1. Dynamic aperture studies for LHC and its upgrade Massimo Giovannozzi CERN – Beams Department Definition and physics/computational issues LHC: layout and key parameters DA studies for LHC LHC upgrade path: HL-LHC DA studies for HL-LHC Outlook Acknowledgements: A. Bazzani, O. Brüning, R. De Maria, S. Fartoukh, E. Laface, F. Lang, Y. Nosochkov, W. Scandale, F. Schmidt, E. Todesco, G. Turchetti, C. Yu.

2. Frascati 18/10/2012 Definition and issues - I Massimo Giovannozzi - CERN2 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

3. Frascati 18/10/2012 Definition and issues - II Massimo Giovannozzi - CERN3 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!

4. Frascati 18/10/2012 Definition and issues - III Massimo Giovannozzi - CERN4 Here are some examples of indicators: Lyapunov exponent Tune difference (this indicator triggered several studies on accurate computation of tunes in numerical simulations).

5. Frascati 18/10/2012 Definition and issues - IV Massimo Giovannozzi - CERN5 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

6. Frascati 18/10/2012 Definition and issues - VII Massimo Giovannozzi - CERN6 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 

7. Frascati 18/10/2012 Definition and issues - VI Massimo Giovannozzi - CERN7 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.

8. Frascati 18/10/2012 Definition and issues - VII Massimo Giovannozzi - CERN8 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

9. Frascati 18/10/2012 Definition and issues - VIII Massimo Giovannozzi - CERN9 Fit of DA vs. time can lead to a number of extensions: Losses in hadron machines due to non-linear effects (single particle).

10. Frascati 18/10/2012 Tevatron data: proton bunch at injection Estimates from purely diffusive model included. Estimates from purely diffusive model included. Massimo Giovannozzi - CERN 10 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.

11. Frascati 18/10/2012 SPS data: proton bunch at 55 GeV in coast Estimates from purely diffusive model included. Estimates from purely diffusive model included. Massimo Giovannozzi - CERN 11 Negative second order derivative cannot be reproduced by diffusive models!

12. Frascati 18/10/2012 Definition and issues - VIII Massimo Giovannozzi - CERN12 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

13. Frascati 18/10/2012 Definition and issues - VIII Massimo Giovannozzi - CERN13 Evolution of luminosity. LHC Tevatron

14. Frascati 18/10/2012Massimo Giovannozzi - CERN The LHC machine has an height-fold symmetry. Eight arcs (arc is the curved periodic part of the machine). Sixteen dispersion suppressors to match the arc with the straight sections (geometry and optics). Eight long straight sections (also called insertion regions). LHC layout - I 14

15. Frascati 18/10/2012Massimo Giovannozzi - CERN LHC layout - II mid-cellsilver mid-cellsilver beam waist H V Six dipoles are located in each cell. Each dipole comprises correctors: Sextupoles Octupoles and decapoles Two quadrupoles are located in each cell. Each quadrupole is equipped with: Beam Position Monitor Dipole corrector (for closed orbit) Sextupoles (for chromaticity) 15

16. Frascati 18/10/2012Massimo Giovannozzi - CERN LHC layout - III Interaction point Low-beta quadrupoles Separation/ricombination dipole Absorber (neutral particles) Towards dispersion suppressor and arc High luminosity insertions 16 To be replaced for upgrade

17. Frascati 18/10/2012Massimo Giovannozzi - CERN Key parameters 17 (Bunch spacing: 25 ns)

18. Frascati 18/10/2012 Intermezzo: overview of beam parameters Massimo Giovannozzi - CERN18 LHC nominal2011 50 ns 2012 50 ns Energy [TeV]7.03.54.0 # Bunches28081380 p/bunch [10 11 ]1.151.41.5-1.6  x,y  [  m] 3.752.5   [cm] (baseline) 5510060 Lumi loss factor (F)0.840.910.81 Peak lumi [10 34 ]1.00.320.6-0.7 # of events per crossing251632

19. Frascati 18/10/2012Massimo 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 19

20. Frascati 18/10/2012 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 - CERN20

21. Frascati 18/10/2012Massimo Giovannozzi - CERN Courtesy E. Todesco - CERN Main dipoles - III Field errors in dipole production: b3 21

22. Frascati 18/10/2012 Main dipoles - IV Massimo Giovannozzi - CERN Systematic field errors in dipoles 22

23. Frascati 18/10/2012 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 - CERN23 Injection Collision Statistical definition of field errors

24. Frascati 18/10/2012 DA studies for LHC - II Massimo Giovannozzi - CERN24 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

25. Frascati 18/10/2012 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 - CERN25 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

26. Frascati 18/10/2012 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 - CERN26 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

27. Frascati 18/10/2012 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 - CERN27 Summary of minimum DA for several running configurations.

28. Frascati 18/10/2012 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 - CERN28 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.

29. Frascati 18/10/2012 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 - CERN29 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.

30. Frascati 18/10/2012 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 - CERN30

31. Frascati 18/10/2012 DA studies for LHC - IX Preliminary results (analysis still in progress) Massimo Giovannozzi - CERN31 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.

32. Frascati 18/10/2012 Upgrade ideas (until 2010) Assumptions (or common belief) Lifetime of triplets under nominal conditions is few years (radiation due to debris) -> they should be replaced Nominal parameters are probably tight and nominal luminosity might be difficult to achieve (triplets’ aperture) Hence, two-stage approach: Phase 1: “Consolidate” the machine with new triplets aiming at reaching ~ 2-3×10 34 cm -2 s -1. Phase 2: “Real” luminosity upgrade aiming at 10 35 cm -2 s -1.. This includes a major upgrade of the detectors. Massimo Giovannozzi - CERN32

33. Frascati 18/10/2012 Phase 1 in short Rough summary of Phase 1 approach Replace “only” triplets with larger aperture magnets to enable reaching smaller  *. Intense studies performed: Minimum  * achievable: ~ 30 cm Limits have been found in other parts of the machine -> much more elements than the triplets should be changed! Very complex optical gymnastics in order to fulfill the correction of chromatic aberrations -> not much operational flexibility left. Massimo Giovannozzi - CERN33 S. Fartoukh at Chamonix 2010 Workshop LS1LS2LS3 Courtesy L. Rossi

34. Frascati 18/10/2012 Scope of High-Luminosity upgrade of LHC Targets: with leveling A peak luminosity of 5×10 34 cm -2 s -1 with leveling An integrated luminosity of 250 fb -1 per year, enabling the goal of 3000 fb -1 in twelve years. Massimo Giovannozzi - CERN 34 Courtesy E. Todesco

35. Frascati 18/10/2012Massimo Giovannozzi - CERN  The peak luminosity depends on  Due to the crossing angle, the geometrical reduction factor F is different from unity and reads Luminosity - I 35 LHC-specific Injectors-specific  NB:  * enters in the factor F via  c and  * : no gain in reducing  * below a certain value. Flat beams represent a mitigation of this effect. L vs.  *

36. Frascati 18/10/2012Massimo Giovannozzi - CERN  Possible strategies for upgrading performance:  Maximize bunch brightness (beam-beam limit)  Minimize beam size (aperture in triplets)  Maximize number of bunches (beam power, e-cloud)  Compensate for F  LHC Upgrade (HL-LHC):  Smaller  * ; new triplets quadrupoles; possibly new technology (Nb 3 Sn instead of Ni-Ti).  Mitigation measures for higher currents (e.g., collimator system upgrade, cooling, beam-beam compensation wires)  Flat beams or crab cavities  Injectors’ Upgrade (LIU):  Increase beam brightness  Mitigation measures for higher currents (e.g., coating SPS vacuum chambers). Luminosity - II 36

37. Frascati 18/10/2012 LHC upgrade: parameter space 25 ns is the baseline, 50 ns is a back-up (e.g. for e-cloud). Parameters still under discussion with the LHC Injector Upgrade project. Relies on crab-cavity Massimo Giovannozzi - CERN LHC nominalHL-LHC 25 ns HL-LHC 50 ns # Bunches2808 1404 p/bunch [10 11 ]1.15 (0.58A)2.0 (1.01 A)3.3 (0.83 A)  L [eV.s] 2.5  z [cm] 7.5   p/p [10 -3 ] 0.1  x,y  [  m] 3.752.53.0   [cm] (baseline) 5515 X-angle [  rad] 285 590 (12.5  )590 (11.4  ) Lumi loss factor (F)0.840.300.33 Peak lumi [10 34 ]1.06.07.4 Virtual lumi [10 34 ]1.220.022.7 T leveling [h] @ 5E34n/a7.86.8 #Pile up @5E3425123247 Courtesy S. Fartoukh 37

38. Frascati 18/10/2012 The proposed optics The Achromatic Telescopic Squeeze optics (invented by S. Fartoukh) enables: Reaching very small  * values At the same time, it enhance the strength of the chromatic sextupoles thus enabling the compensation of the chromatic aberrations that stem from the new triplet quadrupoles. A vigorous experimental programme in the LHC successfully showed the feasibility of the scheme Massimo Giovannozzi - CERN38 Courtesy S. Fartoukh

39. Frascati 18/10/2012 Luminosity levelling Three options at hand: Vary crossing angle (crab cavities help here!) It can be performed with dipoles Easy, but requires aperture in triplets Vary separation It can be performed with dipoles Easy (already tried with LHCb), but requires aperture Vary  * Never tried in existing machines Requires an excellent control of optics and crossing scheme Massimo Giovannozzi - CERN39

40. Frascati 18/10/2012 Crab cavities Massimo Giovannozzi - CERN At IP without crab cavities At IP with crab cavities Crab Anti-crab 40

41. Frascati 18/10/2012 Potential issues of crab cavities RF Noise It could induce emittance growth. So far never used in any proton machine! Design Very limited transverse space in the LHC Imposes creative designs Two types are needed: H and V crossing Machine protection! Field quality? Massimo Giovannozzi - CERN41

42. Frascati 18/10/2012 DA studies for HL-LHC - I Massimo Giovannozzi - CERN42 New magnets to be specified: Triplets Separation dipoles (D1 – now superconducting, and D2) Matching quadrupoles (Q4, Q5) New effects to be included in simulations: Non-linear fringe fields: not completely negligible. Field quality of crab cavities: to be assessed. New simulation campaign for DA evaluation: Also available volunteer computing network LHC@home (for SixTrack). Objectives of Task 2.3 of WP2 of HiLumi project. Activities have started since November 2011. Objectives of Task 2.3 of WP2 of HiLumi project. Activities have started since November 2011.

43. Frascati 18/10/2012 DA studies for HL-LHC - II Massimo Giovannozzi - CERN43 Collision configuration is under consideration Courtesy Y. Nosochkov skewuncertaintyrms normaluncertaintyrms a3100 b3100 a4100 b4100 a520 b520 a620 b650 a710050b750 a810050b860 a910050b970 a1010050b1070 a1110050b1170 a1210050b1270 a1310050b1370 a1410050b1470 DA min =10.7  Error table for triplet quadrupoles: values are relative to the estimated field quality from magnet design.

44. Frascati 18/10/2012 Outlook Massimo Giovannozzi - CERN44 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. It is time to measure the dynamic aperture to assess the accuracy of numerical simulations. The work done seems to have been fruitful: the LHC does not suffer from any single-particle non-linear effects. HL-LHC will require even more sophisticated tools in view of new physics challenges!

45. Frascati 18/10/2012 Thank you for your attention Massimo Giovannozzi - CERN45

46. Frascati 18/10/2012 Spare slides Massimo Giovannozzi - CERN46

47. Frascati 18/10/2012Massimo Giovannozzi - CERN Courtesy E. Métral 47 Trip of the LHC proton beam along the CERN injectors’ chain

48. Frascati 18/10/2012 Nominal LHC beam in PS Massimo Giovannozzi - CERN At PS extraction the bunches have the nominal 25 ns spacing 48

49. Frascati 18/10/2012Massimo Giovannozzi - CERN Triple bunch splitting The stable fixed point bifurcates and three stable ones are generated. 49 Courtesy R. Garoby

50. Frascati 18/10/2012 Nominal LHC filling scheme Massimo Giovannozzi - CERN LHC filling time is about 10 min. 50