Simulation of High Order Short Range Wakefields

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Simulation of High Order Short Range Wakefields
Roger Barlow, The University of Manchester and
Cockcroft Institute, UK Adriana Bungau, The
University of Manchester and Cockcroft Institute,
UK
Abstract
We present a formalism for incorporating
intra-bunch wake fields into particle-by-particle
tracking codes, such as MERLIN. Higher order wake
fields are incorporated in a manner which is
computationally efficient. Standard formulae for
geometric, resistive and dielectric wakefields
can be included for various apertures,
particularly those relevant for ILC collimators.
Implementation in Merlin
Wake effects on a single charge
Consider the effect on a trailing particle at r,
? of a slice of N particles all ahead by the same
distance s. All particles are relativistic. The
effects of transverse velocity and acceleration
during the passage of the particles through the
aperture are ignored. The total effect is given
by simple summation over all particles in the
slice and the combined kick is
The code contained two important and separate
classes relevant to wakefields WakePotentials
and WakeFieldProcess. But the existing standard
implementation only included the monopole
(longitudinal) and dipole (transverse) wakes. To
include higher order modes we must have the
ability to sum over modes. Two new classes were
derived SpoilerWakeFieldProcess and
SpoilerWakePotentials
Wz ? Wm(s) rm Cmcos(m?) Sm
sin(m?) 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)?
WakeFieldProcess
WakePotentials
SpoilerWakeFieldProcess CalculateCm() CalculateSm
() CalculateWakeT() CalculateWakeL() ApplyWakef
ield ()
SpoilerWakePotentials nmodes virtual
Wtrans(s,m) virtual Wlong(s,m)
Where Cm ?rm cos(m?) and
Sm ?rm sin(m?)
r and ? are coordinates of particles in the
leading slice, and Cm and Sm can be calculated
and stored. For a particle in slice i, a
wakefield effect is received for all slices ji.
The total effect is (in x for example)
?j wx ?m m rm-1 cos (m-1)? ?jWm(sj) Cmj
sin (m-1)? ?jWm(sj) Smj
The SpoilerWakeFieldProcess class calculates the
moments Cm and Sm for each slice through new
CalculateSm and CalculateCm routines, calculates
the sums over leading slices j through
CalculateWakeT and CalculateWakeL and has a new
version of ApplyWakefield with an extra loop over
modes.
Similar equations apply for wy and wz the sums
over leading slices j can be calculated
separately and used in the computer code
The virtual functions in SpoilerWakePotentials
will be overriden in child classes like
TaperedCollimatorPotentials
The wake function for a steeply tapered
collimator moving from aperture b to aperture a
is
class TaperedCollimatorPotentialspublic
SpoilerWakePotentials public double a,
b double coeff TaperedCollimatorPoten
tials(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)) TaperedCollimatorPoten
tials()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
Wm(z) 2 (1/a2m - 1/b2m) exp (-mz/a) ?(z)
Merlin Simulations
SLAC beam tests were simulated with the new
additions to the Merlin code. A Gaussian beam
having an energy of 1.19 GeV and 21010 electrons
was sent through a spoiler with a gap of
half-width of 1.9 mm. The lattice functions were
set to ?x 3m, ?y 10m ?x 0.36 mm, ?y
0.16mm. The bunch length was ?z 0.65mm. We were
interested in the deflection in angle of the
particles that emerge from the collimator.
An analysis with higher order modes was performed
for 1.5 mm beam offset. The bunch tail gets a
bigger kick when three modes are considered than
in the case when one mode was included in the
simulation.
Simulations were performed taking the first mode
for the start. The vertical kick varies with the
position along the bunch - the tail is more
affected than the head - which will lead to
non-Gaussian bunch shapes. The effect is small
for one mode and adding m2, 3 etc. does not
change it much.
For a large displacement of 1.5 mm, the bunch
tail gets a bigger kick even when one mode is
considered. Therefore, at large offsets, higher
order modes must be included in the simulation
This work is supported by the Commission of the
European Communities under the 6th Framework
Programme Structuring the European Research
Area, contract number RIDS-011899