1. Jean-Luc Biarrotte, SPL HOM Workshop, CERN, June 25-26, 2009.1 A few longitudinal BBU simulations (in the SPL case) J-Luc Biarrotte CNRS, IPN Orsay

2. Jean-Luc Biarrotte, SPL HOM Workshop, CERN, June 25-26, 2009.2 HOM excitation analyses Beam flying ONCE through 1 cavity –Allows to study when a HOM mode hits a beam spectrum resonance line –Analytical power analysis is possible (&  sufficient) Beam flying ONCE through N cavities –Allows also to study if a Beam Break-Up (BBU) rises « out from the noise » –Simulation is required (with very consuming CPU time) (Beam flying N times through 1 cavity) –Non-Applicable to linacs

3. Jean-Luc Biarrotte, SPL HOM Workshop, CERN, June 25-26, 2009.3 Input parameters for the SPL case PROTON BEAM –352.2 MHz, 50 Hz, 5% d.c. (1ms pulse) –150 MeV input energy –400 mA mean pulse current SC LINAC –150 cavities @704.4 MHz –20.7MV acc.voltage, -15° synch.phase (-> 3.15 GeV) –HOM mode considered: 2000 MHz (far from main beam resonances) Q=1 E 8 (no HOM coupler) R/Q 50 ohms (very high, but will be kept for all simulations )

4. Jean-Luc Biarrotte, SPL HOM Workshop, CERN, June 25-26, 2009.4 BEAM ERRORS –Bunch-to-bunch CURRENT normal fluctuation w/  =10%*I 0 (  120mA@3  ) –Bunch-to-bunch input PHASE normal fluctuation w/  =0.1° (  1.2ps@3  ) LINAC ERRORS –Cav-to-cav HOM frequency spread (normal distribution) w/  =100 kHz (  0.3MHz@3  ) –Bunch-to-bunch cavity acc.voltage & phase normal fluctuations w/  V =0%*V and   =0° Errors distribution

5. Jean-Luc Biarrotte, SPL HOM Workshop, CERN, June 25-26, 2009.5 Run I (Joachim’s more or less) I 0 =400mA  I0 =10%   0 =0.1° f HOM =2GHz  HOM =100kHz Q HOM =1 E 8  Vcav =0%   cav =0°  10 MeV  2.5° max V hom : 3 MV GROWTH  10 MeV  2.5°

6. Jean-Luc Biarrotte, SPL HOM Workshop, CERN, June 25-26, 2009.6 I 0 =40mA : real current (400mA is anyway impossible to accelerate in SPL!)  I0 =10%   0 =0.1° f HOM =2GHz  HOM =100kHz Q HOM =1 E 8  Vcav =0%   cav =0° Run II (meeting the real SPL current)  1 MeV  0.25° max V hom : 12 kV NO CLEAR GROWTH

7. Jean-Luc Biarrotte, SPL HOM Workshop, CERN, June 25-26, 2009.7 I 0 =40mA  I0 =1% : 10% is a lot, especially considering bunch-to-bunch random fluctuations   0 =0.1° f HOM =2GHz  HOM =1MHz : closer to J-Lab or SNS data Q HOM =1 E 8  Vcav =0%   cav =0° Run III (with even more realistic inputs) max V hom : 1 kV NO GROWTH  1 MeV  0.25°

8. Jean-Luc Biarrotte, SPL HOM Workshop, CERN, June 25-26, 2009.8 I 0 =40mA  I0 =1%   0 =0.1° f HOM =2GHz  HOM =1MHz Q HOM =1 E 8  Vcav =0.3%   cav =0.3° : classical errors for (poor) LLRF systems Run IV (the real SPL) max V hom : 4 kV  10 MeV  2.5° The same is obtained @ zero-current (w/o HOMs)

9. Jean-Luc Biarrotte, SPL HOM Workshop, CERN, June 25-26, 2009.9 I 0 =40mA  I0 =1%   0 =0.1° f HOM =2GHz  HOM =1MHz Q HOM =1 E 5  Vcav =0.3%   cav =0.3° Run V (the real SPL w/ HOM couplers ) w/ or w/o HOM couplers: SAME BEAM BEHAVIOR max V hom : 300 V  10 MeV  2.5°

10. Jean-Luc Biarrotte, SPL HOM Workshop, CERN, June 25-26, 2009.10 I 0 =40mA  I0 =1%   0 =0.1° f HOM =2GHz  HOM =0 Q HOM =1 E 8  Vcav =0%   cav =0° Run VI (synchrotron-like case) Always the same exact HOM freq => BEAM is QUICKLY LOST

11. Jean-Luc Biarrotte, SPL HOM Workshop, CERN, June 25-26, 2009.11 I 0 =40mA  I0 =1%   0 =0.1° f HOM =2GHz  HOM =0 Q HOM =1 E 5  Vcav =0%   cav =0° Run VII (synchrotron-like case w/ HOM couplers ) If no HOM spread at all => HOM coupler is required & efficient  1 MeV  0.25°

12. Jean-Luc Biarrotte, SPL HOM Workshop, CERN, June 25-26, 2009.12 Run VIII (meeting a beam line, w/ Joachim’s inputs ) I 0 =400mA  I0 =10%   0 =0.1° f HOM =2113.2MHz : the exact 6th beam line  HOM =100kHz Q HOM =1 E 8  Vcav =0%   cav =0° Complete beam loss after only a few bunch

13. Jean-Luc Biarrotte, SPL HOM Workshop, CERN, June 25-26, 2009.13 Run IX (meeting a beam line, more softly ) I 0 =40mA  I0 =1%   0 =0.1° f HOM =2113.2MHz  HOM =100kHz Q HOM =1 E 8  Vcav =0%   cav =0°  10 MeV  2.5° max V hom : 4 MV

14. Jean-Luc Biarrotte, SPL HOM Workshop, CERN, June 25-26, 2009.14 Run X (meeting a beam line, even more softly ) I 0 =40mA  I0 =1%   0 =0.1° f HOM =2113.2MHz  HOM =1MHz Q HOM =1 E 8  Vcav =0%   cav =0°  10 MeV  2.5° max V hom : 0.7 MV

15. Jean-Luc Biarrotte, SPL HOM Workshop, CERN, June 25-26, 2009.15 Run XI (meeting a beam line, w/ HOM couplers) max V hom : 0.4MV High HOM power has anyway to be handled  10 MeV  2.5° I 0 =40mA  I0 =1%   0 =0.1° f HOM =2113.2MHz  HOM =100kHz Q HOM =1 E 5  Vcav =0%   cav =0° HOM excitation (W) Vs frequency on a beam spectrum line for different Q HOM

16. Jean-Luc Biarrotte, SPL HOM Workshop, CERN, June 25-26, 2009.16 Conclusions (personal draft – for discussion ! ) In SPL-like linacs, BBU « out from the noise » can be damped: –N°1: lowering Q HOM (e.g. using HOM couplers) This is the only solution for extreme conditions (100’s mA with high fluctuations, low HOM dispersion... ) or in circular machines (zero HOM frequency spread) OR –N°2: naturally Simply checking that HOM modes are sufficiently distributed from cavity to cavity (  =100kHz seems even to be enough in the SPL case) Simulating the SPL case without HOM couplers & with realistic input parameters shows apparently no longitudinal Beam Break Up instability rising Cavities should be designed to avoid having a HOM mode exactly matching a beam resonance line. In the case where this is NOT achieved: –Without HOM couplers (high Q HOM ): very low probability to really hit the resonance, but could lead to a cavity quench + beam loss if it happens -> CURE = simply measure all cavities before installation to check –With HOM couplers (low Q HOM ): the probability to hit the resonance is far larger but not catastrophic -> HOM couplers have to be designed to evacuate large HOM powers continuously

17. Jean-Luc Biarrotte, SPL HOM Workshop, CERN, June 25-26, 2009.17 Limits of these non-definitive conclusions... Very poor statistics : more simulations should be performed 10 X pulses should be simulated to confirm the behavior with a « quasi-infinite » pulse train, and to check if any instability linked with the pulse structure is rising (even if for SPL, it is a priori non-relevant for such low pulsing frequencies) The real SPL linac layout should be taken as input, including, for each cavity family, all HOM properties (f, Q, r/Q(  ))... More realistic (non-white) noises should be considered Being aware that all this would imply enormous CPU time...