Simulating Short Range Wakefields

1. Simulating Short Range Wakefields Roger Barlow XB10 1st December 2010

2. Roger Barlow Contents • Collimator Wakefields for new colliders • Higher order (angular) modes • Effective computation • Resistive Wakefields

3. Roger Barlow 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

4. Roger Barlow Beyond the Kick Factor y’=(Nre/ ) t y Many analyses just • Determine t • Apply Jitter Amplification formula This is a simple picture which is not necessarily the whole story

5. Roger Barlow Geometry For large bunch near wall, angle of particle kick not just ‘transverse’ This trignometry should be included Slide 5

6. Roger Barlow 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,... 1+5 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

7. Roger Barlow 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

8. Roger Barlow 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.

9. Roger Barlow 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: 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)]} with Cm = ∑r’m cos(m’) Sm = ∑r’m sin(m’) Using slices, summation is computationally rapid

10. Roger Barlow 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

11. Roger Barlow Wake function formulae: EM simulations Few examples: One is (for taper from a to b) wm(s)=(1/a2m-1/b2m)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 Wm(s) using any symmetry

12. Roger Barlow 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

13. Roger Barlow 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

14. Roger Barlow In frequency space

15. Roger Barlow 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

16. Roger Barlow Cunning (?) trick

17. Roger Barlow

18. Roger Barlow Extensions • Can include higher order modes • Can include AC conductivity (Drude model)

19. Roger Barlow Results

20. Roger Barlow Varying ξ

21. Roger Barlow 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)

22. Roger Barlow 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!

23. Roger Barlow 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!