Single Particle Beam Dynamics Codes

1. Single Particle Beam Dynamics Codes Winni Decking DESY –MPY- HHH Workshop CERN 2004 Winni Decking

2. Overview • Introduction • Code repository • Models used • Programming philosophy • Examples • Summary This is a workshop contribution: The description of methods and their implementation in the various beam dynamics codes is not complete, not always accurate and maybe wrong at all. Winni Decking

3. Definition • Describes the motion of a particle in the 6 dimensional phase space under the influence of external fields • Linear Motion • Fitting of linear optic functions etc. • Definition of magnets and alignment • Definition of geometry • Nonlinear Motion • Nonlinear perturbations • Dynamic Aperture Winni Decking

4. Point of view • The physicist who cares only about the methods/assumptions used • The programmer who wants to implement the newest programming techniques • The user (also a physicist/programmer) who doesn’t care about methods and programming but likes a well documented, usable, cross-checked code to get the work done Winni Decking

5. Program Layout • Get the data/lattice into the code - the lattice parser • Calculate • Linear optics functions • Tracking • Construct Map • Analyze the result • Display optics function • Calculate DA, frequency map, nonlinear distortions Winni Decking

6. A legacy of beam dynamics codes • Many beam dynamics codes written over the years • Here is a – surely – not complete list: AT, BETA, BMAD, COMFORT, COSY-INFINITY, DIMAD, ELEGANT, LEGO, LIAR, LUCRETIA, MAD, MARYLIE, MERLIN, ORBIT, PETROS, PLACET, PTC, RACETRACK, SAD, SIXTRACK, SYNCH, TEAPOT, TRACY, TRANSPORT, TURTLE, UAL Winni Decking

7. Orbit vector Transport through elements using R matrix Linear optics calculations Concatenated by Matrix multiplication Extended to higher order (TRANSPORT) NOT symplectic AT, BETA, BMAD, COMFORT, COSY-INFINITY, DIMAD, ELEGANT, LEGO, LIAR, LUCRETIA, MAD, MARYLIE, MERLIN, ORBIT, PETROS, PLACET, PTC,RACETRACK, SAD, SIXTRACK, SYNCH, TEAPOT, TRACY, TRANSPORT, TURTLE, UAL Model – Ray Tracing Winni Decking

8. Symplecticity Given a transfer Map And its Jacobian J is symplectic if • A Hamiltonian system is symplectic, i.e. a map which fulfills the symplectic conditions describes a Hamiltonian system • Important test to verify validity of chosen approximations • If violated, artificial damping/excitation of motion Winni Decking

9. Model – Element by Element Kick Code • Elements described by thin lens kicks and drifts • Always symplectic • Long elements to be sliced =>slow AT, BETA, BMAD, COMFORT, COSY-INFINITY, DIMAD, ELEGANT, LEGO, LIAR, LUCRETIA, MAD, MARYLIE, MERLIN, ORBIT, PETROS, PLACET, PTC, RACETRACK, SAD, SIXTRACK, SYNCH, TEAPOT, TRACY, TRANSPORT, TURTLE, UAL Winni Decking

10. Model – Lie Maps • Lie map methods allow extension of linear concepts in the non-linear Regime • Lie transformation • Lie maps can be factorized and truncated without loosing symplecticity AT, BETA, BMAD, COMFORT, COSY-INFINITY, DIMAD, ELEGANT, LEGO, LIAR, LUCRETIA, MAD, MARYLIE, MERLIN, ORBIT, PETROS, PLACET,PTC,RACETRACK, SAD, SIXTRACK, SYNCH, TEAPOT, TRACY, TRANSPORT, TURTLE, UAL • Concatenation formulae exist Winni Decking

11. Example – Different Transfer functions in BMAD The computation method used for tracking and calculating transfer maps can be set individually for each element. Winni Decking

12. Model – Differential Algebra • Differential Algebra techniques allow computation of Taylor maps • Basic idea is to “track” a power-series element-by-element • Taylor Map is again not symplectic but can be used to obtain factorized Lie map • AT, BETA, BMAD, COMFORT, COSY-INFINITY, DIMAD, ELEGANT, LEGO, LIAR, LUCRETIA, MAD, MARYLIE, MERLIN, ORBIT, PETROS, PLACET,PTC,RACETRACK, SAD, SIXTRACK, SYNCH, TEAPOT, TRACY, TRANSPORT, TURTLE, UAL Winni Decking

13. Application of One Turn Maps • One turn map tracking • One-turn Taylor Map • Construct mixed-variable generating functions • Cremona maps • Normal form analysis • Extracts higher order lattice function perturbations and their parameter dependence as well as driving terms out of one-turn map • Very useful Winni Decking

14. PTC-TWISS MAD-X Module E. Courant et al, “A comparison of several Lattice Tools for Computation of Orbit Functions of an Accelerator”, published in PAC2003 Portland, shown is nx versus dp/p for a simple cyclotron. Standard MAD-X gives the green curve which deviates since the MAD-X (like MAD8) uses the expanded Hamiltonian. In PTC the exact attribute allows to the treat the true Hamiltonian. Note, that PTC has read-in the structure from MAD-X input. There is now PTC_TWISS as attribute of the PTC MAD-X module (still rudimentary!) that allows to produce the Ripken/ Willeke lattice functions called TWISS3 in MAD8. Example for the Afternoon Normal Form MAD-X Module There is now also a NORMAL attribute of the PTC MAD-X module (still rudimentary!) to calculate dispersion, tune and anharmonicities to high orders and as function of delta. This module will be eventually become the replacement of the DYNAMIC/STATIC of MAD8. Winni Decking

15. Programming Philosophies • Integrate the physics into a high level mathematics software (like Mathematica, MatLab, …) • Many existing codes can be ‘operated’ from high level software • Only few codes are ‘embedded’ into MatLab AT, BETA, BMAD, COMFORT, COSY-INFINITY, DIMAD, ELEGANT, LEGO, LIAR, LUCRETIA, MAD, MARYLIE, MERLIN, ORBIT, PETROS, PLACET, PTC, RACETRACK, SAD, SIXTRACK, SYNCH, TEAPOT, TRACY, TRANSPORT, TURTLE, UAL • Advantages: • Input/output dealt with built in math functions • Easy implementation in control system • Disadvantage: Slow Winni Decking

16. Programming Philosophies • Provide the user with a toolbox (library) rather than an existing program which does contain the needed elements and procedures • Realized in C++, F90, Pascal AT, BETA, BMAD, COMFORT,COSY-INFINITY,DIMAD, ELEGANT,LEGO,LIAR, LUCRETIA, MAD, MARYLIE,MERLIN,ORBIT, PETROS, PLACET, PTC, RACETRACK, SAD, SIXTRACK, SYNCH, TEAPOT,TRACY,TRANSPORT, TURTLE,UAL • Can be tailored to specific problem, should be easy to maintain and extend Winni Decking

17. Programming Philosophies • A more or less advanced process control is implemented in the code itself and allows complicated run logic AT, BETA, BMAD, COMFORT, COSY-INFINITY, DIMAD, ELEGANT, LEGO, LIAR, LUCRETIA, MAD, MARYLIE, MERLIN, ORBIT, PETROS, PLACET, PTC, RACETRACK, SAD, SIXTRACK, SYNCH, TEAPOT, TRACY, TRANSPORT, TURTLE, UAL Winni Decking

18. Programming Philosophies • Programs tailored to more or less specific needs • Not universal, but usually the code is not as complex and allows easy changes • Often only used at only one laboratory and closely linked to the specific facility and control system AT, BETA, BMAD, COMFORT, COSY-INFINITY, DIMAD, ELEGANT, LEGO, LIAR, LUCRETIA, MAD, MARYLIE, MERLIN, ORBIT, PETROS, PLACET, PTC, RACETRACK, SAD, SIXTRACK, SYNCH, TEAPOT, TRACY, TRANSPORT, TURTLE, UAL Winni Decking

19. Input Format • A common format for input and output does not exist • A mad-like description of beam lines seems is the most accepted approach • Much more information on a beam line element needed • Errors, Aperture, Wakefields, …. • Magnet errors • Systematic • Random • Time dependent variation of magnet strength, magnet positions • Correlation between errors important • Girder motion • Correlated systematic errors • Attempt for unified input (SXF) Status? Winni Decking

20. Input Format (Typical) Life Cycle of an Accelerator Project (from N. Walker) • Simple models (sequences of elements) • Tend to work with smaller modules • Fitting, constraints etc. Lattice matching • ‘Generic’ magnet families • Definition of basic parameters • (may have more than one possible optics) Basic Lattice Design Performance Simulation construction Quadrupole L K1 value or range of K1 values Modelling of real machine Winni Decking

21. Input Format (Typical) Life Cycle of an Accelerator Project (from N. Walker) • More complex (complete) models • Tolerance studies • Simulation of a range of “errors” • Refinement of parameter specifications • Definition of prototype component • Tuning algorithms, diagnostics specs. • Power supplies (circuits), Klystrons etc. Basic Lattice Design Performance Simulation construction Quadrupole L Pole tip radius max / min pole-tip field Tolerances (used to generate random errors) Modelling of real machine Winni Decking

22. Input Format (Typical) Life Cycle of an Accelerator Project (from N. Walker) • Engineering design of accelerator components Basic Lattice Design Performance Simulation Quadrupole prototype (family) L Pole tip radius max / min pole-tip field Tolerances Documentation (drawings, cad files etc) construction Modelling of real machine Winni Decking

23. Input Format (Typical) Life Cycle of an Accelerator Project (from N. Walker) • Complete model of real machine • Used for • Online modelling • Continued performance studies • Tuning studies etc. Basic Lattice Design Performance Simulation construction Quadrupole QF10 (now unique) L Pole tip radius max / min pole-tip field Measured field map Other unique physical attributes Modelling of real machine Winni Decking

24. Input Format (Typical) Life Cycle of an Accelerator Project (from N. Walker) Basic Lattice Design Input Format should support all stages of the project Performance Simulation construction Modelling of real machine Winni Decking

25. Example – HERA-e Synchro-Betatron Resonances • Working Point close to integer was found to have low lifetime • Incomplete cancellation of chromatic perturbations in the two IR’s north and south • Analytic treatment with perturbation theory by F. Willeke (EPAC04) Winni Decking

26. Example – HERA-e Synchro-Betatron Resonances • Tracking with SIXTRACK • constant initial amplitude in x, y, d • change of coherent tune with arc FODO quads • Frequency Map analysis of 2048 turns Winni Decking

27. Example – Frequency Map at BESSY II • Model incorporates: • Linear coupled model based on orbit-response measurements • Dipole and quadrupole fringe fields • Longitudinal variation of sextupole field • Systematic octupole in quadrupole • Sextupole and decapole components created by steering magnets Measurement Model Winni Decking from P. Kuske (BESSY) – EPAC04

28. Measurement ALS Off-Momentum Aperture • Excitation of increasing horizontal beam centroid motion with single turn kicker • (Static) momentum variation by change of RF frequency • Single turn BPM’s provide tune of kicked beam (lower figure) • Beam intensity monitor records beam losses (upper figure, dot size corresponds to relative intensity loss) Winni Decking

29. ALS Off-Momentum Aperture • Tracking with ALS model after orbit response measurement • Beam loss on 6th order resonance reproduced Winni Decking

30. Early Indicators – Survival Plots HERA-p , 2000 Model • Huge number of turns • Early indicators for chaotic motion • Frequency maps • Lyapunov coefficient • Survival plot extrapolation • Diffusion rates in amplitude space • Difficulty with time dependent process (ripple, events) Winni Decking

31. Beam Dynamics Measurements with Hadrons • Measurements at SPS, FermiLab, Hera-p, … • Optional control off non-linearity with additional sextupoles and octupoles • Optional additional application of ps-ripple to simulate tune variations • Phase space reconstruction with kicker and single-turn BPM’s • Resonance driving term analysis from single-turn BPM data • Model verified by tune-shift with amplitude measurements (often difficult if aperture restriction does not allow large kick amplitudes) • Measurement of DA agrees with predictions within 20 % Winni Decking

32. Measurements at SPS 2002 Winni Decking

33. Resonance driving terms at HERA • Resonance line amplitudes obtained from single-turn PM data • Line amplitude can be calculated form normal form theory Winni Decking

34. Conclusion • The ideal beam dynamics code looks like • Library type architecture • Linear and nonlinear matching • Symplectic map tracking through arbitrary fields with time and s dependence • Normal form analysis • Element by element kick tracking as alternative to the above • Pre processing • Connection to data base and control system • Uniform format only important for big collaborations • Post processing • Includes FMA, early indicators etc. • Can be done by external math program Winni Decking

35. Conclusion • Single particle dynamics is mature field with many available tools • Codes have been benchmarked against experiments with good agreement if the accelerator model is refined enough • Modern tools like Lie algebra and normal form should be promoted and used more frequently • But: For today’s accelerator physics problems the single particle approach is less and less valid: • Beam-beam, space charge, geometric wakes, ion-and e-cloud, coherent synchrotron radiation, …. Winni Decking

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