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DETECTOR ALIGNMENT with tracks

DETECTOR ALIGNMENT with tracks. OUTLINE. Why do we need software alignment Principle of Chi2 minimization Local, iterative approach with 5-6 parameters solved at a time Example – the experiment NA57 silicon telescope Global alignment method with ALL parameters solved simultaneously

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DETECTOR ALIGNMENT with tracks

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  1. DETECTOR ALIGNMENTwith tracks Adam Jacholkowski

  2. OUTLINE • Why do we need software alignment • Principle of Chi2 minimization • Local, iterative approach with 5-6 parameters solved at a time • Example – the experiment NA57 silicon telescope • Global alignment method with ALL parameters solved simultaneously • Utility of cosmic muons for detector pre-alignment, example of the ALICE ITS detector • Summary and discussion Adam Jacholkowski

  3. TRACK RECONSTRUCTION PRECISION R.L. GLUCKSTERN Nucl. Instr. & Methods 24(1963) 381 Plus (unknown) Mis- align- ment terms !!! σ0 = pitch/sqrt(12) B in kGs, p in GeV/c, L in cm (3 points parabolic approximation) Adam Jacholkowski

  4. Why do we need software alignment • Hardware alignment techniques (optical, photogrametry, X rays etc) have technological limits  50 -100 microns • Geometrical resolution of the modern detectors (like pixels) as good as about 10 -15 microns • High tracking precision necessary in order to reach physics goals of the experiments ( for ex. heavy flavours) • The only practical method to reach the required precision – use of (high momentum) tracks • A general principle – minimize the track residues but keeping some external (physics wise) constraints Adam Jacholkowski

  5. Example - Impact Parameter resolution Impact parameter resolution is crucial for the detection of short-lived particles: charm and beauty mesons and baryons. Determined by pixel detectors: at least one component has to be better than 100 mm (ct for D0 meson is 123 mm) Mass 1.864 GeV/c2 c=124 m impact parameter d0 (rf) better than 40 µm for pT> 2.3 GeV/c ~20 µm at high pT Adam Jacholkowski

  6. track x z y (z) residual (y) LOCAL ALIGNMENT method (1) Adam Jacholkowski

  7. Local sensor alignment(2) Adam Jacholkowski

  8. Local sensor alignment (3) Adam Jacholkowski

  9. Local sensor alignment (4a) Will come back to it Adam Jacholkowski

  10. Local sensor alignment (4b) Adam Jacholkowski

  11. Warning – different rotation conventions Adam Jacholkowski

  12. Slightly more MATH(1) Note – 3rd component of qxc = 0 ! Adam Jacholkowski

  13. Slightly more MATH(2) qw= 0 as we are in the sensor/local reference system ! But tw ≠0 Adam Jacholkowski

  14. Slightly more MATH(3) With good approximation the 2 ratios  ~tanψ and ~tanθ Adam Jacholkowski

  15. Slightly more MATH(4) Adam Jacholkowski

  16. Slightly more MATH(5) Needed for solving the Chi2 minimization problem giving as a solution 6 correction parameters Adam Jacholkowski

  17. Local sensor alignment (4) Adam Jacholkowski

  18. Local sensor alignment (6) Adam Jacholkowski

  19. NA57 SETUP (Pb - Pb run) (~ 1.0 M pixels) 1.4 T Apparatus X 5 cm Target:1% Pb Scintillator Petals: centrality trigger MSD: multiplicity silicon detector Tracking device: silicon pixel planes (5 x 5 cm2 cross section) Lever arm: double side mstrips Adam Jacholkowski

  20. 5 cm 5 cm 30 cm L p- X- HYPERON DETECTION X Plus many other associatedtracks byp byL Adam Jacholkowski

  21. NA57 Alignment plots Z mm Y Z microns Single Y (vertical) ladder Y-plane tilt test Adam Jacholkowski

  22. Y X aspect ratio ≈ 9 ! p-Be 40 GeV/c Ξ event ORHION [cm] Ω3YΩ3ZΩ2YΩ2ZΩ3YΩ2YΩ2ZΩ2Y Ω2Z Ω3YΩ3Y planes sequence Adam Jacholkowski

  23. Z X aspect ratio ≈ 9 ! p-Be 40 GeV/c Ξ event ORHION [cm] Ω3YΩ3ZΩ2YΩ2ZΩ3YΩ2YΩ2ZΩ2Y Ω2Z Ω3YΩ3Y planes sequence Adam Jacholkowski

  24. Mass Resolution: Ξ 158 A GeV/c 40 A GeV/c Adam Jacholkowski

  25. Global Alignment Approach • Limitations of the local method • Correlations not (fully) taken into account • Convergence not always guaranteed • Constraints not easy to be included • Possible solution – simultaneous fit of ALL the parameters (tracks and sensors)  problem of inverting huge matrices ! • Millepede Algorithm developed in DESY by Volker Blobel (http://www.desy.de/~blobel) • Numerical limitations  an attempt to overcome the problem  Millepede II Adam Jacholkowski

  26. Global Alignment (1) Adam Jacholkowski

  27. Global alignment (2) Adam Jacholkowski

  28. Global alignment (3) Adam Jacholkowski

  29. Global alignment (4) Adam Jacholkowski

  30. Global alignment (5) Adam Jacholkowski

  31. Global alignment (6) Adam Jacholkowski

  32. Global alignment (7) Adam Jacholkowski

  33. Global alignment (7) Adam Jacholkowski

  34. A word on constraints Adam Jacholkowski

  35. A simple, explicit example(1) point source of particles (mini telescope of 3 planes) Local & global parameters Millipede Simultaneous fit of global AND local parameters Adam Jacholkowski

  36. Polynomial Parameterization Fit (example) Adam Jacholkowski

  37. A simple, explicit example(2) System of 5 linear equations The resulting matrix equation looks like Adam Jacholkowski

  38. A simple, explicit example(3) No problem to invert 5x5 matrix but let’s see the reduction method 3x3 Adam Jacholkowski

  39. A simple, explicit example(4) The key point is that update of the matrix to be inverted (C11-C12C22-1C21) can be done on the track by track basis due to the quasi diagonal, symmetric form of C22 Adam Jacholkowski

  40. A simple, explicit example(5) Actually the matrix inversion algorithms fail for more than 50000 d.o.f. (even when using quadruple precision !!) The next and the last step would be inclusion of constraints in order to avoid bad collective modes like global displacement and/or shearing Adam Jacholkowski

  41. A simple, explicit example(6): constraints • Forcing the fit to conform to physics principles and/or to external knowledge not known by the internal variables of the fit - 2 methods: • Elimination of unknowns by direct substitution, but equations cannot be always solved analytically, covariance matrix is calculated only for the reduced set of variables • Method of Lagrange multipliers – a preferred one Adam Jacholkowski

  42. A simple, explicit example(7) , WHERE initial set of parameters Let’s assume in our toy example one constraint equation like β1+ β2+ β3 = 0 with d0 = 0 (i. e. no global dis-placement) Adam Jacholkowski

  43. A simple, explicit example(8) Now we have a (final) matrix of (n+r) x (n+r) size, like Adam Jacholkowski

  44. A simple, explicit example(9) Original matrix C11 In our case we have just one extra column and row of 1s, one λ, d0 = 0 !! Adam Jacholkowski

  45. ALICE coordinates Adam Jacholkowski

  46. ALICE Inner Tracking System (ITS) Alignable elements: SPD -- 240 SDD -- 260 SSD - 1698 Total – 2198 * 6 d.o.f + ~12 collective dof Adam Jacholkowski

  47. SUMMARY • All modern particle detectors need software (track) alignment methods in order to reach the design precision • Two main approaches: • Local with many iterations • Global needing inversion of huge matrices • We have looked into MATH involved in these 2 methods, discussed some approximations and tricks • Computing and bookkeeping very challenging in real life, especially in the LHC experiments under preparation • Alignment is part of the art of detector calibration (MATH is not ALL) Adam Jacholkowski

  48. COSMICS as a tool in the detector pre-alignment in ALICE Before the (true) beam becomes available…

  49. Hadronic interaction models in cosmic rays PPR part II $6.11 It is a paragraph describing the importance of the knowledge of hadronic interactions at energies involved in cosmic rays E > 1014 eV The LHC contributions and in particular the ALICE possibilities to study p-p p-A and A-A interactions Adam Jacholkowski

  50. Effects on muons of the Alice environment Location of ALICE set-up Rock composition over Alice  Element H C O Na Mg Al Si K Ca Fe % 0.8 4.3 48.5 0.7 4.2 3.7 21.5 2.3 10.0 4.0 • Nm(Em >15 GeV) • Direction • Energy Adam Jacholkowski

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