Syllabus and slides
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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) r.bartolini1@physics.ox.ac.uk
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
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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).
R. Bartolini, John Adams Institute, 9 May 2013
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Collective 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 beam
saturated injection: difficulty accumulating and storing beam
and/or its properties are compromised
beam oscillations
emittance blow up
Increase effective emittance (via transverse or longitudinal jitter)
Increased bunch length and energy spread
Jitter 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
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Examples of wakefields
The 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
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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
Bunch shape distortion
10. 5mA
10 mA
Bolometer signal (V)
ALS Data
28. 8mA
29 mA
40. 0mA
40 mA
0
20
40
60
80
100ms
Time (m sec )
Time (msec)
R. Bartolini, John Adams Institute, 9 May 2013
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Example: 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 instabilities
Vertical multi bunch instability seen at a pinhole camera (Diamond)
Stable beam
R. Bartolini, John Adams Institute, 9 May 2013
Vertically unstable beam
8/32
Short range and long range wakefields
Broad band parasitic losses
These 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), headtail instabilities (H/V))
Narrow band parasitic losses
These 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.
R. Bartolini, John Adams Institute, 9 May 2013
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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.10/33
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
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

F(r, z, r1 , z1; t )  q E(r, z, r1 , z1; t )  v  B(r, z, r1 , z1; t )

E and B are the electric and magnetic field generated by q1 in the structure
R. Bartolini, John Adams Institute, 9 May 2013
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Wakefields and wake functions (I)
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
p (r, z, r1 , z1 ) 

1
F(r, z, r1 , z1 ; s)ds
c
C
and the energy variation

E ( r , z, r1 , z1 )  F( r , z, r1 , z1 ; s)  ds
C
R. Bartolini, John Adams Institute, 9 May 2013
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Wakefields and wake functions (II)
The longitudinal wake function is the energy variation per unit charge q and q1
w || ( r , z, r1 , z1 ) 
E ( r , z, r1 , z1 )
qq1
longitudinal wake function [V/C]
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

W(r, z, r1 , z1 )  N p eq  (z' ) w (z  z' )dz'

where Np is the total number of particle of charge e in the bunch.
Similarly for the transverse planes we can define
w  ( r , z, r1 , z1 ) 
p ( r , z, r1 , z1 )
qq1
transverse wake function
and the total momentum transfer imparted by a bunch with charge density ρ is given

by
W (r, z, r1 , z1 )  N p eq1  (z' )w  (z  z' )dz'


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Example: wake of a slot cavity like structure
L
1
w || (s)   dz E z (r, z, t  (s  z)/c)
q1 0
t
head: t = z/c
tail: t = (s+z)/c
charge
wake
s
Causality implies that no wake can exist
in front of the bunch
s
z
14/32
Coupling impedance
The wakefields can be described conveniently also in the frequency domain. The
Fourier transform of the wake function is called coupling impedance
1
Z|| (r, r1 ,  ) 
c
1
Z (r, r1 , ) 
c

 w (r, r , z)e
||
i
1


w
 (r, r1 , z)e
z
c dz
i
longitudinal coupling impedance
z
c dz
transverse coupling impedance

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
1
W|| (r, r ' ; z) 
2q1

 Z (r, r' ; )()e
||
i
z
c
d

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.
R. Bartolini, John Adams Institute, 9 May 2013
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Impedance of an accelerator
The 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
R. Bartolini, John Adams Institute, 9 May 2013
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Resonators and broad band impedance (I)
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.
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:
r 
1
LC

r
2Q
QR
C
L
R. Bartolini, John Adams Institute, 9 May 2013
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Resonators and broad-band impedance (II)
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.
Z|| ( ) 
R


1  iQ r  
  r 
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.
R. Bartolini, John Adams Institute, 9 May 2013
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Numerical computation of wakefield (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
R. Bartolini, John Adams Institute, 9 May 2013
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Diamond BPM wakefields
MAFIA provides the wakefield generated by a short pilot bunch
(3mm rms; 1 nC)
-3
Wakefield of standard Diamond BPM
8
0.06
Impedance spectrum of Diamond primary BPM
x 10
7
0.04
0.02
Impedance (Ohm)
Wake potential (V/pC)
6
0
5
4
3
-0.02
2
-0.04
-0.06
1
0
0
0.05
0.1
0.15
0.2
0.25
s (m)
0.3
0.35
0.4
0.45
0.5
0
5
10
15
20
25
30
Frequency (GHz)
35
40
45
50
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
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Impedance 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
Resistive wall
Collimators
….
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.
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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
 yj 

  yj 

n
   M Q 0 (1   y
)   
 y' j 
E 0   y' j  n

  n 1

VRF  
RF 
sin


z

sin



s
n
s

E  
c


z n 1  z n   c cT0 n 1
 n 1   n 
Transverse
Longitudinal
Includes RF nonlinear potential, chromaticity (and simplecticity)
Equation of motion (II)
Radiation damping and diffusion for electrons
2T0
y'n 1  y'0
2T0
2T0
 n 1  2  0
T0
y'n 1  y'n 1 
 n 1   n 1 
y
s
y
s
R
R
Transverse
Longitudinal
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.
R. Bartolini, John Adams Institute, 9 May 2013
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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)
y'n 1  y'n 1 
 n 1   n 1 
Nr0
C
Nr0
C
zi
 dz'  (z' )D (z' )  W (z  z' )
p

Transverse

zi
 dz'  (z' )  W (z  z' )
||
Longitudinal

The kick is computed binning the longitudinal distribution  of the electrons
and computing W and W|| from analytical formula or numerical codes
R. Bartolini, John Adams Institute, 9 May 2013
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Structure of sbtrack - mbtrack
generate the initial 6D particles distribution
apply the one turn map
compute and apply the
kick due to the wakefields
compute the new 6D
particle distribution
wakefields model
• many BBR simultaneously; RW wakefield, CSR, etc
• one turn map extended to consider the full nonlinear motion
R. Bartolini, John Adams Institute, 9 May 2013
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Good wakefields
Not all of these wakefields are bad in accelerator applications. In fact, there
are few examples were wakes play a positive role
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
L’OASIS
Gas
THz
Radiation
Laser
e- bunch
Plasma
channel
Wakefield-based acceleration schemes.
Strong R&D and very promising results.
1 GeV e- beam generated at LBNL in 2006
Gas jet nozzle
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Laser Plasma Acceleration
Tajima & Dawson Phys Rev. Lett. 43 267 (1979)
25 years necessary for developing laser system technology suitable for LPAs
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LPWA (I)
An ionized electron quivers in
the E-field of the laser with a
ponderomotive energy:
Up 
Fp
Fp
1
2
mev 2  0.57 I18μm
mec 2
2
Spatial variation in the
ponderomotive energy gives rise to a
force, the ponderomotive force:
Fp  Up
R. Bartolini, John Adams Institute, 9 May 2013
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LPWA (II)
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
The wakefield is strongest when a
resonance condition is met:
p  1
p 
Ne e 2
me 0
The electric field within the plasma can reach
the wave-breaking limit:
Ewb 
mepc
e
E z  100 GV m-1
Three orders of magnitude larger than the
field used in conventional RF accelerators.
Bubble regime
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.
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
R. Bartolini, John Adams Institute, 9 May 2013
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Wave analogies
R. Bartolini, John Adams Institute, 9 May 2013
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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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