Obtaining short bunch is a very important issue for modern accelerators, especially for colliders and coherent synchrotron radiation (CSR) sources; however the coherent instabilities pose strong limitations on the bunch length and on the current stored in the bunch at a given length. In all the present storage rings the bunch has the same length along the ring; the principle of the Strong RF Focusing (SRFF) consits in realizing a lattice in which bunch length is not constant and we have the so called bunch length modulation. An experiment at DA$Phi$NE, the Frascati $Phi-$factory, has been proposed, in order to test, for the first time, this idea. The subject of this work is the study of the longitudinal single bunch dynamics in the regime of Strong RF focusing. In fact, until now, the principle of bunch length modulation has been studied only from the optical point of view, i.e. without considering the effects of the impedances that pose limitations to the maximum current storable. The thesis is organized as follows. In the first chapter I present the physical interest of DA$Phi$NE collider; in particular, since in a collider the maximum luminosity is related to bunch length, I show the physical advantages to increase the DA$Phi$NE luminosity. In the second chapter the present status of DA$Phi$NE lattice is presented. In the next two chapters the problem of the coherent instabilities is analysed in the DA$Phi$NE case, by considering the interaction of the beam with the vacuum chamber and the effects of the CSR on beam dynamics. In the following chapter the principle of SRFF is reported, in particular it is decsribed the possibility to test it at DA$Phi$NE realizing different lattices able to analyse the different aspects and properties of this new regime In the sixth chapter I describe the C code SPIDER (Simulation Program for Impedances Distributed along Electron Rings) I wrote in order to study the relation between bunch length and current stored in a regime of bunch length modulation. The code, in fact, allows to consider the effect of the impedance of a specific structure of the machine on longitudinal bunch distribution in the structure itself. Therefore, even if the code is able to simulate the behaviour of bunch length of existent machines (it was successfully applied to the present status of DA$Phi$NE), it is the only code able to study the effect of longitudinal bunch stability in a regime of bunch length modulation. Finally, in the last chapter, there are the results of the simulations made by the application of the code SPIDER to the three lattices presented in the fifth chapter.
LONGITUDINAL BEAM DYNAMICS IN SRFF REGIME
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