Contour of Ribbon Beam In Uniform Magnetic Field

In this example, dynamics of a ribbon beam in uniform magnetic field is analyzed and analytical prediction for it's envelope is compared with numerical simulations.

In the case of ribbon beam propagating along the uniform magnetic field, trajectory of the electron on the edge of the beam is given by the following equation (in SI units):

The meaning of the parameters is the same as in Ribbon Beam Contour example: I_0 is linear current density of the beam along the y-axis (i.e. full beam current I divided by beam "height"), \epsilon_0 is vacuum permittivity, v is speed along the z-axis, x_0 and z_0 specify it's initial position, B = B_z is magnetic field amplitude, and omega_g is gyration frequency. Derivation of this equation can be found in supplementary jupyter notebook.

Numerical results can be compared with analytical predictions using the same procedure as in the field-free case.

Simulation parameters are following: total simulation time is 5 [ns], time step is 5 * 10^{-12} [s]. Beam current is 0.008 [A], particles’ speed corresponds to 1 [keV] energy. 500 new macroparticles are added each time step. Magnetic field is B = 0.02 [T].

todo: insert several figures for different values of magnetic field. The beam envelope behavior for several values of the magnetic field can be seen on the following figures...

References:
[Alyamovsky], paragraph 3.1; for trajectory equations: p. 50.