Experimental and Theoretical Investigation into the Effect of the   Electron Velocity Distribution on Chaotic Oscillations in an Electron Beam   under Virtual Cathode Formation Conditions
  The effect of the electron transverse and longitudinal velocity spread at the entrance to the interaction space on wide-band chaotic oscillations in intense multiple-velocity beams is studied theoretically and numerically under the conditions of formation of a virtual cathode. It is found that an increase in the electron velocity spread causes chaotization of virtual cathode oscillations. An insight into physical processes taking place in a virtual cathode multiple velocity beam is gained by numerical simulation. The chaotization of the oscillations is shown to be associated with additional electron structures, which were separated out by constructing charged particle distribution functions. 
Yu. A. Kalinin and A. E. Hramov