1. U. IrisoECLOUD’04 – 21 April 2004 1 ECLOUD’04 April 19-23 2004, Napa, CA Use of Maps for exploration of Electron Cloud parameter space Ubaldo Iriso and Steve Peggs M. Blaskiewicz, A. Drees, W. Fischer, H.C. Hseuh, G. Rumolo, L. Smart, D. Trbojevic, S.Y. Zhang.

2. U. IrisoECLOUD’04 – 21 April 2004 2 Outline 1) Motivation: the bunch to bunch evolution 2) Can the Electron Cloud be represented by maps? 2.1. The first N=N and N=0. 2.2. Examples for the RHIC case 3) N exploration of parameter space 4) Electron Cloud phase transitions at RHIC 5) Conclusion and outlook

3. U. IrisoECLOUD’04 – 21 April 2004 3 1. Motivation: the bunch to bunch evolution After experimental observations at RHIC during Run-3*, the use of gaps along the bunch train is chosen to minimize the detrimental effects of Electron Cloud (EC): ● If EC density does not produce beam instabilities, ● If the flux into the wall does not produce pressure rises above harmful limits, ● If we are below heat load limit, then... who cares if EC is there? QUESTION: How do we evaluate the bunch pattern that minimizes EC density (->maximize luminosity? *BNL C-A/AP/118

4. U. IrisoECLOUD’04 – 21 April 2004 4 1. Motivation: the bunch to bunch evolution ● For a given surface and beam pipe dimensions and an initial electron cloud density (  ), what is the  evolution after a bunch m passes by?

5. U. IrisoECLOUD’04 – 21 April 2004 5 2. Can the EC be represented by maps? For a given surface, for the EC build up the only thing changing between the bunch m and bunch m+1 is  m and  m+1. That is…  ! Plot  m+1 vs  m : –Looks like a parabola that gets to the line y=x for saturation. –The EC build up using 3 rd order fits look quite accurate… Note: I don't show (  m,  m+1 ) corresponding to the first N=0 (first “no-bunch” in the abort gap). I will... and a i (N)!! ; i > 1

6. U. IrisoECLOUD’04 – 21 April 2004 6 2. Can the EC be represented by maps? ● Results for different N using CSEC (M. Blaskiewicz), and ECLOUD (F. Zimmermann). This is, results using different SEY parameterization: ● The first N=0 (first “no bunch” in the abort gap after a bunch train of M bunches with bunch charge N) is determined by another polynomial, and it is independent of N. This point in (rho_m+1, rho_m) is out of the N=0 line due to duration of space charge effects. ECLOUD (Thanks G. Rumolo!) CSEC SEY from Furman & PiviSEY from Cimino & Collins N=0 curve (decay) First N=0 curve N=0 curve (decay) First N=0 curve

7. U. IrisoECLOUD’04 – 21 April 2004 7 2. Can the EC be represented by maps? Once we have a i (i=1,2,3) as a f(N), we just need an algorithm depending on N m, being m the bunch number in the bunch train N=N build up First N=0 N=0 decay (gap) Much faster than following  ns-to-ns using “typical” EC simulation codes (~1h vs ~1ms)  Question: What’s the best way to distribute 68 bunches? Let’s see: We have quite a few possibilities… 110!/(110-68)!68! ~ 10^30

8. U. IrisoECLOUD’04 – 21 April 2004 8 Example: the RHIC application ● At RHIC, a given bunch pattern is determined by the triplet (K s, K b, K g ), where: – K s : bucket spacing (multiple of 3 due to kicker limitations) – K b : number of consecutive bunches with this bucket spacing – K g : bunches not filled with this bucket spacing Example: (3,2,0)(6,4,0) – 3 bunches with 3 buckets spacing, followed by 4 bunches with 6 buckets spacing Some parameters to know about RHIC: Harmonic number, 360. Abort gap: 30 buckets. Bucket length: 35.6 ns. “Bunch harmonic number”: 120. Abort gap, 10 bunches Unless otherwise noted, this is structure is repeated until the abort gap

9. U. IrisoECLOUD’04 – 21 April 2004 9 2. Can the EC be represented by maps? When many successive bunches are filled, this “misalignment” is not significant. The first N=N is needed!! Similarly to what happens with the first N=0,  doesn’t jump from N=0 to N=N in only one bunch. N=0: Decay curve N=N: Build up curve First N=0 curve First N=N curve  evolution from CSEC for the BP (3,2,0)(6,4,0)

10. U. IrisoECLOUD’04 – 21 April 200410 2. Can the EC be represented by maps? ● Complete algorithm then, requires: N=N build up (N,N) First N=0 (0,N) N=0 gap (0,0) First N=N!! (N,0) Evaluating Maps for Electron Cloud (MEC) vs “usual” EC simulations codes (in this case, CSEC) gives now very good agreement. See next slides... Note: MEC requires an initial  0 (seed). (N m, N m-1 )

11. U. IrisoECLOUD’04 – 21 April 200411 2. Can the EC be represented by maps? Bunch pattern: (3,12,8)Bunch pattern:(3,2,0)(6,4,0) 1 st turn2 nd turn3 rd turn 1 st turn2 nd turn3 rd turn

12. U. IrisoECLOUD’04 – 21 April 200412 2. Can the EC be represented by maps? Bunch pattern: (3,4,0)(6,8,0) NO FIRST N=N INCLUDED!! =>  is overestimated FIRST N=N INCLUDED!! => Good agreement

13. U. IrisoECLOUD’04 – 21 April 200413  m+1 (  m ) evolution for BP (3,2,0)(6,4,0) mm  m+1 (0,0): linear coefficient  a 00 < 1 (< a 01 ) (0,N): linear coefficient  a 01 < 1 (N,0): linear coefficient  a 10 >1 (< a11) (N,N): linear coefficient  a 11 > 1 Bunch Number 12345679108 NmNm (N,N) (N,0) (0,N) (0,0) (N m, N m-1 )

14. U. IrisoECLOUD’04 – 21 April 200414 3. N exploration of parameter space All the information for the EC build up can for a “regularly” distributed bunch train can be determined by a i coefficients. ECLOUD (Thanks G. Rumolo!) CSEC δ max =2.3

15. U. IrisoECLOUD’04 – 21 April 200415 3. N exploration of parameter space: a map application Suppose the map: If , remains always small enough, we can use linear approximation. After H possible bunches, having filled up to M bunches and i transitions (from 0 to N, and vice versa), the linear approximation says: We have seen we need four sets of parameters, depending on (N, N) (0, N) (0, 0) (N, 0), where F is:, full bunch follows a full bunch  a 11, b 11, c 11, full bunch follows an empty one  a 10, b 10, c 10, empty bunch follows a full one  a 01, b 01, c 01, empty bunch follows an empty one  a 00, b 00, c 00

16. U. IrisoECLOUD’04 – 21 April 200416 3. N exploration of parameter space: a map application If F > 1;  will increase (up to a saturated value, out of linear regime) If F < 1; the EC disappears. This factor is written as: Minimum F requires  < 1  large values of i !! That is, maximum number of transitions, that is, the most sparse distribution of bunches minimizes EC.  Current way to distribute bunches at RHIC to minimize EC For a given M,  does not blow up if (a 10 ·a 01 )/(a 11 ·a 00 ) < 1

17. U. IrisoECLOUD’04 – 21 April 200417 4. EC phase transitions at RHIC Au 79+ x 10 9 P (Torr) 10 -9 10 -10 10 -9 10 -11 0 50 25 Sudden pressure drop, while beam decays “adiabatically”. Do simulations reproduce this kind of “1 st order phase transition”?

18. U. IrisoECLOUD’04 – 21 April 200418 4. EC phase transitions at RHIC (P, N) diagram for the previous case:

19. U. IrisoECLOUD’04 – 21 April 200419 4. EC phase transitions at RHIC Au 79+ x 10 9 P (Torr) 10 -10 10 -9 10 -11 0 50 25 Not all places show 1 st order phase transitions behavior. 2 nd order types are also present for the same beam. IR10: 1 st order IR12: 2 nd order

20. U. IrisoECLOUD’04 – 21 April 200420 4. EC phase transitions at RHIC (P, N) diagram for the previous case: IR10: 1 st order behaviorIR12: 2 nd order behavior

21. U. IrisoECLOUD’04 – 21 April 200421 4. EC phase transitions at RHIC Simulation results using CSEC for fine  N: 2 nd order behavior, analogy with Type II superconductors) Similar EC behaviors: -D. Schulte P(W/m) vs δ max (in ECLOUD’04) -M. Furman (LHC-Project Report 180) Are the 1 st order phase transitions reproducible with some code?  sat =  (N-N c )   = 0.509 +/- 0.017 Nc = 7.398 +/- 0.005

22. U. IrisoECLOUD’04 – 21 April 200422 5. Conclusions… ● From the EC simulation codes (CSEC), the multi-bunch EC build up for RHIC has been determined using a 3 rd order polinomial map. Preliminary results from ECLOUD are promising. ● A ‘memory’ of two bunches both for the decay as for the build-up is found. With this effect taken into account (“first N=0”, and “first N=N), agreement between MEC and CSEC is very good. ● Given a machine limitation  limit (due to heat load, pressure rise, instabilities…), MEC is useful for RHIC to find out the best way to live with EC by changing the bunch pattern ● Using maps, exploration of ( ,N) is done, and standard maths are used to justify sparse distribution for bunches along a bunch train. ● 1 st order and 2 nd order phase transitions are seen at RHIC, but only 2 nd order phase transitions seems to be reproducible with the codes.

23. U. IrisoECLOUD’04 – 21 April 200423 … and outlook ● How do coefficients vary with SEY, R, etc follows. Can we find some few parameters to describe EC (sic). ● Can we map EC from experimental data? ● Does it work for *your* machine with *your* code? ● Is it an artefact due to long RHIC bunch spacing? Can we go to shorter bunch spacings? B-factories? ● Are the 1 st order phase transitions reproducible with EC codes? … and acknowledgements… M. Blaskiewicz, A. Drees, W. Fischer, H.C. Hseuh, R. Lee, N. Luciano, G. Rumolo, L. Smart, R. Tomás, D. Trbojevic, L. Wang, S.Y. Zhang.