1. Daniel Dobos Seminar: Chaos, Prof. Markus
Accelerator Chaos
Daniel Dobos
Seminar: Chaos, Prof. Markus
Dortmund,

2. 2. nonlinear accelerator optics
Contents
0. Motivation
1. Introduction
1.1 Linear accelerator physics
Hill differential equation - Betatron-oscillation - enveloppe - Courant/Snyder Invariant - Twiss parameters - emittance - working point - acceptance
2. nonlinear accelerator optics
2.1 nonlinear effects optical resonances - coupling resonances - stop bands - energy dependence - chromaticity
2.2 chromaticity compensation
gradient of a sextupole field - Laplace equation - compensation of the chromaticity with sextupoles
3. nonlinear consequences
3.1 structure of the phase space
Henon diagrams - dynamical aperture
3.2 methods to consider nonlinearities
3.3 particle tracking
3.4 KAM-Theory
3.5 stability for N circulations
3.6 backtracking
3.7 Lyapunov Exponent:

3. 0. Motivation

4. 1. Introduction 1.1 Linear accelerator physics
The motion of a charged particle through the electromagnetic fields of a ring accelerator is described in a right-handed, moving coordinates system.
On condition that y << R, you can make a multipole-development of the transversal magnetic filed components of the Maxwell equations. With the quadrupole terms you get the motion equations:
[1.1]
and:
[1.2]
Only a horizontal curve
R = local curve radius of the magnet structure
Coefficient k(s) describes the quadrupole part of the multipole development
´ is the differential along s
Dp/p0 is the relative error of the impulse

5. 1. Introduction
[1.1] is a Hill differential equation and you can write it in the general form:
[1.3]
The general solution of [1.3] can be written in the matrix:
[1.4]
C(s) (cosinus-structure) and S(s) (sinus- structure) vary with the magnet structure and D(s) is the particular solution of Δp/p0 = 1 (dispersion). Nearly all kind of transformation matrices of detector components could be found in tables. With this matrices it is possible to calculate every trajectory along a given magnet structure:
M = MQFMDMEBMBMEBMDMQD [1.5]

6. 1. Introduction
Now we use the case that the impulse error is equal to zero Δp/p0 = 0 → the Hill differential equation describes an oscillation with a variable frequency:
[1.6]
with the Betatron-oscillation called solution:
[1.7]
β(s) is the beta function and:
[1.8]
describes the enveloppe of the oscillation.
In ring accelerators β(s) is periodic with the length of the ring L:
[1.9]
Φ(s) is called the Betatron-phase:
[1.10]

7. 1. Introduction The variation of Φ0 with 2π: [1.11]
describes an ellipse in the phase space (y,y´) with the area:
[1.12]
ε is the Courant/Snyder Invariant:
[1.13]
with the Twiss parameters:
[1.14]
If the phase space ellipse of a particle surround the standard deviation of the Gauß-like beam, ε is called emittance. The number of Betatron- Oscillations per circulation or the phase shift, normalized to 2π is the working point:
[1.15]

8. 1. Introduction The density distribution could be written: [1.16]
The width of the beam is the standard deviation of the density distribution.
The biggest possible emmitance, often restricted by the width of the vacuum chamber, is the acceptance:
[1.17]

9. 2. nonlinear accelerator optics
2.1 nonlinear effects
The linear accelerator theory, which is described in section 1.1, is not able to explain all effects. For example magnetic field errors or resonance from higher terms of the multipole development are not covered by the linear theory. With wrong positions of magnetic components we get a coupling between the Hill differential equations [1.1] and [1.2].
With a field error of a magnetic multipole of the order n we get a resonance of the order n+1 in the motion equations.
field error
optical resonance
dipole error
Q = n
quadrupole error
Q = n + 1/2
sextupole error
Q = n + 1/3
octupole error
Q = n + 1/4

10. 2. nonlinear accelerator optics
We also have coupling resonances, because the Hill differential equations are coupled:
[2.1]
if n, m and r are elements of Z
The width of the stop bands decrease with the order, and is defined as the area in which the motion of the particle is not stable.
In every accelerator machine you try to hold the working point stable in an area as far as possible away from stop bands.
It is also important to understand the energy dependence of every optical component. Quadrupole magnets are focusing particles with two low energy to strong and particles with two high energy to weak. This effect is called chromatic aberration, because of its analogy to light optics.

11. 2. nonlinear accelerator optics
This effect is equivalent to a gradient error of a particle with an optimal energy.
We insert a disturbed quadrupole strength:
K(s) = K0(s) + ΔK
into the Hill differential equations [1.1] and [1.2] and get a energy dependence of the working point:
[2.2]
ΔK is nearly proportional to the relative energy shift - g is the field gradient:
[2.3]

12. 2. nonlinear accelerator optics
and we get [2.2]. This energy dependence of the working point is called chromaticity:
[2.4]
Linear optics have got a natural (negative) chromaticity, because of the natural impulse distribution in a bunch.
2.2 chromaticity compensation:
It is necessary to compensate the natural chromaticity to avoid that the working point gets near a stop band. If this would happen, many particles gets lost in every circulation and the lifetime of the beam becomes very short.
An other effect in the case of negative chromaticity is the Head-Tail-Instability This effect is caused by inter actions between the beam particles among themselves.
This compensation can be realized with some sextupoles.

13. 2. nonlinear accelerator optics
The gradient of a sextupole field grows linearly with the distance from the centre. The magnetic field is given by:
[2.5]
with:
[2.6]
The coupling between the y- and the z-motion could not be avoided, because of the necessary Laplace-equation:
[2.7] The sextupoles with their quadratic fields also have a focusing effect. Particles with low focusing after the quadrupole get a additionally focusing and particles with a to high focusing after the quadrupole are defocused.
Many machines are working with a effective chromaticity of ≈+1 to avoid of getting negative.

14. 2. nonlinear accelerator optics
It is possible two compensate the chromaticity with only a horizontal and a vertical sextupole but this reduce the dynamical aperture. So it is much better to use many weak sextupoles spread over the whole machine.

15. 3. nonlinear consequences
3.1 structure of the phase space:
The solution of the Hill differential equations is an ellipse in the phase space. If we additionally consider the nonlinear effects of a sextupole the phase space is deformed. Other sources of nonlinear effect have got the same effect on the structure of the phase space - it is possible to reduce the examination to sextupoles effects. The deformation of the phase space can be described with the help of Henon diagrams.
In this diagrams you can see a border between the stable and unstable areas. This border is called dynamical aperture. Inside this border the trajectories of particles are closed - outside they are not closed and diverge from the centre of the phase space.
With considering of many nonlinear effects the structure of the phase space gets very complex.

16. 3. nonlinear consequences
3.2 methods to consider nonlinearities:
In many cases the solution of this nonlinear problems could not be found with analytic methods. We need other methods, like numerical simulation, measurements on real machines to estimate and optimize the quality of accelerator machines.
theoretical model
measurements
numerical
simulation
analytical
calculation
single-particle
dynamics
theoretical quality
many-particle
experimental quality
real machines
assessment function
quality of accelerator optics
methods to optimize the quality

17. 3. nonlinear consequences
3.3 particle tracking:
Sextupoles generate a quadratic increasing force with the distance from the centre of this sextupoles - It is not possible to consider the Betatron-oscillation as a harmonic oscillation. The particle dynamic could get chaotic and this means instable. We use the numerical method called particle tracking. We consider the thick- ness of a sextupole as infinitesimal and calculate the track of a particle piece by piece through many circulations of a ring accelerator optic.
We start with a four-dimensional track vector X0. In front of the first sextupole we get:
[3.1]
The magnetic field of the first sextupole at the coordinates y1 and z1 is given by:
[3.2]

18. 3. nonlinear consequences
If l is the effective length of the first sextupole we get a angle change of:
[3.3]
and:
[3.4]
With [3.1] - [3.4] the track vector behind the first sextupole is:
[3.5]
With this method you calculate the track vector from sextupole to sextupole. It is possible that an optic seems to be very stable for many circulations, but after many circulations it becomes unstable and the particle gets lost, if the phase point gets outside the mechanical aperture of the optic phase space.

19. 3. nonlinear consequences
3.4 KAM-Theory:
We want to understand, why it is possible that a nonintegrable system could be stable. The same question for our solar system was answered by the KAM- (Kolmogorov, Arnold, Moser) Theory and it is also possible to use it in this case:
If we have a small enough disturbance of an integrable Hamilton system H0(I):
[3.6]
most of the H0 tori survive the disturbance and we get deformed KAM-tori.
An easy example can be shown with a two dimensional, conservative system. The phase space has got four, the energy surface three and the tori two dimensions.
For all start parameter of this integrable system the phase space trajectories are running on surfaces of tori. The evolution of this system can be calculated precisely and the system is stable.

20. 3. nonlinear consequences
With special start parameter in an nonintegrable system a tori is destroyed by resonances and decays into smaller tori. On the surface of these smaller tori the motion is still stable. The phase space trajectories between the tori of this disturbed, nonintegrable system are running on nonregular tracks and the motion could get unstable. It is possible that a trajectory is running near a torus for a long time - seems to be stable - but diffusing finally away. This is called the Arnold-Diffusion.
3.5 stability for N circulations:
To understand the structure of the Henon diagram it is important to calculate the dynamical aperture of the phase space.
One method is to calculate the stability for N circulations with particle tracking. The trajectory of a particle should stay inside the mechanical aperture for N circulations. If this happens we increase step by step the start distance of the particle from the orbit and check the stability again.

21. 3. nonlinear consequences
If the trajectory does not stay inside the mechanical aperture after N circulations, the dynamical aperture has to be between this and the last stable start coordinates. With decreasing of the interval step we try to find this border.
In this method we conclude that a particle is stable if it is stable for N circulations, but we do not know anything about things that happen after this N circulations and ignore the presence of long time stabilities.
3.6 backtracking:
Calculate the trajectories of particles back ward in time Starting the simulation in areas with unstable trajectories.

22. 3. nonlinear consequences
If we calculate enough trajectories we get a negative image of the dynamical aperture and the number of necessary tracking circulations is reduced.
3.7 Lyapunov Exponent:
We put a cut through the two dimensional phase space and calculate the maximal Lyapunov Exponents. If we examine a non coupled system this cut is sufficient to understand the dynamics of this system, because the L. Exponent describes a trajectory and not a phase space point and all trajectories are registered by this cut. If we have a coupled system we have to calculate the L. Exponents for many cuts or the whole area to register all trajectories. It could not get negative in this case, because we examine a conservative system and this would mean a contraction of the phase space. The increasing of the L. Exponent marks the position of the dynamical aperture, but does not give any information of the structure of the phase space inside it.