Collective effects

1. Collective effects Erik Adli, University of Oslo, August 2014, Erik.Adli@fys.uio.no, v2.01

2. Introduction • Particle accelerators are continuously being pushed to new parameter regimes with higher currents, higher power and higher intensity • Particle accelerator performance is usually limited by multi-particle effects, including collisions and collective instabilities • A core element of particle accelerator physics is the study of collective effects. The understanding of collective instabilities has made it possible to overcome limitations and increase performance significantly • We here describe the two most common effects, space charge and wake fields, in some detail. Example of increase in beam intensity in the CERN PS over the years

3. Intense beams: example Compressed high energy electron beam (at the “FACET” facility at SLAC) : • The beam can be defined by the following parameters : • Charge per bunch: Q = 3 nC (N = 2 x 1010 electrons) • Emittance: enx,y = 100 um • Beam energy: E = 20 GeV (v ≈ c) • Focused size at interaction point : • sx = 20 um,sy = 20 um, sz = 20 um (Gaussian) • Other key beam characteristics deduced from the above accelerator parameters: • Peak current: Ipeak = Qpeak/t = Qc / √2p sz = 19 kA (cf. IAlfven= 17 kA) • Beam density: re≈ N / 4/3p(sxsysz) = 6 x 1017 / cm3 • Plasma description: • Electron transverse energy:Ee= ½ mvx2 = ½ m c2 x’2 ~ 1 meV • Electron (transverse) temperature: Te~ me / kB c2 x’2 ~ 100 K • Debye length (shielding length): lD= √(e0kBT/e2n0) ~ 1 nm • ->lD << sx,sy,sz (collective effects dominate over collisional effects)

4. Space charge – field view point Two-particle interaction : v2 v1 xlab x’ x s z z Fields transforms to the lab frame as : We observe beams, fields and forces in the lab frame. The force on particle 1 is F = e(E+v1 x B). Particle 2 generates no magnetic field in its rest frame, which gives the relation (BT – v2/c2 x E) = 0. The total transverse force on particle 1 in the lab frame is thus F= e(E - v1x (v2/c2 x E)), or for parallel velocities : For v1 = v2 = v we get relativistic space charge surpression: y' y Fx,y= eEx,y (1±v1v2/c2) Fx,y= eEx,y (1-v2/c2) =eEx,y / g2

5. EM fields from a relativistic particle Fields in lab frame Fields in particle’s rest frame Lorentz Tranformations v=0 Ultra-relativistic limit (v=c) : Field compression -> “Pancake field”

6. F FE + + + + F FE Space Charge – lab frame Alternate calculations: Gauss law's. Gauss law is valid also for relativistic moving (or accelerating) charges. In the rest frame, the beam sees an electrostatic field. In the lab frame, the moving charges produce a magnetic field. We assume a uniform density beam in shape of a cylinder, with beam charge density r = Ne / pa2L : Gauss law gives : Ampere’s law gives : Combine terms Fr = e(Er - vBf) : Rest frame Lab frame v FB c FB + c + This is the direct space charge effect. “Direct” : does not take into account the effects of conducting walls surrounding the beam. Lab frame, v = c

7. Space Charge for Guassian Beams KSC ~ Ipeak / (bg)3 “beam perveance” • Typical lattice focusing (FODO with ~10 m between magnets): <b> ~ 10 m -> Kb= 1/100 m-2 • For our example 20 GeVelectron beam (a few slides ago) : KSC = 5 x 10-5 m-2<< Kb • -> space charge completely suppressed at high Lortentz factor • For a 20 MeV electron beam a few slides ago : KSC = 5 x 104 m-2>> Kb

8. Summary K. Schindl Linear defocusing. Gives tune shift in rings. Can be compensated by stronger lattice. Non-linear defocusing. Gives tune shift and tune spreadin rings.

9. Beam-beam effects Relativistic space charge supression is only for equal charge moving at the same velocity, in the same direction (and then only holds fully in free space). These requirement are often violated. An important example is the Beam-Beam interaction : • Two colliding beams see the field of each other before collision. They may be strongly attracted, and may deform. • Important limitation for collider luminosity. Considered the main challenge for LHC luminosity. We will revisit the topic of beam beam effects in the linear collider lectures. For v1 and v2 of opposite sign in our previous calculation (5 slides ago) we get : Fx,y= eEx,y (1+v1v2/c2)

10. Wake fields Most of the material is G. Rumolo’s slides from CAS Course on wake fields

11. Wake fields (general) Source, q1 Witness, q2 z z z 2b L • Whilesource and witness ( qid(s-ct) ), distant by z<0, move centered in a perfectly conducting chamber, the witness does not feel any force (g >> 1) • When the source encounters a discontinuity (e.g., transition, device), it produces an electromagnetic field, which trails behind (wake field) • The source loses energy • The witness feels a net force all along an effective length of the structure, L

12. Wake field characteristics

13. Wake field framework approximations When calculating wake fields in accelerator, the calculations are greatly simplified, allowing for simple descriptions of complex fields, by the following two approximations : 1) Rigid beam approximation: bunch static over length wake field is calculated 2) Impulse approximation: we only care about integrated force over the length wake field is calculated, not the details of the fields in time and space We will define wake function(wake potential) as such integrated quantities.

14. Wake fields (general) Source, q1 Witness, q2 z 2b L • Not only geometric discontinuities cause electromagnetic fields trailing behind sources traveling at light speed. • For example, a pipe with finite conductivity causes a delay in the induced currents, which also produces delayed electromagnetic fields • No ringing, only slow decay • The witness feels a net force all along an effective length of the structure, L • In general, also electromagnetic boundary conditions can be the origin of wake fields.

15. 1. The longitudinal plane dp/p0 z

16. Longitudinal wake function: definition Source, q1 Witness, q2 z 2b L

17. Longitudinal wake function: properties • The value of the wake function in 0,W||(0), is related to the energy lost by the source particle in the creation of the wake • W||(0)>0since DE1<0 • W||(z) is discontinuous in z=0 and it vanishes for all z>0 because of the ultra-relativistic approximation W||(z) z

18. The energy balance What happens to the energy lost by the source? • In the global energy balance, the energy lost by the source splits into • Electromagnetic energy of the modes that remain trapped in the object • Partly dissipated on lossy walls or into purposely designed inserts or higher-order mode (HOM) absorbers • Partly transferred to followingparticles (or the same particle over successive turns), possibly feeding into an instability • Electromagnetic energy of modes that propagate down the beam chamber (above cut-off), which will be eventually lost on surrounding lossy materials

19. The energy balance What happens to the energy lost by the source? • In the global energy balance, the energy lost by the source splits into • Electromagnetic energy of the modes that remain trapped in the object • Partly dissipated on lossy walls or into purposely designed inserts or HOM absorbers • Partly transferred to followingparticles (or the same particle over successive turns), possibly feeding into an instability! • Electromagnetic energy of modes that propagate down the beam chamber (above cut-off), which will be eventually lost on surrounding lossy materials • The energy loss is very important because • It causes beam induced heating of the beam environment (damage, outgassing) • It feeds into both longitudinal and transverse instabilities through the associated EM fields

20. Longitudinal impedance • The wake function of an accelerator component is basically its Green function in time domain (i.e., its response to a pulse excitation) • Very useful for macroparticle models and simulations, because it can be used to describe the driving terms in the single particle equations of motion! • We can also describe it as a transfer function in frequency domain • This is the definition of longitudinal beam coupling impedance of the element under study [W] [W/s]

21. Longitudinal impedance: resonator W|| Re[Z||] Im[Z||] • The frequency wr is related to the oscillation of Ez, and therefore to the frequency of the mode excited in the object • The decay time depends on how quickly the stored energy is dissipated (quantified by a quality factor Q) wr T=2p/wr

22. Longitudinal impedance: cavity • A more complex example: a simple pill-box cavity with walls having finite conductivity • Several modes can be excited • Below the pipe cut-off frequency the width of the peaks is only determined by the finite conductivity of the walls • Above, losses also come from propagation in the chamber Re[Z||] Im[Z||]

23. Single bunch effects Re[Z||] Im[Z||] W||

24. Single bunch effects W||

25. Multi bunch effects Re[Z||] Im[Z||]

26. Multi bunch effects W||

27. Multi bunch effects Dz

28. Example: the Robinson instability • To illustrate the Robinson instability we will use some simplifications: • The bunch is point-like and feels an external linear force (i.e. it would execute linear synchrotron oscillations in absence of the wake forces) • The bunch additionally feels the effect of a multi-turn wake dp/p0 z Unperturbed: the bunch executes synchrotron oscillations at ws

29. The Robinson instability • To illustrate the Robinson instability we will use some simplifications: • The bunch is point-like and feels an external linear force (i.e. it would execute linear synchrotron oscillations in absence of the wake forces) • The bunch additionally feels the effect of a multi-turn wake dp/p0 z The perturbation also changes the oscillation amplitude Unstable motion The perturbation changes ws

30. The Robinson instability • To illustrate the Robinson instability we will use some simplifications: • The bunch is point-like and feels an external linear force (i.e. it would execute linear synchrotron oscillations in absence of the wake forces) • The bunch additionally feels the effect of a multi-turn wake dp/p0 z The perturbation also changes the oscillation amplitude Damped motion

31. 2. The transverse plane

32. Transverse wake function: definition Source, q1 Witness, q2 z 2b L • In an axisymmetric structure (or simply with a top-bottom and left-right symmetry) a source particle traveling on axis cannot induce net transverse forces on a witness particle also following on axis • At the zero-th order, there is no transverse effect • We need to introduce a breaking of the symmetry to drive transverse effect, but at the first order there are two possibilities, i.e. offset the source or the witness

33. Transverse dipolar wake function: definition Source, q1 Witness, q2 Dx1 (or Dy1) z 2b L

34. Transverse dipolar wake function • The value of the transverse dipolar wake functions in 0,Wx,y(0), vanishes because source and witness particles are traveling parallel and they can only – mutually – interact through space charge, which is not included in this framework • Wx,y(0--)<0since trailing particles are deflected toward the source particle (Dx1 and Dx’2 have the same sign) • Wx,y(z) has a discontinuous derivative in z=0 and it vanishes for all z>0 because of the ultra-relativistic approximation Wx,y(z) z

35. Discrete approximation of cavity transverse impedance Impedances in frequency domain calculated using electromagnetic solvers. Impedance can be approximated using a number of discrete modes (above 9 modes used). From CLIC decelerator design

36. Dipole wake instabilities in linacs Single bunch: head drives tail resonantly -> banana shape, beam-break up LinacBeam break-up growth factor: ϒ~NWs/kb N: charge W: dipole wake s: distance kb: betatron k Two-bunches: one bunch drives the second resonantely

37. Rings: A glance into the head-tail modes • Different transverse head-tail modes correspond to different parts of the bunch oscillating with relative phase differences. E.g. • Mode 0 is a rigid bunch mode • Mode 1 has head and tail oscillating in counter-phase • Mode 2 has head and tail oscillating in phase and the bunch center in opposition

38. Calculation of coherentmodesseen at a wide-band pick-up (BPM) • h • The patterns of thehead-tailmodes (m,l) depend on chromaticity Q’=0 Q’≠0 m=1 and l=0  m=1 and l=1  m=1 and l=2 

39. Coherentmodesmeasured at a wide-bandpick-up (BPM) m=1 and l=0 1 Instabilities are a good diagnostics tool to identify and quantify the main impedance sources in a machine.

40. Wake field (impedances) in accelerator ring design Thefull ring isusuallymodeledwith a so calledtotal impedancemade of threemaincomponents: • Resistive wallimpedance • Severalnarrow-bandresonators at lowerfrequenciesthanthepipecutofffrequencyc/b (bbeampiperadius) • One broad band resonator at wr~c/bmodelingtherest of the ring (pipediscontinuities,tapers, othernon-resonantstructureslikepick-ups, kickersbellows, etc.) • Thetotal impedanceisallocated to thesingle ring elementsbymeans of off-linecalculationprior to construction/installation • Total impedancedesigned such thatthe nominal intensityisstable

41. We will talk more about wake fields in the lecture about linear colliders (tomorrow). For more details on wake fields, derived from first physical principles see the excellent book “Physics of Collective Beam Instabilities in High Energy Accelerators”, A. W. Chao (freely available; see course web pages).

42. Part II

43. Overview of multi-particle effects Adapted from G. Rumolo’s slides from USPAS Course on collective effects

44. General definition of multi-particleprocessesin an acceleratororstorage ring • Class of phenomena in whichtheevolution oftheparticlebeamcannotbestudied as ifthebeam was a singleparticle (as isdone in beamoptics), butdepends on thecombination of externalfieldsand interactionbetweenparticles. Particlescaninteractbetweenthemthrough • Selfgeneratedfields: • Directspacechargefields • Electromagneticinteraction of thebeamwiththesurroundingenvironmentthroughthebeam‘sownimages and thewakefields (impedances) • Interaction withthebeam‘sownsynchrotronradiation • Long- and short-rangeCoulombcollisions,associated to intra-beamscattering and Touschekeffect, respectively • Interaction of electronbeamswithtrappedions,proton/positron/ionbeamswithelectronclouds, beam-beamin a collider ring, electroncoolingforions Multi-particleprocessesaredetrimentalforthebeam (degradation and loss, seenextslides)

45. Direct space charge forces (more details later in this lecture) x • Transversespacecharge • Force decayslike 1/g2 • Itisalwaysrepulsive y • Longitudinal spacecharge • Force decayslike 1/g2 • itcanbeattractiveabovetransition dp/p0 z

46. Wake fields, impedances (more details later in this lecture) z W0(z) L s e q Model: A rigid beam withchargeqgoingthrough a deviceoflength L leavesbehindan oscillatingfieldand a probe chargee at distance zfeelsa force as a result. Theintegral of this force overthedevicedefinesthewakefield anditsFouriertransformiscalledtheimpedance of thedevice of length L.

47. Electron cloud Principle of electronmultipacting: Example of LHC • Electronmultiplicationismadepossibleby: • Electrongenerationdue to photoemission, but also residual gas ionization • Electronacceleration in thefield of thepassingbunches • Secondaryemissionwithefficiency larger thanone, whentheelectronshitthe inner pipewallswith high enoughenergy

48. Severalnames to describetheseeffects... • ‚Multi-particle‘ isthemostgenericattribute. ‚High-current‘, ‚high-intensity‘, ‚high brightness‘ are also usedbecausetheseeffectsareimportantwhenthebeam has a high density in phasespace (manyparticles in littlevolume) • Otherlabelsare also used to refer to different subclasses • Collectiveeffects (coherent): • Thebeamresonantlyresponds to a self-inducedelectromagneticexcitation • Are fast and visible in thebeamcentroidmotion(tune shift, instability) • Collectiveeffects (incoherent): • Excitationmoveswiththebeam, spreadsthefrequencies of particlemotion. • Lead to particlediffusion in phasespace and slowemittance growth • Collisionaleffects (incoherent): • Isolatedtwo-particleencountershave a global effect on thebeamdynamics (diffusion and emittance growth, lifetime) • Two-streamphenomena (coherentorincoherent): • Twocomponentplasmasneeded (beam-beam, pbeam-ecloud, ebeam-ions) and thebeamreacts to an excitationcausedbyanother „beam“ The performanceof an acceleratorisusually limited by a multi-particleeffect. Whenthe beam current in a machineispushedabove a certainlimit (intensitythreshold), intolerable lossesor beam qualitydegradationappear due tothesephenomena

49. Example of coherent and incoherent effects Incoherent: intra-beam scattering • Particles within a bunch can collide with each other as they perform betatron and synchrotron oscillations. The collisions lead to a redistribution of the momenta within the bunch, and hence to a change in the emittances. • This is effect is called IBS, intra-beam scattering. If there is a large transfer of momentum into the longidunal plane. It is called Touchek scattering. • May lead to emittance growth and reduced beam life time. Coherent: coherent synchrotron radiation (CSR) • Previous calculations of synchrotron radiation in this course assumed each particle radiates independently  Prad N • If particles are close with respect to the radiation wavelengths, the particles will radiate coherently (as one macro particle),  Prad N2 • Significant effect for short bunch lengths, low energy beams, with large number of particles per bunch • May lead to instabilities

50. More examples Bunch in the CERN SPS synchrotron BNL-RHIC, Au-Au operation, Run-4 (2004) 16h Coherent effects: When the bunch current exceeds a certain limit (current threshold), the centroid of the beam, e.g. as seen by a BPM, exhibits an exponential growth (instability) and the beam is lost within few milliseconds