BETACOOL program for electron cooling simulation basic algorithms

BETACOOL program for electron cooling simulation: basic algorithms and models Anatoly Sidorin JINR cooling group

BETACOOL algorithms - Rms dynamics - Model beam (Monte-Carlo method) - Tracking based on Molecular Dynamics technique Rms dynamics

Model beam blockscheme Generation of the particle array in the selected point of the Ring Matching with the Ring Lattice and RF system IBS: Cycle over dynamics simulation - Calculation of the rms parameters - Calculation of IBS growth rates - Calculation of the square rms scattering angle amplified by integration step cycle Random change of the particle momentum in over particles accordance to the scattering angle ECOOL: cycle over particles Change of the particle momentum in accordance with the friction force value amplified by integration step Bunch rotation using the transformation matrix

Model Beam algorithm Ion beam is presented by array of model particles - Simplest model of the ring – only lattice functions in the location of the effects are necessary. - Realistic model of the ring can be necessary for ion momentum variation due to action of effects. - Each effect calculates a kick of the ion momentum components and changes the particle number Tracking procedure Ion beam is presented by array of real (or macro) particles - Each effect is related with some optic element - The effect works as a transformation map - IBS is calculated as a Coulomb scattering using MD - The ring structure is imported from modified input MAD file

Structure of an effect Transformation MAP Transforms the particle co-ordinates and momentum components from entrance to the exit of optic element, calculates the particle loss probability Kick of model particle momentum Changes of the particle momentum components in accordance with step of the integration over time, Changes the particle number in the total beam and simulates losses in the model beam Growth rate calculation Calculates characteristic times of rms emittance variation and beam life-time

Map of Electron Cooling system Ion co-ordinates at the entrance Model of cooler Solution of the ion motion equations Transformation of the ion co-ordinates Thin lens to the frame referenced to the electron beam orbit Cooler at non zero length Magnetic field errors Electron beam space-charge Electron beam model Transformation of the ion velocity to PRF friction force components to LRF Uniform cilinder Gaussian bunch Calculation of local electron density and temperature Friction force library: Calculation of force components in PRF, d. Ploss/ds Budker’s, by Parkhomchuk, Derbenev-Skrinsky-Meshkov, numerical Ion coordinates at the exit, loss probability

Magnetic field errors Ideal fields Angle between two sections 10 -5 rad in vertical direction

Kick of ion momentum by electron cooling Particle losses in total beam Cooling time calculation -single particle -Monte-Carlo for Gaussian ion beam

Intrabeam scattering simulation Map of the effect can be based on two models: 1. Molecular Dynamics (solution of the particle motion equation) 2. Analytical models (mean diffusion coefficients at given lattice functions) Kick 1. Mean kick for each particle Martini, Jie-Wei, Pivinsky, Gas relaxation 2. Detailed calculation kick depends on particle parameters

Detailed calculation of the IBS effect - Burov’s model: kick for each particle is calculated as a function of betatron and synchrotron amplitudes, but distribution assumed to be Gaussian - “core –tail” model Kick is different for two groups of particles

“Core – tail” model FWHM rms

Luminosity calculation Rms dynamics – analytical formula Map of collision point Calculation of the head beam local areal density i [1/cm 2] Individual luminosity: Li = i/Trev Head beam models for the local density calculation: - Bunch at elliptic cross-section - Arbitrary distribution - taking into account variation of beta function Model Beam algorithm: L = Nb Li Ion loss probability: Ploss = Ncp*Li*

Particle loss simulation in Model Beam 1. Decrease of the model particle number 2. Generation of new co-ordinates for the loosed particle: - Gaussian at current standard deviations - in accordance with current distribution function The current distribution is presented as

Loss simulation Gaussian Real distribution

The goals of the BETACOOL development: -to perform simulations using real electron distribution function calculated with external programs, -to improve the “core – tail” model of intrabeam scattering process simulation on the basis of theory by G. Parzen, -to perform more accurate simulations of intrabeam scattering at coupled transverse motion of the ions, -to prepare a version of the program for multi processor calculations, to provide intrabeam growth rate calculations using Molecular Dynamics technique, -to integrate BETACOOL under UAL framework , -to begin simulations using results of numerical calculation of the electron cooling friction force obtained with external programs, - to develop numerical procedures for beam-beam parameter evaluation and include diffusion due to beam-beam effect into simulations.