ZGOUBI USERS' GUIDE - HEP

4.4 Optical Elements and related numerical procedures 75€€§ÌÌ–4Ì•00 Atg o € – € Ž 4Ì$•¡-, 7 7/‘>s–7::$$7$‘ > 0Ì ÍÍÎÍÍϘ¥¥Š> 4444 ¡., i % % @ 2Finally, the initial conditions for the mapping, at the first turn, are the following- For the (virtual) synchronous particle$]7¥ synchronous phase- For any of the ¤ q , IMAX particles of the beam¥ ¡ ! 1 !7 $ ¡.,synchronous phase¥ ¡ ! 1 !V/O where the quantities ¡ ! 1 ! and O‹ are given in the object description.Calculation of the coordinatesLetŽ€>€>–€>€>> 7/‘> 7/‘be the momentum of particle ¤ at the exit of the cavity, while-, €~> Ž€~> É É• É is its momentum at the entrance. The kick in momentum is assumed to be fully longitudinal,resulting in the following relations between the coordinates at the entrance (denoted by the index zero) and at theÉ -,exitŽ € ,€ € – –and(longitudinal kick)€ • € •ÉÉ " § ÊÉ "‹ É and É (zero length cavity)§ and for the angles (see Fig. 1)€ >w i€ >Éw € >ŽÊÉ s(damping of the transverse motion)¥Ë Atg o ¥ € >€ >ŽIf ¼ °the same simulation of a synchrotron R.F. cavity, as for ’ÁÀ‹Âaà ·, is performed, except that the keywordSCALING (family CAVITE) is not taken into account in this option : the increase in kinetic energy at each traversal, forthe synchronous particle, is’ÁÀ‹ÂaÃ(*)È$where the synchronous ¥ phase is given in the input data. From this, the calculation of the lawµ‚¨ frequency follows, according to the formulae given in IOPT = 1.% ¥ ÐÇ ¡., ¥ and the R.F.If ’ÁÀ‹ÂaÃ_½: acceleration without synchrotron motion. Any particle will be given a kick$˜ where Ç and¥ are input data. ÐÇ ˜%(*)È$NOTE: Calculation of the closed orbit.Due to the fringe fields, the horizontal closed orbit may not coincide with the ideal axis of the optical elements. One wayto calculate it at the beginning of the structure (i.e. where the initial particle coordinates have to be defined) is to ray-trace asingle particle over a sufficiently large number of turns, starting with the initial ¥ C\ condition, andso as to obtain a statistically well-defined phase-space ellipse. The initial conditions of the closed orbit then correspondto the coordinates and of the center of this ellipse. Next, ray-tracing over one turn a particle starting with the initial( condition , †\, ) will provide the length (namely, thecoordinate) of the closed orbit.© ¥< G" q