# Beam dynamics in RF linacs

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16 Νοε 2013 (πριν από 4 χρόνια και 8 μήνες)

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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
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
/
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

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

Example

:
axi
-
symmetric beam

uniform

parabolic

gaussian

Distributions :

Equivalent beams
: Same current, same sigma matrix

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

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,

-

-
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.E-12
1.E-11
1.E-10
1.E-09
1.E-08
1.E-07
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
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
/

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)