The LHC Transverse Coupled-Bunch Instability

In this thesis, the problem of the transverse coupled-bunch instabilities created by the Large Hadron Collider (LHC) beam-coupling impedance, that can possibly limit the machine operation, is addressed thanks to several new theories and tools. A rather complete vision of the problem is proposed here, going from the calculation of the impedances and wake functions of individual machine elements, to the beam dynamics study. Firstly, new results are obtained in the theory of the beam-coupling impedance for an axisymmetric two-dimensional structure, generalizing Zotter's theories, and a new general theory is derived for the impedance of an infinite flat two-dimensional structure. Then, a new approach has been found to compute the wake functions from such analytically obtained beam-coupling impedances, over-coming limitations that could be met with standard discrete Fourier transform procedures. Those results are then used to obtain an impedance and wake function model of the LHC, based on the (resistive-) wall impedances of various contributors (collimators, beam screens and vacuum pipe) and additional estimations of the geometrical impedance contributions. Finally, the existing code HEADTAIL, which is a macroparticle simulation code for beam dynamics studies with wake fields, is improved to make possible the simulation of multibunch trains, and a spectral analysis technique is found to facilitate the analysis of the output given by this code, giving the complex tune shifts of the unstable modes present in a simulation. All those theories and tools are used to obtain new results concerning the LHC transverse coupled-bunch instabilities, demonstrating the rather small impact on coupled-bunch instabilities of the number of bunches in a train when the bunch spacing is fixed, and the existence of coupled-bunch modes with intrabunch motion which are more critical than their single-bunch counterparts. A full verification of the complete procedure (impedance theories, impedance model and simulation code) is also performed by comparing the simulation results with actual measurements in the LHC, giving a very good agreement at injection energy and a correct order of magnitude at 3.5 TeV/c. In the end, several predictions concerning the beam stability at the future 7 TeV/c operation of the machine are performed in the case of 50 ns spacing (1404 bunches), revealing that the coupled-bunch transverse mode coupling instability threshold is far above the ultimate bunch intensity but about 20% smaller than its single-bunch counterpart. Stability studies with Landau octupoles at their maximum currents reveal that the beam remains stable at nominal intensity with Q' = 2 in both planes, provided the particle transverse distributions are Gaussian. At ultimate intensity with either Q' = 0 or Q' = 2, or at nominal intensity when the chromaticity is zero, the beam happens to be unstable, even with the octupoles at their maximum currents.