Offline Data Analysis Software Steve Snow, Paul S Miyagawa (University of Manchester) John Hart (Rutherford Appleton Laboratory) Objectives + requirements Field model + simulation Pre-fit corrections Fit results Post-fit corrections Conclusions

2 November 20052nd ATLAS Magnetic Field Workshop2/12 Objectives Momentum scale will be dominant uncertainty in W mass measurement. Momentum accuracy depends on ∫ r(r max - r)B z dr : –Field at intermediate radii, as measured by the sagitta, is most important. In reality, the limit on silicon alignment under ideal conditions will be 1 μm. At typical sagitta of ~1 mm, we target an accuracy of 0.05% to ensure that B-field measurement is not the limiting factor on momentum accuracy. Field B may be described by a scalar potential  satisfying Laplace’s equation,  2  = 0 Sufficient to measure B on the surface of a cylinder (including the ends) surrounding the tracker

2 November 20052nd ATLAS Magnetic Field Workshop3/12 Mapper Survey Requirements Reinvestigated survey requirements of mapper machine relative to Inner Detector. 1 mm survey error in x (or y) significant for high η tracks. 1 mm survey error in z significant for endcap tracks. 0.1 mrad rotation around x (or y) axis significant for high η tracks.

2 November 20052nd ATLAS Magnetic Field Workshop4/12 Field Model Basis of model is field due to coil of nominal dimensions. Added in field due to magnetised iron (4% of total field). Distribution of iron magnetisation taken from simple FEA model using FlexPDE. Model is symmetric in  and even in z. Field can be displaced and rotated relative to mapping machine.

2 November 20052nd ATLAS Magnetic Field Workshop5/12 Field Mapping Simulation Field model sampled on a grid of 90 z-positions × 8  -angles (defined by encoder values of machine). 4 calibration points (2 near centre, 2 near one end) visited after every 25 measurements. Used current design of machine to determine positions of 48 Hall + 1 NMR probes. At each map point, wrote the following data to file: –Time stamp –Solenoid current –Field modulus measured by 5 NMR probes (1 moving + 4 fixed) –3 field components measured by 48 Hall probes

2 November 20052nd ATLAS Magnetic Field Workshop6/12 Simulated Errors Six cumulative levels of error added to simulated data: 1.No errors. 2.Random errors in each measurement of: Solenoid current, 1 A NMR B-field modulus, 0.1 G Each component of Hall probe B-field, 3 G 3.Drifts by random walk process in each measurement of: Solenoid current, 1 A Each component of Hall probe B-field, 0.1 G 4.Random calibration scale and alignment errors, which are constant for each run: Each component of Hall probe B-field, 0.05% scale Rotation about 3 axes of each Hall probe triplet, 0.1 mrad 5.Symmetry axis of field model displaced and rotated relative to the mapper machine axis. These misalignments are assumed to be measured perfectly by the surveyors. 6.Each Hall probe triplet has a systematic rotation of 1 mrad about the axis which mixes the B r and B z components.

2 November 20052nd ATLAS Magnetic Field Workshop7/12 Simulation of Raw Data Field calculated from solenoid of expected dimensions Magnetisation due to magnetic material from outside Inner Detector Random walk with time for solenoid current and Hall probe measurements Random errors for each measurement

2 November 20052nd ATLAS Magnetic Field Workshop8/12 Correction for Current Drift Average B-field of 4 NMR probes used to calculate “actual” solenoid current. Scale all measurements to a reference current (7600 A). Effect of drift in current removed Calibration capable of coping with any sort of drift.

2 November 20052nd ATLAS Magnetic Field Workshop9/12 Correction for Hall Probe Drift Mapping machine regularly returns to fixed calibration positions –Near coil centre to calibrate B z –Near coil end for B r –No special calibration of B  Each channel is calibrated to a reference time (beginning of run) Offsets from calibration points used to determine offsets for measurements between calibrations

2 November 20052nd ATLAS Magnetic Field Workshop10/12 Geometrical Fit Sum of simple fields known to obey Maxwell’s equations –Long-thin coil (5 mm longer, 5 mm thinner than nominal) –Short-fat coil (5 mm shorter, 5 mm fatter) –Four terms of Fourier-Bessel series (for magnetisation) Use Minuit to minimise a  2 fit between the fitting function and simulated data using 12 free parameters: –1, mixing ratio of long-thin and short-fat solenoids –2, scaling factors for length and field of the combined solenoids –4, coefficients of Fourier-Bessel series –3, offset of field coordinates from mapper coordinates –2, angles between field coordinates and mapper coordinates Fit dominated by large number of Hall probe points. –Could introduce one overall scale factor for all Hall probes as free parameter and take true scale from NMR probes

2 November 20052nd ATLAS Magnetic Field Workshop11/12 Fourier-Bessel Fit General fit able to describe any field obeying Maxwell’s equations. Uses only the field measurements on the surface of a bounding cylinder, including the ends. Parameterisation based on ALEPH (Stephen Thorn) and proceeds in three stages: 1.B z on the cylindrical surface is fitted as Fourier series, giving terms with φ variation of form cos(nφ+α), with radial variation I n (κr) (modified Bessel function). 2.B z meas – B z (1) on the cylinder ends is fitted as a series of Bessel functions, J n (λ j r) where the λ j are chosen so the terms vanish for r=r cyl. The z-dependence is of form cosh(μz) or sinh(μz). 3.The multipole terms are calculated from the measurements of B r on the cylindrical surface, averaged over z, after subtraction of the contribution to B r from the terms above. (The only relevant terms in B z are those that are odd in z.)

2 November 20052nd ATLAS Magnetic Field Workshop12/12 Fourier-Bessel Fit Comments: The fit is very quick since the coefficients are calculated directly as sums over the measured field except for a simple linear fit to the J n coefficients. A typical fit could have up to ~500 terms in the first category taking into account combinations of 25 terms in z, 5 terms in φ and their respective phases, but most of the φ-dependent terms are too small to be measurable and are set to zero. In the second category, up to ~120 terms could be calculated but again most φ-dependent terms are negligible. The I n and cosh or sinh terms decrease more or less exponentially with increasing distance from the outer cylinder or cylinder ends. This means that higher order terms contribute very little over most of the field volume and have little effect on the measured momentum. A poor fit indicates measurement errors rather than an incorrect model.

2 November 20052nd ATLAS Magnetic Field Workshop13/12 Fit Quality with Ideal Performance Quality of fit measured by comparing track sagitta in field model with track sagitta in fitted field: –Used 260 track directions equally spaced in η and . Results shown below are for no errors added (error level 1): –Sagitta error not quite zero due to interpolation errors and finite number of terms. –Error is negligible compared to target level of 5×10 -4.

2 November 20052nd ATLAS Magnetic Field Workshop14/12 Fit Quality with Expected Performance Expected mapper performance corresponds to error level 5. Both fits accurate within target level of 5×10 -4.

2 November 20052nd ATLAS Magnetic Field Workshop15/12 Tables of Results Tables give statistics of Bz error over points and sagitta error over 260 trajectories. Correction procedure does not completely eliminate effects of drifts, so some runs (eg, 24) have normalisation errors of ~ 1× Geometrical fit relatively uninfluenced until error level 6 (systematic probe tilts). Geometrical fit Error level Fourier-Bessel fit * Fit results including probe normalisation and alignment corrections are shown with an asterisk by the file name.

2 November 20052nd ATLAS Magnetic Field Workshop16/12 Probe Normalisation and Alignment Corrections Field should be very uniform near z=0, with Bz at maximum and Br=0. We take advantage of this information to perform two calibrations: 1.Inter-calibration of Bz scale between four Hall probes at common radius: Find Bz value at z=0 using quadratic fit in range |z| < 0.25 m. Normalise each probe to average scale of four at that radius. 2.Removal of z-r rotation: 1.MORE TEXT + PLOTS FOR THIS PAGE!

2 November 20052nd ATLAS Magnetic Field Workshop17/12 Conclusions Both fits have the necessary technical accuracy. Both fits give results within target of 5× Fourier-Bessel fit more sensitive to random measurement errors: –Due to more free parameters. –F-B fit designed for any solenoid-like field, whereas geometrical fit is specifically for the ATLAS solenoid. Probe normalisation + alignment correction helps correct for systematic tilts of Hall probes. With a few refinements, the code will be ready for release.