Post on 19-Jan-2016
description
Simulating Short Range Wakefields
Roger Barlow
XB10
1st December 2010
XB10 workshop Dec 2010 Roger Barlow Slide 2
Contents
• Collimator Wakefields for new colliders
• Higher order (angular) modes
• Effective computation
• Resistive Wakefields
XB10 workshop Dec 2010 Roger Barlow Slide 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 2010 Roger Barlow Slide 4
Beyond the Kick Factor
y’=(Nre/ ) 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 2010 Roger Barlow Slide 5 Slide 5
Geometry
For large bunch near wall, angle of particle kick not just ‘transverse’
This trignometry should be included
XB10 workshop Dec 2010 Roger Barlow Slide 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,... 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
W m s=2 1
a 2m−1
b 2me−m s s
XB10 workshop Dec 2010 Roger Barlow Slide 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 2010 Roger Barlow Slide 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.
Kick=∑ W s , r , r ' , , '
XB10 workshop Dec 2010 Roger Barlow Slide 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:
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)]}withCm = ∑r’m cos(m’) Sm = ∑r’m sin(m’)Using slices, summation is computationally rapid
XB10 workshop Dec 2010 Roger Barlow Slide 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 2010 Roger Barlow Slide 11
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
XB10 workshop Dec 2010 Roger Barlow Slide 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 2010 Roger Barlow Slide 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 2010 Roger Barlow Slide 14
In frequency space
XB10 workshop Dec 2010 Roger Barlow Slide 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
E z=2qb
1ikb2
−k
E z=−2qkb
XB10 workshop Dec 2010 Roger Barlow Slide 16
Cunning (?) trick
XB10 workshop Dec 2010 Roger Barlow Slide 17
XB10 workshop Dec 2010 Roger Barlow Slide 18
Extensions
• Can include higher order modes
• Can include AC conductivity (Drude model) =
1−ikc= 1−iK
XB10 workshop Dec 2010 Roger Barlow Slide 19
Results
XB10 workshop Dec 2010 Roger Barlow Slide 20
Varying ξ
XB10 workshop Dec 2010 Roger Barlow Slide 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 2010 Roger Barlow Slide 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 2010 Roger Barlow Slide 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!