Accelerator Chaos
Daniel Dobos
Seminar: Chaos, Prof. Markus
Dortmund, 01.02.2002
Daniel Dobos – Accelerator Chaos
Contents
0. Motivation
1. Introduction
1.1 Linear accelerator physics
Hill differential equation - Betatron-oscillation -enveloppe - Courant/Snyder Invariant - Twissparameters - emittance - working point -acceptance
2. nonlinear accelerator optics
2.1 nonlinear effectsoptical resonances - coupling resonances - stopbands - 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
3.5stability for N circulations
3.6 backtracking
3.7 Lyapunov Exponent:
Daniel Dobos – Accelerator Chaos
0. Motivation
Daniel Dobos – Accelerator Chaos
1. Introduction
1.1 Linear accelerator physics
The motion of a charged particle through theelectromagnetic fields of a ring accelerator isdescribed in a right-handed, movingcoordinates system.
On condition that y << R, you can make amultipole-development of the transversalmagnetic filed components of the Maxwellequations. With the quadrupole terms you getthe motion equations:
[1.1]
and:
[1.2]
Daniel Dobos – Accelerator Chaos
1. Introduction
[1.1] is a Hill differential equation and you canwrite it in the general form:
[1.3]
The general solution of [1.3] can be written inthe matrix:
[1.4]
C(s) (cosinus-structure) and S(s) (sinus-structure) vary with the magnet structure andD(s) is the particular solution of Δp/p0 = 1(dispersion). Nearly all kind of transformationmatrices of detector components could befound in tables. With this matrices it is possibleto calculate every trajectory along a givenmagnet structure:
M = MQFMDMEBMBMEBMDMQD[1.5]
Daniel Dobos – Accelerator Chaos
1. Introduction
Now we use the case that the impulse error isequal to zero Δp/p0 = 0 → the Hill differentialequation describes an oscillation with avariable 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 thelength of the ring L:
[1.9]
Φ(s) is called the Betatron-phase:
[1.10]
Daniel Dobos – Accelerator Chaos
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 surroundthe standard deviation of the Gauß-like beam, εis called emittance. The number of Betatron-Oscillations per circulation or the phase shift,normalized to 2π is the working point:
[1.15]
Daniel Dobos – Accelerator Chaos
1. Introduction
The density distribution could be written:
[1.16]
The width of the beam is the standard deviationof the density distribution.
The biggest possible emmitance, oftenrestricted by the width of the vacuum chamber,is the acceptance:
[1.17]
Daniel Dobos – Accelerator Chaos
2. nonlinear accelerator optics
2.1 nonlinear effects
The linear accelerator theory, which isdescribed in section 1.1, is not able to explainall effects. For example magnetic field errors orresonance from higher terms of the multipoledevelopment are not covered by the lineartheory. With wrong positions of magneticcomponents we get a coupling between theHill differential equations [1.1] and [1.2].
With a field error of a magnetic multipole of theorder n we get a resonance of the order n+1 inthe 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
Daniel Dobos – Accelerator Chaos
2. nonlinear accelerator optics
We also have coupling resonances, becausethe Hill differential equations are coupled:
[2.1]
if n, m and r are elements of Z
The width of the stop bands decrease with theorder, and is defined as the area in which themotion of the particle is not stable.
In every accelerator machine you try to hold theworking point stable in an area as far aspossible away from stop bands.
It is also important to understandthe energydependence of every optical component.Quadrupole magnets are focusing particleswith two low energy to strong and particles withtwo high energy to weak. This effect is calledchromatic aberration, because of its analogy tolight optics.
Daniel Dobos – Accelerator Chaos
2. nonlinear accelerator optics
This effect is equivalent to a gradient error of aparticle with an optimal energy.
We insert a disturbedquadrupole strength:
K(s) = K0(s) + ΔK
into the Hill differentialequations [1.1] and[1.2] and get a energydependence of theworking point:
[2.2]
ΔK is nearly proportional to the relative energyshift - g is the field gradient:
[2.3]
Daniel Dobos – Accelerator Chaos
2. nonlinear accelerator optics
and we get [2.2]. This energy dependence ofthe working point is called chromaticity:
[2.4]
Linear optics have got a natural (negative)chromaticity, because of the natural impulsedistribution in a bunch.
2.2 chromaticity compensation:
It is necessary to compensatethe natural chromaticity toavoid that the working pointgets near a stop band. If thiswould happen, many particlesgets lost in every circulationand the lifetime of the beambecomes very short.
An other effect in the case ofnegative chromaticity is theHead-Tail-Instability.Thiseffect is caused byinteractions between thebeam particles amongthemselves.
This compensation canbe realized with somesextupoles.
Daniel Dobos – Accelerator Chaos
2. nonlinear accelerator optics
The gradient of a sextupole field grows linearlywith the distance from the centre. The magneticfield is given by:
[2.5]
with:
[2.6]
The coupling between the y- and the z-motioncould not be avoided, because of thenecessary Laplace-equation:
[2.7]The sextupoles with their quadratic fields alsohave a focusing effect. Particles with lowfocusing after the quadrupole get a additionallyfocusing and particles with a to high focusingafter the quadrupole are defocused.
Many machines are working with a effectivechromaticity of ≈+1 to avoid of getting negative.
Daniel Dobos – Accelerator Chaos
2. nonlinear accelerator optics
It is possible two compensate the chromaticitywith only a horizontal and a vertical sextupolebut this reduce the dynamical aperture. So it ismuch better to use many weak sextupolesspread over the whole machine.
Daniel Dobos – Accelerator Chaos
3. nonlinear consequences
3.1 structure of the phase space:
The solution of the Hill differential equations isan ellipse in the phase space. If we additionallyconsider the nonlinear effects of a sextupole thephase space is deformed. Other sources ofnonlinear effect have got the same effect onthe structure of the phase space - it is possibleto reduce the examination to sextupoles effects.The deformation of the phase space can bedescribed with the help of Henon diagrams.
In this diagrams you can see a border betweenthe stable and unstable areas. This border iscalled dynamical aperture. Inside this border thetrajectories of particles are closed - outside theyare not closed and diverge from the centre of thephase space.
With considering of many nonlinear effects thestructure of the phase space gets very complex.
Daniel Dobos – Accelerator Chaos
3. nonlinear consequences
3.2 methods to consider nonlinearities:
In many cases the solution of this nonlinearproblems could not be found with analyticmethods. We need other methods, likenumerical simulation, measurements on realmachines to estimate and optimize the quality ofaccelerator machines.
theoretical model
measurements
numerical
simulation
analytical
calculation
single-particle
dynamics
theoretical quality
many-particle
dynamics
experimental quality
real machines
assessment function
quality of accelerator optics
methods to optimize the quality
Daniel Dobos – Accelerator Chaos
3. nonlinear consequences
3.3 particle tracking:
Sextupoles generate a quadratic increasingforce with the distance from the centre of thissextupoles - It is not possible to consider theBetatron-oscillation as a harmonic oscillation.The particle dynamic could get chaotic and thismeans instable. We use the numerical methodcalled particle tracking. We consider the thick-ness of a sextupole as infinitesimal and calculatethe track of a particle piece by piece throughmany 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 thecoordinates y1 and z1 is given by:
[3.2]
Daniel Dobos – Accelerator Chaos
3. nonlinear consequences
If l is the effective length of the first sextupole weget a angle change of:
[3.3]
and:
[3.4]
With [3.1] - [3.4] the track vector behind the firstsextupole is:
[3.5]
With this method you calculate the track vectorfrom sextupole to sextupole. It is possible that anoptic seems to be very stable for manycirculations, but after many circulations itbecomes unstable and the particle gets lost, ifthe phase point gets outside the mechanicalaperture of the optic phase space.
Daniel Dobos – Accelerator Chaos
3. nonlinear consequences
3.4 KAM-Theory:
We want to understand, why it is possible that anonintegrable system could be stable. The samequestion for our solar system was answered bythe KAM- (Kolmogorov, Arnold, Moser) Theoryand it is also possible to use it in this case:
If we have a small enough disturbance ofan integrable Hamilton system H0(I):
[3.6]
most of the H0 tori survive the disturbanceand we get deformed KAM-tori.
An easy example can be shown with a twodimensional, conservative system. The phasespace has got four, the energy surface three andthe tori two dimensions.
For all start parameter of this integrablesystem the phase space trajectories arerunning on surfaces of tori. The evolution of thissystem can be calculated precisely and thesystem is stable.
Daniel Dobos – Accelerator Chaos
3. nonlinear consequences
With special start parameter in an nonintegrablesystem a tori is destroyed by resonances anddecays into smaller tori. On the surface of thesesmaller tori the motion is still stable. The phasespace trajectories between the tori of thisdisturbed, nonintegrable system are running onnonregular tracks and the motion could getunstable. It is possible that a trajectory isrunning near a torus for a long time - seems tobe stable - but diffusing finally away. This iscalled the Arnold-Diffusion.
3.5 stability for N circulations:
To understand the structure of the Henondiagram it is important to calculate thedynamical aperture of the phase space.
Onemethod is to calculate the stability for Ncirculations with particle tracking. The trajectoryof a particle should stay inside the mechanicalaperture for N circulations. If this happens weincrease step by step the start distance of theparticle from the orbit and check the stabilityagain.
Daniel Dobos – Accelerator Chaos
3. nonlinear consequences
If the trajectory does not stay inside themechanical aperture after N circulations, thedynamical aperture has to be between this andthe last stable start coordinates. With decreasingof the interval step we try to find this border.
In this method we conclude that a particle isstable if it is stable for N circulations, but we donot know anything about things that happen afterthis N circulations and ignore the presence oflong time stabilities.
3.6 backtracking:
Calculate thetrajectories ofparticles back-ward in time.Starting thesimulation inareas withunstabletrajectories.
Daniel Dobos – Accelerator Chaos
3. nonlinear consequences
If we calculate enough trajectories we get anegative image of the dynamical aperture andthe number of necessary tracking circulations isreduced.
3.7 Lyapunov Exponent:
We put a cut through the two dimensionalphase space and calculate the maximalLyapunov Exponents. If we examine a noncoupled system this cut is sufficient tounderstand the dynamics of this system,because the L. Exponent describes a trajectoryand not a phase space point and all trajectoriesare registered by this cut. If we have a coupledsystem we have to calculate the L. Exponentsfor many cuts or the whole area to register alltrajectories. It could not get negative in thiscase, because we examine a conservativesystem and this would mean a contraction of thephase space. The increasing of the L. Exponentmarks the position of the dynamical aperture, butdoes not give any information of the structure ofthe phase space inside it.
