1
Beam dynamics in RF
linacs
Step
2
:
Beam transport
Nicolas PICHOFF
France
CEA

DSM/IRFU/SACM/LEDA
2
Outlines
•
Beam representation
•
Distribution
•
Sigma matrix
•
Emittance
•
Matching
•
Mechanism of emittance growth
3
Beam definition
A
beam
could be defined as a
set of particles
whose maximum average
momentum in one direction (
z
) is higher than its dispersion :
2
2
2
2
z
z
y
x
z
p
p
p
p
p
Beam
representation
z
p
x
y
4
Particle representation
Each particle is represented by a 6D vector :
6
5
4
3
2
1
p
p
p
p
p
p
P
Beam
representation
W
y
y
x
x
z
y
x
p
z
p
y
p
x
or
or …
5
Beam phase

space representation
It is represented by a particle
distribution
in the 6D phase

space (P).
Beam
representation
It can be plotted in 2D sub

phase

spaces :
6
Beam
modelisation
Beam
representation
Beam
: Set of billions
(N)
of particles evolving as a function of
an independent variable
s
Macroparticle
model
:
set of
n
macroparticles
(
n<N
)
macroparticle
: statistic sample of particle
Distribution function model
:
of 6 coordinates
P
d
s
P
f
,
Number
of
particles
Between
and
P
P
d
P
4
3
2
1
0
1
2
3
4
4
3
2
1
0
1
2
3
4
0.2
0.4
0.6
0.8
1.0
N
P
d
s
P
f
,
7
Statistics
Beam
representation
P
d
P
A
P
f
N
P
A
n
P
A
N
P
A
n
j
j
N
i
i
1
1
1
1
1
Average of a function
A
on beam :
8
First order momentum: beam Centre of Gravity (
CoG
)
Beam
representation
P
P
A
6
5
4
3
2
1
0
p
p
p
p
p
p
P
Average :
position,
phase,
Angle,
Energy
…
9
Second order momentum : Sigma matrix
The beam can be represented by a 6
×
6 matrix containing the
second
order momentum
in the 6D phase

space : the
sigma matrix
.
Beam
representation
j
j
i
i
j
i
p
p
p
p
,
.
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.
.
2
1
,
2
1
det
x
4
3
,
4
3
det
y
6
5
,
6
5
det
z
are the beam
2D rms emittances
10
2D RMS Emittance
Beam
representation
2
2
2
~
x
x
x
x
x
x
x
x
x
The statistic surface in 2D sub

phase

space occupied by the beam
Indicator of confinement
(for example)
11
Beam
Twiss parameters
The beam
Twiss parameters
are
:
w
w
w
~
~
2
w
w
w
~
~
2
w
w
w
w
w
w
~
w
w
w
w
w
w
w
w
5
2
2
2
Beam
representation
The goal is to model the beam shape in 2D sub

phase

space with
ellipses
.
5 : uniform elliptic distribution with same rms size.
12
6D transport matrix
The transverse force is generally close to linear.
The
longitudinal
force
can
be
linearized
when
<<
s
.
Linear transport
0
1
6
5
4
3
2
1
0
1
6
5
4
3
2
1
.
.
.
.
.
.
.
.
.
.
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.
.
.
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.
.
.
.
.
/
s
s
p
p
p
p
p
p
s
s
M
p
p
p
p
p
p
The particle transport can be represented by a 6
×
6 transfer matrix :
13
Sigma matrix transport in linear forces
Linear transport
T
s
s
s
s
M
s
s
M
0
1
0
1
/
/
0
1
The
transport
of sigma matrix can be obtained from
particle transfer matrix :
14
RMS emittance evolution in linear, uncoupled forces
Linear transport
x
x
x
x
x
x
x
ds
d
x
2
2
2
~
2
2
2
2
~
x
x
x
x
x
x
x
x
x
x
x
The emittance
is
damped
.
x
z
xn
~
~
, the
normalized
rms emittance
,
is
conserved
.
If
linear
force :
x
k
x
0
~
ds
d
x
x
z
z
x
ds
d
ds
d
~
1
~
x
ds
d
x
k
x
z
z
1
If
acceleration
:
T
he emittance
is
constant
.
15
Linear matching
Linear transport
The linear matching is the
association of 2 notions
:

Slide 36 : In uncoupled, linear & periodic forces, particles are turning
around periodically oscillating ellipses
0
2
2
2
w
s
w
w
s
w
s
wm
wm
wm
Their shapes are given by
Courant

Snyder parameters

Slide 57 : A beam can be represented by 2D

ellipses
2
2
2
w
s
w
w
s
w
s
w
w
w
Their shape are given by
Twiss parameters
16
Beam
linear matched
50% mismatched beam
Beam dynamics
Phase

space trajectory
Phase

space periodic looks
Matched beam
Bigger input beam
Smaller input beam
Phase

space scanned by
the mismatched beams
The beam is
linear

matched
:
wm
w
wm
w
wm
w
Beam Twiss
parameters
(S57)
Channel Courant

Snyder parameters
(S36)
=
The beam second order (or envelope) motion is periodic
17
Linear (rms) matching
Beam dynamics
The matching is done between sections by changing focusing force with
quadrupoles (transverse) and cavities (longitudinal).
Calculations are made with «
envelope codes
» where the beam is modelled by its
sigma matrix.
This type of code calculates automatically the focusing
strength in
elements
that
matches
the beam.
Not
matched beam
Matched beam
18
Space

charge forces
Beam dynamics
Electromagnetic interaction between particles.
It is linear if beam is uniform,
non

linear
otherwise (generally).
0
1
2
3
4
5
0
1
2
0
1
2
3
4
5
0
0.5
1
1.5
Density
Firled
Radial position
Example
:
axi

symmetric beam
uniform
parabolic
gaussian
Distributions :
Equivalent beams
: Same current, same sigma matrix
Radial position
19
Mechanisms of
emittance growth
and particle losses
Introduction
20
The main source of emittance growth is the
beam mismatching in
non linear forces
acting through 3 mechanisms :

The distribution intern mismatching,

The beam
filamentation
,

Resonant interactions
between particle and beam motions
Introduction
Other mechanisms play a (small) role :

Coupling
between directions (x, y, phi),

Interaction with
residual gas
,

Intra

beam
scattering.
21
1
–
Intern mismatching
In an intern matched beam, beam distribution in phase

space
is
constant on particle
trajectory
in
phase

space.
Beam dynamics
H = Cste
r
p
If not, the beam distribution “re

organise” itself.
u
u
H
f
u
u
f
,
,
22
2

Filamentation
When the confinement force is non linear (
multipole
, longitudinal, space

charge
),
the particle
oscillation period depends on its amplitude
:
Particle do not rotate at the same speed in the phase

space : possible
filamentation
Beam dynamics
Linear force
Non linear force
0
2
2
w
w
,
s
k
ds
w
d
w
23
3
–
Space

charge resonance

The space

charge force acting on a particle
depends on beam average size
Beam dynamics

In non

linear forces, the particle
oscillation period
depends on its amplitude

If the beam is mismatched, its average size oscillates with
3 “mismatched” modes

Some particles can have oscillation period being a
multiple of these modes

The amplitude of these particles will
resonantly growth and decreased
Quadrupolar
mode
High

freq
breathing
mode
Low

freq
breathing
mode
24
4

Coupling
The preceding developments assumed that the force along each direction was
depending only on the particle coordinates in this direction (even non

linear).
Beam dynamics
When the
force also
depends on
other
coordinates
Sources of coupling :

Transverse defocusing in cavities depending on phase,

Transverse focusing in quadrupoles depending on energy,

Energy gain in cavities depending on transverse position and slope,

Phase

delay due to transverse trajectory increase,

Skew quadrupoles,

Space charge

force,

…
2 particles
with the same sub

phase

space position
can feel different forces and
get
separated in the sub

phase

space
.
2D emittance growth
25
5
–
Residual gas interaction
Beam dynamics
26
3
2
2
2
10
1
13
2
MeV
cm
E
Z
z
d
d
Cross section :
(Rutherford)
Probability
:
3
2
2
1
31
.
0
)
(
1
MeV
hPa
m
E
P
Z
d
d
P
Atom
nucleus
Particle
Charge : z.e
Mass : m
Energy
: E
Momentum
: p
b
b
max
b
mi
n
b
Charge : +Z.e
Mass : M
min
Electrons
max
26
5
–
Residual gas interaction(2)
Beam dynamics
30
25
20
15
10
5
0
x (mm)
Beam profile
(a) "high" N
2
pressure
(b) "low" N
2
pressure
(a) Simulation result
(b) Simulation result
(a)
(b)
10
1
10
2
10
3
10
4
10
5
10
6
1
Beam
core
Mismatching
Good
agreement
27
6
–
Intra

beam scattering
Beam dynamics
1.E12
1.E11
1.E10
1.E09
1.E08
1.E07
1.E06
1.E05
1.E04
1.E03
1.E02
1.E01
1.E+00
0
0.5
1
1.5
2
2.5
3
n(
x
)
= 1
= 1.5
= 2
= 3
I = 98 mA ; f = 352 MHz
x
0
= y
0
= 2.5 mm ; z
0
= 6.75/
mm
x'
0
= y'
0
= 3.5
7.6MeV
/
mrad
z'
0
=
x'
0
x
1
1.5
2
3
Tails induced by 2 body collision in a uniform proton beam
:
Ratio
between
longitudinal and
transverse
energy
Transfer of energy between 2 directions in a
two

body collision.
Very efficient if different longitudinal

transverse emittances
28
Summary
conclusion
•
Beam is a set of particles
•
Beam
can
be
modeled
with
:
•
macroparticles
,
•
distribution
function
,
•
statistic
properties
•
The simplest is the 2
nd
order momentum : the sigma matrix, including
•
rms emittance (confinement),
•
Twiss parameters and 2D ellipses,
•
Emittance is conserved and damped in linear motion
•
Sigma matrix can be transport with matrix when the force is linear(
ised
)
•
(some) Source of emittance growth and halo are :
•
In mismatched beam in non

linear forces :
filamentation
& resonances
•
forces
coupled
between
directions
•
scattering (intra or with residual gas)
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