Analytical Estimation of Dynamic Aperture Limited by Wigglers

in a Storage Ring

J. GAO 高杰

Institute of High Energy Physics

Chinese Academy of Sciences

ContentsDynamic Apertures of Limited by

Multipoles in a Storage RingDynamic Apertures Limited by Wigglers

in a Storage RingApplication to TESLA damping ringConclusionsReferences

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

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

Amx 34

21

2

)(

LbsB mx

34

2 )(2

m=4 Octupole as an example

Step 4) )97164.0(140 ABK

LbsJ

mx ||)(2

1

3421

2/1

34

2

4

2/1

2/11,, ||

)(2)(

)())(2(

Lbs

s

ssJA mx

mx

x

xxoctdyna

One gets finally

General Formulae for the Dynamic

Apertures of Multipoles

2

1

1

)2(2

1

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,,

)()( xA

ssA

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

Wiggler

Ideal wiggler magnetic fields

)cos()sinh()sinh(0 ksykxkBk

kB yx

y

xx

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

)sin()sinh()cosh(0 ksykxkBkk

B yxy

z

2

222 2

wyx kkk

Hamiltonian describing particle’s motion

)))sin(())sin(((2

1 222 ksApksAppH yyxxzw

))cosh())cosh(1

ykxkk

A yxw

x

where

y

xyx

wy k

kykxk

kA ))sinh())sinh(

1

Particle’s transverse motion after averaging over one wiggler period

)3

2(

22232

22

2

2

2

xykxkxk

k

ds

xdx

w

x

)3

2(

2 2

22232

22

2

2

2

y

xy

w

y

k

kkyxyky

k

k

ds

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

2

3

3 w

wykL

b

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

,

22

,2 2 21 1,

1 1( )

( ) 3 ( )

w w

w y

N Ny w

y i wi ii y w y wN

ks

A s A s N

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

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 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

Application to TESLA Damping Ring

E=5GeV Bo=1.68T 0.4w m

12wN ,1 9y m

,2 15y m (at the entrance of the wiggler)

(at the exit of the wiggler)

The total number of wigglers in the damping ringis 45.

, 2.1total yA cmThe vertical dynamic aperturedue to 45 wiggler is

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.