One-dimensional rigid Gaussian model of the beam-beam interaction

A simple one-dimensional model of the beam-beam interaction (sheet-sheet interaction) and its rigid-Gaussian approximation for the coherent barycentre motion are studied in order to examine the usefulness of the rigidGaussian approximation. Effects of primordial separation between beams are studied in detail. The spontaneous beam separation for the integer resonance can be embedded in a cusp catastrophe structure as a special case. A new technique is introduced to the calculation of the beam-beam force for the multiparticle tracking, without relying on the Gaussian approximation. The rigid-Gaussian approximation is shown to be reliable enough for evaluating closed-orbit effects, except for some extremely exceptional cases

Non-Gaussian Distribution of Electron Beams due to Incoherent

A theory is presented on the equilibrium particle distribution in electron rings, when each electron is being affected by, in addition to the synchrotron radiation effects, incoherent stochastic processes, such as a collision with atoms in the residual gas

Linear Beam-Beam Resonances due to Coherent Dipole Motion

We study the transverse motion of the barycentres of bunches in two beams that circulate in opposite direction in a storage ring, and are coupled by the beam-beam effect. The motion is described by a linear system of oscillators, and represented by a matrix. We distinguish between perfect machines in which the bunch parameters and the arc and interaction point parameters are all equal, and machines with errors in which at least one of these conditions is not satisfied.We determine the regions in tune space where the motion is unstable, analytically for perfect machines, and numerically for machines with errors. We identify these regions as resonances related to the tunes of one of the two beams or to their sum. We establish which resonances are excited under given conditions. We find that more resonances occur in machines with errors than in perfect machines. By multi-particle tracking, we study the instability at finite amplitudes

The Existence of a Static Solution for the Haissinski Equation with

The equilibrium longitudinal distribution of electrons in circular accelerators is discussedfor the case of the 8\u27 wake function. Contrary to the well known fact that the solution does not exist in this case beyond a threshold, it is strongly suggested that the solution actually exists when we regularize the singularity of the 8\u27 wake function in a physical way, therefore the non-existence of the solution has no physical consequence

Gaussian approximation of the bunch lengthening in electron

The validity of the Gaussian approximation of the distribution function in the study of the equilibrium bunch length in electron storage rings is considered with specific reference to the inductive wake. The Gaussian approximation can be used to describe not only localized wakes, but also uniformly distributed wakes, for which the longitudinal profiles at equilibrium are ruled by the potential-well-distortion (Haissinski) equation. Comparison of Gaussian approximation based results with those obtained from Haissinski equation shows good qualitative agreement in their common range of validity, while shedding light on the nature of the unstable regimes

A symplectic beam-beam interaction with energy change

A mapping, called synchro-beam mapping, is proposed which describes the effects of the beam-beam interaction in presence of synchrotron motion. The mapping is symplectic in six dimensional phase space. In addition to the usual transverse kick, this mapping cludes the bunch-length effect on the collision point and the energy variation caused by the electric field of the opposite bunch. The case of weak-strong interaction is investigatedby simulation, showing that the beam-beam energy exchange has an influence on the distribution tail