A 2-D fluid electron and 1-D particle ion hybrid model of ion noise in microwave tubes

Since there is always some ambient gas in electron beam devices, background ionization is ubiquitous. For long pulse times, the electrostatic potentials associated with this ionization can reach significant levels and give rise to such observed phenomena as phase noise in microwave tubes. Observations of noise in microwave tubes such as coupled-cavity traveling wave tubes (CC-TWTs) and klystrons have been discussed in the literature. A hybrid model has been developed in which the electron beam is treated as a 2D fluid using the beam envelope equation, and the ions generated by beam ionization are treated as discrete particles in 1D. The effect of secondary electrons is neglected in the present analysis. The ionization rate depends on the ambient gas pressure and species as well as on the electron beam current and energy. Based on this rate, ions are created and distributed on an axial grid on each time step. The ion charge is then mapped onto the grid, and Poisson´s equation is then solved in 1D under the assumption that the transverse scale lengths are less than the betatron wavelength of the electron beam. The ion charge distribution is then used to integrate the beam envelope equation that updates the beam equilibrium. The ion motion is then integrated subject to the wall potential, the space-charge potential of the electron beam, and the self-consistent ion potential. This process is iterated over any desired pulse time.