STABILISING INTENSE BEAMS
BY LINEAR COUPLING
Elias METRAL




Introduction, observations and motivation
Theory
Experiments
Conclusion
Elias Metral, CERN-PS seminar, 12/04/2000
1
INTRODUCTION
Single-particle trajectory
One particle



Circular design orbit
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)
 Feedback systems
Elias Metral, CERN-PS seminar, 12/04/2000
2
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 :
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
3
THEORY (1/16)

A general formula for the transverse coherent instabilities with

Frequency spreads (due to octupoles)

Linear coupling (due to skew quadrupoles)
x-dispersion
integral
I
1
x ,m
 

 
x
m,m
x
m , m 1
Kˆ 0   l  R 2  02

2  y0
0
x-Sacherer’s
formula
I
1
x , m 1
x
m , m 1
 
x
m 1, m 1
Mode
coupling term
Kˆ 0  l  R 2  02

2  x0
0
Linear
coupling term
0
Kˆ 0  l  R 2  02

2  x0
0
I y,1m   my ,m
  my ,m 1
Kˆ 0   l  R 2  02

2  y0
 my ,m 1
I y,1m 1   my 1,m 1
Elias Metral, CERN-PS seminar, 12/04/2000
4
0
THEORY (2/16)
Uncorrelated distribution
functions (Averaging method)
Coherent frequency
to be determined
I x ,m 
xˆi    yˆ i   

xˆi  0
I y ,m 

yˆ i  0
xˆi    yˆ i   
 
xˆi  0
yˆ i  0
df x 0  xˆi  2
 2
xˆi f y 0  yˆ i  yˆ i
dxˆi
dxˆi dyˆ i
 c   x ,i  xˆi , yˆ i   m  s
2
df y 0  yˆ i  2
 2
yˆ i f x 0  xˆi  xˆi
dyˆ i
dxˆi dyˆ i
 c   y ,i  xˆi , yˆ i   l  0  m  s
2
Near the coupling resonance
Qh  Qv  l
Elias Metral, CERN-PS seminar, 12/04/2000
Kˆ 0  l 
is the lth Fourier coefficient of
the normalized skew gradient
5
THEORY (3/16)
Sacherer’s formula (singlem  ...,  1, 0,1, ... Head-tail modes
and coupled-bunch instabilities)
=> “low intensity” case
n  0,1, ..., M  1 Coupled-bunch modes
k 

x, y
m,m


m 1 
1
j e  Ib
2 m0  Qx 0, y 0  0 L
 Z  h
k 
x, y
k 

k 
Power spectrum
hm,m     
m 0
x, y
k
m,m


x, y
k
  x , y
hm ,m  kx , y   x , y


Pick-up (Beam Position Monitor) signal
-signal
-signal
m 1
m 0
m 1
m 2
  
Elias Metral, CERN-PS seminar, 12/04/2000
Time
One particular turn
Time
6
THEORY (4/16)

Kˆ 0  l   0
Let’s recover the 1D results

In the absence of - Linear coupling
- Mode coupling

mx ,m1  my ,m1  0
In the absence of frequency spreads
I x,1m  c   x 0  m s
Real coherent betatron frequency shift
=> Sacherer’s formula is recovered
Instability growth rate
c   x0  ms  mx ,m  U eqx  j Veqx
Motions
e
j c t
Veqx  0
=>
Instability
Elias Metral, CERN-PS seminar, 12/04/2000
These are the Laslett, Neil
and Sessler (LNS) coefficients
for coasting beams
7
THEORY (5/16)

In the presence of frequency spreads
(1) Lorentzian distribution
(2) Elliptical distribution
Underestimates Landau
Overestimates Landau
damping (sharp edges)
damping (infinite tails)
x
c
c
x
x : HWHH
 x0
1D criterion
x  Veqx
 x : HWB
 x,i
Keil-Zotter’s
stability criterion
Elias Metral, CERN-PS seminar, 12/04/2000
 x,i
 x0
Re

1D criterion

 x2  4U eq2 x  2Veqx
8
THEORY (6/16)

In the absence of linear coupling but in the presence of mode
coupling => “high intensity” case
I x,1m   mx ,m


I
1
x , m 1
 
x
m 1, m 1
0
In the absence of frequency spreads
=> Kohaupt’s stability criterion against Transverse Mode
Coupling Instability (TMCI) is recovered


x
m , m 1
  mx ,m 1
x, y
m , m 1
1

 s   mx ,y1,m 1   mx ,,ym
2
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
9

I
THEORY (7/16)
New 2D results

In the absence of mode coupling only
1
x ,m
  mx ,m
1
y ,m
2

  my ,m 
4  x0  y0
In the absence of frequency spreads

Kˆ 0  l  
I
Kˆ 0  l  R 4  04

2  Q x 0 Q y 0 Veqx Veqy

1/ 2
R 2 0
Necessary condition for stability
Kˆ 0  l 
Stable region
0


V

Stability criterion
(for each mode m)
nx  n y  l for
coupled-bunch modes
(and coasting
beams)
 Veqy   
2
eqx
2
0
 Q h  Qv  l 
2
  Veqx  Veqy 
Veqx  Veqy  0

1/ 2
Transfer of
growth rates
Stability criteria :
Full coupling?
No coupling
Qh  Qv  l
Elias Metral, CERN-PS seminar, 12/04/2000
Full coupling
Veqx  0 Veqy  0
Veqx  Veqy  0
10
C a,    1
THEORY (8/16)
for full coupling
=> Normalised coupling (or sharing) function
1
C  a,    1 
2
a
 1  4 a
1 4 a   
2
2
Kˆ 0  l  R 2 02

2  x 0  y 0 Veqx  Veqy
CCaa 
2

Veqx  Veqy
CCaa 
1
1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
0.5
 1
1
a
0
0.5
  0.25
Elias Metral, CERN-PS seminar, 12/04/2000
 4 2
 0 Qh  Qv  l
CCaa 
a

2 2
1
a
a
0
a
1
0.5
 0
11
a
THEORY (9/16)

In the presence of frequency spreads
(1) Lorentzian distribution
=> Same results with
Veqx , y
replaced by
Veqx , y  x , y
Stability criteria :
No coupling
x  Veqx  y  Veqy
Full coupling
x   y  Veqx  Veqy
Transfer of both instability growth rates and frequency
spreads (Landau damping)
Elias Metral, CERN-PS seminar, 12/04/2000
12
THEORY (10/16)

(2) Elliptical distribution
A particular case : No horizontal tune spread and no vertical wake field
Qh  Qv  l
Stable region
1
 y  2Vx
21/3
Qh  Qv  l

  Kˆ 0  l 
2

1  22 / 3 1  22 / 3


2
max
Elias Metral, CERN-PS seminar, 12/04/2000


3 Vx / 0
13
THEORY (11/16)

Approximate general stability criterion
1) Qh “far from” Qv  l => Transfer of growth rates only
Necessary condition Veqx  Veqy
2) Qh “near” Qv  l
Re
Kˆ 0  l  
 Q
x0

 x2  4U eq2 x   y2  4U eq2 y
 
Q y 0 Re
  4 U
2
x
2
eqx
  2V  Re 
eqx
  2 V
eqx
  4U
2
y
 Veqy 
2
eqy
  2V  
1/ 2
eqy
R 2 0
THE TUNE SEPARATION
Qh  Qv  l
THE ORDER OF MAGNITUDE OF
SHOULD BE SMALLER THAN
 
x
  y  /  0
IN ORDER
TO HAVE THE TRANSFER OF LANDAU DAMPING
Elias Metral, CERN-PS seminar, 12/04/2000
0
14
THEORY (12/16)
On the coupling resonance
H-plane
Qh
Transfer of frequency spread (to Landau damp
Veqx )
V-plane
Qv  l
“One plane is stabilised by Landau damping and the other
one is stabilised by coupling”
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)
U  V

 y  U y
Elias Metral, CERN-PS seminar, 12/04/2000
15
THEORY (13/16)

In the presence of both mode coupling and linear coupling,
neglecting frequency spreads
  
  
c
c
x ,m
y ,m
 
 

 
x
c   x , m 1     m , m 1 
y
c   y , m 1     m , m 1
2
2
Kˆ 0  l  R 4  04
2
4  x0  y0

   c   x ,m    c   y ,m     c   x ,m 1    c   y ,m 1  


2
 Kˆ 0  l  R 4  04

x
y


 2  m ,m 1  m ,m 1
4  x0  y0


 x,m   x0  ms  mx ,m
 y ,m   y 0  l 0  m s  my ,m
=> Necessary condition for stability
 mx ,m 1   my ,m 1 
1
2  s   mx 1,m 1   my 1,m 1   mx ,m   my ,m
2
Elias Metral, CERN-PS seminar, 12/04/2000
16
THEORY (14/16)
Example :

Im Z xBB
,y

h-1,-1

Re Z xBB
,y
h0,0

h-1,-1
 [rad/s]
 rBB
=> Computed gain in intensity of about 50% for the classical ratio of
factor 2 between the transverse sizes of the vacuum chamber
Elias Metral, CERN-PS seminar, 12/04/2000
17
THEORY (15/16)
SHARING OF DAMPING BY FEEDBACKS
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
Electronics

Pick-up
0
Beam
Kicker
1
The stabilising effect of feedbacks can be introduced in the coefficient
Veq
 Its damping effect in one plane, can also be transferred to the other
plane using coupling
Elias Metral, CERN-PS seminar, 12/04/2000
18
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
“Chromaticity sharing” (for Sacherer’s formula)

Transfer of Landau damping for “optimum” coupling
=> Linear coupling is an additional (3rd) method that can be used to
damp transverse coherent instabilities
Elias Metral, CERN-PS seminar, 12/04/2000
19

EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (1/9)
Experimental conditions

High intensity bunched proton beam

1.2 s long flat bottom at injection kinetic energy
Re Z
RW
c
 10
5
 /m


2

- 400
400
-2
-400
M  20

hm,m  

  10 6 rad / s 
2
m 1

I beam  1.5 1013
Ec  1 Gev
Sacherer’s formula
=> coupled-bunch instabilities
Coupled-bunch modes n x , y  13
Most critical head-tail mode
number
m  1 for the horizontal plane
Veqmx1  Veqmy1  0
1
121 s-1
- 40 s-1
 Landau damping is needed
- 400
0
400
Elias Metral, CERN-PS seminar, 12/04/2000
20
EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (2/9)

Observations
1D case 
I skew  0.33 A
See next slides
Spectrum Analyzer
Beam-Position Monitor
(zero frequency span)
(20 revolutions superimposed)
R signal
10 dB/div
Center 360 kHz
RES BW 10 kHz
VBW 3 kHz SWP 1.2 s
Time
rise time  10 ms
One particular turn
Elias Metral, CERN-PS seminar, 12/04/2000
Time (20 ns/div)
m 1
21
EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (3/9)
MEASUREMENT OF THE CERN-PS LINEAR COUPLING



In the PS

Coupling resonance
No solenoid
Qh  Qv  0
In the presence of linear coupling between the transverse planes,
the difference from the tunes of the 2 normal modes is given by
Q  Q 
 Qh  Qv 
2
 CG
2
Guignard’s coupling coefficient
It is obtained from the general formula (in the smooth
approximation used to study instabilities)

Measurement method : For different skew quadrupole currents,
we increase Qh and decrease Qv 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
Elias Metral, CERN-PS seminar, 12/04/2000
22
EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (4/9)


Coupling measurements from mode frequencies by FFT analysis
M  20 I beam  2501010

Low intensity bunched proton beam

1.2 s long flat bottom at injection kinetic energy
“Mountain range” display for the “natural” coupling
Time
Ec  1 Gev
I skew  0 A
FFT Analyzer
 Qv  Qh 
CG  f 0
Frequency
Elias Metral, CERN-PS seminar, 12/04/2000
23
EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (5/9)
=> Modulus of the normalised skew gradient vs. skew quadrupole current

K 0 8105
 m 
2
6
4
2
0
-1.5
-1
-0.5
0
0.5
1
1.5
I skew  A
I skew  0.33 A  0.1A
Elias Metral, CERN-PS seminar, 12/04/2000
24
EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (6/9)

Stabilisation by Landau damping (1D case)

Simplified (elliptical) stability criterion : Keil-Zotter’s criterion
HWHH
 x,y
 3  mx ,,ym

Theoretical frequency spread required
 xHWHH  rad/s   320 I oct  A 
 yHWHH  rad/s   630 I oct  A
 xHWHH  3400 rad/s

Experimental frequency spread required
 xHWHH  1100 rad/s
I oct  3.5 A
 This is less than required by the theory by a factor 3
(without taking into account space-charge non-linearities...)
Elias Metral, CERN-PS seminar, 12/04/2000
25
EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (7/9)

Stabilisation by coupled Landau damping (2D)


K 0 10
5
 m 
Qv  6.22
Qh  6.14
Constant tune separation
2
K0
Measurement
Theory
(Lorentzian vertical distribution)
exp
/ K0
theory
0.7
1.1
1.2
1
0.5
0.3
10
5
0
0
0.5
1
I oct  A
1.5
Elias Metral, CERN-PS seminar, 12/04/2000
2
2.5
3
3.5
26
EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (8/9)

Constant octupole strength
I oct  2 A
Qh  Qv
Qv  Qh
exp
/ Qh  Qv
theory
0.8
1.2
3
Measurement
Theory
(Lorentzian vertical distribution)
I skew  A
Elias Metral, CERN-PS seminar, 12/04/2000
27
EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (9/9)
CONCLUSIONS OF EXPERIMENT-1

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 nonlinearities in the normal modes
Elias Metral, CERN-PS seminar, 12/04/2000
28
EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (1/6)

Single bunch of protons with nominal intensity
N b  1012

1.2 s long flat bottom at injection kinetic energy

Bunch length  b
Ec  1.4 Gev

Transverse tunes
Qh  6.18 Qv  6.21

Transverse chromaticities
 x  0.9  y  1.3
 160 ns
Growth rates [s-1]
50
unstable
0
stable
-50
Sacherer’s formula =>
-100
-150
Horizontal
Vertical
-200
-250
0
1
Elias Metral, CERN-PS seminar, 12/04/2000
2
3
4
5
6
7
8
9
10
Head-tail mode number m
29
EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (2/6)

Observations
1D case 
I skew  0.33 A
Spectrum Analyzer
Beam-Position Monitor
(zero frequency span)
(20 revolutions superimposed)
R signal
10 dB/div
Center 355 kHz
RES BW 10 kHz
VBW 3 kHz SWP 1.2 s
Time
rise time  30 ms
Elias Metral, CERN-PS seminar, 12/04/2000
Time (20 ns/div)
m 6
30
EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (3/6)

since
Stabilisation by linear coupling only
I skew [A]
K0
exp
(10 5 ) [m  2 ]
0.73
-0.07
theory
1.7
1.7
 ~ no emittance blow-up
 The
K0
(10 5 ) [m  2 ]
K0
exp

norm,1
x

  ynorm,1 / 2 1.9 m
I skew  0.02 A

K 0 105

norm,1
y
 m 
2
 / 2  3.2 m
 3 m
(limit)
N b 1.8 1012
7
6
Measurement
5
Theory
4
3
 3 m
2
Qv  Qh
1
but ~ no blow-up in the PS
Elias Metral, CERN-PS seminar, 12/04/2000
theory
1.7
1.7
 ~ no emittance blow-up
norm,1
x
/ K0
1
1
~ same results are obtained for the ultimate beam
I skew  0.68 A

Veqmx6  Veqmy6  0
0
-0.1
-0.05
0
0.05
0.1
0.15
31
EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (4/6)

Voir le file presentation 1
Elias Metral, CERN-PS seminar, 12/04/2000
32
EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (5/6)

N b  1012
8 bunches of protons with nominal intensity
 coupled-bunch instabilities
Growth rates [s-1]
Theoretical stabilising
skew gradient

50
unstable
0
stable
K0
theory
 4.3 10 5 m  2
-50
-100

I skew   0.75 A
-150
or
I skew 1.4 A
-250
Horizontal
Vertical
-200
0
1
2
3
4
5
6
7
8
9
10
Head-tail mode number m
 The
~ same results are obtained for the ultimate beam
Elias Metral, CERN-PS seminar, 12/04/2000
N b 1.8 1012
33
EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (6/6)
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 headtail instabilities)...
* i.e. with neither octupoles
nor feedbacks
Elias Metral, CERN-PS seminar, 12/04/2000
34
OBSERVATIONS OF THE BENEFICIAL EFFECT OF LINEAR
COUPLING IN OTHER MACHINES
 LANL-PSR (from B. Macek)
“Operating at or near the coupling resonance Qh  Qv  1 with a skew
quad is one of the most effective means to damp our 'e-p' instability”
Qh  8.845 Qv  8.890
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)”
Qh  26.62 Qv  26.58
 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
Qh  98.28 Qv  96.26
 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
Elias Metral, CERN-PS seminar, 12/04/2000
35
CONCLUSION
 These results explain why many high intensity accelerators and
colliders work best close to a coupling resonance blablablabla
Qh  Qv  l
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)
THEIR IDEA !
ACK. : R. CAPPI AND D. MOHL, M. MARTINI AND THE OPERATION STAFF
Elias Metral, CERN-PS seminar, 12/04/2000
36