Chaos and Emittance growth due to nonlinear interactions in circular

accelerators

K. Ohmi (KEK)SAD2006

Sep. 5-7. 2006 at KEK

Emittance growth• External incoherent diffusion, radiation, I

ntrabeam etc.• Coherent motion, instability• Nonlinear diffusion• Nonlinearlity coupled to external diffusion

(noise)

1. Incoherent electron cloud2. Beam-beam limit3. Space charge limit

Diffusive or not• Which system does not have emittance gro

wth?1. Integrable system2. System with two degrees of freedom

• Which system can have emittance growth?1. Nonintegrable system with three or more degrees of free

dom2. External diffusion: noise, radiation excitation… The exter

nal diffusion is amplified due to nonlinear interaction

Nondiffusive system• Integrable system – needless to say• System with two degrees of freedom. Particles d

o not across torus layers.

System fall into global stochastic regime may be diffusive even for two degrees of freedom, but the diffusion is limited in the regime.

Diffusion in three or more degrees of freedom

• Motion in a degree of freedom gives modulation

• Particles can get over KAM boundary through additional freedom.

Round beam: Example of non-diffusive system

• Equal tune, no synchrotron oscillation -- equivalent to a system with 2 degrees of freedom (r-s).

2 2

2 21 ( ) ( , )2 2

x yP

p pH x y s U r z

2 22

2 ( ) ( , )2 2r P

p rp s U r zr

sincos' yxr pprp cossin'2

yx pprrp

(x,y)

ctsz

• H does not include , therefore p is a constant of motion.

• Trajectories on a torus r-pr-s• Poincare cross-section is mapped on two dimensional space.

• Motion in r-pr space. max

min

2 2

2 2

1 1 22 2 2

r

r r r

E r EJ p dr drr

2

1

2 2

(2 )cos(2 )

rr

r

r J

J

12

J p d p

Model- round beam interacting with electron cloud

• Transverse beam size depends on z; x(z), y(z).• Assume transverse Gaussian charge distribution. Applica

ble not only beam-beam but also electron cloud and space charge issues.

• Strong localized force x, y ~x, y2 2

2 2

2 20

exp2 2

( , ; ( ), ( ))2 2

x yex y

x y

x yu uNr

U x y z z duu u

( 0) exp : : ( 0)s U s x x

tyx zpypx ),,,,,( x

The force for a round charge distribution• The force depends on z, because of r(z) of strong bea

m or electron cloud.

duuu

rrzrU

r

rer

0 2

2

2

22

exp))(;(

dzdrr

dzdzrU

zzrUF r

rr

er

r

rrz

2

2

2

2

2

2 2exp

212))(;())(;(

2

2

2exp112))(;(

r

err

rr

rrzrUF

• Example, LHC• L=26700 m, x=0.28, y=0.31, s=0.006

• x=y=8x10-9 m, x= x=100 m

• x=y=0.89 mm, z=0.13 m,

Model of cloud• r (cloud) and tune shift• Ne L=1, 2, 4, 6 x1011. Interact at a point in the ring.

Two degree of freedom

• 4 variable (x, px, s, H’). 1 integral for H’, Poincare cross section (certain s) 2 variable.

• Poincare plot, x-px plot at a certain s.

• When one more integral, J(x,px)=constant, the system is solvable. This relation gives a curve in x-px phase space.

Poincare plot• r-pr =0.06, 0.13, 0.27, 0.38

Note r=0.89 mm

Diffusion rate• Diffusion rates are very small compare than those

for 3 degrees of freedom, see later.• T0=89 s.

Resonance overlap• Fourier expansion for r of U.

• Resonance position, Jr,R .

• Motion near the resonance position, pendulum motion, separatrix.

0

cosk rk

U U k

0( )r rr

UJJ

,( ) 2r RnJk

,r r Rr

J JI m

k

,

22 20

2

1 ( ) cos2

r R

r P kr J

UH k I s U

J

Overlap condition, Chirikov criterion

• Resonance width

• Resonance separation

• Overlap condition, width>separation

120

24r kr

UJ UJ

120

, , 2

2( , 1) ( , )r R r Rr

UJ k n J k n

k J

20

2 2kr

Uk UJ

Synchrotron oscillation and symplectic diffusion

• Add synchrotron motion, 3 degrees of freedom • The structures of tori are different for each z. • Are particles return the same torus after one synchrotron period.

s

z

???Are there different worlds for every z?

rpr

prr

22 22 2 2

2 ( ) ( , )2 2 2r z P

p rH p p z s U r zr

Poincare plot for several z

Note r=0.89 mm

Poincare plot for several z• Z=2, 4, 6, 10 cm

Adiabatic invariant?

Solvable or not• 6 variables (x,p,y,p,s,H’)• One integra H’, Poincare cross section at s.• 4 variables• When 2 integrals exist, the system is solvable. The soluti

on is represented by a surface in 3 dimensional space, r-pr-z.

• When a surface is not seen, the system is nonsolvable: i.e., emittance growth occurs.

Solvable

Nonsolvable

Blue: no synchrotron motion at z=0

Separatrix crossing

0 0

( , , , ) cosr z z kl r zk l

U J J U k l

120

, , 2( 1) ( ) zr R r R

r

UJ l J lk J

• U depends on z. Fourier expansion for synchrotron phase.

• Resonance separation (narrower than n-n+1)

20

2

4 1klz r

Uk UJ

Diffusion due to synchrotron motion

• ｚ =0.006 =0.012

Tune difference• 3 degrees of freedom

• p is not constant. Variation of p for small tune difference, x=0.285 & y=0.295.

2 22 2

( ) ( , )2 2

yxx y P

p yp xH s U r z

r

pr

p

2 2 22

2 cos 2 ( ) ( , )2 2r P

p r rp s U r zr

• KAM for various p for equal tune,.

• p =1.2x10-9, 4.8x10-9, 2x10-8.

Note r=0.89 mm

Motion in the phase space and tune space – example I

• Near integrable trajectory---nondiffusive

Motion in the phase space and tune space - example II

• Chaotic trajectory --- diffusive

Tune scan for 3 degrees of freedom

• Tune scan without synchrotron motion

0.05, 0.05

0.45, 0.45

• Pumping machanism• Resonance• Separatrix crossing (~0)

More

• 4 degrees of freedom – actual weak-strong model.

• Colliding beam – 4x2xN+- degrees of freedom• 3 degree of freedom+synchrotron motion. Modul

ation diffusion, stochastic pumping with separatrix crossing.

22 2 22 2 2

2 cos 2 ( ) ( , )2 2 2r z P

p r rH p p z s U r zr

4 degrees of freedom• Tune scan with synchrotron motion, ｚ =0.006.• Vertical emittance growth.• Resonance, m y=n is seen.• Emittance growth is large at cross points of resonances

0.05, 0.05

0.45, 0.45

Importance of Lattice • Nonlinearity of beam-cloud interaction • Integrated the nonlinear terms with multiplying

function and cos (sin) of phase difference23 3 34 45 5 11 12 2 4: : : : : : : : : : : :: : : : : : : :e e e e e e e e e ...e nF U F F U FU F U UM

111 : :: :

1

e exp : ( ) : exp( : :)i

nFF

i Ti

U e M U

x

/ 2 / 21cos( )m m m

i ikx k J m

Nonlinear term should be evaluated with considering the beta function and phase of position where electron cloud exists.

Unphysical cancel of nonlinear term may be caused by simple increase of interaction point.

F: lattice transformation

Beam-beam limit• 4 degree of freedom• Interaction during collision. If z~y, =1 rad,

4th-order term 4.2 2

2 2

2 20

exp2 2

( , ; ( ), ( ))2 2

x yex y

x y

x yu ur

U x y z z duu u

( ) ( , , , , )x x y y z z P T x yH J J J s U x p y p z

111 : :: :

1

e exp : ( ) : exp( : :)i

nFF

i Ti

U e M U

x

Integrability near half integer tune Reduction of the degree of freedom.

• For x~0.5, x-motion is integrable. (work with E. Perevedentsev)

if zero-crossing angle and no error.

• Dynamic beta, and emittance

• Choice of optimum x

,0.5lim 0x

C yD

2 2,00.5

limx

xx

2

0.5limx

xp

1(crossing angle)+(coupling)+(fast noise)x

L

X-px plot near half integer in x

• x=0.503, 0.510, 0.520, 0.540

Crossing angle

• Calculate UT using Taylor map (Diff. algebra)

• Taylor coefficient ~ Fourier coefficient, Uklm.

0

cos( )T klm x y zklm

U U k l m

4-th order Coefficients due to crossing angle• Short bunch z=3mm, x/z=1 at 2x15 mrad, original super KEKB.

4-th order Coefficients as a function of crab sextupole strength, short bunch z=3mm, x/z=1

• H=K x py2/2, theoretical o

ptimum, K=1/xangle.• Clear structure- 220,121• Flat for sextupole streng

th- 400, 301, 040

Crab crossing and crab waist

• Crossing angle induces synchro-beta and odd coupling resonance terms.

• Merit of Crab crossing is the absence of the terms.

• Crab waist reduces the odd coupling resonance term, but keeps the synchro-beta term.

• In the both method, Luminosity performance is improved.

Space charge limit

• Similar as electron cloud, integrate the nonlinear interaction along the ring.

• UT: Gaussian?

111 : :: :

1

e exp : ( ) : exp( : :)i

nFF

i Ti

U e M U

x

( ) ( , , , , )x x y y z z P T x yH J J J s U x p y p z

Summary

Keyword of SAD

Dynamic aperture

Emittance