Syllabus and slides

Syllabus and slidesSyllabus and slides

• Lecture 1: Overview and history of Particle accelerators (EW)• Lecture 2: Beam optics I (transverse) (EW)• Lecture 3: Beam optics II (longitudinal) (EW)• Lecture 4: Liouville's theorem and Emittance (RB)• Lecture 5: Beam Optics and Imperfections (RB)• Lecture 6: Beam Optics in linac (Compression) (RB)• Lecture 7: Synchrotron radiation (RB)• Lecture 8: Beam instabilities (RB)• Lecture 9: Space charge (RB)• Lecture 10: RF (ET)• Lecture 11: Beam diagnostics (ET)• Lecture 12: Accelerator Applications (Particle Physics) (ET)• Visit of Diamond Light Source/ ISIS / (some hospital if possible)

The slides of the lectures are available at

http://www.adams-institute.ac.uk/training/undergraduate

Dr. Riccardo Bartolini (DWB room 622) [email protected]

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Lecture 8: wakefields and basic principles of beam instability

Beam Instabilities

High current operation in storage rings

Wakefields

Single bunch effects

Multi-bunch effects

Numerical analysis of instabilities with tracking codes

“Good wakefields”

Plasma Wakefield accelerators

R. Bartolini, John Adams Institute, 9 May 2013

Wakefields and collective effects

To have high luminosity in colliders and high brilliance in synchrotron light sources, storage rings operate with high circulating currents.

The bunches are intense and short ≤ 1 cm. They can generate a strong e.m. field as they travel down the vacuum pipe.

The electric fields generated by a bunch act back on the bunch itself or on subsequent bunches via the chamber (wakefields), giving rise to current dependent collective phenomena (collective effects).

3/32R. Bartolini, John Adams Institute, 9 May 2013

Collective effectsCollective effects

Main issues

• wakefields interact with the stored beam• RF heating of the vacuum chamber components

The beam can become unstable

beam loss: sudden partial or total loss of the beamsaturated injection: difficulty accumulating and storing beam

and/or its properties are compromised

beam oscillationsemittance blow upIncrease effective emittance (via transverse or longitudinal jitter)

Increased bunch length and energy spreadJitter in arrival times of bunches

These reduce the luminosity of a collider and the brilliance of a light source

R. Bartolini, John Adams Institute, 9 May 2013 4/32

5/32

Examples of wakefieldsThe trailing electric field is termed a wake field. Discontinuities along the vacuum chamber and finite resistivity of the vacuum chamber walls are the main sources for the generation of wake fields in colliders or light sources

Small discontinuities in the chamber act like cavities, where the electron bunch can deposit energy in the form of e.m. fields (the wakefields)

The trapped fields can have a long decay time. The energy goes into the heating of the chamber and the em fields are sensed by following bunches over many turns

Trapped modes in a cavity like structure

6/32

Collective effects: examples

Some instabilities increase gradually with increasing current whilst others have sharp thresholds. Instabilities are of two types, transverse and longitudinal, for both single and multi-bunch filling of the ring

10.5mA

28.8mA

100ms806040200

Time (msec)

40.0mA

10 mA10 mA

29 mA29 mA

40 mA40 mA

Time (msec)Time (msec)

Bo

lom

eter

sig

nal

(V

)B

olo

met

er s

ign

al (

V)

ALS DataALS DataBunch shape distortion


Example: CSR in storage ringsExample: CSR in storage rings

Diamond will operate with short electron bunches for generation of Coherent synchrotron radiation (in the THz region)

This operating regime is severely limited by the onset of the microbunch instabilities.

Sub-THz radiation bursts appeared periodically while the beam was circulating in the ring

A ultra-fast Schottky Barrier Diode sensitive to the radiation with 3.33-5mm wavelength range was installed in a dipole beamport;

1.9 mA 3.0 mA 5.2 mA

Transverse beam instabilitiesTransverse beam instabilities

Stable beam Vertically unstable beam


Vertical multi bunch instability seen at a pinhole camera (Diamond)

9/32

Short range and long range wakefields

Broad band parasitic lossesThese wake fields decay quickly, with a time scale of less than one turn. These fields are produced by:

• vacuum chamber transitions (bellows, tapers, flanges, insertion devices)• vacuum ports• beam position monitors, strip lines• fluorescent screens• injection elements (kickers and their vacuum chamber)

The fields affect charges in the same bunch leading to single bunch instabilities, i.e. electrons at the head of the bunch act on the tail (microwave instability (L), head-tail instabilities (H/V))

Narrow band parasitic lossesThese wake fields ring for a long time affecting the same bunch or another for many turns. The fields are produced by cavity like objects.

Multi-bunch instabilities (H/V) arise from such fields.


10/33

Computation of Wakefields (I)In its full generality, the computation of the wakefields generated by a bunch of charged particles is a very complex electromagnetic problem.

It requires the solution of the Maxwell’s equations with given source terms and boundary conditions imposed by the vacuum chamber

It is a 3D, time dependent problem that should be solved self consistently, i.e. the equation of the e.m. field are coupled to the equation of motion of the charged particles, so that the driving terms (charge and current densities) are also changing with time.

PIC codes runs, for complex EM problems, may take many weeks of CPU time !!! Finite elements codes also require an intensive computational effort despite some simplifying approximations customarily used to deal with this problem:

• the wakefields are computed assuming that the bunch distribution does not change under the action of the wakefields themselves in the structure

• the charge particle beam is assumed to be ultra-relativistic (v ~ c)

• the net effect of the wakefield on each particle is assumed to be small and is computed with the momentum transfer of the particle in the structure

• where possible, symmetric geometry allows the simplification of the problem.

11/32

Computation of wakefields (II)The electromagnetic field generated by the leading particle with charge q1 is computed at a distance Δz where the trailing particle of charge q is located.

The distance Δz is usually considered fixed in the computation of the wakefield. This helps simplifying considerably the task

)t;z,r,z,r(Bv)t;z,r,z,r(Eq)t;z,r,z,r(F 111111

q1 is the leading particle

q is the trailing particle

Δz = z1 – z distance

The force acting on the trailing particle is given by the Lorentz force

E and B are the electric and magnetic field generated by q1 in the structure


12/32

Wakefields and wake functions (I)

C

1111 ds)s;z,r,z,r(Fc

1)z,r,z,r(p

C

1111 ds)s;z,r,z,r(F)z,r,z,r(E

The variation of the momentum and the energy of the trailing particle is computed assuming that these deviations in energy and momentum are small

and

the motion of the trailing particle remains uniform, the relative distance does not change as long as the two charged particles travel within the structure C where the wakefields are generated.

Integrating the expression for the Lorentz force sensed by the trailing particle as it travels along the structure C, we get the momentum variation

and the energy variation


13/32

Wakefields and wake functions (II)

The wake function w|| describes the total energy lost by the test particle q generated by a single point-like charge q1 and therefore is the Green function for the problem: the energy lost by a test particle with charge q due to a collection charge Q = eNp described by a longitudinal charge density ρ is given by the convolution

'dz)'zz(w)'z(eqN)z,r,z,r(W p11

where Np is the total number of particle of charge e in the bunch.

Similarly for the transverse planes we can define

The longitudinal wake function is the energy variation per unit charge q and q1

1

1111|| qq

)z,r,z,r(E)z,r,z,r(w

longitudinal wake function [V/C]

1

1111 qq

)z,r,z,r(p)z,r,z,r(w

transverse wake function

and the total momentum transfer imparted by a bunch with charge density ρ is given by

'dz)'zz(w)'z(eqN)z,r,z,r(W 1p11

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Causality implies that no wake can exist in front of the bunch

Example: wake of a slot cavity like structure

L

0

z1

|| )z)/c(st,z,r(Edzq

1)s(w

head: t = z/ctail: t = (s+z)/c

z

t

s

charge

wake

s

15/32

Coupling impedance

de)();'r,r(Z

q2

1)z;'r,r(W c

zi

||1

||

Since the induced voltage on a trailing particle is given by the convolution of the wake with the charge distribution, in the frequency domain we have

zde)z,r,r(wc

1),r,r(Z c

zi

1||1||

The wakefields can be described conveniently also in the frequency domain. The Fourier transform of the wake function is called coupling impedance

longitudinal coupling impedance

zde)z,r,r(wc

1),r,r(Z c

zi

11

transverse coupling impedance

i.e. the effect of the wakefield depends on the product of the coupling impedance times the Fourier transform of the bunch distribution (the bunch spectrum).

Analogous formulae hold for the transverse plane.


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Impedance of an acceleratorThe impedance is a function of frequency and its spectrum depends on the accelerator components

At low frequencies it is dominated by the resistive wall impedance. High Q resonators (cavities) show up as sharp peaks, and the overall impedance made up of the various components in the ring gives the broadband contribution.

At frequencies beyond the cutoff frequency, the wake field propagates freely along the chamber. This is reflected by the roll-off of the broadband contribution. The cut-off frequency is given by ωc = c/b, b is the chamber radius.

Overlap of the bunch spectrum with the impedance of the machine generate instabilities


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Resonators and broad band impedance (I)

Each mode resembles an RLC - circuit and can, to a good approximation, be treated as such. This circuit has a shunt impedance R, inductance L and capacity C. It can drive multi-bunch instabilities.

In a real cavity these parameters cannot easily be separated and we use others which can be measured directly: The resonance frequency ωr, the quality factor Q and the damping rate α:

LC

1r

L

CRQ

Q2r

There is a strong analogy between wakefields and electronic circuit theories. This can be exploited and wakes can be represented by equivalent circuits.

For example, the impedance of a parallel RLC circuit is often associated to the impedance of the so-called high order modes (HOM), single resonance wakes in the vacuum chamber.


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Resonators and broad-band impedance (II)

r

r||

iQ1

R)(Z

Using this RLC model the HOMs can be classified in two main categories.

Narrow-band impedances. These modes are characterized by relatively high Q and their spectrum is narrow. The associated wake last for a relatively long time making this modes important for multibunch instabilities.

Broad-band impedances. These modes are characterized by a low Q and their spectrum is broader. The associated wake last for a relatively short time making this modes important only for single bunch instabilities.


Numerical computation of wakefield Numerical computation of wakefield (a DLS example)(a DLS example)

For complicated structure the wakefields are computed with numerical codes such as MAFIA (or gdfidl , ABCI, URMEL, ...)

MAFIA model of the Primary BPM block and BPM button for Diamond


MAFIA provides the wakefield generated by a short pilot bunch (3mm rms; 1 nC)

0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

6

7

8x 10

-3 Impedance spectrum of Diamond primary BPM

Frequency (GHz)

Impe

danc

e (O

hm)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-0.06

-0.04

-0.02

0

0.02

0.04

0.06

s (m)

Wak

e po

tent

ial (

V/p

C)

Wakefield of standard Diamond BPM

Diamond BPM wakefieldsDiamond BPM wakefields

The impendance of the structure is obtained dividing the spectrum of the field generated by the pilot bunch by the spectrum of the pilot bunch


Impedance databaseImpedance database

These calculations should be repeated for the main items in the vacuum pipe.

Tapers; Small aperture gap (IDs)

Pumping holes (and grilles); Kickers chambers

Flanges; Cavity like structures

Dipole slot for synchrotron radiation Collimators

Resistive wall ….

The total wakefield is given by the sum of the wakes of each individual item The total wakefield can be approximated as the sum of the wakefields of

many RLC resonators which best fit the total impedance

Next improvement is to use directly the MAFIA outputs as input in the tracking code.


Equation of motion (I)Equation of motion (I)

Particles move in the focussing fields of

quads (transverse motion)cavities (longitudinal motion)

The one turn map for each single particle reads

sn

RFs

RFn1n sinz

csin

E

V

1n0cn1n cTzz

nj

j

0

ny0

1nj

j

'y

y)

E1(QM

'y

y

Transverse

Longitudinal

Includes RF nonlinear potential, chromaticity (and simplecticity)

Equation of motion (II)Equation of motion (II)

Radiation damping and diffusion for electrons

Transverse

Longitudinal

RT2

'yT2

'y'yy

00'y1n

y

01n1n

RT

2T2

s

001n

s

01n1n

These terms guarantee that when tracking a distribution of macroparticles the equilibrium distribution is has the correct equilibrium emittances, beam sizes, divergences, bunch length and energy spread.


Equation of motion (II)Equation of motion (II)

Wakefields apper in the equations of motion as a lumped kick added to the one turn map (both in longitudinal and transverse)

Transverse

Longitudinal

iz

||0

1n1n )'zz(W)'z('dzC

Nr

iz

p0

1n1n )'zz(W)'z(D)'z('dzC

Nr'y'y

The kick is computed binning the longitudinal distribution of the electrons and computing W and W|| from analytical formula or numerical codes


Structure of sbtrack - mbtrackStructure of sbtrack - mbtrack

generate the initial 6D particles distribution

apply the one turn map

compute the new 6D particle distribution

compute and apply the kick due to the wakefields

wakefields model

• many BBR simultaneously; RW wakefield, CSR, etc• one turn map extended to consider the full nonlinear motion


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Good wakefieldsNot all of these wakefields are bad in accelerator applications. In fact, there are few examples were wakes play a positive role

Laser

Gas

Gas jet nozzle

e- bunch Plasmachannel

THzRadiation

L’OASISL’OASIS

Wakefield-based acceleration schemes.Strong R&D and very promising results.1 GeV e- beam generated at LBNL in 2006

Bunches in electron storage rings with longitudinal distribution asymmetrically distorted by wake-fields emit coherent synchrotron radiation at much higher frequencies than bunches with nominal Gaussian distribution.

This can be exploited for designing THz and far-infrared synchrotron light sources with revolutionary performances


Laser Plasma AccelerationLaser Plasma Acceleration

Tajima & Dawson Phys Rev. Lett. 43 267 (1979)

25 years necessary for developing laser system technology suitable for LPAs


LPWA (I)LPWA (I)

An ionized electron quivers in the E-field of the laser with a

ponderomotive energy:

2 2 218 μm

10.57

2p e eU m v I m c

Spatial variation in the ponderomotive energy gives rise to a

force, the ponderomotive force:

p pU F

FpFp


LPWA (II)LPWA (II)

-1100 GVmzE wb

e pm cE

e

Three orders of magnitude larger than the field used in conventional RF accelerators.

The electric field within the plasma can reach the wave-breaking limit:

The wakefield is strongest when a resonance condition is met:

1p

2

0

ep

e

N e

m

The ponderomotive force in a laser pulse with intensity of the order of 1018 W/cm2

expels electrons from the region of the pulse to form a trailing plasma wakefield

Bubble regimeBubble regime

Snap-shot from PIC simulation of bubble acceleration:

electron density map, propagation direction z. The

typical length scale is the plasma wavelength, thus micrometers.

The “bubble” behind the laser can trap nC charge,

thus yielding electron beam currents on the scale of 100 kA

GeV electron beams have been created with this method

At very high laser intensities the wake reach the “blow-out” or “bubble” regime. A cavity with strong electric fields is created after the laser pulse.


Wave analogiesWave analogies


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Bibliography

Wakefields

J. D. Jackson, Classical Electrodynamics, John Wiley & sons.

A. Chao, Physics of collective beam instabilities in High energy Accelerators, John Wiley & Sons

Laser Plasma Wakefield Accelerators

E.Esarey, IEEE Trans. Plasma, 24, 232, (1996)

W. Leemans et al., Phil. Trans. R. Soc. A 2006 364, 585-600


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Syllabus and slides