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 7: Synchrotron radiation (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
http://www.adams-institute.ac.uk/training
Dr. Riccardo Bartolini (DWB room 622)
Lecture 4: Emittance and Liouville’s theorem
Hill’s equations (recap)
More on transfer matrix formalism
Courant-Snyder Invariant
Emittance
Liouville’s theorem
R. Bartolini, John Adams Institute, 1 May 2013
1/28
Linear betatron equations of motion (recap)
In the magnetic fields of dipoles magnets and quadrupole magnets the
coordinates of the charged particle w.r.t. the reference orbit are given by the
Hill’s equations
d2y
 K y ( s) y  0
2
ds
K x ( s) 
K z (s) 
1
1 Bz ( s)

 2 ( s) B x
1 Bz ( s)
B x
weak focussing
of a dipole
quadrupole
focussing
No periodicity is assumed but for a circular machine Kx, Kz and  are periodic
These are linear equations (in y = x, z). They can be integrated.
R. Bartolini, John Adams Institute, 1 May 2013
2/28
Pseudo-harmonic oscillations (recap)
The solution can be found in the form

y(s)   y  y (s) cos  y (s)  

s
 y (s ) 

s0
ds'
 y (s ' )
which are pseudo-harmonic oscillations
The beta functions (in x and z) are
proportional to the square of the
envelope of the oscillations
The functions  (in x and z)
describe the phase of the
oscillations
R. Bartolini, John Adams Institute, 1 May 2013
3/28
Principal trajectories (recap)
The solutions of the Hill’s equation can be cast equivalently in the form of
principal trajectories. These are two particular solutions of the homogeneous
Hill’s equation
y' 'k ( s) y  0
which satisfy the initial conditions
C(s0) = 1;C’(s0) = 0; cosine-like solution
S(s0) = 0;S’(s0) = 1; sine-like solution
The general solution can be written as a
linear combination of the principal trajectories
y(s)  y0C(s)  y'0 S (s)
R. Bartolini, John Adams Institute, 1 May 2013
4/28
Principal trajectories
vs pseudo harmonic oscillations
We can express amplitude and
phase functions
y(s)   (s) cos (s)   
y' (s)  

 (s)
sin (s)      (s) cos (s)   
y(s)  y 0 C(s)  y'0 S(s)
in terms of the principal trajectories
Simple algebraic manipulations yield
C(s) 
 (s)
(cos (s)   0 sin (s))
0
and viceversa
S(s)
 (s)  arctg
 0S(s)   0 C(s)
S(s)   (s)0 sin (s)
2
2
1 
 S (s)   0S(s)   0 C(s) 

 (s) 


 0 
 0S(s)   0 C(s)


or more simply
1
(s) 
0
 S(s) 


sin

(
s
)


 S' (s)
2
R. Bartolini, John Adams Institute, 1 May 2013
(s) 
(s)
 cos(s)
0
sin(s)
5/28
Principal trajectories (recap)
As a consequence of the linearity of Hill’s equations, we can describe the
evolution of the trajectories in a transfer line or in a circular ring by means of
linear transformations
 y(s)   C(s) S(s)  y(s 0 ) 


  

 y' (s)   C' (s) S' (s)  y' (s 0 ) 
C(s) and S(s) depend only on the magnetic lattice not on the particular
initial conditions
 C(s ) S(s ) 

M1 2  
 C' (s ) S' (s ) 
This allows the possibility of using the matrix formalism to describe the
evolution of the coordinates of a charged particles in a magnetic lattice
R. Bartolini, John Adams Institute, 1 May 2013
6/28
Matrices of most common elements
Transfer lines or circular accelerators are made of a series of drifts and
quadrupoles for the transverse focussing and accelerating section for
acceleration.
Each of these element can be associated to a particular transfer matrix
Matrix of a drift space
1 d

M  
0 1
Matrix of a focussing quadurpole

cos( | K |L)

M
  | K | sin( | K |L)


sin( | K |L) 
|K|

cos( | K |L) 
1
Thin lens approximation
L  0, with KL finite
0
 1

M  

|
K
|
L
1


Matrix of a defocussing quadurpole

 cosh( K L)
M
 K sinh( K L)


sinh( K L) 
K

cosh( K L) 
1
R. Bartolini, John Adams Institute, 1 May 2013
 1 0

M  
 KL 1 
7/28
Matrix formalism for transfer lines
For each element of the transfer line we can compute, once and for all, the
corresponding matrix. The propagation along the line will be the piece-wise
composition of the propagation through all the various elements
Q1
L
s1
M 12
Q2
s2
0  1 L  1
0  1  L / f1
L 
 1

  


 

1
/
f
1

1
/
f
1

1
/
f
*
1

L
/
f
0
1

2
1
2


 
1
1 1
L
  
f * f1 f 2 f1 f 2
Notice that it works equally in the longitudinal plane, e.g.
M12
 1
 
 1/ f
0

1 
qVL sin s
1

f
m c3 s3
thin lens quadrupole associate to an RF cavity of voltage V and length L
R. Bartolini, John Adams Institute, 1 May 2013
8/28
Matrix formalism and analogy with geometric
optics
Particle trajectories can be described with a matrix formalism analogous to
that describing the propagation of rays in an optical system.
The magnetic quadrupoles play
the role of focussing and
defocussing lenses, however
notice that, unlike an optical lens,
a magnetic quadrupole is
focussing in one plane and
defocussing in the other plane.
R. Bartolini, John Adams Institute, 1 May 2013
Magnetic field of a quadrupole and Lorentz force
9/28
A slightly more complicated example:
the FODO lattice (I)
Consider an alternating sequence of focussing (F) and defocussing (D)
quadrupoles separated by a drift (O)
The transfer matrix of the basic FODO cell reads
 1
M 1

 f
0 
 1
1 
 0
L  1
 1
2 
1  f
0 
 1
1 
 0
R. Bartolini, John Adams Institute, 1 May 2013

L

1

L
   2f
2 
1    L
 2f 2
L 

L1   
 4f  
L
L2 
1
 2
2f 4f 
10/28
Matrix elements from principal trajectories and
optics functions
In terms of the amplitude and phase function the transfer matrix will read
M s 0 s

 (s)

(cos    0 sin  )
0
S(s)  
 C(s)
  
 
  C' (s) S' (s)    ( (s)   0 ) cos   (1   (s) 0 ) sin 

 (s)  0





0
cos    (s) sin 
 (s)

 (s)  0 sin 
where 0 , 0 and the phase 0 are computed at the beginning of the
segment of transfer line
We still have not assumed any periodicity in the transfer line.
If we consider a periodic machine the transfer matrix over a whole turn
reduces to (put  =  the phase advance in one turn)
M s 0 s 0
 0 sin 
 cos    0 sin 




cos    0 sin  
   0 sin 
0 
1   02
0
This is the Twiss parameterisation of the one turn map
R. Bartolini, John Adams Institute, 1 May 2013
11/28
Stability of motion with the matrix formalism
Consider a circular accelerator with transfer matrix over one turn equal to
M (one turn map). Using the Twiss parameterisation for M
 0 sin 
 cos    0 sin 

  cos   I  sin   J
M  


sin

cos



sin

0
0



J   0
 0
0 

  0 
After n turns, the transformation of the particle coordinates will be given by
the successive application of the one turn matrix n times
x1  Mx 0
x n  Mn x 0
In order for the phase advance  to be real and hence for the motion to be
a stable oscillation, the one turn map must satisfy the condition
| cos  | 
1
| trM |  1
2
It can be proven that (see bibliography)
R. Bartolini, John Adams Institute, 1 May 2013
 0 sin n
 cos n   0 sin n


M n  


sin
n

cos
n



sin
n

0
0


12/28
Example: the FODO lattice (II)
Using the Twiss parameterisation of the matrix or the FODO cell we have

L
1 
2f
M
 L
 2
 2f
hence
L

L1  
 4f 
L
L2
1
 2
2f 4f


   cos    sin 
    sin 


 sin 


cos    sin  
1
L2
cos   trM  1  2
2
8f
The stability requires
| cos  |  | 1 
L2
8f 2
| 1
f
L
4
In a similar way we can compute the optics functions at the beginning of
the FODO cell.
R. Bartolini, John Adams Institute, 1 May 2013
13/28
Optics functions in a transfer line
While in a circular machine the optics functions are uniquely determined by
the periodicity conditions, in a transfer line the optics functions are not
uniquely given, but depend on their initial value at the entrance of the system.
We can express the optics function in terms of the principal trajectories as
 2CS
S2   0 
    C 2
 
 
      CC' CS'SC'  SS'   0 
2 
    C'2
  0 

2
C
'
S
'
S
'
  

This expression allows the computation of the propagation of the optics
function along the transfer lines, in terms of the matrices of the transfer line
of each single element, i.e. also the optics functions can be propagated
piecewise from
 C(s ) S(s ) 

M1 2  
 C' (s ) S' (s ) 
R. Bartolini, John Adams Institute, 1 May 2013
14/28
Examples
In a drift space
2
    1  2s s   0 
 
  
 s   0 
    0 1
  
   0
0
1
  
 0 
1 s

M  
 0 1
The  function evolve like a parabola as a function of the drift length.
In a thin focussing quadrupole of focal length f = 1/KL
   1
  
     KL
    ( KL) 2
  
0   0 
 
1
0   0 
2 KL 1   0 
0
 1 0

M  
 KL 1 
The  function evolve like a parabola in terms of the inverse of focal length
R. Bartolini, John Adams Institute, 1 May 2013
15/28
Diamond LINAC to booster transfer line
Booster optics
functions at the
injection point
Optics functions
from the LINAC
(Twiss
parameters of
the beam)
R. Bartolini, John Adams Institute, 1 May 2013
16/28
Transfer line example: Diamond LTB
R. Bartolini, John Adams Institute, 1 May 2013
17/28
Betatron motion in phase space (recap)
The solution of the Hill’s equations
d2y
ds
2
 K y (s) y  0
y(s)   (s) cos (s)   

sin (s)      (s) cos (s)   
y' (s)  
 (s)
describe an ellipse in phase space (y, y’)
area of the ellipse in phase space (y, y’) is
R. Bartolini, John Adams Institute, 1 May 2013
A(s)  ( y'2 2yy' 2 y 2 ) / 
18/28
Courant-Snyder invariant (I)
Hill’s equations have an invariant
d2y
 K y ( s) y  0
2
ds
A(s)   y'2 2yy' 2 y 2  const.
This invariant is the area of the ellipse in phase space (y, y’) multiplied by .
This can be easily proven by substituting the solutions y, y’
y(s)   (s) cos (s)   
y' (s)  

 (s)
sin (s)      (s) cos (s)   
into A(s). You will get the constant 
A(s) is called Courant-Snyder invariant
R. Bartolini, John Adams Institute, 1 May 2013
19/28
Courant-Snyder invariant (II)
Whatever the magnetic lattice, the area of the ellipse stays constant (if the Hill’s
equations hold)
At each different sections s, the ellipse of the trajectories may change
orientation shape and size but the area is an invariant.
This is true for the motion of a single particle !
R. Bartolini, John Adams Institute, 1 May 2013
20/28
Real beams – distribution function in phase space
A beam is a collection of many charged particles
The beam occupies a finite extension of the phase
space and it is described by a distribution function
 such that
x  (x, p x , y , p y , z,  )
(x, s)d 6 x  1

The beam distribution is characterised by the momenta of various orders

 x  j (s)  x j ( x, s)d 6 x
Average coordinates (usually zero)

R ij (s)  ( x   x ) i ( x   x ) j   ( x i   x  i )(x j   x  j ) ( x, s)d 6 x
The R-matrix also called -matrix describes the equilibrium properties of the
beam giving the second order momenta of the distribution
R11 = bunch H size; R33 = bunch Y size; R55 bunch Z size; R66 = energy spread
R. Bartolini, John Adams Institute, 1 May 2013
21/28
Gaussian beams
In many cases the equilibrium beam distribution is a Gaussian distribution
   ( x , s) 
1
(2 ) 3 det R
1
 R ij1 ( x  x )i ( x  x ) j
e 2
Usually the three planes are independent
hence in each plane
 ( x, x ' ) 
1
2 det R xx '
x '2 2xx 'x 2

2
e
The isodensity curves are ellipses
R. Bartolini, John Adams Institute, 1 May 2013
22/28
-matrix for Gaussian beams
The 6x6 –matrix can be partitioned into nine 2x2 submatrices
  xx

    yx

  zx
 xy  xz 

 yy  yz 

 zy  zz 
Direct computation using
 xx
  x x
 
   x x
with (assuming <x>
= 0 and <x’> = 0)
 ( x, x ' ) 
1
2
  x x 

 x x 

e
x '2 2xx 'x 2
2
 xx
  x 2   x x'  

 
2 
  x x'   x'  
yields


det( xx )   x  x   2x  2x   2x
We can associate an ellipse with the Gaussian beam distribution. The
evolution of the beam is completely defined by the evolution of the ellipse
The ellipse associated to the beam is chosen so that its Twiss parameters
are those appearing in the distribution function, hence, e.g.
 x   x 2  x'2    xx ' 2
R. Bartolini, John Adams Institute, 1 May 2013
x   xx
x'   x  x
23/28
Generic beams – rms emittance
For a generic beam described by a distribution functions  we can still
compute the average size and divergence and the whole -matrix
  x 2   x x'  

 xx  
2 
  x x'   x'  
we associate to this distribution the ellipse which has the same second order
momenta Rij and we deal with this distribution as if it was a Gaussian
distribution
  x x   x x 

 xx  
   x x  x x 
and since
det( xx )   x  x   x  x2   x2
The invariant of the ellipse will be
 x   x 2  x'2    xx ' 2
which is the rms emittance of the beam
R. Bartolini, John Adams Institute, 1 May 2013
24/28
Beam emittance and Courant-Snyder invariant
We have seen that the beam distribution can be associated to an ellipse
containing 66% of the beam (one r.m.s.)
In this way the beam rms emittance is associated with the Courant Snyder
invariant of the betatron motion
This links a statistical property of the beam (rms emittance) with single
particle property of motion (the Courant-snyder invairnat)
In this way the Courant-Snyder invariant acquires a statistical significance as
rms emittance of the beam. Hence the beam rms emittance is a conserved
quantity also for generic beams.
This is valid as long as the Hill’s equations are valid or more generally the
system is Hamiltonian. As such the conservation of the emittance is a
manifestation of the general theorem of Hamiltonian system and statistical
mechanics known as the Liouville theorem
R. Bartolini, John Adams Institute, 1 May 2013
25/28
Liouville’s theorem
Liouville’s theorem: In a Hamiltonian system, i.e. n the absence of
collisions or dissipative processes, the density in phase space along the
trajectory is invariant’.
n
n
df f
f q k
f p k f




 H, f   0
dt t k 1 q k t

p

t

t
k 1
k
Liouville theorem states that volume of 6D phase are preserved during
the beam evolution (take f to be the characteristic function of the volume
occupied by the beam). However if the Hamiltonian can be separated in
three independent terms
H  H(x, p x , y, p y , ,  )  H(x, p x )  H( y, p y )  H( ,  )
The conservation of the phase space density occurs for the three
projection on the
(x, px) plane (Horizontal emittance)
(y, py) plane (Vertical emittance)
(z, pz) = (, ) plane (longitudinal emittance)
R. Bartolini, John Adams Institute, 1 May 2013
26/28
Beam emittance and Liouville’s theorem
The Courant-Snyder invariant is the area of the ellipse phase space.
The conservation of the area is a general property of Hamiltonian systems
(any area not only ellipses !)
The invariance of the rms emittance is the particular case of a very general
statement for Hamiltonian systems (Liouville theorem)
This is valid as long as the motion is Hamiltonian, i.e.
No damping effects, no quantum diffusion, due to emission of radiation
no scattering with residual gas, no beam beam collisions
no collective effects (e.g. interaction with the vacuum chamber, no self
interaction)
R. Bartolini, John Adams Institute, 1 May 2013
27/28
Bibliography
E. Wilson, Introduction to Particle Accelerators
J. Rossbach and P. Schmuser, CAS Lecture 94-01
K. Steffen, CAS Lecture 85-19
R. Bartolini, John Adams Institute, 1 May 2013
28/28