Wakefield Simulations for the ILC-BDS Collimators

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

Introduction - wakefields in collimators

Wakefields in cavities 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 collimators Wakefields cause emittance growth which in turn causes a reduction of the collider’s luminosity. • - 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 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)

r’,’ s s r,  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 • 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

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: 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)]} - 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 }

WakeFieldProcess WakePotentials SpoilerWakeFieldProcess CalculateCm(); CalculateSm(); CalculateWakeT(); CalculateWakeL(); ApplyWakefield (); SpoilerWakePotentials nmodes; virtual Wtrans(s,m); virtual Wlong(s,m); 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

a b 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 ; } ; };

Application to one collimator SLAC beam tests simulated: energy - 1.19 GeV, bunch charge - 2*1010 e- Collimator half -width: 1.9 mm • 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 • large displacement - 1.5 mm • one mode considered • the bunch tail gets a bigger kick

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

ILC-BDS colimators

Geometric wakefields Bunch size • 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;

Geometric wakefields Luminosity - 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

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; c The fields are excited as the beam interacts with the resistive wall surroundings; b c 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

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 Bunch size - 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

Resistive wakes Luminosity - 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!

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