introduction, observations and motivation theory experiments conclusion

Elias Metral, CERN-PS seminar, 12/04/2000

1

Introduction, observations and motivation Theory Experiments Conclusion

STABILISING INTENSE BEAMS

BY LINEAR COUPLING

Elias METRALElias METRAL


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


3

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


4

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


5


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


6


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


7


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


8

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


9


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 presence of frequency spreads

=> A tune spread of the order of the synchrotron tune is needed for stabilisation by Landau damping


10


00

40

42

0

,1,,

1, 4

ˆ

yx

ymmmy

xmmmx

RlKII

New 2D results In the absence of mode coupling only


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)


11


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


12


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)


13


(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


14


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


15


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


16

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


17


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 :


18


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


19


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

=>


20

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


21

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


1D case A33.0skewI See next slides

1m

Time (20 ns/div)


22

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)



23

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


Gev1cE


24

-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



25

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



26

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



27

-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



28

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



29

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


30

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


Observations 1D case A33.0skewI

6m


31

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)


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


32

Voir le file presentation 1



33

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


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


34

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)...


* i.e. with neither octupoles nor feedbacks


35

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


36

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 !

introduction, observations and motivation theory experiments conclusion

Documents

presence of mode coupling

absence of linear coupling

absence of frequency

singlebunch instability

sacherers formula single

coherent instability

dipolar mode type

quadrupolar mode type