Bernstein Modes in a Weakly Relativistic e-e+ Plasma

This thesis examines the behaviour of a homogeneous and quasineutral, equal-mass (electron- positron) plasma in the presence of a constant magnetic field. Two sets of comparisons are made: between e-e+ and e--ion plasmas and between treatments of a progressively more relativistic nature. Fundamental plasma quantities: plasma frequency, op cyclotron frequency, O; and Debye length, lambdaD, are briefly introduced in Chapter 1. The peculiar features of electron-positron plasmas are illustrated in Chapter 2. Quantities defined in Chapter 1 are redefined for e-e+ plasmas. Examples of physical situations where e-e+ plasmas may occur are given. The predictions of cold plasma theory for e-e+ plasmas are summarized at the end of this chapter. Chapter 3 embraces the kinetic theory upon which the remainder of the thesis relies. After explaining the need for a kinetic theory, the development of that theory is reviewed. The first part of the chapter shows the microscopic (Klimontovich) description. Next the necessary concepts from Gibbs' and Boltzmann's statistical theory are presented. These ideas are married in the BBGKY hierarchy. The lowest order expansion of the hierarchy gives the Vlasov equation. A dielectric treatment can then be carried out using the Vlasov equation and Maxwell's equations. At this point the equilibrium conditions are stated. After some analysis general dispersion relations are found for both e- -ion and e-e+ plasmas. These expressions are general in the sense that they permit a choice of momentum distribution. In Chapter 4 the momentum distribution function is the familiar Maxwellian distribution function. The dispersion relations are then used to derive the (electrostatic) Bernstein modes. Bernstein modes propagate perpendicular to the magnetic field and resonate at electron cyclotron harmonics (o ≈ nOe). In an electron-ion plasma, there are seen to be gaps in the frequency spectrum away from nOe where these modes may not propagate. e-e+ plasmas are different: the theory leading to frequency gaps is exact and not a consequence of approximation. These relations are then illustrated. Original work begins in Chapter 5. In this thesis, interest in a weakly relativistic plasma stems from the wish to observe the transition between the existing non-relativistic and fully relativistic kinetic treatments. The relativistic nature of these treatments is governed by the parameter a=moc2/kT: 10 ≤ a ≤ 100 corresponds to weakly relativistic conditions. Chapter 5 is concerned with weakly relativistic e-e+ plasmas. A novel combination of non-relativistic distribution function and otherwise fully relativistic dispersion relation leads to the dispersion relation for weakly relativistic e-e+ plasmas. This expression is then prepared for inclusion in computer code: it is restated in dimensionless units and rearranged so that double quadrature becomes single quadrature with a special function. The design of the computer code is discussed in Chapter 6. The resulting dispersion curves are shown arid described in Chapter 7. It is demonstrated that, as for e--ion plasmas, the introduction of a weakly (or fully) relativistic treatment sees a broadening of the frequencies at which resonance occurs and a downshift in those frequencies. These results have been described briefly in conference proceedings [1,2]. This chapter goes on to consider the possibility of an approximate analytical approach and suggests the direction future work will take. There are three appendices. They deal respectively with: the properties of certain special functions; contour integration; and a listing of the code which is described in Chapter 6