[zipi.tex.cesriv]bfrf.tex CBN 98-08

4/16/1998

STABILITY STUDY OF OFF CENTER BEAM IN THE

SUPERCONDUCTING CAVITY FOR CESR IV

Z. Greenwald, D. L. Rubin

Laboratory of Nuclear Studies, Cornell University, Ithaca, NY 14853

April 16, 1998

Introduction

The electrons and positrons in CESR IV design are guided through separate pipes and share thesuperconducting accelerating cavities. Their orbits inside the cavity are displaced by �4 cm fromthe axis of the cavity. Since this displacement is larger by 2 cm than the present CESR III pretzeledorbit, and the beam current will be about 5 times larger, it is necessary to study the higher ordermode characteristics and the beam stability at this displacement with the higher current.

1 R/Q Calculations of the Higher Order Modes

Calculations of R=Q of the superconducting cavity [1] were done using the code URMEL [2]. Inmodeling the cavity only half the cell was simulated. In order to calculate all possible higher ordermodes, the program was run twice for each �eld mode m = 0; m = 1 using di�erent boundaryconditions at the symmetry line. The �rst one is when the tangential electric �eld is set to zero andthe second is when the tangential magnetic �eld is set to zero. Figures 1-6 show the calculated R=Qof the higher order modes for m = 0; 1 as function of the transverse displacement r. Note, the R=Qof the longitudinal m = 0, Figures 1,2, and the transverse m = 1, Figure 4,5, higher order modeseither stay constant or decrease as the beam is further displaced from the center. On the otherhand, the longitudinal R=Q of all modes with m = 1 increase with the transverse displacement upto r = 6 cm. See Figures 3,4.

2 Stability Calculations

If kb is the number of bunches in the storage ring, there will be kb coupled-bunch modes characterizedby the longitudinal mode number s = 0; 1; 2:::(kb� 1). In addition an individual bunch in the sth

coupled-bunch mode requires an index A = 1; 2; :::etc to describe its motion in the synchrotron phasespace, e:g: the dipole mode A = 1, where the bunches move rigidly as they execute longitudinalsynchrotron oscillation. The quadrupole mode A = 2 where the bunch head and tail oscillatelongitudinally out of phase, etc.

For the transverse case, Two mode numbers s and A are again needed to described transversecoupled-bunch mode. One di�erence in the transverse case is that the index A can assume the valueA = 0, meaning that the bunch moves rigidly as it executes the transverse oscillations; This is called

1

-10

0

10

20

30

40

50

0 0.02 0.04 0.06 0.08 0.1 0.12

SC Cavity Modes m=0 (IZL=1)

499.70958.32977.771010.31064.61142.71238.91347.81463.51561.71600.01711.61829.61942.42018.72065.22165.32205.42261.02305.5

R/Q

(O

hm

s)

Calc

ula

ted b

y U

RM

EL

Transverse Displacement (m)

Figure 1: Longitudinal m = 0 R=Q as function of transverse displacement for the modes withtangential ELECTRIC �eld zero at the leftmost boundary

2

-5

0

5

10

15

20

0 0.02 0.04 0.06 0.08 0.1 0.12

SC Cavity Modes m=0 (IZL=2)����

958.28978.771015.01069.21143.71233.41330.21427.91532.51649.51773.01883.31965.82079.32189.42210.22225.22262.12335.42402.8

R/Q

(O

hm

s) C

lacu

late

d b

y U

RM

EL

Trnsverse displacement (m)

Figure 2: Longitudinal m = 0 R=Q as function of transverse displacement for the modes withtangential MAGNETIC �eld zero at the leftmost boundary

3

-5

0

5

10

15

20

25

30

0 0.02 0.04 0.06 0.08 0.1 0.12

SC Cavity Modes m=1 (IZL=1)

676.31752.45803.74876.28967.991076.51197.41287.41328.51461.61524.61543.91581.11597.31634.31699.21726.81808.21860.71886.1

R/Q

(O

hms)

Cal

cula

ted

by U

RM

EL

Transverse Diplacement (m)

Figure 3: Longitudinal m = 1 R=Q as function of transverse displacement for the modes withtangential ELECTRIC �eld zero at the leftmost boundary

4

-1

0

1

2

3

4

5

6

0 0.02 0.04 0.06 0.08 0.1 0.12

SC Cavity Modes m=1 (IZL=2)

641.16755.38816.66902.701002.81104.41189.71243.31288.71406.51524.31538.41540.71570.01609.31661.71701.11701.11794.61833.4

R/Q

(O

hms)

Cal

cula

ted

by U

RM

EL

Transverse Displacement (m)

Figure 4: Longitudinal m = 1 R=Q as function of transverse displacement for the modes withtangential MAGNETIC �eld zero at the leftmost boundary

5

-2

0

2

4

6

8

10

12

14

0 0.02 0.04 0.06 0.08 0.1 0.12

SC Cavity Modes m=1 (IZL=1)

676.31752.45803.74876.28967.991076.51197.41287.41328.51461.61524.61543.91581.11597.31634.31699.21726.81808.21860.71886.1

(R/Q

)/(k

*r)2

O

hm

s k=

w/c

Transverse Diplacement (m)

Figure 5: Transverse m = 1R=Q as function of transverse displacement for the modes with tangentialELECTRIC �eld zero at the leftmost boundary

6

-0.5

0

0.5

1

1.5

2

2.5

3

0 0.02 0.04 0.06 0.08 0.1 0.12

SC Cavity Modes m=1 (IZL=2)

641.16755.38816.66902.701002.81104.41189.71243.31288.71406.51524.31538.41540.71570.01609.31661.71701.11701.11794.61833.4

(R/Q

)/(k

*r)

2 O

hm

s k

=w/c

Transverse Displacement (m)

Figure 6: Transverse m = 1R=Q as function of transverse displacement for the modes with tangentialMAGNETIC �eld zero at the leftmost boundary

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CESR IV Parameters units

Ring Circumference 756:106 mRF Frequency 499:930 MHzNo. of Cavities 10No. of Bunches 180Current per Bunch 17 maBeam Pipe Radius 12 cmBeam Energy 5289: MeVHorizontal Chromaticity �15:4Horizontal Emittance 1:96 � 10�7

Vertical Emittance 2:38 � 10�9

Momentum compaction 0:01086Horizontal Damping Time 29 msecEnergy Damping Time 14 msecHarmonic No. 1260Synchrotron Tune 0:1125Bunch Length 0:7 cm�H 9:56 m�V 13:942 mHorizontal Tune 12:587Vertical Tune 8:632

Table 1: Summary of CESR IV parameters used in ZAP

the transverse rigid dipole mode. The A = 1 implies the bunch head and tail oscillate transverselyout of phase.

The growth time of the unstable coupled-bunch modes were calculated using the code ZAP [3]with the CESR IV parameters shown in Table 1. The simulation includes 10 cavities. For each

cavity the values of the 20 highest�RQ

���r=4 cm

��Q were used. While the R=Q values were calculated

by URMEL at r = 4 cm transverse displacement, the quality factor values Q for each higher ordermode were taken from measurements on the cavity including the ferrite. See Figure 7 [4].

Figures 8,9 show the growth time of the LONGITUDINAL instability for m = 0 and m = 1�eld modes for various numbers of bunches. Figure 10 shows the TRANSVERSE instability m = 1growth time versus the number of bunches. In each �gure the growth time was calculated for thefollowing two cases:

a: The bunch current is held constant at (17:0 ma=bunch).b: Ib = 17:0 ma�180 bunches

#buncheswhich maintains the designed total current of 3:06 amp=beam at

180 bunches per species.The growth time values shown in Figures 5,6,7 correspond to the fastest growing coupled-bunch

mode s. Because the sth coupled-bunch mode which has the fastest growth time, is di�erent forthe various numbers of bunches, there are some uctuations in the calculated growth time versus#bunches (=kb). In general, in case a, the growth time decreases with increasing number of bunches.And, in case b where the total current is held constant, the growth time decreased for #bunch < 180where the bunch current was increased above 17:0 ma and increased for #bunch > 180 whereIb < 17:0 ma.

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Figure 7: Measured quality factor for the superconducting cavity

9

0

0.05

0.1

0.15

0.2

0.25

0 200 400 600 800 1000 1200 1400

Longitudinal (m=0, A=1) Growth Time (sec) for CESR IV r=4 cm

Long m=0 Ib=const.Long m=0 It=const.

Gro

wth

Tim

e (

sec)

#bunches

Figure 8: Growth Time of Longitudinal m = 0 modes with a. Ib = 17:14 ma=bunch b. Ib =17:0 ma�180 bunches

#buches(It = 3:06 amp = cons:) for various number of bunches

10

0

5

10

15

20

0 200 400 600 800 1000 1200 1400

Longitudinal (m=1,A=1) Growth Time (sec) for CESR IV r=4 cm

Long m=1 Ib=const.Long m=1 It=cost

Gro

wth

Tim

e (s

ec)

#bunches

Figure 9: Growth Time of Longitudinal m = 1 modes with a. Ib = 17:14 ma=bunch b. Ib =17:0 ma�180 bunches

#buches(It = 3:06 amp = cons:) for various number of bunches

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0

0.05

0.1

0.15

0.2

0 200 400 600 800 1000 1200 1400

Transverse (m=1, A=0) Growth Time (sec) for CESR IV r=4cm

tran m=1 Ib=const.tran m=1 It=const.

Gro

wth

Tim

e (

sec)

#bunches

Figure 10: Growth Time of Transverse m = 1 IZL = 1 modes with a. Ib = 17:14 ma=bunch b.Ib =

17:0 ma�180 bunches#buches

(It = 3:06 amp = cons:) for various number of bunches

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3 Conclusion

For CESR IV parameters (see Table 1) the growth time of the unstable modes in both planes,longitudinal and transverse, is larger than the radiation damping time. The instability growth timefor LONGITUDINAL motion m = 0 and m = 1 are 70 msec and 8 sec respectively, compared to theradiation damping time of 14 msec. The TRANSVERSE mode instability growth time is 98 msecwhile the radiation damping time is 29 msec. Thus both modes will be damped. Note, even thoughthe LONGITUDINAL R=Q of the m = 1 modes increase with displacement of the bunch from thecenter (Figures 3,4), their values are still about a quarter of the R=Q of the m = 0 modes, makingthe longitudinal instabilty growth time of m = 1 very slow.

References

[1] H. Padamsee, P. Barenes, C. Chen, W. Hartung, J. Kirchgessner, D.Mo�at, R.Ringrose, D.Rubin, Y. Saed, D. Saraniti, J. Sears, Q. S. Shu, and M. Tigner, "Design Challenges for HighCurrent Storage Rings". PAC-92.

[2] C. Palm, U. van Rienen, T. Weiland, "URMEL User Guide", DESY M-85-11, 1985.

[3] M. S. Zisman, S. Chttopadhyay and J. J. Bisognano, "ZAP User Manual", LBL 21270 ESG-15.

[4] V.Veshcherevich, K axai, P. Barenes, W. Fox, J. Kirchgessner, D. Metzger, D. Mofafat, H.Muller, H. Padamsee, J. Sears, and M. Tigner, "Higher Order Modes Damping in CESR-BCavity", SLAC April 6-10, 1992.

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