Elias Metral, ABP Forum, 26/02/2003
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STABILITY OF THE LONGITUDINAL
BUNCHED-BEAM COHERENT MODES
E. Metral E. Metral (CERN/AB-ABP-HEI)(CERN/AB-ABP-HEI)
Introduction Dispersion relation Stability diagrams
Parabolic distribution Gaussian distribution Distribution used by Sacherer for his stability criterion
Analytical computations for an elliptical spectrum Application to the LHC at top energy
Elias Metral, ABP Forum, 26/02/2003
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INTRODUCTIOINTRODUCTIONN
sSs 0s
It is said that “the coherent synchrotron frequency of the dipole mode does not move”
Incoherent synchrotron frequency shift
The most dangerous longitudinal single-bunch effect in the LHC is the possible suppression of Landau damping at top energy (7 TeV)
Consider the case of the LHC, i.e. an inductive impedance above transition
How can the beam be stable ?
isss 0
Incoh.spread
Elias Metral, ABP Forum, 26/02/2003
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bss I 0
bss I 0 bTRF IVV ˆˆ
bIBB 0 Potential well distortion
Synchronous phase shift
Stationary distribution
Neglected in the following Resistive part of the impedance
DISPERSION RELATION (1/2)DISPERSION RELATION (1/2)
eff
l
sRF
b
s
ss
p
pZj
BVh
I
003
022
0
20
2
cosˆ3
RF
Tss
V
Vˆ
ˆ20
2 3
00
B
B0
3
0
1
0
B
B
B
B
Emittance (momentum spread) conservation for protons (leptons)
PL
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lcmmmI 1
eff
mm
l
sT
sblcmm p
pZ
hVB
Ij
m
m
cosˆ31 3
dr
drrdg
r
drdrrdg
rm
r
Im
s
m
m
0
02
0
02
Dispersion relation
Perturbation (around the new fixed point) Linearized Vlasov equation
Sacherer formula
Dispersion integral
DISPERSION RELATION (2/2)DISPERSION RELATION (2/2)
Elias Metral, ABP Forum, 26/02/2003
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STABILITY DIAGRAMS (1/6)STABILITY DIAGRAMS (1/6) Parabolic Gaussian
S
lcmm
Re
S
lcmm
Im
S
lcmm
Re
S
lcmm
Im
-0.5 0.5 1 1.5 2
-0.5
-0.4
-0.3
-0.2
-0.1
-4 -2 2 4
-5
-4
-3
-2
-1 1m
542 3
0Re
S
lcmm
Capacitive impedance Below Transition or Inductive impedance Above Transition
0 iS
Reminder
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STABILITY DIAGRAMS (2/6)STABILITY DIAGRAMS (2/6)
ss BhS 22
2
16tan
3
51
Approximated full spread between centre and edge of the bunch
0.5 1 1.5 2 2.5 3
0.2
0.4
0.6
0.8
1 0
ˆ
s
s
rad̂
On a flat-top0tan s
Elias Metral, ABP Forum, 26/02/2003
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STABILITY DIAGRAMS (3/6)STABILITY DIAGRAMS (3/6)
Sacherer distribution 220 1 rrg
S
lcmm
Re
S
lcmm
Im
-1 -0.5 0.5 1 1.5 2
-0.8
-0.6
-0.4
-0.2
lcmm
mS
4
sm Sm s
Sacherer stability criterion
Elias Metral, ABP Forum, 26/02/2003
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STABILITY DIAGRAMS (4/6)STABILITY DIAGRAMS (4/6)
sm 2
2
m
m
S
lcmm
Sm s
0Re
S
lcmm
Sm
mm s 1
2
m
m
S
lcmm
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STABILITY DIAGRAMS (5/6)STABILITY DIAGRAMS (5/6)
• Line density• Synchrotron amplitude distribution
-1 -0.5 0.5 1
0.2
0.4
0.6
0.8
1
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1 2
0 2
brg
00 ˆ
s
dg
1 d
2
2
0
2 ̂
s
2/b
z
2/
ˆ
b
r
2bz
r z
P
G
S
P
G
S
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STABILITY DIAGRAMS (6/6)STABILITY DIAGRAMS (6/6)
• What is important for Landau damping (for dipole mode m = 1) is
0
02
02
drdr
rdgr
dr
rdgr
r
0.2 0.4 0.6 0.8 1
1
2
3
4
5 P
G
S
Elias Metral, ABP Forum, 26/02/2003
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ELLIPTICAL SPECTRUM (1/9)ELLIPTICAL SPECTRUM (1/9)
222
202 121 rdr
rdgr
0.2 0.4 0.6 0.8 1
0.2 0.4 0.6 0.8 1
1.2 1.4
2r
1
0
22
202
2
202
drdr
rdgr
drrdg
rS
E
Elias Metral, ABP Forum, 26/02/2003
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ELLIPTICAL SPECTRUM (2/9)ELLIPTICAL SPECTRUM (2/9)
0.2 0.4 0.6 0.8 1
0.5
1
1.5
2
r
1
0
02
02
drdr
rdgr
drrdg
rS
E
S and E curves are very close
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ELLIPTICAL SPECTRUM (3/9)ELLIPTICAL SPECTRUM (3/9)
Case of the dipole mode m = 1
VjUlc 11
22
222
22
222
16
16
16
16
2 VU
VUSVj
VU
VUSU
Ss
lcS 114 Stability criterion
Instability0VMotionstje
Sacherer criterion recovered analytically
U
SSUi
ss 162Re
2
0 Generalization in the presence of frequency spread
VU
Elias Metral, ABP Forum, 26/02/2003
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ELLIPTICAL SPECTRUM (4/9)ELLIPTICAL SPECTRUM (4/9)
Dipole mode
0
ˆ
ˆ91
211
11
011011
RF
Tlc
is
lc
ssscsc
V
V
Neglecting the synchrotron frequency spread
011 sc
Elias Metral, ABP Forum, 26/02/2003
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ELLIPTICAL SPECTRUM (5/9)ELLIPTICAL SPECTRUM (5/9)
Quadrupole mode
is
T
RFis
is
lc
ssscsc
V
V
2
1
ˆ
ˆ
27
42
2
222
2
22
022022
issc
2
12 022
2/346.1
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ELLIPTICAL SPECTRUM (6/9)ELLIPTICAL SPECTRUM (6/9)Taking into account the synchrotron frequency spread
1 2 3 4
-4
-3
-2
-1
1
0
U
s Re
U
S
Incoherent synchrotron frequency spread
s
Ss
0s
0s
Uis
1621
2kk
Uis
1621
2kk
U
Sk
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ELLIPTICAL SPECTRUM (7/9)ELLIPTICAL SPECTRUM (7/9)
Reminder : Besnier’s picture (in 1979 for a parabolic bunch)
No stability threshold due to the sharp edge of the parabolic distribution See stability boundary diagram
Elias Metral, ABP Forum, 26/02/2003
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ELLIPTICAL SPECTRUM (8/9)ELLIPTICAL SPECTRUM (8/9)
eff
mm
l
sTsb
ppZ
BVh
m
mI
5322 cosˆ1
64
3tan
3
51
with bT IV̂ bIB
General stability criterion for mode m
lcmm
mS
4
(Neglecting the synchronous phase shift)
Elias Metral, ABP Forum, 26/02/2003
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ELLIPTICAL SPECTRUM (9/9)ELLIPTICAL SPECTRUM (9/9)
F
p
pZ
BVh
m
mI eff
mm
l
sRFsb
500
32
02 cosˆ1
64
3tan
3
51
with 42
1 2 aaF
eff
mm
l
eff
l
ss
ppZ
ppZ
j
SgnBhm
ma 00
02
02
02 costan
3
51
1
64
9
Elias Metral, ABP Forum, 26/02/2003
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APPLICATION TO THE LHC AT TOP ENERGY (1/2)APPLICATION TO THE LHC AT TOP ENERGY (1/2)
The most critical case is the dipole mode m = 1
F
p
pZ
BVhI eff
l
RFb
11
50
32 ˆ
32
3
32
9 20
2Bha
with
42
1 2 aaF
Elias Metral, ABP Forum, 26/02/2003
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The previous stability criterion is the same as the one used by Boussard-Brandt-Vos in the paper “Is a longitudinal feedback system required for LHC?” (1999), with
Numerical application with the same parameters as the ones used
in the above paper
1F
28.0
1100
eff
l
eff
l
p
pZ
p
pZ
35640hMV16ˆ RFV
cm5.7b 500 101.1 bfB
2105.4 a
01.1F
p/b104.2 11thbN Same value as
the one found by BBV
APPLICATION TO THE LHC AT TOP ENERGY (2/2)APPLICATION TO THE LHC AT TOP ENERGY (2/2)
TeV7E
Elias Metral, ABP Forum, 26/02/2003
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ACKNOWLEDGEMENTACKNOWLEDGEMENTSS
F. Zimmermann, who proposed to look in detail at this mechanism
J. Gareyte, F. Ruggiero, and F. Zimmermann for fruitful discussions
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