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

2. He Zhang ---2--- 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

3. He Zhang ---3--- 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.

4. He Zhang ---4--- 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)

5. He Zhang ---5--- 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.

6. He Zhang ---6--- 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

7. He Zhang ---7--- How BETACOOL performs the simulation

8. He Zhang ---8--- Emittances to Coordinates to Invariants Single particle model: Transversely, Longitudinally,

9. He Zhang ---9--- Emittances to Coordinates to Invariants Monte Carlo model Transversely, Longitudinally, Invariants are calculated statistically.

10. He Zhang ---10--- Friction Force Calculation Many friction force models: Consider Non-magnetic Meshkov model as an example Besides the constants, we need

11. He Zhang ---11--- 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.

12. He Zhang ---12--- Calculate the New Emittance

13. He Zhang ---13---