Abstract
A dispersion relation which includes finite axial wavevectors, k, is derived in the rest frame of the beam where the instability is assumed to be electrostatic. The dispersion relation is obtained in the lab frame by a Lorentz transformation. The instability is found to have a finite k bandwidth with the most unstable mode occurring at k≠0. The growth rate for the most unstable mode is found to be reduced by approximately 1/γ2 from the nonrelativistic result while the real frequency of this mode remains virtually unchanged. The effects of finite vz shear, which are left out of the analysis, are estimated and discussed.
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