C.Limborg, K.Bane
Wakefield computations –
1- ABCI computations for a few structures (Flange, Bellows …)
2- Comparison analytic with ABCI runs
3- Comparison Cross and Cavity using MAFIA
Part 1: ABCI computations for a few structures (Flange, Bellows …)
The structures computed correspond to first drawings provided by M.Palrang. Here we
study the evolution of the loss parameters with rms bunch length. ABCI uses Gaussian
bunches, so we cannot get an accurate computation for our “nearly-square” pulse.
Flange:
g = 2.55

a = 12.79
Transverse
 (mm)
1
0.5
0.25
18.3
Max (V/pC/m)
30.6
31.8
24.9
k (V/pC/m)
9.04
6.97
5.22
Longitudinal
 (mm)
1
0.5
0.25
Min (V/pC)
0.72
1.13
1.72
k (V/pC)
0.521
0.811
1.22
Flange Gun Out:
This flange at the gun exit only differs very slightly in dimensions to the standard flange
geometry.
2.55
6.6
12.79
18.3
20.05
The entrance/exit pipes have the same radius (as a simplification of the structure given by
M.Palrang)
Transverse
 (mm)
1
0.5
Max (V/pC/m)
60.9
61.6
k (V/pC/m)
15.97
11.98
Longitudinal
 (mm)
1
0.5
Min (V/pC)
1.32
2.0
k (V/pC)
0.83
1.43
Welded Bellows
11.5
3
10.5
a = 18.16
23.9
With pipe radius a = 18.16
Transverse
 (mm)
1
0.5
0.25
Max (V/pC/m)
36
39.8
41.2
k (V/pC/m)
12.7
11.33
9.479
Longitudinal
 (mm)
1
0.5
0.25
Min (V/pC)
2.0
3.52
5.71
k (V/pC)
1.44
2.56
4.21
With pipe radius = 12.7
Transverse
 (mm)
1
0.5
0.25
Max (V/pC/m)
103.7
114.8
119
k (V/pC/m)
36.9
32.88
27.9
Longitudinal
 (mm)
1
0.5
0.25
Min (V/pC)
2.9
5
8.3
k (V/pC)
2.07
3.69
6.13
Creneau Bellows
11.5
3
10.5
18.16
With radius a = 18.7
Transverse
 (mm)
1
0.5
0.25
Longitudinal
 (mm)
1
0.5
0.25
With radius a = 12.7
Transverse
 (mm)
1
23.9
Max (V/pC/m)
59
58
k (V/pC/m)
19
14.6
Min (V/pC)
2.91
4.45
k (V/pC)
2.11
3.27
Max (V/pC/m)
174
k (V/pC/m)
56.8
0.5
0.25
169
170
43.1
34.46
Longitudinal
 (mm)
1
0.5
0.25
Min (V/pC)
4.24
6.46
10.0
k (V/pC)
3.08
4.75
7.46
Interpretation and comments:
Fast computation :
A transverse wakefield of 20V/pC/m from head-to-tail will produce
px = 2 V/nC for a 100 m offset
So a kick of  = 0.4 rad at 5MeV
The growth in emittance is given by /t .
t can vary between 100 rad and 3000 rad in the GTL section.
For an offset of 1mm, we would get an emittance growth which varies between 0.0133%
and 4% .
Conclusion:
A more accurate computation is required using local values of betatron functions and
wakefields.
Unfortunately, the wakefield extracted from ABCI is computed from a Gaussian bunch.
To do the convolution accurately, one needs to compute the Green’s function.
Before we use the Green’s function, we compared its accuracy with ABCI results for a
case of a Gaussian bunch.
Part 2 - Comparison ABCI with Analytical formula:
Flange:
For a flange, the diffraction formula [1] is used.
Z c
kloss  4.34 3o 3 g z in V/pC/m
 a
 [mm]
1
 [mm]
1
a [mm]
10
 [mm]
a [mm]
10
 [mm]
5
15
g [mm]
1
kloss ABCI
10.5
kloss theory
15.8
g [mm]
1
kloss ABCI
10.5
kloss theory
15.8
 [mm]
1
 [mm]
2
 [mm]
0.5
a [mm]
10
 [mm]
a [mm]
10
 [mm]
a [mm]
10
 [mm]
5
5
5
g [mm]
10
kloss ABCI
41.89
kloss theory
50.0
g [mm]
1
kloss ABCI
11.98
kloss theory
22.3
g [mm]
1
kloss ABCI
8.36
kloss theory
11.18
The formula gets more accurate for longer gaps or smaller bunch (for large g/ ratio)
Bellows
For a single mode, the wave number is k 
2
a
The wake function for a that mode is then given by Wx  Wxo sin ks
2 Zoc
in V/pC/m2
a4k
The transverse amplitude of the wakefield per unit length is Wxo 
The voltage induced is then given by Vind ( s)  
sp
sp0

1
2 
 s  sp 2
e
2 2
s 
Wx sp dsp

s2
e
Vind s ds
2 
The kloss for a Gaussian bunch can then be computed analytically using Mathematica
(notebook from K.Bane).
The first moment of the loss parameter is given by k loss,1  
1
2 2
s  
Comparison with ABCI :
 [mm]
1
1
1
a [mm]
12.7
16
16
 [mm]
3
3
6
kloss ABCI
165.3
78.2
78.2
kloss theory
191.7
75.5
77.7
Part 3: Comparison cross and cavity using Mafia
We first checked on simple examples similar to those of paragraph 1, that ABCI and
Mafia were giving the same results for cylindrical symmetry structures g~1.5 mm.
We then computed transverse wakefields for a cross and a cavity.
The two structures have geometry shown in the figures below
Cavity
Cross
 [mm]
g [mm]
a [mm]
 [mm]
1
1
1
25.4
5
1.5
12.7
12.7
12.7
12.3
12.3
9
kloss cross
(MAFIA)
20.9
kloss cavity
(MAFIA)
26.5
11.2
5.92
kloss cavity
(ABCI)
34
14
6.76
The agreement between ABCI and MAFIA for cylindrical symmetry structure was better
for narrower cavities (flanges). A 2D MAFIA computation will be done too.
The maximum of the wakefield amplitude is very close (<10%) for MAFIA and ABCI.
The loss parameter is larger for a cavity than for a cross. The 2D computations give an
overestimate. In the future we will assimilate crosses to cavities knowing that we treat the
most pessimistic case.
References:
[1] “Wakefields of Very Short Bunches in an Accelerating Cavity”, K.Bane and Matthew
Sands, SLAC-PUB-4441