issues related to crossing angles

Frank Zimmermann

RF Deflector( Crab Cavity )

Head-onCollision

Crossing Angle (11 x 2 m rad.)

Electrons PositronsLERHER

1.41 MV

1.41 MV

1.44 MV

1.44 MV

Super-KEKB crab cavity scheme

2 crab cavities / beam / IP

voltage of crab cavities

12

2/tan

Re

cEV

rf

cbcrab

crfcrab

x

4max

tolerance on IP offset jitter translatesinto tolerance on left-right crab-cavity phase and crab-main-rf phase

R12 & R22(R11) from MAD

nominal LHC optics

|R12,34|~30-45 m

|R22,44|~1 (from crab cavity to IP)

voltage required for Super-LHC

crossing angle 0.3 mrad 1 mrad 8 mrad

800 MHz 2.1 MV 7.0 MV 56 MV

400 MHz 4.2 MV 13.9 MV 111 MV

200 MHz 8.4 MV 27.9 MV 223 MV

crab cavity voltage for different c’s & rf frequencies

tolerance on R22

|60

12

2

2

2

*12

22

*

2

12

22*

12

22

zc

x

x

zc

x

zzc

cc

RR

R

R

zR

Rz

z-dependent additionalcrossing angle

corresponding Piwinskiangle should be small

not a problem

[for c=1 mrad, x=12 mm, R12=30 m, z=7.55 cm]

KEKB crab cavity • Squashed cell operating in TM2-1-0 (x-y-z)

• Coaxial coupler is used as a beam pipe

• Designed for B-factories (1〜 2A)

Absorbing materialNotch filter

Absorbing material

Squashed Crab cavity for B-factories

Coaxial beam pipeCooling for inner conductor

(axial view)

inner conductor

"Squashed cell"

(K. Akai et al., Proc. B-factories, SLAC-400 p.181 (1992).) Courtesy K. Akai

~1.5 m

K. Ohmi, HHH-2004 ~1.5 MV@500 MHz

longitudinal space required for crab cavities scales roughly linearly with crab voltage; desired crab voltage depends on rf frequency); achievable peak field also depends on rf frequency; 2 MV ~ 1.5 m, 20 MV ~ 15 mfrequency must be compatible with bunch spacing; wavelength must be large compared with bunch length;

1.2 GHz probably too high; 400 MHz reasonable; 800 MHz perhaps ok

longitudinal space & crab frequency

...6

1

22

1)(' 3

2

2

12

z

cz

Rzx

rf

cc

MHz 7752

2

6

z

rf

c

noise

amplitude noise introduces small crossing

angle; e.g., 1% jitter translates into 1%c/2

crossing angle – not critical

phase noise causes beam-beam offset;

→ tight tolerance on left-right crab phase

and on crab-main-rf phase differences

*

2224

revHOrmsIP fxn

t

emittance growth

from turn-by-turnrandom offsets x

requiring less than 10%/hr emittance growth xrms<12 nm ~ 10-3*

<0.012o at c=1 mrad & 500 MHz <0.04o at c=0.3 mrad & 500 MHz

SuperLHC: x,y=0.25 m, nIP=2, HO=0.005,

=7500, =3.75 m

p emittance growth due to random offsets

diffusion rate from strong-strong simulation with BBSS

• x2=x0

2+Dt t: turn

• D~1.4x10-15 x[m]2

z= 0 0.005 0.01

K. Ohmi, HHH-2004

tolerance from Ohmi san’sstrong-strong simulation

• For x=1.6 m (=5 degree) and =100,

　 D~1.4x10-15 x[m]2, where x2=x0

2+Dt, t: turn.

• Tolerance is x=0.016 m, = 0.05 degree for =100, and x=0.0016 m, 0.005 degree for =1, for luminosity life time ~ 1 day

K. Ohmi, HHH-2004

for 300 rad crossing angleand 400 MHz

slightly worse than my“pessimistic estimate”!?

analytic theory of b-b diffusion (T. Sen et al., PRL77, 1051 (1996)

M.P.Zorzano et al., EPAC2000)

2 2 2

0

( ) sinh (2 1) ( )( )

8 4 / cosh cos 2 (2 1)k

xxk

C x k G aD J

k

1 1

1' ' ( 1)k k k k k

aG U U k U kU

a

0 00

1( ) (2 )( 1) ( )

ak w

k k k kU a e I w dww

• Diffusion rate due to offset noise. (round beam)

ln(1 1/ )

*

22p p x

p

N r JC a

K. Ohmi, HHH-2004

comparison with the simulation• D(a=1)=<J2>=1.5x10-25 m2/turn• D(sim)=(-0

2)2/2 =10-28 m2/turn Need to check

K. Ohmi, HHH-2004

analytical diffusion rate from Sen-Ellison-Zorzanomodel even much larger!!

} 3 orders ofmagnitudediscrepancy!

impedance of crab cavities

transverse impedance is an issuedue to large beta function

rise time due to 1 crab cavity =

rise time from ~10 normal rf cavities with the same voltage

dispersion correction

if large crossing angles are realized by placing single D1 dipoles first,

and the triplet between D1 and 2, the dispersion correction could be an issue

to be studied

minimum crossing angle from LR b-b

“Irwin scaling”coefficientfrom simulation

note: there is a threshold - a few LR encounters may have no effect! (2nd PRST-AB paper withYannis Papaphilippou)

minimum crossing angle with wirecompensator

*7

c

m75.3

103223

11*bparda

c

N

x

kd

need dynamic apertureof 5-6 and wirecompensation notefficient within 2 from the beam center