Collective effects

Erik Adli, University of Oslo, August 2015, Erik.Adli@fys.uio.no, v2.12

Introduction• Particle accelerators are continuously

being pushed to new parameter regimes with higher currents, higher power and higher intensity

• Performance is usually limited by multi-particle effects, including collective instabilities and collisions

• 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 Proton Synchrotron the last decades

Compressed high energy electron beam (at the “FACET” facility at SLAC) :

Intense beams: example

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 / √2 p sz = 19 kA (cf. IAlfven = 17 kA)• Beam density: ne ≈ N / 4/3p(sx sy sz) = 6 x 1017 / cm3

Plasma description: • Electron transverse energy: Ee = ½ mvx

2 = 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)

Space chargeA general definition of “Space Charge” : a collection of particles with a net electric charge occupying a region, either in free space or in a device.

In accelerators, we usually use the term space charge to describe the collective field/force in a same-charge particle bunch.

Consider the bunch from the previous example, at rest. We may do a rough estimate of the space charge field using a two-particle model:

This is a very large field (cf. accelerating fields discussed earlier). Using the two particle model, we may estimate the force, and the time scale for the bunch to blow-up due to space charge forces to sub-picosecond. How can we accelerator dense particles bunches at all?

( Later, we give a more detailed calculations, assuming a Gaussian distribution. )

+

+

F

FQ/2

Q/2~2sx,y

Coordinates – speed of light frame

sz = s - vt

x

The beam travels in the s direction, with speed v. The co-moving coordinate z = s-vt is defined in a frame following the beam travelling with speed v , and gives thus the relative position inside the beam. In plasma wakefield applications, the beam is often travelling with velocity v = c. In this case the frame where z = s- ct is defined is called the speed of light frame.

In accelerator physics we usually describe fields and charges in the laboratory frame, however, following the co-moving frame. NB: this is not the same as describing the fields and charges in the rest frame of the beam, as quantities described in the co-moving frame are not Lorenz transformed with respect to the lab frame.

Space charge – moving charges

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 calculate the relativistic space charge suppression :

Fx,y = eEx,y (1±v1v2/c2)

x’

y'

z

v2

x

y

z

v1

Two-particle interaction :

Fx,y = eEx,y (1-v2/c2) = eEx,y / g2

xlab

EM fields from a relativistic particle

v=0

Fields in particle’s rest frame Fields in lab frame

Ultra-relativistic limit (v=c) :

Lorentz Tranformations

Field compression-> “Pancake field”

Spherical symmetrical field

Space Charge – using Gauss’ law

Alternate calculation: 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) :

+

+

F

F

+

+

FE

FE

vFB

FB

+

+

c

c

rEr02

rc

vB

202

Rest frame Lab frame

Lab frame, v = cThis is the direct space charge effect. “Direct” : does not take into account the effects of conducting walls surrounding the beam.

• 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 Lorenz factor• For the same electron beam, but at 20 MeV : KSC = 5 x 104 m-2 >> Kb

KSC ~ Ipeak / (bg)3 “beam perveance”

Space Charge for Gaussian Beams

K. Schindl

Summary

Linear defocusing. Gives tune shift in rings. Can be compensated by

stronger lattice focusing.

Non-linear defocusing. Gives tune shift and tune spread in rings.

Beam-beam effectsRelativistic space charge suppression holds only for equal charge moving at the same velocity, in the same direction (even 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.

Fx,y = eEx,y (1+v1v2/c2)For v1 and v2 of opposite sign in our previous calculation (5 slides ago) we get :

Wake fields

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

13

Wake fields (general)

z z2b z

Source, q1

Witness, q2

– While source and witness ( qi d(s-ct) ), at a distance z, 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)o The source loses energyo The witness feels a net force all along an effective length of the structure, L

L

Wake field characteristics

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 calculated2) 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.

16

Wake fields (general)

2b

Source, q1

Witness, q2

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 fieldso No ringing, only slow decayo 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.

z

z

1. The longitudinal plane

dp/p0

18

Longitudinal wake function: definition

2b z

Source, q1

Witness, q2

L

19

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

20

The energy balance

– In the global energy balance, the energy lost by the source splits into o 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

o Electromagnetic energy of modes that propagate down the beam chamber (above cut-off), which will be eventually lost on surrounding lossy materials

What happens to the energy lost by the source?

21

The energy balance

– In the global energy balance, the energy lost by the source splits into o 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!

o Electromagnetic energy of modes that propagate down the beam chamber (above cut-off), which will be eventually lost on surrounding lossy materials

What happens to the energy lost by the source?

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

22

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] [ /Ws]

23

Longitudinal impedance: resonator

W||Re[Z||]

Im[Z||]

T=2p/wr

wr

– 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)

24

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||]

25

Single bunch effects

W||

Re[Z||]Im[Z||]

26

Single bunch effects

W||

27

Multi bunch effects

Re[Z||]

Im[Z||]

28

Multi bunch effects

W||

29

Multi bunch effects

Dz

30

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

z

dp/p0

Unperturbed: the bunch executes synchrotron oscillations at ws

31

The Robinson instability

z

dp/p0

The perturbation changes ws

The perturbation also changes the oscillation amplitudeUnstable motion

– 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

32

The Robinson instability

z

dp/p0

The perturbation also changes the oscillation amplitude Damped motion

– 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

2. The transverse plane

34

Transverse wake function: definition

2b z

Source, q1

Witness, q2

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

35

Transverse dipolar wake function: definition

2b z

Source, q1

Witness, q2

L

Dx1 (or Dy1)

36

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

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

Dipole wake instabilities in linacs

Linac Beam break-up growth factor:

ϒ~NWs/kb

N: chargeW: dipole wakes: distancekb: betatron k

Single bunch: head drives tail resonantly -> banana shape, beam-break up

Two-bunches: one bunch drives the second resonantely

39

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

40

m=1 and l=0

m=1 and l=1

m=1 and l=2

Calculation of coherent modes seen at a wide-band pick-up (BPM) h

• The patterns of the head-tail modes (m,l) depend on chromaticity

Q’=0 Q’≠0

41

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.

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.)

Þ The total impedance is allocated to the single ring elements by means of off-line calculation prior to construction/installation

Þ Total impedance designed such that the nominal intensity is stable

Wake field (impedances) in accelerator ring design

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).

Part II

Overview of multi-particle effects

Adapted from G. Rumolo’s slides fromUSPAS Course on collective effects

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)

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

z

dp/p0

Longitudinal space charge

• Force decays like 1/g2

• it can be attractive above transition

x

y

Transverse space charge

• Force decays like 1/g2

• It is always repulsive

Direct space charge forces (as discussed in more details earlier)

W0(z)

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.

q

z

e s

Wake fields, impedances (as discussed in more details earlier)

L

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

s

W(z-z‘)l(z‘)dz‘

z

W

s

W(kd) (N kd)

W

d

Wake decays over many bunches Coupled bunch collective interaction

Principle of electron multipacting:

Example of LHC

Electron cloud

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

Example of coherent vs. 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.

More examples

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

BNL-RHIC, Au-Au operation, Run-4 (2004)

16h

Bunch in the CERN SPS synchrotron

Multi-particle effects

CollectiveIncoherent/collisionalTwo-stream

Transverse Longitudinal

Instability/beam loss Beam quality degradation/ emittance growthCoherent tune shift

Head-tail TMCI

E-cloud/trapped ions

Beam/beam

IBS/Touschek

Incoherent tune spreadPotential well

distortion

Microwave/turbulent

Multi- bunch