stability of the longitudinal bunched-beam coherent modes
DESCRIPTION
STABILITY OF THE LONGITUDINAL BUNCHED-BEAM COHERENT MODES. E . Metral (CERN/AB-ABP-HEI). Introduction Dispersion relation Stability diagrams Parabolic distribution Gaussian distribution Distribution used by Sacherer for his stability criterion - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: STABILITY OF THE LONGITUDINAL BUNCHED-BEAM COHERENT MODES](https://reader035.vdocuments.us/reader035/viewer/2022062408/56813abd550346895da2c9f8/html5/thumbnails/1.jpg)
Elias Metral, ABP Forum, 26/02/2003
1
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
![Page 2: STABILITY OF THE LONGITUDINAL BUNCHED-BEAM COHERENT MODES](https://reader035.vdocuments.us/reader035/viewer/2022062408/56813abd550346895da2c9f8/html5/thumbnails/2.jpg)
Elias Metral, ABP Forum, 26/02/2003
2
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
![Page 3: STABILITY OF THE LONGITUDINAL BUNCHED-BEAM COHERENT MODES](https://reader035.vdocuments.us/reader035/viewer/2022062408/56813abd550346895da2c9f8/html5/thumbnails/3.jpg)
Elias Metral, ABP Forum, 26/02/2003
3
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
![Page 4: STABILITY OF THE LONGITUDINAL BUNCHED-BEAM COHERENT MODES](https://reader035.vdocuments.us/reader035/viewer/2022062408/56813abd550346895da2c9f8/html5/thumbnails/4.jpg)
Elias Metral, ABP Forum, 26/02/2003
4
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)
![Page 5: STABILITY OF THE LONGITUDINAL BUNCHED-BEAM COHERENT MODES](https://reader035.vdocuments.us/reader035/viewer/2022062408/56813abd550346895da2c9f8/html5/thumbnails/5.jpg)
Elias Metral, ABP Forum, 26/02/2003
5
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
![Page 6: STABILITY OF THE LONGITUDINAL BUNCHED-BEAM COHERENT MODES](https://reader035.vdocuments.us/reader035/viewer/2022062408/56813abd550346895da2c9f8/html5/thumbnails/6.jpg)
Elias Metral, ABP Forum, 26/02/2003
6
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
![Page 7: STABILITY OF THE LONGITUDINAL BUNCHED-BEAM COHERENT MODES](https://reader035.vdocuments.us/reader035/viewer/2022062408/56813abd550346895da2c9f8/html5/thumbnails/7.jpg)
Elias Metral, ABP Forum, 26/02/2003
7
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
![Page 8: STABILITY OF THE LONGITUDINAL BUNCHED-BEAM COHERENT MODES](https://reader035.vdocuments.us/reader035/viewer/2022062408/56813abd550346895da2c9f8/html5/thumbnails/8.jpg)
Elias Metral, ABP Forum, 26/02/2003
8
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
![Page 9: STABILITY OF THE LONGITUDINAL BUNCHED-BEAM COHERENT MODES](https://reader035.vdocuments.us/reader035/viewer/2022062408/56813abd550346895da2c9f8/html5/thumbnails/9.jpg)
Elias Metral, ABP Forum, 26/02/2003
9
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
![Page 10: STABILITY OF THE LONGITUDINAL BUNCHED-BEAM COHERENT MODES](https://reader035.vdocuments.us/reader035/viewer/2022062408/56813abd550346895da2c9f8/html5/thumbnails/10.jpg)
Elias Metral, ABP Forum, 26/02/2003
10
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
![Page 11: STABILITY OF THE LONGITUDINAL BUNCHED-BEAM COHERENT MODES](https://reader035.vdocuments.us/reader035/viewer/2022062408/56813abd550346895da2c9f8/html5/thumbnails/11.jpg)
Elias Metral, ABP Forum, 26/02/2003
11
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
![Page 12: STABILITY OF THE LONGITUDINAL BUNCHED-BEAM COHERENT MODES](https://reader035.vdocuments.us/reader035/viewer/2022062408/56813abd550346895da2c9f8/html5/thumbnails/12.jpg)
Elias Metral, ABP Forum, 26/02/2003
12
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
![Page 13: STABILITY OF THE LONGITUDINAL BUNCHED-BEAM COHERENT MODES](https://reader035.vdocuments.us/reader035/viewer/2022062408/56813abd550346895da2c9f8/html5/thumbnails/13.jpg)
Elias Metral, ABP Forum, 26/02/2003
13
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
![Page 14: STABILITY OF THE LONGITUDINAL BUNCHED-BEAM COHERENT MODES](https://reader035.vdocuments.us/reader035/viewer/2022062408/56813abd550346895da2c9f8/html5/thumbnails/14.jpg)
Elias Metral, ABP Forum, 26/02/2003
14
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
![Page 15: STABILITY OF THE LONGITUDINAL BUNCHED-BEAM COHERENT MODES](https://reader035.vdocuments.us/reader035/viewer/2022062408/56813abd550346895da2c9f8/html5/thumbnails/15.jpg)
Elias Metral, ABP Forum, 26/02/2003
15
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
![Page 16: STABILITY OF THE LONGITUDINAL BUNCHED-BEAM COHERENT MODES](https://reader035.vdocuments.us/reader035/viewer/2022062408/56813abd550346895da2c9f8/html5/thumbnails/16.jpg)
Elias Metral, ABP Forum, 26/02/2003
16
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
![Page 17: STABILITY OF THE LONGITUDINAL BUNCHED-BEAM COHERENT MODES](https://reader035.vdocuments.us/reader035/viewer/2022062408/56813abd550346895da2c9f8/html5/thumbnails/17.jpg)
Elias Metral, ABP Forum, 26/02/2003
17
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
![Page 18: STABILITY OF THE LONGITUDINAL BUNCHED-BEAM COHERENT MODES](https://reader035.vdocuments.us/reader035/viewer/2022062408/56813abd550346895da2c9f8/html5/thumbnails/18.jpg)
Elias Metral, ABP Forum, 26/02/2003
18
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)
![Page 19: STABILITY OF THE LONGITUDINAL BUNCHED-BEAM COHERENT MODES](https://reader035.vdocuments.us/reader035/viewer/2022062408/56813abd550346895da2c9f8/html5/thumbnails/19.jpg)
Elias Metral, ABP Forum, 26/02/2003
19
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
![Page 20: STABILITY OF THE LONGITUDINAL BUNCHED-BEAM COHERENT MODES](https://reader035.vdocuments.us/reader035/viewer/2022062408/56813abd550346895da2c9f8/html5/thumbnails/20.jpg)
Elias Metral, ABP Forum, 26/02/2003
20
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
![Page 21: STABILITY OF THE LONGITUDINAL BUNCHED-BEAM COHERENT MODES](https://reader035.vdocuments.us/reader035/viewer/2022062408/56813abd550346895da2c9f8/html5/thumbnails/21.jpg)
Elias Metral, ABP Forum, 26/02/2003
21
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
![Page 22: STABILITY OF THE LONGITUDINAL BUNCHED-BEAM COHERENT MODES](https://reader035.vdocuments.us/reader035/viewer/2022062408/56813abd550346895da2c9f8/html5/thumbnails/22.jpg)
Elias Metral, ABP Forum, 26/02/2003
22
ACKNOWLEDGEMENTACKNOWLEDGEMENTSS
F. Zimmermann, who proposed to look in detail at this mechanism
J. Gareyte, F. Ruggiero, and F. Zimmermann for fruitful discussions