1. Lecture 5: Beam optics and imperfections
Errors in magnetic lattices
dipole errors
quadrupole errors
Nonlinear Perturbations
Resonances
Dynamic aperture
R. Bartolini, John Adams Institute, 2 May 2013

2. 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
These are linear equations (in y = x, z). They can be integrated and give
R. Bartolini, John Adams Institute, 2 May 2013

3. 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)
R. Bartolini, John Adams Institute, 2 May 2013

4. Dipole errors (I)
Dipole field errors arise due to calibration errors current vs magnetic fields, misalignment and rolls.
R. Bartolini, John Adams Institute, 2 May 2013
4/26

5. 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
The general solution can be found as the sum of a particular integral and the general solution of the imhogeneous equation
and reads
The effect is stronger where the  functions are larger
The orbit gets unstable if the betatron tune y is close to an integer
R. Bartolini, John Adams Institute, 2 May 2013

6. 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.
R. Bartolini, John Adams Institute, 2 May 2013

7. 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 H V
168*2 rows
168*2 columns
orbit reading at all BPMs
dipole corrector angle kick
R. Bartolini, John Adams Institute, 2 May 2013

8. Orbit correction (III)
Once the matrix R is known we can correct an orbit distortion by inverting the matrix
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. Bartolini, John Adams Institute, 2 May 2013

9. 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.
R. Bartolini, John Adams Institute, 2 May 2013

10. Quadrupole errors (III)
Consider a lattice with a quadrupole distribution K(s) and a gradient error K(s)
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 M0 M
R. Bartolini, John Adams Institute, 2 May 2013

11. Quadrupole errors (IV)
We use the Twiss parameterisation for M and M0
and the thin lens approximation for m and m0
we get to first order in ds
Using again the Twiss parameterisation for the transfer matrix we get the new phase advance
R. Bartolini, John Adams Institute, 2 May 2013

12. Quadrupole errors (V)
If the perturbation is distributed around the ring we have
this can be eventually expressed with an integral over the ring circumference
Therefore a gradient error leads to a betatron tune shift
The effect is stronger where the beta functions are larger (final focus - IPs). A gradient error changes the beta functions as well
The optics get unstable if the betatron tune Qy is close to an half-integer
R. Bartolini, John Adams Institute, 2 May 2013

13. is called the order of the resonance
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.
R. Bartolini, John Adams Institute, 2 May 2013

14. 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)
R. Bartolini, John Adams Institute, 2 May 2013

15. Hill’s equation with nonlinear terms
Including higher order terms in the expansion of the magnetic field
normal multipoles
skew multipoles
the Hill’s equations acquire additional nonlinear terms
No analytical solution available in general:
the equations have to be solved by tracking or analysed perturbatively
R. Bartolini, John Adams Institute, 2 May 2013

16. 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.
R. Bartolini, John Adams Institute, 2 May 2013

17. Sextupole magnets
Nonlinear magnetic fields are introduced in the lattice (chromatic sextupoles)
Normal sextupole
Normal sextupole
Skew sextupole
R. Bartolini, John Adams Institute, 2 May 2013

18. 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
R. Bartolini, John Adams Institute, 2 May 2013

19. Phenomenology of nonlinear motion (I)x x’
Turn 1
Turn 2
Turn 3
The orbit in phase space for a system of linear Hill’s equation are ellipses (or circles)
The frequency of revolution of the particles is the same on all ellipses x’
Turn 1
Turn 2
Turn 3
The orbit in the phase space for a system of nonlinear Hill’s equations are no longer simple ellipses (or circles);
The frequency of oscillations depends on the amplitude x
R. Bartolini, John Adams Institute, 2 May 2013

20. 5-th order resonance phase space plot (machine with no errors)
Nonlinear resonances
m = 5; n = 0; p =1
Turn 1
Turn 2
Turn 3
Turn 4
Turn 5
When the betatron tunes satisfy a resonance relation
the motion of the charged particle repeats itself periodically
If there are errors and perturbations which are sampled periodically their effect can build up and destroy the stability of motion
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)
5-th order resonance phase space plot (machine with no errors)
R. Bartolini, John Adams Institute, 2 May 2013

21. Phenomenology of nonlinear motion (II)
Phase space plots of close to a 5th order resonance
Stable and unstable fixed points appears which 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
trajectories with fast escape to infinity
Qx = 1/5
R. Bartolini, John Adams Institute, 2 May 2013

22. 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
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.
R. Bartolini, John Adams Institute, 2 May 2013

23. 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)
Linear map
Nonlinear map up to third order as a truncated Taylor series
R. Bartolini, John Adams Institute, 2 May 2013

24. Symplectic integrator
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)
M is symplectic transformation
Symplectic integrator
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.
R. Bartolini, John Adams Institute, 2 May 2013

25. 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.
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
Engineering aperture.
R. Bartolini, John Adams Institute, 2 May 2013

26. Bibliography E. Wilson, CAS Lectures 95-06 and 85-19
E. Wilson, Introduction to Particle Accelerators
G. Guignard, CERN 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).
R. Bartolini, John Adams Institute, 2 May 2013