summary of gsi visit july 2013

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Summary of GSI Visit July 2013Summary of GSI Visit July 2013

• Benchmarking of MICROMAP with MAD-X/PTC/SixTrack

• Analysis of Resonances in 4D: In particular non-linear normal Sextupole Coupling Resonance Qx + 2 Qy = n

• Intense Collaboration with Giuliano Franchetti

Frank Schmidt GSI visit

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Benchmarking of MICROMAP Benchmarking of MICROMAP with MAD-X/PTC Iwith MAD-X/PTC I

• MAD-X Twiss: 2nd order terms included agrees with EXACT Hamiltonian up to this order; non-symplectic due to missing higher orders, SC with rescaling turn by turn

• PTC: approximate (kicks), exact or non-exact, always symplectic, very high orders included

• Sixtrack: always symplectic, non-exact, second order

• MICROMAP: always symplectic, non-exact, first order


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Benchmarking of MICROMAP Benchmarking of MICROMAP with MAD-X/PTC IIwith MAD-X/PTC II

• Set-up of the PS an example with minimal set of elements, MAD-X clean-up via seqedit

• Conversion to MICROMAP more efficient now

• Perfect agreement of 4D parameters: tunes, TWISS and dispersion!

• Chromaticity off by 16% in horizontal plan due to lack of EXACT Hamiltonian

➔ SC comparison eagerly awaited!!!


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Resonance CategorizationResonance Categorization

1. Fodo with single Sextupole unstable motion at ➔sextupole resonance, no islands

2. Stabilization via octupole islands➔

3. Sextupole (constant strength) distributed around a ring first order resonance compensated no ➔islands; detuning with amplitude; large scale chaos

4. Random Errors balances resonance and detuning ➔anything possible!


Frank Schmidt Dynamic Aperture 5

Resonances in Tune diagramResonances in Tune diagram

1D Resonance

2D coupled Resonance

Fixlines

2D crossing Resonances

2D Fixpoints


DA in 1DDA in 1D

Par

ticle

Los

s


1D close-up1D close-up

LOSSLOSS


2D Stable and Chaos2D Stable and Chaos


2D Fixpoints2D FixpointsVertical HorizontalHorizontal


2D Fixlines2D Fixlines

Full Projection Cut in Phase Space

Stable

Chaos

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Simple Toy PS Fodo with 1 Simple Toy PS Fodo with 1 Sextupole per FodoSextupole per Fodo


12Frank Schmidt GSI visit

Resonance Qx + 2Qy Large Island Resonance Qx + 2Qy Large Island TubeTube


X-YX-Y PX-PYPX-PY


Stroboscope Large Island Tube Stroboscope Large Island Tube


Resonance Qx + 2Qy Small Island TubeResonance Qx + 2Qy Small Island Tube


X-YX-Y PX-PYPX-PY


Stroboscope Small Island TubeStroboscope Small Island Tube


Resonance Qx + 2Qy on Fix-LineResonance Qx + 2Qy on Fix-Line


X-YX-Y PX-PYPX-PY


Stroboscope on Fix-LineStroboscope on Fix-Line

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Determining the Fix-LineDetermining the Fix-Line

Standard Technique I Standard Technique I

• As in 1D the expectation is that particles are trapped in the island torus around the fix-line.

• To understand this mechanism in detail on has to determine the fixline.

• With this knowledge one can construct a normal plane to the fix-line which will allow to find the extent of the torus structure and the unstable fixlines.

• The motion in the plane can be interpreted as a secondary Poincaré section of motion! In fact, in this plane the motion around the fix-line resembles a 1D resonance.


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Standard Technique I I Standard Technique I I

• First step is to put a particle somewhere into the island structure.

• The method works by restricting the motion in one plane to a wedge say with positive values and best in the linearly normalized phase space (I have called this stroboscoped motion above).

• The opening angle of the wedge is the free parameter in this technique.

• In the other plane one will then find a number of islands according to the nx * Qx + ny * Qy = n resonance.

• The analysis has to be restricted to one of these islands.

• The average of all 4 coordinates represents a good approximation of the stable fix-line.

• The technique is very time consuming since a sufficient amount of data has to be available in the restricted phase space: however with 2000 turns a good approximation of the fix-line could be found in 7 steps.



Determine Fix-lineDetermine Fix-line

X-PxX-Px Y-PyY-Py

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Using artificial Damping Using artificial Damping

• Experiments at LEP some 13 years ago have shown that in the presence of 4d stable fix-points the particle motions is damped to those fix-points rather than (0,0).

• The mechanism works because motion around the fix-point the particles increase there amplitude which can be seen as pseudo energy “gain”.

• Therefore, if the damping is not too strong the motion will zoom into the fix-point until the damping and the energy “gain” balances out.

• This should generally work in higher dimensions.

• One side-effect of this finding is that in presence of weak damping the resonance structure is preserved in all phase space!

• A fine-tuning of the damping as a function of the distance to the resonance and/or fix-line should improve the convergence of the technique.



Lep: Damping at 4Lep: Damping at 4thth order order


Damping in 1D in SimulationDamping in 1D in Simulation


Damping in 2D in SimulationDamping in 2D in Simulation

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Additional Tools Additional Tools

• Originally I had hoped that NormalForm would allow to determine detuning and resonance strength.

• Although true in principle, the trouble is that this technique is divergent in the vicinity of resonance structures. Therefore, despite the valuable information gained (direction of detuning and good prediction far from resonances) one could not determine the precise location of the resonance structures.

• Another tool is the harmonic analysis which decomposes the particle motion into a set of lines.

• Despite best efforts one could not yet determine the fix-lines with the help of this complete decomposion into lines. More effort will be invested.

• However, the island tune can be nicely determine as a set of side-bands.



FFT Large Island TubeFFT Large Island Tube


FFT Small Island TubeFFT Small Island Tube


FFT on Fix-LineFFT on Fix-Line

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Next Steps Next Steps

• The simulations will intensify both with MICROMAP and MAD-X-SC.

• We will have to define and revisit the Random Part of the multipole components newest value from ➔ Simone in collaboration with the magnet experts.

• MAD-X-SC is not yet under MPI but work has been started on that.

• The plan is to finish the analysis of the 2012 PS experiments by the end of the year 2013 and spring 2014 at the latest.

• The 2D resonance analysis will be brought to conclusion including 2D fix-points (2 separate resonance conditions full-filled: E.G. 3Qx and 4Qy) which is an additional challenge ➔publication in planning



Fix-points motion 3Qx AND 4QyFix-points motion 3Qx AND 4Qy


Fix-points motion 3Qx AND 4QyFix-points motion 3Qx AND 4Qy

X-YX-Y PX-PYPX-PY

summary of gsi visit july 2013

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summary of gsi visit

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