ 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: $$ E(z,t) = E (z~ - ~ v_B t)$$. {\bf [More to come...]}