Instability rise-time far above the TMCI
threshold: 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-depabp/LIS/Minutes/2008/20081201/metral1.pdf
• Interesting for instance near g transition, crossing (PS, RHIC) or
isochronous rings (n-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
E. Metral, B. Salvant, G. Rumolo, …
Ith=0.5mA
Nb~7.2 1010
parameters
SPS beam @
26GeV
BB resonator:
1GHz
10 MW/m
Q=1
The instability seen by HEADTAIL is therefore clearly a TMCI!
E.Benedetto, GSI collaboration meeting 19-2-09
The two codes
MOSES
HEADTAIL
(Y.H. Chin, CERN-LEP-Div-Rep-88-005-TH)
(G. Rumolo, F. Zimmermann, SL-Note 2002-036-AP,
CERN 2002)
• It solves Sacherer integrals
• Macroparticle simulations, the
bunch is sliced and interacts
slice-by-slice with the wake-fields.
• Doesn’t know anything about
TMCI or modes
• Mode shifts and coupling due to
the interaction of a bunch with an
impedance (BB resonator)
• It has been developed for the
TMCI
Localized impedance
source
Courtesy G.Rumolo
E.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
Ts
MOSES
• Risetime  TMC 
  Im
  0 x
vs. Ib
s
2 
Linear
Nonlinear
Infinite rise-time
E.Metral, LIS meeting 1/12/08
E.Benedetto, GSI collaboration meeting 19-2-09
MOSES
I bth  0.5 mA
parameters
SPS beam @
26GeV
BB resonator:
1GHz
10 MW/m
Q=1
E.Metral, LIS meeting 1/12/08
E.Benedetto, GSI collaboration meeting 19-2-09
MOSES
 1  18.5
I bth  0.5 mA
I b1  10 mA
parameters
SPS beam @
26GeV
BB resonator:
1GHz
10 MW/m
Q=1
E.Metral, LIS meeting 1/12/08
E.Benedetto, GSI collaboration meeting 19-2-09
MOSES
 1  18.5
I b1  10 mA
I bth  0.5 mA
 2  185
parameters
SPS beam @
26GeV
BB resonator:
1GHz
10 MW/m
Q=1
E.Metral, LIS meeting 1/12/08
I b 2  100 mA
E.Benedetto, GSI collaboration meeting 19-2-09
MOSES
 1  18.5
I b1  10 mA
I bth  0.5 mA
 2  185
parameters
SPS beam @
26GeV
BB resonator:
1GHz
10 MW/m
Q=1
E.Metral, LIS meeting 1/12/08
2
185

 10
 1 18.5
Ib2
 10
I b1

I b 2  100 mA
MOSES
TMC

Ts
2 

Ts
2 Ib
E.Benedetto, GSI collaboration meeting 19-2-09
Simple TMC model with the 2 most
critical modes
Ts
sm
 TMC


with q  [ 0 , 1 ]
q  0 for short bunch, i.e. 2 f r  b  1
q  1 for long bunch, i.e. 2 f r  b  1
I b  I bth and long bunch
Furthermore
E.Metral, LIS meeting 1/12/08
1
I 
Ts
th
b
 Ib
 I

 th  1   thb q  1 
 Ib
  Ib


sm
 TMC
I bth
 
 Ib
Ts
is independent of synchrotron motion as
sm

 TMC
could be anticipated (as the instability
rise-time
is
much
faster
than
synchrotron period)
E.Benedetto, GSI collaboration meeting 19-2-09
HEADTAIL
Nb=0.2 1012
Nb=0.2 1012
• Instability risetime computed by exponential fit over
the horizontal centroid amplitude growth:
1 
ˆx(t )  A exp  t 
 
1e-3<x<10m
E.Benedetto, GSI collaboration meeting 19-2-09
parameters
SPS beam @
26GeV
BB resonator:
1GHz
10 MW/m
Q=1
HEADTAIL
Nb=0.2 1012
Nb=1.0 1012
Ith=0.5mA
Nb,th=~7.2 1010
Qs=10-3
synchr motion OFF
Qs=10-3
synchr motion OFF
•  does not depend on Qs
•  is inversely proportional to Nb
E.Benedetto, GSI collaboration meeting 19-2-09
parameters
SPS beam @
26GeV
HEADTAIL
BB resonator:
1GHz
10 MW/m
Q=1
Ith=0.5mA
Nb,th=~7.2 1010
 (ms)
1 kick/turn
10 kicks/turn
100 kicks/turn
( x Nb)
1 kick/turn
10 kicks/turn
100 kicks/turn
E.Benedetto, GSI collaboration meeting 19-2-09
Some numerical values
• Let’s consider I=100mA
• MOSES:
  185  
MOSES
TMC
Ts
0.0071


 6.1 μs
2   2  185
• Simple TMC model (2 most critical modes)
sm
I bth  0.5 mA   TMC
• HEADTAIL:
I bth 0.0071 0.5
 


 11.3 μs
 Ib

100
Ts
1 
xˆ (t )  A exp  t     (4.4  6.4)μs
 
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 risetime far above threshold…
…since MOSES and HEADTAIL are in very good agreement
for TMCI
doesn’t know TMCI
• 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
sm
 TMC
I bth
 
 Ib
Ts
• The comparison was made for SPS “short” bunches. What
happens for “long” bunches (PS, n-factory proton driver, …)?
E.Benedetto, GSI collaboration meeting 19-2-09