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^{1}and Ronald C. Davidson

^{2}

### Abstract

The linearized Vlasov–Maxwell equations are used to investigate the influence of intense equilibrium self‐fields on the cyclotron maser instability. A uniform density (*n*̂_{ b }) electron beam propagates parallel to an applied axial magnetic field *B* _{0} **e**̂_{ z } with average axial velocity β_{ b } *c*. The particle trajectories are calculated including the influence of the radial self‐electric field and the azimuthal self‐magnetic field. Moreover, the linearized Vlasov–Maxwell equations are analyzed for body‐wave perturbations localized to the beam interior, assuming electromagnetic perturbations about the equilibrium distribution function *f* ^{0} _{ b }=(*n*̂_{ b }/2*p*π_{⊥}) ×δ(*p* _{⊥}−γ_{ b } *m* *V* _{⊥}) ×δ(*p* _{ z }−γ_{ b } *m*β_{ b } *c*). Near the beam axis (ω^{2} _{ p b } *r* ^{2}/*c* ^{2}≪1), it is found that the transverse electron motion is *b* *i* *h* *a* *r* *m* *o* *n* *i* *c*, with oscillatory components at the frequencies ω^{+} _{ b } and ω^{−} _{ b } defined by ω^{±} _{ b } =(ω_{ c b }/2) ×{1±[1−(2ω^{2} _{ p b }/ω^{2} _{ c b }) ×(1−β^{2} _{ b })]^{1/2}}. Similarly, the electromagnetic dispersion relation for waves propagating parallel to *B* _{0} **e**̂_{ z } exhibits *t* *w* *o* types of resonance conditions: a high‐frequency resonance (HFR) corresponding to ω−*k*β_{ b } *c*=ω^{+} _{ b }, and a low‐frequency resonance (LFR) corresponding to ω−*k*β_{ b } *c*=ω^{−} _{ b }.

Both the HFR branch and the LFR branch exhibit instability, with detailed stability properties depending on the value of the self‐field parameter *s*=ω^{2} _{ p b }/ω^{2} _{ c b }. Moreover, the LFR branch is entirely caused by self‐field effects, whereas the HFR branch represents a generalization of the conventional cyclotron maser mode to include self‐field effects. The full dispersion relation is analyzed numerically, and the real oscillation frequency ω_{ r }=Re ω and growth rate ω_{ i }=Im ω are calculated for both types of modes over a wide range of system parameters *s*, β_{⊥}, β_{ b }, and *k* *c*/ω_{ c b }. Analytic estimates are made of the cyclotron maser growth properties in circumstances where β^{2} _{⊥}γ^{2} _{ z }/2 is treated as a small parameter. [Here, γ_{ z }=(1−β^{2} _{ b })^{−} ^{1} ^{/} ^{2}. ] It is found that the maximum growth rate is given by ω_{ i }=(2γ^{2} _{ z })^{−} ^{1} ×[*s*(2β^{2} _{⊥}γ^{4} _{ z }−*s*)]^{1/2}ω_{ c b }, which occurs for wavenumber *k*=*k* _{ m }=γ^{2} _{ z }β_{ b }ω_{ c b }/*c*. As the beam density (*s*) is increased, the growth rate ω_{ i } increases to the maximum value ω^{max} _{ i } =γ^{2} _{ z }β^{2} _{⊥}ω_{ c b }/2 for beam density *s*=*s* _{ m }=β^{2} _{⊥}γ^{4} _{ z }. As *s* is increased beyond *s* _{ m }, the growth rate ω_{ i } decreases to zero for *s*=*s* _{0}=2β^{2} _{⊥}γ^{4} _{ z }. Similarly, the instability bandwidth Δ*k* =(2γ_{ z }ω_{ c b }/*c*)[γ^{2} _{ z }−β^{−1} _{⊥} ×(*s*/2)^{1} ^{/} ^{2}]^{1/2} approaches zero as *s* approaches *s* _{0}.

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