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/|velocity]] ${\overrightarrow{v}}_B$ in a surrounding [[../PlasmaDisplayPanel/|plasma]] of average electron density $n_P$. One is interested in deriving the propagation equations for plasma [[../CosmologicalConstant2/|waves]] with relativistic phase velocities. A simplifying assumption is the presence of relatively slow moving ions at a very small fraction of the [[../CosmologicalConstant/|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 [[../NeutrinoRestMass/|magnetic fields]]; along the plasma containing tube $z$- axis one has a one-dimensional [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|system]] for which [[../FluorescenceCrossCorrelationSpectroscopy/|Maxwell's equations]] can be written in the following simplified form for the [[../FluorescenceCrossCorrelationSpectroscopy/|electrical field]] $\overrightarrow{E}$, average electron velocity in plasma $v$, [[../Charge/|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 $\partial E/\partial Et=-4\pi i$.

The equation of [[../CosmologicalConstant/|motion]] of a plasma electron with [[../Momentum/|momentum]] $p_e$ in the wake of a relativistic electron bunch of average velocity ${\overrightarrow{v}}_B$ can be then written as:

$\partial p_e/\partial t=eE.$

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 [[../AAT/|computer simulations]] that include these equations provide results in the form of numerical data that are consistent with such travelling wave solutions.