RMS Dynamic Simulation for Electron Cooling Using BETACOOL He Zhang Journal Club Talk, 04/01/2013

He Zhang Outline  Basic idea of the RMS Dynamic Simulation  Model of the ion beam  Model of the electron beam  Model of the cooler  How BETACOOL performs the simulation  A brief description of the simulation process  From emittances to coordinates to invariants  Friction force calculation  Transfer map of the cooler  Characteristic time/rate calculate  Emittance calculation

He Zhang Basic idea of the RMS Dynamic Simulation Ion bunch has Gaussian distribution in all directions Solve this equation: In transverse direction, ε i is the emittance in horizontal or vertical direction In longitudinal direction, coasting beam; bunched beam; Ω s is the synchrotron frequency.

He Zhang Model of the ion beam Two models: Single particle model Monte Carlo model Parameters for ion beam:  Horizontal emittance  Vertical emittance  Momentum spread  Number of particles  Model particles (only for Monte Carlo model)

He Zhang Model of the electron beam According to different geometry and different charge distribution, BETACOOL provides the following models: Uniform cylinder, Gaussian cylinder, Hollow beam, Uniform bunch, Gaussian bunch, Electron array, Parabolic, File. Set up the Gaussian bunch model One way: Input bunch size and angle, input number of electrons The other way: Input bunch size and choose from model, imput emittance, temperature, or r.m.s. velocity, input number of electrons.

He Zhang Model of the cooler Parameters for the cooler:  Cooler length  Magnetic field  Section number  Bunch number  Distance between bunches  Cooler model: thin lens, Euler model, Runge Kutta model  Integration steps (for Euler model and Runge Kutta model)  Lattice: β, α, η, and η ́  Shifts

He Zhang How BETACOOL performs the simulation

He Zhang Emittances to Coordinates to Invariants Single particle model: Transversely, Longitudinally,

He Zhang Emittances to Coordinates to Invariants Monte Carlo model Transversely, Longitudinally, Invariants are calculated statistically.

He Zhang Friction Force Calculation Many friction force models: Consider Non-magnetic Meshkov model as an example Besides the constants, we need

He Zhang Friction Force Calculation We have found Many models for electron bunch distribution. Consider the Gaussian bunch as an example: Plug in the ion coordinates into the function above to get n e. Define directly, or define temperature, emittance, velocity spread for the electron bunch, and the program will calculate Now the friction force can be calculated.

He Zhang Calculate the New Emittance

He Zhang