1. Wakefield Simulations for the ILC-BDS Collimators Cockcroft Institute - 3 rd Wakefield Interest Group Workshop Cockcroft Institute, November 2007 Adriana Bungau The University of Manchester

2. Introduction - wakefields in collimators

3. 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

4. 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.

5. 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 !

6. 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

7. Implementation of Higher Order Mode Wakefields in MERLIN for the ILC_BDS collimators

8. 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/

9. 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 (p 0, 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:

10. 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)

11. 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) =  W m (s) r’ m r m cos(m  ) (W m is the ‘wake function’) the transverse and longitudinal wake potentials w L and w T can be obtained from this equation

12. w z = ∑ W’ m (s) r m [ C m cos(m  ) - S m sin(m  )] w x = ∑m W m (s) r m-1 {C m cos[(m-1)  ] +S m sin[(m-1)  ]} w y = ∑m W m (s) r m-1 {S m cos[(m-1)  ] - C m 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: C m = ∑r’ m cos(m  ’) and S m = ∑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 w x = ∑ m m r m-1 { cos [ (m-1)  ] ∑ j W m (s j ) C mj + sin [ (m-1)  ] ∑ j W m (s j ) S mj }

13. 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);

14. Geometric wakefields - Example W m (z) = 2 (1/a 2m - 1/b 2m ) 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

15. 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*10 10 e - Collimator half -width: 1.9 mm

16. 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

17. NoNo NameTypeZ (m)Apertur e 1CEBSY1Ecollimator37.26 ~ 2CEBSY2Ecollimator56.06 ~ 3CEBSY3Ecollimator75.86 ~ 4CEBSY E Rcollimator431.41 ~ 5SP1Rcollimator1066.61x99y99 6AB2Rcollimator1165.65x4y4 7SP2Rcollimator 1165.66x1.8y1.0 8PC1Ecollimator1229.52x6y6 9AB3Rcollimator1264.28x4y4 10SP3Rcollimator1264.29x99y99 11PC2Ecollimator1295.61x6y6 12PC3Ecollimator1351.73x6y6 13AB4Rcollimator1362.90x4y4 14SP4Rcollimator1362.91x1.4y1.0 15PC4Ecollimator1370.64x6y6 16PC5Ecollimator1407.90x6y6 17AB5Rcollimator1449.83x4y4 NoNameTypeZ (m)Aperture 18SP5Rcollimator1449.84x99y99 19PC6Ecollimator1491.52x6y6 20PDUMPEcollimator1530.72x4y4 21PC7Ecollimator1641.42x120y10 22SPEXRcollimator1658.54x2.0y1.6 23PC8Ecollimator1673.22x6y6 24PC9Ecollimator1724.92x6y6 25PC10Ecollimator1774.12x6y6 26ABEEcollimator1823.21x4y4 27PC11Ecollimator1862.52x6y6 28AB10Rcollimator2105.21x14y14 29AB9Rcollimator2125.91x20y9 30AB7Rcollimator2199.91x8.8y3.2 31MSK1Rcollimator2599.22x15.6y8.0 32MSKCRABEcollimator2633.52x21y21 33MSK2Rcollimator2637.76x14.8y9 ILC-BDS colimators

18. 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 -9 m - beam sizes for 4 modes:  x = 0.7*10 -6 m,  y = 0.19*10 -6 m - for small offsets, modes separation occurs at ~10 sigmas; Bunch size

19. Geometric wakefields - at 10 sigmas when the separation into modes occurs, the luminosity is reduced to 20% - for a luminosity of L~10 38 the offset should be less than 2-3 sigmas Luminosity

20. 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; b c c The 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;

21. 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

22. 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);

23. Resistive wakes Benchmark against an SLC result

24. 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 -6 m 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

25. Resistive wakes - at 10 sigmas when the separation into modes occurs, the luminosity is reduced to 10% - for a luminosity of L~10 38 the offset should be less than 1 sigma - the resistive effects are dominant! Luminosity

26. Beam offset in each BDS collimator No wakefields =4.74e-12; Jitter of 1 nm of maximum tolerable bunch-to-bunch jitter in the train with 300 nm between bunches; for 1nm: =8.61e-11 Jitter about 100 nm which intratrain feedback can follow with time constant of ~100 bunches; for 100nm: =5.4e-10 Maximum beam offset is 1 um in collimator AB7 for 1nm beam jitter and 9um for 100 nm jitter

27. 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: =2.37e-9

28. 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

29. 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