BEAM DYNAMICS IN ACCELERATING STRUCTURES
OF THE JLC MAIN LINAC
K.Kubo, T.Higo, K.Takata, T.Taniuchi, (A)M.Yamamoto and K.Yokoya
KEK, National Laboratory for High Energy Physics(A)GUAS，The Graduate University for Advanced Studies
Abstract : Multibunch and single bunch motion with transverse wake field in the X-band
accelerating structures in the JLC main linac was studied by tracking simulation.
1. I n t r o d u c t i o n

In JLC (Japan Linear Collider), 20bunches, L4 nsec spacing, will be accelerated inone RF pulse. Each bunch is to have 2x1010
particles. The vertical and horizontal normalized
emittance is designed to be 3xl0 -8 and 3xl0-6,
respectively.
In section 2, tracking simulation for
multibunch motion is reported. Because of thenarrow spacing between bunches, the beams
may become unstable due to the long range wakefield in the accelerating structures. In thesimulation, each bunch was assumed to becompletely rigid， represented by one
macroparticle. No longitudinal wake, but onlytransverse wake was taken into account. To
suppress the instabilities, two types of cureshave been considered. The first one isaccelerating structures with very low Q-values(called ’damped structure’). The second ismulticell structures with frequency spread (cellto cell frequency differences, called ’detunedstructure'). Details of the structures for JLC aredescribed in other reportsf1]』2]』3]. Only damped
structure was studied in this report.
In section 3, simulation for single bunch
motion is reported. One bunch was divided into
several slices in longitudinal direction and eachslice was represented by a macroparticle. Energy
distribution in a bunch and therefore *BNS
damping' was considered with the accelerating
field and longitudinal short range wake field.
Transverse short range wake function wasassumed to be linear.

Accelerating field : 80 MV/m.
Injection energy :1 0 GeV
Final energy : 750 GeV
Total length : 9.2 Km
Total number of quadrupole magnets : 800
Total number of accelerating cavities :13000
Transverse wake function of a dipole mode
field can be written as follows.
WT(s )= (l/2)(R/Q)cosin(cot)exp[-(co/2Q)t](1)

where co is frequency o f the mode, Q is theloaded Q value of the mode.
Transverse momentum change of n-th bunchin a train can be written as
APin = Emq(l/ 2 )(R/Q)m(com/c)qZ k=1n-1
xksin[com(n-k)tb]exp[-(com/2Q)(n-k)tb](2)

where q is the charge of bunches, xk is thedisplacement of k-th bunch and tb is the time
spacing between bunches, m is an index for
dipole modes. In JLC,q = 2xl010e and tb =1.4 nsec.
Parameters of three modes with large R/Q are as
followst1].
R/Q (M Q /m 2)
221 . Multibunch

0.2
The simulation was performed for a simple
model linac as follows.
Optics : FODO singlet cells, and averaged beta
function is proportional to square root of theenergy
Beta at the injection :3 m
Phase advance per cell: ti/2
Accelerating cavities* length : about 70 cmo.ico/271 ( G H z )162636mode

TM 110
TM ni
TM 121

Because TM110 mode has the largest R/Qvalue and that of the other modes are one ordersmaller, only TMiio mode was considered here.
Equation (2) shows that small Q and/or
nearly integer ftb will suppress the effect of wakefield. The dependence of the beam behaviors on
the Q and f was studied.

一163-Considering cavity alignment error, cavities
were assumed to have vertical offset with respect
to the center axis of the linac. Assuming that theoffset is independent of each other anddistributed randomly in a gaussian distribution
with a standard deviation a y but limited by amaximum 3<Jy. Note that the vertical alignment
tolerance is to be much tight than the horizontal
one。
To measure beam qualities, consider totalluminosity per train (20 bunches). Assume thatthe error distributes as gaussian and the shape ofeach bunch in the phase space is an ellipse of
gaussian distribution which have emittance £q.
Assuming that both electron and positron beams
have the same qualities, then the total luminosity
will be proportional to the inverse of square rootof 'effective emittance'.

16

16.4

16.6

[H( c ) Q=20a y = 3 0 /jjh i

H I i ' i ■I i i i I < i i i ! ' i
16

L - Z (l/8eff)l/2

16.2

16.2

16.4

16.6

16

16.2

16.4

16.6

GHz⑶

where the summation should be taken for allbunches and ’effective emittance1of each bunchis defined as follows.

F ig .1 , Ln vs. frequency o f TM110, (a) Q=20,0 y = lO (im, (b) Q==30? CJy=10 |Lim? (c) Q=20,

ay~30 Jim, (d) Q=30, dy=30 [im⑷where ecm is 'emittance of center o f mass1
defined as
Ecm = jx2 + 2 a x x * + px*2⑶where x and xf are obtained by the tracking
simulation.
Let us define ’normalized luminosity',
L n = Z ( e o /£ e f f ) ^(6)

which will be 20 if £cm=0. In this report, eg isvertical emittance designed as follows.
(E/mc2)£o = 3 x l(h8 rad m ( v e r t i c a l ) .(7)Fig. l(a)-(d) show Ln vs. frequency of themode for Q=20 or 30 and a y=10 jiim or a y=30
|im. Tracking with twenty different sets ofrandom numbers (which were used fortransverse misalignment distribution) wereperformed for each plot. Lines are drown formean values and error bars show nm.s. of thetwenty results.

3, Single bunch
Single bunch motion was studied with shortrange longitudinal and transverse wake field. Abunch was divided into 41 macroparticles withuniform longitudinal spacing. Charge of eachmacro particle was calculated assuming gaussian
like distribution where length between the headmacro particle and the tail one was set to be 5sigma o f bunch length.
Parameters of the model linac are as follows.
Injection Energy
RF peak field
Total length of RF structure
RF frequency
Total bunch charge
Bunch length (az)

10 GeVlOOMeV/m
7.4 Km
11.42 GHz
2xl0 10 e75 jim

Short range wake functions at z (m) were setto be as follows.
Transverse : W j(z) = 2.3xl0 20 z

(V/C/m2)(8)

Longitudinal: WL(z) = 2.6xl0 15- 9.7xlOl0 Vz + L l x l O 1^ z

-1 6 4 -(V/C/m)(9)

To avoid beam breakup due to transverse
wake field, 'BNS damping1 was incorporated.
Energy spread can be controlled by changing thephase of R F field with respect to the bunch. A
parameter 9offcrest (Az)s is defined as theadvanced distance of the bunch center withrespect to the peak of RF. Fig. 2 shows energy
spread at the end of linac (r.m.s.) vs. Az. Notethat minimum energy spread will be obtainedwith positive Az («2 mm), because of thelongitudinal wake field.

Az( mm)Fig. 2, Energy spread at the end of linac(r.m.s.) vs. Az.
(a) Injection error
Fig. 3 shows normalized emittance at theend of the linac vs. Az with vertical injection
error of 1 jLim where the initial normalized
emittance was 3xl0*8 rad®m. It shows Az=1.4
m m is too large and 1.2 m m or less gives enoughenergy spread for B N S damping.
(日でo(b) Cavity misalignment
With vertical misalignment of the accelerating
structures, emittance at the end of the linac werecalculated (Offcrest was chosen as Az=L0 mm.
Because the misalignment is random and is muchlarger than beam oscillation amplitude, ’BNS
damping* will not be effective).10 different setsof random numbers for misalignment were usedfor CTy=10jim and SOjim. The results were as
follows (where the initial normalized emittancewas 3xlOg rad@m).
CJy=10}im
0y~3O}xm

3.94 ±0.58
11.3 ±4.9(xl0'8rad®m)(xl0°8rad«m)

It is shown that the structures should bealigned within an accuracy of 10 |im with respect
to the beam orbit
4, summary

Simulation of multibunch motion showedthat damped structure of Q~20 for the mostimportant dipole mode ( T M 1 10) is acceptable,
and that alignment tolerance of the acceleratingstructures is about 10 (im with respect to thebeam.

s ®丨6<Dnd(D.sdu

Energy spread should not be so large at theend of the linac, because energy acceptance ofthe final focus system will be -±0.6 %•Considering bunch to bunch energy difference,
energy acceptance of a single bunch will be about±0.45 %. As it is necessary to reduce theenergy spread only at the end9Az can be changed
along the linac. Because beams with higher
energy are more rigid, Az can be set larger at thelast part of the linac than at the lower energy part.

Simulation of single bunch motion showedthat ’B N S damping* is effective to suppress
beam break up due to injection error, and thatalignment tolerance of the accelerating structures
is about 10 jim with respect to the beam.
ReferencesAz(mm)名Fig. 3, Normalized emittance at the end of thelinac vs. Az[1]T.Higo et al 1991 IEEE Particle Accelerator
Conference. K E K preprint 91-32.
[2] M.Yamamoto et al. This workshop.
[3] T.Taniuchi et al. This workshop.