1. Elias Métral, LIS meeting, 29/01/2008 1/26 GAMMA TRANSITION JUMP FOR PS2 W. Bartmann, M. Benedikt, E. Métral and D. Möhl u Introduction with the case of the PS u Equations in the case with NO SC (or BB imp.) and NO  t jump re-derived and solved analytically u Equations in the case with both SC (and/or BB imp.) and  t jump re-derived and solved numerically u Application (and comparison with Dieter’s theory of 1969!  CERN-ISR/300/GS/69-62) to the cases below:  nTOF in the present PS  nTOF in (PS2, 10 MHz)  FT in (PS2, 10 MHz) and (PS2, 40 MHz)  LHC in (PS2, 10 MHz) and (PS2, 40 MHz) u Conclusion, outlook and appendices

2. Elias Métral, LIS meeting, 29/01/2008 2/26 Introduction with the case of the PS (1/7) Thanks to M. Martini for the data!   t = - 1.24 in  t jump = 500  s (see also next slides)

3. Elias Métral, LIS meeting, 29/01/2008 3/26 Introduction with the case of the PS (2/7) Zoom

4. Elias Métral, LIS meeting, 29/01/2008 4/26 Introduction with the case of the PS (3/7)  Currently, the nTOF bunch is the most critical at transition due to a TMCI which develops near transition crossing Measurements in 2000 Simulation by Giovanni with HEADTAIL (ICAP06) Without space charge With space charge  > 2.1 eVs are needed for ~ 7  10 12 p/b (PS/RF Note 2002-198)

5. Elias Métral, LIS meeting, 29/01/2008 5/26 Introduction with the case of the PS (4/7) 1 st trace = turn 1Last trace = turn 130Every turn shown Flat chamber HeadTail HEADTAIL SIMULATION - Constant energy - Bunch in the centre of the PU

6. Elias Métral, LIS meeting, 29/01/2008 6/26 Introduction with the case of the PS (5/7) Turn 130 shown Δz  40 cm  f  750 MHz Δz  120 cm  f  250 MHz HEADTAIL SIMULATION

7. Elias Métral, LIS meeting, 29/01/2008 7/26 Introduction with the case of the PS (6/7)  Measurements from Rende in 2007 on the AD beam with low longitudinal emittance (no precise beam parameters)

8. Elias Métral, LIS meeting, 29/01/2008 8/26 Introduction with the case of the PS (7/7) PSPS2 nTOF 10 MHz FT 10 MHz LHC 10 MHz FT 40 MHz LHC 40 MHz R [m]100214.3  [m]70100 Bdot [T/s]2.21.5 Vrf [kV]200500 1500 h81545 180 pp 0.0270.0076  L [eVs]2/2.3/2.52.51.5 0.40.6 Bunch area A (Dieter’s units) 0.051/59/64 0.0560.101 0.1070.161 N b [10 10 p/b]80010003201908048  t (1 , norm.) [  m] 5662.56  t average [m] 16*15* Pipe [cm  cm]3.5 / 7 QtQt 6.2511.25** ** Assumption: Q t ~  t = 1 /   p = 11.47 * R / Q t

9. Elias Métral, LIS meeting, 29/01/2008 9/26 General result for the nonadiabatic transition region (with neither space charge, or BB imp., nor ) General result for the nonadiabatic transition region (with neither space charge, or BB imp., nor  t jump) Nonadiabatic timeTransition  Same picture as the one obtained by K.Y. Ng in his book “Physics of Intensity Dependent Beam Instabilities” (2006), p. 707  The bunch is tilted near transition

10. Elias Métral, LIS meeting, 29/01/2008 10/26 T c  1.9 ms Case of the nTOF bunch in the PS WITHOUT space charge (1/2)

11. Elias Métral, LIS meeting, 29/01/2008 11/26 Case of the nTOF bunch in the PS WITHOUT space charge (2/2)

12. Elias Métral, LIS meeting, 29/01/2008 12/26 Case of the nTOF bunch in the PS WITH space charge (1/5) STATIC (MATCHED) CASE

13. Elias Métral, LIS meeting, 29/01/2008 13/26 Case of the nTOF bunch in the PS WITH space charge (2/5) DYNAMIC (“REAL”) CASE WHEN TRANSITION IS CROSSED

14. Elias Métral, LIS meeting, 29/01/2008 14/26 Case of the nTOF bunch in the PS WITH space charge (3/5) DYNAMIC (“REAL”) CASE WHEN TRANSITION IS CROSSED

15. Elias Métral, LIS meeting, 29/01/2008 15/26 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

16. Elias Métral, LIS meeting, 29/01/2008 16/26 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

17. Elias Métral, LIS meeting, 29/01/2008 17/26 (Asymmetric)  t jump for nTOF in the PS (considering SC and TMCI) T c  1.9 ms

18. Elias Métral, LIS meeting, 29/01/2008 18/26 PSPS2 nTOF (2 eVs) nTOF (2.3 eVs) nTOF (2.5 eVs) nTOF (10 MHz) FT (10 MHz) LHC (10 MHz) FT (40 MHz) LHC (40 MHz) Tc [ms]1.86 4.292.97 1.27 g0g0 3.85 3.73 4.63.734.6 SC imp. [  ]- 19.85  j - 5.36  j - 6.62  j- 5.36  j- 6.62  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 = -1.50 if PS2 BB imp. 6 times smaller than PS Summary table for PS and PS2 (considering SC and TMCI) It is the  t of the present PS

19. Elias Métral, LIS meeting, 29/01/2008 19/26 Case of the nTOF bunch in the PS with longitudinal inductive BB impedance (neglecting SC)  t jump  In this case, the asymmetry of the  t jump is in the other direction

20. Elias Métral, LIS meeting, 29/01/2008 20/26 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!

21. Elias Métral, LIS meeting, 29/01/2008 21/26 Case of the nTOF bunch in the PS with both longitudinal inductive BB impedance and SC and 2 times bigger transverse emittance

22. Elias Métral, LIS meeting, 29/01/2008 22/26 Conclusion and outlook (1/3)  t u Transition crossing with a  t jump looks possible in PS2 for the densities foreseen with the FT and LHC beams  t u It its more difficult for conditions corresponding to the present nTOF bunch. In this case a strong reduction of the BroadBand impedance is necessary to keep the required  t jump  ~ - 2 * u 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  Q t  0  t - 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

23. Elias Métral, LIS meeting, 29/01/2008 23/26  t u Future work: What about the time needed to perform the  t jump in PS2?  Compute the instability rise-time for  Negative-mass instability * (and longitudinal microwave instability with BB impedance  Could be increased by reducing the BB impedance)  TMCI  Could be increased by reducing the BB impedance  t * A preliminary estimate reveals that the time needed to perform the  t jump of the present PS (i.e. ~ 500  s) could be enough as the negative-mass instability rise-time in the PS2 is larger than the one in the PS for the nTOF bunch Conclusion and outlook (2/3) See Appendices

24. Elias Métral, LIS meeting, 29/01/2008 24/26 u 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  … Conclusion and outlook (3/3)

25. Elias Métral, LIS meeting, 29/01/2008 25/26 APPENDIX A: NEGATIVE MASS (SC) INSTABILITY RISE TIME FOR nTOF IN THE PS Keil-Schnell

26. Elias Métral, LIS meeting, 29/01/2008 26/26 APPENDIX B: VERTICAL MICROWAVE INSTABILITY RISE TIME FOR nTOF IN THE PS Keil-Schnell (transverse)