Elias Metral, LCE meeting, 31/01/2003 1

Stability of the longitudinal coherent modes

• General stability criterion

• Application to the LHC at top energy

Elias Metral, LCE meeting, 31/01/2003 2

General stability criterion (1/6)

bss I 0

bss I 0 bTRF IVV ˆˆ

bIBB 0

• Stationary distribution

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

Potential well distortion

• Perturbation (around the new fixed point)

Dispersion relation

Sacherer formulaDispersion integral

Synchronous phase shift

Elias Metral, LCE meeting, 31/01/2003 3

General stability criterion (2/6)

Distribution of the normalised synchrotron amplitudes

220 1 rrg

-1 -0.5 0.5 1

0.2

0.4

0.6

0.8

1

Stability diagram

S

lcmm

Im

-1 -0.5 0 0.5 1

-0.8

-0.6

-0.4

-0.2

S

lcmm

Re

1m

54

2

3

Elias Metral, LCE meeting, 31/01/2003 4

General stability criterion (3/6) The stability boundary can be approximated by semicircles

(following the example of Keil-Schnell for coasting beams)

lcmm

mS

4

sBhS 22

16

is the approximated full spread between centre and edge of the bunch on a flat-top

0.5 1 1.5 2 2.5 3

0.2

0.4

0.6

0.8

1 0

ˆ

s

s

rad̂

Sacherer stability criterion

Elias Metral, LCE meeting, 31/01/2003 5

General stability criterion (4/6)

Dipole mode

011

011011

is

lc

ssscsc

eff

l

sRF

sbis p

pZ

BhV

I

00032

0 Imcosˆ2

3

is the incoherent small-amplitude frequency shift

Quadrupole mode

is

is

lc

ssscsc

2

1

2

222

22

022022

At low intensity and neglecting synchrotron frequency spread

Elias Metral, LCE meeting, 31/01/2003 6

General stability criterion (5/6) Stability of the dipole mode

lcS 114

is

lc 11

isS 4

eff

l

sTb

ppZ

BVhI

11

532 cosˆ

32

3

Stability criterion discussed during the last LCE meeting

lcS 114

with

bs I

bT IV̂

bIB

Elias Metral, LCE meeting, 31/01/2003 7

General stability criterion (6/6)

F

ppZ

BVhI eff

l

sRFthb

11

500

32 cosˆ

32

3

with baaF 42

1 2

eff

l

eff

l

s

p

pZ

p

pZj

SgnBh

a

11

000

20

2

cos32

9

s

sb

cos

cos 0

Elias Metral, LCE meeting, 31/01/2003 8

Application to the LHC at top energy

4.1F

• 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 (0.28 for 00 and 11, …)

p/b104.3 11thbN

1F

F 11104.2 with

Value found by BBV