3.The residual B r on the cylindrical surface is represented by multipole terms The results from the combined geometrical + general Maxwell fit show that the residuals between the measured and fit B-field values are within 4 Gauss rms for each component. All geometrical fit parameters (length, position, angles) are consistent with expected results. The relative sagitta error is within the target. Implementation in ATLAS software The Inner Detector tracking algorithms typically require the magnetic field at hundreds of different points per event. While exact results could be obtained by performing a Biot-Savart calculation on the fly, this is impracticably slow (~2+ hours for points). Therefore, a table of field map values on a grid is used with interpolation between grid points. Software Representation of the ATLAS Solenoid Magnetic Field Stephen Snow, Paul S Miyagawa (The University of Manchester); John C Hart (Rutherford Appleton Laboratory) Many thanks to the CERN mapping project team: Martin Aleksa, Felix Bergsma, Laurent Chevalier, Pierre-Ange Giudici, Antoine Kehrli, Marcello Losasso, Xavier Pons, Heidi Sandaker The ATLAS experiment is one of four experiments based at the Large Hadron Collider at CERN. The ATLAS Inner Detector records particle tracks near the interaction point to measure their curvature. The ATLAS solenoid encloses the Inner Detector and produces an axial magnetic field of ~2 Tesla in this region. The solenoid magnetic field was measured using the field mapping machine. The mapping machine consisted of a carriage which scanned in the axial (z) direction, four Corrections to mapper data A number of corrections were applied to the field map data to remove any effects induced by the mapping machine. These include geometrical effects, such as tilting of the carriage or individual probes; relative and absolute probe normalisations; and magnetisation of machine components. Carriage tilts B x and B y on the central axis were calculated from low-radii probes by averaging over φ. Plotting these versus the z-position of the arm shows that each arm follows the same pattern, but out of phase with each other. However, plotting against the z-position of the carriage shows a Field fit to within 4 Gauss The field map data was fit using field models satisfying Maxwell’s equations. As there were no currents or magnetic materials in the region of interest (Inner Detector volume), Maxwell’s equations reduce to the form   B = 0,   B = 0. The fit used Minuit to minimise The fit quality was defined as the relative sagitta error δS/S where (c r and c z are direction cosines) Magnetic components The mapping machine was built as much as possible using non-magnetic materials. Despite these efforts, some spikes could Probe alignments The probe tilt angles A φr and A φz mix the B- field components such that the measured. The tilts for each probe can be calculated by integrating (derived from   B = 0) over one revolution in φ at fixed z. Evaluating at several z values gives a system of equa- tions which can be solved with a least squares fit. The alignment angle A zr and B r normalisa- tion for each probe are found by applying Gauss’s theorem (consequence of   B = 0) to a cylinder centered on the central axis with radius equal to the probe radius and thickness corresponding to the z-spacing between measurements: where Probe normalisations For each probe the general Maxwell fit was applied to the measurements by that Applying this to a series of such cylinders gives a system of equations which can be solved with a least squares fit. 96% of the field is produced directly by the solenoid current. The geometrical fit models this field by integrating the Biot- Savart law (dB = μ 0 I dl  r / 4πr 3 ) over the current path. The conductor geometry was determined from engineering surveys of the solenoid taken as it was built. There were 7 free parameters: 2 length scales, 3 positions, 2 tilt angles. Additionally, 4% of the field is induced by magnetised iron outside the solenoid (Tile Calorimeter, girders, shielding discs, etc). This is parametrised using Fourier-Bessel terms: There are two features for which there is inadequate information to model them accu- rately in the geometrical fit: the coil winding density is modelled as constant, whereas reality is that the density varies; the coil cross section is modelled as circular, whereas it is actually elliptical due to gravity. These features are instead modelled using the general Maxwell fit, which is a solution of Laplace’s equation  2 B = 0 (derived by combining Maxwell’s equations). This means that measurements on a bounding surface (in this case, a cylinder) determine the field within the enclosed volume. The fit proceeds in three stages: 1.B z on the cylindrical surface is represented by Fourier-Bessel terms 2.The residual B z on the cylinder ends is represented by hyperbolic terms interpolation order interpolation time (arbitrary units) z, r, φ φ, r, z z 2, r, φ φ, r, z z 2, r 2, φ φ, r 2, z BzBz Grid format and spacing A cylindrical grid with 113 z  25 r  21 φ = points was chosen as the field is nearly φ- symmetric. (A Cartesian grid would have required 130  130  111 = points to achieve the same precision.) The grid density was increased near spe- cial features of the solenoid (e.g., BrBr BφBφ propeller arms which rotated in φ, and 12 Hall probes per arm mounted at varying radii. The field mapping targets are motivated by the measurement of the W-boson mass, which is determined using the momentum of high-p T muons. As the track momentum accuracy depends on the magnetic field as ∫ r(r max - r)B z dr, the field mapping target is 0.05% accuracy on the track sagitta. coherent pattern for all arms. This is inter- preted as evidence for the entire carriage tilting as it travels in z along the rails. The degree of tilt can be calculated by integrating the following equation (derived from   B = 0) to find the expected value of B x or B y. be seen in the data as the mapper arms scanned in φ which were attributable to components on the mapping machine. To remove these perturba- tions, a dipole was sub- tracted from the data. probe. The fit was then used to calculate the field at the origin. The average over all probes defined the central field value. The relative probe normalisations were deter- mined by normalising each probe to this central value. welds between coil sections, ends of solenoid, return conductor) where the field would vary rapidly. The grid points were chosen such that field residuals due to interpolation were less than 3 Gauss for linear interpolation. Linear interpolation Quadratic interpolation in z, r Order of interpolation Different orders of the interpolation (e.g., zrφ vs φrz) were tested for both linear and quadratic interpolation to check whether they would affect the calculated values. Apart from rounding error, the order of interpolation was found not to affect the precision. Linear vs quadratic interpolation The accuracy and timing performance of different methods for interpolating the field between grid points were studied. The accuracy was determined by the difference between the interpolated field value and the fit field value. The baseline method was linear interpolation in all three coordinates. Using quadratic interpolation in either the z- or r-direction (or both) yielded much more accurate values, but was substantially slower. (For quadratic interpolation, one quadratic was fit with the cell on the left, another with the cell on the right, and the final value was a proportional mixture of the two quadratic values.) As speed was the primary concern and the accuracy of the linear interpolation was felt to be adequate, linear interpolation in all coordinates was the method chosen for implementation.