Dispersion of Waves in Cyclotron Harmonic Resonance Regions in Plasmas

The dispersion of waves near electron cyclotron harmonics is investigated including to first order wavenumbers parallel to the magnetic field. The proper relativistic form for the dielectric tensor elements is applied. The relativistic formulation is compared with the nonrelativistic one applicable for larger wavenumbers parallel to magnetic field and the differences are noted. In either analysis, different behaviors result depending on whether the wavenumber or frequency is taken to be complex. In the former case, the waves near the Appleton-Hartree values are localized and a gap exists between them and the higher-order cyclotron harmonic plasma waves. In the latter case, this gap does not occur but the relativistic analysis shows a rapid rise in frequency above the harmonic as one tends to zero wavenumber perpendicular to magnetic field. It is also shown that the large argument expansion in terms of inverse argument of the nonrelativistic Z function produces incorrect dispersion results in certain regions.