STABILITY OF THE LONGITUDINAL BUNCHED-BEAM COHERENT MODES

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


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


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


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)


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


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


7


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


8


sm 2

2

m

m

S

lcmm

Sm s

0Re

S

lcmm

Sm

mm s 1

2

m

m

S

lcmm


9


• 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


10


• 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


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


12


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


13


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


14


Dipole mode

0

ˆ

ˆ91

211

11

011011

RF

Tlc

is

lc

ssscsc

V

V

Neglecting the synchrotron frequency spread

011 sc


15


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


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


17


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


18


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)


19


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


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


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


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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

Documents

STABILITY OF THE LONGITUDINAL BUNCHED-BEAM COHERENT MODES