Speaker
Description
Space-charge effects in material-filled beam regions are relevant for ionization cooling in a future muon collider, where low-$Z$ absorbers are used to reduce the beam emittance. We extend the conventional cylindrical space-charge impedance formalism to the case of a beam region filled with a material of arbitrary electromagnetic properties and surrounded by a perfectly conducting boundary. Analytical expressions are obtained for the longitudinal and transverse space-charge impedances, including both direct and indirect contributions. Compared with the conventional vacuum case, the material modifies both contributions. In particular, the usual relativistic factor $1/\gamma^2$ is replaced by the material-dependent quantity $F=1/\varepsilon_1-\mu_1\beta^2$, which modifies the electromagnetic response of the beam region. These changes lead to qualitatively different impedance regimes. A finite conductivity of the beam-region material gives rise to a nonzero real part of the impedance and can also change the sign of its imaginary part. For sufficiently large dielectric permittivity, the beam velocity can exceed the phase velocity of electromagnetic waves in the material, allowing the excitation of resonant modes. The origin and characteristic frequencies of these resonances can be related to the electromagnetic properties of the material and the transverse geometry. The results provide a framework for understanding space-charge effects in absorber regions and form a basis for subsequent beam-dynamics studies of ionization-cooling channels.