PlanetPhysics/Plasma Wave Excitation

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 {\mathbf 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:

$$E(z,t) = E (z~ - ~ v_B t)$$.

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.