Simulating Short Range Wakefields Roger Barlow XB10 1 st December 2010

XB10 workshop Dec 2010Roger BarlowSlide 2 Contents Collimator Wakefields for new colliders Higher order (angular) modes Effective computation Resistive Wakefields

XB10 workshop Dec 2010Roger BarlowSlide 3 Wakefields at the ILC and CLIC Short Range Wakefields in non- resonant structures (collimators) may be important like never before Luminosity is everything Charge densities are high Collimators have small apertures

XB10 workshop Dec 2010Roger BarlowSlide 4 Beyond the Kick Factor  y’=(Nr e /  )  t y Many analyses just 1)Determine  t 2)Apply Jitter Amplification formula This is a simple picture which is not necessarily the whole story

XB10 workshop Dec 2010Roger BarlowSlide 5 Geometry For large bunch near wall, angle of particle kick not just ‘transverse’ This trignometry should be included

XB10 workshop Dec 2010Roger BarlowSlide 6 Head – tail difference: Banana bunches Particles in bunch with different s get different kick. No effect on start, bigger effect on centre+tail Modes 1, 1+2, Offsets 0.3 mm, 0.6 mm, 0.9 mm 28.5 GeV electrons in 1.4mm aperture in 10 mm beam pipe Use Raimondi formula

XB10 workshop Dec 2010Roger BarlowSlide 7 Non-Gaussian bunches Kick factor assumes bunch Gaussian in 6 D Contains bunch length  z in (some) formulae Even if true at first collimator, Banana Bunch effect means it is not true at second Replace  t =   W(s-s’)  (s)  (s’) ds’ ds by numerical sum over (macro)particles and run tracking simulation

XB10 workshop Dec 2010Roger BarlowSlide 8 Computational tricks Effect of particle at (r,ө) of a particle s ahead, at (r',ө') That's N (N-1)/2 calculations Make possible by binning in s and expanding (ө-ө') in angular modes.

XB10 workshop Dec 2010Roger BarlowSlide 9 Wake functions Integrated effect of leading particle on trailing particle depends on their transverse positions and longitudinal separation. Dependence on transverse positions restricted by Laplace’s equation and parametrisable using angular modes Dependence on longitudinal separation s much more general. Effect of slice on particle : 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)  ]} with C m = ∑r’ m cos(m  ’) S m = ∑r’ m sin(m  ’) Using slices, summation is computationally rapid

XB10 workshop Dec 2010Roger BarlowSlide 10 Merlin Basic MERLIN Dipole only and ‘Transverse’ wakes New features in MERLIN Arbitrary number of modes Correct x-y geometry Easy-to-code wake functions Still only for circular apertures at present

XB10 workshop Dec 2010Roger BarlowSlide 11 Wake function formulae: EM simulations Few examples: One is (for taper from a to b) w m (s)=(1/a 2m -1/b 2m )e (-mz/a)  (z) Raimondi Need to use EM simulation codes and parametrise Run ECHO2D or GdfidL or … Has to be done with some bunch: point in transverse coordinates, Gaussian in z. Need to do this several times with different transverse positions: extract modal bunch wake functions W m (s) using any symmetry

XB10 workshop Dec 2010Roger BarlowSlide 12 Does it matter? First suggestions are that effects of high order modes etc are small This is not sufficiently solid to spend $N Bn of taxpayers’ money Plot by Adriana Bungau using MERLIN

XB10 workshop Dec 2010Roger BarlowSlide 13 Resistive Wakes Circular (thick) pipe radius a, conductivity σ Work in frequency space diffn → multn Find Longitudinal wake, get transverse from Panofsky-Wenzel theorem Solve Maxwell's equations Decompose into angular modes Match boundary conditions Back-transform

XB10 workshop Dec 2010Roger BarlowSlide 14 In frequency space

XB10 workshop Dec 2010Roger BarlowSlide 15 Back to real space Approximate Long-range (Chao) More accurately (Bane and Sands) General technique: make no approximations and integrate numerically. Separation into even and odd parts helps

XB10 workshop Dec 2010Roger BarlowSlide 16 Cunning (?) trick

XB10 workshop Dec 2010Roger BarlowSlide 17

XB10 workshop Dec 2010Roger BarlowSlide 18 Extensions Can include higher order modes Can include AC conductivity (Drude model)

XB10 workshop Dec 2010Roger BarlowSlide 19 Results

XB10 workshop Dec 2010Roger BarlowSlide 20 Varying ξ

XB10 workshop Dec 2010Roger BarlowSlide 21 Implementation Write 3D table – function of s,ξ,Γ – for each mode, evaluated using Mathematica. Do this once (or get them from us) At start of simulation form 1D table for each collimator, at appropriate ξ and Γ Use this table in Merlin. (Also usable in other codes)

XB10 workshop Dec 2010Roger BarlowSlide 22 Relevance Bane and Sands (ξ=0) is fine for conventional structures as radius >> scaling length But we have the technique ready for small apertures in low-conductance materials!

XB10 workshop Dec 2010Roger BarlowSlide 23 Conclusions Nobody has all the answers The physics is complicated (and interesting) Plenty of room for exploring different approaches in computation, maths, and experiment ILC/CLIC requires relaxing some standard approximations. This can be done. There’s more to Wakefields than Kick factors!