Abstract
The longitudinal space charge effect is simulated by binning the longitudinal beam profile in order to calculate the force on the bins using the binned particle distribution via FFT, and applying momentum kick based upon this space charge force to macro-particles. Usually, the longitudinal space charge kick is calculated once per turn since the longitudinal profile doesn't change much in a single turn. Besides, the longitudinal profile is used as a weighting factor for the transverse space charge force. The transverse space charge effect is simulated by projecting the 3-D beam to a 2-D Gaussian distribution in order to use the complex error function to compute the transverse space charge force, and applying this space charge force to macro-particles. One transverse space charge calculation per scale length of the beam shape variation requires at least ten transverse space charge force calculations per betatron oscillation.
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