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6. Benchmarking of the stand alone space charge algorithm without rotation

In order to validate the stand alone space charge algorithm, comparisons have been done with theoretical expressions of space charge fields. For cylindrical and ellipsoidal distributions, analytical expressions of the electric fields are available in the absence of external forces and when all particles are at rest. Comparisons with other algorithms have also been considered, but confronted to a short period of time and satisfied by the good agreement of our results with the theory, it has been eventually rejected to focus on the simulation of beam line components. On top of all the graphical conparisons made for all the fields components, we focused our comparisons on the longitudinal field . and reported our results in this chapter. We will first derive the expression of the theoretical potentials and fields for a cylindrical distribution, in order to graphically compare the resulting fields for different values of the bunch size. In a second time, the same thing is done for an ellipsoidal distribution. Eventually, the calculaion of the RMS value of the longitudinal fields on all the particles is done numerically and analytically for further comparisons with previous work of H. Edwards and Y-E Sun[12].

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Emmanuel Branlard
http://emmanuel.branlard.free.fr/