introduction, observations and motivation theory experiments conclusion
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STABILISING INTENSE BEAMS BY LINEAR COUPLING. Elias METRAL. Introduction, observations and motivation Theory Experiments Conclusion. INTRODUCTION. Single-particle trajectory. One particle. Circular design orbit. Low intensity Single-particle phenomena - PowerPoint PPT PresentationTRANSCRIPT
Elias Metral, CERN-PS seminar, 12/04/2000
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Introduction, observations and motivation Theory Experiments Conclusion
STABILISING INTENSE BEAMS
BY LINEAR COUPLING
Elias METRALElias METRAL
Elias Metral, CERN-PS seminar, 12/04/2000
2
INTRODUCTIONINTRODUCTION
Low intensity Single-particle phenomena High intensity Collective effects
2 stabilising mechanisms against transverse coherent instabilities :
Landau damping by non-linearities (space-charge and octupoles)
Non-linearities Perturbations of the single-particle motion (resonances)
Single-particle trajectory
Circular design orbit
!
One particle
Feedback systems
Elias Metral, CERN-PS seminar, 12/04/2000
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OBSERVATIONS OBSERVATIONS
In 1989, a coherent instability of the quadrupolar mode type driven by ions from the residual gas has been observed by D. Mohl et al. in the CERN-AA and successfully cured by adjusting both tunes close to 2.25
In 1993, a single-bunch instability of the dipolar mode type driven by the resistive wall impedance has been observed by R. Cappi in the CERN-PS and “sometimes cured” by adjusting both tunes close to 6.24
THE IDEA (from R. CAPPI and D. MOHL) WAS TO :THE IDEA (from R. CAPPI and D. MOHL) WAS TO :
USE LINEAR COUPLING TO “TRANSFER DAMPING” FROM THE STABLE TO THE UNSTABLE PLANE, IN ORDER TO REDUCE THE EXTERNAL NON-LINEARITIES
Elias Metral, CERN-PS seminar, 12/04/2000
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THEORY (1/16) THEORY (1/16)
A general formula for the transverse coherent instabilities with Frequency spreads (due to octupoles) Linear coupling (due to skew quadrupoles)
0
2
ˆ0
02
ˆ2
ˆ0
02
ˆ
1,11
1,1,0
20
20
1,,1,
0
20
20
0
20
20
1,11
1,1,
0
20
20
1,,1,
ymmmy
ymm
y
ymm
ymmmy
y
x
xmmmx
xmm
x
xmm
xmmmx
IRlK
IRlK
RlKI
RlKI
x-dispersion integral
x-Sacherer’s formula
Mode coupling term
Linear coupling term
Elias Metral, CERN-PS seminar, 12/04/2000
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THEORY (2/16) THEORY (2/16)
i
i
i
i
y
y
iisiiixc
iiyii
ixx
x
mx ydxdmyx
yyfxxd
xdf
Iˆ
0ˆ ,
0202
ˆ
0ˆ
, ˆˆˆ,ˆ
ˆˆˆˆ
ˆ2
i
i
i
i
y
y
iisiiiyc
iixii
iyx
x
my ydxdmlyx
xxfyyd
ydf
Iˆ
0ˆ 0,
0202
ˆ
0ˆ, ˆˆ
ˆ,ˆ
ˆˆˆˆ
ˆ2
Near the coupling resonance
lQQ vh
Uncorrelated distribution functions (Averaging method)
lK 0ˆ is the lth Fourier coefficient of
the normalized skew gradient
Coherent frequency to be determined
Elias Metral, CERN-PS seminar, 12/04/2000
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THEORY (3/16) THEORY (3/16)
k
k
yxkmm
k
k
yxkmm
yxkyx
yx
byxmm
yx
yx
h
hZ
LQm
Iejm
,
,
,,
,,
,,
00,00
1,, 2
1
Sacherer’s formula (single-and coupled-bunch instabilities) => “low intensity” case
...,1,0,1..., m
1...,,1,0 Mn
Head-tail modes
Coupled-bunch modes
Power spectrum Pick-up (Beam Position Monitor) signal
mmh ,
One particular turn
Time
0m 1m0m
1m
2m
-signal -signal
Time
Elias Metral, CERN-PS seminar, 12/04/2000
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THEORY (4/16) THEORY (4/16)
In the absence of - Linear coupling
- Mode coupling
Let’s recover the 1D results 0ˆ
0 lK
01,1, y
mmx
mm
=> Sacherer’s formula is recovered
Instability
xxx
mmsxc VjUm eqeq,0
0eq xVMotions =>
Instability growth rate
Real coherent betatron frequency shift
tj ce
In the absence of frequency spreads
sxcmx mI 0
1,
These are the Laslett, Neil and Sessler (LNS) coefficients for coasting beams
Elias Metral, CERN-PS seminar, 12/04/2000
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THEORY (5/16) THEORY (5/16) In the presence of frequency spreads
(1) Lorentzian distribution
0x
x
ix,
HWHH:x
c
1D criterion
xx Veq
(2) Elliptical distribution
x
0x ix,
HWB:x
c
1D criterion Keil-Zotter’s stability criterion
Overestimates Landau damping (infinite tails)
Underestimates Landau damping (sharp edges)
xxx VU eq2eq
2 24Re
Elias Metral, CERN-PS seminar, 12/04/2000
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THEORY (6/16) THEORY (6/16)
In the absence of linear coupling but in the presence of mode coupling => “high intensity” case
01,1
11,1,
1,,1,
xmmmx
xmm
xmm
xmmmx
I
I
=> Kohaupt’s stability criterion against Transverse Mode Coupling Instability (TMCI) is recovered
yxmm
yxmms
yxmm
,,
,1,1
,1, 2
1
In the absence of frequency spreads
In the presence of frequency spreads
=> A tune spread of the order of the synchrotron tune is needed for stabilisation by Landau damping
Elias Metral, CERN-PS seminar, 12/04/2000
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THEORY (7/16) THEORY (7/16)
00
40
42
0
,1,,
1, 4
ˆ
yx
ymmmy
xmmmx
RlKII
New 2D results In the absence of mode coupling only
In the absence of frequency spreads
0eqeq yx VV
yx
vhyxyxyx
VV
lQQVV
R
VVQQlK
eqeq
2/122
0
2
eqeq
02
2/1
eqeq000
2ˆ
Necessary condition for stability Transfer of growth rates
0
lK 0ˆ
lQQ vh
Stable regionNo coupling
Full coupling
Stability criteria :
0eq xV 0eq yV
0eqeq yx VV
Full coupling?
Stability criterion (for each mode m)
lnn yx for coupled-bunch modes (and coasting beams)
Elias Metral, CERN-PS seminar, 12/04/2000
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THEORY (8/16) THEORY (8/16)
222222 441412
11, aaaC
yxyx VV
RlKa
eqeq00
20
20
2
ˆ
yx
vh
VV
lQQ
eqeq
0
=> Normalised coupling (or sharing) function
0 0.5 1 a
0.2
0.4
0.6
0.8
1
aC
0 0.5 1 a
0.2
0.4
0.6
0.8
1
aC
0 0.5 a
1
0.2
0.4
0.6
0.8
1
aC
1 25.0 0
aC aC aC
a a a
1, aCfor full coupling
Elias Metral, CERN-PS seminar, 12/04/2000
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THEORY (9/16) THEORY (9/16)
In the presence of frequency spreads
(1) Lorentzian distribution
=> Same results with replaced by yxV ,eq yxyxV ,,eq
No coupling
Full coupling
Stability criteria :
yxyx VV eqeq
xx Veq yy Veq
Transfer of both instability growth rates and frequency spreads (Landau damping)
Elias Metral, CERN-PS seminar, 12/04/2000
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THEORY (10/16) THEORY (10/16)
(2) Elliptical distribution A particular case : No horizontal tune spread and no vertical wake field
xy V2 3/23/20max
2121/3 xvh VlQQ
1 21/3 2
lQQ vh
Stable region
2
0ˆ lK
Elias Metral, CERN-PS seminar, 12/04/2000
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THEORY (11/16) THEORY (11/16)
1) “far from”
2) “near”
hQ lQv
yxyyxx VVUU eqeq2eq
22eq
2 244Re
02
2/1
eq2eq
2eq
2eq
200
0
24Re24Reˆ
R
VUVUQQlK
yyyxxxyx
lQQ vh
0/ yx
THE TUNE SEPARATION SHOULD BE SMALLER THAN
THE ORDER OF MAGNITUDE OF IN ORDER
TO HAVE THE TRANSFER OF LANDAU DAMPING
hQ lQv 0eqeq yx VV
Approximate general stability criterion
=> Transfer of growth rates only
Necessary condition
Elias Metral, CERN-PS seminar, 12/04/2000
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THEORY (12/16) THEORY (12/16)
lQv
hQ
H-plane
V-plane
Transfer of frequency spread (to Landau damp )xVeq
Same result obtained considering both non-linear space-charge forces and octupoles for coasting beams => D. Mohl and H. Schonauer’s 1D stability criterion (gain of factor ~2)
On the coupling resonance
VU “One plane is stabilised by Landau damping and the other one is stabilised by coupling”
yy U
Elias Metral, CERN-PS seminar, 12/04/2000
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THEORY (13/16) THEORY (13/16) In the presence of both mode coupling and linear coupling, neglecting frequency spreads
ymm
xmm
yx
mycmxcmycmxc
yx
ymmmycmyc
xmmmxcmxc
RlK
RlK
1,1,00
40
42
0
1,1,,,
00
40
42
02
1,1,,
2
1,1,,
24
ˆ
4
ˆ
xmmsxmx m ,0, y
mmsymy ml ,00,
=> Necessary condition for stability
ymm
xmm
ymm
xmms
ymm
xmm ,,1,11,11,1, 2
2
1
Elias Metral, CERN-PS seminar, 12/04/2000
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THEORY (14/16) THEORY (14/16)
rB B
[ ra d /s ]
h -1 , -1
BByxZ ,Im BB
yxZ ,Re
h -1 , -1 h 0 ,0
=> Computed gain in intensity of about 50% for the classical ratio of factor 2 between the transverse sizes of the vacuum chamber
Example :
Elias Metral, CERN-PS seminar, 12/04/2000
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THEORY (15/16) THEORY (15/16)
SHARING OF DAMPING BY FEEDBACKSSHARING OF DAMPING BY FEEDBACKS
The stabilising effect of feedbacks can be introduced in the coefficient
eqV
An electronic feedback system can be used to damp transverse coherent instabilities. Its action on the beam can be described in terms of an impedance, which depends on the distance between pick-up and kicker, and the electronic gain and time delays
Kicker
Electronics
Beam
Its damping effect in one plane, can also be transferred to the other plane using coupling
0 1Pick-up
Elias Metral, CERN-PS seminar, 12/04/2000
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THEORY (16/16) THEORY (16/16)
SUMMARY OF THEORY 1 general formula for transverse coherent instabilities in the presence of Frequency spreads (due to octupoles) Linear coupling (due to skew quadrupoles) In the absence of coupling the well-known 1D results are recovered as expected Effects of linear coupling (skew quadrupoles and/or tune distance from coupling resonances) : Transfer of growth rates for “any” coupling
Transfer of Landau damping for “optimum” coupling
“Chromaticity sharing” (for Sacherer’s formula)
Linear coupling is an additional (3rd) method that can be used to damp transverse coherent instabilities
=>
Elias Metral, CERN-PS seminar, 12/04/2000
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Experimental conditions
High intensity bunched proton beam
1.2 s long flat bottom at injection kinetic energy13105.1 beamI20M
EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (1/9) EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (1/9)
- 400 400
- 2
2
-400
- 400 400
1
2
0
1m mmh ,
m/10Re 5 RWcZ
s/rad10 6
Gev1cE
Sacherer’s formula
=> coupled-bunch instabilities Coupled-bunch modes Most critical head-tail mode
number
for the horizontal plane
13, yxn
1m
Landau damping is needed
01eq
1eq m
ym
x VV
121 s-1 - 40 s-1
Elias Metral, CERN-PS seminar, 12/04/2000
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Observations
ms10timerise
10 dB/div
SWP 1.2 s
R signal
Spectrum Analyzer(zero frequency span)
Beam-Position Monitor(20 revolutions superimposed)
One particular turn
Center 360 kHz RES BW 10 kHz VBW 3 kHz Time
EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (2/9) EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (2/9)
1D case A33.0skewI See next slides
1m
Time (20 ns/div)
Elias Metral, CERN-PS seminar, 12/04/2000
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MEASUREMENT OF THE CERN-PS LINEAR COUPLING MEASUREMENT OF THE CERN-PS LINEAR COUPLING
In the presence of linear coupling between the transverse planes, the difference from the tunes of the 2 normal modes is given by
22Gvh CQQQQ
Measurement method : For different skew quadrupole currents, we
increase and decrease in the vicinity of the coupling resonance
and we measure the 2 normal mode frequencies using a vertical kicker,
a vertical pick-up and a FFT analyzer
hQ vQ
In the PS Coupling resonance No solenoid
0 vh QQ
Guignard’s coupling coefficient
It is obtained from the general formula (in the smooth approximation used to study instabilities)
EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (3/9) EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (3/9)
Elias Metral, CERN-PS seminar, 12/04/2000
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Coupling measurements from mode frequencies by FFT analysis
Low intensity bunched proton beam
1.2 s long flat bottom at injection kinetic energy20M 1010250beamI
“Mountain range” display for the “natural” coupling
Frequency
Time
A0skewI
FFT Analyzer
0fCG
hv QQ
EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (4/9) EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (4/9)
Gev1cE
Elias Metral, CERN-PS seminar, 12/04/2000
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-1.5 -1 -0.5 0 0.5 1 1.50
2
4
6
8
AskewI
250 m10 K
A0.1A33.0 skewI
=> Modulus of the normalised skew gradient vs. skew quadrupole current
EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (5/9) EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (5/9)
Elias Metral, CERN-PS seminar, 12/04/2000
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Stabilisation by Landau damping (1D case)
rad/s1100HWHH x
Theoretical frequency spread required
This is less than required by the theory by a factor 3 (without taking into account space-charge non-linearities...)
A320rad/sHWHHoctx I A630rad/sHWHH
octy I
Simplified (elliptical) stability criterion : Keil-Zotter’s criterion
yxmmx,y
,,
HWHH 3
rad/s3400HWHH x Experimental frequency spread required
A5.3octI
EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (6/9) EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (6/9)
Elias Metral, CERN-PS seminar, 12/04/2000
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Stabilisation by coupled Landau damping (2D)
0 1 2 3 0
2 4
6
8 10
AoctI
250 m10 K
Measurement
Theory(Lorentzian vertical distribution)
0.71.11.21
0.50.3
0
5
10
0 0.5 1 1.5 2 2.5 3 3.5
I oct [A]
#REF!
#REF!
0
5
10
0 1 2 3
I oct [A]
|K0
| (*1
0-5) [
m-2
]
#REF!#REF!Poly. (#REF!)Poly. (#REF!)
theory0
exp0 / KK
Constant tune separation 14.6hQ 22.6vQ
EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (7/9) EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (7/9)
Elias Metral, CERN-PS seminar, 12/04/2000
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-1.2 -0.7 -0.20
0.02
0.04
0.06
0.08
0.10
0.12
0.14
AskewI
hv QQ
Measurement
Theory
(Lorentzian vertical distribution)
Constant octupole strength
theoryexp/ vhvh QQQQ
0.81.23
A2octI
EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (8/9) EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (8/9)
Elias Metral, CERN-PS seminar, 12/04/2000
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The experimental results confirm the predicted beneficial
effect of coupling on Landau damping
Using coupling, a factor 7 has been gained in the octupole
current (for this particular case) => Less non-linearities
Difference between theoretical predictions and experiments
Space-charge non-linearities, impedance and tune spread
models…
Further theoretical work => More precise treatment of the non-
linearities in the normal modes
CONCLUSIONS OF EXPERIMENT-1
EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (9/9) EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (9/9)
Elias Metral, CERN-PS seminar, 12/04/2000
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EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (1/6) EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (1/6)
1.2 s long flat bottom at injection kinetic energy Bunch length Transverse tunes Transverse chromaticities
18.6hQ 21.6vQ
9.0x 3.1y
1210bN
Gev4.1cEns160b
Head-tail mode number m
Growth rates [s-1]
-250
-200
-150
-100
-50
0
50
0 1 2 3 4 5 6 7 8 9 10
Horizontal
Vertical
unstable
stable
Sacherer’s formula =>
Single bunch of protons with nominal intensity
Elias Metral, CERN-PS seminar, 12/04/2000
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ms30timerise Time
10 dB/div
SWP 1.2 s (20 ns/div)
R signal
Spectrum Analyzer(zero frequency span)
Beam-Position Monitor(20 revolutions superimposed)
Center 355 kHz RES BW 10 kHz VBW 3 kHz Time
EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (2/6) EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (2/6)
Observations 1D case A33.0skewI
6m
Elias Metral, CERN-PS seminar, 12/04/2000
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Stabilisation by linear coupling only
~ no emittance blow-up m9.12/1norm,1norm, yx
0.73 1.7 1 1.7-0.07 1.7 1 1.7
]A[skewI ]m[)10( 25exp
0K ]m[)10( 25theory
0K
theory
0
exp
0 / KK
m3 (limit)
EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (3/6) EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (3/6)
since 06eq
6eq m
ym
x VV
The ~ same results are obtained for the ultimate beam12108.1 bN
~ no emittance blow-up
m2.32/1norm,1norm, yx
but ~ no blow-up in the PS
A68.0skewI A02.0skewI
m30
1
2
3
4
5
6
7
-0.1 -0.05 0 0.05 0.1 0.15
Measurement
Theory
250 m10 K
hv QQ
Elias Metral, CERN-PS seminar, 12/04/2000
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Voir le file presentation 1
EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (4/6) EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (4/6)
Elias Metral, CERN-PS seminar, 12/04/2000
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8 bunches of protons with nominal intensity
Theoretical stabilising
skew gradient
coupled-bunch instabilities
25theory
0 m103.4 K
A75.0skewI
A4.1skewI
or
1210bN
EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (5/6) EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (5/6)
Head-tail mode number m
Growth rates [s-1]
-250
-200
-150
-100
-50
0
50
0 1 2 3 4 5 6 7 8 9 10
Horizontal
Vertical
unstable
stable
The ~ same results are obtained for the ultimate beam12108.1 bN
Elias Metral, CERN-PS seminar, 12/04/2000
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CONCLUSIONS OF EXPERIMENT-2
The stability criterion for the damping of transverse head-tail
instabilities in the presence of linear coupling only has been
verified experimentally and compared to theory, leading to a
good agreement (to within a factor smaller than 2)
The CERN-PS beam for LHC (nominal or ultimate intensity)
CAN BE STABILISED using linear coupling only* (skew
quadrupoles and/or tune separation). Furthermore, this result
should be valid for “any” intensity (as concerns pure head-
tail instabilities)...
EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (6/6) EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (6/6)
* i.e. with neither octupoles nor feedbacks
Elias Metral, CERN-PS seminar, 12/04/2000
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OBSERVATIONS OF THE BENEFICIAL EFFECT OF LINEAR OBSERVATIONS OF THE BENEFICIAL EFFECT OF LINEAR COUPLING IN OTHER MACHINESCOUPLING IN OTHER MACHINES
LANL-PSR (from B. Macek)“Operating at or near the coupling resonance with a skew quad is one of the most effective means to damp our 'e-p' instability”
BNL-AGS (from T. Roser)“The injection setup at AGS is a tradeoff between a 'highly coupled' situation, associated with slow loss, and a 'lightly coupled' situation where the beam is unstable (coupled-bunch instability)”
CERN-SPS (from G. Arduini)“A TMCI in the vertical plane with lepton beams at 16 GeV is observed. Using skew quads ('just turning the knobs'), gains in intensity of about 20-30%, and a more stable beam, have been obtained”=> MDs are foreseen to examine these preliminary results in detail
CERN-LEP (from A. Verdier)“The TMCI in the vertical plane at 20 GeV sets the limit to the intensity per bunch. The operation people said that it's better to accumulate with tunes close to each other” => MDs are foreseen to examine these preliminary results in detail 1 vh QQ
845.8hQ 890.8vQ
62.26hQ 58.26vQ
28.98hQ 26.96vQ
Elias Metral, CERN-PS seminar, 12/04/2000
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CONCLUSIONCONCLUSION These results explain why many high intensity accelerators and colliders
work best close to a coupling resonance blablablabla and/or using skew
quadrupoles. They can be used to find optimum values for the transverse
tunes, the skew quadrupole and octupole currents, and the chromaticities (=>
sextupoles)
The CERN-PS beam for LHC can be stabilised by linear coupling only
Linear coupling is also used at BNL and LANL, and seems to be helpful in
SPS and LEP => See future MDs
Using this “simple” formalism, the following results are also obtained:
Coherent beam-beam modes => Decoupling the 2 beams by making the
tune difference much larger than the beam-beam parameter (A. Hofmann)
2-stream instabilities => Same stability criterion with negative coupling
(Laslett, Mohl and Sessler)
lQQ vh
ACK. : R. CAPPI AND D. MOHL, M. MARTINI AND THE OPERATION STAFF
THEIR IDEA !