1. Managed by UT-Battelle for the Department of Energy Electromagnetic wake fields and impedances in particle accelerators From the primary concept of coupling impedance to the most recent generalizations V. Danilov SNS AP group Erice, Italy, April 24-28, 2014

2. 2Managed by UT-Battelle for the Department of Energy Presentation_name Talk outline  Primary impedance concept  Transition to wake field concept  Standard properties  Unconventional longitudinal wakes  Transverse conventional case  Important additions. Examples.  What can’t be described by generalized wakes and impedances?

3. 3Managed by UT-Battelle for the Department of Energy Cases considered  Longitudinal (m=0) case – no dependence on transverse coordinates  Transverse dipole fields only  The reasons – the generalizations are obvious  The practical use of higher order modes is negligible Presentation_name

4. 4Managed by UT-Battelle for the Department of Energy Presentation_name Longitudinal case. The first introduction of impedance, wakes – conventional theory. February 6, 1967

5. 5Managed by UT-Battelle for the Department of Energy Transition to wake field concept Presentation_name  Here and below all conventional definitions from A. Chao, “Physics of Collective Beam Instabilities in High Energy Accelerators”, Wiley & Sons, Inc., (1993)  Wake fields emerge naturally when one considers space-time domain (instead of frequency domain)

6. 6Managed by UT-Battelle for the Department of Energy Properties (ultra relativistic case) Ultra relativistic case is of most importance since electrons almost instantly become relativistic and for slow protons their space charge is way more important 1) Wake standard properties: a) W(z)=0, z>0; b) W(0 - )≥ |W(z)| for all z; etc. 2) The most important recently investigated wake that deviates from these properties is (author’s opinion) is Coherent Synchrotron Radiation (CSR) wake. Acts forward. It is equal to: Presentation_name

7. 7Managed by UT-Battelle for the Department of Energy Its importance. Naive TLEP estimations  The wake is related to fields of tail particles catching up head particles (Ya. S. Derbenev, et al, DESY report, 1995-05, J. B. Murphy et al, Part. Acc., 57, 9, (1997))  G. Stupakov and S. Heifets, PRSTAB, 054402 (2002) – microbunching instability from this wake  Energy loss with F 0 and its derivative are around 1. For gradient calculation we expand F 0 around s=-1 and take the second term Presentation_name Needed voltage for equilibrium distribution

8. 8Managed by UT-Battelle for the Department of Energy Naïve TLEP-H estimations  TLEP-H – Higgs factory under development at CERN  Its circumference is about 80 to 100 km  Beam length is about 1 mm  The formula gives equivalent voltage for TLEP-H (100 km) around 23 GV (compared to V RF total of 6 GV) Presentation_name

9. 9Managed by UT-Battelle for the Department of Energy CSR wake screening  The effect of VC screening (Warnock, SLAC- PUB-5375, (1990)) is reducing the number  The effect of VC screening helps – only 0.3 mm of beam out of 1mm are not screened. It reduces the effect by factor three, but it still larger than the longitudinal design focusing (8 GV vs 6GV) – the beam will significantly reshape itself, not talking about instabilities (more from A. Novokhatsky talk, this workshop) Presentation_name

10. 10Managed by UT-Battelle for the Department of Energy Summary for Longitudinal Case  Sessler-Vaccaro formalism works very well  It works even for unusual wakes like CSR one  It is possible to find more exotic cases, but nothing of significance is known (to the author)  Please, inform if anything interesting pops up Presentation_name

11. 11Managed by UT-Battelle for the Department of Energy Transverse impedances and wakes and their properties - standard approach Presentation_name Ultra relativistic case again a) W(z)=0, z>0; b) Two particle picture – F-force acting on a trailing particle, r 0 -displacement of a head particle, L – circumference or element length.

12. 12Managed by UT-Battelle for the Department of Energy Why the transverse case is so different?  Fast oscillations – large betatron tunes  Two dimensions (important for detuning wakes that will be discussed in next slides)  It turns out conventional wakes are not enough to describe the collective phenomenon first seen in Novosibirsk by the author – the Fast Damping !!! It was observed in VEPP-2, BEP, etc. Presentation_name

13. 13Managed by UT-Battelle for the Department of Energy Long strip lines in VEPP-2. Fast Damping Part I Presentation_name The fields in matched line in standard theory are generated at the ends. It produces a field splash with the length of the edge area length. This wake can’t produce any imaginary part for the dipole mode (zero chromaticity), observed at VEPP2. There appeared beam damping proportional to beam current! Besides, the effect of FAST DAMPING is proportional to the strip line length – any hint on this is not present in conventional picture

14. 14Managed by UT-Battelle for the Department of Energy Long strip lines in VEPP-2. Fast Damping Part II Presentation_name Additional fields are generated when particle’s velocity has projection on E fields of strip line TEM mode. The real action on particles is + - e-e- Angular wake Conventional wake - + Λ(x)=xΘ(x) is Heaviside function times its argument

15. 15Managed by UT-Battelle for the Department of Energy The effect of the angular wakes  They produce fast damping – typically, much faster than the SR damping for medium energy machines (VEPP-2, BEP, etc.)  It is proportional to beam current  Its absolute value ratio to that of the real tune shift due to edges is strip line length (l / beta function)  Derbenev’s formula was almost reproduced (discussion 3 slides later) Presentation_name

16. 16Managed by UT-Battelle for the Department of Energy Fields that not fitting the picture – tail particle displacement comes into play  History started with work on special feedback to mitigate TMCI  Continued with probing other methods to increase TMCI thresholds Danilov, RF quad vs TMCI, PRSTAB 1, 041301 (1998)  RF quad could be generated by collective fields! S. Heifets, B. Zotter, A. Wagner, SLAC/AP110, 1998.  A. Burov and V. Danilov, “Suppression of Transverse Bunch Instabilities by Asymmetries in the Chamber Geometry”, PRL, 82, 11 (1999) Presentation_name

17. 17Managed by UT-Battelle for the Department of Energy Detuning Wakes Presentation_name Round chamber – no detuning for resistive wake 1 2 Particles 1(head) and 2 (tail) with the same coordinate have zero force for trailing particle. If you shift them in flat chamber, the force remains zero It means W = - D (see formulas above) for horizontal case and W=D for vertical one Important property D x = - D y They are just quadrupole fields, generated by the beam in VC

18. 18Managed by UT-Battelle for the Department of Energy What difference does it make? Part I  It introduces large spread of betatron frequencies – instabilities thresholds for flat vacuum chamber become significantly higher. Round chamber TMCI (top), and flat chamber (bottom) (A. Burov, V. Danilov, PRL 82, 11, (1999)) Presentation_name

19. 19Managed by UT-Battelle for the Department of Energy What difference does it make? Part II Presentation_name Unfortunately, the spread can be so large that some strong resonances can be inevitably crossed. TLEP-Z example. 45 GeV Beam-beam tune shift is around 0.1 in both planes. Synchrotron tune >0.1, coherent tune shift and incoherent spread is 0.1 near the threshold of TMCI Their sum is more than 1/5. It means some particles cross as a minimum 1/5 (or stronger resonance) in both directions. Typically, those particles gets lost. Solution – round vacuum chamber

20. 20Managed by UT-Battelle for the Department of Energy The last term – does it exists? Presentation_name Physically important cases with nonzero G are not known? All this is collected in the paper V. Danilov, “ Extended Definitions of Wake Fields and Their Influence on Beam Dynamics”, PRST-AB 3, 014201 (2000)

21. 21Managed by UT-Battelle for the Department of Energy Where the above wakes don’t work?  D. Pestrikov, N. Dikansky book (AIP Press, 1994) gives slightly different answer for the strip lines when we take into account all wakes (angular and conventional one);  This book starts from first principles, is obviously more involved then A. Chao’s book;  For the strip lines the difference in answer for zero airbag harmonic is a factor J 2 0 (ζ), where ζ is the betatron oscillation phase advance on the bunch length. Presentation_name

22. 22Managed by UT-Battelle for the Department of Energy Where all the above wakes don’t work II Presentation_name A long element with the large angle and coordinate change What coordinate or angle do we use for a long element? The problem is solvable by averaging when particles don’t move significantly in longitudinal direction When they move significantly in longitudinal direction in the element, then the whole approach has to be reconsidered

23. 23Managed by UT-Battelle for the Department of Energy Where all the above wakes don’t work III Long rings with high synchrotron tune – the bulk definition for the ring doesn’t produce the correct description of SBR. The remedy – dividing the ring to many pieces with own impedances to calculate step by step the transverse mode transformations;  In general, wake fields has to be expanded into more general entity Presentation_name

24. 24Managed by UT-Battelle for the Department of Energy How we define the needed entity?  Physics of collective phenomena drives finding the answer  The correct linear equation (linearized Vlasov equation) tells us what we need  The force (not integrals, not other linear operators of it) enters the equation  The EM force can be defined via Green functions Presentation_name

25. 25Managed by UT-Battelle for the Department of Energy Most general approach – Green function Presentation_name 1)The beam dynamics has to be adjusted appropriately; 2) It has to resemble solving a differential equation in this case; 3) Leap-frog method is one of the methods to find the solutions for collective tune shifts (next slide)

26. 26Managed by UT-Battelle for the Department of Energy Accurate approach with Green functions Presentation_name 1)Many approaches to solve for the modes (started from F. Sacherer using the Hermite functions) 2) The approach based on the division of the synchrotron phase space (left) was used by V. Danilov and E. Perevedentzev, CERN Report No. CERN-SL-92-57-AP (1992); 3) Similar approach described by A. Burov in PRST-AB 17, 021007 (2014) paper Synchrotron phase space D

27. 27Managed by UT-Battelle for the Department of Energy Nonrelativistic wakes. β <<1  The wake fields grow forward – leads to immediate break-up instability (V. Danilov, ibid)  But, their strength falls: for strip line up to β 5 for some elements  In general, nonrelativistic case collective phenomena are dominated by space charge Presentation_name

28. 28Managed by UT-Battelle for the Department of Energy Space charge Part I  It has two distinct features – an incoherent part (inside the beam interactions) and a coherent part (the interaction with VC)  The coherent part can be, in principle, described as an impedance  The incoherent part affects the collective phenomena a lot – it separates coherent and incoherent tunes of a coasting beam, for example, reducing the instabilities threshold  The bunched beam TMCI may benefit from space charge (M. Blaskiewicz, PRST-AB, 1, 044201, (1998)) Presentation_name

29. 29Managed by UT-Battelle for the Department of Energy Space charge Part II  Space charge of e-cloud works well as impedance for coasting beam  For a bunched beam it is problematic to use e-cloud space charge force as something similar to electromagnetic impedance – it depends on bunch shape, intensity, etc. (SNS case)  More info – F. Zimmermann talk at this workshop Presentation_name

30. 30Managed by UT-Battelle for the Department of Energy Presentation_name Conclusion  The Sessler-Vaccaro formalism works well in the longitudinal direction  With the addition of some special wakes it will work in the transverse direction as well  The modern machines like TLEP need to consider mentioned additions to the wakes to accurately predict coherent and incoherent tunes  The cases when we have to employ the more detailed Green function approach are still rare that is why Sessler-Vaccaro approach is still a major tool for collective tune estimations  Thanks for your attention!!!