Analytical Treatment of Some Nonlinear Beam Dynamics Problems in Storage Rings

J. GaoLaboratoire de L’Accélérateur Linéaire

CNRS-IN2P3, FRANCE

Journées Accélérateurs

Porquerolles, Oct. 5-7, 2003.

Contents

Dynamic Apertures of Multipoles in a Storage Ring

Dynamic Apertures limited by WigglersLimitations on Luminosities in Lepton

Circular Colliders from Beam-Beam Effects

Nonlinear Space Charge Effect Nonlinear electron cloud effect

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

kLsLxx

BBmx

sKpH )*(!1

2)(

2 1

12

2

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 Mapping

Near the nonlinear resonance, simplify the difference equations to the form of STANDARD MAPPING

sin0KII

I

Stochastic motions

When 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

General Formulae for the Dynamic

Apertures of Multipoles 2

1

1

)2(21

2,||))2((

1)(2

m

m

m

mx

xmdynaLbmsm

sA

kkdecadyna

i jjoctdynaisextdyna

totaldyna

AAA

A...111

1

2,,

2,,

2,,

,

2,,

2

1

1,,

)()( xA

ssA

xsextdynay

xysextdyna

Single octupole limited dynamic aperture simulated by using BETA

x-y plane x-xp phase plane

Comparisions between analytical and numerical results

Sextupole Octupole

Wiggler

Ideal wiggler magnetic fields

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

y

xx

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

)sin()sinh()cosh(0 ksykxkBkk

B yxy

z

2222 2w

yx kkk

One cell wiggler

One cell wiggler Hamiltonian

One cell wiggler limited dynamic aperture

iw

yw iLsykyHH )(

1241 4

2

22

201,

2/1

2

2

,13

)()(

)(

wy

w

wy

yy ks

ssA

Full wiggler and multi-wigglersDynamic 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

NLs

sk

AsA)(

)(31

)(1

,2

1 2

2

1 2,

2

,

wy

w

my

y

ywN LkssA

2,,

)(3)(

my,

Full wiggler and multi-wigglers

Many wigglers (M)

Dynamic aperture in horizontal plane

M

j ywjy

ytotal

sAsA

sA

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 ny034.0)(, mA ay17584.0)( mlw

067.0)(, mA ny067.0)(, 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

Maximum Beam-Beam Tune Shift in Circular Colliders

Luminosity of a circular collider

ee

IPye

yce

yx

ce

rfNfNL

242

)(2,

yxy

IPyeey

rN

where

Beam-beam interactionsKicks from beam-beam interaction at IP

),,( ,'' yxee yxfrNxiy

22

2),,,(yx

yxyxf

)(222exp

)(2 222

2

2

2

22 yx

y

x

y

y

yxyx

yixw

yxiyxw

Beam-beam effects on a beam

We study three cases

2

2

'4

exp12 r

rrNr ee

xx

xx

ee duuxrNx

2

0

22

2

' exp4

exp2

yx

xx

ee yerfxrNy

22exp2

2

2

'

(RB)

(FB)

(FB)

Round colliding beam

Hamiltonian

22)(

22

2eeyy rNyskpH

kkLsyyy )(......

11521

641

41 6

64

42

2

Flat colliding beams

Hamiltonians

kxxxkLsxxx )(......

1801

1211 6

64

42

2

kyxyxyxkLsyyy )(......

1201

1211 6

54

32

22)(

22

2eexx

xrNxskpH

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,,

,s

sAs

sAy

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

The Limitation from Space Charge Forces to TESLA Dog-Borne Damping Ring

Total space charge tune shift

Differential space charge tune shift

Beam-beam tune shift

zyxy

yaveesc

LNr

22

,

2)(2

zyxy

yeesc

ssssNrs

22000

00

21

))()()((2)()('

))()()((2)()(

IPyIPxIPy

IPyeeIPbb

ssssNrs

Space charge effect

Relation between differential space charge and beam-beam forces

Gsfds

sdfIPbb

sc )()(

zG

2221

Space charge effect limited dynamic apertures

GrNssssA

ee

yx

y

yysc

s

)()(23)()())'(( 0

30

20

2 ,

L

s ysc

ysctotal

sA

sA

002,

,,

)')((1

1)(

scy

ysctotaly

sAR 23)(

2

,,2

Dynamic aperture limited by differential space charge effect

Dynamic aperture limited by the total space charge effect

Space charge limited lifetime

Space charge effect limited lifetime expressions

Particle survival ratio

2122 , exp2

yyy

ysc RR

yscysc

yysc

,

1

,,

23exp

23

2

)(exp1)(

, scysc

stscR

TESLA Dog-Borne damping ring as an example

Particle survival ratio vs linear space charge tune shift when the particles are ejected from the damping ring.

TESLA parameters

Nonlinear electron cloud effect

Relation between differential electron cloud and beam-beam forces

)21(

))()()((2)(

0,0,0,

0,'

LSSS

SNr

yxy

yeeec

)(21)(

IPbbec sF

Ldssdf

Nonlinear electron cloud effect

Normalized dynamic aperture due to electron cloud

Lyavxav

eec

N

,,,,2

Lecyavey

yecyec

r

AR

,,

,2

2,

23)(

Combined nonlinear beam-beam and electron cloud effectNormalized dynamic aperture due to

combined beam-beam and electron cloud effects

ybbIPy

IPybbybb

AR

,,,,

,,2

2,,

23)(

RR

R

yecybb

ytotal

2,,

2,,

2,, 11

1

Combined nonlinear beam-beam and electron cloud effectBeam lifetime due to the combined

effect

where is the damping time of positron in the vertical plane

)exp(2

2,,

2,,

1,,, RR ytotalytotal

yytotal

y,

PEP-II positron ring as an example

Machine parameter

msy 30, 6067

kmL 2.2

myav 18,,

msy 30,

PEP-II positron ring as an example

Machine parameter

6067

kmL 2.2myav 18,,

msy 30,

PEP-II positron ring as an example

If the beam-beam alone limited maximum beam-beam tune shift is

with

the maximum beam-beam tune shift will be reduced to

045.0,max,, ibby

015.0,max,, ibby

101225.3 ec

Conclusion

Various nonlinear effects are the main limiting factors to the performance of storage rings.

In addition to numerical simulations, analytical treatments are very helpful in understanding the physics behind the phenomena, are very economic.