Analytical Estimation of Dynamic Aperture Limited by Wigglers

in a Storage Ring

高傑 J. Gao弘毅

Laboratoire de L’Accélérateur LinéaireCNRS-IN2P3, FRANCE

KEK, Feb. 24 2004

ContentsDynamic Apertures of Limited by

Multipoles in a Storage RingDynamic Apertures Limited by Wigglers in a Storage RingDiscussionsPerspectiveConclusionsReferencesAcknowledgement

Dynamic Aperturs of Multipoles

Hamiltonian of a single multipole

Where L is the circumference of the storage ring, and s* is the place where the multipole locates (m=3 corresponds to a sextupole, for example).

k

mm

zm

kLsLxxB

BmxsKpH )*(!1

2)(

2 1

12

2

Eq. 1

Important Steps to Treat the Perturbed Hamiltonian

Using action-angle variablesHamiltonian differential equations should be

replaced by difference equations

Since under some conditions the Hamiltonian don’t have even numerical solutions

pH

dtdq

qH

dtdp

Standard MappingNear the nonlinear resonance, simplify

the difference equations to the form of STANDARD MAPPING

sin0KII

I

Some explanationsDefinition of TWIST MAP

)1)(mod(xg

)(Kfxx

)()1( ff

xdx

xdg ,0)(

where

Some explanationsClassification of various orbits in a Twist Map, Standard Map is a special case of a Twist Map.

Stochastic motions

For Standard Mapping, when global stochastic motion starts. Statistical descriptions of the nonlinear chaotic motions of particles are subjects of research nowadays. As a preliminary method, one can resort to Fokker-Planck equation .

97164.00K

m=4 Octupole as an example

Step 1) Let m=4 in , and use canonical variables obtained from the unperturbed problem.

Step 2) Integrate the Hamiltonian differential equation over a natural periodicity of L, the circumference of the ring

Eq. 1

m=4 Octupole as an example

Step 3) 111 4sin AJJ

111 JB

ABK 40

LbsJ

A mx 342

1

2)(

LbsB mx3

42 )(2

m=4 Octupole as an example

Step 4) )97164.0(140 ABK

LbsJ

mx ||)(21

3421

2/1

34

2

4

2/12/1

1,, ||)(2

)()(

))(2(

Lbs

ss

sJA mxmx

xxxoctdyna

One gets finally

General Formulae for the Dynamic Apertures of Multipoles

21

1

)2(21

2, ||))2((1)(2

m

m

m

mx

xmdyna LbmsmsA

i j kkdecadynajoctdynaisextdyna

totaldyna

AAA

A...111

1

2,,

2,,

2,,

,

2,,2

1

1,, )(

)( xAssA xsextdynay

xysextdyna

Eq. 2

Eq. 3

Super-ACOLattice Working point

Single octupole limited dynamic aperture simulated by using BETA

x-y plane x-xp phase plane

Comparisions between analytical and numerical results

Sextupole Octupole

2D dynamic apertures of a sextupole

Simulation result Analytical result

WigglerIdeal wiggler magnetic fields

)cos()sinh()sinh(0 ksykxkBkkB yx

y

xx

)cos()cosh()cosh(0 ksykxkBB yxy

)sin()sinh()cosh(0 ksykxkBkkB yx

yz

2222 2

wyx kkk

Hamiltonian describing particle’s motion

)))sin(())sin(((21 222 ksApksAppH yyxxzw

))cosh())cosh(1 ykxkk

A yxw

x

where

y

xyx

wy k

kykxkk

A ))sinh())sinh(1

Particle’s transverse motion after averaging over one wiggler period

)32(

22232

22

2

2

2

xykxkxk

kds

xdx

w

x

)32(

2 2

22232

22

2

2

2

y

xy

w

y

kkk

yxykyk

kds

yd

In the following we consider plane wiggler with Kx=0

One cell wigglerOne cell wiggler Hamiltonian

After comparing with one gets

one cell wiggler limited

dynamic aperture

i

wy

w iLsyk

yHH )(124

1 42

22

201,

2/1

2

2

,13

)()()(

wy

w

wy

yy ks

ssA

Eq. 4

Eq. 4 Eq. 1

2

23

3 w

wykLb

Using one getsEq. 2

A full wiggler Using one finds dynamic aperture for a

full wiggler

or approximately

where is the beta function in the middle of the wiggler

w

wwiy

wN

i yw

ywN

i yiywN

NLss

kAsA )()(31

)(1

,2

1 2

2

1 2,

2,

wy

w

my

y

ywN LkssA

2,,

)(3)(

my,

Eq. 3

Multi-wigglers

Many wigglers (M)

Dynamic aperture in horizontal plane

M

j ywjy

ytotal

sAsAsA

1 2,,

2

,

)(1

)(1

1)(

2,,2

,

,,, yAA ywigldynamx

myxwigldyna

Numerical example: Super-ACO

Super-ACO lattice with wiggler switched off

Super-ACO (one wiggler)7.2)( mw 017.0)(, mA ny 019.0)(, mA ay

13)(, mmy 17584.0)( mlw 5168.3)( mLw

Super-ACO (one wiggler)3)( mw 023.0)(, mA ny 024.0)(, mA ay

7.10)(, mmy 17584.0)( mlw 5168.3)( mLw

Super-ACO (one wiggler)4)( mw 033.0)(, mA ny 034.0)(, mA ay

5.9)(, mmy 17584.0)( mlw 5168.3)( mLw

Super-ACO (one wiggler)

4)( mw

016.0)(, mA ny 017.0)(, mA ay

5.9)(, mmy

08792.0)( mlw

5168.3)( mLw

033.0)(, mA ny 034.0)(, mA ay17584.0)( mlw

067.0)(, mA ny 067.0)(, mA ay35168.0)( mlw

Super-ACO (two wigglers)6)( mw 032.0)(, mA ny 03.0)(, mA ay

75.13)(, mmy 17584.0)( mlw 5168.3)( mLw

Discussions The method used here is verygeneral and the analytical resultshave found many applications insolving problems such as beam-beameffects, bunch lengthening, haloformation in proton linacs, etc…

Maximum Beam-Beam Parameter in e+e- Circular Colliders

Luminosity of a circular collider

IPye

yce

yx

ce

rfNfNL

242

)(2,

yxy

IPyeey

rN

where

Beam-beam interactionsKicks from beam-beam interaction at IP

),,( ,'' yxee yxfrNxiy

222),,,(

yxyxyxf

)(222exp)(2 222

2

2

2

22 yx

y

x

y

y

yxyx

yixwyxiyxw

Beam-beam effects on a beam

We study three cases

2

2' 4exp12

rr

rNr ee

xx

xx

ee duuxrNx

2

0

22

2' exp4exp2

yx

xx

ee yerfxrNy

22exp2

2

2'

(RB)

(FB)

(FB)

Round colliding beam

Hamiltonian

22)(

22

2eeyy rNyskpH

kkLsyyy )(......1152

164

14

1 66

44

22

Flat colliding beams

Hamiltonians

kxxx

kLsxxx )(......1801

1211 6

64

42

2

kyxyxyx

kLsyyy )(......1201

1211 6

54

32

22)(

22

2eexx

x rNxskpH

22

)(2

22

eeyyy rNyskpH

Dynamic apertures limited by beam-beam interactions

Three cases

Beam-beam effect limited lifetime

)(

16)(

)( 2

2

2,8,

IPyee

ydyna

srNssA

)(

6)(

)( 2

2

2,8,

IPxee

x

x

xdyna

srNssA

)(

23)(

)(2

2,8,

IPyee

yx

y

ydyna

srNssA

(RB)

(FB)

(FB)

)()(exp)(

)(2 2

2,,

1

2

2,,

, ssA

ssA

y

ybbdyna

y

ybbdynayybb

Recall of Beam-beam tune shift definitions

)(2,

yxx

IPxeex

rN

)(2,

yxy

IPyeey

rN

Beam-beam effects limited beam lifetimes

Round beam

Flat beam H plane

Flat beam V plane

xx

xxbb 3exp32

1

,

yy

yybb 4exp42

1

,

yy

yybb

23exp

23

2

1

,

Important finding

Defining normalized beam-beam effect limited beam lifetime as

An important fact has been discovered that the beam-beam effect limited normalized beam lifetime depends on only one parameter: linear beam-beam tune shift.

y

bbbbn ,

Theoretical predictions for beam-beam tune shifts

msy 30

FByFByRBy ,max,max,max 89.1324

0843.0)1(,max hourbbRBy

0447.0)1(,max hourbbFBy

For example

Relation between round and flat colliding beams

First limit of beam-beam tune shift (lepton machine)

or, for an isomagnetic machine

whereHo=2845

*These expressions are derived from emittance blow up mechanism

NrH

IP

ey R 60,max

NT

IPy

Hy

0

20

,max

Second limit of beam-beam tune shift (lepton machine)

Flat beam V plane

0447.02

3exp

0447.02

32

,

1

,,

y

Maxy

y

Maxyyybb

Some Examples

DAFNE: E=0.51GeV,ymax,theory=0.043,ymax,exp=0.02 BEPC: E=1.89GeV,ymax,theory=0.039,ymax,exp=0.029 PEP-II Low energy ring:

E=3.12GeV,ymax,theory=0.063,ymax,exp=0.06 KEK-B Low energy ring:

E=3.5GeV,ymax,theory=0.0832,ymax,exp=0.069 LEP-II: E=91.5GeV,ymax,theory=0.071,ymax,exp=0.07

Some Examples (continued)

PEP-II High energy ring: E=8.99GeV,ymax,theory=0.048,ymax,exp=0.048

KEK-B High energy ring: E=8GeV,ymax,theory=0.0533,ymax,exp=0.05

Beam-beam effects with crossing angle

Horizontal motion Hamiltonian

Dynamic aperture limited by synchro-betatron coupling

22)(

22

2eexx

x rNxskpH

kxxx

kLszxzxzx )(......)(1801)(12

1)(1 66

44

22

zee

x

IPx

xxbetasyn rNs

ssA4

2/1

3,2

)(3)(2)(

Crossing angle effect

Dynamic aperture limited by synchro-betatron coupling

Total beam-beam limited dynamic aperture

2

22

2,

,1

32

)()(

xx

xbetasynxbetasyn s

sAR

x

zWhere

is Piwinski angle

xbetasynFByxbetasynFBy

yxtotalbb

RRRR ,,8,

1

,,8,

, 111exp

111

4

KEK-B with crossing angle KEK-B luminosity reduction vs Piwinski

angle

Perspective

It is interesting and important to study the tail distribution analytically using the discrete time statistical dynamics, technically to say, using Perron-Frobenius operator.

Conclusions1) Analytical formulae for the dynamic apertures limited by multipoles in general in a storage ring are derived.2) Analytical formulae for the dynamic apertures

limited by wigglers in a storage ring are derived.

3) Both sets of formulae are checked with numerical simulation results.4) These analytical formulae are useful both for

experimentalists and theorists in any sense.

References1) R.Z. Sagdeev, D.A. Usikov, and G.M. Zaslavsky,

“Nonlinear Physics, from the pendulum to turbulence and chaos”, Harwood Academic Publishers, 1988.

2) R. Balescu, “Statistical dynamics, matter our of equilibrium”, Imperial College Press, 1997.

3) J. Gao, “Analytical estimation on the dynamic apertures of circular accelerators”, NIM-A451 (2000), p. 545.

4) J. Gao, “Analytical estimation of dynamic apertures limited by the wigglers in storage rings, NIM-A516 (2004), p. 243.

Acknowledgement

Thanks go to Dr. Junji Urakawa forinviting the speaker to work on ATFat KEK, and to have this opportunityto make scientific exchange with youall, i.e. 以文会友 .