Waves in Magnetized Plasmas

for both ions and electrons, where and are the perturbed electric and magnetic fields, respectively. Likewise, is the perturbed distribution function, and the equilibrium distribution function.

In order to have an equilibrium state at all, we require that

(1037) |

Let the trajectory of a particle be , . In the
unperturbed state

(1038) | |||

(1039) |

It follows that Eq. (1036) can be written

where is the total rate of change of , following the unperturbed trajectories. Under the assumption that vanishes as , the solution to Eq. (1040) can be written

where , is the unperturbed trajectory which passes through the point , when .

It should be noted that the above method of solution is valid for any set of equilibrium electromagnetic fields, not just a uniform magnetic field. However, in a uniform magnetic field the unperturbed trajectories are merely helices, whilst in a general field configuration it is difficult to find a closed form for the particle trajectories which is sufficiently simple to allow further progress to be made.

Let us write the velocity in terms of its Cartesian components:

(1042) |

where is the cyclotron frequency. The above expression can be integrated to give

Note that both and are constants of the motion. This implies that , because is only a function of and . Since , we can write

Let us assume an
dependence of all perturbed quantities, with lying in the - plane.
Equation (1041) yields

Making use of Eqs. (1043)-(1049), and the identity

(1051) |

where

(1053) |

Maxwell's equations yield

(1054) | |||

(1055) |

where is the perturbed current, and is the dielectric permittivity tensor introduced in Sect. 4.2. It follows that

where is the species- perturbed distribution function.

After a great deal of rather tedious analysis, Eqs. (1052) and (1056) reduce to
the following expression for the dielectric permittivity tensor:

(1058) |

(1059) | |||

(1060) | |||

(1061) |

The argument of the Bessel functions is . In the above, denotes differentiation with respect to argument.

The dielectric tensor (1057) can be used to investigate the properties of waves in just the same manner as the cold plasma dielectric tensor (485) was used in Sect. 4. Note that our expression for the dielectric tensor involves singular integrals of a type similar to those encountered in Sect. 6.2. In principle, this means that we ought to treat the problem as an initial value problem. Fortunately, we can use the insights gained in our investigation of the simpler unmagnetized electrostatic wave problem to recognize that the appropriate way to treat the singular integrals is to evaluate them as written for , and by analytic continuation for .

For Maxwellian distribution functions, we can explicitly perform the velocity
space integral in Eq. (1057), making use of the identity

(1062) |

(1063) |

Here, , which is the argument of the Bessel functions, is written

whilst and represent the plasma dispersion function and its derivative, both with argument

Let us consider the cold plasma limit,
. It follows from
Eqs. (1065) and (1066) that this limit corresponds to
and
. From Eq. (1030),

as . Moreover,

as . It can be demonstrated that the only non-zero contributions to , in this limit, come from and . In fact,

(1070) | |||

(1071) | |||

(1072) |

and . It is easily seen, from Sect. 4.3, that the above expressions are identical to those we obtained using the cold-plasma fluid equations. Thus, in the zero temperature limit, the kinetic dispersion relation obtained in this section reverts to the fluid dispersion relation obtained in Sect. 4.