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^{1}and Han S. Uhm

^{2}

### Abstract

The thermal equilibrium properties of an intense relativistic electron beam with distribution function *f* ^{0} _{ b }=*Z* ^{−1} _{ b }exp[−(*H*−β_{ b } *c* *P* _{ z }−ω_{ b } *P* _{ϑ}) /*T*] are investigated. This choice of *f* ^{0} _{ b } allows for a mean azimuthal rotation of the beam electrons (when ω_{ b }≠0), and corresponds to an important generalization of the distribution function first analyzed by Bennett. Beam equilibrium properties, including axial velocity profile *V* ^{0} _{ z b }(*r*), azimuthal velocity profile *V* ^{0} _{ϑb }(*r*), beam temperature profile *T* ^{0} _{ b }(*r*), beam density profile *n* ^{0} _{ b }(*r*), and equilibrium self‐field profiles, are calculated for a broad range of system parameters. For appropriate choice of beam rotation velocity ω_{ b }, it is found that radially confined equilibrium solutions [with *n* ^{0} _{ b }(*r*→∞) =0] exist even in the absence of a partially neutralizing ion background that weakens the repulsive space‐charge force. The necessary and sufficient conditions for radially confined equilibria are ω^{−} _{ b }<ω_{ b }<ω^{+} _{ b } for 0⩽ (2ω̂^{2} _{ p b } /ω^{2} _{ c b }) (1−*f*−β^{2} _{ b }) ⩽1, and 0<ω_{ b }<ω_{ c b } for (2ω̂^{2} _{ p b }/ω^{2} _{ c b }) (1−*f*−β^{2} _{ b }) <0. Here, ω_{ c b }=*e* *B* _{0}/γ_{ b } *m* *c* is the relativistic cyclotron frequency, ω_{ p b }= (4πn̂_{ b } *e* ^{2}/γ_{ b } *m*)^{1/2} is the on‐axis (*r*=0) plasma frequency, *f*=*n* ^{0} _{ i }(*r*)/*n* ^{0} _{ b }(*r*) = const is the fractional charge neutralization, β_{ b } *c*= (1−1/γ^{2} _{ b })^{1/2} *c* is the mean axial velocity of the beam, and ω^{±} _{ b }= (ω_{ c b }/2) {1±[1−(2ω̂^{2} _{ p b }/ω^{2} _{ c b }) (1−*f*−β^{2} _{ b })] ^{1/2}} are the allowed equilibrium rotation frequencies in the limit of a cold electron beam( *T*→0) with uniform density n̂_{ b }.

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