49th icfa advanced beam dynamics workshop electron cloud induced instabilities… · 2010-10-13 ·...

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


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.


Accelerators

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

high-energy particles

September 24, 2010 ECLOUD`10 - Cornell University 3


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


•The focusing magnets

(quadrupoles) provide the

restoring forces needed for

stable oscillations.

•The RF cavity provides

energy to the beam.


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:

Circular orbit (~ 760 m in CESR)

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


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

of the potential energy function,

determines the amplitude of the

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determines the amplitude of the

oscillation.

•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”.


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.


Dots represent the position and velocity

of a beam particle at point A in the ring,

on each turn

Area=emittance

A

Beam particlePhase plot

at point A


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


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

oscillation 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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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”.


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

oscillation 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.


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

of 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


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


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 Bunches

generated by previous

bunches.

• Hence the effects of the cloud

on 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


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

oscillations by looking at its vertical position at

one point in the ring on subsequent turns:Measurement pointKICK

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


Measurement of the tune-II

We then Fourier analyze the motion to extract the characteristic

oscillation 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.


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


A 21 bunch train, followed by witnesses

SEY=2.0SEY=.2.2SEY=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.


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

oscillators.

September 24, 2010 19ECLOUD`10 - Cornell University


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

of phase.

m=0


of 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

oscillating electron cloud, one

or both of the modes may be

unstable.

m=1


-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 numberSnapshot of the most unstable mode


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.



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

HEADTAIL

Direction

of motion

than the “head”.

September 24, 2010 23ECLOUD`10 - Cornell University


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.

TAILHEAD

Beam number density

Electron cloud density near the beam


•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.


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

oscillations 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.


HEAD

TAIL

After ½ of a

synchrotron

period


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.


m=1


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 oscillation amplitude of the bunch at the synchrotron frequency. •The Fourier spectrum of


•The Fourier spectrum of this modulated motion exhibits modulation sidebands which are the signals for the presence of head-tail motion.

Oscillation amplitude vs. time of a

bunch undergoing head-tail motion, as

observed 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.


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


sideband


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

observed,

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


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.


• Instabilities can involve the coupling of one bunch in a train to the

other (multi-bunch instabilities), or the coupling of particles in the tail

of 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


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

of single-bunch instabilities.

• In some future machines, non-linear emittance growth may also be

important. This is a subject of active study.

49th icfa advanced beam dynamics workshop electron cloud induced instabilities… · 2010-10-13 ·...

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