Lecture 6: beam optics in Linacs
LINAC overview
Acceleration
Focussing
Compression
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LINAC overview
A LINAC is an accelerator consisting of several subsystems
Gun (particle source)
Accelerating section (and RF sources)
Magnetic system (focussing and steering)
Diagnostics – Vacuum – etc
Depending on the application a LINAC might have
bunch compression system (radiation sources, FELs, colliders)
beam delivery systems (medical linacs, colliders)
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A 100 MeV LINAC (at Diamond Light Source)
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Acceleration
Acceleration is achieved with RF cavities, using e.m. modes with the electric
field pointing in the longitudinal direction (direction of motion of the charged
particle)
The RF electric field can be provided by travelling wave structure or standing
wave structure
Ez
Ez
c
c
ct  
2
c
z
z
Travelling wave: the bunch sees
a constant electric field
Standing wave: the bunch sees
a varying electric field
Ez=E0 cos()
Ez=E0 cos(t+)sin(kz)
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Travelling wave and standing wave structures
The wave velocity and the particle velocity have to be equal hence we need
a disk loaded structure to slow down the phase velocity of the electric field
To achieve synchronism vp< c
Slow down wave using irises.
In a standing wave structure the electromagnetic field is the sum of two
travelling wave structure running in opposite directions.
Only the forward travelling wave
takes part in the acceleration process
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Beam dynamics during acceleration (I)
Consider a particle moving in the electric field of a travelling wave

E z  E 0 cos(t  kz) with a phase velocity v f 
k
The equations used to describe the motion in the longitudinal plane are
dpz
 eE 0 cos(t  kz)
dt
Define the synchronous particle as
d
 eE 0 z cos(t  kz)
dt
d s
 eE 0 v s cos  s
dt
For the generic particle, using as coordinates the deviation from the
energy and time from the synchronous particle, we have
  s  W
z  zs  u
and changing variable to
  kz  t   s 
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
vs
u
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Beam dynamics during acceleration (II)
We get the system of equations
dW
 eE 0 cos   cos s 
ds
d

W
 3 3
ds
 s  s c mc 2
These describe the usual RF bucket in the longitudinal phase space (, W)
We assumed here that the acceleration is adiabatic i.e. ds/ds  0. If this in
not true, numerical integration shows that the RF bucket gets distorted into
a “golf club”
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RF technology
Usual operating frequencies for RF cavities for Linear accelerators are
Warm cavities
gradient
repetition rate
S-band (3GHz)
15-25 MV/m
50-300 Hz
C-band (5-6 GHz)
30-40 MV/m
<100 Hz
X-band (12 GHz)
100 MV/m
<100 Hz
Superconducting cavities
L band (1.3 GHz)
< 35 MV/m
up to CW
The main RF parameters associated to the RF cavity, such as shunt
impedance quality factor will be discussed in the Lecture 10 on RF.
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Particle sources and Gun
Electrons
Thermionic gun
Photocathode guns
Protons and Hplasma discharge
Penning ion sources
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Thermionic gun (I)
Electrons are generated by thermionic emission from the cathode and
accelerated across a high voltage gap to the anode. A grid between anode
and cathode can be pulsed to generate a train of pulses suitable for RF
acceleration
cathode assembly
BaO/CeO-impregnated
tungsten disc is heated
and electrons are
emitted
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Thermionic gun (II)
Electrons are generated by thermionic emission tend to repel therefore an
advance e.m. design is envisaged to control the beam dynamics and reduce
the emittance of the beam.
This requires solving Laplace equation for
the potential of the e.m. field in the given
geometry
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Photocathode guns (I)
Electrons are generated with a laser
One and half cell RF photocathode gun
field by photoelectric effect
High voltage at the cathode is
delivered by the RF structure
50-60 MV/m in L-band
100-140 MV/m in S-band
Higher gradients are useful to
accelerate the particle fast and
reduce the effect of space charge
(scales as 1/E2)
Electron pulses can be made short
(as the laser pulse - few ps)
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Photocathode guns
BNL /SLAC/UCLA RF gun
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Photocathode guns
Photoemission with a pulsed laser
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Photocathode guns
.. and RF acceleration
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Photocathode guns
.. and RF acceleration
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Photocathode guns
.. and RF acceleration
The emittance and the energy spread are determined by the laser parameters
and the properties of the cathode material.
The emittance can be tens of times better than in a thermionic guns (< 1 m)
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Photocathode guns
RF signal distribution for an RF photocathode gun (5-cells )
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Focussing system in long LINACs
In a long linac we need a magnetic channel to keep the beam focussed in
the transverse dimension.
This can be accomplished with a FODO lattice
e.g.
SCSS Japan
or with a doublet structure
Twiss Parameters
90
Beta X (m)
Beta Y (m)
Dispersion (cm)
80
70
Amplitude
60
50
40
30
20
10
0
-10
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2013
0 Institute,
20
403 May
60
80
100
S (m)
120
140
160
180
200
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A doublet channel
In a FODO channel the RF cavities are placed in the drift sections.
To create longer straight section a double (or triplet) channel is envisaged.
A doublet channel is a series of pairs of quadrupoles F and D with long drift
sections between the pairs. the RF cavities are placed in the drift sections
short drift d
long drift 2L
 1 L  1

M  
 0 1   1 / f
0  1 d  1


1  0 1   1 / f
0  1 L 


1  0 1 
We can compute in the usual way the phase advance and the optics function
for the basic cell, assuming it is repeated periodically
dL
m11  m22
dL and putting
x 2
cos 
 1 2
f
2
f
m11  m 22
1
m12
d  L( 2  x )



x


2 sin 
2
2
sin 
2L
1
2x  x
x
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The focussing effect of the cavity is usually added in refined calculations
Beam dynamics issues: wakefields
The interaction of the charged beam with the RF cavity and the vacuum
chamber in general generate e.m. fields which act back on the bunch itself
Dtb
In the RF cavity these fields can build up resonantly and disrupt the bunch
itself in the so called single beam break up or multi bunch break up
t0
t1
t2
t3
t4
t5
t6
More on lecture 8 on instabilities
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Bunch Compression (I)
In many applications the length of the bunch generated even by a photoinjector (few ps) is too long. Tens of fs might be required.
The bunch length needs to be shortened. This is usually achieved with a
magnetic compression system.
A beam transport line made of four equal dipole with opposite polarity is used
to compress the bunch. In this chicane the time of flight (or path length) is
different for different energies
blue = low energy
red = high energy
The time of flight of the high
energy particle is smaller
(v  c ...but it travel less !)
This effect can be used to compress the bunch length
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Bunch compression (II)
To exploit the dependence of the time of flight (or path length) for different
energies we need to introduce an energy-time correlation in the bunch.
This is done using the electric field of an RF cavity with as suitable timing
tail
head
An energy chirp is required for
the compression to work
The high energy particle at the tail travels less and catches up the
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synchronous
particle.
The
is a the compression of the bunch
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Institute,
13net
May result
2011
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Bunch compression (III)
Bunch compression can be computed analytically. Inside the RF cavity the
energy changes with the position z0 as
z1  z 0
eVRF


cos  k RF z 0 
E0
2

In the linear approximation in (z, )
1   0 
 z1   1
   
 1   R 65
0  z0 
   
1   0 
eVRF
R65 
sin  RF k RF
E0
In the chicane the coordinate changes as
z2  z1  R561  T56612  U 566613 
 2  1
In the linear approximation
 z 2   1 R56   z1 
   
   
  2   0 1   1 
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Bunch compression (IV)
The full transformation is, as usual, the composition of the matrices of each,
element and reads
z 
 z2 
   M   0 
 2 
 0 
1  R65 R56
M  
 R65
R56 

1 
For a given value of R65 (energy chirp induced), the best compression that
can be achieved is
 z2  | 1  R 65 R 56 |  z0 
 z0
C
C is the compression factor. It can be a large number!
Since the transformation is symplectic (i.e. area preserving  Liouville
theorem) the longitudinal emittance is conserved
   z2 2   z2
The minimum reachable bunch length is limited to the product of the energy
spread times R56
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Bunch compression (V)
Further limitations to the achievable compression comes from the high
current effect that we have neglected in the linear approximations.
These are longitudinal space charge, wakefields and coherent synchrotron
radiation (CSR) – more on lecture 7
When taken into account, these effects can produce serious degradation of
the beam qualities, e.g in simulations
Longitudinal phase space of a
disrupted beam
over
compressed
under
compressed
10 e- bunches
with different
compression C
superimposed
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Linear Colliders
Linear accelerators are at the heart of the next generation of linear colliders
ILC (International Linear Collider)
CLIC (Compact Linear Collider)
L-band SC cavities
X-band NC cavities
30 MV/m
100 MV/m
500 GeV (36 km overall length)
3 TeV (48 km overall length)
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Fourth generation light sources
Linear accelerators are at the heart of the next generation of synchrotron
radiation sources, e.g. the UK New Light Source project was based on
High brightness electron gun operating (initially) at 1 kHz
2.25 GeV SC CW linac L- band
experimental stations
photoinjector
laser heater
BC1
BC2
IR/THzundulators
diagnostics
accelerating modules
spreader
collimation
BC3
FELs
to feed 3 FELS covering the photon energy range 50 eV – 1 keV
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Bibliography
M. Conte, W.W. MacKay,
The physics of particle accelerators, World Scientific (1991)
P. Lapostolle
Theorie des Accelerateurs Lineaires, CERN 87-10, (1987)
J. Le Duff
Dynamics and Acceleration in linear structures, CERN 85-19, (1985)
T.P. Wangler
RF Linear Accelerators, Wiley, (2008)
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Syllabus 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
Dr. Riccardo Bartolini (DWB room 622) r.bartolini1@physics.ox.ac.uk