Elias Métral, RLC meeting, 21/10/2005

1/16

E. MétralE. Métral Animation of the BBU theory (direction of propagation solved!)

Review of Laclare’s theory (azimuthal and radial modes mq)

Animation of the TMC theory (direction of propagation not solved!)

Coupling between modes -33 and -44 (bad direction!)

Coupling between modes -24 and -35 (good direction!)

Animation of SPS measurements

With a relative (measured) chromaticity of 0.14

With a relative (measured) chromaticity of 0.54

With a relative (measured) chromaticity of 2.04

Animation of the BBU theory for the PS compared to measurements

FOLLOW-UP OF THE FAST VERTICAL SINGLE-

BUNCH INSTABILITY AT SPS INJECTION

Elias Métral, RLC meeting, 21/10/2005

2/16

BBU THEORYBBU THEORY

1st trace = turn 1 Last trace = turn 115 Every turn shown

ns7.0t

/mM10 yR

GHz1rf

1Q

BB resonatorimpedance

Elias Métral, RLC meeting, 21/10/2005

3/16

Follow-up of last RLC meeting when I said that the spectrum of

mode mq (σmq) is peaked at and extend

2 modes with same q are peaked at the same frequency (same line

density), and therefore have ~ the same sensitivity in the same

frequency range. They however correspond to entirely different

patterns in phase space…

bq

qf

2

1

1 b

LACLARE’S FORMALISM (1/6)LACLARE’S FORMALISM (1/6)

Elias Métral, RLC meeting, 21/10/2005

4/16

LACLARE’S RESULTS – LACLARE’S RESULTS – CERN 87-03, p. 315 (1/2)CERN 87-03, p. 315 (1/2)

Q

not specified

and

kmq 2

k0with

22bb

2ˆ0 b

(2/6)(2/6)

Elias Métral, RLC meeting, 21/10/2005

5/16

(3/6)(3/6)

LACLARE’S RESULTS – LACLARE’S RESULTS – CERN 87-03, p. 316 (2/2)CERN 87-03, p. 316 (2/2)

Elias Métral, RLC meeting, 21/10/2005

6/16

MY RESULTS (1/2)MY RESULTS (1/2)

f

00

S00

X00

f

02

S02

X02

f

04

S04

X04

f

11

S11

X11

S00

S02

S04

S1110 b

13.xQ

0

revfQf

(4/6)(4/6)

Elias Métral, RLC meeting, 21/10/2005

7/16

MY RESULTS (2/2)MY RESULTS (2/2)

f

13

S13

X13

f

22

S22

X22

f

24

S24

X24

f

33

S33

X33

S13

S22

S24

S33

(5/6)(5/6)

Elias Métral, RLC meeting, 21/10/2005

8/16

I have the same results as Laclare (I use his formalism…) except

for σ02 , σ11 , σ33 , where I have an opposite sign

I also have an opposite sign for X02 , X11 , X33

However, it has no effect on the signal observed at a Pick-Up

And it has also no effect in the computation of the tune shift as it

is σmq2 which enters into the formula

LACLARE’S FORMALISM (6/6)LACLARE’S FORMALISM (6/6)

Elias Métral, RLC meeting, 21/10/2005

9/16

TMC THEORY: coupling between modes -33 and -44 TMC THEORY: coupling between modes -33 and -44

1st trace = turn 1 Last trace = turn 115 Every turn shown

ns8.24 tb

turns SPS 55MOSESTMC

Elias Métral, RLC meeting, 21/10/2005

10/16

TMC THEORY: coupling between modes -24 and -35 (1/2) TMC THEORY: coupling between modes -24 and -35 (1/2)

1st trace = turn 1 Last trace = turn 115 Every turn shown

ns8.24 tb

turns SPS 55MOSESTMC

Elias Métral, RLC meeting, 21/10/2005

11/16

I asked to Elena Shaposhnikova if one can identify which (radial)

modes couple: -2? with -3? (In her paper CERN/SL/93-43 (RFS) on

MOSES’s results, it is said that if coupling of mode m is important,

radial modes up to k ~ |m| / 2 have to be considered)

If k = 1 then it is the same case as the one of the previous page

/mM10 yR

GHz1rf

1Q

BB resonatorimpedance

p102.7

mA5.010

thb

thb

N

I

mA83.0

p102.1 11

b

b

I

N

turnsSPS552TMC

sT

9.0

mA5.0thbI

mA83.0bI

TMC THEORY: coupling between modes -24 and -35 (2/2) TMC THEORY: coupling between modes -24 and -35 (2/2)

Elias Métral, RLC meeting, 21/10/2005

12/16

1st trace (in red) = turn 2 Last trace = turn 150 Every turn shown

MEASUREMENTS WITH A MEASUREMENTS WITH A CHROMATICITY OF 0.14CHROMATICITY OF 0.14

Elias Métral, RLC meeting, 21/10/2005

13/16

MEASUREMENTS WITH A MEASUREMENTS WITH A CHROMATICITY OF 0.54CHROMATICITY OF 0.54

1st trace (in red) = turn 2 Last trace = turn 150 Every turn shown

Elias Métral, RLC meeting, 21/10/2005

14/16

MEASUREMENTS WITH A MEASUREMENTS WITH A CHROMATICITY OF 2.04CHROMATICITY OF 2.04

1st trace (in red) = turn 2 Last trace = turn 150 Every turn shown

Elias Métral, RLC meeting, 21/10/2005

15/16

BBU THEORY FOR THE PS AT TRANSITION (1/2)BBU THEORY FOR THE PS AT TRANSITION (1/2)

1st trace = turn 1 Last trace = turn 90 Every turn shown

ns5.7t

/mM3 yR

MHz700rf

2Q

BB resonatorimpedance

Elias Métral, RLC meeting, 21/10/2005

16/16

, R, V signals

Time (10 ns/div)~ 700 MHz

MEASUREMENTS IN 2000MEASUREMENTS IN 2000 BBU THEORY BBU THEORY AFTER 90 TURNS (~ 200 AFTER 90 TURNS (~ 200 μμss))

BBU THEORY FOR THE PS AT TRANSITION (2/2)BBU THEORY FOR THE PS AT TRANSITION (2/2)

Seem very close except head and tail exchanged!!! Does it start only

for the maximum peak intensity, i.e. in the middle???