1. W. Bartmann, M. Benedikt, E. Métral, D. Möhl, G. Rumolo and B. Salvant
TRANSITION CROSSING
REQUIREMENTS FOR PS2
W. Bartmann, M. Benedikt, E. Métral, D. Möhl, G. Rumolo and B. Salvant
Space Charge
Broad-Band
Introduction with the case of the PS
Equations in the case with NO SC (or BB imp.) and NO t jump re-derived and solved analytically + HEADTAIL simulations
Equations in the case with both SC (and/or BB imp.) and t jump re-derived and solved numerically + some HEADTAIL simulations
Application (and comparison with Dieter’s theory of 1969!  CERN-ISR/300/GS/69-62) to the cases below:
nTOF in the present PS and in (PS2, 10 MHz)
FT in (PS2, 10 MHz) and (PS2, 40 MHz)
LHC in (PS2, 10 MHz) and (PS2, 40 MHz)
Conclusions and outlooks + proposed solution + work missing for the review
Elias Métral, PS2 WG meeting, 15/05/2008

2. Introduction with the case of the PS (1/9)
Thanks to M. Martini for the data! 
t =
in tjump = 500 s (see also next slides)
Elias Métral, PS2 WG meeting, 15/05/2008

3. Introduction with the case of the PS (2/9)
Zoom
Zoom
Elias Métral, PS2 WG meeting, 15/05/2008

4. Introduction with the case of the PS (3/9)
Courtesy of M. Martini
Elias Métral, PS2 WG meeting, 15/05/2008

5. Introduction with the case of the PS (4/9)
 Currently, the nTOF bunch is the most critical at transition due to a TMCI which develops near transition crossing
Simulation by Giovanni with HEADTAIL (ICAP06)
Without
space charge
Measurements in 2000
With
space charge
 > 2.1 eVs are needed for ~ 71012 p/b
(PS/RF Note )
Elias Métral, PS2 WG meeting, 15/05/2008

6. Introduction with the case of the PS (5/9)
BB only
Introduction with the case of the PS (5/9)
HEADTAIL SIMULATION
Flat chamber
Head
Tail
- Constant energy
- Bunch in the centre of the PU
1st trace = turn 1
Last trace = turn 130
Every turn shown
Elias Métral, PS2 WG meeting, 15/05/2008

7. Introduction with the case of the PS (6/9)
Δz  120 cm
 f  250 MHz
Δz  40 cm
 f  750 MHz
HEADTAIL SIMULATION
Turn 130 shown
Elias Métral, PS2 WG meeting, 15/05/2008

8. Introduction with the case of the PS (7/9)
BB + SC
Introduction with the case of the PS (7/9)
HEADTAIL SIMULATION
Flat chamber
Head
Tail
1st trace = turn 1
Last trace = turn 200
Every turn shown
Elias Métral, PS2 WG meeting, 15/05/2008

9. Introduction with the case of the PS (8/9)
 Measurements from Rende in 2007 on the AD beam with low longitudinal emittance (no precise beam parameters)
Elias Métral, PS2 WG meeting, 15/05/2008

10. Introduction with the case of the PS (9/9)
nTOF
10 MHz FT
LHC
40 MHz
R [m]
100
214.3
 [m] 70
Bdot [T/s]
2.2
1.5
Vrf [kV]
200
500
1500 h 8 15 45
180 p
0.027
0.0076
L [eVs]
2/2.3/2.5
2.5
0.4
0.6
Nb [1010 p/b]
800
1000
320
190 80 48
t (1, norm.) [m] 5 6
t average [m]
16*
15*
Pipe [cm  cm]
3.5 / 7 Qt
6.25
11.25**
* R / Qt
Elias Métral, PS2 WG meeting, 15/05/2008
** Assumption: Qt ~ t = 1 /  p = 11.47

11. is tilted near transition
General analytical result for the nonadiabatic transition region (with neither SC, or BB imp., nor t jump)
 The bunch
is tilted near transition
Transition
Nonadiabatic time
 Same picture as the one obtained by K.Y. Ng in his book “Physics of Intensity Dependent Beam Instabilities” (2006), p. 707
Elias Métral, PS2 WG meeting, 15/05/2008

12. … Confirmed by HEADTAIL simulations
Residual mismatch
Elias Métral, PS2 WG meeting, 15/05/2008

13. Case of the nTOF bunch in the PS WITHOUT space charge (1/3)
Tc  1.9 ms
Elias Métral, PS2 WG meeting, 15/05/2008

14. Case of the nTOF bunch in the PS WITHOUT space charge (2/3)
Analytical result for the evolution of the longitudinal ellipse when crossing transition
Elias Métral, PS2 WG meeting, 15/05/2008

15. Case of the nTOF bunch in the PS WITHOUT space charge (3/3)
… Confirmed by HEADTAIL simulations
Only the blue particles are displayed to assess the tilt of the ellipse
Elias Métral, PS2 WG meeting, 15/05/2008

16. Case of the nTOF bunch in the PS WITH space charge (1/5)
STATIC (MATCHED) CASE
Elias Métral, PS2 WG meeting, 15/05/2008

17. Case of the nTOF bunch in the PS WITH space charge (2/5)
DYNAMIC (“REAL”) CASE WHEN TRANSITION IS CROSSED
Elias Métral, PS2 WG meeting, 15/05/2008

18. Case of the nTOF bunch in the PS WITH space charge (3/5)
DYNAMIC (“REAL”) CASE WHEN TRANSITION IS CROSSED
Elias Métral, PS2 WG meeting, 15/05/2008

19. Case of the nTOF bunch in the PS WITH space charge (4/5)
EVOLUTION OF THE DYNAMIC CASE WHEN SPACE CHARGE IS INCREASED THROUGH INTENSITY
Elias Métral, PS2 WG meeting, 15/05/2008

20. Case of the nTOF bunch in the PS WITH space charge (5/5)
EVOLUTION OF THE DYNAMIC CASE WHEN SPACE CHARGE IS DECREASED THROUGH LONGITUDINAL EMITTANCE
Elias Métral, PS2 WG meeting, 15/05/2008

21. (Asymmetric) t jump for nTOF in the PS (considering SC and TMCI)
Tc  1.9 ms
Elias Métral, PS2 WG meeting, 15/05/2008

22. Summary table for PS and PS2 (considering SC and TMCI)
nTOF
(2 eVs)
(2.3 eVs)
(2.5 eVs)
(10 MHz) FT
LHC
(40 MHz)
Tc [ms]
1.86
4.29
2.97
1.27 g0
3.85
3.73
4.6
SC imp. []
 j
 j
 j
t SC from x = - 2
-0.72
-0.65
-0.61
-1.50
-0.88
-0.79
-0.50
-0.34
t with SC & TMCI
-1.58
-1.35
-1.23
-6.85
-2.91
-2.97
-2.68
-1.08
It is the t of the present PS
= if PS2 BB imp. ~ 6 times smaller than PS
Elias Métral, PS2 WG meeting, 15/05/2008

23.  In this case, the asymmetry of the t jump is in the other direction
Case of the nTOF bunch in the PS with longitudinal inductive BB impedance (neglecting SC)
 In this case, the asymmetry of the t jump is in the other direction
Elias Métral, PS2 WG meeting, 15/05/2008

24. Case of the nTOF bunch in the PS with both longitudinal inductive BB impedance and SC
 Caution: Here the long. BB impedance cancels the SC for the bunch length matching but will lead to a more critical long. microwave instability!
Elias Métral, PS2 WG meeting, 15/05/2008

25. Case of the nTOF bunch in the PS with both longitudinal inductive BB impedance and SC and 2 times bigger transverse emittance
Elias Métral, PS2 WG meeting, 15/05/2008

26. Longitudinal negative mass (SC) instability rise time (1/4)
PS nTOF (2 eVs)
Using coasting-beam formalism with peak values
(constant energy)
Keil-Schnell
Elias Métral, PS2 WG meeting, 15/05/2008

27. Longitudinal negative mass (SC) instability rise time (2/4)
PS nTOF (2 eVs)
PS nTOF (2.5 eVs)
PS2 nTOF (10 MHz)
PS2 nTOF (10 MHz)
Elias Métral, PS2 WG meeting, 15/05/2008

28. Longitudinal negative mass (SC) instability rise time (3/4)
PS2 LHC (10 MHz)
PS2 FT (10 MHz)
PS2 LHC (40 MHz)
PS2 FT (40 MHz)
Elias Métral, PS2 WG meeting, 15/05/2008

29. Longitudinal negative mass (SC) instability rise time (4/4)
PS2 LHC (10 MHz)
PS2 FT (10 MHz)
PS2 LHC (40 MHz)
PS2 FT (40 MHz)
Elias Métral, PS2 WG meeting, 15/05/2008

30. Vertical microwave (BB) instability rise-time (1/4)
PS nTOF (2 eVs)
Using coasting-beam formalism with peak values
(constant energy)
Keil-Schnell (transverse)
HEADTAIL simulations
Elias Métral, PS2 WG meeting, 15/05/2008

31. Vertical microwave (BB) instability rise-time (2/4)
th (Coasting-beam formalism with peak values )  12.5  10-3
More on
PS nTOF (2 eVs) ?
HEADTAIL simulations  th (HEADTAIL)  8.5  10-3
th (TMCI)  6  This formula was used to obtain the summary table of page 22
 Slightly optimistic!
Elias Métral, PS2 WG meeting, 15/05/2008

32. Vertical microwave (BB) instability rise-time (3/4)
PS nTOF (2 eVs)
PS nTOF (2.5 eVs)
PS2 nTOF (10 MHz)
Elias Métral, PS2 WG meeting, 15/05/2008

33. Vertical microwave (BB) instability rise-time (4/4)
PS2 LHC (10 MHz)
PS2 FT (10 MHz)
PS2 LHC (40 MHz)
PS2 FT (40 MHz)
Elias Métral, PS2 WG meeting, 15/05/2008

34. Conclusions and outlooks (1/7)
Transition crossing with a t jump looks possible in PS2 for the densities foreseen with the FT and LHC beams
It its more difficult for conditions corresponding to the present nTOF bunch. In this case a strong reduction of the Broad-Band impedance is necessary to keep the required t jump  ~ - 2 *
Further improvement of the longitudinal density beyond that of nTOF seems excluded, even if only the (unavoidable!) space-charge impedance of the beam is taken into account
* - It is believed that t until ~ - 2 can be performed
- It should be done with Qt  0
- During the t jump the dispersion has the tendency to increase  This can lead to an increase of the horizontal beam size and subsequent beam losses
Elias Métral, PS2 WG meeting, 15/05/2008

35. Conclusions and outlooks (2/7)
Reminder on the BB in the PS and SPS (measured from coherent vertical tune shift vs. intensity):
PS: ~ 3 M/m
SPS: ~ 22 M/m in 2007 (i.e. ~ 7 times higher for a machine 11 times bigger)
The PS2 will be more than 2 times bigger than the present PS and will operate at higher energy (more powerful kickers needed etc.)
 Will be challenging to build the PS2 machine within the required (transverse) impedance budget (~ 6 times smaller than the present PS!)
Elias Métral, PS2 WG meeting, 15/05/2008

36. Conclusions and outlooks (3/7)
Concerning the time needed to perform the t jump in PS2
Longitudinal negative-mass (SC) instability rise-times are larger in the PS2 compared to the present PS
Vertical microwave (BB) instability rise-times are larger in the PS2 compared to the present PS
 The time needed to perform the t jump of the present PS (i.e. ~ 500 s) should be enough for the PS2
Elias Métral, PS2 WG meeting, 15/05/2008

37. Conclusions and outlooks (4/7)
Proposed MDs in the PS in 2008 to check our estimates (fast analog signals are available in the CCC since the end of 2007):
Movie of the TMCI to compare with the HEADTAIL simulations (same data as the ones taken by Rende in 2007)
Remove the t jump and measure the evolution of the longitudinal, horizontal and vertical signals vs. transverse bunch emittance (to see the relative effect of the SC impedance, which is emittance-dependent, and the BB impedance, which is constant
Same measurements scanning the amplitude of the t jump
Same measurements scanning the transition timing (symmetric vs asymmetric jump…) for different transverse bunch emittances …
S. Aumon and W. Bartmann will be the contact persons for these studies
Elias Métral, PS2 WG meeting, 15/05/2008

38. Conclusions and outlooks (5/7)
Perspectives with HEADTAIL (which was upgraded recently for these studies)
Possibility to compute the transverse instability rise-time with acceleration
Possibility to include a t jump
Possibility to include the transverse and longitudinal space charge
Possibility to take into account the nonlinear momentum factors
 It should be possible to benchmark all the measurements (which will be performed in 2008) against HEADTAIL
Elias Métral, PS2 WG meeting, 15/05/2008

39. Conclusions and outlooks (6/7)
Proposition to increase the intensity threshold when transition crossing is limited by TMCI as in the PS (see for instance RLC meeting of 19/09/2006)
Since the work of Pellegrini&Sands1969 on the (low-intensity) Head-Tail instability, the sign of the H and V chromaticities is changed when the transition is crossed (i.e. when the sign of the slip-factor  changes)  The chromatic frequency is kept positive
Before, the Head-Tail instability above transition was damped (e.g. in the PS) using Landau octupoles
It is proposed now not only to change the sign of the chromaticity when the transition is crossed but to correlate its variation with the one of the slip-factor (due to TMCI reason)
 Could/should be checked first with HEADTAIL…
Elias Métral, PS2 WG meeting, 15/05/2008

40. Conclusions and outlooks (7/7)
TMCI intensity threshold
Consider the case below transition ( < 0 and  < 0) where a TMCI is stabilized by decreasing  and let’s call
 = 0 in the stable situation below transition
During transition crossing (and after), one wants to keep   0
Increasing the longitudinal emittance (e.g. in the PS) would not be required anymore
 = 0 
Elias Métral, PS2 WG meeting, 15/05/2008

41. Still to be done for the review (1/3)
M. Benedikt sent us new beam parameters on 12/05/08
 Re-do some computations for the new beam parameters (see next slides in red)
Elias Métral, PS2 WG meeting, 15/05/2008

42. Still to be done for the review (2/3)
Elias Métral, PS2 WG meeting, 15/05/2008

43. Still to be done for the review (3/3)
Elias Métral, PS2 WG meeting, 15/05/2008