Schematic diagram of the charge densities and rotation frequency in a non-neutral plasma column consisting of three species. Cylindrical conductors bound the plasma at and rw ; in many experiments, the inner conductor is not present. Centrifugal and/or charge separation 25 can cause the species to separate radially, in order of largest to smallest charge to mass ratio.
Imaginary part of the admittance versus frequency for an wall perturbation, in the cold fluid limit, at different collision frequencies ν (measured in units of for the density profile of Eq. (79) with , where rw is the wall radius).
Imaginary part of the admittance for an cold-fluid cyclotron mode in a plasma with an inner conductor of radius , for two values of the profile width (in units of rw ) and for , 1/100 and 1/1000 (in order from broadest to sharpest admittance curves). The arrow shows the frequency for a step profile, Eq. (84) .
Schematic of a plasma for which there is a single upper hybrid cutoff. Varying ω moves the profile vertically and changes the location of the cutoff.
Schematic of a plasma with parameters chosen so that there is a single resonance.
Perturbed potential of Bernstein modes for two different temperatures but the same frequency, and assuming uniform , with , and density given by Eq. (79) . Solid lines: numerical solution of Eq. (85) . Dashed lines: WKB approximation, Eqs. (147) , (150) , (138) , (101) , and (107) . The dashed vertical line shows the location of the upper hybrid cutoff. (a) and (b) The WKB approximation improves for smaller rc , and breaks down as expected near r = 0 and .
Admittance versus frequency for uniform α, density given by Eq. (79) , and for , at three temperatures, with corresponding cyclotron radii rc measured in units of rw . As rc decreases, the admittance approaches cold fluid theory, Eq. (70) . For larger rc , Bernstein mode peaks are evident. Curves: WKB theory given by Eq. (151) , (138) , and (127) . Dots: numerical solution of Eq. (85) , shown only in the regime of validity for Bernstein solutions of Eq. (85) , . The arrow at shows the prediction of Eq. (77) for the frequency of the surface cyclotron wave.
Frequency spectrum of internal Bernstein modes for two values of the cyclotron radius (measured in units of rw ). Plasma parameters are chosen, so that there is a single cutoff: the profile is assumed to be uniform, and density is given by Eq. (79) . Crosses: numerical solution of Eq. (85) , valid for the low order modes with . Dots: WKB approximation, Eq. (155) , valid for .
The five lowest order internal Bernstein potential eigenmodes for for the same plasma parameters as in Fig. 12 , taking .
α and β profiles for a case with two upper hybrid cutoffs.
β and representative α profiles (in units of ) for a single species plasma with monotonically decreasing density . Frequencies are chosen so that there is one cutoff and one or more resonances. (a) and (b) .
Orbit coefficients in Eqs. (13) and (14) .
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