Lecture 5: Beam optics and imperfections
Errors in magnetic lattices
dipole errors
quadrupole errors
Nonlinear Perturbations
Resonances
Dynamic aperture
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Linear betatron equations of motion (recap)
In the magnetic fields of dipoles magnets and quadrupole magnets (without
imperfections) the coordinates of the charged particle w.r.t. the reference orbit
are given by the Hill’s equations
2
d y
 K y ( s) y  0
2
ds
1
1 Bz ( s)

 2 ( s) B x
1 Bz ( s)
K z (s) 
B x
K x ( s) 
These are linear equations (in y = x, z). They can be integrated and give
y(s)   y  y (s) cosx (s)  x 0 
s
 y (s ) 

s0
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ds'
 y (s ' )
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Errors in lattices
Transfer lines or circular accelerators are made of a series of drifts and
quadrupoles for the transverse focussing and accelerating section for
acceleration.
Real lattices will always unavoidably be different from the model lattices.
This is due to
• misalignments and rolls of the magnetic elements
• gradient error
• additional unwanted magnetic field multipoles (stray fields)
• time varying magnetic errors (due to change in magnetic fields, ground
motion, etc)
d2y
 K y ( s) y  0
2
ds
R. Bartolini, John Adams Institute, 2 May 2013
M
d2y ~
k

K
(
s
)
y

F
(
s
)

c
(
s
)
y
y
k

ds2
k 1
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Dipole errors (I)
Dipole field errors arise due to calibration errors current vs magnetic fields,
misalignment and rolls.
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Dipole errors (II)
Dipole field errors and misalignment of the magnetic elements produce orbit
distortions. They generate an additional term in the Hill’s equation
y' ' K (s) y 
 B(s )
B0 
The general solution can be found as the sum of a particular integral and the
general solution of the imhogeneous equation
s
y(s)  aC(s)  bS(s)  u p (s)
and reads
y(s) 
 y (s)
2 sin y
sL

s
s


s0
s0
u p (s)  S(s) F(s' )C(s' )ds'  C(s) F(s' )S(s' )ds'
B(s' )
 y (s' ) cos[ y (s)   y (s' )   y ] ds'
B0 
The effect is stronger where the  functions are larger
The orbit gets unstable if the betatron tune y is close to an integer
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Orbit corrections (I)
With many distributed dipole errors the orbit will be distorted. The beam
position monitors (BPMs) will read a non zero orbit.
To correct the orbit we use a system of horizontal and vertical corrector
dipoles.
We need to know what is the effect of each correctors on the orbit itself,
first. Then we can use this information to correct orbit distortion.
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Orbit correction (II)
The orbit response matrix collects all this information in one matrix
The orbit response matrix R is the change in the orbit at the BPMs as a
function of changes in the steering magnets strength.
orbit RM at diamond
168*2 rows
x
  orbit reading at all BPMs
 y
 x 
  dipole corrector angle kick
 
 y
 x 
x
   R model  
 
measured  y
 y
 
168*2 columns
V
V
H
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H
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Orbit correction (III)
Once the matrix R is known we can correct an orbit distortion by inverting
the matrix
  x 
 x 
1

  Rmodel
 
  
measured  y 
 y
The matrix R generally is not a square matrix so it might be non-invertible
but we can use the Singular Value Decomposition (SVD) of the response
matrix R to invert it and correct the closed orbit distortion
R  UV t
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R 1  V1U t
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Quadrupole errors (I)
Quadrupole magnet are due to calibration errors in current vs field gradient,
misalignment and rolls. A misaligned quadrupole adds a dipole kick.
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Quadrupole errors (III)
Consider a lattice with a quadrupole distribution K(s) and a gradient error K(s)
y' '(K(s)  K(s)) y  0
This is still a homogeneous Hill’s equation and can be solved with the matrix
method
Let us consider a localised quadrupole error and let us put
M0 the transfer matrix without error
M the transfer matrix with the error
m the transfer matrix of the localised section where the error is
m0 the transfer matrix of the same section without error
We can compute the matrix M with the relation
m0
M  m  m01  M0
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M0
m
M
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Quadrupole errors (IV)
We use the Twiss parameterisation for M and M0
 cos  0   sin  0
M 0  
  sin  0

 sin  0

1 0

  cos  0 
  sin  0 
cos  0   sin  0 
0 1

 
  cos  0  I  sin  0  J
  
and the thin lens approximation for m and m0
0
 1


m0  1
1

 Kds 
1
0



1
m
1
 (K  K )ds

we get to first order in ds
0
 1

  cos   
M 1
0 
1

  Kds  
 Kds 


  sin  0
 Kds   
Using again the Twiss parameterisation for the transfer matrix we get the new
phase advance
cos  
1
1
traceM  cos 0  sin 0  Kds  (s1 )
2
2
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Quadrupole errors (V)
If the perturbation is distributed around the ring we have
1
 cos    sin 0  Kds  (s1 )
2
1
 cos    sin 0 
2
n
 K ds   (s )
j
j
j
j1
this can be eventually expressed with an integral over the ring circumference

1
 cos    sin 0   (s)K(s)ds
2
Therefore a gradient error leads to a betatron tune shift
Q 

(cos  )
1


2
2 sin  0 4
  (s)K(s)ds
The effect is stronger where the beta functions are larger (final focus - IPs).
A gradient error changes the beta functions as well
 (s)
 (s) 
 (s' )K (s' ) cos2( (s)   (s' ))  2Qds'
2 sin(2Q)

The optics get unstable if the betatron tune Qy is close to an half-integer
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Resonances
The shift in the betatron tunes has to be controlled to avoid creating
resonance conditions between the horizontal and the vertical tune
m Qx + n Qz = p
|m| + |n|
is called the order of the resonance
We have already met two example of resonances: integer resonances which
make the closed orbit unstable and half integer resonances which make the
optics unstable. More examples when we will consider nonlinear elements.
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Nonlinear magnetic errors
Multipolar expansion of the magnetic field inside a magnet
The on axis magnetic field can be expanded into multipolar components (dipole,
quadrupole, sextupole, octupoles and higher orders)
1 Bz
1  2 Bz 2 1  3 Bz 3
Bz  B0 
x
x 
x  ...
2
3
1! x
2! x
3! x
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Hill’s equation with nonlinear terms
Including higher order terms in the expansion of the magnetic field
 k (s)  ijn (s)

Bz  iBx  B0 0  n
( x  iz ) n 
n!
 n1

M
kn 
1
 n By
B0  0 x n
 n Bx
jn 
B0  0 x n
normal multipoles
( 0, 0 )
1
skew multipoles
( 0,0)
the Hill’s equations acquire additional nonlinear terms

d 2x  1
 M kn (s)  ijn (s)
n




k
(
s
)
x

Re
(
x

iz
)
1



ds2   2 (s)
n!
 n2


d 2z
 M kn (s)  ijn (s)
n

k
(
s
)
z


Im
(
x

iz
)
1


ds2
n!
 n2

No analytical solution available in general:
the equations have to be solved by tracking or analysed perturbatively
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Example: nonlinear errors in the LHC
main dipoles
Finite size coils
reproduce only partially
the cos- desing
necessary to achieve a
pure dipole fields
LHC main dipole cross section
Multipolar errors up to very high order have a significant impact on the
nonlinear beam dynamics.
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Sextupole magnets
Nonlinear magnetic fields are introduced in the lattice (chromatic sextupoles)
Normal
sextupole
1  2 Bz 2
2
Bz 
(
x

z
) Normal sextupole
2
2 x
1  2 Bx
Skew sextupole
Bz 
 2 xz
2
2 x
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Example: nonlinear elements in small
emittance machines
Small emittance  Strong quadrupoles  Large (natural) chromaticity
 Strong sextupoles (sextupoles guarantee the focussing of off-energy particles)
strong sextupoles have a significant impact on the electron dynamics
 additional sextupoles are required to correct nonlinear aberrations
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Phenomenology of nonlinear motion (I)
x’
The orbit in phase space for a system of linear
Hill’s equation are ellipses (or circles)
Turn 1
Turn 2
The frequency of revolution of the particles is
the same on all ellipses
x
Turn 3
x’
The orbit in the phase space for a system of
nonlinear Hill’s equations are no longer simple
ellipses (or circles);
Turn 2
Turn 1
x
The frequency of oscillations depends on the
amplitude
Turn 3
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Nonlinear resonances
m = 5; n = 0; p =1
When the betatron tunes satisfy a resonance relation
Turn 2
Turn 1
mQ x  nQz  p
the motion of the charged particle repeats itself
periodically
Turn 5
Turn 3
Turn 4
If there are errors and perturbations which are
sampled periodically their effect can build up and
destroy the stability of motion
5-th order resonance phase space plot
(machine with no errors)
The resonant condition defines a set of lines in the
tune diagram
The working point has to be chosen away from the
resonance lines, especially the lowest order one
(example CERN-SPS working point)
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Phenomenology of nonlinear motion (II)
th
Stable and unstable fixed points appears which Phase space plots of close to a 5
order resonance
are connected by separatrices
Islands enclose the stable fixed points
On a resonance the particle jumps from one
island to the next and the tune is locked at the
resonance value
region of chaotic motion appear
The region of stable motion,
called dynamic aperture, is limited by the
appearance of
unstable fixed points and
Qx = 1/5
trajectories with fast escape to infinity
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Phenomenology of nonlinear motion (III)
The orbits in phase space of a non linear system can be broadly divided in
•
Regular orbit  stable or unstable
•
Chaotic orbit  no guarantee for stability but diffusion rate may be very small
The particle motion on a regular and stable
orbit is quasi–periodic
n
i
z (n)   c k e  2 i k n ck  ak e k
k 1
The betatron tunes are the main frquencies
corresponding to the peak of the spectrum
in the two planes of motion
The frequencies are given by linear
combination of the betatron tunes.
Only a finite number of lines appears
effectively in the decomposition.
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Tracking (I)
Most accelerator codes have tracking capabilities: MAD, MADX, Tracy-II,
elegant, AT, BETA, transport, …
Typically one defines a set of initial coordinates for a particle to be tracked
for a given number of turns.
The tracking program “pushes” the particle through the magnetic
elements. Each magnetic element transforms the initial coordinates
according to a given integration rule which depends on the program used,
e.g. transport (in MAD)

P


X  ( x, x , y, y , z,  ) ,  
P0


X f  RX i
Linear map
x j , f   R jk x j,i   Tjkl x j,i xl,i   U jklmx j,i xl,i xm,i  
k
kl
klm
Nonlinear map up to third order as a truncated Taylor series
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Tracking (II)
A Hamiltonian system is symplectic, i.e. the map which defines the
evolution is symplectic (volumes of phase space are preserved by the
symplectic map)
x f  M ( xi ) M is symplectic transformation
Symplectic
integrator
J ab ( xi ) 
xa, f
xb,i
J T SJ  S
non
symplectic
If the integrator is not
symplectic one may found
artificial damping or excitation
effect
The well-known Runge-Kutta integrators are not symplectic. Likewise the
truncated Taylor map is not symplectic. They are good for transfer line but
they should not be used for circular machine in long term tracking analysis
Elements described by thin lens kicks and drifts are always symplectic:
long elements are usually sliced in many sections.
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Dynamic Aperture and Frequency Map
Strongly excited resonances can destroy the Dynamic Aperture.
The Frequency Map Analysis is a technique introduced in Accelerator Physics
form Celestial Mechanics (Laskar).It allows the identification of dangerous non
linear resonances during design and operation.
Engineering aperture.
To each point in the (x, y) aperture
there corresponds a point in the (Qx,
Qy) plane
The colour code gives a measure of
the stability of the particle (blu =
stable; red = unstable)
The indicator for the stability is given
by the variation of the betatron tune
during the evolution: i.e. tracking N
turns we compute the tune from the
first N/2 and the second N/2
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Bibliography
E. Wilson, CAS Lectures 95-06 and 85-19
E. Wilson, Introduction to Particle Accelerators
G. Guignard, CERN 76-06 and CERN 78-11
J. Bengtsson, Nonlinear Transverse Dynamics in Storage Rings, CERN 88-05
J. Laskar et al., The measure of chaos by numerical analysis of the fundamental
frequncies, Physica D65, 253, (1992).
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