Synchro-betatron Resonanzen:eine Einführung und Berechnung der

Resonanzstärken für verschiedene HERA Optiken

F. WillekeBetriebsseminar Salzau 5-8. Mai 2003

•Einführung und Definition•Hamilton Funktion für die Bewegung mit 3 Freiheitsgraden•Einführung der Dispersion und Enkopplung von transversalen und longitudinalen Schwingungen•Behandlung von Resonanzen mit 2Freiheitsgraden•Diskussion der Ergebnisse für HERA

Einführung

Es gibt große Probleme die Polarizations- tunes bei HERA einzustellen:

fx= 6.5 kHz und fz=9kHz

Der horizontale Tune liegt zwischen dem 2-fachen und dem 3-fachen der Synchrotronfrequenz

fs=2.5kHz

Verdacht: Der Bereich der gewünschten Arbeitspunkte ist durch starke

Synchrobetaronresonanzen eingeschränkt.

Synchrobetatron-Resonanzen wurden bei DORIS I entdeckt (Piwinski 1972)

Ursache: Starker vertikaler Kreuzungswinkel der kollidierenden Elektron und Positronstrahlen: Die transversale Strahl-Strahl-Kraft hängt bei einem Kreuzungswinkel von der longitudinalen Position im Bunch ab

DORIS

Allgemein

Der Strahl kann in 3 Ebenen oszillieren.

1) Hängt die triebende Kraft in einer Ebene von der Koordinate oder dem Impuls in der anderen Ebene ab, sind die jeweiligen Schwingungsebenen gekoppelt.

2) Wie alle Kräfte können koppelnde Kräfte mit der gleichen Frequenz oszillieren wie der Strahl selbst:

Dann kommt es zu einer resonanzartigen Verstärkung selbst sehr kleiner Kräfte. Resonanzen führen zum Energieaustausch zwischen den Schwingungsebenen

oder zu Instabilität

Synchro-Betatron Resonances in HERASynchro-Betatron Resonances in HERAIntroduction to the Theory and Recent Introduction to the Theory and Recent

EvaluationsEvaluationsHERA Betriebsseminar Salzau,

5-7 May 2003

Synchro-Betatron Resonances in HERASynchro-Betatron Resonances in HERAIntroduction to the Theory and Recent Introduction to the Theory and Recent

EvaluationsEvaluationsHERA Betriebsseminar Salzau,

5-7 May 2003

•Coupled Synchro-betatron Motion•Decoupling of Synchro-betatron Oscillation•Non-linear Coupling between Synchrotron and Betatron Oscillations•Width of multi-dimensional Nonlinear Resonances•Comparison of the width of Satellite Resonances in HERA for various Beam Optics

•Coupled Synchro-betatron Motion•Decoupling of Synchro-betatron Oscillation•Non-linear Coupling between Synchrotron and Betatron Oscillations•Width of multi-dimensional Nonlinear Resonances•Comparison of the width of Satellite Resonances in HERA for various Beam Optics

Synchrobetatron-Synchrobetatron-ResonancesResonances

Synchrobetatron-Synchrobetatron-ResonancesResonances

Coupling between transverse and longitudinal oscillations gives rise to excitation of resonances for tunes which satisfy

Qx+mQs+q=0

Such resonances can be driven by • Dispersion in the cavities• Dispersion in sextupoles• Chromaticity• A crossing angle or Dispersion a the Collision

Point• Wakefields• RF Quadrupoles• …

Coupling between transverse and longitudinal oscillations gives rise to excitation of resonances for tunes which satisfy

Qx+mQs+q=0

Such resonances can be driven by • Dispersion in the cavities• Dispersion in sextupoles• Chromaticity• A crossing angle or Dispersion a the Collision

Point• Wakefields• RF Quadrupoles• …

Coupled Synchro-betatron OscillationsCoupled Synchro-betatron OscillationsCoupled Synchro-betatron OscillationsCoupled Synchro-betatron Oscillations

Horizontal Betatron Oscillations and Synchrotron oscillations are strongly coupled

by a term

xx··// x is the horizontal coordinate, is the

relative energy deviation from nominal and is he curvature of he design orbit)

This is shown in the following slides

Lagrangian for charged relativistic particle using the Lagrangian for charged relativistic particle using the accelerator coordinate systemaccelerator coordinate system

Lagrangian for charged relativistic particle using the Lagrangian for charged relativistic particle using the accelerator coordinate systemaccelerator coordinate system

eAdt

rd

cdt

rdcmL

22

0 1 eAdt

rd

cdt

rdcmL

22

0 1

''1

)(0

yexeex

sdt

rd

yexesrr

yxs

yx

''1

)(0

yexeex

sdt

rd

yexesrr

yxs

yx

m0c2 rest mass, r is the position vector, A is the vector potential is the scalar potential

Expressing L in accelerator coordinates

One obtains

eAyAxAx

c

seyx

x

c

scmL yxs

''1''11(

2/1

22

222

0

eAyAxAx

c

seyx

x

c

scmL yxs

''1''11(

2/1

22

222

0

Hamiltonian pictureHamiltonian pictureHamiltonian pictureHamiltonian picture

eAc

e

x

pA

c

epA

c

epcmcH

ecm

H

z

Lp

LspypxpH

ss

yyxx

z

syx

2

2222

0

2

20

1

1

eAc

e

x

pA

c

epA

c

epcmcH

ecm

H

z

Lp

LspypxpH

ss

yyxx

z

syx

2

2222

0

2

20

1

1

b= v / c ˜= 1

z=x,y,s

Path length s as independent variablePath length s as independent variablePath length s as independent variablePath length s as independent variable

The Hamiltonian is symmetric in all coordinatesThe Hamiltonian is symmetric in all coordinates

s

yyxx

syyxxs

syx

syx

axapapx

K

EEH

Ax

Ac

epA

c

epcm

c

eHxKp

Hppypxds

dsdsctcdt

Hpspypxdt

111

1111

1

0

11

0''

'

0

2

2

2

2

0

2222

0

2

s

yyxx

syyxxs

syx

syx

axapapx

K

EEH

Ax

Ac

epA

c

epcm

c

eHxKp

Hppypxds

dsdsctcdt

Hpspypxdt

111

1111

1

0

11

0''

'

0

2

2

2

2

0

2222

0

2

(Variation principle)(Variation principle)

m0c2/E0<<1

(gauge)as=e/cAs/E0

Hamiltonian for motion in x-s planeHamiltonian for motion in x-s plane

s

xx axapx

K

1

11111 2

2

s

xx axapx

K

1

11111 2

2

Expanded and without solenoid fieldsExpanded and without solenoid fields

sx ax

px

K

11

2

1111 2 sx a

xp

xK

11

2

1111 2

The term px2x/is considered small and has been droppedThe term px2x/is considered small and has been dropped

Cavity FieldCavity FieldCavity FieldCavity Field

30

0

22

00

00

00

00

00

cos2

6

1cos

2

2

1

sin2

cos2

sin2

sin

E

eU

L

h

E

eU

L

ha

E

eU

L

h

E

eU

h

La

Potential

E

eU

L

h

E

eU

s

s

30

0

22

00

00

00

00

00

cos2

6

1cos

2

2

1

sin2

cos2

sin2

sin

E

eU

L

h

E

eU

L

ha

E

eU

L

h

E

eU

h

La

Potential

E

eU

L

h

E

eU

s

s

Expanded and without constants, energy loss concentrated at cavity, damping neglected

as=1/2 V · 2 + 1/6 W · 3as=1/2 V · 2 + 1/6 W · 3

Hamiltonian with cavities and sextupolesHamiltonian with cavities and sextupolesHamiltonian with cavities and sextupolesHamiltonian with cavities and sextupoles

322322

2

6

1

2

1

2

1

6

11

2

1

2

1

WV

xpxmxkpH 32232

22

6

1

2

1

2

1

6

11

2

1

2

1

WV

xpxmxkpH

Strong linear coupling between horizontal and longitudinal motion

Strong linear coupling between horizontal and longitudinal motion

chromaticschromatics

NonlinearitiesTransverse motionNonlinearitiesTransverse motion

Nonlinearities longitudinal motionNonlinearities longitudinal motion

Linear optics Linear opticsLongitudinal focussingLongitudinal focussing

Approximations:

v=c

p2x/neglected

Square root expanded

1/(1+) expanded into 1-

Approximations:

v=c

p2x/neglected

Square root expanded

1/(1+) expanded into 1-

Linear Decoupling of Synchrobetatron OscillationsLinear Decoupling of Synchrobetatron OscillationsLinear Decoupling of Synchrobetatron OscillationsLinear Decoupling of Synchrobetatron Oscillations

01

kDD 0

1

kDD

Introduction of the

dispersion function

Introduction of the

dispersion function

transformationtransformation

s

FKK

DxDp

Dpp

Dxx

DDxDDxpF

Dppp

Dxxx

2

2

1

s

FKK

DxDp

Dpp

Dxx

DDxDDxpF

Dppp

Dxxx

2

2

1Generating function

Decoupled HamiltonianDecoupled HamiltonianDecoupled HamiltonianDecoupled Hamiltonian

3332

3

2222

222

22

222

6

1

6

1

2

16

12

1

2

12

1

2

1

2

1

2

1

2

12

1

2

1

2

1

mWD

xm

xDDpWxmDpD

xDDpWxmDp

xDDpV

VD

xDDpVxkp

K

3332

3

2222

222

22

222

6

1

6

1

2

16

12

1

2

12

1

2

1

2

1

2

1

2

12

1

2

1

2

1

mWD

xm

xDDpWxmDpD

xDDpWxmDp

xDDpV

VD

xDDpVxkp

K

Transverse linear optics

Longitudinal linear optics

Linear coupl. by dispersion in cavities

Chromatics

2nd satellite driving terms

Nonlinearities trans.

Nonlinearities lon.

Integer Satellite Driving TermsInteger Satellite Driving TermsInteger Satellite Driving TermsInteger Satellite Driving Terms

Qx+Qs+p=0

-DVp + D’Vx

Qx+2Qs+p=0

-D’ p 2 + ½ mD2 2x + ½ WD 2p

Qx+Qs+p=0

-DVp + D’Vx

Qx+2Qs+p=0

-D’ p 2 + ½ mD2 2x + ½ WD 2p

Chromatics sextupole contribution dispersion in cavities Chromatics sextupole contribution dispersion in cavities

Resonances in x-s Phase SpaceResonances in x-s Phase SpaceResonances in x-s Phase SpaceResonances in x-s Phase Space

K = Klin + KnlK = Klin + Knl

))(sin(/2

))(cos(2

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

))(cos(2

ssss

ssss

xxxxx

xxxx

sJ

sJ

ssJp

sJx

))(sin(/2

))(cos(2

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

))(cos(2

ssss

ssss

xxxxx

xxxx

sJ

sJ

ssJp

sJx

Linear opticsLinear optics

Variation of constant: Keep the form for x, p but vary the invariants Jx,s and x,s to solve for the nonlinearities and coupling

(transformation to action and angle variables)

Result Hamiltonian form of e.o.m. with Knl as new Hamiltonian

Variation of constant: Keep the form for x, p but vary the invariants Jx,s and x,s to solve for the nonlinearities and coupling

(transformation to action and angle variables)

Result Hamiltonian form of e.o.m. with Knl as new Hamiltonian

0)cos(2 0

20

smc

s eUh

LE

0)cos(2 0

20

smc

s eUh

LE

Smooth rf model

∂J/ ∂s = - ∂Knl/ ∂∂ / ∂s = ∂Knl/ ∂J∂J/ ∂s = - ∂Knl/ ∂∂ / ∂s = ∂Knl/ ∂J

Procedure to calculate resonance widthsProcedure to calculate resonance widthsProcedure to calculate resonance widthsProcedure to calculate resonance widths

• Express Knl in J- coordinates• Factorise into ring periodic and nonperiodic terms• Express periodic forms in Fourier Series• Realise, that only slow terms can affect a change

in invariants• Drop nonresonant terms• Transform into a rotating system to get a time-

indpendent system• Calculated fixpoints• Find distance from resonance to reach the fix

points for a given amplitude

• Express Knl in J- coordinates• Factorise into ring periodic and nonperiodic terms• Express periodic forms in Fourier Series• Realise, that only slow terms can affect a change

in invariants• Drop nonresonant terms• Transform into a rotating system to get a time-

indpendent system• Calculated fixpoints• Find distance from resonance to reach the fix

points for a given amplitude

Perform this for the term term ½ ½ WDWD22ppPerform this for the term term ½ ½ WDWD22pp

.).(.).(.).(2)(sin)cos(8

.).(.).(.).(2)(sin)sin(8

)sin()sin(882

)sin()cos(882

2

1

)2()2()(2

)2()2()(2

22/1

2/1

22/1

2

cceccecceyx

cceccecceyx

JJWD

JJWD

K

pWDK

yxiyxixi

yxiyxixi

xxxxsx

xs

x

xxxxsx

s

xx

.).(.).(.).(2)(sin)cos(8

.).(.).(.).(2)(sin)sin(8

)sin()sin(882

)sin()cos(882

2

1

)2()2()(2

)2()2()(2

22/1

2/1

22/1

2

cceccecceyx

cceccecceyx

JJWD

JJWD

K

pWDK

yxiyxixi

yxiyxixi

xxxxsx

xs

x

xxxxsx

s

xx

Since we are only near one resonance at a time, we are only interested in one of the terms x+2y

ccL

sQQiiiJJ

L

sQQisisi

WDK

cciiiiJJWD

K

sxsxsxsxsx

x

xsc

sxsxsx

x

xsc

.)2

2(2exp()2

2()(2)(exp(82

).22exp(82

2/112

2/112

ccL

sQQiiiJJ

L

sQQisisi

WDK

cciiiiJJWD

K

sxsxsxsxsx

x

xsc

sxsxsx

x

xsc

.)2

2(2exp()2

2()(2)(exp(82

).22exp(82

2/112

2/112

Periodic factor non-periodic factor

qqcsxsxsxqc

qqcsxsxsxqc

qqcsxsxsxqc

sxsx

x

xsqc

qQQJJkK

L

sqQQJJk

LK

ccL

sqQQiJJk

LK

L

sqQQi

WDdsk

122/1

12

122/1

12

122/1

12

12

)2(2cos

2)2(2cos

2

.))2

)2(2(exp(2

2

))2

)2(2(exp(82

1

qqcsxsxsxqc

qqcsxsxsxqc

qqcsxsxsxqc

sxsx

x

xsqc

qQQJJkK

L

sqQQJJk

LK

ccL

sqQQiJJk

LK

L

sqQQi

WDdsk

122/1

12

122/1

12

122/1

12

12

)2(2cos

2)2(2cos

2

.))2

)2(2(exp(2

2

))2

)2(2(exp(82

1

Fourier Series for periodic partFourier Series for periodic part

Select only the one resonant term and ‘drop’ all the others, replace 2s/L by

(change independent variable from s to

Select only the one resonant term and ‘drop’ all the others, replace 2s/L by

(change independent variable from s to

Resonance HamiltonianResonance HamiltonianResonance HamiltonianResonance Hamiltonian

qcsxsxqcsqsxqxqcqc

xx

ss

qsssxss

qxxsxxx

sxsssxxx

qcsxxsxqcqc

IIkIIF

KR

JI

JI

qQQ

qQQ

qQQIqQQIF

qQQJJkK

122/1

1212121212

12

12

122/1

1212

2cos

)2(5

2

)2(5

1

)2(5

2)2(

5

1

)2(2cos

qcsxsxqcsqsxqxqcqc

xx

ss

qsssxss

qxxsxxx

sxsssxxx

qcsxxsxqcqc

IIkIIF

KR

JI

JI

qQQ

qQQ

qQQIqQQIF

qQQJJkK

122/1

1212121212

12

12

122/1

1212

2cos

)2(5

2

)2(5

1

)2(5

2)2(

5

1

)2(2cos

Transformation into rotating system via generating function F

Note there is a similar term denoted by “s” which comes from the sine part of p

Note there is a similar term denoted by “s” which comes from the sine part of p

qssxsxqssqsxqxqs IIkIIR 122/1

12121212 2sin qssxsxqssqsxqxqs IIkIIR 122/1

12121212 2sin

The two driving terms k12qs and k12qc are combined into a single oneThe two driving terms k12qs and k12qc are combined into a single one

)2cos(

)sin(2

122/1

12121212

121222

12 12121212

qsxsxqsqsxqxq

qsqcq

IIkIIR

kkkkkqsqcqsqc

)2cos(

)sin(2

122/1

12121212

121222

12 12121212

qsxsxqsqsxqxq

qsqcq

IIkIIR

kkkkkqsqcqsqc

Resonance WidthResonance WidthResonance WidthResonance Width

00'' 220''2

2

1sxsxsxs

sx

x

IIIIIIIR

IR

00'' 220''2

2

1sxsxsxs

sx

x

IIIIIIIR

IR

2Ix-Is is an invariant and we can reduce the system to a 1-dim system

Note: sum resonances are instable and difference resonances stable!!

2Ix-Is is an invariant and we can reduce the system to a 1-dim system

Note: sum resonances are instable and difference resonances stable!!

qQQ

IIIIIkIR

sx

sxxxxqx

2

)cos()22( 02/1

02/12/3

12

qQQ

IIIIIkIR

sx

sxxxxqx

2

)cos()22( 02/1

02/12/3

12

R

I

Separatrix, unstable trajectory

I0=unstable fix point

I

2/1001212

00

2/100

2/101212

2/10

2/10

2/112

2

1

)2

12(

)2

13(

0

xsqq

xs

xsxqq

xsxxxq

IIk

II

IIIk

IIIIIk

R

I

R

2/1001212

00

2/100

2/101212

2/10

2/10

2/112

2

1

)2

12(

)2

13(

0

xsqq

xs

xsxqq

xsxxxq

IIk

II

IIIk

IIIIIk

R

I

RCondition for unstable fix point

Evaluate this for Evaluate this for IIxx = Ix0

12q12q==½½ k k12q12q I Is0s0 I Ix0x0-1/2-1/212q12q==½½ k k12q12q I Is0s0 I Ix0x0-1/2-1/2

5 104

0 5 104

0.001 0.00150.001

0

0.0017.878 10

4

7.878 104

I j sin j

1.093 1031.118 10

4 I j

cos j

0.003 0.002 0.001 0 0.001 0.002 0.0030.003

0.002

0.001

0

0.001

0.002

0.003

Ixj

sin j

Ixj

cos j

5 104

0 5 104

0.001 0.00150.001

0

0.0017.878 10

4

7.878 104

I j sin j

1.093 1031.118 10

4 I j

cos j

0.003 0.002 0.001 0 0.001 0.002 0.0030.003

0.002

0.001

0

0.001

0.002

0.003

Ixj

sin j

Ixj

cos j

Hi j

2

J si

1

2h J si

J s0

2J x0

1

2 J si cos j H

i j2

J si

1

2h J si

J s0

2J x0

1

2 J si cos j

0 5 100

1 107

2 107

1.671 107

0

Hi j

100 J si

J s0

0 5 100

1 107

2 107

1.671 107

0

Hi j

100 J si

J s0

5 107

0 5 1071 10

61 10

6

0

1 106

I j sin j

I j cos j 5 10

60 5 10

61 10

55 10

6

0

5 106

4.805 106

4.805 106

I xj

sin j

5.261 1064.392 10

6 I xj

cos j

5 107

0 5 1071 10

61 10

6

0

1 106

I j sin j

I j cos j 5 10

60 5 10

61 10

55 10

6

0

5 106

4.805 106

4.805 106

I xj

sin j

5.261 1064.392 10

6 I xj

cos j

Hi j

5

J xi 2

5 J si h J xi

1

2 J si cos j H

i j5

J xi 2

5 J si h J xi

1

2 J si cos j

Satellite Resonance with

h=0.1m-1

Satellite Resonance with

h=0.1m-1

Hamilonian vs actionHamilonian vs action

Horizonal projection of separatrixHorizonal projection of separatrix Longitudinal projection of separatrixLongitudinal projection of separatrix

EvaluationsEvaluationsEvaluationsEvaluations Integrals are replaced by sums, optical functions are replaced by

their integrated value over the elements, then a thin lens treatment is applied

Optics Qx+Qs Qx+2Qs

Cavities Chrom. sext total

helum72gj 19 297 487 747 1306

helum72sm 22 276 476 405 948

helumv6 24 93 222 250 458

Results:Results: Resonance width for a 10 sigma particle in Hz (nx

2+ns2=100)

optics mco [10-4] s/m s[10-6m] x[10-9m]

helumv6 6.8 11.5 9.0 42

helum72gj 4.75 9.9 14.5 22

helum72sm 4.75 9.9 14.5 22

Beam Parameters used

Resonance

Optics

3Qx Qx Qx+2Qy Qx-2Qy Qx+0Qy 2Qx+Qs

Helum72gj 77 63 167 2289 223 2231

Helum72sm 252 73 303 983 210 1470

Helumv6 88 161 759 239 492 990

Comparison of sextupole driven transverse resonances

for one sigma transverse, full coupling

and width of horizontal half integer stopband for 10 sigma long.

Comparison of sextupole driven transverse resonances

for one sigma transverse, full coupling

and width of horizontal half integer stopband for 10 sigma long.

ConclusionsConclusions

• The 3rd order transverse and satellite resonances are stronger in the 72deg optics compared to the old 60 degree optics

• The stronger satellite resonances are due to more unfavorable ratio of Longitudinal and transverse emittance

• The SM optics has lower contributions from sextupole driven satellites

• The 3rd order transverse and satellite resonances are stronger in the 72deg optics compared to the old 60 degree optics

• The stronger satellite resonances are due to more unfavorable ratio of Longitudinal and transverse emittance

• The SM optics has lower contributions from sextupole driven satellites