Synchro-betatron Resonanzen: eine Einführung und Berechnung der Resonanzstärken für verschiedene HERA Optiken F. Willeke Betriebsseminar 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
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Einführung und Definition Hamilton Funktion für die Bewegung mit 3 Freiheitsgraden
Synchro-betatron Resonanzen: eine Einführung und Berechnung der Resonanzstärken für verschiedene HERA Optiken F. Willeke Betriebsseminar Salzau 5-8. Mai 2003. Einführung und Definition Hamilton Funktion für die Bewegung mit 3 Freiheitsgraden - PowerPoint PPT Presentation
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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
•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
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
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
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
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