Dispersion in the interaction point

Hybrid computation of normal mode tune shifts in rounded-rectangular pipes

“…Computing the related betatron tune-shifts, resulting from collective (space-charge and image) effects is a key problem to prevent resonant betatron excitations leading to potentially harmful beam instabilities. The normal mode coherent and incoherent 1 tune-shifts can be written in terms of the normal mode Laslett coefficients as follows [2]:…”

“…‡ e-mail stefania@kekvax.ac.jp 1 The incoherent and coherent regimes correspond to r = r b = r eq and r = r b = r eq , respectively, r eq denoting the beam center of charge equilibrium position [2]. 2 The pipe-shape independent space-charge contribution to the tuneshift is neglected here for simplicity. the machine radius, r 0 is the classical particle radius, L is a scaling length (usually, the maximum pipe diameter), ν is the nominal tune, and…”

Coupling impedances for corrugated beam pipes from impedance boundary conditions

“…According to [1] the longitudinal and transverse beam coupling impedances Z 0, (ω) andZ 0,⊥ (ω) of a simple, unperturbed pipe (e.g., circular, perfectly conducting) assumed known, can be related to those Z (ω),Z ⊥ (ω) of another pipe differing from the former by some perturbation in the boundary geometry and/or constitutive properties, as follows (beam at r = 0) 1 :…”

“…In this paper we estimate the longitudinal and transverse coupling impedances for a pipe with corrugated walls using the general framework presented in [1] and summarized below, using an impedance boundary condition (b.c.) of the Leontóvich type, to account for the corrugations.…”