Beam dynamics in RF linacs
Step 2 : Beam transport
Nicolas PICHOFF
France
CEA-DSM/IRFU/SACM/LEDA
1
Outlines
• Beam representation
• Distribution
• Sigma matrix
• Emittance
• Matching
• Mechanism of emittance growth
2
Beam representation
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 :
pz
x
2
 px2  p y2   pz  pz

2
p
z
y
3
Beam representation
Particle representation
Each particle is represented by a 6D vector :
 p1 
 
 p2 
  p3 
P 
 p4 
p 
 5
p 
 6
x
 
 x 
y
 
 y 
 
 
W 
 
or
 x 
 
 px 
 y
 
 py 
 z 
 
p 
 z
or …
4
Beam phase-space representation
Beam representation
It is represented by a particle distribution in the 6D phase-space (P).
It can be plotted in 2D sub-phase-spaces :
5
Beam representation
Beam modelisation
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
 
 


Number ofparticles
f P, s  dP


Between P and P  dP


 f P , s  dP  N
1.0
0.8
0.6
0.4
0.2
-4
4
-3
3
-2
2
-1
1
0
0
1
-1
2
-2
3
-3
4
-4
6
Statistics
Beam representation
Average of a function A on beam :

 


1 N
A P    A Pi
N i 1

1 n
   A Pj
n j 1

 
1
   f P  A P  dP
N
 
 
7
Beam representation
First order momentum: beam Centre of Gravity (CoG)

 
AP P



 
P0  





p1
p2
p3
p4
p5
p6










Average :
position,
phase,
Angle,
Energy
…
8
Beam representation
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.
.

.
.
   
.
.

.

. . . . .

. . . . .
. . . . .

. . . . .
. . . . .
. . . . .

 i , j   pi  pi  p j  p j
 x  det12,12   y  det 34,34 

 z  det 56,56 
are the beam 2D rms emittances
9
2D RMS Emittance
2
2
2
~
 x  x  x   x  x   x  x  x  x 
Beam representation
(for example)
 The statistic surface in 2D sub-phase-space occupied by the beam
 Indicator of confinement
10
Beam Twiss parameters
Beam representation
The goal is to model the beam shape in 2D sub-phase-space with ellipses.
The beam Twiss parameters are :
~2
~2
w
w
w  ~
w  ~


w
w  
w 
w
w  w  w
~

w
 w  w2  2 w  ww   w  w2  5   w
5 : uniform elliptic distribution with same rms size.
11
Linear transport
6D transport matrix
The transverse force is generally close to linear.
The longitudinal force can be linearized when  << s.
The particle transport can be represented by a 6×6 transfer matrix :

 p1 
.
 


 p2 
.

p 
.

 3   M s1 / s0   
 p4 
.

p 
.

 5


.
p 


 6  s1 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.   p1 
  
.    p2 
.   p3 
   
.    p4 
 
.   p5 
  

.   p6  s
0
12
Sigma matrix transport in linear forces
Linear transport
The transport of sigma matrix can be obtained from
particle transfer matrix :
 s  M s1 / s0   s  M s1 / s0 
T
1
0
13
Linear transport
RMS emittance evolution in linear, uncoupled forces
~x2  x 2  x2  x  x

2

x  x  x


 x  x  x

d~x2
 2  x  x  x 2  x  x  x  x
ds
If linear force :

d~x
0

ds
x  k  x
 The emittance is constant.
dz

 x
If acceleration : x  k  x 
z ds
1
d~x
1 dz ~


 x

ds
z ds
 The emittance is damped.
~xn  z  ~x
, the normalized rms emittance, is conserved.
14
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
Their shapes are given by Courant-Snyder parameters
 wm s   w2  2 wm s  ww   wm s  w2   0
- Slide 57 : A beam can be represented by 2D-ellipses
Their shape are given by Twiss parameters
 w s w2  2 w s ww  w s w2  
15
Beam dynamics
Beam linear matched
The beam is linear-matched :
Beam Twiss
parameters
(S57)
Phase-space trajectory
Phase-space periodic looks
Channel Courant= Snyder parameters
(S36)
 w   wm
 w   wm
 w   wm
Matched beam
Bigger input beam
Smaller input beam
Phase-space scanned by
the mismatched beams
50% mismatched beam
 The beam second order (or envelope) motion is periodic
16
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
17
Beam dynamics
Space-charge forces
Electromagnetic interaction between particles.
It is linear if beam is uniform, non-linear otherwise (generally).
Example : axi-symmetric beam
2
Firled
Density
1.5
1
0
1
0.5
0
1
2
3
4
Radial position
Distributions :
uniform
5
0
0
1
2
3
Radial position
parabolic
4
5
gaussian
Equivalent beams: Same current, same sigma matrix
18
Introduction
Mechanisms of
emittance growth
and particle losses
19
Introduction
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
Other mechanisms play a (small) role :
- Coupling between directions (x, y, phi),
- Interaction with residual gas,
- Intra-beam scattering.
20
1 – Intern mismatching
Beam dynamics
In an intern matched beam, beam distribution in phase-space is constant on particle

trajectory in phase-space.
p
H = Cste 

r
f u, u  f H u, u
If not, the beam distribution “re-organise” itself.
21
2 - Filamentation
Beam dynamics
When the confinement force is non linear (multipole, longitudinal, space-charge),
the particle oscillation period depends on its amplitude :
d 2w
 k w s , w  w  0
2
ds
Particle do not rotate at the same speed in the phase-space : possible filamentation
Linear force
Non linear force
22
Beam dynamics
3 – Space-charge resonance
- In non-linear forces, the particle oscillation period depends on its amplitude
- The space-charge force acting on a particle depends on beam average size
- If the beam is mismatched, its average size oscillates with 3 “mismatched” modes
Quadrupolar
mode
High-freq
breathing
mode
Low-freq
breathing
mode
- Some particles can have oscillation period being a multiple of these modes
- The amplitude of these particles will resonantly growth and decreased
23
4 - Coupling
Beam dynamics
The preceding developments assumed that the force along each direction was
depending only on the particle coordinates in this direction (even non-linear).
When the force also depends on other coordinates
 2 particles with the same sub-phase-space position can feel different forces and
get separated in the sub-phase-space.
 2D emittance growth
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,
- …
24
5 – Residual gas interaction
Beam dynamics
Atom nucleus
Charge : +Z.e
Mass : M
bmin
b
Particle
Charge : z.e
Mass : m
Energy : E
Momentum: p
Cross section :
(Rutherford)
Probability :
b
bmax
Electrons
d cm2
d
min
13 z 2 Z 2 1
 26



10
2
EMeV
3
dPm1 ( )
d
max
0.31 Z 2 PhPa 1

 3
2
EMeV

25
Beam dynamics
5 – Residual gas interaction(2)
-30
-25
-20
-15
-10
-5
0
1
(a) "high" N2 pressure
10-1
Beam profile
(a) Simulation result
Beam
core
(b) "low" N2 pressure
(b) Simulation result
10-2
10-3
Good agreement
Mismatching
(a)
10-4
10-5
(b)
x (mm)
10-6
26
Beam dynamics
6 – Intra-beam scattering
Transfer of energy between 2 directions in a
two-body collision.
Very efficient if different longitudinaltransverse emittances
1.E+00
=1
 = 1.5
=2
=3
I = 98 mA ; f = 352 MHz
x0 = y0 = 2.5 mm ; z0 = 6.75/ mm
x'0 = y'0 = 3.57.6MeV/ mrad
z'0 = x'0
1.E-01
1.E-02
1.E-03
 : Ratio between
longitudinal and
transverse energy
1.E-04
n(x)
1.E-05
1.E-06
1.E-07
1.E-08
1.E-09
1.E-10
1.E-11
1
1.E-12
0
0.5
1
1.5
1.5
x
2
2
3
2.5
3
Tails induced by 2 body collision in a uniform proton beam
27
Summary
conclusion
• Beam is a set of particles
• Beam can be modeled with:
• macroparticles,
• distribution function,
• statistic properties
• The simplest is the 2nd 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)
28