Beam-Cyclotron Instabilities by Moderately Relativistic Electron Ring-Beam

High frequency (ω>ω_e) beam-cyclotron instabilities excited by moderately Jelativistic electron ring-beam are studied in regions where the electron plasma fre-quency wc is much higher than the electron cyclotron frequency Q_e(i.e.ω_e²>>M_e²).It is found that the beam-cyclotron modes of energetic electrons are unstable in the cases of N²cos²θ<1 and N²cos²θ>and stable in the case of N²cos²θ=1,where N is the refractive index and d is the propagation angle.When the frequency of beam-cyclotron mode is nearly one of the eigenmodes of cold background plasma,i.e.to ≈ω_e,the growth rate is the largest.Numerical solutions of the dispersion equation are also presented and discussed in this paper.