Transcript Basics_Ghana_LRivkinx

Particle Accelerators: An introduction
Lenny Rivkin
Swiss Institute of Technology Lausanne (EPFL)
Paul Scherrer Institute (PSI)
Switzerland
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Applied relativity
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
FOR THE SAME ENERGY EXTRACTED FROM THE FIELD,
A PARTICLE WITH LOWER MASS IS MORE RELATIVISTIC
CATHODE
v
c
-
U
+
1

v
1
 1 2

c
0.8
v/c
0.6
e
p
0.4
eU
  1
Eo
0.2
0
0.001
0.01
0.1
1
10
100
1000 10000
eU [MeV]
Beams of ultrarelativistic particles:
e.g. a race to the Moon
An electron with energy of a few GeV emits a photon...
a race to the Moon!
∆𝑡 =
𝐿
𝛽𝑐
𝐿
𝑐
- =
𝐿
𝛽𝑐
1−𝛽
𝐿
~
𝛽𝑐
∙
1
2𝛾2
𝐿
Electron will lose
∆𝐿 = 𝐿 1 − 𝛽 = 2
2𝛾
 by only 8 meters
 the race will last only 1.3 seconds
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Transformation of angles: collimation
 1
q =   qe
qe
q
v~c
Sound waves (non-relativistic)
q
qe
v
v
 vs
vs
q = v + v = v  1 v  qe  1 v
s||
s|| 1 +
1+ v
v
s
Doppler effect
(moving source of sound)
s
heard

v
 emitted 1  
 vs 
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Time compression
Electron with velocity emits a wave with period Temit
while the observer sees a different period Tobs because
the electron was moving towards the observer
n
q

Tobs  (1  n  β) Temit
The wavelength is shortened by the same factor
obs  (1   cosq ) emit
in ultra-relativistic case, looking along a tangent to the
trajectory

2
1
–

1

1
1
–

=

since
obs = 2 emit
1 +  2 2
2
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Electromagnetism
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Maxwell equations (poetry)
War es ein Gott, der diese Zeichen schrieb
Die mit geheimnisvoll verborg’nem Trieb
Die Kräfte der Natur um mich enthüllen
Und mir das Herz mit stiller Freude füllen.
Ludwig Boltzman
Was it a God whose inspiration
Led him to write these fine equations
Nature’s fields to me he shows
And so my heart with pleasure glows.
translated by John P. Blewett
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Field of a charge
At rest: Coulomb field
Moving with constant velocity
v = const.
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Charge in an external electromagnetic field
𝐹 = 𝑒(𝐸 + [𝑣x𝐵])
Lorentz force
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Particle motion in electromagnetic fields
Lorentz force
F = e E + v  B
Same force for:
 Magnetic field B = 1 Tesla
(typical for magnets)
 Electric field E = 3·108 V/m
(presently out of
reach)
Magnetic fields are used exclusively to bend and
focus ultra-relativistic particles
B
 Constant magnetic field
 Magnetic rigidity

p
B = e
or, in practical units

Fm
e

Fc
1
T  m B =
p GeV
c
0.29979
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
v
Roller derby in Los Angeles on 7 July 2012
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Under the sign of the Higgs on 7 July 2012
after
the announcement on July 4, 2012
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Fields of a long bunch (linear charge density )
Transverse electric field: from Gauss law
Er

2 r  Er 
0

Er 
2 0 r
Iv
Transverse magnetic field: from Ampere law
0 

v
Bq 
v
 2
2 r
2 0 r c
2 r  Bq  0 I
1 V 
Bq T   Er
c  m 
Bq
1
0 0  2
c
0  4  107 V s Am
 0  8.85  10
12 C
V m
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Fields in the bunch
 Round uniform distribution
1.0
eN  1
Er  2 
l r
0
eN 
Er  2 
0l
r
a2
ra
ra
Fields
0.8
0.6
0.4
0.2
0.0
0
2
4
6
8
10
6
8
10
r
 Round Gaussian distribution
0.5
Fields
2
 12 r  

eN 1  e


Er 
r
2 0l 




0.4
0.3
0.2
0.1
0.0
0
2
4
r in standard
Introduction to Accelerators, African School of Physics, KNUST, Kumasi,
Ghana;deviations
L. Rivkin, PSI & EPFL
Using large magnetic fields of electron beam
10


GV
eN
r
N
10
rm


Er

 14.4
2
 m  4 0l
l mm   2 m
Er
N 1010 rm
Bq T  
 50
c
l mm  2 m
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Force seen by a test particle
 “Fellow-traveler”: E and B nearly cancel
 v2  1
Force  eE r  evBq  eE r 1  2   2 eE r
 c  
 Particle travelling in the opposite direction:
contributions from E and B add
Force  eE r  evBq  2eE r
 For round Gaussian distribution
2
e2 N 1 
 12 r  
F
  1  e

 0l r 

Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Synchrotron radiation
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
THEORETICAL UNDERSTANDING 
1873
Maxwell’s equations
 made evident that changing charge densities would
result in electric fields that would radiate outward
1887 Heinrich Hertz demonstrated such waves:
….. this is of no use whatsoever !
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Accelerated charges radiate EM waves
An electron of energy E in a magnetic field B
Power emitted is proportional to:
2

2
PE B
4
cC E
P 
 2
2 
re
–5
m
C = 4
=
8.858

10
3 mec 2 3
GeV 3

Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
The power is all too real!
4
cC E
P 
 2
2 
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Synchrotron radiation power
Power emitted is proportional to:
cC  E 4
PSR =

2  2

re
–5
4
m
C =
=
8.858

10
3 mec 2 3
GeV 3
P  E 2 B 2

4
2
2
PSR = hc 2
3


Energy loss per turn:

4
E
U0 = C  
 1
=
137
 = 197 Mev  fm
hc

4
4
U0 = hc 
3
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Why do they radiate?
Charge at rest: Coulomb field, no radiation
Uniformly moving charge
does not radiate
v = const.
But! Cerenkov!
Accelerated charge:
fields separate from the charge
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Bremsstrahlung
or
breaking radiation
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Liénard-Wiechert potentials

1
q
t =
40 r 1 – n  

At =
ret
q
v
40c 2 r 1 – n  
ret
and the electromagnetic fields:

1 
 A + 2
=0
c t
(Lorentz gauge)
B =  A

A
E = –  –
t
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Fields of a moving charge

Et =
q
n–
1

4 0 1 – n   3 2 r 2

q n  n –  
3 2
40c
1–n 
+
ret
 1r
ret

1
B t = n E
c
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Transverse acceleration
a
v
Radiation field quickly
separates itself from the
Coulomb field
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Longitudinal acceleration
a
v
Radiation field cannot
separate itself from the
Coulomb field
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
High Energy Storage Ring
To store relativistic particles (v ≈ c) in a ring for ~ 10h
they travel a distance of diameter of Pluto’s orbit
 Trajectories are bent into a closed path
 Beams need to be focused to keep
particles close to ideal orbit
(stability questions)
Ideal orbit (usually in horizontal plane)
 Smooth, roughly circular shape closed curve,
consisting of arcs and straight sections
 Magnets are placed along the ideal orbit, design
fields adjusted, so that particles of nominal
energy follow the ideal orbit for ever and ever
and ever ...
Storage ring layout
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Particle motion in electromagnetic fields
Lorentz force
F = e E + v  B
Same force for:
 Magnetic field B = 1 Tesla
(typical for magnets)
 Electric field E = 3·108 V/m
(presently out of
reach)
Magnetic fields are used exclusively to bend and
focus ultra-relativistic particles
B
 Constant magnetic field
 Magnetic rigidity

p
B = e
or, in practical units

Fm
e

Fc
1
T  m B =
p GeV
c
0.29979
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
v
Bending magnets (iron dominated)
Iron saturates at 2 T
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
SLS dipole
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Bending magnets (dipoles)

B dl = 2NI

Iron dominated magnets (B < 2 Tesla)

h
 0 = 4  10 – 7 m

N turns coil
h
B
Iron
yoke
BT  
2 0 N I  Amp  turns 
hm
 for h = cm
e.g.
N I = 20'000 Amp  turns
B = 1.6 Tesla
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
System of coordinates
z
Design orbit in horizontal plane
 consists of arcs and straight segments
x
Local curvilinear coordinates:
 x, z
 s
 (s)

s
transverse displacements from design orbit
measured along the design orbit
local radius of curvature (depends on field)
 Length element
2


x
dl 2  dx 2  dz2  1   ds2
  
dl  ds
dl
x

ds

x
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Quadrupole lens
Focusing in one plane
Defocusing
N
S
Defocusing in the other
plane
Focusing
S
N

Bz Bx
 B = 0 
=
x
z
Linear restoring force
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Theoretical magnetism (after Bruno Touschek)
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Focusing elements
Focal length of a thin quad
Particle off-axis in a quad
 sees constant magnetic field (thin lens!)
and is bent by it
 dx
 the slope x 
ds
changes by

eBz
l
x = –  = – l p
 Defining the focal length

1 = e g l
f p
with gradient
Bz
g
x
x

x
f
s
x  – x
l
f
1 Dioptre = 1 m-1

1 = g l
-1]
[m
f
B
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Hamiltonian dynamics
(brief reminder)
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
A dynamical system
q
H q, p, t  p
t
is described by a Hamiltonian
coordinate
canonical momentum
independent variable (time)
The equations of motion: Hamilton‘s equations
dq H

,
dt p
dp
H

dt
q
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Phase space
 x, p x 
– canonical variables
 x , x 
– a point in 2-d phase space
px
x 
p
x´
x  xo  s xo'
x '  xo'
x
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Conservative
Hamiltonian systems
2
p
H
 V q, t 
2m
The equations of motion are:
H p
q 

p m
H
V
p  

 F q, t 
q
q
The Hamiltonian is conserved, ist value – energy
dH H
H
H H H  H 

q 
p 


0
dt
q
p
q p p  q 
dH
0
dt
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Phase space
q, p 
 H H 
v
,

 p p 
 H H 
H q, p   
,

 p p 
– a point in 2-d
phase space
– velocity vector in 2-d
phase space
– the gradient of the
Hamiltonian,
orthogonal to velocity
The motion is along the curves of H = const
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
DRIFT SECTION
CHANGE OF PARTICLE DISTRIBUTION IN PHASE SPACE
The initial coordinates of a particle ensemble in the
transverse phase plane are contained in the ellipse:
x  xo  s xo'
dx
x' 
ds
INITIAL COORDINATES
( xo , x'o )
x '  xo'
( x , x' )
x
FINAL COORDINATES
Focusing is needed to avoid beam blow up !
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
FOCUSING QUADRUPOLE
CHANGE OF PARTICLE DISTRIBUTION IN PHASE SPACE
x' 
x  xo
dx
ds
x' 
( xo , x'o )
INITIAL COORDINATES
x'o
xo

f
x
( x , x' )
FINAL COORDINATES
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
PHASE SPACE: angle – action variables

x
A
A
2
x


Linear transformation is a simple
rotation in these coordinates
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
PHASE SPACE TRANSFORMATION in NON-linear element
(sextupole magnet)
2

  m


A



Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Conservation of phase space: emittance
Canonical transformations
preserve phase space areas
x'
x'
x

x
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
The language of
Accelerator Physics
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Matrix notation
Transfer matrices (as in geometric optics)
 Describe canonical transformations
i.e. phase space area is preserved
(symplectic matrices)
 Thin focusing lens
 Drift of length L

x
x
out
det M = 1
=
x
x
1
0
1
 x
x
in
= 1L  x
01
x
out
in
–1f
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Strong focusing example
f
Mx =
–f
1 0
1f 1
–f
1 0
–1 f 1
f
1L 1 0
0 1 –1 f 1
Mz =
1–Lf L
Mx =
– L f2 1 + L f
focusing for L « f
1L
01
1 0
1f 1
This lens doublet focuses in both planes
 The focal length is, of course, the good
old lens makers equation
1
1
f*
=
+ 1 – L = L2
f1 f2 f1  f2 f
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
SUMMARY:
Strong (Transverse)Focusing –
Alternating Gradient Principle
A sequence of focusingdefocusing fields provides a
stronger net focusing force.
Quadrupoles focus
horizontally, defocus vertically
or vice versa. Forces are
proportional to displacement
from axis.
A succession of opposed
elements enable particles to
follow stable trajectories,
making small oscillations about
the design orbit.
Technological limits on
magnets are high.
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Equation of motion
dl
x
In horizontal plane
x  q  q0
dq 0  
and
x 
d q  q 0 
ds
ds

ds
0
By
dq    
dl

( B )
dl
x  x  dl
q
x

dl  ds  1  
 
d q  q 0 
x 1
1

 1

x  
    k x  1       2  k  x
ds

   


x  K x  x  0 where K x 
1

2
k
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Equations of motion
In individual elements K = const. : Harmonic Oscillator
x + K  x = 0
(KISS principle of accelerator building)
z – K  z = 0
Overall, K(s) is a piecewise constant, periodic function
x + K s  x = 0
K s =K s+C
K(s)
Hill equation
 p2 K s  x2
H=
+
2
2
(C - circumference or period)
C s
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Hill equation
First used by an astronomer G. Hill in his studies of the
motion of the moon, a motion under the influence
of periodically changing forces
1838 -- 1914
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL

u + Ku = 0
Harmonic oscillator
Solution:
+A
u
u s = A  cos K  s
s
-A
Amplitude:
constant A
Phase advance:
uniform:  s = K  s
Phase space:
u = – A K  sin K  s
Independent of s:
2
A = u 2 + u
K
u’
A K
A u
2

u + Ku = 0
Solutions of the Hill equation
u + k s  u = 0
“Pseudo-harmonic oscillator” solutions (here u stands for x or z)

s
ds
s =
u s = a  s cos  s – 0
s
0

We have introduced a periodic “envelope” function (s)
the amplitude of the betatron oscillation is modulated by
 s

the rate of phase advance at any point along the
accelerator is related to the value of the  function at that
point
 1
 =

Hill equation (pseudo-harmonic oscillations)
Solution:
u
 = A (s) cos (s)
u(s)
Amplitude:
s
modulated with s: A (s)

Phase advance: non-uniform: (s) =
s
0
ds
(s)

Phase space: u s = –  u – 1 A  sin 



where  s  – 1  s
2
Depends on s:
2
A  s = u 2 + u + u
2
A

u’
A (s)
u

u + K s u = 0
Sphere rolling in a gutter analogy
Turn, after turn, after turn…
Betatron oscillations within an envelope
x s    s  sin  s    0 
Turn, after turn, after turn…
Betatron oscillations within an envelope
x s    s  sin  s    0 
Harmonic oscillator solutions (K = const.)
For K(s) constant, “principal” solutions are (harmonic oscillator)
 case of K > 0
1
sin( Ks)
K
for K  0
1
C(s)  cosh( K s) and S(s) 
sinh( K s)
K
for K  0
C(s)  cos( Ks) and S(s) 


case of K < 0
these are linearly independent solutions with initial conditions:
dC
dS
C(0)  1; C (0) 
 0 and S(0)  0 ; S(0) 
1
ds
ds

any other solution is a linear combination of these:
u(s)  C(s)u0  S(s)u0
u(s)  C (s)u0  S (s)u0
u(s)  C(s) S(s) u0 

  
 
u(s) C (s) S (s)u0
Harmonic oscillator solutions (matrix form)
Transfer matrices for particular cases:
 drift space ( K=0 )
x ' ' K x ( s ) x  0
1 L u 
u 
 
   
uout 0 1 u in

focusing magnet ( K > 0, const. ) of length l

u 
cos( Kl )


 
uout 
 K sin( Kl )

1

sin( Kl )u 

K

u


cos( Kl )  in
defocusing magnet ( K < 0, const. ) of length l

u 
 cosh( K l )

 
uout 
 K sinh( K l)


y ' ' K y ( s ) y  0
the thin lens limit:

1
sinh( K l)u 
K
 

u 

cosh( K l )  in
lim l  0 keeping K l 
1
= const.
f
TRANSVERSE MOTION SUMMARY
x ' ' K x ( s ) x  0
y ' ' K y ( s ) y  0
DRIFT:
k=0
1 L u 
u 
 
   
uout 0 1 u in
QUADRUPOLE:
K > 0  focusing
K < 0  defocusing

u 
cos( Kl )
   
uout 
 K sin( Kl )

u 
cosh( K l )

   
uout  K sinh( K l)

1

sin( Kl )u 

K

u

cos( Kl )  in

1
sinh( K l)u 
K
 

u 

cosh( K l )  in
Stability of transverse (betatron) oscillations
The transfer matrix of a beamline that consists of elements
with individual matrices M1 , M2 , ... Mn M tot  Mn  ...  M 2  M1
(N.B. the order in which matrices are multiplied!)

Full turn matrix M
x
x

= Mn x
x
n
0
After n turns must remain finite for arbitrarily large n
Stability condition
Let v1 and v2 be eigenvectors and 1 and 2 eigenvalues of M
x
x


0
= Av1 + Bv2
0
= A1nv1 + B2nv2
n
n

,

For stability 1 2 must not grow with n
since the product of eigenvalues is unity:
 M = 1    = 1
det
1 2
we can write in general

n x
M
x
1 = e i,  2 = e – i
For stability µ should be real!
 M =  + = 2 cos 
Tr
1
2

1
– 1  Tr M  1
2
Example
Consider one period of FODO lattice:

1 0
1 0
M= 1L 
 1L 
0 1 1f 1
01
–1f 1
M=
1– L– L
f
f
– L2
f
2
f
–f
L
2
L
2L +
f
L
1+L
f

–1  1– 1 L
2 f

applying the stability condition

The motion is stable, provided the focal
length > 1/2 the lens spacing
2
L
1
2f
 1
Solutions of the Hill equation
u + k s  u = 0
“Pseudo-harmonic oscillator” solutions (here u stands for x or z)

s
ds
s =
u s = a  s cos  s – 0
s
0

We have introduced a periodic “envelope” function (s)
the amplitude of the betatron oscillation is modulated by
 s

the rate of phase advance at any point along the
accelerator is related to the value of the  function at that
point
 1
 =

Courant - Snyder invariant
At any point s along the accelerator for a given betatron
oscillation the following combination of u and u’ has the same
value

2
2

u
a2 =
+  u – u

2
 1
  – 
2
 1 + 2
Introducing some additional notation:


 2
2
2
 = a = u + 2uu + u
Describing an ellipse in phase space {u,u’} with area ·
 The parameters ,, vary along the machine
 The phase space area remains constant
Betatron oscillation solution
xs    s  cos s   0 
 Displacement
x s   

Slope

combining the two

x s    s  sin  s   0 

x 2   x   x2   
   x  2 xx   x
2
equation of an ellipse with area = 
2
Phase space ellipse: Courant – Snyder Invariant
Single particle motion
 At a place with
Courant-Snyder
parameters
( , ,  )

xmax
= 
 2
2
 = x + 2xx +x
x’

 s  – 1  s
2
 1+s 2
s 
(s)

slope
=– 

xmax = 
x

at a given point s
xn   cos s  n  L  0 
Beam centroid
Simple case: Upright ellipse
For the simple case when
 0   
2
x
2
    x

the ellipse is upright
1 2


x’
x
1

Beam emittance

Area
= 
Betatron oscillations
• Particles in the beam execute betatron oscillations with
x’
different amplitudes.
Transverse beam distribution
• Gaussian (electrons)
• “Typical” particle: 1 -  ellipse
(in a place where  = ’ = 0)

 x2
Emittance 


Units
of  m  rad

x =  
 x =  / 
x’
x
x

 =  x   x
 x
=
x
2-D Gaussian distribution
x
Electron rings emittance definition
 1 -  ellipse
x’

n x dx =
1 e –x 2 / 2 2dx
2 
x
x
Area = x


Probability to be inside 1- ellipse
Probability to be inside n- ellipse
P1 = 1 – e – 1 2 = 0.39
Pn = 1 –
n2 2
–
e
Courant - Snyder parameters: transfer matrices
Consider a transfer matrix M for a full turn starting at some point

We know that it is symplectic, i.e. det M = 1

Any such matrix with unit determinant can be
parameterized:

cos +  sin 
sin 
M=
–  sin 
cos  –  sin 
where in order to satisfy the condition of unit determinant
 –  2 = 1
we can regard this now as just a formal parameterization
Phase space ellipse

xmax
= 
 2
2
 = x + 2xx +x
at smax/min where = 0
x’

slope
=– 

xmax = 
x

x

Beam centroid
 1+s

 s  – 1  s  s  (s)
2
x 
2
circle of radius 
everywhere
in the machine
Tune
Transfer matrix for one complete turn:

cos +  sin 
sin 
M=
–  sin 
cos  –  sin 
Phase advance over one turn is independent of location

=
ds
s
Tune Q is the number of betatron oscillations in one revolution

ds
Q 1
2  s
Transfer matrix between two points
The transfer matrix between two arbitrary point in the
machine
x
= M 12 x
x 2
x 1
in terms of Courant - Snyder parameters at these points and
the phase advance between them

2
cos  +  1sin 
 1 2 sin 
1
1 +  1 2 sin  +  2 –  1 cos 
1
–
cos  –  2 sin 

 1 2
2
Transforming C-S parameters between two points
The transfer matrix between two arbitrary point in the machine
x
x
2
= M 12
x
x

m11 m12
M 12 = m m
21 22
1
The Courant - Snyder parameters at those points are

2
2

m
–
2m
m
m
2
11 12
11
12
2 =
– m11  m21 1 + 2m12 m21 – m12  m22
2
m2
– 2m m
m2
21
21
22
22
And the phase advance between the points

tan  =
m12
m11   1 – m12   1

or sin  =
m12
 1 2
related by

1
1
1

 =
2
1
ds
s
Full turn transfer matrix
Transfer matrix for one complete turn:
 sin 
 cos    sin 

M 

cos    sin  
   sin 
Tune Q is the number of betatron oscillations in one revolution

1
ds
Q

2 2   s 
If the tune is an integer, i.e.
  2  n
 1 0
M 

 0 1
Some simple cases and their phase advance
s1  s2
Thin lens
‘point to point imaging’
r12  0
 x2   r11 0  0   0 
 
   

 x2   r21 r22  x1   r22 x1 
‘parallel to point imaging’
r11  0
 x2   0 r12  x1   0 
 
   

 x2   r21 r22  0   r21 x1 
drift of length L
  0
r12  L
Proton therapy Gantry at PSI: point to parallel!
  n  
tan   
1
1
for 10   n 
sin  
L
1 2

2
Courant - Snyder parameters: transfer matrices
The eigenvalues of this matrix are related to µ:
 = e i  Tr M = 2cos
We can also write our matrix M as
M = I cos + J sin 

J
M
=
e

where J =   and J 2 = – I
– –
The powers of matrix M can be written simply as:
M k = I cos k + J sin k
And the elements of Mk are bounded for all k if µ is real
 M 2
Tr
FODO cell lattice
ACCELERATING CAVITY
BENDING MAGNET
DIPOLE
CIRCULAR ACCELERATOR
FOCUSING/DEFOCUSING MAGNET
QUADRUPOLE
SECTOR BENDING MAGNET
xo
 xo
B
Positive displacement xo of the initial coordinate from the center axis
leads to a longer path inside the magnet, i.e. more deflection
Negative displacement -xo of the initial coordinate from the center axis
leads to a shorter path inside the magnet, i.e. less deflection
In both cases the trajectory comes closer to the central orbit  FOCUSING
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Off-energy particles
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Off-momentum particles
Particle with an energy deviation d
Design orbit
d0
 Will be bent and focused differently
d0
 The equation of motion: non-homogeneous Hill equation
x  k ( s ) x 
δ
ρ
Off-momentum particles
Particle with an energy deviation d
Design orbit
d0
 Will be bent and focused differently
 The equation of motion: non-homogeneous Hill equation
x  k ( s ) x 
 The motion is a sum of the
d0
δ
ρ
solution of homogeneous equation + a particular solution
Dispersion function
Particle deviation from ideal orbit
x = x + x = x + D s  d
D(s) - dispersion function

Periodic solution of the inhomogeneous Hill equation

D + k s D = 1
s


= 0 in straights
= 1 in bends
New equilibrium orbit of a particle with energy deviation d
Betatron oscillations are executed around this new
equilibrium
Matrix notation: extended to 3 by 3 case
Taking into account particle energy deviation, particle position
x
  
x   x 
d 
 
and
 x s  
 x0 


 

x s    xs   M   x0 
 d 
d 


 
 C s  S s  D s  


M   C s  S s  Ds 
 0

0
1


where D and D’ are the solutions of inhomog. equation
we usually assume
that d
does not change
Examples of 3 by 3 transfer matrices
For simple cases of piece-wise constant K(s), (s)
 cos 


  K sin 


0


 cosh


  K sinh 


0


1
sin 
K
cos 
0
1
sinh 
K
cosh
0
1
1  cos  
K

1
sin  

 K

1


1
cosh  1
  K 

1
sinh  

 K

1


K 0
  K s  s0 
K 0
   K s  s0 
Bending magnet transfer matrix
Pure dipole field:
 k = 0
 q – bending angle
 cosq
 1
M    sin q
 

0

In the vertical plane - drift
K
1

2
   K s 
 sin q
s

q
 1  cosq 
cosq
sin q
0
1





Dispersion: periodic solution
Let the matrix for one full period be
Dispersion being a periodic solution:
D 
m13m21  1  m11 m23
1  m11 1  m22   m21m12
m13
m12

D
D 
1  m11
1  m11
 m11 m12

M   m21 m22
 0
0

m13 

m23 
1 
D
D
 
 
 D   M   D 
1
1
 
 
FODO cell lattice
Beam size

When the beam energy spread is d
x’
d0
d0
x
2
 =
2

+
2

2 2
=   + D d
Full turn transfer matrix
Transfer matrix for one complete turn:
 sin 
 cos    sin 

M 

cos    sin  
   sin 
Tune Q is the number of betatron oscillations in one revolution

1
ds
Q

2 2   s 
If the tune is an integer, i.e.
  2  n
 1 0
M 

 0 1
Full turn transfer matrix: special cases
The tune is half-integer, i.e. {Q}  0.5
 1 0 
M 
  I
 0  1
M2  I
The tune is quarter-integer, i.e. {Q}  0.25

M 

 
J
 
M 2  J 2  I
M4  I
Errors, errors, errors
Suppose at some point along the accelerator
 extra field B over some length l
 it will kick a particle by an angle q
 B  l
q=
B
 10
If the tune Q is close to an integer, M  0 1
the kicks will add up in phase each turn
driving the particle out of the machine
Integer resonance
v’
q
q
q
v
Betatron oscillation solution

x s    s  cos s   0 
Displacement
x s   

Slope

combining the two

x s    s  sin  s   0 

x 2   x   x2   
   x  2 xx   x
2

2
e.g. if we start a particle with x0‘ at a place whith 0
   0 x0
2
x s   x0  0  s  cos s   0 
Field error
In the presence of such a kick
 x = 0 is no longer a solution
 there will be a new closed orbit

q  s q
xs =
cos  s – Q
2 sin Q
Particles perform betatron oscillations around this new
closed orbit

 function is a measure of sensitivity to errors

when Q approaches an integer value, the new closed
orbit becomes very large
Focusing error
A gradient error over a short distance
 1

 1 f
0

1

a thin lens

transfer matrix for full turn becomes
 1
M  M0 
 1 f
0

1
Focusing error (algebra)
 1
M  M0 
 1 f
 sin 
 cos    sin 



cos    sin  
   sin 
 cos 0   sin 0

   sin 0
1

 0 sin 0
  1
 
cos 0   sin 0   f

0

1

Comparing the traces of the two matrices, the new tune:
1 0
cos 2 Q  cos 2 Q0 
sin 2 Q0
2 f
0

1
Stability
The motion remains stable in the presence of focussing errors,
if the new tune remains a real number, i.e.
1 0
cos 2 Q  cos 2 Q0 
sin 2 Q0  1
2 f
and when the unperturbed tune is not near an integer or
half-integer resonance and the perturbation is sufficiently
small
1 
Q  Q0  d Q  Q0 

4 f
Focusing error
v’
Tune near half-integer: {Q}  0.5
7
5
 1 0 
M 
  I
 0  1
3
1
2
4
6
8
the kicks will add up in phase every two turns
driving the particle out of the machine
Half-integer resonance
v
Tune shift
 A small gradient error leads to a change in tune:
1 
dQ  
4 f
 A distribution of gradient errors leads to a tune shift
1 g s    s 
Q 
ds

B 
4
This is how the tunes are adjusted
Chromaticity
Focusing depends on particle energy
Equivalent to an error in gradient
k = – k d
g
g
k

 k 1  d 
B  B 0 1  d 
Causes a tune shift of:
We define chromaticity x

Q = 1 k  ds = – 1 k ds  d
4
4

Q = – x d
In strong focusing rings x ~ -100 ! For energy spread d ~1%
Q ~ 1 !!
Need positive chromaticity to prevent “head-tail” instability
Chromaticity correction
How can we adjust chromaticity? We need
gradients (focusing) that changes with energy deviation d
Sextupole magnets
In horizontal plane
B  m  x2
B   2m  x  2m D  d
1 BL 1
m
 g L
2  B  2
2
B
'
'
[
T
/
m
]L[m]
m[m 2 ]  0.2998
po [GeV / c ]
Sextupoles to correct chromaticity
Two ingredients are needed:
 Sextupoles placed in a region of
finite dispersion: sort particles
according to their energy deviation
x  x  Dd
 Gradients that depend
on particle position
x  mx 2  m( x  Dd ) 2
 mx2  2mDd  x  mD 2d 2
quadrupole term
DYNAMIC APERTURE
Having corrected chromatic aberrations we introduced
geometric aberrations:

we increased energy acceptance

but particles with large transverse amplitudes are no
longer stable!
Linear
A
Non-Linear
A
d
d
Longitudinal dynamics
Phase stability
Introduction to Accelerators, African School of Physics, KNUST, Kumasi, Ghana; L. Rivkin, PSI & EPFL
Longitudinal motion:
compensating radiation loss U0
f RF  h  f 0
 RF cavity provides accelerating field
with frequency
VRF
• h – harmonic number
U0
 The energy gain:
U RF  eVRF  
 Synchronous particle:
• has design energy
• gains from the RF on the average as
much as it loses per turn U0

Longitudinal motion:
phase stability
VRF
U0
 Particle ahead of synchronous one
• gets too much energy from the RF
• goes on a longer orbit (not enough B)
>> takes longer to go around
• comes back to the RF cavity closer to synchronous part.
 Particle behind the synchronous one
• gets too little energy from the RF
• goes on a shorter orbit (too much B)
• catches-up with the synchronous particle

Orbit Length
Length element depends on x
dl = 1 + x ds
Horizontal displacement has two parts:
dl
ds

x
x = x + x


To first order x does not change L
x – has the same sign around the ring
Length of the off-energy orbit

L  = dl =

Ds
p E
L =d 
ds where d = p =
E
s
x
1 +  ds = L 0 +L
L =  d
L
Something funny happens on the way around the ring...
 L
T0 = 0
c
Revolution time changes with energy

T = L – 
T
L


d
dp
1
= 
 Particle goes faster (not much!)
 2 p



while the orbit length increases (more!)

The “slip factor”    since
T =  – 1  dp = dp
p
T
2 p
Ring is above “transition energy”
 >> 12

 1
 2
 tr
isochronous ring:  = 0 or  =  tr
(relativity)
L =  dp
p
L
Longitudinal motion: energy-time oscillations
energy deviation from the design energy, or
the energy of the synchronous particle


longitudinal coordinate measured from the
position of the synchronous electron
Longitudinal motion:
damping of synchrotron oscillations
P  E 2 B 2
During one period of synchrotron oscillation:
 when the particle is in the upper half-plane, it loses more
energy per turn, its energy gradually reduces

U > U0
U < U0

 when the particle is in the lower half-plane, it loses less
energy per turn, but receives U0 on the average, so its
energy deviation gradually reduces
The synchrotron motion is damped
 the phase space trajectory is spiraling towards the origin