wakefield simulations for the ilc-bds collimators cockcroft institute - 3 rd wakefield interest...

Wakefield Simulations for the ILC-BDS Collimators

Cockcroft Institute - 3rd Wakefield Interest Group Workshop

Cockcroft Institute, November 2007

Adriana Bungau

The University of Manchester

Introduction - wakefields in collimators

Extensive literature for wakefield effects and many computer

codes for their calculations:• concentrates on wake effects in RF cavities

(axial symmetry)• only lower order modes are important • only long-range wakefields are considered

Wakefields in collimators - different than in cavities!

Wakefields in cavities

Wakefields in collimators

- collimators generate short-range wakefields

- the longitudinal wakes increase the energy spread

- the transverse wakes cause emittance growth

- we consider only the intra-bunch wakefields - the time between the

bunches is believed to be long enough for the wake currents to damp

down

Wakefields cause emittance growth

which in turn causes a reduction of the

collider’s luminosity.

Wakefields in collimators

- tapering relaxes the wakefields

- near-wall wakefields play considerable role

in single bunch dynamics

- for bunches close to the axis: the

longitudinal effect is dominated by the

monopole mode (m=0) and the transverse

effect is dominated by the dipole mode

(m=1)

For near-wall wakefields higher order modes must be considered and the total wakefield effect is a sum over all multiple contributions !

Wakefield Theory for Collimators

Two main contributions to wakefields:

- Geometric wakefields

1. change in the vacuum chamber section at the collimator

2. walls assumed perfectly conducting

3. fairly complicated esp. for flat collimator (x gap>>y gap)

4. only the tapering part contribute to the geometric wakefields

- Resistive wakefields

1. due to finite resistivity of the collimator material

2. both the flat and the tapered part contribute to resistive wakes

The ILC-BDS collimation system raised new questions about wakefields and required an extension to the existing simulation tools

Implementation of Higher Order Mode Wakefields in MERLIN for the

ILC_BDS collimators

The MERLIN code

- is a set of software libraries for the simulation of charged particle acceleration (N.Walker and A.Wolski)

- written in C++, has been developed for both UNIX/LINUX and WINDOWS platforms

- purpose: to study the beam dynamics of high energy colliders

- has the capability to simulate storage ring accelerators in principle

- continuously evolving

http://www.desy.de/~merlin/

Tracking with MERLIN

• Accelerator Model

- containes classes for modelling accelerator components (magnets, drifts etc)

- common features: length, aperture, EM field, geometry, wakefield potential

- for multi-component beamlines, it is designed to read MAD optics tables which

contains a sequential list of accelerator components and their attributes defined by

the user (MAD Interface class)

• Beam Model

- define the beam parameters and use them in the construction of a particle bunch

(p0, beta func., alpha func., gamma func., emittances, coupling and dispersion

factors, rms energy spread, bunch length and particle number) - BeamData struct.

- values are used only in the initial construction of the particle bunch

- the initial phase space vectors for each particle in a bunch are given values by the

following routine:

Tracking with MERLIN

= D B A C e

e is generated using uncorelated random no.for emittances, dp and ct, sampled from

a Gaussian distribution

• Tracking Procedure

- the particles’ phase vectors are propagated along the beamline and altered

accordingly

- the propagation is in steps corresponding to increments of distance along the

beamline

- includes physical processes (collimation, wakefields, synchrotron radiation, ground

motion etc)

Wakefield Theory : the Effect of a Single Charge

• Investigate the effect of a leading unit charge on a trailing unit charge separated by distance s

r’,’

s

r,

s

• the change in momentum of the trailing particle is a vector w called ‘wake potential’

• w is the gradient of the ‘scalar wake potential’: w=W

• W is a solution of the 2-D Laplace Equation where the coordinates refer to the trailing

particle; W can be expanded as a Fourier series:

W (r, , r’,s) = Wm(s) r’m rm cos(m) (Wm is the ‘wake function’)

• the transverse and longitudinal wake potentials wL and wT can be obtained from this equation

wz = ∑ W’m(s) rm [ Cmcos(m) - Sm sin(m)]

wx = ∑m Wm(s) rm-1 {Cmcos[(m-1)] +Sm sin[(m-1)]}

wy = ∑m Wm(s) rm-1 {Sm cos[(m-1)] - Cm sin[(m-1)]}

The Effect of a Slice

- the effect on a trailing particle of a bunch slice of N particles all ahead by the same

distance s is given by simple summation over all particles in the slice

- if we write: Cm = ∑r’m cos(m’) and Sm = ∑r’m sin(m’) the combined kick is:

- for a particle in slice i, a wakefield effect is received for all slices j≥i:

∑j wx = ∑m m rm-1 { cos [ (m-1) ] ∑jWm(sj) Cmj +

sin [ (m-1) ] ∑jWm(sj) Smj }

Wakefield Implementation in MERLIN

Previously in Merlin:• Two base classes: WakeFieldProcess and

WakePotentials

- transverse wakefields ( only dipole mode)

- longitudinal wakefields

Changes to Merlin• Some functions made virtual in the base

classes• Two derived classes:

- SpoilerWakeFieldProcess - does the

summations

- SpoilerWakePotentials - provides

prototypes for W(m,s) functions (virtual)• The actual form of W(m,s) for a collimator

type is provided in a class derived from SpoilerWakePotentials

WakeFieldProcess WakePotentials

SpoilerWakeFieldProcess

CalculateCm();CalculateSm();

CalculateWakeT();CalculateWakeL();ApplyWakefield ();

SpoilerWakePotentials

nmodes;virtual Wtrans(s,m);virtual Wlong(s,m);

Geometric wakefields - Example

Wm(z) = 2 (1/a2m - 1/b2m) exp (-mz/a) (z)

Class TaperedCollimatorPotentials: public SpoilerWakePotentials

{ public:

double a, b;

double* coeff;

TaperedCollimatorPotentials (int m, double rada, double radb) : SpoilerWakePotentials (m, 0. , 0. )

{ a = rada;

b = radb;

coeff = new double [m];

for (int i=0; i<m; i++)

{coeff [i] = 2*(1./pow(a, 2*i) - 1./pow(b, 2*i));} }

~TaperedCollimatorPotentials(){delete [ ] coeff;}

double Wlong (double z, int m) const {return z>0 ? -(m/a)*coeff [m]/exp (m*z/a) : 0 ;} ;

double Wtrans (double z, int m) const { return z>0 ? coeff[m] / exp(m*z/a) : 0 ; } ; };

b a

Application to one collimator

• large displacement - 1.5 mm• one mode considered• the bunch tail gets a bigger kick

• small displacement - 0.5 mm• one mode considered• effect is small• adding m=2,3 etc does not change much the result

• large displacement - 1.5 mm• higher order modes considered (ie. m=3)• the effect on the bunch tail is significant

SLAC beam tests simulated: energy - 1.19 GeV, bunch charge - 2*1010 e-

Collimator half -width: 1.9 mm

Extension to the ILC - BDS collimators

- beam is sent through the BDS off-axis (beam offset applied at the end of the linac)

- parameters at the end of linac:

x=45.89 m x=2 10-11 x = 30.4 10-6 m

y =10.71 m y =8.18 10-14 y = 0.9 10-6 m

- interested in variation in beam sizes at the IP and in bunch shape due to wakefields

No

Name Type Z (m) Aperture

1 CEBSY1 Ecollimator 37.26 ~



4 CEBSYE

Rcollimator 431.41 ~

5 SP1 Rcollimator 1066.61 x99y99

6 AB2 Rcollimator 1165.65 x4y4

7 SP2 Rcollimator 1165.66 x1.8y1.0

8 PC1 Ecollimator 1229.52 x6y6






14 SP4 Rcollimator 1362.91 x1.4y1.0




No Name Type Z (m) Aperture



20 PDUMP Ecollimator 1530.72 x4y4


22 SPEX Rcollimator 1658.54 x2.0y1.6




26 ABE Ecollimator 1823.21 x4y4




30 AB7 Rcollimator 2199.91 x8.8y3.2

31 MSK1 Rcollimator 2599.22 x15.6y8.0

32 MSKCRAB Ecollimator 2633.52 x21y21

33 MSK2 Rcollimator 2637.76 x14.8y9

ILC-BDS colimators

Geometric wakefields

- beam parameters at the end of linac:

x = 30.4 10-6 m, y = 0.9 10-6 m

- beam size at the IP in absence of wakefields:

x = 6.51*10-7 m, y = 5.69*10-9m

- beam sizes for 4 modes: x = 0.7*10-6 m,

y = 0.19*10-6m

- for small offsets, modes separation occurs at

~10 sigmas;

Bunch size

Geometric wakefields

- at 10 sigmas when the separation into modes occurs, the luminosity is reduced to 20%

- for a luminosity of L~1038 the offset should be less than 2-3 sigmas

Luminosity

Resistive wall

• pipe wall has infinite thickness; it is smooth;

• it is not perfectly conducting

• the beam is rigid and it moves with c;

• test charge at a relative fixed distance;

bc

cThe fields are excited as the beam interacts with the resistive wall surroundings;

For higher moments, it generates different wakefield patterns; they are fixed and move down the pipe with the phase velocity c;

General form of the resistive wake

• Write down Maxwell’s eq in cylindrical coordinates• Combined linearly into eq for the Lorentz force components and

the magnetic field• Assumption: the boundary is axially symmetric (

are ~ cos mθ and are ~ sin mθ )• Integrate the force through a distance of interest L• Apply the Panofsky-Wenzel theorem

sr eBFFF ,|| ,,

rFF ,||

sBF ,

Lz

c

bzW

mmm 2/1

012

1

)1(

2)(

δπ +−= +

Lz

c

bzW

mmm 2/3

012

' 1

)1(

1)(

δπ += +

Implementation of the Resistive wakes

WakeFieldProcess WakePotentials

SpoilerWakeFieldProcess

CalculateCm();CalculateSm();

CalculateWakeT();CalculateWakeL();ApplyWakefield ();

SpoilerWakePotentials

nmodes;virtual Wtrans(s,m);virtual Wlong(s,m);

ResistiveWakePotentials

Modes;Conductivity;pipeRadius;

Wtrans(z,m,AccComp); Wlong(z,m, AccComp);

Resistive wakes

• Benchmark against an SLC result

Resistive wakefields

- beam size at the IP in absence of wakefields:

x = 6.51*10-7 m, y = 5.69*10-9 m

- beam sizes for 4 modes: x = 1.2*10-6 m,

y = 3.5*10-6m

• For small offsets the mode separation starts at ~10 sigmas

• At larger offsets (30-35 sigmas) there are

particles lost in the last collimators

The increase in the bunch size due to resistive wakefields is far greater than in the geometric case

Bunch size

Resistive wakes

- at 10 sigmas when the separation into modes occurs, the luminosity is reduced to 10%-for a luminosity of L~1038 the offset should be less than 1 sigma- the resistive effects are dominant!

Luminosity

Beam offset in each BDS collimator

• No wakefields <y>=4.74e-12;

• Jitter of 1 nm of maximum tolerable bunch-to-bunch jitter in the train with 300 nm between bunches; for 1nm: <y>=8.61e-11

• Jitter about 100 nm which intratrain feedback can follow with time constant of ~100 bunches; for 100nm: <y>=5.4e-10

• Maximum beam offset is 1 um in collimator AB7 for 1nm beam jitter and 9um for 100 nm jitter

Beam offset in each collimator

• Beam jitter of 500 nm of train-to-train offset which intratrain feedback can comfortably capture

• The maximum beam offset in a collimator is 40 um (collimator AB7) for a 500nm beam jitter

• For 500nm: <y>=2.37e-9

Bunch Shape Distortion

• The bunch shape changes as it passes through the collimator; the gaussian bunch is distorted in the last collimators

• But the bunch shape at the end of the linac is not a gaussian so we expect the luminosity to be even lower than predicted

Summary

• The higher order wakefield modes play a significant role if the bunch

is close to the collimator edges

• For small offsets, we could use safely one mode in our wakefield

calculations

• For small (realistic) offsets the separation into modes start at about 10

sigmas

• Resistive wakefields seem to be dominant

• For other (complicated) collimator geometries the wakes will have to

be read by MERLIN as tables created with other codes like ECHO-2D

or GDFIDL

• The changes to the MERLIN core and the newly created libraries are

now being added to the CVS repository at DESY

wakefield simulations for the ilc-bds collimators cockcroft institute - 3 rd wakefield interest...

Documents

longrange wakefields

shortrange wakefields

introduction wakefields

collimators collimators

intrabunch wakefields

collimators tapering

collimators different

wakefield effects