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 S C A T T E R I N G F O R A N G L E S By Blair A l e x Jamieson B . A . S c . , T h e University of B r i t i s h Columbia, 1997 A T H E S I S S U B M I T T E D T H E I N P A R T I A L R E Q U I R E M E N T S M A S T E R F U L F I L L M E N T F O R T H E D E G R E E O F O F S C I E N C E in T H E F A C U L T Y D E P A R T M E N T O F O F G R A D U A T E P H Y S I C S A N D S T U D I E S 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 © Blair A l e x Jamieson, 1999 O F S M A L L In presenting this thesis i n partial fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the L i b r a r y 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 T h e University of B r i t i s h C o l u m b i a 6224 A g r i c u l t u r a l R o a d Vancouver, C a n a d a V 6 T 1Z1 Date: Abstract For measurements of pion-nucleon scattering i n the Coulomb-Nuclear Interference ( C N I ) 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 C a n a d i a n H i g h 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. T h e 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 ( W C 2 ) cathode strips and anode wires, and wire chamber four's ( W C 4 ) two resistive wires. T h e 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 . T h e fringe field of C H A O S acts like a lens, i n that it has a slight focusing effect on charged particles. T h e fringe field correction moves the W C 4 (x,y,z) data point so that it is on the linear object ray. In this paper I will present the model for the fringe field correction. I w i l l explain how the three dimensional scattering angle resolution was determined, and as the final test of the scattering angle I will present the cross-section for fiC scattering, obtained using C H A O S data. ii Table of Contents Abstract ii List of Tables vi List of Figures vii Acknowledgments xiv 1 Introduction to Pions 1 1.1 T h e P i o n Beam, at T R I U M F 3 1.2 T h e C H A O S Spectrometer 6 1.2.1 T h e Proportional W i r e Chambers W C 1 a n d W C 2 6 1.2.2 T h e Drift W i r e Chamber W C 3 9 1.2.3 T h e Vector W i r e Chamber W C 4 1.3 2 10 1.2.4 T h e C H A O S Fast Trigger Blocks 12 1.2.5 The 13 ITLL Stack Coulomb-Nuclear Interference Region Measurements 18 Using C H A O S at Small Scattering Angles 22 2.1 F u l l Scattering Angle Versus In-Plane Scattering Angle 23 2.2 Determining the Out-of-Plane Scattering Angle 26 2.2.1 T h e C H A O S Fringe F i e l d 28 2.2.2 Fringe F i e l d Correction M o d e l 30 2.2.3 G E A N T Simulation Used to F i n d M o d e l Parameters 34 iii 2.2.4 2.3 3 5 6 34 T h e Interaction Vertex at Small Scattering Angles 37 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 4 Determining Fringe F i e l d Correction M o d e l Parameters A n g u l a r Resolution 49 3.2.1 G E A N T Simulation of Angular Resolution 51 3.2.2 Importance of Angular Resolution i n Cross Section Determination 53 Cross Section Considerations 56 4.1 Particle Identification 57 4.2 E l i m i n a t i o n of P i o n Decay Events 59 4.2.1 Use of Different M o m e n t u m Calculations 62 4.2.2 Use of the TT/J, Stack 4.2.3 Use of Track Projections 64 70 4.3 A n g u l a r Dependence of Solid Angle 74 4.4 Effects of B i n n i n g D a t a 76 4.5 Scattering Angle Offset Effects 77 The LLC Cross Section in the C N I Region 79 5.1 Theoretical Shape of Cross Section 79 5.2 T h e LLC Cross Section Obtained from C H A O S D a t a 83 Conclusion 90 6.1 Determining Fringe F i e l d Correction Parameters w i t h C H A O S D a t a 6.2 Use of the Coordinates of the Projection to the Target Plane iv . . 91 92 Bibliography 94 A Trace Program Used to Generate Tracks Through C H A O S 96 A.l C H A O S Materials File 96 A.2 Trace Code 99 B Root Macro Used to Find Residual in Projected Z at W C 4 v 131 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 z , and z' are in millimeters. The paa a rameters d and e have units of (Mev/Tesla) 32 2 a 3.3 a Resolution of picket fence bars for run 6257 at scattering angles greater than 30 degrees 3.4 44 Resolution of picket fence bars for run 6258 at scattering angles greater than 30 degrees 3.5 45 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 5.8 Fourier-Bessel series coefficients for 6.9 Resolution of picket fence bars for run 6257 at small scattering angles (less 1 2 C nuclear charge density [18]. 67 . . . than 30 degrees), and at large scattering angles (greater than 30 degrees.) vi 83 93 List of Figures 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. T h e ions travel i n circular trajectories w i t h radius based on the momentum of the ions. 4 1.3 T h e M 1 3 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. 1.5 7 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. T h e circle of dots i n the center of the figure represent the wires i n W C 1 . T h e two circles of dots around W C 1 are the wires in W C 2 and W C 3 respectively. T h e 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. T h e in-plane (x,y) C H A O S coordinate system is also shown i n this figure 1.6 8 A cell of W C 4 showing anodes ( A ) , guard wires (G) and resistive wires ( R ) . A l l units i n this figure are i n millimeters 1.7 11 Resistive wire of length L w i t h charge injected at a distance D from the centre of the wire 1.8 12 Sketch of the n/i stack showing five layers of scintillators and four layers of a l u m i n i u m absorbers 1.9 14 Simulation of pion decay kinematics for 55 M e V pions. T h e x axis is the scattering angle i n degrees and the y axis is the momentum i n M e v / c . vii . 16 1.10 Illustration from raw C H A O S data of why the stack is important for eliminating pion decay events. T h e scatterplot shown at top is from a 55 M e V C a r b o n target run, and the lower scatterplot is the calculated kinematics. B o t h scatterplots have scattering angle i n degrees on the y axis, and momentum i n M e V / c on the x axis 1.11 M a n d e l s t a m plot of v versus t. 17 T h e 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 i n , y is perpendicular to x, and z is the out-of-plane axis. Note that the scattering vector shown i n this figure is the difference between the beam tangent vector, and scattered track tangent vector taken at the origin. T h e y axis is actually drawn as -y to make drawing the vector components easier 24 2.13 Angle definitions i n the C H A O S coordinate system, and i n the physical coordinate system. T h e y axis is drawn as -y to make vector components easier to draw 25 2.14 Geometry used i n finding pathlengths to W C 1 and W C 2 . Note that the wire chambers are not drawn to scale 27 2.15 Vertical track profiles show the focusing effect of the fringe field. Note the change i n scale from vertical to horizontal. T h e uppermost track has a solid straight line overlaid, illustrating how far off the object line the W C 4 hits are 29 viii 2.16 Particle tracks at small scattering angles i n C H A O S . T h e vertical profile for each track is shown below each in-plane track. T h e in-plane tracks shown at top axe display events described i n Section 1.2. In the vertical profiles, the points represent hits i n the wire chambers i n 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 2.17 Az 33 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. T h e upper four plots are residuals for W C 4 inner, and the bottom four plots are for W C 4 outer. T h e y axes are all Az i n millimeters. T h e 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 w i t h the uncorrected z, and the product of the track slope w i t h the uncorrected z squared. T h e upper two rows are for the inner resistive wire, and the bottom two rows are for the outer resistive wire. 2.18 Az . 36 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 i o n decays were disabled i n the G E A N T run 2.19 D i a g r a m showing how the in-plane scattering angle is found 37 38 3.20 Sketch of copper picket fence shows location of horizontal pickets, labelled A through F . T h e figure is not drawn to scale ix 41 3.21 Picket fence reconstruction from: run 6257 w i t h pickets A , B , and C clearly identified, and run 6258 w i t h pickets C , D , E , and F . Refer to text for more details 43 3.22 Picket fence reconstructed using small scattering angle data from r u n 6257, w i t h pickets A , B , and C clearly identified 46 3.23 All But One fits used to get an estimate of the resolution i n 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 6 d showing that the angular resolution is constant 3 over the angles generated in G E A N T 3.26 Correction factors (e ) conv 52 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- T h e plot i n the lower left is a ratio of the theoretical cross section to the theoretical cross section convolved w i t h a Gaussian of 1° standard deviation 55 4.27 T i m e 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 . T h e humps on the right sides of the pion peaks, in the spectra i n the second and t h i r d rows, are due to pions which decay i n 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 . T h e dotplots shown are from 17 M e V pion kinetic energy runs 60 x 4.29 M o m e n t u m 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 i n C H A O S . T h e dots at momentum higher than the channel momentum (135 M e V / c ) must be pion decays 61 4.30 Histogram used i n eliminating some pion decay events. W h e n the difference i n two different momentum calculations is much bigger than zero, it is tagged as a decay event. T h e histogram is from a C a r b o n target r u n w i t h 55 M e V incident pions 63 4.31 Histogram showing the range of pions and muons i n the TV/J stack. T y p i cally we try to stop pions i n layer two or three, and muons i n layer four or five. The histograms going clockwise from the upper left corner are of a l l events, events tagged as pions i n T O F , events tagged as electrons in T O F , and events tagged as muons i n T O F 65 4.32 Track profiles drawn out to the stack generated using trace. T h e 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 w i t h 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. T h e plot at the left is for electrons, and the plot at the right is for pions 70 xi 4.36 Out-of-plane angle residuals generated using 40 M e V incident pion G E A N T runs w i t h horizontal r o d targets. Results w i t h 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 i n target is also used to eliminate decay events. T h i s dotplot came from a C H A O S 55 M e V incident pion run, w i t h 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. T h e dotplot on the left came from a C H A O S run w i t h 33 M e V pions incident on a plastic sewer pipe. T h e dotplot on the right is at the same energy but w i t h no target i n 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 M a p p i n g 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 4.41 Correction to cross section measurement for binning of data. 76 Correction is for a b i n w i d t h of two degrees. Correction is for LIC —> LLC scattering at an incident muon momentum of 135 M e V / c 78 5.42 C a r b o n nuclear charge density 81 5.43 C a r b o n form factor squared 82 xii 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 i n the left half of the figure. T h e cross section using the new solid angle, and w i t h the correction for binning is shown i n the left half of the figure 85 5.45 R a t i o 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 w i d t h i n fad, and solid angle correction using a strip i n fad 87 5.47 LIC cross section w i t h positive and negative three degree in-plane scattering angle offsets applied. Results are from C H A O S data runs w i t h 66 M e V muons incident on a C a r b o n target 89 6.48 Simulated resolution of z\ and z' found using W C 1 , W C 2 and a fibre 0 scintillator before W C 3 91 6.49 Picket fence reconstruction using small scattering angle data from r u n 6257, where the track reconstruction includes the use of the incoming beam z_proj 93 xiii Acknowledgment s W i t h o u t 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 m y supervisor, D r . Greg S m i t h , for providing me w i t h an interesting and challenging project to work on. His ideas always pointed out a new way of looking at a problem. T h a n k s are due to D r . K e l v i n R a y wood for his many ideas on how to improve my algorithms and on how to proceed w i t h my project. I would also like to thank D r . R o m a n 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. xiv 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 w i t h 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. T h e pion was found by Lattes et al i n 1947 from cosmic ray tracks on photographic plates [2]. F r o m the Particle D a t a Book, the pion has a mass m , = 1 3 9 . 5 6 6 9 M e V / c 2 . By Heisenberg's uncertainty principle [3] this mass corresponds to an interaction range of he d « 5 t - c « - ^ « 1 . 4 / m . mc z (1.1) T h i s 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 Q u a n t u m C h r o m o d y n a m i c s ( Q C D ) . In Q C D the pions are made of quark antiquark pairs (TT = ud, u~ = ud). + T h e lightest pair of quarks are called up (u) w i t h a charge + | and down (d) w i t h a charge — | . T w o heavier pairs of quarks have been observed; the first pair are charm (c), and strange (s); the second pair are top (t) and b o t t o m (b). In direct analogy w i t h 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 i a the strong force is shown i n Figure 1.1. T h e reason for the name Chromodynamics comes from the colour quantum number of the quarks. Quarks that make up Hadrons (protons, neutrons, etc.) come i n colourless 1 2 Chapter 1. Introduction to Pions Figure 1.1: A n Example of Strong Force Interaction p n -> n p. combinations of the three quark colours. T h e colour quantum number explains the following problem. B y Pauli's exclusion principle, only one Fermion can be i n 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 i n the same state. M a n y combinations of quarks are not observed i n nature. For example a combination of two up quarks (uu) w i t h a charge of + | is not observed. Requiring only colourless combinations of quarks solves this problem. Three kinds of pions exist (IT , 7r°, and 7r~), a n d are found to conserve isospin ( T , T ) + z in nuclear reactions. T h e isospin of pions is T = l ; the isospin of 7 r is T =+1, the isospin + z of 7T° is T =0, and the isospin of 7r~ is T =—1. T h e conservation of isospin i n the strong z z force interaction i n Figure 1.1 can be verified knowing that a proton has T — + | , a n d z a neutron has T = —\. T h e fact that protons and neutrons have nearly the same mass, z 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 3 Chapter 1. Introduction to Pions in electromagnetic interactions is a small difference i n the pion masses (m^± = 139.5669 MeV/c , 2 and m , » = 134.9745 MeV/c .) 2 Pions have been used as probes into nuclear structure for many years. M o r e recently attention has been focussed on using the pion to determine parameters i n Q C D . In the followings sections I will discuss how pions are created at the T r i - University Meson Facility ( T R I U M F ) , introduce the Canadian H i g h Acceptance O r b i 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 T h e 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. T h e T R I U M F cyclotron has a large dipole magnet, between the poles of which ions are accelerated by a radio frequency ( R F ) electric field oscillating at 23 M H z . T h e basic operation of a simple cyclotron is illustrated i n 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 w i t h 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. E a c h sector has a high magnetic field, and between each sector there is no magnetic field. T h e alternating sectors provide focusing of the beam w i t h carefully designed azimuthal components of the field i n between sectors. Cyclotrons w i t h this type of design are called azimuthally varying field ( A V F ) cyclotrons. A s the H~ accelerates, its radius i n the cyclotron grows, i n proportion to its speed. W h e n the ions get to the outer edge of the Chapter 1. Introduction 4 to Pions X X X X X / X X \ X X X X X X X X Magnetic Field Figure 1.2: Basic operation of a cyclotron showing the path of an ion. T h e ions travel i n circular trajectories w i t h radius based on the momentum of the ions. cyclotron they are extracted by sending them through a thin piece of foil. T h e 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 i n 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 w i t h a target. Since the primary beam has a higher energy t h a n the mass of a pion, pions are ejected from the target. T h e pion production target is usually made of C l2 or Be. 9 For example pions are produced by bombarding C 12 following reactions: P + 1 2 C- P + C l2 p + B + 7T+ p + N + TT" 12 12 w i t h protons i n the Chapter 1. Introduction horizontal vertical 5 to Pions vacuum valvft beam blocker a horizontal sjit absorb er , .vert!COl slit jaws target ^ IATI X F 3 final foe diagnostics' j vertical slit horizontal slit J 0 1 2 3 feat Figure 1.3: T h e M 1 3 beam line layout [4]. In the Coulomb-Nuclear Interference (CNI) experiment being done w i t h C H A O S , the M 1 3 channel at T R I U M F is being used. T h e M 1 3 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. M u o n s in the channel are produced by pions that are stopped near the surface of the production target, and by pions that decay to muons i n flight along the channel. Pions decay into muons and muon neutrinos (TT —* p + + + and n~ —> p~ + T h e M 1 3 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]. T h e beam line elements in M 1 3 are shown i n Figure 1.3. T h e beam line elements are tuned to produce a focus at the C H A O S target 80 c m downstream of F 3 . 6 Chapter 1. Introduction to Pions 1.2 The C H A O S Spectrometer The C H A O S spectrometer is a 360 degree pion spectrometer w i t h a momentum resolution A P / P of about 1% cr, and an out-of-plane acceptance of ± 7 ° [5]. T h e spectrometer consists of a 55 ton dipole magnet between the poles of which lies four cylindrical rings of wire chambers. T h e inner three wire chambers lie i n the uniform magnetic field of the dipole magnet, while the outer most wire chamber lies i n the fringe field of the magnet. In a cylindrical ring 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 i n 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. T h e path of the incident pion is recorded by hits i n each of the wire chambers. T h e path pion can later be reconstructed using the wire chamber hits (refer to Section 2.2.) T h e pion scatters off an atom i n the target and travels along another approximately circular path w i t h a new radius. A g a i n hits i n the wire chambers are recorded, allowing the pion's path to be reconstructed. A typical C H A O S event is shown i n Figure 1.5. The following sections contain details on the operation of the wire chambers, a description of the C F T blocks, and introduces the TILL stack used to distinguish pions and muons. 1.2.1 The Proportional Wire Chambers W C 1 and W C 2 The two inner most wire chambers ( W C 1 and W C 2 ) are proportional chambers that are used to track b o t h incident and scattered tracks. W C l and W C 2 each contain 720 wires w i t h 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. 8 Chapter 1. Introduction to Pions -500 0 500 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 i n millimeters. T h e circle of dots i n the center of the figure represent the wires i n W C 1 . T h e two circles of dots around W C l are the wires i n W C 2 a n d W C 3 respectively. T h e circle of radial dashes represent the W C 4 anode wires and resistive wires. T h e circular arrangement of blocks represent the C F T blocks. T h e in-plane (x,y) C H A O S coordinate system is also shown i n this figure. Chapter 1. Introduction to Pions 9 respectively. T h e proportional chambers are filled w i t h a gas mixture of 80% CF4 a n d 20% isobutane. A high voltage is applied to the wire chamber anodes. T h e 360 strips of cathode on the chamber wall are inclined at 30° from the anode wires. T h e cathode strips are on the larger radius wall of the wire chamber, while the smaller radius wall has a single cathode foil. W h e n 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 a n d cathode is large enough to accelerate the ions and electrons to the point where secondary ionisations occur. T h i s amplification of ionisations is called a Townsend avalanche [6]. T h e result is that larger voltage pulses can be obtained on the wires. In principle the voltage of the pulses i n a proportional chamber is proportional to the energy that the particle loses i n 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 i n Section 2.1. T h e x and y coordinates are given by the centroid of the cluster of anode wires that fire. T h e z coordinate requires the use of the cathode strips which are inclined at 30 degrees w i t h respect to the anodes. Intersecting the centroid of the cluster of anode hits w i t h 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 W i r e chamber three is a cylindrical drift chamber located at a radius of 343.77 m m . 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. T h e relation is kept simpler, i n the absence of a magnetic field, i f there is 10 Chapter 1. Introduction to Pions a uniform electric field. T h e electric field is kept nearly uniform by placing thick field shaping cathode wires between each anode wire. T h e 144 anode wires are equally spaced 15mm apart around the circumference of the chamber, and each drift cell is equipped w i t h 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 i n 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 W C 4 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. E a c h of the 100 trapezoidal cells i n W C 4 contains eight anode wires, four guard wires, and two resistive wires. A cell of W C 4 is shown i n 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. A n o d e wires provide drift times, and the anode staggering resolves the left-right ambiguity. T h e drift time to distance relation is kept reasonably linear by having cathode strips on a l l sides of the chamber walls. T h e 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. W h e n 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 w i l l be the same, resulting i n 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 w i 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 i n this figure are i n millimeters. 12 Chapter 1. Introduction to Pions L -«£ D ^ t Q ADC A ADC B Figure 1.7: Resistive wire of length L w i t h 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 = In equation 4A + — +2 B (1.2) v ; 1.2, A and B are the amount of charge collected by the A n a l o g to D i g i t a 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 i n drift chambers w i t h 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 a l l the detector information to tape. T h e 20 C F T blocks are a crucial element i n making this decision. E a c h C F T block contains two layers of N E 1 1 0 Scintillators and a lead glass Cerenkov detector. These scintillators, along w i t h a scintillator situated where the beam enters C H A O S are used i n 13 Chapter 1. Introduction to Pions coincidence as a first level trigger for readout. T h e 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 i n the Coulomb-Nuclear interference ( C N I ) experiment are so low, there is a considerable background due to pion decays. Thus a method of identifying pions from muons is required. T h e TTU stack, designed by the Italian collaborators, does this by exploiting the different 7r a n d p energy loss, and by using range characteristics of 7T and /x. T h e stack consists of five layers of a l u m i n i u m absorbers sandwiched by six layers of scintillators. O p t i m u m particle identification at each energy of interest is determined using stack calibration data that are fed through a neural network. T h e first set of scintillators consists of eight vertical bars that are read out using photomultiplier tubes on either end. T h e photomuliplier signals are digitised using A n a l o g to D i g i t a l Converters ( A D C s ) , a n d T i m e to D i g i t a l Converters ( T D C s ) . U s i n g a difference over the sum of these A D C s provides a relative height at which a particle passes through the stack. T h e other five layers of scintillators each consist of two horizontal scintillator bars. A sketch of the m a i n elements of the stack are shown i n Figure 1.8. T h e center of the stack is at 209 c m from the center of C H A O S , a n d 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]. T h e stack is particularly important i n 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 w i l l scatter into a cone. T h e 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 a l u m i n i u m absorbers. 15 Chapter 1. Introduction to Pions _ l+ l m rpCM m T h e direction i n the C M of the decay muon is arbitrary. T o Lorentz boost the muon into the laboratory frame of reference we need to know which direction the muon decays in. T o do a simulation of the decay kinematics, we need to know the original kinetic energy of the pion (T ), a n d generate a random in-plane angle (6CM) measured from the n direction of motion of the pion. T h e Lorentz boost i n the x direction, defined here to be the direction of motion of the pion, is given by: — MK. run 1 The y direction of the momentum requires no Lorentz boost since it is perpendicular to the motion of the C M . T h e 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^ versus 9 BAB AB for 55 M e V (135.4 M e V / c ) pions is shown i n Figure 1.9. In real data, the pions and muons w i l l be much more spread out than is shown i n Figure 1.9, because of quantum mechanical effects and detector resolution. Some raw d a t a from C H A O S of the pfc AB shown i n Figure 1.10. versus 9 BAB data for 55 M e V (135.4 M e V / c ) pions are Chapter 1. Introduction to Pions 16 Figure 1.9: Simulation of pion decay kinematics for 55 M e V pions. T h e x axis is the scattering angle i n degrees and the y axis is the momentum i n M e v / c . 17 Chapter 1. Introduction to Pions 2Q | •oi.VB.Bcat i i i _ l Dotplot 18 Run 6852 TEST r e g . e v e n t 1—rjl i i I i—i—i—i—I—i—i——•—•- reg.event 3000 dots 15H 5-{ —I 1 1— -20 1 1 1 -IS MOMT pliunu 20 I I 1 I 1—I r -10 l_^J (x10 M + Dotplot 1 1 I 23 Run 1 1 L. 8852 TEST _J i i i 2000 dot* 3000 doti 15H .10 H 5H -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 eliminating pion decay events. T h e scatterplot shown at top is from a 55 M e V C a r b o n target run, and the lower scatterplot is the calculated kinematics. B o t h scatterplots have scattering angle i n degrees on the y axis, and momentum i n M e V / c on the x axis. 18 Chapter 1. Introduction to Pions 1.3 C o u l o m b - N u c l e a r Interference Region Measurements Coulomb-Nuclear Interference (CNI) measurements are n^p —» ir^p cross section measurements 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 r y i n g to extract threshold parameters, we get data at low incident pion energies and low scattering angles where the Coulomb interaction becomes more important. T h i s section contains a simple incomplete description of how the low energy, and small scattering angle d a t a 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 ) + Cheng-Dashen point (v—0, t=2u ). Here 2 i n the (z/,t) plane to the 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') u = -(p-q') 2 2 u = (p' ~q) 2 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 i n 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. T h e physics of interest is measuring the explicit breaking of chiral symmetry i n Q C D . C h i r a 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: M a n d e l s t a m plot of v versus t. T h e hatched regions show where physical measurements for s-channel and u-channel scattering are made. 7rp scattering is an s-channel process. 20 Chapter 1. Introduction to Pions symmetric it w i l l have b o t h 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 a n d left handed quarks. T h e consequence of spontaneous symmetry breaking was studied by Goldstone, and was found to result i n massless particles (the Goldstone Bosons). Since quarks have a small mass, the right a n d left handed quark states m i x , explicitly breaking chiral symmetry, a n d resulting i n Goldstone Bosons (TT, K , 77). T h e 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]. T h e sigma term defined i n E q u a t i o n 1.3 is related through some algebra to the strange quark content of the proton (y). rh _ o = ^ <p\uu + dd\p> ^ ~ _ " (1.3) 35±5MeV 1-2/ 2<p\ss\p> <p\uu+dd\p> In E q u a t i o n 1.3, rh is half the sum of the u and d quark masses (rh — \(m u + m )). T h e d sigma term is related to the forward scattering amplitudes i n a complex way through the capital sigma term given by: ^ = FlD (v + = 0,t = 2u ) (1.4) 2 where F = 9 2 A M e V is the pion decay constant. n A n extrapolation of carefully chosen C N I cross section measurements to threshold w i l l be used to find the pion S and P wave scattering lengths and . T h e scattering lengths appear i n the threshold subtraction constants which parameterise the dispersion relations used i n 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. 21 Chapter 1. Introduction to Pions Finally, the C N I data will be used to get a better measure of the TTN partial 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 O b t a i n i n g good small scattering angle data are of fundamental importance for the C N I experiment. T h i s chapter explains the methods used to get the best possible estimate of small scattering angles. T h e 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 i n 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 i n the target using vertex reconstruction. B y inspecting the section titles i n this chapter, one can see that the m a i n difficulty i n determining the small scattering angles is i n decoding what happens i n 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. K n o w i n g the out-of-plane coordinate (z) of the W C 4 hit w i l l improve the efficiency of determining the out-of-plane angle. Recognising out-of-plane pion decays w i 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. 23 Using CHAOS at Small Scattering Angles having an extra point will 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 U n t i l recently, C H A O S experiments d i d not go down to low scattering angles, so it was sufficient to use the in-plane scattering angle. In this coordinate system 0 d is defined as 2 the angle i n the horizontal x y 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 i n w i t h BQHAOS — 0, w i t h no rotation of C H A O S , and no translation of C H A O S . T h e in-plane C H A O S coordinates can be seen i n Figure 1.5. T h e in-plane scattering angle, 02d, and the full scattering angle, 9 d are defined i n Figure 2.12. 3 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, w i t h 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 (6 , 3d fad)- T h e unit coordinates i n terms of the angles i n each system are: C H A O S Coordinates x = cos(6>2d) sm(fa ) (2.5) y = sm(6 ) sm(fa ) (2-6) 2d d z = cos(fad) d (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 i n , y is perpendicular to x, and z is the out-of-plane axis. Note that the scattering vector shown i n this figure is the difference between the beam tangent vector, and scattered track tangent vector taken at the origin. T h e y axis is actually drawn as -y to make drawing the vector components easier. Chapter 2. 25 Using CHAOS at Small Scattering Angles Physical Coordinates (2.8) x = cos(# ) 3d (2-9) y = sm(6 ) cos(4>3d) 3d (2-10) z = sm(9 ) sin(<^3d) 3d Physical Coordinates CHAOS Coordinates z Figure 2.13: Angle definitions i n the C H A O S coordinate system, a n d i n the physical coordinate system. The y axis is drawn as -y to make vector components easier to draw. Using 2.5 a n d 2.8 we find, for our example track, 9 3d = 16.5 degrees. T h i s 1.5° difference is crucial since the cross section is inversely proportional to s i n 0/2, resulting 4 in a difference of nearly 50% i n the cross section. A t 6 2d difference from 6 3d greater than twenty degrees, the is of the order of one degree or less. A diagram of the two coordinate systems are i n Figure 2.13. To accurately estimate 9 3d we need to determine (p2d- Chapter 2. 2.2 26 Using CHAOS at Small Scattering Angles D e t e r m i n i n g the Out-of-Plane Scattering A n g l e To find an out-of-plane scattering angle, fad, we need to look at the profile of a track i n 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 fadOut-of-plane information i n C H A O S can be obtained from W C 1 cathodes a n d anodes, W C 2 cathodes and anodes, and W C 4 resistive wires. T h e out-of-plane wire chamber hit coordinates are labelled z[ and z' for W C 1 and W C 2 incident beam tracks; scattered track 2 coordinates are labelled z and z for W C 1 and W C 2 hits respectively. T h e coordinates 2 x from inner and outer W C 4 resistive wire scattered track hits are labelled Zi and z . 0 Note that the W C 4 cell i n the incident beam, and cell i n 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 i n plane hits. T h e radius R of the circle is given by the momentum of the track, as shown i n the following equation: R = mv mPc qB qBE In the final result of E q u a t i o n 2.11, P xy P-, 0.29979^5 (2.11) is the in-plane momentum i n M e v / c , B is the C H A O S magnetic field i n Tesla, z is the number of electron charges including polarity, and R is the radius i n 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. T h e 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 i n K e r m a n i ' s M . S c . thesis [7]. A pathlength from W C 1 or W C 2 to the interaction vertex is given by: Figure 2.14: Geometry used i n finding pathlengths to W C 1 and W C 2 . Note that the wire chambers are not drawn to scale. Chapter 2. 28 Using CHAOS at Small Scattering Angles 0,arc 2 yj(xi - x ) 2 v = arcsm + (yi - y) 2 v 2R Pathlength — S = 9 arc * R (2.12) (2.13) where (x;, y^) are the in-plane coordinates of the hits i n W C l or W C 2 , and (x , v y) v are the in-plane coordinates of the interaction vertex. T h e geometry of the wire chambers and track is shown i n 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 i n determining the particle momentum. In a uniform magnetic field i n the z direction, the vertical component of the scattered track profile is a straight line given by z(S) = m S + b. T h e slope, m, and the intercept, b, are determined using linear regression on the four wire chamber hits for the scattered track i n C H A O S . We label the points as S , a and z , a 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 i n the fringe field of C H A O S , which means the field is no longer uniform. T h i s results i n non-linear track profiles, as is shown i n 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. T h e following two sections give some detail on how particle tracks i n 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 i n the C H A O S fringe field was done using a routine, trace, developed by R o m a n Tacik [13]. Trace uses a fifth order R u n g e - K u t t a method [14] to step particles i n time through C H A O S . T h e differential equation being Chapter 2. 29 Using CHAOS at Small Scattering Angles Tracks for P / B = 167 M e V T - 1 A —i r—7— 1 1 . •• ' ' ^_ - 5 3 o o o —v 1 -* - — - -AM - ' ~~ - " • •**™'" - 20 - ^ ^ ^ 40 60 S - Path Length i n c m 80 100 Figure 2.15: Vertical track profiles show the focusing effect of the fringe field. Note the change i n scale from vertical to horizontal. T h e 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. F = dP d = — (7m '«) = 7 m x = qv x B 0 0 [2.14) Initial conditions supplied to trace are the starting position, the momentum, and the direction (Q d, fad) of the track. F r o m equation 2 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. T h e z component of the track acceleration is given by: Q 7m X By X XyB X (2.15) 0 A n example of how the fringe field affects the profile of tracks i n C H A O S is shown i n Figure 2.15. A listing of the trace program used to generate tracks i n C H A O S is i n Chapter 2. 30 Using CHAOS at Small Scattering Angles 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 i n a magnetic field. To add an element of realism a simple energy loss calculation was done between each time step. T h e model for energy loss uses a C H A O S materials file used i n Greg Smith's eloss program. T h e energy lost i n material is given by the Bethe-Bloch equation [15] as follows: dE _ Kz Z 1. Hx~ ~ pA 2 2 2 2m c p -f T 2 2 2 e max o 2 (2.16) P l n 2m c /?V 2 T = max e 1 + 2>ym /M + e (2.17) (m /M) 2 e Symbol Definition Units or Value E Incident particle energy Avogadro's number 47rA Ar m c Charge of incident particle A t o m i c number of medium MeV 6.0221 0.3071 A t o m i c mass of medium Electron mass xc M e a n excitation energy Incident particle mass gmol~ N A K ze Z A r 2 e mc 2 e I M 2 2 xl0 molMeVg- cm 23 1 l 2 x 0.511 M e V MeV Table 2.1: Definition of variables i n Bethe-Bloch equation. Tmax is the most kinetic energy that could be transferred to a free electron i n a single collision. T h e variables i n the Bethe-Bloch equation are summarised i n Table 2.1. 2.2.2 Fringe Field Correction Model Since the fringe field vertically focuses particle tracks, a simple lens model was adopted. T h i s model corrects hits i n W C 4 to put them on the object line. In our model the object Chapter 2. 31 Using CHAOS at Small Scattering Angles line is the straight line i n the pathlength (S) versus height (z) plane that projects wire chamber hits back to where the particle hit the target. T h e corrected z values, labelled z' , were assumed to be a function of: the uncorrected a z value, the slope i n 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' = f(z ,m,p/B)=mS a a + h a (2.18) T h e exact form of f was chosen empirically as w i l l be described i n section 2.2.4. T h e result is shown i n equation 2.19. z' = (a z a a a + b zl + c m) a a + (2-19) g In order to do the fit to determine the constants i n 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. T h e constants i n the model make the model overdetermined. T h i s choice of constants was made so that a fit to E q u a t i o n 2.19, w i t h the term involving the ratio of p / B set to one, could be done first. B y fitting for the constants a ,b ,c , a a a the approximate value of some of the constants could be used as starting points i n subsequent fits involving all five parameters. I found this desirable because it took a lot less time to converge on the parameters when starting w i t h a good guess for some of the parameters. T h e initial fit to find a ,b ,c a a a was fast since a linear regression was used. F i t t i n g to get a l l the parameters took more time since a non-linear fitting algorithm h a d to be used. Values obtained for the parameters are summarised i n Table 2.2 To use the model to find where a hit i n W C 4 (S ,z ) a a is moved we rearrange E q u a - tion 2.19 so that it is of the form z" = mS'^ + h. T h e corrected wire chamber hits are a given by the coordinates z = z", and S = S". T h e result is given by: 32 Chapter 2. Using CHAOS at Small Scattering Angles a 1 a 1 2 i 1 1.08 1.10 0 c d 6« 0 0 -12.6 -26.8 0 0 355000 273000 0 0 339000 243000 a a 0 0 2.44xl0" 2.36xl0" 6 6 a Table 2.2: Fringe field correction parameters determined from G E A N T simulations. T h e G E A N T simulation used is described i n the following section. T h e parameters have units such that z , and z' are i n millimeters. T h e parameters d and e have units of a a a a (Mev/Tesla) . 2 z'l = Z\ S'[ = Si S'— S z'2 = z z'l = (a + kzfMp/B) lZl S? = Si- 4 = ( o o + b z )g (p/B) a 2 2 2 z where, g = a SZ = S - 3 0 0 0 0 (p/B)c gi (2.20) t g {p/B)c 0 0 B)* {p/ +ea So long as we get two of a possible four (S , z ) points, we can use E q u a t i o n 2.20 to a a find the track slope (m) and intercept (h). B y inspecting the parameters i n Table 2.2 and looking at the equations for (v ,w ) a a we can say that the correction moves a hit i n 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 i n Figure 2.16. In the vertical profiles shown i n Figure 2.16, the points represent hits i n 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 event t e s t , r e g . e v e n t Pathlength YBS. EVNTi (mm) 425 33 event t e s t , r e o - e v e n t YBS. EVNTi 468 P a t h l e n g t h (mm) Figure 2.16: Particle tracks at small scattering angles i n C H A O S . T h e vertical profile for each track is shown below each in-plane track. T h e in-plane tracks shown at top are display events described i n Section 1.2. In the vertical profiles, the points represent hits in the wire chambers i n 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 2.2.3 34 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 . T h e version of G E A N T that I used was already coded w i t h C H A O S geometry, and targets. R u n n i n g G E A N T generates data i n 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 i n G E A N T . There was no target i n the runs used for determining the model parameters, allowing b o t h incident and outgoing hits to be used i n 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 i n finding the model parameters. Resolution of the in-plane wire chamber hits i n C H A O S were already coded into G E A N T . T h e out-of-plane resolution of the wire chamber hits was not included i n G E A N T . G E A N T d a t a were fuzzed up i n 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. T h e 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 i n the parameter estimation using real data was too large because of the difficulty i n eliminating pion decays. In order to do the fit to the constants i n 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 to get the constants a_, b , and c i n z' = a z a a a 35 Angles a a + b z^ + c was done first. For tracks a a produced i n 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 a , b , and c . B y finding a , b , a n d c first we can use a a a a a a them as initial guesses i n subsequent fits to get a l l the constants. T o get the constants d and e , the value of a is set to be the value found i n our first fit. T h e n , a non-linear a a a fit to E q u a t i o n 2.19 to find the constants b , c , d , and e is done using a Levenberga a a a M a r q u a r d t algorithm i n Mathematica [14]. T h e resulting fits for b o t h 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' ) were calculated. Residuals (Az) to the fits using trace a data show that, neglecting the detector resolution, multiple scattering, a n d pion decays, the correction to the W C 4 z coordinates is good to w i t h i n 0.4 m m (a). T h e fit residuals from trace data are shown i n 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 w i l l have to travel further, resulting i n a smaller signal. B y examining the Az residuals we see that as we get to larger z coordinates the model fit is worse. T h e residual i n all the cases tested showed no additional dependence on the z coordinate or the slope of the track. T h e 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 w i t h i n 3.5 m m (a). T h e fit residuals from trace d a t a are shown i f Figure 2.18. T h e effect of turning on a l 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 36 1.0 1 J 0.5 0.0 | -0.5 8 Uncorrected J - 0 . 2 0 - 0 . 1 5 - 0 . 1 0 - 0 . 0 5 0.00 0.05 Track Slope (mm) 0.10 0.15 0.20 00 S a: a; -10 -t5 2 X) 12 (mm) 4 6 8 Track Slope • Uncorrected z 14 16 -1500 -1000 - 5 0 0 0 500 1000 Track Slope • z ^ (mm*) 1500 1 0 . 2 0 - 0 . 1 5 - 0 . 1 0 - 0 . 0 5 0 . 0 0 0 . 0 5 0.10 0.15 0 . 2 0 Track Slope 50 , (mm) 15- '•;!, • . * 0.5- 1j -0.5- 0.0- - -1.0- - £ -is- - • —2.0- 5 0 5 10 Track Slope • z ^ (mm) 15 20 -2000-1500-1000-500 6 5 0 0 1000 1500 2 0 0 0 Track Slope * z ^ (mm ) 1 1 Figure 2.17: Az residuals for WC4 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 WC4 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 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. Run Using CHAOS SD002 aO-ltor-Ha 11|21|2I P i l l - at Small Scattering 122.1 a t 111 48 f l t a - -I.I 37 Angles H i * MI02 M-tlai-99 1 1 l 2 1 i 4 2 Ptafc- 122.1 I t 111 -1.7 47 l a t a - 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 i o n decays were disabled i n the G E A N T run. 2.3 T h e Interaction Vertex at Small Scattering Angles To find 0 , the full three dimensional incident beam tangent vector p1 , and the scattered 3d track tangent vector po n Ut need to be found at the interaction vertex. T h e n the true 3d scattering angle is given by E q u a t i o n 2.21. Pin ' Pout n C 0 S ^ = | -|hn- I \Pm\\Pout\ p , n ^ \ 0 1 2 1 ) T h e vertical component of the momentum is related to the in-plane components of the momentum by E q u a t i o n 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. T h e tangents are found using the tangents to the circles, at the intersection closest to the target, of two circles. T h e circles are found by Chapter 2. Using CHAOS at Small Scattering Finding Large Scattering Angles Angles 38 Finding SmaU Scattering Angles Figure 2.19: D i a g r a m 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 w i l l be used as a point on the circle. E q u a t i o n 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 i n Section 2.2 is used for the scattered tracks. T h e geometry of the hits and circle fits is shown i n 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 i n the W C hit coordinates results i n a large error i n the longitudinal vertex coordinates. To solve this problem we use the intersection of the circles w i t h a fixed target plane. T h e n the intersection of the incident track w i t h the target plane gives us p~i , the intersection of the scattered track w i t h the target plane n 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 i n the target. See Figure 2.19 for a diagram of a b a d event w i t h a large A at a small scattering angle. W h e n 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 i n material surrounding the target. For further discussion of the use of A refer to section 4.3. Chapter 3 A n g u l a r and Coordinate Resolution Having good estimates of the angular resolution and coordinate resolutions i n C H A O S are important i n extracting the experimental results, and i n comparing results to theoretical models. T h i s chapter explains how the out-of-plane angular resolution (A(f)) and outof-plane coordinate resolutions (Az) the Az at the vertex (Az ) 0 are found. T h e following sections discuss: how 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 reconstructing a horizontal picket fence target. Six horizontal pickets, made w i t h four ^ T " diameter C u rods above two | " diameter rods, are attached to a copper sheet w i t h the shape of an upside-down A m e r i c a n football goal post. Going from top to b o t t o m , the thinner upper C u rods are spaced 1 cm, 2 cm, and 1 c m apart. T h e first thicker C u r o d is 1.5 c m below the lowest t h i n C u rod, and the last C u r o d is 1.5 c m lower. A sketch of the picket fence target, also showing where the beam spot was for two different runs, is shown i n Figure 3.20. The picket fence can be reconstructed from C H A O S wire chambers using b o t h incident and scattered tracks. For events which triggered the detector, and thus presumably interacted w i t h 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 . T h e figure is not drawn to scale. Chapter 3. Angular and Coordinate Resolution 42 hits i n W C 1 and W C 2 , along w i t h 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 w i t h pathlengths, are also used to determine the height at the target plane. In the following sections I w i 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 w i t h cuts to eliminate TT decay events are used. T h e vertex cut is a box i n A versus the in-plane scattering angle 8 d, which is particularly useful at small scattering angles. Further discussion of how TT 2 decay events are eliminated can be found i n Section 4.2. For r u n 6257, a 40 M e V TT~ run, the beam illuminated picket fence bars A , B , and C . T h e resulting histogram for height of the incident track at the target (z_proj), for a l l scattering angles, without any cuts on it is shown i n Figure 3.21(a). Z_proj w i t h cuts on the vertex, cuts on TT decay, and a cut requiring scattering angles greater than 30 degrees, is i n 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. T h e relative heights of A , B , and C when comparing between the figure i n (a), and the figure i n (b) depends on the heights at which the muons and electrons hit the target. Intersection of the outgoing track w i t h the target plane, including a l l cuts, gives a height at the target shown i n 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). T h e resolution of the bars is summarised i n 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 w i t h pickets A , B , and C clearly identified, and r u n 6258 w i t h pickets C , D , E , and F . Refer to text for more details. Chapter 3. Angular and Coordinate Resolution Bar A B C Z-proj a (mm) 1.49 1.47 1.27 ZJWC1WC2 a 44 (mm) Z-all a 2.14 2.38 1.61 (mm) 1.83 2.28 1.83 Table 3.3: Resolution of picket fence bars for run 6257 at scattering angles greater than 30 degrees. of difficulties i n eliminating all of the TT decay events. W h e n the TT/J, stack is finally calibrated, the resolution on the way out should be equal to the resolution on the way i n . 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 i n 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 i n 3.3 we see that the resolution using just W C l , and W C 2 , to when all the wire chambers are used are about the same. A g a i n , it must be pion decays which are limiting 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 i n the lower half of Figure 3.21. For run 6258 the beam illuminated picket fence bars C , D , E , and F . T h e histogram for height of the incident track at the target (z_proj), for all scattering angles, without any cuts on it is shown i n Figure 3.21(e). Z_proj w i t h cuts on the vertex, cuts on TX decay, and a cut requiring scattering angles greater than 30 degrees, is i n 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. T h e relative heights of C , D , E , and F when comparing between the figure i n (e), and the figure i n (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 w i t h the target plane, including a l l cuts, gives a height at the target shown i n the last row of Figure 3.21. In Figure 3.21, (g) uses only W C l a n d W C 2 on the way out (z_wclwc2), and (h) uses a l l of the wire chambers (z_all). T h e resolution of the bars is summarised i n Table 3.4. Bar Z-proj a (mm) C D E F 1.96 1.60 1.89 1.78 Z-wdwc2 a (mm) 2.34 2.52 3.31 3.38 Z-all a (mm) 2.10 4.31 3.39 3.11 Table 3.4: Resolution of picket fence bars for r u n 6258 at scattering angles greater than 30 degrees. T h e results of r u n 6258 indicate that we see a ^ " r o d as being almost as thick as a | " r o d , thus making our vertical coordinate resolution at the target about 3 mm. Note that i n r u n 6258 our elimination of pion decay events must not be as good as i n r u n 6257, since a l l the resolutions are worse. In particular, bar D must include some pion decays, making its resolution found using a l l the wire chambers worse than it should be. T h e horizontal r o d 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 r u n 6257 w i t h cuts on A versus # <i, 6^ i n the range 7° — 30°, and w i t h cuts to eliminate pion 2 decays gave the picket fence reconstruction shown i n Figure 3.22. T h e resolution of the bars using small scattering angle data is summarised i n 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 w i 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. 46 Chapter 3. Angular and Coordinate Resolution Run 1 20 6257 1 1 sum= 1 01-Rpr"99 I 1 248. 1 1 lean 1 D : 3 5 : 3 3 Peak= 1 I 1 1 1 1 I 1 7. Q a t 1 1 1 I 1 3 9 . 8 6 7 Mean c a t a = bin= bin 1 40 d a t a = 1 1 -9.633 1 I 1 -9.5 1 sigma= 1 1 12.421 B A 1 5— - l LO -I— 1 c ZD CD CJ 1 D LI j 5 i i i -60 Un=1. nn i i -40 i H i 924 i h i | -20 i r i i i sh_ail_nd2 | 0 i jn nqn 20 p 1 1 1 40 [h_a11/sm_scat_nd2] 1 1 1 1 60 0v=0. Figure 3.22: Picket fence reconstructed using small scattering angle data from r u n 6257, w i t h pickets A , B , and C clearly identified. Chapter 3. 47 Angular and Coordinate Resolution Bar ZJWC1WC2 a A B C 2.11 2.25 3.53 (mm) Z-all a (mm) 2.15 2.46 3.22 Table 3.5: Resolution of picket fence bars for run 6257 at scattering angles i n 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 w i t h 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 i n existing G E A N T simulations of C H A O S . Out-of-plane resolutions were initially chosen to be normally distributed w i t h 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 i n W C 2 incident (s2i,z2i), the hit i n W C l scattered ( s l o . z l o ) , the hit i n W C 2 scattered (s2o,z2o), the corrected hit i n W C 4 inner scattered (sio,zio), and the corrected hit i n W C 4 outer scattered (soo,zoo). F r o m the linear regression we find an estimate of the height at W C l incident ( h l i ) . T h e All But One resolution (a) of W C l incident is then given by A z i z = zli — hli. R e a l 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 r u n w i t h 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 i n 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 i n each wire chamber. Chapter 3. Angular and Coordinate Resolution 49 than for incident tracks, because of difficulties i n eliminating TT decay events. A g a i n it is expected that the incident and scattered track resolutions w i 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 w i d t h horizontal rod i n G E A N T as we do from real data. T h e G E A N T run used was a 40 M e V pion run, scattered at any scattering angle, w i t h 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 . T h e real data 16 run being compared to is the picket fence run 6257, which is described i n Section 3.1.1. After a few iterations of changing the a values of the out-of-plane chamber resolutions i n G E A N T from the nominal values, we get nearly identical results for z_proj i n simulated and real data. T h e sigma values for the wire chamber out-of-plane resolutions used i n G E A N T are: sigma W C l is 0.8 m m , sigma W C 2 is 2.0 m m , and sigma W C 4 is 1.0 m m . T h e 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 w i l l give reasonable estimates of the resolution we w i l l get from C H A O S . 3.2 Angular Resolution T h e angular resolution is found using G E A N T simulations. Setup of G E A N T for use i n three dimensions of C H A O S was discussed i n the previous section. T h e following section details how the angular resolution of C H A O S is found. F i n d i n g detector resolutions is important by itself, but further use of the resolution needs some explanation. T h e final section of this chapter w i l l outline the importance of finding the angular resolution i n particular. 50 Chapter 3. Angular and Coordinate Resolution Run 300 30008 I 8 - F e b " 9 9 sua- 1 7: OBi 28 P e a k - 1366. Mean b i n - 266.0 a t b i n 34.544 Mean d a t a - 35 d a t a - -14.956 a l g a a - -14. 5 3.512 ^simulated = 2 4 m m 200 100 50 H ' -BO 1 I " -40 Un=1. r i " 1 - I 1 700. i i i | i i nd_ali [h_all/no Run 6257 1 8 - F e b - 3 9 , . . . I . . . . 50 i '| i 0 -20 H i 85 17: 19: 27 P e a k I . . . . I Mean b i n - 46, 0 a t b i n • • I . . 3 9 . 2 6 3 Mean d a t a - i i | 20 ch_decayl . 40 60 Ov=Q. 41 d a t a * -8. 5 • I • • • • -10.237 s l g a a - 13.354 c 7 = 2 . 4 m m l a e r ° 20 J A L 10 I -60 Un=1. -40 1 1 1 H i 898 1 I 1 -20 1 1 1 I 1 1 0 20 40 nd_kz_ail [h_all/no_ch_decay3] Ov=1. 60 Figure 3.24: Comparison of horizontal r o d z_proj using simulations (top) a n d real d a t a (bottom). Result is shown for ^ " rods. 51 Chapter 3. Angular and Coordinate Resolution 3.2.1 G E A N T Simulation of Angular Resolution Once we have our simulation calibrated to work i n 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 i n G E A N T scattering angle that G E A N T generated (9 ). (9 d), 3 and the T h e angular resolution is thus defined geant as: A9 = 9 id —9 (3.23) geant 9 d is the full three dimensional angle between the incident beam, and the scattered track. 3 To get a realistic estimate of the angular resolution the G E A N T runs used material files for the C N I liquid Hydrogen target. These G E A N T runs included multiple scattering and energy loss, but had pion decays turned off. Three runs w i t h incident pion energies of 40 M e V , 55 M e V , and 67 M e V were analysed, w i t h the out-of-plane wire chamber resolutions simulated i n the analysis. Since we expected our resolution (A9), at scattering angles i n the range 0° to 40°, to worsen at large scattering angles, we looked at histograms of AO w i t h cuts on different five degree angular regions. T h e result was that the resolution was fairly constant over different scattering angles. A dotplot of A9 versus 9 , i n Figure 3.25, 3d shows that the angular resolution is constant over the angles generated i n 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 67 M e V 55 M e V 40 M e V Resolution a in degrees 0.8 0.85 1.0 Table 3.6: Angular resolutions found at different energies. Chapter 3. Angular and Coordinate Resolution dth_vs_th_sang _i i i_ reg.event A 9 Dotplot _i 5000 i i i_ 17 Run 30040 _L i I TEST reg_event i i i i i i i r I i i_ dots o J -2H 10 i 1 1 e r 20 n i i r 30 40 3d Figure 3.25: A dotplot of A9 versus 6 over the angles generated i n G E A N T . 3d showing that the angular resolution is 53 Chapter 3. Angular and Coordinate Resolution 3.2.2 Importance of Angular Resolution in Cross Section Determination T h e experimental cross section that we measure is related to the actual theoretical cross section, by a convolution w i t h a detector response function H(t9 — 9'). dcr f da <0) = J^ff)H(8-ff)dff dQ, exp (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 w i l l apply a correction similar to that done by J o r a m 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 i n Equation 3.25. ± do ^ ( 7 Nr (9)R,da tt ± r In E q u a t i o n 3.25, N^ {9) catt r p ) = N r u ( 9 ) R ^ e ( 3 catt n et ™ *> p)e - is the number of pions scattered at angle 9, N^ (9) the number of muons scattered at angle 9, R and muons i n the beam, e^ ± 2 5 ) is and R^ are the relative fraction of pions is the detector efficiency, and e conv is the correction due to the angular resolution. T h e correction (e ) is given by E q u a t i o n 3.26. conv Using the Karlsrhue-Helsinki (KH80) calculation of the 7 r p cross sections and the ± u^p M o t t cross sections, we can calculate the correction factor. T h e correction factor and cross sections calculated for 1° detector resolution at an incident pion energy of 40 M e V are shown i n Figure 3.26. For the scattering angles that we are interested i n , 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. T h e 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 i n a supposed experimental cross section that is much lower than i n reality. Chapter 3. Angular and Coordinate Resolution 10* II 1 1 1 1 1 1 1 1 I 1 1 1 1 I 1 1 10 _J J 1 1 1 55 10'-J _Q \\ \\ \\ \\ 10' M - - - 7T I I I I L_ + 3 b \ • \ fe- io--d \ I I 10 I 4.5 I V . fe- io"-J \ 10 G a u s s i a n A0 = 1° cr io -d V_ tO'-J \\ I withp fe- 10*-J TJ I I ' 20 I I I ' I I I ' I I 10 ' 30 I I "I 1 1 I 1 1 1 10 1.15 l_ 1 1 40 _l I I I I 1 1 1 1 1 I I LJ I 1 1 20 1 1 - 40 30 I I I I I I I I l_ Correction Factor CL X b TJ I 7T~ \\ \.\ \\\ 10' C X! \ I I I LJ -^da/df) c o n v o l v e d l\ to I I I 3.5 1.10 H t (e H ) 1-05 H b TJ 1.00 1.5 • D 0.5 1 1 ' I 10 'I 20 1 1 1 1 I 1 1 1 30 0 LAB 0.95 1 40 0 'I 10 1 1 1 1 I 1 20 1 1 ' I 1 1 30 1 40 (degrees) Figure 3.26: Correction factors (e ) to the ir^p cross sections for detector angular resolution of one degree (a). Results are shown for 40 M e V pions, and p = p^. T h e plot in the lower left is a ratio of the theoretical cross section to the theoretical cross section convoluted w i t h a Gaussian of 1° standard deviation. C(mv n Chapter 4 Cross Section Considerations T h e 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^. (9 )), catt sd • the number of incident pions on the Hydrogen target ( A ^ ) , n c • the angle of the target relative to the beam (9 t), tg • the number of target Hydrogen atoms per unit area (N t), tg • the solid angle subtended by the detector (dfl), and • the efficiency of the detector (e ). det T h e cross section is given by Equation 4.27. do_ dtt Discussion of what 9 3d Nr (e )cos9 tt = 3d tgt N™N dne tgt 1 det ' 1 a particle was scattered at can be found i n Chapter 2. T h e result is used i n counting pions scattered to different angular bins giving a value of N° (9 ). catt 3d T h e incident number of pions is counted using time of flight w i t h 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 liquid H target for C H A O S w i l l be carefully recorded, allowing the target density p tgt 56 to be calculated. 57 Chapter 4. Cross Section Considerations N-tgt can then be found using a measurement of the target thickness t, the atomic weight A , and Avogadro's number NANtg = p tAtN t tg A (4.28) The following sections contain an explanation of: • how particles are identified i n 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 i n the measurement of target angle has. In order to get a good yield, N^ (6- d), catt i we have to be sure we are counting only pions. Accounting for decays i n 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. T h e time of flight ( T O F ) is measured using the finger counter scintillator at the entrance of C H A O S , timed w i t h respect to the 23 M H z cyclotron R F . T i m e of flight spectra at different beam energies, showing the separation of pions, muons, and electrons are shown i n Figure 4.27. A t low energies, below 25 M e V , it is no longer possible to separate pions a n d 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 w i t h 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 Run 6569 23"Ftb-39 15:25:17 Peak- 58 Considerations 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 Un=0. Hi 1107 Full.tcap I f ull _t cap/r eg .event) Ov-U. 1000 1 100 1200 1300 14DQ 1500 1600 1700 Un=0. Hi 1116 Fuii.tcap I full_tcap/reg_event! Ov30. Run 6449 23-Feb-99 15t 20i 13 Ptalt- 4043. D i t bin 1 Bl data- 646.5 50U 0n=0. Hi 6DD11D6 70 0 B O O 9D0 10 DO 1100 120D Fuli.tcap 1ful1_tcap/reg_event! Ow»132. 1000 11 DO 1200 1300 MOD 15D0 1600 1700 Un-0. Hi 1107 full.tcap I full_tcap/reg_event) Ov-17. 1000 11 DO 1200 1300 14DD 1500 1600 1700 Un-D. Hi 1107 Fuli_tcap [full_tcap/reg_event) Ov-7. Run 6434 23-Feb~99 15i 21 • 4B P-ak- 903.0 at bin 157 data- B3D.S 50Un»0. 0 H60 0 700 BOO 9D0 1 000 1100 1 200 i t)06 Full„tcap 1full_tcap/reg_eventJ Ov-45. Figure 4.27: T i m e 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 . T h e humps on the right sides of the pion peaks, i n the spectra i n the second and t h i r d rows, are due to pions which decay i n flight from the production target to the C H A O S finger counter. Chapter 4. Cross Section Considerations 59 w i l l get wrapped around to having the slowest T O F . For example, at a pion energy of 1 7 M e V , the T O F between pions and electrons is greater than 43 ns. T h e 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. T o differentiate between pions and electrons, which overlap i n T O F at low energies, we need to look at a combination of the T O F information w i t h the energy deposited by the particle (AE). Energy deposited is also measured using the finger scintillation counter, whose pulse height is digitised using an A n a l o g to D i g i t a l Converter ( A D C ) . T h e 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 i n Figure 4.28. Using information about the particles flight time, and how much energy it loses i n matter, we can clearly identify pions. However, at higher energies, the muons are still difficult to identify using this information alone. T h e much larger number of pions as compared to muons at higher energies, makes the tail of pions i n the T O F spectrum a significant amount at the muon T O F . Since we w i 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. T h e dotplot of P versus T O F at 55 M e V is shown i n 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 i n 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 f_pid ' 60 Considerations ' Dotplot 1 Run 6448 TEST reg.event I I reg.event 3000 dots i i . . I Li 1 000 900 7T e 700 600 H 111 500 T—T—r-r-T—t—T—T—i—i—i—I—i 200 250 300 s 2a adc Dotplot F.pid 1 i 1 350 6434 TEST reg.event 1 Run _L 1 200 reg^event 300D dots + LL 900 e + + 700 600 ~r I 50 200 250 adc 300 350 400 s2a 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 m•m_vs_tcap _i 1 450 i 1 400 Dotplot 2000 44 Run 6 8 6 5 TEST L _i i_ reg_event 61 Considerations _l I I I i reg_event I I L_ T = 55Me dots 7Y muons H euro CJ I 1 350 1 300 H 1 250 "~i 100 pion pions 1 1 1 1 1 20 1 | r 1 1 1 1 40 r ~i 60 r decays ~~I 1 80 r 200 momentum Figure 4.29: M o m e n t u m 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 i n C H A O S . T h e dots at momentum higher than the channel momentum (135 M e V / c ) must be pion decays. Chapter 4. Cross Section 62 Considerations m) is ~ 14 ns. In this time about 37% of the pions w i 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 w i l l decay into a cone whose angle is dependent on the pion energy. Details of the pion decay kinematics can be found i n Section 1.2.5. T h e 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 w i l l be discussed. T h e methods used are: comparing different momentum calculations of a track, identifying particles i n the iru. stack, and using projections of the track to different points i n C H A O S . 4.2.1 Use of Different Momentum Calculations Track momentum can be estimated using the hits i n wire chambers one, two and three. W i r e chamber four can also be used by itself, or together w i t h the inner chambers i n 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 i n momentum characteristic of a pion decaying. Since only the in-plane information of the wire chambers is used i n momentum reconstruction this method w i l l only tell us if a pion decayed w i t h 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 a l l wire chambers is shown i n Figure 4.30. T h i s method can not eliminate a l l decay events. Even good events w i l l have some uncertainty i n their momentum calculation, so we can only cut away some decays. T h e method also relies on having hits i n all the wire chambers; i n 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 6871 24-Feb-99 8Q000 63 Considerations I It 07i 06 Peak" 71 2604. Mean b i n - 7B008. 0 at bin 102 d a t a i i i i t i L • i I • 102. 388 Milan l^a n data5. 776 s l g n a - 60000 - \ Figure 4.30: Histogram used i n eliminating some pion decay events. W h e n the difference in two different momentum calculations is much bigger than zero, it is tagged as a decay event. T h e histogram is from a C a r b o n target run w i t h 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 a l 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. T h e stack was designed to identify pions from muons using the different ranges, energy losses, and T O F of pions and muons i n matter. A n introduction to the stack can be found i n Section 1.2.5. F r o m test runs done i n the summer of 1998, a histogram of the range i n the stack is shown i n Figure 4.31. Results shown i n Figure 4.31 are not using the final version of absorber thicknesses, and analysis methods. W o r k is currently being done by the Italian part of the collaboration to implement a neural network to decide if an event is a pion or a muon i n the stack. T h e stack can also be used to obtain an out-of-plane coordinate that can be used i n 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 w i t h the W C 4 out-of-plane information, the z coordinate of the stack (z k) st is affected by the magnetic fringe field of C H A O S . T h e profiles, out to the stack, of several random tracks i n C H A O S are shown i n 4.32. In the following section projections, as a method of identifying pion decays, w i l l be discussed. Here I w i l l present the fringe field correction model applied to projecting wire chamber out-of-plane coordinates to the stack. T h e projection is an estimate of the height at which the stack is hit, including the effect of the focusing of the fringe field. T h e projection can later be compared directly w i t h the out-of-plane coordinate found Chapter 4. Cross Section Run 65 Considerations Run 8 5 B 4 U O R p r 3 9 1 2 i 2 6 t 2 7 Peik- 6 5 B 4 Q B R p r 9 9 5511. t.BSS n e t n dill- 1 2 i 2 6 i I B Pelk- n e t n bin- 2 7 3 4 0 . II A Un-0. HI 1141 s t k . r a n g e TT E S t k . r a n g e / r e g _ e v e n t ] Ov-D. Run SSB4 G B f l p r 9 9 13.26.19 Pcik52.) it bin 4 dit«... I . . . . I . . . . I . . . . 1 • • • • um114. M E M bin2.93' linn dit»* 3.430 llgn- bin 1. 766 linn dm- 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 ] O v - 0 . Run 4.5 6 5 S 4 0B-f*»"93 204. 1. 4D0|- 1 2 . 2 8 . 3D Puk- 59.0 it bin 6 dita- 6.5 n e t n bin- e I Un-0. HI 1143 stk_range_mu t Stk_ranQe/Q_3tk_ranp,e_inu] O v - 0 . Un-0. Hi I 44 11 I ' I 2 3 4 5 6 7 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 i n the ir/j, stack. T y p i c a l l y we t r y to stop pions i n layer two or three, and muons i n layer four or five. T h e histograms going clockwise from the upper left corner are of a l l events, events tagged as pions i n T O F , events tagged as electrons i n T O F , and events tagged as muons i n T O F . Chapter 4. Cross Section Considerations 66 Figure 4.32: Track profiles drawn out to the stack generated using trace. T h e straight line shows the image line to the stack. 67 Chapter 4. Cross Section Considerations from the stack to see i f any pion decays occurred. The model presented here takes the z coordinate of the hits i n W C l , W C 2 a n d W C 4 and corrects them to put them on the image line. A n image line is shown i n Figure 4.32; it is the straight line w i t h slope equal to the slope of the track at the stack. T h e correction is given by: z' = (apZp + bpzl + cpm) ^ 0 = rriiSp + t\ (4.29) [PiBy + ep where B = (1, 2, i , o) are the hits i n W C l , W C 2 , W C 4 inner, a n d W C 4 outer. T h e 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. T h e model parameters to put the hits on the image line are given i n Table 4.7. p dp h 1 2 i 1.37 1.25 1.03 1.02 15.3xl0" -2.67xl0~ -1.98xl0~ -1.46xl0 0 C 6 6 6 - 6 /3 189 103 6.02 3.49 dp e 53800 81000 964000 1530000 22400 49200 933000 1500000 /3 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 i n 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' ) we stk need to plug the track length to the stack (s k) into z' st stk = miS k + hi. T h e pathlength is st 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 i n 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 x coordinate in mm 300 400 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. T h e 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 W C 4 270 255 260 265 270 Theta at W C 4 275 280 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 w i t h random starting positions, angles, a n d 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 i n 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 ). A n stfc 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 9 k, as shown i n Figure 4.34. T h e relation between st the two angles is given by 9 k = 1.006 9 CA st W — 5.287, where the angles are i n degrees. Chapter 4. Cross Section zstk proj e l 0 i I Dotplot i . i i 70 Considerations 3 Run I i zetk_proj_pl_0 685B TEST t c e p . e l . t e e t i i I i—i—i—i i Dotplot 7 Run 6858 TEST t c s p . p l . t e o t tcip_pl_tcit 1000 dot* tcip_el_teit 1DO0 dot! 1950 I -1 • —•—•—I 0 1 1 stk_z_proJ_1 — 1 1 I 1 -1 1 0 I stk_z_proJ_1 ( x 10* ) 2 (Xl0* ) 2 Figure 4.35: Dotplot of the projected height at the stack versus the actual height at the stack. T h e 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 w i l l be presented. T h e 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 i n the previous section and the actual z that we get from the stack. If we get points that are far off from the m a i n correlation line, then we can conclude that there was an out-of-plane pion decay. C u r r e n t l y the resolution on the T D C s used i n the stack is too poor to make any useful conclusions w i t h this information. For example the dotplots of the stack projection versus the stack height from hardware for b o t h electrons and pions are shown i n 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 m o m e n t u m i n the out-of-plane direction, then we can look for large changes i n the out-of-plane angle. For Chapter 4. Cross Section R u n 3 0 0 O S 2 4 F e b " 9 9 I B t 2 7 i 4 1 PeikI . , , , I 1 Dt. D it bin , R u n 3 0 D O 5 2 4 F r b ~ 9 9 1 9 1 2 3 1 .0 Peik125,0 . • . • 1 I 1543. P l e i n bin3 9 . 2 0 1 lie i n dn 40 diti- 3.055 Hem d i l l - I 5 3 S . M e n i bln- decays offn -B Un»ll. R u n -4 -2 D 2 4 5 8 H i 39B p h i _1 2 _ l o _ 1 I p h i _ 1 2 _ i o _ l / r eg . e v e n t ) Ov-16. 3 0 0 0 6 2 4 F B b " 9 9 1 8 i 2 9 . 33 Pctk1D0D . it bin 4 D dm0.1 1426. He in bin- 3 9 . 5 1 1 LJ_I decays on 0 . 0 2 2 tlOll- i •0 -6 H i 398 phl_l2_lo_1 I phi_12_ia_1/reg_eventl Ov-33. 1 • i 1 1 40 dita, , , I 0 . 0 5 3 ilgit- t.621 1i-2o • • 1 1 -2 • • i' 1 1 0 2 1 4 6 Un-S. H i 336 phl_1l_2o [ p h i _ l l _ 2 o _ 1 / r e g _ e v e n t 1 Ov-8. R u n 3 0 0 0 B 2 4 F e b 9 9 1 B . 2 9 . 5 2 Peik105.0 tt bin 40 d l t n 0.1 I . . . . I . . . . I . . . . 1 1459. Rem bin39.306 D u n dttb- 2 3 .B 9 12—io 1 -4 ' -0.039 i l g i t - l.710h decays o I I ' -6 - 4 - 2 H i 336 phi 1 l_2o 1 Un-36. it bin decays of 12—io •! ^ n -8 71 Considerations -B Un-19. 1 1 1 1 1 1 I I 0 2 [phl_1l_2oJ/reg_eventl 1 1 1 1 1 1 Ov-17 Figure 4.36: Out-of-plane angle residuals generated using 40 M e V incident pion G E A N T runs w i t h horizontal r o d targets. Results w i t h pion decays turned o n a n d pion decays turned off are shown. example we can get an out-of-plane angle using just W C l a n d W C 2 (fin), a n d we c a n get an out-of-plane angle using just W C 4 inner and W C 4 outer (cf> ). If the track decays io somewhere between W C l and W C 4 then we would expect to get a large residual i n the difference between 0 12 a n d fa (4>i2-i )- One problem w i t h this scheme is that the W C 4 0 0 hits are close together in-plane, so a small error i n Z{ or z will result i n a large error i n Q 4>i . One possibility is to look at the difference between angles 4>u a n d fao giving 4>u-2o0 G E A N T simulations were used to look at these angle residuals w i t h pion decays turned off and on; results are shown i n Figure 4.36. Chapter 4. Cross Section vsep_vs_scat+ J I 72 Considerations I I I I Dotplot I I I I 9 Run I I I 6870 TEST r e g . e v e n t I I I I , I I I I L SEPR A ( m m ) Figure 4.37: Dotplot of A verses 6 used to decide if interaction occurred i n 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, w i t h a 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. 2d 1 2 T h e resolution i n the out-of-plane scattering angle factors into where we can place cuts on the angle residuals. A s can be seen i n 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 projections onto the target plane. T h e in-plane projection of tracks onto the target plane is the same as what is done i n deciding if an interaction occurred i n the target. Refer to Section 2.3 for the method used i n finding the intersection of tracks w i t h the target plane. A dotplot of the separation between intersections of incident and scattered tracks w i t h the target planes ( A ) versus the in-plane scattering angle (9 ) 2d is shown i n Figure 4.37. T h e C a r b o n target is clearly identified as having a small A . P i o n decays and scattering Chapter 4. Cross Section Considerations 73 vsep_vs_scat+ _i I I i vsep_cut_pl I Dotplot i . I 15 Run i 1 685 dots I 6785 TEST . I i reg.event t L no target SEPR Figure 4.38: Dotplot of A on the x axis verses 6 d on the y axis for two different running conditions. T h e dotplot on the left came from a C H A O S r u n w i t h 33 M e V pions incident on a plastic sewer pipe. T h e dotplot on the right is at the same energy but w i t h no target i n 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. 2 events from wire chambers are clearly seen by the points w i t h 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 r u n where the target is a cylinder of plastic. N o part of the cylindrical sewer pipe target is near the origin, so we should expect to get large values of A . T h e results from a C H A O S r u n w i t h 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 r u n to a r u n w i t h no target, we can see that scattering events w i t h vertices i n W C l , which are further from the center of C H A O S than the sewer pipe, result i n larger values of A . T h e dotplot on the right side of Figure 4.38 is for a C H A O S r u n w i t h no target. Chapter 4. 4.3 Cross Section 74 Considerations 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 w i t h 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. T h e solid angle using C H A O S coordinates is given by E q u a t i o n 4.30. (4.30) A typical two degree wide b i n i n 9 d is then associated w i t h a fixed element of solid angle 2 (dQ ~ (2°)(14°) ~ 8.5 msr.) T h e trouble w i t h 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 w i l l scatter into a cone w i t h a scattering angle 9 d- Near 0° and 3 180° the entire cone falls into the detector acceptance, whereas near 90° only a small fraction does (~ 8%.) T h e solid angle using the physical coordinate system is given by E q u a t i o n 4.31. (4.31) Solid angle calculations i n 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 (9 d,(p2d) distribution to (9 d,4>3d)- T h i s mapping is shown i n 2 3 Figure 4.39. T h e physical solid angle is proportional to the w i d t h i n <p d at the scattering, and also 3 has a factor that depends on the b i n w i d t h being used. For example a two degree b i n w i d t h has the b i n w i d t h factor of (cos(9 d 3 — 1°) — cos(9 d 3 + 1°))- C o m p a r i s o n of cross sections using b o t h solid angle calculations w i l l be done i n Chapter 5. Chapter 4. Cross Section 75 Considerations i 30 CHAOS e 2d Figure 4.39: M a p p i n g 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 76 Considerations d Q V Mrue Angle Figure 4.40: Effect on cross section measurement of binning data. 4.4 Effects of 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. W h e n we put the yield of the cross section i n two degree bins, we assume the angle of the b i n is at the middle of the bin. For example a b i n 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 i n Figure 4.40. If our assumed b i n position is Qun a n d our b i n w i d t h is A 0 & i , then the cross section n Chapter 4. Cross Section 77 Considerations at the true b i n position is given by: do . \ _ Je' =0 -O.5Ae dQ^rue)\ytrue) — bin bm bin dU^bin) d' 9' =e =8 +o.5Ae +0.5A9 bin bin bin bin bin bin •l0'^=e -0.5A8 bin T h e angle 9 true s . u aa V^- J oz, {9 in) bin Sm bin K bin) bin sm M b is the weighted mean of the b i n . T h e correction (eu ) is the ratio of n the cross section at 0 e to the cross section at QuntrU **n = j (9true) (4-33) For correcting data we divide by the correction. Instead of correcting the data, we w i l l be m u l t i p l y i n g the theoretical cross section by eu to compare w i t h our data. n T h e binning correction is only important at very forward angles where the cross section changes most rapidly. T h e 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 b i n correction. 4.5 S c a t t e r i n g A n g l e Offset E f f e c t s Part of finding the cross section is i n knowing the angle of the target relative to the incident beam. A n error i n our estimation of the target angle would also introduce an offset i n our scattering angle. W e use b o t h the intersection of the incident track a n d scattered track w i t h the target plane i n our calculation of the scattering angle. Since we do this an error i n the target angle estimation becomes a second order effect, not just an offset. T h e effect of an error i n the target angle estimate on the cross section is considered i n 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 b i n w i d t h 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 C N I Region T h e LL^C —> LL^C cross section can be calculated theoretically to very high accuracy. For this reason, and because there is less problem w i t h muon decay than there is w i t h pion decay, the LIC cross section calculation is a good test of the new scattering angle algorithms for C H A O S . T h e following section w i l l present the theoretical model used i n calculating the uC cross section. 5.1 Theoretical Shape of Cross Section T h e theoretical cross section is found using the M o t t cross section for a point-like target nucleus, which is modified using a form factor. T h e M o t t cross section for \iC —> LIC is given by [17]: i da^ , =( aZhc 2 2 2 2 I* \£V . , 0 ) ^ ( l - ^ s m ^ ) 2 : (5.34) where p , i n M e V / c , is the momentum of the incident muon, Z = 6 is the charge of the M C a r b o n nucleus, and the ratio of energies arises from the recoil of the 1 2 C . T h e ratio of energies is given i n Equation 5.35. 1 E' (5.35) In E q u a t i o n 5.34 the total energy of the incident muon is given by E i n M e V , and the total energy of the muon after scattering is E'. T h e cross section is i n units of (fm) . 2 79 The 80 Chapter 5. The uC Cross Section in the CNI Region momentum transfer from the muon (q) is given by kinematics i n the following equation. Q= T h e form factor we used for 1 2 nc / 2p sin ^ u , (5-36) , v ^ ^ 1+ sin2 C was one experimentally determined by Reuter et. a l . [18]. F r o m electron scattering measurements, the experimental \iC cross section is given by the product of the form factor squared w i t h the M o t t cross section. T h e method used by Reuter et. a l . to determine the form factor led to a result for the nuclear charge density i n terms of a Fourier-Bessel series. T h e form factor i n terms of the charge density is given by a Fourier transform: „ /)— ^ -^4irr ( ? ) dr „„'2. F(q) = - /f p( Z J qr 1 s i n r y (5.38) 2 r T h e 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 " ™I 1 T 0 (5.39) R r > R where R = 8 fm is a cutoff radius for the charge density. T h e coefficients a n were determined out to n=15, and are summarised i n Table 5.8. A plot of the nuclear charge density versus radius shows that the C a r b o n nucleus has a radius of about 2.5 fm. T h e C a r b o n nuclear charge density calculated using the Fourier-Bessel relations given above is shown i n Figure 5.42. Theoretical models of the C a r b o n charge density give results similar to the experimental result. T h e form factor calculated using the charge density of Reuter et. a l . is shown i n Figure 5.43. A drop i n 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: C a r b o n 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 a 0.4 fe o 5-1 fe 0.2 0 20 40 60 Scattering Angle in Degrees Figure 5.43: C a r b o n form factor squared. 80 83 Chapter 5. The pC Cross Section in the CNI Region n a 1 3 5 7 9 11 13 15 1.5737 x I O " 3.7085 x 10~ - 4 . 4 8 3 0 x 10~ -6.8695 x I O " -7.7228 x I O " 1.0636 x I O " -5.0134 x I O " -4.7686 x I O " n 2 2 3 3 4 4 6 n a 2 4 6 8 10 12 14 3.8896 x I O " 1.4795 x I O " -1.0057 x I O " -2.8813 x I O " 6.6907 x I O " -3.6864 x I O " 9.4548 x 10~ n 2 2 2 3 5 5 6 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 w i t h the experimental data i n 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 i n finding a cross section. Here I w i 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 C a r b o n target. Cuts made on the data are: • a b o x cut on the track momentum versus T O F to the C H A O S finger counter measured w i t h respect to the cyclotron R F , • a b o x cut on A versus 62a, • a cut requiring that there be only one scattered track, • a cut on the range i n 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 w i t h the target plane, • a cut on the out-of-plane angle, and • a cut on the difference between the momentum calculated using a l l the wire chambers to using only W C l , W C 2 , and W C 3 . W i t h a l 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. T h e resulting cross section, without applying any binning corrections, or solid angle corrections is shown i n Figure 5.44. The cross sections i n Figure 5.44 look rather bad, however that is only because the events shown are required to have triggered i n 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. T h e stack physically ends at scattering angles near 25 degrees. Similarly, at the smallest scattering angles the cross section drops off because i n the test r u n that the data comes from the first stack bar from 1° to 6° was not operational. D a t a to be taken this summer w i l l include events i n the first stack bar region. In Figure 5.44 we see that adding i n the proper solid angle does not have as big an effect as one might expect. A ratio of the solid angle calculated using the o l d C H A O S coordinates to the solid angle calculated using the new physical coordinates is shown i n 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. T h e events i n Figure 5.44 are missing many events i n the first stack bar region; when we get data i n the first stack bar region this summer, we w i 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 i n the left half of the figure. T h e cross section using the new solid angle, and w i t h the correction for binning is shown i n the left half of the figure. Chapter 5. The \iC Cross Section in the CNI Region 0 . 0 111111111111 86 1111111111111111111 0 5 10 15 2 02 5 3 0 Lab Scattering Angle (degrees) Figure 5.45: R a t i o of the solid angle calculated using the o l d C H A O S coordinates to the solid angle calculated using the new physical coordinates. 87 Chapter 5. The pC Cross Section in the CNI Region T h e pC cross sections presented i n this section are absolutely normalised. T h e value of »r- cos fttgt ,r (- ) 5 40 from E q u a t i o n 4.27 was chosen arbitrarily to fit the data to the theoretical curves. T h e value of the arbitrary normalisation i n Equation 5.40 is 1/270. W h e n the efficiency of the stack has been better analysed, the true normalisation can be used. 100 ~o ~o CO ^%o o theoretical CL using strip 1?5 0 -| 0 1 1 1 1 r 10 20 30 40 50 transformed theta 60 Figure 5.46: Comparison of solid angle correction using theoretical w i d t h i n fa solid angle correction using a strip i n fa . d) and d T h e solid angle correction was made using the theoretical w i d t h 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 i n statistics. Throwing away data that has a magnitude of fa d greater than 7° has the same effect of applying the solid angle correction. T o verify Chapter 5. The pC Cross Section in the CNI Region 88 that the theoretical correction and strip i n fad method give the same results, b o t h were calculated using Monte-Carlo techniques. T h e verification showed that b o t h methods give about the same result. T h e theoretical correction is shown as a solid line, a n d the correction using a strip i n fad is shown as a dashed line i n Figure 5.46. Testing the effect of giving the in-plane scattering angle an offset required modifying the analysis program. T h e data were rerun w i t h b o t h positive and negative scattering angle offsets. O n l y the i n plane scattering angle was given an offset, since only the i n plane angle would be affected by an error i n our knowledge of the angle of the target plane. T h e cross sections w i t h +3 and -3 degree i n plane scattering angle offsets are shown i n Figure 5.47. Increasing the in-plane scattering angle slightly lowers the low scattering angle cross section. Decreasing 6 d raises the low scattering angle cross section. 2 Chapter 5. The LIC Cross Section in the CNI Region 89 Figure 5.47: uC cross section w i t h positive and negative three degree in-plane scattering angle offsets applied. Results are from C H A O S data runs w i t h 66 M e V muons incident on a C a r b o n 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 i n this thesis. T h e scattering angle resolution was found to be better than 1° for the energies of interest i n 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 w i l l be taken i n the fall of 1999, and again i n i n the summer of 2000. T h e stack w i l l be fully calibrated for eliminating pion decay events from CHAOS. D a t a collected i n the summer of 1999 will subsequently be analysed to extract ir^p —> ir^p cross sections. Analysis of the data will 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. T h e method used to do this is described i n the introductory chapter. M o r e detail on the analysis method can be found i n 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 i n fits to find the incident beam angle, or the scattered track angle. The methods for doing these two improvements, and what effect they w i l l have are described i n the following two sections. 90 Chapter 6. 91 Conclusion Z Projection of WC1, WC2 and Fibre Scintillator to WC4 dz at WC4 Outer dz outer Nent = 5000 200 r 180 Mean =-0.00465117] Mean = -0.0054811 RMS =2.06441 RMS =1.9117 160 '140 r 120^ 100 r 80^ 604020-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 10 15 20 25 Figure 6.48: Simulated resolution of z\ and z' found using W C l , W C 2 and a fibre scin0 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 i n 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 F S is i n a region after the particle has been deflected by the fringe field. Using the fibre scintillator before W C 3 along w i t h W C l and W C 2 information would give a reasonable error i n our knowledge of z\ and z' . A simulation using root was done 0 to see how well we could know the corrected z coordinates in W C 4 using the F S ; the simulation of the resolution of the projected z coordinates i n W C 4 is shown i n Figure 6.48. Chapter 6. Conclusion 92 The fibre used i n this simulation was assumed to be 3 m m i n diameter. A listing of the root macro used to generate the histogram i n Figure 6.48 can be found i n A p p e n d i x 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. W i t h o u t any decay events, the resolution from C H A O S should be much better. In principle the stack w i l l be able to eliminate over 98% of the pion decay events, so data w i t h cuts using information from the stack w i l l be useful i n 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 i n 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. W h e n 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 i n analysis code is a recalculation of the incident beam angles when one of the hits i n W C l or W C 2 i n the incident beam region is missing. The recalculation is done using the coordinates of the scattered track's intersection w i t h the target plane along w i t h the wire chamber hit that we do obtain i n the incident beam region. W h e n we do get b o t h wire chamber hits i n the incident beam, we could use the z_proj as an additional point i n our fit to find the parameters of the scattered track. A quick trial, 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 i n Figure 6.49. T o compare the results to where z_proj was not used i n the picket fence reconstruction see Figure 3.22. 93 Chapter 6. Conclusion Run 6257 08-flpr"99 SUB= I2I19J15 Peak- 252. Mean bin= 23.0 at bin 41.214 Mean data= 52 data- "8.286 signa= 2.5 12.551 A 20 • B c /I 60 him Un=0. i ' ' ' * i ' 1 • ' i -40 -2D 0 20 40 60 Hi 924 s h _ a l l _ n d 2 I h _ a l l / s m _ s c a t _ n d 2 ] Ov = 0. 1 1 1 Figure 6.49: Picket fence reconstruction using small scattering angle data from r u n 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. T h e resolutions of picket fence bars using z_proj as a point, for run 6257, are summarised i n Table 6.9. One reason for the considerable improvement i n the resolution is that any pion decay events which occur i n the outgoing beam that are not identified w i l l be corrected slightly by the extra point used i n the fit. Bar Z_all a (mm) for small 9 Z-all a (mm) for large 9 A B C 1.49 2.40 2.19 1.15 1.80 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, P r o c . Phys. M a t h . Soc. Japan 17, 48 (1935). [2] C . M . G . Lattes, M . M u i r h e a d , G . P . S . Occhialini a n d C . F . Powell, Nature 1 5 9 , 694 (1947). [3] W . Heisenberg, Z . Physik 43, 172 (1927). [4] C . J . O r a m et. al, N u c l . Inst. M e t h . 179, 95 (1981). [5] G . R . S m i t h et al, N u c l . Inst. M e t h . A362, 349 (1995). [6] K . S . K r a n e , Introductory Nuclear Physics. John W i l e y & Sons, New York, 1988. [7] M . K e r m a n i , M . S c . Thesis, University of B r i t i s h Columbia, Unpublished, 1993. [8] G . C . Barbarino et al, N u c l . Inst. M e t h . 179, 353 (1981). [9] C . B i i n o et al, N u c l . Inst. M e t h . 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 B a r r y H . Holstein eds., P14. [12] H . W i n d , N u c l . Inst. M e t h . 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 A r t of Scientific Computing. Cambridge University Press, Cambridge, 2nd edition, (1992). [15] Particle D a t a Book, Phys. Rev. D54, 575 (1996). [16] C . J o r a m et. al, Phys. Rev. C51, 2144 (1995). [17] Halzen, F . and M a r t i n , A , Quarks and Leptons. John W i l e y & 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. 1 1 3 , 1640 (1959). [21] T R I U M F Users Handbook. [22] Gertjan Hofman, M . S c . Thesis, University of B r i t i s h Columbia, Unpublished, 1992. [23] Sheila Mcfarland, M . S c . Thesis, University of B r i t i s h Columbia, Unpublished, 1993. [24] T R I U M F Kinematics Handbook. [25] J . Lange, P h . D . Thesis, University of B r i t i s h Columbia, Unpublished, 1997. [26] D . Halliday, Introductory Nuclear Physics, John W i l e y & Sons, New York, 1955. [27] G . Hofman, P h . D . Thesis, University of B r i t i s h 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 i s the material d e f i n i t i o n f i l e f o r CHAOS c n i expt. t o t a l ! UNITS ARE cm ! ! F i r s t go from target to Q7 window i n M13 ! i n and out of the spectrometer, ie to upstream face of the stack. !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! next go from target to face of stack: MM!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! *_wcl_out air 11 .0 ! a i r to anodes (11.3) - gap-roha aluminum 1. 0e-4 ! r f s h i e l d 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" mylar 2. 54e-3 ! inner wall 2 96 thick*(width/pitch) (1 to elo 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 s h i e l d aluminized (guess 1 micron) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! *_wc2_out air 11.3 ! a i r to anodes (22.6) - .3 -(wcl+.3) (2 to aluminum 1.0e-4 ! r f s h i e l d 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 0.5e-5 ! 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. 0e-4 ! r f s h i e l d aluminized (guess 1 micron) !!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! *_wc3_out air II. 1 ! a i r wc3 rad (34.4)-gap(.4)-(wc2+.3) (3 to aluminum 1.0e-4 ! r f s h i e l d aluminized (guess 1 micron) Appendix A. Trace Program Used to Generate Tracks Through CHAOS 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" mylar 2 54e-3 ! inner wall 2 copper 1 5e-5 rohacell 0 1 ! rohacell mylar 1 25e-3 ! window (should be kapton) mylar 2 54e-3 ! r f shield aluminum 1 Oe-4 ! r f s h i e l d aluminized (guess 1 micron) thick*(width/pitch) ! 1500 angstroms copper (ECN) i i i i i i i ii i i i I I I I i i i i I ii i i i ii 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 ii *_wc4_out air 24.7 20. ! a i r (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 i s 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) 98 Appendix A. Trace Program Used to Generate Tracks Through 99 CHAOS ! rib: !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 1 5e-5 65. ! 1500 angstroms copper (ECN) i i i i i i i i i i i i M M 1 M M 1 M 1 M 1 M 1M M 1M 1M M M M M 1 M M 1M 1 *_air:wc4_to_stack air 140. mylar 2.54e-3 air 1000. end of files Trace A.2 Code program track c track.f c c This program generates out of plane tracks i n CHAOS. c at CHAOS wire chambers 1, 2 and 4 are output. Hit coordinates The deviation of the c out-of-plane h i t s i n WC4 from a straight l i n e i s also output. c files c c of data produced by t h i s program are: pathXXXX.dat p a r t i c l e track coordinates XXXX=track number (pathlen,x,y,z) The Appendix A. Trace Program Used to Generate Tracks Through CHAOS c hitXXXX.dat coordinates of WC h i t s (pathlen,x,y,z) c wc4zin.dat z _ i n , pathlen_in, p/B r a t i o , z _ i n ' c wc4zout.dat z_out, pathlen_out, p/B r a t i o , 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 logical hit_del integer*4 ndat common / t r a c e _ i n / 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') 100 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 s t a r t i n g coordinates 20 x = -1.25 + 2.5 * random(iseed) y = -1.25 + 2.5 * random(iseed) r = if sqrt(x**2+y**2) (r.gt.2.5) go to 20 z = -2.5 + 5.0 * random(iseed) c vertex(l) = x vertex(2) = y vertex(3) = z Then choose s t a r t i n g 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 Call "trace" to track p a r t i c l e through the CHAOS f i e l d call trace(hit_del) ngen = ngen + 1 if (.not.hit_del) nacc = nacc + 1 goto 25 Appendix A. Trace Program Used to Generate Tracks Through CHAOS c Extrapolate z at wc4 inner and outer wire chambers c using track data ndat=40 call f it(pdat,zdat,ndat,sig,0,a,b,siga,sigb,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 if ( ngen . I t . file 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 if (pdat(kk).eq.0.0) write goto 101 (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) 102 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) if (nacc.lt.ntot) close (20) write (6,*) go to 20 ' Number of generated and accepted events: ngen,nacc end <fc ^t* *fc ^t* *fc *^ *^ *f* *^ ^1^ ^ ^* ^ *fc *k ^t* ^ 4^ ^ ^ ''f* ^* ^ *^ ^ subroutine init_mat i m p l i c i t none c material info arrays real*4 ang(200), dx(200), dist(200), integer*4 mater(200), rad(200) max_mat character*20 label(200) c liline variables real*4 rpar(10) integer*4 npar, partyp(10), character*132 character*80 line spar(10) character*20 item c other variables integer*4 str_len, i, jj character*80 s t r i n g character*40 mat_name,get_mat_name ilen,ist,ipar(10) *^ Appendix A. Trace Program Used to Generate Tracks Through CHAOS l o g i c a 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) ilen,string do i=l,10 spar(i)=' ' rpar(i)=0.0 end do call liline(string,ilen,npar,spar,rpar,ipar.partyp,ist) if (spar(l)(1:1) .eq. '*') then l i t i s a l a b e l to remember do j j = l , 2 0 item(jj:jj)=' ' end do s t r _ l e n = index(spar(1),' ') str_len = str_len - 1 item(l:str_len) else i f + ( = spar(l)(1:str_len) spar(l)(l:l) .ne. '!' .and. s p a r ( l ) ( l : l ) .ne. ' ') max_mat = max_mat + 1 dist(max_mat) ang(max_mat) = rpar(2) = rpar(3) then 104 Appendix A. Trace Program Used to Generate Tracks Through dx(max_mat) = dist(max_mat) 105 CHAOS / cosd( ang(max_mat) ) rad(max_mat) = rad(max_mat-l) + dx(max_mat) s t r _ l e n = index( s p a r ( l ) , ' ' ) str_len = str_len - 1 label(max_mat) = item ! f l i p the s t r i n g to uper case do i = l , s t r _ l e n if (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) if ( mat_name(l:str_len) .eq. spar(1)(1:str_len) ) then found = .true, end i f end do if ( .not.(found) write(6,*) end i f ) then 'Material ' , spar(1)(1:20),' not i n data base' Appendix A. Trace Program Used to Generate Tracks Through CHAOS 106 end i f end do 300 continue return end subroutine trace(hit_del) c T h i s s u b r o u t i n e t r a c e s a charged p a r t i c l e (assumed t o be a p i o n ) c t h r o u g h t h e CHAOS f i e l d , 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 s e t t o t r u e i f t h e p a r t i c l e hits c the d e l counter. the c p a r t i c l e i s probably s p i r a l i n g i n s i d e the f i e l d , c returns with hit_del false, c Inputs i n s m a l l s t e p s g i v e n by d t (in ns). I f t h e number of s t e p s t a k e n exceeds 2 0 0 , and t h e routine ( v i a common b l o c k t r a c e _ i n ) : c fs magnetic f i e l d strength i n Tesla c vertex(3) initial (x,y,z) c mom_vec(3) initial ( p , t h , p h i ) w i t h p i n MeV/c, coordinates ( i n cm) c 0 < t h < 360, t h e i n - p l a n e CHAOS a n g l e , c 0 < p h i < 180, t h e o u t - o f - p l a n e CHAOS a n g l e c (phi=90 i n t h e m i d - p l a n e of CHAOS) c Outputs ( v i a common b l o c k trace_out): c wcl(4) ( p a t h l e n , x , y , z ) c o o r d i n a 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 ) c o o r d i n a t e s of h i t i n wc2 c wc4(8) (pathlen i n n e r , x , y , z inner, c c pathlen o u t e r , x , y , z outer) c o o r d i n a 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 r a v e l l i n g c through CHAOS. i m p l i c 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 p i , t h l , p h i , e , g a m , b e t , c , v , x O , y O , z O , r O 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 d e , f r a c , a n g l , z d e l integer*4 n l s t e p s , cur_mat, kk, ccount real*4 xy_to_phi_conv logical hit_del c material info arrays real*4 ang(200), dx(200), dist(200), integer*4 mater(200), max_mat rad(200) Appendix A. Trace Program Used to Generate Tracks Through CHAOS character*20 label(200) c common blocks common / t r a c e _ i n / 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 S t a r t i n g 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 = p l / e c = 0.6439458 / gam v = 29.979 * bet xO = vertex(l) yO = vertex(2) zO = vertex(3) rO = sqrt(x0**2+y0**2) xi = xO yi = yo zi = zO ri = rO vxi = v * sind(phl) * cosd(thl) v y i = v * sind(phl) * s i n d ( t h l ) v z 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 do kk = zeros 1,200 pdat(kk) = 0 xdat(kk) = 0 ydat(kk) = 0 zdat(kk) = 0 end do c Start of main t r a c i n g loop hit_del = .false, nlsteps = 0 cur_mat = 1 50 nlsteps = nlsteps + 1 if (nlsteps.gt.200) go to 60 call & runge_kutta(fs,c,xi,yi,zi,vxi,vyi,vzi,dt, x,y,z,vx,vy,vz) prevpathlen = pathlen pathlen = pathlen + sqrt( pdat(nlsteps) (x-xi)**2 + (y-yi)**2 ) = pathlen xdat(nlsteps) = x ydat(nlsteps) = y zdat(nlsteps) = z r = sqrt(x**2+y**2) c Lets subtract the c that we just took energy l o s t over small step t h l = atand( vy / vx ) 110 Appendix A. Trace Program Used to Generate Tracks Through CHAOS phi = atand( vy / vz / s i n d ( t h l ) ) tpi = e - mpi if ( r if .gt. ( r r i ) then .It. r a d ( cur_mat ) ) t h e n ! t h i s i s t h e easy ! - we're s t i l l case i n same m a t e r i a l de = 0 . 0 rpar = r - r i call calc_e_loss( mater(cur_mat),rpar, m p i , t p i , de ) e = e - de e l s e i f ( cur_mat . e q . max_mat ) t h e n ! w e ' r e o u t s i d e s p e c t r o m e t e r now, so use air de = 0 . 0 rpar = r - r i call c a l c _ e _ l o s s ( 1 , r p a r , m p i , t p i , de ) e = e - de else ! we have t o do s t e p t h r o u g h s e v e r a l m a t e r i a l s rcur = r i do w h i l e ( rcur .It. r ) de = 0 . 0 if ( rad(cur_mat) .gt. r ) then ! t h i s i s l a s t m a t e r i a l to step through rpar = r - r c u r c a l l c a l c _ e _ l o s s ( mater(cur_mat), e = e - de r p a r , m p i , t p i , de ) Appendix A. Trace Program Used to Generate Tracks Through CHAOS tpi 111 = e - mpi rcur = r else ! step through next m a t e r i a l rpar = rad(cur_mat)-rcur c a l l c a l c _ e _ l o s s ( mater(cur_mat), e = e - de tpi = 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 e n e r g y , so r e c a l c u l a t e pi = sqrt( vx,vy,vz e*e - mpi*mpi ) bet = p i / e v = b e t * 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 s u r e we g e t t h e s i g n s r i g h t . if ( (vx . I t . 0.0) .and. ( v x i . g t . 0.0) .and. + (vy . I t . 0 . 0 ) . a n d . ( v y i . I t . 0 . 0 ) . a n d . + ( v x i . g t . abs( v y i ) ) ) then vx = - v x r p a r , m p i , t p i , de ) Appendix A. Trace Program Used to Generate Tracks Through CHAOS 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 if ( (vx .It 0 0) . and. (vxi •gt. 0 0) ) ccount = if ( (vx •gt 0 0) . and. (vxi .It. 0 0) ) ccount = if ( (vy .It 0 0) . and. (vyi •gt- 0 0) ) ccount = if ( (vy •gt 0 0) . and. (vyi .It. 0 0) ) ccount = if ( ccount eq 2 ) then vx = -vx vy = -vy end i f c Record coordinates of h i t i n wcl if ((ri.It.rwcl).and.(r.ge.rwcl)) then frac = ( r w c l - r i ) / ( r - r i ) xhit = x i + f r a c * ( x - x i ) yhit = y i + frac*(y-yi) zhit = z i + frac*(z-zi) angl = atan2d(yhit,xhit) if (angl.It.0.) wcl(l) angl = 360. + angl = prevpathlen + frac*(pathlen-prevpathlen) wcl(2) = xhit 112 Appendix A. Trace Program Used to Generate Tracks Through CHAOS wcl(3) = yhit wcl(4) = z h i t end i f c Record coordinates of h i t i n wc2 if ((ri.It.rwc2).and.(r.ge.rwc2)) then frac = ( r w c 2 - r i ) / ( r - r i ) xhit = x i + f r a c * ( x - x i ) yhit = y i + frac*(y-yi) zhit = z i + frac*(z-zi) angl = atan2d(yhit,xhit) if ( a n g l . I t . 0 . ) angl = 360. + angl wc2(l) = prevpathlen + frac*(pathlen-prevpathlen) wc2(2) = xhit wc2(3) = yhit wc2(4) = z h i t end i f c Record z-coordinate of h i t i n f i r s t r e s i s t i v e if ((ri.It.rwc4_l).and.(r.ge.rwc4_l)) wire of wc4 then frac = ( r w c 4 _ l - r i ) / ( r - r i ) wc4(l) = prevpathlen + frac*(pathlen-prevpathlen) xhit = x i + f r a c * ( x - x i ) yhit = y i + frac*(y-yi) zhit = z i + frac*(z-zi) wc4(2) = xhit wc4(3) = yhit wc4(4) = z h i t 113 Appendix A. Trace Program Used to Generate Tracks Through CHAOS end i f c Record z-coordinate of h i t i n second r e s i s t i v e wire of wc4 if ((ri.It.rwc4_2).and.(r.ge.rwc4_2)) then frac = ( r w c 4 _ 2 - r i ) / ( r - r i ) wc4(5) = prevpathlen + frac*(pathlen-prevpathlen) xhit = x i + f r a c * ( x - x i ) yhit = y i + frac*(y-yi) zhit = z i + frac*(z-zi) wc4(6) = xhit wc4(7) = y h i t wc4(8) = z h i t end i f c Check v e r t i c a l p o s i t i o n at del radius if ((ri.It.rdel).and.(r.ge.rdel)) frac = ( r d e l - r i ) / ( r - r i ) zhit = z i + frac*(z-zi) zdel = abs(zhit) if (zdel.le.8.9) then h i t _ d e l = .true. end i f go to 60 end i f xi = x yi = y zi = z ri = sqrt(xi**2+yi**2) then 114 Appendix A. Trace Program Used to Generate Tracks Through CHAOS v x i = vx v y i = vy v z 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 p a r t i c l e (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 c xi,yi,zi i n i t i a l coordinates c v x i , v y i , v z i i n i t i a l v e l o c i t i e s (in cm/ns) c dt c Outputs x , y , z (6439.5/gamma) (in cm) time i n t e r v a l coordinates after step c vx,vy,vz v e l o c i t i e s after step c Written by: Roman Tacik real*4 k l x , k l y , k l z , k l v x , k l v y , k l v z real*4 k2x, k2y, k2z, k2vx, k2vy, k2vz real*4 k3x, k3y, k3z, k3vx, k3vy, k3vz real*4 k4x, k4y, k4z, k4vx, k4vy, k4vz call klx bfield(xi,yi,zi,fs,bx,by,bz) = dt * v x i 115 Appendix A. Trace Program Used to Generate Tracks Through CHAOS kly = dt * v y i klz = dt * v z i klvx = c * (kly*bz - klz*by) klvy = c * (klz*bx - klx*bz) klvz = c * (klx*by - kly*bx) call bfield(xi+klx/2.,yi+kly/2.,zi+klz/2.,fs,bx,by,bz) k2x = dt * (vxi+klvx/2.) k2y = dt * (vyi+klvy/2.) k2z = dt * (vzi+klvz/2.) k2vx = c * (k2y*bz - k2z*by) k2vy = c * (k2z*bx - k2x*bz) k2vz = c * (k2x*by - k2y*bx) call bfield(xi+k2x/2.,yi+k2y/2.,zi+k2z/2.,fs,bx,by,bz) 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) call bf i e l d ( x i + k 3 x , y i + k 3 y , z i + k 3 z , 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 = xi + ( k l x +2.*k2x +2.*k3x +k4x)/6. y = yi + ( k l y +2.*k2y +2.*k3y +k4y)/6. z = zi + ( k l z +2.*k2z +k4z)/6. vx = v x i + +2.*k3z (klvx+2.*k2vx+2.*k3vx+k4vx)/6. vy = v y i + ( k l v y + 2 . * k 2 v y + 2 . * k 3 v y + k 4 v y ) / 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 ) Bl(61,20,2),B2(501,20,2),B3(61,20,2) REAL*4 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 OPEN & (10) (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) B3(I,J,1),B3(I,J,2) END DO END DO CLOSE (10) END I F R=SQRT(X**2+Y**2) IF (X.EQ.O.) IF THEN (Y.GT.O.) THEN TH=90. ELSE I F ( Y . E Q . O . ) THEN BX=0. BY=0. BZ=B1(1,1,1) GO TO 70 ELSE IF ( Y . L T . O . ) TH=270. END IF ELSE TH=ATAND(Y/X) THEN Appendix A. Trace Program Used to Generate Tracks Through CHAOS IF ((Y.GE.O.).AND.(X.LT.O.)) TH=180.+TH IF ((Y.LT.O.).AND.(X.LT.O.)) TH=180.+TH IF ((Y.LT.O.).AND.(X.GT.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) DI=(R-0.5*(I-l.))/0.5 J=2.*(AZ+0.5) DJ=(AZ-0.5*(J-l.))/0.5 T1=B1(I,J,1) +DI*(B1(I+1,J,1) -BKI.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) -B1(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 BX=-BX BY=-BY END I F ELSE I F ((R.GT.30.).AND.(R.LE.80.)) THEN RB=R-30. I=10.*(RB+0.1) DI=(RB-0.1*(I-1.))/0.1 J=2.*(AZ+0.5) DJ=(AZ-0.5*(J-l.))/0.5 T1=B2(I,J,1) +DI*(B2(I+1,J,1) -62(1,3,1)) T2=B2(I,J+1,1)+DI*(B2(I+1,J+1,1)-B2(I,J+1,1)) BR=T1+DJ*(T2-T1) T1=B2(I,J,2) +DI*(B2(I+1,J,2) -B2(I,J,2)) T2=B2(I,J+1,2)+DI*(B2(I+1,J+1,2)-B2(I,J+1,2)) BZ=T1+DJ*(T2-T1) BX= BR*COSD(TH) BY=-BR*SIND(TH) IF (Z.LT.O.) THEN BX=-BX BY=-BY END I F ELSE IF ((R.GT.80.).AND.(R.LE.140.)) RB=R-80. I=RB+1.0 DI=RB-I+1. J=2.*(AZ+0.5) THEN Appendix A. Trace Program Used to Generate Tracks Through CHAOS DJ=(AZ-0.5*(J-1.))/0.5 T1=B3(I,J,1) +DI*(B3(I+1,J,1) -B3(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) -B3(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.LT.O.) THEN BX=-BX BY=-BY END I F ELSE IF ( R . G T . 1 4 0 . ) THEN BX=0. BY=0. BZ=0. END I F 70 BX=FS*BX*1.0648/10000. BY=FS*BY*1.0648/10000. BZ=FS*BZ*1.0648/10000. END c FUNCTION MAT NAME c h a r a c t e r * 4 0 f u n c t i o n get_raat_name(jmat) Appendix A. Trace Program Used to Generate Tracks Through CHAOS i m p l i c 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 122 Appendix A. Trace Program Used to Generate Tracks Through CHAOS c======================================================== subroutine calc_e_loss(mat,dist,Mpr,Tpr,de_tot) i m p l i c i t none c t h i s routine handles to fact that f o r a large loss the medium has c to be s l i c e d up. real*4 d i s t , M p r , T p r , d d e , d e , d e _ t o t , T p r _ l o c a l integer*4 m a t , i , j , k Tpr_local=Tpr ! dont want to change Proton enery f o r main c a l l absorb (mat, d i s t , Mpr, 1., 0, T p r _ l o c a l , de, dde) c check i f the energy loss was a small f r a c t i o n of the energy c bigger then say 10% if (abs(de/Tpr).It. 0.05) then de_tot=de return end i f c i f not, s l i c e up the material i n say 100 s l i c e s do i = l , 100 c a l l absorb (mat, d i s t / 1 0 0 . , Mpr, 1., 0, T p r _ l o c a l , de, dde) de_tot=de_tot+de Tpr_local=Tpr_local-de c now we could s t i l l be i n trouble - when the proton i s so slow that c s l i c e s 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 i s off the c order of the energy i t s e l f . if end do ( T p r _ l o c a l . l t . .01 . o r . ( T p r _ l o c a l - d e ) . I t . 0.) return 124 Appendix A. Trace Program Used to Generate Tracks Through CHAOS return end c=========================================================== c SUBROUTINE to calculate the energy loss (Greg Smith) c=========================================================== SUBROUTINE absorb (JMAT, xO, xmass, z i n c , i s t e u , c usage: c c a l l absorb (JMAT, xO, xmass, z i n c , i s t e u , p t b , c index, thk(cm) -1 c index, thk(cm) 0 c index, thk(cm) +1 character*40 mat_nam(200),mater_nam c param( p r o p e r t i e s , #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 ) = A i r 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 p t b , de, dde) de, momentum eloss(meV) dde) uncert. K.E. eloss(meV) uncert. beta eloss(meV) uncert. Appendix A. Trace Program Used to Generate Tracks Through CHAOS c param( i , 7 ) = Iron c param( i , 8 ) = Helium gas c param( i , 9 ) = S i l i c o n c param( i,10)= Magic Gas c param( i , l l ) = Helium l i q u i d c param( i,12)= butanol l i q u i d c param( i,13)= Ice c param( i,14)= Copper c param( i,15)= MYLAR c param( i,16)= glO c param( i,17)= LH2 c param( i,18)= r o h a c e l l common/mat/ param(7,20) c in the data statement, each l i n e f o r each m a t e r i a l . c 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, 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., 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 1 , 14.0,28.6,172.0, 2.33, 0.2182, 1 , 22.8,45.4, 266.8, 0.00204, 12. 14.4 ! a i r ! carbon ,2.0 , 4.0 ! He gas 14.0, 28.6 ! S i l i c o n 121.5, 22.8, 45.4 ! Magic Gas 125 Appendix A. Trace Program Used to Generate Tracks Through CHAOS 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 , 2 9 . , 64.0 , 320. , 8 . 9 6 0 , 0.0143, 29.0 , 64.0 ! copper 1 , 3 . 7 7 8 , 7.12085 , 65.2 , 1.39, 0.287, 3.778 , 7.12085 ! Mylar 1 , 10.0, 20.0, 150.0 , 1 ,2., 2 . , 21.8, 0.0708, 8.65, 1., 1 , 6., 12., 80.3, 50.0E-3, 7.93, 6 . , 1 ,6., 12., 80.3, 1.74, 0.245, 6 . , 12. 1 , 6., 12., 80.3, 1.74, 0.245, 6., 12. / 1.7, 0.194, 10.0, 20.0 ! glO 1. ! LH2 12. ! Rohacell ! carbon ! 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) write(6,*)' then Insufficient energy i n absorb: ' , ptb c a l l exit endif IF(ISTEU) 30,40,50 c i s t e u = -1 —> ptb i s momentum 30 126 E=PTB*PTB+XMASS*XMASS P=PTB T=E-XMASS BETA=PTB/E Appendix A. Trace Program Used to Generate Tracks Through CHAOS BEGA=PTB/XMASS GO TO 60 c i s t e u = 0 —> ptb i s k i n e t i c energy 40 E=PTB+XMASS T=PTB P=SQRT(PTB*(PTB+2.*XMASS)) BETA=P/E BEGA=P/XMASS GO TO 60 c i s t e u = +1 —> ptb i s 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) — - ; : 127 Appendix A. Trace Program Used to Generate Tracks Through CHAOS c========================================================== SUBROUTINE C ELOSSTOF(DEDRHOX,DDEDRH0X2,DD,CC,BETA,BEGA,ZINC2) 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 C DD : CONST 0.31007*Zmed/Amed C CC : CONST 2*Me/Imed C BETA : v / c OF INCOMMING PARTICLE [MeV**2*cm**2/g] 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 XME=0.511 0.3070*1.01(NU-PART) ! " [MeV*cm**2/Mol] MASS(ELEKTRON) B2=BETA*BETA BG2=BEGA*BEGA DEDRHOX=0. DDEDRH0X2=0. I F C C C . L T . 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 [Me Appendix A. Trace Program Used to Generate Tracks Through CHAOS c Gertjan J Hofman Triumf '90 c This function returns atan(x/y) c This i s used by MANY of the u s e r l i b _ f routines c GJH 31/8/95 modified to use atan instead. between 0—>360 deg. real*4 function xy_to_phi_conv(x,y) i m p l i c i t none real*4 x , y real*4 rad_to_deg parameter(rad_to_deg=57.29578) if (abs(x).lt. l.e-10 .and. y . l t . O . ) then xy_to_phi_conv = 270. return end i f if (abs(x).lt. l.e-10 .and. y.ge.O.) then xy_to_phi_conv = 90. return end i f if (x.lt. - l . e - 1 1 ) then xy_to_phi_conv = atan(y/x)*rad_to_deg return end i f if + l.e-11 .and. y.ge.O.) xy_to_phi_conv=atan(y/x)*rad_to_deg if + (x.gt. (x.gt. l.e-11 .and. y . l t . O . ) xy_to_phi_conv=atan(y/x)*rad_to_deg+360. + 180. Appendix A. Trace Program Used to Generate Tracks Through CHAOS return end subroutine yget_runno( convert irunno, crunno ) c Purpose: integer run number (irrunno) c character array run number c Input: c integer*4 irunno c Output: c character crunno(4) into a four d i integer containing run number 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 W C 4 // 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; if (npts <= 2 ) return; for ( i = 0 ; i < npts ; i++ ) { sx = sx + x[i] ; sy = sy + y [ i ] ; } sxoss = sx/( for (Double_t)npts ( i = 0 ; i < npts ); ; i++ ) { t= x[i] - sxoss; 131 Appendix B. R o o t 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 i b r e // scintillator to WC4 r e s i s t i v e wires look l i k e / / Create a new canvas TCanvas * c3 = new TCanvas("c3","Z P r o j " , // 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, pd2 = new TPad("pd2","WC4 outer", 0.03, 0.97,0.9 ); pdi 0.52, 0.9 ); -> 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 , h 2 _ l , h i l ; Double_t hfibreO, d_hi, d_ho, h o i , h t l , slop; const Double.t deg2rad = 3.141597 / 180.0; const Double_t rwcl const Double.t rwc2 = 114.59; = 229.18; const Double_t rwc4in = 617.5; const Double_t rwc4ou = 672.5; const Double_t r f i b r e = 325.0 const Int_t kUPDATE = 250; f o r ( Int_t i = 0; i< 5000; i++ ) { angO = 14.0 * ( gRandom->Rndm( 1 ) - 0.5 hfibreO = 40.0 + 3.0 * gRandom->Rndm(); htO = hfibreO - r f i b r e * 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; ) * deg2rad; Appendix B. Root Macro Used to Find Residual in Projected x[l] = rwc2; x[2] = rfibre; y[0] = hl_l; y[l] = h2_l; y[2] = hfibreO; / / Do f i t Unfit( 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 ); hist2->Fill( } pd2->cd(); hist->Draw(); pdl->cd(); hist2->Draw(); pdl->Modified(); pd2->Modified(); pdl->Update(); pd2->Update(); d_ho ); Z at WC4 134
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Calibration of the chaos spectrometer for small scattering...
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
Calibration of the chaos spectrometer for small scattering angles Jamieson, Blair Alex 1999
pdf
Page Metadata
Item Metadata
Title | Calibration of the chaos spectrometer for small scattering angles |
Creator |
Jamieson, Blair Alex |
Date Issued | 1999 |
Description | For measurements of pion-nucleon scattering in the Coulomb-Nuclear Interference (CNI) region, it is implicit that we are looking at small scattering angles. At small scattering angles the in-plane (x,y coordinate) scattering angle is not the true scattering angle. Since the Canadian High Acceptance Orbit Spectrometer (CHAOS) 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 CHAOS 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 WC4 requires a fringe field correction, because it lies outside the uniform magnetic field of CHAOS. The fringe field of CHAOS acts like a lens, in that it has a slight focusing effect on charged particles. The fringe field correction moves the WC4 (x,y,z) data point so that it is on the linear object ray. In this paper I will present the model for the fringe field correction. I will explain how the three dimensional scattering angle resolution was determined, and as the final test of the scattering angle I will present the cross-section for fiC scattering, obtained using CHAOS data. |
Extent | 5376707 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-06-12 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0099335 |
URI | http://hdl.handle.net/2429/9101 |
Degree |
Master of Science - MSc |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 1999-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
Download
- Media
- 831-ubc_1999-0205.pdf [ 5.13MB ]
- Metadata
- JSON: 831-1.0099335.json
- JSON-LD: 831-1.0099335-ld.json
- RDF/XML (Pretty): 831-1.0099335-rdf.xml
- RDF/JSON: 831-1.0099335-rdf.json
- Turtle: 831-1.0099335-turtle.txt
- N-Triples: 831-1.0099335-rdf-ntriples.txt
- Original Record: 831-1.0099335-source.json
- Full Text
- 831-1.0099335-fulltext.txt
- Citation
- 831-1.0099335.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0099335/manifest