STUD% OF RESISTI,E-WALL BEA2 BREA3UP
N. Nakamura
ISSP, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan
Abstract
Beam breakup (BBU) caused by the long-range
transverse resistive-wall wake in ERLs is studied mainlywith a developed simulation program. The simulation
program is useful and essential for understanding theresistive-wall BBU behaviours especially in the parameter
ranges and initial conditions where the asymptotic
expressions analytically derived are invalid. The
simulation results show that the position displacement ororbit distortion due to the wake increase in proportion to
the square root of the time and the wake strength in theearly stage of the resistive-wall BBU and that, in case of
the test ERL in Japan, the maximum position
displacements reach 3% and 1% of the initial position
offset with all the quadrupole magnets off and on at 77 !s.
Effects of a small-gap insertion device(ID) duct and ID
focusing are also investigated.

I5TRODU6TIO5
In storage rings such as synchrotron light sources,
transverse coupled-bunch instabilities due to long-range
resistive-wall wake as well as other instabilities due to RF
cavities and ion trapping are sometimes observed against
radiation damping[1]. It is expected that transverse multibunch resistive-wall beam breakup can be a serious
problem in high-current ERLs. While the BBU due to
HOMs of superconducting cavities has been muchinvestigated, the resistive-wall beam breakup (RWBBU)
has been hardly studied and only the asymptotic
expressions were analytically derived in simple andlimited conditions[2]. In this paper, more detailed studyof the RWBBU with a simulation program is presented.

RESISTI,E-WALL WA3E
The long-range transverse resistive-wall wake-function
for a round pipe with a radius b, length L and electric
conductivity !c is expressed by
W! ! "

Z$
cL,#b ! z"/ # #$ c

(1)

where c, Z0 and z are the velocity of light, the vacuum
impedance and the distance from the wake source such asa preceding bunch. Eq. (1) is valid under the following
condition:3!2!! $ !! ! 2! " # 0! " #0(2)

An electron is kicked by such wake and the transverse
kick by a preceding bunch with the position y and the
distance z is given by
+0
" 2#
" 2#
&'
(3)
!" ! = #%$ % ! =%%!,$$ &( 3 )1* 2 &' &where E is the beam energy and N is the electron numberin the bunch. The wake-field is easily cumulated in themulti-bunch beam, because it is slowly decayed following
Eq. (1). It also depends on pipe characteristics such as
pipe radius, electric conductivity and length. On the other
hand, the long-range longitudinal wake-function is given
by
#$
'#
(4)
! "" ! !
(
!" "
"%&
"# #
In this case, cumulation of the resistive-wall wake-field isrelatively smaller than in the transverse case. Thus only
the transverse beam breakup is treated here.

AS%2PTOTI6 E7PRESSIO5
Here the asymptotic expressions[2] for the RWBBU isreviewed. The asymptotic expressions were analyticallyderived from the following equation:
M "$
a
(5)
y M !! !s" + k y# y M !s" = #y N !s" ( M $ #)
N =$
M"N
!!

" !#
c
" "
$ #b !c$ ' "#/ !

#/ !

Z$
"* !c$ ' "
# '"
#% c
$
#b "

Z$
#% c

where "B and ky are bunch time separation and external
focusing strength, and IB is the average beam current
defined by eN/"B. Eq. (5) shows equation of motion for
the M-th bunch (M # 2) for the case where an electron
beam consisting of a series of identical point-like bunches
passes through a round pipe with uniform external
focusing. For two extreme cases, no focusing(NF) caseand strong focusing(SF) case, the asymptotic expressions
are given by
1 ,$
1
)$
"$ $ %$5 .$! 00 "$$ %& %$10+$
(6)
! " ! #" #$$ '$ ( '()+$$$ '$ .$$2! #$ $ &$#$ %& &$ .$+$*$-$t NF !

"" B $!! '! #&! )! $ !
! # %!" (! a s

(7)

and
1 +$
1
1
($
($
+$
"$ $ %$3 -$ *$ "$ $ %$3! 00 "$$ %& %$6! -$*$! " ! #" =$$ '$ '()*$$$ '$ -$c+#*$ 3$$ '$ . , ! # . -$6
2! #$ $ &$#$$ %& &$ -$ *$ #$$ %& &$
*$
-$
)$
,$ )$
,$

(8)
! "# !

2y &$

1"k " $!2 '! 1&! )!
# %!$ (! a 2 s2

(9)respectively. The initial condition is that all the bunches
have the same position offset yM(0)=…y2(0)=y1(0)$y00 at
the pipe entrance and the time t from the injection of theinitial bunch is approximately equal to M"B for infinite
bunch number (M%&). The tNF and tSF are the growthtime for the two cases.

These expressions require some conditions to be valid.
These conditions are as follows:(i) NF casea "s>> 1 , t >> , t >> t NFck y 2!12 / 5

(10)

2

!2

<< 1 ,

1/ 3

a "

s
>> 1 , t >> , t >> t SF
c

(11)

where
!1 =

a "s
a "s
, !2 =.
4M
4 Mky

cos(kys)

0.5
0
-0.5

-1.5

ky ! 2

Simulation(ND=10)-1(ii) SF case2/ 3

Simulation(ND=1)

1

0

5

(12)

Thus the asymptotic expressions are valid only for thelimited parameter ranges satisfying (10) or (11). Initialconditions are also limited. Therefore a simulation
program is needed to study RWBBU more minutely andgenerally.

10

15

20

s[m]

25

30

Figure 1: Simulation results (ND=1,10) and analytic
solution cos(kys) for no wake case with external focusing
ky=3 m-1.
2.5
M=2(simulation)
M=3(simulation)
M=2(analytic)
M=3(analytic)2

yM(s)/y00!14 / 5 << 1 ,

1.5

1.5

1

RWBBU SIMULATION
0.5

A simulation program has been developed to study the
RWBBU behaviors and to evaluate effects of the RWBBU
on ERLs. Features of this program are as follows:•

Long-range transverse resistive-wall wake
Lowest mode (dipole mode)
Rigid and point-like bunches
1-D tracking
Single pass
Combination of round pipes with/without external
focusing or defocusing
Basic time increment is the bunch seperation !B
divided by an integer ND (division number)

In order to check the simulation program, simulations
are compared with analytic solutions for some special
cases. The first is no wake case with uniform external
focusing. In this case, the transverse position along thelongitudinal location s is just sine-like for all the bunches.
Figure 1 shows an example with ky=3 m-1. The position
displacement is normalized by the initial offset y00. The
simulation result for ND=10 is consistent with the exact
solution. It should be noted that the division number must
be carefully selected for strong focusing, because an
insufficient division number leads to a wrong result asshown for ND=1 in the figure. The second is non-zerowake case without external focusing for a small bunch
number. In this case, Eq. (5) can be analytically solved
and the exact solutions are given by1(13)y 2 ( s) / y 00 = 1+ as2
2
and1!
1 $ 2 1 2 4(14)y 3 ( s) / y 00 = 1+ #1+
as& as +
2"
24
2%
for the second and third bunches (M=2 and 3),
respectively. As shown in Fig. 2, the simulation results
are in good agreement with the analytic solutions.

0

0.6

s[m]

1.2

1.8

Figure 2: Simulation results and analytic solutions for thesecond and third bunches (M=2 and 3) in non-zero wakecase (a=0.5 m-1) without external focusing.

Next simulations are compared with the asymptotic
expressions. Figures 3 and 4 show examples of NF andSF cases, respectively. Parameter values were properlychosen for each case so that they should almost satisfy thecondition (10) or (11). The blue broken lines indicate theasymptotic expressions and the red solid lines simulation
results. The simulation results are also in agreement withthe asymptotic solutions. From these comparisons, it isconcluded that the simulation program works well.
30
Simulation(ND=10)
25

Asymtotic expression

20

yM(L)/y00

•
•
•
•
•
•

0

15

10

5

0

0

5000

10000

15000

20000

M

Figure 3: Simulation result (ND=10) and asymptotic
expression for no focusing case (a=2.62x10-5 m-1, L=50
m).

4
3

Simulation(ND=10)
Asymptotic expressionyM(L)/y00

2
1
0
-1
-2
-3
-4

0

5000

10000

15000

20000

M

Figure 4: Simulation result (ND=10) and asymptotic
expression for strong focusing case (a=2.62x10-3 m-1, ky=3
m-1, L=10 m).

SIMULATION IN THE TEST ERL
The simulation program was applied to the test ERL in
Japan, which is being designed[3]. Figure 5 shows layoutof the test ERL. Simulation conditions are as follows:•
Beam energy E=60 MeV, average beam current
IB=100 mA for uniform bunch filling, and bunch
separation !B=0.769 ns (RF frequency fRF=1.3
GHz).
•
All the vacuum ducts are assumed to be Al pipes
with b=25 mm and "c=3.5x107 #-1 m-1.
•
The simulation start and end points are the exit andentrance of the superconducting cavity section asshown in Fig. 5 and the total path length L is 56.44m.
•
All the bunches have the same initial position
offset y00 at the simulation start point.
•
1-D tracking in the vertical direction for t $ 77 µs(M $ 100000).
•
Focusing and defocusing of 37 quadrupole(Q)magnets in the beam path are considered.
•
Sextupole magnet fields, magnet and ductalignment errors, orbit correction by correctors are
not considered.
•
When the insertion device(ID) is considered, its
vacuum duct is assumed to be a stainless-steel(SS)pipe with a radius of 10 mm, a conductivity of
1.4x106 #-1 m-1 and a length of 10 m.
First a situation with all the Q-magnets off and without
the ID is considered for simulation. In this situation thetest ERL just equals the Al pipe with a constant radius of
25 mm and a total length of 56.44 m and without external
focusing and defocusing. Figure 6 shows position
displacement due to resistive-wall wake at the simulation
end point (s=L) normalized by the initial position offset
y00 as function of time t or bunch number M. The
normalized position displacement does not depend on thevalue of y00. The position displacement obtained from thesimulation is proportional to t1/2 and increase up to 3% of
the initial position offset at 77 µs. The asymptotic
expression of NF case indicated by the blue broken line in

Fig. 6 monotonously decreases with the time and it is notin agreement with the simulation result. For the parameter
range, the asymptotic expression is invalid and thesimulation is essential for obtaining correct results.
Figures 7 (a) and (b) show orbit distortions yM(s)-y1(s)due to the RW wake along the beam pipe with all the Qmagnets off and on, together with initial orbits y1(s).
These initial orbits and orbit distortions are also
normalized by the initial position offset y00 in the figures.
Since every bunch except the first bunch are continuously
kicked by the RW wake in the same direction, the orbit
distortion monotonously increases with the longitudinal
distance s in Fig. 7(a). On the other hand, in Fig. 7(b), theorbit distortion is oscillatory along the beam pipe because
the initial orbit is also oscillatory around the pipe centerby the Q-magnet focusing. As a result, the maximum orbit
distortion is reduced down to 1% of the initial position
offset from 3%. Orbit correction is expected to beeffective for suppressing the growth of the resistive-wall
BBU, though pipe alignment error may be still left evenafter perfect COD correction. Figures 8 (a) and (b) show
the conductivity and current dependences of the RWBBU
with all the Q-magnet on. In Fig. 8(a), position
displacements at the exit for two cases of Al and SS pipes
and their ratio are shown as a function of time. The ratio
is almost constant and just equals the ratio of the electric
conductivities of SS and Al. In Fig. 8(b), position
displacements for the average currents of 100mA and
10mA and their ratio are shown as function of time. The
ratio is also constant and equals the ratio of the average
currents. These results suggest that position displacement
due to the RW wake is simply proportional to the wakestrength a within the simulated bunch number or time.
INSERTION DEVICEQM
CHICANEBMSXBMSUPERCONDUCTING CAVITYMERGER

Simulation end pointeEXTRACTOR

Simulation start point

Figure 5: Layout of the test ERL in JapanBM

Initial Orbit
y1(s)/y00

1.5

0.03

Initial Orbit

1

0.025

0.5

0.02

M=100000
M=10000

0

0.015

-0.5

0.01

-1

0.005

-1.5
-2

0
0

8

16

24

32

40

s[m]

48

56

-0.005

Orbit Distortion due to Wake{yM(s)-y1(s)}/y00

0.035

"b$
Initial Orbit

0.015

M=100000
M=10000

1

0.01
0.005

0.5

0

0
-0.5

-0.005

-1

-0.01

-1.5

-0.015

-2

0

8

16

24

32

40

s[m]

48

56

-0.02

&'()*e!,-!I/'0'a1!2*b'03!a/4!2*b'0!4'302*0'2/3!4)e!02!05e!67
8a9e! :2*! !! ;! <====! a/4! <=====! 8'05! a11! 05e! >?@a(/e0"a$!2::!a/4!"b$!2/! '/! 05e!0e30!E6BC!D5e!b)/c5!/)@be*3!!
;! <====! a/4! <=====! c2**e3F2/4! 02! 05e! 0'@e3! "#;! ,C,! a/4,,!!3C"a$

100000
8
7
6

-0.02

5

Ratio(SS/Al)
-0.03

4

-0.04

-3

10

20

30

40

2
50

Time t[µs]60Position Displacement at Exit due to Wake{yM(L)-y1(L)}/y00

"b$
0.002

20000

40000

60000

10000013
12

Ratio(100mA/10mA)

11-0.00410-0.0069

IB=100mA
IB=10mA-0.008

16

24

32

s[m]

8
7

10

20

30

56

40

50

-0.06

0.04
0.02
0

0
-1

-3

0

8

16

24

32

s[m]

-0.02

M=100000
M=10000

ID Section

40

48

-0.04
56

-0.06

&'()*e!Q-!I/'0'a1!2*b'03!a/4!2*b'0!4'302*0'2/3!4)e!02!05e!67
8a9e!:2*!!!;!<====!a/4!<=====!""#;!,C,!a/4!,,!!3$!8'05
IR!:2c)3'/(!30*e/(05!"a$!'(!;!=!@?<!a/4!"b$!'(!;!<!@?<!'/!a3@a11?(aF!IR!F'Fe!a0!IR!3ec0'2/C

DISCUSSIONS
I/!05e!F*eJ'2)3!3ec0'2/T!05e!3'@)1a0'2/!*e3)103!3)((e30e4
05a0! 05e! F23'0'2/! 4'3F1ace@e/0! 4)e! 02! 05e! 67BBX! '3
F*2F2*0'2/a1! 02! "</\! a/4! *C! Ve*e! 05e! 67BBX! be5aJ'2*! '3
c5ec9e4! 8'05! a/! a/a1S0'c! @e0524C! D5e! ba3'c! e])a0'2/3! 2:
05e! 67BBX! :2*! 05e! '/'0'a1! "!;<$! a/4! !?05! "!! #! \$b)/c5e3!a*e!('Je/!bS!1!! (s) ! # ! (s)!1 (s) " 0

"<O$

%!s"
y & !s" ( M $ 2) "<L$
M"&
E]C! "<L$! '3! '4e/0'ca1! 8'05! E]C! "O$! eIceF0! 05a0! )/':2*@:2c)3'/(! 30*e/(05! '(3! a/4! 8a9e! 30*e/(05! *! '/! E]C! "O$! a*e
eI0e/4e4! 02! /2/?)/':2*@! :2c)3'/(! 2*! 4e:2c)3'/(! 30*e/(05
4("3$! a/4! 8a9e! 30*e/(05! *"2$C! &*2@! 05e3e! 082! e])a0'2/3T
05e!:21128'/(!e])a0'2/!'3!2b0a'/e4M #1
a!s" y N !s""<,$! M "" !s" ! K y !s"! M !s" " $N "1
M # N y ##y M !! !s" + $ y !s"y M !s" = #

& =1

0

48

Initial Orbit

M "1

-0.01
-0.01240)*+*,-"-+#.-/-01-0$-2
80000

0-0.002

8

-0.04

"b$

1

70

Bunch Number M
00

ID Section-2

3

Al(σc=3.5x107)
SS(σc=1.4x106)0

Initial Orbit

1

80000

-0.01

-0.06

-0.02

-1
-2

Initial Orbit
y1(s)/y00

60000

Ratio(SS/Al)

Position Displacement at Exit due to Wake{yM(L)-y1(L)}/y00

40000

0

-0.05

0

Orbit Distortion due to Wake{yM(s)-y1(s)}/y00

20000

0.02

0

2

Bunch Number M
0

M=100000
M=10000

1

"a$
0.01

0.04

2

Orbit Distortion due to Wake{yM(s)-y1(s)}/y00

Initial Orbit
y1(s)/y00

1.5

Orbit Distortion due to Wake{yM(s)-y1(s)}/y00

0.02

2

&'()*e3! Q! "a$! a/4! "b$! 3528! e::ec03! 2:! a! 3@a11?(aF! IR4)c0!a/4!05e!IR!:2c)3'/(!*e3Fec0'Je1SC!P3!3528/!'/!&'(C!Q"a$T! 3'@)1a0e4! 2*b'0! 4'302*0'2/! 4)e! 02! 05e! 67! 8a9e3'(/':'ca/01S! '/c*ea3e! '/! 428/30*ea@! 2:! 05e! IR! 3ec0'2/beca)3e! 2:! 30*2/(! *e3'30'Je?8a11! 8a9e! 4)e! 02! 05e! 3@a11?
(aF! IR! 4)c0C! U/! 05e! 205e*! 5a/4T! '/! &'(C! Q"b$T! 05e! 2*b'0
4'302*0'2/! '3! 8e11! *e4)ce4! beca)3e! 05e! '/'0'a1! 2*b'0! '3
23c'11a0e4!a*2)/4!05e!F'Fe! ce/0e*!bS!05e!IR!:2c)3'/(!"'(;<@?<$! '/! 05e! IR! 3ec0'2/C! V28eJe*! 05e! IR! :2c)3'/(! 30*e/(05'3! c5a/(eab1e!'/! 05e! )3e*! 2Fe*a0'2/! a/4! ca//20! a18aS3! bee::ec0'Je! :2*! 3)FF*e33'2/! 2:! 05e! 67BBXC! P! c2FFe*?
c2a0e4! MM! 4)c0! ca/! be! e::ec0'Je! :2*! *e4)c'/(! 05e! 8a9e
30*e/(05!'/!05e!IR!3ec0'2/YKZC

Initial Orbit
y1(s)/y00

!"a$
2

60

70

6

Time t[µs]&'()*e! G-!H23'0'2/! 4'3F1ace@e/03! a0! eI'0!4)e! 02!67! 8a9e
"a$!:2*!e1ec0*'c!c2/4)c0'J'0'e3!2:!F'Fe3! !$!;!<CKI<=L!"?<!@?<"MM-! 30a'/1e33! 30ee1$! a/4! NCOI<=,! "?<! @?<! "P1-! a1)@'/)@$
a/4!"b$! :2*! 05e!aJe*a(e! c)**e/03!%&!;! <==! @P!a/4!<=!@P
8'05! a11! 05e! >?@a(/e0! 2/! a3! a! :)/c0'2/! 2:! 0'@eC! D5e'**a0'23!a*e!a132!3528/C

85e*e

! M (s) "y M (s) - y1 (s) .
y 00

(18)

The symbol !M is the position displacement of the M-thbunch from the initial bunch position normalized by the
initial position offset y00 and it is caused by the cumulated
resistive-wall wake. In the early stage of the RWBBU (!M<< 1), this equation is approximately rewritten by
M $1
a(s) y1 (s)! M "" (s) # %(19)
M $ N y 00
N =1
y1 (s)# 2 M a(s)( M >> 1)y 00
For yM’(0)= y1’(0), yM(0)= y1(0), and a(s)=a, the following
relation is derived from Eq. (19):! M (s) " at1/ 2
(20)
1/ 2
1/ 2
" # c b$3 IB E $1% B t1/ 2This relation is consistent with the simulation results in
Figs. 5 and 8. In addition, dependences of the RWBBU onother parameters such as the beam energy, the pipe radius
and the bunch separation are forseen.

Validity of Wakefunction
The validity of the simulation results depends on that ofEq. (1). From the condition of (2), one can estimate thedistance z where the wakefunction of Eq. (1) is valid.
When all the vacuum ducts of the test ERL are Al pipes
with a radius b = 25 mm and an electric conductivity "c =3.5x107 #-1 m-1, the valid distance range is given by(21)
3.62 !10"5 << z[m] << 8.24 !10 6 .
The above distance range is easily converted to the valid
time range as(22)
1.21!10"13 << t[sec] << 2.75 !10"2 .
The upper time limit for the validity of Eq. (1) is 27.5 ms,
which corresponds to M=3.575x108 and 3575 times larger
than the maximum simulation time, 77 µs. Similarly thevalid time ranges for the stainless steel(SS) pipe case ("c= 1.4x106 #-1 m-1) are given by(23)
3.53 !10"13 << t[sec] << 1.10 !10"3 (b = 25 mm)
and1.92 !10"13 << t[sec] << 1.76 !10"4(b = 10 mm) .

(24)

The upper time limits for the SS pipe case are 1.1 ms and176 µs for b=25 mm and b=10 mm. They are still larger
than the maximum simulation time and Eq. (1) is valid forboth cases. Therefore the parameter values used for thesimulations almost satisfy the condition of (2).
Next effects of the pipe thickness are considered. Eq.
(1) is derived under the assumption that the pipe thickness
is infinite. However such assumption is not always
appropriate. Here the upper time limit is roughly
estimated as the inverse of the frequency of which theskin depth is equal to the pipe thickness d. The upper
time limit tul is given by(25)
t ul ! "µ0# c d 2 ,where µ0 is the permeability of vacuum. For the Al pipecase with d = 5 mm, the upper time limit is 3.45 ms andlarger than the maximum simulation time. On the other
hand, for the SS pipe case with d = 2 mm, the upper time
limit is 22 µs and smaller than the maximum simulation
time of 77 µs. In this case, the wakefunction of Eq. (1)may be somewhat modified. If the simulation time ismuch increased even for the Al pipe case, thewakefunction should be modified. The effects of the pipe
thickness as well as the pipe conductivity and radius on
the wakefunction should be further investigated to know
the RWBBU behaviors at larger times.

SUMMARY AND CONCLUSION
A simulation program of the RWBBU has beendeveloped because the asymptotic expressions are valid
only in the limited parameter ranges and initial conditions.
The simulation program is useful for studying thetransverse RWBBU more minutely and generally. The
simulation program was applied to the test ERL in Japan.
In the simulations, the maximum position displacementswith all the Q-magnets off and on are about 3% and 1%of the initial position offset at 77 µs. The Q-magnet
focusing contributes to slowing the RWBBU growth andorbit correction is also expected to be effective. A smallgap ID duct significantly increases the orbit distortion,and the ID focusing well suppresses the RWBBU growth,though it is changeable. The RWBBU should be further
studied with simulations and the simulation program
should also be developed to introduce a new definition of
the wakefunction for wider time range, pipe alignment
errors, orbit correction due to correctors, non-linear field
elements such as sextupole magnets, copper-coating
effects on the ID duct wake and so on. Furthermore
simulation results should be compared experimentally andquantitatively with operational data of the planned test
ERL and/or the existing ERLs to confirm the validity ofthe simulations. Finally schemes of overcoming theRWBBU must be surveyed and studied, if necessary.

REFERENCES[1] See, for example, J. M. Filhol et al., “Overview of the
Status of the Soleil Project”, EPAC’06, Edinburgh,July 2004, p. 249; J. L. Revol and R. Nagaoka,“Observation, modelling and cure of transverse
instabilities at the ESRF”, PAC’01, Chicago, June
2001, p. 1930.
[2] J. M. Wang and J. Wu, “Cumulative beam breakup
due to resistive-wall wake”, PRST-AB 7034402(2004).
[3] K. Harada et al., “Lattice and Optics Designs of the
Test ERL in Japan”, these proceedings.
[4] N. Nakamura et al., “Reduction in Resistive-Wall
Impedance of Insertion-Device Vacuum Chamber byCopper Coating”, EPAC’98, Stockholm, August
1998, p984; H. Sakai et al., “Development of Copper
Coated Chamber for Third Generation Light
Sources”, PAC’05, Knoxville, May 2005, p2633