Electron cloud Induced Instabilities, Non-Linear Beam Dynamics, and - - PowerPoint PPT Presentation

49th ICFA Advanced Beam Dynamics Workshop

Electron cloud Induced Instabilities, Non-Linear Beam Dynamics, and Emittance Growth

• G. Dugan, Cornell University

ECLOUD’10 WORKSHOP 10/8/10

10/13/2010

49th ICFA Advanced Beam Dynamics Workshop

Electron clouds and particle beams

• You have heard about how electron clouds are formed and can

build up in the vacuum chambers of accelerators.

• The high-energy particle beam in the accelerator has to share

the “vacuum” chamber with the electron cloud, and does not like it.

• Note that the dominant effects are present for positively

charged beams (e.g., protons, positrons), since in these cases

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charged beams (e.g., protons, positrons), since in these cases the beam attracts the electron cloud and can be strongly influenced by it.

• In this talk, we will discuss the effects that electron clouds can

have on the dynamics of particle beams in accelerators. The emphasis will be on positron beams and experiments at CESR.

49th ICFA Advanced Beam Dynamics Workshop

Accelerators

• An accelerator is a device used to produce a beam of

high-energy particles

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49th ICFA Advanced Beam Dynamics Workshop

Cornell Electron Storage Ring

• The Cornell Electron Storage Ring (CESR) is a circular

accelerator of a type called a “synchrotron”.

• The particles in the beam (positrons) travel at very close to

light speed in roughly circular orbits of circumference about 760 m.

• The bending magnets

(dipoles) provide the bending needed for a circular orbits.

• The focusing magnets

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• The focusing magnets

(quadrupoles) provide the restoring forces needed for stable oscillations.

• The RF cavity provides

energy to the beam.

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Particle beam oscillations

• The particles in the beam (positrons), traveling in

approximately plane circular orbits within a vacuum chamber, execute small-amplitude oscillations about the plane:

Beam particle

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Circular orbit (~ 760 m in CESR) Oscillation (amplitude << 1 mm)

In the accelerator, the focusing forces which provide the stability for the oscillations are provided by the quadrupole magnets, arranged in a “lattice”.

Vacuum chamber not shown

49th ICFA Advanced Beam Dynamics Workshop

Particle beam dynamics

• Here is a very simple analogy for the transverse motion of a beam

particle: a particle in a quadratic potential well.

• A beam particle oscillates in the

potential well. The total energy of the particle, together with the curvature

• f the potential energy function,

determines the amplitude of the

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determines the amplitude of the

• scillation.
• The (linear) forces responsible for the

quadratic potential energy are provided by quadrupole magnets in the accelerator.

• The frequency of the oscillations is

called the “tune”.

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Phase space and emittance

• A plot of the position vs. velocity of the particle as measured at some

point in the ring, on subsequent turns, is called a “phase space” plot.

• Over many cycles of the oscillation, if the forces are linear, the particle

will trace out an ellipse in phase space.

• The area contained within the ellipse is related to the amplitude of

the oscillations, and is called the “emittance”.

• At different points in the ring, the orientation and shape of the ellipse

will be different, but the area will always be the same. will be different, but the area will always be the same.

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Dots represent the position and velocity

• f a beam particle at point A in the ring,
• n each turn

Area=emittance A Beam particle Phase plot at point A

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Beam quality: emittance

• When we have a collection of

beam particles with different emittances in the accelerator, the projection of the collection’s points onto the position axis is the beam position distribution at that point in the ring. It is determined by the average emittance.

• For many applications, the beam

Phase plot at point A

• For many applications, the beam

size should be kept as small as possible: hence the average emittance should be as low as possible.

• The most important reason to

control the electron cloud in accelerators is to prevent any growth in the emittance.

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Beam size at point A Beam distribution at point A

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Effects of the electron cloud-I

• The electron cloud constitutes a

charge distribution which can exert forces on the beam. This changes the effective potential energy curve.

• If these forces are linear with

displacement, the change is reflected in a change in the

• scillation frequency, but no

change in the emittance. change in the emittance.

• If the forces are non-linear, the

shape of the potential well is distorted, chaotic motion ensues, and the beam emittance increases.

• This source of emittance growth is

usually small, but it could be important for future accelerators which require very low emittance beams.

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49th ICFA Advanced Beam Dynamics Workshop

Effects of the electron cloud-II

• The electron cloud can move under

the influence of the fields of the beam, so that the beam and the cloud interact dynamically.

• The beam-cloud interaction will result

in each system undergoing oscillations driven by the other system.

• If this mutual interaction is strong

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• If this mutual interaction is strong

enough, unstable motion of the beam can result, increasing its emittance and possibly even driving it into the vacuum chamber walls.

• In plasma physics terminology, this is

called a “two-stream instability”.

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Effects of the electron cloud

Summarizing the effects of the electron cloud on the beam:

• 1. At low cloud densities, the linear forces exerted by the cloud change the
• scillation frequency (tune) of the beam.
• 2. At higher densities, the smaller non-linear forces can cause chaotic motion of

the beam particles, leading to growth in the emittance (and size) of the beam.

• 3. At still higher densities, the mutual dynamic interaction between the beam

and the cloud can cause the beam’s motion to become unstable, leading to very large growth in the emittance and possibly also loss of the beam from the

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large growth in the emittance and possibly also loss of the beam from the vacuum chamber.

• Effect 2 can be observed by precise measurements of the beam size in the

presence of the cloud.

• We will look a little more closely at effects 1 and 3, and how they can be

measured in CESR-TA.

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Bunches and trains

• It is important to understand how the beam is

“formatted” in CESR.

• The circulating beam is a string of “bunches”

which form a “train”. Each bunch is a collection

• f around 1010 beam particles. The length of a

bunch is about 2 cm (~ 0.1 x 10-9 s), and they are typically separated by about 4.2 m (14 x 10-9 s).

• The bunches can be “loaded” to form trains of

Bunch length

• from 1 to ~ 500 bunches.
• In a given experiment, the number of particles in

a bunch, the bunch spacing, and the number of bunches in a train can be varied.

• In the experiments I will describe at CESR, the

length of the train is much less than the circumference, so there is a big “gap” after each train.

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Bunches in the train z

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Electron cloud build-up

• As you have heard from previous speakers,

each bunch emits synchrotron radiation in each bunch emits synchrotron radiation in the bending magnets, which strikes the vacuum chamber wall, creating photoelectrons.

• These photoelectrons, together with

secondary electrons they produce, form the electron cloud.

• The cloud from each bunch decays as

electrons are absorbed on the vacuum chamber walls.

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Bunch Bunch

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Electron cloud build-up

• If the bunches are closer

together than the cloud decay time, the electron cloud builds up along the train, so that each later bunch in the train sees the electron cloud generated by previous generated by previous bunches.

• Hence the effects of the cloud
• n the beam are greater for

bunches later in the train, than for earlier bunches.

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• The change in oscillation

frequency (“tune shift”) due to the electron cloud is related to the cloud density, and it will increase for each later bunch in the train.

Bunches

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Measurement of the tune-I

To measure the frequency of oscillation of a bunch, we “kick” the bunch (give it an initial vertical oscillation amplitude using a time-varying magnetic field) and then observe its subsequent

• scillations by looking at its vertical position at
• ne point in the ring on subsequent turns:

Measurement point

KICK

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Bunch vertical position at measurement point vs. turn number after “kick”

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Measurement of the tune-II

We then Fourier analyze the motion to extract the characteristic

• scillation frequency:

Vertical oscillation frequency

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We do this for each bunch in the train. The change in frequency along the train is due in part to the influence of the electron cloud’s electric fields, which are building up during the train.

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Cloud buildup along a train at CesrTA

The electron cloud builds up during a 45-bunch train, resulting in an increase in the vertical oscillation frequency with bunch number. We can compare the data (black) with a calculation (red) from an electron cloud simulation program (POSINST). Electron cloud builds up during the train

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A 21 bunch train, followed by witnesses

SEY=2.0 SEY=.2.2 SEY=1.8

Electron cloud decay Electron cloud builds up during the train Electron cloud decay after the train is measured using “witness bunches” placed at different times after the train, to probe the cloud at that time.

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Multi-bunch instabilities

• Instabilities can be caused by the

dynamic interplay between the cloud and the bunches in the train.

• The electron cloud can couple

the transverse motion of bunch N+1 to bunch N (and all previous N+1 to bunch N (and all previous bunches in the train).

• The motion of the bunches in

the train will be very similar to the motion of any set of coupled

• scillators.

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Normal modes for 2 bunches

• For the simplest case of two

evenly-spaced bunches in the ring, there will be two normal modes of oscillation: the “m=0” mode in which the two bunches oscillate in phase, and the “m=1” mode in which the two bunches oscillate out

• f phase.

m=0

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• f phase.
• Each mode will have a

specific frequency, determined in part by the electron cloud’s electric fields.

• Depending on the relative

phase of the bunches and the

• scillating electron cloud, one
• r both of the modes may be

unstable. m=1

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• Example of a multi-bunch instability

Here is an example of a calculation of an electron-cloud driven horizontal multi-bunch instability in Cesr-TA. The estimate is done for a train of 63 bunches with 42 ns spacing, 0.5 mA/bunch, positrons. The most unstable mode has a growth time of about 8.5 ms.

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Horizontal growth rate vs. mode number Snapshot of the most unstable mode

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Single-bunch instabilities-I

• To control instabilities, we can use feedback systems, in which we

detect the motion of each bunch and apply a “kick” to cancel out the detected motion.

• Multi-bunch instabilities have a growth time which is low enough

that they can typically be stabilized by state-of-the-art transverse feedback systems.

• For this reason, they are not a major concern.
• For this reason, they are not a major concern.
• However, the electron cloud can also give rise to single-bunch

instabilities, which occur within a single bunch.

• In positron storage rings, the short bunches mean that these

instabilities have a very high frequency spectrum, with rapid growth times, and typically cannot be controlled by conventional feedback.

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49th ICFA Advanced Beam Dynamics Workshop

Single-bunch instabilities-II

• In a single-bunch instability, the electron

cloud couples the motion of the “head” of the bunch with the “tail”.

• A unique feature of this interaction is that

the “head” of the bunch “pinches” the cloud due to the attraction between the beam and the cloud.

• The “tail” then gets a much stronger kick

than the “head”.

Bunch length HEAD TAIL Direction

• f motion

than the “head”.

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Cloud pinch

Here is a result from a cloud simulation program (POSINST) showing the increase in cloud density during the passage of bunch 17 (in a 20 bunch train) through a slice of the cloud. TAIL HEAD Beam number density Electron cloud density near the beam

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• Thus the tail of the bunch can be driven by motion of the head, amplified

by the “pinched” electron cloud.

• Since the tail and head have the same vertical oscillation frequency, the

tail is driven on resonance and a large amplitude could be expected to develop rapidly.

• However, in a circular machine, the beam particles in the head and the tail

exchange places due to their synchrotron motion.

49th ICFA Advanced Beam Dynamics Workshop

Synchrotron motion

• The beam particles have a range of beam

energies typically ~ 0.1% about the mean energy.

• The transit time of a beam particle around

the ring depends on its energy.

• The radiofrequency (RF) cavities in the ring

provide a time-dependent energy gain or loss which results in stable oscillations of energy of each particle about the mean

Energy- time phase space TAIL HEAD

energy of each particle about the mean energy of the beam. The frequency of these

• scillations is called the “synchrotron

frequency”.

• The dependence of the transit time on

energy and the energy oscillation results in the exchange of particles between the head and the tail at twice the synchrotron frequency.

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HEAD TAIL After ½ of a synchrotron period

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Head-tail modes

• The head-tail exchange tends to

stabilize the motion, especially if the tune also has some dependence on the energy (“chromaticity”), but with sufficiently high density clouds, instability can still occur.

• The head and tail motion is

m=0

• The head and tail motion is

described in terms of normal modes, just as for two bunches. In one mode, the head and tail are in-phase; in the other mode, they are out-of-phase.

• One or both of the normal

modes may be unstable.

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m=1

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Head-tail motion

• When the head and tail

are oscillating in both the m=1 mode and the m=0 mode, this is manifested as a modulation of the

• scillation amplitude of

the bunch at the synchrotron frequency.

• The Fourier spectrum of

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• The Fourier spectrum of

this modulated motion exhibits modulation sidebands which are the signals for the presence

• f head-tail motion.

Oscillation amplitude vs. time of a bunch undergoing head-tail motion, as

• bserved on a dipole pickup

This head-tail motion leads to an increase in the effective beam size, and is the dominant cause of beam emittance growth due to the electron cloud.

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Synchrotron sidebands

• Fourier spectrum of a bunch undergoing head-tail motion

Spectral line due to vertical oscillation; frequency ~ 222 kHz Synchrotron frequency ~ 25 kHz Synchrotron (modulation) sideband

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sideband

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When the electron cloud induces head-tail motion, we see the appearance and subsequent growth of these modulation sidebands along the bunch train

Synchrotron sideband growth along the train

Beam size growth is also

• bserved,

typically coincident with the sidebands.

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• At low cloud densities, the linear forces exerted by the electron

cloud change the oscillation frequency (tune) of the beam. We can use the measured tune shifts to test models which predict the growth of the electron cloud.

• At higher cloud densities, the smaller non-linear forces can cause

chaotic motion of the beam particles, leading to growth in the emittance of the beam. This emittance growth may be small, but

Conclusions - I

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emittance of the beam. This emittance growth may be small, but could be important for future accelerators which require very low emittance beams.

• At still higher densities, the mutual dynamic interaction between

the beam and the cloud can cause the beam’s motion to become unstable, leading to very large growth in the emittance and possibly also loss of the beam from the vacuum chamber.

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• Instabilities can involve the coupling of one bunch in a train to the
• ther (multi-bunch instabilities), or the coupling of particles in the tail
• f one bunch to those in the head of the same bunch (single-bunch

instabilities). Both varieties will be produced by the electron cloud.

• Multi-bunch instabilities typically have slow growth times and can be

controlled by conventional beam feedback systems.

• Single-bunch instabilities are more dangerous and not easily
• controllable. They can lead to rapid emittance growth.

Conclusions

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• controllable. They can lead to rapid emittance growth.
• In the design of measures to mitigate the development of the

electron cloud in existing or future accelerators, the design requirement usually taken is to be below the threshold for the onset

• f single-bunch instabilities.
• In some future machines, non-linear emittance growth may also be
• important. This is a subject of active study.