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