Implementation of Collimator Wakefields in MERLIN

1
Implementation of Collimator Wakefields in MERLIN
MERLIN Developers Meeting - DESY
Hamburg, February 2008

• Adriana Bungau
• The University of Manchester, UK

2
The physics case

• The literature is mainly concerned with
  wakefields in RF cavities with axial symmetry and
  with bunches near the axis -gt only lower order
  modes are important. Cavities ring at particular
  frequencies -gt long range wakefields are
  considered.
• For collimators
• The collimators are not characterized by
  resonances
• The system have no axial symmetry
• Only short range wakefields are important
• Particle bunches are distorted from their
  original Gaussian shape
• Higher order modes have significant effect
  (bunches close to the collimator edges)

3
Kick Factors

• The existing literature on collimator wakefields
  concentrates mainly on kick factors but kick
  factors are not enough!
• ? y K
  y
• they only look at the incoming and outgoing
  angular jitter
• include only the lowest order term (dipole
  mode)
• they do not describe what happens near the
  collimator edges
• kick factors depend on components and bunch
  shape (bunches usually treated as Gaussian)
• they do not describe the change in bunch shape

4
Goals

• development of a mathematical formalism for
  higher order collimator wakefields
• implementation of higher order mode collimator
  wakefields
• in MERLIN
• benchmark with other codes/experiments
• study one collimator for the start
• extend the study to all ILC_BDS collimators
• study of emittance increase at the IP
• study of luminosity reduction due to collimator
  wakefields
• study of bunch shape distorsion

5
Mathematical formalism

• The change in momentum of the trailing particle
  given by the integration over effects of E and B
  fields is

• The results depend on the bunch shape and a
  Gaussian shape is assumed
• In numerical simulations the integrals are
  replaced by sums and for N particles in n bunch
  slices the bunch potential is

• The wake potential can be found
  for any aperture, analytically or numerically
• The total effect on a charge is given by
  superposition of the contribution of all other
  particles and the bunch potential is

6
Wake effects from 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) rm rm cos(m?) (Wm is the wake
  function)
• the transverse and longitudinal wake potentials
  wL and wT can be obtained from this equation

7
Wake effects of many bunch slices
?j wx ?m m rm-1 cos (m-1)? ?jWm(sj) Cmj
sin (m-1)?
?jWm(sj) Smj
8
The initial wakefield implementation
Defined in BeamDynamics/ParticleTracking

• create an instance of WakeFieldProcess
• add the wakefield proces in the list for
  tracking
• invokes InitializeProcess and splits the bunch
  into slices
• in components, the pointer is set to the
  WakePotentials
• invokes DoProcess which calls ApplyWakefield
  (this sums the earlier contributions)

9
The initial wakefield implementation
Defined in AcceleratorModel

• each component may contain a WakePotentials
  object
• assigned to a particular accelerator component
  by a call to AcceleratorComponentSetWakePotentia
  ls (WakePotentials)

10
New classes

• contains a new data member int nmodes
• contains new versions for ApplyWakefield and
  CalculateWakeT and CalculateWakeL
• calculation of moments Sm and Cm is done through
  new CalculateSm /Cm routines

• contains the no of modes nmodes
• contains virtual functions Wtrans and Wlong
  which will be overriden in child classes

11
Accelerator model

• Constructed with the MADInterface using the
  lattice file from 2006 provided by F.Jackson
  ebds1.optics (ILC2006e)
• Difficulties with this lattice
• - several components in the beamline were not
  recognized by the MAD Interface and were treated
  as Drifts (INSTRUMENT, WIRE)
• - the lattice component MULTIPOLE is not
  defined in MAD Interface (the highest K magnet is
  OCTUPOLE)
• - the lattice does not end at the IP but also
  contains the beamline components to the beam dump
  -gt the lattice had to be split with MAD into two
  parts and the end-of-linac to the IP part was
  kept for tracking simulations

12
A first try

• The nominal linac exit parameters were defined as
  BeamData and the Gaussian Distribution was
  assumed for the bunch core
• Initially the beam was tracked from the first
  element in the beamline to the IP
• Tracking was aborted several times the
  character in the spoilers aperture in the MAD
  optics file was causing a break -gt problem solved
  by opening the apertures to X99Y99 for
  collimators CEBSY1, CEBSY2, CEBSY3, CEBSY
• The final beam parameters were recorded in a file
  but there seemed to be no wakefield effects on
  the bunch tail even at large offsets and using
  higher order modes !

13
More Difficulties

• In MADInterface the collimators were treated as
  Drifts
• In MADInterface.cpp
• - changed the collimator type to
  typeSPOILER
• - read the name, length and X0
• - aperture type is easily read from the MAD
  optics deck
• aptype X99Y99
• - conversion of the aperture dimentions from
  mm (MAD file) to m (required by most wakefield
  formulae)

14
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 i0 iltm i)
• coeff i 2(1./pow(a, 2i) -
  1./pow(b, 2i))
• TaperedCollimatorPotentials()delete
  coeff
• double Wlong (double z, int m) const
  return zgt0 ? -(m/a)coeff m/exp (mz/a) 0
  
• double Wtrans (double z, int m) const
  return zgt0 ? coeffm / exp(mz/a) 0
  

15
Application to one collimator
SLAC beam tests simulated energy - 1.19 GeV,
bunch charge - 21010 e- Collimator half -width
1.9 mm

• large displacement - 1.5 mm
• one mode considered
• the bunch tail gets a bigger kick

• small displacement - 0.5 mm
• one mode considered
• effect is small
• adding m2,3 etc does not
• change much the result

• large displacement - 1.5 mm
• higher order modes considered
• (ie. m3)
• the effect on the bunch tail
• is significant

16
Emittance increase

• 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 6.5110-7 m, ?y 5.6910-9m
• beam sizes for 4 modes ?x 0.710-6 m, ?y
  0.1910-6m
• for small offsets, modes separation occurs at
  10 sigmas

17
Luminosity reduction
at 10 sigmas when the separation into modes
occurs, the luminosity is reduced to 20 - for
a luminosity of L1038 the offset should be less
than 2-3 sigmas
18
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
19
General form of the resistive wake

• Write down Maxwells 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 m? and are sin m? )
• Integrate the force through a distance of
  interest L
• Apply the Panofsky-Wenzel theorem

20
Emittance increase

• - beam size at the IP in absence of wakefields
  ?x 6.5110-7 m, ?y 5.6910-9 m
• - beam sizes for 4 modes ?x 1.210-6 m, ?y
  3.510-6m
• 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

21
Luminosity reduction

• at 10 sigmas when the separation into modes
  occurs, the luminosity is reduced to 10
• for a luminosity of L1038 the offset should be
  less than 1 sigma
• the resistive effects are dominant!

22
Beam jitter in collimators

• No wakefields ltygt4.74e-12
• Jitter of 1 nm of maximum tolerable
  bunch-to-bunch jitter in the train with 300 nm
  between bunches for 1nm ltygt8.61e-11
• Jitter about 100 nm which intratrain feedback
  can follow with time constant of 100 bunches
  for 100nm ltygt5.4e-10
• Maximum beam offset is 1 um in collimator AB7
  for 1nm beam jitter and 9um for 100 nm jitter

23
Beam Jitter in collimators

• 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 ltygt2.37e-9

24
Bunch shape distorsion

• 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

25
More difficulties

• Few analytical calculations of the bunch
  potential corresponding to collimator geometries
  can be found in literature
• The calculations are for very simple geometries
• To deal with more complicated geometries MERLIN
  should read wake tables provided by Gdfidl or
  ECHO

26
Conclusions

• The new classes in MERLIN are uploaded in the
  MERLIN CVS repository in the newwake branch
• Emittance increase and luminosity reduction was
  studied for geometric and resistive wakefields
• It was found that at several sigma offset from
  the main axis the higher order mode contribution
  is not significant
• The contribution from higher order modes becomes
  important near the collimator edges away from the
  main axis