# Lecture 6: beam optics in Linacs

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16 Νοε 2013 (πριν από 4 χρόνια και 5 μήνες)

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R. Bartolini, John Adams Institute, 3 May 2013

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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)

R. Bartolini, John Adams Institute, 3 May 2013

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A 100 MeV LINAC (at Diamond Light Source)

R. Bartolini, John Adams Institute, 3 May 2013

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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

E
z

z

c

Travelling wave: the bunch sees
a constant electric field

E
z
=E
0

cos(

)

E
z

z

c

c

2
ct

Standing wave: the bunch sees
a varying electric field

E
z
=E
0

cos(

t+

)sin(kz)

R. Bartolini, John Adams Institute, 3 May 2013

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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
v
p
<
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

R. Bartolini, John Adams Institute, 3 May 2013

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Beam dynamics during acceleration (I)

Consider a particle moving in the electric field of a travelling wave

)
kz
t
cos(
E
E
0
z

k
v
f

with a phase velocity

The equations used to describe the motion in the longitudinal plane are

)
kz
t
cos(
eE
dt
dp
0
z

)
kz
t
cos(
z
eE
dt
d
0

s
s
0
s
cos
v
eE
dt
d

Define the synchronous particle as

For the generic particle, using as coordinates the deviation from the
energy and time from the synchronous particle, we have

W
s

u
z
z
s

u
v
t
kz
s
s

and changing variable to

R. Bartolini, John Adams Institute, 3 May 2013

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Beam dynamics during acceleration (II)

We get the system of equations

s
0
cos
cos
eE
ds
dW

2
3
s
3
s
mc
W
c
ds
d

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”

R. Bartolini, John Adams Institute, 3 May 2013

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RF technology

Usual operating frequencies for RF cavities for Linear accelerators are

Warm cavities

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.

R. Bartolini, John Adams Institute, 3 May 2013

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Particle sources and Gun

Electrons

Thermionic gun

Photocathode guns

Protons and H
-

plasma discharge

Penning ion sources

R. Bartolini, John Adams Institute, 3 May 2013

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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

R. Bartolini, John Adams Institute, 3 May 2013

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Thermionic gun (II)

Electrons

are

generated

by

thermionic

emission

tend

to

repel

therefore

an

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

R. Bartolini, John Adams Institute, 3 May 2013

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Photocathode guns (I)

One and half cell RF photocathode gun

Electrons are generated with a laser
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

accelerate the particle fast and
reduce the effect of space charge

(scales as 1/E
2
)

Electron pulses can be made short
(as the laser pulse
-

few ps)

R. Bartolini, John Adams Institute, 3 May 2013

Photocathode guns

BNL /SLAC/UCLA RF gun

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R. Bartolini, John Adams Institute, 3 May 2013

Photocathode guns

Photoemission with a pulsed laser

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R. Bartolini, John Adams Institute, 3 May 2013

Photocathode guns

.. and RF acceleration

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R. Bartolini, John Adams Institute, 3 May 2013

Photocathode guns

.. and RF acceleration

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R. Bartolini, John Adams Institute, 3 May 2013

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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R. Bartolini, John Adams Institute, 3 May 2013

Photocathode guns

RF signal distribution for an RF photocathode gun (5
-
cells )

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R. Bartolini, John Adams Institute, 3 May 2013

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

or with a doublet structure

e.g.

SCSS Japan

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0
20
40
60
80
100
120
140
160
180
200
-10
0
10
20
30
40
50
60
70
80
90
S (m)
Amplitude
Twiss Parameters

Beta X (m)
Beta Y (m)
Dispersion (cm)
R. Bartolini, John Adams Institute, 3 May 2013

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A doublet channel

1
0
L
1
1
f
/
1
0
1
1
0
d
1
1
f
/
1
0
1
1
0
L
1
M

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

We can compute in the usual way the phase advance and the optics function
for the basic cell, assuming it is repeated periodically

2
22
11
f
dL
1
2
m
m
cos

1
x
2
L
2
1
sin
2
m
m
22
11
x

2
12
x
x
2
)
x
2
(
L
d
sin
m



2
f
dL
x

and putting

The focussing effect of the cavity is usually added in refined calculations

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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

D
t
b
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

More on lecture 8 on instabilities

t
0

t
1

t
2

t
3

t
4

t
5

t
6

R. Bartolini, John Adams Institute, 3 May 2013

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Bunch Compression (I)

In many applications the length of the bunch generated even by a photo
-
injector (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

This effect can be used to compress the bunch length

blue = low energy

red = high energy

The time of flight of the high

energy particle is smaller

(v

c ...but
it travel less !)

R. Bartolini, John Adams Institute, 3 May 2013

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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

R. Bartolini, John Adams Institute, 13 May 2011

An energy chirp is required for
the compression to work

The high energy particle at the tail travels less and catches up the
synchronous particle. The net result is a the compression of the bunch

tail

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Bunch compression (III)

Bunch compression can be computed analytically. Inside the RF cavity the
energy changes with the position z
0

as

0
RF
0
RF
0
1
0
1
z
k
2
cos
E
eV
z
z

In the linear approximation in (z,

)

0
0
65
1
1
z
1
R
0
1
z

RF
RF
RF
k
E
eV
R

sin
0
65

In the chicane the coordinate changes as

1
2
3
1
5666
2
1
566
1
56
1
2

U
T
R
z
z
In the linear approximation

1
1
56
2
2
1
0
1

z
R
z
R. Bartolini, John Adams Institute, 3 May 2013

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Bunch compression (IV)

The full transformation is, as usual, the composition of the matrices of each,

1
1
65
56
56
65
0
0
2
2
R
R
R
R
z
z
M
M

Since the transformation is symplectic (i.e. area preserving

Liouville
theorem) the longitudinal emittance is conserved

2
2
2

z
z

For a given value of R
65

(energy chirp induced), the best compression that
can be achieved is

C
|
R
R
1
|
0
0
2
z
z
56
65
z

C is the compression factor. It can be a large number!

The minimum reachable bunch length is limited to the product of the energy
56

R. Bartolini, John Adams Institute, 3 May 2013

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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

more on lecture 7

When taken into account, these effects can produce serious degradation of
the beam qualities, e.g in simulations

10 e
-

bunches
with different
compression C
superimposed

under
compressed

over
compressed

Longitudinal phase space of a
disrupted beam

R. Bartolini, John Adams Institute, 3 May 2013

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Linear Colliders

ILC (International Linear Collider)

L
-
band
SC cavities

30 MV/m

500
GeV

(36 km overall length)

CLIC (Compact Linear Collider)

X
-
band
NC cavities

100 MV/m

3
TeV

(48 km overall length)

Linear accelerators are at the heart of the next generation of linear colliders

R. Bartolini, John Adams Institute, 3 May 2013

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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

photoinjector

BC1

BC2

BC3

laser heater

accelerating modules

collimation

diagnostics

FELs

IR/THzundulators

experimental stations

High brightness electron gun operating (initially) at 1 kHz

2.25 GeV SC CW linac L
-

band

to feed 3 FELS covering the photon energy range 50 eV

1 keV

R. Bartolini, John Adams Institute, 3 May 2013

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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)

R. Bartolini, John Adams Institute, 3 May 2013

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 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