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 Z[over ˜](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_{z}z-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}=ω[over ^]_{pb}^{2}/2γ_{b}^{2}ω_{β⊥}^{2} in the interval 0<s_{b}