and . Here,

where is the species- Larmor radius.

The first root of the dispersion relation (1074) is

(1089) |

Note that the cyclotron harmonic resonances appearing in the dispersion
relation (1088) are of *zero width* in frequency space: *i.e.*, they are
just like the resonances which appear in the cold-plasma limit.
Actually, this is just an artifact of the fact that the waves we are studying
propagate *exactly perpendicular* to the equilibrium magnetic field. It is
clear from an examination of Eqs. (1064) and (1066) that the cyclotron
harmonic resonances originate from the zeros of the plasma dispersion
functions. Adopting the usual rule that substantial damping takes place
whenever the arguments of the dispersion functions are less than or of
order unity, it is clear that the cyclotron harmonic resonances lead to
significant damping whenever

(1090) |

The appearance of the cyclotron harmonic resonances in a warm plasma is of great practical importance in plasma physics, since it greatly increases the number of resonant frequencies at which waves can transfer energy to the plasma. In magnetic fusion these resonances are routinely exploited to heat plasmas via externally launched electromagnetic waves. Hence, in the fusion literature you will often come across references to ``third harmonic ion cyclotron heating'' or ``second harmonic electron cyclotron heating.''

The other roots of the dispersion relation (1074) satisfy

with the eigenvector . In the cold plasma limit, , this dispersion relation reduces to that of the

However, another mode also exists. In fact, if we look for a mode with a
phase velocity much less than the velocity of light (*i.e.*,
) then it is clear from (1083)-(1086) that
the dispersion relation is approximately

Let us consider electron Bernstein waves, for the sake of definiteness.
Neglecting the contribution of the ions, which is reasonable provided that
the wave frequencies are sufficiently high, the dispersion relation (1092)
reduces to

(1093) |

(1094) |

At small values of , the phase velocity becomes large, and it is no longer legitimate to neglect the extraordinary mode. A more detailed examination of the complete dispersion relation shows that the extraordinary mode and the Bernstein mode cross over near the harmonics of the cyclotron frequency to give the pattern shown in Fig. 39. Here, the dashed line shows the cold plasma extraordinary mode.

In a lower frequency range, a similar phenomena occurs at the harmonics of the ion cyclotron frequency, producing ion Bernstein waves, with somewhat similar properties to electron Bernstein waves. Note, however, that whilst the ion contribution to the dispersion relation can be neglected for high-frequency waves, the electron contribution cannot be neglected for low frequencies, so there is not a complete symmetry between the two types of Bernstein waves.