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Einführung und Definition Hamilton Funktion für die Bewegung mit 3 Freiheitsgraden

Synchro-betatron Resonanzen: eine Einführung und Berechnung der Resonanzstärken für verschiedene HERA Optiken F. Willeke Betriebsseminar Salzau 5-8. Mai 2003. Einführung und Definition Hamilton Funktion für die Bewegung mit 3 Freiheitsgraden

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Einführung und Definition Hamilton Funktion für die Bewegung mit 3 Freiheitsgraden

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  1. Synchro-betatron Resonanzen:eine Einführung und Berechnung der Resonanzstärken für verschiedene HERA OptikenF. WillekeBetriebsseminar Salzau 5-8. Mai 2003 Einführung und Definition Hamilton Funktion für die Bewegung mit 3 Freiheitsgraden Einführung der Dispersion und Enkopplung von transversalen und longitudinalen Schwingungen Behandlung von Resonanzen mit 2Freiheitsgraden Diskussion der Ergebnisse für HERA

  2. Einführung Es gibt große Probleme die Polarizations- tunes bei HERA einzustellen: fx= 6.5 kHz und fz=9kHz Der horizontale Tune liegt zwischen dem 2-fachen und dem 3-fachen der Synchrotronfrequenz fs=2.5kHz Verdacht: Der Bereich der gewünschten Arbeitspunkte ist durch starke Synchrobetaronresonanzen eingeschränkt. Synchrobetatron-Resonanzen wurden bei DORIS I entdeckt (Piwinski 1972) Ursache: Starker vertikaler Kreuzungswinkel der kollidierenden Elektron und Positronstrahlen: Die transversale Strahl-Strahl-Kraft hängt bei einem Kreuzungswinkel von der longitudinalen Position im Bunch ab DORIS

  3. Allgemein Der Strahl kann in 3 Ebenen oszillieren. 1) Hängt die triebende Kraft in einer Ebene von der Koordinate oder dem Impuls in der anderen Ebene ab, sind die jeweiligen Schwingungsebenen gekoppelt. 2) Wie alle Kräfte können koppelnde Kräfte mit der gleichen Frequenz oszillieren wie der Strahl selbst: Dann kommt es zu einer resonanzartigen Verstärkung selbst sehr kleiner Kräfte. Resonanzen führen zum Energieaustausch zwischen den Schwingungsebenen oder zu Instabilität

  4. Synchro-Betatron Resonances in HERAIntroduction to the Theory and Recent EvaluationsHERA Betriebsseminar Salzau,5-7 May 2003 Coupled Synchro-betatron Motion Decoupling of Synchro-betatron Oscillation Non-linear Coupling between Synchrotron and Betatron Oscillations Width of multi-dimensional Nonlinear Resonances Comparison of the width of Satellite Resonances in HERA for various Beam Optics

  5. Synchrobetatron-Resonances Coupling between transverse and longitudinal oscillations gives rise to excitation of resonances for tunes which satisfy Qx+mQs+q=0 Such resonances can be driven by • Dispersion in the cavities • Dispersion in sextupoles • Chromaticity • A crossing angle or Dispersion a the Collision Point • Wakefields • RF Quadrupoles • …

  6. Coupled Synchro-betatron Oscillations Horizontal Betatron Oscillations and Synchrotron oscillations are strongly coupled by a term x·e /r xis the horizontal coordinate, e is the relative energy deviation from nominal and r is he curvature of he design orbit) This is shown in the following slides

  7. Lagrangian for charged relativistic particle using the accelerator coordinate system m0c2 rest mass, r is the position vector, A is the vector potential, f is the scalar potential Expressing L in accelerator coordinates One obtains

  8. Hamiltonian picture z=x,y,s b= v / c ˜= 1

  9. Path length s as independent variable The Hamiltonian is symmetric in all coordinates (Variation principle) (gauge) as=e/cAs/E0 m0c2/E0<<1

  10. Hamiltonian for motion in x-s plane Expanded and without solenoid fields The termpx2x/r is considered small and has been dropped

  11. Cavity Field Expanded and without constants, energy loss concentrated at cavity, damping neglected as=1/2 V · s 2 + 1/6 W · s 3

  12. Hamiltonian with cavities and sextupoles Longitudinal focussing Linear optics NonlinearitiesTransverse motion chromatics Approximations: v=c p2x/r neglected Square root expanded 1/(1+e) expanded into 1-e Strong linear coupling between horizontal and longitudinal motion Nonlinearities longitudinal motion

  13. Linear Decoupling of Synchrobetatron Oscillations Introduction of the dispersion function Generating function transformation

  14. Decoupled Hamiltonian Transverse linear optics Longitudinal linear optics Linear coupl. by dispersion in cavities Chromatics 2nd satellite driving terms Nonlinearities trans. Nonlinearities lon.

  15. Integer Satellite Driving Terms Qx+Qs+p=0 -DVs p + D’Vs x Qx+2Qs+p=0 -D’ p e2 + ½ mD2 e2x + ½ WD s2p Chromatics sextupole contribution dispersion in cavities

  16. Resonances in x-s Phase Space K = Klin + Knl Linear optics Smooth rf model Variation of constant: Keep the form for x, p s,e but vary the invariants Jx,s and fx,sto solve for the nonlinearities and coupling (transformation to action and angle variables) Result Hamiltonian form of e.o.m. with Knl as new Hamiltonian ∂J/ ∂s = - ∂Knl/ ∂f ∂ f / ∂s = ∂Knl/ ∂J

  17. Procedure to calculate resonance widths • Express Knl in J-f coordinates • Factorise into ring periodic and nonperiodic terms • Express periodic forms in Fourier Series • Realise, that only slow terms can affect a change in invariants • Drop nonresonant terms • Transform into a rotating system to get a time-indpendent system • Calculated fixpoints • Find distance from resonance to reach the fix points for a given amplitude

  18. Perform this for the term ½ WDs2p Since we are only near one resonance at a time, we are only interested in one of the terms x+2y

  19. Periodic factor non-periodic factor Fourier Series for periodic part Select only the one resonant term and ‘drop’ all the others, replace 2ps/L by q (change independent variable from s to q)

  20. Resonance Hamiltonian Transformation into rotating system via generating function F

  21. Note there is a similar term denoted by “s” which comes from the sine part of p The two driving terms k12qs and k12qc are combined into a single one

  22. Resonance Width 2Ix-Is is an invariant and we can reduce the system to a 1-dim system Note: sum resonances are instable and difference resonances stable!! R DI Separatrix, unstable trajectory I0=unstable fix point I

  23. Condition for unstable fix point Evaluate this for Ix = Ix0 D12q=½ k12q Is0 Ix0-1/2

  24. Hamilonian vs action Satellite Resonance with h=0.1m-1 Horizonal projection of separatrix Longitudinal projection of separatrix

  25. Evaluations Beam Parameters used Integrals are replaced by sums, optical functions are replaced by their integrated value over the elements, then a thin lens treatment is applied Results:Resonance width for a 10 sigma particle in Hz (nsx2+nss2=100)

  26. Comparison of sextupole driven transverse resonances for one sigma transverse, full coupling and width of horizontal half integer stopband for 10 sigma long.

  27. Conclusions • The 3rd order transverse and satellite resonances are stronger in the 72deg optics compared to the old 60 degree optics • The stronger satellite resonances are due to more unfavorable ratio of Longitudinal and transverse emittance • The SM optics has lower contributions from sextupole driven satellites

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