Instability rise-time far above the TMCIthreshold: Comparison between simple

theory, MOSES and HEADTAIL

E. Benedetto, E. Metral

Acknowledgements: G. Rumolo, D. Quatraro, B. Salvant

(CERN)

19/2/09

CERN/GSI beam dynamics and collective effects collaboration meeting

Outline

• Motivation• TMC theory to compute rise-time far above threshold• Simple TMC model, MOSES, HEADTAIL:

– Qualitative– Quantitative

• Conclusions and discussion

E.Benedetto, GSI collaboration meeting 19-2-09

Transverse Instability for high-intensity single-bunch beams

• In the past, studies have been done for what concerns finding the instability threshold

• Different approaches:– Beam Break-up

– TMC theory

– Coasting beam with peak value

– post Head-Tail

– fast blow-up

Unified the different approaches and formalisms to compute instability threshold

→ E.Metral, 2004

E.Benedetto, GSI collaboration meeting 19-2-09

Transverse Instability for high-intensity single-bunch beams

• Next step:

• for intensities far above the TMCI intensity threshold

• i.e. instability risetime much faster then synchrotron period

• How to evaluate the risetime?

Can we still use the concept of modes and modes coupling?

→ Follow-up discussion with W. Fisher and G. Rumolo at the CARE-HHH workshop (24-25/11/08, Chavannes-de-Bogis)→ E.Metral, LIS meeting 1/12/08, https://ab-dep-abp.web.cern.ch/ab-dep-abp/LIS/Minutes/2008/20081201/metral1.pdf

• Interesting for instance near transition, crossing (PS, RHIC) or isochronous rings (-factory proton driver accumulator)

E.Benedetto, GSI collaboration meeting 19-2-09

TMC theory and intensity threshold

• Comparison HEADTAIL vs. MOSES approaching Ith

• Very good agreement between the 2 codes for what concerns mode shifts and instability threshold

The instability seen by HEADTAIL is therefore clearly a TMCI!

parameters SPS beam @26GeV

BB resonator:1GHz10 M/mQ=1

E. Metral, B. Salvant, G. Rumolo, …

Ith=0.5mANb~7.2 1010

E.Benedetto, GSI collaboration meeting 19-2-09

The two codes

MOSES(Y.H. Chin, CERN-LEP-Div-Rep-88-005-TH)

• It solves Sacherer integrals

• Mode shifts and coupling due to the interaction of a bunch with an impedance (BB resonator)

• It has been developed for the TMCI

HEADTAIL(G. Rumolo, F. Zimmermann, SL-Note 2002-036-AP,

CERN 2002)

• Macroparticle simulations, the bunch is sliced and interacts slice-by-slice with the wake-fields.

• Doesn’t know anything about TMCI or modes

Localized impedance source

Courtesy G.RumoloE.Benedetto, GSI collaboration meeting 19-2-09

TMC theory and intensity threshold

• Extension of TMCI theory far above TMCI threshold

• Comparison theory - HEADTAIL – MOSES for I>>Ith

Courtesy B. Salvant

MOSES

• Imaginary part of the modes shift vs. Ib

• Risetime

Nonlinear

Infinite rise-time

2MOSESTMC

sTs

x

0Im

Linear

E.Metral, LIS meeting 1/12/08 E.Benedetto, GSI collaboration meeting 19-2-09

mA5.0thbI

E.Metral, LIS meeting 1/12/08

parameters SPS beam @26GeV

BB resonator:1GHz10 M/mQ=1

MOSES

E.Benedetto, GSI collaboration meeting 19-2-09

mA5.0thbI

5.181

mA101 bI

E.Metral, LIS meeting 1/12/08

parameters SPS beam @26GeV

BB resonator:1GHz10 M/mQ=1

MOSES

E.Benedetto, GSI collaboration meeting 19-2-09

1852

mA1002 bI

mA5.0thbI

5.181

mA101 bI

E.Metral, LIS meeting 1/12/08

parameters SPS beam @26GeV

BB resonator:1GHz10 M/mQ=1

MOSES

E.Benedetto, GSI collaboration meeting 19-2-09

1852

mA1002 bI

mA5.0thbI

105.18

185

1

2

101

2 b

b

I

I

b

ss

I

TT

22MOSESTMC

5.181

mA101 bI

E.Metral, LIS meeting 1/12/08

parameters SPS beam @26GeV

BB resonator:1GHz10 M/mQ=1

MOSES

E.Benedetto, GSI collaboration meeting 19-2-09

is independent of synchrotron motion as

could be anticipated (as the instability

rise-time is much faster than

synchrotron period)

11

TMC

qII

II

T

thb

bthb

b

ssm

with ]1,0[q1 2 i.e. bunch,shortfor0 brfq 1 2 i.e. bunch,longfor1 brfq

b

thbssm

I

IT

TMCbunch long and thbb II

Furthermore

s

thb TI

1 sm

TMC

Simple TMC model with the 2 most critical modes

E.Metral, LIS meeting 1/12/08 E.Benedetto, GSI collaboration meeting 19-2-09

HEADTAIL

• Instability risetime computed by exponential fit over the horizontal centroid amplitude growth:

tAtx1

exp)(ˆ 1e-3<x<10m

Nb=0.2 1012 Nb=0.2 1012

E.Benedetto, GSI collaboration meeting 19-2-09

HEADTAIL

• does not depend on Qs

• is inversely proportional to Nb

Nb=0.2 1012 Nb=1.0 1012

parameters SPS beam @26GeV

BB resonator:1GHz10 M/mQ=1

Ith=0.5mANb,th=~7.2 1010

Qs=10-3

synchr motion OFF

Qs=10-3

synchr motion OFF

E.Benedetto, GSI collaboration meeting 19-2-09

HEADTAIL

( x Nb)

1 kick/turn 10 kicks/turn 100 kicks/turn

s

1 kick/turn 10 kicks/turn 100 kicks/turn

parameters SPS beam @26GeV

BB resonator:1GHz10 M/mQ=1

Ith=0.5mANb,th=~7.2 1010

E.Benedetto, GSI collaboration meeting 19-2-09

Some numerical values

• Let’s consider I=100mA

• MOSES:

• Simple TMC model (2 most critical modes)

• HEADTAIL:

μs 1.61852

0071.0

2 185 MOSES

TMC

sT

μs 3.11100

5.00071.0 mA 0.5 TMC

b

thbssmth

b I

ITI

μs)4.64.4(1

exp)(ˆ

tAtx

E.Benedetto, GSI collaboration meeting 19-2-09

Conclusion• Answer to the question (of W.Fisher and others) is:

Yes! We can still use the concept of modes and modes coupling to deduce the rise-time far above threshold…

…since MOSES and HEADTAIL are in very good agreement

• Far above threshshold a simple formula, (TMC model with only the 2 most critical modes) gives good approx:– independent of Ts (as expected)

– proportional to 1/Ib

• The comparison was made for SPS “short” bunches. What happens for “long” bunches (PS, -factory proton driver, …)?

b

thbssm

I

IT

TMC

for TMCI doesn’t know TMCI

E.Benedetto, GSI collaboration meeting 19-2-09