Accelerator Physics Topic III Perturbations and Nonlinear Dynamics

J. J. Bisognano

Topic Three: Perturbations & Nonlinear Dynamics

1

UW Spring 2008

Accelerator Physics

Accelerator PhysicsTopic III

Perturbations and Nonlinear Dynamics

Joseph BisognanoSynchrotron Radiation Center

University of Wisconsin

J. J. Bisognano


2

UW Spring 2008

Accelerator Physics

Chromaticity

•

pppxyKxKH

etcdsdxp and pp-(p with and p,by Dividing

xpp2pp

2pp

yKxKpH

is equationsorder first gives that nHamiltonia The

yx

x0

2y

2x

2ˆ

2ˆ]

22)1)[(1(

.,/ˆ/)

)(

]22

)1[(

2222

2

0

0

22

20

From form, it’s clear tune will depend on momentum

J. J. Bisognano


3

UW Spring 2008

Accelerator Physics

Sextupoles

• A sextupole field can remove much of this

))(()1()1(

)33(6

)()1()1(

2ˆ

2ˆ)3(

6)()1(

)1(]22

)1)[(1(

)3(6

)(

22

23

22

2

230

xysSyKy

yxsSxKx

pppxxyxsS

yKxKHH

xyxsSpc

eA

y

x

yx

s

s

Tune change

J. J. Bisognano


4

UW Spring 2008

Accelerator Physics

Natural Chromaticity

][

)()(

)()(

minmax41

,1,1,41

,

1141

ffC

lattice FODO a For

CsKs

dssks :Recall

yxyxyxyx

J. J. Bisognano


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UW Spring 2008

Accelerator Physics

Dispersion to the Rescue

dependence weak withfunction, periodic a is Dρ

)(DSD)(D)K--(D

satisfiesD dispersion Recall

x

12

11 2

We can move to orbit at energy offset by canonical transform

))(ˆ))((()ˆ,(2 sDpsDxpxFLet

J. J. Bisognano


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UW Spring 2008

Accelerator Physics

Chromaticity Correction

)ˆˆ(2

)ˆˆ3ˆ(6

)(2ˆ)(

2ˆ)[(

)ˆˆ3ˆ)((2ˆ

2ˆ

2ˆ

2ˆˆ

ˆ

)(ˆ);(ˆ

ˆ

2222322

232222

yxSDyxxsSyKSDxKSD

yxxsSyKxKppH

sFHH

sDpxFpsDx

pFx

yx

yxyx

s

ss

Judicious choice of SD vs K’s can cancel chromaticityPrice: NONLINEARITY

J. J. Bisognano


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UW Spring 2008

Accelerator Physics

Linear Coupling

])[()(

))(())(([21~

)(;)()(

222222

xzzxsgxzsp

ysgKxsgKyxH

xsBgAzsBgA :SolenoidsxzsBpA :quads Skew

Potentials Vector

yx

zx

s

J. J. Bisognano


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UW Spring 2008

Accelerator Physics

Linear Coupling with Skew Quads

1,

)(

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

))()(cos()(2)(

,

))()((21

yx

ll

lliyxyx 1

1

uuuu

uu

ll1- where

eaaspH

have weterm, quad skewthe beH Lettingcanonical"" be to out turns this y; or xu where

sssssasu

sssasuconstants of variationtry wesolutions,

uncoupled the of form the Following

yx

yyyxxx

See Wiedemann II

J. J. Bisognano


9

UW Spring 2008

Accelerator Physics

Linear Coupling/cont.

rNlterm slow

qNl where

laaH

dsesp

with

eaa

espsH

length) period lattice Ls/L;2 (withterms varying slowout separatetoStrategy

yx

yx

yxyxq

ql

LqNlllli

yxql

lllliyx

lllli

llyx1

yyxxyyxx

yyxxyyxx

yyxxyyxx

yx

00

00

0

)(

)(

)(

,21

)cos()(~

)(21

)()(

:

00

00

00

periodic

J. J. Bisognano


10

UW Spring 2008

Accelerator Physics

Linear Coupling/cont.

)~~cos(~

);~~sin(

)~~cos(~

);~~sin(

)~~cos()(

~~~~

~~)(~)(~

21

21

21

21

21

21

21

21

21

21

21

yxy

xrlr

yyxyxrl

y

yxx

yrlr

xyxyxrl

x

yxyxrlyrxr

yrxrr

yyryy

xxrxx

ryyrxx

laallaal

a

laa

laaa

laaala

alaHH

aa l

aa

laaG Using

J. J. Bisognano


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UW Spring 2008

Accelerator Physics

Difference Resonance l=-1

;sin2

)(;sin2

cos)(

)(2

);(2

;

0)(

000

~~

iwviwww

w with startsnoscillatio ssumingA

vkwiddvwkvi

ddw

eavea wLet

widthStopband

constant or

STABLEaadd

r

0

rr

iy

ix

yx

yx

yx

J. J. Bisognano


12

UW Spring 2008

Accelerator Physics

Difference Resonance/cont.

220,0,

22

2

22

2

0222

20 )sin4

);cos(41

rryxIII

ry

x

xyrxx aaaa

Implies measurement scheme for

tunes

quad

J. J. Bisognano


13

UW Spring 2008

Accelerator Physics

Sum Resonance

2221

;0)(

r

iy

ix

yx

with

eaieauform the of are Solutions

YINSTABILIT ofy possibilit constant, remains

differenceonly sinceaadd

J. J. Bisognano


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UW Spring 2008

Accelerator Physics

Action Angle Variables

222

1

2

1

1

22

tan22

/

tan2

),(

/;tan)sin()();cos()(

)sin();cos(2

xxFJ

xxF

xFpx-p SinceJapJax

for Looktaptax

x2pH

Oscillator Harmonic2

Ruth/Wiedemann

J. J. Bisognano


15

UW Spring 2008

Accelerator Physics

Action Angle Variables/cont.

]2

[tan2

),(

]cos2

[sin)(

cos)(

)])(cos(2

))([sin(

))(cos(2)(

sin2;cos/2

)(21

2

1

2/1

2/1

2/1

2/1

2

222

zzF

Japp

JazWant

ssap

saz

zsK2pH

have weproblem, raccelerato for NowJK

JpJx

pxJ

2

J. J. Bisognano


16

UW Spring 2008

Accelerator Physics

Action Angle Variables/cont.

JRsFHH

JJssd

Cs

Jssd

CsJJF withFinally

JpJxssds

sJsFHK

zpzFJ

s

s

s

ˆ)/(/ˆ

ˆ;)(

2ˆ

ˆ])(

2[ˆ)ˆ,(,

)cos(sin/2;cos2)(

)(

)(//2

)2

(2

2

0

02

2

00

22

J. J. Bisognano


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UW Spring 2008

Accelerator Physics

Canonical Perturbation Theory

θG/HH

ΦG/JJ

JG/ΦΦ

θ)J,ΦG(JΦθ),J(ΦΦF

tiontransforma canonical Consider

Φθ, over 0V of average Assume2π period has H Assume

JH(J) nonlinear bemay (J)Hθ)J;V(Φ((J)HHan of Form HamiltoniStart with

2

00

0

ˆ

ˆˆ

ˆˆ

ˆˆˆ

/

Following R. Ruth

J. J. Bisognano


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UW Spring 2008

Accelerator Physics

Canonical Perturbation Theory/cont.

, of tindependen left everythingGJVGJ

IfGJVGJ

JVGJV

GJJHGJHJHH

GGJVGJHH

0),ˆ,()(

),ˆ,()(

)],ˆ,(),ˆ,([

])()ˆ()ˆ([)ˆ(ˆ),ˆ,()ˆ(ˆ

000

0

J. J. Bisognano


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UW Spring 2008

Accelerator Physics


nm

inimnm

m

nmnm

inim

mnm

inim

mnm

nmeV

iG

VGinimThen

eJGJG

eJVJV

solutionsperiodic for Look

,

,

,,

,

,

][

)ˆ(),ˆ,(

)ˆ(),ˆ,(

J. J. Bisognano


20

UW Spring 2008

Accelerator Physics


JV)J(ν)Jν(

spreadfrequency gives Vagain over game repeat can We

order secondV whereVJHH

tiontransforma this With

0

ˆ/ˆˆ

)ˆ(ˆ

0

,

J. J. Bisognano


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UW Spring 2008

Accelerator Physics

Octopole

dependent amplitude JdsBchangetune

JB1//dsdhave weterm constant forSo

BJsJHJBsJH

JBsJH

BxxsKpH

C

]23[

3,

)32cos44(cos)(/)cos4()(/

)cos2()(/

\2

)(2

0

2

2

22

4422

4

422

J. J. Bisognano


22

UW Spring 2008

Accelerator Physics

Isolated Resonance

mn and mJfJJJ mnHH

JJmn

JmnF

)n-f(J)cos(m(J)JHE.g.,

tionstransforma canonical more Rather,apply tdoesn'theory onperturbati

singular,getting rsdenominato Sincenm whenoccur can Resonances

2

/)cos()ˆ()ˆ(ˆˆ/ˆ

ˆ;/

ˆ)/(

1

1

J. J. Bisognano


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UW Spring 2008

Accelerator Physics

Fixed Points

stabilitydetermines m of Sign

mJfJ

m

HJH

ˆcos

0ˆcos)ˆ()ˆ(

0ˆsin

ˆ/ˆ0ˆ/ˆ

J. J. Bisognano


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UW Spring 2008

Accelerator Physics

Island Structure

From Ruth

J. J. Bisognano


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UW Spring 2008

Accelerator Physics

Resonance Widths

• Expanding around unstable fixed point at a resonance action Jr yields an equation for the separatrix, and, on expanding, a “bucket height” or width

small assuming,)()(2)(

cos)()(cos)()(

2 fJJfJJ

mJfJJmJfJJ

r

rr

rrrr

J. J. Bisognano


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UW Spring 2008

Accelerator Physics

R Ruth

Avoiding Low Order Resonances

J. J. Bisognano


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UW Spring 2008

Accelerator Physics

Dynamic Aperture

resonances sumfor saw weas , stabilityfor ntinsufficie

is that but integrals, Poincare volumes, spacephase preserves Map

0000

0000010

J

e.g., is, J where,JJM MThen,

jacobian be MLetpq

MPQ

mapping a yield equations sHamilton

T

0110

1

;~

'

J. J. Bisognano


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UW Spring 2008

Accelerator Physics

Eigenvalues

buys condition symplecticthe whatsthat' *;*,1/,,1/

quadruple in comethey thatcondition satisfy Mof seigenvalue The

J. J. Bisognano


29

UW Spring 2008

Accelerator Physics

Surface of Section• For an nD time independent Hamiltonian, energy is

conserved, and motion is on shell, a (2n-1)D set• Condition qn=constant gives (2n-2) surface, a surface

of section• Let’s take a look at Henon map, with the Hamiltonian

having a cubic nonlinearity, sort of sextupole like

)2()( 323

22

21

22

212

121

212

1 qqqqqppH

J. J. Bisognano


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UW Spring 2008

Accelerator Physics

Position Plot of Henon Map

J. J. Bisognano


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Accelerator Physics

E=1/12

J. J. Bisognano


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Accelerator Physics

E=1/8

J. J. Bisognano


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E Almost 1/6

J. J. Bisognano


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UW Spring 2008

Accelerator Physics

Dynamic Aperture

• Determines usable aperture of accelerator, which must be consistent with emittance, injection gymnastics

• Determines whether intrabeam scattered particles survive and be damped in electron machines

• Definition: Region in phase space where particles have stable motion, will be stored indefinitely

• More practically, will particles remain in the machine for the planned storage time; e.g., 107-109 turns in proton accelerators, or synchrotron damping times (104 turns in electron storage rings

• For higher dimensional systems Arnold diffusion adds further complications, but we will take a practical approach

J. J. Bisognano


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UW Spring 2008

Accelerator Physics

Tools

• Tracking (approximate computer mapping) is primary game

• But tracking for “storage time” is still beyond computational limits, so some “numerically derived” criteria to extrapolate are essential

• Since systems are “chaotic,” they are very sensitive to initial conditions and numerical error, so one has to be careful

Scandale, et al.

J. J. Bisognano


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UW Spring 2008

Accelerator Physics

Tracking Tools• Work-horse programs such as MAD, SIXTRACK use

transfer maps for linear part of mapping, but “thin lens” approximation for nonlinearities. This maintains symplecticity of transforms

• Extensions of transfer maps of finite length (or turn) for nonlinearities using differential algebra techniques with Taylor expansions, etc. used for “analysis.”

• “Symplectification” is issue that limits initially perceived advantages of maps over element by element approach

J. J. Bisognano


37

UW Spring 2008

Accelerator Physics

Indicators of Chaos

)()()(

)0()(log1

21

0

tdtdtd

dtd

tlim

Exponent Lyapunov

td(0)

J. J. Bisognano


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UW Spring 2008

Accelerator Physics

Survival Plots

• Plot maximum number of turns that survive as function of starting amplitude

• Plots are interpolated with fitting on functional form

4(1997))-65 Accel Part. al. et zi,(Giovannozanalysisby suggesteddwith

NNbDND

2/)1(

))/(log

1()(0

J. J. Bisognano


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UW Spring 2008

Accelerator Physics

A Survival Plot

Scandale, Todesco

J. J. Bisognano


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UW Spring 2008

Accelerator Physics

Implications of Dynamic Aperture Studies• Sources of nonlinearities: chromatic sextupoles,

multipoles in dipoles, multipoles in lattice quads, multipoles in low- quads, long-range beam-beam kicks

• For hadron colliders, multipoles of dipoles can dominate at injection; at collision, low- quads can dominate

• Target aperture roughly 12 at 105, which implies a 6 with safety margins

• Yields limits on multipole content, suggests multipole correction schemes, optimized optics, beam separation

J. J. Bisognano


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UW Spring 2008

Accelerator Physics

Homework for Topic III

• From S.Y. Lee– 2.5.1– 2.5.3– 2.5.8– 2.6.1– 2.6.2– 2.7.3

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Accelerator Physics Topic III Perturbations and Nonlinear Dynamics