1. Collective effects
Erik Adli, University of Oslo, August 2014, 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(sx sy sz) = 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 : x’ y' z v2
xlab x y z v1 s
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 - v1 x (v2/c2 x E)), or for parallel velocities :
For v1 = v2 = v we get relativistic space charge surpression :
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. Space Charge – lab frame+ FE v FB + F
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 c + 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 GeV electron 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. Non-linear defocusing. Gives tune shift and tune spread in rings.
Summary
K. Schindl
Linear defocusing. Gives tune shift in rings. Can be compensated by stronger lattice.
Non-linear defocusing. Gives tune shift and tune spread in rings.

9. Beam-beam effects Fx,y = eEx,y (1+v1v2/c2)
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) 2b z z z L
Source, q1
Witness, q2 z z z 2b L
While source and witness ( qi d(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) 2b z L
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.
This is like a Green function, we need to define which is the pulse response (in time domain), in which we are interested!

15. 1. The longitudinal planez
dp/p0
1. The longitudinal plane

16. Longitudinal wake function: definition
Source, q1
Witness, q2 z 2b L
This is like a Green function, we need to define which is the pulse response (in time domain), in which we are interested!

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)>0 since 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 following particles (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 The energy loss is very important because
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 following particles (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
T=2p/wr wr
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)

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
This is like a Green function, we need to define which is the pulse response (in time domain), in which we are interested!
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
This is like a Green function, we need to define which is the pulse response (in time domain), in which we are interested!

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--)<0 since 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
Linac Beam 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 coherent modes seen at a wide-band pick-up (BPM)
The patterns of the head-tail modes (m,l) depend on chromaticity h
Q’=0
Q’≠0
 m=1 and l=0 
 m=1 and l=1 
 m=1 and l=2 

39. Coherent modes measured at a wide-band pick-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
The full ring is usually modeled with a so called total impedance made of three main components:
Resistive wall impedance
Several narrow-band resonators at lower frequencies than the pipe cutoff frequency c/b (b beam pipe radius)
One broad band resonator at wr~c/b modeling the rest of the ring (pipe discontinuities, tapers, other non-resonant structures like pick-ups, kickers bellows, etc.)
The total impedance is allocated to the single ring elements by means of off-line calculation prior to construction/installation
Total impedance designed such that the nominal intensity is stable

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-particle processes in an accelerator or storage ring
Class of phenomena in which the evolution of the particle beam cannot be studied as if the beam was a single particle (as is done in beam optics), but depends on the combination of external fields and interaction between particles. Particles can interact between them through
Self generated fields:
Direct space charge fields
Electromagnetic interaction of the beam with the surrounding environment through the beam‘s own images and the wake fields (impedances)
Interaction with the beam‘s own synchrotron radiation
Long- and short-range Coulomb collisions, associated to intra-beam scattering and Touschek effect, respectively
Interaction of electron beams with trapped ions, proton/positron/ion beams with electron clouds, beam-beam in a collider ring, electron cooling for ions
Multi-particle processes are detrimental for the beam (degradation and loss, see next slides)

45. Direct space charge forces (more details later in this lecture)x
Transverse space charge
Force decays like 1/g2
It is always repulsive y
Longitudinal space charge
Force decays like 1/g2
it can be attractive above transition
dp/p0 z

46. Wake fields, impedances (more details later in this lecture)z
W0(z) L s e q
Model:
A rigid beam with charge q going through a device of length L leaves behind an oscillating field and a probe charge e at distance z feels a force as a result. The integral of this force over the device defines the wake field and its Fourier transform is called the impedance of the device of length L.

47. Electron cloud Principle of electron multipacting: Example of LHC
Electron multiplication is made possible by:
Electron generation due to photoemission, but also residual gas ionization
Electron acceleration in the field of the passing bunches
Secondary emission with efficiency larger than one, when the electrons hit the inner pipe walls with high enough energy

48. Several names to describe these effects...
‚Multi-particle‘ is the most generic attribute. ‚High-current‘, ‚high-intensity‘, ‚high brightness‘ are also used because these effects are important when the beam has a high density in phase space (many particles in little volume)
Other labels are also used to refer to different subclasses
Collective effects (coherent):
The beam resonantly responds to a self-induced electromagnetic excitation
Are fast and visible in the beam centroid motion (tune shift, instability)
Collective effects (incoherent):
Excitation moves with the beam, spreads the frequencies of particle motion.
Lead to particle diffusion in phase space and slow emittance growth
Collisional effects (incoherent):
Isolated two-particle encounters have a global effect on the beam dynamics (diffusion and emittance growth, lifetime)
Two-stream phenomena (coherent or incoherent):
Two component plasmas needed (beam-beam, pbeam-ecloud, ebeam-ions) and the beam reacts to an excitation caused by another „beam“
The performance of an accelerator is usually limited by a multi-particle effect. When the beam current in a machine is pushed above a certain limit (intensity threshold), intolerable losses or beam quality degradation appear due to these phenomena

49. Example of coherent and incoherent effects
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
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.

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

51. Single bunch versus multi-bunch effets
Single or multi-bunch behavior depends on the range of action of the wake fields
Single bunch effects are usually caused by short range wake fields (broad-band impedances)
Multi bunch or multi-turn effects are usually associated to long range wake fields (narrow-band impedances)
Wake decays over a bunch length  Single bunch collective interaction W
Wake decays over many bunches  Coupled bunch collective interaction W
W(z-z‘)l(z‘)dz‘
W(kd)N(kd) z s d s

52. Multi-particle effects
Collective
Two-stream
Incoherent/collisional
Transverse
Longitudinal
Beam/beam
E-cloud/trapped ions
IBS/Touschek
Multi- bunch
Microwave/turbulent
Potential well distortion
Head-tail
TMCI
Incoherent tune spread
Instability/beam loss
Coherent tune shift
Beam quality degradation/ emittance growth