1. 1 LHCb B Field Map Géraldine Conti IT Survey EPFL, the 21 th of May 2007 Monday Seminar

2. 2 Parameterized LHCb B Field Map

3. 3 Outline … Analytical Field Service Reminder of the Goal May 21, 2007Monday Seminar, EPFL Géraldine Conti MC Parameterization Iterative Polynomial Fitting Method One-go 3D Fitting Method Choice of Regions Results in the Acceptance Region Residual Parameterization Measurement Matching Method Preliminary Results Analytic Parameterization

4. 4 B measurement Campaign (Dec. 2005) B Measurements done… by 3D Hall probes arranged in a fixed configuration on a movable support obtained for the two polarities cover most of LHCb acceptance: Upstream : x = [-1.0, 1.0] m y = [-0.4, 0.4] m Magnet : x = [-2.7, 2.4] m y = [-1.0, 1.0] m Downstream : x = [-2.5, 2.5] m y = [-1.7, 1.7] m (with demagnetization cycle) (fine grid of 8x8x10 cm) May 21, 2007Monday Seminar, EPFL Géraldine Conti

5. 5 Reminder of the Goal 1) TOSCA simulated B field map (grid of 10x10x10 cm) 2) Measurements (They don’t cover all the acceptance) Why is a parameterized B field map needed ? 1) The real measurements can be compared with MC data to model the residuals : Residual parameterization. Provide an accurate determination of the B field map as close as possible to the measurements What is available to perform the parameterizations? Goal 2) The extrapolation of the Residual parameterization can be realized for the regions where no measurement is available. May 21, 2007Monday Seminar, EPFL Géraldine Conti

6. 6 B-Field Map based on MC data (MC Parameterization)

7. 7 Principle of the Method (eg. B y component) : Iterative Multipolynomial Fitting Method (1) Fit B y as a function of y with x,z fixed. B y (y) for x=10cm and z=830cm Fit A i as a function of z with x fixed. Fit B j as a function of x. At the end: N·M·P coefficients to cover the map. May 21, 2007Monday Seminar, EPFL Géraldine Conti N coefficients A i for each slice. M coefficients B j for each slice. P coefficients C k. Pol(4) A0A0 A1A1 A2A2 A3A3 A4A4 B0B0 B1B1 B2B2 B3B3 B4B4 z z z z zx x x x x A i (z) for x=210cm : B j (x) for A 0 : Pol(4)

8. 8 Iterative Multipolynomial Fitting Method (2) May 21, 2007Monday Seminar, EPFL Géraldine Conti The method works well, but not easy to achieve the final precision because of the difficulty of optimizing iteratively. Downstream region is … Magnet region… Residuals ∆(B MC - B analytic ) at z=830cmB x (y) for x=90cm and z=[350,680] cm … reasonably described. … needs Fourier parameterization due to oscillatory pattern.

9. 9 One-go 3D Fitting Method (1) Least square procedure : Determine the c n coefficients and the F n functions, such that : Modified Gram-Schmidt orthogonalisation algorithm to keep the functions F n that significantly reduce S : Basis change to have W n functions which are orthogonal between them.  WnWn FnFn  WnWn FnFn To decide if the N th function is to be kept, the projection of F n on W n is measured and should be greater than a given value to contribute significantly to the reduction of S (the angle should be greater than a given value). See H. Wind, Yellow report, vol.72-21,CERN,1972 ; H. Wind, Yellow report EP/81-12,CERN,1981 May 21, 2007Monday Seminar, EPFL Géraldine Conti is minimal, with F n are not orthogonal between them.

10. 10 Implementation of the method (2) An implementation of the method is available in ROOT (TMultiDimFit Class). Some technical problems have been encountered, but solved thanks to René Brun. Configuration of the optimization : A)Type of Polynomials : Monomials, Legendre, Chebyshev C) Main limits to the number of terms in the parameterization : 1)Max. of terms in the final parameterization 2)Max. of powers for each variable x,y,z to be considered 3)Min.  angle B) Relative error accepted : May 21, 2007Monday Seminar, EPFL Géraldine Conti

11. 11 Compromise between relative precision, small number of regions, small number of terms in the parameterization. Regions Definition for the B field Maps (1) Regions definitions as simple as possible. May 21, 2007Monday Seminar, EPFL Géraldine Conti acceptance anglesz coordinate x and y coordinates Lots of different cuttings tested : Cuttings with respect to B field gradient have been tested too, but the geometry of the cuttings was too complicate.

12. 12 Regions Definition for the B field Maps (2) Cutting depends mainly on only one variable (z) Downstream Magnet 8m Upstream 3m 10m- 0.5m 4m 11.522.4 4.1 4.7 5.16.16.77.38.59 z(m) y(m) May 21, 2007Monday Seminar, EPFL Géraldine Conti The same cutting is chosen for B x,B y and B z components.

13. 13 Regions Definition for the B field Maps (3) However, in the magnet region, [y max -30cm,y max ] values have been fitted separately, but with the same z cuts to improve the fits. Bx for x=1.3m and z=4.8m By for x=1.3m and z=4.8m MC Parameterizations involve 50 to 150 terms, depending on the B field fluctuations. May 21, 2007Monday Seminar, EPFL Géraldine Conti

14. 14 B y map Result (x=0m, y=0m) (1) May 21, 2007Monday Seminar, EPFL Géraldine Conti MC parameterized B y TOSCA simulated B y Very good matching !

15. 15 B y map Result : Relative Precision (2) UpstreamMagnetDownstream Relative precision on B y < 0.001 inside the three regions Relative precision : May 21, 2007Monday Seminar, EPFL Géraldine Conti

16. 16 B x map Result (x=0.4m, y=0.4m) May 21, 2007Monday Seminar, EPFL Géraldine Conti MC parameterized B y TOSCA simulated B y

17. 17 B z map Result (x=0.4m, y=0.4m) May 21, 2007Monday Seminar, EPFL Géraldine Conti MC parameterized B y TOSCA simulated B y

18. 18 Continuity at the region boundaries The relative discrepancy between MC parameterizations at the boundary of two regions is in the same order of the fit precision (~10 -3 ). Discrepancy between B y parameterizations at the 14 boundaries Discrepancy between the two B y parameterizations at the z=610cm boundary Relative Discrepancy : May 21, 2007Monday Seminar, EPFL Géraldine Conti By boundary at z=610cm

19. 19 Out of acceptance regions The Downstream region has been already parameterized. Problems (peaks) are encountered for the Upstream and Magnet regions to find a good parameterization, because we are inside material. May 21, 2007Monday Seminar, EPFL Géraldine Conti UpstreamMagnetDownstream Parameterization needed, but with a less acurate precision.

20. 20 Matching with the measurements (Residual Parameterization)

21. 21 Clean-up of the measurements Clean-up of the measurements done in the three regions (started by Adlene Hicheur). Clean-up Some non-physical behaviours observed in the measurements, which can perturb the parameterizations. May 21, 2007Monday Seminar, EPFL Géraldine Conti

22. 22 Matching Method Parameterize the residuals (B measurements - B analytic values) : Analytic Parameterization = MC Parameterization + Residual Parameterization Extrapolate the Residual Parameterization to regions where no measurement is available. Calculate the B values with the MC Parameterizations at the same measurement coordinates (x,y,z). MeasurementsAnalytic valuesResiduals May 21, 2007Monday Seminar, EPFL Géraldine Conti

23. 23 B y Residual Results (1) May 21, 2007Monday Seminar, EPFL Géraldine Conti x=4cm, y=4cm, Magnetx=4cm, y=4cm, Upstream Interpolation MC Parameterization Analytic Parameterization Measurements (MC + Residual Parameterization) Non negligible corrections for the most important B component (B y ) near the centre (x~0cm and y~cm) !

24. 24 B y Residual Results (2) May 21, 2007Monday Seminar, EPFL Géraldine Conti x=100cm, y=4cm, Magnet x=204cm, y=164cm x=140cm, y=140cm Interpolation MC Parameterization Analytic Parameterization Measurements (MC + Residual Parameterization) The most important discrepancies between MC data and measurements are found for big x and/or y values

25. 25 Reverse Polarisation B y Residuals (3) May 21, 2007Monday Seminar, EPFL Géraldine Conti x=100cm, y=4cm, Magnet Interpolation MC Parameterization Analytic Parameterization -(Reverse Polarisation Measurements) (MC + Residual Parameterization) The values obtained with the Analytic Parameterization (found for positive polarisation measurements) seems to match well with the reverse polarisation measurements. More comparisons still have to be done…

26. 26 Analytic Field Service

27. 27 Analytic Field Service Speed tests forseen with tracks  Several scenarios : May 21, 2007Monday Seminar, EPFL Géraldine Conti It has been implemented in Det/Magnet, but is not available in CVS yet. However, according to preliminary tests, the speed of the B value access seems to be an issue… 1) Analytic Parameterization is faster or of the order of time of the interpolation method :  Best scenario : faster access and more accurate B values. 2) Analytic Parameterization is slower than the interpolation :  Only the Residual Parameterization could be used with the interpolation method to give more accurate B values.  A new file with B values used by the interpolation could be generated with the help of the Analytic Parameterization.

28. 28 Conclusions and Perspectives Residual parameterizations for B x and B z. Extrapolate the Residual parameterizations to regions where no measurements are available. Parameterization based on MC simulation data has been succesfully performed in the acceptance region by the One-go 3D Fitting Method and reaches a rel. Prec. < 10 -3 for the B y component in all the 3 regions. MC parameterization of the « Out of acceptance » Magnet and Upstream regions. On-going / to do : May 21, 2007Monday Seminar, EPFL Géraldine Conti Speed tests of the Analytic Service with tracks. Parameterization of the B y residuals gives the expected more accurate B values with respect to the measurements.