1. Controlling non-linear effects in circular particle accelerators and the dynamic aperture saga: the case of the CERN LHC Massimo Giovannozzi CERN – Beams Department Introduction (long) Definition and physics/computational issues DA studies for LHC Outlook Acknowledgements: A. Bazzani, O. Brüning, R. De Maria, S. Fartoukh, E. Laface, F. Lang, S. Nagaitsev, W. Scandale, F. Schmidt, E. Todesco, G. Turchetti, C. Yu.

2. Colloquium 28/05/2013 Introduction - I Are non-linear effects good or bad for beam dynamics? Standard answer: they are bad! It reflects the classical approach to charged particle beam dynamics. LHC falls in this category. New answer: they might be useful! New ideas are coming: Possible applications of non-linear machines. Possible way to design non-linear machines. Massimo Giovannozzi - CERN2

3. Colloquium 28/05/2013 Introduction - II 1952 ground breaking paper by Courant and Snyder (Christofilos also prepared the ground, but did not publish his results): strong focusing principle. By alternating the sign of the focusing strength along a circular machine it is possible to focus the charge particles in both planes. Massimo Giovannozzi - CERN3

4. Colloquium 28/05/2013 Introduction - III Equation of motion: Hill equation. Floquet theory applies. Immediate consequence of strong focusing: smaller machines! Massimo Giovannozzi - CERN4 Cosmotron CERN PS

5. Colloquium 28/05/2013 Introduction - IV Since the beginning of the history of accelerators: the unperturbed system is linear. Non-linear effects are considered as perturbation (hence bad). They are: Needed: e.g., sextupoles are imposed to correct chromaticity (all effects related with differences in energy of charged particles). Unavoidable: field imperfections in magnets. This is particularly the case of superconducting magnets. The advent of superconducting accelerators triggered an interest in non-linear dynamics. Massimo Giovannozzi - CERN5

6. Colloquium 28/05/2013Massimo Giovannozzi - CERN Main elements are the 2-in-1 superconducting dipoles (1232) and quadrupoles (392) operating in superfluid helium at a temperature of 1.9 K Main dipoles - I 6

7. Colloquium 28/05/2013 Main dipoles - II Field is generated by current layers -> unavoidable field errors. Multipole expansion: b n : normal multipoles (n=1 dipole) a n : skew multipoles Hard work of specification during design stage Massimo Giovannozzi - CERN7

8. Colloquium 28/05/20138Massimo Giovannozzi - CERN Introduction - V Why non-linear dynamics should be useful? The non-linear motion introduces a dependence of the particle frequency in phase space on amplitude. The frequency spread stabilises collective instabilities (Landau damping). A non-linear system might be more robust than a linear one (stable under perturbation). However, it should be integrable to ensure stable motion Typical non-linear motion with regular and chaotic orbits.

9. Colloquium 28/05/2013 Introduction - VI Interest in determining Maxwellian fields (magnets) leading to integrable configurations. Massimo Giovannozzi - CERN9 Integrable (left) and non-integrable (right) systems. V. Danilov, Phys. Rev. ST Accel. Beams, 11, 114001 (2008)

10. Colloquium 28/05/2013 Introduction - VII In spite of these recent results, some studies, have been carried out already in the 60s to find classes of integrable systems. In this framework McMillan found the following map: Important remark: unlike multipolar non-linearities (divergent to infinity), these non-linearities are bounded. A special configuration with two invariant of motions is the basis of the accelerator to be built in US. Massimo Giovannozzi - CERN10

11. Colloquium 28/05/2013 Introduction - VIII Integrable Optics Test Accelerator – IOTA To be built in Fermilab 160 MeV electrons 36 m circumference Massimo Giovannozzi - CERN11 Courtesy S. Nagaitsev

12. Colloquium 28/05/2013 Introduction - IX Back to the LHC approach… Even if a more classical approach has been used, the design of the LHC triggered a wealth of studies in the domain of non-linear dynamics: Numerical simulations Symplectic tracking Symplectification of truncated maps Refined tools for analysis of simulations data (e.g., harmonic analysis). Perturbative theory Transfer map approach Poincaré section Poincaré- Birkhoff normal forms Massimo Giovannozzi - CERN12 Nothing completely new, but everything never applied to accelerators!

13. Colloquium 28/05/2013 Definition and issues - I Massimo Giovannozzi - CERN13 The dynamic aperture is the region in phase in which bounded motion occurs. So far only tracking allows computing the DA of a given system. From a numerical point of view: A volume should be evaluated. This entails a scan over the phase space variables. An appropriate choice of the steps in the variables is required. NB: I will deal with protons -> symplectic dynamics! The tracking has to be long-term

14. Colloquium 28/05/2013 Intermezzo: DA for 2D systems Maybe not so useful for a real accelerator, but the DA for a 2D system can be found in terms of invariant manifolds of low order unstable fixed points. Massimo Giovannozzi - CERN14 Poincaré knew already the relevance of these manifolds… Homoclinic tangle for the Hénon map (left) and comparison between the DA computed numerically and with invariant manifolds (right)

15. Colloquium 28/05/2013 Definition and issues - II Massimo Giovannozzi - CERN15 DA computations are CPU intensive. Fast tracking tools are required: Optimised codes (e.g., kick codes) Parallel approach (this is only possible over the initial conditions). As an alternative (maybe a dream…): find a dynamical quantity with a good correlation with DA, but less expensive in terms of CPU. A trade-off between number of turns and number of initial conditions might be possible (e.g., use a dense set of initial conditions iterated a small number of turns). Do not forget stable chaos and intermittency!

16. Colloquium 28/05/2013 Definition and issues - III Massimo Giovannozzi - CERN16 Here are some examples of indicators: Lyapunov exponent Tune difference (this indicator triggered several studies on accurate computation of tunes in numerical simulations).

17. Colloquium 28/05/2013 Definition and issues - IV Massimo Giovannozzi - CERN17 Another strategy could be: is there a model to describe DA vs. time? In mathematical sense DA does not depend on time. Numerical simulations are performed with a specific maximum number of turns (N max ): the computed DA does depend on N max How does DA depend on N max in numerical simulations? ). Studies have been performed recently to review the functional dependence on  of fit model

18. Colloquium 28/05/2013 Definition and issues - VII Massimo Giovannozzi - CERN18 Dynamic aperture of a model of the LHC ring (left) in physical space: The red points represent the initial conditions stable up to 10 5 turns The blue points represent unstable conditions and their size is proportional to the number of turns by which their motion is still bounded. The time-evolution of the DA is shown on the right. The markers represent the numerical results The continuous line shows the fitted inverse logarithmic law. The dotted line represents D 

19. Colloquium 28/05/2013 Definition and issues - VI Massimo Giovannozzi - CERN19 Is this a purely phenomenological fit? In fact not quite. The physical picture is: For r < D  The motion is governed by KAM theorem. Fully stable region (only Arnold diffusion for a set of initial conditions of small measure -> irrelevant from the physical point of view). For r > D  The motion follows Nekhoroshev theorem, i.e., the stability time N(r) of a particle at radius r is given by This provides a pseudo-diffusion.

20. Colloquium 28/05/2013 Definition and issues - VII Massimo Giovannozzi - CERN20 Two regimes found in 4D simulations: D , b,  are always positive. This implies a stable region for arbitrary times. In 4D simulations with tune ripple or 6D simulations: There could be situations in which no stable region for arbitrary times exists. This corresponds to

21. Colloquium 28/05/2013 Definition and issues - VIII Massimo Giovannozzi - CERN21 Fit of DA vs. time can lead to a number of extensions: Losses in hadron machines due to non-linear effects (single particle).

22. Colloquium 28/05/2013 Tevatron data: proton bunch at injection Estimates from purely diffusive model included. Estimates from purely diffusive model included. Massimo Giovannozzi - CERN 22 Nice agreement for all models! Experimental data from: T. Sen et al. “Beam Losses at Injection Energy and During Acceleration in the Tevatron”, IPAC03, p. 1754.

23. Colloquium 28/05/2013 SPS data: proton bunch at 55 GeV in coast Estimates from purely diffusive model included. Estimates from purely diffusive model included. Massimo Giovannozzi - CERN 23 Negative second order derivative cannot be reproduced by diffusive models!

24. Colloquium 28/05/2013 Definition and issues - VIII Massimo Giovannozzi - CERN24 Evolution of DA in presence of beam-beam effects. N b =0.10×10 11 N b =1.15×10 11 N b =1.70×10 11

25. Colloquium 28/05/2013 Definition and issues - VIII Massimo Giovannozzi - CERN25 Evolution of luminosity. LHC Tevatron

26. Colloquium 28/05/2013 DA studies for LHC - I Injection energy: single-particle non-linear dynamics used to derive bounds on field quality. Criterion: 12  DA as target. NB: a reduction by a factor of 2 (simulations/reality) would lead to a DA of 6 , still compatible with collimators’ performance. Collision energy: single-particle non-linear dynamics less relevant. Weak-strong beam-beam simulations are used to assess the actual situation. Massimo Giovannozzi - CERN26 Injection Collision Statistical definition of field errors

27. Colloquium 28/05/2013 DA studies for LHC - II Massimo Giovannozzi - CERN27 Single particle DA studies were used to: Specify field quality of main dipoles (including feedback). NB: some low-order field components have been bounded using analytical criteria (e.g., linear coupling correction, chromaticity correction, tune spread etc.). Verify expected field quality of other superconducting magnets (excluding feedback). Verify expected field quality of normal conducting magnets (including feedback). E.g., MQWs in collimation insertions

28. Colloquium 28/05/2013Massimo Giovannozzi - CERN Courtesy E. Todesco - CERN Main dipoles - III Field errors in dipole production: b3 28

29. Colloquium 28/05/2013 Main dipoles - IV Massimo Giovannozzi - CERN Systematic field errors in dipoles 29

30. Colloquium 28/05/2013 DA studies for LHC - III After specification studies: Production of magnets: follow up of field quality, including corrective actions (change of cross-section of main dipoles to control field quality). Massive campaign of magnetic measurements: All measured warm Some measured cold Sorting algorithm for installation: Main dipoles: optimisation of: geometry b1, a2, b3. Main dipoles: optimisation of: geometry, a2, b2. Other elements: optimisation: geometry, transfer function. Massimo Giovannozzi - CERN30 Determination of warm/cold correlation to extrapolate field quality at cold conditions Possibility to move from LHC description based on statistical errors to machine as-built

31. Colloquium 28/05/2013 DA studies for LHC - IV Possibility to evaluate the DA for the machine as-built Possibility to evaluate impact of sorting on DA Massimo Giovannozzi - CERN31 Summary of DA at injection energy. The error bars represent the effect of 60 seeds Summary of DA at injection energy. The error bars represent the effect of 60 seeds

32. Colloquium 28/05/2013 DA studies for LHC - V Possibility to evaluate the DA for the machine as-built Possibility to evaluate impact of sorting on DA Massimo Giovannozzi - CERN32 Summary of minimum DA for several running configurations.

33. Colloquium 28/05/2013 DA studies for LHC - VI Possibility to evaluate the DA for the machine as-built Possibility to evaluate impact of sorting on DA Massimo Giovannozzi - CERN33 Impact of sorting. Selected generic seedsSelected generic seeds Each sequence of errors is re-ordered.Each sequence of errors is re-ordered. The various dynamical quantities are computed.The various dynamical quantities are computed. Yellow: all seeds (initial and re-ordered) Blue: selected seeds. Impact of sorting. Selected generic seedsSelected generic seeds Each sequence of errors is re-ordered.Each sequence of errors is re-ordered. The various dynamical quantities are computed.The various dynamical quantities are computed. Yellow: all seeds (initial and re-ordered) Blue: selected seeds.

34. Colloquium 28/05/2013 DA studies for LHC - VII Possibility to evaluate the DA for the machine as-built Possibility to evaluate impact of sorting on DA Massimo Giovannozzi - CERN34 Impact of sorting. Selected generic seedsSelected generic seeds Each sequence of errors is re-ordered.Each sequence of errors is re-ordered. The various dynamical quantities are computed.The various dynamical quantities are computed. Yellow: all seeds (initial and re-ordered) Blue: selected seeds. Red: average DA for as- built machine. Impact of sorting. Selected generic seedsSelected generic seeds Each sequence of errors is re-ordered.Each sequence of errors is re-ordered. The various dynamical quantities are computed.The various dynamical quantities are computed. Yellow: all seeds (initial and re-ordered) Blue: selected seeds. Red: average DA for as- built machine.

35. Colloquium 28/05/2013 DA studies for LHC - VIII What is the DA of the real machine? No lifetime problems or slow losses at injection. During aperture measurements (with beams probing high amplitudes) no sign of slow losses was found. This observation indicates that DA should be of the same order of mechanical aperture, i.e., about 10 . Measurement campaign launched: Two MD sessions organised (2011, 2012). Objective: benchmark numerical simulations against measurements (e.g., for HERA a factor of two was found). Strategy: reduce DA by means of the octupolar spool pieces. Fit intensity vs. time with inverse logarithm model Massimo Giovannozzi - CERN35

36. Colloquium 28/05/2013 DA studies for LHC - IX Preliminary results (analysis still in progress) Massimo Giovannozzi - CERN36 Inverse logarithm model provides a very good agreement with intensity evolution! Beam is blown-up to enhance losses. Scan over the octupoles’ strength. Beam is blown-up to enhance losses. Scan over the octupoles’ strength.

37. Colloquium 28/05/2013 Outlook Massimo Giovannozzi - CERN37 The LHC project has been a trigger for several studies in non-linear beam dynamics. Several tools and techniques have been devised to achieve the goal of estimating the dynamic aperture. The work done seems to have been successful: the LHC does not suffer from any single-particle non-linear effects! The dynamic aperture has been probed in the LHC and detailed analysis should be carried out to assess the accuracy of numerical simulations. It is time to probe new situations in which the beam dynamics is intrinsically non-linear albeit integrable!

38. Colloquium 28/05/2013 Thank you for your attention Massimo Giovannozzi - CERN38