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Calibration of the chaos spectrometer for small scattering angles Jamieson, Blair Alex 1999

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C A L I B R A T I O N O F T H E C H A O S S P E C T R O M E T E R F O R S M A L L S C A T T E R I N G A N G L E S B y Blai r A lex Jamieson B . A . S c . , The University of Br i t i sh Columbia, 1997 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S D E P A R T M E N T O F P H Y S I C S A N D A S T R O N O M Y We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A A p r i l 1999 © Blai r A lex Jamieson, 1999 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Br i t i sh Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics and Astronomy The University of Br i t i sh Columbia 6224 Agricul tura l Road Vancouver, Canada V 6 T 1Z1 Date: Abstract For measurements of pion-nucleon scattering in the Coulomb-Nuclear Interference (CNI) region, it is implicit that we are looking at small scattering angles. A t small scattering angles the in-plane (x,y coordinate) scattering angle is not the true scattering angle. Since the Canadian High Acceptance Orbit Spectrometer ( C H A O S ) has a vertical extent of plus or minus seven degrees, an in-plane scattering angle of zero degrees could, i n three dimensions, be seven degrees. The C H A O S detector's out-of-plane (z coordinate) information comes from three wire chambers: wire chamber one ( W C l ) cathode strips and anode wires, wire chamber two (WC2) cathode strips and anode wires, and wire chamber four's (WC4) two resistive wires. The out-of-plane information from W C 4 requires a fringe field correction, because it lies outside the uniform magnetic field of C H A O S . The fringe field of C H A O S acts like a lens, in that it has a slight focusing effect on charged particles. The fringe field correction moves the W C 4 (x,y,z) data point so that i t is on the linear object ray. In this paper I wi l l present the model for the fringe field correction. I wi l l explain how the three dimensional scattering angle resolution was determined, and as the final test of the scattering angle I wi l l present the cross-section for fiC scattering, obtained using C H A O S data. i i Table of Contents Abstract ii List of Tables vi List of Figures vii Acknowledgments xiv 1 Introduction to Pions 1 1.1 The P ion Beam, at T R I U M F 3 1.2 The C H A O S Spectrometer 6 1.2.1 The Proportional Wire Chambers W C 1 and W C 2 6 1.2.2 The Drift Wire Chamber W C 3 9 1.2.3 The Vector Wi re Chamber W C 4 10 1.2.4 The C H A O S Fast Trigger Blocks 12 1.2.5 The ITLL Stack 13 1.3 Coulomb-Nuclear Interference Region Measurements 18 2 Using C H A O S at Small Scattering Angles 22 2.1 Fu l l Scattering Angle Versus In-Plane Scattering Angle 23 2.2 Determining the Out-of-Plane Scattering Angle 26 2.2.1 The C H A O S Fringe Fie ld 28 2.2.2 Fringe Fie ld Correction Model 30 2.2.3 G E A N T Simulation Used to F i n d Model Parameters 34 i i i 2.2.4 Determining Fringe Fie ld Correction Model Parameters 34 2.3 The Interaction Vertex at Small Scattering Angles 37 3 Angular and Coordinate Resolution 40 3.1 Resolving a Horizontal R o d 40 3.1.1 C H A O S Horizontal R o d Resolution 42 3.1.2 G E A N T Simulation of Spectrometer Resolution 47 3.2 Angular Resolution 49 3.2.1 G E A N T Simulation of Angular Resolution 51 3.2.2 Importance of Angular Resolution in Cross Section Determination 53 4 Cross Section Considerations 56 4.1 Particle Identification 57 4.2 El iminat ion of P ion Decay Events 59 4.2.1 Use of Different Momentum Calculations 62 4.2.2 Use of the TT/J, Stack 64 4.2.3 Use of Track Projections 70 4.3 Angular Dependence of Solid Angle 74 4.4 Effects of Binning Data 76 4.5 Scattering Angle Offset Effects 77 5 The LLC Cross Section in the CNI Region 79 5.1 Theoretical Shape of Cross Section 79 5.2 The LLC Cross Section Obtained from C H A O S Data 83 6 Conclusion 90 6.1 Determining Fringe Fie ld Correction Parameters wi th C H A O S Data . . 91 6.2 Use of the Coordinates of the Projection to the Target Plane 92 iv Bibliography 94 A Trace Program Used to Generate Tracks Through C H A O S 96 A . l C H A O S Materials Fi le 96 A . 2 Trace Code 99 B Root Macro Used to Find Residual in Projected Z at WC4 131 v List of Tables 2.1 Definition of variables in Bethe-Bloch equation 30 2.2 Fringe field correction parameters determined from G E A N T simulations. The G E A N T simulation used is described in the following section. The parameters have units such that za, and z'a are in millimeters. The pa-rameters da and ea have units of (Mev/Tesla)2 32 3.3 Resolution of picket fence bars for run 6257 at scattering angles greater than 30 degrees 44 3.4 Resolution of picket fence bars for run 6258 at scattering angles greater than 30 degrees 45 3.5 Resolution of picket fence bars for run 6257 at scattering angles in the range seven to thirty degrees 47 3.6 Angular resolutions found at different energies 51 4.7 Fringe field correction parameters to put wire chamber hits on the image line to the stack. Parameters determined from trace simulations 67 5.8 Fourier-Bessel series coefficients for 1 2 C nuclear charge density [18]. . . . 83 6.9 Resolution of picket fence bars for run 6257 at small scattering angles (less than 30 degrees), and at large scattering angles (greater than 30 degrees.) 93 vi L i s t o f F i g u r e s 1.1 A n Example of Strong Force Interaction p n - > n p 2 1.2 Basic operation of a cyclotron showing the path of an ion. The ions travel in circular trajectories with radius based on the momentum of the ions. 4 1.3 The M13 beam line layout [4] 5 1.4 Isometric view of the C H A O S Spectrometer. For clarity, the top magnet pole, several the C F T blocks, and parts of the wire chambers are removed. 7 1.5 In-plane scattering of a pion reconstructed from C H A O S wire chamber hits. B o t h x and y axes are i n millimeters. The circle of dots i n the center of the figure represent the wires in W C 1 . The two circles of dots around W C 1 are the wires in W C 2 and W C 3 respectively. The circle of radial dashes represent the W C 4 anode wires and resistive wires. The circular arrangement of blocks represent the C F T blocks. The in-plane (x,y) C H A O S coordinate system is also shown in this figure 8 1.6 A cell of W C 4 showing anodes (A) , guard wires (G) and resistive wires (R) . A l l units in this figure are in millimeters 11 1.7 Resistive wire of length L wi th charge injected at a distance D from the centre of the wire 12 1.8 Sketch of the n/i stack showing five layers of scintillators and four layers of a luminium absorbers 14 1.9 Simulation of pion decay kinematics for 55 M e V pions. The x axis is the scattering angle in degrees and the y axis is the momentum in M e v / c . . 16 v i i 1.10 Illustration from raw C H A O S data of why the stack is important for elim-inating pion decay events. The scatterplot shown at top is from a 55 M e V Carbon target run, and the lower scatterplot is the calculated kinemat-ics. Bo th scatterplots have scattering angle in degrees on the y axis, and momentum in M e V / c on the x axis 17 1.11 Mandelstam plot of v versus t. The hatched regions show where phys-ical measurements for s-channel and u-channel scattering are made. 7rp scattering is an s-channel process 19 2.12 Definition of the in-plane and full scattering angles. In C H A O S x is defined as the direction the beam comes in, y is perpendicular to x, and z is the out-of-plane axis. Note that the scattering vector shown in this figure is the difference between the beam tangent vector, and scattered track tangent vector taken at the origin. The y axis is actually drawn as -y to make drawing the vector components easier 24 2.13 Angle definitions in the C H A O S coordinate system, and in the physical coordinate system. The y axis is drawn as -y to make vector components easier to draw 25 2.14 Geometry used in finding pathlengths to W C 1 and W C 2 . Note that the wire chambers are not drawn to scale 27 2.15 Vert ical track profiles show the focusing effect of the fringe field. Note the change in scale from vertical to horizontal. The uppermost track has a solid straight line overlaid, illustrating how far off the object line the W C 4 hits are 29 v i i i 2.16 Particle tracks at small scattering angles in C H A O S . The vertical profile for each track is shown below each in-plane track. The in-plane tracks shown at top axe display events described in Section 1.2. In the vertical profiles, the points represent hits in the wire chambers in pathlength versus height. W C 4 has four points; the two solid W C 4 points below the track line are the actual hits as recorded by the wire chamber, and the two open W C 4 points on the track line are the points corrected using the fringe field model 33 2.17 Az residuals for W C 4 inner and outer resistive wires. Residuals were generated from trace data which do not account for multiple scattering, detector resolution, or pion decay. The upper four plots are residuals for W C 4 inner, and the bottom four plots are for W C 4 outer. The y axes are all Az in millimeters. The x axes going clockwise from the upper left (for the top four plots) are: the uncorrected z, the track slope, the product of the track slope wi th the uncorrected z, and the product of the track slope wi th the uncorrected z squared. The upper two rows are for the inner resistive wire, and the bottom two rows are for the outer resistive wire. . 36 2.18 Az residuals for W C 4 inner and outer resistive wires, generated from G E A N T data which included multiple scattering and detector resolution. P ion decays were disabled in the G E A N T run 37 2.19 Diagram showing how the in-plane scattering angle is found 38 3.20 Sketch of copper picket fence shows location of horizontal pickets, labelled A through F . The figure is not drawn to scale 41 ix 3.21 Picket fence reconstruction from: run 6257 wi th pickets A , B , and C clearly identified, and run 6258 with pickets C , D , E , and F . Refer to text for more details 43 3.22 Picket fence reconstructed using small scattering angle data from run 6257, wi th pickets A , B , and C clearly identified 46 3.23 All But One fits used to get an estimate of the resolution in each wire chamber 48 3.24 Comparison of horizontal rod z_proj using simulations (top) and real data (bottom). Result is shown for ^ " rods 50 3.25 A dotplot of AO versus 63d showing that the angular resolution is constant over the angles generated in G E A N T 52 3.26 Correction factors (econv) to the ir^p cross sections for detector angular resolution of one degree (a). Results are shown for 40 M e V pions, and Pn = Pjx- The plot in the lower left is a ratio of the theoretical cross section to the theoretical cross section convolved wi th a Gaussian of 1° standard deviation 55 4.27 Time of flight spectra at different energies. T O F shows its usefulness for identifying 7r, / i , and e at energies above 20 M e V . The humps on the right sides of the pion peaks, in the spectra in the second and third rows, are due to pions which decay in flight from the production target to the C H A O S finger counter 58 4.28 T O F on the y axis versus AE on the x axis used for identifying TT, p, and e at energies below 20 M e V . The dotplots shown are from 17 M e V pion kinetic energy runs 60 x 4.29 Momentum on the x axis versus T O F on the y axis used for separating 7r, and \x for 55 M e V incident pions. Since the track momentum is being used, the events shown are required to have scattered, and to have made a trigger in C H A O S . The dots at momentum higher than the channel momentum (135 M e V / c ) must be pion decays 61 4.30 Histogram used in eliminating some pion decay events. When the differ-ence in two different momentum calculations is much bigger than zero, it is tagged as a decay event. The histogram is from a Carbon target run wi th 55 M e V incident pions 63 4.31 Histogram showing the range of pions and muons in the TV/J stack. T y p i -cally we try to stop pions in layer two or three, and muons in layer four or five. The histograms going clockwise from the upper left corner are of al l events, events tagged as pions in T O F , events tagged as electrons in T O F , and events tagged as muons in T O F 65 4.32 Track profiles drawn out to the stack generated using trace. The straight line shows the image line to the stack 66 4.33 In-plane tracks generated using trace. Note that the scales of the axes are different so that the angles of track slopes to the stack are exaggerated. The heavy lines show the difference between the in-plane slope of the track at W C 4 , and the in-plane slope of the track at the stack 68 4.34 Dotplot of the in-plane track angle at W C 4 to the in-plane track angle at the stack. Angles were found using tracks wi th random starting positions, angles, and momenta generated using trace 69 4.35 Dotplot of the projected height at the stack versus the actual height at the stack. The plot at the left is for electrons, and the plot at the right is for pions 70 x i 4.36 Out-of-plane angle residuals generated using 40 M e V incident pion G E A N T runs wi th horizontal rod targets. Results with pion decays turned on and pion decays turned off are shown 71 4.37 Dotplot of A verses 02d used to decide if interaction occurred in target is also used to eliminate decay events. This dotplot came from a C H A O S 55 M e V incident pion run, with a 1 2 C target as indicated by the box. Dots falling outside of the box are from events that scattered off one of the wire chambers, or decayed somewhere other than at the target 72 4.38 Dotplot of A on the x axis verses #2d on the y axis for two different running conditions. The dotplot on the left came from a C H A O S run wi th 33 M e V pions incident on a plastic sewer pipe. The dotplot on the right is at the same energy but with no target in C H A O S . Note that this data has been cut on events that trigger the stack, meaning that we won't get any dots at scattering angles above about 30 degrees 73 4.39 Mapping of C H A O S angles to physical angles shows how different the solid angle is at small scattering angles 75 4.40 Effect on cross section measurement of binning data 76 4.41 Correction to cross section measurement for binning of data. Correction is for a bin width of two degrees. Correction is for LIC —> LLC scattering at an incident muon momentum of 135 M e V / c 78 5.42 Carbon nuclear charge density 81 5.43 Carbon form factor squared 82 x i i 5.44 jiC —> / i C cross section for 66 M e V incident muons calculated using the old solid angle, and without binning correction is shown in the left half of the figure. The cross section using the new solid angle, and wi th the correction for binning is shown in the left half of the figure 85 5.45 Ratio of the solid angle calculated using the old C H A O S coordinates to the solid angle calculated using the new physical coordinates 86 5.46 Comparison of solid angle correction using theoretical width in fad, and solid angle correction using a strip in fad 87 5.47 LIC cross section wi th positive and negative three degree in-plane scattering angle offsets applied. Results are from C H A O S data runs wi th 66 M e V muons incident on a Carbon target 89 6.48 Simulated resolution of z\ and z'0 found using W C 1 , W C 2 and a fibre scintillator before W C 3 91 6.49 Picket fence reconstruction using small scattering angle data from run 6257, where the track reconstruction includes the use of the incoming beam z_proj 93 x i i i Acknowledgment s Without the support of others, the past year and a half I have been working on this project would not have been possible. In particular, I would like to thank my supervisor, Dr . Greg Smith, for providing me with an interesting and challenging project to work on. His ideas always pointed out a new way of looking at a problem. Thanks are due to Dr . K e l v i n Ray wood for his many ideas on how to improve my algorithms and on how to proceed wi th my project. I would also like to thank Dr . Roman Tacik for his experienced and insightful thoughts on various aspects of my project. Outside of work I would like to thank my wife, Natalie, for her understanding and support. Finally, I would like to thank my family for their encouragement and financial help. x iv Chapter 1 Introduction to Pions For over fifty years, people have been studying pions, the exchange particle responsible for the strong force. In 1935 Hideki Yukawa postulated that a particle wi th a mass of ~ 200 M e V , passed between nucleons, was what caused them to be attracted to one another [1]. It took twelve years for the pion to be correctly identified as the Yukawa particle. The pion was found by Lattes et al in 1947 from cosmic ray tracks on photographic plates [2]. From the Particle Data Book, the pion has a mass m , = 139 .5669MeV/c 2 . B y Heisenberg's uncertainty principle [3] this mass corresponds to an interaction range of he d « 5 t - c « - ^ « 1 . 4 / m . (1.1) mcz This picture of the nucleus sees it held together by pions passed between nucleons, making the distance between nucleons approximately 1.4 fm. Yukawa's simple picture has been replaced by Quantum Chromodynamics ( Q C D ) . In Q C D the pions are made of quark antiquark pairs (TT+ = ud, u~ = ud). The lightest pair of quarks are called up (u) wi th a charge + | and down (d) wi th a charge — | . Two heavier pairs of quarks have been observed; the first pair are charm (c), and strange (s); the second pair are top (t) and bottom (b). In direct analogy wi th Yukawa's picture, the force between nucleons can be understood as quarks being shuffled between nucleons. A n example of a proton (p) and neutron (n) interacting v ia the strong force is shown in Figure 1.1. The reason for the name Chromodynamics comes from the colour quantum number of the quarks. Quarks that make up Hadrons (protons, neutrons, etc.) come in colourless 1 Chapter 1. Introduction to Pions 2 Figure 1.1: A n Example of Strong Force Interaction p n -> n p. combinations of the three quark colours. The colour quantum number explains the following problem. B y Pauli 's exclusion principle, only one Fermion can be in a particular state at one time. Physically we observe that: a proton has two up quarks and a down quark, a neutron has one down quark and two up quarks, and a A + + - B a r y o n has three up quarks. Quarks being spin | , and thus Fermions, must have a quantum number (red, green, or blue) differentiating quarks that are otherwise in the same state. Many combinations of quarks are not observed in nature. For example a combination of two up quarks (uu) wi th a charge of + | is not observed. Requiring only colourless combinations of quarks solves this problem. Three kinds of pions exist (IT+ , 7 r ° , and 7r~) , and are found to conserve isospin (T, Tz) in nuclear reactions. The isospin of pions is T = l ; the isospin of 7 r + is Tz=+1, the isospin of 7T° is Tz=0, and the isospin of 7r~ is Tz=—1. The conservation of isospin in the strong force interaction in Figure 1.1 can be verified knowing that a proton has Tz — + | , and a neutron has Tz = —\. The fact that protons and neutrons have nearly the same mass, and come from the same isospin quantum number T = | indicates that they come from the same family of particle (nucleons). A consequence of the non conservation of isospin Chapter 1. Introduction to Pions 3 in electromagnetic interactions is a small difference in the pion masses (m^± = 139.5669 MeV/c2, and m,» = 134.9745 MeV/c2.) Pions have been used as probes into nuclear structure for many years. More recently attention has been focussed on using the pion to determine parameters in Q C D . In the followings sections I wi l l discuss how pions are created at the Tr i - University Meson Facil i ty ( T R I U M F ) , introduce the Canadian High Acceptance Orbi t Spectrometer ( C H A O S ) , and motivate the need for a carefully calibrated measurement of small angle scattering. 1.1 The Pion Beam at T R I U M F The primary beam created at T R I U M F is a 520 M e V proton beam. A t the start H~ ions are injected into the center of the cyclotron. The T R I U M F cyclotron has a large dipole magnet, between the poles of which ions are accelerated by a radio frequency (RF) electric field oscillating at 23 M H z . The basic operation of a simple cyclotron is illustrated in Figure 1.2. Classically the radius of the ion orbit depends only on its velocity, but as the particle becomes relativistic there is a need for a higher magnetic field as the particle mass increases (R = jg). One problem with increasing the magnetic field as the radius of the orbit increases is that the curvature of the magnetic field lines causes a defocusing of the H~ ions. A t T R I U M F the magnetic field is broken up into six crescent shaped sectors. Each sector has a high magnetic field, and between each sector there is no magnetic field. The alternating sectors provide focusing of the beam with carefully designed azimuthal components of the field in between sectors. Cyclotrons wi th this type of design are called azimuthally varying field ( A V F ) cyclotrons. As the H~ accelerates, its radius in the cyclotron grows, in proportion to its speed. When the ions get to the outer edge of the Chapter 1. Introduction to Pions 4 X X X / X X \ X X X X Magnetic Field X X X X X X Figure 1.2: Basic operation of a cyclotron showing the path of an ion. The ions travel in circular trajectories wi th radius based on the momentum of the ions. cyclotron they are extracted by sending them through a thin piece of foil. The foil strips the electrons off the H~ ions turning them into bare protons. Since the charge on the proton is positive, the polarity of the circle it was following in the cyclotron changes. The protons are then collected from the edge of the cyclotron and fed along an evacuated beam pipe. Pions are produced at T R I U M F by colliding the typically 140 pA current of 520 M e V protons wi th a target. Since the primary beam has a higher energy than the mass of a pion, pions are ejected from the target. The pion production target is usually made of l2C or 9Be. For example pions are produced by bombarding 12C wi th protons in the following reactions: P + 1 2 C - p + 12B + 7T+ P + l2C p + 12N + TT" Chapter 1. Introduction to Pions 5 h o r i z o n t a l a v e r t i c a l jaws target ^ IATI vacuum valvft beam blocker horizontal sjit absorb er , .vert!COl slit d iagnos t ics ' j v e r t i c a l s l i t horizontal s l i t J X F 3 f i na l foe 0 1 2 3 feat Figure 1.3: The M13 beam line layout [4]. In the Coulomb-Nuclear Interference (CNI) experiment being done wi th C H A O S , the M13 channel at T R I U M F is being used. The M13 channel is a 20 M e V / c to 130 M e V / c pion and muon channel, which is at 135 degrees from the primary proton beam. Muons in the channel are produced by pions that are stopped near the surface of the production target, and by pions that decay to muons in flight along the channel. Pions decay into muons and muon neutrinos (TT+ —* p+ + and n~ —> p~ + The M13 channel consists of two 60 degree beam bends, seven quadrupole magnets used for focusing, a slit for vertical acceptance, and a slit for horizontal acceptance [4]. The beam line elements in M13 are shown in Figure 1.3. The beam line elements are tuned to produce a focus at the C H A O S target 80 cm downstream of F3 . Chapter 1. Introduction to Pions 6 1.2 The C H A O S Spectrometer The C H A O S spectrometer is a 360 degree pion spectrometer wi th a momentum resolution A P / P of about 1% cr, and an out-of-plane acceptance of ± 7 ° [5]. The spectrometer consists of a 55 ton dipole magnet between the poles of which lies four cylindrical rings of wire chambers. The inner three wire chambers lie in the uniform magnetic field of the dipole magnet, while the outer most wire chamber lies in the fringe field of the magnet. In a cylindrical r ing outside of the wire chambers are 20 adjacent C H A O S fast trigger ( C F T ) blocks, each of which is 18 degrees wide. A n isometric view of C H A O S is shown in Figure 1.4. A typical C H A O S event is useful in demonstrating how the spectrometer works. A pion enters C H A O S and follows an approximately circular path, passing through the wire chambers, to the center of the spectrometer where it scatters from a target. The path of the incident pion is recorded by hits in each of the wire chambers. The path pion can later be reconstructed using the wire chamber hits (refer to Section 2.2.) The pion scatters off an atom in the target and travels along another approximately circular path wi th a new radius. Again hits in the wire chambers are recorded, allowing the pion's path to be reconstructed. A typical C H A O S event is shown in Figure 1.5. The following sections contain details on the operation of the wire chambers, a de-scription of the C F T blocks, and introduces the TILL stack used to distinguish pions and muons. 1.2.1 The Proportional Wire Chambers WC1 and WC2 The two inner most wire chambers (WC1 and W C 2 ) are proportional chambers that are used to track both incident and scattered tracks. W C l and W C 2 each contain 720 wires wi th active lengths of 70mm, and are located at radii of 114.59 m m and 229.18 m m Chapter 1. Introduction to Pions 7 Figure 1.4: Isometric view of the C H A O S Spectrometer. For clarity, the top magnet pole, several the C F T blocks, and parts of the wire chambers are removed. Chapter 1. Introduction to Pions 8 - 5 0 0 0 5 0 0 Figure 1.5: In-plane scattering of a pion reconstructed from C H A O S wire chamber hits. B o t h x and y axes are in millimeters. The circle of dots in the center of the figure represent the wires in W C 1 . The two circles of dots around W C l are the wires in W C 2 and W C 3 respectively. The circle of radial dashes represent the W C 4 anode wires and resistive wires. The circular arrangement of blocks represent the C F T blocks. The in-plane (x,y) C H A O S coordinate system is also shown in this figure. Chapter 1. Introduction to Pions 9 respectively. The proportional chambers are filled wi th a gas mixture of 80% CF4 and 20% isobutane. A high voltage is applied to the wire chamber anodes. The 360 strips of cathode on the chamber wall are inclined at 30° from the anode wires. The cathode strips are on the larger radius wall of the wire chamber, while the smaller radius wall has a single cathode foil. When a charged particle passes through the chamber, it ionises the gas mixture. These ions then drift toward the cathode, and the free electrons drift toward the wires. Since these are proportional chambers, the voltage between the anodes and cathode is large enough to accelerate the ions and electrons to the point where secondary ionisations occur. This amplification of ionisations is called a Townsend avalanche [6]. The result is that larger voltage pulses can be obtained on the wires. In principle the voltage of the pulses in a proportional chamber is proportional to the energy that the particle loses in the chamber. A three dimensional coordinate for a particle passing through the wire chamber can be determined. Definition of the C H A O S coordinate system can be found in Section 2.1. The x and y coordinates are given by the centroid of the cluster of anode wires that fire. The z coordinate requires the use of the cathode strips which are inclined at 30 degrees wi th respect to the anodes. Intersecting the centroid of the cluster of anode hits wi th the centroid of the charge distribution of cathode hits gives a value for the z coordinate [7]. 1.2.2 The Drift Wire Chamber W C 3 Wire chamber three is a cylindrical drift chamber located at a radius of 343.77 mm. Drift chambers time how long it takes to get a signal on the anode to estimate how far a particle track was from the anode wire. A n accurate x,y coordinate on a particle track can be obtained using the drift time, but only if the drift time to distance relation is known. The relation is kept simpler, in the absence of a magnetic field, if there is Chapter 1. Introduction to Pions 10 a uniform electric field. The electric field is kept nearly uniform by placing thick field shaping cathode wires between each anode wire. The 144 anode wires are equally spaced 15mm apart around the circumference of the chamber, and each drift cell is equipped wi th four cathode strips. These cathode strips are not at an angle, so no z coordinate information can be obtained. Instead, four cathode strips for each anode are placed on the chamber walls, arranged so that one strip is in each quadrant around an anode wire. The signals from the cathode strips are used to decide which side of the wire the track passed on. 1.2.3 The Vector Wire Chamber WC4 W C 4 is used to determine the direction that a particle track is going before it leaves the spectrometer. In providing this vector, it determines the position out-of-plane at two points. Each of the 100 trapezoidal cells in W C 4 contains eight anode wires, four guard wires, and two resistive wires. A cell of W C 4 is shown in Figure 1.6 The in-plane part of the track vector is obtained from the eight 5 m m spaced anode wires, which are staggered ± . 2 5 m m perpendicular to the radius of the spectrometer. Anode wires provide drift times, and the anode staggering resolves the left-right ambigu-ity. The drift time to distance relation is kept reasonably linear by having cathode strips on a l l sides of the chamber walls. The high voltage is stepped between each cathode to compensate for the trapezoidal shape of the cell. Resistive wires are used to get the out-of-plane position of the track. When electrons from the track ionisation get to the resistive wire they can travel either direction along the wire. If the electrons arrive at the centre of the resistive wire, the resistance to either end wi l l be the same, resulting in equal amounts of charge traveling to either end of the wire. If the charge arrives somewhere other than the centre of the wire, then different amounts of charge wi l l travel to either end of the wire. Using the method of charge Chapter 1. Introduction to Pions 11 Figure 1.6: A cell of W C 4 showing anodes (A) , guard wires (G) and resistive wires (R) . A l l units in this figure are in millimeters. Chapter 1. Introduction to Pions 12 L -«£ D ^ t Q A D C A A D C B Figure 1.7: Resistive wire of length L with charge injected at a distance D from the centre of the wire. division, the height at which the charge entered the resistive wire is given by 7J = 4 — + (1.2) A + B 2 v ; In equation 1.2, A and B are the amount of charge collected by the Analog to Dig i ta l Converters ( A D C ) at either end of the resistive wire, L is the electrical length of the wire, and O is an offset. Resistive wires have been operated in drift chambers wi th resolutions of the order of 1% of the length of the wire [8] [9]. A diagram of a resistive wire is shown in Figure 1.7. 1.2.4 The C H A O S Fast Trigger Blocks A decision has to be made by electronics when to read al l the detector information to tape. The 20 C F T blocks are a crucial element in making this decision. Each C F T block contains two layers of NE110 Scintillators and a lead glass Cerenkov detector. These scintillators, along wi th a scintillator situated where the beam enters C H A O S are used in Chapter 1. Introduction to Pions 13 coincidence as a first level trigger for readout. The C F T blocks are also used for particle identification of 7T, e, p, and d by looking at scatter plots of the energy deposited versus the momentum of the particle track. 1.2.5 The TT/JL Stack Since the energies of interest in the Coulomb-Nuclear interference (CNI) experiment are so low, there is a considerable background due to pion decays. Thus a method of identifying pions from muons is required. The TTU stack, designed by the Italian collaborators, does this by exploiting the different 7r and p energy loss, and by using range characteristics of 7T and /x. The stack consists of five layers of aluminium absorbers sandwiched by six layers of scintillators. Opt imum particle identification at each energy of interest is determined using stack calibration data that are fed through a neural network. The first set of scintillators consists of eight vertical bars that are read out using photomultiplier tubes on either end. The photomuliplier signals are digitised using Analog to Digi ta l Converters ( A D C s ) , and Time to Digi ta l Converters (TDCs) . Using a difference over the sum of these A D C s provides a relative height at which a particle passes through the stack. The other five layers of scintillators each consist of two horizontal scintillator bars. A sketch of the main elements of the stack are shown i n Figure 1.8. The center of the stack is at 209 cm from the center of C H A O S , and extends from a scattering angle of about zero degrees to thirty degrees. In the region of the TILL stack, it replaces the C F T blocks, providing both one end of the first level trigger and TILL identification [10]. The stack is particularly important in C N I since kinematics alone cannot eliminate all the pion decays. Kinematics of pion decay into muons and muon neutrinos dictates that the muons wi l l scatter into a cone. The center of mass ( C M ) energy and momentum of the decay muon are given by the following equations involving pion and muon masses. Chapter 1. Introduction to Pions 14 Figure 1.8: Sketch of the TT/J, stack showing five layers of scintillators and four layers of aluminium absorbers. Chapter 1. Introduction to Pions 15 rpCM _ ml+ml The direction in the C M of the decay muon is arbitrary. To Lorentz boost the muon into the laboratory frame of reference we need to know which direction the muon decays in . To do a simulation of the decay kinematics, we need to know the original kinetic energy of the pion (Tn), and generate a random in-plane angle (6CM) measured from the direction of motion of the pion. The Lorentz boost in the x direction, defined here to be the direction of motion of the pion, is given by: — MK. 1 run The y direction of the momentum requires no Lorentz boost since it is perpendicular to the motion of the C M . The total lab muon momentum and cone angle are therefore: 6™* = a r c t a n ^ / p ^ The muon lab momentum and cone angle for 5000 randomly chosen C M decay angles was generated. A plot of p^AB versus 9BAB for 55 M e V (135.4 M e V / c ) pions is shown in Figure 1.9. In real data, the pions and muons wi l l be much more spread out than is shown in Figure 1.9, because of quantum mechanical effects and detector resolution. Some raw data from C H A O S of the pfcAB versus 9BAB data for 55 M e V (135.4 M e V / c ) pions are shown in Figure 1.10. Chapter 1. Introduction to Pions 16 Figure 1.9: Simulation of pion decay kinematics for 55 M e V pions. The x axis is the scattering angle in degrees and the y axis is the momentum in M e v / c . Chapter 1. Introduction to Pions 17 • o i . V B . B c a t Dotplot 18 Run 6852 TEST reg.event 2Q | i i i _ l 1—rjl i i I i — i — i — i — I — i — i — — • — • -15H 5-{ reg.event 3000 dots — I 1 1— 1 1 1 1—I r -20 -IS -10 MOMT (x10 +M pl iunu Dotplot 23 Run I I 1 I l _ ^ J 1 1 I 1 1 L. 20 15H .10 H 5H 8852 TEST _ J i i i 2000 dot* 3000 doti -20 "15 - i — i — r -x (xlO*1) Figure 1.10: Illustration from raw C H A O S data of why the stack is important for elim-inating pion decay events. The scatterplot shown at top is from a 55 M e V Carbon target run, and the lower scatterplot is the calculated kinematics. Bo th scatterplots have scattering angle in degrees on the y axis, and momentum in M e V / c on the x axis. Chapter 1. Introduction to Pions 18 1.3 Coulomb-Nuclear Interference Region Measurements Coulomb-Nuclear Interference (CNI) measurements are n^p —» ir^p cross section mea-surements at small scattering angles where the cross section for scattering by the nuclear or by the electromagnetic interaction are of the same order of magnitude. In t rying to extract threshold parameters, we get data at low incident pion energies and low scattering angles where the Coulomb interaction becomes more important. This section contains a simple incomplete description of how the low energy, and small scattering angle data are used to extract physical information about the structure of protons. Low energy measurements are important, since the interesting physics we can learn comes from extrapolating forward scattering amplitudes (D+) in the (z/,t) plane to the Cheng-Dashen point (v—0, t=2u2). Here we use the variables: q for the pion four-momentum, p for the proton four-momentum, M for the proton mass, \x for the pion mass, and the Mandelstam variables (s,t,u) and v. s = ~{p + q)2 t = -(Q-d')2 u = -(p-q')2 = (p' ~q)2 u AM To make the distance of extrapolation to threshold from physical regions of the (i/,t) plane as small as possible, we need good low energy measurements. A plot of t versus v in Figure 1.11 shows the physical regions where measurements are made. For pion proton scattering we get data in the s-channel. Notice that the closest data to the Cheng-Dashen point are at low v and low t, which corresponds to low beam energies and small angles. The physics of interest is measuring the explicit breaking of chiral symmetry in Q C D . Chi ra l symmetry is a symmetry of handedness of a particle. If a set of particles is chirally Chapter 1. Introduction to Pions 19 Figure 1.11: Mandelstam plot of v versus t. The hatched regions show where physical measurements for s-channel and u-channel scattering are made. 7rp scattering is an s-channel process. Chapter 1. Introduction to Pions 20 symmetric it wi l l have both right and left handed particles. Quarks are approximately chirally symmetric; if they were massless they would be chirally symmetric, but because they have mass there is a small coupling between right and left handed quarks. The consequence of spontaneous symmetry breaking was studied by Goldstone, and was found to result in massless particles (the Goldstone Bosons). Since quarks have a small mass, the right and left handed quark states mix, explicitly breaking chiral symmetry, and resulting in Goldstone Bosons (TT, K , 77). The low value of the n mass results from the fact that the chiral symmetry of quarks is only approximate. A n empirical measurement of the chiral symmetry breaking is given by the cr term [11]. The sigma term defined in Equation 1.3 is related through some algebra to the strange quark content of the proton (y). rh _ -o = ^ <p\uu + dd\p> (1.3) ^ 3 5 ± 5 M e V ~ 1-2/ _ 2<p\ss\p> " <p\uu+dd\p> In Equat ion 1.3, rh is half the sum of the u and d quark masses (rh — \(mu + md)). The sigma term is related to the forward scattering amplitudes in a complex way through the capital sigma term given by: ^ = FlD+(v = 0,t = 2u2) (1.4) where Fn = 92AMeV is the pion decay constant. A n extrapolation of carefully chosen C N I cross section measurements to threshold wi l l be used to find the pion S and P wave scattering lengths and . The scattering lengths appear in the threshold subtraction constants which parameterise the dispersion relations used in extrapolating below threshold to the Cheng-Dashen point. It is proposed to find the scattering lengths using a measurement of Re(D+) at t=0 for several energies. Chapter 1. Introduction to Pions 21 Finally, the C N I data wi l l be used to get a better measure of the TTN part ial waves, which relies on a good measure of Re(D+) at t=0. Chapter 2 Using C H A O S at Small Scattering Angles Obtaining good small scattering angle data are of fundamental importance for the C N I experiment. This chapter explains the methods used to get the best possible estimate of small scattering angles. The following sections explain: • why the full three dimensional scattering angle is required, • how the out-of-plane scattering angle is found using C H A O S , • how out-of-plane information is obtained from W C 4 even though it is in the C H A O S magnetic fringe field, • how parameters in the fringe field correction model are determined, and • how to decide if an interaction took place in the target using vertex reconstruction. B y inspecting the section titles in this chapter, one can see that the main difficulty in determining the small scattering angles is in decoding what happens in the fringe field of W C 4 . In principle the out-of-plane scattering angle can be found using only W C 1 and W C 2 . We need to do the W C 4 out-of-plane correction for the following three reasons. Knowing the out-of-plane coordinate (z) of the W C 4 hit wi l l improve the efficiency of determining the out-of-plane angle. Recognising out-of-plane pion decays wi l l be aided by a knowledge of z at W C 4 , since this is essentially the z coordinate at the end of the track. Finally, 22 Chapter 2. Using CHAOS at Small Scattering Angles 23 having an extra point wi l l improve the resolution of the full three dimensional scattering angle, and of the out-of-plane coordinate of the vertex. 2.1 Full Scattering Angle Versus In-Plane Scattering Angle Unt i l recently, C H A O S experiments did not go down to low scattering angles, so it was sufficient to use the in-plane scattering angle. In this coordinate system 02d is defined as the angle in the horizontal xy plane of C H A O S measured from the x axis. In C H A O S the x-axis is defined as the direction that the beam comes in wi th BQHAOS — 0, wi th no rotation of C H A O S , and no translation of C H A O S . The in-plane C H A O S coordinates can be seen in Figure 1.5. The in-plane scattering angle, 02d, and the full scattering angle, 93d are defined in Figure 2.12. Physically, a particle scattered at some 62d c a n also have an out-of-plane component. C H A O S has an out-of-plane acceptance of ± 7 degrees, hence the in-plane component of the scattering angle is sufficient for angles larger than 20 degrees. For example suppose a track scatters at an in-plane scattering angle of 9 2d = 15 degrees, wi th an out-of-plane component of fad = 83 degrees. We can then use a transformation of coordinate systems to go from the C H A O S system (6 2d, fad) to the physical system (63d, fad)- The unit coordinates in terms of the angles in each system are: C H A O S Coordinates x = cos(6>2d) sm(fad) (2.5) y = sm(62d) sm(fad) (2-6) z = cos(fad) (2.7) Chapter 2. Using CHAOS at Small Scattering Angles 24 z Figure 2.12: Definition of the in-plane and full scattering angles. In C H A O S x is defined as the direction the beam comes in, y is perpendicular to x, and z is the out-of-plane axis. Note that the scattering vector shown in this figure is the difference between the beam tangent vector, and scattered track tangent vector taken at the origin. The y axis is actually drawn as -y to make drawing the vector components easier. Chapter 2. Using CHAOS at Small Scattering Angles 25 Physical Coordinates x = cos(#3 d) y = sm(63d) cos(4>3d) z = sm(93d) sin(<^3d) (2.8) (2-9) (2-10) CHAOS Coordinates z Physical Coordinates Figure 2.13: Angle definitions in the C H A O S coordinate system, and in the physical coordinate system. The y axis is drawn as -y to make vector components easier to draw. Using 2.5 and 2.8 we find, for our example track, 93d = 16.5 degrees. This 1.5° difference is crucial since the cross section is inversely proportional to s in 4 0/2, resulting in a difference of nearly 50% in the cross section. A t 62d greater than twenty degrees, the difference from 63d is of the order of one degree or less. A diagram of the two coordinate systems are in Figure 2.13. To accurately estimate 93d we need to determine (p2d-Chapter 2. Using CHAOS at Small Scattering Angles 26 2.2 Determining the Out-of-Plane Scattering Angle To find an out-of-plane scattering angle, fad, we need to look at the profile of a track in C H A O S . We require a knowledge of the length travelled along a particle track to the wire chambers, and a knowledge of the out-of-plane coordinate of the track at the wire chambers to find fad-Out-of-plane information in C H A O S can be obtained from W C 1 cathodes and anodes, W C 2 cathodes and anodes, and W C 4 resistive wires. The out-of-plane wire chamber hit coordinates are labelled z[ and z'2 for W C 1 and W C 2 incident beam tracks; scattered track coordinates are labelled zx and z 2 for W C 1 and W C 2 hits respectively. The coordinates from inner and outer W C 4 resistive wire scattered track hits are labelled Zi and z0. Note that the W C 4 cell in the incident beam, and cell in the unscattered beam have been removed since they cannot handle the high rate. In-plane pathlengths to W C 1 and W C 2 are found using arcs of a circle fit to the in-plane hits. The radius R of the circle is given by the momentum of the track, as shown in the following equation: In the final result of Equation 2.11, Pxy is the in-plane momentum in M e v / c , B is the C H A O S magnetic field in Tesla, z is the number of electron charges including polarity, and R is the radius in millimeters. For the incident track, the beam momentum is used. For the scattered track the Quintic Spline method is used to fit W C 1 , W C 2 , W C 3 , and W C 4 to find the track momentum. The Quintic Spline method was developed by W i n d [12], and a good description of the method applied to C H A O S can be found in Kermani 's M.Sc . thesis [7]. A pathlength from W C 1 or W C 2 to the interaction vertex is given by: mv mPc P-, (2.11) R = qB qBE 0.29979^5 Figure 2.14: Geometry used in finding pathlengths to W C 1 and W C 2 . Note that the wire chambers are not drawn to scale. Chapter 2. Using CHAOS at Small Scattering Angles 28 0, arc 2 = arcsm yj(xi - xv)2 + (yi - yv)2 2R (2.12) Pathlength — S = 9arc * R (2.13) where (x;, y^) are the in-plane coordinates of the hits in W C l or W C 2 , and (xv, yv) are the in-plane coordinates of the interaction vertex. The geometry of the wire chambers and track is shown in Figure 2.14. Pathlengths to W C 4 inner and outer resistive wires are found by doing numerical line integrals along the Quintic Splines that result from fits that are done in determining the particle momentum. In a uniform magnetic field in the z direction, the vertical component of the scattered track profile is a straight line given by z(S) = m S + b. The slope, m, and the intercept, b, are determined using linear regression on the four wire chamber hits for the scattered track in C H A O S . We label the points as Sa, and za, where a = (l,2,i,o) labels points from W C l , W C 2 , W C 4 inner resistive wire, and W C 4 outer resistive wire respectively. W C 4 is in the fringe field of C H A O S , which means the field is no longer uniform. This results in non-linear track profiles, as is shown in Figure 2.15. In order to fit a straight line to the track profile, we need to find a fringe field correction for the W C 4 resistive wire hits. The following two sections give some detail on how particle tracks in the fringe field were generated, and how the fringe field correction is made. 2.2.1 The C H A O S Fringe Field A n investigation of what happens to particles in the C H A O S fringe field was done using a routine, trace, developed by Roman Tacik [13]. Trace uses a fifth order Runge-Kut ta method [14] to step particles in time through C H A O S . The differential equation being Chapter 2. Using CHAOS at Small Scattering Angles 29 5 3 o o o Tracks for P / B = 167 M e V T A - 1 — i 1 r—7—1 1 —v . • • ' ' ^_ --* - — --AM - ' ~~ -" • •**™'" - - ^ ^ ^ 20 40 60 80 S - Path Length in cm 100 Figure 2.15: Vert ical track profiles show the focusing effect of the fringe field. Note the change in scale from vertical to horizontal. The uppermost track has a solid straight line overlaid, illustrating how far off the object line the W C 4 hits are. solved is given by the following force balance. dP d F = = — (7m 0 '«) = 7 m 0 x = qv x B [2.14) Init ial conditions supplied to trace are the starting position, the momentum, and the direction (Q2d, fad) of the track. From equation 2.14 the force responsible for the non-linear pathlength to track height is given by components of the magnetic field that are in-plane. The z component of the track acceleration is given by: Q 7 m 0 XXBy XyBX (2.15) A n example of how the fringe field affects the profile of tracks in C H A O S is shown in Figure 2.15. A listing of the trace program used to generate tracks in C H A O S is in Chapter 2. Using CHAOS at Small Scattering Angles 30 Appendix A . The trace routine does not account for multiple scattering, particle decays or any other process; trace generates simple classical tracks of charged particles in a magnetic field. To add an element of realism a simple energy loss calculation was done between each time step. The model for energy loss uses a C H A O S materials file used in Greg Smith's eloss program. The energy lost in material is given by the Bethe-Bloch equation [15] as follows: dE _ Kz2Z Hx~ ~ p2A 1. 2mec2p2-f2Tmax o 2 T = max 2 l n P 2m e c 2 /?V 1 + 2>yme/M + (me/M)2 (2.16) (2.17) Symbol Definition Units or Value E Incident particle energy M e V NA Avogadro's number 6.0221 xl023mol-1 K 47rA rAr 2m ec 2 0.3071 MeVg-lcm2 ze Charge of incident particle Z Atomic number of medium A Atomic mass of medium gmol~x mec2 Electron mass xc2 0.511 M e V I Mean excitation energy M e V M Incident particle mass Table 2.1: Definition of variables in Bethe-Bloch equation. Tmax is the most kinetic energy that could be transferred to a free electron in a single collision. The variables in the Bethe-Bloch equation are summarised in Table 2.1. 2.2.2 Fringe Field Correction Model Since the fringe field vertically focuses particle tracks, a simple lens model was adopted. This model corrects hits in W C 4 to put them on the object line. In our model the object Chapter 2. Using CHAOS at Small Scattering Angles 31 line is the straight line in the pathlength (S) versus height (z) plane that projects wire chamber hits back to where the particle hit the target. The corrected z values, labelled z'a, were assumed to be a function of: the uncorrected z value, the slope in S versus z of the scattered track (m), and the ratio of the C H A O S magnetic field to track momentum ( p / B ) . Since the result has to fit on the object line, we also find that: z'a = f(za,m,p/B)=mSa + h (2.18) The exact form of f was chosen empirically as wi l l be described in section 2.2.4. The result is shown in equation 2.19. z'a = (aaza + bazl + cam) + g (2-19) In order to do the fit to determine the constants in equation 2.19, the slope (m), and target intersection height (h) had to be estimated using only W C l and W C 2 track hits. The constants in the model make the model overdetermined. This choice of constants was made so that a fit to Equation 2.19, with the term involving the ratio of p / B set to one, could be done first. B y fitting for the constants aa,ba,ca, the approximate value of some of the constants could be used as starting points in subsequent fits involving all five parameters. I found this desirable because it took a lot less time to converge on the parameters when starting with a good guess for some of the parameters. The ini t ia l fit to find aa,ba,ca was fast since a linear regression was used. F i t t ing to get al l the parameters took more time since a non-linear fitting algorithm had to be used. Values obtained for the parameters are summarised in Table 2.2 To use the model to find where a hit in W C 4 (Sa,za) is moved we rearrange Equa-tion 2.19 so that it is of the form z"a = mS'^ + h. The corrected wire chamber hits are given by the coordinates z = z", and S = S". The result is given by: Chapter 2. Using CHAOS at Small Scattering Angles 32 a aa ca da 6 « 1 1 0 0 0 0 2 1 0 0 0 0 i 1.08 2 . 4 4 x l 0 " 6 -12.6 355000 339000 0 1.10 2 . 3 6 x l 0 " 6 -26.8 273000 243000 Table 2.2: Fringe field correction parameters determined from G E A N T simulations. The G E A N T simulation used is described in the following section. The parameters have units such that za, and z'a are in millimeters. The parameters da and ea have units of (Mev/Tesla)2. z'l = Z\ S'[ = Si z'2 = z2 S2' — S2 z'l = (alZl + kzfMp/B) S? = Si- gi(p/B)ct (2.20) 4 = (aozo + b0z30)g0(p/B) SZ = S0- g0{p/B)c0 where, ga = {p/B)*+ea So long as we get two of a possible four (Sa, za) points, we can use Equat ion 2.20 to find the track slope (m) and intercept (h). B y inspecting the parameters in Table 2.2 and looking at the equations for (va,wa) we can say that the correction moves a hit in W C 4 away from the S and z axes. A n example of where the W C 4 hits are moved by the model correction is shown in Figure 2.16. In the vertical profiles shown in Figure 2.16, the points represent hits in the wire chambers. W C 4 has four points; the two solid W C 4 points below the track line are the actual hits as recorded by the wire chamber, and the two open W C 4 points on the track line are the points corrected using the fringe field model. Chapter 2. Using CHAOS at Small Scattering Angles 33 event t e s t , reg .event YBS. EVNTi 425 event t e s t , reo -event YBS. EVNTi 468 P a t h l e n g t h (mm) P a t h l e n g t h (mm) Figure 2.16: Particle tracks at small scattering angles in C H A O S . The vertical profile for each track is shown below each in-plane track. The in-plane tracks shown at top are display events described in Section 1.2. In the vertical profiles, the points represent hits in the wire chambers in pathlength versus height. W C 4 has four points; the two solid W C 4 points below the track line are the actual hits as recorded by the wire chamber, and the two open W C 4 points on the track line are the points corrected using the fringe field model. Chapter 2. Using CHAOS at Small Scattering Angles 34 2.2.3 G E A N T Simulation Used to Find Model Parameters G E A N T is a detector description and simulation tool developed at C E R N . The version of G E A N T that I used was already coded wi th C H A O S geometry, and targets. Running G E A N T generates data in the same ( Y B O S ) format as real data, allowing the same analyser to be used for the simulated data as for real data. The G E A N T run used to determine the fringe field parameters included multiple scattering and energy loss calculations. To simplify the parameter estimation, pion decay was turned off in G E A N T . There was no target in the runs used for determining the model parameters, allowing both incident and outgoing hits to be used in finding the track slope, and the height at the target (as if it were there). Since there was no target, unscattered tracks had to be used in finding the model parameters. Resolution of the in-plane wire chamber hits in C H A O S were already coded into G E A N T . The out-of-plane resolution of the wire chamber hits was not included in G E A N T . G E A N T data were fuzzed up in the out-of-plane direction by adding a routine into the analysis code. 2.2.4 Determining Fringe Field Correction Model Parameters The fringe field correction model parameters can be determined using tracks produced by trace, tracks produced by G E A N T , or by wire chamber hits from real data. The first two of these methods yield sensible results. M u c h time and effort was spent trying to extract parameters from C H A O S data, but when parameters were being found, the algorithms for eliminating pion decays had not been refined. Uncertainty in the parameter estimation using real data was too large because of the difficulty in eliminating pion decays. In order to do the fit to the constants in equation 2.19, the slope (m), and target intersection height (h) had to be estimated using only W C l and W C 2 track hits. A fit Chapter 2. Using CHAOS at Small Scattering Angles 35 to get the constants a_, ba, and ca in z'a = aaza + baz^ + ca was done first. For tracks produced in G E A N T or trace we know the slope (m), and target intersection height (h), since they are known exactly without reading any wire chamber hits. Thus linear regression can be done to find aa, ba, and ca. B y finding aa, ba, and ca first we can use them as ini t ia l guesses in subsequent fits to get all the constants. To get the constants da and ea, the value of aa is set to be the value found in our first fit. Then, a non-linear fit to Equation 2.19 to find the constants ba, ca, da, and ea is done using a Levenberg-Marquardt algorithm in Mathematica [14]. The resulting fits for both sets of simulated tracks give the same constants. Residuals (Az) between the actual coordinate of the W C 4 hit from the simulation (z^m), and the coordinate found using wire chamber hits corrected using the model (z'a) were calculated. Residuals (Az) to the fits using trace data show that, neglecting the detector resolution, multiple scattering, and pion decays, the correction to the W C 4 z coordinates is good to within 0.4 m m (a). The fit residuals from trace data are shown in Figure 2.17. Notice that the resolution is worse as tracks hit further from the center of the resistive wire. Resistive wire hits further from the center of the wire are expected to have worse resolution because the charge going to one end wi l l have to travel further, resulting in a smaller signal. B y examining the Az residuals we see that as we get to larger z coordinates the model fit is worse. The residual in all the cases tested showed no additional dependence on the z coordinate or the slope of the track. The more realistic Az residuals to the fits using G E A N T data show that the correction to the W C 4 z coordinates is good to within 3.5 mm (a). The fit residuals from trace data are shown if Figure 2.18. The effect of turning on al l the realism gave the same model parameters as trace, but, as expected, made our resolution worse. Chapter 2. Using CHAOS at Small Scattering Angles 3 6 1.0 1 0.5 J 0.0 Uncorrected (mm) | -0.5 8 -0 .20-0.15-0.10-0.05 0.00 0.05 0.10 0.15 0.20 Track Slope J 0 0 S a: a; -10 - t 5 2 4 6 8 X) 12 14 16 Track Slope • (mm) - 1 5 0 0 - 1 0 0 0 - 5 0 0 0 5 0 0 1000 1500 Track Slope • z ^ 1 (mm*) Uncorrected z 5 0 , (mm) 0 . 2 0 - 0 . 1 5 - 0 . 1 0 - 0 . 0 5 0 .00 0 . 0 5 0.10 0.15 0 .20 Track Slope 15-* 0.5-0.0--'•;!, • . 1j - 0 . 5 - -• -1.0- -£ -is- -—2.0-5 0 5 10 15 20 Track Slope • z ^ (mm) - 2 0 0 0 - 1 5 0 0 - 1 0 0 0 - 5 0 0 6 5 0 0 1000 1500 2 0 0 0 Track Slope * z ^ 1 (mm 1) Figure 2.17: Az residuals for WC4 inner and outer resistive wires. Residuals were gener-ated from trace data which do not account for multiple scattering, detector resolution, or pion decay. The upper four plots are residuals for WC4 inner, and the bottom four plots are for WC4 outer. The y axes are all Az in millimeters. The x axes going clockwise from the upper left (for the top four plots) are: the uncorrected z, the track slope, the product of the track slope with the uncorrected z, and the product of the track slope with the uncorrected z squared. The upper two rows are for the inner resistive wire, and the bottom two rows are for the outer resistive wire. Chapter 2. Using CHAOS at Small Scattering Angles 37 Run SD002 aO-ltor-Ha 1 1 | 2 1 | 2 I P i l l - 122.1 a t 111 48 f l t a - - I . I H i * M I 0 2 M - t l a i - 9 9 1 1 l 2 1 i 4 2 Ptafc- 122.1 It 111 47 l a t a - - 1 . 7 Figure 2.18: Az residuals for WC4 inner and outer resistive wires, generated from G E A N T data which included multiple scattering and detector resolution. P ion decays were disabled in the G E A N T run. 2.3 The Interaction Vertex at Small Scattering Angles To find 03d, the full three dimensional incident beam tangent vector p1n, and the scattered track tangent vector poUt need to be found at the interaction vertex. Then the true 3d scattering angle is given by Equation 2.21. n Pin ' Pout ,n 0 1 \ C 0 S ^ = | p - | h n - I ^ 2 1 ) \Pm\\Pout\ The vertical component of the momentum is related to the in-plane components of the momentum by Equation 2.22. (2.22) A t large scattering angles, the in-plane components of the track momentum vectors are found using the tangents to circles. The tangents are found using the tangents to the circles, at the intersection closest to the target, of two circles. The circles are found by Chapter 2. Using CHAOS at Small Scattering Angles 38 Finding Large Scattering Angles Finding SmaU Scattering Angles Figure 2.19: Diagram showing how the in-plane scattering angle is found. fitting to the appropriate track momentum, the W C l and W C 2 hits for incident tracks, and the W C l and W C 2 hits for scattered tracks. For the scattered track if one of W C l or W C 2 is missing, then a W C 3 hit wi l l be used as a point on the circle. Equat ion 2.11 gives the radius of the circles; the momentum of the beam is used for the incident track, and the track momentum found using the Quintic Spline method described in Section 2.2 is used for the scattered tracks. The geometry of the hits and circle fits is shown in Figure 2.19. A t small scattering angles the tangents of the two circles that are found are nearly parallel. A s a result, a small error in the W C hit coordinates results in a large error in the longitudinal vertex coordinates. To solve this problem we use the intersection of the circles wi th a fixed target plane. Then the intersection of the incident track wi th the target plane gives us p~in, the intersection of the scattered track wi th the target plane Chapter 2. Using CHAOS at Small Scattering Angles 39 gives us p^ut, and the distance (A) between the two intersection points is used to decide if the interaction was in the target. See Figure 2.19 for a diagram of a bad event wi th a large A at a small scattering angle. When A is large the event is tagged as being bad. A large A can occur if the pion decays inside, or if the interaction occurred in material surrounding the target. For further discussion of the use of A refer to section 4.3. Chapter 3 Angular and Coordinate Resolution Having good estimates of the angular resolution and coordinate resolutions in C H A O S are important in extracting the experimental results, and in comparing results to theoretical models. This chapter explains how the out-of-plane angular resolution (A(f)) and out-of-plane coordinate resolutions (Az) are found. The following sections discuss: how the Az at the vertex (Az0) is found, how the Az at the wire chambers are estimated, how G E A N T is used to replicate the Az of the wire chambers, how G E A N T is used to determine the A(j), and why the A<p is particularly important. 3.1 Resolving a Horizontal R o d The z coordinate resolution of the vertex can be determined from C H A O S data by re-constructing a horizontal picket fence target. Six horizontal pickets, made wi th four ^ T " diameter C u rods above two | " diameter rods, are attached to a copper sheet wi th the shape of an upside-down American football goal post. Going from top to bottom, the thinner upper C u rods are spaced 1 cm, 2 cm, and 1 cm apart. The first thicker C u rod is 1.5 cm below the lowest thin C u rod, and the last C u rod is 1.5 cm lower. A sketch of the picket fence target, also showing where the beam spot was for two different runs, is shown in Figure 3.20. The picket fence can be reconstructed from C H A O S wire chambers using both incident and scattered tracks. For events which triggered the detector, and thus presumably interacted wi th the target, we try to determine the height at the target. Incident track 40 Chapter 3. Angular and Coordinate Resolution 41 1 cm 2 cm 1 cm 1.5 cm 1.5 cm Figure 3.20: Sketch of copper picket fence shows location of horizontal pickets, labelled A through F . The figure is not drawn to scale. Chapter 3. Angular and Coordinate Resolution 42 hits in W C 1 and W C 2 , along wi th pathlengths, are projected along a straight line to determine the height at the target plane. Scattered track hits from W C l , W C 2 and W C 4 resistive wires, along with pathlengths, are also used to determine the height at the target plane. In the following sections I wi l l present: the picket fence resolution using real data, and the calibration of a Monte-Carlo simulation of C H A O S for out-of-plane data. 3.1.1 C H A O S Horizontal Rod Resolution A histogram of the height at the target is used to determine the resolution at the vertex. To get a cleaner histogram, a box cut on the target vertex, along wi th cuts to eliminate TT decay events are used. The vertex cut is a box in A versus the in-plane scattering angle 82d, which is particularly useful at small scattering angles. Further discussion of how TT decay events are eliminated can be found in Section 4.2. For run 6257, a 40 M e V TT~ run, the beam illuminated picket fence bars A , B , and C . The resulting histogram for height of the incident track at the target (z_proj), for al l scattering angles, without any cuts on it is shown in Figure 3.21(a). Z_proj wi th cuts on the vertex, cuts on TT decay, and a cut requiring scattering angles greater than 30 degrees, is in Figure 3.21(b). In going from Figure 3.21(a) to Figure 3.21(b), any electron or muon events are removed, which makes each of the peaks (A, B , and C) lower. The relative heights of A , B , and C when comparing between the figure in (a), and the figure in (b) depends on the heights at which the muons and electrons hit the target. Intersection of the outgoing track wi th the target plane, including al l cuts, gives a height at the target shown in the second row of Figure 3.21. In Figure 3.21, (c) uses only W C l and W C 2 on the way out (z_wclwc2), and (d) uses all of the wire chambers (z_all). The resolution of the bars is summarised in Table 3.3. Resolutions for the outgoing track are worse than for the incoming track because Figure 3.21: Picket fence reconstruction from: run 6257 wi th pickets A , B , and C clearly identified, and run 6258 wi th pickets C, D , E , and F . Refer to text for more details. Chapter 3. Angular and Coordinate Resolution 44 Bar Z-proj a (mm) ZJWC1WC2 a (mm) Z-all a (mm) A 1.49 2.14 1.83 B 1.47 2.38 2.28 C 1.27 1.61 1.83 Table 3.3: Resolution of picket fence bars for run 6257 at scattering angles greater than 30 degrees. of difficulties in eliminating al l of the TT decay events. When the TT/J, stack is finally calibrated, the resolution on the way out should be equal to the resolution on the way in . We also see that the use of W C 4 resistive wires improves the statistics, because adding W C 4 can give us two extra out-of-plane points to use in finding fad, and the height at the target. If one of the W C l or W C 2 hits was missing, then using the W C 4 hits w i l l allow us to calculate the height at the target. In principle using W C 4 should also improve the resolution of the picket fence. B y inspecting resolutions in 3.3 we see that the resolution using just W C l , and W C 2 , to when al l the wire chambers are used are about the same. Again , it must be pion decays which are l imit ing the picket fence resolution on the way out. Histograms of the picket fence for run 6258, another 40 M e V TT~ run, are shown in the lower half of Figure 3.21. For run 6258 the beam illuminated picket fence bars C, D , E , and F . The histogram for height of the incident track at the target (z_proj), for al l scattering angles, without any cuts on it is shown in Figure 3.21(e). Z_proj wi th cuts on the vertex, cuts on TX decay, and a cut requiring scattering angles greater than 30 degrees, is in Figure 3.21(f). In going from Figure 3.21(e) to Figure 3.21(f), any electron or muon events are removed, which makes each of the peaks (C, D , E , and F) lower. The relative heights of C, D , E , and F when comparing between the figure in (e), and the figure in (f) depends on the heights at which the muons and electrons hit the target. Intersection of Chapter 3. Angular and Coordinate Resolution 45 the outgoing track with the target plane, including all cuts, gives a height at the target shown in the last row of Figure 3.21. In Figure 3.21, (g) uses only W C l and W C 2 on the way out (z_wclwc2), and (h) uses all of the wire chambers (z_all). The resolution of the bars is summarised in Table 3.4. Bar Z-proj a (mm) Z-wdwc2 a (mm) Z-all a (mm) C 1.96 2.34 2.10 D 1.60 2.52 4.31 E 1.89 3.31 3.39 F 1.78 3.38 3.11 Table 3.4: Resolution of picket fence bars for run 6258 at scattering angles greater than 30 degrees. The results of run 6258 indicate that we see a ^ " rod as being almost as thick as a | " rod, thus making our vertical coordinate resolution at the target about 3 mm. Note that in run 6258 our elimination of pion decay events must not be as good as in run 6257, since al l the resolutions are worse. In particular, bar D must include some pion decays, making its resolution found using al l the wire chambers worse than it should be. The horizontal rod resolution for real data cut on small scattering angles (7° — 30°) was found to be of the same order as that found for larger scattering angles. Using run 6257 wi th cuts on A versus #2<i, 6^ in the range 7° — 30°, and wi th cuts to eliminate pion decays gave the picket fence reconstruction shown in Figure 3.22. The resolution of the bars using small scattering angle data is summarised in Table 3.5. A n explanation for the slightly poorer resolutions of the picket fence bars using small scattering angle data is that most of the muons coming from pion decay wi l l scatter at small angles. In order to get resolutions similar to those at larger scattering angles, we need to use more effective pion decay removing cuts. Chapter 3. Angular and Coordinate Resolution 46 Run 6 2 5 7 0 1 - R p r " 9 9 1 D : 3 5 : 3 3 P e a k = 1 7. Q a t b i n 40 d a t a = - 9 . 5 20 1 5 LO -I—1 c 1 D ZD CD C J 5 1 1 1 1 I 1 1 1 1 I 1 1 1 1 I 1 1 1 1 I 1 1 1 1 I 1 1 1 1 — sum= 2 4 8 . nn l e a n b i n = 3 9 . 8 6 7 B l LI j h Mean c A r j a t a = - 9 . 6 3 3 s i g m a = 1 2 . 4 2 1 n nqn p -i i i i i i i i | i i i i | i 1 1 1 1 1 1 1 - 6 0 -40 -20 0 20 40 60 Un=1. H i 924 s h _ a i l _ n d 2 [ h _ a 1 1 / s m _ s c a t _ n d 2 ] 0v=0. Figure 3.22: Picket fence reconstructed using small scattering angle data from run 6257, wi th pickets A , B , and C clearly identified. Chapter 3. Angular and Coordinate Resolution 47 Bar ZJWC1WC2 a (mm) Z-all a (mm) A 2.11 2.15 B 2.25 2.46 C 3.53 3.22 Table 3.5: Resolution of picket fence bars for run 6257 at scattering angles in the range seven to thirty degrees. 3.1.2 G E A N T Simulation of Spectrometer Resolution Scattering angle resolution could not be determined using C H A O S data alone, because we have no angle ruler like we had with the picket fence for the z coordinate resolution. To solve this problem we need to use a Monte-Carlo simulation, G E A N T , where we know the scattering angle a priori . The detector resolution for in-plane wire chamber hits was already included in existing G E A N T simulations of C H A O S . Out-of-plane resolutions were ini t ial ly chosen to be normally distributed wi th a a determined using All But One fits for each wire chamber. All But One fits are done using real C H A O S data from runs that had no target. For example an All But One fit for an incident track at W C l is done by linear regression on: the hit in W C 2 incident (s2i,z2i), the hit in W C l scattered (s lo .z lo) , the hit in W C 2 scattered (s2o,z2o), the corrected hit in W C 4 inner scattered (sio,zio), and the corrected hit in W C 4 outer scattered (soo,zoo). From the linear regression we find an estimate of the height at W C l incident (h l i ) . The All But One resolution (a) of W C l incident is then given by A z i z = zli — hli. Real C H A O S data used to do the All But One fits was taken from, run 6507, a 45 M e V 7 r + run wi th no target. In run 6507, unscattered pion beam was directed at the fourth stack bar. Results of the All But One fits for run 6507 are shown in Figure 3.23. Notice that the resolution of the wire chambers appears to be worse for scattered tracks Chapter 3. Angular and Coordinate Resolution 48 Figure 3.23: All But One fits used to get an estimate of the resolution in each wire chamber. Chapter 3. Angular and Coordinate Resolution 49 than for incident tracks, because of difficulties in eliminating TT decay events. A g a i n it is expected that the incident and scattered track resolutions wi l l be the same when decays are eliminated using the final 7r/x stack calibrations. To check our simulation results using the a values for each wire, we can see if we get the same width horizontal rod in G E A N T as we do from real data. The G E A N T run used was a 40 M e V pion run, scattered at any scattering angle, wi th multiple scattering, energy loss, and pion decays enabled. In the G E A N T run used, pion tracking was started at the T ^ " horizontal rod located 15 millimeters below the center of C H A O S . The real data 16 run being compared to is the picket fence run 6257, which is described in Section 3.1.1. After a few iterations of changing the a values of the out-of-plane chamber resolutions in G E A N T from the nominal values, we get nearly identical results for z_proj in simulated and real data. The sigma values for the wire chamber out-of-plane resolutions used in G E A N T are: sigma W C l is 0.8 mm, sigma W C 2 is 2.0 mm, and sigma W C 4 is 1.0 mm. The final z_proj that results from these out-of-plane wire chamber resolutions is shown in Figure 3.24. W i t h this final choice of wire chamber resolutions, we can be confident that our G E A N T simulation wi l l give reasonable estimates of the resolution we wi l l get from C H A O S . 3.2 Angular Resolution The angular resolution is found using G E A N T simulations. Setup of G E A N T for use in three dimensions of C H A O S was discussed in the previous section. The following section details how the angular resolution of C H A O S is found. F inding detector resolutions is important by itself, but further use of the resolution needs some explanation. The final section of this chapter wi l l outline the importance of finding the angular resolution in particular. Chapter 3. Angular and Coordinate Resolution 50 300 Run 30008 I8 -Feb"99 1 7: OBi 28 Peak- 266.0 at b in 35 d a t a - -14 . 5 s u a - 1366. Mean b i n - 34.544 Mean d a t a - -14 .956 a l g a a - 3.512 ^simulated = 2-4mm 200 100 50 H ' 1 I " r i " 1 - 1 I i '| i i i i | i i i i | -BO -40 -20 0 20 40 60 Un=1. Hi 85 n d _ a l i [ h _ a l l / n o ch_decayl Ov=Q. 50 Run 6257 18 -Feb-39 17: 19: 27 Peak -, . . . I . . . . I . . . . I 46, 0 at b in 41 d a t a * - 8 . 5 • • I . . . • I • • • • 700. Mean b i n - 39.263 Mean d a t a - - 10 .237 s l g a a - 13.354 ° 20 A 10 J c7real=2.4mm L I 1 1 1 1 I 1 1 1 1 I 1 1 -60 -40 -20 0 20 40 60 U n =1 . Hi 898 n d _ k z _ a i l [h_a l l / no_ch_decay3] O v = 1 . Figure 3.24: Comparison of horizontal rod z_proj using simulations (top) and real data (bottom). Result is shown for ^ " rods. Chapter 3. Angular and Coordinate Resolution 51 3.2.1 G E A N T Simulation of Angular Resolution Once we have our simulation calibrated to work in three dimensions it is easy to find the angular resolution of C H A O S . A l l we need to do is take the difference between the full three dimensional scattering angle that we reconstruct in G E A N T (93d), and the scattering angle that G E A N T generated (9geant). The angular resolution is thus defined as: A9 = 9id — 9geant (3.23) 93d is the full three dimensional angle between the incident beam, and the scattered track. To get a realistic estimate of the angular resolution the G E A N T runs used material files for the C N I l iquid Hydrogen target. These G E A N T runs included multiple scattering and energy loss, but had pion decays turned off. Three runs wi th incident pion energies of 40 M e V , 55 M e V , and 67 M e V were analysed, with the out-of-plane wire chamber resolutions simulated in the analysis. Since we expected our resolution (A9), at scattering angles in the range 0° to 40°, to worsen at large scattering angles, we looked at histograms of AO with cuts on different five degree angular regions. The result was that the resolution was fairly constant over different scattering angles. A dotplot of A9 versus 93d, in Figure 3.25, shows that the angular resolution is constant over the angles generated in G E A N T . Resolutions found got worse as the beam energy was reduced, as expected due to multiple scattering. Table 3.6 summarises the angular resolutions found at different beam energies. Energy Resolution a in degrees 67 M e V 55 M e V 40 M e V 0.8 0.85 1.0 Table 3.6: Angular resolutions found at different energies. Chapter 3. Angular and Coordinate Resolution d t h _ v s _ t h _ s a n g D o t p l o t 17 Run 30040 TEST r e g _ e v e n t _i i i_ _i i i i_ _L i I i i i i I i i_ reg.event 5000 dots A 9 o J -2H i 1 1 r n i i r i i i r 10 20 e 30 40 3d Figure 3.25: A dotplot of A9 versus 63d showing that the angular resolution is over the angles generated in G E A N T . Chapter 3. Angular and Coordinate Resolution 53 3.2.2 Importance of Angular Resolution in Cross Section Determination The experimental cross section that we measure is related to the actual theoretical cross section, by a convolution with a detector response function H(t9 — 9'). dcrexp f da dQ, <0) = J^ff)H(8-ff)dff (3.24) In the previous section I described how the angular resolution was determined. Since the histograms of A6> were approximately Gaussian, we can use a Gaussian for our detector response function. Rather than by de-convolution of our experimental data, which would amplify any errors, we wi l l apply a correction similar to that done by Joram et. al. [16]. The method involves normalising the cross section to a well known cross section. For example if we want to find the cross section for 7r ±p scattering we can normalise it to the easily calculated / i ± p scattering cross section as in Equation 3.25. do ± Nrtt(9)R,dar ± ^ ( 7 r p ) = N r u ( 9 ) R ^ p)e™e*> ( 3 - 2 5 ) In Equat ion 3.25, N^catt{9) is the number of pions scattered at angle 9, N^catt(9) is the number of muons scattered at angle 9, Rn and R^ are the relative fraction of pions and muons in the beam, e^et is the detector efficiency, and econv is the correction due to the angular resolution. The correction (econv) is given by Equation 3.26. Using the Karlsrhue-Helsinki (KH80) calculation of the 7 r ± p cross sections and the u^p Mot t cross sections, we can calculate the correction factor. The correction factor and cross sections calculated for 1° detector resolution at an incident pion energy of 40 M e V are shown in Figure 3.26. For the scattering angles that we are interested in , the Chapter 3. Angular and Coordinate Resolution 54 correction is less than or equal to 2%. A t scattering angles less than two degrees, the correction is nonsense. The reason for this is that we haven't continued the cross section to negative angles; the convolution of the resolution function at around zero degrees results in a supposed experimental cross section that is much lower than in reality. Chapter 3. Angular and Coordinate Resolution 55 10* to _Q C X ! \ b TJ 10'-J 10' tO'-J 10 1 1 1 1 1 1 1 1 1 I 1 1 1 1 I 1 1 1 1 II l\ \\ \\ M \\ - - - 7T + \\ \\ 7T~ \.\ \\\ V _ \ \ • \ I I I ' ' ' ' 4.5 CL X b TJ 3.5 H b TJ D 1.5 • 0.5 1 1 ' I 10 10J 10' io3-d fe- 10*-J fe- io"-J fe- io--d 10 _J I I I I I I L J I I I I I L_ -^da/df) c o n v o l v e d withp \\ G a u s s i a n A0 = 1° cr V . \ 10 20 30 40 I I I I I I I I I I I I I I I l _ "I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -10 20 30 40 1.15 1.10 H t 1-05 H 1.00 0.95 _l I I I I I I I L J I I I I I I I I l_ Correction Factor (e ) 'I 1 1 1 1 I 1 1 1 1 20 30 40 0 'I 1 1 1 1 I 1 1 1 ' I 1 1 1 10 20 30 40 0 L A B (degrees) Figure 3.26: Correction factors (eC(mv) to the ir^p cross sections for detector angular resolution of one degree (a). Results are shown for 40 M e V pions, and pn = p^. The plot in the lower left is a ratio of the theoretical cross section to the theoretical cross section convoluted wi th a Gaussian of 1° standard deviation. Chapter 4 Cross Section Considerations The cross section measures the probability for a reaction to occur. Measuring the up —> ixp cross section requires a measurement of: • the number of pions scattered at a particular angle (N^.catt(9sd)), • the number of incident pions on the Hydrogen target ( A ^ n c ) , • the angle of the target relative to the beam (9tgt), • the number of target Hydrogen atoms per unit area (Ntgt), • the solid angle subtended by the detector (dfl), and • the efficiency of the detector (edet). The cross section is given by Equation 4.27. do_ = Nrtt(e3d)cos9tgt dtt N™Ntgtdnedet 1 ' 1 Discussion of what 93d a particle was scattered at can be found in Chapter 2. The result is used in counting pions scattered to different angular bins giving a value of N°catt(93d). The incident number of pions is counted using time of flight wi th respect to the cyclotron R F , and energy loss information from a scintillator near at the entrance of C H A O S (the finger counter). Pressure and temperature of the proposed l iquid H target for C H A O S wi l l be carefully recorded, allowing the target density ptgt to be calculated. 56 Chapter 4. Cross Section Considerations 57 N-tgt can then be found using a measurement of the target thickness t, the atomic weight A , and Avogadro's number NA-Ntgt = ptgtAtNA (4.28) The following sections contain an explanation of: • how particles are identified in C H A O S , • how pion decays are eliminated from our count of the yield, • how the solid angle of the detector is found, • what the effect of binning the data has, and • what effect offsets in the measurement of target angle has. In order to get a good yield, N^catt(6-id), we have to be sure we are counting only pions. Accounting for decays in our yield is thus a very important part of our cross section measurement. 4.1 Particle Identification A t most energies, particles can be identified using their time of flight. The time of flight ( T O F ) is measured using the finger counter scintillator at the entrance of C H A O S , t imed wi th respect to the 23 M H z cyclotron R F . Time of flight spectra at different beam energies, showing the separation of pions, muons, and electrons are shown in Figure 4.27. A t low energies, below 25 M e V , it is no longer possible to separate pions and muons using the T O F . In principle the lower energy T O F should be easier to use since the T O F between particles is greater. However, the T O F is measured wi th respect to the cyclotron R F , meaning that if the particles are spread out by more than 43 ns, the fastest particles Chapter 4. Cross Section Considerations 58 Run 6569 23"Ftb-39 15:25:17 Peak- 490.0 «t bin 320 dan- 12B2.S Run 6SB4 23"Feb-39 15i 25; 55 P t . . - 5D2.0 it bin 321 dm- 12BG.5 ,.=,5 1Q00 110D 1200 1300 14DD 1500 1600 1700 1000 11 DO 1200 1300 MOD 15D0 1600 1700 Un=0. Hi 1107 Full.tcap I f ull _t cap/r eg .event) Ov-U. Un-0. Hi 1107 full.tcap I full_tcap/reg_event) Ov-17. 1000 1 100 1200 1300 14DQ 1500 1600 1700 1000 11 DO 1200 1300 14DD 1500 1600 1700 Un=0. Hi 1116 Fuii.tcap I full_tcap/reg_event! Ov30. Un-D. Hi 1107 Fuli_tcap [full_tcap/reg_event) Ov-7. Run 6449 23-Feb-99 15t 20i 13 Ptalt- 4043. D i t bin 1 Bl data- 646.5 Run 6434 23-Feb~99 15i 21 • 4B P-ak- 903.0 at bin 157 data- B3D.S 500 6DD 700 BOO 9D0 10 DO 1100 120D 500 600 700 BOO 9D0 1 000 1100 1 200 Un=0. Hi 11D6 Fuli.tcap 1ful1_tcap/reg_event! Ow»132. Un»0. Hi t)06 Full„tcap 1full_tcap/reg_eventJ Ov-45. Figure 4.27: T ime of flight spectra at different energies. T O F shows its usefulness for identifying 7r, p, and e at energies above 20 M e V . The humps on the right sides of the pion peaks, in the spectra in the second and third rows, are due to pions which decay in flight from the production target to the C H A O S finger counter. Chapter 4. Cross Section Considerations 59 wi l l get wrapped around to having the slowest T O F . For example, at a pion energy of 17MeV, the T O F between pions and electrons is greater than 43 ns. The electrons, which are faster than the electrons get wrapped around to a slower T O F . Physically this means that we see the electrons from the previous beam burst. To differentiate between pions and electrons, which overlap in T O F at low energies, we need to look at a combination of the T O F information wi th the energy deposited by the particle (AE). Energy deposited is also measured using the finger scintillation counter, whose pulse height is digitised using an Analog to Digi ta l Converter ( A D C ) . The pulse height is proportional to the energy loss, thus the A D C value gives an indication of which type of particle is entering C H A O S . A dotplot of T O F versus AE for an incident pion kinetic energy of 17 M e V , showing the separation of pions and electrons is shown in Figure 4.28. Using information about the particles flight time, and how much energy it loses in matter, we can clearly identify pions. However, at higher energies, the muons are st i l l difficult to identify using this information alone. The much larger number of pions as compared to muons at higher energies, makes the tai l of pions in the T O F spectrum a significant amount at the muon T O F . Since we wi l l be calculating the pion cross section and normalising to the muon cross section, we need to correctly separate muons from pions. A method for identifying the muons at low energies is to look at the momentum (P) that is reconstructed using the wire chamber hits. A dotplot of P versus T O F then gives a reasonable separation of pions and muons. The dotplot of P versus T O F at 55 M e V is shown in Figure 4.29. 4.2 Elimination of Pion Decay Events The half-life of a charged pion is T I = 26.0 ns. A t the lowest pion beam energy being used in the C N I experiment (20 M e V ) , the time it takes a pion to traverse C H A O S (2.0 Chapter 4. Cross Section Considerations 60 1 000 900 700 600 H 500 1 200 f_pid Dotplot 1 Run 6448 TEST reg.event ' ' I I i i . . I reg.event 3000 dots Li e 7T 1 1 1 T—T—r-r-T—t—T—T—i—i—i—I—i 200 250 300 a d c s 2a 1 i 1 350 900 700 600 F.pid Dotplot 1 Run 6434 TEST reg.event _L reg^event 300D dots + LL + + e ~r I 50 200 250 300 a d c s 2 a 350 400 Figure 4.28: T O F on the y axis versus AE on the x axis used for identifying TT, JI, and e at energies below 20 M e V . The dotplots shown are from 17 M e V pion kinetic energy runs. Chapter 4. Cross Section Considerations 61 m•m_v s_ t cap D o t p l o t 44 Run 6865 TEST r e g _ e v e n t 1 4 5 0 _ i i i _ _ i L _ l I I I i I I L_ r e g _ e v e n t 2 0 0 0 d o t s 1 4 0 0 H eu-ro CJ I 1 3 5 0 1 300 H p i o n s 1 2 5 0 T7Y = 55Me m u o n s 1 0 0 "~i 1 1 1 1 1 r 1 20 p i o n d e c a y s | 1 1 1 r 1 40 m o m e n t u m ~ i r 60 ~~I r 1 80 200 Figure 4.29: Momentum on the x axis versus T O F on the y axis used for separating TT, and / i for 55 M e V incident pions. Since the track momentum is being used, the events shown are required to have scattered, and to have made a trigger in C H A O S . The dots at momentum higher than the channel momentum (135 M e V / c ) must be pion decays. Chapter 4. Cross Section Considerations 62 m) is ~ 14 ns. In this time about 37% of the pions wi l l decay into muons. Since this is a three body decay, the decay products can go into a continuum of different directions in the center of mass frame of reference. In the lab frame of reference, the muons wi l l decay into a cone whose angle is dependent on the pion energy. Details of the pion decay kinematics can be found in Section 1.2.5. The change of direction requires us to identify pion decay events. Counting a pion decay as a pion would give an extra count at the wrong angle or momentum, which may or may not cause it to be rejected. In the following sections the methods used to identify pion decay events wi l l be discussed. The methods used are: comparing different momentum calculations of a track, identifying particles in the iru. stack, and using projections of the track to different points in C H A O S . 4.2.1 Use of Different Momentum Calculations Track momentum can be estimated using the hits in wire chambers one, two and three. Wi re chamber four can also be used by itself, or together wi th the inner chambers in the momentum calculation. If we compare the momentum we get using just W C l , W C 2 , and W C 3 to what we get using all the wire chambers, we can see if there was a change in momentum characteristic of a pion decaying. Since only the in-plane information of the wire chambers is used in momentum reconstruction this method wi l l only tell us if a pion decayed wi th a significant momentum change in-plane. A histogram of the difference in momentum using only W C l , W C 2 , and W C 3 to using al l wire chambers is shown in Figure 4.30. This method can not eliminate al l decay events. Even good events wi l l have some uncertainty in their momentum calculation, so we can only cut away some decays. The method also relies on having hits in all the wire chambers; in cases where one or two of the hits are missing we need other methods. Also for decays close to W C l or W C 4 , the Chapter 4. Cross Section Considerations 63 6871 24-Feb-99 I It 07i 06 Peak" 8Q000 7B008. 0 at bin i i i i t i L 102 data-• i I • 71 2604. Mean b i n - 102. 388 Milan data-l^a 5. 776 s l g n a -60000 - \ Figure 4.30: Histogram used in eliminating some pion decay events. When the difference in two different momentum calculations is much bigger than zero, it is tagged as a decay event. The histogram is from a Carbon target run with 55 M e V incident pions. Chapter 4. Cross Section Considerations 64 momentum difference is too small to tell us anything. Finally, the momentum calculated using only W C l , W C 2 and W C 3 has a much larger relative uncertainty than using al l four wire chambers, making the difference difficult to interpret. 4.2.2 Use of the n/j, Stack A t small scattering angles we can use the ixp stack to eliminate pion decay events. The stack was designed to identify pions from muons using the different ranges, energy losses, and T O F of pions and muons in matter. A n introduction to the stack can be found in Section 1.2.5. From test runs done in the summer of 1998, a histogram of the range in the stack is shown in Figure 4.31. Results shown in Figure 4.31 are not using the final version of absorber thicknesses, and analysis methods. Work is currently being done by the Italian part of the collabo-ration to implement a neural network to decide if an event is a pion or a muon in the stack. The stack can also be used to obtain an out-of-plane coordinate that can be used in deciding if a pion decayed dut-of-plane. Out-of-plane information comes from the front eight stack bars. Since each bar has a time to digital converter ( T D C ) at either end, the difference between the signals at either end gives a measure of how high a particle went through the stack. A s wi th the W C 4 out-of-plane information, the z coordinate of the stack (zstk) is affected by the magnetic fringe field of C H A O S . The profiles, out to the stack, of several random tracks in C H A O S are shown in 4.32. In the following section projections, as a method of identifying pion decays, wi l l be discussed. Here I wi l l present the fringe field correction model applied to projecting wire chamber out-of-plane coordinates to the stack. The projection is an estimate of the height at which the stack is hit, including the effect of the focusing of the fringe field. The projection can later be compared directly wi th the out-of-plane coordinate found Chapter 4. Cross Section Considerations 65 Run 85B4 UO-Rpr-39 12i26t27 Peik-t.BSS netn dill-A Run 65B4 QB-Rpr-99 12i26iIB Pelk- 2734.0 II bin 5511. netn bin- 1. 766 linn dm-TT Un-0. HI 1141 stk . range E Stk . range / reg_event ] Ov-D. Run SSB4 GB-flpr-99 13.26.19 Pcik- 52.) it bin 4 dit«- 4.5 . . . I . . . . I . . . . I . . . . 1 • • • • um- 114. M E M bin- 2.93' linn dit»* 3.430 llgn- 1. 4D0|-Un-0. HI 1)42 5tk_rangc_pi t S t k _ r a n o c / g _ s t k _ r a n g c _ p i ] Ov-0. Run 65S4 0B-f*»"93 12.28. 3D Puk- 59.0 it bin 6 dita- 6.5 204. netn bin-e Un-0. HI 1143 stk_range_mu t Stk_ranQe/Q_3tk_ranp,e_inu] Ov-0. Un-0. Hi 11 I ' I I I 2 3 4 5 6 7 44 s t k _ r a n g e _ e l [ S t k _ r a n g e / Q _ s t k _ r a n Q C _ e l ] Ov-0. Figure 4.31: Histogram showing the range of pions and muons in the ir/j, stack. Typica l ly we try to stop pions in layer two or three, and muons in layer four or five. The histograms going clockwise from the upper left corner are of all events, events tagged as pions in T O F , events tagged as electrons in T O F , and events tagged as muons in T O F . Chapter 4. Cross Section Considerations 66 Figure 4.32: Track profiles drawn out to the stack generated using trace. The straight line shows the image line to the stack. Chapter 4. Cross Section Considerations 67 from the stack to see if any pion decays occurred. The model presented here takes the z coordinate of the hits in W C l , W C 2 and W C 4 and corrects them to put them on the image line. A n image line is shown in Figure 4.32; it is the straight line wi th slope equal to the slope of the track at the stack. The correction is given by: z'0 = (apZp + bpzl + cpm) ^ = rriiSp + t\ (4.29) [PiBy + ep where B = (1, 2, i , o) are the hits in W C l , W C 2 , W C 4 inner, and W C 4 outer. The slope of the image line is m,j, and hi is the height of the image line at the target plane. Note that this is exactly the same model being used to correct the W C 4 hits; refer to Section 2.2.2 for more details on this method. The model parameters to put the hits on the image line are given in Table 4.7. p dp h C / 3 dp e / 3 1 1.37 1 5 . 3 x l 0 " 6 189 53800 22400 2 1.25 - 2 . 6 7 x l 0 ~ 6 103 81000 49200 i 1.03 - 1 . 9 8 x l 0 ~ 6 6.02 964000 933000 0 1.02 - 1 . 4 6 x l 0 - 6 3.49 1530000 1500000 Table 4.7: Fringe field correction parameters to put wire chamber hits on the image line to the stack. Parameters determined from trace simulations. Using the parameters, and the actual z coordinates of the hits in the wire chambers, a fit can be done to find m,j and hi. To get an estimate of the height at the stack (z'stk) we need to plug the track length to the stack (sstk) into z'stk = miSstk + hi. The pathlength is found by adding the pathlength out to wire chamber four to the pathlength from W C 4 to the stack. A plot of some in-plane tracks drawn out to the stack, shown in Figure 4.33, shows that we can approximate the track from W C 4 out to the stack as a straight line. Chapter 4. Cross Section Considerations 68 Path of In Plane Tracks -200-100 0 100 200 300 400 x coordinate in mm Figure 4.33: In-plane tracks generated using trace. Note that the scales of the axes are different so that the angles of track slopes to the stack are exaggerated. The heavy lines show the difference between the in-plane slope of the track at W C 4 , and the in-plane slope of the track at the stack. Chapter 4. Cross Section Considerations 69 Theta at Stack vs Theta at WC4 270 255 260 265 270 275 280 Theta at WC4 Figure 4.34: Dotplot of the in-plane track angle at W C 4 to the in-plane track angle at the stack. Angles were found using tracks with random starting positions, angles, and momenta generated using trace. In real data we can easily find the in-plane slope of the track at W C 4 , but what we really want is the slope at the stack. In principle we could determine the in-plane slope of the stack using the photomultipliers on either end of the scintillators in layers two through six of the stack, but the angular resolution obtained may be poor. A method of finding the in-plane slope of the track at the stack, using the slope at W C 4 , is described here. If we use the trace simulation, we can make a dotplot of the in-plane angle of the track at W C 4 (Owe*) versus the in-plane angle of the track at the stack(f9 s t f c). A n empirical relation between the angle at the stack, and the angle at W C 4 is found by looking at the dotplot of Owe* versus 9stk, as shown in Figure 4.34. The relation between the two angles is given by 9stk = 1.006 9WCA — 5.287, where the angles are in degrees. Chapter 4. Cross Section Considerations 70 zstk proj el 0 Dotplot 3 Run 685B TEST tcep .e l . t ee t i I i . i i I i i i i I i—i—i — i zetk_proj_pl_0 Dotplot 7 Run 6858 TEST t c s p . p l . t e o t tcip_el_teit 1DO0 dot! I • 1—•—•—I 1 — 1 1 1 I -1 0 1 stk_z_proJ_1 (X l0* 2 ) 1950 tcip_pl_tcit 1000 dot* -1 0 I stk_z_proJ_1 ( x 10*2) Figure 4.35: Dotplot of the projected height at the stack versus the actual height at the stack. The plot at the left is for electrons, and the plot at the right is for pions. 4.2.3 Use of Track Projections In this section three methods of using track projections to look for pion decays wi l l be presented. The first method uses the stack out-of-plane information, the second method looks at out-of-plane scattering angles found using different combinations of wire chambers, and the final method uses projections back to the target plane. We can look at the correlation between z'stk found in the previous section and the actual z that we get from the stack. If we get points that are far off from the main cor-relation line, then we can conclude that there was an out-of-plane pion decay. Currently the resolution on the T D C s used in the stack is too poor to make any useful conclusions wi th this information. For example the dotplots of the stack projection versus the stack height from hardware for both electrons and pions are shown in Figure 4.35. Using just wire chambers we can look at the out-of-plane angle we get using different combinations of wire chambers. If a decay occurs changing the track momentum in the out-of-plane direction, then we can look for large changes in the out-of-plane angle. For Chapter 4. Cross Section Considerations 71 Run 300OS 24-Feb"99 IBt27i41 Peik- 1 Dt. D it bin 40 diti-I . , , , I , I53S. Mein bln- 3.055 Hem d i l l -decays offn 12—io •! ^  n -8 -B -4 -2 D 2 4 5 8 U n » l l . H i 39B p h i _1 2 _ lo _1 I ph i _ 1 2 _ i o _ l / r eg . e v e n t ) Ov-16. Run 30006 24-FBb"99 18i29. 33 Pctk- 1D0.D it bin 4D dm- 0.1 1426. He in bin- 39.511 decays on LJ_I 0.022 t lO l l - 2.3B9 12—io Run 30DO5 24-Frb~99 19123.10 Peik- 125,0 it bin 40 dita-. • . • 1 I , , , I 1543. Plein bin- 39.201 lie in dn decays of -0.053 i l g i t - t.621 1i-2o i 1 1 1 • i 1 • • 1 1 1 1 • • i ' 1 •0 -6 -4 -2 0 2 4 6 Un-S. H i 336 phl_1 l_2o [ p h i _ l l _ 2 o _ 1 / r e g _ e v e n t 1 Ov-8. Run 3000B 24-Feb-99 1B.29.52 Peik- 105.0 tt bin 40 dltn- 0.1 I . . . . I . . . . I . . . . 1 ' 1459. Rem bin- 39.306 Dun dttb- -0 .039 i l g i t - l .710h decays o Un-36. H i 398 p h l _ l 2 _ l o _ 1 I p h i _ 1 2 _ i a _ 1 / r e g _ e v e n t l Ov-33 . I 1 1 1 1 I 1 1 ' 1 I 1 1 1 1 I 1 1 -B -6 - 4 - 2 0 2 Un-19. H i 336 p h i 1 l _2o [ p h l _ 1 l _ 2 o J / r e g _ e v e n t l Ov-17 Figure 4.36: Out-of-plane angle residuals generated using 40 M e V incident pion G E A N T runs wi th horizontal rod targets. Results wi th pion decays turned on and pion decays turned off are shown. example we can get an out-of-plane angle using just W C l and W C 2 (fin), and we can get an out-of-plane angle using just W C 4 inner and W C 4 outer (cf>io). If the track decays somewhere between W C l and W C 4 then we would expect to get a large residual in the difference between 01 2 and fa0 (4>i2-i0)- One problem with this scheme is that the W C 4 hits are close together in-plane, so a small error in Z{ or zQ wi l l result in a large error in 4>i0. One possibility is to look at the difference between angles 4>u and fao giving 4>u-2o-G E A N T simulations were used to look at these angle residuals wi th pion decays turned off and on; results are shown in Figure 4.36. Chapter 4. Cross Section Considerations 72 vsep_vs_scat+ Dotp lo t 9 Run 6870 TEST r eg . even t J I I I I I I I I I I I I I I I I , I I I I L SEPR A ( m m ) Figure 4.37: Dotplot of A verses 62d used to decide if interaction occurred in target is also used to eliminate decay events. This dotplot came from a C H A O S 55 M e V incident pion run, wi th a 1 2 C target as indicated by the box. Dots falling outside of the box are from events that scattered off one of the wire chambers, or decayed somewhere other than at the target. The resolution in the out-of-plane scattering angle factors into where we can place cuts on the angle residuals. A s can be seen in Figure 4.36 only a few pion decay events can be eliminated using out-of-plane angle differences. One of the best methods of removing pion decay events is by looking at track pro-jections onto the target plane. The in-plane projection of tracks onto the target plane is the same as what is done in deciding if an interaction occurred in the target. Refer to Section 2.3 for the method used in finding the intersection of tracks wi th the target plane. A dotplot of the separation between intersections of incident and scattered tracks wi th the target planes (A) versus the in-plane scattering angle (92d) is shown in Figure 4.37. The Carbon target is clearly identified as having a small A . P ion decays and scattering Chapter 4. Cross Section Considerations 73 v s e p _ v s _ s c a t + Dotp lo t 15 Run 6785 TEST r e g . e v e n t _ i I I i I i . I i 1 I . I i t L vsep_cut_pl 685 dots no target SEPR Figure 4.38: Dotplot of A on the x axis verses 62d on the y axis for two different running conditions. The dotplot on the left came from a C H A O S run wi th 33 M e V pions incident on a plastic sewer pipe. The dotplot on the right is at the same energy but wi th no target in C H A O S . Note that these data have been cut on events that trigger the stack, meaning that we won't get any dots at scattering angles above about 30 degrees. events from wire chambers are clearly seen by the points wi th large A values. To verify that we get a large value of A when the target is far from the center of C H A O S , we can look at data from a C H A O S run where the target is a cylinder of plastic. No part of the cylindrical sewer pipe target is near the origin, so we should expect to get large values of A . The results from a C H A O S run wi th 33 M e V pions incident on a plastic sewer pipe are shown on the left side of Figure 4.38. B y comparing the sewer pipe run to a run wi th no target, we can see that scattering events wi th vertices in W C l , which are further from the center of C H A O S than the sewer pipe, result in larger values of A . The dotplot on the right side of Figure 4.38 is for a C H A O S run wi th no target. Chapter 4. Cross Section Considerations 74 4.3 Angular Dependence of Solid Angle In Section 2.1 we discussed the difference between the C H A O S coordinate system, and the more physical coordinate system with a full three dimensional scattering angle. If we use the C H A O S coordinate system to determine the detector solid angle (dQ) we get the same answer for every scattering angle. The solid angle using C H A O S coordinates is given by Equation 4.30. A typical two degree wide bin in 92d is then associated wi th a fixed element of solid angle (dQ ~ (2°)(14°) ~ 8.5 msr.) The trouble wi th this solid angle calculation is that it does not account for how the particles scatter. Physically, particles are not forced to scatter just in-plane; particles wi l l scatter into a cone wi th a scattering angle 93d- Near 0° and 180° the entire cone falls into the detector acceptance, whereas near 90° only a small fraction does (~ 8%.) The solid angle using the physical coordinate system is given by Equat ion 4.31. Solid angle calculations in the physical coordinate system yield different dQ values at different scattering angles. Graphically the angular dependence of the solid angle can be seen by mapping a uniform (92d,(p2d) distribution to (93d,4>3d)- This mapping is shown in Figure 4.39. The physical solid angle is proportional to the width in <p3d at the scattering, and also has a factor that depends on the bin width being used. For example a two degree bin width has the bin width factor of (cos(93d — 1°) — cos(93d + 1°))- Comparison of cross sections using both solid angle calculations wi l l be done in Chapter 5. (4.30) (4.31) Chapter 4. Cross Section Considerations 75 i 30 CHAOS e 2d Figure 4.39: Mapping of C H A O S angles to physical angles shows how different the solid angle is at small scattering angles. Chapter 4. Cross Section Considerations 76 d Q V M r u e Angle Figure 4.40: Effect on cross section measurement of binning data. 4.4 Effec ts o f B i n n i n g D a t a A t small scattering angles, where the cross section is changing rapidly, we need to consider the effect of binning data. When we put the yield of the cross section in two degree bins, we assume the angle of the bin is at the middle of the bin. For example a b in from 12° to 14° is assumed to be at 13°. In reality the theoretical cross section should be lower, since we need to consider the weight of the cross section. A diagram showing why the binning correction needs to be done is in Figure 4.40. If our assumed bin position is Qun a n d our bin width is A0&i n , then the cross section Chapter 4. Cross Section Considerations 77 at the true bin position is given by: do . \ _ Je'bin=0bm-O.5Aebin dU^bin) smKubin)aabin dQ^rue)- 9'bin=8bin+0.5A9bin \ytrue) — d bin ebin o. ebin . s V^- o z ,J •l0'^=ebin-0.5A8bin Sm{9bin)Mbin The angle 9 t r u e is the weighted mean of the bin. The correction (eun) is the ratio of the cross section at 0trUe to the cross section at Qun-j (9 true) **n = (4-33) For correcting data we divide by the correction. Instead of correcting the data, we wi l l be mult iplying the theoretical cross section by eun to compare wi th our data. The binning correction is only important at very forward angles where the cross section changes most rapidly. The binning correction for the [iC —> \iC cross section was calculated using the theoretical pC cross section. In Figure 4.41, the binning correction is shown. A t angles greater than about ten degrees, the correction is negligible. A lowering of the measured cross section results from the bin correction. 4.5 S c a t t e r i n g A n g l e Offset Effec ts Part of finding the cross section is in knowing the angle of the target relative to the incident beam. A n error in our estimation of the target angle would also introduce an offset in our scattering angle. We use both the intersection of the incident track and scattered track wi th the target plane in our calculation of the scattering angle. Since we do this an error in the target angle estimation becomes a second order effect, not just an offset. The effect of an error in the target angle estimate on the cross section is considered in Section 5.1. Chapter 4. Cross Section Considerations 78 Figure 4.41: Correction to cross section measurement for binning of data. Correction is for a bin width of two degrees. Correction is for uC —• fiC scattering at an incident muon momentum of 135 M e V / c . Chapter 5 The aC Cross Section in the CNI Region The LL^C —> LL^C cross section can be calculated theoretically to very high accuracy. For this reason, and because there is less problem wi th muon decay than there is wi th pion decay, the LIC cross section calculation is a good test of the new scattering angle algorithms for C H A O S . The following section wi l l present the theoretical model used in calculating the uC cross section. 5.1 Theoretical Shape of Cross Section The theoretical cross section is found using the Mot t cross section for a point-like target nucleus, which is modified using a form factor. The Mot t cross section for \iC —> LIC is given by [17]: da^ , a2Z2h2c2 \ £ V . , 0: i =( I* ) ^ ( l - ^ s m 2 ^ ) (5.34) where p M , in M e V / c , is the momentum of the incident muon, Z=6 is the charge of the Carbon nucleus, and the ratio of energies arises from the recoil of the 1 2 C . The ratio of energies is given in Equation 5.35. E' 1 (5.35) In Equat ion 5.34 the total energy of the incident muon is given by E in M e V , and the total energy of the muon after scattering is E'. The cross section is in units of (fm)2. The 79 Chapter 5. The uC Cross Section in the CNI Region 80 momentum transfer from the muon (q) is given by kinematics in the following equation. 2pu sin ^ , Q = / , (5-36) n cv 1 +^ s i n 2^ The form factor we used for 1 2 C was one experimentally determined by Reuter et. al . [18]. From electron scattering measurements, the experimental \iC cross section is given by the product of the form factor squared with the Mot t cross section. The method used by Reuter et. al . to determine the form factor led to a result for the nuclear charge density in terms of a Fourier-Bessel series. The form factor in terms of the charge density is given by a Fourier transform: 1 f „ / ^ s i n ( ? r ) „„ '2 . F(q) = - / p(r)—y-^4irr2dr (5.38) Z J qr The charge of the nucleus is given by Z, and r is the radius from the center of the nucleus. A Fourier-Bessel series is of the following form: p(r) = { Z j n " 1 ™TIR (5.39) 0 r > R where R = 8 fm is a cutoff radius for the charge density. The coefficients an were determined out to n=15, and are summarised in Table 5.8. A plot of the nuclear charge density versus radius shows that the Carbon nucleus has a radius of about 2.5 fm. The Carbon nuclear charge density calculated using the Fourier-Bessel relations given above is shown in Figure 5.42. Theoretical models of the Carbon charge density give results similar to the experimental result. The form factor calculated using the charge density of Reuter et. al . is shown in Figure 5.43. A drop in the point cross section at larger scattering angles is the result of the nucleus having a form. Chapter 5. The pC Cross Section in the CNI Region Figure 5.42: Carbon nuclear charge density. Chapter 5. The pC Cross Section in the CNI Region 12C Form Factor at Tmu=66 MeV T3 g 0.8 -S 0.6 -»—> o d fe a 0.4 5-1 o fe 0.2 0 20 40 60 80 Scattering Angle in Degrees Figure 5.43: Carbon form factor squared. Chapter 5. The pC Cross Section in the CNI Region 83 n an n an 1 1.5737 x I O " 2 2 3.8896 x I O " 2 3 3.7085 x 10~ 2 4 1.4795 x I O " 2 5 -4.4830 x 10~ 3 6 -1.0057 x I O " 2 7 -6.8695 x I O " 3 8 -2.8813 x I O " 3 9 -7.7228 x I O " 4 10 6.6907 x I O " 5 11 1.0636 x IO" 4 12 -3.6864 x I O " 5 13 -5.0134 x IO" 6 14 9.4548 x 10~ 6 15 -4.7686 x IO" 6 Table 5.8: Fourier-Bessel series coefficients for 1 2 C nuclear charge density [18]. The final form of the theoretical cross section can be seen plotted wi th the experi-mental data in the following Section. 5.2 The pC Cross Section Obtained from C H A O S Data In Chapter 4 I explained the many considerations that had to be made in finding a cross section. Here I wi l l use these methods to make a procedure for finding the pC cross section from C H A O S data. For the 66 M e V incident muon data being considered many cuts are applied to the data to ensure most of the events are muons scattering from the Carbon target. Cuts made on the data are: • a box cut on the track momentum versus T O F to the C H A O S finger counter mea-sured wi th respect to the cyclotron R F , • a box cut on A versus 62a, • a cut requiring that there be only one scattered track, • a cut on the range in the stack corresponding to muons, Chapter 5. The uC Cross Section in the CNI Region 84 • a cut on the distance from the origin of C H A O S of the intersection of the incident track wi th the target plane, • a cut on the out-of-plane angle, and • a cut on the difference between the momentum calculated using al l the wire cham-bers to using only W C l , W C 2 , and W C 3 . W i t h al l these cuts, we have eliminated most of the pions from the data. However since the stack was not fully optimised at the time these data were being analysed, the elimination of pions was not complete. The resulting cross section, without applying any binning corrections, or solid angle corrections is shown in Figure 5.44. The cross sections in Figure 5.44 look rather bad, however that is only because the events shown are required to have triggered in the stack. A t scattering angles above 25 degrees, we see the cross section drop off because we have made a cut requiring muons in the stack. The stack physically ends at scattering angles near 25 degrees. Similarly, at the smallest scattering angles the cross section drops off because in the test run that the data comes from the first stack bar from 1° to 6° was not operational. Da t a to be taken this summer wi l l include events in the first stack bar region. In Figure 5.44 we see that adding in the proper solid angle does not have as big an effect as one might expect. A ratio of the solid angle calculated using the old C H A O S coordinates to the solid angle calculated using the new physical coordinates is shown in Figure 5.45. Figure 5.45 shows us that there is little difference between the two solid angles, except at scattering angles less than ten degrees. The events in Figure 5.44 are missing many events in the first stack bar region; when we get data in the first stack bar region this summer, we wi l l be able to see the effect of using the new solid angle. Chapter 5. The pC Cross Section in the CNI Region 85 Figure 5.44: pC —> pC cross section for 66 M e V incident muons calculated using the old solid angle, and without binning correction is shown in the left half of the figure. The cross section using the new solid angle, and wi th the correction for binning is shown in the left half of the figure. Chapter 5. The \iC Cross Section in the CNI Region 86 0 . 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 5 1 0 1 5 2 0 2 5 3 0 L a b S c a t t e r i n g A n g l e ( d e g r e e s ) Figure 5.45: Rat io of the solid angle calculated using the old C H A O S coordinates to the solid angle calculated using the new physical coordinates. Chapter 5. The pC Cross Section in the CNI Region 87 The pC cross sections presented in this section are absolutely normalised. The value of cos ft tgt »r- ,r (5-40) from Equat ion 4.27 was chosen arbitrarily to fit the data to the theoretical curves. The value of the arbitrary normalisation in Equation 5.40 is 1/270. When the efficiency of the stack has been better analysed, the true normalisation can be used. 100 ~o ~o CO %^o o CL 1 ? 5 theoretical using strip 0 - | 1 1 1 1 r 0 10 20 30 40 50 60 t r a n s f o r m e d theta Figure 5.46: Comparison of solid angle correction using theoretical width in fad) and solid angle correction using a strip in fad. The solid angle correction was made using the theoretical width of fad for a C H A O S out-of-plane extent of ± 7 ° . Equivalently we could have done this using the data, but at the cost of some loss in statistics. Throwing away data that has a magnitude of fad greater than 7° has the same effect of applying the solid angle correction. To verify Chapter 5. The pC Cross Section in the CNI Region 88 that the theoretical correction and strip in fad method give the same results, both were calculated using Monte-Carlo techniques. The verification showed that both methods give about the same result. The theoretical correction is shown as a solid line, and the correction using a strip in fad is shown as a dashed line in Figure 5.46. Testing the effect of giving the in-plane scattering angle an offset required modifying the analysis program. The data were rerun wi th both positive and negative scattering angle offsets. Only the in plane scattering angle was given an offset, since only the in-plane angle would be affected by an error in our knowledge of the angle of the target plane. The cross sections wi th +3 and -3 degree in plane scattering angle offsets are shown in Figure 5.47. Increasing the in-plane scattering angle slightly lowers the low scattering angle cross section. Decreasing 62d raises the low scattering angle cross section. Chapter 5. The LIC Cross Section in the CNI Region 89 Figure 5.47: uC cross section wi th positive and negative three degree in-plane scattering angle offsets applied. Results are from C H A O S data runs wi th 66 M e V muons incident on a Carbon target. Chapter 6 Conclusion The C H A O S detector is now calibrated for particles scattering at small scattering angles in the range 6° to 25°. Algorithms used to extract small scattering angle data have been described in this thesis. The scattering angle resolution was found to be better than 1° for the energies of interest in the C N I experiment. C N I data w i l l be collected starting on A p r i l 7, 1999, and ending on August 18, 1999 at T R I U M F . Further data wi l l be taken in the fall of 1999, and again in in the summer of 2000. The stack wi l l be fully calibrated for eliminating pion decay events from C H A O S . D a t a collected in the summer of 1999 wi l l subsequently be analysed to extract ir^p —> ir^p cross sections. Analysis of the data wi l l allow us to get a better measurement of the nN S and P wave scattering lengths, the Re(D+) at t=0, the low energy TIN phases, and the strange quark sea content of the proton. The method used to do this is described in the introductory chapter. More detail on the analysis method can be found in the C N I experiment proposal [19]. Two follow-ups to this thesis are: • finding the fringe field correction parameters using C H A O S data, and • using the projection to the target as a point in fits to find the incident beam angle, or the scattered track angle. The methods for doing these two improvements, and what effect they wi l l have are described in the following two sections. 90 Chapter 6. Conclusion 91 dz at WC4 Outer 200 r 180 160 '-140 r 120^ 100 r 80^ 6 0 -4 0 -2 0 -Z Projection of WC1, WC2 and Fibre Scintillator to WC4 dz outer Nent = 5000 Mean =-0.00465117] RMS =2.06441 -25 -20 -15 -10 -5 0 5 10 15 20 25 Mean = -0.0054811 RMS =1.9117 -25 -20 -15 -10 10 15 20 25 Figure 6.48: Simulated resolution of z\ and z'0 found using W C l , W C 2 and a fibre scin-tillator before W C 3 . 6.1 Determining Fringe Field Correction Parameters with CHAOS Data A follow-up to this thesis would be to determine the fringe field correction parameters using real C H A O S data. One method of doing this involves installing a horizontal fibre scintillator (FS) before W C 3 to get an accurate height reading that can be used in finding the parameters for the object line. Technically this is difficult to do because of space constraints between the wire chambers. It would be much easier to put the fibre scintillator outside of C H A O S . A fibre scintillator outside of C H A O S is not useful for finding estimates of the object line slope and height at the target, because then the FS is in a region after the particle has been deflected by the fringe field. Using the fibre scintillator before W C 3 along wi th W C l and W C 2 information would give a reasonable error in our knowledge of z\ and z'0. A simulation using root was done to see how well we could know the corrected z coordinates in W C 4 using the FS; the simulation of the resolution of the projected z coordinates in W C 4 is shown in Figure 6.48. Chapter 6. Conclusion 92 The fibre used in this simulation was assumed to be 3 m m in diameter. A listing of the root macro used to generate the histogram in Figure 6.48 can be found in Appendix B . A second method of finding the model parameters using real data involves using C H A O S data that we are sure contain no decay events. Without any decay events, the resolution from C H A O S should be much better. In principle the stack wi l l be able to eliminate over 98% of the pion decay events, so data wi th cuts using information from the stack wi l l be useful in finding model parameters. 6.2 Use of the Coordinates of the Projection to the Target Plane In a perfect C H A O S event we get all of the hits in wire chambers, allowing us to do two separate projections to the target; one projection to the target is done using hits from W C l and W C 2 on the incident beam, and the other projection to the target is done using W C l , W C 2 , and W C 4 from the scattered track. When one of the wire chamber hits on the incident beam is missing, we can no longer determine, using only incident wire chamber hits, the tangent vector to the beam at the target plane. Already implemented in analysis code is a recalculation of the incident beam angles when one of the hits in W C l or W C 2 in the incident beam region is missing. The recalculation is done using the coordinates of the scattered track's intersection wi th the target plane along with the wire chamber hit that we do obtain in the incident beam region. When we do get both wire chamber hits in the incident beam, we could use the z_proj as an additional point in our fit to find the parameters of the scattered track. A quick tr ial , looking at the projection of the scattered track back to the target plane, using z_proj as a point on the scattered track is shown in Figure 6.49. To compare the results to where z_proj was not used in the picket fence reconstruction see Figure 3.22. Chapter 6. Conclusion 93 Run 6257 08-flpr"99 I2I19J15 Peak- 23.0 at bin 52 data- 2.5 SUB= 252. Mean bin= 41.214 Mean data= "8.286 signa= 12.551 A 20 • c /I B him i ' ' ' * i 1 1 ' 1 1 • ' i 60 -40 -2D 0 20 40 60 Un=0. Hi 924 sh_a l l_nd2 I h_al l /sm_scat_nd2] Ov = 0. Figure 6.49: Picket fence reconstruction using small scattering angle data from run 6257, where the track reconstruction includes the use of the incoming beam z_proj. Resolutions of the picket fence bars using z_proj as a point i n the fit are considerably better than the resolutions found without using z_proj as a point. The resolutions of picket fence bars using z_proj as a point, for run 6257, are summarised in Table 6.9. One reason for the considerable improvement in the resolution is that any pion decay events which occur in the outgoing beam that are not identified wi l l be corrected slightly by the extra point used in the fit. Bar Z_all a (mm) for small 9 Z-all a (mm) for large 9 A 1.49 1.15 B 2.40 1.80 C 2.19 2.40 Table 6.9: Resolution of picket fence bars for run 6257 at small scattering angles (less than 30 degrees), and at large scattering angles (greater than 30 degrees.) Bibliography [1] H . Yukawa, Proc. Phys. Ma th . Soc. Japan 17, 48 (1935). [2] C . M . G . Lattes, M . Muirhead, G.P.S . Occhialini and C . F . Powell, Nature 159, 694 (1947). [3] W . Heisenberg, Z. Physik 43, 172 (1927). [4] C . J . Oram et. al, Nucl . Inst. Meth . 179, 95 (1981). [5] G . R . Smith et al, Nucl . Inst. Meth . A362, 349 (1995). [6] K . S . Krane, Introductory Nuclear Physics. John Wiley & Sons, New York, 1988. [7] M . Kermani , M.Sc . Thesis, University of Br i t i sh Columbia, Unpublished, 1993. [8] G . C . Barbarino et al, Nucl . Inst. Meth . 179, 353 (1981). [9] C . Bi ino et al, Nuc l . Inst. Meth . A271, 417 (1988). [10] R . R u i , Internal C H A O S document: Preliminary TILL Separator Gameplan, (1998). [11] H . Leutwyler, Chiral Dynamics: Theory and Experiment, Proceedings of the Workshop held at M I T , Cambridge, M A , U S A , 25-29 July 1994. A r o n M . Bernstein and Barry H . Holstein eds., P14. [12] H . W i n d , Nuc l . Inst. Meth . 115, 431 (1974). [13] R . Tacik, Subroutine trace, (1998). [14] W i l l i a m H . Press it et al., Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press, Cambridge, 2nd edition, (1992). [15] Particle Da ta Book, Phys. Rev. D54, 575 (1996). [16] C. Joram et. al, Phys. Rev. C51, 2144 (1995). [17] Halzen, F . and Mar t in , A , Quarks and Leptons. John Wi ley & Sons, New York, 1984. [18] W . Reuter, et. al, Phys. Rev., C26, 806 (1982). 94 Bibliography 95 [19] G . Smith, et. al, T R I U M F Research Proposal, Experiment 778, unpublished, 1996. [20] G . F . Chew and F . E . Low, Phys. Rev. 113, 1640 (1959). [21] T R I U M F Users Handbook. [22] Gertjan Hofman, M.Sc . Thesis, University of Br i t i sh Columbia, Unpublished, 1992. [23] Sheila Mcfarland, M.Sc . Thesis, University of Br i t i sh Columbia, Unpublished, 1993. [24] T R I U M F Kinematics Handbook. [25] J . Lange, P h . D . Thesis, University of Bri t ish Columbia, Unpublished, 1997. [26] D . Halliday, Introductory Nuclear Physics, John Wi ley & Sons, New York, 1955. [27] G . Hofman, P h . D . Thesis, University of Br i t i sh Columbia, Unpublished, 1997. Appendix A Trace Program Used to Generate Tracks Through C H A O S A . l C H A O S Materials File ! This is the material def ini t ion f i l e for CHAOS cni expt. to ta l elo ! UNITS ARE cm ! !Firs t go from target to Q7 window in M13 !in and out of the spectrometer, ie to upstream face of the stack. ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! next go from target to face of stack: M M ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! *_wcl_out a i r 11 .0 ! a ir to anodes (11.3) - gap-roha (1 to aluminum 1. 0e-4 ! r f shield aluminized (guess 1 micron) mylar 2. 54e-3 ! r f shield mylar 1. 25e-3 ! window (should be kapton) rohacell 0. 1 ! rohacell mylar 2. 54e-3 ! inner wall 1 aluminum 1. 0e-4 ! inner wall 1 aluminized (guess 1 micron) magigas 0. 4 ! iron 1. Oe-5 ! anode wires" thick*(width/pitch) mylar 2. 54e-3 ! inner wall 2 96 Appendix A. Trace Program Used to Generate Tracks Through CHAOS copper 1.5e-5 ! 1500 angstroms copper (ECN) rohacell 0.1 ! rohacell mylar 1.25e-3 ! window (should be kapton) mylar 2.54e-3 ! r f shield aluminum 1.0e-4 ! r f shield aluminized (guess 1 micron) ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! *_wc2_out a i r aluminum mylar mylar rohacell mylar aluminum magigas iron mylar copper rohacell mylar mylar aluminum ! ! ! ! ! ! ! ! ! ! ! ! ! *_wc3_out a i r aluminum 11.3 ! a i r to anodes (22.6) - .3 -(wcl+.3) (2 to 1.0e-4 ! r f shield aluminized (guess 1 micron) 2.54e-3 ! r f shield 1.25e-3 ! window (should be kapton) 0.1 ! rohacell 2.54e-3 ! inner wall 1 1.0e-4 ! inner wall 1 aluminized (guess 1 micron) 0.4 ! 0.5e-5 ! anode wires" thick*(width/pitch) 2.54e-3 ! inner wall 2 1.5e-5 ! 1500 angstroms copper (ECN) 0.1 ! rohacell 1.25e-3 ! window (should be kapton) 2.54e-3 ! r f shield I. 0e-4 ! r f shield aluminized (guess 1 micron) ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! II. 1 ! a i r wc3 rad (34.4)-gap(.4)-(wc2+.3) (3 to 1.0e-4 ! r f shield aluminized (guess 1 micron) Appendix A. Trace Program Used to Generate Tracks Through CHAOS 98 mylar 2. 54e-3 ! r f shield mylar 1. 25e-3 ! window (should be kapton) rohacell 0 1 ! rohacell mylar 2 54e-3 ! inner wall 1 copper 1 5e-5 ! 1500 angstroms copper (ECN) magigas 0 7 iron 0 75e-4 ! anode wires" thick*(width/pitch) mylar 2 54e-3 ! inner wall 2 copper 1 5e-5 ! 1500 angstroms copper (ECN) rohacell 0 1 ! rohacell mylar 1 25e-3 ! window (should be kapton) mylar 2 54e-3 ! r f shield aluminum i i i i i i i i i 1 i i i I I I I Oe-4 i i i i I i i ! r f i i i i i shield aluminized (guess 1 micron) i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i *_wc4_out a ir 24.7 20. ! a ir (wc3+.4) to wc4 inner (59.5) mylar 1 25e-3 25. ! kapton inner window rohacell 0 1 25. i mylar 2 54e-3 25. ! kapton aluminum 1 0e-4 25. ! inner wall 1 aluminized (guess 1 micron) magigas 10. 25. ! copper 1 5e-5 25. ! 1500 angstroms Cu (glO is copper coated) glO 0 025 25. ! back wall 250u ! now either the track crosses a spine, or a r i b . ! Spine: (with x=t/cos(65), l/cos(65)=2.37) glO 0 16 65. ! 1.6mm spine at 65deg, x=t/cos(ang) Appendix A. Trace Program Used to Generate Tracks Through CHAOS 99 ! r i b : !mylar 2 54e-3 65. ! inner wall 2 !copper 1 5e-5 65. ! 1500 angstroms copper (ECN) !rohacell 0 1 65 ! rohacell !mylar 2 54e-3 65 ! inner wall 2 !copper i i i i i i i i i i i i 1 M M 5e-5 1 M M 1 65. M 1 M ! 1500 angstroms 1 M 1 M M 1 M 1 M M copper M M M 1 (ECN) M M 1 M 1 *_air:wc4_to_stack a ir 140. mylar 2.54e-3 a ir 1000. end of f i l e s A.2 Trace Code program track c t rack . f c c This program generates out of plane tracks in CHAOS. Hit coordinates c at CHAOS wire chambers 1, 2 and 4 are output. The deviation of the c out-of-plane hits in WC4 from a straight l ine is also output. The c f i l e s of data produced by this program are: c pathXXXX.dat part ic le track coordinates (pathlen,x,y,z) c XXXX=track number Appendix A. Trace Program Used to Generate Tracks Through CHAOS 1 0 0 c hitXXXX.dat coordinates of WC hits (pathlen,x,y,z) c wc4zin.dat z_in , pathlen_in, p/B ra t io , z_ in' c wc4zout.dat z_out, pathlen_out, p/B ra t io , z_out c Written by, B l a i r Jamieson (1998) real*4 a,b,siga,sigb,chi2,q,sig(200) real*4 fs,vertex(3),mom_vec(3),pdat(200) real*4 xdat(200),ydat(200),zdat(200) real*4 wcl(4),wc2(4),wc4(8) character*20 fname character*4 run_num log ica l hit_del integer*4 ndat common / trace_ in / fs,vertex,mom_vec common /trace_out/ wcl,wc2,wc4,pdat,xdat,ydat,zdat external random c I n i t i a l i z e random number generator iseed = 123456789 dummy = random(iseed) c More i n i t i a l i z i n g stuff fs =0.67 ntot = 5000 ph_min = 82. ph_max = 98. fac = cosd(ph_min) - cosd(ph_max) open (unit=20,file='wc4zin.dat',status='unknown') open (unit=22,file='wc4zout.dat',status='unknown') Appendix A. Trace Program Used to Generate Tracks Through CHAOS c I n i t i a l i z e CHAOS Materials arrays c a l l init_mat c GENERATE EVENTS: ngen = 0 nacc = 0 c F i r s t choose starting coordinates 20 x = -1.25 + 2.5 * random(iseed) y = -1.25 + 2.5 * random(iseed) r = sqrt(x**2+y**2) i f (r .gt .2.5) go to 20 z = -2.5 + 5.0 * random(iseed) vertex(l) = x vertex(2) = y vertex(3) = z c Then choose start ing momentum and angles 25 p = 95.0 + 40.0 * random(iseed) th = 360.0 * random(iseed) ph = acosd(cosd(ph_min)-random(iseed)*fac) mom_vec(l) = p mom_vec(2) = th mom_vec(3) = ph c C a l l "trace" to track part ic le through the CHAOS f i e l d c a l l trace(hit_del) ngen = ngen + 1 i f ( .not.hit_del) goto 25 nacc = nacc + 1 Appendix A. Trace Program Used to Generate Tracks Through CHAOS 102 c Extrapolate z at wc4 inner and outer wire chambers c using track data ndat=40 c a l l f i t (pdat ,zdat ,ndat , s ig ,0 ,a ,b ,s iga ,s igb,chi2 ,q) zwc4in = a + b * wc4(l) zwc4out = a + b * wc4(5) c Get dz values dz_in = zwc4in - wc4(2) dz_out = zwc4out - wc4(4) write(20,*) wc4(4)*10.0,b,p/fs,zwc4in*10.0 write(22,*) wc4(8)*10.0,b,p/fs,zwc4out*10.0 c Write out f i r s t few tracks to f i l e i f ( ngen .It . 8 ) then write (6,*),'Momentum=',p,' Rad. Curv.=',p/29.979/fs c a l l yget_runno( ngen, run_num ) fname = 'path' / / run_num / / ' .dat ' open(unit=23,file=fname,status='unknown') do kk=l,200 i f (pdat(kk).eq.0.0) goto 101 write (23,*),pdat(kk),xdat(kk),ydat(kk),zdat(kk) end do 101 continue close(23) fname = 'h i t ' / / run_num / / ' .dat ' open(unit=23,file=fname,status='unknown') write(23,*),wcl(l),wcl(2),wcl(3),wcl(4) Appendix A. Trace Program Used to Generate Tracks Through CHAOS 103 write(23,*),wc2(l),wc2(2),wc2(3),wc2(4) write(23,*),wc4(l),wc4(2),wc4(3),wc4(4) write(23,*),wc4(5),wc4(6),wc4(7),wc4(8) end i f i seed=random(iseed) i f (nacc. lt .ntot) go to 20 close (20) write (6,*) ' Number of generated and accepted events: ngen,nacc end <fc ^ t* *fc t^* *fc *^  *^  *f* *^ 1^^  ^ ^* ^ *fc *k t^* ^ 4^ ^ ^ ''f* ^* ^ *^ ^ *^ subroutine init_mat impl ic i t none c material info arrays real*4 ang(200), dx(200), dist(200), rad(200) integer*4 mater(200), max_mat character*20 label(200) c l i l i n e variables real*4 rpar(10) integer*4 npar, partyp(10), i l en , i s t , ipar(10) character*132 l ine character*80 spar(10) character*20 item c other variables integer*4 str_len, i , j j character*80 s tr ing character*40 mat_name,get_mat_name Appendix A. Trace Program Used to Generate Tracks Through CHAOS 1 0 4 log ica l found integer*4 max_mat_known parameter (max_mat_known=18) c common block common /mat_info/ ang,dx,dist,rad,mater,max_mat,label open(unit=10,file='chaos .mat',status='unknown') max_mat = 0 do while ( .true.) read (10,'(q,80a)',end=300) i l e n , s t r i n g do i=l,10 spar(i)=' ' rpar(i)=0.0 end do c a l l l i l i n e ( s t r i n g , i l e n , n p a r , s p a r , r p a r , i p a r . p a r t y p , i s t ) i f (spar(l)(1:1) .eq. '*') then l i t i s a label to remember do jj=l,20 i tem(jj: j j )=' ' end do str_len = index(spar(1),' ') str_len = str_len - 1 item(l:str_len) = spar(l)(1:str_len) else i f ( spar ( l ) ( l : l ) .ne. ' ! ' + .and. spar ( l ) ( l : l ) .ne. ' ') then max_mat = max_mat + 1 dist(max_mat) = rpar(2) ang(max_mat) = rpar(3) Appendix A. Trace Program Used to Generate Tracks Through CHAOS 105 dx(max_mat) = dist(max_mat) / cosd( ang(max_mat) ) rad(max_mat) = rad(max_mat-l) + dx(max_mat) str_len = index( spar( l ) , ' ' ) str_len = str_len - 1 label(max_mat) = item ! f l i p the str ing to uper case do i=l , s tr_len i f ( ichar(sparCl)( i : i ) ) .gt .64) then spar ( l ) ( i : i ) = + char( ichar(spar(1)( i : i ) ) -+ iand(ichar(spar( l ) ( i : i ) ) ,32)) end i f end do i=0 found = . false. do while (i.It.max_mat_known .and. .not.(found) ) i = i + 1 mater(max_mat) = i mat_name = get_mat_name(i) i f ( mat_name(l:str_len) .eq. spar(1)(1:str_len) ) then found = .true, end i f end do i f ( .not.(found) ) then write(6,*) 'Material ' , spar(1)(1:20),' not in data base' end i f Appendix A. Trace Program Used to Generate Tracks Through CHAOS 106 end i f end do 300 con t inue r e t u r n end sub rou t ine t r a c e ( h i t _ d e l ) c T h i s sub rou t ine t r a c e s a charged p a r t i c l e (assumed to be a p ion ) c th rough the CHAOS f i e l d , i n s m a l l s teps g i v e n by dt ( i n n s ) . c The l o g i c a l v a r i a b l e h i t _ d e l i s set t o t r u e i f the p a r t i c l e h i t s c the d e l coun te r . I f the number of s teps t aken exceeds 200, the c p a r t i c l e i s p robab ly s p i r a l i n g i n s i d e the f i e l d , and the r o u t i n e c r e t u r n s w i t h h i t _ d e l f a l s e , c Inputs ( v i a common b l o c k t r a c e _ i n ) : c f s magnetic f i e l d s t r e n g t h i n T e s l a c v e r t e x ( 3 ) i n i t i a l ( x , y , z ) coo rd ina t e s ( i n cm) c mom_vec(3) i n i t i a l ( p , t h , p h i ) w i t h p i n MeV/c , c 0 < t h < 360, the i n - p l a n e CHAOS a n g l e , c 0 < p h i < 180, the o u t - o f - p l a n e CHAOS angle c (phi=90 i n the mid-p lane of CHAOS) c Outputs ( v i a common b l o c k t r a c e _ o u t ) : c wc l (4 ) ( p a t h l e n , x , y , z ) coo rd ina t e s of h i t i n w c l c wc2(4) ( p a t h l e n , x , y , z ) coo rd ina t e s of h i t i n wc2 c wc4(8) (pa th l en i n n e r , x , y , z i n n e r , c p a t h l e n o u t e r , x , y , z ou te r ) c coo rd ina t e s of h i t s i n wc4 r e s i s t i v e w i r e s Appendix A. Trace Program Used to Generate Tracks Through CHAOS c pdat pathlength data points c xdat x coordinate associated with pdat c ydat y coordinate associated with pdat c zdat z coordinate associated with pdat c Modifications: c July 24, 1998 (ABJ) c Now takes into account e-loss while t rave l l ing c through CHAOS. impl ic i t none real*4 pathlen,prevpathlen real*4 mpi/139.57/,dt/0.05/ real*4 rwcl/11.459/,rwc2/22.918/,rwc3/34.377/ real*4 rwc4_l/61.75/,rwc4_2/67.25/,rdel/72.2/ real*4 fs,vertex(3),mom_vec(3),pdat(200) real*4 xdat(200),ydat(200),zdat(200) real*4 wcl(4),wc2(4),wc4(8) real*4 x , y , z , v x , v y , v z , r , t p i , r p a r real*4 pi , thl ,phi ,e ,gam,bet ,c ,v ,xO,yO,zO,rO real*4 x i , y i , z i , r i , v x i , v y i , v z i , x h i t , y h i t , z h i t , r c u r real*4 de,frac,angl ,zdel integer*4 nlsteps, cur_mat, kk, ccount real*4 xy_to_phi_conv log ica l hit_del c material info arrays real*4 ang(200), dx(200), dist(200), rad(200) integer*4 mater(200), max_mat Appendix A. Trace Program Used to Generate Tracks Through CHAOS character*20 label(200) c common blocks common / t race_ in / fs,vertex,mom_vec common /trace_out/ wcl,wc2,wc4,pdat,xdat,ydat,zdat common /mat_info/ ang,dx,dist,rad,mater,max_mat,label c Starting conditions pi = mom_vec(l) t h l = mom_vec(2) phi = mom_vec(3) e = sqrt(pl**2+mpi**2) gam = e/mpi bet = pl /e c = 0.6439458 / gam v = 29.979 * bet xO = vertex(l) yO = vertex(2) zO = vertex(3) rO = sqrt(x0**2+y0**2) x i = xO y i = yo z i = zO r i = rO vxi = v * sind(phl) * cosd(thl) vyi = v * sind(phl) * sind(thl) vz i = v * cosd(phl) pathlen =0.0 Appendix A. Trace Program Used to Generate Tracks Through CHAOS prevpathlen =0.0 c I n i t i a l i z e arrays to zeros do kk = 1,200 pdat(kk) = 0 xdat(kk) = 0 ydat(kk) = 0 zdat(kk) = 0 end do c Start of main tracing loop hit_del = . false, nlsteps = 0 cur_mat = 1 50 nlsteps = nlsteps + 1 i f (nlsteps.gt.200) go to 60 c a l l r u n g e _ k u t t a ( f s , c , x i , y i , z i , v x i , v y i , v z i , d t , & x ,y ,z ,vx,vy,vz) prevpathlen = pathlen pathlen = pathlen + sqrt( (x-xi)**2 + (y-yi)**2 ) pdat(nlsteps) = pathlen xdat(nlsteps) = x ydat(nlsteps) = y zdat(nlsteps) = z r = sqrt(x**2+y**2) c Lets subtract the energy lost over small step c that we just took t h l = atand( vy / vx ) Appendix A. Trace Program Used to Generate Tracks Through CHAOS 110 p h i = atand( vy / vz / s i n d ( t h l ) ) t p i = e - mpi i f ( r . g t . r i ) then i f ( r . I t . rad( cur_mat ) ) then ! t h i s i s the easy case ! - we ' re s t i l l i n same m a t e r i a l de = 0 .0 r p a r = r - r i c a l l c a l c _ e _ l o s s ( ma te r ( cu r_ma t ) , rpa r , m p i , t p i , de ) e = e - de e l s e i f ( cur_mat .eq . max_mat ) then ! we ' re o u t s i d e spect rometer now, so use a i r de = 0 .0 r p a r = r - r i c a l l c a l c _ e _ l o s s ( 1 , r p a r , mp i , t p i , de ) e = e - de e l s e ! we have t o do s tep through s e v e r a l m a t e r i a l s r c u r = r i do w h i l e ( r c u r . I t . r ) de = 0 .0 i f ( rad(cur_mat) . g t . r ) then ! t h i s i s l a s t m a t e r i a l t o s tep th rough r p a r = r - r c u r c a l l c a l c _ e _ l o s s ( mater (cur_mat) , r p a r , mp i , t p i , de ) e = e - de Appendix A. Trace Program Used to Generate Tracks Through CHAOS 111 t p i = e - mpi r c u r = r e l s e ! s tep through next m a t e r i a l r p a r = r a d ( c u r _ m a t ) - r c u r c a l l c a l c _ e _ l o s s ( mater (cur_mat) , r p a r , mp i , t p i , de ) e = e - de t p i = e - mpi r c u r = rad(cur_mat) cur_mat = cur_mat + 1 end i f end do end i f end i f c Now we have new energy, so r e c a l c u l a t e v x , v y , v z p i = s q r t ( e*e - mpi*mpi ) bet = p i / e v = bet * 29.979 vx = v * s i n d ( p h i ) * cosd( t h l ) vy = v * s i n d ( p h i ) * s i n d ( t h l ) vz = v * cosd( p h i ) c Make sure we get the s i g n s r i g h t . i f ( (vx . I t . 0 .0) .and. ( v x i . g t . 0 .0) .and. + (vy . I t . 0 .0) .and. ( v y i . I t . 0 .0) .and. + ( v x i . g t . abs( v y i ) ) ) then vx = - v x Appendix A. Trace Program Used to Generate Tracks Through CHAOS 112 vy = -vy else i f ( (vx . I t . 0.0) .and. (vxi .gt. 0.0) .and. + (vy .gt. 0.0) .and. (vyi .gt. 0.0) .and. + (vxi .gt. vyi) ) then vx = -vx vy = -vy end i f ccount = 0 i f ( (vx .It 0 0) . and. (vxi •gt. 0 0) ) ccount = i f ( (vx •gt 0 0) . and. (vxi . I t . 0 0) ) ccount = i f ( (vy .It 0 0) . and. (vyi •gt- 0 0) ) ccount = i f ( (vy •gt 0 0) . and. (vyi . I t . 0 0) ) ccount = i f ( ccount eq 2 ) then vx = -vx vy = -vy end i f c Record coordinates of hi t in wcl i f ( (r i . I t . rwcl ) .and. (r .ge .rwcl ) ) then frac = ( r w c l - r i ) / ( r - r i ) xhit = x i + frac*(x-xi) yhit = y i + frac*(y-yi) zhit = z i + frac*(z-zi) angl = atan2d(yhit,xhit) i f (angl .It .0 . ) angl = 360. + angl wcl(l) = prevpathlen + frac*(pathlen-prevpathlen) wcl(2) = xhit Appendix A. Trace Program Used to Generate Tracks Through CHAOS 113 wcl(3) = yhit wcl(4) = zhit end i f c Record coordinates of hi t in wc2 i f ((ri .It .rwc2).and.(r.ge.rwc2)) then frac = (rwc2-r i ) / ( r -r i ) xhit = x i + frac*(x-xi) yhit = y i + frac*(y-yi) zhit = z i + frac*(z-zi) angl = atan2d(yhit,xhit) i f (angl .It .0 . ) angl = 360. + angl wc2(l) = prevpathlen + frac*(pathlen-prevpathlen) wc2(2) = xhit wc2(3) = yhit wc2(4) = zhit end i f c Record z-coordinate of hi t in f i r s t res is t ive wire of wc4 i f ( (r i .It .rwc4_l) .and.(r .ge.rwc4_l)) then frac = ( rwc4_ l -r i ) / ( r -r i ) wc4(l) = prevpathlen + frac*(pathlen-prevpathlen) xhit = x i + frac*(x-xi) yhit = y i + frac*(y-yi) zhit = z i + frac*(z-zi) wc4(2) = xhit wc4(3) = yhit wc4(4) = zhit Appendix A. Trace Program Used to Generate Tracks Through CHAOS 114 end i f c Record z-coordinate of h i t in second res is t ive wire of wc4 i f ((ri.It.rwc4_2).and.(r.ge.rwc4_2)) then frac = (rwc4_2-ri) / (r-ri) wc4(5) = prevpathlen + frac*(pathlen-prevpathlen) xhit = x i + frac*(x-xi) yhit = y i + frac*(y-yi) zhit = z i + frac*(z-zi) wc4(6) = xhit wc4(7) = yhit wc4(8) = zhit end i f c Check ver t i ca l position at del radius i f ( ( r i . I t . rde l ) . and . ( r . ge . rde l ) ) then frac = ( r d e l - r i ) / ( r - r i ) zhit = z i + frac*(z-zi) zdel = abs(zhit) i f (zdel . le .8.9) then hit_del = .true. end i f go to 60 end i f x i = x y i = y z i = z r i = sqrt(xi**2+yi**2) Appendix A. Trace Program Used to Generate Tracks Through CHAOS 115 vxi = vx vyi = vy vz i = vz go to 50 60 continue return end subroutine r u n g e _ k u t t a ( f s , c , x i , y i , z i , v x i , v y i , v z i , & dt ,x ,y ,z ,vx ,vy ,vz) c This subroutine steps a charged part ic le (assumed to be a pion) c through the CHAOS f i e l d c Inputs: fs magnetic f i e l d strength c c constant (6439.5/gamma) c x i , y i , z i i n i t i a l coordinates (in cm) c v x i , v y i , v z i i n i t i a l ve loci t ies (in cm/ns) c dt time interval c Outputs x ,y , z coordinates after step c vx,vy,vz velocit ies after step c Written by: Roman Tacik real*4 klx , k ly , k l z , klvx, klvy, klvz real*4 k2x, k2y, k2z, k2vx, k2vy, k2vz real*4 k3x, k3y, k3z, k3vx, k3vy, k3vz real*4 k4x, k4y, k4z, k4vx, k4vy, k4vz c a l l b f i e l d ( x i , y i , z i , f s , b x , b y , b z ) klx = dt * vxi Appendix A. Trace Program Used to Generate Tracks Through CHAOS k l y = dt * v y i k l z = dt * v z i k l v x = c * (kly*bz - k lz*by) k l v y = c * (klz*bx - klx*bz) k l v z = c * (klx*by - kly*bx) c a l l b f i e l d ( x i + k l x / 2 . , y i + k l y / 2 . , z i + k l z / 2 . , f s , b x , b y , b z ) k2x = dt * (vx i+klvx /2 . ) k2y = dt * (vy i+k lvy /2 . ) k2z = dt * ( v z i + k l v z / 2 . ) k2vx = c * (k2y*bz - k2z*by) k2vy = c * (k2z*bx - k2x*bz) k2vz = c * (k2x*by - k2y*bx) c a l l b f i e l d ( x i + k 2 x / 2 . , y i + k 2 y / 2 . , z i + k 2 z / 2 . , f s , b x , b y , b z ) k3x = dt * (vxi+k2vx/2.) k3y = dt * (vyi+k2vy/2.) k3z = dt * (vzi+k2vz/2 . ) k3vx = c * (k3y*bz - k3z*by) k3vy = c * (k3z*bx - k3x*bz) k3vz = c * (k3x*by - k3y*bx) c a l l bf i e ld (x i+k3x ,y i+k3y , z i+k3z , f s , b x , b y , b z ) k4x = dt * (vxi+k3vx) k4y = dt * (vyi+k3vy) k4z = dt * (vzi+k3vz) k4vx = c * (k4y*bz - k4z*by) k4vy = c * (k4z*bx - k4x*bz) k4vz = c * (k4x*by - k4y*bx) Appendix A. Trace Program Used to Generate Tracks Through CHAOS x = x i + (klx +2.*k2x +2.*k3x +k4x)/6. y = y i + (k ly +2.*k2y +2.*k3y +k4y)/6. z = z i + ( k l z +2.*k2z +2.*k3z +k4z)/6. vx = v x i + (klvx+2.*k2vx+2.*k3vx+k4vx)/6. vy = v y i + (klvy+2.*k2vy+2.*k3vy+k4vy)/6. vz = v z i + (klvz+2.*k2vz+2.*k3vz+k4vz)/6. end SUBROUTINE B F I E L D ( X , Y , Z , F S , B X , B Y , B Z ) REAL*4 B l ( 6 1 , 2 0 , 2 ) , B 2 ( 5 0 1 , 2 0 , 2 ) , B 3 ( 6 1 , 2 0 , 2 ) LOGICAL F I R S T / . T R U E . / IF (FIRST) THEN FIRST=.FALSE. OPEN (UNIT=10,FILE='BFIELD/xz30.nfmt' , & STATUS='OLD',F0RM='UNFORMATTED') DO I = 1, 61 DO J = 1, 20 READ (10) B 1 ( I , J , l ) , B 1 ( I , J , 2 ) END DO END DO CLOSE (10) OPEN (UNIT=10,FILE='BFIELD/xz80.nfmt' , & STATUS='OLD ;,FORM='UNFORMATTED') DO I = 1, 501 DO J = 1, 20 READ (10) B 2 ( I , J , l ) , B 2 ( I , J , 2 ) Appendix A. Trace Program Used to Generate Tracks Through CHAOS END DO END DO CLOSE (10) OPEN (UNIT=10,FILE='BFIELD/xzl40.nfmt' , & STATUS='OLD',FORM='UNFORMATTED') DO I = 1, 61 DO J = 1, 20 READ (10) B 3 ( I , J , 1 ) , B 3 ( I , J , 2 ) END DO END DO CLOSE (10) END IF R=SQRT(X**2+Y**2) IF ( X . E Q . O . ) THEN IF ( Y . G T . O . ) THEN TH=90. ELSE IF ( Y . E Q . O . ) THEN BX=0. BY=0. BZ=B1(1,1,1) GO TO 70 ELSE IF ( Y . L T . O . ) THEN TH=270. END IF ELSE TH=ATAND(Y/X) Appendix A. Trace Program Used to Generate Tracks Through CHAOS IF ( ( Y . G E . O . ) . A N D . ( X . L T . O . ) ) TH=180.+TH IF ( ( Y . L T . O . ) . A N D . ( X . L T . O . ) ) TH=180.+TH IF ( ( Y . L T . O . ) . A N D . ( X . G T . O . ) ) TH=360.+TH END IF TH=-TH AZ=ABS(Z) IF (AZ.GT.9 .5 ) THEN BX=0. BY=0. BZ=0. RETURN END IF IF ( R . L E . 3 0 . ) THEN I=2.*(R+0.5) D I = ( R - 0 . 5 * ( I - l . ) ) / 0 . 5 J=2.*(AZ+0.5) D J = ( A Z - 0 . 5 * ( J - l . ) ) / 0 . 5 T1=B1(I ,J ,1) +DI*(B1(I+1,J,1) - B K I . J . I ) ) T2=B1(I,J+1,1)+DI*(B1(I+1,J+1,1)-B1(I,J+1,1)) BR=T1+DJ*(T2-T1) T1=B1(I ,J ,2) +DI*(B1(I+1,J,2) - B 1 ( I , J , 2 ) ) T2=B1(I,J+1,2)+DI*(B1(I+1,J+1,2)-B1(I,J+1,2)) BZ=T1+DJ*(T2-T1) BX= BR*COSD(TH) BY=-BR*SIND(TH) IF ( Z . L T . O . ) THEN Appendix A. Trace Program Used to Generate Tracks Through CHAOS B X = - B X B Y = - B Y E N D I F E L S E I F ( ( R . G T . 3 0 . ) . A N D . ( R . L E . 8 0 . ) ) T H E N R B = R - 3 0 . I = 1 0 . * ( R B + 0 . 1 ) D I = ( R B - 0 . 1 * ( I - 1 . ) ) / 0 . 1 J = 2 . * ( A Z + 0 . 5 ) D J = ( A Z - 0 . 5 * ( J - l . ) ) / 0 . 5 T 1 = B 2 ( I , J , 1 ) + D I * ( B 2 ( I + 1 , J , 1 ) - 6 2 ( 1 , 3 , 1 ) ) T 2 = B 2 ( I , J + 1 , 1 ) + D I * ( B 2 ( I + 1 , J + 1 , 1 ) - B 2 ( I , J + 1 , 1 ) ) B R = T 1 + D J * ( T 2 - T 1 ) T 1 = B 2 ( I , J , 2 ) + D I * ( B 2 ( I + 1 , J , 2 ) - B 2 ( I , J , 2 ) ) T 2 = B 2 ( I , J + 1 , 2 ) + D I * ( B 2 ( I + 1 , J + 1 , 2 ) - B 2 ( I , J + 1 , 2 ) ) B Z = T 1 + D J * ( T 2 - T 1 ) B X = B R * C O S D ( T H ) B Y = - B R * S I N D ( T H ) I F ( Z . L T . O . ) T H E N B X = - B X B Y = - B Y E N D I F E L S E I F ( ( R . G T . 8 0 . ) . A N D . ( R . L E . 1 4 0 . ) ) T H E N R B = R - 8 0 . I=RB+1.0 DI=RB-I+1. J = 2 . * ( A Z + 0 . 5 ) Appendix A. Trace Program Used to Generate Tracks Through CHAOS D J = ( A Z - 0 . 5 * ( J - 1 . ) ) / 0 . 5 T1=B3(I ,J ,1) +DI*(B3(I+1,J,1) - B 3 ( I , J , 1 ) ) T2=B3(I ,J+l ,1 )+DI*(B3(I+1 ,J+l ,1 ) -B3(I ,J+l ,1 ) ) BR=T1+DJ*(T2-T1) T1=B3(I ,J ,2) +DI*(B3(I+1,J,2) - B 3 ( I , J , 2 ) ) T2=B3(I,J+1,2)+DI*(B3(I+1,J+1,2)-B3(I ,J+l ,2)) BZ=T1+DJ*(T2-T1) BX= BR*COSD(TH) BY=-BR*SIND(TH) IF ( Z . L T . O . ) THEN BX=-BX BY=-BY END IF ELSE IF (R.GT.140. ) THEN BX=0. BY=0. BZ=0. END IF 70 BX=FS*BX*1.0648/10000. BY=FS*BY*1.0648/10000. BZ=FS*BZ*1.0648/10000. END c FUNCTION MAT NAME character*40 f u n c t i o n get_raat_name(jmat) Appendix A. Trace Program Used to Generate Tracks Through CHAOS 122 impl ic i t none character*40 mat_nam(200) integer*4 jmat mat _nam( 1) = 'AIR' mat _nam( 2) = 'SCINTILLATOR' mat _nam( 3) = 'CH2' mat _nam( 4) = 'CD2' mat _nam( 5) = 'CARBON' mat _nam( 6) = 'ALUMINUM' mat _nam( 7) = 'IRON' mat _nam( 8) = 'HELIUM_GAS' mat _nam( 9) = 'SILICON' mat _nam( 10) = 'MAGIGAS' mat _nam( 11) = 'HELIUM_LIQ' mat _nam( 12) = 'BUTANOL_LIQUID' mat _nam( 13) = 'ICE' mat _nam( 14) = 'COPPER' mat _nam( 15) = 'MYLAR' mat _nam( 16) = 'GlO' mat _nam( 17) = 'LIQHYD' mat _nam( 18) = 'ROHACELL' get_mat_name=mat_nam(jmat) return end c===============================================: c SUBROUTINE CALCULATE E LOSS FOR THE MEDIUM Appendix A. Trace Program Used to Generate Tracks Through CHAOS c======================================================== subroutine calc_e_loss(mat,dist,Mpr,Tpr,de_tot) impl ic i t none c th is routine handles to fact that for a large loss the medium has c to be s l iced up. real*4 dist ,Mpr,Tpr,dde,de,de_tot ,Tpr_local integer*4 m a t , i , j , k Tpr_local=Tpr ! dont want to change Proton enery for main c a l l absorb (mat, d i s t , Mpr, 1., 0, Tpr_local , de, dde) c check i f the energy loss was a small fract ion of the energy c bigger then say 10% i f (abs(de/Tpr).It. 0.05) then de_tot=de return end i f c i f not, s l i ce up the material in say 100 s l ices do i= l , 100 c a l l absorb (mat, dist /100. , Mpr, 1., 0, Tpr_local , de, dde) de_tot=de_tot+de Tpr_local=Tpr_local-de c now we could s t i l l be in trouble - when the proton is so slow that c s l ices s t i l l get below 0. de c so check i f the energy <0.01 Mev OR i f the energy loss is off the c order of the energy i t self . i f (Tpr_ loca l . l t . .01 .or. (Tpr_local-de) .It . 0.) return end do Appendix A. Trace Program Used to Generate Tracks Through CHAOS 124 return end c=========================================================== c SUBROUTINE to calculate the energy loss (Greg Smith) c=========================================================== SUBROUTINE absorb (JMAT, xO, xmass, z inc, isteu, ptb, de, dde) c usage: c c a l l absorb (JMAT, xO, xmass, z inc, isteu, ptb, de, dde) c index, thk(cm) -1 momentum eloss(meV) uncert. c index, thk(cm) 0 K . E . eloss(meV) uncert. c index, thk(cm) +1 beta eloss(meV) uncert. character*40 mat_nam(200),mater_nam c param( properties, #of materials) c param(l, i) = Z c param(2, i) = A c param(3, i) = I c param(4, i) = rho(g/cm**3) c param(5, i) = radlength(m) c param(6, i) = Zmoli c param(7, i) = Amoli c param( i , l ) = Air c param( i ,2) = S c i n t i l l a t o r c param( i ,3) = CH2 c param( i ,4) = CD2 c param( i ,5) = Carbon c param( i ,6) = Aluminum Appendix A. Trace Program Used to Generate Tracks Through CHAOS 125 c c c c c c c c c c c c c c param( i ,7) = Iron param( i ,8) = Helium gas param( i ,9) = S i l i con param( i,10)= Magic Gas param( i , l l ) = Helium l i q u i d param( i,12)= butanol l iqu id param( i,13)= Ice param( i,14)= Copper param( i,15)= MYLAR param( i,16)= glO param( i,17)= LH2 param( i,18)= rohacell common/mat/ param(7,20) in the data statement, each l ine for each material. Mat. prop. : (Z, A, I, rho(g/cm**3), radlength(m), Zmoli, Amoli) data param / 1 36., 72., 94.7, .001205, 304.2, 7.2, 14.4 ! a i r 1 , 7. , 13., 63.2, 1.032, 0.424, 6., 12. ! scint 1 , 8., 14., 52.8, 0.926, 0.484, 6., 12. ! CH2 1 , 8., 16., 52.8, 1.065, 0.421, 6., 12. ! CD2 1 , 6., 12., 80.3, 1.74, 0.245, 6., 12. ! carbon 1 , 13., 26.98, 160.9, 2.70, 0.089, 13., 26.98 ! ALUM 1 , 26., 55.85, 300.0, 7.87, 0.0176, 26., 55.85 ! Iron 1 , 2., 4.0 , 29.86 , 0.000178,5298.88 ,2.0 , 4.0 ! He gas 1 , 14.0,28.6,172.0, 2.33, 0.2182, 14.0, 28.6 ! S i l i c o n 1 , 22.8,45.4, 266.8, 0.00204, 121.5, 22.8, 45.4 ! Magic Gas Appendix A. Trace Program Used to Generate Tracks Through CHAOS 126 1 , 2. , 4.0 , 41.7 , 0.125, 7.55, 2.0 , 4.0 ! He l i q 1 , 2 . 8 , 4.9416 , 57.4 , 0.99, 0.34, 2.8 , 4.9416 ! Butanol l i q . 1 , 3.333, 6.005 , 70.3 , 1.00, 0.36, 3.333 , 6.005 ! ice 1 , 29., 64.0 , 320. , 8 .960 , 0.0143, 29.0 , 64.0 ! copper 1 , 3 .778 , 7.12085 , 65.2 , 1.39, 0.287, 3.778 , 7.12085 ! Mylar 1 , 10.0, 20.0, 150.0 , 1.7, 0.194, 10.0, 20.0 ! glO 1 , 2 . , 2., 21.8, 0.0708, 8.65, 1., 1. ! LH2 1 , 6., 12., 80.3, 50.0E-3, 7.93, 6., 12. ! Rohacell 1 , 6 . , 12., 80.3, 1.74, 0.245, 6., 12. ! carbon 1 , 6., 12., 80.3, 1.74, 0.245, 6., 12. / ! carbon zmed=param( 1, jurat) amed=param( 2, jmat) aioni=param( 3, jmat) rhomed=param( 4, jmat) 10 CONTINUE 20 CONTINUE IF(PTB.LT. l .E-5) then write(6,*)' Insufficient energy in absorb: ' , ptb c a l l exit endif IF(ISTEU) 30,40,50 c isteu = -1 —> ptb is momentum 30 E=PTB*PTB+XMASS*XMASS P=PTB T=E-XMASS BETA=PTB/E Appendix A. Trace Program Used to Generate Tracks Through CHAOS 127 BEGA=PTB/XMASS GO TO 60 c isteu = 0 —> ptb is kinet ic energy 40 E=PTB+XMASS T=PTB P=SQRT(PTB*(PTB+2.*XMASS)) BETA=P/E BEGA=P/XMASS GO TO 60 c isteu = +1 —> ptb is beta 50 BETA=PTB BEGA=BETA/SQRT(1.-BETA*BETA) P=BEGA*XMASS E=P/BETA T=E-XMASS 60 CONTINUE dd=.3070*1.01*zmed*zinc*zinc/amed cc=l.022e6/aioni CALL ELOSStofC DEDRHOX,DDEDRH0X2, dd,cc, BETA,BEGA, zinc) DEDX=DEDRH0X*RH0MED DE=DEDX*X0 DDE=SQRT(DDEDRH0X2*RH0MED*X0) return END ; : - — - ; : c (GS) Appendix A. Trace Program Used to Generate Tracks Through CHAOS c========================================================== SUBROUTINE ELOSSTOF(DEDRHOX,DDEDRH0X2,DD,CC,BETA,BEGA,ZINC2) C CALCULATES THE ENERGY LOSS OF AN HEAVY PARTICLE IN MATTER C DEDROHX : MEAN STOPPING POWER dE/d(Rho*x) [MeV*cm**2/g] C DDEDRH0X2 : VARIANCE OF DEDRHOX [MeV**2*cm**2/g] C DD : CONST 0.31007*Zmed/Amed C CC : CONST 2*Me/Imed C BETA : v/c OF INCOMMING PARTICLE C BEGA : BETA*GAMMA OF INCOMMING PARTICLE C ZINC2 : (CHARGE OF INCOMMING PARTICLE)**2 C C. OTTERMANN 25.1.85 C D=0.31007 ! CERN BOOKLET 0.3070*1.01(NU-PART) [MeV*cm**2/Mol] XME=0.511 ! " MASS(ELEKTRON) [Me B2=BETA*BETA BG2=BEGA*BEGA DEDRHOX=0. DDEDRH0X2=0. IFCCC.LT. l .E-5) RETURN DEDRH0X=DD/B2*(ALOG(CC*BG2)-B2) AK=2.6667 CCI0B2=CC*B2 DDEDRH0X2=DD*XME*(1.+AK/CCI0B2*AL0G(CCI0B2) ) *ZINC2 RETURN END c FUNCTION convert x,y to a phi between 0—>360 deg Appendix A. Trace Program Used to Generate Tracks Through CHAOS c Gertjan J Hofman Triumf '90 c This function returns atan(x/y) between 0—>360 deg. c This is used by MANY of the userlib_f routines c GJH 31/8/95 modified to use atan instead. real*4 function xy_to_phi_conv(x,y) impl ic i t none real*4 x,y real*4 rad_to_deg parameter(rad_to_deg=57.29578) i f (abs(x) . l t . l .e-10 .and. y . l t . O . ) then xy_to_phi_conv = 270. return end i f i f (abs(x) . l t . l.e-10 .and. y.ge.O.) then xy_to_phi_conv = 90. return end i f i f ( x . l t . - l .e-11) then xy_to_phi_conv = atan(y/x)*rad_to_deg + 180. return end i f i f (x.gt. l.e-11 .and. y.ge.O.) + xy_to_phi_conv=atan(y/x)*rad_to_deg i f (x.gt. l.e-11 .and. y . l t . O . ) + xy_to_phi_conv=atan(y/x)*rad_to_deg+360. Appendix A. Trace Program Used to Generate Tracks Through CHAOS return end subroutine yget_runno( irunno, crunno ) c Purpose: convert integer run number (irrunno) into a four d i c character array run number c Input: c integer*4 irunno integer containing run number c Output: c character crunno(4) character array to hold run number integer*4 irunno character crunno(4) integer*4 j j , tmp_runno, runno runno = irunno do 33=1 A tmp_runno = runno/10 tmp_runno = tmp_runno*10 crunno(5-jj) = char( runno - tmp_runno + 48 ) runno = runno/10 end do return end Appendix B Root Macro Used to Find Residual in Projected Z at WC4 // void U n f i t ( Double_t * x, Double_t * y, Int_t npts, Double_t & a, Double_t & b ) / / Linear regression macro { Int_t i ; Double_t sx = 0.0 , sy = 0.0; Double_t sxoss =0.0 , st2 =0.0 , t; a = 0.0; b = 0.0; i f (npts <= 2 ) return; for ( i = 0 ; i < npts ; i++ ) { sx = sx + x[i] ; sy = sy + y[i] ; } sxoss = sx/( (Double_t)npts ); for ( i = 0 ; i < npts ; i++ ) { t= x[i] - sxoss; 131 Appendix B. Root Macro Used to Find Residual in Projected Z at WC4 st2 = st2 + t*t; b = b + t * y[i] ; } b = b/st2; a = (sy - sx*b) / ( (Double_t)npts ); return; } // void zwc4fibr() { / / This macro shows what projection of WCl, WC2, and f ibre / / s c i n t i l l a t o r to WC4 res ist ive wires look l ike / / Create a new canvas TCanvas * c3 = new TCanvas("c3","Z Proj", 10, 10, 1000, 500 ); / / Create T i t l e TPaveLabel * t i t l = new TPaveLabeK 0.35, 0.99, 0.65, 0.91, "Z Projection of WCl, WC2 and Fibre S c i n t i l l a t o r to WC4"); titl->Draw(); / / Create 2 pads pdi = new TPad("pdl","WC4 inner", 0.03, 0.03, 0.48, 0.9 ); pd2 = new TPad("pd2","WC4 outer", 0.52, 0.03, 0.97,0.9 ); pdi -> DrawO ; pd2 -> DrawO ; / / Create a histogram TH1F* hist=new THlF("dz inner","dz at WC4 Inner",250,-25.0,25.0) TH1F* hist2=new THlF("dz outer","dz at WC4 Outer",250,-25,250 ); Appendix B. Root Macro Used to Find Residual in Projected Z at WC4 / / Use Global Random Number Generator to generate data gRandom->SetSeed(); Double_t x[3] ,y[3] ; Double_t htO, angO, hl_0,h2_0, hiO, hoO, h l _ l , h2_l , h i l ; Double_t hfibreO, d_hi, d_ho, hoi , h t l , slop; const Double.t deg2rad = 3.141597 / 180.0; const Double_t rwcl = 114.59; const Double.t rwc2 = 229.18; const Double_t rwc4in = 617.5; const Double_t rwc4ou = 672.5; const Double_t r f ibre = 325.0 const Int_t kUPDATE = 250; for ( Int_t i = 0; i< 5000; i++ ) { angO = 14.0 * ( gRandom->Rndm( 1 ) - 0.5 ) * deg2rad; hfibreO = 40.0 + 3.0 * gRandom->Rndm(); htO = hfibreO - r f ibre * tan( angO ); hl_0 = htO + rwcl * tan( angO ); h2_0 = htO + rwc2 * tan( angO ); hiO = htO + rwc4in * tan( angO ); hoO = htO + rwc4ou * tan( angO ); h l _ l = gRandom->Gaus( hl_0, 2.42 ); h2_l = gRandom->Gaus( h2_0, 0.71 ); h i l = gRandom->Gaus( hiO, 2.28 ); hoi = gRandom->Gaus( hoO, 2.28 ); x[0] = rwcl; Appendix B. Root Macro Used to Find Residual in Projected Z at WC4 134 x[l] = rwc2; x[2] = r f ibre ; y[0] = h l _ l ; y[l] = h2_l; y[2] = hfibreO; / / Do f i t U n f i t ( x, y, 3, h t l , slop ); d_hi = hiO - h t l - rwc4in * slop; d_ho = hoO - h t l - rwc4ou * slop; h i s t - > F i l l ( d_hi ); h i s t2 ->Fi l l ( d_ho ); } pd2->cd(); hist->Draw(); pdl->cd(); hist2->Draw(); pdl->Modified(); pd2->Modified(); pdl->Update(); pd2->Update(); 

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