The δf algorithm for beam dynamics

An algorithm is developed to study particle dynamics of beams including collective interaction with high accuracy and low noise. Particle dynamics with collective interactions is treated through particle simulation, where the main or average distribution f0 and the deviation away from it δf are separately followed. The main distribution f0 is handled by an analytic equilibrium solution and the perturbation away from it δf is followed by the method of characteristics. We call this the δf algorithm. We specifically model a synchrotron collider which includes the collision section where collective effects of collisions are simulated by this δf algorithm and the rest of the collider where single particle dynamics are treated by simple harmonic transport. The most important target of this simulation is to understand and predict the long-time behavior of the beam luminosity and lifetime. The δf method allows the study the effect of small perturbations over long timescales on beam lifetime by eliminating the numerical noise problem inherent in Particle-in-Cell techniques. In the δf code using the reference parameters of the SSC (Superconducting Super Collider), beam blow-up near resonances and oscillations in the tune shift, Δν, far from resonances are observed. In studying long timescale particle diffusion in the phase space of the beams away from resonances, the δf code performance is compared with a tracking code which does not incorporate collective interaction