1. “two beam” impedances Frank Zimmermann AP Forum, 28 September 2012

2. outline motivation - 2012 LHC observations historical studies resistive wall 2-beam impedance - review of work from 2006 open questions TCT – Alexej Grudiev Y chamber - Bruno Spataro

3. decreasing octupoles' current from ~ 450 A to 97 A (on flat-top before the squeeze)  beam dump due to losses estimated Q’ + 5 units single-beam instability threshold 4 TeV Elias Metral, Alexey Burov, Nicholas Mounet, Giulia Papotti consistent with impedance model from summary of MD block #2

4. single beam instability data from MD block no 2. After squeeze: 1)Q'x,y = + 5, ADT as in physics: decreased octupole current down to 0; at 0 the beam became unstable in Vplane (still stable at ~ 20 A in the D octupoles). Increased octupoles to save beam (also Q'x,y +2). 2)Ioct = 450 A, ADT as in physics: decreased Q'x,y from + 7 to - 10 by steps of 1 unit. The beam was always stable. Then we moved in 1 step to Q'x,y = + 10 and the beam was still stable. 3)Q'x,y = - 5, ADT as in physics: decreased the octupole current down to ~ 220 A, when the beam became unstable in Vplane. Just after the instability, increased again the octupole current to save beam. 4)Q'x,y = + 2, ADT as in physics: we decreased the octupole current down to ~60 A, when the beam became unstable in Hplane. Just after the instability, increased again octupole current to save beam. 5)Q'x,y = + 2, Ioct = 450 A: decreased the ADT gain by steps of 20% and after having reached the limit of the lower value, switched ADT off. The beam was stable. BBQ signal in H increased ~ linearly ~ 3 times above the noise level. 6)Q'x,y = + 2, ADT switched off: decreased the octupole current by 1 step to ~ 400 A and the beam was unstable in Vplane and finally dumped. A. Burov, M. Pojer, Y., G. Trad, Y. Le Borgne, T. Pieloni, E. Metral from summary of MD block #2

5. 2-beam machine settings (ramp) until August tunes: 0.28 (H)/0.31(V) constant chromaticity: close to 0 (+1-2 units) to minimize growth rate of higher order head-tail modes and possible lifetime problems observed at Tevatron with high (~20 units) chromaticity. octupoles: increasing at the end of the ramp when collimators are closing (~5.3 sigmas for   =2.5  m at the TCPs). ~450 A at 4 TeV (negative for ROF/positive for ROD) to damp higher order head-tail modes. Polarity to maximize damping for given current. Current scaled from 2011 experience taking into account increased impedance of the collimators (cfr. E. Métral). ROF.B1 Energy ROF.B2 Observations on instabilities - G. Arduini, LMC #145

6. 2-beam stability level investigations on the ramp Fill Q’Oct [A] (ramp) Instability end of ramp (first) Comment 29081.512/2/2/2 -417 NoneOK 29111.512/2/2/2 -417 B2HLosses at the end of flat-top Saved by raising Q’ (+3H/+3 V)(EM,LP) 29121.527/7/7/7 -417 B1H or B2H OK 29131.567/7/7/7 -417 B2H starting first Ramp OK Losses in collision BP 29151.527/7/7/7 -330 B1H starting first Ramp OK Losses in collision BP 29171.487/7/7/7 -150 B1H starting first 29181.487/7/7/7 -150 B2H Saved by raising Q’ (+2 H/+3 V) 29191.477/7/7/7 -200 B2H 29201.467/7/7/7 -200 B1HSaved by raising Q’ (+2 H) Observed instabilities for nominal chromaticity - had to increase it Based on these observations inverted octupole polarity with 417 A and chromaticity of 5 Observations on instabilities - G. Arduini, LMC #145

7. octupole reverse polarity – 2 beams Fill Q’Oct flat top/squeeze Instability (first) Comment 29261.395/5/5/5 417 None480b 29271.435/5/5/5 417 B1VEnd of squeeze 29281.486/7/6/7 475 B1HBeginning of squeeze 29291.459/10/9/10 510 B2VSqueeze at 1.5 m 29321.449/10/9/10 510 B2HFlat-top 29331.4611/11/11/11 510 NoneIncoherent losses QV on 3 rd integer 29341.4611/11/11/11 510 NonePhysics 29571.5511/11/11/11 510 NoneNew record luminosity Transverse feedback unchanged (factor 2 to 4 below maximum) all through flat-top and squeeze Instability developing at high energy (flat-top and squeeze) H and V Always cured by increasing chromaticity No instability observed during adjust. Observations on instabilities - G. Arduini, LMC#145

8. LHC observations in a nutshell with old octupole polarity single beam was stable with I oct =100 A (before squeeze) or 20 A (after squeeze) two-beam stability required I oct >300 A

9. around the four interaction regions, the two LHC beams share a common beam pipe, including a number of tertiary collimators protecting the low-beta quadrupoles against beam loss here the resistive-wall wake fields induced by one of the two beams can excite oscillations in the other beam wake fields acting between bunches propagating in opposite direction could be important whenever two colliding beams pass through a common region of beam pipe (not only for the LHC but for all storage-ring colliders or even for the interaction regions of linear colliders) two-beam cross talk

10. coupling via resonator wake fields instabilities involving the coupling of two counter-propagating oppositely charged beams via resonator wake fields discussed in: [1] A. Renieri and C.Pellegrini, Longitudinal Instabilities of the Dipole Oscillation Modes for Two Beams Many Bunches Storage Ring, Report Adone T-64, Laboratori Nazionali di Frascati (1974). [2] C.Pellegrini, Longitudinal Coupled Bunch Instability for Two Counter Rotating Beams, CERN-LEP/TH/86-17 (1986) [3] J.M.Wang, Transverse Two Beam Instability, CERN-LEP/TH/87-65 (1987) [9] T.-S. Wang, Coherent Synchro-Betatron Effects in Counter-Rotating Beams, CERN/SL/90-09 (AP) (1990)

11. J.M. Wang 1987

12. model I - without frequency spread → narrow band high-Q resonance could explain difference between LHC 1 & 2-beam stability

13. two-beam resistive-wall impedance an activity in the former CERN AB-ABP-RLC section literature search for 2-beam wake fields, action of 18.11.2005; assigned to Elias Metral, Frank Zimmermann; high priority; COMPLETED on 16.12.2005 develop theory for 2-beam resistive-wall wake field, action of 17.02.2006, assigned to Frank Zimmermann; high priority; DONE on 03.03.2006 (presentation at RLC section meeting) poster & paper “Two beam resistive wall wake field”, by F. Zimmermann, at PAC07, Albuquerque, Proc. P. 4237; LHC- Project-Report -1021LHC- Project-Report -1021

14. I am grateful to the late Francesco Ruggiero (1957-2007) for encouraging this study. The PAC07 article was dedicated to him. I also thank Karl Bane and Katsunobu Oide for helpful discussions about wake-field calculations in the complex plane and the Panofsky-Wenzel theorem. Elias Metral & Bruno Zotter kindly provided references [2] & [3]. Boris Podobedov has later sent me some interesting questions. acknowledgements

15. coupling via resistive wall Case of resistive-wall impedance differs from resonator case. Longitudinal electric field excited by a bunch moving in the opposite direction changes sign with respect to that of a bunch moving in the same direction. At larger distances the field excited by the other bunch becomes decelerating. The total wake-field effect on a “probe” particle (bunch) must be obtained by integrating over the distance z from the ``driving'' particle (or “driving” bunch) in the opposite beam, instead of considering a constant value of z as is common for single-beam wake-field calculations. As a consequence of the change in sign for the longitudinal electric field, the coupled-beam longitudinal resistive-wall impedance also changes sign. Transverse resistive-wall wake field acting between counter- propagating beams or particles has electric & magnetic components. The transverse electric force stays the same as for the conventional single-beam wake field, but magnetic Lorentz force inverts its sign.

16. not necessarily good assumption for LHC collimators! I drop r 2 term (“sextupolar” wake)

17. [4] A.W. Chao, Physics of Collective Instabilities in High Energy Accelerators, John Wiley, 1993. [5] P.L. Morton, V.K.Neil, A.M. Sessler, “Wake Fields of a Pulse of Charge Moving in a Highly Conducting Pipe of Circular Cross Section,” J.Appl.Phys.37, 10, 3875-3883, 1966. [6] K.L.F. Bane, M. Sands, “The Short-Range Resistive-Wall Wake Fields,” AIP Conf. Proc. 367: 131-149, 1996. [7] W.K.H. Panofsky, W.A. Wenzel, “Some Considerations Concerning the Transverse Deflection of Charged Particles in Radio-Frequency Fields,” Rev.Sci.Instrum. 27, 967 (1956). [8] K. Oide, private communication (2007). standard resistive wall - references

18. firstlast

20. Complex w plane and the two k-plane Riemann sheets with branch cuts corresponding to the complex function w=k 1/2, and pole of the function k 1/2 /(k 1/2 +(1+i)/(2bX)).

22. Parentheses ( Approach follows Bane and Sands’ treatment for the single beam wake field [SLAC-Pub- 95-7064]: )

23. The equality of the transverse single-beam and two-beam wake fields, except for the sign, can be attributed to the Panofsky-Wenzel theorem [7], which relates the transverse and longitudinal wake fields as and which should hold for any force that can be derived from a Hamiltonian [8]. For a single beam the total net wake effect is obtained by multiplying the wake field with the length L of the resistive beam pipe considered, where z refers to the distance e.g. between the source bunch and a later bunch moving in the same direction, which does not change. In the case of the two-beam wake field, we must integrate with respect to z, and compute expressions of the form discussion

24. Total net force acting during the passage of two bunches moving in opposite direction = resistive-wall component and + direct long-range beam-beam force. Consider the situation that two proton beams are offset from the center of a round vacuum chamber, by ±a, so that the beams are transversely separated by 2a. The total beam-beam deflection experienced by a proton at the center of one beam after passing a bunch of the second beam and then traversing a common circular vacuum chamber of conductivity  and length L, equals where N b denotes the number of protons per bunch, and e is the elementary charge. The second term represents the long-range beam-beam force in free space. The positive sign of the two-beam Green function wake field indicates that the wake force points towards the center of the chamber, i.e. the two-beam wake tends to counteract the direct long-range beam-beam force.

25. Examples for parameters typical of the LHC inner triplet – b~30mm, L~50 m, b/a~5 –, force cancellation for  ~4x10 10 s -1, about seven orders of magnitude smaller than the conductivity of copper or aluminum with approximate parameters for an LHC two-beam tertiary tungsten collimator, – b~1.5 mm, b/a~5,and L~1 m –, the force cancellation occurs for  ~4x 10 11 s -1, which is still six orders of magnitude below the conductivity of tungsten

26. We have derived the transverse two-beam resistive-wall wake function describing the coupling of two beams moving in opposite direction via the impedance of a round resistive vacuum chamber. The Green function wake field equals that of a single beam, but it has the opposite sign. The modification of the long-range beam-beam deflection by the resistive-wall wake field is small for typical LHC parameters. Larger effects would arise either for smaller beam pipes or for chamber walls of higher resistivity. Conclusions (2006)

27. questions from Boris Podobedov (2009) concerning res.-wall wakefields for a possible small proton ring at BNL with counter-propagating beams sharing the same small-aperture chamber - two questions: “ 1) How to estimate the multi-turn two-beam wakefield kick? For the standard co-propagating beam situation I would just sum contributions from all turns until the field diffuses through the thickness of the vacuum chamber. Of course, at each subsequent turn the transverse wake would be sqrt(z/(z+C)) smaller. Is this approach directly transferable to the two-beam wakefield? 2) A possibly related question is as follows. Now I go back to the single- turn wake, and I am still in the long-range regime |z|>>\kappa^{1/3}b. What happens if there are two identical resistive elements in the ring instead of one? Is the total two-beam resistive wall kick a factor of 2 or a factor of sqrt(2) times larger than if there was just one element? In other words, could resistive pieces be considered independent as long as they are somewhat apart?” 16.07.2009

28. comment diverging for large L

29. possible further work on two-beam resistive-wall wake field multiturn wake? (Boris I) additivity? (Boris II) sextupolar wake component drop assumption |  >>1/b