Wall-impedance-driven collective instability in intense charged particle beams

The linearized Vlasov-Maxwell equations are used to investigate detailed properties of the wall-impedance-driven instability for a long charge bunch (bunch length ℓ_b ≫ bunch radius r_b) propagating through a cylindrical pipe with radius r_w and wall impedance Z̃(ω). The stability analysis is carried out for perturbations about a cylindrical Kapchinskij-Vladimirskij beam equilibrium with a flattop density profile in the smooth-focusing approximation. The perturbations are assumed to be of the form δψ(x, t) = δψ^ℓ(r) exp(iℓθ + ik_zz - iωt), where (r, θ) are the radial and azimuthal coordinates in the transverse direction, and z is the coordinate in the longitudinal direction. Here, ℓ = 1, 2, ... is the azimuthal mode number of the perturbation in the transverse direction, k _z is the wave number in the longitudinal direction, and ω is the oscillation frequency. As an example, detailed stability properties are determined for dipole-mode perturbations (ℓ = 1) assuming negligibly small axial momentum spread of the beam particles. The stability analysis is valid for a general value of the normalized beam intensity s_b = ω̂_pb²/2γ_b²ω _β⊥² in the interval 0 < s_b < 1, where ω̂_pb = (4πn̂_be_b ²/γ_bm_b)^1/2 is the relativistic plasma frequency and ω_β⊥ is the applied focusing frequency.