# Presentation: 056149

## BETACOOL program 

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

## Rms dynamics

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

## Model beam 

**block- **
**scheme**

**Generation of the particle array in the selected point of the Ring**

**Matching with the Ring Lattice and RF system**

**- Calculation of the rms parameters**
**- Calculation of IBS growth rates**
** ****Calculation of the square rms scattering angle    **
**   ****amplified by integration step**

**   ****cycle**
**   ****over **
** ****particles**

**Cycle**
**over**
**dynamics**
**simulation**

**Bunch rotation using the transformation matrix**

**IBS:**** **

**ECOOL:**** **

- Random change of the particle momentum in accordance to the scattering angle

-  Change of the particle momentum
-  in accordance with the friction force
-  value amplified by integration step

**  ****cycle**
**   ****over **
** ****particles**

## Model Beam algorithm

**- 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**

**Ion beam is presented by array of model particles**

**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

- Friction force library:
- Calculation of
- force components in PRF,
- dPloss/ds

- Electron beam model
- Transformation of the ion velocity to PRF
- friction force components to LRF
- Calculation of local electron density and temperature

- Budker’s,
- by Parkhomchuk,
- Derbenev-Skrinsky-Meshkov,
- numerical

- Uniform cilinder
- Gaussian cilinder
- Gaussian bunch

- Model of cooler
- Solution of the ion motion equations
- Transformation of the ion co-ordinates
- to the frame referenced to the electron beam orbit
- Magnetic field errors
- Electron beam space-charge

- Thin lens
- Cooler at non zero length

- Ion co-ordinates at the entrance

- 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

**Cooling time**** calculation**
- single particle
- Monte-Carlo for Gaussian ion beam

- Particle losses in total beam

## Intrabeam scattering simulation

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

**       ****Model Beam algorithm**:
-                     L = N*b*L*i*
- Ion loss probability: P*loss* = N*cp**L*i**

**                           ****Map of collision point**
- Calculation of the head beam local areal density *i* [1/cm2]
- Individual luminosity: L*i* = *i*/T*rev*
- Head beam models for the local density calculation:
- - Bunch at elliptic cross-section
- - Arbitrary distribution
- - taking into account variation of beta function

## Particle loss simulation in Model Beam

- Decrease of the model particle number
- 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. **