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A precision measurement of the neutron-neutron scattering length from the reaction [pi]-d -> [gamma]… Saliba, Michael Angelo 1998

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A P R E C I S I O N M E A S U R E M E N T OF T H E N E U T R O N - N E U T R O N S C A T T E R I N G L E N G T H F R O M T H E R E A C T I O N TT~~d 7 r m B y M I C H A E L A N G E L O S A L I B A B.Mech.Eng.(Hons), University of Mal ta , 1986 M . A . S c . , University of Br i t i sh Columbia, 1991 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y O F G R A D U A T E S T U D I E S Department of Physics and Astronomy We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y OF B R I T I S H C O L U M B I A A p r i l 1998 © Michael Angelo Saliba, 1998 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of ^British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of IHY6l£S / W h AsmoNOMV The University of British Columbia Vancouver, Canada Date A3 April m * DE-6 (2/88) 11 A b s t r a c t A measurement of the x5o neutron-neutron scattering length ann has been carried out at T R I U M F by studying the shape of the photon energy spectrum from the reaction ir~d —» 77m in the region near the endpoint. A 40.5 M e V pion beam was degraded and stopped in a l iquid deuterium target and al l three final state particles from the reaction were detected in triple coincidence. The photon was detected in a large Na l (T l ) crystal, while the neutrons were detected in a 2 m x 2 m position-sensitive array of plastic scintillation counters, located at a distance of 3 m from the target. The experimental photon energy spectrum was reconstructed to a resolution of 40 keV F W H M from the measured momenta of the two neutrons, and contains 123,000 counts in the top 450 keV region near the endpoint after background subtraction. The value of ann is determined from a comparison of this experimental spectrum to simulated spectra that are being developed simultaneously at the University of Kentucky. These spectra are derived from a new model of this reaction that is based on a half off-shell N N T matrix and the elementary JTT operator due to Lee and Nozawa. The experimental geometry and resolution are taken into account using Monte Carlo techniques. A comparison of our final experimental spectrum to a preliminary set of the simulated spectra has yielded the provisional result of ann — —21.8 ± 0.3 fm (theoretical errors excluded) before correction for electromagnetic effects. This preliminary result is in disagreement wi th the currently accepted experimental value of ann = —18.5 ± 0.3 fm, however we stress that the theoretical model is s t i l l under development. We anticipate that our final result wi l l make a significant contribution to the discussion of charge symmetry breaking in the strong interaction, particularly wi th regard to the current uncertainty that surrounds the contribution of the (p — u) mixing term in standard meson-theoretic potentials. T a b l e o f C o n t e n t s A b s t r a c t i i L i s t o f Tab l e s i x L i s t o f F i g u r e s x S u m m a r y x v i i A c k n o w l e d g e m e n t x x 1 I n t r o d u c t i o n 1 1.1 General Overview 1 1.2 The Nucleon-Nucleon Interaction 3 1.2.1 The Range of the Nuclear Force 3 1.2.2 The Deuteron 4 1.2.3 N - N Scattering at Low Energies 6 1.2.4 N - N Scattering at Intermediate Energies 12 1.3 Measurement of the Neutron-Neutron Scattering Length 13 1.3.1 Previous Measurements 13 1.3.2 Extract ion of ann from ix~d - » jnn 15 1.3.3 Extract ion of ann from nd —> nnp 21 1.4 A n Introduction to T R I U M F Experiment E661 23 1.4.1 Pr imary Considerations 23 1.4.2 The E661 Collaboration 26 i i i iv 2 M e s o n T h e o r y a n d C h a r g e S y m m e t r y B r e a k i n g 27 2.1 Introduction and Preview 27 2.2 Meson Theory 30 2.2.1 The Yukawa Potential 30 2.2.2 The One-Boson Exchange Potential 31 2.2.3 Advanced Meson Exchange Models 34 2.2.4 Phenomenological and Hybr id Models 37 2.3 Charge Independence Breaking and Charge Symmetry Breaking 39 2.3.1 The Isospin Formalism 39 2.3.2 Definitions of C I B and C S B 42 2.3.3 Classification of Isospin Dependent Forces 43 2.3.4 Physical Manifestations of C I B and C S B 44 2.3.5 Sources of C I B and C S B 45 2.3.6 The Contr ibut ion of (p - co) M i x i n g to C S B 47 2.3.7 Current Status of Experiment and Theory 50 2.3.8 C S B from Q C D 51 3 D e s c r i p t i o n o f t h e E x p e r i m e n t 53 3.1 Beam Production and the M13 Channel 53 3.2 The Experimental Set-Up 55 3.2.1 General Layout 55 3.2.2 Electronics Set-Up 58 3.2.3 The Wire Chambers 63 3.2.4 The L iqu id Deuterium Target Assembly 65 3.3 Signal Measurement 66 3.3.1 The C A M A C Signal Sources 66 V 3.3.2 Energy Measurement and the A D C s 67 3.3.3 T ime Measurement and the T D C s 67 3.4 The Experimental R u n 68 3.4.1 Al locat ion of the Available Beam Time 68 3.4.2 The Production Runs 69 3.4.3 The Calibrat ion Runs 70 3.5 Overview of the On-line and Off-line Analyses 71 4 Data Analysis I: Equipment Calibration 74 4.1 Objectives 74 4.2 Cal ibrat ion of the Neutron Bar T D C s 76 4.2.1 Measurement of the Differential Non-Lineari ty of the T D C s . . . 76 4.2.2 Measurement of the Integral Non-Linearity, Ga in , and Relative Ef-ficiency of the T D C s 78 4.2.3 Time and Y-Pos i t ion Cal ibrat ion of the T D C s : Concept and Deriva-tion 82 4.2.4 The T D C Singles Spectra 86 4.2.5 The Y-Pos i t ion Distributions 93 4.2.6 The T D C Calibrat ion Procedure 94 4.2.7 T D C Calibrat ion Results 97 4.2.8 Measurement of the Time Walk of the T D C s 97 4.2.9 Neutron Bar T O F and Y-Pos i t ion Resolutions 100 4.3 Cal ibrat ion of the Neutron Bar A D C s 101 4.3.1 Introduction 102 4.3.2 A Summary of the Fu l l A D C Calibrat ion Procedure 103 4.3.3 The E G S Simulation 104 vi 4.3.4 The Experimental 6 0 C o Energy Spectra 107 4.3.5 The A D C Pedestal Distributions 108 4.3.6 Determination of the A D C Gains 108 4.3.7 Light Attenuation in the Neutron Bars 110 4.4 Cal ibrat ion of the T I N A A D C s 113 4.5 Cal ibrat ion of the T I N A Wire Chamber (WC3) 118 4.6 Cal ibrat ion of the Pion-Tracking Wire Chambers (WC1 and W C 2 ) . . . 122 5 Data Analysis II: Reconstruction of the Photon Energy Spectrum 124 5.1 Introduction 124 5.2 Processing and Sorting of the Da ta 124 5.3 Other E661 Kinemat ic Parameters 129 5.4 The Accidental Background 135 5.5 The Diagonal Background 144 5.6 The Experimental Photon Energy Spectrum 149 6 The Theoretical Models 152 6.1 Introduction • 152 6.2 Phenomenological N N Potential Models 152 6.2.1 General Derivations 152 6.2.2 Notes on N N Potentials 154 6.2.3 The New "High-Precision" N N Potentials 156 6.2.4 The Argonne u 1 8 Potential 157 6.3 The Reaction Model for E661 159 6.4 The Monte Carlo Model for E661 161 v i i 7 R e s u l t s a n d D i s c u s s i o n 165 7.1 Fi ts to the Preliminary Monte Carlo Spectra 165 7.2 Analysis of Systematic Error 167 7.2.1 Overview 167 7.2.2 Da ta Reduction 169 7.2.3 The Vertical Conversion Factor 169 7.2.4 The T D C Pedestals 171 7.2.5 The T D C Gains 173 7.2.6 The T D C Time Walk 174 7.2.7 The A D C Pedestals and Gains 174 7.2.8 The Light Attenuation Correction 176 7.2.9 The T I N A Energy Thresholds 176 7.2.10 Background Subtraction 177 7.2.11 Geometry Measurements 178 7.2.12 Known Defects 179 7.3 Evaluat ion of the Simulated Spectra 183 7.3.1 Error Analysis and Tests of the Monte Carlo 183 7.3.2 Evaluation of the E661 Theoretical Model 189 7.4 Comparison to Previous Experiments 193 7.4.1 The Salter et al. Measurement 194 7.4.2 The Gabioud et al. Measurement 195 7.4.3 The Schori et al. Measurement 199 7.4.4 The Advantages and Disadvantages of E661 202 7.5 Remaining Work 203 7.6 Conclusion 204 viii B i b l i o g r a p h y 208 A p p e n d i x 219 A T h e E 6 6 1 C o l l a b o r a t i o n 219 B D e t e c t o r S p e c i f i c a t i o n s 221 B . l Photon A r m 221 B.2 Beam Path 222 B.3 Neutron A r m 224 B.4 Gamma-tag Detectors 226 C P h o t o g r a p h s 227 D C i r c u i t D i a g r a m s 231 E T h e C A M A C S i g n a l Sources used i n E 6 6 1 237 F A n E s t i m a t e o f t he P h o t o n E n e r g y R e s o l u t i o n i n E 6 6 1 239 L i s t o f T a b l e s 1.1 Summary of selected recent measurements of ann 14 2.1 The main contributions from the various field couplings to the N - N potential. 34 3.1 M13 bending and quadrupole magnet settings for E661 68 3.2 Typica l set of visual scaler readings 69 4.1 T D C test results for Bars 1 to 10 80 4.2 T D C test results for Bars 11 to 20 81 4.3 T O F s and T D C channels for particles detected by the E661 neutron bars. 87 4.4 T D C calibration results for Bar 10, R u n Group 1 97 6.1 Comparison between the new N N potentials 157 6.2 Values of PqJ „„ and the corresponding values of ann in a model based on the Argonne v\% potential 160 7.1 M i n i m u m xl achievable for various theoretically generated spectra. . . . 166 7.2 Neutron bar A D C hardware thresholds 185 7.3 A summary of the experimental systematic errors in the E661 result. . . 205 A . l Members of the E661 collaboration 219 A . 2 Da ta analysis: division of responsibilities 220 B . l The x- and z- coordinates of the neutron bar longitudinal axes. 224 E : l E661 C A M A C signal sources (1-64) \ 237 E.2 E661 C A M A C signal sources (65-122) 238 ix L i s t o f F i g u r e s 1.1 The deuteron wave function for R = 2.1 fm 6 1.2 The basic model for scattering 7 1.3 The wave function for triplet np scattering 8 1.4 Wave function exhibiting a negative scattering length 10 1.5 Summary of p-p cross-section measurements at a laboratory angle of 45° covering the energy range from 300 to 420 keV 11 1.6 The 1S0 phase shift for neutron-proton scattering at intermediate energies. 12 1.7 Previous measurements of ann 14 1.8 Kinematics for ir~d —> jnn 16 1.9 Graph of the reaction n~d —> jnn according to the impulse approximation. 17 1.10 Graph for the pion rescattering correction 19 1.11 Sensitivity of the photon energy spectrum to ann in the reaction ir~d —> jnn. 20 1.12 Sensitivity of the photon energy spectrum to r n n in the reaction ir~d —> jnn. 20 1.13 Sensitivity of the proton spectrum to ann, in the reaction nd —> nnp. . . . 21 2.1 Feynman diagram for the one-boson-exchange contribution to N N scatter-ing in the cm frame 31 2.2 A pictorial representation of the 2n exchange contribution to the N N in-teraction as viewed by dispersion theory 35 2.3 Field-theoretic model for the 2TT exchange 36 2.4 A comparison between the N N scattering phase shifts, as predicted by various potential models, and the experimental data 38 x x i 2.5 General graph for the 27r-exchange three-body potential, used in the Tucson-Melbourne model 39 2.6 A n illustrative representation of a rotation of ir about the y-axis of isospin space 41 2.7 One-pion exchange contributions to (a) pp (nn) and (b) pn scattering. . . 46 2.8 Contribution of electromagnetic meson mixing to C S B 47 2.9 The decay process co —> p —» 2ir 48 2.10 The (p - co) contributions to the N - N potential 49 3.1 The T R I U M F cyclotron and beam-lines : ' 54 3.2 The M13 beam line layout 55 3.3 E661 experimental set-up: plan view (schematic) 56 3.4 General schematic for the E661 electronics set-up: Product ion of the four E661 trigger types 59 3.5 Discrimination of the neutron bar P M T pulses 63 3.6 Beam's view of the L D target 64 3.7 Horizontal sectional view of the L D target 65 3.8 Da ta analysis of E661 using the D I S P L A Y program 73 4.1 Differential non-linearity test result for T D C lOt 77 4.2 Differential non-linearity test result for T D C 4b 77 4.3 Integral non-linearity test result for T D C lOt (raw spectrum) 78 4.4 Graph used for the extraction of the integral non-linearity, and gain, of T D C lOt 79 4.5 T O F and y-position measurement of the neutrons: conceptual sketch. . . 82 4.6 T O F S2 - Converter, Runs 346, 347 and 352 (Run Group 1) 88 x i i 4.7 Bar 10 T D C mean time singles spectra for gtof events, R u n Group 1 (corrected for path length) 89 4.8 Bar 10 T D C mean time singles spectra for ntof events, R u n Group 1 (corrected for path length) 89 4.9 Bar 10 T O F : Peaks l b , 2b and 3 (Run Group 1) 92 4.10 Y-posi t ion distribution for Bar 10, R u n Group 1 93 4.11 T d i f f distribution for Bar 10, R u n Group 1 94 4.12 T O F pedestal vs. run group for Bar 10 98 4.13 Y-posi t ion pedestal vs. run group for Bar 10 98 4.14 Illustration of T D C time walk for a leading edge discriminator 99 4.15 Geometry for the neutron bar E G S simulation 105 4.16 The Monte Carlo energy spectrum of the electrons in a plastic scintillator, obtained from the E G S simulation 106 4.17 The relative importance of the three major photon interactions wi th matter.106 4.18 The A D C calibration spectrum for A D C lOt (Run Group 2). . 108 4.19 The pedestal distribution for A D C lOt 109 4.20 The simulated E G S spectrum after convolution with a Gaussian resolution function, rebinning, and convolution with the A D C pedestal 110 4.21 Variat ion of A D C lOt gain wi th time during E661 I l l 4.22 Variat ion of A D C lOt pedestal with time during E661 I l l 4.23 Light yield functions for Bar 11 112 4.24 Energy distribution of the E661 photons that were incident on the T I N A collimator, as obtained from the E661 Monte Carlo [63] 114 4.25 E G S result for 7-ray energy deposition in T I N A 115 4.26 E G S result for 7-ray energy deposition in the converter 115 4.27 E G S result for the total 7-ray energy deposition in T I N A and the converter. 116 4.28 Calibrated energy distribution in T I N A for the E661 data 117 4.29 Calibrated energy distribution in the converter for the E661 data 117 4.30 Calibrated energy distribution in T I N A and the converter for the E661 data. 118 4.31 Calibrated x-position distribution in W C 3 120 4.32 Calibrated y-position distribution in W C 3 121 4.33 X Y density plot in W C 3 for E661 events 121 4.34 Number of individual W C 3 x-plane wire hits per event 122 4.35 Number of individual W C 3 y-plane wire hits per event 123 5.1 Flow chart for P R E S O R T $ U S E R . F O R 125 5.2 E661 E1 spectrum, before background subtraction 129 5.3 E661 t2 vs. ti density plot 130 5.4 Magnitude of the relative momentum vector q 131 5.5 Momentum distribution of the neutrons 131 5.6 Distr ibut ion of the x-component of p r e 5 i d 132 5.7 Dist r ibut ion of the y-component df p r e s i d 133 5.8 Distr ibut ion of the z-component of p r e s i d 133 5.9 Bar number hit pattern 134 5.10 Y-posi t ion distribution in Bar 10 134 5.11 Energy deposited by the neutrons in the bars 135 5.12 Bot tom vs. top T D C calibrated times, for Bar 10 single-hit events 136 5.13 X-posi t ion distribution for single-hit events 137 5.14 The E661 accidental background windows 138 5.15 M i n i m u m xl achievable for various ( R - S , S-S) combination models. . . . 140 5.16 The simulated accidental background t2 vs. t\ spectrum 141 5.17 Sectional views of Figure 5.3 and of Figure 5.16 after optimal scaling. . . 142 XIV 5.18 Contr ibut ion of accidental background events to the E1 spectrum, as de-rived from the Monte Carlo simulation 142 5.19 Comparison between the raw E1 spectrum and the accidental background spectrum 143 5.20 Comparison between the raw q spectrum and the accidental background spectrum 143 5.21 The region of the t2 vs. t\ spectrum that satisfies the E661 momentum constraints (from the Monte Carlo simulation) 144 5.22 The profile of the diagonal background 145 5.23 The t2 vs. t\ spectrum after subtraction of al l adjacent bar hit events. . . 146 5.24 The profile of the diagonal background, after subtraction of all adjacent bar hit events 146 5.25 A simulation using the main E661 Monte Carlo that shows the high energy neutron bar-to-bar scattering events 147 5.26 The low energy photon diagonal background events, after transposition into the E661 region of interest 148 5.27 Contr ibut ion of diagonal background events to the E1 spectrum 149 5.28 The final experimental photon energy spectrum as extracted from E661. . 150 5.29 The distribution of the estimated uncertainty aE of the calculated photon energy, for al l events in the top 450 keV of the photon energy spectrum. . 151 6.1 The Reaction Model for E661 159 6.2 Simulated x-position distribution of the pion stops in the L D target. . . . 162 6.3 Simulated z-position distribution of the pion stops in the L D target. . . . 162 6.4 The E7 spectrum from the E661 Monte Carlo simulation for ann = —18.5 fm.164 7.1 The E661 experimental E1 spectrum and the a^n = —21.8 fm Monte Carlo spectrum 166 X V 7.2 The E661 experimental En spectrum and the ann = —18.5 fm Monte Carlo spectrum 167 7.3 The experimental Ey spectrum, as obtained from analysis using the correct value for V (= 1.99 cm/ch) in the P R E S O R T program; and using V = 2.09 cm/ch 170 7.4 Experimental E1 spectra obtained for different settings of the high energy software thresholds 176 7.5 T d i f f distribution ( 6 0 C o events) for Bar 12, R u n Group 1 179 7.6 Y-posi t ion distribution in Bar 12 for valid E661 events 180 7.7 Integral non-linearity test result for T D C 12t (raw spectrum) 181 7.8 T d i f f distribution ( 6 0 C o events) for Bar 10, R u n Group 1, with an artificially induced error in the T D C readings 181 7.9 E661 photon spectra (i) after removal of al l events involving Bar 12, and (ii) after removal of all events involving Bar 9 182 7.10 F i t to the low energy hardware cut-off for A D C 10b 186 7.11 A comparison between the E1 spectra from the Base Monte Carlo model and the new, independently written model 188 7.12 The photon energy distribution as a function of the n-n relative momentum q as derived from the E661 theoretical model for different values of ann. . 190 7.13 The photon energy distribution as a function of the n-n relative momentum as derived from different theoretical models 191 7.14 A comparison between the photon energy distributions, as functions of the n-n relative momentum q, as derived from de Teramond and from the E661 theoretical model 191 7.15 The E7 distributions, obtained using the new Monte Carlo, for ann = — 18.0 fm: from de Teramond, and from the E661 model 192 x v i 7.16 The polar angle distribution of q 193 7.17 Experimental set-up for the Salter et al. measurement 195 7.18 F ina l fits for the Salter et al. data 196 7.19 Experimental set-up for the Gabioud et al. measurement 197 7.20 Experimental photon spectra for the Gabioud et al. measurement 198 7.21 Experimental set-up for the Schori et al. measurement 199 7.22 Raw neutron T O F spectrum from the Schori et al. data 200 7.23 Corrected neutron T O F spectra from the Schori et al. data 201 7.24 F i n a l neutron T O F spectrum from the Schori et al. data 201 B . l The T I N A N a l ( T l ) crystal 221 B.2 Location of the equipment along the beam path 222 B.3 Drawing of the L D target used for E661 223 B.4 A n isometric view of the aluminum target housing 223 B.5 Drawing of the base plate of the neutron bar supporting stand 225 B . 6 Drawings of the neutron bar light guide and its attachment to the P M T . 226 C . l Photograph 1: Beam path equipment 227 C.2 Photograph 2: Beam path equipment and scintillator S4 228 C.3 Photograph 3: Front view of the neutron bar array 229 C. 4 Photograph 4: Rear view of the neutron bar array 230 D . l T I N A and converter circuitry 232 D.2 Beam telescope and L A M circuitry 233 D.3 Neutron bar and gamma-tag detector circuitry 234 D.4 S4 veto counter and random pulser circuitry 235 F . l Cyl indr ica l coordinate system for the neutron direction vector 240 Summary x v i i S u m m a r y This thesis involves experiment E661 at the Tri-Universi ty Meson Facil i ty ( T R I U M F ) in Vancouver, Canada, in which a high-precision measurement of the neutron-neutron scattering length ann was carried out via the reaction 7 r _ d —^  77m. It contains a review of the theoretical motivation for this measurement, a description of the experimental procedure, and a report on the data analysis and results. The thesis is organized as follows: In Chapter 1 we give an introduction to the various aspects of this project. This includes a general overview of the motivation and objectives of this experiment (Sec-tion 1.1), an introduction to our present knowledge on the nucleon-nucleon interaction (Section 1.2), a summary of the methods available for the extraction of a n n , wi th partic-ular emphasis on the use of the reaction ir~d —> ^ynn (Section 1.3), and an introduction to the particular features of experiment E661 (Section 1.4). Chapter 2 delves deeper into the underlying theory of this experiment. After a short preview (Section 2.1), we give an introduction to meson theory (Section 2.2) and to charge independence breaking (CIB) and charge symmetry breaking (CSB) in the strong interaction (Section 2.3). The quantitative discussions on C I B and C S B are l imited mainly to their manifestations in differences between the three S-wave nucleon-nucleon scattering lengths. In Section 2.3.7 we summarize the current status of both experiment and theory on these issues. In Chapter 3 we describe the E661 experiment. This includes a brief introduction to the T R I U M F cyclotron and the M13 beam channel (Section 3.1), a description of the equipment set-up and the electronic circuitry used for this experiment (Sections 3.2 and 3.3), a summary of the experimental runs (Section 3.4), and an overview of the Summary x v i i i on-line and off-line analyses (Section 3.5). Chapter 4 contains a detailed description of the off-line equipment calibration that was carried out for E661. After a brief summary of the objectives (Section 4.1), we describe the calibration of the neutron detector T D C s (Section 4.2), the neutron detector A D C s (Section 4.3*), the photon detector A D C s (Section 4.4*), the photon-tracking wire chamber (Section 4.5), and the beam tracking wire chambers (Section 4.6*). In Chapter 5 we describe the manner in which the endpoint region of the experimental photon energy spectrum was reconstructed for the reaction %~d —>• jnn. This spectrum constitutes the main experimental result of this research. After a brief introduction (Section 5.1), we describe the manner in which the data were sorted to produce the raw photon energy spectrum (Section 5.2). In Section 5.3 we discuss the distributions of other kinematic parameters of this reaction, and this is followed by the analysis of the accidental background (Section 5.4) and the bar-to-bar scattering background (Section 5.5*) that were present with our data. In Section 5.6 we present the final version of our photon energy spectrum. Chapter 6 deals wi th the production of the simulated photon energy spectra for E661, for comparison to the experimental spectrum. After a brief introduction (Section 6.1), we give a brief overview of the various phenomenological nucleon-nucleon potentials that are available in the literature (Section 6.2). A t the end of this section (Section 6.2.4) we introduce the Argonne Vi$ potential, which is the one selected for our theoretical analysis. In Section 6.3* we give a brief outline of the theoretical reaction model that was developed for E661, while in Section 6.4* we give a summary of the G E A N T - b a s e d Monte Carlo simulation program that was used (in conjunction wi th the results of Section 6.3) to generate the simulated photon energy spectra for various theoretical values of ann. *The major contribution to the work reported in these marked sections was made by other members of the E661 collaboration. Refer to Section 1.4.2 and to Appendix A. Summary xix In Chapter 7 we present, analyze and discuss the results of experiment E661. Fi ts to the preliminary Monte Carlo spectra are given in Section 7.1. This is followed by an in-depth error analysis of our experimental photon energy spectrum (Section 7.2) and of the simulated spectra (Section 7.3). In Section 7.4 we compare the E661 experimental set-up, procedure and results to those of previous experiments that measured ann from the reaction ir~d —¥ jnn. This is followed by a summary of the work that remains to be done before the result of our experiment can be finalized (Section 7.5). F ina l ly in Section 7.6 we give a conclusion to this thesis. XX Acknowledgement I would like to express my gratitude and appreciation to my supervisor Prof. David Measday, for his constant guidance, advice and support throughout this work. I would also like to thank Prof. Mike Kovash who acted as the spokesman for this experiment, and was my unofficial second supervisor. Thanks are also due to Chenhong Jiang and Dr . Br i an Doyle for their contribution to the data analysis, and to the whole E661 collab-oration for their help during the experiment. I would also like to thank Professors Dave Axen , Doug Beder, Bertrand Clarke, W i l l i a m Falk, Harold Fearing, Richard Johnson and J . Ke l ly Russell for their advice and detailed reading of the manuscript. I would like to express my appreciation to the many other members of the faculty, staff and student bodies of the U B C Physics department, wi th whom I have interacted during my studies here, as well as to al l of the staff at T R I U M F who have contributed to this research in any way. Special thanks to Prof. Malcolm M c M i l l a n for his early support for my studies. I would like to thank my family at home, especially my parents, for their constant support and encouragement of my studies. Final ly, thanks L . C . C h a p t e r 1 I n t r o d u c t i o n 1.1 G e n e r a l O v e r v i e w The nuclear scattering parameter a, called the scattering length, was introduced by Fermi and Marshal l in 1947 [1] to describe the phase shift of a low energy neutron wave function due to scattering off various nuclei. Later in the same year, it was re-introduced in the context of low energy nucleon-nucleon scattering by Schwinger [2, 3], in the formulation of his Effective Range Theory. Schwinger's phenomenological theory is able to describe nucleon-nucleon scattering in the low energy regime (up to about 10 M e V ) in terms of only two parameters (the zero-energy-limit scattering amplitude a, and the effective range r 0 ) , in a manner that does not require the assumption of a specific theoretical model for the shape of the nucleon-nucleon potential. Thus, three nucleon-nucleon scattering lengths are defined in the literature. These are the neutron-proton scattering length anp, the proton-proton scattering length app, and the neutron-neutron scattering length ann. The subject of this thesis is the experimental measurement of the S-wave neutron-neutron scattering length ann from the reaction ir~d - 4 77m. The experiment was carried out at Canada's National Meson Facility, T R I U M F , in the spring of 1994 (Experiment E661). The remainder of this section is dedicated to a brief description of the importance of this experimental measurement. The reader may be surprised to learn that, despite the fact that they are very basic nuclear physics parameters that were defined half a century ago, anp, app and ann are sti l l 1 Chapter 1. Introduction 2 the subjects of lively research efforts in today's nuclear and particle physics communities. O n the theoretical side, a fundamental, and universally accepted, theory of the strong force, that completely (and correctly) describes the nucleon-nucleon interaction, and that would therefore predict the values of the nucleon-nucleon scattering lengths with high accuracy, is st i l l out of reach. In particular, there is much discussion regarding the expected differences between a%p and a%n (or app) (charge independence breaking), and between a%n and app (charge symmetry breaking). 1 It has therefore become extremely important to obtain experimental measurements of these parameters that are as precise and accurate as possible, in order to shed light on the validity of the various theoretical models that are currently in existence, and on our general understanding of the strong interaction. For neutron-proton scattering there are excellent experimental results, and the scat-tering lengths are well established. Reliable measurements of ann and app, however, have been difficult to obtain, for different reasons. In the case of app, in spite of the availability of accurate scattering data, there are (relatively) large uncertainties that are due to the model dependence that is involved in the subtraction of electromagnetic effects from the measured values of app. In the case of ann, the main problem is the unavailability of a free neutron target for use in direct scattering experiments. The neutron-neutron scattering length must therefore be measured through indirect means, with results that are inher-ently less accurate than would be obtained from a direct particle scattering experiment. The derivation of ann from the measurement of the energy/momentum spectra of the final state particles in the reaction n~d —> jnn is currently regarded as the most reliable method of measuring the parameter. 1The superscripts ./V denote that electromagnetic effects have been removed for these definitions, i.e. these are the predicted values for the scattering lengths if the strong force were acting alone. Through-out the remainder of this thesis (unless noted otherwise) the superscript N will be assumed. Where electromagnetic effects are included, the superscript E will be used, e.g. app. Chapter 1. Introduction 3 The objective of the experiment that is described in this thesis, is to obtain a measure-ment of ann that is higher in precision and accuracy than any that has been previously made. The result is expected to provide an important contribution to the ongoing inves-tigation of charge symmetry breaking effects in the nucleon-nucleon interaction. 1.2 T h e N u c l e o n - N u c l e o n I n t e r a c t i o n W i t h i n the confines of present knowledge, the nuclear, or strong, force is known to be one of the four fundamental forces of nature. It acts between all hadrons, and specifically between the neutrons and protons that form the atomic nucleus. A serious investigation into the physical characteristics of this force started shortly after the discovery of the neutron by Chadwick in 1932 [4], following the subsequent suggestion that the neutron and the proton were the fundamental constituents of the nucleus. Al though much has already been learned about the strong interaction, the investigation s t i l l goes on today. Many of the more fundamental properties of the nucleon-nucleon interaction can be surmised from the observation and study of atomic nuclei, as well as from experiments involving nucleon-nucleon (N-N) scattering. In this section, these properties are summa-rized and discussed. 1.2.1 T h e R a n g e o f t he N u c l e a r F o r c e Unlike the electromagnetic and gravitational forces, the nuclear force has a finite, and short, range. This is evident from several observations. A t the molecular level, the electromagnetic force alone is sufficient to explain al l of the known phenomena, suggesting that the range of the strong force is smaller than interatomic distances. Furthermore, for nuclei wi th A > 4, the binding energy per nucleon and the nuclear density remain roughly constant as the atomic number increases, suggesting that each nucleon is only Chapter 1. Introduction 4 interacting wi th the other nucleons that are in very close proximity to it. Based on these observations, Wigner suggested, in 1933, that the range of the nuclear force appeared to be roughly equal to 1.7 fm, the radius of the alpha particle [5]. The nuclear force is traditionally divided into three areas of effectiveness [6]: the classical, or long-range, region, with r > 2 fm (where r is the distance between the centres of the two nucleons); the dynamical, or intermediate, range (1 fm < r < 2 fm); and the core, or short-range, region (r < 1 fm) (ranges of r are approximate). Different effects become dominant in each of these three regions (see Chapter 2). Since the nucleus is bound, it is clear that the N - N force must be attractive in the intermediate range. In the core region, however, the interaction is repulsive (see Section 1.2.4). 1.2.2 The Deuteron The simplest bound state of nucleons is the deuteron, which consists of one proton and one neutron. A n understanding of this atomic nucleus can therefore provide important preliminary information regarding the nature of the nucleon-nucleon interaction. The deuteron is the only two-nucleon bound state that exists in nature, and does so only in the ground state, as a weakly bound nucleus. It has a binding energy of 2.224573(2) M e V [7], even parity, and a total spin J = 1 [8]. It has a magnetic dipole moment /id = +0.857438230(24) pN, and an electric quadrupole moment Q d = +0.002860(15) barns [9].2 The rms charge radius of the deuteron is 2.106(11) fm [11]. The parity of the combined neutron-proton wavefunction is given by Pari ty [ip(n,p)] = {-1)L Par i ty [ib(n)] Pari ty [ib(p)] (1.1) Since the proton and the neutron each have the same parity (positive by convention), it follows that the orbital angular momentum L of the deuteron must be even, in order 2HN is the nuclear magneton, equal to 3.15245166(28) x 10" 1 4 MeV/T [10]. Chapter 1. Introduction 5 to give the deuteron its even parity. Furthermore, since the deuteron has J = 1, where J = L + S, the only possible values of (L,S) for the deuteron are (0,1) and (2,1), or, in standard 2S+1Lj notation, 3 Si and ZD\. The non-zero value of Qd indicates that the deuteron cannot be in a pure, spherically-symmetric S-state. A comparison between the values of the magnetic dipole moments of the deuteron and of the proton and neutron then indicates that the deuteron ground state is predominantly 3 Si , wi th a small admixture (~ 4%) of the 3Di state [12]. The isospin of the deuteron (T = 0) can be inferred either from symmetry arguments [12], or from the fact that the stable di-neutron and di-proton, which would be the other manifestations of a T = 1 isotriplet, have not been observed experimentally. Knowledge of the binding energy and radius of the deuteron can be used to obtain a preliminary estimate of the strength of the nucleon-nucleon potential. If a spherical square-well potential, of depth Vo and range R = 2.1 fm, is used as a first approximation of the nuclear potential, a solution of the Schrodinger equation for the radial part of the deuteron wave function, ~2%idT^v{r)u{r) = E u { r ) ( L 2 ) yields a well depth of V0 « 35 M e V , which is in fact a reasonable estimate for the strength of the nucleon-nucleon potential [13]. The deuteron ( B E = —2.225 M e V ) therefore lies very close to the top of this potential well, and, as illustrated in Figure 1.1, its interior wave function is barely able to connect smoothly to a decaying exponential outside the range r = R and make a bound state. Apar t from a general indication of its strength, two other important pieces of in-formation, regarding the nature of the nucleon-nucleon potential, can be obtained from the deuteron. Firstly, the presence of a non-zero electric quadrupole moment, i.e. a D -wave component in the wave function, indicates that the nuclear potential must have a Chapter 1. Introduction 6 r(fm) Figure 1.1: The deuteron wave function for R = 2.1 fm [13]. non-central component, or, more specifically, a tensor component [14]. Secondly, an in-compatibil i ty between the measured binding energy of the deuteron and the extrapolated zero-energy cross-section data for n-p scattering [15] establishes spin-dependence of the nuclear force (see Section 1.2.3). This spin dependence can also be inferred from the fact that there is no S = 0 bound state of the deuteron. 1.2.3 N - N S c a t t e r i n g at L o w E n e r g i e s T h e Ef fec t ive R a n g e A p p r o x i m a t i o n A full derivation of the effective range approximation for low energy nucleon-nucleon scattering can be found in most standard textbooks of introductory nuclear physics (e.g. [13, 14]). The original derivation by Schwinger [2, 3] used the method of variational principles, however most texts prefer the somewhat less complicated derivation due to Bethe [16]. The elastic scattering of two nucleons is modeled as an incoming plane wave function (incident beam of momentum k), and a scattered spherical wave that moves away from the scattering centre (Figure 1.2). A t a large distance r from the scattering Chapter 1. Introduction 7 sca t te r ing c e n t r e \ V > / > 9 (so l id ang le ) ^ - i > i - l - i + ---• s c a t t e r e d s p h e r i c a l w a v e s \ / / / / / / ' i / / / Figure 1.2: The basic model for scattering. centre, the wave function ip(r) = ip(x,y,z) (1.3) has the form ip(r) = eikz + fid)6-^- (1.4) where f(9) is the scattering amplitude. From a partial wave decomposition of the wave function, it is shown that for low energy (S-wave) scattering, the scattering amplitude is independent of the direction 9 of the scattering, and is given by f = e i S 0 f S ° ( 1 . 5 ) where <50 is the S-wave phase shift of the scattered wave. B y defining the scattering length as a = — hm — - — , (1.6) /c->o k the scattering can be parametrized by the effective range approximation kcotS0 = ~ + K 0 k 2 + O{k4) (1.7) where r 0 is a constant 3 and is called the effective range. r 0 = 2 J^°(VQ — UQ) dr, where uo(r) is the radial wave function at zero energy and vo(r) = 1 — ^ . Chapter 1. Introduction 8 Figure 1.3: The wave function for triplet np scattering [13]. This approximation is valid for incident energies for which the scattering can be considered to be predominantly S-wave, i.e. up to incident energies of approximately 10 M e V . If the higher order terms in k are neglected, any reasonable shape for the nucleon-nucleon potential can be made to fit the scattering parameters a and r 0 , as long as the well depth and range parameters of the model are adjusted accordingly. A s k approaches zero, the wave function, at large r, can be written as 0ikr r ^( r ) = elkz - a— (1.8) so that a, effectively, gives the amplitude of the scattered wave. In this l imi t , the scat-tering cross-section is given by limcr = 47ra 2 (1.9) It can be shown that the numerical value of a is given by the point at which the radial part of the zero-energy wave function, u(r), passes through zero (Figure 1.3). Spin Dependence In neutron-proton scattering, the nucleon spins (each | ) can combine in any of four ways. The 5 = 1 combination can have three orientations (i.e. a spin triplet): Chapter 1. Introduction 9 \S, Ms) 1, 1), |1, 0) or | 1 , - 1 ) while the S = 0 combination can only occur in one state (i.e. a spin singlet): S, Ms) |0, 0) Under the general assumption that the scattering may be spin dependent, the total cross-section for n-p scattering is therefore given by where ot and os are the cross-sections for scattering in the triplet and singlet states respectively. A comparison between the measured values of o [18, 19] and the value of ot as inferred from the deuteron binding energy [13], shows that ot and os do in fact have significantly different values, and therefore that the nucleon-nucleon interaction is spin dependent. The corresponding triplet and singlet scattering lengths at and as can be measured by scattering neutrons from para and ortho hydrogen [17], or by combining measurements of the zero-energy-limit scattering cross-section with measurements of the hydrogen coherent scattering length [19]. The experiments are very precise and there has been little change in the last two decades. The present recommended values are [20] Note that these two values are very different numerically, and are even of opposite sign. W i t h respect to the scattering wave function, a negative value of a corresponds to an extrapolation of u(r), from the point r = R, down to u(r) — 0 (Figure 1.4). The negative value of as indicates that a stable bound state is not formed in the singlet state. 3 1 (1.10) at = +5.424 ± 0.004 fm as = -23.748 ± 0.009 fm Chapter 1. Introduction 10 Figure 1.4: Wave function exhibiting a negative scattering length [13]. In neutron-neutron (or proton-proton) low-energy scattering, the triplet state com-binations are precluded by the Paul i exclusion principle, since the required asymmetry in the combined wave function must now come from the intrinsic spin portions of the constituent fermion wave functions (i.e. the spin orientations of the interacting nucleons must be opposite to one another). Thus, only the singlet spin state is allowed for the low energy (L = 0) scattering. If the nucleon-nucleon interaction were isospin (charge) independent, then it would be expected that ann (and app) would be equal to the singlet scattering length as ob-tained for neutron-proton scattering. As wi l l be discussed in Chapter 2, however, charge independence breaking effects cause ann and app to differ significantly from the singlet n-p scattering length anp. M e a s u r e m e n t o f t he P r o t o n - P r o t o n S c a t t e r i n g L e n g t h The proton-proton scattering length, obtained from low energy proton-proton scattering, is known to a high accuracy [20] af p = -7.8098(23) fm (1.11) In order to obtain a value of the nuclear proton-proton scattering length, Coulomb and vacuum-polarization effects must be subtracted from ap . The resulting value of app Chapter 1. Introduction 11 * 1111111111111111111111111111111111111111111111111111111111111 300 310 320 330 340 350 360 370 380 390 400 410 420 energy (keV) Figure 1.5: Summary of p-p cross-section measurements at a laboratory angle of 45° covering the energy range from 300 to 420 keV (reproduced from [24]). The data labelled present work refer to the results of Dombrowski et al. [24]. The other references are S A I D : computer code by Arnd t [25], Thomann et al. [26], and Brolley et al. [27]. has been, shown to be strongly dependent on the model used for the (unknown) nuclear potential at short distances [21, 22]. The currently accepted value is [23] app = -17 .3 ± 0.4 fm (1.12) A t low energies (< 500 keV) the (repulsive) Coulomb phase and the (attractive) nuclear phase interfere wi th each other, resulting in a minimum of the cross-section near 380 keV (Figure 1.5). The position of this min imum can be used to obtain further information re-garding the nuclear potential, through the effective range approximation (Equation 1.7). A recent experiment [24] has remeasured proton-proton scattering around this minimum with modern detectors. The deep min imum in the data that is seen in Figure 1.5 has Chapter 1. Introduction 12 80 -20 1 ' 1 ' 1 ' 1 ' 1 ' 1 ' 1 ' 1 0 50 100 150 200 250 300 350 Energy (MeV) Figure 1.6: The 1So phase shift for neutron-proton scattering at intermediate energies [28]. not been satisfactorily fitted by either the S A I D or the Nijmegen 4 phase-shift analyses, however any impact that this uncertainty may have on the value of the p-p scattering length is expected to be considerably less than that due to the removal of the Coulomb effects. 1.2.4 N - N Scattering at Intermediate Energies Two further important properties of the nucleon-nucleon interaction can be inferred from the partial wave decomposition of N - N scattering data at intermediate energies. The first of these is that the N - N interaction becomes repulsive at short (core) distances. The evidence for this is the fact that the S-wave phase shift for the scattering turns negative at around 250 M e V (see Figure 1.6), indicating a change from an attractive to a repulsive force. The second property, that is needed to explain the polarization of the scattered 4The Nijmegen phase-shift analysis will be introduced in Section 6.2.2. Chapter 1. Introduction 13 protons in p-p scattering at intermediate energies [29], takes the form of an L - S term in the N - N potential, and gives rise to a spin-orbit force. The spin-orbit force is present only for scattering states wi th non-zero orbital angular momenta. 1.3 M e a s u r e m e n t o f t he N e u t r o n - N e u t r o n S c a t t e r i n g L e n g t h 1.3.1 P r e v i o u s M e a s u r e m e n t s Since direct neutron-neutron scattering is not feasible (see Section l . l ) 5 , the neutron-neutron scattering length is measured through indirect techniques that involve the final state interaction (FSI) of two neutrons in a reaction of the type A + B -> X + 2n where A, B and X are systems of one or more particles. The attractive final state interaction between the two neutrons has the effect of distorting the phase space of the reaction products towards a smaller relative momentum q between the two neutrons. This effect can be detected and measured either through the direct observation of the |q| spectrum, or through observation of the consequent effects on the energy spectrum of the third reaction product X. Measurements of ann have been carried out using the reactions d(ir~,j)2n, d(n,p)2n, 3 H ( n , d ) 2 n , 3 H(d , 3 He)2n and 3 H ( 3 H , 4 H e ) 2 n . The preferred reactions, particularly in more recent years, have been the first two, due to a simpler underlying theory that does not require the description of a system consisting of four or more nucleons. The results from previous measurements of ann are shown in Figure 1.7 and a few of the more recent are given in Table 1.1. We shall discuss these in Chapter 7. 5 The measurement of the n-n elastic scattering cross-section using underground nuclear explosion (s) or an intense neutron reactor has also been discussed [30]. Chapter 1. Introduction 14 • n " d - * n n 7 ; n - n — 7 • TT*d - * n n 7 ; y o n d - * p n n ; p • n d - p n n ; p n o r n n Figure 1.7: Previous measurements of ann (from [31]). A compilation and summary, with references, of the experiments carried out prior to 1975 can be found in [32]. See [33] for additional measurements. The more recent results are listed in Table 1.1. A new measurement using ir~d —> 'ynn is also currently being undertaken at L A M P F [34]. Reaction Method ann (fm) -K~d —>• 'ynn n~d —>• jnn ir~d —> jnTi nd —> nnp L B L , 3-fold coincidence SIN, 7-ray singles SIN, 2-fold coincidence compl. k in . , 10.3 M e V neutrons -16.7 ± 1.3 [32] -18.5 ± 0.4 [35] -18.7 ± 0.6 [36] -17.0 ± 1.0 [37] Table 1.1: Summary of selected recent measurements of a. Chapter 1. Introduction 15 1.3.2 Extraction of ann from 7T d —> -ynn When a slow negative pion (with an energy of less than approximately 10 M e V ) enters l iquid hydrogen or deuterium, it is very quickly slowed down [38, 39] unti l it is captured by a hydrogen nucleus to form an excited 7r~-H + system (a pionic atom). The atom then rapidly de-excites, unti l the pion is absorbed by the proton v ia the strong interaction, wi th final absorption occurring mainly from an S-state in the n = 3 or n = 4 orbits [40]. The entire process takes place in a time of about 2.3 x 10~ 1 2 seconds [41], which is much shorter than the lifetime of the ir~ to p~ decay (2.6 x 10~ 8 seconds). In deuterium, the nuclear capture can occur v ia one of four possible reactions: 7r~ + d —>• n + n + 7 ( R l ) 7T" + d - » n + n (R2) TT~ + d - » n + n + e+ + e~ (R3) IT- + d -> n + n + ir0 (R4) Reaction (R3) 6 is an order a less probable than reaction ( R l ) , while reaction (R4) is strongly suppressed by conservation of parity and angular momentum, and has a branching ratio of only 1.45(19) x 10~ 4 [43]. The ratio 5, given by s = + + ( L 1 3 ) U(TT- + d - » n + n + 7) is equal to 2.83(4) [44]. Thus, capture v ia the reaction TX~d —> jnn occurs about 26% of -the time. Kinematics The pion can be considered to be essentially at rest when it is absorbed, and the small effect of the internal motion of the proton within the deuteron is negligible [45]. The 6The use of reaction (R3) as an experimental method for determining alternative to reaction (Rl), has also been suggested [42]. Chapter 1. Introduction 16 laboratory frame is therefore also the centre of mass frame for the reaction, and the kinematics, for the reaction products, are as shown in Figure 1.8. The Q-value for the reaction is equal to Q = - (mn - mp) - BEd = 136.052 M e V (1.14) where mn, mn and mp are the masses of the pion, neutron and proton respectively [10], and BEd is the binding energy of the deuteron. The binding energy of the pion in the n = 4 level is approximately 0.2 keV [47] and can be neglected. Conservation of momentum gives k + p i + p 2 = 0 (1.15) and conservation of energy gives E-r + [y/pi + m* - mn] + [ y ^ + ml - mn] = Q (1.16) The energy of the photon can therefore be reconstructed from a knowledge of the mo-menta of the two neutrons. For three-body phase space, the maximum momentum for particle 3, |p>31 ( m a x ) ' 1 S obtained when |p x | = |p 2 | . For reaction ( R l ) , the maximum momentum (energy) of the photon is obtained when the two neutrons recoil wi th equal momentum vectors Chapter 1. Introduction 17 Figure 1.9: Graph of the reaction 7r d -> ynn according to the impulse approximation. P i = P 2 = P n - Thus ^ 7 ( m a x ) = 2 I P n l (1-17) Combining equations 1.16 and 1.17 gives the maximum permissible energy for the photon, i.e. the endpoint of the photon energy spectrum, viz. £ 7 ( m a x ) = 1 3 1 - 4 5 9 M e V (1-18) T h e T r a n s i t i o n M a t r i x From Fermi's golden rule, the reaction transition rate is given by N = ^\M\2F (1.19) where M. is the reaction amplitude and F is the phase space. For the reaction -K~d —» ynn, the dominant graph is that shown in Figure 1.9. This term corresponds to the impulse approximation, in which it is assumed that (i) the recoil momentum of the photon is ini t ial ly absorbed solely by the proton, and (ii) the wave function of the proton in the deuterium is identical to the wave function of a free proton. Thus (iii) any momentum that the spectator neutron picks up comes only from its interaction wi th Chapter 1. Introduction 18 the other neutron, in a final state interaction. The transition matrix A4 is in general a function of the particle momenta, and is therefore sensitive to any interaction between the final state particles. Since the 7-n interaction is very weak, any deviation in iV from regular three-body phase space can be attributed to the effect of the n-n interaction. The transition matrix element is [35, 45, 46] M = j(e-jk-(ri+r2Vnn(r))*e-jk-riT^(r)^(r1) d 3 n d 3 r 2 (1.20) where r x and r 2 are the positions of the proton and the neutron, r = r i — r 2 , and ipnn, ipd and i/v are the wave functions for the n-n system, the deuteron and the pion respectively. In the l imit of zero kinetic energy of the pion, the transition operator is given by T = a • e, where a is the Paul i spin vector and e represents the photon polarization [48]. The sensitivity of A4 to the neutron-neutron interaction has been calculated and corrected by several authors [45, 46, 49, 50, 51, 52, 53]. Historically, the first calculation for reaction ( R l ) was carried out by Watson and Stuart [45] (see also [54]). The original model assumed the validity of the impulse approximation, and within this framework, the g-dependent part of M. was found to vary as for small relative neutron momentum q. This result was used to estimate the dependence of the endpoint region of the E1 spectrum on ann. M c V o y [46] used a similar analysis to calculate the neutron spectra, at specific values of the angle ^1, for different values of ann. The model was improved by Bander [49] through the inclusion of the pion rescat-tering correction (Figure 1.10), and the inclusion of corrections related to the potentials between the particles of the scatterer. Gibbs, Gibson and Stephenson [50] improved the models that were used for the pion, deuteron and short-range n-n wave functions (see Equation 1.20), and calculated the theoretical uncertainty in the determination of ann from ir~d —¥ jnn to be about 0.3 fm. De Teramond analyzed the n-n final state in-teraction using dispersion relations [51], and included off-shell rescattering corrections Chapter 1. Introduction 19 Figure 1.10: Graph for the pion rescattering correction. and higher n-n partial waves to the reaction transition operator [52, 53]. The latter two effects are mainly applicable to higher n-n relative momenta, and have little effect on the extraction of ann from reaction ( R l ) . T h e P h o t o n S p e c t r u m The sensitivity to ann of the region near the endpoint of the photon energy spectrum, as calculated in [35], is illustrated in Figure 1.11. The spectrum is only sensitive to ann in the top 600 keV region. This sensitivity manifests itself as a shift in the position and relative magnitude of the energy peak, and in the slope of the curve immediately leading up to the peak. It is shown in [35] that the shape near the endpoint in not sensitive to the choice of theoretical model that is used to calculate the final state n-n interaction. The sensitivity of the photon spectrum to rnn is shown in Figure 1.12. The shape of the top 600 keV region is quite insensitive to rnn. It should be noted that the portion of the spectrum that is sensitive to rnn involves larger values of the relative n-n momenta, and that therefore some of the assumptions made in references [45], . . . , [51] are less valid. Consequently, the extraction of rnn from ir~d —>• ynn is presently more susceptible to Chapter 1. Introduction 20 P h o t o n E n e r g y ( M e V ) 1 . 0 0.8 0 . 8 0 . 2 0 . 0 T - T— i — i — i — i — f — i — r - | — i — i — i — i — | — i — i — i — i — | — i — i — i — i — | — i — i — i — r ' • • • . ' • • . • ll 129.0 129.5 130.0 130.6 131 .0 131 .6 132.0 Figure 1.11: Sensitivity of the photon energy spectrum to ann in the reaction % d —» ynn (solid line: -20 fm, dotted line: -18 fm, dot-dashed line: -16 fm) [35]. P h o t o n E n e r g y ( M e V ) — 1 1 1 1 0.2 (-o.o I i i_ i I . I i I L I 127 12B 129 130 131 132 Figure 1.12: Sensitivity of the photon energy spectrum to rnn in the reaction TT d —> ynn (solid line: 3.2 fm, dotted line: 2.8 fm, dot-dashed line: 2.4 fm) [35]. Chapter 1. Introduction 21 30 25 > 20 ^ .5 a "O € 10 5 0 8.0 9.0 10.0 11.0 12.0 E p (MeV) Figure 1.13: Sensitivity of the proton spectrum to ann, in the reaction nd —» nnp [55]. theoretical uncertainty than is the extraction of ann. 1.3.3 Extraction of a n n from nd —> nnp A significant number of past experimental measurements of ann have been carried out using the deuteron breakup reaction nd —> nnp (see Section 1.3.1). In this section, the merits and demerits of measuring ann in this manner are briefly discussed. The endpoint region of the proton energy spectrum from the neutron - deuteron breakup reaction exhibits sensitivity to ann in a manner that is similar to that of the photon spectrum in ix~d —>• ynn (Figure 1.13). The derivation of this theoretical spectrum typically involves a rigorous solution of the Faddeev three-body equations [56, 57, 58], and requires the assumption of a specific shape for the two-nucleon potential. Experiments involve the use of a mono-energetic neutron beam that strikes a deuteron target, and the measurement of the outgoing proton energies at specific scattering angles. The main advantages of the deuteron breakup reaction over the radiative pion capture reaction are that neutrons can be easily produced by small laboratories (pion beams of Chapter 1. Introduction 22 sufficient intensities can only be produced in meson factories such as T R I U M F ) , and also that the proton, being a charged particle, is more easily detectable than the neutron. The currently predominant view in the nuclear and particle physics communities, however, is that ann measurements from nd —> nnp are less reliable than those obtained from Tr~d —> 777,n (e.g. [23, 55, 59]). This is due to a number of reasons. From the experimental side, an accurate knowledge of the incident neutron energy distribution is required, since this has been shown to have a significant effect on the endpoint of the proton energy spectrum [55]. From the theoretical side, the choice of n-n potential for use in the analysis, as well as the effect of higher partial waves (often neglected in the analyses) in the n-n final state interaction, have also been shown to affect the shape of the endpoint. Bo th these latter effects become more pronounced at higher incident neutron energies [55]. A major cause of uncertainty, regarding the values of ann that are extracted from nd —» nnp, stems from the current lack of knowledge regarding the existence and strength of irreducible three-body forces in the N - N interaction. Since the deuteron breakup reaction involves a three-nucleon system in its final state, it would be subject to the influence of such forces. Phenomenologically, the existence of a measurable three-body force is currently the most widely accepted explanation for the observed values of the binding energies of bound three- and four-nucleon systems, which would otherwise be irreconcilable with our knowledge of the two-body N - N potential as obtained from N - N scattering data. Theoretically, a three-body force would result from certain two-meson exchange diagrams in a meson-theoretic N - N potential (see Chapter 2). The forces that are calculated from such diagrams, however, are not alone sufficient to reconcile the observed discrepancies in the binding energies of 3 H , 3 H e and 4 H e . It has been suggested that the difference in the values of ann that are extracted from reactions ir~d —> jnn and nd —> nnp can be cited as evidence for irreducible three-body Chapter 1. Introduction 23 forces in the N - N interaction [60]. In this regard, the main significance of the deuteron breakup experimental results may be in their contribution towards our knowledge of three-body nuclear forces, rather than towards our knowledge of the precise value of ann. 1.4 A n Introduction to T R I U M F Experiment E661 1.4.1 Pr imary Considerations The objective of T R I U M F Experiment E661 was to produce an experimental version of the endpoint of the photon energy spectrum from the reaction w~d —> jnn (refer to Figure 1.11 on page 20 for the theoretical spectrum), and then to extract ann by comparing the experimental spectrum to a number of simulated spectra produced using various (input) values of ann. The goal of the experiment was an uncertainty of about ± 0.3 fm in the measured value of ann. A n accuracy of this order is needed to provide experimental confirmation of the existence of charge symmetry breaking effects in the strong interaction. In the experiment, negative pions from a high intensity pion beam were brought to rest and captured in a l iquid deuterium target. A l l three final state particles from the reaction 7r~d —> 7 7 m (refer to Figure 1.8, page 16) were detected in triple coincidence. The photon was detected in a sodium iodide crystal, while the two neutrons were detected in an array of position-sensitive plastic scintillators. The neutron energies were calculated from their times of flight, and the energy of the photon, for every valid event, was reconstructed from a knowledge of the neutron momentum vectors p i and p 2 (refer to Equations 1.15 and 1.16 on page 16). 7 In order to achieve the desired accuracy in the measurement of ann, it was necessary, from the onset, to ensure that certain primary conditions would be met. The more 7The direct measurement of the photon energy would give us insufficient resolution. Chapter 1. Introduction 24 important of these conditions are listed below: (i) Since the photon spectrum is only sensitive to ann in the top 600 keV energy region, it was very important to be able to achieve high energy resolution for the photon. A s wi l l be shown in Chapter 5 the realistic energy and position resolutions of the neutron detectors resulted in an uncertainty (a) of about 17 keV in the reconstructed photon energy. This was far superior to any that had been achieved in previous measurements. (ii) It was also very important to reduce the statistical uncertainty in the number of counts in each individual bin of the reconstructed photon energy spectrum by as much as was reasonably possible. This was done by running the experiment to high statistics. In the final analysis, E661 produced about 123,000 valid events in the top 450 keV region of the photon energy spectrum. Using 5 keV bins, this resulted in about 1400 events per bin at the peak of the spectrum, giving a (Poisson) statistical uncertainty of about 2.7% per bin. Since the final experimental curve consisted, effectively, of a best fit through 90 such points (bins), the statistical uncertainty in the final curve was extremely small. (iii) The third primary consideration was the need to minimize, or to at least un-derstand, the systematic errors in the experimental photon energy spectrum. Since the spectrum was reconstructed from the momenta of the neutrons, most of the (experimen-tal) systematic errors that could affect the final result would originate in the detection of the neutrons. In order to minimize these sources of systematic error, both the energy and the time calibrations of the neutron detectors were monitored continuously, throughout the experiment, at the same time that the data were being taken. Dur ing the off-line analysis, any drifts in the neutron energy and time calibrations were taken into account. Another source of systematic error was the presence of background data in the photon energy spectrum. Al though the amount of background in E661 was greatly reduced due to the kinematically-overdetermined nature of the experiment, extensive background analysis was carried out off-line, and the small residual backgrounds from different sources Chapter 1. Introduction 25 were subtracted from the final spectrum. The impact of each individual source of systematic error on the final (measured) value of ann was directly evaluated by first making a careful estimate of the possible magnitude of the error, and then artificially inducing this error in the analysis programs and re-analyzing the data. The greatest contribution was found to be that due to the A D C low energy threshold determination, resulting in an uncertainty of about 0.2 fm in the reported value of ann. (iv) Final ly, it was very important to compare the experimentally determined photon spectrum to theoretical spectra that were accurate and reliable. The theoretical spectra were derived from a model based on a half off-shell N N T matr ix and the elementary 77r operator due to Lee and Nozawa. The nucleon-nucleon potential that was used to describe the nn final state interaction is the Argonne vi$ potential [61]. This potential is partly field-theoretic (in the description of the long-range region of the N - N interaction), and partly phenomenological (in the description of the intermediate and short-range regions). The E661 theoretical model w i l l be described more fully in Chapter 6. In order to enable meaningful comparison between the experimental and theoretical spectra, the theoretical spectra needed to include the effects of the detector acceptances and resolutions that were present during the experiment. This is particularly important for E661 due to the practical l imitations that are imposed on the experiment by the detector system. 8 The theoretical photon energy spectra were therefore produced using Monte Carlo techniques that included a careful simulation of the experimental set-up, as well as of the individual detector acceptances. The detector acceptances were obtained either from actual experimental measurement or from other Monte Carlo calculations. The Monte Carlo simulations were pushed to high statistics in order to reduce the effects of statistical uncertainties in the simulation. 8For example, the neutron detector only covers a limited solid angle, and moreover has an absolute detection efficiency of 40-45% for neutron detection in our energy range of interest [31]. Chapter 1. Introduction 26 1.4.2 The E661 Collaboration It should be emphasized that, as is normal in nuclear or particle physics experiments, the E661 experiment involved a collaboration of scientists, and that the author of this thesis has carried out only a part of the total research effort for this measurement. In particular, the analysis of the experimental data was shared between two P h . D . students (the author, and C . Jiang [62]). The derivation of the theory, as well as the production of the simulated spectra, were carried out mainly by B . Doyle [63]. Throughout this thesis, those parts of the research to which the author did not contribute the major effort w i l l be clearly indicated, and wi l l also, in most cases, be described in somewhat lesser detail. The spokesperson for the E661 experiment is M . A . Kovash [31]. The members of the collaboration, as well as the formal division of responsibilities for the data analysis, are given in Appendix A . Experiment E661 is described in detail in Chapter 3, and the analysis is described and discussed in the subsequent chapters. The next chapter (Chapter 2) is dedicated to a brief description of the meson theory of the strong interaction, leading up to a discussion of charge independence breaking and charge symmetry breaking effects. C h a p t e r 2 M e s o n T h e o r y a n d C h a r g e S y m m e t r y B r e a k i n g 2.1 I n t r o d u c t i o n a n d P r e v i e w Soon after the existence of the strong force had been established in the early 1930s, it became highly desirable to try to describe this new interaction by means of a fundamen-tal theory, in the same manner that the electromagnetic force had been so successfully described by the theory of electromagnetism and the early work on quantum electrody-namics ( Q E D ) . In Q E D , the electromagnetic interaction is described by the exchange of massless field quanta (photons) between charged particles. The massless nature of the photon gives the electromagnetic force its infinite range. In 1935 Yukawa [64] suggested that the strong force might be similarly mediated by means of an "intermediate mass" particle, later called a meson. The mass of such a particle would be responsible for the short-range nature of the strong interaction. This marked the emergence of the meson theory of the strong interaction. The discovery of the muon in 1937 [65, 66], and its (incorrect) interpretation as the Yukawa particle, led to increased activity in the development of a meson theory for the nuclear force. 1 Thus, when the pion was finally discovered in 1947 [68, 69, 70], a sizeable theoretical framework for its adoption as the quantum, in a field-theoretic meson theory, of the strong interaction already existed. Attempts to describe the strong interaction by means of one-pion exchange ( O P E ) alone, however, failed to reproduce many of the more basic properties of the strong interaction. In particular, the short and intermediate 1 Refer to [67, pp. 190-205] for a more detailed history of the development of meson theory. 27 Chapter 2. Meson Theory and Charge Symmetry Breaking 28 range properties of the nuclear force could not be described by one-pion exchange. The discoveries of the p [71] and co [72, 73] vector (i.e. spin 1) mesons in 1961 prompted the development of one-boson exchange ( O B E ) descriptions of the nuclear force, involving the exchange of various mesons that have different masses and quantum numbers (e.g. spin, isospin, parity). In this way further N - N properties, such as the short-range repul-sion and the spin-orbit force, were derived from meson theory. 2 Since then, during these last thirty years, the meson theory of the strong interaction has been further developed and refined, mainly by means of the inclusion of more complex multi-meson and meson-photon exchange forces, and the inclusion of intermediate A resonances. The Yukawa, O P E and O B E nucleon-nucleon potentials, as well as more advanced meson-exchange potentials, wi l l be discussed in greater detail in Section 2.2. In Section 2.2.4 (and later in Chapter 6) we shall also discuss the development of phenomenological descriptions of the N - N interaction, which occurred, at least in the earlier years, independently of meson theory. We now come to an important property of the strong interaction, that of its charge independence. This notion has been around almost since the very beginning. In the mid-thirties, the analysis of np and pp scattering data [75], as well as studies of the masses of 3 H and 3 H e [76], strongly indicated that the nuclear force did not differentiate between neutrons and protons, i.e. that it was charge independent. This led to the introduction of a new quantum number, isospin, by which it was postulated that the neutron and the proton were identical particles that had different orientations in isospin space (see below). Al though this concept is st i l l a very good and useful one today, some experimental evidence over the years has suggested that the strong interaction does in fact appear to differentiate slightly between neutrons and protons, and that it may therefore 2 The suggestion that these N-N properties could be explained in terms of vector boson fields had already been made by Breit in 1960 [74], prior to the discovery of the p and to mesons. Chapter 2. Meson Theory and Charge Symmetry Breaking 29 not be completely charge independent. Interestingly, a certain (small) degree of charge dependence is also predicted by meson theory. Charge independence breaking (CIB) , and the more specific charge symmetry break-ing ( C S B ) , of the strong interaction wi l l be formally introduced and classified in Sec-t ion 2.3. The physical manifestations of C I B and C S B , as well as their derivations from meson theory and Q C D (see below), wi l l also be discussed (the differences between the values of anp, app and ann constitute one of the more important of these physical mani-festations). A brief description of the isospin formalism wi l l be given early in the section. Final ly, a word about Q C D . Since the early 1980s, meson theory has been somewhat displaced from centre stage due to the emergence of quantum chromodynamics ( Q C D ) and quark theory as the preferred candidate for a fundamental theory of the strong interaction. Today, it is generally believed that nucleons are made up of smaller particles called quarks, and are therefore not fundamental in nature (see, for example, [77] for some compelling reasons that lead us to believe in quarks). In Q C D , it is gluons (massless vector bosons like the photon) that are the fundamental field quanta exchanged between quarks. Thus meson theory is no longer thought of as a fundamental theory, but rather as a useful and effective model for the analysis and prediction of the nucleon-nucleon inter-action in the low energy regime. To wit, it is the only theoretical model that can make correct predictions in this energy regime, since Q C D has as yet been rather unsuccessful in this regard (see Section 2.3.8). Since the mid-seventies, following the emergence of accelerators capable of producing high intensity beams of protons and of mesons (the meson factories), a large amount of experimental data regarding N N scattering at low and intermediate energies has become available. These data have served to confirm the status of meson theory as a successful and relevant model for the strong interaction at these energies. Chapter 2. Meson Theory and Charge Symmetry Breaking 30 2.2 Meson Theory 2.2.1 The Yukawa Potential Yukawa [64] ini t ia l ly described the field between a neutron and a proton by drawing an analogy to the classical wave equation for the scalar potential of the electromagnetic field. He added an extra (mass) term to the wave equation, in order to get a solution that reflected the short range nature of the nuclear interaction. Specifically, the electromagnetic wave equation, has a solution, in the static case, that has a central symmetry K B y adding a term A 2 to the wave equation, i.e. { v 2 - ^ - A 2 } c , = °- <2-2> where U — U(x,y, z,t), a solution of the form C ^ r - is obtained, i.e. the potential decreases rapidly with distance r, as required by the phenomenology. Yukawa later revised the proposal and derived the potential from a quantized theory [78], however the general structure of the solution remained unchanged. The Yukawa potential is in fact the static solution to the Kle in-Gordon equation for a scalar (spin-0) field, for a point source with strength g located at the origin of coordinate space. In its most familiar form, it is given by ^ = i f ^ ' ( 2' 3 ) where </>(r) is the field generated by a single nucleon, of infinite mass, situated at the origin, and g is a coupling constant. From a simple approximation [79] using Heisenberg's uncertainty relation AE • At h, the range of a particle exchange potential would be given by R « which is just the Compton wavelength of the particle. A particle that Chapter 2. Meson Theory and Charge Symmetry Breaking 31 E', q' E', - q ' E , q (E-E'.q-tf) m 'a ® E , - q Figure 2.1: Feynman diagram for the one-boson-exchange contribution to N N scattering in the cm frame (from [80]). would satisfy the observed range of about 1.7 fm, for an exchange potential for the strong force, would need a mass of about 120 M e V . 2.2.2 The One-Boson Exchange Potential For a detailed review and description of the extraction of the N - N potential from a quantized meson exchange theory, the reader is referred to [67, 80]. Here we give a brief summary of the methods and results. First of a l l , the setting up of meson theory as a quantum field theory needs some justification, since, due to the strengths of the coupling constants involved, perturbation theory would be expected to give diverging results. This is partly offset by the massive nature of the exchanged particles, in that, in the long range, only lower order exchanges are allowed. The lowest order contributions involve the exchange of one meson between the nucleons, and can be represented by the Feynman diagram shown in Figure 2.1, where ma represents the mass of the meson that is exchanged. It should therefore be emphasized that a potential that is derived solely from such one boson exchanges could only be valid Chapter 2. Meson Theory and Charge Symmetry Breaking 32 in the long range region of the N-N interaction (and, with some reservations, in the intermediate range; see below). In the core region of the strong interaction, multi-pion exchanges, as well as single or multiple exchanges of heavier mesons, and the quark-gluon exchanges predicted by Q C D , al l play important roles. (Essentially, meson theory breaks down in the short range, due to various reasons such as the extended quark structure of the nucleons and the failure of perturbation theory to yield convergent results. This shortcoming is circumvented by applying form factors (cutoffs) to each meson-nucleon vertex). In the derivation of a simple one-boson exchange potential ( O B E P ) , the potentials that would result from the exchange of various mesons are evaluated individually, and they are then added up at the end to yield the final potential. Each individual potential is derived by evaluating the amplitude for the appropriate O B E Feynman graph (shown generically in Figure 2.1), which is of the form [80] where the subscript a stands for the particular meson under consideration, Pa/(q2—m2a) is the (meson) propagator, g\ and g2 are the vertex coupling constants, the u(g)'s and u{qYs are Dirac spinors and adjoint spinors, and Ti and T 2 are vertex properties (matrices) that are obtained from the appropriate interaction Lagrangian that couples the nucleon to the meson field. The Lagrangian for the pion field, for example, is [80] where \& represents the nucleon Dirac spinor field, $ n is the pion field, r is the isospin operator (see Section 2.3.1), and ry 5 gives the Lagrangian its pseudoscalar character (We know from experimental observations [81, pp. 32-39] that the pion has spin 0, negative intrinsic parity, and isospin 1, making it a pseudoscalar, isovector particle). The fields giU1(q')T1u1(q)Pag2U2(-q')r2U2(-q) q2 - ml (2.4) (2.5) Chapter 2. Meson Theory and Charge Symmetry Breaking 33 that are relevant to the O B E P are the pseudoscalar (ps), scalar (s), and vector (v) fields. A pseudovector (pv) coupling is also commonly considered as a derivative coupling of the ps-field. The final result for the pion exchange, after a Fourier transform into coordinate space, is [80] = s s s ^ • A*1 •CT2 + Sl2(i) t1 + + R^) 1 ^ (2-6) where 5 i 2(x) = 3 cr x • x <x2 • x — o"! • er2, (2.7) x = *, and CTJ and cr2 are the Paul i spin vectors for nucleons 1 and 2 respectively. The scalar product T% • r 2 w i l l be described in Section 2.3.1. Equation 2.6 describes a weak spin-dependent central force and a strong tensor force. The more important mesons that have been used to derive O B E P s are the ix, n, p, cu, S, and o mesons. These have al l been observed experimentally, with the exception of the cr-meson. (The cr-meson is necessary in a O B E P to provide the intermediate-range attractive behavior of the N - N and 7r-N interaction. Al though this meson has not been observed directly, a broad resonance that is seen in TVTT scattering, in the region 400-1200 M e V , is normally identified with this a-meson. Refer to [10, pp. 355-356] for a concise note on the present status of this and other scalar mesons.) The individual main contributions of these mesons towards the properties of the N - N interaction are summarized in Table 2.1. In a more systematic derivation of a O B E P in g-space, the relativistic potentials are derived within the framework of 3-dimensional reductions of the 4-dimensional Bethe-Salpeter equation [82] (see [80, Section 5]). O B E P s that are constructed in momentum space using the full, relativistic Feynman amplitudes for the various one-boson exchanges are commonly referred to as relativistic O B E P s (e.g. [83, 84, 85, 86]). Earl ier O B E P s Chapter 2. Meson Theory and Charge Symmetry Breaking 34 Coupl ing Mesons M a i n forces created pseudoscalar 77, 7T tensor scalar er, 8 central (attractive); spin-orbit vector w, p central (repulsive); spin-orbit tensor u, p tensor (opposite to pseudoscalar) Table 2.1: The main contributions from the various field couplings to the N - N potential. The dominant contributing meson for each coupling is shown in bold. (From [67].) were represented in coordinate space, and are referred to as non-relativistic O B E P s 3 (e.g. the Bryan-Scott potential [89], and the original version of the Nijmegen potential [90]). The success of (any) N - N potential model is gauged by comparing the predictions that are made by the model, for certain specific observable effects and parameters, to experimental measurement. Parameters that are commonly used include, for example, the deuteron properties and the nucleon-nucleon scattering parameters. Examples of such comparisons are given in Reference [67, Section 4 and Appendix A ] . Of particular relevance to this thesis is the fact that, where there is an analytical expression for the potential, equations such as the ones given in Section 1.2.3 could be solved exactly (or at least numerically), to yield the predicted shape of the scattering wave function and the value of the scattering length. 2.2.3 Advanced Meson Exchange Models In more advanced meson exchange models, that include two-meson exchange contribu-tions (primarily the 2n exchange potential, or T P E P ) , the N - N potential can be derived without the need to include the a-meson. Two main approaches have been used to derive the 2ir exchange potential. In the 3Non-relativistic OBEPs are less accurate than their relativistic counterparts, due to approximations that must be taken to convert the Feynman amplitudes from momentum to coordinate space. See, e.g., [87, 88] Chapter 2. Meson Theory and Charge Symmetry Breaking 35 M Figure 2.2: A pictorial representation of the 2TT exchange contribution to the NN inter-action as viewed by dispersion theory (from [67]). dispersion-theoretic approach, the total diagrams are broken down into sections that are then analyzed individually. Thus, in Figure 2.2, the 2TV exchange diagram is broken into two halves, where the hatched ovals represent all the possible processes that two pions and a nucleon can undergo. Each of these halves can be further broken into its several constituent processes, and the amplitude for each member of the set of possible processes is calculated, or deduced, from ivN and KIT experimental scattering data. The total amplitude for the 2ir exchange diagram is obtained by summing up the individual amplitudes. Examples of dispersion-theoretic potentials are the Stony-Brook [91] and Paris [92] potentials. In the field-theoretic approach, each possible Feynman graph that constitutes a net exchange of two pions between the nucleon is written down, and its amplitude is calcu-lated using standard relativistic quantum field theory techniques. A n example is shown in Figure 2.3. The first six graphs (top three rows) represent the uncorrelated 2n ex-change. The fourth row shows three graphs where the two pions interact in relative S-wave (Durso et al. [93] have shown that these graphs can be approximated by the ex-change of a scalar-isoscalar boson, called the o'). The final graph in Figure 2.3 shows the two interacting pions in relative P-wave, resulting in the p-meson resonance. The model shown in Figure 2.3 is the one that is used to construct the 2ix exchange part of the Bonn Chapter 2. Meson Theory and Charge Symmetry Breaking 36 Figure 2.3: Field-theoretic model for the 2ir exchange (from [67]). Chapter 2. Meson Theory and Charge Symmetry Breaking 37 potential model [94] for the N - N interaction. Other graphs that play an important role in the construction of the Bonn potential include np exchange diagrams, as well as further irreducible 37r and 47r exchanges. A comparison between the N N scattering data (from Arnd t et al. [95]) and the predictions from the Bonn [94], Paris [92], Nijmegen [90], and Bryan-Scott [89] potential models, for the 1 P 1 , 3 D 2 , and 3 D 3 partial waves, are shown in Figure 2.4. Three-body forces (previously mentioned in Section 1.3.3) have been derived from 2TY exchanges (mainly the Tucson-Melbourne model, see Figure 2.5), and from irp exchanges [100, 101, 102, 103]. 2.2.4 P h e n o m e n o l o g i c a l a n d H y b r i d M o d e l s A different approach, independent of meson theory, that has been used to construct the N - N potential is the purely phenomenological approach. This method involves wri t ing down the most general form that a (reasonable) analytical expression for the N - N po-tential might have, in terms of the nucleon spins, relative momenta, isospins, etc. (see, for example, [104] for a general derivation), and then obtaining the coefficient of each term in the expression by fitting the model to experimental data. Such potential models have been very successful, even as early as the 1950s. In the last decade or so, there has been a trend to construct "high precision" N - N potentials that are hybrids between meson-theoretic and phenomenological. The phenomenological part of these models is needed mainly to describe the short range part of the interaction. This approach wi l l be discussed in greater detail in Chapter 6. Chapter 2. Meson Theory and Charge Symmetry Breaking 38 i.O Lab. Energy (MeV) Figure 2.4: A comparison between the NN scattering phase shifts, as predicted by various potential models, and the experimental data. (B-S refers to the Bryan-Scott potential of 1969). (Reproduced from Machleidt and L i [96].) Chapter 2. Meson Theory and Charge Symmetry Breaking 39 TC Figure 2.5: General graph for the 27r-exchange three-body potential, used in the Tuc-son-Melbourne model [97, 98]. The shaded area represents anything except a forward propagating nucleon state (in which case the process would simply be the iteration of a two-body force [99]). 2.3 C h a r g e I n d e p e n d e n c e B r e a k i n g a n d C h a r g e S y m m e t r y B r e a k i n g 2.3.1 T h e I s o s p i n F o r m a l i s m The isospin4 operator was first introduced by Heisenberg in 1932 [105]. In 1936 Cassen and Condon [106] applied the operator to describe the charge independence of the nuclear force. In the isospin formalism 5 , the proton and the neutron are considered to be different states of the same particle (the nucleon). A new quantum number mT is introduced, wi th the proton being assigned mT = | and the neutron 6 mT — —\- The mathematical structure of isospin is identical to that of intrinsic spin, and in this context the quantum number mT can be viewed as the z-component (Iz) of the total isospin I of the nucleon, in a three-dimensional "isospin space". The proton is therefore represented by the state and the neutron by the state 4Also called isotopic spin or isobaric spin. 5For more detailed introductions to the isospin formalism, the reader is referred to [107, 108]. 6In nuclear physics texts these assignments are reversed. 2 ' 2 I) \P) Chapter 2. Meson Theory and Charge Symmetry Breaking 40 \I, Iz) = |I, -I> = |n) The 2 x 2 isospin operator matrices TX, TV, and rz (identical to the Paul i spin matrices o~x, oy, and oz) therefore represent the three components of the total isospin vector r , in an SU(2) group representation. In this notation, the nucleon isospin wave functions are represented by the column vectors \P) = 0 \n) = I and we have Tz \p) — \p) and rz \n) = — \n). We can also construct the ladder operators r + = \{TX + iTy) and r_ = \{rx — iry), which, respectively, convert a neutron to a proton and vice versa. In a system of two nucleons, the four possible combined isospin wavefunctions are p + p = \P>1 |P>2 n + n ^ i , - i = \n)l \N)2 p + n # 1 , 0 = 75 [ IP>I l n ) 2 + \ n)l b> 2 ] *o,o = 71 [ l n > 2 - \ n)i b > 2 . where the wavefunctions are written in the notation ^I;IZ, and the subscripts 1 and 2, in the nucleon wavefunctions, refer to nucleon 1 and nucleon 2 respectively. The scalar product of the two isospin vectors, T\ • T2 , is given by 7 T i • T 2 = TXLTX2 + TYLTY2 + T 2 l T 2 2 = 2 ( r i + r 2 - + r i _ r 2 + ) + TZ1TZ2 (2.8) This operator therefore operates on the four two-nucleon wavefunctions as follows 7In this representation, the subscripts 1 and 2 of the r operators indicate the specific particle (1 or 2) on which that particular operator acts. Chapter 2. Meson Theory and Charge Symmetry Breaking 41 (neutron) Figure 2.6: A n illustrative representation of a rotation of rr about the y-axis of isospin space. T i - T 2 # i , i = forz = - 1 , 0 , o r l (2.9) r i • r 2 ^o,o = - 3 ^o,o (2.10) The operator T\ » T 2 can therefore distinguish between the iso-triplet and iso-singlet states, however it does not distinguish between the members of the iso-triplet state. In the SU(2) description of the isospin matrix group, a rotation of TV about the "y-axis" of isospin space is given by P = emTy (2.11) Al though isospin space bears no physical relation to regular Cartesian coordinate space, the analogy between the mathematical structures of the isospin and spin formalisms can be used, as in Figure 2.6, to show that a rotation of n about the ry axis would result in a conversion of all protons to neutrons and vice versa. Chapter 2. Meson Theory and Charge Symmetry Breaking 42 2.3.2 D e f i n i t i o n s o f C I B a n d C S B Charge independence of the strong interaction means that p-p, n-n, and n-p interactions are equivalent for systems that have the same total isospin / . In the isospin space representation of a two-nucleon system, this means that the N - N potential is independent of the direction in which the total isospin vector T is pointing. In the strictest sense, a completely isospin independent potential would also be unaffected by whether the N - N system is in an iso-triplet (I = 1) or iso-singlet (/ = 0) state. This is however manifestly not true, and the term charge independence is therefore normally used to describe a N - N potential that does not distinguish between the three members of the 1 = 1 iso-triplet state [109]. A N - N potential that exhibits charge independence breaking (CIB) , therefore, is one that distinguishes between any two of the p-p, n-n, and 1 = 1 n-p interactions. Charge symmetry of the strong interaction refers to the equivalence of the p-p and n-n interactions. In the isospin space representation, this means that the N - N potential is not affected if there is a rotation of TT, about the y-axis of isospin space, of the total isospin vector r . As shown in the previous section, such a rotation converts all protons to neutrons and vice versa. Charge symmetry is therefore a weaker symmetry than charge independence, since it only requires the N - N potential to remain unchanged for one specific kind of rotation of r in isospin space. If the potential breaks charge symmetry it wi l l also, by the above definition of C I B , break charge independence. A N - N potential that exhibits charge symmetry breaking (CSB) , therefore, is one that either (i) distinguishes between the n-n and p-p interactions in any respect, or (ii) is sensitive to an interchange of particles in a n-p system. A n example of case (ii) would be a potential that distinguishes between n-p and p-n interactions, where ft or p refers to a polarized nucleon. C I B and C S B are normally quantified in terms of an appropriate observable (such Chapter 2. Meson Theory and Charge Symmetry Breaking 43 as the scattering length, as we shall see below). Since C S B effects are found to be small, it has become common in this context to redefine " C I B " to refer to the measured or predicted difference between the value of the observable in the n-p system and its (average) value in the p-p and n-n systems. 2.3.3 C l a s s i f i c a t i o n o f I s o s p i n D e p e n d e n t Fo rces Isospin dependent forces have been classified into four distinct groups [110] : C l a s s (I) These are forces that are charge independent. They are of the general form Vj = a + 6 n • T 2 (2.12) where a and b are reasonable isospin independent operators. C l a s s (II) These are forces that exhibit C I B but no C S B . They are normally represented in the form Vn = C[TZ1TZ2 - • r 2 ] (2.13) C l a s s ( I I I ) These are forces that exhibit both C I B and C S B , and that are symmetric under the interchange 1 -f-> 2 Vni = d[rzl + rz2] (2.14) These forces distinguish between n-n and p-p systems, but go to zero in an n-p system. C l a s s ( I V ) These are forces that exhibit both C I B and C S B , and that are anti-symmetric under the interchange 1 -H- 2 Viv = e[r 2 l - TZ2] + / [ n x r 2 ]Z (2.15) These forces vanish in the n-n and p-p systems. In the n-p system they are sensitive to an interchange of the nucleons, and also cause isospin mixing between the 1 = 1 and 1 = 0 states. Chapter 2. Meson Theory and Charge Symmetry Breaking 44 As an example, the manner in which the Class (III) C S B term operates on the n-n, p-p, and n-p wavefunctions is illustrated below: 8 \{rzi+Tz2)\n)l\n)2 = ^ [rzl\n)l\n)2 + rz2\n)l\n)2 = \ [ - \ n ) i \ n ) 2 - \n ) i \ n ) 2 ] = - » » 2 ( 2 - 1 6 ) \ {rz\ +rz2)\p)1\p)2 = ^ [ r z i |p> 1 | p ) 2 - r r z 2 | p> 1 | p>2_ = | [ | p > i l p > 2 + b > i l p > 2 ] = |P>llP>2 ( 2 - 1 7 ) 1 1 (TZ1 + TZ2) [\p)l\n)2 + |n) 1 |p) 2 ] = ^ 7 | [ r z i | p ) 1 | n ) 2 + rzl\n)x\p)2 + Tz2\p)l\n)2 + rz2\n)l\p)2] = ^q[\p)i\ n)2 - \n)ilP)2 - \p)i\n)2 + | n ) i b ) 2 ] = 0 (2.18) 2.3.4 Physical Manifestations of C I B and C S B C I B and C S B manifest themselves in various small, observable effects, that can be mea-sured experimentally. A s stated earlier, one of these effects is the difference between the nucleon-nucleon scattering lengths. Other effects include the binding energy difference between the 3 H e and 3 H nuclei [111]; the binding energy differences, minus electromag-netic effects, that are observed for mirror nuclei (the Nolen-Schiffer anomaly) [112]; and the differences between the neutron and proton analyzing powers in intermediate energy n-p scattering (e.g. [113]). The measurement of ann, in experiment E661, is being carried out in order to look for a difference between the value of this parameter and the value of app. The primary 8Recall that r is analogous to the Pauli matrix, and represents twice the isospin. The CIB properties of Class (I), (II) and (IV) forces can be similarly illustrated. Chapter 2. Meson Theory and Charge Symmetry Breaking 45 objective of E661, therefore, is to try to detect the presence of a Class III C S B force in the N-N interaction. 2.3.5 Sources of C I B and C S B A large effort has been made over the years to try to derive the C I B and C S B of the strong interaction from meson-theoretic considerations ([109],[114]-[128]), as well as from Q C D ([129]-[138]) and from other sources. The literature has also been reviewed com-prehensively by several authors [23, 59, 67, 109, 139]. In this section (2.3.5), and in Sections 2.3.6, 2.3.7, and 2.3.8, the various sources of C I B and C S B , that come from the theory, wi l l be discussed. The causes of C I B are generally categorized into two types: effects that break charge independence but preserve charge symmetry, and effects that break both charge indepen-dence and charge symmetry. The theoretical investigation of C I B , in most cases, involves the derivation of expressions for the difference in N-N potentials (V — V, see below) that arises from particular C I B or C S B effects that are being studied. This difference in po-tentials is then quantified in terms of an observable effect. A s indicated in Section 2.3.4, C I B can be quantified in terms of several physical effects, however in this thesis we shall l imit our discussion to the quantification of C I B and C S B in terms of their effect on the differences between the nucleon-nucleon scattering lengths. The change in scattering length that results from a change in potential V — V can be calculated from [122, 123, 140] where M J V is the average nucleon mass and u and u' are 1SQ asymptotic radial wave functions normalized such that u(r —> oo) = 1 — ^ . The following quantities are therefore (2.19) Chapter 2. Meson Theory and Charge Symmetry Breaking 46 p n n p p(n) p(n) p n p n (a) (b) Figure 2.7: One-pion exchange contributions to (a) pp (nn) and (b) pn scattering, defined: A a C i B = a - anp (2.20) A & C S B = Q>pp " n n (2-21) where a = \{ann + app). The largest source of C I B that is derivable from meson theory is the (7r° - TT^ mass difference in the One P ion Exchange Potential ( O P E P ) . This effect is illustrated in F i g -ure 2.7, and occurs because a proton and a neutron can exchange both charged and neutral pions, whereas two protons (or two neutrons) can only exchange the neutral vari-ety. This has been calculated to give a A a C I B contribution of 3.11 ± 0.1 fm [123]. Meson mass splitt ing also causes C I B (to a lesser extent) in the 2ir Exchange Potential ( T P E P ) ( A a C I B = 0 . 8 8 ± 0 . 1 fm [122]; 0 . 8 0 ± 0 . 1 fm [123]), and in np, TVO and ircu exchange contri-butions ( A a C I B = 0.68 ± 0.2 fm [123]). C I B also results from iry exchange contributions ( A a C I B = 1.10 ± 0.4 fm [122]). The total theoretical contribution to A a C I B from these calculations is 5.69 ± 0.5 fm [67]. A n obvious source of C S B is the effect of the n-p mass difference on the kinetic energy of the nucleon in scattering experiments: for the same momentum k, a proton and a neutron have different kinetic energies, resulting in slightly different solutions to Chapter 2. Meson Theory and Charge Symmetry Breaking 47 H em co 7L U H em Figure 2.8: Contr ibut ion of electromagnetic meson mixing to C S B . the Schrodinger equation for the neutron and for the proton in a central potential V(r). In [109] this is calculated to give a A a C S B contribution of 0.30 ± 0.10 fm. The n-p mass difference is also responsible for a C S B contribution from (meson-theoretic) 2TT exchange, also calculated to be of the order of 0.3 fm [121]. The largest source of C S B is calculated to be due to the electromagnetic mixing of neutral mesons with the same spin and parity, but different isospin (Figure 2.8). O f these, the largest contribution comes from the (p° — to) graph ( A a C S B ~ 1 fm [126]). The combined contribution to A a C S B from the (ir° — rf) and (TT0 — n') mixing graphs has been calculated to be less than 0.03 fm [120]. Another source of C S B has also been shown to be that derived from j7r° exchange diagrams, with a A a C S B contribution of the order of 0.15 fm [115, 117].9 The total theoretical contribution to A a C S B from these calculations is 1.53 ± 0 . 5 fm [67]. 2.3.6 T h e C o n t r i b u t i o n o f (p - to) M i x i n g t o C S B The (p — to) mixing term (leftmost graph in Figure 2.8) has been calculated by several authors [114, 116, 118, 119, 125, 126] to be responsible for the largest single contribution to the C S B of the strong interaction. Here we look at this contribution in some detail. Firs t of al l , the reason that the meson mixing graphs contribute at al l to the N-N potential 9The CIB and CSB that arise from JTT exchanges have not yet been fully investigated [67]. For example, contributions of as much as —1.31 fm to Aa C sB have been obtained [124]. Chapter 2. Meson Theory and Charge Symmetry Breaking 48 CO Hem P 7C Figure 2.9: The decay process cu —)• p —> 2rr. is due to the (empirical) presence of mixing between iso-vector and iso-scalar mesons of the same spin and parity. Thus, for example, the physical eigenstates of the p° and the cu° mesons are [114] 1 0 |p°) = |p°) - e\cu°) (2.22) \u>°) = \cu°) + e\p°) (2.23) where e is a small dimensionless parameter determined by electromagnetic effects of order a. The extent of this mixing can be determined from the observed decay rate T(cu —>• 27r), which must, due to G-parity conservation in the strong interaction, occur v ia the production of the p meson (Figure 2.9). This decay rate can best be observed from studies of the e+e~ —¥ TV+TT~ cross section at q2 ~ m 2 [141, 142]. The currently accepted value of the (p — cu) mixing matrix element is [126] 1 1 (p\H\u)\ « 0.00452 ± 0.0006 G e V 2 (2.24) The (p — cu) mixing graph shown in Figure 2.8 actually consists of two contributions (Figure 2.10). The contribution to the N - N potential that comes from these graphs is, in momentum space [116], 1 0There is also mixing with the (f>° meson, see [109, 127] 1 1 The interaction Hamiltonian is made up of a strong and an electromagnetic contribution (H = HsiT + Hem, see [23]), with, respectively, short and long range interaction effects. Chapter 2. Meson Theory and Charge Symmetry Breaking 49 1' 2' 2' H CO em + co Figure 2.10: The (p — to) contributions to the N-N potential. VNN = -^9p9u(p \Hem\u) u{p[)rz [7" + ^ ^ ( P ' l ~ Pi)a] u(Pl)u(p'2) [7fl + ^lo-PjPi - P2)p] u(p2) (ml-t)(ml-t) + (1 i—> 2) (2.25) where t = (p[ — pi)2 = (p'2 —P2)2, KV and KS are the iso-vector and iso-scalar anomalous magnetic moments of the nucleon, and the other symbols have their usual meaning. (The lSo potential in r-space is obtained from a Fourier transform of the above expression for V^N, see [118, 126]). The crit ical factor for C S B is the r z operator, that appears at only one of the vertices in each of the two terms in Equation 2.25. The evolution of the R H S of Equation 2.25 yields terms of the forms given in Equations 2.14 and 2.15 (page 43), i.e. both Class (III) and Class (IV) C I B forces are produced. In the n-n interaction, Class (IV) forces vanish, and thus only the Class (III) term (i.e. the term in [TZ\ + rz2\) in the above expression for is relevant to the C S B measurement objectives of experiment E661. The potential terms, analogous to (2.25), for (7r° — 77) and (vr° —77') mixing are simpler in structure 1 2 , due to the scalar nature of the mesons (and therefore the couplings) 1 2In QCD-based studies, (TT° — r] — 77') mixing is treated in the context of mixing between the SU(3) singlet and octet representations of the basic quarks of the lightest pseudoscalar meson octet [131, 132, 139, 143]. Chapter 2. Meson Theory and Charge Symmetry Breaking 50 involved, and they only contribute Class (III) C I B forces [120]. It should be noted that the extent of the electromagnetic mixing between meson pairs varies as the reciprocal of A m 2 , where Am is the difference in mass between the two mesons involved. It is because the p° and co mesons overlap in mass, that the (p — co) mixing graphs give rise to a A V ^ that is greater in magnitude than that due to the other meson mixing diagrams. 2.3.7 C u r r e n t S t a t u s o f E x p e r i m e n t a n d T h e o r y The currently accepted experimental values for the nucleon-nucleon scattering lengths 1 3 are [23] anp = -23.748 ± 0.009 fm ann = -18 .8 ± 0.3 fm app = -17 .3 ± 0.4 fm This gives A a C i B ( e x P ) ~ 5.7 ± 0.3 fm AacsB(exp) ^ 1.5 ± 0.5 fm These measurements are, in general, consistent wi th the theoretical predictions sum-marized in the previous sections. The sources of experimental uncertainty have already been outlined in Chapter 1; from the theoretical side, the main problem appears to be that of calculating the contribution from the (p — to) mix ing term wi th sufficient ac-curacy. The calculation is clearly model dependent (e.g. it depends on the choice of potential model that is used to compute the wavefunctions in Equation 2.19 (page 45)) 13The value for ann includes electromagnetic corrections to the experimentally measured value. These corrections are calculated to be of the order of —0.3 fm. Chapter 2. Meson Theory and Charge Symmetry Breaking 51 [126]. The theory has recently been further called into question (see, for example, [128]) due to the fact that several quark-meson models [144, 145, 146, 147, 148] have shown that the (p — to) mixing amplitude may be momentum dependent. 1 4 This may have repercussions in the C S B calculations, due to the contribution of off-shell (p — to) mixing in both QCD-based and meson-theoretic models of the nucleon-nucleon potential. 2.3.8 C S B from Q C D Ideally, one would like to be able to derive the phenomenological characteristics of the nucleon-nucleon interaction from Q C D , since this theory is generally accepted, as part of the Standard Model , to be the fundamental theory of the strong interaction. Al though it has been reasonably successful in describing high energy phenomena, Q C D has however fared significantly worse in its attempts to describe the strong interaction in the low energy (nuclear physics) regime. The essential reason for this (see, e.g. [151, 152]) is that Q C D is designed to describe the interaction between point particles (as is Q E D ) , in this case quarks, that couple to gluons by means of a single coupling constant as- Unlike the electromagnetic coupling constant (a), however, the strong constant as is highly dependent on momentum transfer at long distances (it increases rapidly with distance, and is referred to as a running coupling constant). This effect leads to a breakdown of perturbation theory, since the Feynman graphs of increasing order cannot be ignored as in Q E D . A t high momentum transfers (small distances) as converges to a small constant value (zero; this is called asymptotic freedom), and perturbative Q C D can be applied. For this reason Q C D , in its basic form, can be used to describe hadronic interactions in the high energy regime, but not in the low energy regime. In spite of the problem described above, however, there have been many attempts to 14See also [23, 129, 130, 149, 150] for additional contributions to this discussion. Chapter 2. Meson Theory and Charge Symmetry Breaking 52 construct models, derived from or inspired by Q C D principles, that describe, at least qual-itatively, some of the features of nucleon structure and the N - N interaction. These include the bag models of the nucleon (most notably the M I T bag model [153]), leading to studies of the N - N interaction via analysis of the six-quark system (e.g. [154]); lattice gauge the-ory [155] (see e.g. [156]); and Q C D sum rules [157]. The Skyrme model [158] incorporates the chiral symmetry of Q C D in the l imi t Nc —» oo (Nc = number of quark colours), and works as an effective meson field theory. The linear [159] and non-linear [160] a-models are meson-based theories that involve the breaking of chiral symmetry in a strong in-teraction Lagrangian, and they have also been used in hybrid models involving coupling between mesons and quarks (e.g. [161, 162]). Introductions and descriptions of many of the above mentioned models can be found in specialized Q C D / n u c l e a r physics texts [163, 164, 165, 166]. In a quark picture, C S B arises from the difference in the electric charges of the u and d quarks (electromagnetic origin), and from the difference in their current masses (mu ^ m<i [10, p. 307]). The derivation of C S B from Q C D has been attempted within the framework of several of the nuclear models listed above, and is reviewed in [23, 139]. A a C S B has been calculated from one-gluon exchange ( O G E ) and quark exchange [133, 134, 136]; from Q C D sum rules [137]; and from lattice Q C D [138]; while mechanisms, stemming from Q C D , that cause C S B in the meson-nucleon coupling constants are discussed in [143, 167]. Most of these computations give predictions for C S B effects that are of the correct order (i.e. they are consistent, in a broad sense, wi th experimental observation), however they are all as yet rather inconclusive. For this reason, QCD-based derivations of C S B should, for the present, be considered to be only exploratory. C h a p t e r 3 D e s c r i p t i o n o f t h e E x p e r i m e n t The experiment described in this thesis, E661, was proposed in July 1992 [31], and was carried out using the beam from the M l 3 pion and muon channel of the Tri-University Meson Facil i ty ( T R I U M F ) in Vancouver, Canada. F ina l data collection occurred during M a y and June, 1994, and util ized six weeks of beam time. 3.1 B e a m P r o d u c t i o n a n d t he M 1 3 C h a n n e l The T R I U M F [168] experimental facility is based on a six-sector isochronous cyclotron that accelerates H ~ ions to a peak energy of 520 M e V . Proton beams of continuously variable energy are obtained by stripping the electrons from the ions using adjustable carbon foils. The general layout of the T R I U M F facility is shown in Figure 3.1. Two main proton beams are extracted from the cyclotron. One of these beams (BL4) feeds experimental areas in the Proton Hall wing of the laboratory, while the other (BL1) feeds experimental areas in the Meson Hall. The latter beam can attain intensities of 140/iA. These primary proton beams are normally delivered in 3 nsec pulses every 43 nsec, with a 99% macroscopic duty factor. Protons that travel along beam line 1A ( B L 1 A ) in the Meson Ha l l interact with meson production targets at stations 1AT1 and 1AT2, to produce secondary beams of pions and muons. Three meson channels are fed from each production target. The M13 beam line is a low momentum (5 to 150 M e V / c ) pion and muon channel that views the 1AT1 production target at an angle of 135° relative to the direction of 53 Chapter 3. Description of the Experiment Figure 3.1: The T R I U M F cyclotron and beam-lines [169]. Chapter 3. Description of the Experiment 55 [Vacuum valve beam blacker horizontal 9 vertical jaws horizontal slit I absorber I / .vertical slit X F 3 f i n a l f o c u s target I A T I-horizontal slit-' 1 metre scale 0 1 2 3 feet Figure 3.2: The M13 beam line layout [170]. the primary 500 M e V proton beam [170]. A sketch of the beam layout, from [170], is shown in Figure 3.2. The channel contains two bending magnets (producing two 60° bends) and seven quadrupole magnets. Two sextupole magnets (not shown in the figure) are installed at the entrance and exit of the field lens that consists of (Q3,Q4,Q5). The channel length (to F3) is 9.4 m. Dur ing the E661 run, a third bending magnet was installed downstream of Q7, inside the M13 area, to direct the beam towards the l iquid deuterium experimental target. The target 1AT1 , which feeds the M13 channel, was a 12 mm thick beryl l ium target. 3.2 T h e E x p e r i m e n t a l S e t - U p 3.2.1 G e n e r a l L a y o u t The set-up of the experimental equipment is shown schematically in Figure 3.3. More detailed specifications are can be found in Appendix B . Photographs of the apparatus are given in Appendix C . The plastic scintillation counters S i , S2 and S3, shown in Figure 3.3 constituted a conventional beam telescope. A coincidence of S i and S2, with anti-coincidence in S3 Chapter 3. Description of the Experiment 56 Neutron Detectors Figure 3.3: E661 experimental set-up: plan view (schematic). Chapter 3. Description of the Experiment 57 (S1-S2-S3), defined a pion stop in the l iquid deuterium (LD) target. The pion beam, set at a mean energy of 40.5 M e V , was directed towards the target v ia the scintillator S i , an aluminum degrader (thickness 24.8 mm), beam-tracking wire chambers W C 1 and W C 2 , and plastic scintillator S2. The photon arm of the apparatus consisted of a lead collimator, a N a l scintil lation counter (the converter), a wire chamber (WC3) , and a large N a l crystal ( T I N A ) . The purpose of the converter was to convert the incident photons into electron-positron show-ers, for detection by W C 3 . The energy lost by the photon inside the converter would later be added to the coincident energy losses in T I N A , in order to get an estimate of the energy of the incident photon. The T I N A crystal was fitted with seven photomultiplier tubes ( P M T s ) on its back side, while two P M T s were mounted onto the edge of the converter disk. The main component of the neutron arm of the apparatus consisted of an array of twenty 2 m x 10 cm x 10 cm BC408 plastic scintillators, to detect neutrons from the E661 reaction (ir~d —> ynn). These detectors were set up vertically, adjacent to each other in a circular arc configuration, so that they were al l equally distant from the centre of the L D target. For reference purposes, the bars were numbered 1 to 20 from right to left as indicated in Figure 3.3. A light guide and a P M T were mounted at each end of each neutron bar. The aluminum stand that was used to hold the neutron bars in place was designed so as to minimize the incidence of particle reflections (off the stand) onto the bars. The thin plastic scintillation counter, S4, was placed between the L D target and the neutron bar array, so that it covered the same solid angle as the neutron detectors. The purpose of this counter was to detect charged particles, and the signals were later used, off-line, as a software veto. The light produced in S4 was collected by two P M T s , installed on light guides at the top and bottom ends of the scintillator. Chapter 3. Description of the Experiment 58 Two photon calibration sources ( 6 0 C o and 2 2 N a ) were positioned about 1 m behind the neutron bar array, such that each source i l luminated the entire array. A small N a l detector, called a gamma-tag detector, was placed behind each of these sources. The gamma-tag detectors were shielded from each other such that each one was only visible to radiation from its associated photon calibration source. T h e L o n g - a n d S h o r t - F l i g h t - P a t h L a y o u t s Experiment E661 was divided into two main parts that had different objectives. For the first portion of the experiment, the entrance face of the neutron bar array was placed at a radial distance of 3.00 m from the centre of the L D target (long-flight-path). This allowed us to get the momentum resolution, for the detected neutrons, that was required for our measurement of ann. For the second portion, the neutron bar array was moved to a distance of 1.50 m from the target (short-flight-path). This increased the solid angle covered by our neutron detector, and would enable us to construct the E 7 spectrum, for the reaction 7r~d —> ynn, down to energies of about 2 M e V from the reaction endpoint, where the spectrum is sensitive to the effective range parameter r n n (see Section 1.3.2). The objective of this second part of the experiment was to obtain a measurement of r n n . The rest of the equipment set-up, as well as the electronics set-up, were the same for both parts of the experiment. This thesis deals wi th the data analysis and results of only the long-flight-path portion of experiment E661. 3.2.2 E l e c t r o n i c s S e t - U p A diagram showing the general layout of the major constituents of the electronic circuitry for the experiment is given in Figure 3.4. More detailed schematics and specifications are given in Figures D . l , D.2, D.3, and D.4 in Appendix D . Chapter 3. Description of the Experiment 59 TINA 1 TINA 2 ... 7 TINA S u m C o n v 1 C o n v 2\ i r i i S1 S 2 H S 3 Pulser C o n v S u m B a (to r 1 P) Bar 1 (bottom) pulseheight |_ , threshold TINA, C o n v s u m S1»S2»S3 A N D "Type IV" "Type III" A N D Pu lser Delay pulseheight threshold A N D Bars 2 ... 20 O R A N D ~ L 7 "Type I" O R ( T R I G G E R ) L A M A N D O R "Type II" A N D [ 1 Pu lser A N D Nal ( 6 0 C o ) Na l ( Z 2 N a ) Ana log S igna ls Logic S igna ls Figure 3.4: General schematic for the E661 electronics set-up: Production of the four E661 trigger types. Chapter 3. Description of the Experiment 60 T h e E v e n t T r i g g e r s For E661, there were four different hardware triggers that signalled the data acquisition system to read and record the detector information for an event. Figure 3.4 illustrates the manner in which these four event trigger signals were formed. The E661 triggers are listed and described below. T y p e I t r i g g e r : A h i g h e n e r g y T I N A event This kind of event required the coincidence of the following three conditions: (i) A signal from one or more of the T I N A and/or converter P M T s , with the hardware sum of all nine analogue pulses being higher than a preset hardware threshold. This condition would be satisfied when a particle that passed through the T I N A collimator resulted in the deposition of a total energy (in the converter and T I N A ) greater than approximately 40 M e V . (ii) A signal from one or both of the converter P M T s , with the hardware sum of the two analogue pulses being higher than a preset hardware threshold. This requirement was made in order to exploit the fact that the converter efficiency for neutron detection is low. It resulted in a reduction in the number of high energy neutrons in T I N A that triggered data acquisition. (iii) A n S1-S2-S3 logic signal from the beam telescope circuitry. This indicated a stop in the L D target. The valid ir~d —» ynn events in E661 would be events of Type I. Type I events wi l l be referred to as "TINA events" in the remainder of this thesis. The detailed circuitry for the formation of the T I N A event signals is given in Figures D . l and D.2. T y p e I I t r i g g e r : A c a l i b r a t i o n source ( g a m m a - t a g ) event The primary decay modes for both 6 0 C o and 2 2 N a result in the simultaneous emission of photon pairs of well known energies, making these sources useful for use in the energy Chapter 3. Description of the Experiment 61 calibration of radiation detectors. 1 In E661, a gamma-tag event required the coincidence between a signal from either of the two gamma-tag detectors and a signal, from both P M T s , from at least one neutron bar. During the experiment, two kinds of events were initiated by radiation from either of the two calibration sources (see Figure D.3). These were the prompt gamma-tag events, resulting from a coincidence between the appropriate gamma-tag detector and a neutron bar; and the delayed gamma-tag events, resulting from a coincidence between a neutron bar signal and a delayed gamma-tag N a l signal. The prompt events were used in the energy calibration of the neutron bars. The delayed events were used to measure the accidental coincidence event rates among the prompt coincidence signals. They were also used in the setting up of the Type III event triggers (see below). T y p e III t r i g g e r : A r a n d o m pu l s e r event The circuitry that was used to generate this type of event is shown in Figure D.4. A coincidence was formed between a delayed gamma-tag signal from the 2 2 N a N a l detector and a pulser signal, wi th the t iming of the logic coincidence pulse being determined by the gamma-tag signal. This pulse was used to start the neutron bar T D C s , whereas the pulser signal was used to stop the T D C s . This resulted in a flat time distribution of signals to the T D C s , since the physical time difference between the accidental gamma-tag signal and the pulser signal was randomly distributed. The random pulser events were used to measure the differential non-linearity of the neutron bar T D C s (Section 4.2.1). T y p e I V t r i g g e r : A b e a m s a m p l e event (Refer to Figures 3.4 and D.2.) A coincidence was formed between the STS2-S3 logic signal and a pulser signal, such that the t iming was determined by S2. This coincidence was used as the fourth type of trigger for E661, and generated the beam sample events. 1For 6 0Co the 1.173 MeV and 1.333 MeV 7-rays are always emitted simultaneously. For 2 2Na the /3+ decay always produces a 1.275 MeV 7-ray plus two back-to-back 0.511 MeV annihilation quanta. Chapter 3. Description of the Experiment 62 These events were useful for the evaluation of beam-correlated backgrounds. Dur ing the experiment, the trigger rates for event types II, III and I V were prescaled down, in order to make up only a small percentage of the total number of detected events. T h r e s h o l d s The discriminator thresholds for the SI and S2 pulses were set to just below the pion energy band of the M13 beam. The threshold for S3 was set lower, since this counter was intended to detect, as efficiently as possible, pions that had passed through the L D target without stopping. Similarly, the threshold for S4 was set to a low value, since this scintillation counter was very thin, and was intended to detect small energy losses from charged particles that traversed it. The hardware thresholds on the individual T I N A and converter pulse-height discrim-inators were set to low levels. The discriminator threshold on the hardware-summed analogue signal from the seven T I N A and two converter pulses, however, was set to the equivalent of about 40 M e V of energy. Thus, a relatively high value of the summed T I N A/converter P M T pulse energy was required to generate a Type I trigger that would signal the data acquisition system to register and record the event. The discriminator thresholds for the forty neutron bar P M T s were adjusted to very low settings. This enabled low energy neutrons (down to about 1 M e V energy) to be detected. In order to minimize time walk problems with the T D C s 2 , two discriminators were used wi th each neutron bar P M T . This can be seen in Figure D.3, and is also illustrated schematically in Figure 3.5. The t iming for the T D C reading was determined by the logic pulse from the constant fraction discriminator ( C F D ) , whereas the leading edge discriminator ( L E D ) , which was set to a somewhat higher threshold, ensured that 2Refer to Section 4.2.8. Chapter 3. Description of the Experiment 63 Analog Signal from PMT - * \ CFD To TDC Stop Figure 3.5: Discrimination of the neutron bar P M T pulses. all the accepted signals were ones that were sufficiently above the C F D threshold so as to ensure correct functioning of the internal C F D time walk compensation circuitry. The high voltage supplies to the neutron bar P M T s were set close to the highest recommended values for the tubes, in order to push the low energy l imit on detectable neutrons as far down as possible. This resulted in hardware thresholds for the neutron bars of about 50 keVee. 3 3.2.3 T h e W i r e C h a m b e r s Three wire chamber assemblies were utilized in E661 ( W C 1 , W C 2 and W C 3 ; see Sec-tion 3.2.1). W C l and W C 2 were positioned within the beam line, 7.5 cm apart, upstream of scintillator S2. The purpose of these wire chambers was to track the pions before they entered S2 and the target, and therefore to give an indication of the region, wi thin the target cross-section, where the pion stopped. This information was used to determine whether the pion had stopped within the right-hand half of the L D target (with respect to the beam direction) or wi thin the left-hand half (refer to Figure 3.6). This distinction was relevant because neutrons that originated from pion stops on the left side of the target would have a greater probability of scattering inside the l iquid deuterium, before 3The electron-equivalent energy unit keVee is explained in Section 4.3.1 (page 102). A recoil proton of energy 450 keV would deposit approximately 50 keVee of energy in the plastic scintillator. Chapter 3. Description of the Experiment 64 y n Target z Back half Front half of target of target Figure 3.6: Beam's view of the L D target. proceeding towards the neutron bar array, than would neutrons originating from pion stops on the right-hand side. W C 3 was placed in the photon arm of the apparatus, between the converter and T I N A . The purpose of this wire chamber was to give an indication of the position at which the detected photon had entered the collimator. Each wire chamber consisted of two planes of wires (to give the x- and y- coordinates of the hits) laid on, and separated by, 25-micron sheets of mylar. The chamber walls also consisted of 25-micron mylar sheets. The wires were made of gold-plated tungsten, and had a diameter of 20 microns. The chambers W C l and W C 2 were filled with a gas consisting of 80% C F 4 and 20% isobutane. Wire spacing was 0.762 mm. W C l had 96 wires in each of its two planes, while W C 2 had 64 wires in each plane. W C 3 was larger, with 256 wires per plane. For E661, some of the peripheral wires on W C 3 were not used: 224 wires were connected in the x-coordinate plane, and 192 wires in the y-coordinate plane. The W C 3 chamber was filled with magic gas (65% A r , 35% isobutane, 0.5% freon), and wire spacing was 1.0 mm. The individual wires were connected to a data register and controller which, for each Chapter 3. Description of the Experiment 65 Beam Window (0.005" Kapton) . Four layers superinsulation ~7 (0.00025" Myler) Target capsule (0.002" Ni) Photon Window (0.005" Kapton) Aluminum housing Beam Window (0.005" Kapton) Figure 3.7: Horizontal sectional view of the L D target. hit, provided an encoded number ( P C O S code) for the data acquisition system, indicating the wire that had been hit. For multiple hits in the same event, the controller passed on a stream of P C O S codes, in series, indicating the addresses of the different individual wire hits. 3.2.4 T h e L i q u i d D e u t e r i u m T a r g e t A s s e m b l y The l iquid deuterium target vessel assembly was designed and constructed specifically for E661 [171]. A horizontal sectional view is shown in Figure 3.7. The assembly consisted of a capsule-shaped nickel target vessel (4 cm in diameter, 7 cm in length; wall thickness 0.002" (0.051 mm)) contained within a vacuum compartment of rectangular cross-section (16.03 x 9.84 x 24.13 cm). The main structure of the vacuum compartment was made out of aluminum. The compartment had 0.005" (0.127 mm) kapton windows in the walls that faced the incident pion beam and T I N A , and a 0.002" (0.051 mm)) stainless steel K Beam Neutron Window (0.002" stainless steel) Chapter 3. Description of the Experiment 66 window in the wall that faced the neutron detectors. A number of layers of 0.00025" (0.00635 mm) mylar superinsulation sheets were installed wi th in the vacuum chamber. L iqu id deuterium was fed into the target vessel through an elaborate circulation system, within which the correct temperature and pumping pressure of the L D were maintained by equipment that was situated outside the M13 experimental area. The aluminum column that supported the target assembly from above was itself supported on rollers on a frame, about 3 m above ground level. The whole assembly could therefore be rolled out of position whenever this was required. The L D was 99.8% isotopically pure, wi th the remaining 0.2% being l iquid hydrogen (protium). 3.3 S i g n a l M e a s u r e m e n t 3.3.1 T h e C A M A C S i g n a l Sources The data acquisition for E661 was carried out using the V D A C S software [172]. When-ever conditions were satisfied that required an event to be saved (i.e. whenever there was a valid event trigger), a set of readings, originating from the detectors inside the exper-imental area, was taken by V D A C S and encoded to tape. For E661, the readings from 112 CAMAC signal sources, together wi th the P C O S wire chamber data, were recorded per event. The 112 C A M A C signal sources consisted mostly of the digital outputs from various analog-to-digital converters ( A D C s , measuring energies) and time-to-digital converters ( T D C s , measuring time differences). The full list of the E661 C A M A C sources is given in Appendix E . In summary, these consisted of energy and time readings from the forty neutron bar P M T s and from the five plastic scintillator P M T s on S i , S2, S3, S4(top) and S4(bottom); energy readings from the seven T I N A N a l detector P M T s , the two converter P M T s , and the two gamma-tag detector P M T s ; time readings from the Chapter 3. Description of the Experiment 67 hardware-summed T I N A pulse, the hardware-summed converter pulse, the hardware-summed T I N A and converter pulses, the S4 mean-timer pulse, and the beam-cap signal; and energy readings from the hardware-summed T I N A , and the hardware summed T I N A and converter pulses. The remaining two C A M A C signal sources were a neutron array coincidence register and a gamma coincidence register. These two registers recorded the type of trigger that had caused that particular event to be recorded. In the case of a gamma-tag trigger, the neutron array coincidence register was able to differentiate between the 2 2 N a prompt, 2 2 N a delayed, 6 0 C o prompt and 6 0 C o delayed events. 3.3.2 E n e r g y M e a s u r e m e n t a n d t he A D C s Pulse height (energy) measurement of signals that came from the neutron bars, the beam telescope counters, scintillator S4, and the gamma-tag detectors was carried out using LeCroy model 2249A A D C s . Energy measurements for the T I N A and converter signals were effected using LeCroy model 2249W A D C s . Where required, signals were attenuated before they were routed to the appropriate A D C channels (refer to the circuit drawings in Appendix D) . The A D C gates for the different kinds of events were generated using separate gate generators, so that the gate width could be more effectively customized to the event type. 3.3.3 T i m e M e a s u r e m e n t a n d t he T D C s Signal t iming for all T I N A and beam sample events was measured relative to the lead-ing edge of the S2 logic discriminator pulse. T i m i n g for the random pulser events was measured relative to the 2 2 N a N a l detector signal, as described in Section 3.2.2. No time measurements were needed for the gamma-tag events. A l l the T D C s used in E661 were LeCroy model 2228A T D C s . Chapter 3. Description of the Experiment 68 Magnet D A C Setting Amps B I 10140 155.8 B2 11297 127.4 Q l 2550 621 Q2 30750 234 Q3 2497 453 Q4 2884 351 Q5 2497 458 Q6 1500 184 Q7 1300 157 Table 3.1: M13 bending and quadrupole magnet settings for E661. (Refer to Figure 3.2, page 55, for magnet configuration. Quadrupole currents are accurate to within ± 5%.) 3.4 T h e E x p e r i m e n t a l R u n 3.4.1 A l l o c a t i o n o f the A v a i l a b l e B e a m T i m e The official beam time for E661 ran from the 19th of May to the 28th of June, 1994. The first (long-flight-path) part of the experiment ran unti l the 18th of June. The second (short-flight-path) part of the experiment was performed during the remainder of the beam time. The first two and a half weeks of the experiment were dedicated to the tuning of the equipment and electronics. Ac tua l production running only started on the 5th of June. Throughout the experiment, data acquisition was continuously monitored and scrutinized using on-line sorting and analysis software (see Section 3.5). The quadrupole and bending magnet settings for the M13 beam-line, as well as the thickness of the aluminum degrader that was positioned just downstream of S i , were tuned to maximize the rate of high energy events (photons) detected in T I N A and the converter. The final settings that were selected for the experiment are shown in Table 3.1. The bending magnet settings represented a mean pion beam energy of 40.5 M e V . The Chapter 3. Description of the Experiment 69 Visua l scaler readings (counts per sec): SI 4,450,600 S1-S2 770,600 Neutron bar array 1,709,600 TINA+Conver te r 1,536 L A M S 157 Table 3.2: Typ ica l set of visual scaler readings. (Run 460; Readings taken at the begin-ning of the run; proton beam current ~ 150 fiA). optimum thickness of aluminum degrader was empirically found to be 24.8 mm. 3.4.2 T h e Production Runs After al l the channel settings had been finalized, and the acquisition hardware and soft-ware calibrated and debugged, the production runs proceeded quite smoothly, and there were very few problems during the data-taking stage of E661. A net total of about 172 hours of long-flight-path production data were taken during the experiment. It was esti-mated, on-line, that an average of about 1670 counts of va l id 4 E661 data (before accurate calibration and without background subtraction) were collected per hour. This gave a preliminary estimate of a total of about 228,000 raw counts for the long-flight-path part of the experiment. A typical E661 run was about two hours long, and in al l , 90 separate runs made up the 172 hours of long-flight-path production data. The beam current and a small set of C A M A C visual scaler data were recorded at the start of each run, and a typical set of these values is given in Table 3.2. 4In this context, valid refers to triple-coincidence events for which momentum conservation require-ments were approximately satisfied. Ultimately, after the off-line data analysis had been completed, only about half of these events were found to lie within the top 600 keV of the photon energy spectrum. Chapter 3. Description of the Experiment 70 3.4.3 T h e C a l i b r a t i o n R u n s Dur ing E661, much of the data that were required for calibration purposes were collected during the production runs themselves, at the same time that the production data were being collected. These calibration events included the prescaled data that resulted from the gamma-tag, random pulser and beam sample events. The prolific it~d —> nn reaction in the L D was suppressed by the requirement of a converter signal in the hardware T I N A trigger (refer to Section 3.2.2); however a sufficient number of these mono-energetic, back-to-back neutron events were detected and recorded, during the production runs, to enable them to be util ized for calibration purposes. Apar t from the in-situ calibration data described above, further calibration data was collected from a number of additional runs, dedicated solely to calibration, that were carried out over the course of the experiment. These runs included: 1. Runs, wi th beam-on, in which the L D target was replaced by a L i H target. These runs enabled detection of the photons resulting from the decay reaction TT° —> 7 7 , following the charge-exchange reaction ir~p —> 7r°n on the protons in the L i H . The final state particles resulting from the radiative capture reaction w~p —> 777, were also useful for calibration purposes. 2. Runs, wi th beam-off, using (non-prescaled) gamma-tag events as the sole triggers for data recording. 3. Runs, wi th beam-off, using (non-prescaled) random pulser events as the sole triggers for data recording. 4. T D C integral-linearity calibration runs, wi th beam-off, in which a pulser signal was delayed using a precisely precalibrated delay box, and fed into the E661 T D C s . Chapter 3. Description of the Experiment 71 5. Runs, wi th beam-off, using a collimated 6 0 C o source, and its associated N a l detec-tor, placed at specific positions along each of the twenty neutron bars. 6. Runs, with beam-on, during which the L D target was empty. These calibration runs were only carried out during the short-flight-path portion of E661. Chapter 4 of this thesis is dedicated, in its entirety, to a description of the manner in which the E661 calibration data were analyzed and interpreted. 3.5 O v e r v i e w o f t he O n - l i n e a n d Off - l ine A n a l y s e s On-line analysis during the E661 experiment involved: (i) preliminary calibration of al l the equipment, (ii) continuous monitoring of equipment performance and data acquisi-tion, (iii) preliminary sorting of the data, and (iv) preliminary estimates of the quantity and quality of the data being collected. The on-line analysis for E661 was carried out on the 3200 V A X s t a t i o n (named M 1 3 D A C ) that is installed in the M13 counting room at T R I U M F . The data from E661 were stored, for later off-line analysis, on 8 mm magnetic tape (data cartridges) in V D A C S format. Every saved event contained the digital readout from the 112 C A M A C signal sources, and from the P C O S wire chamber controller. Both on-line and off-line analyses of the data were carried out using the sorting and analysis program called D I S P L A Y [173]. D I S P L A Y is a software package, writ ten in F O R T R A N , that can sort data, both on- and off-line, into customized histograms, as specified by the user. Some of the program routines are user-written, and can be customized to carry out extensive analysis of the data before final sorting. The D I S P L A Y sorting routines also allow cuts and conditions (e.g. Boolean logic) to be imposed on the data to select events for inclusion into specific histograms. Chapter 3. Description of the Experiment 72 In the data analysis of E661, most of the customized user-written software was con-tained in the program module P R E S O R T S U S E R . E X E , an executable subroutine com-piled from a F O R T R A N source code. This presorting routine is called by D I S P L A Y for every event in the data, and is able to read in al l the C A M A C and P C O S input values for that event, perform the required analysis, and output a number of new derived signal sources back to the main data-replaying program. These extra signal sources can then be sorted and plotted by D I S P L A Y in the same manner as the original C A M A C signal sources. A general representation of the way in which D I S P L A Y was used to analyze the E661 data is given in Figure 3.8. For an event to be included in the Reconstructed E1 Endpoint spectrum, it had to satisfy the rigorous analysis routines in the module P R E S O R T S U S E R , as well as survive the subsequent cuts and conditions imposed by D I S P L A Y . If, at any stage during the P R E S O R T analysis, an event was determined to be invalid, a logical variable ("skip-flag") was set to u.true.", and D I S P L A Y carried out no further processing of that particular event. During the off-line analysis, almost 99% of the events that were stored in the original data files were found to be invalid, and eliminated using the skip-flag variable. It is in fact typical of most nuclear and particle physics experiments, that only a small percentage of the stored data is used for the physics goal of the experiment. The off-line analysis for E661 was carried out using the V A X cluster of computers at the University of Br i t i sh Columbia 's Subatomic Physics lab (a V A X 3100 M76 and a V A X 3100 M38 named, respectively, U B C V S A and U B C V S B ) , the T R I U M F Data Analysis Centre V A X cluster of computers (mainly two V A X 4000/100s named E R I C H and R E G ) , and the V A X cluster of computers at the University of Kentucky's Intermediate Energy Nuclear Physics Lab (a p V A X I I named N L A B 1 , two 3100 VAXsta t ions named N L A B 2 and Z E P P O , two 3200 VAXsta t ions named H A R P O and C H I C O , and a V A X 3000 named M O E ) . Chapter 3. Description of the Experiment 73 Start D I S P L A Y H a s the whole buffer been sorted? Start Data Acqui re or Data Rep lay Program R e a d in a buffer of V D A C S formatted data Identify data for next event PRESORT PROGRAM Initialize the var iables that are used in the P R E S O R T code A d d new "der ived" signal sources to the C A M A C signal sources Ana lyze the wire chamber data Ana lyze the T INA time and energy data Ana lyze the neutron bar time and energy data Reconstruct the photon energy spectrum Sort the signal sources , using user-def ined select ion rules * Update the histograms i G o to the next event Figure 3.8: Data analysis of E661 using the D I S P L A Y program. Chapter 4 Data Analysis I: Equipment Calibration 4.1 Objectives The first portion of the data analysis for E661 involved the calibration of the various signal analyzers that were used during the experiment. Al though some preliminary cali-brations were done on-line during the experiment, most of the work involved the correct interpretation, off-line, of the various calibration data that were collected over the course of the experimental runs. The main components that needed calibration were the neutron bar T D C s and A D C s , the three wire chambers, and the T I N A A D C s . For every valid E661 event, the energy of the outgoing photon, from the reaction ir~d —r 'jnn, was calculated from the momenta of the two neutrons (Equations 1.15 and 1.16, page 16). The momentum vectors of the neutrons were calculated from the neutron times of flight (TOFs) and hit positions. For every neutron that was detected, the neutron T O F was calculated from the bar top and bottom T D C mean time wi th respect to the time of the pion signal in S2. The neutron hit position along the bar was calculated from the difference between the two T D C readings. A n accurate calibration of the neutron bar T D C s was therefore extremely important for the success of E661. The final, reconstructed photon spectrum represented the experimental measurement of part of the phase space of the photon from the E661 reaction, and the shape of this spectrum was used to determine ann. It was therefore very important for the experimental 74 Chapter 4. Data Analysis I: Equipment Calibration 75 spectrum to reflect the true phase space of the photon, and for this reason the relative neutron detection efficiencies of al l twenty bars, as well as the variation of these efficiencies wi th neutron energy, needed to be taken into account, in order to avoid the inclusion of systematic biases into the final result. The variation of detector efficiency with energy was simulated in the E661 Monte Car lo (Chapter 6); from the experimental side the important thing was to have an accurate knowledge of the low energy threshold and the A D C gains for each of the twenty bars. The accurate calibration of al l the neutron bar A D C s , and the continuous monitoring of this calibration over the analysis of the entire experiment, was therefore crucial to E661. "Calibrat ion" of the three wire chambers involved the decoding of the P C O S informa-tion in order to obtain a physical wire address for each hit, and the wri t ing of computer algorithms that would assign the best coordinate positions in the case of multiple hits in the same plane. The time and energy information from the T I N A and converter P M T s was used, during the off-line analysis, to select those events that satisfied (broadly) the conditions for a high energy photon in T I N A . 1 The converter T O F spectrum was used to differentiate between photons and other particles. The T I N A and converter energy spectra were used to make a software cut at a lower l imi t of photon energy, and therefore needed some calibration. C o o r d i n a t e S y s t e m s A Cartesian coordinate system was adopted, and used consistently, throughout the entire analysis of E661. The directions of the x- and z-axes are shown in Figure 3.3 (page 56). The positive direction of the y-axis pointed upwards. The origin O of the coordinate *As described in Chapter 3, the summed TINA+converter signal was also used as the hardware trigger for the recording of E661 events during the experiment. Chapter 4. Data Analysis I: Equipment Calibration 76 system was the centre of the L D target. Dur ing the course of the analysis it was sometimes convenient to temporarily use a local coordinate system that was valid for a particular neutron bar that was under consideration. In such cases, the bar centre was taken to be the origin O' of the local coordinate system, and the axes x', y', and z' were taken to be parallel to the faces of the bar. 4.2 C a l i b r a t i o n o f t he N e u t r o n B a r T D C s 4.2.1 M e a s u r e m e n t o f the D i f f e r e n t i a l N o n - L i n e a r i t y o f t h e T D C s The differential non-linearity tests for the forty neutron bar T D C s were carried out during the experiment, during beam downtime. For these tests, the electronics (prescaler) were set up such that the sole trigger for data acquisition was the Type III trigger (the random pulser; see Section 3.2.2, page 60). These events produced a time input to all the T D C s that was randomly distributed over the whole range of sensitivity of the T D C s . The calibration run was left to run overnight. The result for the T D C for the top P M T of bar 10 ( T D C lOt) is shown in Figure 4.1. The figure shows a constant response (within statistical errors) of the T D C for its entire range of operation. The only T D C that showed any differential non-linearity was that for the bottom P M T of bar 4 ( T D C 4b), and this is shown in Figure 4.2. This defect corresponded to a variation of 2% in the T D C efficiency over the time range that was relevant to the E661 result. The error due to this non-linearity w i l l be discussed in Section 7.2.12, and was found to be negligible. Chapter 4. Data Analysis I: Equipment Calibration 77 o o h o o C\2 O O O - i 1 r - - i 1 r - i 1 1 1 1 1 r -_! 1 l _ 500 1000 1500 TDC Reading (channels) 2000 Figure 4 . 1 : Differential non-linearity test result for T D C lOt. 500 1000 1500 TDC Reading (channels) 2000 Figure 4.2: Differential non-linearity test result for T D C 4b. Chapter 4. Data Analysis I: Equipment Calibration 78 o X m T—I m a o o u o o o m ~ i 1 1 r - i 1 1 r -500 1000 1500 TDC Reading (channels) 2000 Figure 4.3: Integral non-linearity test result for T D C lOt (raw spectrum). 4.2.2 M e a s u r e m e n t o f t he I n t e g r a l N o n - L i n e a r i t y , G a i n , a n d R e l a t i v e E f f i -c i e n c y o f t he T D C s The integral non-linearity tests for the forty neutron bar T D C s were also carried out during the E661 experiment. For these tests, a pulser signal was delayed using a precisely pre-calibrated delay box, and fed simultaneously into al l of the neutron bar T D C s . This was done for various values of time delay. The raw data ( T D C digital output) for T D C lOt is shown in Figure 4.3 (the differences in the peak heights are due to different time durations for which the pulser signal was left on, and are inconsequential). The centroid positions of these peaks were graphed against the input delay values (Figure 4.4), and the resultant curves were fitted to polynomials of the form a + bt + ct2. For T D C lOt the results were a = 4.89 ± 0.10 ch b = 3.9546 ± 0.0009 ch/ns c = 9.2 x 1 0 - 6 ± 1.8 x 10" 6 ch/ns 2 Chapter 4. Data Analysis I: Equipment Calibration 79 0 100 200 300 400 500 Input Delay (ns) Figure 4.4: Graph used for the extraction of the integral non-linearity, and gain, of T D C lOt. The coefficient c is a measure of the T D C non-linearity, and was found to be negligible for all forty T D C s . The coefficient b represents the gain of the T D C (the T D C s were set to a nominal gain of 250 ps/ch). The coefficient a is an unimportant offset. The coefficients c and b, for all forty T D C s , are given in Tables 4.1 and 4.2. The raw histograms (Figure 4.3) were also used during the data analysis to measure the relative efficiencies of the T D C s . This was done by first matching the peaks in the forty histograms 2 in order to find an average number of counts per T D C for each peak; then identifying those peaks in the histogram that fell in the E661 range of interest of the T D C spectrum (this could be done due to our knowledge of the T D C pedestals, refer to Section 4.2.6); and finally extracting the percentage deviation, from the average, in the efficiency of each T D C in the E661 range of interest. The final results are included in Tables 4.1 and 4.2. The variations in efficiency among the forty T D C s were considered negligible. 2For example, the 5th peak would contain the same number of counts for each of the forty TDCs if the TDCs were all equally efficient, since the pulser signal was fed simultaneously to all forty TDCs; the Chapter 4. Data Analysis I: Equipment Calibration 80 Non-linearity Ga in Relative Efficiency T D C (quadratic coefft.) ( x l 0 ~ 6 ch/ns 2 ) (ch/ns) (percentage deviation from average) It 7.5 ± 2.9 4.0861 ± 0.0015 0.1 l b 8.8 ± 1.5 4.0010 ± 0.0008 0.2 2t 7.1 ± 2.2 4.0001 ± 0.0012 0.2 2b 2.7 ± 3.6 4.0059 ± 0.0018 0.2 3t 3.6 ± 2.7 4.0173 ± 0.0014 0.1 3b 6.2 ± 2.2 4.0038 ± 0.0011 0.2 4t 8.1 ± 1.5 3.9960 ± 0.0008 0.1 4b 8.0 ± 2.4 3.9843 ± 0.0012 -1.5 5t 10.0 ± 1.9 3.9815 ± 0.0010 -0.5 5b 5.9 ± 1.8 4.0038 ± 0.0009 0.3 6t 9.3 ± 0.8 3.9926 ± 0.0005 0.2 6b 6.6 ± 1.6 3.9951 ± 0.0008 0.2 7t 9.0 ± 1.7 3.9830 ± 0.0009 0.1 7b 7.8 ± 1.8 4.0197 ± 0.0009 0.2 8t 10.4 ± 2.6 3.9980 ± 0.0013 0.0 8b 12.2 ± 2.8 3.9943 ± 0.0015 0.1 9t 5.2 ± 1.9 3.9796 ± 0.0010 0.0 9b 8.2 dz 1.6 3.9974 ± 0.0008 0.1 lOt 9.2 ± 1.8 3.9546 ± 0.0009 0.1 10b 7.9 ± 1.8 3.9512 ± 0.0010 0.1 Table 4.1: T D C test results for Bars 1 to 10. Chapter 4. Data Analysis I: Equipment Calibration 81 Non-linearity Ga in Relative Efficiency T D C (quadratic coefft.) (ch/ns) (percentage deviation ( x l O - 6 ch/ns 2 ) from average) l i t 7.8 ± 1.6 4.0193 ± 0.0008 0.1 l i b 7.6 ± 1.9 4.0311 ± 0.0010 0.2 12t 10.4 ± 2.3 4.0554 ± 0.0012 -0.3 12b 8.1 ± 2.7 3.9678 ± 0.0014 0.0 13t 1.5 ± 1.9 4.0743 dz 0.0010 -0.8 13b 1.8 ± 2.7 3.9312 ± 0.0014 0.1 14t 4.2 ± 1.9 4.0969 ± 0.0010 0.0 14b - 2 . 9 ± 2.9 3.9168 ± 0.0015 0.1 15t 3.5 ± 2.0 3.9549 ± 0.0010 0.0 15b 12.8 ± 2.1 3.9853 ± 0.0011 0.1 16t 10.2 ± 2.5 3.9935 ± 0.0013 0.1 16b 1.3 ± 3.3 3.9748 ± 0.0017 0.0 17t 8.1 ± 1.6 4.1102 ± 0.0008 -0.1 17b 8.2 ± 1.5 4.1340 ± 0.0008 0.0 18t 10.0 ± 1.3 4.1182 ± 0.0007 -0.1 18b 3.4 ± 4.2 4.1562 ± 0.0022 0.1 19t 8.1 ± 4.5 4.1516 ± 0.0023 -0.2 19b 6.7 ± 2.3 4.1588 ± 0.0012 0.0 20t 3.0 ± 3.1 4.1614 ± 0.0016 -0.3 20b 4.0 ± 3.5 4.1180 ± 0.0018 0.0 Table 4.2: T D C test results for Bars 11 to 20. Chapter 4. Data Analysis I: Equipment Calibration 82 KS2 r-i Signal Processing Top Light Guide and PMT Bottom Light Guide and PMT Figure 4.5: T O F and y-position measurement of the neutrons: conceptual sketch. 4.2.3 T i m e a n d Y - P o s i t i o n C a l i b r a t i o n o f t he T D C s : C o n c e p t a n d D e r i v a -t i o n Figure 4.5 is a conceptual sketch of the relevant components and variables that determine the T D C output when a hit is detected in a neutron bar. A pion passes through the scintillator S2, and, a time tw- later, stops in the target. The signal from S2, after appropriate processing (refer to Figure D.2), is used to start the appropriate T D C (the time delay caused by the circuitry between S2 and the T D C is labeled tKs2)- A neutron of energy En (momentum p n ) is produced in the L D target via the reaction ir~d —y/ynn, and heads towards the neutron bar. The neutron has a T O F tn and scatters against a proton in the bar at a position y' relative to the bar centre. The proton rapidly loses its energy in the scintillator, and the resulting light travels upwards and downwards towards procedure described here uses the deviations from this equality in the number of counts as a measure of the relative efficiencies of the TDCs. Chapter 4. Data Analysis I: Equipment Calibration 83 the P M T s , taking time tlt to reach the top P M T and ttb to reach the bottom P M T . 3 The signals from the P M T s , after appropriate processing, are fed to the T D C Stops. The time delay caused by the circuitry between the top (bottom) P M T and the T D C is tj<t (tKb). We need to determine the magnitude and direction of the momentum vector p n of the neutron. The x- and z-coordinates of the neutron hit are determined by the physical location of the bar that was hit (we use the coordinates of the axial centre-line of the bar). The y-coordinate of the hit is derived from the time difference between the top and bottom T D C signals (see below). The energy and momentum of the neutron can then be inferred 4 from its T O F tn. In the remainder of this section we derive the equations for tn and y. Let t t o p represent the actual time that elapses between the T D C Start signal and the top P M T T D C Stop signal. Then we have ttop = tn- + tn + tit + txt — txS2 (4-1) If we assume that the statistical variations in tv-, txs2, and t^t are small compared to tn, then we have ttop = tn + tu + tKt + tK (4.2) where tK = tn- — tK$2] tu is a function of y; and tKt, tK are constant for al l events. The top T D C wi l l register a reading Tt, determined from t t o p = At x {Tt + B[) (4.3) where At is the gain of the T D C and B't is an offset. We therefore have, for the top T D C A{Ft = tn + tH + tKt + tK- AtB't (4.4) 3The light is attenuated as it travels along the neutron bar. This is discussed in Section 4.3.7. 4This is true within certain limitations: e.g. there are position uncertainties due to the bar thickness and the extended target geometry. This will be discussed in Section 4.2.9. Chapter 4. Data Analysis I: Equipment Calibration 84 Similarly, for the bottom T D C AbTb = tn + ttb + tKb + tK- AbB'b (4.5) A d d i n g Equations 4.4 and 4.5 gives AtTt + AbTb = 2tn + {tu + ttb) - AtB[ - AbB'b + 2tK + tKt + tKb (4.6) where {tu + to,) is now independent of y and is therefore a constant. If we let K2 we have AtTt + AbTb where Bt {= B't — ^ ) and Bb are the pedestals for the top and bottom T D C s respectively. We therefore get t, = MTl + Bt) + M T i + Bb) ( 4 n ) where At{Tt + Bt) is the calibrated time (for this event) for the top T D C and Ab{Tb + Bb) is the calibrated time for the bottom T D C . The difference between the top and bottom T D C calibrated times can be used to give the y-position of the hit as follows: {AbTb 4- AbBb) - {AtTt + AtBt) = {AbTb + AbB'b - K2) - {AtTt + AtB't - Kx) = ( < b o t - K2) - ( t t o p - Kx) = ( t b o t - t t o p ) + {Kx - K2) 2 (*/t + ^ft) + IK + tret 2 {tit + Mb) +tx + tKb (4.7) (4.8) = 2tn - At{B't Ab{B'b - ^) = 2t„. — AfBt — AbBb (4.9) (4.10) Chapter 4. Data Analysis I: Equipment Calibration 85 {tlb — t u + t K b — t K t ) + { t K t — t K b ) — tib — tu [(2 +y')-(2-y1)] (4.12) where L is the length of the neutron bar and vi is the effective speed of the light along the bar. We therefore get The gains At and Ab were known for every bar (see Section 4.2.2), and the value of y0 was also known, from actual measurement, to be 13.3 cm. The factor V was measured at the University of Kentucky [62, 31] and was found to be equal to 7.96 cm/ns (1.99 cm/ch) (this w i l l be described in Section 7.2.3). Thus, for any neutron that originated in the target and was detected by the neutron bar, tn and y could be computed from the T D C readings Tt and Tb using Equations 4.11 and 4.13, as long as the T D C pedestals Bt and Bb were known. 5 The off-line time and y-position calibration of the T D C s mainly involved the development of a method to compute the values of the twenty sets of pedestals (Bt, Bb) for the neutron bar T D C s , and to keep track of any shifts in these values that may have occurred over the course of the experiment. This work is described in the following sections. 5It should be noted that the quantities Bt and Bb are not electronic pedestals in the usual sense. Rather, they represent constant offsets that are to be added to the raw TDC readings in order to obtain correctly calibrated values of tn and y. y = V[Ab{Tb + Bb) - At(Tt + Bt)} + y0 (4.13) where V (= ^-) is the vertical conversion factor and y0 is the y-coordinate of the centre of the bar. Chapter 4. Data Analysis I: Equipment Calibration 86 4.2.4 T h e T D C S ing le s S p e c t r a We start by listing the reactions that may be induced by the pion beam within the L D target or target assembly. The ir~d reactions have already been introduced in Sec-tion 1.3.2 (page 15). Of these, the reactions that have a large enough branching ratio to be significant are the first two: 1) 7 T _ + d—*n-\-n + 'y This is the E661 reaction. We expect photons that have an energy distibuted over the range 0 - 131.4594 M e V , and neutrons that have an energy that is distributed over the region 0 - 68.026 M e V (refer to Equations 1.14 and 1.18 in Section 1.3.2). 2) n~ + d —¥ n + n This is the non-radiative pion capture reaction. Since the centre of mass frame for the reaction is also the lab frame, the two neutrons are emitted back to back, each wi th an energy of 68.026 M e V . Since hydrogen is present as an impurity inside the L D target, and is a constituent of some of the components of the target assembly (e.g. the scintillators and mylar windows), we also expect to see reactions due to pion capture on hydrogen: 3) ir~ +p —> 7 r ° + n This is the charge-exchange reaction. A 0.4 M e V neutron and a 2.9 M e V neutral pion are emitted back to back. The 7r° quickly decays by the reaction 7T° —> 27, and, due to the small velocity of the n°, these photons are approximately equal in energy, constituting a box spectrum from 55 to 83 M e V . 4) 7r~ + p —Y n + 7 This is the radiative capture reaction, and produces an 8.87 M e V neutron and a 129.40 M e V photon, back to back. The ratio of yields for reactions (3) and (4) is the Panofsky ratio for hydrogen [174], and is equal to 1 .533±0.021 [175]. The radiative capture reaction can also occur on the proton within a deuterium nucleus, as the quasi-free process n~p(n) —)• 771(71). In this case, the in i t ia l momentum of the proton wi l l not be zero, and consequently the neutron wi l l have an energy spread Chapter 4. Data Analysis I: Equipment Calibration 87 Detected Particle T O F for T D C channel in E661 3.05 m (ns) ( T D C gain is 0.25 ns/ch) Photon 10.17 40.7 68 M e V n from T T d —> nn 28.17 112.7 8.9 M e V n from TT ~P ~^ ln 74.59 298.4 0.4 M e V n from 7r" ~p —> 7r°n 341.0 1364 Table 4.3: T O F s and T D C channels for particles detected by the E661 neutron bars. in the lab frame around 8.87 M e V . The T O F s for the above reaction products to travel the distance between the centre of the L D target and the centre of a neutron bar, in E661, are listed in Table 4.3. Apar t from the reactions listed above, the stopped pions could also induce reactions due to absorption by other nuclei that were present in the target assembly, such as 1 2 C and 2 7 A 1 [176, 177]. Such reactions can result in the emission of neutrons from (7r~,2n) reactions, as well as photons and fast nucleons from the de-excitation of the residual nuclei. Before looking at the neutron bar T O F spectra, we wi l l first look at the T O F spec-t rum for the N a l converter. This is shown in Figure 4.6. Recall that the converter was positioned behind the lead collimator, in the photon arm of the apparatus (see Figure 3.3 on page 56). The leftmost peak in Figure 4.6 represents signals from photons that were incident upon T I N A . Al though the converter was rather inefficient at neutron detection, due to its small thickness and high hardware threshold (about 2.7 MeVee [62]), a signif-icant number (rate) of high energy neutrons did deposit enough energy in the converter to trigger data acquisition. These signals can be seen in the slower "peak" in the F i g -ure 4.6. This neutron "peak" is actually caused by a cut-off that occurs due to the energy threshold in the converter. The histogram in Figure 4.6 was not calibrated; however, this raw spectrum was used Chapter 4. Data Analysis I: Equipment Calibration 88 j I 800 1000 1200 1400 TOF (channels) Figure 4.6: T O F S2 - Converter, Runs 346, 347 and 352 (Run Group 1). extensively in the E661 analysis to distinguish between two classes of events, as shown in the figure. These events were labeled gtof events (gamma T O F in T I N A ) , and ntof events (neutron T O F in T I N A ) . The events resulting from the reaction ir~d —> ynn were a subset of the gtof events. The ntof events, however, were very important for the T D C calibrations, as.will be explained below. The neutron bar T D C singles spectra for Type I triggers for Bar 10 are shown in F i g -ures 4.7 and 4.8. These spectra represent the data taken from the first three production runs (about six hours of data acquisition). The T O F s in these spectra were calculated using Equation 4.11, where the pedestals Bt and Bb were, in the first instance, set to val-ues that were obtained from preliminary on-line calibrations. These T O F s are corrected for differences in the path lengths of the detected particles that result from different hit positions along the neutron bar. This correction could be made due to our knowledge of the y-position of the hit (refer to Section 4.2.5). Thus, all the T O F s in Figures 4.7 and 4.8 represent the time taken to travel the nominal distance of 305 cm. Chapter 4. Data Analysis I: Equipment Calibration 89 500 1000 1500 TOF ( c h a n n e l s ) Figure 4.7: Bar 10 T D C mean time singles spectra for gtof events, R u n Group 1 (corrected for path length). o o o o •(lb)(2b)@ 1 1 1 r-I . , L.J I . . I. L l , 500 1000 TOF ( c h a n n e l s ) „.I.L 1500 Figure 4.8: Bar 10 T D C mean time singles spectra for ntof events, R u n Group 1 (corrected for path length). Chapter 4. Data Analysis I: Equipment Calibration 90 The features of these two spectra are interpreted as follows: Peak la: This is the photon, or prompt, peak. The events in this peak are believed to originate mainly from the decay 7T° —>• 2y. The other photon from the decay is responsible for the T I N A trigger. Peak lb: This peak occurs at the same position ( T O F ) as Peak l a . These photons are believed to originate mainly from de-excitations of the residual nucleus following capture of a negative pion on a light nucleus, such as 1 2 C or 2 7 A l , in the target assembly. A high energy neutron from the (7r~,2n) reaction in the nucleus, triggers T I N A . This hypothesis has not been examined in great detail, however it is at least partly supported by the fact that Peak l b was also present during runs in which the deuterium was removed from the target, but disappeared when the large target assembly was completely removed for the L i H calibration runs. Peak 2a: This gtof peak is displaced by 6.8 ns from Peak l a . These events are too fast to be neutrons. They correspond to the time taken by a photon to travel an extra distance of about 2.04 m through the air (this is approximately twice the distance between the L D target and T I N A ) . We believe that Peak 2a is a secondary (cascade) photon from the T I N A or converter crystals, following shower production in the crystal by a high energy photon from ir~d —>• ynn or from 7r° —> 2y. The primary photon would be responsible for the T I N A trigger. Peak 2b: This small ntof peak is about 11.8 ns later than Peak l a . Its location in the T D C spectrum corresponds to the time taken by a 68 M e V neutron to travel to the N a l converter added to the time taken by a photon to travel from the converter to a neutron bar. The peak is therefore believed to originate from a mechanism similar to that postulated for Peak 2a, except that in this case it is a 68 M e V neutron that is responsible for the T I N A trigger and for the secondary photon that later strikes the neutron bars. Al though we were not able to test our hypotheses for the origins of Peaks Chapter 4. Data Analysis I: Equipment Calibration 91 2a and 2b after the experiment, we believe that the presence of both peaks in the data lends credence to both hypotheses. Our high sensitivity to the cascade photons from the T I N A and converter N a l detectors is attributed to the very low energy thresholds on the neutron bars. Peak 3: This is the 68 M e V neutron from 7r~d —¥ nn. It appears mainly in the ntof spectrum (the bl ip that appears in the gtof spectrum is due to a slight overlap between the gtof and ntof peaks in Figure 4.6). Since this peak originates from a back to back reaction in the L D target, it is only visible in the spectra of Bars 6 to 15, due to the acceptance of the T I N A collimator. This peak wi l l be discussed further below. Peak 4: This is the 8.9 M e V neutron peak from the reaction ix~d = -n~p(ns) —>• jn(ns), where ns refers to the spectator neutron wi thin the deuteron. The energy spread is due to the quasi-free reaction of the pion wi th the proton in the deuteron. Range 5: This corresponds to a neutron energy range of 1-4 M e V , and therefore represents the energy range of the two neutrons from the E661 reaction n~d —> ynn near the photon energy endpoint. The 0.4 M e V neutrons from n~p —> 7r°n did not deposit enough energy to enable them to be visible in Figure 4.7. In spite of the fact that it was only present in ten of the twenty neutron bars, we decided to use the 68 M e V peak (Peak 3) as our time standard. This selection was made for the following three reasons: 1. It was the cleanest peak in the spectrum, with a full width at half max ( F W H M ) of about six channels (1.5 ns) (see Figure 4.9). 2. We knew exactly what the origin of the peak was. 3. The peak originated exclusively within the L D target, and was therefore the most appropriate to use as a calibration peak for E661. Chapter 4. Data Analysis I: Equipment Calibration 92 ^ i U i - i i I i iZ i i I i i i i I 50 100 150 TOF ( c h a n n e l s ) Figure 4.9: Bar 10 T O F : Peaks l b , 2b and 3 (Run Group 1). The time resolution of the T D C s wi l l be discussed in Section 4.2.9. Regarding the fitting of the peak, we tried both a simple Gaussian and a Gaussian/ ta i l fit, and finally opted for a simple Gaussian (and a constant background) fit to the peak since this gave more reliable (and consistent) results with the automated software. In order to obtain a correctly calibrated T D C spectrum, Peak 3 needed to be placed at the correct channel that corresponded to the known time of flight of 28.17 ns (see Table 4.3 on page 87). The gain in al l the T D C spectra was accurately known to be 0.25 ns/ch, since the small deviations of the individual T D C s from this nominal value had already been corrected using the measured gain values in Tables 4.1 and 4.2 (page 80). The required channel for Peak 3 was therefore 112.7 channels. In order to enable calibration of Bars 1 to 5 and 16 to 20 (where Peak 3 was absent) the position of Peak l a relative to Peak 3 was also utilized. The full calibration procedure is described in Section 4.2.6. Chapter 4. Data Analysis I: Equipment Calibration 93 i 1 1 1 1 r v t • ,. .ill m.f« • I - 2 0 0 - 1 0 0 0 100 200 Y — p o s i t i o n ( cm) Figure 4.10: Y-pos i t ion distribution for Bar 10, R u n Group 1. 4.2.5 The Y-Posit ion Distributions In order to carry out the position calibration of the neutron bar, we required a source that i l luminated each neutron bar over its entire length, with an intensity that was approximately constant over the length of the bar. For this purpose we selected the 6 0 C o gamma-tag signals. 6 The Bar 10 y-position spectrum for production runs 346, 347 and 352, calibrated using Equation 4.13 (page 85), is shown in Figure 4.10. The large variation in counts between adjacent channels is a binning effect, and is a l imi t of our resolution. In order to obtain correctly calibrated y-position measurements for this bar, the centroid of the distribution shown in Figure 4.10 would have to coincide with the y-coordinate of the centre of the bar (i.e. X/Q = 13.3 cm, see Appendix B ) . 7 In order to circumvent the binning 6The 6 0Co 7-rays are not, in general, emitted back to back, as is the case for the 0.511 MeV 7-rays from 2 2 Na. 7A small error due to the slight difference in height between the 6 0 Co source and the bar centre was found to be negligible [62]. Chapter 4. Data Analysis I: Equipment Calibration 94 o CO o CO w -)-> a 3 o O xr1 u o C\2 i 1 1 r~ i 1 1 r -- 1 0 0 - 5 0 0 50 TDC t i m e d i f f e r ence ( c h a n n e l s ) Figure 4.11: T d i f f distribution for Bar 10, R u n Group 1. problem of Figure 4.10, we decided to plot the variable T d i f f = (y~^°'>; i.e. T d i f f = [Ab(Tb + Bb) - At(Tt + Bt 100 (4.14) The T d i f f spectrum for Bar 10 is shown in Figure 4.11. The figure shows a Gaussian fit super-imposed on the data. Al though the data in Figure 4.11 were not expected to be purely Gaussian [62] we found that due to the symmetry in the data, a Gaussian fit provided an effective method of determining the centroid YQ of the T d i f f distribution. In a correctly calibrated spectrum, this centroid would be located at channel ^ . 4.2.6 T h e T D C C a l i b r a t i o n P r o c e d u r e The ninety E661 long-flight-path runs were divided into twenty-six groups, and the T D C s (and also the A D C s ) were calibrated separately for each group. This was done in order to keep track of any drifts that occurred in the T D C pedestals, and in the A D C gains and pedestals, over the course of the experiment. The groups were selected in a manner such 3Refer to Equation 4.13. Chapter 4. Data Analysis I: Equipment Calibration 95 that any adjustments or settings that had been made in the equipment or electronics during the experiment, and any long time lapses, occurred between run groups, rather than between runs in the same group. Each run group represented about six to eight hours of production running. W i t h regard to the time calibration, there was a problem due to the fact that the 68 M e V neutron peak, our selected time standard, was only present in the central ten bars (refer to page 91). This was dealt wi th as follows. We noted that the distance (in channels) between the Peak l b and Peak 3 centroids was very consistent for all of Bars 6 to 15. It was therefore possible to calculate a single best (average) position T i for Peak l b , that would place Peak 3 at the required channel of 112.7 in the T O F spectra of these bars. The T O F calibration procedure then involved placing Peak l b at channel T i for each of the twenty bars. The dual constraints on the respective positions of T i and Yc in the T O F and T d i f f spectra, in conjunction with Equations 4.11 and 4.14, were used to calculate the pedestals Bt and Bb for each bar for every run group. The full calibration procedure ran as follows: 1. Replay the data in the run group (from tape), and plot the T O F and T d i f f spectra for each bar i (i = 1 . . . 20), using Equations 4.11 and 4.14, and using the preliminary values B'ti and B'bi that were obtained during the on-line calibrations. 2. For each bar i, carry out a Gaussian fit to Peak l b to find the centroid T'u of the peak. 3. For each of Bars 6 to 15, carry out a Gaussian fit to Peak 3, and find the centroid T'Zi of the peak. Using the results from Step 2 above, calculate the best (average) channel position T i . The equation is Ti = H 2 . 7 - I f ^ - T ^ ) (4.15) i U i=6 Chapter 4. Data Analysis I: Equipment Calibration 96 4. For each bar i, find the centroid Y'Ci of the T d i f f distribution. 5. For each bar i, calculate the adjustments 5Bti and 5Bbi that would have to be made to B'ti and B'bi in order to bring Peak l b to channel T\ whilst simultaneously bringing the T d i f f centroid to channel Yc (note: the value Yc is a constant for al l the bars for the whole experiment). The equations are 5Btl = B'H[ru - 71 - \{YC% - Yc)] (4.16) 5Bbl = B ' ^ - T . + ^ - Y c ) } (4.17) 6. The correct pedestals for Bar i, for this run group, are then given by Bti = B'ti + SBti (4.18) Bbi = B'bl + SBbl (4.19) 7. Repeat Steps 1 to 6 for each of the 26 run groups. Step 1 above was carried out using the D I S P L A Y program, while Steps 2 to 6 were fully automated in separate software. A l l fits were done using the M I N U I T [178] function minimizat ion and error analysis software (the software was used to minimize the x2 °f the fits). The fitting parameters for Peaks l b and 3 were the amplitude, peak centroid, standard deviation, and constant background. The parameters for fitting the T d i f f distri-bution were the vertical gain, peak centroid and standard deviation. Typ ica l M I N U I T fitting errors were ± 0 . 0 5 ch in the Peak l b centroid, ±0 .02 ch in the Peak 3 centroid, and ± 0 . 5 ch in the T d i f F centroid. The uti l izat ion of a single average value of T i for al l the twenty bars caused an error of about ± 0 . 6 ch in the final position of Peak 3. The calibration software incorporated automatic monitoring of the results of each fit, and gave warning messages whenever any of the fitting parameters went outside certain prespecified limits — in these cases the fitting was redone manually. Chapter 4. Data Analysis I: Equipment Calibration 97 T O F Spectrum (1 ch = 0.25 ns): Peak Position Error from F W H M (ch) nominal (ch) (ch) l b 41.1 +0.4 8.1 3 112.0 - 0 . 7 5.5 T d i f f Spectrum: Centroid Position Error from Nominal (ch) (cm) (ch) (cm) 6.3 12.3 - 0 . 5 - 1 . 0 Table 4.4: T D C calibration results for Bar 10, R u n Group 1. 4.2.7 T D C C a l i b r a t i o n R e s u l t s The T D C singles spectra shown on page 89, and the y-position and T d i f f spectra shown on page 93, are in fact fully calibrated spectra, and were produced using the correct final values of the pedestals Bt and Bb for Bar 10 for R u n Group 1. The results from these spectra are tabulated in.Table 4.4. These results are typical. The variations in the terms ^[AtBt + AbBb] and V [AbBb — AtBt] over the course of the experiment, for Bar 10, are plotted in Figures 4.12 and 4.13. Similar data were generated for each of the twenty bars. In all cases the T D C pedestals remained fairly consistent over the course of the experiment. 4.2.8 M e a s u r e m e n t o f t he T i m e W a l k o f t he T D C s Time walk refers to an error in the T D C reading that is caused by a dependence of the reading on the pulse height of the analog input signal. This is a common fault that occurs when leading edge discriminators are used to time analog pulses. The source of the problem is illustrated in Figure 4.14. In Figure 4.14a, when the two pulses are detected, the higher energy pulse (pulse B) would register a faster arrival time in the Chapter 4. Data Analysis I: Equipment Calibration 98 CD „ — _ m "tO -i-> w - 0) C\2 T 3 CM •' &> fe fe O O E-i 0 0 T -4 —'—r ~i 1 1 V 10 15 Run Group 20 25 Figure 4.12: TOF pedestal vs. run group for Bar 10. The T O F Pedestal is the collective term \[AtBt + AbBb] in Equation 4.11. to •+-> o OJ L T 3 OJ fe • X CO w O P H I >--1 1 1 1 r-_ i i i_ 10 15 Run Group 20 25 Figure 4.13: Y-position pedestal vs. run group for Bar 10. The Y-posi t ion pedestal is the collective term V [AbBb — AtBt] in Equation 4.13. Chapter 4. Data Analysis I: Equipment Calibration 99 Pu lse Height ( A D C Reading) (a) (b) Figure 4.14: Illustration of T D C time walk for a leading edge discriminator. Figure (a) shows the variation in discriminator response for two pulses that arrive simultaneously but have different heights. Figure (b) shows the resultant response of the T D C as a function of pulse height (time walk is exaggerated). T D C than would pulse A (i.e. tB < £ 4 ) , even though the pulses both start and reach their maximum height at the same time. For equal-time signals, the general response for the T D C would be a function of the pulse height, as shown in Figure 4.14b. In E661, the time walk problem was controlled by using constant fraction discrimina-tors (CFDs) , as described in Section 3.2.2. The data, however, st i l l needed to be tested for residual time walk effects. This was done by searching for a shift in the position of the 68 M e V neutron peak (Peak 3), in the T D C singles spectra, when different cuts were applied to the energy deposited in the neutron bars. Three energy cuts were used. These were 120-500 keVee, 500-1000 keVee, and 1000-1500 keVee. The test was carried out on the data in R u n Groups 1 and 2. The position of Peak 3 for Bars 6 to 15 shifted by an average of about 0.5 ns over the full energy range (120-1500 keVee) (the amount of shift varied between bars, but was close to this average value in all cases). A similar shift was seen in the position of Peak l b in all twenty bars. A n average correction was calculated for al l twenty bars, and applied to the time calculation in the P R E S O R T program during Chapter 4. Data Analysis I: Equipment Calibration 100 the data analysis of E661. A s wi l l be shown in Chapter 7, the amount of time walk present in E661 was too small to affect the final result for ann. 4.2.9 N e u t r o n B a r T O F a n d Y - P o s i t i o n R e s o l u t i o n s The neutron bar TOF resolution for E661 can be estimated from the F W H M of Peak 3 as listed in Table 4.4 (page 97). This width is about 6 channels (1.5 ns), and includes a contribution of 0.9 ns that is due to the 10 cm thickness of the neutron bar, as well as a contribution of about 0.4 ns due to the 4 cm thickness of the target. The remaining 1.0 ns in the width of this peak (the resolutions are added in quadrature) comes mainly from the intrinsic time resolution of the s c i n t i l l a t o r / P M T / T D C set-up (e.g. time spread due to different path lengths of light traveling along the tube, time spread in P M T , time resolution of the T D C module), as well as from the stopping time distribution of the pions after they cross the scintillator S2. These latter effects would be independent of the energy of the detected neutron, and we therefore define the constant contribution to the T O F resolution by F W H M C 1.0 ns n A . . otc = - « w 0.4 ns (4.20) 2.355 2.355 k ' A t the lower end of the E661 neutron energy range of interest, a 1 M e V neutron requires about 220 ns to traverse the nominal 3.05 m distance from the target to the bar centre, and would therefore require about 3 ns (i.e. 1.4% of the T O F ) to traverse the full width of the L D target and about 7 ns (3.2% of the T O F ) to traverse the 10 cm bar thickness. A t the upper end of the energy range, a 4 M e V neutron requires about 110 ns to travel 3.05 m, and would therefore traverse the bar in about 3.5 ns. The variable contribution to the T O F resolution is therefore given by ^0.0322 + 0.014 2 m ^ 0.035 x T O F n m ^ ° t v = 2 355 X ~ 2 355 ~ X > ° t c ( ^ Chapter 4. Data Analysis I: Equipment Calibration 101 The T O F resolution for the E661 neutrons is therefore dominated by the uncertainty in the depth within the bar at which the neutron energy is deposited, and is given by <rt = yjal + ol « 0.015 x T O F (4.22) The neutron bar time resolution for E661 neutrons therefore ranges from about 1.7 ns (for 4 M e V neutrons) to about 3.3 ns (for 1 M e V neutrons). The y-position resolution of the bars was measured prior to the experiment [62] and was found to vary wi th the hit position along the bar, as well as with the amount of energy deposited in the bar. The resolution oy varied from about 5 cm (bar centre, high energy deposited) to about 10 cm (close to bar ends, low energy deposited). We are also able to derive an independent estimate of the y-position resolution from the T O F resolution, since from Equations 4.11 and 4.13 (page 84) we expect 9 oy = 2Votc « 2 x 7.96 x 0.4 « 6.4 cm (4.23) in good agreement wi th the experimentally measured values. 1 0 4.3 C a l i b r a t i o n o f t he N e u t r o n B a r A D C s The neutron bar A D C calibrations for E661 were carried out by C. Jiang, and a more complete report on the calibration results can be found in her P h . D . thesis [62]. In this section we shall give only a summary of the procedure. 9We assume that atc is dominated by the intrinsic scintillator/PMT/TDC resolution over the pion stopping time distribution in the LD target. This is probably reasonable in view of the short distance between scintillator S2 and the target, and is in any case the conservative estimate. 10We expect the 68 MeV neutrons from jv~d —• nn to give a time resolution that is close to the higher (i.e. better) end of the measured range, since (i) these neutrons hit only the central region of the bars, and (ii) they deposit a higher average energy in the bars. We can work back from the measured y-resolutions and estimate the worst expected value of oic to be about 0.6 ns. Chapter 4. Data Analysis I: Equipment Calibration 102 4.3.1 Introduction The objective of the A D C calibration was to enable us to have as accurate a measurement as possible of the energy that was deposited by the neutrons in the neutron bar. A s indicated in the introduction to Section 4.2.3, a neutron must scatter off a proton in order to be detected in the plastic scintillator. In general, the neutron wi l l transfer only part of its energy to the proton during such a col l i s ion . 1 1 Moreover, a proton that starts off wi th a specific amount of kinetic energy in the scintillator w i l l deposit less light (per unit of energy loss) than would an electron of the same energy. 1 2 We therefore refer to the electron-equivalent energy of the detected proton, i.e. the energy that a fast electron would need to lose in order to deposit as much light in the scintillator as the detected proton. For a neutron that scatters in the centre of the bar (i.e. y' = 0), the electron-equivalent energy of the recoil proton (e t o p) can be obtained from the top A D C reading Et v ia the equation e t o p = Gtx (Et - Pt) (4.24) where Gt is the gain factor that is associated with the top A D C measurement, and Pt is the pedestal value of the top A D C reading. The same energy can be derived from the bottom A D C reading Eb v ia e b o t = Gb x (Eb - Pb) (4.25) where we must have (at least nominally) e t o p = e b o t since both A D C s are measuring the energy of the same proton. The A D C calibration consisted in large part of determining the gains (Gt and Gb) and pedestals (Pt and Pb) for all of the twenty bars, and to keep track of any drifts in 11See, for example, Section 2.1 of Allen [179] for an introduction to the kinematics of the (n,p) and similar reactions. 1 2This is called ionization quenching. See, for example, Section 6.1 of Birks [180]. Chapter 4. Data Analysis I: Equipment Calibration 103 these values over the course of the experiment. It is important to note that the gains are a very strong function of hit position along the bar, since the scintillator light undergoes a considerable amount of attenuation as it travels along the bar. Thus the gains Gt and Gt, are defined for hits that occur at the bar centre. A separate calibration was carried out to determine the corrections that would be needed for hits that occurred away from the bar centre. 4.3.2 A S u m m a r y o f t he F u l l A D C C a l i b r a t i o n P r o c e d u r e The energy calibration of the neutron bars was carried out using the 1.173 M e V and 1.333 M e V photons from the 6 0 C o gamma-tag source. The set-up and electronics for this gamma-tag source are described in Section 3.2.2 (page 60) and Section 4.2.5 (page 93). The photons are detected in the plastic scintillator mainly v ia the kinetic energy that they impart to atomic electrons when they Compton scatter inside the detector (see, for example, [181, Section 4.8] and [182]). The energy of such electrons is of course directly equal to our earlier definition of "electron-equivalent" energy. The full energy calibration for E661 consisted of the following steps: 1. The theoretical energy distribution of the electrons in a plastic scintillator of per-fect resolution, after interaction with a mono-energetic photon beam, was simulated using E G S (Electron-Gamma Shower) [183] Monte Carlo Software. The simulation was carried out separately for photons of 1.173 and 1.333 M e V , and the two spec-tra were later added together, to produce a theoretical energy spectrum for the detector. 2. For each E661 run group, the raw energy spectra from 6 0 C o events for each of the forty A D C s were plotted. The plots were done wi th a y'-position cut between —20 and +20 cm, so that they contained only those events that fell in the central Chapter 4. Data Analysis I: Equipment Calibration 104 40 cm region of the bars. The pedestal spectrum for each A D C was also plotted. This was done by plott ing the A D C readings for events in which the particular bar that was associated with the A D C did not register a hit. The centroid of the pedestal distribution was later used as the nominal pedestal value (for that particular A D C and run group) during the E661 analysis. 3. For each A D C for each run group, the theoretical and experimental energy spectra were fit to each other using the M I N U E T software. The M I N U I T fitting parameters were a Gaussian resolution, the energy binning ( A D C gain), and an amplitude factor, that were imposed on the theoretical spectrum. The theoretical spectrum was also convoluted with the appropriate measured pedestal distribution during the fitting process. The final value for A D C gain that was returned by M I N U I T in the above analysis was the one used in the final E661 analysis. 4: The variation of gain with hit position, for each of the forty A D C s , was deter-mined. This was done using calibration data that had been taken, during the E661 experiment, for this specific purpose. 5. The results from Steps 3 and 4 above were combined, such that the energy deposited in the bar by a detected neutron could be accurately determined from a knowledge of the top and bottom A D C readings and the y-position of the hit. Each of the steps listed above wi l l be described in greater detail in the following sections. 4 .3.3 T h e E G S S i m u l a t i o n E G S is a program that uses Monte Carlo techniques to simulate the transport of elec-trons and photons through various materials and geometries. In our case we needed to simulate the energy deposited by a mono-energetic photon beam that was incident onto Chapter 4. Data Analysis I: Equipment Calibration 105 10 cm 0 Polystyrene Lead • Air Y-ray source Figure 4.15: Geometry for the neutron bar E G S simulation. a specific portion of the plastic scintillator. The geometry that was constructed, in the E G S software, for this simulation is illustrated in Figure 4.15. A round source (1 cm in radius) of mono-energetic (1173 keV) photons was simulated at the centre of the lowest plane, as shown, and i l luminated a lead collimator from below. The E G S code was then allowed to simulate the various interactions of the photons, and of their interaction products, wi thin the defined geometry, and to output a spectrum of the total energy that was deposited in the scintillator. The resultant energy spectrum is shown in Figure 4.16. A similar spectrum was generated for photons of 1333 keV. The E G S spectrum is interpreted as follows. First of al l we refer to Figure 4.17, which shows the relative importance of the three major interactions of photons with matter, for various values of E1 (incident photon energy) and Z (atomic number of the stopping material). The atoms that are present in the plastic scintillator are carbon (Z = 6) and hydrogen (Z = 1), so that for photons of approximately 1 M e V the main mechanism for interaction is Compton scattering. The main Compton edge in Figure 4.16 can be seen Chapter 4. Data Analysis I: Equipment Calibration 106 o X tf o £ r FWHM c tf S o o o o m -1 1 r-Energy 1.173 MeV 1.0% 200 400 600 800 Energy (keV) 1000 1200 1400 Figure 4.16: The Monte Carlo energy spectrum of the electrons in a plastic scintillator, obtained from the E G S simulation (from [62]). N 80 60 40r-20 .01 Photoelectric effect is predominant 0.1 Compton effect is predominant 1 10 100 £ Y (MeV) Figure 4.17: The relative importance of the three major photon interactions wi th matter (from [181]). Chapter 4. Data Analysis I: Equipment Calibration 107 at about 960 keV. The small peak at 151 keV corresponds to the electron energy that results from photon absorption by electron-positron pair production wi th the loss of both annihilation quanta (another at 662 keV wi th a single escape is hard to see), while the peak at 1173 keV results from absorption by the photoelectric effect. The energy ridge between 960 and 1080 keV is due to photons that Compton scatter twice or more. 4.3.4 T h e E x p e r i m e n t a l 6 0 C o E n e r g y S p e c t r a In E661 we were faced with two main problems in the extraction of the neutron bar energy spectra for the 6 0 C o gamma-tag events. The first of these was due to the fact that our 6 0 C o source i l luminated the entire neutron bar array. Since the A D C gains were highly sensitive to the y-position of the hit wi thin the bar, we needed a beam that was collimated onto a small section of each neutron bar . 1 3 This problem was offset by using our knowledge of the y-position of the photon hit (from the already calibrated T D C s ) to make a software cut on the data, and to accept only those photons that scattered in the central portion (—20cm <y'< 20cm) of the bar. The second problem was the low number of 6 0 C o events that were recorded during each E661 run. This problem was exacerbated by the y-position cuts mentioned above. It was for this reason that we decided to combine the E661 runs into groups of three or four for the A D C (and hence also the T D C ) calibrations. A typical A D C calibration spectrum for 6 0 C o triggers, that was so obtained, is shown in Figure 4.18. Al though the number of counts in this spectrum is st i l l rather low, it was considered sufficient to enable us to obtain a reliable fit to the simulated spectrum. 1 3 A small amount of such collimated data was available from dedicated calibration runs. However we preferred to use data that had been taken in a consistent manner throughout the experiment, at the same time that the E661 production data was being taken. Chapter 4. Data Analysis I: Equipment Calibration 108 o u LO 200 400 600 E n e r g y ( c h a n n e l s ) 800 1000 Figure 4.18: The A D C calibration spectrum for A D C lOt (Run Group 2). 4.3.5 The A D C Pedestal Distributions Although A D C pedestals would in theory be only one channel wide (they would repre-sent a constant offset on each of the A D C readings), in practice these pedestals have a distribution of readings due to variations in the voltage supply to the electronics and other transient effects. In E661 the pedestal distributions were obtained by plott ing the A D C readings for those events in which the associated bar did not fire. The pedestal distribution for A D C lOt is shown in Figure 4.19. The centroid of each pedestal distr i-bution was used as the variable Pt (or Pb) in Equations 4.24 and 4.25 (page 102) during the E661 analysis. 4.3.6 Determination of the A D C Gains For each A D C , for each run group, the A D C gains were calculated in the following manner: 1. Start with the simulated E G S spectrum for 1.173 and 1.333 M e V photons, described Chapter 4. Data Analysis I: Equipment Calibration 109 1 1 1 1 1 1 1 1 1 , 1 1 1 1 ; i ; 50 100 150 E n e r g y ( c h a n n e l s ) Figure 4.19: The pedestal distribution for A D C lOt. in Section 4.3.3. 2. Convolute the spectrum wi th a Gaussian resolution function. The standard devia-tion of the Gaussian distribution was the first fitting parameter for M I N U I T , and was of the order of 200 keV. 3. Rebin the spectrum to a new gain (i.e. keV/ch) . This was the second fitting pa-rameter for M I N U I T . 4. Convolute the spectrum wi th the appropriate pedestal spectrum for the A D C . A n example of the resulting spectrum is shown in Figure 4.20. 5. Mul t ip ly the spectrum by an amplitude factor. This was the third fitting parameter for M I N U I T . 6. Calculate the x2 °f the fit between the resulting spectrum and the appropriate experimental spectrum for the A D C (Figure 4.18). Chapter 4. Data Analysis I: Equipment Calibration 110 o .—1 X CD •* O , — i X o u o X o I i i i i L2J i i i i i i i , i i i , i i : — . • I 0 100 200 300 400 500 Channel Figure 4.20: The simulated E G S spectrum after convolution wi th a Gaussian resolution function, rebinning, and convolution with the A D C pedestal (for A D C l i t ) [62]. 7. Use M I N U I T to vary the fitting parameters and minimize the x2 °f the fit. The final value of the gain found from Step 3 above was used as the variable Gt (or Gb) in Equations 4.24 and 4.25 during the final E661 analysis. The drifts in the values of Gt and Pt for Bar 10, over the course of the experiment, are shown in Figures 4.21 and 4.22 respectively. Similar plots were made for each of the forty A D C s , and the gains and pedestals were found to have remained reasonably constant during the experiment in all cases. 4.3.7 L i g h t A t t e n u a t i o n i n t he N e u t r o n B a r s In order to obtain the light attenuation functions for the neutron bars, we used the data from the calibration runs that had been collected with non-prescaled gamma-tag events as the sole triggers for data recording. During these runs, the entire neutron detector was i l luminated by the 6 0 C o and 2 2 N a gamma-ray sources. We used only the data from the 6 0 C o triggers, since there were insufficient 2 2 N a data. Chapter 4. Data Analysis I: Equipment Calibration 111 10 15 R u n G r o u p Figure 4.21: Variat ion of A D C lOt gain wi th time during E661. CTi CO OT cu CD Pu <! CD -1 1 1 1 1 - ~ i 1 r~ 1 ^ - | 1 r -^ ^ ^ ^ ~ e ^ - ^ - © &^-e-^-Q 0 0 Q - Q 9 0 Q 9 O _ J l l _ 10 15 R u n G r o u p 20 25 Figure 4.22: Variat ion of A D C lOt pedestal wi th time during E661. Chapter 4. Data Analysis I: Equipment Calibration 112 o o . Yl=a*exp(-xl(l/b + cxl/200 + d(xl/200)**2)) S '. Y2 = e*exp(-x2(l/b + cx2/200+d(x2/200)**2)) '. a = 270.38116 b = 321.48460 c = 0.00122 •0.00131 50 100 200 Distance From Tl lb (cm) Figure 4.23: Light yield functions for Bar 11 (from [62]). For the light attenuation analysis, these data were replayed using twenty windows on the y-position for each bar, each window covering a 10 cm portion of the bar. The A D C gain for data that fell in each window (for each bar) was computed in a manner similar to that described in Section 4.3.6. The reciprocal of the gain (i.e. the light yield) was then plotted for each A D C as a function of hit position. The data points for the top and bottom A D C s for Bar 11 are shown in Figure 4.23. Each data set was fit to a function of the form where b, c and d are constants for each bar, a is an amplitude, and yb is the distance in cm from the bottom P M T . A geometric mean function was calculated for each bar, and the result for Bar 11 is included in Figure 4.23. For each bar, this mean function was very nearly flat with respect to the hit position in the bar, (4.26) (4.27) Chapter 4. Data Analysis I: Equipment Calibration 113 and this greatly reduced our susceptibility to errors in energy measurement that were due to the hit position along the bar. In the final E661 analysis, the energy that was deposited by the neutron in the bar was obtained using (4.28) where e t o p and e b o t were calculated using Equations 4.24 and 4.25 (page 102), and g(yb) was a correction obtained from Equat ion 4.27, normalized such that g(100) = 1, viz. , } = Y i e l d s ( y 6 ) 9 [ V b ) Y ie ld m e a n (100 ) ( 4 - ' 9 j 4.4 C a l i b r a t i o n o f t h e T I N A A D C s A full account of the A D C calibration of the T I N A and converter crystals is given in [62]. The simulation was carried out using E G S , in a manner similar to that used for the neutron bar A D C s , and using the appropriate geometry for T I N A . The simulated photon source in this case was not mono-energetic, but was sampled, using Monte Carlo techniques, from a distribution that was obtained from the main E661 Monte Carlo (Chapter 6). The photon energy distribution that was sampled is shown in Figure 4.24. The hardware threshold 1 4 on the converter signal was also included in the simulation. The E G S results are shown in Figures 4.25, 4.26, and 4.27. The calibration was completed by comparing the simulated spectra to those obtained experimentally, and forcing the peaks in the experimental spectra to coincide with those obtained in the simulation. The final calibrated experimental spectra are shown in F i g -ures 4.28, 4.29, and 4.30. Dur ing the final E661 analysis, software energy limits of 90 M e V and 140 M e V where placed on the total energy spectrum (Figure 4.30) based 14Refer to Section 3.2.2, page 62. Chapter 4. Data Analysis I: Equipment Calibration 114 E n e r g y ( 1 0 0 k e V / c h ) Figure 4.24: Energy distribution of the E661 photons that were incident on the T I N A collimator, as obtained from the E661 Monte Carlo [63]. Chapter 4. Data Analysis I: Equipment Calibration oo -t-> G o o 1000 1200 1400 E n e r g y (100 k e V / c h ) Figure 4.25: E G S result for 7-ray energy deposition in T I N A [62]. 0 200 ,400 600 800 1000 1200 1400 E n e r g y (100 k e V / c h ) Figure 4.26: E G S result for 7-ray energy deposition in the converter [62]. Chapter 4. Data Analysis I: Equipment Calibration 116 o 0 200 400 600 800 1000 1200 1400 E n e r g y ( 1 0 0 k e V / c h ) Figure 4.27: E G S result for the total 7-ray energy deposition in T I N A and the converter [62]. Chapter 4. Data Analysis I: Equipment Calibration 50 100 Energy (MeV) 150 Figure 4.28: Calibrated energy distribution in T I N A for the E661 data. 50 100 Energy (MeV) 150 Figure 4.29: Calibrated energy distribution in the converter for the E661 data. Chapter 4. Data Analysis I: Equipment Calibration 118 50 100 150 Energy (MeV) Figure 4.30: Calibrated energy distribution in T I N A and the converter for the E661 data. on this calibration, in order to eliminate a portion of the background events. As wi l l be shown in Section 7.2.9 however, our final result for ann turned out to be insensitive to the imposition of these thresholds. 4.5 C a l i b r a t i o n o f t he T I N A W i r e C h a m b e r ( W C 3 ) The purpose of W C 3 was to enable us to obtain more accurate information regarding the trajectory of the E661 photon. It was hoped that this would enable us to tighten the momentum conservation constraints on the reaction products, and would therefore be useful for the elimination of some of the unwanted background. The hardware set-up for W C 3 has been described in Sections 3.2.1 and 3.2.3. The calibration of W C 3 involved an interpretation of the data that had been stored by the P C O S software during the experiment, with the aim of determining the centroid (x,y) coordinates of the photon shower as it crossed the wire chamber planes. After the calibration had been completed, it was discovered that the use of the W C 3 Chapter 4. Data Analysis I: Equipment Calibration 119 information would force us to reject more than a third of our raw (triple-coincidence) data, due to low detection and resolving efficiencies of the chamber. We decided that this was too high a price to pay for the benefits that the W C 3 information would provide. We therefore did not utilize the W C 3 data during the final E661 analysis. 1 5 Instead, we assumed that al l the detected photons had passed through the centre of the collimator, and relaxed our conservation of momentum constraints by a small amount in order to accommodate this assumption. The same procedure was followed in the E661 Monte Carlo simulation. Al though the W C 3 information was not utilized in the final E661 analysis, a brief description of the calibration is given in the remainder of this section. The P C O S information was encoded at the end of the V D A C S data record for every event, and contained the addresses of al l the chamber wires (in W C 1 , W C 2 and W C 3 ) that registered a hit during that event. Every single wire in the six wire chamber planes had a unique address, and the correspondence between the wires and the addresses is annotated in the program P R E S O R T S U S E R . F O R . The use of the W C 3 data presented two main problems. The first of these concerned the relatively large percentage of high energy photons that were detected in T I N A (or in the converter) without firing at least one wire in each of the x- and y- W C 3 planes. The efficiency of W C 3 in this regard, for E661 photons, was about 72%. The second problem involved the interpretation of sometimes ambiguous information from W C 3 (e.g. two or more wires that were physically very far apart would fire during the same event). For each of the x- and y- W C 3 planes, the interpretation of such multi-wire hits was done as follows: 1. F i n d the average position of al l hits that were registered by W C 3 during this event. 15Note that due to the triple-coincidence constraint on the E661 data, the presence of large back-grounds was NOT a major problem in this experiment. Chapter 4. Data Analysis I: Equipment Calibration 120 -200 -100 0 100 200 X—position in TINA collimator (mm) Figure 4.31: Calibrated x-position distribution in W C 3 . 2. Remove any of the above hits that had occurred at a distance that was unreasonably far from this average pos i t ion . 1 6 3. Check that at least one valid hit remains after the eliminations carried out is Step 2. Otherwise reject this event. 4. Recalculate the average position of the remaining valid hits, and accept this value as the coordinate of the photon. The efficiency of the above algorithm, for all E661 events in which at least one wire in each of the x- and y- W C 3 planes had fired, was of approximately 89%. The total efficiency of the W C 3 data acquisition and interpretation package was therefore equal to 64%. Figures 4.31 and 4.32 show the calibrated x- and y- position distributions in W C 3 for valid E661 events in which these coordinates could be successfully calculated. A two-dimensional density plot of these coordinates is shown in Figure 4.33. The number 1 6A distance of 32 mm was chosen, somewhat arbitrarily, as the limit for a "reasonable" distance. Chapter 4. Data Analysis I: Equipment Calibration 121 O O - 2 0 0 - 1 0 0 0 100 200 Y — p o s i t i o n i n TINA c o l l i m a t o r ( m m ) Figure 4.32: Calibrated y-position distribution in W C 3 . S ° B q to G • i—I T J S-O O o I o o CO u n 1 1 1 1 1 1 r~ - 1 0 0 0 100 WC3 x — c o o r d i n a t e ( m m ) Figure 4.33: X Y density plot in W C 3 for E661 events. Chapter 4. Data Analysis I: Equipment Calibration 122 10 Number of Hits 15 Figure 4.34: Number of individual W C 3 x-plane wire hits per event. of individual wire hits per event, for each of the x- and y- W C 3 planes, are shown in Figures 4.34 and 4.35. 4.6 Calibration of the Pion-Tracking W i r e Chambers ( W C l and W C 2 ) The hardware set-up for W C l and W C 2 , as well as the purpose of these wire chambers in E661, are explained in Sections 3.2.1 and 3.2.3. The track that would be left by the pion, as it crossed W C l and W C 2 , would be projected into the target, thereby giving an estimate of the (y, z) coordinates of the pion stop position wi thin the L D target. A n account of the interpretation of the W C l and W C 2 data for E661 is given in [62]. It turned out that the efficiency of the tracking wire chambers was also very low (~ 55%), and we decided not to require W C l and W C 2 information in our final E661 analysis. This was partly justified by the fact that the main E661 Monte Carlo was able to simulate scattering of the outgoing neutrons wi thin the L D target. Chapter 4. Data Analysis I: Equipment Calibration 10 Number of Hits 15 Figure 4.35: Number of individual WC3 y-plane wire hits per event. C h a p t e r 5 D a t a A n a l y s i s I I : R e c o n s t r u c t i o n o f t he P h o t o n E n e r g y S p e c t r u m 5.1 I n t r o d u c t i o n The primary objective of the data analysis of E661 was to produce a calibrated spectrum of the energy distribution of the photons from the reaction ir~d —> ynn free of background contamination. In this chapter we describe the manner in which this spectrum was obtained. The chapter is organized as follows. First , in Section 5.2 we describe the manner in which the E661 data were sorted, and the conditions that the events were required to satisfy in order to be included in the final E1 spectrum. In Section 5.3 we then look at the distributions of various E661 kinematic parameters, and discuss the shapes of these distributions. In Sections 5.4 and 5.5 we discuss the manner in which background events were subtracted from the E1 spectrum. The final experimental E1 spectrum is presented in Section 5.6. 5.2 P r o c e s s i n g a n d S o r t i n g o f t he D a t a In Section 3.5 we gave an introduction to the off-line analysis of the E661 data, and a general flow chart for the analysis was given in Figure 3.8. In this section we shall examine the main components of this flow chart (i.e. the presorting code and the D I S P L A Y ' selection rules) in greater detail. A detailed flow chart of the presorting program is given in Figure 5.1. The actual 124 Chapter 5. Data Analysis II: Reconstruction of the Photon Energy Spectrum 125 Read in the digital data from the 112 CAMAC Signal Sources. Initialize all the relevant variables and set the selection criteria. Do you want to use the wire chamber data? Is this a gtof event? (If not, reject this event). • Obtain a calibrated value of the energy deposited by the photon in TINA and the converter. Check against the TINA energy threshold. If photon fails, reject this event. Calculate the WC1, WC2, WC3 coordinates for this event. If any of these cannot be calculated, reject this event. 1 • Check for a veto signal in the S4 TDCs. If it is present, reject this event. Calculate the direction cosines of the photon. I Did only two bars register a hit that was on-scale on the TDCs? (If not, reject this event). Obtain a calibrated value of the y-position of each hit. Obtain a calibrated value of the TOF of each neutron. Calculate the magnitudes and directions of the neutron momentum vectors. Obtain a calibrated value of the energy deposited by each neutron in the bar. Compare against the predetermined software energy thresholds. If either neutron fails, reject this event. Calculate the reconstructed photon energy. Calculate the x,y,z residual momenta for this event. If all three lie within the prespecified constraints, include this event in the final photon energy spectrum. Figure 5.1: Flow chart for P R E S O R T S U S E R . F O R . Chapter 5. Data Analysis II: Reconstruction of the Photon Energy Spectrum 126 code is called P R E S O R T S U S E R . F O R , and wi l l be referred to often in the remainder of this thesis. The presorting routine was applied to every single event in the E661 data. The flow chart in Figure 5.1 represents the final version of the code that was used in the production of the experimental E661 E1 spectrum. The analyses of the various calibration data were carried out using different variations of this code. Figure 5.1 is meant to be as self-explanatory as possible. The main inputs to the pre-sorting code were the 112 C A M A C signal sources and the P C O S wire chamber addresses (refer to Section 3.3.1), that constituted the relevant data for every recorded event. For every event, P R E S O R T $ U S E R . F O R initialized all variables, and set selection criteria such as thresholds and limits . It then rejected any events that were not gtof events, by-passed the wire chamber analysis, and calculated the calibrated energy deposited in T I N A and the converter. If this energy fell outside the threshold l imits of 90-140 M e V , the event was rejected. The direction cosines of the photon were then calculated. Since the wire chambers were not used in the final calculations, the same direction cosines were applied to the photon for every event. The program then checked for a veto signal from the S4 veto counter A D C s . A window had been set on the S4 T D C s that was consistent wi th the T O F s to S4 of any particle that could originate in the target and be detected in the neutron bars, and a time signal in this window defined an S4 veto. The program then examined the T D C signals from all twenty bars, and only accepted the event i f it found exactly two hits that were on scale in the T D C s . P R E S O R T S U S E R . F O R then calculated the y-position of each of the two hits, and used these results and the A D C readings to get calibrated values of the energy deposited by the neutrons in the bars. If either of these energies fell outside the threshold limits of 120-1500 keVee, the event was rejected. The T O F s of the neutrons were then calculated, and used in conjunction wi th the y-position information to obtain the magnitudes and Chapter 5. Data Analysis II: Reconstruction of the Photon Energy Spectrum 127 directions of the neutron momentum vectors p i and p 2 . The x- and z- coordinates of the neutron hits, needed for these calculations, were taken to be the coordinates of the central longitudinal axes of the bars that were hit. This introduced a certain degree of discretization to the data, that, as wi l l be seen, was st i l l visible in the final E^ spectrum. The program then calculated the reconstructed photon energy using Equations 1.15 and 1.16 (page 16). The fully relativistic calculation of the photon energy from these equations involved the solution of a fourth-order equation in E7 in terms of the variables q and cosf9, i.e. 1 E* + bE* + c £ 2 + dE1 + e = 0 (5.1) where b = 8{Q + 2 m ) / t a n 2 f? (5.2) c = - 4 [ 2 g 2 t a n 2 c ? - 4 m 2 + 5(Q + 2m) 2 ] / tan 2 r9 (5.3) d = 16[(C? + 2m) 3 - 2(<2 + 2 m ) ( m 2 + (7 2)]/tan 2r9 (5.4) e = [ 1 6 g 4 - 4 ( Q + 2m) 4 + 16(m 2 + g 2 ) (Q + 2m) 2 ] / tan 2 c? (5.5) It is worth noting that a non-relativistic solution for E1 can be obtained in terms of q alone, viz. q2 £ 7 = 131.4564 - (MeV) (5.6) 7 1007.5975 K ' K ' where q must be entered in units of M e V / c . In our final analysis, we used the relativistic equation for E1. The derivations for both equations are documented in [62]. It can be shown that the use of Equation 5.1 to calculate E1, rather than the simple use of the conservation of energy Equat ion 1.16 on page 16, greatly increases the accuracy of the calculated E1 value. This is because Equation 5.1 has a lower sensitivity to slight 1 Refer to Figure 1.8 on page 16. 6 is defined here by 6 = tpi + ip2 • Also note that q is defined to be equal to half the relative nn momentum. We have defined q = |q| and m = mn. Chapter 5. Data Analysis II: Reconstruction of the Photon Energy Spectrum 128 errors that may have been present in the measured values of the neutron momentum magnitudes p± and p2-P R E S O R T S U S E R . F O R finally calculated the components of the residual momentum vector p r e s i d for the event, given by Presid = k + P i + p 2 (5.7) where k is the momentum vector of the photon. The calculated value of E7 would only be included in the final spectrum if al l three components of p r e S i d fell wi th in their respective preset l imits. The limits on the residual momentum were set at - 2 0 M e V / c < Preside < 20 M e V / c (5.8) - 2 0 MeV/c < Presidy < 20 M e V / c (5.9) - 5 M e V / c < PresidZ < 5 M e V / c (5.10) The residual momentum distributions are shown in the next section. The resulting Ey spectrum, before background subtraction, is shown in Figure 5.2, wi th a binning of 5 keV/ch . The energy distribution rises to a peak at about 131.37 M e V , and then falls rapidly to zero at the endpoint. The periodic sub-structure in the distri-bution, mostly evident in the region of the peak, is due to our discretization in the x-coordinates of the neutron hits mentioned above. The small peak at the endpoint is due to a particular source of background, and wi l l be discussed in Section 5.5. This spectrum contains 131,000 counts in the top 450 keV region. The resolution, and the statistical and systematic errors, that are associated wi th this spectrum wi l l be discussed in Section 5.6 and in Chapter 7. Chapter 5. Data Analysis II: Reconstruction of the Photon Energy Spectrum 129 1 3 0 . 6 1 3 0 . 8 1 3 1 1 3 1 . 2 1 3 1 . 4 P h o t o n e n e r g y (MeV) Figure 5.2: E661 E7 spectrum, before background subtraction. 5.3 O t h e r E 6 6 1 K i n e m a t i c P a r a m e t e r s A density plot of the T O F of the second neutron hit vs. the T O F of the first neutron hit ( i 2 vs. ti) is shown in Figure 5.3. This plot contains the events that passed al l of the cuts and conditions (with the exception of the residual momentum cuts) described in the previous section. The T O F s in this plot have been corrected for differences in the distance traveled to the bar (based on the y-position calculations), so that they reflect the neutron energies. The crescent shaped cluster in the centre of the plot is the valid E661 reaction region, and represents pairs of neutrons wi th a combined energy of about 4.5 to 5.5 M e V . A s wi l l be shown in Section 5.4, the momentum-conservation constraints served to restrict the accepted events to this crescent region. The extra events on the diagonal (i.e. wi th ti m t2) are mainly a result of bar-to-bar scattering of the same particle, and are discussed in Section 5.5. The remainder of the events in this plot are accidental background events, and these wi l l be examined in detail in Section 5.4. It is important to note that both the diagonal and the accidental backgrounds extend into the E661 Chapter 5. Data Analysis II: Reconstruction of the Photon Energy Spectrum 130 20 40 60 80 TOF of h i t 1 (1 c h = 4 ns) Figure 5.3: E661 t2 vs. t\ density plot (the graph axes are labeled in channels). region, beneath our valid data, and it was therefore very important to understand the nature of these backgrounds. The measured distr ibution of the relative momentum q between the two neutrons is shown in Figure 5.4. This distribution peaks at about 15.3 M e V / c . The high momentum ta i l is background (see page 143). A s can be seen from Equat ion 5.6, there is a one-to-one correspondence between E1 and q (at least in the non-relativistic approximation), and events in the top 450 keV region of the Ey spectrum can be shown to be kinematically restricted to those in which q is less than approximately 21 M e V / c . The magnitude pn of the neutron momentum vector, as calculated in the presort program, has the measured distribution shown in Figure 5.5. The neutron momenta for valid E661 events range from about 45 to 90 M e V / c , representing an energy range 2 of 1.0 M e V <En< 4.3 M e V . The distribution peaks at about 66 M e V / c (En w 2.3 M e V ) . The x-component of the residual momentum, for the E661 events that satisfied the 2The events outside this range are due to background, and they are eliminated when a cut of i? 7 > 131.0 MeV is taken on the spectrum in Figure 5.2. Chapter 5. Data Analysis II: Reconstruction of the Photon Energy Spectrum 131 - i 1 1 1 1 1 1 r~ "T I I I I I J I I I I I I T" o § 2 o 0 10 20 30 40 50 Relative m o m e n t u m of the two neutrons (q) (MeV/c) Figure 5.4: Magnitude of the relative momentum vector q. 50 100 150 Momentum d i s t r ibu t ion of the neutrons (MeV/c) Figure 5.5: Momentum distribution of the neutrons. Chapter 5. Data Analysis II: Reconstruction of the Photon Energy Spectrum 132 o Ln n r-1 o -50 0 50 X - c o m p o n e n t of the r e s i d u a l m o m e n t u m ( M e V / c ) Figure 5.6: Distr ibut ion of the x-component of p, r e s i d ' y- and z- residual momentum cuts, is plotted in Figure 5.6. The dashed lines in this plot represent the limits that were set for acceptance into the E7 spectrum (see previous section). Similar plots for the y- and z-components of the residual momentum are given in Figures 5.7 and 5.8 respectively. The shapes of these distributions are a result of the time and position resolution functions of the neutron detector, the uncertainty in the assumed direction of the photon in T I N A , and the inclusion of background events. The discrete nature of the distribution of the x-component of presid is due to the discrete nature of our x-position measurement in the neutron detector. The x-position distribution (bar number hit pattern) for al l the E661 events, as well as the y-position distribution in Bar 10, are shown in Figures 5.9 and 5.10 respectively. The bias of these distributions towards the centre of the neutron detector is due to the acceptance of the T I N A collimator. Finally, Figure 5.11 shows the distribution of the energy deposited by the neutrons in the neutron bars, before the imposition of the software thresholds, as obtained from Chapter 5. Data Analysis II: Reconstruction of the Photon Energy Spectrum ti 2 -50 0 50 Y—component of the residual momentum (MeV/c) Figure 5.7: Distribution of the y-component of presid. -50 0 50 Z-component of the residual momentum (MeV/c) Figure 5.8: Distribution of the z-component of p r e Sid-Chapter 5. Data Analysis II: Reconstruction of the Photon Energy Spectrum o p i 1 1 1 1 1 1 1 1 1 1 1 1 r-Tf o OT -r-i PI o -1 1 1 1 1 1 1 1 r~=\ 100 - 5 0 0 50 X — p o s i t i o n ( c m ) Figure 5.9: Bar number hit pattern. 100 - 1 0 0 -50 0 50 Y — p o s i t i o n ( cm) 100 Figure 5.10: Y-posi t ion distribution in Bar 10. Chapter 5. Data Analysis II: Reconstruction of the Photon Energy Spectrum 135 o o -C D o o o o o o o 500 1000 1500 2000 Energy deposited in the neut ron bars (keV) Figure 5.11: Energy deposited by the neutrons in the bars. the calibrated A D C readings. Due to the geometry of the apparatus, we were sensitive to ir~d —> ynn neutrons of energies of up to about 1500 keVee (~ 4 M e V ) . Events in the spectrum in Figure 5.11 beyond this value are due to various backgrounds. A t the lower end, the cut-off at about 100 keVee is due to the hardware thresholds of the neutron detectors, and this is discussed further in Section 7.3.1. The E7 spectrum shown in Figure 5.2 was produced with cuts of 120 keVee (low software threshold) and 1500 keVee (high software threshold) on Figure 5.11. 5.4 The Accidental Background A significant effort was made to understand the structure of the accidental background events that can be seen scattered over the entire region of Figure 5.3 (page 130). It was immediately clear that the surface formed by this background was not flat, but rather that it had a definite structure that was correlated to the T O F s of the two hits in the neutron bars. As would be expected, the distr ibution appeared to be symmetrical about Chapter 5. Data Analysis II: Reconstruction of the Photon Energy Spectrum 136 o ^ o ' CD 3 o S o o pq 20 40 60 80 100 Top TDC t i m e (1 c h = 4 ns) Figure 5.12: Bot tom vs. top T D C calibrated times, for Bar 10 single-hit events, the ti = t2 diagonal. It was noted that the narrow ridges at i j « 2 ch and at t\ ~ 4 ch in Figure 5.3, as well as wider ridges at t\ ~ 19 ch and at t\ ~ 34 ch, corresponded to the locations of the peaks in the T D C singles spectra of the neutron bars (refer back to Figure 4.7, page 89). It was therefore decided to try to generate an accidental background from these T D C 7 — n coincidence distributions, using Monte Carlo techniques. The manner in which this was done is explained in the remainder of this section. We first replayed the entire E661 data, and generated a two-dimensional plot of the top T D C calibrated time vs. the bottom T D C calibrated time for the singles events in each of the twenty bars. The plot for Bar 10 is shown in Figure 5.12. We also generated the x-position distribution for single-hit events, and this is shown in Figure 5.13. In this latter histogram, the x-position for every recorded event was assigned a value that was sampled from a random distribution across the width of the bar that had been hit, and this resulted in a continuous x-position distribution, rather than a discrete distribution Chapter 5. Data Analysis II: Reconstruction of the Photon Energy Spectrum 137 o L _ J i U i i i i i i i i i i . i i . i l ' i I - 1 0 0 - 5 0 0 50 100 X — p o s i t i o n ( cm) Figure 5.13: X-posi t ion distribution for single-hit events. similar to the one shown in Figure 5.9. We next defined an "E661 background window" (this actually consisted of two windows) in Figure 5.3, as shown in Figure 5.14. The background generating procedure then consisted of the following main steps: 1. Generate the x-position of the first hit, by sampling the distribution shown in Figure 5.13 using Monte Carlo techniques. Determine which of the twenty bars would be hit. 2. Generate a value for the top T D C calibrated time, by sampling the projection of the data in Figure 5.12 (for the appropriate bar) on the horizontal axis. 3. Generate the bottom T D C calibrated time, by sampling the data in Figure 5.12 (for the appropriate bar) that corresponds to the already established top T D C time. These first three steps establish the relevant parameters for the first hit in this event. 4. Generate the x-position of the second hit, as in Step 1. If the same bar as in Step 1 Chapter 5. Data Analysis II: Reconstruction of the Photon Energy Spectrum 138 20 40 60 80 TOF of h i t 1 (1 c h = 4 ns) Figure 5.14: The E661 accidental background windows, would be hit, reject this event. 5. Generate the mean T O F for the second hit from a uniform distribution. 6. Generate the y-position for the second hit from a uniform distribution. 7. Calculate the top and bottom T D C calibrated times from the results of Steps 5 and 6. The relevant parameters for the second hit in this event are thereby deter-mined. 8. Process the generated top and bottom times, and the bar number information for both hits in a manner that is identical to the way in which it is done in P R E -S O R T S U S E R . F O R . If this event does not get rejected during this process, a value of uE-j" for this event wi l l be calculated, and the event wi l l be included in the simulated (t 2 vs. t\) plot. If the "momentum conservation" constraints are met, the calculated value for E7 w i l l be included in the final simulated Ey accidental background spectrum. Chapter 5. Data Analysis IT. Reconstruction of the Photon Energy Spectrum 139 9. Check to see whether this event would fall in the E661 background window. If it does, increment the Monte Carlo event counter. If this counter exceeds a pre-specified l imi t proceed to Step 10. Otherwise return to Step 1 to generate the next event. 10. Scale the Monte Carlo generated data in the (t2 vs. ti) histogram, in order to obtain the best possible x2 fit to the experimental data within the E661 background window. 11. Output the scaled accidental background E1 spectrum. We called the generator described above our " R - S " model, since one of the hits in each event had been generated from a fiat (random) distribution, while the other had been generated from the T D C singles spectra. We obtained a best-xl fit, between the simulated (after scaling down from 10 6 events using the best possible scale factor) and experimental data in the E661 background window, of 2.91. This was considered unacceptably high. We then modified Steps 5 to 7 in the above procedure such that the T D C calibrated times for the second hit were generated in exactly the same way as those for the first hit (i.e. both hits were generated from the T D C singles spectra). We called this the "S-S" model Monte Carlo, and obtained a best-xl fit of 4.69 to the data. We then proceeded to try different combinations of the above two models, in which a specific percentage of the Monte Carlo events were generated using the R - S method, while the remainder were generated using the S-S method. The best-x2, values that could be achieved for each model are plotted in Figure 5.15. 3 A l l of the Monte Carlo runs that are represented in this plot were run to 10 6 events in the E661 background window before scaling. The best fit that could be obtained to the data was from a model 3We also tried combinations between the R-S method and an "R-R" method, in which both hits were generated from fiat distributions. All such models, however, gave us higher \\ fits to the experimental data. Chapter 5. Data Analysis II: Reconstruction of the Photon Energy Spectrum 140 0.2 0.4 0.6 0.8 Fraction of S—S events Figure 5.15: M i n i m u m xl achievable for various ( R - S , S-S) combination models. in which 67.5% of the events were generated using the R - S method and 32.5% of the events were generated using the S-S method. We called this our "0.325SS" model, and were able to get a best-x2, fit to the data of 1.8.4 We later re-ran the 0.325SS Monte Carlo to 10 7 events in the E661 background win-dow, and were able to improve our xl fit to the data to 1.64. Although this value of xl was sti l l rather high, we decided to accept this as our model for the generation of the E661 accidental background events. 5 The simulated (£ 2 vs. ti) spectrum, before scaling, is shown in Figure 5.16. Figure 5.17 shows sectional views of this histogram, in which the 4This result suggests that about 67.5% of the accidental background events stem from an (accidental) coincidence between a beam-correlated neutron and a beam-uncorrelated particle (neutron or photon) in the M13 area. The remainder come from events where both neutrons are beam correlated. A rough calculation shows that for about 36% of E661 events, at least one of the two neutrons will scatter inside the LD target. This scattered neutron may still be detected by the neutron array, however it will be unlikely to satisfy the momentum conservation constraints. These latter coincidences are probably the main source of our S-S events (both hits would be beam-correlated). A small number of S-S events may also stem from cases where we detect the neutrons from two separate pions (from the same beam bucket) that react in the LD target. 5As will be seen in Section 7.2.10, our final value for ann was in fact less sensitive to the exact shape of the accidental background E7 histogram than we had originally thought that it might be. Chapter 5. Data Analysis II: Reconstruction of the Photon Energy Spectrum 141 o CV! o CO o CD O •4-1 O O C\2 fe o E - 1 20 40 60 80 TOF of h i t 1 (1 c h = 4 ns) Figure 5.16: The simulated accidental background t2 vs. tx spectrum, for the 0.325SS Monte Carlo. generated background data is compared to the actual experimental data. These graphs show a very good agreement between the data and the Monte Car lo simulation for events that lie within the E661 background window. Figure 5.18 shows the scaled accidental background Ey spectrum, while Figure 5.19 is a comparison between the Ey spectrum on page 129 and the background spectrum to be subtracted. From Figure 5.19 it was estimated that only about 5.9% of the events in the top 450 keV of the raw Ey spectrum came from accidental events. Figure 5.20 shows a comparison between the q spectrum on page 131 and the back-ground spectrum to be subtracted. Figure 5.21 shows the simulated accidental back-ground ( £ 2 vs. t{) plot after the residual momentum constraints used in E661 had been applied, i.e. the shaded region in this plot corresponds to the space that would satisfy the momentum conservation constraints of the E661 reaction. The middle portion of this long crescent corresponds to the space occupied by E661 events near the endpoint of the Chapter 5. Data Analysis IT. Reconstruction of the Photon Energy Spectrum 142 TOF of hit 3 (1 ch = 4 ns) TOF of hit 2 (1 ch = 4 ns) Figure 5.17: Sectional views of Figure 5.3 (solid curves) and of Figure 5.16 (dashed curves) after optimal scaling, for various values of Hi t 1 T O F . T i i i i i i i , i i L 130.6 130.8 131 131.2 131.4 Reconst ructed photon energy (MeV) Figure 5.18: Contr ibut ion of accidental background events to the Ey spectrum, as derived from the Monte Carlo simulation. Chapter 5. Data Analysis II: Reconstruction of the Photon Energy Spectrum 143 i • r — i l i l i I ~ -130.6 130.8 131 131.2 131.4 Energy (MeV) Figure 5.19: Comparison between the raw Ey spectrum from Figure 5.2 (solid curve) and the accidental background spectrum (dashed curve). Momentum (MeV/c) Figure 5.20: Comparison between the raw q spectrum from Figure 5.4 (solid curve) and the accidental background spectrum (dashed curve). Chapter 5. Data Analysis II: Reconstruction of the Photon Energy Spectrum 144 o 00 2 «M O O C\2 o fc o 20 40 60 80 TOF of h i t 1 (1 c h = 4 ns) Figure 5.21: The region of the t2 vs. t\ spectrum that satisfies the E661 momentum constraints (from the Monte Carlo simulation). photon energy spectrum (q < ~ 25 M e V / c ) . 6 5.5 The Diagonal Background Figure 5.22 shows a profile of the data that lies on the diagonal in Figure 5.3 (page 130) (i.e. the manner in which the height of this diagonal ridge varies along its length). The high peak near channel 35 (140 ns) is the central part of the crescent in Figure 5.3, and contains a portion of the valid E661 data. The features of the diagonal background are a somewhat constant (i.e. time-uncorrelated) background that can be seen to the right of the data, and a time-correlated background that peaks to the left of the data, at about 24 ch (96 ns). The ripple that can be seen in the constant background has a periodicity of 43 ns, and corresponds to the duty cycle of the T R I U M F cyclotron. It should be 6The two small unphysical regions in the lower left hand portion of this density plot represent events outside the crescent that also accidentally satisfy the E661 residual momentum constraints. These events, however, return values for "J37" below 131 MeV, and therefore do not affect our analysis. Chapter 5. Data Analysis II: Reconstruction of the Photon Energy Spectrum 145 T P r o f i l e of t he E661 d i a g o n a l b a c k g r o u n d (ch) Figure 5.22: The profile of the diagonal background. stressed that these backgrounds are only present in a small portion of the E661 region of interest, namely where the crescent crosses the diagonal in Figure 5.3. Figure 5.23 was produced from a replay of the data in which the events with adjacent bar hits were rejected. The profile of the diagonal region of this histogram is shown in Figure 5.24. This graph contains only about 27% of the events in the diagonal region outside the E661 region in comparison to before the cut (see Figure 5.22). This indicated that most of the diagonal background came from events in which the same particle had scattered between adjacent bars, triggering both bars in the process; presumably a proportion of the few events that remain on the diagonal in Figure 5.24 would have come from data in which the same particle has triggered two bars that were not adjacent. The diagonal background was investigated quite extensively by Doyle [63] and Jiang [62]. From a simulation using the main E661 Monte Carlo [63], the time-correlated diagonal background was found to be due to bar-to-bar scattering of single, higher energy (> 4 M e V ) neutrons that originated from the E661 reaction. The Monte Carlo spectrum Chapter 5. Data Analysis II: Reconstruction of the Photon Energy Spectrum 146 •4-1 O O CM fe o E-•1 , L i I i I , 20 40 60 80 TOF of h i t 1 (1 c h = 4 ns) Figure 5.23: The t2 vs. t\ spectrum after subtraction of al l adjacent bar hit events. Compare to Figure 5.3, page 130. 40 60 80 P r o f i l e of t he d i a g o n a l b a c k g r o u n d (ch) Figure 5.24: The profile of the diagonal background, after subtraction of al l adjacent bar hit events (compare to Figure 5.22). Chapter 5. Data Analysis II: Reconstruction of the Photon Energy Spectrum 147 I , I , , I , I , , I , , I , I I 0 50 100 150 200 250 300 350 400 timel vs. time2 (ns) Figure 5.25: A simulation using the main E661 Monte Carlo that shows the high energy neutron bar-to-bar scattering events [63]. is illustrated in Figure 5.25. Since this background was reproducible in the main Monte Carlo, we did not need to subtract it from the experimental spectrum. W i t h regards to the constant diagonal background, it was postulated that this could be due to the bar-to-bar scattering of either (i) high energy neutrons, or (ii) low energy neutrons, or (in) low energy photons that were present in the experimental area during the experiment. Each of these three possibilities was investigated by simulating the scattering using the main E661 Monte Carlo, and comparing the energies deposited in the bars in the simulation to the actual energies that were deposited in the bars by the constant diagonal events during the experiment. The experimental energy distribution for these events exhibited a sharp drop over the region 0.2 - 4.0 M e V , and this test ruled out the high energy neutrons as being the source of this background. A second test was 3 0 0 2 b 0 203 150 100 Chapter 5. Data Analysis II: Reconstruction of the Photon Energy Spectrum 148 =tfc 00 O O O 20 40 60 80 100 120 T i m e - o f - f l i g h t for hi t #1 (4 n s / c h ) Figure 5.26: The low energy photon diagonal background events, after transposition into the E661 region of interest (from [62]). carried out by investigating the relative T O F s (|£i — t 2 | ) registered by the two bars in these bar-to-bar scattering events. This was done, once again, by comparing Monte Carlo simulations to the actual data. From the (narrow) width of the experimental | t i — t2\ distribution it was concluded that the constant diagonal background events were due to the bar-to-bar scattering of low energy photons. The events in the upper portion (ii = £ 2 > 40 ch) of the constant diagonal back-ground in Figure 5.3 were transposed to the E661 kinematic region of interest, as shown in Figure 5.26. The contribution of these diagonal background events to the Ey spec-trum, after the residual momentum cuts had been applied, and after subtraction of the additional accidental background that has been described in Section 5.4, is plotted in Figure 5.27. A s can be seen, these events only affected the spectrum in the region very close to the endpoint. In our final extraction of ann we therefore ignored the top 25 keV Chapter 5. Data Analysis II: Reconstruction of the Photon Energy Spectrum 149 130.5 131 131.5 R e c o n s t r u c t e d p h o t o n e n e r g y (MeV) Figure 5.27: Contribution of diagonal background events to the E7 spectrum [62]. of the E1 spectrum 7 , so the low energy photon diagonal background is not considered to have contributed to any errors in our final result for ann. 5.6 The Experimental Photon Energy Spectrum The final experimental photon energy spectrum that was extracted from E661 is pre-sented in Figure 5.28. This spectrum was obtained by subtracting the general simulated background (Figure 5.18) and the estimated photon scattering effect (Figure 5.27) from Figure 5.2. The spectrum peaks at E~, ~ 131.37 M e V , and rises by a factor of 1.89 between 131.10 and 131.37 M e V . It contains 123,000 events in the top 450 keV region. Energy Resolution The instrumental resolution, OE, of the reconstructed photon energy in Figure 5.28 can be estimated from the uncertainties in the measured quantities x, y, z, and t for the two 7This was simply a precautionary measure. The data in Figure 5.27 were subtracted from our experimental photon energy spectrum. Chapter 5. Data Analysis II: Reconstruction of the Photon Energy Spectrum 150 130.6 130.8 131 131.2 131.4 E661 e x p e r i m e n t a l p h o t o n e n e r g y s p e c t r u m (MeV) Figure 5.28: The final experimental photon energy spectrum as extracted from E661. neutron hits in the neutron detector. As we have seen in Section 4.2.9, the time and position resolutions of the neutrons are not fixed, but depend on the particular values of the neutron T O F s and hit positions for each particular event. In general, therefore, the uncertainty OE varies from event to event. The distribution of the uncertainties OE for the events in the top 450 keV of our experimental spectrum is shown in Figure 5.29. The mean resolution in the photon energy is therefore about 17 keV (40 keV F W H M ) . The derivation of O~E is outlined in Appendix F . Chapter 5. Data Analysis II: Reconstruction of the Photon Energy Spectrum 151 Figure 5.29: The distribution of the estimated uncertainty OE of the calculated photon energy, for al l events in the top 450 keV of the photon energy spectrum. Chapter 6 The Theoretical Models 6.1 Introduction The objective of this chapter is to briefly describe the major theoretical models that were used to analyze the E661 data, as well as the manner in which the simulated photon energy spectra were generated. The chapter is organized as follows: In Section 6.2 we give a short history of the development of phenomenological and hybrid N N potential models (already introduced in Section 2.2.4, page 37), leading to a description of the Argonne vis potential, which is the one that was selected for use in E661. Then in Section 6.3 we describe the manner in which the Tv~d —> 7 7 m reaction was modeled in E661, in order to produce the theoretical distributions of the n-n relative momentum vector q for various t r ia l values of ann. In Section 6.4 we describe the Monte Carlo model that was used to simulate the E661 geometry and the detector resolutions and acceptances, and to produce the theoretical photon energy spectra based on the theoretical distributions of q. 6.2 Phenomenological N N Potential Models 6.2.1 General Derivations The first step in the construction of a phenomenological nuclear N N potential involves wri t ing down the most general analytical form of the potential function, in terms of the nucleon attributes (positions: rl7 r2; momenta: pi, p2] spins: <Xi, <T 2 ; and isospins: T x , r 2 ) . Ear ly derivations were made in [104, 184, 185, 186], and they yielded similar, 152 Chapter 6. The Theoretical Models 153 although not identical, results. In general, the vectors listed above could be combined in quite a large number of ways, yielding a potential function that would consequently contain a large number of terms. The imposit ion of some necessary conditions, however, served to rule out certain terms from the onset. These conditions are: 1. The potential can depend only on the relative position vector r = ri—r2 (translation invariance) in order to ensure the conservation of momentum. 2. The potential can depend only on the relative momentum vector p = p i — p 2 (Galilean invariance), in order to ensure independence of the potential on the ve-locity of the frame of reference. 3. The potential must be invariant under space rotations, in order to ensure the con-servation of angular momentum. 4. The potential must be invariant under space reflection, in order to ensure the conservation of parity. 5. The potential must be invariant under time reversal, in order to ensure the conser-vation of time-parity. 6. The potential must be invariant under the interchange of the two particles, in order to ensure the conservation of statistics. The early derivations assumed charge independence of the nuclear force, and for this reason, as indicated in Section 2.3.3, the isospin operators could appear only in the dot product T\ • r2.1 The result from [104], that satisfies al l of the above conditions, is a nucleon-nucleon potential of the form 1 Note that combinations such as T\ X r or T2 • p are invalid since r is not a vector in Cartesian space. Chapter 6. The Theoretical Models 154 /3o"i • r o~2 • f \ V = VCL + l / C S 0 - i • cr2 + VT I — CTX . o-2 ) + VLS (o-i + cr2) • r X p + Ks2 [(o-i + tr2) • r X p ] 2 (6.1) where V c l is the spin independent central potential, VCS is the spin dependent central po-tential, VT is the tensor potential, V L S is the spin-orbit potential, and V L S 2 is the quadratic spin-orbit potential. The term that is multiplied by VT is the tensor operator Si2 already introduced on page 33. Each of the V's can be of the form V + WT\ • r 2 , so that there are ten terms in al l . These coefficients V are normally taken to be functions of r 2 , al-though they may sometimes also be taken to be functions of p2 and L2 (see [104] and Section 6.2.3 below). The construction of a simple phenomenological potential would be completed by fitting Equation 6.1, v ia phase shift analysis, to the data, and obtaining the best values for the coefficients V. A comparison between Equation 6.1 above and Equation 2.6 on page 33 w i l l show that the correct partit ioning of VCS and VT (into "long-range" and "short-range" portions) can reproduce the one pion exchange potential as part of the overall N N potential. In fact most phenomenological potentials include the O P E P as part of the overall model. 6.2.2 N o t e s o n N N P o t e n t i a l s A large number of attempts have been made, since the early sixties, to develop a N N potential that can reproduce al l of the p-p and n-p scattering data, as well as the low energy n-n scattering parameters, the deuteron properties, and (with considerably less success) the three-nucleon binding energies. The earliest models that achieved a certain degree of success were the Hamada-Johnston [187], Yale [188]2, and Reid [189] potentials. The Reid potential, in particular, became the potential of choice during the 1970s, as 2Refer also to reference 16 in [189]. Chapter 6. The Theoretical Models 155 a useful and effective model of the N N interaction. It was constructed by carrying out separate fits for each N N partial wave, using a different V = Vc(r) (6.2) for each uncoupled state (states wi th J — L and the 3 P 0 state) and V = Vc{r) + VT(r)S12 + VLS(r)L • S (6.3) for each coupled state. 3 There were two versions of the Reid potential: the hard core version used an infinitely hard short range repulsion, while the "soft" core version used a Yukawa short range repulsion. The Reid , as well as the Hamada-Johnston and Yale potentials, gave reasonably good fits to the p-p and n-p scattering data that existed during the time of their development. During the 1970s and 1980s many N N potential models were developed that were mainly phenomenological. (See, for example, the Mongan [191]; Graz [192] and Graz II [193]; Urbana [194], and Argonne and « 2 8 [195] potentials; as well as the Moscow [196] potential which is a partly phenomenological potential based on a six-quark model with one-pion exchange.) A t the same time, N N potentials that were derived from pure meson theory 4 , such as the O B E P s (including the original version of the Nijmegen potential [90]), the original version [197] of the Paris potential, and the Bonn Potential [94, 198] (refer to Sections 2.2.2 and 2.2.3), were being developed. In 1975 the Paris group published a semi-phenomenological potential [199] that used a slightly updated version of their original model ( O B E P , T P E P , and u;-exchange) to describe the N N interaction for r > « 0.8 fm, added to a phenomenological description (for r < « 0.8 fm) that contained six free parameters (central, spin-spin, spin-orbit, tensor, quadratic spin-orbit, 3Coupled states are states with the same S and J, but different L. There can be mixing between these states, e.g. the 3 S i and 3 D i states, as measured for the deuteron (see Section 1.2.2). In the original versions of the Reid potential, the parametrizations were only carried out on states with J < 2. Parametrizations for higher J can be found in [190]. 4 Albeit with the phenomenological fitting of a few coupling constants. Chapter 6. The Theoretical Models 156 and velocity-dependent central) for each of the isospin states 1 = 0 and 1 = 1 that were fit to the data. The potential was later parametrized into a simpler analytical form [92].5 During the late 1980s and early 1990s the Nijmegen group carried out a partial wave analysis of al l the published p-p and n-p scattering data below T l a b = 350 M e V [201, 202, 203], and incorporated new phenomenological parametrizations in the Nijmegen potential that resulted in a x2 P e r degree of freedom fit to the data of 1.08 (the Nijm93 potential). This spurred the development of the new "high precision" N N potentials that are described in the next section. 6.2.3 T h e N e w " H i g h - P r e c i s i o n " N N P o t e n t i a l s The N N potential models that were discussed in the previous section typically contained 10-15 free parameters that were available for fitting to the data, and in most cases gave a X 2 / d a t u m that was considerably larger than l . 6 The mid 1990s have seen the emergence of new N N potentials that employ about 50 parameters to fit the data, and that have a X 2 / d a t u m fit to the world bank [206] of p-p and n-p data that is approximately equal to the "perfect" value 7 of 1.0. These potentials are the Nijml, Nijmll and Reid93 models [207]; the Argonne u 1 8 model [61]; and the CD-Bonn O B E P [208]. Their properties are tabulated in Table 6 .1 . 8 , 9 5This is the version of the Paris potential that is commonly quoted in the literature. Refer also to [200] for a separable representation of this potential. 6Typically x2/datum = 2 ~ 20. For discussions regarding the proper ways in which to obtain these X2 fits and quantitative comparisons between the various NN potentials refer to [204, 205]. 7Although the number of fitted parameters is quite large, the number of data is in the thousands, so that this is in fact quite a significant achievement. 8A non-local potential is one in which the potential term in the Schrodinger equation is not just a function of the point under consideration, but involves integration over the whole space [104, Section 2.3]. A momentum-dependent potential will give rise to such a structure in configuration space. For treatments of such potentials see [209, 210]. 9The Nijml potential is an updated version of the partial wave analysis carried out in [203], in which the fitted parameters were adjusted separately in each partial wave to optimize the x2 fit to the data. Nijmll was derived in a similar manner, except that the momentum-dependent terms were omitted, giv-ing a potential that has a local structure in configuration space, while Reid93 uses parametrizations that Chapter 6. The Theoretical Models 157 Model No. of Parameters fit X 2 / d a t u m Remarks N i j m l 41 1.03 non-local N i j m l l 47 1.03 local Reid93 50 1.03 local Argonne vX8 40 1.09 local C D - B o n n 45 1.04 non-local Table 6.1: Comparison between the new N N potentials. A s a general comment regarding phenomenological N N potentials, it is important to point out that the significance of these models is not so much in their fundamental physics content 1 0 as it is in their applicabili ty to the analysis of problems in which an accurate analytical description of the N N interaction is required. Our analysis of the E661 data is an example of such an application. 6.2.4 T h e A r g o n n e v 1 8 P o t e n t i a l Since the Argonne vx8 potential is the one that was used in the analysis of the E661 data, we give here a summary of its general structure. Fu l l details can be found in [61]. In this model, the nucleon-nucleon potential is written down as v(NN) = vEM(NN) + vn(NN) + vR(NN) (6.4) where vEM(NN) is an electromagnetic part, v77(NN) is a O P E part, and vR(NN) is an intermediate- and short-range phenomenological part. The E M interaction for p-p are similar to those used in the original Reid potential. The Argonne vis potential contains an updated version of the v\4 potential [195], together with additional charge dependent and charge asymmetric operators and a complete electromagnetic interaction. The CD-Bonn potential is a relativistic OBEP in which extra fitting parameters were created by adjusting the rj-boson individually in each partial wave. 1 0For example, one shortcoming of these models, in spite of their accuracy in reproducing the exper-imental phase shifts, is that they fix the potential only on-shell. For a discussion on the derivation of off-shell amplitudes from phenomenology, refer to [211]. Among the many NN potentials that give good fits to the data, the full Bonn model of 1987 perhaps retains the distinction of being the least reliant on phenomenology in its derivation. Chapter 6. The Theoretical Models 158 scattering includes one- and two-photon Coulomb terms, the Darwin-Foldy term, and vacuum polarization and magnetic moment interactions. For n-p scattering the E M interaction includes only the one-photon and magnetic moment interactions, while for n-n scattering it includes only the magnetic moment interaction. The O P E part of the potential is constructed in the usual manner, with the irNN coupling constants taken to be equal for all the three cases (f2 = 0.075, as recommended in [203], and as has subsequently been accepted by near consensus in [212]), so that the only charge dependence in this part of the potential arises from the mass difference between the charged and the neutral pions. The phenomenological part of the potential is expressed as v§T(NN) = v^NN{r)+vllNN{r)L2+ v^TtNN(r)S12 +vLslNN(r)L • S + vLs^NN(r)(L • S ) 2 (6.5) where the subscripts ST, NN refer to the spin state, isospin state, and nucleons under consideration, and where each term is given the general form 4 r , jv ;v( r ) = IST,NNT2(r) + [PST,NN + ^QST,NN + (vr)2RST,NNW(r), (6.6) where / i is the average pion mass, T^(r) is the tensor function wi th an experimental cut-off, and W(r) is a Woods-Saxon function that provides the short-range core. The four sets of constants PST,NNI ^ST,NN^ QST,NN-> a n d TLLSTNN are, bar some minor constraints, the free parameters that are fit to the data. In particular, the parameter P^X NN is set to be slightly different from PQ{VP in order to reproduce the singlet n-n scattering length as determined by de Teramond and Gabioud [53]. Chapter 6. The Theoretical Models 159 6.3 T h e R e a c t i o n M o d e l for E 6 6 1 The main reaction model that was used to analyze the E661 data was developed by Doyle and Lee [214]. Here we briefly summarize its main features. The reaction model is illustrated in Figure 6.1. The transition matrix Tfi is given by (cimslms2\r\cfm*slrn*2) Mjdm>slm's2m*sl„ «53(q*-q) + a*2 , • - h it x<¥Mjd(K m* 1 „m* 2 )(kApim* 1 |J|p ' 1 'm* 1„u) (6.7) In the above equation, ^ is the deuteron wave-function, with K the relative p-n momen-tum in the deuteron, and Mjd the spin state of the deuteron; q* and q are respectively the off-shell and on-shell relative n-n momentum vectors; m*sl„ is the magnetic quantum number of the proton in the deuteron, and p'-[ is the momentum of this proton, wi th P i = Q* + | ~ u ; J refers to the electromagnetic (7, IT) operator, due to Nozawa and Lee [213]; A is the polarization of the photon; m*sl and p x refer to the off-shell neutron that is outgoing from the (7,7r) vertex, wi th P i = q* — |; m*s2 refers to the neutron in the deuteron before the FSI; and m s 2 refers to this neutron after FSI . In Figure 6.1 Chapter 6. The Theoretical Models 160 ann (fm) Polnn (MeV) ann (fm) P^nn (MeV) -17.02 3346.69 -21 .04 3336.95 -18.00 3344.61 -21 .40 3336.18 -18.42 3343.40 -21 .59 3335.77 -18.54 3342.34 -21 .80 3335.36 -18.76 3341.74 -22 .00 3334.95 -18.84 3341.27 -22.19 3334.57 -19.16 3340.69 -22.54 3333.89 -19 .40 3340.08 -22 .90 3333.23 -19.64 3339.48 -23.28 3332.53 -20.18 3338.88 Table 6.2: Values of P 0i,nn a n d the corresponding values of a^n in a model based on the Argonne Vis potential [63]. P 2 = -q* - |. The tw operator describes the n-n final state interaction, and was derived from a Fourier transform of the Argonne vis potential into momentum space, using standard techniques. 1 1 The value of Po{nn m the Argonne potential (refer to Section 6.2.4) was varied to represent the FSI for various values of ann: for each t r ia l value of Po{nn, the "correct" corresponding values of ann and rnn were obtained by calculating the theoretical n-n phase shift as a function of q, plott ing the resulting theoretical curve of qcot50 vs. q2 (refer to Equation 1.7, page 7), and fitting the low-g part of this curve to a straight line. This correspondence is given in Table 6.2. 1 2 The S3(q* — q) term in Equation 6.7 represents the quasi-free process (i.e. no n-n FSI) . The cross-section of the ir~~d —> ynn reaction as a function of q was described by da rn2lq2k £ / dnq \TSl\2 (6.8) dq 127r8r?v mslms2 The theoretical E7 spectra were obtained by sampling the ^ distribution for q < i i Refer, for example, to [215]. 1 2 The ann values in Table 6.2 include E M effects. Chapter 6. The Theoretical Models 161 25 M e V / c and by sampling the distribution wi th fixed q. 6.4 The Monte Carlo M o d e l for E661 The entire E661 geometry, and the detector response to the final state particles from the E661 reaction, were modeled using Monte Carlo techniques and using the G E A N T [216] software package. This work was carried out by B . Doyle [63] at the University of Kentucky. A brief summary of this simulation is given below: Firstly, the stopping distribution of the pions inside the L D target was obtained by simulating the degradation of the 40.5 M e V pion beam between the beam window and the target, taking into account the various detectors and the target equipment that lay along the beam path. This work was done early on in the development of the E661 simulation, and the resulting pion-stop distribution was used in al l subsequent E661 simulation runs. The distributions of the x- and z-coordinates of the pion stop positions within the target are shown in Figures 6.2 and 6.3. 1 3 The E661 geometry was also constructed, wi thin the G E A N T software, early on in the analysis, and used throughout the subsequent simulations. A typical E661 simulation run (using a particular set of q and cos(6) distributions, as explained in Section 6.3, wi th q and 6 defined on page 127) involved the main steps given below for each generated event: 1. Select a reaction vertex position wi thin the L D target by sampling the pion-stop distribution. 2. Sample the q and cos(9) curves to determine the values of these kinematic param-eters for this event. 13The y-position distribution is similar to the 2-position distribution. Chapter 6. The Theoretical Models 162 - 2 0 2 X — p o s i t i o n i n t a r g e t ( c m ) Figure 6.2: Simulated x-position distribution of the pion stops in the L D target [63]. - 4 - 2 0 2 4 Z—pos i t ion i n t a r g e t ( c m ) Figure 6.3: Simulated z-position distribution of the pion stops in the L D target [63]. Chapter 6. The Theoretical Models 163 3. Generate the photon direction for the 7r d —> jnn reaction, by sampling the solid angle between the reaction vertex position and the T I N A collimator. 4. Generate the azimuthal angle of the reaction plane. Steps 1 to 4 completely deter-mine the kinematics for the E661 reaction in this event. 5. Convert the neutron momentum vectors from the local coordinate system (in which the origin is the reaction vertex, and the z-direction is the photon trajectory) to the master coordinate system that was used for the G E A N T simulation. 6. Al low G E A N T to track the neutrons through the simulated geometry. If two neu-tron hits are registered by the neutron bar array (in different bars), proceed to Step 7; otherwise return to Step 1 to generate another event. 7. Record the energies that are deposited by the two neutrons in the bars (these values are returned by the internal G E A N T simulation software). Calculate the energy at the P M T s using Equat ion 4.26 (page 112) and compare to the appropriate P M T hardware thresholds (Table 7.2, page 185) to determine whether the P M T s would fire. If all four of the relevant P M T s fire, the event is allowed to proceed to Step 8. 8. Smear the energy deposited, y-position, and neutron T O F values by sampling Gaus-sian functions, using o values that were previously obtained from dedicated tests on the neutron bars at the University of Kentucky. 1 4 9. Analyze this event in a manner that is identical to the way in which the real experimental data is analyzed (i.e. using routines that are similar to those used in P R E S O R T S U S E R . F O R ) . "Refer to Sections 4.2.9 and 7.3.1. Chapter 6. The Theoretical Models 164 TC o X CO Tf o o o X Tf o 131.1 131.2 131.3 131.4 Photon energy (MeV) Figure 6.4: The E1 spectrum from the E661 Monte Carlo simulation for ann = —18.5 fm. This simulation was run to 15 x 10 6 events. Note the periodic sub-structure in this distribution that is caused by the discretization in the x-coordinates of the neutron hits, as in the experimental spectrum of Figure 5.28 (page 150). 10. Unless a preset number of events has already been generated, return to Step 1 to generate a new event. A typical simulation run was allowed to generate 8 x 10 6 events. The theoretical E1 spectrum for ann = —18.5 fm (including electromagnetic effects) is shown in Figure 6.4. C h a p t e r 7 R e s u l t s a n d D i s c u s s i o n 7.1 F i t s t o t h e P r e l i m i n a r y M o n t e C a r l o S p e c t r a The min imum chi-squared values achievable for fits between the experimental spectrum and the Monte Carlo generated spectra for various values of ann are given in Table 7.1. The Monte Carlo spectra were run to about eight times the number of statistics in the experimental spectrum. The best fit was obtained for the Monte Carlo generated curve for a^n = —21.8 fm (ann ~ —22.1 fm), with a chi-squared of 147.0 for 86 data points (bins) 1 , giving a chi-squared per degree of freedom of xl — 1-73. The fit is shown in Figure 7.1. For values of ann outside the range given in Table 7.1, the xl °f the fits became progressively higher. In particular, a comparison of our experimental spectrum to the Monte Carlo spectrum that was derived from the presently accepted value of ann = — 18.5 fm gave a best xl value of 5.20. This fit is shown in Figure 7.2. The Monte Carlo spectra are st i l l preliminary, and wi l l be evaluated in Section 7.3. The primary focus of this chapter w i l l be Section 7.2 in which we show that, to our best knowledge, there are no systematic effects in the experimental spectrum that are greater than the order of ± 0 . 2 fm. Later in Section 7.4 we wi l l compare the E661 experiment to a number of previous experiments that were carried out to measure ann. In Section 7.5 we summarize the work that remains to be done before a final result for the scattering 1The last five channels near the endpoint were not fitted for reasons given in Section 5.5, although they are included in the experimental data that are presented throughout this chapter. 165 Chapter 7. Results and Discussion 166 Value of ann used to generate theoretical curve X2 of fit to data Xl of fit to data -21 .0 fm 168.3 1.98 -21 .4 fm 169.2 1.99 -21 .8 fm 147.0 1.73 -22 .2 fm 182.8 2.15 -22 .5 fm 201.5 2.37 -22 .9 fm 191.3 2.25 -23 .3 fm 215.9 2.54 Table 7.1: M i n i m u m xl achievable for various theoretically generated spectra. (The a. values include E M effects.) j I i i i i I i i i i I i i i i I i_ 131.1 131.2 131.3 131.4 Energy (MeV) Figure 7.1: The E661 experimental E1 spectrum (solid curve) and the ann = —21.8 fm Monte Carlo spectrum (dashed curve). The Monte Carlo spectrum was scaled for best fit. Chapter 7. Results and Discussion 167 J i i i i i i i i i i i i i i i i i_ 131.1 131.2 131.3 131.4 Energy (MeV) Figure 7.2: The E661 experimental Ey spectrum (solid curve) and the ann = —18.5 fm Monte Carlo spectrum (dashed curve). The Monte Carlo spectrum was scaled for best fit. length can be achieved, and the concluding comments are given in Section 7.6. 7.2 A n a l y s i s o f S y s t e m a t i c E r r o r 7.2.1 O v e r v i e w A n extensive systematic error analysis was carried out on the E661 result, and this is documented in detail in Sections 7.2 and 7.3. We start by listing all of the (known) factors in our experiment that, through incorrect measurement or treatment, may have contributed to a systematic error in our final result: 1. Vert ical conversion factor V in the neutron bars (i.e. effective speed of light in the bars). 2. T D C pedestals: T O F calibration and y-position calibration. 3. T D C gains. Chapter 7. Results and Discussion 168 4. T D C time walk. 5. A D C pedestals and gains: determination of the low and high energy thresholds in the bars. 6. Light attenuation correction. 7. T I N A energy thresholds. 8. Subtraction of the background events. 9. Geometry measurements. 10. Known defects. 11. Energy resolution in the neutron bars. 12. The T O F and y-position resolutions in the neutron bars. 13. Errors in the E661 theoretical model. 14. Errors in computer programming. For each of items 1 to 10 in the above list, we first estimated the amount of sys-tematic error that could be present in the parameter(s) under consideration, then ar-tificially induced an error of the same magnitude in the data sorting program P R E -S O R T $ U S E R . F O R , then reprocessed the E661 data in order to obtain the "erroneous" experimental curve, and finally fit to the theoretical spectra in order to estimate the effect on our result for ann. For items 11 and 12, the evaluation was carried out in a similar manner except that the artificial error was induced in the main Monte Carlo program rather than in the experimental data sorting program. Item 13 was evaluated by comparing the theoretical (/-distributions obtained using our reaction model to those Chapter 7. Results and Discussion 169 obtained from a previously and independently developed model for specific values of ann. Item 14 was checked by methodical debugging of al l the software written for our analy-sis, by checking that al l of the intermediate calculations in our program were producing results that made sense, and in many cases by rewriting entire portions of computer code independently. Each of the items listed above wi l l be discussed in greater detail in the sections that follow. 7.2.2 D a t a R e d u c t i o n In order to facilitate the systematic error analysis, the raw E661 data were reduced to a compact format 2 that contained only those events (i) that satisfied the gtof condition in T I N A (refer to Section 4.2.4), A N D (ii) in which exactly two bars registered hits that were on-scale on the T D C s . The "valid" T D C scale for the data reduction was equivalent to a time range of —30 ns to +380 ns. The known small differences in T D C pedestals were corrected for in the final determination of these ranges in order to reduce any possible bias towards specific bars. The entire reduced E661 data set could be replayed for re-analysis in a time of approximately two hours. 7.2.3 T h e V e r t i c a l C o n v e r s i o n F a c t o r The Vertical Conversion Factor V has been introduced in Equation 4.13 (page 85) and is a measure of the effective speed of light in the neutron bars. It is used in the determination of the y-coordinates of the neutron hits in the bars. A n error analysis of this parameter revealed that an error of 5% in its value would have a significant effect on the shape of our experimental E1 spectrum (Figure 7.3). Our previous measurement of V was subsequently rechecked on Bar 16 at the University of Kentucky [217] using a collimated 2The original version of the DISPLAY data-sorting program adopted for E661 did not contain a data reduction, or skimming, facility. This facility was developed by the author and incorporated into a new version of the software package. Chapter 7. Results and Discussion 170 o o O l—i i i i I i i i i I i i i i I i i i i i 131.1 131.2 131.3 131.4 E n e r g y (MeV) Figure 7.3: The experimental Ey spectrum, as obtained from analysis using the cor-rect value for V (= 1.99 cm/ch) in the P R E S O R T program (solid curve); and using V = 2.09 cm/ch (dashed curve). Note the slight difference in the shape of the spectra (the dashed curve has a flatter peak). This difference causes an error in the result for ann of the order of 1 fm. Chapter 7. Results and Discussion 171 P u B e neutron source that was moved along the bar in 20 cm increments. The linear correlation between the position y of the source along the bar and the top-bottom T D C time difference (i.e. position of the neutron peak in the T d i f f spectrum) was of 0.9997, and this linearity was observed over the whole length of the bar. The accuracy of the gains of the T D C s used for this test were verified using a programmable pulse generator and a digi tal oscilloscope. We believe that our final value of V = 1.99 cm/ch is accurate to wi thin about 1%. A n error of this magnitude would not affect our result for ann.3 7.2.4 T h e T D C P e d e s t a l s Systematic errors from the neutron bar T D C pedestals could result from errors in the method used to fit the 68 M e V neutron peak in the T D C singles spectra (Figure 4.9, page 92), and/or from errors in the method used to locate the centroid of the T d i f f dis-tr ibution (Figure 4.11, page 94). These two sources of T D C calibration error wi l l be discussed separately below. E r r o r s i n t he T O F C a l i b r a t i o n The T O F to travel 3.05 m, for a 68.026 M e V neutron from %~d —¥ ynn, was calculated relativistically, and is equal to 28.17 ns (see Table 4.3, page 87). The average interaction distance of a 68 M e V neutron (with a hydrogen atom) within 10 cm of plastic scintillator was calculated to be wi thin 0.04 cm of the mid-depth of the bar, so that our use of a nominal travel distance of 3.05 m was well justified. Also, the probabili ty of scattering of the neutron in 2 cm of L D (prior to leaving the target) was estimated to be only about 1.5%. This latter scattering, as well as the possible scattering of the neutron against a 3 A simulated error of 1% in V does not result in any visible change to the shape of the Ey spectrum, i.e. the solid curve in Figure 7.1, as binned (bin size is 5 keV), remains unchanged. We estimate the associated systematic error in ann for such a situation to be less than about ±0.05 fm. Chapter 7. Results and Discussion 172 carbon atom in the scintillator, would have contributed to the appearance of the slight ta i l to the Gaussian peak in the T O F spectrum. The decision to ignore this ta i l in fitting the 68 M e V peak contributed a systematic error of about 0.6 ch (0.15 ns) in the T O F calibration. A s wi l l be shown shortly, an error of this magnitude is negligible in our analysis. A separate systematic error of about 0.6 ch (in the opposite direction) was later discovered to have been incurred due to saturation of the input voltages to the LeCroy 429F linear fan-in-fan-out modules and/or the TC455 A D C s , for a portion of the 68 M e V neutron signals used for the T D C calibration (the high thresholds in the software on the A D C signals were turned off for the calibration analysis, in order to increase the statistics in the calibration peak). Since the fitting of the photon peak (Peak l b in Figure 4.9) was also directly involved in the T D C calibrations, we decided to make a (very conservative) estimate for a possible systematic error in the T O F calibrations equal to about one standard deviation in the Gaussian fit to this peak. From Table 4.4 (page 97) this is equal to about 4 channels (1.0 ns). The values of the T D C pedestals for all the bars were therefore raised 4 channels above their nominal values, and the E661 data reprocessed. The data was later also reprocessed with the pedestals set at 4 channels below their nominal values. In each case there was no visible change in the shape of the experimental photon spectrum, and the result for ann (from fits to our grid of Monte Carlo spectra; refer to Table 7.1) remained unchanged. This indicated that any systematic error originating from the neutron bar T O F calibrations was negligible. E r r o r s i n t he Y - P o s i t i o n C a l i b r a t i o n Since the vertical conversion factor V is equal to 1.99 cm/ch , the horizontal scale in the histogram in Figure 4.11 is very nearly given by 1 ch : 2 cm. It was observed during the T D C calibration procedure that binning effects could cause an uncertainty of up to 4 cm Chapter 7. Results and Discussion 173 in the location of the y' = 0 (distribution centroid) position. Al though this uncertainty-was statistical in nature, and would be expected to balance out over the course of the experiment, we decided to test for a possible (conservative) systematic error of twice this amount (8 cm: this simulated error is of the order of the vertical asymmetries in our apparatus). This was done by setting al l the top T D C pedestals to 2 channels above nominal, and setting all bottom T D C pedestals to 2 channels below nominal 4 , and then reprocessing the E661 data. Al though we did observe a very slight change in the shape of the photon spectrum in each of these tests, the best result for ann from fits to our Monte Carlo spectrum grid remained unchanged. These results indicated that we had a possible systematic error of up to about ± 0 . 1 fm in our result for ann that was due to the y-position calibration. 7.2.5 The T D C Gains The T D C gains were very carefully measured during the experiment over the entire range of the time scales (see Section 4.2.2, page 78). Firstly, the effect of the small quadratic (non-linearity) coefficients c in the fits to the calibration spectra (Figure 4.4) would result in a totally negligible error of the order of 0.05% in the T D C readings in the E661 range of interest. The highest statistical errors in the b coefficients were of the order of 0.055% (Tables 4.1 and 4.2). We simulated a systematic error of 0.2% in al l of the T D C gains and replayed the E661 data. This had no effect on the shape of the E1 spectrum. Changes in the photon spectrum shape only became significant for induced errors in the T D C gains of more than about 0.4%. 4This has the effect of increasing all the measured y-position values by 8 cm, while leaving the TOF measurements unchanged. A similar test was carried out in which all the measured y-position values were decreased by 8 cm. Chapter 7. Results and Discussion 174 7.2.6 T h e T D C T i m e W a l k The time walk correction (see Section 4.2.8, page 97) was removed from the time cal-culation in the P R E S O R T program, and the E661 data was reprocessed. There was no detectable effect on the shape of the experimental photon energy spectrum, indicating that we were not sensitive to small amounts of time walk of the order that were present in our data. 7.2.7 T h e A D C P e d e s t a l s a n d G a i n s The possible systematic errors in our neutron bar energy calibrations were rather difficult to quantify. We first sought to isolate the specific assumptions that were implici t in the A D C calibration procedure (Section 4.3) that, although justified, were not immediately verifiable using independent tests. These were: 1. We relied on a geometric model, used in the E G S simulation, that could not be made exactly identical to that of our experimental calibration; 2. We relied on the accuracy of the E G S software; 3. We relied on accurate fitting to A D C calibration spectra that contained a rather low number of statistics (Figure 4.18, page 108); and 4. We gave M I N U I T the freedom of three parameters to use for the fitting process (Gaussian resolution, A D C gain, and amplitude), and relied on the abili ty of this software to make good fits to the data without a consistent (systematic) error. In spite of the above, however, a combination of factors worked strongly in our favour. A preliminary error analysis revealed that we were very insensitive to errors in the high energy software threshold used in the data analysis, as long as this was set to a high Chapter 7. Results and Discussion 175 enough value. The reason for this was that the reaction kinematics and the E661 geometry set a definite upper l imit on the maximum energy of a neutron that could form part of a valid triple-coincidence event for which the photon energy would be wi th in the top 450 keV of the spectrum. 5 We therefore needed to be concerned only with the accuracy of the low thresholds in our neutron energy measurements. The locations and distributions of the A D C pedestals were known very accurately (Section 4.3.5, page 108). Thus, even a systematic error as large as 100 keV in the location of the Compton edge in Figure 4.18 6 would result in "only" a 10 keV error down at our 120 keV low software threshold. In Figure 4.18, 100 keV is equivalent to about 25 channels. Art i f ic ia l ly induced systematic errors of 10 keVee in the low energy software thresholds in our analysis resulted in a small change in the shape of our photon spectrum, that corresponds to an error of about 0.2 fm in a n n . A n attempt to verify this systematic error quantitatively was carried out by repro-cessing the experimental data using various values of the high energy software threshold (starting at 1250 keVee, and increasing in 50 keVee increments), and recording the ap-proximate value of this threshold for which the E7 spectrum stopped changing shape. The "saturation value" for this threshold, at about 1350 keVee, was well wi th in 100 keVee of the value predicted by the main E661 Monte Car lo . 7 5This upper limit on the neutron energy was about 3.8 MeV. If such a neutron transferred all of its energy to a proton in the scintillator, this would result in an energy deposited of less than 1400 keVee. In our analysis, we set our high thresholds at 1500 keVee (Section 5.2). 6This Compton edge, resulting from a superposition of the edges from the 1.17 and 1.33 MeV photons from the 6 0Co source, is in the region of 1000 keV. 7Note that the value of this high threshold "saturation value" is derived from limits that are imposed by the reaction kinematics and by the E661 geometry and detector resolutions. In particular, it would not be expected to be strongly dependent on the value of ann. A qualitative illustration of the way in which the Ey spectrum changes shape with high threshold is given in Figure 7.4. Chapter 7. Results and Discussion 176 T 1 [ 1 1 1 r 131.1 131.2 131.3 131.4 131.5 E n e r g y (MeV) Figure 7.4: Experimental E1 spectra obtained for different settings of the high energy soft-ware thresholds: 1000 keVee (lowest curve), 1200 keVee (second lowest curve), 1500 keVee, and 2000 keVee (highest curve). Note that an increase in the high threshold beyond 1500 keVee does not significantly affect the shape of the spectrum. 7.2.8 T h e L i g h t A t t e n u a t i o n C o r r e c t i o n The factor g(yb) in Equat ion 4.28 (Section 4.3.7, page 110) was removed from the energy-deposited calculation in the P R E S O R T program, and the E661 data was reprocessed. There was no detectable effect on the shape of the experimental photon energy spectrum, indicating that we were not sensitive to the small corrections to the geometric means of the calibrated A D C readings in our data analysis. Our susceptibility to light attenuation effects in the neutron bars is further discussed in Section 7.3.1. 7.2.9 T h e T I N A E n e r g y T h r e s h o l d s The E661 data were reprocessed without the imposition of the T I N A energy thresholds for event acceptance, in order to assess our susceptibility to an error in the energy calibration of T I N A or of the converter. The shape of the experimental E1 spectrum remained Chapter 7. Results and Discussion 177 unchanged, indicating that we were insensitive to these thresholds. 7.2.10 B a c k g r o u n d S u b t r a c t i o n We assessed our susceptibility to a systematic error from the accidental background subtraction by adopting many of the background models in Figure 5.15 (page 140), including the pure R - S and S-S models, in place of our optimal 0.325SS model, and investigating the effect on our result for ann. We found that we were extremely insensitive to variations in the accidental background model, the xl fits to the Monte Carlo curves (Table 7.1) being affected only in the thi rd place of decimal. This was due to the small percentage of background events in our raw E7 spectrum. Our susceptibility to errors from the low energy (time-uncorrelated) photon diagonal background subtraction has already been assessed in Section 5.5, and is also considered to be negligible. W i t h respect to the high energy neutron (time-correlated) diagonal background (from Tr~d —> ynn events in which we only detect one neutron that scatters between two bars, see Section 5.5), we were somewhat concerned that this might not have been completely reproduced in the Monte Carlo simulation. This is due to the fact that in the simulations we only sampled the relative momentum distributions for q < 25 M e V / c (see Section 6.3) (i.e. only part of the phase space was sampled). In order to check for this possibility, we carried out a separate analysis of the data in which al l events that involved adjacent bar hits were omitted from the experimental E7 spectrum, and compared the experimental spectrum to simulated spectra obtained using the same condit ion. 8 We obtained fits that were consistent with our preliminary result {xl = 6.45 for ann = —18.5 fm, and xl = 1-94 for ann = —21.8 fm). A t the end of the data analysis however, before the 8The inspection of Figures 5.22 and 5.24 in Section 5.5 indicate that if there were a significant systematic effect in our result that originated from bar-to-bar scattering events, then this effect would be greatly reduced when adjacent bar hits were omitted from the data. Chapter 7. Results and Discussion 178 result is finalized, it may be useful to generate a few Monte Carlo spectra, close to the measured value of ann, in which the phase space is sampled in its entirety (or at least to a significantly higher q), and then to make a final assessment of our result. 7.2.11 Geometry Measurements Although measurements of the experimental geometry were carried out very carefully both before and after the experiment, we st i l l tested our susceptibility to geometry er-rors during the final stage of the analysis. This was done mainly by simulating various geometry errors in our main Monte Carlo program. In al l cases we found that we were insensitive to geometry errors of the order of about 2 cm. Neutron Detection in the Light Guides W i t h reference to the detector geometry, we needed to test for susceptibility to neutron detection in the light guides that were attached to the two ends of each neutron bar. This was done using the collimated P u B e source, at the same time that the vertical conversion factor V was being measured (Section 7.2.3). The source was moved beyond the plastic scintillator material onto the light guide, and there was no further movement of the neutron peak in the T d i f f spectrum beyond the position that corresponded to the edge of the scintillator. A t a distance of about 5 cm beyond the plastic scintil lating material at either end, the peak was no longer discernible. Since the P u B e source has a neutron emission energy spectrum that goes well beyond 4 M e V , this indicated that we were not susceptible to neutron detection in the light guides during the E661 experiment. This was further confirmed by increasing the y-position acceptence limits from ± 1 0 5 cm to ± 1 1 0 cm in our data analysis software, and noting that there was no change in the shape of the experimental photon spectrum. Chapter 7. Results and Discussion 179 o CO 3 o o u O C\2 o o -100 -50 0 50 100 T D C t i m e d i f f e r e n c e ( c h a n n e l s ) Figure 7.5: T d i f f distribution ( 6 0 C o events) for Bar 12, R u n Group 1 (compare to F i g -ure 4.11, page 94). 7.2.12 Known Defects A small number of defects in our experimental equipment were discovered during the data analysis of E661, and needed to be investigated. The first of these has already been reported in Section 4.2.1 (page 76), and involved a small differential non-linearity in the response of T D C 4b. We quantified the effect of this error by simulating (in the data analysis software) the same amount of non-linearity in the T D C for the symmetrically located P M T 17b, and reprocessing the E661 data. The photon spectrum remained unchanged, indicating that the slight non-linearity of T D C 4b was negligible. The second and third known defects involved Bar 12, and their manifestations can be seen in the T d i f f and y-position distributions for this bar (Figures 7.5 and 7.6) The two anomalies in the spectrum in Figure 7.5 are (i) a peaking of a number of events in a specific region of the spectrum, corresponding to a position y ?n —20 cm along the bar, and (ii) a, small ta i l at the extreme right hand side of the distribution. We had a further Chapter 7. Results and Discussion 180 -100 -50 0 50 100 Y — p o s i t i o n ( c m ) Figure 7.6: Y-posi t ion distribution in Bar 12 for valid E661 events (compare to F ig -ure 5.10, page 134). indication of a problem with Bar 12 from the raw integral non-linearity test spectrum for T D C 12t (Figure 7.7). A n investigation of the events in the peaks of Figure 7.7 revealed that there was a problem in the electronics whereby, about 13% of the time, the T D C 12t gave an output signal that was exactly 35 channels too early. We simulated the same effect in the P R E S O R T program for T D C lOt, and reprocessed the 6 0 C o events for R u n Group 1, as well as the entire valid E661 data. These tests gave us two important results: 1. (Refer to Figure 7.8.) We reproduced one of the two defects in Figure 7.5, indicating that we had understood its source. 2. There was negligible change in the shape of the experimental photon spectrum for the valid E661 data, which gave us a preliminary indication that the T D C problem seen in Figure 7.7 did not present a serious problem. It turned out that the strange peaking in Figures 7.5 and 7.6 was unrelated to the T D C Chapter 7. Results and Discussion 181 500 1000 1500 Input Delay (ns) 2000 Figure 7.7: Figure 4.3, Integral non-linearity test result for T D C 12t (raw spectrum). Compare to page 78. o 00 o CO w -)-> d o o u o -] 1 1 1 - - l 1 1 -100 - 5 0 0 50 TDC t ime difference (channels) 100 Figure 7.8: T d i f f distribution ( 6 0 C o events) for Bar 10, R u n Group 1, with an artificially induced error in the T D C readings. Notice the small ta i l at the extreme right hand side of the distribution, and compare to Figure 7.5. Chapter 7. Results and Discussion 182 o -o -10 - 1 .9 o 130.5 130.6 130.7 130.8 Energy (MeV) Figure 7.9: E661 photon spectra (i) after removal of al l events involving Bar 12 (solid curve), and (ii) after removal of al l events involving Bar 9 (dashed curve). problem depicted in Figure 7.7 (at least as far as we could determine), and we have in fact been unable to determine its cause with cer ta inty . 9 , 1 0 We made a further assessment of the impact of both Bar 12 defects by reprocessing the E661 data twice more: Dur ing the first run we rejected al l events that involved Bar 12, and during the second run, for the purpose of comparison, we rejected all events that involved Bar 9. The reconstructed Ey spectra from both these runs had the same shape (see Figure 7.9), indicating that the contribution from Bar 12 was similar to that of the symmetrically placed Bar 9, and that therefore the defects in Bar 12 did not affect our final result. As a final test of these Bar 12 defects, we generated Monte Carlo spectra for ann values of —18.5 fm and of —21.8 fm in which all events that involved Bar 12 were excluded, and 9This strange peak in the Bar 12 y-position distribution was probably caused by a local defect in the bar, such as a crack or a light leak. 10We have also considered the possibility of a large systematic error in the y-position calibration of Bar 12, that could have occurred due to the anomalous features of Figure 7.5. We have simulated a systematic error of 14 cm in the Bar 12 y-position in the PRESORT program and reprocessed the E661 data. This did not affect the shape of the Ey spectrum. Chapter 7. Results and Discussion 183 fitted these spectra to the corresponding experimental spectrum described in the previous paragraph (solid curve in Figure 7.9). The xl of these fits were 6.72 (for ann = —18.5 fm) and 1.57 (for ann = —21.8 fm), and were therefore consistent wi th our result. It is relevant to point out that the amount of affected data in Figure 7.6 is estimated to contribute to only about 2.5% of the events in our final photon spectrum, and it is for this reason that the Bar 12 problems did not turn out to be more serious. 1 1 7.3 E v a l u a t i o n o f t h e S i m u l a t e d S p e c t r a 7.3.1 E r r o r A n a l y s i s a n d Tes t s o f t he M o n t e C a r l o A brief outline of the main Monte Carlo simulation program that was used to analyze the E661 data has been given in Section 6.4. The possibility of systematic errors in our result that could originate from this simulation program needed to be investigated and quantified. We classify the possible sources of error into two broad groups: (i) errors that stemmed from experimentally measured parameters that were then used as inputs to the Monte Carlo (e.g. detector resolutions), and (ii) errors in the Monte Carlo program itself. We wi l l look at these two classes of errors in turn. E r r o r s f r o m E x p e r i m e n t a l I n p u t s t o t he M o n t e C a r l o The main experimentally measured inputs to the main Monte Carlo program were the energy, y-position, and T O F resolutions of the neutron bars, as well as the light attenu-ation functions for the bars, and the values of the hardware energy thresholds at the bar P M T s . The T O F and y-resolutions have already been discussed in detail in Section 4.2.9. We found that changes of about 10% in these inputs to the Monte Carlo did not affect nThe rejection of all events involving Bar 12 from our valid data would result in the loss of about 16% of our statistics. Chapter 7. Results and Discussion 184 the shapes of the theoretical E7 curves. Regarding the energy resolution of the bars, we use a commonly derived value for al l the forty P M T s in the present version of our Monte Carlo, given by as inferred from the A D C calibrations. We found that a systematic error of 100 keV in this smearing (in all forty tubes) could result in a change in our result for ann of as much as about 0.3 fm. Al though we consider an error of this magnitude to be highly unlikely, this source of error may need further investigation, and wi l l be referred to again in Section 7.5. The hardware energy thresholds of the P M T s were estimated by reprocessing the E661 data with the low energy software thresholds turned off, and by then fitting the low-energy cut-offs of the appropriately binned calibrated A D C spectra to a step (or steep ramp) function that was convoluted with the appropriate A D C pedestal distribution. A typical fit is shown in Figure 7.10 1 2 , and the hardware energy thresholds for all forty P M T s are listed in Table 7.2. In the analysis of E661, our low energy software threshold of 120 keVee was sufficiently high such that we were not susceptible to small errors in the hardware threshold values that were utilized in the Monte Carlo program. The light attenuation functions, described in Section 4.3.7 (page 110), were used in the main Monte Carlo program for the purpose of checking the energy deposited by the (simulated) neutrons (at the hit position) against the hardware thresholds at the appropriate P M T s , and later for the rescaling of this energy-deposited to the mid-bar value, which was the scale that was used in the experimental analysis. The Monte Carlo then used the same Equations 4.26 to 4.29 (with the appropriately smeared value of yb) 12The energy deposited by a neutron on the "mid-bar" scale refers to the energy that would have to be deposited by a neutron at the bar centre (y1 = 0) in order to produce the same energy signal (after attenuation) at the PMT as the detected neutron. (7.1) er 7. Results and Discussion Tube Hardware Threshold Tube Hardware Threshold on mid-bar scale at P M T on mid-bar scale at P M T (keVee) (keVee) (keVee) (keVee) It 24 17 l b 17 12 2t 27 20 2b 37 27 3t 41 28 3b 15 10 4t 32 23 4b 35 25 5t 28 17 5b 07 04 6t 41 30 6b 30 22 7t 48 38 7b 65 52 8t 38 27 8b 43 30 9t 63 42 9b 65 43 lOt 52 36 10b 43 30 l i t 79 56 l i b 92 66 12t 48 35 12b 51 37 13t 29 22 13b 44 33 14t 43 33 14b 29 22 15t 39 30 15b 47 36 16t 43 30 16b 41 29 17t 25 19 17b 32 24 18t 33 22 18b 47 31 19t 41 27 19b 30 19 20t 28 17 20b 14 09 Table 7.2: Neutron bar A D C hardware thresholds. Chapter 7. Results and Discussion 186 o 1% in 50 100 150 200 Energy (1 ch = 4 keVee) Figure 7.10: F i t to the low energy hardware cut-off for A D C 10b (energy scale is the "mid-bar" scale). that were used in the analysis of the expermental data. Due to the facts that (i) the light attenuation functions (Equation 4.26) were derived through careful measurement for each individual bar [62] (see Section 4.3.7); (ii) the Y i e l d m e a „ functions for each bar (Equation 4.27) were found to be nearly flat; and (iii) the energy rescaling calculation in the Monte Carlo, mentioned above, in general only needed to be done over a small distance along the bar (from the smeared hit position to the bar centre), we do not immediately expect large errors in our result for ann that are due to light attenuation in the neutron bars. The fact, however, that there is a lot of light attenuation in the bars (witness Figure 4.23), coupled with our already demonstrated sensitivity to errors in the low energy threshold values in the E661 analysis (Section 7.2.7) could make this factor a potential candidate for further investigation. This may need to include a study of the variation of the energy resolution in the bars with y-position, that may result in a modification to Equation 7.1 above. We wi l l refer back to this issue in Section 7.5. Chapter 7. Results and Discussion 187 Evaluation of the Monte Carlo Program The unexpected nature of our preliminary result has forced us to re-examine every aspect of our experiment and data analysis, including the accuracy of the G E A N T simulations, and the possibility that a serious programming error could be concealed somewhere in the Monte Carlo code. We developed various strategies to make tests that were as conclusive as possible on the validity of each portion of the simulation, as follows: We first sought to understand the effects that the various individual sections of the Monte Carlo had on the shape of the Ey spectrum. We did this by first downgrading the simulation to a simpler version that lacked the following features: energy, time and y-position smearing in the neutron bars, neutron scattering against carbon nuclei in the bars, multiple neutron scattering in the bars, non-zero hardware thresholds in the bars, neutron scattering in the l iquid deuterium inside the target, reaction vertex distribution, and vertical offsets (in the geometry) between the L D target, neutron bars, and T I N A col l imator . 1 3 Al though the combined effect of removing alloi these contributions changed the shape of the photon spectrum considerably, we found that the individual effect of most of the above factors was quite small . The exception to this was the presence of carbon in the bars, which had a rather significant effect on the shape of the E7 spectrum. In particular, the scattering of neutrons inside the L D target, and the multiple scattering of neutrons in the bars were both found to have negligible effect, in agreement wi th the findings in the previous measurement by Salter et al. [32]. In order to test many of the basic calculations in the main E661 Monte Carlo program (in particular, Steps 2-6,8-10 in the procedure that is outlined in Section 6.4), a separate program was written by the author to simulate conditions that were almost identical to those of the Base version of our main program. A l l the routines in this new program were 13We will refer to this downgraded version of the simulation as our Base version in the remainder of this section. Chapter 7. Results and Discussion 188 T 1 1 1 j 1 , 1 , j 1 , 1 , 1 1 1 , 1 1 1 1 1 j-131.1 131.2 131.3 131.4 E n e r g y (MeV) Figure 7.11: A comparison between the E7 spectra from the Base Monte Carlo model (solid curve) and the new, independently written model (dashed curve). The xl of the fit is 1.27. The very small difference in shapes is due to minor differences between the two simulations (mainly simplifications in the new model). written independently of their counterparts in the main E661 Monte Carlo. In particular, this program did not utilize G E A N T , and the neutron scattering cross-sections were estimated using an empirical relation found in [179, Appendix I]. The new program also used a different random number generator in the Monte Carlo sampling routines. 1 4 The agreement obtained between the shapes of the E7 spectra from the Base model and from the new model is shown in Figure 7.11, and helps to validate many of the routines in the main E661 Monte Carlo. We also tested for the presence of compensating errors in either of the two programs by comparing intermediate outputs from various stages of the simulations. Later, the energy, time, and y-position smearings, as well as the vertical geometry offsets, were introduced into the new Monte Carlo, and we found that these factors had very little effect on the shape of the E7 spectrum, in agreement with the 1 4 The main E661 Monte Carlo used the RANDI routine [218], while the new model used RAN2 [219]. We have also tried the use of RAN2 in the full version of the main E661 Monte Carlo simulation. Chapter 7. Results and Discussion 189 findings from the main E661 Monte Carlo. We believe that the results from the main E661 Monte Carlo are accurate, as long as: 1. The theoretical q distributions are correct; 2. The experimentally measured detector resolutions are correct; and 3. G E A N T is simulating the neutron scattering correctly. Regarding Point 3 above, this version of G E A N T [216] uses the C A L O R 8 9 software package, which calls the M I C A P simulation package [220] for the neutron energy regime in which we operate. Al though we would expect these simulation programs to be accurate, it may be prudent to make an experimental measurement of the neutron efficiency curves for our bars using a neutron source, or using neutron beams of known energy, in order to confirm this accuracy. We wi l l refer back to this in Section 7.5. Point 2 above has already been discussed in the previous section, while Point 1 is addressed in the following section. 7.3.2 E v a l u a t i o n o f t h e E 6 6 1 T h e o r e t i c a l M o d e l A more complete analysis of possible systematic error in the theoretical model used for E661 can be found in [63]. 1 5 In the present version of our model, the quasi-free process (the term in (53(q* — q) in Equation 6.7, page 159) is neglected, and so is the contribution from the pion-rescattering graph (Figure 1.10, page 19). In this section we shall attempt to briefly assess the impact of these approximations, and to evaluate the reliability of the present version of our theoretical model for E661. Refer first to the photon energy distribution spectra shown in Figure 7.12. Note the 1 5 The development of the theoretical model and of the simulated photon energy spectra at the Uni-versity of Kentucky has recently been temporarily interrupted, however it is expected to resume in late 1998. Chapter 7. Results and Discussion 190 o 0 10 20 30 n —n r e l a t i v e m o m e n t u m ( M e V / c ) Figure 7.12: The photon energy distribution as a function of the n-n relative momen-tum q as derived from the E661 theoretical model for ann = —21.8 fm (solid curve), a n „ = —18.0 fm (dashed curve), and ann = —17.0 fm (dotted curve). (The vertical scale is arbitrary.) manner in which the peak of the distribution moves to a higher q as the theoretical value of \ann\ is reduced. 1 6 Next, in Figure 7.13, we show a similar set of curves, derived from different theoretical models, for a common value of ann ( — 18.0 fm). Note that differences in the various models only become significant for larger values of q (p in Figure 7.13). In Figure 7.14 we compare the photon energy distribution spectra for ann = —18.0 fm, as derived from de Teramond [51] (Curve A of Figure 7.13), and from the E661 model. Note that the two curves agree very well for q < 18 M e V / c , then diverge slightly as q increases further (note that our experimental data extend to about 21 M e V / c ) . A comparison between simulated E1 spectra obtained using the new Monte Carlo model, using each of the two distributions of Figure 7.14, is shown in Figure 7.15. The very good agreement between these two curves indicates that our preliminary theoretical model is 1 6In the main Monte Carlo program we sample the absolute q distribution in phase space, rather than these photon energy distribution spectra. There is a one-to-one correspondence between the two sets of spectra. Chapter 7. Results and Discussion 191 Figure 7.13: The photon energy distribution as a function of the n-n relative momentum q as derived from different theoretical models: Curve A , de Teramond; Curve B , Gibbs, Gibson, and Stephenson; Curve C , Bander; and Curve D , de Teramond, S-wave only. A l l curves were calculated with rnn = 2.8 fm and ann = —18.0 fm, except the Bander curve where ann = —18.3 fm. (Reproduced from Gabioud et al. [221].) 0 10 20 n - n relat ive m o m e n t u m (MeV/c) Figure 7.14: The photon energy distribution as a function of the n-n relative momen-tum q for ann = —18.0 fm, as derived from de Teramond (solid curve, extracted from Figure 7.13), and from the E661 model (dashed curve). (The vertical scale is arbitrary.) Chapter 7. Results and Discussion 192 T i i i j i i i ~i | r~ i i i j i ' r™ i i | i i i 1 r 131.1 131.2 131.3 131.4 Energy (MeV) Figure 7.15: The E7 distributions, obtained using the new Monte Carlo, for o-nn = —18.0 fm: from de Teramond (solid curve), and from the E661 model (dashed curve). The xl °f the fit is 1.02. already roughly in accordance with that of de Teramond, at least for ann = —18.0 fm. It does not appear, from Figure 7.15, that our theoretical approximations would cause an error in ann of greater than a few tenths of a fermi. In Figure 7.16 we show the polar angle distribution of the n-n relative momentum vector q,17 calculated for ann = —18.5 fm at a value of q « 10 M e V / c . These polar angle distributions are in general functions of both ann and q. They are sufficiently flat, however, so as to have a negligible effect on the shape of the simulated E7 spectra that are obtained from the Monte Carlo programs [218]. 1 8 The present version of our theoretical simulation sti l l shows some minor inconsis-tencies in the shapes of the generated ^-distributions when we attempt to make small 1 7In this context, the polar angle is defined as the angle between the relative momentum vector q and the photon momentum vector k in Figure 1.8, page 16. 18The simulated E7 spectra that are represented in Table 7.1 (page 166) were produced using flat distributions of this polar angle. Chapter 7. Results and Discussion 193 o CO o CO 3 o o ^ U o C\2 ~i 1 1 r i 1 1 r -Q I I 1 I I I I I I I I I I I I I I 1 I 1 I 0.5 0 0.5 Cosine of the polar angle of the q vector Figure 7.16: The polar angle distribution of q, as calculated from the E661 theoretical model for ann = —18.5 fm at a value of q ~ 10 M e V / c . adjustments of the order of 0.2 fm to the input value of ann. Before our result can be finalized, the E661 theoretical model w i l l certainly have to be upgraded and tested further. It would also be important to analyze our data using a number of other inde-pendent models, and to make a more exact quantification of the possible systematic error in our result that originates from the theory. We note that the best theoretical models that currently exist in the literature claim errors of the order of 0.3 fm [50, 51, 52, 53]. 7.4 Comparison to Previous Experiments In this section we wi l l briefly discuss the three most accurate measurements of ann from ir~d —>• jnn that are found in the literature, and compare specific features of these experiments to those of E661. The results of these three experiments have already been given in Table 1.1 (page 14). Different approaches to the experimental set-up, as well 1 9It is for this reason that we have employed a grid of simulated curves in increments of 0.4 fm in the extraction of our preliminary result (see Table 7.1). The final result should be extracted from a grid of curves that are incremented in steps of 0.2 fm or less. Chapter 7. Results and Discussion 194 as to the data analysis, were taken in each of these measurements. The experiment by Salter et al. [32] was carried out at the Lawrence Berkeley Laboratory ( L B L ) and is the most similar to E661, with al l three final state particles from the reaction being detected in triple coincidence. The data were analyzed, however, by studying the neutron energy spectra for specific values of the lab opening angle between the two neutrons, rather than by studying the reconstructed Ey spectrum. In the experiment by Gabioud et al. [35] the photon energy spectrum was measured directly in singles mode, while the experiment by Schori et al. [36] involved a study of the neutron T O F spectra for 7 — n coincidence events. The latter two experiments were carried out at the Swiss Institute for Nuclear Research (S IN) . 2 0 Each of these three experiments is discussed in greater detail below. 7.4.1 T h e S a l t e r et al. M e a s u r e m e n t The experimental set-up is shown in Figure 7.17. The neutron counters contained NE224 liquid scintillating material, while the 7-counters were each composed of a sandwich of eight sheets of 0.64 cm plastic scintillator wi th 0.45 cm lead sheets interspersed. In the reconstruction of the kinematics, the reaction was assumed to have occurred in the plane perpendicular to the incoming beam line, with the final state particles originating in the centre of the target and striking the centres of the counters. For each event the following information was recorded: (1) the addresses of the n- and 7-counters; (ii) the corresponding neutron T O F s relative to the 7-counter signal; (ii) the pulse heights of the respective n- and 7-signals photographically recorded from high speed oscilloscope traces; and (iv) an event number to correlate photographs and digital data. The neutron detectors were calibrated separately on the Los Alamos Van de Graff Accelerator at several energies in the range 1-6 M e V with neutrons of a known flux [222]. 2 0 Now the Paul Scherrer Institute (PSI). R E A C T I O N P L A N E Figure 7.17: Experimental set-up for the Salter et al. measurement (from [32]). The data were analyzed using the theoretical model due to Bander [49] (see Sec-tion 1.3.2) assuming r n n = 2.65 fm, and by uti l izing Monte Carlo techniques to simulate the experimental conditions. The final results are shown in Figure 7.18. The data for 9 = 27.2° were thrown out due to a calculated lower sensitivity to ann at large 9. A total of 4200 events were analyzed, and the final result from this experiment was of o„„ = —16.7 ± 1.3 fm (errors from the theoretical model are excluded). Al though the E1 spectrum was not used in the final analysis, the resolution in the reconstructed photon energy that could be extracted from the data in this experiment was reported in [32] to be around 50 keV. 7.4.2 The Gabioud et ai. Measurement The set-up for this experiment is shown in Figure 7.19. Photons from the target were converted to e +e~ pairs in a 88 //m-thick gold foil 92 cm away from the target, and Chapter 7. Results and Discussion 196 Figure 7.18: F ina l fits for the Salter et al. data: (a) 9 = 6.8°, (b) 9 = (d) 9 = 27.2° (from [32]). 10.2°, (c) 9 = 13.6°, Chapter 7. Results and Discussion 197 Magnet gap B = 0 . 8 T C o n v e r t e r ] MWPC 3 3 MWPC 2 3 MWPC 1 \imuiiiuuiinimnm • willllllllllHIlHIIHh A i B i Lead s h i e l d i n g AC1 . A C 2 7-1 me t e r Degrader beam T a r g e t e P 4 P 3 Figure 7.19: Experimental set-up for the Gabioud et al. measurement (from [35]). the trajectories of both particles were bent by a 0.8 T magnetic field. Three multi-wire proportional chambers, one before and two after the converter, detected the charged par-ticles. The scintillators A C l and A C 2 vetoed charged particles entering the spectrometer. The target could alternatively be filled with l iquid hydrogen or deuterium. The experimental photon spectra from ir~p —>• yn ( L H target, 158,400 events) and from TT~d ->• ynn ( L D target, 542,400 events) are shown in Figure 7.20. The 129 M e V photon from the radiative capture on hydrogen was used for the energy calibration, and for the determination of the resolution function (which had a F W H M of 720 keV). This resolution was folded onto the theoretical photon spectrum for comparison with the experimental spectrum. The theoretical model for the reaction was due to de Teramond [53]. This experiment obtained results of —18.5 ± 0.4 fm for ann, and 2.80 ± 0.11 fm for r. nn • Chapter 7. Results and Discussion 198 Figure 7.20: Experimental photon spectra for the Gabioud et al. measurement: 7r~~d —> jnn (solid line), n~p —> jn (dashed line, number of counts divided by four). The theoretical spectra for the same reactions are also shown (dots and dot-dashed) (from [35]). Chapter 7. Results and Discussion 199 Figure 7.21: Experimental set-up for the Schori et al. measurement (from [36]). 7.4.3 T h e S c h o r i et a l . M e a s u r e m e n t This experiment was performed by the same group that carried out the previously de-scribed (Gabioud et al.) measurement, and, for part of the experiment, used the same photon detector as before. For the remainder of the experiment the pair spectrometer was replaced by a lead-glass detector (described in [223]). The full set-up is shown in Figure 7.21. The neutron detector consisted of a double layer of ten plastic scintillators (NE110) with dimensions 10 x 100 x 5 c m 3 (in the final analysis data from only the front layer were utilized). Each of these bars was fitted with two P M T s , one at each end, and the y-position of the hits was measured from the appropriate T D C time differences, as in E661. The energy calibrations were carried out in a separate experiment at the Lausanne University neutron generator for neutrons in the energy region 0.5 to 2.2 MeVee [224]. Chapter 7. Results and Discussion 200 events per 2ns 800 600 400 200 t —neutron — r 1 o 10Q 200 300 t„(ns) Figure 7.22: Raw neutron T O F spectrum taken in coincidence wi th a photon in the pair spectrometer, from the Schori et al. data (from [36]). The raw neutron T O F spectrum, taken in coincidence wi th a photon in the pair spec-trometer, is shown in Figure 7.22. The background underneath the data were obtained from an extrapolation of the events outside the region of interest (dotted curve). Theo-retical spectra, based on the reaction model of de Teramond, were generated for various values of ann, and Monte Carlo techniques were used in order to compare these spec-tra to the background-subtracted experimental spectrum. After background subtraction, 40,000 (90,000) events remained from the pair spectrometer (lead-glass) data sample, and the results are shown in Figures 7.23 and 7.24. This experiment obtained a final value for ann of —18.7 ± 0.6 fm. The equivalent reconstructed photon energy resolution in this experiment was of about 80 keV F W H M . Chapter 7. Results and Discussion 201 Events per 7.9ns 4000 l ' 1 ' ' i ' 1 1 1 i 1 ' 1 1 i 50 100 150 200 , (, Events per 7.9ns h » » i i i i i » j i i i i | i ns) 50 100 150 200 t (ns) Figure 7.23: Corrected neutron T O F spectra (random coincidences eliminated) taken in coincidence wi th a photon in the pair spectrometer (left) and the lead-glass Cerenkov detector (right), from the Schori et al. data. The solid curves show the best fit to the data (in the region between the vertical lines) corresponding to ann = —18.4 ± 0.8 fm (left) and ann = -18 .8 ± 0.5 fm (right) (from [36]). Figure 7.24: F i n a l neutron T O F spectrum taken in coincidence with a photon in the pair spectrometer wi th a measured energy above 130 M e V , from the Schori et al. data. The dashed line represents the fit to the data wi th ann = —18.3 ± 1.1 fm (from [36]). Chapter 7. Results and Discussion 202 7.4.4 T h e A d v a n t a g e s a n d D i s a d v a n t a g e s o f E 6 6 1 Experiment E661 was explicitly intended to measure ann to a precision that exceeded that of its predecessors, and was therefore designed to have certain features that were superior to those of the previously described experiments. The first of these is the higher resolution in our photon energy spectrum (40 keV F W H M , see Section 5.6). W i t h reference to the Salter et al. experiment, we also note the far greater number of statistics in our experiment, which is expected to result in a greatly improved statistical uncertainty in our measurement. W i t h reference to the Gabioud et al. and Schori et al. experiments, we note that the kinematically over-determined nature (triple-coincidence measurement) of E661 served to practically eliminate problems associated with background events, and this is expected to significantly improve the systematic uncertainty in our final result . 2 1 In the case of the Gabioud et al. experiment we further note that the energy resolution of 720 keV in the Ej spectrum may have caused appreciable difficulty in accurately measuring small differences in the spectrum that occur in the top 450 keV of the distribution. The authors, however, were aware of this weakness and took great care to work in stable conditions. We note also a possible disadvantage in the E661 set-up, in that the large length of the neutron bars (2 m) may have introduced new problems, due to somewhat excessive and statistically widely distributed light attenuation,.that were not a factor in the Gabioud et al. experiment, and that may have been less extensive in the Salter et al. and Schori et al. measurements. W i t h reference to the methods used to analyze the data, we note that in both of the above discussed previous measurements that involved the detection of neutrons, the 21Refer, for example, to Figure 7.22 for an indication of the large amount of background that needed to be subtracted from the Schori et al. data. These background data were assumed to have no TOF correlation other than the 50 MHz rf of the SIN accelerator. Chapter 7. Results and Discussion 203 efficiency of the detector as a function of neutron energy was measured in separate exper-iments using neutron beams of known energy (in E661 we have made use of the G E A N T software to reproduce these efficiencies). We also note that in both of these previous ex-periments, an analysis of the neutron T O F (or energy) spectra was selected in preference to an analysis of the (reconstructed) photon energy spectrum. In [32] the impact of the experimental geometry, on the observed distribution of Ey is cited as the reason for this choice. Al though in E661 we already have a very good knowledge of the extent to which the various relevant factors contribute to changes in the shape of the photon spectrum (as discussed in Sections 7.2 and 7.3), it may be instructive to undertake a separate study of our data based on an analysis of the neutron T O F spectra for specific bars. 2 2 We wi l l refer back to this in Section 7.5. 7.5 Remaining Work The remaining work for E661, as discussed in Sections 7.2, 7.3, and 7.4, is summarized below: 1. A n upgrading of the theoretical model to include the quasi-free process and the pion rescattering correction in the E661 reaction. 2. A reanalysis of the data using one or more other independent theoretical models. 3. A n experimental measurement of the absolute and relative efficiencies of the neutron bars, using neutron beams of known energies. 4. A dedicated measurement of the energy resolutions of the neutron bars, and of the dependence of these resolutions on the y-position along the bars, using collimated photon sources such as 6 0 C o or 2 2 N a . 22We would then, however, have to deal with spectra having far less statistics than does our full E7 spectrum. Chapter 7. Results and Discussion 204 5. A final assessment of the manner in which the high energy neutron bar-to-bar scattering background is generated in the main Monte Carlo program. 6. The generation of a final grid of simulated E7 spectra, in increments of 0.2 fm or less in ann, and to considerably higher statistics than those reported in Table 7.1. 7. A final assessment of the errors in our final result. The work in Sections 7.1 through 7.3 indicates that (i) the experimental statistical error in our final re-sult is negligible (less than ± 0 . 0 5 fm in ann), due to the high number of events in our experimental photon spectrum 2 3 ; (ii) the experimental systematic error is of the order of ± 0 . 3 fm, with the largest contribution (±0 .2 fm) coming from the A D C low threshold determination; and (iii) a theoretical systematic error of ± 0 . 3 fm is easily achievable. A summary of the experimental systematic errors is given in Table 7.3. 2 4 8. (Possibly) A reanalysis of the data based on the neutron T O F spectra for specific bars. 7.6 Conclusion This thesis constitutes a major contribution to a large experiment that is st i l l ongoing. The work that has been accomplished includes: 1. A n active participation in the preparation and execution of the E661 experimental run at T R I U M F ; 2 3 See Table 7.1 for an indication of the extent to which the x2 of the fits change with ann near the minimum. 24The total error in Table 7.3 was obtained by adding the maximum values of the individual errors in quadrature. Chapter 7. Results and Discussion 205 Source Estimated Error Vertical conversion factor < ± 0 05 fm T D C pedestals ± 0 1 fm T D C gains < ± 0 05 fm T D C time walk < ± 0 05 fm A D C pedestals and gains ± 0 2 fm Light attenuation correction < ± 0 05 fm T I N A energy thresholds < ± 0 05 fm Accidental background subtraction < ± 0 05 fm Diagonal background subtraction < ± 0 05 fm (?) Geometry measurements < ± 0 05 fm Known defects < ± 0 1 fm Energy resolution in the bars < ± 0 1 fm (?) T ime resolution in the bars < ± 0 05 fm Software (including G E A N T ) errors < ± 0 05 fm (?) Total ± 0 3 fm Table 7.3: A summary of the experimental systematic errors in the E661 result. The question marks indicate where a final assessment st i l l needs to be made. The theoretical error is not included in this table. Chapter 7. Results and Discussion 206 2. The calibration of the forty neutron bar T D C s for T O F and y-position measure-ment; 3. The calibration of the T I N A wire chamber for photon tracking; 4. The reconstruction of the photon energy spectrum for the reaction -n~d —¥ ynn near the endpoint from the selected data (with C . Jiang); 5. The modeling of the accidental background that was present during the experiment; and 6. A n exhaustive error analysis of al l aspects involved in the generation of the ex-perimental spectrum, and of the main Monte Carlo simulation (the latter with B . Doyle). We present an experiment that has a number of improvements over previous attempts to measure the neutron-neutron scattering length. These are mainly in the areas of event statistics, energy resolution of the photon spectrum, and the elimination of background. The preliminary result from experiment E661 is an unexpected one, since it is in disagreement with the more recent measurements of a n n , that are considered to be the most accurate results currently available. We are also presently in disagreement with the results from theoretical calculations of charge symmetry breaking in the strong interac-tion. We note however that several outstanding issues in the theoretical calculations of C S B st i l l remain, such as the contributions from yn exchanges (see Section 2.3.5), and from (p — co) mixing (see Section 2.3.7). The best xl fit between our experimental spectrum and the theoretical spectrum for ann = —21.8 fm is st i l l rather high at 1.73. This may not be particularly surprising, given the complexity of the experiment, and given the assumptions and approximations made in the theoretical model. Under normal circumstances, the remaining work that Chapter 7. Results and Discussion 207 has been listed in Section 7.5 would not be expected to change our final value of ann by more than a few tenths of a fermi. In the light of our result, however, we note that there is certainly a possibility that these remaining tests and analyses may uncover a large systematic error in our work that has hitherto been missed. It is therefore important to emphasize that our result can only be finalized after the work listed in Section 7.5 has been completed. Experiment E661 is the only high statistics triple-coincidence (kinematically over-determined) measurement of ann from the reaction n~d —>• -ynn that has been made to date. Moreover, the error analysis that is contained in this thesis has enabled us to acquire a very good understanding of al l of the systematic effects that need to be taken into consideration when ann is extracted in this manner. 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Schori, Ph.D. Thesis, Universite de Lausanne (1985) (unpublished). Appendix A The E661 Collaboration Name Institution B . Bassalleck University of New Mexico M . E . Christy University of Kentucky B . C . Doyle University of Kentucky H . Fischer University of Freiburg T . P . Gorringe University of Kentucky C . Jiang University of Kentucky E . Korkmaz University of Alber ta M . A . Kovash University of Kentucky T - S . H . Lee Argonne National Laboratory K . L i u University of Kentucky D . F . Measday University of Br i t i sh Columbia A . Opper University of Alber ta D . Ottewell T R I U M F M . A . Saliba University of Br i t i sh Columbia K . S im Korea University J . Stasko University of New Mexico T . J . Stocki University of Br i t i sh Columbia D . Wolfe University of New Mexico Table A . l : Members of the E661 collaboration. 219 Appendix A. The E661 Collaboration D a t a A n a l y s i s : D i v i s i o n o f R e s p o n s i b i l i t i e s M . A . Saliba Cal ibrat ion of the neutron bar T D C s Calibrat ion of the W C 3 T I N A wire chamber Reconstruction of the E1 spectrum (with C . Jiang) Subtraction of the accidental background Da ta reduction Error analysis C . Jiang Calibrat ion of the neutron bar A D C s Cal ibrat ion of the W C 1 / W C 2 tracking wire chambers Calibrat ion of the T I N A A D C s Reconstruction of the E1 spectrum (with M . Saliba) Subtraction of the bar-to-bar scattering background B . C . Doyle E661 reaction theory (with T - S . H . Lee) Production of the theoretical q distributions Development of the Monte Carlo simulation for E661 M . A . Kovash Experiment spokesperson Table A . 2 : Data analysis: division of responsibilities. A p p e n d i x B D e t e c t o r S p e c i f i c a t i o n s This appendix provides information that complements the contents of Sections 3.2, 3.3 and 3.4. A l l of the detectors that are described below are also shown on the general schematic diagram that is given in Figure 3.3 (page 56). B . l P h o t o n A r m T I N A : A single Na l (T l ) crystal, cylindrical shape. Dimensions 46 cm in diameter x 51 cm long. Viewed at the rear face by seven R C A 4522 P M T s , as shown in Figure B . l . 00 41" 0 X Ts" optical windows (7 off) T3i" 0 ' 20" Figure B . l : The T I N A Na l (T l ) crystal, showing the seven optical windows on which the P M T s are mounted. Converter: A Na l (T l ) detector, disk shape. Dimensions 28.5 cm in diameter x 4 cm thick. Viewed from the edge by two P M T s . Located 3 | " (9.8 cm) behind the rear face of the lead collimator. 221 Appendix B. Detector Specifications 222 Coll imator: Lead with straight bore, 6]J" (17.62 cm) in diameter and 10" (25.4 cm) in length. Front face located 58 cm away from L D target centre. Centreline vertically offset by —1.5 cm with respect to L D target centre. B.2 Beam Path Bending Magnet Wire Chambers Degrader! LD Target Housing 220 230 Figure B.2: Location of the equipment along the beam path (Plan view) (dimensions in mm). SI : Plastic scintillation counter (BC412). Dimensions 6" (15.24 cm) x 6" x | " (3.18 mm). S2: Plastic scintillation counter (BC412). Dimensions 2" (5.08 cm) in diameter x ^ r " (1.59 mm). S3: Plastic scintillation counter (BC412). Dimensions 6" x 6" x | " (6.35 mm). L D Target: Description and sectional view given in Section 3.2.4. Fu l l details and draw-ings are available in [171]. A drawing of the nickel target capsule, and an isometric sketch of the aluminum housing in which it was contained, are given in Figures B.3 and B.4 respectively. Appendix B. Detector Specifications 223 , 1 cm length wall -T- th ickened to 0.010" Figure B.3: Drawing of the LD target used for E661 (from [171]). The capsule-shaped vessel was constructed out of nickel. Appendix B. Detector Specifications 224 B.3 Neutron A r m S4: Plastic scintillation counter (BC412). Dimensions 24" (60.96 cm) x 24" x i " (6.35 mm). Located 70 cm away from L D target centre. S4 centre vertically offset by +2.8 cm wi th respect to L D target centre. Neutron bar supporting stand: The stand was designed by the author and custom built for E661. The drawing for the base plate is shown in Figure B.5 . This plate was supported on concrete blocks during the experiment. The bars were further supported by lateral beams and a ceiling plate, as shown in Figures C.3 and C.4. The x- and z-coordinates of the holes in Figure B.5 (in which the bars were supported), wi th respect to the L D target centre, are given in Table B . l (The base plate was positioned such that these holes formed an arc of radius 3.05 m wi th respect to the L D target centre). Bar x-coord. z-coord Bar x-coord. z-coord (cm) (cm) (cm) (cm) 1 97.76 288.91 11 -5.24 304.95 2 87.79 292.09 12 -15.71 304.60 3 77.70 294.94 13 -26.16 303.88 4 67.53 297.43 14 -36.58 302.80 5 57.28 299.57 15 -46.96 301.36 6 46.96 301.36 16 -57.28 299.57 7 36.58 302.80 17 -67.53 297.43 8 26.16 303.88 18 -77.70 294.94 9 15.71 304.60 19 -87.79 292.09 10 5.24 304.9.5 20 -97.76 288.91 Table B . l : The x- and z- coordinates of the neutron bar longitudinal axes. Neutron bars: Plastic scintillation counters (BC408). Each bar 10 cm x 10 cm x 200 cm, wi th U V T transmitting acrylic light guides and Thorn EMI Electron Tubes 9821KB P M T s attached at each end (Figure B.6). Mid-length horizontal section of each bar was vertically offset by +13.3 cm wi th respect to the L D target centre. Appendix B. Detector Specifications 225 LU h-< _ l Q. CO CO LU z o X h-LO O *c C O o co I 3 ! —1 o < ? ^ E o 5 CD g CO n 03 i 0) ci) E CO _CD o x: CD CD O X CD E E d +1 co CD o c CO o C O C O LO CM Q CD CO CD -*—1 t o CD CO > CD co co 0) c E" o co g co c CD E LO d Q LO CM .£ d co CD ^ O CO CO CD > CD i _ CO CO jD O . C CD CO CD CO .!= CO c i — o o O Figure B.5: Drawing of the base plate of the neutron bar supporting stand. Appendix B. Detector Specifications 226 T 4.0" Figure B.6: Drawings of the neutron bar light guide (left) and its attachment to the P M T (right) (from [62]). B.4 Gamma-tag Detectors Each detector N a l ( T l ) , dimensions 3" (7.62 cm) in diameter x 3" long. 7-sources were placed 1 m behind neutron bar array, 6 cm in front of respective detector, and at ap-proximately mid-bar height. Lead shielding between detectors was 6.35 cm thick. Appendix C Photographs Figure C . l : Beam path equipment, showing (left to right) the bending magnet, the W C 1 / W C 2 wire chamber assembly, the L D target housing, scintillator S3, and the T I N A collimator. 227 Appendix C. Photographs 228 Figure C.2: (Short-flight-path configuration) L D target has been moved out of the beam path towards the collimator. Scintillator S2 can be seen in front of the tracking wire chamber assembly. Also visible are scintillator S4 (left) and the T I N A collimator (far right). Appendix C. Photographs 229 Figure C.3: (Short-flight-path configuration) Front view of the neutron bar array. Figure C.4: (Short-flight-path configuration) Rear view of the neutron bar array. The gamma-tag detectors can be seen on the right hand side of the picture. A p p e n d i x D C i r c u i t D i a g r a m s Detailed schematics of the E661 electronic circuitry are shown on the following pages. An index of the C A M A C modules that were used is given at the end of this appendix. 231 Appendix D. Circuit Diagrams 232 LO LO < O 1-co CO 1 c CM en CM CO 3 O b a) > c o o T> C (0 < Z 428F L L u. CO co CM CM 00 c L L co co LO CM <M < . z 1' u. 00 C\J 1 i Is6 Figure D . l : T I N A and converter circuitry. Appendix D. Circuit Diagrams 233 3 O T3 C CO V Q. O O </> O 0) I-E (0 a> m If p i > > * 0. S ftttt cn c • CO T f 1CM CO CM CM CM CO CM o CM TO CM CM " CM CO TO CM C CM TO CM CO < C\J T f CM CM ' CM CO t: o co =*K3TO —I K D < • CM CM CM CO TO CM c n T f CM CM < CD T f CM CM < cn T f CM CM < cn T f CM CM CM < c n T f CM CM < cn T f CM CM Figure D.2: Beam telescope and L A M circuitry. Appendix D. Circuit Diagrams 234 Figure D.3: Neutron bar and gamma-tag detector circuitry. Appendix D. Circuit Diagrams 235 O ob cn o C\J co CO 3 U i_ o 0> w 3 Q. E o T3 C (0 QC « o Figure D.4: S4 veto counter and random pulser circuitry. Appendix D. Circuit Diagrams 236 I n d e x o f E l e c t r o n i c M o d u l e s Mode l No. Description C212 Coincidence buffer 222 Dua l gate generator 364AL, 365AL 4-fold logic gate 365, 465 Coincidence unit 428F Linear fan-in-fan-out 429, 429A Logic fan-in-fan-out 454, 474 T i m i n g filter amplifier TC455 Constant fraction discriminator 621, 821 Quad discriminator 622 Quad coincidence unit 624 Octa l mean timer 630 Quad discriminator 1320 Output Register 2228A T D C 2249A, 2249W A D C 2341, 2341S 16-fold register Abbreviat ion At ten Attenuator 200ns 200 ns delay S18-1 Scaler module V S Visua l scaler A p p e n d i x E T h e C A M A C S i g n a l Sources u s e d i n E 6 6 1 Source No. Assignment Source No. Assignment 1 Bar It ADC 33 Bar 17t ADC 2 Bar lb ADC 34 Bar 17b ADC 3 Bar 2t ADC 35 Bar 18t ADC 4 Bar 2b ADC 36 Bar 18b ADC 5 Bar 3t ADC 37 Bar 19t ADC 6 Bar 3b ADC 38 Bar 19b ADC 7 Bar 4t ADC 39 Bar 20t ADC 8 Bar 4b ADC 40 Bar 20b ADC 9 Bar 5t ADC 41 10 Bar 5b ADC 42 11 Bar 6t ADC 43 12 Bar 6b ADC 44 13 Bar 7t ADC 45 14 Bar 7b ADC 46 15 Bar 8t ADC 47 Nal (22Na) ADC 16 Bar 8b ADC 48 Nal (60Co) ADC 17 Bar 9t ADC 49 Bar it TDC 18 Bar 9b ADC 50 Bar lb TDC 19 Bar lOt ADC 51 Bar 2t TDC 20 Bar 10b ADC 52 Bar 2b TDC 21 Bar l i t ADC 53 Bar 3t TDC 22 Bar l i b ADC 54 Bar 3b TDC 23 Bar 12t ADC 55 Bar 4t TDC 24 Bar 12b ADC 56 Bar 4b TDC 25 Bar 13t ADC 57 Bar 5t TDC 26 Bar 13b ADC 58 Bar 5b TDC 27 Bar 14t ADC 59 Bar 6t TDC 28 Bar 14b ADC 60 Bar 6b TDC 29 Bar 15t ADC 61 Bar 7t TDC 30 Bar 15b ADC 62 Bar 7b TDC 31 Bar 16t ADC 63 Bar 8t TDC 32 Bar 16b ADC 64 Bar 8b TDC Table E . l : E661 C A M A C signal sources (1-64). 237 Appendix E. The CAMAC Signal Sources used in E661 238 Source No. Assignment Source No. Assignment 65 Bar 9t TDC 97 66 Bar 9b TDC 98 S4 mean TDC 67 Bar lOt TDC 99 S4t TDC 68 Bar 10b TDC 100 S4b TDC 69 Bar l i t TDC 101 70 Bar l i b TDC 102 TINA 1 ADC 71 Bar 12t TDC 103 TINA 2 ADC 72 Bar 12b TDC 104 TINA 3 ADC 73 Bar 13t TDC 105 TINA 4 ADC 74 Bar 13b TDC 106 TINA 5 ADC 75 Bar 14t TDC 107 TINA 6 ADC 76 Bar 14b TDC 108 TINA 7 ADC 77 Bar 15t TDC 109 Converter 1 ADC 78 Bar 15b TDC 110 Converter 2 ADC 79 Bar 16t TDC 111 TINA sum ADC 80 Bar 16b TDC 112 TINA+Conv sum ADC 81 Bar 17t TDC 113 SI - S2 TDC 82 Bar 17b TDC 114 S2 - S2 TDC 83 Bar 18t TDC 115 S3 - S2 TDC 84 Bar 18b TDC 116 TINA+Conv - S2 TDC 85 Bar 19t TDC 117 Beam-cap - S2 TDC 86 Bar 19b TDC 118 Beam-cap+21 - S2 TDC 87 Bar 20t TDC 119 TINA - S2 TDC 88 Bar 20b TDC 120 Converter - S2 TDC 89 SI ADC 121 n-array coinc. reg. 90 S2 ADC 122 7 coinc. reg. 91 S3 ADC 92 S4t ADC 93 S4b ADC 94 95 96 Table E.2: E661 C A M A C signal sources (65-122). A p p e n d i x F A n E s t i m a t e o f t h e P h o t o n E n e r g y R e s o l u t i o n i n E 6 6 1 We use the non-relativistic expression for E1 given in Equation 5.6 (page 127), where q is given by 1 q2 = \{p2 + pl-2plP2cos6) ( F . l ) We wi l l define the cylindrical coordinates r and cf> for each neutron hit as shown in Figure F . l , such that r 2 = x2 + z2 = (305 cm) 2 (nominal value) (F.2) x (p = arctan - (F.3) The path length R for the neutron is given by R2 = r2 + y2 (FA) The neutron momentum is approximately given by |p | = mn x neutron velocity R(cm) 10 9 = m„ x — x t where t is the neutron T O F and c is the speed of light 100 i(ns) 939.57 - R / A r w / N x 10 7 x — ( M e V / c ) c t 31.34 x - ( M e V / c ) (F.5) 1 Refer to Figure 1.8 on page 16. 6 is defined on page 127. 239 Appendix F. An Estimate of the Photon Energy Resolution in E661 240 From Equations F.5 and F.4 we therefore have, for the two neutrons p\ P\ 982.2-982.2 (r22 + V\) t2 l2 (F.6) (F.7) To calculate cost9 we note that the direction cosines for each hit, for the coordinate system that we have chosen, are given by I --m n z r cos <p R R X r sin (j) R R y R so that cost9 I1I2 + miW2 + n\U2 rxr2 cos 4>\ cos c/>2 rxr2 sin <j>i sin 4>2 Ri~R~2 + " rxr2 \ / r i +y2\Jr2 + y2 RXR2 COS (0i - fa) + + ym R1R2 ym' r\r2. ( F i (F.9) Appendix F. An Estimate of the Photon Energy Resolution in E661 241 We substitute the expressions in Equations F.6, F .7 and F.9 into Equation F . l , to get q2 « 191 1 ( + V ^ I ^ 2 + ^ r i r 2 C 0 S ^ l ~ ^ y m ] \ 2t 2 2t2 t\t2 t\t2 J = 491.1 q'2 (F.10) where q' is defined to be the expression in the curly brackets. From Equation 5.6 (page 127), we have dE7 _ 1 d{q2) 1007.5975 so that the photon energy resolution is given by (F.ll) cr 2 491 1 oE = ^ = * CTV2 = 0.487 oQn (F.12) 1007.5975 1007.5975 q q y ' The problem is therefore reduced to calculating the resolution oqn, which is given by an equation of the form ( W 2 ) \ \ 2 , (HQ'2)W 2 , (HQ'2)\\ 2 / \2 " \1 I \ Ji i W I ^ 2 , 1 ) 2 '1 W2)\2 2 , (H<i'2)Y 2 , W T 2 2 '%,2)V 2 , W T 2 + -J7- a h + 1 7 - a t 2 ( F - 1 3 ) where o~ri,..., ot.2 are the resolutions of the measured values r 1 ; . . . , t2. From the detector geometry, we have 10 cm 2.355 0.0343 radians 2.355 4.25 cm (F.14) 0.015 radians (F.15) Appendix F. An Estimate of the Photon Energy Resolution in E661 242 The y-position resolutions vary between 5 and 10 cm, and are discussed in Section 4.2.9 (page 100): oyi and oy2 for each event can be estimated from the y-positions of the hits and the energies deposited in the bars, using appropriate interpolations of the results given in [62]. The time resolutions otl and ot2 are given by Equations 4.20 to 4.22 in Section 4.2.9. We note that r and t are strongly correlated, and, since the uncertainty in the neutron T O F s is dominated by the uncertainty in the depth wi th in the bar at which the neutrons scatter, we can say to a reasonable accuracy that t ~ KR ~ Kr (F.16) where K is a constant. Equation F.10 is therefore effectively independent of rx and 7-2, so that the terms in of and erf vanish from Equation F.13. The remaining partial derivatives are evaluated below: %'2) nr2 501 tit2 5(q'2) nr2 s i n ( ^ - 0 2 ) (F.17) o(f>2 txt2 %'2) yi V2 sin {<f>x - <h) (F.18) (F.19) 5yx t\ txt2 6y2 t\ txt2 'o~{q'2)\ = vm , **1 Jtv *?*2 '5{q'2)\ _ vm , fa ) t v txt2 '6{p\ = _ ( r [ + y D + 1 [ ^ c o s ^ - ^ - h ^ ] (F.23) , oti J t c li tih 'Hq'2)\ _ (4 + y22) +JL[nr2 c o s _ 0 2 ) + ym] ( F . 2 4 ) (F.20) (F.21) (F.22) 5t2 )tc t32 txi2 Note that we have evaluated two sets of partial derivatives with respect to time. The first three terms in q'2 (Equation F.10) do not contribute to the derivatives due Appendix F. An Estimate of the Photon Energy Resolution in E661 243 to the correlations outlined in Equation F.16. These derivatives w i l l be associated wi th the uncertainties otv, given by Equation 4.21. The last two derivatives in the above list wi l l be associated wi th the uncertainties otc, where otc — 0.4 ns from Equation 4.20. It should be noted that there is a small contribution to the y-position uncertainties oy that is correlated to r . This contribution ranges from zero at the bar centres to about ± 1 . 6 cm at the ends of the bars. We have neglected this correlation since it only contributes to a small fraction of the total y-position resolution. Our final expression for oqi2 is therefore 5 fa ) °* + [ SVl I w) ~ r x H °<h + I T - °<h  I T - A : + + ( R 2 5 ) The erg distribution in Figure 5.29 (page 151) was generated from a dedicated replay of the reduced E661 data set, in which the Ey resolution for each valid event was computed using Equations F.25 and F.12 above. 

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