Waves in Non-Neutral Plasmas - Nonneutral Plasma Group

December 21, 2012 6:0 World Scientific Review Volume - 9in x 6in anderegg˙waves˙nnpsWaves in Non-neutral Plasma 9identical to the image charge model for infinite length and small amplitude,Eq. (1.6).One can see from Eq. (1.14) that for the m θ = 1 mode, the diocotronfrequency is the plasma rotation frequency reduced by (R p /R w ) 2 . Also fora small radius plasma, the m θ = 2 diocotron frequency is almost at theplasma rotation frequency.1.2. Plasma WavesWe are considering a long plasma in a conducting cylinder with a magneticfield B aligned with the trap axis. The magnetic field is strong enough tomake the cyclotron frequency much larger than the plasma frequency, butthe exact magnitude of the magnetic field is not important.To derive the frequency of plasma waves in a trap, we start by writingthe continuity equation,and Newton equation∂n∂t + ∂ ∂z n · v z = 0 (1.15)m ∂v z∂t = qE z = −q ∂ φ. (1.16)∂zNote that we have kept only the z dynamics in these two equations, but wewill keep all components for the Poisson equation:∇ 2 φ = −4πqδn. (1.17)We will now assume that the density perturbation is δn(r) exp{i(m θ θ +k z z − ωt)}, and the above equations become−iωδn + n ik z δv z = 0; (1.18)−miωδv z = −qik z δφ; (1.19)−k 2 δφ = −4πqδn. (1.20)Combining these 3 equations, one gets the Trivelpiece-Gould 12 mode dispersionrelation for a cold plasma,where ω p is the plasma frequency.ω 2 = k2 z 4πq 2 nk 2 m= k2 zk 2 ω2 p (1.21)