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˜(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=ω-^ _pb²/2γ_b²ω_β⊥² in the interval 0<s_b<1, where ω-^ _pb=(4πn-^ _be_b²/γ_bm_b)^1/2 is the relativistic plasma frequency and ω_β⊥ is the applied focusing frequency.