0.1 Electron Acceleration by Non-linear Plasma Wave Excitation
Consider an electron pulse (or “bunch”) of average density ρB and average bunch velocity vB in a
surrounding plasma of average electron density nP . 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 E, average electron velocity in plasma v, charge density ρ = ρB + δnP , current
density
![i = [(n + δn )v + n v ]e
P P B B](https://images.physicslibrary.org/cache/objects/791/make4ht/PlasmaWaveExcitation0x.png)
and perturbed electron density +δnP :
and
.
The equation of motion of a plasma electron with momentum pe in the wake of a relativistic
electron bunch of average velocity vB can be then written as:
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:
.
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.
