Beam dynamics in crab collision

K. Ohmi (KEK)

IR2005, 3-4, Oct. 2005

FNAL

Thanks to K. Akai, K. Hosoyama, K. Oide, T. Sen, F. Zimmermann

Contents

Introduction of crab cavity Effect on the Beam-beam performance.

Crossing angle and symplectic diffusion

Luminosity degradation due to noise

Introduction

Half crossing angle 0.15 mrad. Other possibilities are 0.225, 0.5 and 4 m

rad. E=7 TeV. Bunch population 1.15x1011

Bunch spacing 25 ns, RF=400.8 MHz. Number of bunch 2808 I = 0.584 A L=26,016m

Crabbing voltage

Deflecting RF voltage, : half crossing angle

*=0.5m =4000 m, fRF=400 MHz

V=2.8 MV is required for =0.15 mrad. V=75 MV for 4 mrad

*

tan

RF x x

cEV

18.7 [mrad] MVV

KEKB type crab cavity TM110 500 MHz TM010 324 MHz V=1.44 MV Need 2x2 cavities for = 0.15 mrad. Need more cavities 0.225, 0.5 and 4 mrad.

How is multi-cell cavity? Coupled bunch instability issue.

Impedance of KEKB crab cavity

Z()L=13 k.GHz/cav.

Z()T=0.025 M/m/cav.

KEKB type single cell

TESLA type multi-cell

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).)

Coupled bunch instability caused by the parasitic modes

Longitudinal

f ZL,peak (KEKB) =13 [k GHz/cav] , =1.5 sec /cav@injection　　　　　　　　　　　　　　　　　 : Growth time (sec)

Transverse

Zt,peak (KEKB) =0.025 [M/m/cav], =1.5 sec /cav (KEKB) @injection,

Zt,peak (TESLA) > 1 [M/m/cav],

0 0 020

( ) ( )2

em s s

s p

MNri pM m Z pM m

T

020

( )2

em

p

MNri Z pM m

T

Effect of the crab cavity on beam-beam performance (Symplectic diffusion)

Optics error at the collision point determines the beam-beam performance in lepton colliders with high beam-beam parameter.

Crossing angle is a kind of optics error, =x/z, (=x/pz).

Symplectic diffusion is enhanced by the optics error, with the result that the luminosity degrades 　 in lepton colliders.

Is optics error at the collision point important for hadron colliders? If important, crab cavity may improve the beam-beam performance.

Crab cavity always compensate the geometrical reduction.

Vertical dispersion (KEKB) Diffusion behavior due to

dispersion in a system without synchrotron radiation.

Luminosity and beam size are degraded.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

y (mm)

L/L0

0

1

2

3

4

5

6

0 0.2 0.4 0.6 0.8 1 1.2

y (mm)

s y/s

y0e-e+

Gaussian approx.

PIC simulation

X-y coupling (KEKB)

Diffusion due to x-y coupling.

Luminosity and beam size degradation.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15 0.2 0.25

r4

L/L0

0

1

2

3

4

0 0.05 0.1 0.15 0.2 0.25

r4

s y/s

y0e-e+

Gaussian approx.

PIC simulation

Crossing angle (KEKB)

Crossing angle is equivalent to x-z coupling. Diffusion and luminosity degradation due to crossing

angle

Gaussian approx. PIC simulation

Is the Symplectic diffusion important for LHC?

Not seen in the short time tracking. How about long turn tracking? It is difficult to disting

uish with diffusion due to artifact in computer.

L sx

The beam size with crab is larger, but is pretense, <xx>c=<xx>+2<zz>. Note that the luminosity is higher.

Effect on beam-beam performance of the crab cavity - Fluctuation in collision due to the crab cavity and cavity noise -

Noise of RF system. Deviation of RF phase, .

Phase error between two crab cavities.

tanRF

RF

cx

cos( ( *, ))tan tan

2sinx c

c cRF x RF

s sc cx

Fluctuation in collision due to the crab cavity noise

Random fluctuation of beam offset at the collision point.

Example to sketch rough behaviors x=1.6 m for =5 degree (z=1 cm) and =0.15

mrad. Note sx=17 m. Correlation of the fluctuation. <x(n) x(n+m)>=e-m/, where n, m are turn. z=1, 0.5, 0.2, 0.1 cm at =1, 100 were examined.

A Strong-strong simulation was executed including the fluctuation.

Diffusion due to RF phase error, z

L sx

x is raised by dispersion x=z induced by the crab cavity.

Diffusion rate given by the simulation

sx2=sx0

2+Dt t: turn D~1.4x10-3 x2 [m2]

z= 0 0.005 0.01

No crab cavity、 RF phase error

Diffusion without crab cavity was weak. Noise of transverse offset is origin of the diffusion.

L sx

Diffusion due to phase error of crab cavity

x=1.7 m and dz=1 cm (x =1.7 m) Similar diffusion rate

L sx

Coherent motion is induced by the noise.

Analytic theory of beam-beam diffusion (T. Sen et al., PRL77, 1051 (1996), M.P.Zorzano et al., EPAC2000)

( ) ( , ) x x y y P x yH J J U J J

( ) ( ) '( ) U x x U x U x x

0

'( )( ) (0) ( )

N

x x

l

U lJ N J x l

Diffusion rate due to offset noise. (round beam)

2 2

2 20

exp(2 ) 2 s s

eNr du x y

Uu u

2

0 0

( ) (0) '( ) '( )( ) ( ) ( )

N N

J xx xl m

J N J U l U mD J x l x m

N

*** D~x2

Diffusion rate due to offset noise. (round beam)

2 2 2 2 2

0

( ) (0) ( ) sinh (2 1) ( )( ,0)

8 4 / cosh cos 2 (2 1)

x x kJ x

xk

J N J C x k G aD a

N k

1 1

1' ' ( 1)k k k k k

aG U U k U kU

as s

0 00

1( ) (2 )( 1) ( )

ak w

k k k kU a e I w dww

ln(1 1/ )

**

2 20

2 2

s s

p p yxx y

p

N r JJC a a

Comparison with the simulation

DJ(a=1)=<J2>=1.5x10-25 m2/turn for x=1.7 m and t=100. DJ(sim)=2JJ=2 D / =2x3.5x10-15x5x10-10/0.5 =7x10-24 m2/turn. (missed at HHH04). This value is somewhat larger than analytical estimation. Coherent motion and chaotic

(resonance) behavior seem to make enhance the diffusion.

0.5 1 1.5 2a

510-26

110-25

1.5 10-25

210-25

2.5 10-25

310-25

D

Tolerance

For x=1.7 m (=5 degree) and =100,

　 D~1.4x10-3 x2 [m2], where sx2=sx0

2+Dt, t: turn. Tolerance is x=0.017 m(s/1000), = 0.05 degree

for =100, and x=0.0017 m (s/10000), 0.005 degree for =1, if luminosity life time ~ 1 day is required.

We extrapolate the diffusion rate using x2 scaling. Simulation for noise s/1000 requires >106 macro-particle.

Luminosity degradation due to noise in KEKB-Feedback noise and beam-beam effect-

In 2005 spring operation, luminosity boosted up 1.35x1034 to 1.58x1034 cm-2s-1.

It is due to that the gain of the transverse bunch-by-bunch feedback system was optimized (weakened but kept a sufficient strength to suppress the coupled bunch instability).

0dB

4.5dB

0dB-1.5dB -3dB

3dB

1.5dB

Specific luminosity

Specific luminosity and feedback gain (Funakoshi)

External diffusion: Vertical offset noise　(simulation)

Since the beam-beam system is chaotic, such noise enhances the diffusion of the system.

Luminosity degradation for the noise without correlation between turns.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.02 0.04 0.06 0.08 0.1y/ sy

L/L0

Tcor=1

0

1

2

3

4

5

6

0 0.02 0.04 0.06 0.08 0.1

y/ sy

s y/s

y0e-e+

2( ) ( ') ( ')y t y t y t t

Orbit offset (static) (simulation)

Static vertical offset. Tolerance is easier than the fast noise. For slower variation than radiation damping time, emittance

can be an adiabatic invariant.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1y/ sy

L/L0

Offset

0

0.5

1

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1

y/ sy

s y/s

y0

e-e+

2( ) ( ') exp( ) 2y t y t y t

1/20 compare than that for fast noise

Estimation of feedback noise(Hiramatsu, K.O. & Tobiyama)

Twp-tap filter and vector composition with two position monitors

Phase space position at kicker, vector composition with two position monitor

Offset noise due to kicker error (E) and monitor error(P(X1,X2)))

2 1

2 121

sin sin1

cos cossin

K K

K K

N1

2K

XXN

XP

2 2 2 21 2 1 22

1 2

3 (3 3 2 cos )

6 cos 2 cos 2

E K K K K PX

K K

Kicker noise measurement (LER)

(7/14/05) Kicker output depending on feedback gain.

Feedback Gain set Out Vp-p OutVrms kick(Vrms) E(x10-7)

-13.4dB(3.77 oper.) 107mV 16.3mV 100V 0.62

-14.3dB(final value) 87.3mV 13.3mV 83V 0.51

-10.4dB(Bad lum.) 196.9mV 30mV 188V 1.16

FB switch OFF 4.2mV 0.6mV 4.1V 0.025

E=1/2 k/E0 E0=3.5 GeV

Speculated beam noise for the kicker noise

0

0.005

0.01

0.015

0.02

0.025

0.03

-15 -14 -13 -12 -11 -10

LER

K=0.1 at -10.4dB

K=0.05 at -10.4dB

(<dX

2 >/

x)1/2

dB

K = K0 10dB/20

Effect on the beam-beam performance of the phase jitter of cavity and crab RF’s in KEKB

0

0.5

1

1.5

2

2.5

0 0.02 0.04 0.06 0.08 0.1 0.12

x/ sx

s x/s

x 0e-e+e-e+

Luminosity and beam size as functions of x. Correlation time of the jitter, 1 or 10 turns, is important for the

degradation. Since Q=200,000 and H=5120, the correlation time will be lar

ger than 10 turns. Tolerance is 0.05 degree.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.02 0.04 0.06 0.08 0.1 0.12x/ sx

L/L0

Tcor=10Tcor=1

Summary Crab cavity is expected to reduce the sympletic diffusion in KEKB. The symplectic diffusion seems to be weak for hadron machines

with low beam-beam parameter. Since there is no damping mechanism, it is difficult to conclude whether the crab cavity improve luminosity more than the geometrical effect.

800 MHz crab cavity may be possible if geometrical loss is small. Tolerance for collision offset noise induced by RF phase modulati

on is severe. The correlation time, t=100, may be optimistic. Luminosity degradation due to the noise (mainly due to feedback

noise) has been observed in many machines, KEKB, DAFNE, HERA, RHIC.