The authors carried out systematic investigations of the dynamics of relativistic electron and nonrelativistic ion beam propagation in both an electric field and an axisymmetric nonuniform magnetic field. In previous papers, the investigation results of the acceleration, and the charge and current compensation of HHCIBs in one and two linac cusps are reported. These results have shown that both in the presence and in the absence of an accelerating electric field, the following effects take place: (1) charge and current compensation of HHCIB in the accelerating gaps, (2) stability of the ion beam during times that substantially exceed the inverse ion Langmuir and Larmor frequencies. The performed numerical simulations have also shown that a required charge neutralization of the ion beam is only achieved within accelerating gaps of linac. This investigation is directed towards: (1) reaching of the optimal relation between the external electric field parameters and the compensating electron beam parameters; (2) defining of the optimal thickness of annular beam at which the ion beam is effectively accelerated and at a time is remaining compensated and not losing a stability.
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