Simulating Short Range Wakefields

Roger Barlow

XB10

1st December 2010

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Contents

• Collimator Wakefields for new colliders

• Higher order (angular) modes

• Effective computation

• Resistive Wakefields

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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

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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

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Geometry

For large bunch near wall, angle of particle kick not just ‘transverse’

This trignometry should be included

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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

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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

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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 ' , , '

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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

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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

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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

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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

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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

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In frequency space

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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

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Cunning (?) trick

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Extensions

• Can include higher order modes

• Can include AC conductivity (Drude model) =

1−ikc= 1−iK

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Results

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Varying ξ

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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)

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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!

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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!