Kinetic stability properties of an intense relativistic electron ring in a high-current betatron accelerator

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Abstract
The kinetic stability properties of an intense relativistic electron ring located at the midplane of an externally applied betatron field are investigated within the framework of the linearized Vlasov–Maxwell equations, including the important influence of electromagnetic effects and surface‐wave perturbations. Stability properties are calculated for eigenfrequency ω near harmonics of the relativistic cyclotron frequency ωcz in the applied betatron field. Making use of the large‐aspect‐ratio assumption (R0≫ac), a closed algebraic dispersion relation is obtained for the longitudinal instability, assuming that the electron ring is located inside a perfectly conducting toroidal shell. Several points are noteworthy in this analysis. First, transverse electromagnetic effects can provide complete stabilization provided the ring current is sufficiently large. Second, for the case where there are no ions (f=0) and the betatron focusing force exceeds the self‐field defocusing force (2ω2β =ω2cz −ω2Pb/γ2b >0), it is found that stabilization occurs at sufficiently low transverse temperature of the beam electrons. Third, for the case where ω2β <0 and the transverse temperature of the beam electrons is sufficiently low, it is found that surface perturbations on the electron beam drive a radial kink instability.