Proceedings of the 1st Annual Meeting of Particle Accelerator Society of Japanand the 29th Linear Accelerator Meeting in Japan (August 4 - 6, 2004, Funabashi Japan)

A feasibility study for manufacturing of the wake-compensated detuned
accelerator tube with the brazing and dimpling method
M. Kuriki , H. Hayano
High energy accelerator research organization, Tsukuba,Ibaraki,JapanINTRODUCTION
In the high-energy physics, the collider based on a storage ring has been a most powerful tool. It is considered,however, that LEP at CERN would be the last ring based
collider because of the synchrotron radiation loss that diverges as forth-order of the beam energy. GLC can breakthe energy limit of the ring collider because of no synchrotron radiation. On the other hand, GLC may have adifficulty on the luminosity, since the beam pass the interaction region only once.
Large luminosity on LC can be taken with extremely low
emittance beam. Since there is no convergence effect on theemittance in the linac, it is important to preserve it during
the acceleration. The main phenomena to spoil the beamemittance is transverse kick by the beam induced dipole
mode in the accelerator tube, wake field.
Let us assume the leading particle with charge  travels in a cylindrically symmetrical structure with a displacement from the center axis,  . The field observed by a witness particle delayed with time  from the leading particle, is the wake
 , where is expressed as   field. The wake field in a accelerating tube is expressed as
the sum of the each cell’s contributions
as;



 




   "! #%$ ' &
 







where ( is the number of cells in a tube,
*)   eter given by,


)



+







+

, &



(1)
is loss param(2)andare the shunt impedance and the quality factor
for the dipole modes, is speed of light,  is frequency ofthe dipole mode, and  is iris radius of the cell. The shunt
impedance is defined along an axis parallel to the center
axis, at the iris edge.
There are three methods to compensate the wake field,
damping, detune, and the hybrid. Here we employ the detuned method because the dimpling frequency tuning, thatis a popular way to manufacture the tube, can not be usedwith the damping method.

-masao.kuriki@kek.jp

/ .



 

 . . 

 ! #%$  .  103214 65 8. 7:9 ;  <&(3)

where the variables with bars mean its average over thetube, 7=96; is the sigma of the Gaussian distribution. Thisequation means the wake is “damped” quickly by the exponential term. This effect can be explained as the cancellation among the dipole modes, so that the sum of the wakelooks small, but it is not damped at all.
In a reality, the number of modes is finite, so that thecancellation is not perfect. After some period, the detuned
modes are re-synchronized and the amplitude of the sum
grows up again like FM wave.
If the modes are distributed linearly, the damping speed
is much slower than that with Gaussian, but the re too. The wake function is
synchronization is slow
! #%$



> .

 

 . .

"! #%$  .  !@#A$   . 7 96;1?
.  7:96; ?  (B &(4)the last term shows the slow damping.
In GLC, the beam is accelerated in 195 bunches in a train
with 1.4 or 2.8 ns spacing. If the sinusoidal oscillation termof the wake field synchronizes with the bunch spacing, thewake felt by the beam could be much lower than its envelope. This Bunch spacing Harmonized Detuned Structure,
BHDS originally proposed by Bane[2].

DESIGN
Cell geometry was determined by MAFIA 2.5 dimension eigen mode solver. Because the solution depends onthe mesh size in MAFIA calculation, the final value wasobtained with the extrapolation toward the zero mesh size.
The phase advance per cell on the acceleration mode was? ,ED to decrease the group velocity. The cell radius,set
F to Cwas determined giving 2856 MHz resonant frequency at,GDphase advance for the various HJI H6K radius,  as shown
C ?
in Figure 1.
x 10

-1

R (Mohm/m)

Global linear collider, GLC project[1] have to be anepoch to open the new horizon of the particle physics.
To obtain a large luminosity in GLC, the low emittancebeam is one of the most important feature. In this article, we discuss the detuned method on the S-band acceleration tube for GLC pre-linac to preserve the beam emittance.
The manufacturing of the tube with the standard dimpling
method is also considered.

In the detuned method, the cell geometry is varied so thatthe accelerating modes are kept and the dipole modes are
spread widely. If the infinite number of dipole modes are shape, the sum of them becomes
distributed as a Gaussian

0.435

b (m)

Abstract

0.433
0.43

100
80

0.428

60

0.425
0.423

40

0.42
0.418

20

0.415
0.413
0.41
0.012

0.014

0.016

0.018
a (m)

Figure 1: Cell radius L as function of iris radius M for 2856MHzN3OQP'R mode cells.

341

0
0.012

0.014

0.016
0.018
iris radius a (m)Figure 2: Shunt impedance perunit as function of iris radius M for
N3OQP'R mode cells.
2856MHzProceedings of the 1st Annual Meeting of Particle Accelerator Society of Japanand the 29th Linear Accelerator Meeting in Japan (August 4 - 6, 2004, Funabashi Japan)

There are still an ambiguity to determine the tube design;
33
which  should be used. In the conventional design of accelerator tubes, there are two styles to determine the series;
constant impedance, CI and constant gradient, CG. On the
detuned tube,  s are distributed so that the wake field iscompensated. The design procedure of the detuned tube issimilar to CG, but the purpose is different.
The dipole mode driven by the beam is a synchronousmode to the light. To obtain this synchronous frequency,
we have to model the dispersion curve of the dipole mode.
Since the lowest dipole mode is strongly coupled to the second dipole mode[3], the actual mode is a mixed state of thetwo modes. It can be modeled by the equivalent circuit as
fllows;

 

 





(5)

( ) 

value
76
3m

HI

DE@F GHI

Last iris radius

0.017044 m
0.044 c0.015850 m
0.035 c0.035
60.25 MV @ 100MW input
20.38 MV/m
19.89 MV/mJ

Tilt parameter
Total acceleration
Highest field
Lowest field          
of the loss parameter. There is some optimum value of 3      for given   and  pair. The initial choice on   fixes thewhere"! are the band width for TE and TM modes,#  and ! # "  andare the fundamental
mode frequencies for TE and#TM modes, $ and are the phase advance per cell and thefrequency of the oscillating mode respectively.
The crossDterm can be vanished with $ &% or . In that case, themodes
are isolated and the formulas becomes independent.
! , ! " ,#  , and # "  are thendetermined from the mode frequenD
cies at $ '% and respectively. Once the parameters are
determined, the dispersion curve is drawn by Eq. 5. The
cross point of the curve and the light line corresponds tothe synchronous mode.
The wake for a whole tube are obtained by Eq. 1. The
synchronous dipole mode frequency is evaluated assuming
an infinite series of the identical cells. Eq. 1 means that thedipole mode behavior in an actual tube is same as this ideal
case. In a real tube, the next cell’s geometry is slightly different. This calculation ignores this small difference. Even
the difference is small, the impact to the final result could
be large. This is one of the next issues.
From the wake function, the beam averaged wake ( *)for a beam train is calculated as follows;



item
Number of cells
Total length
Phase advanceFirst iris radius

,+

 

.-


 H0/21
( )

&

-

(6)

where ( ) is number of bunches in a train,  is the bunch
spacing in time. If the oscillation period of the wake func
tion is harmonized with the bunch spacing, the averaged wake ( *) is minimized. Other words, the tube geometry is optimized to minimize the averaged wake.
As mentioned, we employ the linear detuning. The linear detuning is parameterized the initial point   , the end
point   , and the tilt parameter, 3 . The tilt parameter 3
54 564 ? 74
4 where 4  and 4 are /is defined as 


 4
4
the mode densities at   and   respectively. ?  and 
4
4
98;:
 :
can be expressed as   <>=@?A<@B and   <>=C?A<@B , so themode distribution is determined identically with the threeparameters.
If some   is taken,   is naturally determined giving themean dipole frequency harmonized with the bunch spacing.
The purpose of 3 is the compensation of the  dependencesize of the detuning. The size of the detuning determinedthe damping speed on the wake, but the speed is not critical for this harmonized detuning. Therefore, there is someambiguity on the   selection. This time,   is set to makethe tube semi-CG behavior, i.e. the field gradient along thetube becomes constant.
According to these procedures, a S-band tube is designed. The total length is set to 3m that is limited bysize of the brazing oven. The phase advance per cell was,ED
set to C ?
to decrease the group velocity of the accelerating mode because the tube optimized for the lowest wake
function was organized with larger  s giving higher groupvelocity. The number of cells in a tube was 76. The specifications were summarized in Table 1. The acceleration
is estimated without beam-loading. The field uniformity is,E,5LK
nearly 1%. The%NMPO ?RQ , but shunt impedance isthe group velocity is still high due to the large iris radius,%TS %GC K 5 %TS % ,E, resulting this low acceleration efficiency.
The wake function is calculated according to Eq. 1 forthe designed tube. The result is shown in Fig. 3. The
horizontal and vertical axes are time in ns and the wake in
MPU ?RQ  4;V . The wake function is plotted as a curve, but
it looks like a shaded area due to its high density. The wakeat the beam position (every 1.4ns) is marked as the closed
circles. The toleranceontransverse wake is estimatedK
4;V the. Allless than %WS %EC MPU ?XQ of the points are below thethreshold.
wake (MV/m.m.nC) 

Table 1: Specifications of the designed tube. The acceleration is estimated without beam-loading.

1

10

10

10

10

-1

-2

-3

-4

0

50

100

YZ\[5]

150

200

250
time(ns)

Figure 3: The wake function
for the designed tube as funtion oftime is plotted. The closed circles show the wake where the beam isplaced with 1.4 ns spacing. All circles are positioned below the threshold,[^_ ^'N>`Cab Odc  ef

342

averaged wage (MV/m.m.nC)

In the previous section, it is demonstrated that the designed tube fits to the GLC requirement. It is a human nature that a question arises; is it difficult to manufacture it?
The tolerance on the averaged wake against errors were
examined. The random error corresponds to the machining
error and the systematic error corresponds to some scaleerror on the dipole mode frequency like temperature shift,systematic error on the EM simulation were assumed.

10

10

10

-1

-2

-3

0

0.1

0.2

0.3

0.4

0.5

0.6 0.7 0.8 0.9
1size of error (MHz)Figure 4: The averaged wake felt by the bunched beam as function of
the size of errors on the dipole frequency. The solid and dashed line show
those by the systematic error and the random error. The random error'NO,
more than 0.2 MHz makes it above the threshold,but the averaged wake is kept less than the threshold even with 1 MHz
systematic error.

^_ ^ >`Cab dc  ef

Fig. 4 shows the averaged wake as function of the sizeof errors. The solid and dashed lines are those with the systematic and random errors respectively. For the systematic
error, the dipole frequencies for all cells is shifted towarda same direction with a same size. On the other hand, forthe random error, the size and direction of the frequency
shift is determined randomly for each cell according to a
Gaussian distribution. The size of error means sigma of theGaussian distribution. One datum was obtained as mean of
100 trials.
From the simulations, the wake does not exceed the tolerance even with the 1MHz systematic shift. On the other
hand, more than 0.2 MHz random shift destroys the wakecompensation. As the summary, the systematic and random frequency errors on the dipole mode have to be lessthan 1MHz and 0.2 MHz to keep the detuning respectively.
A conventional method for the accelerating tube manufacturing is the brazing and dimpling. The copper cells are
unified by brazing that moves the acceleration mode. The
reason is not understood well, but the amount of the drift
is about 0.3 MHz. To absorp this drift, the cell is designed
slightly lower than the target frequency and adjusted by thedimpling that is a deformation of the cavity wall by an external force.
To examine the feasibility of the brazing method for the
detuned tube, we simulated the dimpling with 3D MAFIA
eigen mode solver. The absolute frequency resolution isvery limited in this 3D code, but only the frequency shift
is matter for this study. Using the calculated EM fieldsof the acceleration and dipole modes, the frequency shiftby the dimpling was estimated with the field perturbation
method[4]. Fig. 5 and 6 show the frequency shifts(%) of
the acceleration and dipole modes as function of the dim-df/f (%)

MANUFACTURING

df/f (%)

Proceedings of the 1st Annual Meeting of Particle Accelerator Society of Japanand the 29th Linear Accelerator Meeting in Japan (August 4 - 6, 2004, Funabashi Japan)
0.035

0.04

0.035

0.03

0.025

0.03
0.025

0.02

0.02

0.015

0.015

0.01

0.01

0.005

0.005
0

0

-0.005-0.005

0

10

20

30

40
50
dimple size mm3

Figure 5:

Relative furequency
shift of the acceleating mode in %as function of the dimpling size in. The curve is drawn for M.

c c ^_ ^



0

10

20

30

40
50
dimple size mm3

Figure 6: Relative furequency
Rshift of the first dipole mode in% as function of the dimpling size
. The curve is drawn for
in
.
M

c c 
 ^_ ^pling size respectively. From these plots, the frequency
shift on the dipole is roughly 80% of that on the acceleration.
Including the dimpling effect, the tube is designed oncewith 2855 MHz acceleration mode frequency. The dipole
frequency is also evaluted with this geometry. The dimpling effects for various iris size are parameterized withquadratic curves each for the accelerating mode, first andD
respectively.
second dipole modes at $  % and $ The amount of the dimpling is determined for each cell giving +1MHz frequency shift on the accelerating mode. The
dipole modes are also shifted for each cell with this dimpling amount according to the dimpling cureves. Using thedispersion curves defined by these dipole modes frequencies, the synchronous frequency for each cell is obtained.
The wake is then estimated.
As mentioned already, the frequency shift by the brazing process is about 0.3 MHz. If this shift is made bysome deformation on the cavity wall that is equivalent to
the dimpling deformation, the dipole mode frequency after
the dimpling adjustment is exactly same as expected.
Even this shift is made by some other reason,
the random# F , where
7 ; ) iserror caused by the brazing process is %TS 7
sigma of distribution of the frequency shift for all cells. The
frequency shift by the brazing is distributed around 0.2- 0.3
MHz empiricaly, so the sigma should be much smaller than
0.2 Mhz. By this fact, the detuned tube can be manufactured by the brazing and dimpling method.

SUMMARYFor GLC pre-linac, a wake-compensated S-band acceleration tube with the detuning technique is considered. We,GDphase advance with the detundesigned a tube with C ?
ing wake compensation mechanism. The wake compensation was enough for GLC requrement. The acceleration
efficiency is slightly lower,
D but it can be recovered by thecell, e.g. ?
. Assuming the estimated tolslower
erances against the random and systematic errors on the
dipole frequency, the conventional method, brazing anddimpling, can be applied to manufacture the detuned tube.

REFERENCES[1] http://lcdev.kek.jp/[2] K. Bane and Z. Li, “Dipole mode detuning in the NLC injector
linac”,arXive/physics0008167, 2000
[3] K. Bane and R. Gluckstern, Part. Accel. 42,123, 1994
[4] P.B. Wilson, “High energy electron linacs: Applications to storage
ring RF systems and linear colliders”, SLAC-PUB-2884, 1982

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