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Spin-transfer measurement for the [pi] d --> pp reaction at energies spanning [delta] resonance Feltham, Andrew G. 1992

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S P I N - T R A N S F E R M E A S U R E M E N T S F O R T H E ird pp R E A C T I O N A T E N E R G I E S S P A N N I N G T H E A R E S O N A N C E B y Andrew G . Feltham B.Sc. (Hons), Carleton University, 1986 M . S c . University of B r i t i s h Co lumbia , 1988 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 We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A M a y 1992 © Andrew Fel tham, 1992 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 The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract We describe the first spin-transfer experiment performed for the pi to pp reaction. Three spin-transfer parameters were measured: KJ^s'^ ^'ss'^ ^'NN^ each, at a single angle for a number of energies spanning the A resonance of this system. T h e apparatus employed i n this experiment consisted of established systems, including a dynamical ly polarized deuteron target and a proton polarimeter which uti l ized the well known proton-carbon analyzing powers. T w o arms of detectors were used to minimize the background contribution by preferentially selecting those two-body final states corresponding to the nd pp reaction kinematics. We compare our results w i th the predictions of published part ia l wave amplitude fits pertaining to existing data of the time-reversed pp —> dir reaction, and demonstrate the inadequacy of these established fits i n providing a complete description of this fundamental process. In addit ion, our data are compared w i th the predictions of two current theories. T h e failure of these theories to describe the fundamental features of this reaction clearly demonstrates the need for further theoretical work i n this area. The data presented i n this thesis are essential for the unique determination of the par t ia l wave amplitudes characterizing the pp ^ dw reaction. The accurate experimental determination of these amplitudes wi l l provide an important framework for testing further theoretical developments. Table of Contents Abstract ii List of Tables vi i List of Figures vii i Acknowledgements x I Introduction 1 1.1 Introduction to the pp *-* dir Reaction 2 1.2 Introduction of Important Concepts 5 1.3 Introduction to the Experiment 10 1.4 Summary of Thesis 12 II T h e o r y 13 II. 1 Histor ica l Developments 14 11.2 Coupled Channel Models 16 11.3 Relat iv ist ic Calculations 19 11.4 Faddeev Type Calculations 23 11.5 Introduction to Ampl i tudes 24 11.5.1 Hel ic i ty and Helic ity P a r t i a l Wave Ampl i tudes 26 11.5.2 1-s P a r t i a l Wave AmpHtudes 29 III Previous W o r k 32 111.1 History of the Reaction 32 111.2 The Status of the Current D a t a Set 35 111.3 Previous Spin-transfer Experiments 37 111.3.1 The Geneva Experiment 38 111.3.2 The T R I U M F Experiment 40 111.3.3 The L A M P F Experiment 42 III. 4 AmpUtude Analysis Work 43 111.4.1 Bugg's 1-s P a r t i a l Wave Analys is 43 111.4.2 T h e W a t a r i 1-s P a r t i a l Wave Analys is 47 IIL4.3 The Russian Hehcity P a r t i a l Wave AmpUtudes 48 I V Motivat ion 49 I V . 1 Status of Ampl i tude Analyses 49 IV.2 W h y Spin-Transfer Observables? 51 IV .3 Advantages of this Experiment 54 I V . 4 Exper imenta l Philosophy 55 V T h e Polarimeter 57 V . l Polar izat ion Theory 57 V . 2 Fourier Techniques 63 V I T h e Experiment 67 V I . 1 P r i m a r y Detector Construction 68 V I . 1.1 The Scintillators 71 V I . 1.2 The W i r e Chambers 74 VI .2 The Target 81 VI.2.1 Target Construct ion 81 VI.2.2 Polar izat ion of Target 83 VI .3 Peripheral Detectors 86 VI .4 T h e D a t a Acquis i t ion System 88 VI.4.1 T h e J - 1 1 89 VI.4.2 The On-Hne Software 92 V I . 5 Exper imenta l Summary 93 V I I Off-line Analysis 94 V I I . 1 Replay Software 95 VII .2 CaHbration of Software 99 VII .3 Software Checks 101 VII .4 The P O L A R software 105 V I I . 5 Background Subtraction 108 VIII Polarization Formsdism and Spin-Transfer Results 117 V I I I . 1 General Description of a Reaction Involving Polarized Particles . 117 VIII .1.1 Aspects of Time-Reversal 126 V I I I . 1.2 Observable Nomenclature 127 V I I I . 2 Systematic Checks of Polar izat ion 129 I X Extract ion of Polarization Observables 131 I X . 1 Complicat ions 132 I X . 1.1 Equations of M o t i o n 133 IX .2 Pro ton Trajectory Mode l 135 I X . 3 Polar izat ion Extract ion Algebra 139 X E r r o r Evaluation 148 X . l Systematic Errors 148 X . 1.1 Scintil lator Gap Prob lem 159 X . 2 Confidence i n Results 161 X I Discussion and Conclusions X I . 1 D a t a Comparison and Evaluat ion 165 166 XI .2 Future Work 168 Bibliography 173 A A p p e n d i x 178 A . l Sub-Atomic Physics Pr imer 179 List of Tables I Nomenclature for 1-s part ia l wave amplitudes 31 II Ampl i tude constraints of U B C fitting package 51 III Detector angles and distances for various configurations 68 I V Differences for various target configurations 82 V Table defining essential angles and kinematic quantities calculated i n R E P L A Y 115 V I Typ i ca l software efficiencies 116 V I I Transformation between tensor and Cartesian polar izat ion and analysing power observables 119 V I I I Relationship between tensor and Cartesian spin-transfer observables. 121 I X Polar izat ion components wi th respect to the axis of quantization . 122 X Transformation between analysis and Mad i son coordinate systems of Cartesian spin-transfer observables 126 X I Transformation between analysis, Mad i son and time-reversed co-ordinate systems of Cartesian spin-transfer observables 127 X I I Spin-transfer results w i th statistical errors 144 X I I I Comparison of measured and expected normal polarization at the target 146 X I V Comparison of s imilar spin-transfer quantities obtained from dif-ferent r u n sets 150 X V Analys ing power errors resulting from uncertainties i n beam energy 152 X V I Comparison of measured normal polarization of protons at the target w i th expected value using an unpolarized deuteron target. . 154 X V I I Quantitat ive summary of systematic error contributions 163 X V I I I Spin-transfer results w i th experimental errors 164 X I X Spin-transfer results for the pp —* d-n reaction 166 X X Table of nature's constituent particles 181 X X I Properties of important particles in nuclear physics 184 List of Figures 1 One pion exchange i n Nucléon-Nucléon interaction 3 2 Basic p ion production mechanisms 15 3 M u l t i p l e scattering i n hadronic interactions 16 4 P i o n product ion i n the coupled channels model 17 5 NA transit ion potential of the coupled channels model 18 6 Comparison of existing experimental data and current theoretical predictions 20 7 Leading terms of relativistic calculations 21 8 Three-body NA interaction term 25 9 Energy dependence of the wd —* pp total cross-section 34 10 A r g a n d plots depicting the P W A ' S of the Bugg fit 46 11 Insensitivity of the tensor analysing powers to various P W A sets . . . 52 12 Predictions of spin-transfer observables for various P W A solutions . . 53 13 Energy and angle dependence of proton-carbon analysing powers and cross section 61 14 Layout of experimental area 69 15 Scinti l lator signal flow 72 16 Energy loss i n scintillators and a r m B T O F 73 17 Schematic diagram of a mult i -wire drift chamber 75 18 F low of wire chamber electronic signals 76 19 T y p i c a l wire chamber histograms 78 20 P lot indicat ing typical resolution of wire chambers 80 21 E V E N T logic 89 22 Effect of J-11 cut on accepted data 91 23 F low of off-line replay software 97 24 Definit ion of coordinate system and angles used i n analysis 98 25 Examples of traceback histograms produced by R E P L A Y 103 26 Examples of histograms produced by R E P L A Y 104 27 Examples of systematic checks of polarization 107 28 Sl ic ing technique used in background fits of kinematic data 109 29 F i t of Gaussian curve to background kinematics d istr ibut ion I l l 30 Technique for determining the ratio of background under the fore-ground peak 112 31 Kinemat ics dependence of polarizat ion wi th background target. . . . 115 32 Comparison of coordinate systems for the analysis and M a d i s o n frames. 124 33 Comparison of coordinate systems for time reversed reactions 128 34 Definitions of angles for boosts from centre of mass frame to labora-tory frame 136 35 Progression of spin rotations applied i n model 136 36 Comparison of Pjv and ANO for each target configuration 147 37 P - A results for data obtained w i th an unpolarized deuteron target . 155 38 Effect of J-11 on 9carbon d istr ibution 159 39 I l lustration of scintillator gap effect on dtarget d istr ibution of PN • • • 160 40 Scinti l lator x-projection, i l lustrating gap 161 41 Comparison of new spin-transfer data w i th the prediction of the Bugg P W A ' s 167 42 Comparison of new spin-transfer data w i t h several theoretical pre-dictions 169 43 Hierarchy of scientific disciplines 180 44 Schematic representation of nucleon-nucleon interaction i n terms of quarks 182 Acknowledgements T h i s thesis would not have been possible without the valuable contributions of many people. I would foremost like to acknowledge a l l the participants of T R I U M F experiment 331. In particular I would like to thank my supervisor G a r t h Jones for his guidance throughout the course of the experiment and the wr i t ing of this thesis. I would also like to acknowledge the contributions of Marcel lo Pavan , Peter Trelle, Ted Math ie , and Peter Weber which were made on several occasions from beginning to end of this project. In addit ion, I thank G a r t h and Reena Meijer Drees for the many hours spent i n reading the "intermediate" versions of this thesis and offering valuable suggestions. F ina l ly , one does not spend three years working on a thesis alone. Thus I would like to thank a l l the people who have made this t ime an interesting experience for me and have managed to put up w i t h me (they know who they are.... I hope). Chapter I Introduction T h e discipHne of physics has without question been greatly advanced by two circumstances: the development of new technology; and the re-evaluation of old problems by new approaches. A s examples, one need only cite the progress which resulted from the cyclotron technology of Lawrence and Liv ingston [1] or the new understanding of the world which resulted from Einstein 's theory of relativity. It is i n this spirit that we have returned to an old problem of sub-atomic physics: the precise quantitative description of the fundamental p ion absorpt ion/product ion reaction, pp ^ d'K. To do so, we have invoked relatively new, yet well developed technologies associated w i th polarized targets and polarimeters. W h a t is novel about the measurements described i n this thesis is that they are the first spin-transfer measurements to be performed for the ird pp reaction (the few others have been performed on the time-reversed pp d-K reaction). In addit ion, this experiment is the only spin-transfer measurement to encompass the range of energies which span the A resonance of this system. T h e ult imate goal of such work is to obtain sufficient data to permit a unique, unbiased determination of the part ia l wave amplitudes characterizing this important reaction. These amplitudes axe essential for the detailed testing of theories which form the basis for our understanding of the fundamental processes responsible for the production and absorption of pions. T h e remainder of this introduct ion is intended to prepare the reader for topics which are discussed later i n this thesis. Section I . l treats the importance of the pp ^ div reaction, section 1.2 introduces important concepts used throughout the thesis, section 1.3 presents details of the experiment and section 1.4 reviews the contents of the complete thesis. A n y readers unfamil iar w i t h the field of sub-atomic physics are referred to the primer given i n appendix A . l . I.l Introduction to the pp dir Reaction Of the many processes occurring i n nuclear physics, one of the most interesting is that of p ion absorption by, and pion production i n , nuclear systems. In fact the appearance and disappearance of this unique particle has fed the curiosity of physicists for many decades. Of part icular importance is the role of the pion as a mediator of the strong nuclear force. Th i s concept, introduced i n appendix A . l , is diagrammatical ly represented i n the interaction between two nucléons as i n F i g . 1. In such an interaction, the existence of the pion is not detected by the observer^, as it is both produced and absorbed completely w i t h i n the two nucléon system. Another important process is the absorption or product ion of a free pion which can be observed directly i n an experiment^. A s suggested by F i g . 1, such a process is int imately related to the basic interaction between baryons. In addit ion, the absorbtion or production of a p ion (which has significant mass) involves a large change i n the momentum of the associated nuclear systems. According to the Heisenberg uncertainty principle, a large change i n the momentum of a particle is associated w i t h an interaction occurring over a short distance: fi ApAx > -In other words, free p ion absorption or production processes are sensitive to ^This is often referred to as a virtual pion. ^Production can only occur when two nuclear systems collide with centre of mass energy greater than the mass of the pion. Figure 1: One pion exchange in Nucléon-Nucléon interaction. interactions which occur at very short ranges i n the nuclear system. Such short range nuclear processes might be expected to display sensitivity to effects which can only be described i n terms of quark interactions^, as discussed in appendix A . l . Thus the study of p ion absorption and production can offer insight into physics at both the nuclear and sub-nuclear (quark) levels. In the spirit of few-body nuclear physics, it is of interest to identify the simplest system in which pion production and absorption can be observed. M o m e n t u m and energy conservation dictate that a free p ion cannot be absorbed or produced by a single free nucléon. Hence one must resort to a system consisting of at least two nucléons i n order to produce or absorb a free pion. This can be done using a reaction of the form: where N represents a nucléon, either a proton or a neutron. Th i s general reaction expression can manifest itself i n several forms, for various charge states (ie. w i th neutrons, protons and 7r*'°). A particularly important one is the pp ^ dn reaction, where the deuteron (d) is a neutron-proton bound state w i th total spin 1^. T h i s is an especially useful reaction from an experimental point of view because both the in i t i a l and final states are two-body i n nature and thus the ^(which only become important over distances much shorter than the diameter of the nucléon) particles have well-defined kinematics. In addit ion, a l l particles i n the in i t ia l a j i d final states are charged, making them easy to detect. F r o m the viewpoint of theory, the two-body final states are also easier to work w i th , especially since the deuteron bound state has been studied theoretically for many years and is wel l understood. Another important feature is the dominance of the A particle at intermediate energies through the transitions: pp ^ NA ^ dn. The reaction therefore provides a mechanism for studying the production and propagation of this important nuclear particle. In the same manner, this is also the simplest inelastic process i n which one may look for an elusive di-baryon^ (^B) resonance (ie. pp B ^ dir). The unique nature of such a particle has made its potent ial existence the object of much work. For these reasons, the pp ^ dn reaction has been of great interest to experimentalists and theorists alike for many years. A s a result of the invariance of strong interactions under time-reversal, the product ion of a p ion involves the same physics as does its absorption. The one is s imply a time-reversed version of the other (as watching the same event proceed both forwards and backwards i n time^. Th is concept is known as time reversal invariance and is of fundamental importance i n nuclear physics. As wi l l be discussed i n later chapters, many pp ^ dn experiments have been performed over the last few decades, most of which have been done i n the pp dir direction. Since p ion absorption and production are essentially the same process, i n principle the same information can be obtained from experiments performed in either direction. Indeed some experiments, including the work described i n this thesis, have been done i n the inverse ird —> pp direction. A l though the implications of this are discussed throughout the thesis, the reader should be aware of the distinctions between the two directions from the outset. '*A di-baryon is a tightly bound state of two nucléons which forms a single entity of baryon number 2. ^(similar to watching a movie run forwards and backwards) 1.2 Introduction of Important Concepts T h i s section is intended to introduce the reader who is unfamihar w i th the details of the field of sub-atomic physics to most of the important terms found throughout this thesis. Experiments throughout this century have shown that quantum mechanics [2] is successful i n describing the many features of the sub-atomic world . In quantum mechanics a system of one or more particles is described i n terms of the states which the system may occupy. These states, l ike the colour, model and manufacturer of a car, provide a unique label for the particles. Some properties, such as the momentum or position of the system, can take a continuous range of values, whereas others, like mass, charge, and spin can only assume discrete (or quantized) values. The related quantities spin, orbital angular momentum and total angular momentum are of fundamental importance to this work. Sp in is a property of an ind iv idua l particle having no exact macroscopic equivalent, although it is intuit ive ly related to the spinning of a r ig id body. O r b i t a l angular momentum describes the angular motion of two or more bodies (particles) about one another. B o t h spin and orbi ta l angular momentum are vector quantities described by three characteristics: the magnitude of the spin vector (\Js(s + 1)^.)^ or angular momentum vector (yfl(l + 1)^), expressed i n terms of the quantum number s (or /) which characterizes the system; the axis of quantization, an arbitrary direction i n space to which the spin or angular momentum vectors may be referred; and a projection which is a scalar representing the extent to which the direction of the spin or angular momentum is aligned w i th the axis of quantization. The projections, often denoted by m , can take on 2^ -|- 1 values ranging over is a fundamental quantum of angular momentum where ft = and h is Planck's constant. m = —s, —s + 1 , . . . + s. Tota l angular momentum is the vector sum of the total spin^ and orb i ta l angular momentum vectors of a system. It also is characterized by a quantization axis and projection. Orb i ta l angular momentum states are frequently referred to as: s-wave, p-wave, d-wave, etc. states representing respectively angular momenta of 0^, Ih, 2% W h e n studying a reaction between two or more particles, the in i t i a l and f inal states can be characterized i n terms of those properties described above. Of part icular interest when studying a reaction is the probability that a system i n one of the possible in i t i a l states w i l l end up i n a specific f inal state. Th i s is known as the transition amplitude between the two states. Specification of a l l the transit ion amplitudes provides a complete quantitative description of the reaction. A n observable is a measurable property of a system. Experiments are performed to measure observables. In general, a system is prepared repeatedly i n a particular in i t i a l state, the reaction occurs each time, and the experimenter measures an observable characterizing the resulting final states. In this manner, one (or a combination of) transit ion amplitude(s) is determined statistically. T h e design of the experiment depends on which in i t ia l and final states are being studied. For pract ical reasons, it is often simpler to prepare a system in a specific in i t i a l spin state rather than i n a known angular momentum (etc.) state. However, i n practice, it is generally impossible to prepare a system of particles (such as a beam or target) wi th a l l of their spins point ing i n the same direction. It is therefore useful to introduce the statistical quantity, polarization, which describes the number of particles having a given spin projection relative to the total niimber of particles. Polar izat ion is identified by two quantities: a magnitude describing the relative population of states; and a direction relating to the spin Total spin refers to the vector sum of spins of a system of particles. axis of quantization to a coordinate system of interest. The polarization tensor is typical ly represented i n one of two coordinate systems: Cartesian , or spherical. In general, Cartesian tensors lend themselves to a simple physical interpretation i n an experiment, whereas spherical tensors enjoy simple rotational properties and are more straightforward to employ when describing systems wi th many possible spin-projections. The work i n this thesis is concerned only w i th particles of s p i n - | (protons) and spin-1 (deuterons)^. Following the prescription discussed earlier i n this section, these particles can have two and three projection states, respectively. For the protons, these projections are denoted by m = ± | , and for the deuteron, m = — 1 , 0 , I n this thesis (following common conventions), vector polarization refers to the relative numbers of particles whose projection is parallel or anti-parallel w i t h the axis of quantization (m = ± 5 ) . Quantitatively, the magnitude of the vector polarization is given by [3]: _ n+-n. -^ 1 ~ \ i where n± refers to the numbers of particles i n the ± 3 states (aligned or anti-aligned) and UQ refers to the number of particles (deuterons) i n the orthogonal m = 0 state. The tensor polarization provides information concerning the relative number of particles i n the m = 0 state and is defined as follows [3]: Pn = « 4 . - f rzo + n_ Obviously only the spin-1 deuterons can experience a tensor polarization. F ina l ly , one can introduce polarization dependent observables. Three types of such observables are discussed i n this thesis. Perhaps the conceptually simplest *In addition there are pions which have spin 0 and thus cannot be spin-polarized. experiment which can be performed is the measurement of the differential cross-section ( ^ ) . Th i s measurement is merely a comparison of the number of particles i n the i n i t i a l state of a reaction to the number which arrive in a particular region of phase space of the final state (denoted by dÇl), averaged over any unspecified spin degrees of freedom of the in i t ia l state. If one of the in i t ia l particles is polarized, the angular dependence of the differential cross-section is modified. For a vector polarized in i t ia l state, an azimuthal asymmetry (e) of the differential cross-section is observed. Th i s asymmetry is quantitatively defined as [4]: ' NL + NR where (NL) and (NR) represent the numbers of particles scattering to the left or right of the incident beam. If the in i t i a l particles experience a tensor polarization, a more complicated modification to the angular distr ibut ion results. In general, the quantity which characterizes the change i n the cross-section due to a polarization (/>,) i n the in i t i a l state is known as an analysing power (Ai). dcTpol daunpol ^ l + J2Ap] i I dQ da where i refers to the projection of the in i t ia l polarization i n the relevant coordinate system. Of part icular importance to this thesis is the analysing power related to vector polarization i n the in i t ia l state^. This can be determined from the ratio of the scattering asymmetry (e) to the magnitude of the polarization ( P ) : - = l T h e analysing power provides a relationship between the simple cross-section and the in i t ia l spin states of the reaction. As wi th polarization, we w i l l also distinguish between tensor and vector analysing powers throughout the thesis. These terms ^For a more complete discussion of the analysing power associated with tensor polarization in the initial state, see Ref. [5]. indicate whether the analysing power is associated w i t h tensor or vector polarization i n the in i t ia l state. A lso associated w i th analysing powers are the observables which describe the cross-section modification due to the polarization of bo th in i t ia l state particles. These are known as spin-correlation parameters. A l though these quantities have no direct relevance to this work^°, they are introduced here since many such observables have been measured for the pp dx reaction. Another important experiment is one where the polarization of one or more particles of the final state is measured. Such experiments are performed using a polarization measuring device known as a polarimeter. The principle by which most polarimeters operate is also based on the asymmetry of scattering result ing from the analysing power of a reaction. In this case, the known analysing power of a "secondary" reaction is used to evaluate the polarization of the final state particles (of the studied reaction) through the measurement of the scattering asymmetry i n the secondary reaction. T h e last polarization observable which is discussed is the spin-transfer observable. Th i s quantity relates the polarization of a final state particle to the polarization of those i n the in i t ia l state. Spin-transfer observables are described i n detail i n the next section. T h e notation used i n this thesis is as follows: polarization observables are always expressed i n lower case, w i th a t representing the spherical tensor and p the Cartesian tensor; the spherical tensors representing analysing powers and spin-transfer quantities are represented by either upper or lower case t or T (specific conventions are given i n section V I I I . 1.2); the Cartesian tensor representing analysing powers is given by an upper case A and spin-transfer observables an upper case K. F ina l ly , the term analysing power w i l l be used in ^°Only the initial state deuteron can be polarized for the wd —>• pp reaction. two contexts: as an experimental quantity of pr imary interest, revealing directly properties of the reaction being studied and therefore the object of an experiment; and also as an experimental quantity of secondary interest, used as a tool , i n conjunction w i t h a polarimeter to determine the polarization of the system of particles i n the final state of a pr imary reaction. The measured polarization is i n t u r n related to the properties of the studied reaction. In the latter context, the analysing power is preferably known i n advance of, or measured i n conjunction w i t h the experiment. 1.3 Introduction to the Experiment The experimental work described i n this thesis involved the measurement of three spin-transfer observables of the ird —> pp reaction. A n intuit ive definition of a spin-transfer observable is: a parameter which describes how the spin of an in i t i a l state particle affects (is transferred to) the spin of a final state particle. In this part icular experiment, we studied the transfer of deuteron spin to the spin of a final state proton i n the reaction: Trd^ pp (1) In expression 1, the arrow over a particle indicates its polarization was known or measured. Th i s experiment was the f i r s t spin-transfer measurement performed for the ird—*^ pp reaction. The three separate parameters which were measured are denoted by K'j^g, K'ss and K'^j^ where the primes indicate the Trd pp direction, i n contrast to similar observables describing the pp —> di: reaction (which are wr i t ten without primes). The subscripts refer to, i n order, the direction of the target vector polarization and the measured direction of the polarization of the final state proton. E a c h direction is defined i n an appropriate coordinate system. S denotes sideways, N normal, and L is longitudinal. These correspond to the direction of x, y and z respectively in either of two ut i l ized coordinate systems (see F i g . 24 and F i g . 32). These systems are referred to i n this thesis as the Madison [6] and analysis conventions. B o t h coordinate systems follow the " r ight -hand" rule w i t h the i - ax i s defined along the trajectories of the incident particles. The difference between these two systems involves the definition of the positive direction of the 2-axis. In the Madison convention the positive z direction is that of the deuteron's motion i n the centre of mass frame. We employ this convention to be consistent w i th other published polarization measiirements. In the analysis convention the positive i - a x i s is that of the direction of the incident pion beam i n the laboratory. Th is system is a more natura l one to use from the analysis point of view. In both systems the y-axis is defined by the normal to the scattering plane which can be determined from the in i t i a l (A;,) and final (kj) momentum vectors: ki x kj. The apparatus used i n this experiment was situated i n a p ion beam-line which delivered pions incident on a target containing deuterons. T w o arms of detectors were used to identify both final state protons of the 7rd pp reaction. T h e arms were placed at conjugate angles defined by the kinematics of the reaction. The forward a r m doubled as a polarimeter which served to measure the polarization of the protons entering this device. E a c h a r m contained position-sensitive instruments for providing the trajectory information essential to both the kinematics and polarization analyses. The system of polarized deuterons was contained i n a target vessel, w i th the polarization axis defined by a 2.5 Tesla magnetic field i n this region. T h e trajectory and polarization of the forward traveling proton produced i n the pion absorption reaction was measured whereas only trajectory information was obtained for the accompanying proton. The axis of quantization used to describe the polarization of the protons was referred to the coordinate systems defined above, i n terms of the trajectory information provided by the detector arms, together w i th that of the incident pion beam. 1.4 Summary of Thesis The contents of this thesis are as follows. Chapters II and III present the history and status of theoretical and experimental progress i n describing the pp ^ d-rr reaction. In addit ion, some discussion is devoted to the formalism used to describe the amplitudes and the efforts to define these amplitudes experimentally. Chapter I V reviews the reasons and motivations for the spin-transfer measurement undertaken. The remaining chapters pertain directly to the description of the experimental measurements and the associated results. In chapter V the theory and implementation of a polarimeter designed to measure the polarization of protons is discussed. Chapters V I and V I I describe the experimental apparatus and the computer software used i n the "on-l ine" and "off-line" analyses of the data. The formalism for defining spin-transfer observables is presented i n chapter V I I I . In addit ion an approach for identifying possible systematic errors is introduced. In chapter I X , several experimental complications are discussed, and the measures developed to deal w i t h them are given. Chapter X serves to discuss and evaluate the experimental errors. In addit ion , techniques are described which were used to reduce or eliminate various systematic errors which were identified. A summary and conclusions are given i n chapter X L Chapter II Theory Al though the aims of this thesis are experimental i n nature, the results of an experiment are of l imited value i f no theory exists to interpret the data. A theory offers a quantitative description of the process being studied and thus provides predictions of those quantities which are observed i n an experiment. Theory and experiment should complement each other. Frequently new experimental results w i l l motivate a new theory. Alternatively, the desire to test a new theory may prompt a new experiment. It is the a im of this chapter to outline the history and status of the relevant theories which attempt to describe pion production and absorption. In addit ion, the final section w i l l discuss the parameterization of reaction observables and how they can be used by both theorists and experimentalists alike to identify inconsistencies or uncertainties i n theory or experiment. W h e n developing a theory to describe a nuclear reaction, one must select those microscopic processes which are l ikely to occur i n order to produce the effects which are observed i n an experiment. These processes are given mathematical descriptions and are often found to obey various physical postulates such as conservation laws^. Theories can exist at various levels ranging from very fundamental quark calculations to ideas based on crude approximations. Ideally, a theory should involve the interactions between fundamental particles, like quarks and leptons. However, knowledge i n these areas is s t i l l insufficient for yielding ^ (unless the theory is assuming the violation of those conservation laws) meaningful results at the nuclear level. Thus , it is usual to rely on simpler approximate models to identify the important features of a reaction. There are several excellent reviews presenting the history and status of theoretical efforts to describe the pp ^ div reaction [7,8,9,10]. A brief summary of the major developments of the theoretical approaches is presented in the following sections. II. 1 Historical Developments E a r l y work i n the theory of pion production was quite simple i n nature. T h e first models relied on a naive pion production mechanism known as the impulse approximation. This approach is based upon the matr ix element of a Hami l t on ian describing the non-relativistic coupling of a p ion to a nucléon between in i t i a l and final state nucléon wave functions: Th i s simple process, shown diagrammatical ly i n F i g . 2a, greatly underestimated the cross-section^. It was soon recognized that inclusion of the TTN s-wave and p-wave pion rescattering diagrams of F i g . 2b and F i g . 2c was required i n order to describe the gross features of the data. The s-wave mechanism was important for describing the threshold behaviour of p ion production whereas explicit inclusion of p-wave rescattering was essential because of the strong TTAT resonance (the A ) i n this channel. The effects of in i t ia l and final state interactions amongst the nucléons could be introduced through distortions i n the incoming plane waves and the outgoing spherical waves. More recent improvements to this model have included the addit ion of p and 27r exchange terms in the intermediate state [11 . These theories tend to involve several adjustable parameters, such as ^Early work focused on describing the total and differential cross-sections of the reaction. Figure 2: Basic p ion production mechanisms: a) Impulse approximation, b) s-wave rescattering; the blob represents the s-wave irN rescattering. c) p-wave rescattering; the solid box represents the p-wave TTN rescattering, (which includes the possibility of A formation). four-body state N , N , J L 1 \ ' ' / 1 \ ' 1 1 \ ' . / 1 \ • 1 / / 1 , \ / 1 , \ / \ , • / \ ' / N . Figure 3: T y p i c a l mult iple scattering contributions which were not explic it ly han-dled i n the older theoretical calculations of section II. 1. The dotted box outlines a four-body contribution to the mult iple scattering. coupling strengths and momentum cut-offs. A s a result most theories are capable of f i tt ing the basic cross-section data. However, it is generally accepted that this approach is incapable of providing a good description of the complicated multiple scattering processes characteristic of hadronic interactions (such as those suggested i n F i g . 3). T h e more modern approaches, which have benefitted greatly from the earlier models, attempt to treat the mult iple scattering which is prominent in the intermediate state i n a more realistic fashion. These theories are discussed i n brief detail i n the following sections, w i th the important features of each emphasized. II.2 Coupled Channel Models This model was developed several years ago by the Hels inki group of Green et a l . [12,13,14] i n an effort to describe more correctly the product ion and rescattering of the pion i n the nuclear environment. They felt that since pion production from the A was a significant process it should be treated w i th the same importance as production from the nucléon. T h e diagrams representing the production process are given in F i g . 4. In this case, the p ion product ion operator N NN N NA N Figure 4: P i o n production i n the coupled channels model. E q u a l weight is given to p ion production by A and nucléon. is sandwiched between two baryon wave functions where the in i t i a l state consists of NN or i V A and the final state a deuteron (also NN') and a free pion. The impulse approximation is again used as the basic product ion mechanism, s-wave rescattering of the pion is included as i n F i g . 2b to account for the threshold behaviour of the reaction. The fundamental improvement of this model over the earlier calculations is that the A is now introduced at the same level as the nucléon. The A ' ' A system is created i n a consistent approach which couples the NN, NA, and A A channels using NN NN, NN NA, NN A A etc. transit ion potentials^. Hence the creation of the NA wave function arises through solving a coupled set of "Schrodinger-like" differential equations which contain the various transit ion potentials. ^In real intermediate energy calculations the A A channels are often assumed negligible, for energetic reasons, and hence omitted. N N Figure 5: NA transit ion potential of the coupled channels model . B o t h forward and backward scattered mesons (7r ,p) are included. A n attractive feature of this model is that the NN scattering potential resulting from the solution of the coupled channels can be compared w i t h known NN phase shifts^. This l ink wi th the well known NN elastic scattering process provides a useful constraint for the theory. T h e transit ion potential used by Niskanen [14] is shown i n F i g . 5. The interaction, based on the simple one boson exchange ( O B E ) theory [15], readily permits the inclusion of both pions and rho mesons as exchange particles. In general, Niskanen's results have been the most successful at describing the pp —»• dw reaction. In particular , the description of some polarization observables (see F i g . 6), which are sensitive to more subtle effects of the interaction, are qualitatively reasonable. However, despite its strengths, this approach st i l l fails i n a number of important aspects. In part icular , it is unable to correctly predict the ''(obtained from parameterizations of the considerable NN scattering data set) angular dependence of the differential cross-section and it has completely missed for the spin-correlation parameter ANN- A S w i th many other models, this theory is for the most part non-relativistic, part icularly in its dependence on potentials^. Hence one might expect this approach to break down at higher energies where relativistic effects are more important . Another issue which is not fully addressed is that of unitarity^. This can only be properly included i n a model which includes a l l possible channels of the NN ^ NNTT reaction. Another important concern is the restriction to two-body interactions (through the potentials) which means that the more complex interactions between the NNTT, J V A T T etc. systems are not rigorously included. Other theoretical approaches which attempt to address some of the above issues are discussed i n the following sections. However it should be emphasized that at present, none of these theories enjoy the same success at describing the data as does the coupled channels model . A s an example, results from this theory are compared i n F i g . 6 wi th results of some other approaches (which are discussed i n the following sections), together w i t h experimental data. II.3 Relativistic Calculations The advent of higher energy data has prompted questions regarding the validity of various theoretical approaches at A-resonance energies and above. One must ask i f a theory which is not Lorentz invariant can adequately describe a relativistic process. In the past decade several new calculations have appeared which have attempted to describe the pion product ion process i n a completely covariant manner [16,17,18]. Some of the important features of these models are given in this section. ^These potentials are not Lorentz invariant. ^Unitarity implies conservation of probability, such that all possible transitions are accounted for. 1.0 0.8 0.6 0.4 0.2 0.0 1.0 0.6 0.2 -0.2 -0.6 -1.0 1.0 0.8 0.6 0.4 0.2 0.0 2 2 COS 9 COS 0 COS d Figure 6: Comparison of existing experimental differential cross-section, analysing power and spin-correlation data of the pp —> dir reaction at Tp— 580 M e V , to current theoretical predictions. The solid lines represent Niskanen's predictions (see section II.2), the dashed lines those of Locher (see section II.3) and the dash-dot lines are the results of Blankleider (see section II.4). a ) P ^ b) dpn V n V dpn Figure 7: Leading terms of relativistic calculations: a) the pion rescattering d ia -gram; b) the second leading term, neutron exchange. The leading term of a relativistic calculation is given i n F i g . 7a. The important terms of this diagram are the interaction vertices VT^NN-, Vdpn and the off-shell -KN elastic scattering ampHtude T-^N- Another important contribution is given i n F i g . 7b where the VT^NN vertex produces a real p ion and the neutron becomes the exchange particle. In each of these diagrams the vertices and propagators are treated covariantly which allows for the exchange of anti-particles. A formalism is provided which allows the masses to go off their mass-shelF and the A is treated as an off-shell effect i n the -KN interaction. For a starting point i n the off-shell prescription, the on-shell values for the scattering amplitudes are taken from phase shift analyses of experimental data. In following this approach, one can study the ^The mass shell is defined by the kinematic relation: vn? = E"^ -p^. Only virtual particles in the intermediate states can deviate from this relationship. poorly understood ofF-shell behaviour of nuclear interactions. Conversely since the off-shell behaviour is not well known, some uncertainty is introduced to the theory. W i t h the basic features of the reaction mechanism described, as i n F i g . 7, the effects of i n i t i a l and f inal state interactions can be introduced to the scattering matr ix ( M ) following the Sopkovich formalism [19]: where J is the total angular momentum of a part icular reaction channel, and ^pp/Trd ^ scattering matrices of the pp and TTC? elastic scattering channels. It is through these S matr ix elements that one describes distortions to the incoming and outgoing waves. T h i s model has had some success i n predicting the cross-section data of the pp —> dir reaction (for example, the Locher curves of F i g . 6). However it can c la im only very l imited success i n predicting the polarizat ion observables of this reaction. A l though it describes the general shape of many of these observables, it is quantitatively incorrect. M u c h of the model's failure has been attr ibuted to its underestimate of the magnitude of those amplitudes related to the reaction channels involving the pp triplet spin state [18]. In addit ion, models such as these are subjected to "double counting" difficulties, where a reaction mechanism has been included more than once through different diagrams. For example, it can be shown that the neutron exchange diagram of F i g . 7b can be fundamentally included i n the leading diagram 7a [20]. Th i s latter concern is dealt w i th to a certain extent i n the more rigorous three-body theories discussed i n the next section. II.4 Faddeev Type Calculations This approach to the NNTT system is based upon the Faddeev technique for solving three-body interactions. The calculation involves the summation over a l l mult iple scattering diagrams of the type represented in F i g . 3 (except for those requiring four-body states such as the one identified i n F i g . 3). Th i s is a very strong feature of this technique. Such a model also includes the explicit calculation of interaction potentials (VNN^NN, VArjv,AfAj etc.) from the appropriate meson-baryon vertices, and hence avoids the two-body phenomenological potentials used i n the coupled channels model . The three-body approach has probably been the most extensively studied model over the past decade w i t h various calculations being performed by several groups [21,22,23,24] (and see review [25]). The pr ime incentive for these calculations is that by applying a coupled channels approach, one can i n principle solve the entire set of NNTT systems: NN 1 ( NN ird (2) TTNN Such a model is attractive because it satisfies unitar i ty and provides "global" predictions for a variety of different reactions. A l though it is not inherently a covariant theory, sophisticated relativistic corrections can be included i n the calculations without significant difficulty [7]. Some of the difficulties w i t h this model include its complexity which requires use of various assumptions to be implemented i n any practical calculation. The calculation of Blankleider [26], which follows a Faddeev approach, is shown in F i g . 6. Other more fundamental difficulties are also present. A satisfactory mechanism for pion production or absorption does not yet exist, as the complete picture would require contributions from more than three-body terms which are necessary to describe a l l aspects of mult ip le scattering (see F i g . 3). Th i s is beyond the realm of regular Faddeev theory. A s well , this approach is not very sensitive to short range interactions, which are an essential component of the product ion/absorpt ion process. This results again from the complexity of the mathematics which prohibits the inclusion of effects such as /?-meson exchange [7]. One of the groups attempts to deal w i th the lack of short range sensitivity by retaining some phenomenological potentials. The model of Lee and M a t s u y a m a [21,27,28,29] includes transitions such as NN NN and iVA'' -> NA through potential models, such as the Paris potential , which impl i c i t ly contain short range effects. Hence, as w i th the coupled channels model of section n . 2 , comparisons can be made to known phase shifts. The three-body interactions are explic it ly included for transitions such as NA NA as i n F i g . 8, which describe the propagation and interactions of the A i n the nuclear system. Unfortunately, the results of this work offer l itt le improvement to the calculations of Blankleider . T h e Faddeev type models have seen only qualified success considering the amount of work being done i n this area. Perhaps their strongest feature is their abil ity to simultaneously provide a reasonable description of different reaction channels. However, most of the calculations encounter difficulties predicting the observed magnitude of the reaction cross sections, as well as i n describing the polarization observables. Interestingly, this approach seems to underestimate the tr iplet -spin pp channels i n a manner s imilar to the relativistic models of section 11.3. II.5 Introduction to Amplitudes It is the fundamental goal of a theory to describe the transit ion of a system from an in i t i a l state to its final state. Th i s is most conveniently depicted in terms of a Figure 8: Three-body A ' 'A interaction term, TNA,NA of the Lee and M a t s u y a m a model, describing the interaction and propagation of the A i n a simple nuclear environment. The circles represent rescattering mechanisms. complex transit ion amplitude: a = {f\T\i) (3) which represents the projection onto a part icular final state |/) of a specific i n i t i a l state |z) which has been transformed by the operator T containing the physics of the reaction. The in i t ia l and final state bases \i) and |/) can take various forms. Natura l ly the form of these amplitudes w i l l depend on the choice of representation these bases take. For example, as w i l l be demonstrated i n this chapter, amplitudes can be constructed between i n i t i a l and final spin states or between total angular momentum states, etc. In an experiment, one measures observables which can be expressed as a sum of bi-l inear products of the complex amplitudes. Conversely a theory attempts to describe the properties of the transformation operator T. Hence, one can see f rom equation 3 that the set of amplitudes form the common ground between theory and experiment. T H E O R Y <—> A M P L I T U D E S <—^ E X P E R I M E N T In this section the general forms of the most commonly used amplitudes are introduced and their relationship to each other is discussed. II.5.1 Helicity and Helicity Partial Wave Ampli tudes I shal l begin by introducing a representation for the in i t ia l and final states which is easily related to measurable quantities. G iven that many aspects of a reaction are dependent on the spins of the reaction particles, one can choose spin states as a representation for the amplitudes. A very useful basis by which the spin states of a system can be represented is known as the helicity representation. For ind iv idua l particles, the i - ax i s is defined i n the direction of the particle 's momentum and the helicity is the projection of the particle 's spin along this axis. T h i s basis is well suited for use i n relativist ic or non-relativistic applications because it is s imply defined i n any reference frame. Hel ic i ty amplitudes represent the transit ion probabilities between in i t i a l and final states of well-defined helicity. These amplitudes are functions of energy (E) and polar reaction angle (9). For the reaction a + b —>• c -\- d, the helicity amplitudes (F) can be given as [4]: F^f{e, E; bia, c)d} = K< j6\T\a^ > (4) where ( 7^ ) and (ajS) are the final and in i t i a l helicities of the reaction particles, T is the transit ion matr ix for the reaction and K is a, normalization constant containing kinematic information. In this representation, the entire description of the reaction is contained in the set of these amplitudes. In the case of the pp dw reaction, this implies there are only 4 x 3 = 12 complex amplitudes required*. B y applying symmetry properties (parity and time-reversal invariance) to these amplitudes, this can be reduced to 6 complex numbers or 11 real numbers i f one ignores the overall phase [5]. In terms of the well-defined spin states of the helicity representation, it is t r i v ia l to relate the experimental observables to these amplitudes. For example, the unpolarized cross section i n terms of helicity amplitudes is: dQ ( 2 . , - M ) ( 2 . , + 1) J ; ' , ' where an average over the total number of possible in i t i a l spin states and Sy) has been carried out. Th i s expression is dominated by the larger amplitudes and is insensitive to relative phases of amplitudes. Sp in observables can also be expressed i n terms of the helicity amplitudes. The tensor polarization of the final state deuteron for the pp dir reaction is ^(corresponding to the 4 possible spin projection combinations of the 2 protons and the 3 possible spin projections of the deuteron) expressed as 5 where the subscript S representing the spinless pion has been suppressed. Th i s expression demonstrates the sensitivity of polarization observables to the relative phases of amplitudes as well as sensitivity to those smaller amplitudes which cannot be distinguished through a cross-section measurement. A smaller amplitude can contribute most significantly when occurring as a product w i t h a larger amplitude. Other examples of spin observables have been given elsewhere [30,17,5 . G iven such relationships between amplitudes and observables, it is i n principle only necessary to measure 11 observables to define the 6 complex helicity amplitudes of the pp —> dir reaction^. In practice this is not a t r i v ia l task, as they must a l l be measured at the same energy and angle^°. Due to experimental complications^^, it is frequently difficult to measure a l l the required observables at exactly the same angle. It is therefore useful to define a set of amplitudes which are independent of reaction angle. Such amplitudes would permit the comparison of experiments performed at different scattering angles. In this sense, a natura l extension of the helicity amplitudes is the helicity par t ia l wave amplitudes. Here the helicity amplitudes F"f are expanded i n terms of the Wigner-c? functions. The J-dependent (total angular momentum) coefficients which are termed the helicity par t ia l wave amplitudes are defined following the derivation of Jacob and ^(one has 11 equations and 11 unknowns) ^°In the end the amplitudes are only defined at those energies and angles at which the experiments have been performed. ^^Such experiments are rarely done at the same time with the same apparatus. W i c k [30], F^s\0,E) = Y.f"e{J.E)di_,^_,{e) (6) J The price to be paid for these angle-independent amplitudes ( / ° f ) is that one has now introduced an infinite number of them corresponding to a l l possible angular momentum quantum numbers. O f course one can argue that for a reaction of finite energy, the centrifugal barrier l imits the number of J-values that can significantly contribute. Thus , the sum can be truncated at some upper value of J . Unfortunately, i n terms of these total angular momentum J states, the appropriate truncation point is not clearly defined. W i t h this i n m i n d , we now discuss a t h i r d expansion of the reaction amplitudes, which is i n terms of angular momentum states. II.5.2 1-s Part ial Wave Ampli tudes A n expansion of the angle independent helicity part ia l wave amplitudes i n terms of states of well-defined relative orbi ta l angular momentum (/) and total spin {s) yields the following expression [30]: f^^{J,E)= Yl <JlfSf\JjO><Jhsi\Ja^>l'^;:^{J) (7) . Th i s expansion defines a new set of amplitudes |^^ *'^ ( J ) , frequently referred to as part ia l wave amplitudes ( P W A ' S ) , which are dependent on J , and 5 , / / , the total angular momentum, the in i t ia l and final relative angular momenta and total spins, respectively. In expression 7 the quantity < Jls\Ja^ > is defined i n terms of standard Clebsch-Gordan coefficients: < Jls\Jal3 >= (-1)^-*^+" < s^a, Sk - ^\sa -^><sl3-a,Ja- I3\10 > Use of the 1-s part ia l wave amplitudes is popular i n nuclear physics since kinetic energy and central potentials (characteristic of nuclear interactions) are diagonal i n the 1-s basis. A direct result of this diagonalization is the concept of the centrifugal barrier, which classically relates the kinetic energy of the reaction particles to their relative angular momentum: E ^ 1(1 + l)/r'^. Quantum mechanically, this provides an estimate of the most probable angular momentum value (/). Th i s relationship thus defines a natural truncation for the number of amplitudes necessary to describe a reaction at some given energy. One should note however, that the 1-s P W A ' S are of l i t t le value at high energies where the number of amplitudes which contribute becomes prohibit ively large. A lso , since the /-values characterizing a reaction are not expl ic it ly measurable observables, in the sense that polarization is, complicated expressions must be developed to relate these amplitudes to the more accessible spin observables. T h e 1-s par t ia l wave amplitudes are of part icular importance i n this thesis since they are the basis of most of the discussion pertaining to the pp ^ dn reaction parameterization. The nomenclature used to describe the indiv idual 1-s transitions was originally suggested by M a n d l and Regge [31] and was subsequently expanded by Blankleider [32] to include higher angular momentum terms. These amplitudes are listed i n Table I. The explicit normalization of these amplitudes varies amongst the authors who have used them. T h e relations between these normalizations are documented elsewhere [5 . Table I: Definit ion of 1-s part ia l wave amplitudes for well-defined i n i -t ia l and final relative angular momentum (/,,//) and total spin states (si,Sf) of the pp —>• dn reaction. The p-p states are de-scribed i n terms of the standard nomenclature ^^'^^Ij for the p-p system. refers to the total angular momentum and par-i ty of the reaction charmel. This set of amplitudes is truncated at Trd relative h-waves (/^  = 5), since theoretically it has been determined that higher-order amplitudes are insignificant at intermediate energies {Tp < 800 M e V ) . ^/ h 5,- p-p ai i^d) (pp) (pp) state Co 0+ 1 1 0 0 'So ai 1- 0 1 1 1 'Pi «2 2+ 1 1 2 0 «3 1- 2 1 1 1 'Pi 04 2 - 2 1 1 1 'P2 «5 2 - 2 1 3 1 «6 3 - 2 1 3 1 'Fs a? 2+ 3 1 2 0 'D, «8 4+ 3 1 4 0 'G, «9 3 - 4 1 3 1 'Fs OlO 4 - 4 1 3 1 'F, «11 4 - 4 1 5 1 'H, «12 5 - 4 1 5 1 'H, «13 4+ 5 1 4 0 'G, «14 6+ 5 1 6 0 Chapter III Previous Work This chapter provides a brief experimental history of the pp ^ d-K reaction and how this process has been instrumental i n our understanding of the pion and its interactions. In the last two decades there has been a great profusion of experimental data which has permitted several groups to attempt the experimental determination of the amplitudes characterizing this fundamental reaction. A summary of the data now available, as well as a description of the variovis efforts i n parameterizing the data are discussed here. III.l History of the Reaction The first evidence of the pion was seen i n cosmic ray emulsion experiments of the late 1940's [33]. However it was not unt i l the early 1950's w i th progress i n particle accelerators that pions could be produced i n sufficient numbers that experiments could be performed to study their properties. T h e early cross-section measurements of the pp —>• d-K and ivd —> pp reaction were instrumental i n determining the spin of the pion to be zero [33]. T h i s was accomplished using the principle of detailed balance which relates the ratio of the total cross-sections of the two reactions to the spins of the constituent particles and a kinematic factor: a{pp ^ ofTr) 2(25, + 1)(26, + 1) pi a(Trd ->• pp) (2sp -f-1)2 p, (8) where 5 , , Sd and Sp are the spins of the pion, deuteron and proton respectively and p„ and Pp are the pion and proton centre of mass momenta. T h e factor of 2 is required because the two protons are indistinguishable. Later experiments concentrated on understanding the processes of p ion production and absorption. B y the mid-1950's the results of several experiments provided pp —> differential cross-sections at several energies near the production threshold. A n early parameterization of the total cross-section by G e l l - M a n n and Watson [34] was of the following form: (Tpp^d, ^ar] + ^rf (9) where fl — ^ (PTT is the pion momentum i n the centre of mass system and m„ the rest mass of the pion) , and a and are the respective n — d s-wave and p-wave contributions to the total cross-section. Th i s simple parameterization suggested that only pions of relative angular momentum 0 and 1^ (s-wave and p-wave) were important i n describing the production process i n the energy region studied. Th i s model suggested that the gross features of the reaction result from its kinematics (the r) dependence) and that the physics which defined the reaction (contained w i t h i n a and /3) was essentially independent of energy. These features are more clearly i l lustrated when equation 9 is expressed i n terms of the wd pp total cross-section using equation 8: ^Trd—tpp — „ 2 3 The ^ dependence of the a term is clearly seen i n F i g . 9 at p ion energies below 10 M e V and the /S term, linear in rj, dominates above 10 M e V . W i t h increasing energy it became obvious that higher pion angular momentum terms would be necessary to properly describe the reaction [35]. In addit ion, suggestions [16,36 that the pp ^ dn reaction is a likely candidate for possible signatures of di-baryon resonances have required that an accurate quantitative description of this process be obtained (to validate or disprove this issue). 15 4 ° 0 50 100 150 200 250 300 Pion Kinetic Energy (MeV) Figure 9: The energy dependence of the ird —>• pp total cross-section. The points represent data from recent measurements and the solid l ine is a parameterization of this data i n the form of SpuUer and Measday [37 . The advent of highly accurate cross-section data i n the region of the A resonance (see F i g . 9) offered information regarding those par t ia l waves which coupled to the A . However due to the large relative dominance of the A , information regarding amplitudes which do not couple to the A is not readily accessible through differential cross-section data, since cross-sections are only sensitive to dominant contributions of a reaction. It has long been recognized that polarization observables are sensitive to the smaller amplitudes as well as the relative phases of a l l the parameters [31] (for example see equation 5). Therefore it was realized that i n order to obtain a complete description of the reaction and hence provide strict tests for theories it would be necessary to undertake a series of polarization dependent measurements. Such an endeavour has only become possible in the last couple of decades as a result of appropriate technological developments i n experimental apparatus. For example, the construction of meson factories (such as TRIUMF, L A M P F and PSi), which provide users w i th intense, high quality beams of both protons and pions, and the development of highly polarized beams, targets and polarimeters. As a result copious amounts of polarizat ion dependent data have become available in the last 15 years. These data are discussed i n the next section. III.2 The Status of the Current Data Set The status of the existing data set is discussed i n several recent review articles [38,39]. References to the experiments as well as highlights of their impact are provided in these publications and hence a similar review w i l l not be given here. Instead a discussion of the types of observables which have been measured w i l l be presented. For a reasonably complete tabulation of the existing data, the reader is referred to the compilation of Strakovsky et al [40 . T h e experimental data which are available to date generally fal l into three categories; • unpolarized total and differential cross-sections • differential cross-sections associated w i th polarized i n i t i a l states • polarization measurement i n the final state w i t h polarized or unpolarized i n i t i a l states The first k i n d of data hsted is similar to that discussed i n the previous section. Such observables convey no information regarding the polarization of the particles i n the in i t i a l or final state and only offer sensitivity to the dominant amplitudes. The experimental measurement of the differential cross-section is expressed in the following manner: da events dÇt niTittùSl where n, is the number of incident beam particles, rit is the number of target particles/cm^, AO, is the acceptance of the detector and e is the detection efficiency of the apparatus. One can obtain the total cross-section by using a detector w i th "47r acceptance", by a transmission experiment, or by integrating the differential cross-section: da JATT dW The inherent difficulties w i th precision cross-section measurements lie i n counting the number of beam particles and knowing the acceptance and efficiency of the detector system. Good quality differential cross-section data exist at a l l energies of concern to this experiment. The second k ind of observable listed above also involves cross section measurements but they are obtained w i th one or both particles of the in i t i a l state polarized. Such observables are known to be sensitive to weaker amplitudes because they result from interference between large and small amplitudes. Those observables involving only one of the in i t i a l particles being polarized are analysing powers and those wi th both particles polarized are referred to as spin-correlation observables. Obviously spin-correlation observables can only be measured i n the pp dn direction due to the spin-less nature of the pion. A considerable number of measurements of the analysing powers and spin correlation observables have been made for the pp —> dir reaction over the energy range of this experiment. However, measurements of only a single analysing power exists currently for the Ttd pp reaction (this observable, iTn, has been measured by several groups [41,42]). Ana lys ing powers and spin-correlation observables are measured by observing the az imuthal asymmetry introduced into the differential cross-section by the polarization of the in i t i a l state. One s imply measures the relative changes i n cross-section which result from a change of the i n i t i a l polarization. B y normalizing to the relative unpolarized cross-section, one can cancel uncertainties resulting from the acceptance of the apparatus and the target thickness. The th i rd type of observable l isted requires the use of a polarimeter to measure the polarization of one of the final state particles. These experiments tend to be more difficult and t ime consuming than their spin-dependent cross-section counterparts since a double scattering experiment is required. A s a result only a few such experiments have ever been performed. In fact the polarization observables involving an unpolarized in i t ia l state are related i n a straightforward way to the above-mentioned analysing powers as a result of t ime reversal invariance [4] (discussed i n section VIII.1.1) and hence are more easily measured as an analysing power. Those experiments which i n addit ion have one of their in i t ia l state particles polarized, are spin-transfer experiments. As they are the basis of this work, they wi l l be discussed i n greater detai l i n the next section. A more complete discussion of the nomenclature used for the observables is given i n section V I I I . 1.2. III.3 Previous Spin-transfer Experiments P r i o r to this experiment there existed three spin-transfer measurements, a l l of which were performed i n the pp — > dix direction. T w o of these used a similar polarimeter technique involving deuteron-carbon scattering. T h e t h i r d used a novel approach which required deuteron break-up followed by a proton polarimeter. The following sections wi l l briefly describe these experiments and discuss their merits and shortcomings. III.3.1 T h e Geneva Experiment T h i s work, pubHshed i n 1987 by Cantale et a l . [5], is the most extensive spin transfer measurement performed to date. The project was a culmination of several experiments performed at PSI by the Geneva group to determine the six complex helicity amplitudes (discussed i n section II.5.1) of the pp dir reaction. The ir earlier experiments consisted of several spin dependent and independent measurements of the k i n d discussed i n the previous section. Us ing a deuteron-carbon polarimeter, scattering asymmetries (at the polarimeter) were obtained at three proton beam energies ( T p = 447, 515 and 580 M e V ) , and several centre of mass angles ranging from 6deut.= 50° to 6deut.= 150°. Scattering asymmetries at the polarimeter which depend on the spin-transfer observables Kss, KNN, KLS and the polarization itn were obtained by respectively polariz ing the incident beam i n three directions ( ± S , ± N , ± L ) as well as unpolarized. The scattering asymmetries were obtained by f itt ing the coefficients A , B , C , D and E of the following expression to the az imuthal scattering distr ibut ion i n <^ c-n{<t>c) = j-{A +B sin (f>c+ C cos + Dsm2<l)c + Ecos2<l)c) =-^(1 +es sin(l)c + ec cos (j)c + e2ssin2<f)c+ e2c cos2(j)c) ^ ' One of the foremost advantages of a deuteron-carbon polarimeter experiment is its potential to measure simultaneously both tensor and vector components of the deuteron's polarization^. Unfortunately, at the t ime the experiment was carried out, very l i tt le information was available regarding deuteron-carbon analysing powers. As a result, the Geneva group was unable to extract the deuteron polarization and hence the spin-transfer quantities explicitly. In order to overcome this l imi ta t i on , the group performed a helicity ampl i tude fit (as was the ^However these components are significantly mixed when Lorentz boosted from their measured value in the lab to the centre of mass system. a im of their experimental program) to a l l their data including the asymmetries described i n equation 10, w i th the deuteron-carbon analysing powers as addit ional parameters i n the fit. Such a fit was possible since the large number of asymmetries measured, i n addit ion to their existing data, were sufficient to over-constrain the parameters. W i t h this procedure, the desired helicity amplitudes as well as predictions for the deuteron-carbon analysing power were obtained. Some interesting comments can be made about the results. F i r s t of a l l , the deuteron-carbon analysing powers predicted by this experiment were i n good agreement w i th those measured i n a subsequent experiment at S A C L A Y [43] using a s imi lar polarimeter. However the results indicated that the analysing powers were extremely smal l (iTn ~ .1-.3, T20 ~ 0, T21 ~ 0 and T22 ~ .05)^. Fortunately, the large number of events obtained wi th this experiment meant that the asymmetries were well defined. However the small analysing powers makes the results very susceptible to systematic errors. Another comment pertains to the value of the fit. O n l y data previously obtained by the Geneva group were incorporated into their fit. Thus their results d id not benefit from the wealth of other data available which would have enhanced the quality of the fit and also served as a systematic check of their data. The importance of such a systematic check has been i l lustrated following the in i t ia l publ icat ion of their spin-transfer results. A significant sign error was discovered i n their asymmetries [44] after an independent measurement (see section III.3.2) had been performed by a different group. The following list describes the strengths and weaknesses of the Geneva work: • Strengths - very high statistics ^The magnitude of these analysing powers is a measure of the ability to measure the deuteron polarizations itn, <20, <2i •^nd <22 (which are discussed in section VIII.1.2) respectively. — large range of angles {Odeut.= 50°- 150°) — ful l range of incident beam polarization states ( ± S , ± N , ± L ) — sensitivity to both vector and some tensor deuteron polarization — two-arm experiment detecting bo th f inal state deuteron and pion (extremely low background contamination) • Weaknesses — lack of knowledge of deuteron-carbon analysing powers hampered their abi l i ty to extract results — complicated mix ing of deuteron tensor components when boosting from lab to centre of mass system — subsequent measurements have shown the deuteron-carbon analysing powers i T i i and T22 to be smal l and strongly energy dependent and the others to be negligible making the results susceptible to systematic errors — despite the high statistics, the energy dependence of the analysing powers suggests that the results at large centre of mass angles {ddeut. > 90°), where the analysing powers are extremely smal l , provide l i tt le real information about the deuteron polarization — the Geneva helicity amplitude fits, as published, do not include data from other experiments (with different systematic errors) III.3.2 T h e T R I U M F Experiment T h e results of these experiments were published i n 1989 by Hutcheon et a l . [45 and 1991 by Abegg et al . [46]. The i r experiment was similar to the Geneva experiment i n that it was also performed i n the pp —> dn direction and used a deuteron-carbon polarimeter. Results were obtained at a single energy ( T p = 507 M e V ) and several centre of mass angles ranging from 6deut.= 20°-155°. The major changes i n the experimental design which they used were: i) the deuterons were detected i n a momentum-selecting spectrometer^ which however meant that the final state deuterons suffered a 60° vertical bend, and ii) a t h i n plate of steel was placed before the carbon analyzer as a "hardener" i n an attempt to increase the vector analysing power of the deuteron-carbon reaction. The first difference led to a somewhat more complicated analysis, than that employed by the Geneva group as the spin precession due to the interaction of the deuteron magnetic moment w i th the magnet field of the spectrometer had to be accounted for i n addit ion to the Lorentz boost. The second feature (the steel hardener) led to a small but not a dramatic improvement i n the analysing power. A n important point however, was that this work did include known deuteron-carbon analysing powers i n the analysis. Asymmetries were obtained for both sideways and longitudinal ly ( ± S , ± L ) polarized proton beams by fitting the azimuthal scattering distr ibut ion using equation 10. Such asymmetries are associated wi th the observables Kss and KLS-Their results are i n good agreement w i t h the Tp= 515 M e V data of the Geneva group"*. Results were obtained using both a one arm and two a r m system. Backgrotind separation was good even wi th a one a r m system due to the momentum definition obtained from the spectrometer. A note of interest regarding their data acquisition: i n a manner similar to that of this thesis experiment, a J-11 preprocessor was successfully used i n their data acquisition system to reject the smal l angle Coulomb scattering. ^ ( T R I U M F ' S Medium Resolution Spectrometer (MRS) with the polarimeter at the focal plane) ''(following the correction of the Geneva group's sign error) III.3.3 T h e L A M P F Experiment This work was presented by T u r p i n et a l . at the Karlsruhe Few-Body conference i n 1984 [47]. The measurement involved a novel technique to determine the deuteron's vector polarization. Th i s method involved the dissociation i n a th in foi l , of the final state deuteron from the pp —> dn reaction. Th i s was followed by the measurement of the polarization of the resulting proton i n a standard proton polarimeter. The experiment was performed at a single energy (Tp= 800 M e V ) for several centre of mass angles {O^eut. ~ 15° - 60°). The deuterons were detected i n a broad range spectrometer which had a carbon dissociator placed at its mid-plane. A proton polarimeter following the spectrometer measured the polarization of the dissociated protons^. The relationship Pp ~ (1 — ^p)Pd [48], where p is the deuteron D-state probabil i ty (p= 0.07), describes the proton polarization (Pp) i n terms of the deuteron polarization (Pd). A l l deuteron tensor polarization information is lost in such an experiment. The v iabi l i ty of such a technique was later demonstrated by Bugg and W i l k i n [49,50]. A l though the observables K^N, itu, and hnear combinations of Kss, KSL, KLS and KLL were measured, only the values for KNN and itu were reported at the conference [47]. Unfortunately the results have never been published i n a refereed journal , and are only available through private communications. One should note however, that their it-^ data are i n good agreement w i th the iTn data of S m i t h et a l . [41] (refer to section V I I I . 1.2 for the relationship between itu and i T n ) . ^(which could be separated from any undissociated deuterons using spectrometer information which distinguished the different charge-to-mass ratios) III.4 Amplitude Analysis Work T h e parameterization of experimental variables i n terms of the amplitudes described i n section II. 5 has long been recognized as a useful way of extracting the underlying physics of the pp ^  c?7r reaction. E a r l y attempts at describing the energy dependence of the total cross-section (see equation 9) by G e l l - M a n n and Watson and later SpuUer and Measday [37] and Ritchie [51] were instrumental i n determining the s and p wave components of the TTC? system near the production threshold. In order to obtain information concerning the relative phase of these amplitudes, M a n d l and Regge [31] indicated how measurement of the analysing power, ANO, could be parameterized i n terms of the sines and cosines of the scattering angle, 6, using bi-linear products of the 1-s amplitudes (see reference [31]) listed i n Table I. A t that t ime, a l l theoretical and experimental parameterization of the differential cross-sections and analysing powers were i n terms of cos 6 and sin 6. T h i s expansion was used for many years unt i l it was suggested by Niskanen that it would be more efficient to expand i n terms of orthogonal Legendre polynomials [52], which axe more closely related to the P W A expansions discussed i n section II.5. The new expansion, also discussed i n section II.5.2, forms the basis for two of the groups which have attempted modern parameterizations of the pp diT reaction. A t h i r d type which is only briefly discussed involves an expansion i n terms of helicity part ia l waves, which were introduced i n section II.5.1. III.4.1 Bugg's 1-s Part ial Wave Analysis O f a l l the parameter fits describing the pp —> d'K reaction that have been published, the work by Bugg [44,53,54,55] is currently the most relevant for the simple reason that these fits are the most up to date. His latest fit includes a l l available data up to 1988. For this reason the other published fits, which only include data up t i l l the early 1980's, are described i n less detail than the following discussion of Bugg's work. A key feature of Bugg's work is that the data are in i t ia l ly fit as freely as possible to identify those dominant amplitudes which are part icularly constrained by the data. For those amplitudes which are less wel l defined, Bugg invokes ideas from available theories to assist i n constraining them. These theoretical aids are l isted as follows (Table I provides a definition of the amplitudes): • the higher order amplitudes (as and above) are fixed to the theoretical predictions of Blankleider • the relative phase of the dominant 02 amplitude as compared to ag is fixed to the value provided by Blankleider • poorly defined phases are constrained using Watson's theorem (see below) T h e first of these is justified by the fact that the higher order amplitudes (ie. larger values of angular momentum) are small i n magnitude because of the centrifugal barrier and it is generally accepted that theory predicts these amplitudes well , i n terms of one pion exchange potentials. T h e second i tem is essential to ensure continuity of the amplitudes from energy to energy through the energy dependence of the theoretical amplitudes^. The last i tem is invoked when the phase of an amplitude is st i l l not fixed by the above constraints. Watson's theorem [54] states that the phase of an inelastic channel can be obtained from the elastic phases describing the in i t i a l and final channels assuming only weak ®(by fixing the relative phase of 02 to theory, and allowing the interference of the smaller ampli-tudes with 02 to determine their relative phases) coupling is present between the channels: where S^pj and are the phases taken from elastic scattering. Such an approximation is only val id near threshold, where contributions from the N N T T channel are not significant. The constraints mentioned above are invoked by Bugg by means of a penalty function which is added to the overall of the fit: 2 _ V-^ / 't>exvt. — <t>iheory \ ^ 8<f> ) where the Scj) is chosen by h i m to reflect the uncertainty i n the quantities being compared. A n addit ional penalty function is also added to the overall to take into account normalization error for the various experiments: where a,- is the normalization constant for each measured observable, and ^ a , is its uncertainty. Th i s feature helps account for and identify systematic errors i n different experiments. T h e most recent published fits of Bugg [44] are i l lustrated i n the A r g a n d plots of F i g . 10 for the energy range related to this experiment. These plots depict the real and imaginary parts of each amplitude as function of reaction energy. The work of Bugg is generally accepted as the status quo for the ampl i tude description of the pp —» dir reaction. He has published fits on several occasions [54,55,53,44] following the appearance of new data, and has tried to keep his fits up to date. Bugg's amphtudes are frequently referred to i n papers as the standard to which new data are compared. In addit ion, theorists are also start ing to compare their calculations w i th the amplitude determinations of Bugg [21 . >,o.5^ «s ^0.5] -1.Q a 4 X .0 ' - d . 6 • IS + 0.0 R e a l 0.6 1.0 0.4H ^0.2 B-0.2 -0.4 - C • 0.1 0.3 0.5 Real u •2 c + -J 1 0 X 1 1 r • .0 u ta 53 a y • i X 0 Real Real Figure 10: A r g a n d plots of the Bugg P W A fit [44]. The symbols refer to fits per-formed to data obtained at part icular energies. These are, i n terms of proton lab energies: 493 M e V (•); 578 M e V ( x ) ; 650 M e V ( x w i t h i n • ) ; 700 M e V (+); 800 M e V (O) . III.4.2 T h e Watar i 1-s Part ia l Wave Analysis A Japanese group also performed an 1-s par t ia l wave analysis of the pp —»• d'K were reaction i n the early 1980's [56,57,58]. The first two published results [56,57 performed w i t h l i tt le or no theoretical constraints involved, leading to several plausible, yet widely varying amplitude sets. In their t h i r d publication [58] they agreed that insufficient data existed to perform a fit independent of theory. In that work, they followed Bugg's example by using theory to fix the higher order par t ia l waves. A single result was obtained which was similar to Bugg's [54,55] for the larger amplitudes, but which had some significantly different smaller amplitudes. The i r definition of the amplitudes fit was similar to that of Bugg (ie. the M a n d l and Regge amplitudes (defined i n Table I), except for the following change: where and S^p are the ^-matrices for TTO? and pp elastic scattering respectively and are introduced to account for in i t ia l and final state distortion effects. In general these 5-matrices are close to unity i n the energy range of interest. A key feature of the Watar i fit was the inclusion of addit ional parameters to account for the energy dependence of the amplitudes, as indicated i n equation 11. This allowed a global fit to observables at a number of energies. It was hoped that the inclusion of data from many energies might make up for the sparseness of the data at single energies. a,- "{cii+aiQ + aiq + a^q ) (11) where q is the centre of mass pion momentum i n proton mass units and L' is the final orb i ta l angular momentum. A penalty function to account for relative normalizat ion errors of the experimental data was also included in their fit. Recent spin-transfer data [45,46] have been more consistent w i th the most recent Bugg solution then w i th the Watar i fit. Th i s is not surprising given that Bugg's solution includes significantly more data, including the Geneva spin-transfer results [5], which were not included i n the Watar i fit. III.4.3 T h e Russian Helicity Part ial Wave Amplitudes This work [59,60] suffers from age as it was performed using only data which were available i n the early 1980's. The fit differs from the other published works on several counts. First of a l l , helicity part ia l waves (discussed i n section II.5.1) were used instead of the more usual 1-s part ia l waves. L ike the Watar i approach, these amplitudes were given an energy dependence. The possibility of formation of di -baryon resonances, i n the standard Bre i t -Wigner form, was also included i n the energy dependence of these parameters. Due to inconsistencies [61] i n the way their amplitudes are defined, as compared to the more usual definitions of helicity par t ia l waves [62], it has not been possible to compare the Russ ian amplitudes w i t h the work of Bugg or to any more recently obtained data. Chapter IV Motivation A s was discussed i n section I . l the pp ^ dw reaction is of great interest i n the field of intermediate energy physics. The preceding two chapters have summarized the tremendous theoretical and experimental advances made over the last three decades towards an understanding of this fundamental process. In this chapter, however, evidence w i l l be presented to demonstrate that despite the considerable body of experimental data already i n existence, the quantitative description of this reaction is st i l l far from complete. It w i l l be shown that spin-transfer data can be instrumental i n removing remaining uncertainties. F ina l ly , reasons for the part icular choice of observables and a summary of experimental philosophy are given. I V . l status of Amplitude Analyses Due to the significant increase i n available data over the past decade, much is now known of the pp —> dir amplitudes. U p o n inspection of these amplitudes (see F i g . 10), a great deal can be said of the intermediate physics which takes place w i t h this reaction. A n obvious example is the domination of the amplitude i n the region around 600 M e V which is consistent w i t h a significant product ion of the relative s-wave i V A system in the intermediate state. The magnitudes of other dominant amplitudes such as (24 and OQ are also well determined by the data i n this energy range. The assumptions and theoretical guidance used to arrive at the amplitudes of F i g . 10 were presented in section III.4.1. Tl ie end result is several smoothly varying amplitudes which offer insight to the physics involved. Unfortunately the strength of this insight is di luted by the artif icial constraints required to obtain the fit. In addit ion, despite these applied constraints, the energy dependence of several of the amplitudes remains questionable. Part icular attention should be directed to amplitudes O i and 0 7 , whose energy dependence is unclear, especially i n the region of 600 to 750 M e V where less data currently exist. Other amplitudes such as «5 have been strongly constrained by Bugg to follow part icular energy dependences. Hence, despite the smooth energy dependent behaviour of most of the amplitudes, it is not clear if their dependence tru ly describes the reaction or has been "forced" to appear as if they do. These issues and others can only be satisfactorily addressed by experimental data which are sensitive to the remaining uncertainties. We, at U B C , have implemented an amplitude fitting package which permits "Bugg type" fits at single energies. Th i s routine is an amalgamation of the observable calculating routine P I N N O B [63] and the C E R N parameter opt imizat ion package M I N U I T [64]. Th i s routine permitted various "theoretical" constraints to be imposed on the fit as desired, thus allowing investigation of the impact of these constraints on the amplitudes. Table II shows the constraints used i n the various fits performed using this package. The degree to which some of the fits were unique was useful for identifying those observables which are sensitive to different constraints imposed on the amplitudes. (It should be noted that the x^/degrees of freedom for each of the fits were comparable at each of the energies studied). Clearly, it is important to measure those observables which are most sensitive to the remaining discrepancies of the fits. O n l y then w i l l the theorists be provided w i th an unbiased quantitative description of this fundamental reaction. Table II: Constraints imposed on the amplitudes for the U B C ampl i -tude fitting package. The fit denoted by " B u g g " was similar to that performed by Bugg, whereas fits " A " and " B " were modifications of the Bugg approach. F i t name phase angles constrained magnitudes constrained B u g g 4>2 fixed to theory «8 and above are fixed to theory A (}>Q fixed to theory og and above are fixed to theory B (f)Q fixed to theory lag] is fixed to theory ag and above are fixed to theory IV.2 W h y Spin-Transfer Observables? A n y general survey of the available set of data for pp ^ dir observables, indicates a distinct lack of deuteron spin dependent observables. In fact, despite the considerable collection of spin-correlation parameters, iTn is the only deuteron spin observable of significance i n the energy range applicable to this thesis. In this sense, then, there is a significant region of "observable space" which has not been studied and it is worthwhile to investigate the influence of these observables on the ambiguities which exist in the determination of the amplitudes. T h e contribution of the iTn analysing power data was essential for constraining previously undefined amplitudes [53,42], part icular ly OQ. One can then ask how other analysing power observables such as T21 and T22 (which depend solely on the deuteron spin) might impact upon the fits. The predictions for these observables, for the various U B C fits of Table II, are given i n F i g . 11. It is seen that the different fits give very s imilar predictions for each observable. In fact, Blankleider [32] has indicated that since these observables are dominated by the well known term [02 P they are therefore largely determined by the existing data. These conclusions are also confirmed by Bugg [44]. As a result, the measurement of such observables would yield l itt le or no new information -1.50-' ' • ' I ' ' • ' I ' • • ' I l I l . M l l l f l l l l l l l • • • • I . . • • I • . . . I • . . 0. 0. 0. 45. 90. 135.18C ^ d e u t . (degrees) Figure 11: Insensitivity of the tensor analysing powers to different P W A sets. regarding the poorly determined amplitudes^. Next , the impact of the spin-transfer observables on the amplitudes w i l l be considered. Blankleider points out [32] that none of the spin-transfer parameters depend on contributions from | a 2 p , but depend instead on terms which consist of bi-l inear products of smaller amplitudes w i t h larger, well known amplitudes such as 0 2 . Hence the values of the smaller amplitudes are "ampli f ied" by 02 as far as their influence on the spin-transfer quantities is concerned. Figure 12 demonstrates the spin-transfer observables predicted by the U B C fits. It is clear that s imilar qual ity fits to the existing data can predict drastically different results for the spin-transfer parameters. Therefore the spin-transfer observables do provide sensitivity to the remaining ambiguities of the amplitude fit. In addit ion to our own work, others have suggested that the addit ion of spin-transfer parameters to the existing data base is essential for a complete determination of the amplitudes [44,32,53]. Not only would they help p in down the remaining uncertain amplitudes but they would serve as a useful check of systematic errors for some of the existing data. ^ However these observables could provide a useful systematic check of the existing spin-correlation observables. 45^ ' 90. 135. 180. -1 I I I I I I I I I I I I 1 I I I i_ -1—I—I—I—I—I—r 0. 45. 90. 135. 18 _ —1 1 1 1 1 1 1 1 1 1 1 1 L - J 1 1 1 1 1 : ^ ^ / \ : / \ / \ : / \ / \ : • ^ W \ -J I 1 1 I 1 I I i I 1 I I I L - o . 3o : 0 L S 0. 45 . 90. 135. 180. ^ d e u t . (degrees) 1—1—I—I—I—I—I—I—I—I—I—I—I—I—I—I—I—I—r-45. 90. 135. 180. Figure 12: Predict ions of spin-transfer observables of the pp dn reaction at Tp = 500 M e V , for different P W A solutions. T h e solid corresponds to a "Bugg-type" solution and the dashed line to a " B - t y p e " solution as defined i n Table IL IV.3 Advantages of this Experiment Despite the fact that most experiments have been performed i n the pp —> dir direction, there are several good reasons for doing the experiment i n the time reversed ird —> pp direction. As no spin-transfer measurement had ever been made i n the p ion absorption direction, barring unreasonable experimental difficulties, an entirely new approach to the measurement of these observables becomes available. In fact this choice of reaction direction affords us several advantages. 1. A proven technique of proton polarimetry is available to measure polarization components orthogonal to the proton momentum. 2. A fixed direction polarized deuteron target had already been developed at T R I U M F . W i t h this target, a well known vector polarizat ion wi th smal l but calculable tensor polarization is readily available. 3. The polarization technique was "self-calibrating" through comparison of the normal polarization results for the nd pp reaction to the well known analysing powers, ANQ of the time-reversed reaction. 4. The fact that the centre-of-mass system is almost at rest i n the lab for this reaction permitted us to span the A resonance i n the ird —>• pp reaction using pion beams available at T R I U M F . Such an energy range woii ld not have been possible at T R I U M F for the pp dn reaction. In addit ion, the small relative velocity of the centre of mass i n the lab minimized the effect of the Lorentz boost on the proton's spin. Item 4 was of particular importance i n that it enabled this work to be the first spin-transfer measurement, in fact one of the very few polarization experiments, to span the A resonance of the pp ^ dn reaction w i th a single experiment. T h i s feature serves also to check relative systematic errors i n other experiments which were measured i n different energy regions. In addit ion, the systematic errors experienced by this experiment were very different from those of other spin-transfer measurements (see section III.3). The results of this experiment therefore provide an important check of the various systematic errors at common energies. Difficulties associated w i th our experiment axise due to the low fluxes typical of p ion beams, background resulting from pions interacting w i t h the complex nuclei of the polarized deuterated butanol target and precession of the proton's spin due to the strong magnetic field i n the region of the target. The measures which have been taken to deal w i th these problems are discussed i n later chapters. IV.4 Experimental Philosophy In this experiment we measured the vector polarization of protons coming from a polarized deuteron target. A s mentioned i n the previous section, the deuteron target is only capable of producing large vector polarization^. Hence the determination of vector spin-transfer coefficients is natura l for this configuration. The significantly lower event rates for p ion beams, coupled w i t h the need for a "second scattering" to measure the proton polarization, necessitated designing the experiment to make the most of the statistics available. It was decided to perform high statistics measurements of three spin-transfer observables at single angles, but over several energies spanning the A resonance. The angle chosen was one where the observable i n question was most sensitive i n the pr ior studies involving the U B C amplitude predictions (see section IV.1) . These studies indicated that it was unnecessary to measure each observable over a significant angular range, because i n most cases, the angular dependence is already highly constrained by •^The tensor polarization, although small, is not insignificant and does impact somewhat upon the results as shall be seen in later chapters. the existing data set. The part icular observables we chose to measure were selected both for their experimental v iabi l i ty and their sensitivity to the amplitude ambiguities. The target was operated w i th the polarization axis projected along one of three Cartes ian directions ( longitudinal , sideways and normal , a l l defined w i th respect to the incident p ion beam^), so that the spin-transfer parameters K'j^si I^'ss K'pfN were the ones measured (recall that the first subscript corresponds to the target polarization and the second to the final proton polarization^). B y measuring several different parameters, the impact of a single systematic error characteristic of a particular configuration is reduced. The predicted values of these observables are shown i n F i g . 12 (expressed i n terms of the pp —> d-K reaction) for equivalent p ion energies, = 105 M e V {Tp = 500 M e V ) . ^Both sideways and normal polarizations were transverse to the pion beam. Sideways was to the right and left of the incident beam, in the lab, and normal was up and down with respect to this beam. ''Only sideways and normal proton polarization could be measured directly in the polarimeter. Longitudinal polarization could only have been measured by precessing that component into a trans-verse direction using a magnetic field. Chapter V The Polarimeter A polarimeter, as its name suggests, is a device which measures the polarization of a system of particles. It is not unlike a thermometer which measures the temperature of a system. B o t h rely on well defined physical effects and each must be cal ibrated w i th respect to a known state. Th i s chapter w i l l outline the underlying physical principles which allow one to observe polarization and w i l l describe the procedure through which an absolute polarization measurement is obtained. O f interest to this thesis is the polarization measurement of protons, charged nucléons of intrinsic sp in-^^. V . l Polarization Theory W h e n designing an instrument to measure an observable quantity, one must search for a physical effect which enhances or is sensitive to the observable of interest. For a proton polarimeter, one utilizes the inherent spin dependence of the proton-nuclear interaction. The scattering of nucléons by nuclei is characterized by a complicated spin-dependence [65]. Amongst these are interactions which depend on nucléon spin, the spin of the nucleus as well as the relative angular momentum of these bodies. The net result for the scattering of polarized protons is a spin-dependent asymmetry of the azimuthal distr ibution of the scattered particles. This spin dependent effect is characterized by the analysing power of the reaction. Thus if the analysing power of a part icular reaction is known, it may be used to relate an observed scattering asymmetry to the polarization of the in i t ia l particles (this w i l l be discussed further, later i n this section). Over the energy range of interest to this experiment, the nucleus appears as a composite body of nucléons. Hence, the incident proton can scatter i n a variety of ways, w i th a l l or only part of the nucleus. As a result, the final state can consist of elastic scattering (p + X —> p + X), or one of a variety of forms of inelastic scattering (p -\- X p -\- X'). E a c h channel w i l l exhibit a different spin dependence characterized by its own analysing power. Quantitat ive knowledge of this spin dependence is essential for its use as a gauge of polarizat ion. The analysing powers of a number of particular reaction channels have been measured and evaluated (for example see references [66,67]). However such experiments involve a complicated apparatus to specifically select the scattering of interest, and often yield results which are strongly dependent on energy and scattering angle [68]. These are not ideal properties for a general purpose instrument. It has been demonstrated however, that inclusive^ spin-dependent measurements tend to dampen the strong dependencies characteristic of the ind iv idua l channels [68]. Inclusive measurements can be performed wi th a much simpler apparatus i f it doesn't have to select a specific scattering channel. These features are considerably more favourable for a general purpose instrument. Therefore several groups have performed inclusive analyzing power measurements on many potential analyzers [69,70,71]. As a general measurement tool , it is important to investigate the sensitivity of analysing powers to the properties of the apparatus. Problems may arise as a result of a different acceptance for the various scattering channels due to dissimilar detector thresholds and target thicknesses. If the sensitivity to such effects is too large, polarization measurements would be intolerably vulnerable to drift of the ^(inclusive meaning, including all detectable final states) electronics, etc. over the course of an experiment. Fortunately it has been observed that inclusive measurements tend to y ie ld results which are independent of the detailed configuration of the apparatus and target thickness [69]. It is this feature which permits the application, to the apparatus used i n this experiment, of information obtained from previous inclusive analysing power measurements. In the experiment described i n this thesis, the inclusive proton-carbon (p-C) reaction was used to measure the polarization of protons. Th i s reaction was chosen due to its reasonably large analysing powers which are slowly varying as a function of energy and angle, and which have been well-determined by several independent measurements [69,70,71]. The polarimeter was designed to resemble as closely as possible those other polarimeters on which the analysing power calibrations were performed (thereby removing gross acceptance differences). A s an addit ional check of consistency, this experiment also had recourse to a cahbration using a source of protons of known polarization (see section VIII .2) . T h e scattering distr ibut ion of polarized protons from an unpolarized target can be described i n the following manner [72]: 1(6, (j>) = Io(e) [1 + A(9)(-Ps s in (J> + PN COS </.)] Acc(e, <^ ) (12) where Io(6) is the unpolarized scattering cross section, A(9) is the p - C analysing power, and Acc(6, (f>) is the effective acceptance^ of the protons scattered wi th in the polarimeter and the polar and azimuthal angles 6 and (j> are respectively identical to the angles 6c and in Table V and F i g . 24. In this chapter, however, a l l angles refer to scattering i n the carbon and hence the subscript " c " is dropped. Pj\f and Ps are the projections of the proton polarization on, respectively, the normal (N) axis, defined by the normal to the ird pp reaction plane, and the sideways (S) axis, which is orthogonal to both N and the longitudinal axis (L) ^This quantity is a property of the apparatus. (defined by the momentum of the proton incident on the carbon) following the "r ight-handed" convention. It should be noted that equation 12 contains no term which depends on longitudinal polarization ( P L ) - Th i s is because no asymmetry can be introduced due to without vio lat ing parity conservation [4], and par i ty is known to be conserved to 1 part i n 10^ for strong interactions. Equat ion 12 indicates that the scattering of polarized protons is characterized by an asymmetry (e) i n the azimuthal distr ibution of the proton scattering, the size of which is related to the magnitude of the polarization and the analysing power of the scattering: e5(^)= - Aie)Ps e^iO) = A{9)PN ^^'^> where the negative sign i n equations 12 and 13 is required to relate the sign of the scattering asymmetry to the positive direction of the sideways polarization. In this manner, the analysing power is a measure of the abi l i ty of a reaction to measure polarization. In vector form, equation 13 can be written: A{6) The typica l energy and angle dependences of the p - C analysing powers are shown in F i g . 13a,b. The data was extracted from published [71] fits to existing data obtained from an apparatus very similar to the one used i n this experiment. Superimposed on the energy dependent curve are are arrows indicat ing typica l ranges of proton energy produced, for each incident pion energy investigated i n this experiment. Th i s demonstrates the typical range of analysing powers required i n this experiment for each configuration. For the two lowest energies (T^= 105, 145 M e V ) , the analysing powers axe highly energy dependent. It was therefore important that the energies of the protons be well known, i n order to minimize the introduction of errors due to this energy dependence. a) b ) c) .55-.50-u .45-O IX .40-N .35-.30-< .25-.20-180 MeV •255 MeV 100 T 250 .6-1 CD S .4 1—I 80ff 600-g 400H > 200-0 150 200 Proton Energy at Carbon Centre (MeV) J I I I I I I I L 300 Ccnilomb Scatter ing Nuclear Scattering 0. 5. 9 carbon (degrees) 20 Figure 13: a) the energy dependence of proton-carbon analysing powers. A lso i n d i -cated i n this plot are the proton energy distributions for each pion energy employed i n the K^N configuration, b) the angular dependence of the analysing power for pro-ton energies of 200 M e V . c) the angular dependence of the scattering cross-section. Figure 13b shows the angular dependence of the p - C analysing powers. In general the magnitude of A{d) drops off rapidly below 8 degrees. Th i s is pr imar i ly due to the diminishing contribution of the nuclear interaction. Coulomb scattering dominates at these smal l angles. F igure 13c, taken from an unbiased sample of events obtained i n our experiment, demonstrates this fact. There is a large contribution at angles less than 5 degrees due to Coulomb scattering. On ly those events at larger angles, due to nuclear interactions, are of use to this experiment. Another design consideration is that of the thickness of the carbon "analyzer" . Three physical effects must be considered when choosing this op t imum thickness. First of a l l , the scattering probabi l i ty is increased w i th greater thickness, resulting i n more useful statistics. Secondly, a thicker analyzer w i l l decrease the average energy of the protons when they interact. Depending on the energy of the incident proton, this effect could improve or worsen the analysing powers (see F i g . 13a). F inal ly , smal l angle mult iple Coulomb scattering is worsened w i t h a thicker target, thereby degrading the angular definition of the scattering, and the knowledge of the interaction energy (the means by which this quantity was calculated is discussed i n section IX.2) . In addit ion , because some of the Coulomb scattering w i l l be spread into the region of useful nuclear scattering, a degradation of the analyzing power w i l l result. T h e opt imum carbon thickness can be determined by considering a Figure of Merit Th i s parameter weighs the analysing power w i t h the usefuP cross-section at each scattering angle. j^^ = A\e). § ^ . de (14) J6min. incident where N'"':J^\ is the number of useful scattering events at a part icular angle normalized to the total number of incident particles, a ratio proport ional to the ^ Useful implies (spin-dependent) nuclear scattering as opposed to coulomb scattering. value of the inclusive cross-section at the angle B. Unfortunately such information was not available when this experiment was being planned. Carbon thicknesses for this experiment were therefore chosen on the basis of published data from similar polarimeters [70,71]. Measurements of ^ have since been made for our polarimeter at several proton energies and carbon thicknesses [73]. V.2 Fourier Techniques Using the relations of 13, one can rewrite equation 12 as follows; 1(9, <t>) = loiO) [(1 + es{e) s in <f> + eAr(^) cos (/>)] Acc{e, cf,) (15) As mentioned i n the previous section, i f prior knowledge of the analysing powers is available, measurement of the quantities es and is sufficient to enable extraction of the incident proton polarization, P. Several different methods have been described [69,72] for obtaining these quantities. Choice of the opt imum one depends on the knowledge of the acceptance function Acc{6, <j)). A n obvious way of determining these parameters is to s imply perform a least squares fit to the observed cross section I{d, <f)) of equation 15, w i t h es(^) and e^(d) as variable parameters (the e's are a function of 6 since they depend on A(ê) as indicated i n equation 13). This technique is complicated by the nature of the acceptance term Acc(9, (j)) which i n general is not cyl indrical ly symmetric . Th i s term could, i n principle, be defined by the scattering distr ibut ion of unpolarized protons i n the polarimeter. However, such a source was not available i n this experiment. Alternatively , a Monte Car lo could be performed to model the acceptance. However, this requires considerable knowledge of the geometry and performance of each detector i n the apparatus. Alternatives to the least squares fitting method are available. Equat ion 15 can be rewritten i n the form 1(6, 4») = Io(6)f{9, (j)), where the distr ibution f{9, </>) contains a l l the az imuthal scattering information: f{9, (I>) = [1 + esie) s in <!> + e^iO) cos cl>] Acc{$, <t>) (16) If the acceptance function Acc{9, (f)) was always cyl indrical ly symmetric for each scattering through 6, one could employ Fourier analysis techniques to extract the asymmetry parameters as a function of 6: f t f{0,<l>) sin cl>dci>^ Acc{ey-f^ f t f (9, <f>) COS <j>d<l>= Acciey-^ ftf{9A)d4>= Acc{9) This technique assumes that any event corresponding to a scattering through {9, (f)) would also have been accepted, w i th equal probabil ity, at {9, (/>'), where —TT < (/)' < -\-TT. Since this would be essentially impossible to achieve instrumental ly for a device w i th a broad acceptance of incident protons, it would have to be carried out by only analysing that subset of events which are known to satisfy this condition. Th i s approach is readily implemented provided that a good knowledge of the polarimeter geometry is available. However it results i n a significant reduction of the data set (with accompanying loss i n statistical precision) due to the stringency of the cuts. A t h i r d approach involves simply ensuring that the acceptance function is TT-symmetric (ie. Acc(9, (f)) = Acc{9, (j) + TT)). For such a case a Fourier analysis provides the following expressions [72,74]: f t f{9A)cos9dcj>= eNftf(0,<l>)cos'H<l> + es f t f{0,<t>)sin<l>cos<f>d<l> f t f (9, <f>) sin 9d<l>= eNftf{d,<t>)sin<l>coscf>d<t> + es f t f{0,<f>) sin^ <i>d<t> ^ Here the acceptance function appears on both sides of the equation and hence cancels. A g a i n , achievement of such a criterion is essentially impossible to atta in instrumentally for an extended incident beam of protons. Thus , the pract ical solution involves the selection of the subset of data which satisfies the TT-symmetric requirement. Besset et a l . [72] have demonstrated that this approach is significantly more efficient than the 27:-symmetric method, pr imari ly because fewer statistics are removed from the data sample without introducing bias. T h e TT-symmetric approach was the method used i n our experiment. T h e algebra referred to here was developed by Besset et a l . [72] and has been discussed i n previous work [74], so only a summary of the important points is given here. The integrals of equation 17 were replaced w i t h summations and expressed i n matr ix notat ion: B{e) Zi wi{9) cos\<j>,) M^) sm(< /^) cos(< ;^) J2i wt(e) sin((^/) cos(< /^) El wi{9) sin^(< ;^) f{e) eN{e) (18) where corresponds to the / t h (j) b in and wi{6), the niimber of events scattering through angles 6 and U p o n weighting each of the summations i n equation 18 by the analysing power A{9, E) i n order to remove their 6 and E (proton energy at carbon centre) dependence, the following expressions are obtained: ^ Y:iwi{e)A{e,E)cos{4>i) ^ B\0) = EiWi{9)A{d,E)s\n{<i>i) (19) and El wi{e)A(e, Ef cos\<f>i) El M^)A(e, Ef sm{ct>i) cos(<^0 \ (20) V E / wi{e)A{e, Ef sin(<^,) cos{<t>i) Ei wi{0)A(e, Ef sm\cf>,) where the addit ional A(6, E) term i n P has been brought from e{6) = A{d, E)P. These equations can be expressed i n the matr ix notation: B'{d) = F {0)Pg. Here the polarization vector, Pe is given the subscript "e", denoting the average polarization of those events which had a scattering angle of 6"*. The energy dependence is no longer considered in this development, assuming the correct energy dependence is accounted for, when applying the p - C analysing powers. ''One should note that no dependence of P on ^  is expected. Using the method of matr ix inversion, one may now solve for the polarization parameters: P, = f \d)-^Ê\9) (21) It can be shown [72,75] that the variance of these parameters, V (Pe), is well estimated by the covariance matr ix : V{Pe) ~ F ' " > ) (22) A weighted average polarization. Pave, can now be obtained from each of the Pe sample populations: J^ave — „ 1 ^« V\e) or since ?'{9) = F '~\e) and B'{e) = # \6)Pe: Plue — ~i (23) E . P ( ^ ) It has been shown by Besset et a l . [72] that this set of statistical estimators is efficient, unbiased, and robust^. To extract the polarizations, the five summations introduced i n equations 19 and 20 must be carried out, using the appropriate energy dependent p - C analysing powers, A{d,E), relevant to the polarimeter, together w i th a knowledge of the polarimeter's acceptance (to the level of TT-symmetry). The software used to carry out these calculations is discussed i n section VII .4 . ^Robust implies insensitivity to the knowledge of the distribution which one is fitting. This is an important point, since the acceptance function {Acc{6,(j))) is not completely known. Chapter VI The Experiment T h e experiment was performed on the M-11 beamline at T R I U M F . Th i s beamHne was designed to produce a high flux of pions over the intermediate energy range associated w i t h this experiment. The beam was brought to an achromatic focus^ at the centre of the target vessel. The beam properties of M-11 had been determined previously [76]. The beam profile at the focus has a typical horizontal w i d t h of 1.5 cm ( F W H M ) and vertical w id th of 1.0 cm ( F W H M ) . The beam divergences are typical ly ±0.67° i n the horizontal plane and ±3.2° i n the vertical . T h e target was situated at the centre of the lab system as i l lustrated i n F i g . 14. A two arm detection system was used. B y requiring a coincidence between events i n each arm, the number of background events which triggered the apparatus could be significantly reduced. The forward a r m (A) defined the acceptance of the system, while the backward arm (B) was placed at the conjugate angle defined by the two-body kinematics of the ird —> pp reaction. The angular and radial positions of each arm as defined i n F i g . 14 are l isted i n Table III for each observable measured. The positions at which the detectors were placed were determined on the floor wi th respect to the centers of the carts which held them. The angle at which each cart was placed was known to w i t h i n a half degree. The distance of the cart from the centre of the target was measured to w i t h i n a half centimetre. The position of the detectors w i th in the cart was known to a much greater accuracy. The acceptance of the apparatus was roughly determined by the ^The momentum of the particle is independent of its position within the beam envelope. Table III: L is t of detector angles and distances for the various configura-tions. The distances l isted are from the centre of the target to the first chamber of each arm. Config. A r m A A r m A A r m B A r m B approx. angle dist. angle dist. accept. (degrees) (cm) (degrees) (cm) (msrad.) 27 150 135-140 75 40 ^'ss 60 150 90-96 75 40 13 145 150 85 43 t h i r d wire chamber of arm A . The approximate acceptances of the system for each configuration is given i n Table III. A lso situated i n the experimental area were several secondary detectors used to monitor incident beam flux and beam profile. They had no direct bearing on the polarization measurement but were used as checks of systematics when necessary. Most of the apparatus used i n this experiment has been described previously [74]. However, for completeness, a significant summary of the set up wi l l be provided i n this thesis. Further details can be obtained from the sources referred to, i f so desired. VI . 1 Primary Detector Construction E a c h pr imary detector arm consisted of scinti l lation counters and wire chambers supported by carts of a lumini irm frame construction. The scintillators provided fast trigger information, crucial to the event definition and t iming , whereas the wire chambers contributed the posit ion information required for the reconstruction of a particle's trajectory. The wire chambers chosen for this experiment were Multi-wire delay-line Drift Chambers ( M W D C ' S ) , which offer high resolution over a large area wi th S3A "wc6 T e l e s c o p e A r r a y wc5 S5A ^ P . / / S4A •wc4 S1A A r m A S2A M u o n C o u n t e r s wc3 C a r b o n ~ - - . M 7T wc2 A r m B " P C O S " > wcl W i r e C h a m b e r s _ ; i ^ o l a r i z e d T a r g e t %c8 ' wc7 B . wc9 S c i n t i l l a t o r s / D H o d o s c o p e Figure 14: Layout of experimental area (not to scale). minimal associated electronics. The latter feature was essential to reduce the cost of the overall apparatus. The size of the wire chambers was pr imari ly determined by the length of printed circuit board (for the delay lines), which was readily available at the time (this length was 60 cm). The wire chambers were grouped into sets of three^. The polarimeter axm (A) contained a set of large and a set of small chambers^, whereas a r m B contained only a smal l chamber set. In addit ion a carbon analyzer, required for proton-carbon scattering, was situated i n the polarimeter a r m midway between the smal l and large chambers. The analyzer consisted of 30x30 cm^ graphite^ sheets 1 cm i n thickness. A number of sheets were stacked along the longitudinal axis of the cart producing a total thickness of 5 or 7 cm, a figure which was selected according to the average proton energy. The wire chambers i n each set were separated by 21.0 cm, whereas the separation between the large and small set was 42.0 cm as measured from the centers of the nearest chambers. The centre of the carbon analyzer was placed midway between the two sets of chambers, 21.0 cm from the centre of the chambers on either side. The wire chambers and carbon block were positioned on their carts by snug fitting frames, such that the knowledge of the aforementioned distances was l imited only by the accuracy of the frame construction (±0.1 mm) . T h e height of the carts could be adjusted by "screw" legs so that the vertical centers of the chambers could be positioned to match the height of the incident pion beam. Independent leg adjustment also allowed compensation for the less than perfectly flat meson hal l floor. Pcdrs of scintillators were placed on each end of the wire chamber set of ^Although only two wire chambers are required for the definition of a trajectory, the existence of the third increased the overall efficiency of a set since only 2 out of 3 then had to "fire". ^The small chambers had an active area of 30x30 cm^ and the large chambers 60x60 cm^. ^The graphite had a density of 1.71±0.05 g/cm^. a r m B and of the large chamber set of arm A (the first pair of scintillators was placed immediately after the carbon block). A s their pr imary function was to quickly signal the passage of a charged particle, accurate knowledge of their longitudinal positions was not of great importance. However it was essential for the lateral coverage of the scintillators to be well known, since this would influence the acceptance of each detector arm. Th is was part icularly important for the polarimeter scintillators as an uncertainty i n the acceptance could introduce a systematic error i n the polarization (see section V . l ) . As a result, care was taken to ensure that the scintillators were placed symmetrical ly about the longitudinal axis of the wire chambers. In addit ion a t h i r d scinti l lator, S 5 A , was placed behind the last pair ( S 3 A and S 4 A ) on arm A situated so that it overlapped. The pr imary purpose of S 5 A was to check the relative efficiencies of S 3 A and S 4 A and hence ensure an acceptable acceptance probability. V I . 1.1 T h e Scintillators T h e pr imary function of the scintillators was to provide fast triggering information for the event definition (see section VI.4) and to define the t iming for the wire chambers. In addit ion, the energy loss of the particles i n , and their time of flight ( T O F ) to, the scintillators provided useful information for particle identification. L ight was collected from the scintillators and converted to electrical signals using photomultipl ier tubes. On ly one such tube was used on each of the smaller a r m B scintillators. In order to increase the efficiency of the larger arm A scinti l lators, a tube was place on each end so that the signal could be detected from either or both ends. Th i s also improved the t iming definition by removing the vertical position dependence of the t iming by averaging the times from each end of the scintil lator. The exact dimensions of the scintillators are described in previous work [74]. FIFO discriminator scintillator signals -• to event logic - > to TDC -• to ADC Figure 15: Dis tr ibut ion of scintil lator signals from analog input through Fan-In -Fan-Out ( F I F O ) and split into analog and digital signals. For the arm A scintil lators, the discriminated signals were sent to a mean-timer (to average the signal times from the top and bot tom photomultipliers) and the output of the mean-timer was sent to the T D C and event logic. The signals coming from the photomultipliers were split into two identical pulses^ w i t h one signal going to a discriminator to be used i n the master trigger and for the T O F , and the other was fed into an analog-to-digital converter ( A D C ) for pulse integration. The flow of these signals is i l lustrated i n F i g . 15. The t iming for the wire chambers, as well as the "s tart " for the a r m B T O F was defined by the forward scintillators of arm A ( s l A - f S 2 A ) , i n coincidence wi th the master trigger (defined i n F i g . 21). The T O F t iming resolution was hmited by the resolution of the t ime-to-digital converters ( T D C ' S ) , which were set to 50 ps /channel . A typical T O F d istr ibution is shown i n F i g . 16b. Figure 16a shows the energy deposition distribution of the arm B scintillators used for particle identification. Two peaks are evident, w i th the lower energy peak being that of the pions, and to its right the larger proton peak. Th is diagram demonstrates the abi l i ty of the energy loss distr ibut ion i n the scintillators to distinguish protons from pions i n the event sample. The thresholds of the discriminators fed by the scintillators were set to ehminate most of the pion events i n this d is tr ibut ion , but not cut into the proton distr ibution (see F i g . 16a). ^This was accomplished using a linear "Fan-in-Fan-out" ( F IFO ) . a) 2000 1500-m SioocH > 500-pions \ protons "Tt—'—'—I—'—'—'—'—I—' threshold 250 500 ADC Channels 750 lOOO 2500 2000 TDC Channels Figure 16: a) T y p i c a l energy loss i n t h i n scintillators. The two observed peaks can be ascribed to pions and protons. T h e large "spike" and zero is due to events which had passed through the adjacent scintillator. The discriminator threshold is also indicated, b) T O F d istr ibution as meastired w i t h a r m B scintillators. Th i s d istr ibut ion is broadened due to position dependence of a r m B scintil lator t iming . The large arrows on both diagrams a) and b) il lustrate typica l software cuts placed on these distributions. VI.1.2 T h e W i r e Chambers T h e sole function of the wire chambers was to provide position information for the reconstruction of the particle's trajectory. A n indiv idual chamber could only provide lateral coordinates (" x and y " ) , w i th remaining " z " information which is required to define a trajectory being provided by the known longitudinal positions of the chambers. A t least two of the three chambers i n a set had to "see" the event i n order to successfully determine a trajectory. The M W D C ' S used i n this experiment, were bui lt at the T R I U M F detector faci l ity based on a design developed at L A M P F [77]. E a c h chamber consisted of two independent planes, rotated 90° to each other, containing bo th anode and cathode wires and separated from each other by an aluminized mylar ground plane. A s shown i n F i g . 17, a l l the anode wires are connected to a common delay l ine, whereas the cathode wires are alternatively connected to an O D D or E V E N bus. Electronic signals were read from both ends of the anode's delay line, but from only one end of the O D D and E V E N cathode buses. The flow of the wire chamber signals is i l lustrated i n F i g . 18. The anode-anode and cathode-cathode wire spacings were 8.128 m m . The gas contained w i t h i n the chambers was a mixture of 65% A r g o n and 35% isobutane. Positive voltages (typically 2.0 to 2.1 k V ) were applied to the anode wires w i th the cathodes held at ground potential . The voltages were set to obtain m a x i m u m efficiency and performance from the chambers without inducing uncontrollable breakdown i n the gas. Th i s procedure is described elsewhere [73 . The operation of the wire chambers may be briefly summarized as follows. The passage of a charged particle through the chamber results i n the ionization of the chamber gas. The electric field due to the positive potential on the anode wires causes the electrons to drift towards the anodes and the positive ions to drift o d d / e v e n bus EVEN delay line Figure 17: Schematic diagram of a mult i -wire drift chamber. Shown is a single plane for acquiring " x " information (right to left). The complete chamber also includes aji identical plane rotated by 90 degrees to provide " y " information (up and down). The two planes are separated by an aluminized mylar sheet which is held at ground potential . In addit ion the aluminized mylar sheets provided the ground potential on the outer sides of each plane. d i s c r i m i n a t o r TDC a t t e n u a t o r ODD ADC s p l i t t e r d i s c r i m i n a t o r TDC E V E N ADC Figure 18: F low of anode {tn/h) and cathode ( O D D / E V E N ) wire chamber signals. The cathode ( O D D / E V E N amplifiers are inverting to provide the correct signal po-larity for the N I M electronics. towards the cathodes. As the electrons approach the nearest anode wire they experience a rapidly increasing (^) electric field and so experience greater acceleration. W i t h their increased kinetic energy, subsequent collisions w i t h other gas molecules w i l l result i n significant ionization of the gas. The resulting avalanche effectively amplifies the signal. It is the motion of the resulting large clusters of electrons and positive ions which induces detectable negative and positive pulses respectively on the nearest anode and cathode wires. The position information was obtained by combining knowledge of the anode wire which the event had passed closest to, w i th the t ime required for the electrons produced by the ionizing particle to drift to that wire^. B o t h of these quantities were obtained from the measurement of the arrival t ime of the pulses to both ends of the anode delay line (tfl ,tL) w i t h respect to a start signal defined by the scintillators. The anode pulses were amplified and sent to a discriminator, whose threshold was set just above the noise level of the amplifiers. The discriminator pulses then fed a TDC which recorded the t iming information. It can be shown [74] that the wire position, wire, is given by the difference i n tR and t ,^ whereas the drift t ime, drift, is determined by the sum of these quantities. wire= (tR-tL)Djiff, + Cdiff. drift = {tR + tL)D sum "l~ Csum Where Ddi//., Dsumi Gdi/f. and Csum are cal ibration constants which relate the measured times (in TDC channels), to appropriate scales and offsets. Since the positions of the wires were well known, one could calibrate wire i n units of distance. The drift time however, was not directly proportional to the drift distance due to the non-imiformity of the electric fields around the wires. As a result, a look-up table was used to relate the drift t ime to drift distance. The cal ibration of this lookup table is discussed elsewhere [73]. T y p i c a l wire "combs" ^This time is known as the drift time. Figure 19: Typ i ca l wire chamber histograms: a) Checksums; b) Combs (ifj —t/,); c) Dr i f t - t ime (tR + ti); d) O D D - E V E N . The arrows on a) and d) indicate the positions of the software cuts placed on these distributions. and drift distributions are shown i n F i g . 19b and 19c. T h e trajectory information is s t i l l not completely defined by this information, since one does not know on which side of the anode wire the particle passed. In order to determine whether the drift distance was to be added to or subtracted from the wire posit ion, information from the cathode wires was used. It has been shown [78] that for an appropriate range of high voltages, the signal induced on the cathode wire closest to the event w i l l be 10-15% larger than that induced on the opposite cathode. Hence by integrating the pulses arr iv ing on the O D D and E V E N buses one could determine on which side of the anode wire the event had occurred. Th i s was done using A D C ' s . The quantity " O D D minus EVEN''' ( O M E ) determined whether the drift distance was added or subtracted. ODD-EVEN , , ODD + EVEN (2^) where O M E was normalized by the sum of the signals to remove the energy dependence. A typical O M E d istr ibut ion is shown i n F i g . 19d. T h e overall position of the ion t ra i l i n a M W D C plane can be summarized i n the following expression: position = wire + {phase x (drift distance) x sign(OME)} (26) where phase ensures that the drift distance was added i n the same relative manner for each wire chamber. T h e resolution of a set of wire chambers was determined by comparing the actual position of a trajectory i n the central chamber w i th the predicted position calculated from information obtained from the outer two chambers (for those events for which a l l three chambers had "fired") : resolution = position2 — {position^ + positioni)/2 (27) A typica l resolution distr ibution is shown i n F i g . 20 for the smal l wire chambers (for the large chambers, the plot is somewhat broader). If one assumes that each chamber has the same resolution then the indiv idual chamber resolution can be expressed as width/\/3, or ±140 and ±250 microns respectively [73] for the smal l and large chambers. In order to obtain the chamber efficiencies, one must first define a good event. T h i s was done using the check sum (chksum) calculation. Th is is essentially a comparison of the drift times as measured by the anode t iming and the cathode t iming^ (recall that the cathode signal is induced by the same phenomena as '^Similar to the anode timing, the ODD cathode bus signal was fed to a discriminator and then to a T D C (see Fig. 18). Resolution Plot (microns X 100) Figure 20: Resolution of set of three smal l wire chambers. which the anode signal is induced). chksum — {tji + tif) — drift time toDD is the arrival t ime of the pulse on the O D D cathode bus w i th Drei. converting it to the same units as the drift t ime. The last term is required to remove a position dependent delay line effect of the O D D / E V E N bus. Note that wire was used to define the position along the cathode bus at which the signal originated. The quantity chksum is necessarily a constant. Hence if the event produces a chksum value which deviates from the expected value it is considered to be a "bad event". A typical distr ibution is shown i n F i g . 19a. Most bad events result from more than one event passing through the chamber at the same time, thereby producing a tn and t ,^ originating from different wires^. The a r m B wire chambers had a typ ica l efficiency of 92%. For a r m A the large and small chambers had typical 75% and 80% respectively. The chambers of A r m A were generally less efficient because they were downstream from the target and at a smaller angle to * M W D C ' s are only useful in lower rate experiments due to their inabihty to handle more than one event at a time. Their "dead-time" is that characteristic of the large delay time of the anodes. Drel.toDD + Dcorr. {tR " *Z,) (28) the beam. A s a resiolt they were subjected to a high rate of muons from the beam as wel l as background events scattered from the target and surrounding hardware. In part icular , for the K'j^^^ configuration the efficiencies of the large chambers were as low as 40%-50% due to the close proximity of a r m A to the pion beam. VI.2 The Target The polarized deuteron target was a well established facihty at T R I U M F and had been previously used i n several other experiments. A s a result, the behaviour of the target system was well understood at the t ime of the experiment [79,80,81 VI.2.1 Target Construction T h e target consisted of a d i lut ion refrigerator, superconducting Helmholtz coils, a microwave source for dynamic nuclear polarization, an N M R circuit for measuring the target polarization and the target material itself [82]. T h e target material consisted of deuterated butanol (C^DgOD) and 5% D2O doped w i t h 6 x 10^^ a t o m s / m L of deuterated E H B A - C r ^ complex. T h i s material was frozen into beads of 1 m m diameter and held i n a rectangular container which also supported the copper N M R coil . The target beads and their container were immersed i n a bath of 94% ^ i f e and 6% ^He a l l of which was enclosed i n a cy l indr ica l a lumin ium cup. Surrounding the cup were several walls of mater ia l which served as heat and vacuum barriers. The nature of the material i n these walls as well as the dimensions of the target container varied from configuration to configuration. The basic differences are summarized i n Table I V . T h e superconducting Helmholtz coils were capable of producing a 2.5 Tesla magnetic field which was uniform to 1 part in 10^ i n the region about the target. B o t h coils were built into the structure of the target cryostat. The d i lut ion refrigerator maintained the target temperature close to 1 K e l v i n . The target Table I V : Lis t of differences i n target dimensions (depthxwidthxhe ight ) , material surrounding the target, and target angle for the var i -ous configurations employed. Config. Targ . D i m . ( d e p . x w i d . x h t . ) (mm^) Container M a t e r i a l Shielding W a l l M a t e r i a l Targ . Rot . Angle (degrees) K'LS 1 0 x 2 0 x 2 0 teflon ( F E P ) C u , mylar , st. steel 0 2 0 x 1 0 x 2 0 teflon ( F E P ) C u , mylar , st. steel 0 1 9 x 1 9 x 1 4 mylar C u , mylar , K a p t o n 12 temperature was monitored using two separate thermometers, both of which had been calibrated to a 1% accuracy prior to the experiment [82 . A 100 m W impatt microwave source capable of a range of frequencies from 69 to 72 G H z provided the power for polarizing the target. A copper wavegmde carried the microwaves from their source above the cryostat to the target microwave cavity. The source was typically operated at 1 m W . A n E I P model 548A microwave frequency counter monitored the microwave frequency and through a computer feedback control loop, this frequency was stabil ized to order of 1 M H z [82]. The N M R coil about the target was connected i n series to a voltage controlled capacitor which was used to tune the resonance frequency of the circuit. A constant current radio-frequency ( R F ) signal was supplied to the resonant circuit . The voltage appearing across the circuit was sensitive to the change of impedance when the target became polarized. The R F signal frequency was varied so to span the resonant frequencies associated w i th the projection states of the deuteron (as w i l l be discussed i n the next section). The signal was externally amplified and its shape analysed by a computer to determine the polarizat ion of the target [79,82 . VI.2.2 Polarization of Target The implementation and determination of the target polarization was pr imari ly the task of the target group at T R I U M F [82]. The target polarization was continually monitored by the computers belonging to the target group. This value was i n t u r n recorded on the experiment's magnetic tapes as a scaler reading every five minutes. As well the errors associated w i th the target polarization were provided by the target group. The following is merely a brief review of concepts related to the target polarization and the N M R techniques used to evaluate i t . In the absence of a magnetic field, the deuteron exists i n a bound state of well defined energy Ud- The presence of a homogeneous magnetic field serves to split the possible energies of the deuteron system through the coupling of the deuteron magnetic moment (JJ.) to the magnetic field. Th i s results in three separate energy levels for m = - l , 0 , + l (see section 1.2) depending on the alignment of the deuteron's spin: U'd = Ud + Um where Um = —fJ-\B\m is the change i n the energy of the deuteron due to the couphng of its magnetic moment to the magnetic field. T h e lowest energy state is for m = + l . Th i s state is preferentially occupied according to the Maxwel l -Bo l t zmann distr ibution: n+ no (dE ; ^ = ; ^ = ^ " p ^ ^ ^ where dE is the transit ion energy between levels, k is the Bo l t zmann constant, T is the target temperature and n , are the relative populations of the corresponding energy levels. Th i s populat ion differential is known as natural polarization, g iving rise to a vector polarization defined as: n^. — n _ n+ + no + n_ 83 The "posit ive" direction of this natura l polarization corresponds to the direction of the magnetic field. T h e natura l polarization is extremely smal l and is only observable at temperatures well below 1 K .^ A s a result, one relies on dynamic polarization of the target to realize a significant target polarization. T h e tensor polarization of the target is obtained from the following expression [81]: PT = 2 - \/4 - 3 P 2 (29) T h e principle of solid state dynamic polarization [83] relies on the weak spin-spin coupling between the magnetic moments of molecular electrons and the magnetic moments of the deuterons. T h e target material is doped w i t h paramagnetic ions {Cr^) whose electron spins essentially become 100% aligned wi th the large magnetic field about the target. Deuterons whose nuclear spins are aligned or anti-aligned w i t h the external magnetic field can couple to the electrons of the par-ions to form a system wi th a net magnetic moment which can be expressed as: U p o n excit ing this system wi th electromagnetic radiat ion the appropriate L a r m o r frequency, which can loosely be expressed as: one can preferentially induce spin flips of the (fi^ + fJ^d) or (fie — Hd) systems. It is important that the relaxation time^° of the electrons be much shorter than that of the deuterons, since, because of the weak coupling, the electron w i l l soon realign w i t h the magnetic field^^ leaving the deuteron in the " f l ipped" state. If the bandwidth of this Larmor frequency (u>i) is much narrower than Ud then a 0.0005 for a temperature of IK and a field of 2.5 Tesla. ^°(the average time before the spin flips back to its natural state) ^^This provides an inexhaustible source of electrons. deuteron left in the " f l ipped" state w i l l not be induced back to its original state by the exciting frequency. B y this means, one can preferentially enhance the populat ion of a part icular state by selecting the appropriate Larmor frequency. A s a restilt, it is not necessary to change the direction of the magnetic field i n order to change the "s ign" of the deuteron polarization but simply to select a different frequency of radiation. Thus , data could be obtained corresponding to runs w i t h opposite deuteron polarization without having to change the magnetic field which would i n turn have required changing the kinematics and acceptance of the system. T h i s feature proved to be very useful for minimiz ing the contribution of systematic errors i n the measurement (see chapter I X ) . T h e magnitude of the target polarization was measured using N M R techniques. A smal l t ime-oscil lating magnetic field of variable frequency was introduced orthogonal to the large homogeneous field. W h e n the frequency of osci l lation was brought close to the L a r m o r frequency of the deuterons, transitions were induced between the deuteron's energy levels. These transitions manifested themselves as changes to the absorptive component of the complex magnetic susceptibility of the target material . The N M R circuit , which includes an inductive coil about the target, was able to sense this change. As a result power was absorbed from or added to the circuit depending on the sign of the target polarizat ion. Th i s power change can be visualized i n the following manner: for a positively polarized deuteron, the oscil lating magnetic field induces more transitions from the lowest energy state to higher energy states, hence energy must be acquired from the N M R circuit ; conversely for a negatively polarized target, the transitions go from higher to lower energy states, thereby releasing energy to the N M R system. To obtain the value of the polarization attained, the power absorption from or by the N M R circuit was integrated w i th respect to the varying excitation frequency. This integral is proportional to the target polar izat ion. The proportionality factor was determined by measuring the known natural polarization i n an identical manner before the dynamic polar izat ion was introduced. The polarization of the dynamical ly polarized target could then be obtained by simply scaling: App l i cab i l i ty of this relationship relies on the fact that the amplifiers of the N M R circuit must remain linear for input signals varying over several orders of magnitude. Errors associated w i th the measurement of the dynamic polarization pr imar i ly result from the calculation of the natura l polarization (due to the uncertainty i n the target temperature) and the integration of Areanat. (due to poor statistics). A n addit ional relative error of 5% was added i n quadrature [84 . Th i s error represented the systematic discrepancy between a test polarization measurement of the known neutron-proton spin-correlation parameter A^N [85 and a s imilar N M R system at T R I U M F . VI.3 Peripheral Detectors A d d i t i o n a l detectors were included i n the experiment for diagnostics purposes. They were used to monitor the pion beam incident on the target and served to indicate i f the beam had strayed from the target or its intensity had fluctuated. The ir approximate locations are i l lustrated in F i g . 14. T w o sets of "muon counters" were placed at the exit of the M-11 beam pipe. A set of muon counters consisted of two smal l scintillators placed at an angle to the pion beam line so that they intercept the cone of muons originating from the pions decaying i n the beam line. To count a "muon" a coincidence between the pair of scintillators i n a set was required. Since the two sets of muon counters were placed symmetrical ly on opposite sides of the beam-line, and asymmetry i n the muon count rate from the two sets would indicate a shift i n the pion beam posit ion (thereby changing the acceptance of the muon counters). In addit ion, the sum of the count rates from the two sets provided an indicat ion of the pion beam flux w i t h i n the channel. A "telescope counter", placed at an arbitrary angle to the beam-line, was adjusted to point at the target. T h i s counter consisted of a set of three small scintil lators i n a row wi th a good event requiring the coincidence of al l three detectors. The small directional acceptance of this device ensured detection of particles from the region of the target. Since the number of particles originating i n the target is proportional to the number of pions actually h i t t ing the target, this detector was particularly sensitive to any shift of the beam off the target^^. A "hodoscope" was placed i n the pion beam downstream from the target. T h i s instriiment consisted of four identical scintillators (4cm x 4cm) grouped together to form four quadrants of a square. This was centred on the middle of the pion beam. Relative shifts i n the beam position could then be determined by comparing the combined rates of the upper scintillators w i t h those of the lower, or conversely, the left rate w i th the right rate. Th i s detector was only used on the K'l^s runs where the incident pion beam was not deflected by the magnetic field of the target. For the other configurations, the energy dependence of the deflected beam unduly complicated the use of this instrument. The final peripheral detector was a fast in-beam wire chamber constructed w i t h 0.76 m m wire spacing. This standard T R I U M F faci l i ty [86,87] was placed i n the beamline just before the target structure. Using ind iv idua l wire readout, the horizontal beam profile could be monitored, providing information regarding shifts ^ ^ However the absolute rate is affected by the target polarization due to the analysing power of the associated reactions. or broadening i n the beam. N o information from these detectors was directly used i n the polarization analysis. However they were frequently checked so that any degradation of the beam properties could be quickly detected. VI.4 The Data Acquisition System O u r data acquisition system was based on a well proven system developed at TRIUMF. It consisted of a C E S Starhurst " J - H " front end preprocessor, connected to a C A M AC crate and a Dig i ta l Equipment P D P - 1 1 / 3 4 computer. The J-11 was resident inside the C A M A C crate which was i n turn connected, v ia the crate controller, to the memory bus of the P D P - 1 1 computer. T h e function of the J-11 was to read the ADC, TDC and scaler units contained i n the C A M A C crate and to transfer this information to an event buffer i n the memory of the J -11 . In addit ion , the J-11 also performed calculations on and made decisions about the data (see section VI.4.1) before passing the information on to the P D P - 1 1 . The function of the P D P - 1 1 was to write a l l data to magnetic tape and to perform some rudimentary analysis of a fraction of the data for diagnostic purposes (see section VI.4.2) . T h e J-11 was prompted to look at the data upon receiving a LAM {look at me) interrupt from the hardware logic. The definition of L A M , pictured i n F i g . 21, consisted of a EVENT "seen" in the scintillators of a r m A and B in coincidence, together w i t h the requirement that the computer was not busy dealing w i th a previous EVENT. E V E N T = ( s l B 0 S2B) 0 (S3B ® S4B) 0 [ (s lA ® S2A) 0 (S3A ® S4A © S5A)' Those events which approximately satisfied the two-body kinematics of the •nd pp reaction were preferentially selected by this EVENT definition. The Timing Definition Event LAM= EVENT ©BUSY A r m B Event Figure 21: F low of logic defining an E V E N T from the two detector arms. L A M is defined as a coincidence between E V E N T and B U S Y (computer N O T busy). discriminators and logic units which were used i n selecting events were of the common fast N I M variety [88]. VI.4.1 T h e J-11 A s discussed i n section V . l , Coulomb scattering dominates the smal l angle region of the proton-carbon interaction. Since these interactions have the largest cross-section, yet contain essentially no polarization information, it is advantageous from the point of view of economy and dead time reduction to reject as many of these events as possible at the front end of the data acquisition system. Such a rejection has become a common feature of most polarimeter systems. E a r l y versions performed their rejection at the hardware level [89,90]. In these cases, the E V E N T definition included a requirement that the proton had scattered i n the carbon by more than a predefined m i n i m u m angle, w i th the scattering angle calculated using t iming information from the wire chambers. In this experiment the J-11 preprocessor was used to provide the small angle rejection. U p o n receiving a L A M interrupt , the J-11 woiold read the instruments i n the C A M A C crate and reconstruct the particle 's trajectory based on information from wire chambers W C l , W C 3 , WC4 and W C 6 . In order to avoid possibly biasing the accepted events, no additional cuts were placed on the wire chamber information (like C H E C K S U M ) . A s a result, a number of "bad events" were accepted by the J -11 , to be later rejected by the off-line analysis. The scattering angle calculation used only wire information (no drift information included) and employed a small angle approximation i n order to reduce calculation time. T h i s was carried out for the x and y planes separately. e. Az Ay Az In the case of the K'j^g and K'gg measurements, either 6^ or 6y was required to be 6 degrees or larger. Th i s produced a square cut i n the {Ox,Oy) domain. For the K'^j^f runs, the cut was improved to a more efficient circle cut, imposed i n the following manner: T h e effects of such cuts are shown i n F i g . 22. It was important that the wire chambers were properly calibrated during data acquisition. It is obvious that any shift i n the positions of the angle cuts could introduce an asymmetric acceptance at the smaller angles. Hence the position of these cuts were frequently monitored throughout the experiment. As w i l l be discussed i n section VI I .2 , the J-11 cut was turned of for a smal l fraction of the data acquisition time which provided an unbiased data sample to be used for software calibration purposes. 30O 150-Figure 22: The upper plot shows the scattering angle distributions for data acquired w i t h no J-11 cut, the middle plot shows the effect of the circular J-11 cut on the accepted data, and the lower plot shows the effect of the square J-11 cut on the accepted data. VI.4.2 T h e On- l ine Software The pr imajy function of the on-Hne data acquisit ion software was to take the raw data from the J-11 event buffer (if it had passed the J-11 cut) and orchestrate the wr i t ing of this data to magnetic tape. In addit ion , a fraction of these events were analysed (using on-Hne software) i n an effort to identify any problems wi th the apparatus a.s they occurred. The STAR system [91] is the F O R T R A N based software package employed for the acquisition. Th i s package was i n common use at T R I U M F at the time of this experiment. It is a flexible system which allows the user to define both one and two dimensional histograms from a globally defined array of variables. In addi t ion , it permits the appl icat ion of tests to these variables, the results of which could be subjected to logical manipulations. For example, the checksum test was defined by a gate around the checksum peak (see F i g . 19a). T h e test was "true" i f an event lay inside this gate. T h e result of this test could be combined wi th others following standard Boolean algebra. T h e simplicity of the package allowed one to easily customize the software for a part icular appl icat ion through the addit ion of user-written F O R T R A N subroutines. For diagnostics purposes, the following quantities were monitored throughout the experiment: • the checksum distributions, for providing information concerning the relative efficiency of each chamber • 6x and By distributions, to check for shifts in the J-11 acceptance • the position distr ibution of the fast in-beam wire chamber, to check for shifts of the incident beam on the target • the ^ distr ibution of the scintillators, to check that the gains were set correctly and that the discriminators were not cutt ing into the proton distr ibution • any of the raw T D C / A D C distributions to identify problems wi th ind iv idua l channels • the distributions of O D D , E V E N and O M E = ( O D D - E V E N ) / ( O D D - | - E V E N ) , to ensure that the cathode O D D / E V E N system was working properly T h e on-line analysis was easily adapted to the T R I U M F V A X for the off-line analysis and hence the routines used to calculate the on-line quantities are basically the same as those discussed i n section V I I . 1 for the off-line analysis. VI.5 Experimental Summary The experimental effort was carried out over three separate running periods. In January 1987 a two week test r u n was made to evaluate the performance of the system. Over a six week running period i n M a y / J u n e 1987 the K'j^g and K'gg data were obtained. The necessary target modifications to change from the K'j^g to the K'gg configuration, as described i n section V I . 2 , were performed midway through that r u n , during a short down time i n the beam. The final data set, K'j^^i ^as obtained during a three week r u n i n September 1988. The major target modifications required for this r u n were performed during the months preceding. Chapter VII Off-line Analysis Most of the off-hne data analysis used i n this experiment is based on software which has already been described [74]. This chapter provides a sunmiary of the important aspects of the analysis procedure. The data was fed through three levels of analysis. Each level selected that subset of the input data which satisfied the good event cr iteria defined at that level. T h e data of that subset was then transformed into the composite quantities required by the next level of analysis. Such data is wri t ten to a file which is read by the software of the next level. The three levels of analysis are summarized as follows: 1. T h e R E P L A Y software takes raw data from tape, rejects events which are pion-like i n arm B and also those events w i th incomplete wire chamber information (the required m i n i m u m of 2 of 3 wire chambers had not fired in at least one of the chamber sets). It then produces an event file where each event is characterized by the scattering angles at both the target and the carbon analyzer as well as kinematic information required for background discr imination (these quantities are defined in section V I I . 1). 2. The P O L A R software distinguishes background and foreground events using the kinematic information provided by R E P L A Y . In addit ion , it calculates the polarization for each foreground/background subset as a function of several selected independent quantities (to look for systematic dependences). The program outputs the normal and sideways components of the polarization measured at the polarimeter , w i th statistical errors, for both the foreground and background events (as defined i n section VII .4 ) . 3. The S T _ E X T software takes the polarizations measured at the polarimeter and transforms them into the centre of mass values, at the target, of the quantities of interest. These quantities are then used to calculate the appropriate spin-transfer observables. Only the first two of these routines are discussed i n this chapter, w i th the last i tem being discussed i n chapter I X . VII. 1 Replay Software This software was taken directly from the on-line P D P - 1 1 software and adapted for use on a V A X . A l l the events were read from files which had been writ ten onto magnetic tape dur ing the acquisition of data and then processed i n the manner discussed below. T w o types of data buffers (each of record length 4 K-bytes) were wri t ten to tape dur ing the experiment. One was a table of the cumulative scaler values which were recorded at five minute intervals throughout a run . The other buffer type contained a l l the relevant A D C and T D C values provided by the C A M A C instrumentation for each event stored on an event by event basis. E a c h event contain 130 words (4 bytes/word) . Th i s information was processed to produce the desired scattering angle and kinematic quantities. Figure 23 describes the hierarchy of subroutines and logic which select the good events and perform the calculations. A summary of the routines is provided below: • C B L K 2 and W C P O S calculated the wire chamber information: checksums, wire positions, drift t ime etc.^ • C O D E - C H A M determined which wire chambers had good information and which d id not. It then rejected those events having inadequate wire chamber information (at least 2 of 3 chambers had to have fired i n each set). • T R A J E C T O R Y calculated from the wire chamber information vectors describing the particle's direction of motion. These vectors were used i n the routines which followed. • X Y Z T R B K and C A R B O N T R B K calculated the "traceback" to the scattering vertices at the deuteron and carbon targets respectively using the trajectory information from the previous routine. The scattering vertex was defined as the average coordinate (x, y, z) of the points of closest approach of the two traced back vectors. The m i n i m u m distance of approach of each pair of traced back trajectories was also determined. These distances are termed: root difference squared (RDS ) (see reference [74]) and are a measure of the quality of the traceback. Those events w i th poor traceback information were rejected. G o o d traceback events were defined as those ly ing between the arrows displayed on the histograms of F i g . 25. It was insisted that for the carbon traceback: RDS < 10 m m ; and for the deuteron traceback: RDS < 30-50 m m . The larger value of RDS i n the deuteron traceback was allowed because of the larger lever arm i n the traceback and the distortion of the trajectories due to the target magnetic field. • C O P L and C _ A N G calculated the scattering angles and the kinematic quantities defined in Table V and i l lustrated in F i g . 24. • A C _ T S T and S C I N T _ A C C were subroutines which selected events which ^Details of these calculations are given in section VI.1.2. / g o o d scintil lator^ V i n f o r m a t i o n ± CBLK2 WCPOS CODE-CHAM TRAJECTORY 1 COPL C_ANG V AC_TST SCINT_TST f good events \ w r i t t e n to file R e t u r n R e t u r n XYZTRBK CARBGNTRBK R e t u r n R e t u r n Figure 23: F low of the off-line R E P L A Y software. R e t u r n paths to the right indicate rejected events. satisfying the 7r-symmetric condition required by the polarization analysis (see section V . l ) . The former invoked an acceptance defined by the geometry of the last wire chamber, whereas the latter reflected the acceptance characteristics of the scintillators S l A and S 2 A . Events not satisfying the "7r-symmetric" condition were rejected. The R E P L A Y software cJso contained several peripheral routines to provide the user w i th diagnostic tools to check for problems w i t h the calibrations, or Figure 24: Definit ion of coordinate system and angles used i n analysis. The f -ax i s , not shown, is defined from the y and z axes by the " r ight -hand" rule. instrumental malfunctions which may have gone unnoticed during the data acquisition. These included one and two dimensional histograms of the data as well as logical tests applied to the data. These checks are discussed i n greater detai l i n section VI I . 3 . VII.2 Calibration of Software The cal ibrat ion of the R E P L A Y software was pr imari ly performed w i t h the T R I U M F plott ing and data manipulat ion package P L O T D A T A [92]. The cal ibration steps are l isted below. Details of these steps have been described previously [74,73]. The cal ibrat ion was only carried out for those events wi th the J-11 cut turned "off". Th is was accomplished for the A'^^ and K'^g data by turning off the J-11 cut for a " ca l ibrat ion" r u n for each run set (every time the energy was changed or a wire chamber changed etc.). For the K'j^j^ data a sample "uncut" event was automatical ly allowed by the data acquisition system every 100 events. The cal ibrat ion parameters obtained i n the following discussion were applied to a l l the runs of that particular r u n set: 1. Gates were placed around the "proton" peaks i n the ^ distr ibution of the scintillators of arm B (see F i g . 16) as well as the two body correlation peak in the a r m B T O F d istr ibution. The scintillators of arm A were not subjected to this cut to avoid an arti f ic ial bias i n the post-carbon acceptance. 2. The wire chamber checksums were next calibrated and gates were placed about their peak (shown i n F i g . 19a). This defined the good wire chamber events. 3. The drift t ime was cahbrated using the statistical average of a l l the wire chamber drift times which had good checksums. T h i s has been described earlier [73]. A drift distance lookup table was created f rom this average. 4. The indiv idual wire chamber "hit positions" (tR — ti from equation 24) were calibrated so that the central wire was the "zeroeth wire" and the adjacent wires were at integer locations on either side. These iniegerized wires could then be s imply converted to a real posit ion by mult ip ly ing by the well known wire spacing. 5. The cathode O D D / E V E N A D C distributions were calibrated by adjusting the relative gains of the O D D and E V E N signals (in software) to be the same and then by means of an offset, shifting the valley of the = ODD+EVEN-Ga^ d istr ibution to be at zero (see F i g . 19d). 6. The relative phase (see equation 26) of the O D D / E V E N d istr ibution was determined so that each wire chamber corrected the "hi t pos i t ion" for the drift distance to the associated anode wire i n a manner consistent w i t h the others of the set. The phase cal ibrat ion has been discussed previously [73,74 . 7. The relative positions of a l l chambers were aligned w i th respect to wire chamber 3 of a r m A using those "J-11 off" events originating i n the target. W i r e chamber 7 of a r m B was also aligned w i th respect to wire chamber 3, i n this case using the correlated events of the nd —* pp reaction. However the remaining chambers of a r m B were aligned wi th respect to wire chamber 7. 8. Cuts were placed around the traceback and R D S distributions at the target and carbon analyzer. This el iminated those events which d id not scatter from the intended target or were of poor quality. T y p i c a l distributions are shown i n F i g . 25. 9. T h e kinematics variables, 6A + OB and coplanarity were corrected to remove any inherent dependence on scattering angle^, and thereby to narrow their respective distributions. Th i s has been described previously [74 . 10. F i n a l l y the az imuthal d istr ibut ion of the smal l angle mult iple Coulomb scattering events i n the polarimeter was evaluated using Fourier analysis as a check of the overall cal ibration. A cal ibration was considered successful i f the cos (p and sin (f) components were less than 0.1%. It has been shown that such a Fourier check is extremely sensitive to any misalignment of the chambers due to cal ibration error [73]. The adequacy of, or any subsequent shifts i n , the cal ibration could be checked on a run-by -run basis using the software checks of the next section. VII.3 Software Checks Several routines were included to ensure that the data was behaving well on a run-by-run basis. The most important of these was a set of histograms containing quantities known to be sensitive to changes i n calibration or other instrumental problems. A series of such histograms were generated for each r u n . A list of these histograms is given below, w i th examples i l lustrated i n figures 20,25 and 26. • kinematic quantities - OA + ÔB — coplanarity • a r m A acceptance ~ ^target ^This dependence comes from the kinematics of the reaction. "~ 4>target • carbon scattering "carbon • resolution plots for each set of three wire chambers • scattering angles, identical to those calculated on-line by the J-11 • target traceback and R D S distributions — deuteron target traceback — carbon analyzer traceback • in-beam wire chamber profile Another useful diagnostic feature was a data-testing facil ity which listed the rates at which the data had lay wi th in the "gate cuts" , described i n section VII .2 . The results of these gate tests ( T R U E / F A L S E ) could be combined, using standard Boolean algebra, w i th other tests (on an event-by-event basis) to obtain the net efficiency of groups of tests. Gate tests which were of part icular importance (showed sensitivity to potential problems i n the analysis) are listed below: • the scintil lator distributions (fraction of events satisfying "good proton" requirements) • C H E C K S U M distributions (relative efficiency of the chambers and chamber sets) • traceback distributions (what fraction appeared to come from the target) • acceptance tests (what fraction were accepted by these tests) -400 -200 Ô iÛO 4Ô0 Carbon X Vertex (mm) 800 -400 -200 0 2 0 0 4Û0 Carbon Z Vertex (mm) 3500-1 -100 -50 6 5b ido Target X Vertex (mm) -50 6 5b . arget Z Vertex (mm) -TO -400 -200 0 200 400 Carbon Y Vertex (mm) 12000-10000 8000: 6000-4000 2000^ i , ^ ^ Carbon RDS (mm) 9b -100 -50 ô 5b ido Target Y Vertex (mm) 1600^  4D 90 140 Target RDS (mm) Figure 25: Examples of traceback histograms produced by R E P L A Y . The arrows indicate the boundaries of the cuts placed on these distributions. A n event ly ing outside any of these cuts was rejected. Figure 26: Examples of histograms produced by R E P L A Y . O n l y the 6carbon d istr ibu-t ion had a cut placed on i t . A n event whose 9carbon was less than 6° was rejected. • polar angle scattering distr ibution in carbon (what fraction has scattered more than 6°) The numbers of events "passing" these tests were normalized to the number of "proton l ike" events^ in the r u n . Table V I provides some typica l efficiencies (good events /No. of "proton-l ike" events) for the tests listed above. These values are energy and configuration dependent. Other information available included the incremental and total scaler readings which counted rates i n various detectors such as the scintillators, wire chambers, monitor telescope and muon counters etc. In general this information was not required for the analysis but was useful for flagging or confirming any possible equipment failure or shift in operating conditions. VII.4 The P O L A R software The function of this program was to estimate the proton polarizat ion from the carbon scattering angles obtained from R E P L A Y . The method used for the polarization extraction follows the discussion in section V . 2 . The proton energy at the centre of the carbon (required for the energy dependent p - C analyzing powers) was obtained from a look-up table i n terms of the proton trajectory incident upon the carbon. Th is trajectory was defined by the angles (6t,(f)t) (see Table V ) . The energy look-up table was produced by the trajectory modell ing program described i n section IX .2 . The analysing powers were tabulated as a function of both the proton energy at the centre of the carbon and the carbon scattering angle (Ocarbon) using the parameterization of McNaughton et a l . [71] (see F i g . 13). In this manner, the correct p - C analyzing power was applied on an event-by-event basis. The polarization was evaluated as a function of dcarbon for the background and foreground data separately, then averaged over Ocarbon to obtain a mean value ^(as defined by the arm B scintillator tests and discussed in section Vn.2) for bo th the sideways and normal polarizations. In addit ion the foreground and background data were further divided into subsets as functions of a number of independent quantities. B y evaluating the polarization as a function of these quantities, any undesirable systematic effect present i n the data can readily be distinguished. The quantities investigated were: • carbon polar scattering angle (Bcarbon) • target scattering angles {Otarget,<f>target) • r u n number • kinematic radius where: kinematic radius = \/[(0A + OB) — (OA + BB)Y + [coplanarity — coplanarity^ (31) OA + OB and coplanarity are the means, or peak positions, of their respective distributions. The kinematic radius is a measure of the angular deviation of an event from the two-body kinematic peak. B y selecting those events less than or greater than a predefined kinematic radius, P O L A R effectively distinguishes foreground events from background events (although some background events remain under the foreground peak). A n example of this kinematic radius is given i n F i g . 28. The kinematic quantities introduced here, are discussed in greater detai l i n the next section. T h e typica l functional dependence of polarization w i th respect to the quantities l isted above are shown i n F i g . 27. Such plots were readily produced using executable codes written for T R I U M F ' S P L O T D A T A graphics package [92 . 1.00 o (d N .1-1 t-, "o a, o o ( H -1.00 H ' î I I 10 11 g 0.75-®Deut. degrees) - 5 - 4 - 3 - 2 -1 0 T 2 3 4 5 9 D e u t . (degrees/2) 1.00 g 0.75-% 0.50--3 0.00-^-0.25-5-0.50-o (i;-o-75--1.00-I ] I I I I I I — I — I — I — I — I — I — I — I — I — I — [ — 10 12_ 14 16 18 20 0 , 1.00-- a o 0.75--l-> N 0.50-• F N ( H 0.25-(0 l - H O 0.00--0.25-O -0.50-o u -0.75--1.00-14 16 18 20 ^Carbon (degrees) 1 . 0 0 -...= 255 M e V C o n f i g . 0.75H o 0.50H N 0.25^ t-, cO 3 0.00-g - 0 . 2 ^ 2-0.50H Oh -0.75-_L_ i J I I I I I I L F o r e g r o u n d / Background cut ~ 1 - 0 t H I f 1 I I I 1 I I r 0 1 2 3 4 5 6 7 8 9 10 11 Kinematics Radius (degrees) Figure 27: Examples of checks for systematic dependences i n the polarization. T h e above four plots include only "foregroiind" data as selected by the cut depicted i n the lower plot. VII.5 Background Subtraction In the previous section, a procedure for distinguishing background events from the foreground events was discussed. T h i s procedure separated from the foreground data set those events which d id not satisfy the two-body kinematic correlation characteristics of the ird —> pp reaction. However, those background events which lie under the foreground kinematics peaJf are not removed by this cut. Thus it is important to determine the relative number of background events under this kinematic peak and evaluate their contribution to the observed polarization. The majority of the background comes from two-body absorption on "quasi-deuterons" w i t h i n nuclei such as carbon or oxygen, which are also present i n the target material . These reactions, which also produce two protons in the final state (as well as other particles, such as the "spectator" nucleus), have a kinematic behaviour similar to the free ird —> pp reaction except that their kinematic properties are greatly broadened due to Fermi motion (and momentum transferred to the other particles i n the final state). W h e n analysing the free Trd —> pp data , two independent kinematic variables were available: first, the opening angle between the final state protons, which described the angular correlation of the outgoing particles w i t h i n the scattering plane; second, the coplanarity of the in i t i a l and final state particles measured their deviation from the common reaction plane. Two-body kinematics requires the constituent reaction particles to lie w i th in the same plane and that the opening angle of the protons at a given reaction angle to be defined by the angular correlation. Th i s two-body correlation is clearly seen in F i g . 28. T h e circle superimposed on this plot represents the kinematic radius which was chosen in P O L A R to separate foreground from background. To determine the fraction of background events which remained inside the 110 100 H -p •H u 90H rH 0 u 80-70 data used in coplanarity fit data used in fit ffl kinematic radius • 140 150 60 170 Figure 28: Sl ic ing technique used i n background fits of coplanarity and OA + OB projections. For example, only the data ly ing wi th in the vert ical box were used i n the coplanarity projection of F i g . 30. kinematic radius, a fit to the background, outside the kinematic radius, was independently performed for each of the two kinematic variables. The data were sliced such that only those events ly ing along the opening angle or coplanarity axes were used i n each fit, as shown i n F i g . 28. Us ing the information of the projection distr ibut ion outside the kinematic radius, one can interpolate the shape of the background inside the kinematic radius i f its shape is known. T h e observed shape of the background distr ibution is largely determined by the acceptance of the experimental apparatus. For the purposes of this analysis, the characteristics of the background were determined by analysing data from a non-deuterated butanol target^. It was found that the background distr ibution was well described by a Gaussian shape (as shown i n F i g . 29). It was also determined that by fitting this Gaussian to the " ta i l s " ^ of the distr ibution one could reproduce the magnitude of the central regions to high accuracy. For example, i n F i g . 29 the fitting procedure determined that 99.1% of the events w i th in the foreground peak area were background (obviously 100% was expected for a background run) . Th i s is an absolute error of 0.9%. As a result, the shape of the background under the foreground peak (and hence the relative ntimber of background events) could be obtained w i t h confidence by fitting a Gaussian to those events i n the tails of the kinematics distr ibution. Th i s procedure is demonstrated i n F i g . 30. T h e relative amount of background w i t h i n the kinematic radius, as separately determined by each orthogonal fit was, as expected, always consistent. For example, for the 145 M e V unpolarized deuteron data, a background of 5.9% was determined wi th the opening angle fit compared w i th 6.3% for the coplanarity fit. This agreement gives addit ional credibil ity to the technique. In general the contribution of the background to the polarization was found ' 'The molecular structure was identical to the regular target except that was replaced by ^H. ^The "tails" imphes events outside the kinematic peak. 200-"foreground events" K> 150-tt O o 100-50 -0 70 80 90 100 110 Coplanarity Figure 29: F i t of background curve to kinematics d istr ibut ion using a non-deuterated butanol target. The shaded area corresponds to data which were outside the "kinematics radius" and the unshaded area was inside the kinematics radius. to be fairly smal l , typical ly less than 10%. A s a result an approximate approach was found adequate to account for its contribution. For a s p i n - | particle, the relationship between polarizat ion, analysing power and the scattering asymmetry is e(^) = A(6)P (from equation 13). The asymmetry (e) can be defined as e = or the difference i n the numbers of events scattering right and left, normalized by the total events. No = ri(L) + n{R). For a given scattering angle 9 the observed asymmetry can be considered to be the sum of contributions from foreground and background events: The total observed asymmetry can be expressed i n terms of sums of those scattering to the left and right due to foreground and background events: eo = ef + 66 (32) {n,{L)+nt,(L))-(nf(R)+n,(R)) Region used i n f i t to b a c k g r o u n d 80 90 100 Coplanarity 110 Figure 30: Technique for determining the ratio of background under the foreground peak. The upper plot is an expanded version of the lower plot. The shading is as defined i n F i g . 29. These data were taken from the K'j^j^ configuration at T^,— 145 M e V . where the indiv idual asymmetries are expressed as: _ n,(L)-n,{R) _ nb(L)-nb{R) w i t h Nf/b representing the total foreground/background events respectively. Solving for the foreground asymmetry, ey, the quantity of interest: Thus Nb Nj r Nb No Nb '"Nj ''NJ Co -Nb -''No No Nj [Po- -BPb] 1 F (35) where i n the last expression (35), the analysing power has been incorporated to y ie ld the polarization assuming an average analysing power which is common to both background and foreground. F and B {F = 1 — B) are the fraction of foreground and background events w i t h i n the kinematic radius, as determined by the above f itt ing procedure. A l though it would be more correct to weight the asymmetries at each polar angle b in by the analysing power, this would only unduly complicate the analysis since the background pr imar i ly serves to "d i lute" the foreground polarization w i th events of much less polarization. For example, i n the extreme case where the background is completely unpolarized, equation 35 indicates that the true polarization is simply di luted by the addit ional background events: Po = P/ • F = Pf • ]y^^f^. O n the other hand, if the background had the same polarization as the foreground, equation 35 yields: Po = P/, as expected. It is clear from equation 35 that the dominant contribution to the foreground polarization comes from the relative fraction of foreground events (F ~ 0.9). Knowledge of the background polarization adds only a small correction (B ~ 0.1). Th i s point is important since the background polarization is not as well known as that of the foreground. The background polarizat ion has to be obtained by evaluating the asymmetry of the events i n the ta i l region of the kinematics distr ibution. However, analysis of data from non-deuterated background runs demonstrates that the polarization of the events i n the ta i l agrees well w i th those events ly ing under the kinematic peak, as seen i n F i g . 31. A further complication arises, however, from the fact that the energy of the background protons was not well known due to the greater uncertainty in the relevant kinematics. Thus the energy-dependent analysing powers which should be employed for the background events would be expected to be somewhat different from the values actually used i n the analysis. In the above, the assumption applied was that the average proton energy was similar to the two-body reaction and that therefore the average analysing power was the same. The sensitivity of this assumption varies wi th the range of proton energies as demonstrated in F i g . 13 and thus is also related to the incident pion energy. However, since the contribution of the background polarization to the determination of the foreground polarization is typical ly less than 10%, some uncertainty i n its value does not have a significant impact on the value extracted for the foreground polarization. It w i l l be demonstrated i n chapters I X and X that no significant problem exists w i t h the polarization extraction procedure. 1.00 0.75 -O 050-.1-1 s) N 0-25-0.00-- 0 . 2 5 -a r—t O a o O - 0 . 5 0 -u eu -0.75--1.00 -(S PL, 0 1 J I l_l I I I L •3 El No -mal Polarization 1 I I I I I I r 2 3 4 5 6 7 8 9 1011 Sid T J I I I I I I L rays Polarizat ion 0 1 Kinematics Radius (degrees) T—I—I—r 2 3 4 5 6 n—I—I—r 7 8 9 1011 Figure 31: Kinematics dependence of polarization using a non-deuterated butanol target. The results obtained at large kinematic radi i agree well w i t h the results at smal l rad i i . Table V : Definitions of quantities calculated i n COPL and C_ANG. ^^,,,/,s are the respective momenta for the incident p ion , i n i t i a l and final proton trajectories i n arm A and the proton of a r m B . y is " u p " i n the laboratory and Ut is the normal to the scattering plane at the deuteron target { fit = ( K x x Ot - l i t , Be - E ! ^ cos(^t) = n t - y <i>c c o s ( . ^ o ) - ( | j ; , j ; i ) - . i , sign of (pt sign([y X fit] • K) sign of (})c cosieA + eB)-:^:l, Coplanari ty Table V I : T y p i c a l software efficiencies: a) checksums for ind iv idua l cham-bers; and a set of three chambers (at least 2 of 3 had fired); b) traceback to carbon scattering vertex; and deuteron target vertex; c) acceptance tests (introduced i n section V I I . 1); d) f inal good events, before and after Ocarbon cut. a) chambers ind iv idua l at least 2 of 3 had fired i n set 1,2,3 ~ 80% ~ 85% 4,5,6 ~ 75% ~ 80% 7,8,9 ~ 92% ~ 98% b) tracebacks carbon deuteron 20%-40% 30%-60% c) acceptance tests A C _ T S T 30%-50% S C I N T _ A C C 20%-45% d) final good events al l 10%-25% ^carbon > 6° 3%-20% Chapter VIII Polarization Formalism and Spin-Transfer Results A t this point it is useful to digress from the details concerning the experiment and its analysis to introduce the formalism which describes polarization and spin-transfer observables. The concepts described i n this chapter are central to the topics discussed i n the remainder of the thesis. Emphasis is given to the relationship between the two important observable representations: spherical and Cartes ian tensors. In terms of this formalism, the relationship of the spin-transfer observables to the measured data is presented. Furthermore, the behaviour of these observables under the operation of time-reversal is discussed. T h e latter is the basis of the connection between the observables of this experiment and those obtained from the pp —> dn reaction. VIII . l General Description of a Reaction Involving Polarized Particles The following discussion is based on the formalism defined by the Madison convention (see section 1.2). The Madison convention provides no specific recommendations for spin-transfer observables, so we use the convention of Simonius [4], a consistent extension of the Madison Convention. T h e development which follows is adapted from the work of Simonius [4] and is essentially identical to results presented elsewhere [32,93,94 . Consider ( in the centre-of-mass system) the general reaction: {d;a(b,c)d} where 6 is the angle between the incident particle h and the outgoing particle c. T h e M a d i s o n convention defines the i -ax i s appropriate to the frame of the in i t ia l or final particles (to be denoted as "frame I" and "frame H " respectively) to be i n the direction of momentum of the relevant polarized particle. The y-axis is defined by the normal to the reaction plane (in terms of the in t ia l and final momenta, k ^ x kc) and hence is common to both coordinate systems. A prime (') is used to denote those observables appropriate to the final state coordinate system (frame II). Fol lowing the very standard approach of density matrices [4], one can describe, i n the spherical tensor representation, the final state polarization as a function of the in i t i a l state polarization. io(o)ZkqhqT^q " m<i>) ^''^ where 1 ( 6 , <j)) is the differential cross-section for the polarized target, i ' ^ ,^ , and tkq are the polarizations of the final and in i t ia l particles respectively, Tkq are analysing powers and T^^' are spin-transfer observables, w i th the asterisk (*) representing complex conjugation. Spherical tensor observables have simple properties under rotations (similar to the rotat ion properties of spherical harmonics) and are therefore useful for the discussion which follows. In the ird —» pp reaction, the deuteron i n the in i t i a l state has spin 1 and the final state proton has spin |. T h i s implies that the tensor representing the deuteron polarization has rank 2 (Â; = 0,1,2) and the tensor for the proton rank 1 (A; = 0,1). It is useful to also express the spherical tensor observables i n terms of Cartesian tensors, since operators representing the Cartes ian observables are Hermi t ian and hence are easily related to the observables of an experiment. The transformation between spherical tensor and Cartesian polarization observables is Table V I I : List of transformations between tensor and Cartesian observ-ables. The vector observables have a simple Cartesian analogy and apply to observables of both rank 1 and 2, where as the ten-sor observables apply only to rank 2 observables. For analysing powers, substitute P,t y-^ T and the same relations apply. s represents the spin of the of the particle whose polarization is being described. vector observables (rank 1 & 2) tensor observables (rank 2) Cartes ian spherical tensor Cartesian spherical tensor Px Pxx f ( t 2 2 + t 2 _ 2 ) - ^ Py Pyy - f ( i 2 2 + < 2 - 2 ) - ^ Pz Pzz Pxy = Pyx -^^(<22 - ^2-2) Pxz ~ Pzx - f (<21 -<2-l) Pyz = Pzy Ï ^ ( i 2 1 + i 2 - l ) presented i n the Madison convention [6] and is summarized i n Table V I I . These transformations apply also to the analysing powers. In the development of this formalism, the Cartesian axes are referred to by y and z (instead of the usual S , N and L ) i n the appropriate coordinate system. The " S - N - L " notation wi l l be reintroduced at the end of this discussion. Us ing the information of Table V I I one can describe the vector components of the f inal proton polarization i n terms of the spherical tensor components of equation 36. For example, [ <io(T/J '^- no^-J + Uni'^- Th-'*) + h.,{Th-^- TU) +^20(^20* ~ ^20 *) + ^2i{Tll* — Til + ^2-1(221 — Til) +^22(^22* "~ ^ 22 *) + *2-2(—Î22 ^ + Til) ] where the Hermit i c i ty condition [4] for spin-transfer tensors Tt:r: = {-^r^''Tt:y,: m has been used. A further relationship which combines parity conservation with the Hermit i c i ty condition, permits wr i t ing the expression for p^' as [ t i o O T - Tlô') + (<n - t i - i ) ( T / / - T / r ^ ) F ina l ly , by using Table V I I , the spherical tensor target polarizations can be replaced w i t h their Cartesian equivalents. To obtain the relationship between spherical tensor and Cartesian tensor spin-transfer observables, the prescription, as provided by Simonius [4] is followed: Us ing the relationship between upper and lower case observables (which is further explained i n section V I I I . 1.2): ttlllie- ail c)d} = T!::^:''{6; a{b, c)d} (39) one can now construct, for example, the hybr id observable i n a manner similar to the polarization observables of Table V I I : Us ing equation 39 one obtains and taking the complex conjugate of both sides. Now specific Cartesian spin-transfer observables can be obtained again using the relations defined i n Table V I I . For example, using the notation K- for a Cartesian spin-transfer observable, one obtains: = v f r a - r , \ - ^ ) 120 Table V I I I : Relationship between spherical tensor and Cartesian spin-transfer observables as expressed i n the coordinate system de-fined by the Madison convention (see F i g . 32). vector to vector tensor to vector Cartesian spherical tensor Cartesian spherical tensor _2_nnll ly/liTll-Tl,-') K'ss = KZ' ^/m|-T,\-') K '^\/î(T'll — ') -t2Tll py' - ; ^ œ + roV )^ Ry' - Ry' ^^xx ^^yy -zVëiTll + Tl^-') where s = ^ and s' = 1 for the spins of the proton and deuteron respectively. This is the observable termed R'gg in this experiment. Similar expressions can be developed for the other variables of interest to this experiment. The ful l list is provided i n Table V I I I . The term Poo i n Table V I I I is an inherent polarization of the final particle which is not associated wi th the polarization of the target. It is therefore not given the spin-transfer notation of " / iT" . The importance of this quantity is discussed i n section VIII .1 .1 . Us ing the relations of tables V I I and V I I I equation 36 can be expressed i n terms of purely Cartesian quantities. O f part icular importance to this experiment are the expressions for p^' and pyi [93]: xy (40) Py' = W ) i{e, <i>) where I{d, (j)) is: i{9,<f>) = h{d) Pi + \pyKl' + \pzzKi + \p..Rt + \{p.. - Pyy){Rt - Ri) 1 + \pyPy + ^P^^ + \pxzPxz + ^{Pxx - Pyy){Pxx " Pyy) (41) (42) The components of the fixed target polarization w i t h respect to the quantization axis ( i -axis ) (defined by the reaction plane) and the magnitude of the polarization, px is the total vector Table I X : polarizat ion and pxr is the total tensor polarization. /? is the angle between the fixed polarization axis (5) and the z-axis, and 7 is the angle between the vector S y. z and the i - a x i s of the right-handed coordinate system. Vector Project ion Tensor Project ion Component Component Px —pT s in l3 s in 7 Pxy — f P T T sin^ ^ sin 7 cos 7 Py PT sin /3 cos 7 Pxz — ^PTT sin ^ cos /3 s in 7 Pz PT COS 13 Pyz I P T T s in ^  cos jS cos 7 \{Pxx - Pyy) — JPTT sin^ /3 cos 2 7 Pzz | p r T ( 3 c o s 2 ^ - l ) Further , the Cartesian components of the target polarization can be expressed i n terms of the total polarization and its alignment w i th respect to the quantization axis [93] of interest ( in this case the z—axis) as is listed i n Table I X . T h e angles of Table I X are only useful for relating the fixed target polarization to the variable reaction plane. The directions of the positive/negative deuteron polarizations are determined by the target magnetic field, as w i l l be discussed later i n this section. Since a l l quantities obtained from the experiment are expressed i n terms of the analysis convention (described in F i g . 24), it is necessary to develop a transformation which relates a l l observables obtained i n this system to the system of the Madison convention (the form i n which we would ult imately like to represent a l l observables). In addit ion, the target polarizat ion, whose quantization axis is defined by the magnetic field and therefore is independent of the above conventions, must also be related to these systems. To start this discussion, it is useful to clearly identify the similarities and differences between the relevant coordinates. The coordinate system used i n the analysis of the proton polarization is defined i n F i g . 24. The pr imary difference between this system (the "analysis" convention) and the Madison convention is the definition of the z-axis in the in i t ia l state (frame I of F i g . 32). Since the deuteron is the particle whose polarization is being measured, its mot ion i n the centre of mass is, by definition, the i - ax i s of the Madison convention. T h i s is opposite to that used i n the analysis of this experiment, where the pion momentum is the intrinsic definition of the i - ax i s . However the £-axis of the final state (frame II of F i g . 32) is unambiguously defined by the momentum of the proton i n the polarimeter. Th i s results i n an inverted definition of the y-axis for both in i t ia l and final particles, since y is defined by ki x kj and is obviously inverted by the change i n sign of ki. The "r ight-hand rule" therefore implies that the definition of the x-axis is i n the same direction for the analysis and Madison conventions i n frame I, but i n the opposite direction for frame II. These axes are clearly denoted i n F i g . 32. The natura l definition of "posit ive" polarization of the deuteron (see section VI .2 ) , defined by the target magnetic field direction, was always opposite to that of the Madison convention (target frame versus frame I (Madison) in F i g . 32). In the case of the analysis convention, the target frame, describing the positively polarized deuteron d id agree i n direction w i th that of the analysis frame i n the definition of the y and i -axes (target frame versus frame I (analysis) in F i g . 32), but were opposite for the positive direction of the f -ax i s . The directions of the target magnetic field were chosen to safely deflect the unscattered pion beam away from the detectors or " inhabited" areas of the meson ha l l . The differences between the Madison coordinate system, the deuteron spin directions, and the analysis coordinate system are i l lustrated i n F i g . 32. The only axes i n agreement between the two conventions are the final state (frame II) i -axes which are described by the momentum of the outgoing proton, and the Figure 32: Comparison of coordinate systems for the analysis and M a d i s o n frames. x-axes of frame I (which in t u r n disagrees w i th the " n a t u r a l " positive deuteron polarizat ion of the target frame). It is now helpful to take advantage of the well defined symmetry properties of the tensor observables to introduce the transformations between the various representations. The spin-transfer observables can be transformed by performing rotations equivalent to separate rotations on the in i t ia l and final state polarizations [4]. In the in i t ia l state the polarization can be transformed from the analysis basis to the Madison basis by a 180° rotation about the z-axis followed by another 180° rotation about the y-axis. For the final state, a single 180° rotation about the z is sufficient. A l though Wigner-c? functions are employed for these rotations, the results can be simplified i n the following manner. Rotations about the z-axis result i n : t'kç = i - ^ k , (43) and rotations about the y-axis which are similar to a parity inversion, produce: 4, = ( -1) '^"^. - , (44) Thus , the spin-transfer coefficients of the the analysis frame can be related to the Mad i son frame by using the above expressions together w i th the Hermit ic i ty condit ion, giving: Tt2{Madison) = {-if ^^'^^T',;'''^ {analysis) (45) W h e n applied to the Cartesian expressions of Table V I I I , this transformation yields the relationships between the analysis and Madison observables which are listed i n Table X . It should be noted that the definition of the scattering angle, is defined differently in the two frames as shown in F i g . 32: ^ " " " ' J / ^ " = TT - B^'"^''"''. Table X : Transformation between the analysis and Madison coordinate systems of Cartesian spin-transfer observables. The indicated angle 9 is from the analysis system. Analysis Frame Madison Frame Kisi^ - 9) -K'ssin - 9) K'!^N(9) PoNi9) Pm(^-9) VIII.1.1 Aspects of Time-Reversal A crucial assumption employed i n this experiment is the invariance under t ime reversal of the strong interaction. Th i s assumption is the basis of the principle of detailed balance [33] which indicates that the matr ix elements describing a reaction between states a + b ^ c + d are identical i n either direction (at the same centre of mass energy): |2 This principle has been tested i n several cross-section and polarization experiments w i t h no evidence of its v io lat ion being observed (to the significance level of less than 1% [95]). It is just this invariance which allows the measurement of observables of interest in the pp —>• dn reaction to be carried out i n the nd pp reaction. Simonius provides a simple prescription for relating polarization observables of time-reversed processes [4]. T h i s involves a 180 degree rotat ion about the y-axis to reverse the momentum directions and then a s imilar rotat ion about the .j-axis to correct the definition of the y-axis: ttim die, b)a} = (_i)('^ -^<.)+(^c-,c)y^^^.,.|^/. (^^ ^ (46) where 9' is the Madison angle defined in F i g . 33, a figure which illustrates the relative transformation between the time-reversed Madison conventions of the Table X I : Transformation between the analysis, Madison and time-reversed coordinate systems of Cartesian spin-transfer observ-ables. Note that the polarization PQO becomes an analysing power ANO and that the subscripts of K'j^g and KSL are re-versed. =» Trd pp =1 pp —>• dTT Analys is Frame Madison Frame Madison Frame K'LS{^ - 9) -KSL(^ - 9) -K'ssiir - 9) -Kss{^-9) KNN{-^ - 0) P^(^ - 0) Am{9,) Trd ^ pp reactions. Table X to can now be expanded to include the variables of the time-reversed pp —>• dTr reaction as listed i n Table X I . O f part icular importance to this experiment, i n addit ion to the spin-transfer observables, is the transformation of the observable PQO under time reversal. App l i ca t i on of equation 46 to the appropriate expression from Table V I I I , yields: ^0^0 = - ; ^ ( ^ o o + no') - ^ ( C + O = Ay = ANO (47) ANO is perhaps the most thoroughly studied polarization observable which exists for the pp dir reaction, w i t h high accuracy measurements available at many energies and angles [40]. It should be noted that A^o is tradit ional ly measured wi th respect to the pion scattering angle 9'^. As seen i n F i g . 33, 9'.^ is equivalent to ^analysis i n the Centre of mass system. This explains the change of angles for A^Q i n Table X I . VIII.1.2 Observable Nomenclature W i t h many different spin observables i n existence for both directions of the pp ^ dTT reaction, it is important to clearly distinguish between each of them while recognizing the fact that each observable has a t ime reversed equivalent. B y p p ^ d r r Figure 33: Comparison of_the Madison coordinate systems for time reversed tions; Trd pp and pp dTv. its definition, the nomenclature of Simonius [4] applies symmetrical ly for either direction of a time reversible process. That is, when referring to a spherical tensor observable i n lower case, t, the upper index denotes the incoming particle and the lower index corresponds to the outgoing particle , whereas for those observables represented by an upper case T , the reverse is true, thus: .kiqi rpkjqj ^kfqj •> -^kiqi In principle, these parameters can represent quantities i n either kinematic direction. In the l iterature, the practise is to assign lower case to those spin-transfer and analysing powers of the pp —> d-n reaction and upper case to the corresponding observables of the ird —> pp reaction. Furthermore, the lower case is always assigned to statistical tensor quantities such as the polarization of beams or target. In the case of Cartesian tensor notation, the s ituation is not so rat ional . Upper case letters are used to denote analysing powers of bo th directions. A P is —* used for analysing powers of the ird —> pp reaction and an A is used for the pp d-K reaction. The spin-transfer notation is not so consistent i n the l iterature. In this thesis, they are referred to (in either direction) w i t h an upper case K w i th the first index of the subscript indicat ing the polarization of the in i t ia l state particle and the second index the final state particle. The spin-transfer quantities for the ird pp reaction are primed to distinguish them from the time reversed quantities. A lower case p is used to represent the statistical polarization of a beam or target. VIII.2 Systematic Checks of Polarization The fact that the parameter PQQ (^ ^ discussed i n section VIII.1.1) has been extensively measured by many groups provides an extremely useful check of the polarization measurement procedure employed i n this thesis. In the t r i v ia l instance where the deuteron target is unpolarized, the normal polarization of the final state proton, is, from equation 41, = P^Q. Hence data obtained wi th an unpolarized target, provides a direct check for systematic errors i n the procedure for measuring the normal polarization. Due to the kinematic spin coupling (discussed i n section IX.3 ) , this approach also serves to check for systematic errors i n the sideways polarization measurement (although wi th less sensitivity). In the situation where the target is polarized, such an approach st i l l provides a check for systematic errors. However the expression for p'y (equation 41) is now complicated by the presence of addit ional terms involving spin-transfers and analysing powers, quantities which have not been previously measured. Fortunately their contribution is weighted by the typical ly smal l value of target tensor polarization (<0.10). Hence w i th reasonable estimates of the magnitudes of these additional terms, a check of the measurement technique can be obtained for all configurations. These systematic checks are discussed further i n sections IX .3 and X . l . Chapter IX Extraction of Polarization Observables Return ing to the description of the analysis used i n our experiment, the m a i n task is to now relate the expressions developed i n the previous chapter to the various components of the proton polarization measured at the polarimeter. These quantities must be i n terms of the polarimeter polarization components which have been determined using the techniques discussed i n sections V . 2 , VII .4 and V I I . 5 . Before beginning, it is important to recall that: • it is common to express a l l polarization observables i n a centre-of-mass frame, as defined by the Madison convention; • a l l polarization components measured at the polarimeter are expressed i n the laboratory frame, as defined by the analysis convention (see F i g . 24). In the previous chapter, the transformation between the centre of mass analysis convention and the centre of mass Mad i son convention was discussed. Here, centre-of-mass refers to the frame describing the nd —> pp reaction at the polarized target. T h i s chapter describes the complications encountered together w i t h the resulting expressions used to relate the polarization at the polarimeter to the relevant quantities in the centre-of-mass analysis frame. IX. 1 Complications To begin this section, we recall two important properties of the proton, its magnetic moment fl and its spin s. These properties axe closely related: where e and m are the charge and mass of the proton, c the speed of light and g is the Lande g-factor [33]. The magnetic moment is a magnetic dipole strength which, then, describes the interaction of the proton w i th a magnetic field. Spin is an intrinsic quantum mechanical property to which polarization is related. B o t h and 5* are vectors^. The complications related to this experiment arise from two effects. F i rs t ly , i n the rest frame of the proton, the coupling of the magnetic moment to a magnetic field induces Larmor precession resulting i n a rotation of the spin vector about an axis defined by the field direction. In this experiment, the magnetic field is that used i n the polarization of the deuteron target. Secondly, i f the proton rest frame is moving w i th respect to the frame i n which the particle is observed (the laboratory frame), the observed spin direction w i l l , i n general, not be parallel to its direction i n the rest frame. Th is is a result of the Lorentz boost between coordinate systems. B o t h effects can be described i n terms of appropriate rotations of the spin direction. It is important to discuss the impact of these rotations on the proton polarization, the statistical parameter describing particles w i t h spin. Recal l that the polarization is given by: P = (49) n + - ( - n -where and n _ are the numbers of protons which occupy the aligned and anti-aligned states respectively. The system of protons may therefore be ^Obviously from equation 48 the two are (anti-)parallel. considered a system of spin vectors which are parallel or anti-parallel to the axis. If a single rotat ion is carried out on two vectors which are in i t ia l ly anti-parallel , they continue to be anti-parallel after the rotation. A p p l y i n g this argument to the system of protons (which is subjected to Larmor precession or a Lorentz boost), the axis along which they are aligned w i l l rotate i n the manner of a single spin. Thus the polarization vector behaves as a spin vector under the influence of rotations, w i th its magnitude, defined by equation 49, remaining constant. In addit ion to rotation of the polarization axis, it is also important to correctly define the kinematics of the reaction. A l though the trajectories of the incident p ion beam and outgoing proton beams can be experimentally determined outside the region of the target, the presence of the magnetic field distorts the trajectories i n the neighbourhood of the target itself. A s a result, crucial information such as the reaction angle and scattering plane must be inferred from the trajectory information distant from the target, together w i th corrections based on the equations of motion for a charged particle in a magnetic field. IX.1.1 Equations of M o t i o n T h e classical equations of mot ion for a charged particle in a magnetic field are well known at the undergraduate level and are treated i n many textbooks such as Jackson [96]. G iven knowledge of the particle 's in i t i a l momentum^, one can determine the momentum at any later time assuming that the magnetic force acting on the particle is known throughout. ^(in this case either the initial pion beam momentum, as defined by the magnet elements of the beam line, or the proton momentum, as defined by reaction kinematics) T h i s can be expressed relativistical ly dv , _ -J, 7 ^ 0 - ^ = q{v X B) where rrio is the rest mass, 7 = ^^^^^ is the relativistic factor, q and v are the charge and velocity of the particle, respectively, and B the magnetic field. A l l quantities, except m,,, are expressed i n the laboratory frame. The equations of mot ion for the spin vector are not so simple, but are indeed straightforward and are described, for example i n Jackson [96], following the development of Bargmann, Miche l and Telegdi [97] ( B M T equation). ds e _ -SX ^ - 1 + -dt rrioC [\2 -y J (50) 7 Terms involving electric field have been dropped from the B M T expression since no such fields are present for our situation i n the lab frame. T h e spin vector s is defined i n the rest frame of the particle, whereas the remaining components are those i n the lab frame. I3 is the velocity of the particle i n the lab frame and g is the Lande factor. The first term inside the square brackets describes the Larmor precession of the spin vector and the second term the rotat ion of the rest frame coordinate system relative to the direction of motion of the particle, due to the bending of the particle 's trajectory i n the magnetic field. A description of the boost of the spin vector between the lab and rest frames of the particle is also required. T h i s can be obtained i n terms of relativistic boosts [96]: where s is the spin vector i n the rest frame and S the vector i n the laboratory. F ina l ly , the value of the spin vector i n the centre-of-mass must be linked to the value observed i n the lab. Th i s can be done by performing a simple rotation about the normal to the reaction plane, which is a common axis i n both systems. The angle of rotat ion, a;, is given by [32]: coso; = ( cos^cos^i + 7 C M s in ^  s in ^i,) s ina ; = (^ ) (s in^cos^£, — 7cA/cos^sin^£,) where 7CA / describes the relative motion of the centre of mass i n the lab system and E and mo are the total energy and mass of particle c. The angles, d and OL, are defined i n F i g . 34. T h e spin vector shown i n this diagram represents its projection onto the reaction plane, since any component of the spin vector normal to the reaction plane is unaffected by the boost. In practice, this boost only rotates the vector into a frame which is stationary i n the lab but is defined by the reaction plane. A n addit ional rotation about the incident pion beam is required to express this vector i n the analysis frame of the polarimeter which is paral lel to the floor. A g a i n these rotations are just as applicable to the vector which describes the polarization axis as they are to the spin vector. The magnitude of polarization is unaffected by the rotations, however. The succession of transformations given to a spin vector i n going from the centre of mass system to the polarimeter frame is described i n F i g . 35. IX.2 Proton Trajectory Model Despite the existence of well-defined relations for the equations of mot ion of spin and momentum vectors in a magnetic field, it was not possible to obtain a simple analytical expression for the spin precession and trajectory bending due to the complicated non-uniform magnetic field at the target. Fortunately an accurate field map for the Helmholtz coils of the target was available i n the form of a data base. It was therefore possible to model the trajectories of the protons and their spin i n this magnetic field by numerical methods. II om Il lab Figure 34: Definit ion of rotation angles for the spin vector Sc (of the reaction a + 6 —> c + d), required for boosts from centre of mass frame to laboratory frame. The frame I represents the in i t ia l state and II the final state (which is different for the lab compared to centre of mass). Note that: u = \a — ai,\ = \6 — ÔL\ / ' ' a n a l y s i s (Centre of Mass) X ^ S y s t e m / L a b F r a m e ( Paral le l to \ ^ Floor Figure 35: Progression of spin rotations applied i n model . For this purpose, an existing code, F I N D A N G [98], designed to trace the trajectories of charged particles i n a magnetic field and calculate their energy loss i n target materials, was adapted to model the quantities of interest to this experiment. Th i s routine incorporates the same basic equations of motion that were discussed i n section I X . 1.1. Three basic frames of reference were of interest: 1. centre-of-mass system at the target ( C M ) 2. laboratory frame at the target for the -nd —»• pp reaction (L'^) 3. laboratory frame defined by the observed trajectories (outside the influence of the target magnetic field) of the incident pion and outgoing proton (LP°0 The coordinate system i n each of these frames was defined i n terms of the analysis frame convention (see F i g . 24). Init ial ly the pion's trajectory was traced into the target to determine its deflection for a given incident energy. In the case of the longitudinal target field associated w i th K'j^g, there was no deflection of the pion beam. In the case of the K'gg measurement, the transverse horizontal field caused a vertical deflection (9°-13° downward at the target) of the incident beam. Simi lar ly the vertical field for K'J^N resulted i n a horizontal beam deflection (10°-17°). B o t h deflections produced a rotation of the reaction plane i n the lab thereby causing a component of the deuteron polarization (which was expected to he along the x or y axes for sideways or normal polarization) to be projected onto an orthogonal axis^. The final direction of the momentum of the pion at the centre of the target defined the z-axis of the 17 frame. Relative to this axis, several proton trajectories were generated to span the acceptance of the ^This rotation of the reaction plane was only orthogonal to the pion beam resulting in a coupling between the sideways and normal components for K'^g and K'f^j^. polarimeter after the protons left the magnetic field region. For each of these trajectories, several "reaction" quantities were calculated: 1. centre-of-mass reaction angle 2. the f inal proton lab energy at the centre of the carbon analyzer i n the polarimeter 3. For spin vectors in i t ia l ly produced along the x, y and z axes of the C M , the f inal components i n the L ' ' " ' system (after a l l the rotations of section IX.1.1 had occurred) were determined. 4. the angle representing the rotat ion of the x and y-axes of the reaction plane w i t h respect to the stationary axis defining the sideways or normal polarization i n the lab (this is the angle 7 of Table IX ) 5. the lab reaction angles 9t and (f)i (defined i n Table V ) as observed i n the W' frame E a c h trajectory was thus characterized by a 9t ^-nd (f>t of the L*""' frame (these angles are the same reaction angles as determined from the data analysis and are defined i n Table V ) . Thus a "look-up table" was generated, from which, for each real data event, any of the "reaction" quantities could be obtained. Us ing the real data {6t,(f>t) distributions, weighted averages were obtained for each of the quantities calculated i n the above list. The routines were checked using a uni form field map"*, for which analytic expressions for spin precession and trajectory bending i n a uni form magnetic field [76] are available. The energy loss routines were checked both w i t h the T R I U M F program ELOSS and the Bethe-Bloch equation [76]. The target materials ^Any arbitrary field map could be read by the program. i n which energy loss were considered are described i n Table I V . In addit ion, energy loss i n the air and in-beam scintillators was also included. IX.3 Polarization Extraction Algebra T h e following discussion describing the technique for relating the polarization observables at the target to the polarization components measured at the polarimeter. For each configuration (L ,S ,N) data were obtained for both (opposite) directions of the deuteron polarization. Expressing the polarization measured at the polarimeter i n terms of the centre-of-mass components at the target^: P ; ; + = / A T P ^ + ONP^- + hNPt + PN= INPN + 9NPS + h^PE + SM Ps''= fsP^ + 9sPÈ + hsPt + Ss ^^'> Ps~= ÎSPN + 9sPs + hsPE + Ss T h e superscript ^ denotes the direction of deuteron polarization (relative to its natura l "posit ive" polarization) , the pr imed (') quantities Pj^ g are the polarizat ion components measured at the polarimeter, and the unprimed PN,L,S are the centre-of-mass proton polarizations at the target. The coefficients /,-, Çi, and hi represent the coupling^ respectively between the N, S, L target components and the ith (i Ç N, S) polarimeter component. These spin coupling coefficients are calculated by the computer model discussed i n the previous section. T h e quantities represent a possible systematic error, independent of the sign of the vector polarization, which may be present i n the measurement. These quantities w i l l be discussed i n more detail later i n this section. T h e basic form of the proton polarization at the target, as it depends on the deuteron polarization (given by equations 40 and 41) can be summarized as ^Recall that the polarimeter can only measure transverse components of polarization. ^"Coupling" accounts for the net effect of all spin rotations discussed in section IX.1.1. follows: oc ^ P , - a ^ (52) « ^ P . - oc ^ where D± is a common denominator term, which is basically the polarized differential cross section of equation 42. A g a i n ± represents the sign of the deuteron vector polarization, whereas the distinctions between the deuteron polarizations PD\\ and P D J . represent the components of the deuteron polarization i n the reaction plane (either S, or L depending on the configuration) and normal (N) to the reaction plane respectively. Note that where P j " and P^ are directly proportional to the deuteron vector polarization P^y, P^ is given by a constant (-PA/") plus a term proportional to the normal component of the deuteron vector polarization (JCP^j^). There is no transfer of spin between the normal deuteron polarization and the sideways or longitudinal proton polarization (or vice versa) as a result of parity conservation [4]. In the expressions of 52, the terms relating to tensor polarization spin-transfer have been ignored. Such quantities have not been measured experimentally and are poorly defined by the existing par t ia l wave amplitudes. Further, ignoring these terms is justifiable on two accounts: firstly, they are weighted by the typical ly smal l deuteron tensor polarizations (<0.10) (see equations 40 and 41); and secondly these quantities do not change sign when the deuteron vector polarization is reversed, and thus can be grouped w i t h the systematic error terms Sj^/s of equation 51, terms which are shown to cancel in the development which follows. The terms involving tensor analyzing powers, however, are retained i n D^. In this case, the analysing powers, although not previously measured, are heavily constrained by existing data (as was discussed i n section IV.2) . Therefore their contributions can be reasonably taken into account. Using relations 52, expressions 51 can be rewritten i n the following manner: D+P'^= ÎNiPsT + K^Pèi.) + ONKSP^^^ + h^K^Pèii + D~PN= fN{P^ + KNPB±) + 9NKSPB\\ + h^KLP^^^ + Sj^ D^Ps''= fsiPAr + KMPèJ + 9sKsPèi\ + hsK^P^^^ + Ss D-P'g-^ fs(PAr + KNPB±) + 9sKsPû\\ + hsKLP^w + Ss so that , for example, the term g^KsP^^ represents the transfer of the deuteron polarization in the reaction plane to the centre of mass sideways proton polarization at the target which is then coupled to the measured normal polarization at the polarimeter, etc. A t this point the reader is reminded that the "iiiTPD" terms so far are general and apply to a l l three experimental configurations. The i r specific definitions w i l l follow later i n this discussion. A further substitution relates both signs of the deuteron polarizations to the common term, P p : PD = -PB and PD = RP^ D — where R ~ Thus : D+P'^= fN{Pu + KN^) + ^KSPDW + ^KLPDW + S^r D-PN= fN{P^-KNPDL) - 9NKSPD\\ - HNKLPDW + SM D+P's+= fsiP^ + KN^) + "iKsPow + ^KLPDW + Ss D-Ps= fs{PM-KNPD±) - 9SKSPD\\ - HSKLPDW + Ss (53) To simplify these expressions, the terms involving the coupling coefficients, hi are dropped. Th i s is justified as these coefficients were found to be about two orders of magnitude smaller than the corresponding fi and y, coefficients^. The result is four equations w i th the five unknown quantities P ^ , KSPD\\I KNPDL-, S^f and Ss- A solution is not difficult as reasonable assumptions can be made about some of these quantities thus permitt ing extraction of the quantities of interest. ^There is very little coupling between the longitudinal polarization at the target and the measured sideways and normal polarizations at the polarimeter. First of a l l , Sj\f and Ss can be eliminated by taking the difference of both pairs of the above equations: D+P's'-D-p's- = fsKNPD±{i+l) + gsKsPD\\{^ + l) ^ ^ From these, it is straightforward to solve for the essential parameters: = ^ ( F ï ï (^ )^ s imilar ly p -D-P'^)-{D+P',+ -D-P's-)f^ 1 KNPDI = j^^zTrm ^ ( T ^ (56) A n important feature of equations 55 and 56 is that the systematic error terms i^s/Af) have been el iminated. Such a situation results when the systematic errors axe instrumental i n nature and constant in time. A s stated earlier, these expressions are general and apply to a l l configurations of the experiment. In order to reduce these expressions so that the desired spin-transfer parameters can be obtained for the experimental quantities, it is necessary to consider how the terms on the left hand side of equations 55 and 56 depend on the configuration employed. In the case of K'^g, the deuteron is polarized along an axis which is parallel to the pion beam. As a result the pions travel paral lel to the magnetic field of the target axid are not deflected by i t . Therefore the target deuterons have a simple longitudinal polarization. However, for the transversely polarized targets used for measurements of K'gg and K'j^jyj, the appropriate normal and sideways axes as defined by the in i t i a l and final particle trajectories (ki x kj) are not identical w i th the "sideways" and "normal " directions defined by the magnetic field of the target. Thus projections of the fixed target polarizat ion onto the coordinate systems defined by the possible reaction planes had to be made. Th i s is further complicated by the bending of the incident p ion beam i n the magnetic field of the target. T h e angle 7 of Table I X , is the angle by which the target orientation and the reaction orientation differ. T h i s angle was one of the quantities calculated i n the spin-precession and trajectory model of section IX .2 . The contribution of this coupling differed for each configuration. In the case of the iiT^^^ configuration, the acceptance of the polarimeter was such that the deuteron polarization projection onto the sideways axis of the reaction frame cancels due to equal projections i n opposite directions. Thus the net effect is a reduced normal polarization by a smal l factor (~ 0.99). However, for the K'^g configuration, the acceptance was such that there was a net projection onto the normal axis (~ 15% of magnitude). T h i s resulted i n a reduced magnitude for the sideways polarization (~ 0.98) and a smal l "normal to normal " spin-transfer component which was ignored. However, i n none of the configurations was there a coupling between the longitudinal and transverse components. W i t h the above considerations, it is now possible to adapt equations 55 and 56 to the specific quantities of interest. Us ing equations 52, 40 and 41 and making the appropriate substitutions: • for K'LS-. Poll ^ PD and Ks f A ^ s • for K'ss'. PD\\ ^ PD C O S 7 , PD± PD s i n 7 , Ks =^ ^K'ss and KN =^ fi^A^jv Thus the following expressions now relate the spin-transfer parameters to the experimental quantities: • for K', NN : PD± =^ PD COS 7 and KN =^ ^I^NN , _ 2 1 jD'-P^-' - D-P's-) - (D^PN^ - D-PN)J^ 1 ^PD 9S-9N^ + 2 1 {D^P's^-D-p's-)-{D^P'rt-D-P'N-)^ 1 (57) 3 P D C O S 7 9s-9NJf; (^ + 1) * Although there is some sensitivity, this term was not considered in the K'gg analysis Table X I I : L is t of spin-transfer results i n the Madison frame. The errors quoted include statistical errors from the proton polarization measurement and both statistical and systematic errors of the deuteron polarization measurement. T , ( M e V ) e' (degrees) 6' (degrees) 6' (degrees) 105 147±3 -0 .115±°^ N . A . N . A . 137*3 0 .243 * °«2 145 146±3 -0.144±"'»i 106±2 0.202*°^^ 141*3 0.318*°™ 180 146±3 -0 .302*°^ 106±3 0.152*°'''' 143*3 0 37g±.059 205 146*3 -0 .253±°^ 105*3 0.216*°"^ 144*3 0 . 2 8 2 * ° " 255 145±3 -0 .212*°^ 104±3 0.305*°^'' 145*3 0.095*- i°^ 2 1 {D^P'^-D-P'N)-{D^P'S'^-D-P'S-)^ 1 ^ ^ ^ " 3 P c c o s 7 (59) fN-fs^ ( è + 1) Values for these spin-transfer parameters are listed i n Table X I I for the variovis energies and configurations studied i n the experiment. The errors shown include both the statistical uncertainty (see chapter V ) and the errors associated w i th the target polarization (see section VI .2) . Also included are errors associated w i t h the tensor analyzing power terms wi th in £ )* , which were assigned relative errors of 6%. These errors d id not contribute significantly to the uncertainty of £ )* since they were weighted by the tensor polarization of the target. T h e quantity, P ^ , can also be extracted from the experimental data. Start ing w i t h equations 53 and adding the pairs of equations yields: D+PI+ + D-P'N= 2hPx + ÇNPsii-l) + h^PLi^-l) + 2 5 ^ D+P's+ + D-P's-= 2fsPM + gsPs(^-l) + / I S P L ( ^ - 1 ) + 2Ss A g a i n ignoring the terms involving / i , , we can solve for Pj^: jD^P'^ -f D-P'N) - iD^P's" + D-P's-)^ - iK'^NPpd^ - 1) - 2 5 ^ + 2 ^ . (60) P^f = 2ifN - fs^) (61) However, unlike the situation for equations 55 and 56, the expression for Pj^ does depend on systematic errors. However, since the value of Pj^ (Poo) is known to reasonable accuracy through data already i n existence (as discussed i n section VIII .1.1) , the extraction of Pj^ i n the manner described here provides information concerning the size of possible systematic errors characterizing the experimental arrangement. T h i s feature w i l l be discussed further i n chapter X . In equation 61, the coefficient of the K'^^j^ term is smal l since i2 ~ 1.1, so that ^ — 1| ~ 0.1 . Hence the value of P v obtained is not very sensitive to the value of K'NM assumed. A s a result, the value obtained from equation 59 was used. Table X I I I provides a list of the values of Pj^ obtained by this means compared to corresponding values of A^o from the pp —> di: reaction. T h e errors quoted are statistical i n nature. There is no contribution from the target polarizat ion i n equation 61. The errors for the expected value of A^o include a 4% uncertainty which is typical of many measurements of this observable. A s well some uncertainty was included to account for the variation of this parameter over the acceptance of our detectors. G o o d agreement is obtained between the measured normal polarization and the expected value determined by ANO, as i l lustrated i n F i g . 36. Table X I I I : Comparison of measured and expected normal polarization at the target. The expected value was taken directly from a pa-rameterization of the ANO data [99] and the errors given are taken from typical values of an A^Q measurement. Configuration P i o n Energy Expected Value Measured Value •Am PM 105 0.225=^°" 0 213±020 145 0.399±°26 0.43o±o23 180 0_442±-023 0455±.o32 205 0.388±°^^ 0 397±.o25 255 0 .280±°" 0.302±°3'» K'ss 145 0376±o2o 0.361±°24 180 0.4i6±.oi8 0 392±o22 205 0.352*°^'' 0.356=^°24 255 0_i89±oo9 0.198±°28 K'NN 105 0 237±oo9 0.281±-°25 145 0.426±-°2i 0,492±-029 180 0.454±.o22 0452±.o28 205 Q394±.oi8 0.388= -^°25 255 0.288±-°^2 0.308±°^^ 5-4-3-2-.1- K ' LS i l 100 120 140 160 180 200 220 240 260 280 Nominal Pion Lab Energy (MeV) 100 120 UO 160 180 200 220 240 260 280 Nominal Pion Lab Energy (MeV) 100 120 140 160 180 200 220 240 260 280 Nominal Pion Lab Energy (MeV) Figure 36: Comparison of the measured values of (•) w i t h the expected value from the observable ANO of the pp dir reaction (o) for data obtained i n each target configuration. Note the latter values have been shifted by +3 M e V on the X-axis for c larity of presentation. Chapter X Error Evaluation T h e results presented so far i n this thesis have been assigned errors which are statistical only (excepting systematic contributions from the target polarization uncertainty) . However i n any experimental measurement there always exists the possibil ity of systematic errors. Systematic errors result from bias introduced through incorrect cal ibration of, or unexpected behaviour by, the apparatus. Such errors tend to be reproducible (as opposed to randomly fluctuating) and, i n many cases the results can be "corrected" through appropriate measures, or the errors estimated through a detailed analysis of the data. In this chapter the methods used to search for the presence of systematic errors i n this experiment are discussed. In addit ion, the procedures used to evaluate the contributions of, or if possible, correct for the systematic errors are described. F ina l ly , such analysis, when combined w i th the statistical errors establishes an overall uncertainty for the measurement. X . l Systematic Errors Three types of systematic errors are considered i n this discussion: • any shift i n the properties of the apparatus which varies i n time • measurement errors which scale w i t h the magnitude of polarizat ion • bias which is independent of the magnitude of polarization T h e consideration of time-dependent errors is essential for this experiment due to its long duration. A s discussed i n section V I . 5 , data were obtained during three separate running periods, each lasting many weeks at a t ime. A l though many time-dependent effects could be considered, only three were felt to have any real potential impact on this measurement. T h e first i tem considered was a "dr i f t " of the polarimeter electronics, such as changes i n the gains of the signal amplifiers or shifts i n the threshold levels of discriminators, etc. To accommodate such a possibility, the software cal ibration, discussed i n section VI I .2 , was performed for each r u n set^, or more frequently s t i l l i f a component of the apparatus was knowingly changed (such as the replacement of a T D C un i t , wire chamber etc.). In addit ion, the on-line diagnostics discussed i n section V I I . 3 were instrumental i n helping to identify shifts i n the performance of the electronics. In order to identify any time-dependent effect which was not accounted for by the frequent cal ibration or software diagnostics, the proton polarization was calculated on a run-by-run basis to identify any trends. In addit ion , selected r u n sets were repeated several days after their first measurement. A l l pairs of "ear ly / la te " results, shown i n Table X I V , agree well w i t h i n their assigned statistical error. Unfortunately t ime constraints d id not permit this check to be done for a l l configurations. However the results l isted i n Table X I V indicate that time-dependent effects due to shifts i n the electronics were negligible. A second time-dependent effect which is clearly important is that of variations i n time of the deuteron target polarization. These variations resulted from changes i n the target temperature, the de-tuning of the polariz ing microwave frequency, etc. (see section VI.2.2) . As such fluctuations were expected, the target polarization was monitored at five minute intervals and the values stored on magnetic tape. The average polarization for a r u n set was obtained by weighting run set consists of consecutive runs between changes in beam energy or target polarization. Table X I V : Comparison of spin-transfer quantities obtained w i th similar beam energy and target polarization but from different r u n sets. Th i s data is useful for identifying any possible t ime-dependent fluctuations i n the measurement. parameter beam energy polarization first set second set A time ( M e V ) sign (days) A l s 205 -ve -0.270±'^« -0 .253±"^ ~10 K'LS 255 -ve -0.212±«^« -0.266=^°™ ~ 9 K'ss 180 -ve 0.152^°'''' ~ 3 K'ss 205 -ve 0.237^°"'' 0.216±°^^ ~ 3 K'NN 205 -ve 0 . 282±° " 0 254±o6o ~11 each "five minute" polarization sample by the number of events accumulated over that period (recorded by the scalers). Included in the standard deviation of the value of the average polarization was the variation of the target polarization throughout the r u n set. Runs i n which a rapid depolarization of the target occurred (which happened only rarely) were excluded from the analysis. Absolute fluctuations of the deuteron polarization throughout a r u n set were typical ly less than 0.02 . These errors are included w i t h the ones discussed i n section VI.2.2 and listed w i t h the average target polarization for each energy and configuration i n Table X V I I . The t h i r d time-dependent effect considered was a possible shift i n the beam energy throughout the r u n . This would affect any energy dependent quantity (such as the p - C analysing powers) used i n the calculations. T h e beam energy was determined by a dipole magnet, situated i n the first half of the M-11 beamline, which selected particles of a well defined momentum. T h e magnetic field of the dipole was monitored on a run-by-run basis using an N M R probe. The beam energy proved to be very stable, w i th typical fluctuations of the N M R probe reading being on the order of 1 part i n 10^. Hence time-dependent beam energy fluctuations were considered negligible. T h e next type of systematic errors considered i n this experiment are those which scale w i t h the magnitude of the polarization being measured. Such an error is usually associated w i th the cal ibration of the apparatus. As an i l lustration of such contributions, we recall the relationship between polarizat ion, scattering asymmetry and analysing power i n the polarimeter: P = ^. In this s ituation, the inferred polarization is proportional to the asymmetry measured, w i t h a proportionality constant (^) which itself is subject to error. A n y systematic shift (from the true value) of the value of the analysing power used i n this expression w i l l thus result i n an absolute polarization error whose size depends on the magnitude of the asymmetry (e). Two sources of error for the analysing power were considered: an error i n the determination of the proton energy at the carbon centre, resulting i n erroneous values of the analysing power due to the energy dependence of this parameter (see F i g . 13); and the possibility that the values for the analysing powers used i n this analysis are not applicable to our apparatus (recall that the analysing powers used here were measured w i t h a different experimental apparatus). T h e proton energies were determined from the proton trajectory model discussed i n section IX .2 . As this routine has been rigorously checked, the only possibility for a concern i n this area would be due to errors i n the values of the parameters which relate the model to the real experiment. For example, uncertainty i n the beam energy, or detector positions (particularly the angle at which the polarimeter was placed), could result i n the trajectory model yielding incorrect proton energies. These possibilities w i l l now be considered i n detail . A l though the mean p ion energy was very constant i n t ime, it was also characterized by a momentum spread of ^ ~ 5% (full width)^ which natural ly introduces an uncertainty to the reaction energy. Such an uncertainty also affects 2For the K'^^j^ runs at T„ = 145 MeV, 180 MeV and 205 Mev ^ was ~ 1.5%. Table X V : E r r o r i n proton-carbon analysing powers resulting from uncer-tainties i n the reaction energy. Configuration N o m i n a l Beam Energy ( M e V ) True Energy at Target ( M e V ) Pro ton Energy Uncertainty ( M e V ) Analys ing Power Uncertainty (relative) (%) K'LS 105 104.3±^-2 126.3*3-3 4.5 145 144 2±5.5 162.3*^-3 2.5 180 179.0^^-^ 192.3*^-^ 0.8 205 203.7±^-2 213.6±^-^ 0.3 255 250.3=t«-« 253.0±«-^ 1.9 K'ss 145 143.3±^-5 122.1±3-2 4.6 180 178.2±«-5 145_4±3.8 3.5 205 202.9±^-2 161.4±''-i 2.5 255 249.6±«« 190.4±4« 0.7 K'NN 105 103.7±''-2 129.8*3-3 4.0 145 143.7±i-^ 166.3*1-3 0.7 180 178.7±2-o 190.0*1-^ 0.3 205 203.3±i-^ 212.0*11 0.1 255 246.7±«-^ 250.5*^-^ 0.9 the knowledge of the f inal state proton energies which is essential to determine the p -C analysing powers. The final mean energy determined for the pions at the centre of the target together wi th the associated width of its d istr ibut ion at these energies is given i n Table X V . Also listed i n that table are the average proton energies at the centre of the carbon and the approximate widths of their distributions resulting from the pion energy widths and assuming the relationship Tp ~ (0.55 — 0.80)r^ , a relationship determined from the reaction kinematics. The associated error i n the analysing power ( ^ ) as determined from the slope of the analysing power versus energy curve of F i g . 13, is also l isted in Table X V . A l though the angle dependence of the proton energy is taken into consideration by the trajectory model, the impact of any uncertainty i n this angle must also be discussed. As mentioned i n section V I . 1, the angle at which the polarimeter was located was known to w i th in a half degree. Th i s results i n a proton energy uncertainty of ~ 0.2 — 1.0 M e V (from reaction kinematics) , and was generally negligible compared to the spread of the beam energy. It should be noted that this uncertainty could be enhanced by uncertainties i n the amount of bending of the trajectories i n the target magnetic field. A l though the trajectory model was thoroughly checked, unexpected behaviour i n the field (particularly i n the fringe field) could contribute some small effects. Such an effect was ignored, however, since an absolute check of the polarization analysis was available, as we now discuss. In order to verify that the proton-carbon analysing powers used i n the analysis were representative of those which characterized our polarimeter, and that no significant energy dependent errors remained undetected, a known polarizat ion was measured w i th the apparatus. As discussed i n section V I I I . 1.1, the normal polarization of protons resulting from the TTC? —> pp reaction, w i th the deuteron target unpolarized, is equal to the analysing power, AJ^Q for the time-reversed pp —> dir reaction and these analysing powers are well known over the kinematic range of interest i n our experiment. Hence for each configuration data were obtained for at least one beam energy w i th the target unpolarized. Th i s served to check the validity of the analysing powers used i n the polarization measurement. The normal polarization of the protons at the target was simply obtained from the polarization components measured at the polarimeter using a reduced form of equation 51: where the sideways and longitudinal components at the target are necessarily zero for an unpolarized target. The normal polarizations calculated from the data, are presented i n Table X V I , and compared w i th expected values obtained from an Table X V I : Lis t of the measured normal polarization of the protons pro-duced from an unpolarized deuteron target as compared w i t h the expected value determined by the quantity A^o- Also listed wi th each measurement are the average energies of the protons at the centre of the carbon. This demonstrates the range over which this systematic check was performed. Conf iguration Beam Energy ( M e V ) Expected Value (from ANO) Measured Pv" Average Pro ton Energy ( M e V ) 205 0.388*°^^ 0_438±.O39 213.6 205 0.353*°^'' 0.325*°^^ 161.4 105 0.237±-oo9 Q 254±.o32 130.1 145 0.426*°2i 0.474±.037 166.2 255 0.288*°^2 0 248±059 250.4 energy and angle dependent parameterization of the existing 7^^ 0 data [99]. The values agree wi th in the statistical uncertainty of the measurement as demonstrated i n F i g . 37 where the difference between the measured and expected values of normal polarization ( P - A ) has been plotted as a function of the mean proton energy at the carbon centre. Unfortunately t ime constraints d id not permit this check to be performed at a l l energies for each configuration. However this check was done over a sufficient range of proton energies involved i n this experiment that we have confidence that any systematic malfunction of the polarimeter at any of the configurations and beam energies studied would have been revealed (see F i g . 37). In addit ion, these checks suggest that the p - C analysing powers used i n this experiment are indeed appropriate for our apparatus. However, an error which refiects the level of agreement, as a function of proton energy, between two recent p - C analysing power measiirements [70,71], has been added to the overall error. These parameterization errors, which are listed i n Table X V I I , tend to be larger for lower proton energies. 120 140 160 180 200 220 240 260 Proton Energy at Carbon Centre (MeV) Figure 37: The difference between the proton normal polarization (P) of the ird pp reaction and the corresponding analysing power ( A ) of the pp —» JTT reaction as a function of the proton's energy at the carbon centre. These results span the range of protons energies applicable to this experiment. T h e f inal category of systematic errors which was considered i n this analysis was the type independent of the magnitude of the polarization being measured. Such errors arise pr imari ly from bias i n the apparatus which introduces artificial asymmetries to the measurement, thereby erroneously shifting the perceived value of polarization. In our case it is the scattering asymmetry e of the expression P = "I which is of pr imary concern. Three types of artif icial asymmetry were considered: a misalignment of the wire chambers, which would result i n an angular bias for the trajectory reconstruction and thus an incorrect calculation of scattering angles; a possible spatial dependence of the detection efficiency for particles scattering i n the latter half of the polarimeter, a dependence which would, i f ignored i n the acceptance function characterizing the polarimeter (see chapter V ) , lead to the generation of spurious values of e; and a possible acceptance bias due to the front-end data selection of the J-11 preprocessor. In principle, wire chamber misalignments were removed i n the software cal ibration (see section VII .2 ) , w i th the result that the chambers were aligned to the degree permitted by their resolution. In addit ion, after each cal ibration, a software check was carried out to ensure that no significant misalignment was present. Th i s task was performed by looking for structure i n the azimuthal d istr ibut ion of events which had experienced l itt le or no scattering i n the carbon {^carbon < 3°). For thcsc "smal l angle" scattering events, the p - C analysing power is essentially zero. In addit ion, the az imuthal angle is undefined for those events whose polar angle (dcarbon) is zero leaving the calculated 4>carbon d istr ibution flat for those trajectories (each b in in (j)carbon having equal probabil i ty) . Possible deviations from this flat az imuthal distr ibution were sought by fitting the following function to the smal l angle scattering data , w i th A, B and C as variable parameters (for a flat distr ibution S = C = 0): /((^) = A - f .Bcos<^-|-Csin<^ (62) Th is expression follows directly from equation 15 w i th the parameters A, B, and C related to the standard asymmetries (e) through: ^iv = f and es = j . The asymmetries determined through this procedure were typical ly less that 0.03 % demonstrating that the chambers were indeed well aligned. Asymmetries introduced by spatial inefficiency were more elusive i n nature and steps had to be taken to reduce their l ikelihood. Such asymmetries could be introduced by detectors located after the carbon i n the polarimeter, either scintillators or the wire chambers. Knowledge of the scintillator acceptance and detection efficiency was essential because they defined the hardware E V E N T (see section VI.4) thereby determining if the particle trajectories were even examined by the electronics. For this reason, these detectors (which covered a reasonably large area) were placed i n pairs w i t h photomultipliers situated on each end. Since only one of a pair of photomultipliers had to detect the particle, these instruments operated at close to 100% efficiency. To check this , a t h i r d scintil lator was placed behind the latter scintil lator pair ( S 3 A and S 4 A ) and overlapping both , thus allowing a comparison of the relative efficiencies of the accompanying pair (see section V I . 1.1). To boost the efficiency of the wire chambers, three chamber redundancy was used. Thus only two of three chambers had to "fire" i n order to yield the required trajectory information. A n y inefficiency of a chamber i n one part icular region, would be made up for by the presence of the t h i r d chamber. Unfortunately, the large wire chambers following the carbon were the least efficient of a l l (see Table V I ) . A s a result an alternative approach was used to check for the possibi l ity of spatial ly dependent asymmetries. This was accomplished by calculating the polarization obtained from different spatial regions of the polarimeter (as was discussed i n section VII .4) and searching for systematic differences between these regions. T h e polarization was calculated as a function of the target scattering angles {dtargeti'jitaTget)- On ly oue systematic trend was identified, which resulted from the existence of a gap between the scintillators ( s l A and S 2 A ) which immediately followed the carbon analyzer. Th i s is discussed i n section X . 1 . 1 . As discussed i n section VI.4.1 a J-11 preprocessor was used to select only those events which scattered by more than a predefined m i n i m u m angle (~ 6° — 7°). If for any reason this angular definition shifted so that more events scattering to one side were accepted than those scattering to the other, an artif icial asymmetry i n the recorded events would occur. To ensure that this feature of the data acquisition system d id not impact upon the polarizat ion measurement, a further software cut was imposed on the data to remove those events which lay in the region which may have been influenced by such an on-line o a N ca 'o Oh tn • r H • en 1.00 0.75-0.50-0.25-0.00--0.25--0.50--0.75--1.00 J I I I I I I I I I I I L Off-line 9 eut i 1 i i 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 0 Carbon (degrees) Figure 38: The dependence of the proton polarization on the polar scattering angle i n the carbon. A s a constant result is expected, any systematic shift at smaller angles represents the impact of bias introduce by the J -11 . J-11 cut. Th is off-line cut rejected those events associated w i t h an angular scattering of less than that deemed to represent the "circle of influence" of the J-11 . T h i s circle of influence included effects due to the poorer resolution of the on-line software (which used only wire positions obtained from 4 of the 6 chambers to calculate coordinates). The cut (applied i n R P O L A R ) typical ly removed events scattered by less than dcarbon = 8 ° . To check the val idity of this approach, the polarization was calculated as a function of Ocarbon • T h e systematic deviation of the polarization from the 'resulting average value for small angle scattering would indicate a bias being introduced by the J -11 , thus requiring the cut to be adjusted accordingly^. Th i s is i l lustrated i n F i g . 38 where the off-line cut is also indicated. ^The measured polarization is expected to be independent of the angle of scattering in the carbon. 1.00 -0.75-systematic bump T Z * I T I T T . I J I - I Foreground Normal Polarization M 1 i ^ i ' } ' 1 Foreground Sidevajrs Polarization 1 ! I : . I . i I r I , ? I I . f ' • Î I J - I Foreground Normal Polarization f—f- f t ' 1 ' { ' j Foreground Sidevayi Polarization I-0.50 -''-0.75 --1.00 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 11 e^, (degrees +22) Figure 39: a) Impact of scintil lator cut on "normal " components of polarization as a function of Otarget- b) I l lustration of the impact of the "7r-symmetric" acceptance test on the same distr ibution. X.1 .1 Scintillator G a p P r o b l e m Another i l lustrat ion of the effectiveness of the techniques presented for identifying spatial ly dependent false asymmetries, is that of the effect of a gap between scintillators S l A and S 2 A . After an in i t i a l analysis of the data, it was found that a part icular point of the versus t^arget plot was consistently high for a l l r u n sets, as demonstrated in F i g . 39a. The effect was only observed for "normal " polarizations. Further investigation revealed that a smal l gap, a few mill imeters wide, existed between scintillators S l A and S 2 A . Th i s gap, shown i n F i g . 40, ran vertically^ and thus only impacted on the "left-right" asymmetry resulting from normally polarized protons. This potential problem was treated by invoking the useful 7r-symmetry ''In fact it is not clear if a physical gap existed or that the detection efficiency dropped off at the extreme edge of the scintillators. Since they were wrapped in aluminized mylar and additional light-tight material, the existence of a physical gap between the scintillators was difficult to ascertain. Figure 40: T h e "gap" between the S 1 A / S 2 A scintillators as determined by the "x-project ion" of the proton trajectories onto the "z-posit ion" of the scintillators (obtained from the wire chamber information). acceptance test, discussed i n section V . 2 . Since the position of the gap was known (from distributions of the type shown i n F i g . 40), one could remove from the event sample those trajectories whose "TT -symmetric" event would have passed through the gap and thus l ikely would not have been detected. That this was indeed the source of asymmetry which produced the "bump" in F i g . 39a, F i g . 39b, shows how effectively the acceptance test correction to the data d id indeed remove such "bumps" from the distr ibution. Throughout the tests which the gap problem entailed, no significant shift i n the sideways polarization was ever observed, an observation consistent w i t h the lack of any vertical non-uniformity of the detectors. In general, a l l aspects of the gap problem and its solution exhibited consistent behaviour. We are therefore confident that any false asymmetries introduced by the gap between scintillators S l A and S 2 A have been corrected by the technique presented here. X.2 Confidence in Results In this section a quantitative summary of the systematic errors discussed i n the previous sections is presented. U p o n combining these errors w i th the statistical errors of section V . 2 , and uncertainties associated w i t h the target polarization (discussed i n the previous section and i n section VI.2.2) , an overall uncertainty is obtained for the spin-transfer parameters measured i n this experiment. These contributions are assumed to be uncorrected and thus are combined i n quadrature. The systematic errors calculated as described i n the previous section are l isted i n Table X V I I . The largest systematic contribution comes from the uncertainty of the proton-carbon analysing powers due pr imar i ly to uncertainties i n the proton energies at the centre of the carbon and the p - C analysing power parameterization itself. However, these errors are somewhat smaller than the statistical errors associated w i th the measurement of the proton polarization, thus there is no reason to assume a systematic bias i n the result. Confidence i n the quantitative treatment of the errors is provided by the inherent normal polarizat ion ( i V ) result obtained for the reaction. As indicated i n Table X I I I , these results agree well w i th in errors w i th their expected values determined by the experimental measurement of ANQ of the pp dir reaction. In the case of the K'^^g and K'gg measurements, this quantity serves to check both the sideways and normal polarization measurements at the polarimeter (due to the spin precession coupling), whereas i n the A'j^^^ configuration, it serves solely as a check of the normal polarization measurement. The good agreement between measured and expected values indicates both that no significant systematic contributions have been overlooked i n the analysis, and that the error analysis yields values of the appropriate magnitude. Table X V I I : Summary of systematic errors considered i n error analysis. A lso l isted are the average magnitudes of target vector po-larizations obtained i n this experiment. Config. Beam Energy ( M e V ) Target Polar izat ion Target Polar izat ion w i th Fluctuat ions Analys ing Power Uncertainty due to Pro ton Energy (%) Analys ing Power Uncertainty due to Parameterization (%) K'LS 105 -f ve -ve 0.2886*°!^^ 0.3757*-°225 4.5 5.0 145 -f-ve -ve 0.2812*-°!^^ 0.3523±°266 2.5 2.0 180 -fve -ve 0.2929*°^^^ 0 . 4 0 0 7 * ° 2 « 0.8 2.0 205 -|-ve -ve 0.2879*°^^^ 0.3887*°^'*^ 0.3 2.0 255 -|-ve -ve 0.2879*°^^^ 0.3830±°326 1.9 2.0 K'ss 145 -(-ve -ve 0.3286*-°277 0.3775*-°^°^ 4.6 5.0 180 -|-ve -ve 0.3401*-"2O5 0.3815*-°33i 3.5 2.0 205 -|-ve -ve 0.3282*°2O4 0.3942±°337 2.5 2.0 255 -f-ve -ve 0.3369*°23i 0.3944*°242 0.7 2.0 K'NN 105 -|-ve -ve 0.2837*-'^i9^ 0.2846*°i^i 4.0 5.0 145 -f-ve -ve 0.2926*°i^^ 0.2855*°i^* 0.7 2.0 180 -|-ve -ve 0.3134*°2^2 0.3640±°22o 0.3 2.0 205 -|-ve -ve 0.2816±-"207 0.3340*°2oo 0.1 2.0 255 -fve -ve 0.2877*°i^5 0.2894±-°i^« 0.9 2.0 Table X V I I I : L i s t of spin-transfer results for the ird —y pp reaction in the Mad i son frame. The errors quoted here include both statistical and systematic contributions. T . ( M e V ) 0' (degrees) K'LS 6' (degrees) K'ss 9' (degrees) 105 147±3 -0 .115*°39 N . A N . A . 137*3 Q 243±.064 145 146*3 -0.144*°' '2 106*2 0.202*°^3 141±3 0.318*°^2 180 146*3 -0.302*°« ' 106*3 0.152*-°''5 143*3 0.376±.o62 205 146*3 -0.253*°^^ 105*3 0.216*°''« 144*3 0.282*-°59 255 145*3 -0 .212*°^ 104*3 0.305*°^^ 145*3 0.095*-^°^ It is now possible to rewrite Table X I I , which lists the spin-transfer parameters measured i n this experiment, to include a meaningful uncertainty representative of both the statistical and systematic errors present i n the analysis. T h i s , presented i n Table X V I I I , represents the final quantitative results of this experiment. Chapter XI Discussion and Conclusions The underlying goal of this thesis has been to describe the first experiment ever carried out to measure spin-transfer observables of the fundamental wd —>• pp reaction. In addit ion, this is the first experiment to obtain spin-transfer data over a range of energies spanning the A resonance of the pp ^ dn system. T h i s data compliments the considerable set of spin-dependent data which already exists, but which has been obtained pr imari ly using the pp —»• dn reaction. Such spin-transfer data is essential for the unambiguous determination of the part ia l wave amplitudes which characterize this reaction, amplitudes which i n t u r n provide the essential testing ground for theories developed to describe this important pion product ion/absorption process. The equivalence of observables obtained from either direction of the pp ^ dir reaction results from the invariance of strong interactions under time-reversal. Not only does this principle allow a simple transformation of the Trd —>• pp spin-transfer observables to the time-reversed frame, but it allows data from the pp dir reaction to be used as a valuable check of the entire measurement procedure (ANO ^ PA/") employed i n this thesis. A summary of the data obtained i n this experiment is given i n Table X I X , where, to be consistent w i th the vast majority of published data , the results have been expressed i n terms of the time-reversed (pp —^dn) observables. They are obtained from Table X V I I I using the transformations indicated in Table X I . The uncertainties associated wi th these results contain contributions from statistical as Table X I X : List of spin-transfer results for the pp —> dir reaction i n the centre of mass frame i n accordance to the Madison Conven-t ion. The results quoted here included errors which are both statistical and systematic i n nature. ( M e V ) ^deut. (degrees) KSL 9deut. (degrees) Kss Odeut. (degrees) KNN 105 147±3 0.115*°39 N . A N . A 137*3 0 243±o64 145 146*3 0.144±.042 106*2 0.202*°53 141*3 0.318*-°^2 180 146*3 0302±-O6o 106*3 0.152*°"^ 143*3 0 376±062 205 146*3 0.253*°^^ 105*3 0.216*°''« 144*3 Q 2 8 2 ± ° 5 9 255 145*3 Q212±069 104*3 0.305*°56 145*3 0.095*-^°^ well as systematic errors. A l though both types of errors are of the same order of magnitude, the statistical errors generally dominate. X I . l Data Comparison and Evaluation To properly evaluate the contribution of this new spin-transfer data to the knowledge of the part ia l wave amplitudes characterizing the pp dir reaction, it would be necessary to perform a detailed PWA fit w i t h this new data included i n the global data set. U n t i l such a program is carried out, one can get an idea of the impact of this data by comparing it w i t h the existing predict ion of Bugg's PWA fit. Such an i l lustrat ion is presented i n F i g . 41, where the energy dependence of our data and the corresponding predictions of Bugg's amplitudes [44] (at comparable angles) are depicted for a l l three observables. In general, qualitative agreement is observed between the predictions of the Bugg amplitudes and the data. However there are some significant quantitative differences. Most notable is the energy dependent behaviour of the observable Kss, part icular ly between Tp= 580 M e V to 700 M e V . Th is energy range spans the peak of the A resonance and thus the structure observed may indicate some 4 0 0 ^5Ô ëSÔ TSÔ 850" 9 0 0 Proton Lab Energy (MeV) 0.5-0 5Ô0 6Ô0 7Ô0 8Ô0 9Ôi Proton Lab Energy (MeV) 0 5Ô0 6Ô0 7Ô0 8Ô0 90i Proton Lab Energy (MeV) Figure 41: pp dit spin-transfer observables as a function of proton lab energy. The solid l ine represents the predictions of Bugg [44] at comparable centre of mass angles. interesting behaviour of the amplitudes associated w i th this transition, a behaviour which is not predicted by the existing fits. In addit ion, the systematic difference between the KSL and K^N data and their amplitude predictions suggests a general discrepancy which may result from errors of either phase or magnitude i n one or more of Bugg's amplitudes. O f part icular interest are those energies where Bugg's fit has included spin-transfer data from previous experiments (see section III.3, namely T p = 500, 580 and 800 M e V ) ^ . In general no significant discrepancy is observed, thus i l lustrat ing the overall level of consistency between our results and previous spin-transfer results^. In F i g . 42, the new spin-transfer data is compared wi th the predictions of the two theories presented i n chapter II. None of the theories successfully predict the new data presented here, w i th the possible exception of the theory of Niskanen [100]. In addit ion to its success w i th other polarization observables (see section II.2), Niskanen's theory is somewhat successful at predicting the energy dependence of KSL and KNN- However, both theories fail i n their predict ion of Kssi even to the level of getting the correct sign! It is difficult however to suggest improvements to theoretical calculations based s imply on their inabi l i ty to predict a single observable (due to the many contributing amplitudes to such an observable). Clear ly a comprehensive understanding of the behaviour of the amplitudes is essential. XI.2 Future Work A t this point, it is relevant to ask: is further experimental work needed for the accurate determination of the part ia l wave amplitudes describing the pp ^ dir ^Only the Geneva [5] and L A M P F [47] spin-transfer data were included in the Bugg fit [44]. ^However, recall that the other spin-transfer measurements were performed in a different scat-tering angle region (see section III.3) and for the inverse reaction. 0.5 400 500 600 TOO 800 900 Proton Lab Energy (MeV) 0.5-4Ô0 5Ô0 6Ô0 7Ô0 8Ô0 900 Proton Lab Energy (MeV) 0.5-40 0 0 5Ô0 6Ô0 73Ô Mo 90i Proton Lab Energy (MeV) Figure 42: The new spin-transfer data is compared w i t h the predictions of various theoretical models. The solid line represents the theory of Blankleider (see sec-t ion II.4) and the dashed line represents the theory of Niskanen (see section II.2). reaction? W i t h o u t the detailed PWA fit discussed i n the previous section, it is not possible to answer this question completely. However, it is reasonable to suggest that the new experimental results presented here w i l l at least restrict the region of observable space i n which any poorly defined amplitudes could show sensitivity. In other words, any further ambiguity i n the amplitudes should manifest itself i n a part icular observable(s) over a well defined energy and angular region. For instance, it is possible that some addit ional sensitivity may exist for the spin-transfer observables associated w i th the tensor polarization of the deuteron [44]. However, such quantities are notoriously difficult to measure using existing techniques. For example, as discussed i n chapters V I I I and I X , the tensor polarization of a deuteron target is quite smal l (therefore one is not very sensitive to this polarization). A lso using the standard d-C polarimeter techniques discussed i n section III.3, it was seen that there is httle sensitivity to the deuteron's tensor polarization. Therefore current technology dictates that only "vector" spin-transfer measurements are feasible. W h a t about future polarimeter experiments? I suggest that no further experiments involving the pp ^ dw reaction should be done pending the outcome of an updated detailed PWA analysis. However, i f further spin-transfer measurements were deemed necessary, suggestions can be made regarding further experiments of the type discussed i n this thesis: • Due to the domination of statistical errors i n this work, it is v i ta l that one optimize those parameters associated w i th the proton polar izat ion measurement. This could be attained through careful studies of the figure of merit, discussed i n section V . l . Some of these measurements have since been performed for the polarimeter used i n this experiment [73 . • Some design modifications to the polarimeter could reduce some of the systematic uncertainty associated w i th the acceptance of the latter half of the polarimeter. For example by placing the first scintillators of the polarimeter S l A and S 2 A upstream of the carbon analyzer one removes any acceptance effects associated w i t h these instruments as was the case w i t h the "gap" , (see section X.1.1) . In our work the problem was solved at the cost of lost statistics. • Studies should be done to determine a means to increase the efficiency of the large wire chambers of the polarimeter. Due to their large size, these chambers are more sensitive to background radiat ion i n the experimental area and thus experience more multiple events. The i r lower efficiency obviously cuts into the overall statistics which can be evaluated and does introduce some uncertainty to the polarimeter's acceptance. A n expensive solution to this problem would be to replace the low cost M W D C ' s w i th ind iv idual wire read-out drift chambers, thereby providing some abil ity to discriminate mult iple events. • Independent measurement of the proton-carbon analysing powers for the polarimeter used i n this experiment would remove any systematic uncertainty arising due to the use of analysing powers obtained from a different apparatus. This is not a difficult task given the intense, highly polarized proton beams currently available [101]. • The use of a "frozen-spin" polarized deuteron target would reduce complications due to trajectory bending and spin-precession, and introduce new possibilities to this measurement procedure. A frozen-spin target requires a magnetic field whose magnitude is about half that of a dynamical ly polarized target. This would result i n less spin-precession and bending of the particle trajectories. In the case of the K'gg measurement, this would reduce the out-of-plane projection of the sideways deuteron polarization by roughly half. For the K'j^^ measurement one could obtain data at a significantly more " forward" angle (~ 10° wi th a set up similar to this experiment) w i th less horizontal bending of the incident p ion beam. In conclusion, much is currently known about the features ( including subtle ones) of the fundamental pion absorption/product ion reaction, pp ^ dn. Physicists are now on the verge of providing an accurate quantitative description of this important process after th ir ty years of experimental work. W h a t is perhaps most disappointing about this progress is that the theoretical models have not kept up w i t h the advances of the experiments. Despite the considerable effort expended i n developing a good theoretical description, vast discrepancies lie between the quantitative picture and the abil i ty of models to understand this observed behaviour. It is hoped that the impending completion of an accurate description of the pp ^ dn reaction shall provide incentive and encouragement to those workers attempting the difficult calculations required for the understanding of this simplest of pion absorption/production processes. Bibliography [1] E . O . Lawrence and M . S . Liv ingston . Phys. Rev., 40:19, 1932. 2] C . Cohen-Tannoudji , B . D i u , and F . Laloe. Quantum Mechanics. John Wi l ey and Sons, New York , U S A , 1977. [3] G . R . S m i t h et a l . Phys. Rev., €38 :251, 1988. [4] M . Simonius. Polarization in Nuclear Physics, page 38. Volume 30, Springer, B e r l i n , 1974. 5] G . Cantale et a l . Helv. Phys. Acta, 60:398, 1987. [6] Mad i son Convention. Polarization Phenomena in Nuclear Reactions, page XXV. U n i v . of Wisconsin Press, Madison , U S A , 1971. 7] B . Blankleider. Conference on particle production near threshold. Nashvil le, U S A , 1991. A I P conference proceedings, No. 221. 8] H . Garci lazo and T . 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Healey. Summary of Deuteron Target Polarization Analysis for Experiment 331. Technical Report , T R I U M F , 1987. unpublished. 83] M . Borgh in i . Phys. Rev. Lett, 20:419, 1968. [84] D . Healey. 1991. Private Communications. [85] R . A . Abegg et a l . Nucl. Instr. Meth., A306 :432, 1991. 86] R . Henderson et a l . IEEE Trans. Nucl. Sci., 34:528, 1987. [87] R . Henderson et a l . IEEE Trans. Nucl. Sci., 35:477, 1988. 88] W . R . Leo. Techniques for Nuclear and Particle Physics Experiments. Springer-Verlag, B e r l i n , 1987. 89] R . D . Ransome et a l . Nucl. Instr. Meth., 201:309, 1982. [90] D . Besset et a l . Nucl. Instr. Meth., 184:365, 1981. [91] G . R . S m i t h . STAR system on-line manual. Technical Report , T R I U M F , 1987. unpublished. [92] J . L C h u m a . P L O T D A T A Command Reference Manual. Technical Report , T R I U M F , 1990. unpublished. 93] G . L . G a m m e l , P . W . Keaton J r . , and G . G . Ohlsen. Polarization Phenomenon in Nuclear Reactions, page 411. U n i v . Wiscons in Press, Mad i son , U S A , 1971. 94] F . Sperisen, W . Gruebler, and V . Koenig . Nucl. Instr. Meth., 204:491, 1983. [95] R . Handler et a l . Phys. Rev. Lett, 19:933, 1967. 96] J . D . Jackson. Classical Electrodynamics. John Wi l ey and Sons, New York , U S A , 1975. 97] V . Bargmann, L . M i c h e l , and V . L . Telegdi. Phys. Rev. Lett, 2:435, 1959. [98] M . E . Sevior. 1988. Private Communicat ion . [99] P . Walden . 1988. Private Communicat ion . [100] J . A . Niskanen. 1990. Private Communications. [101] Cern Courier. December 1991. pg. 10. Appendix A Appendix In the latter part of the nineteenth century the world seemed a relatively uncomplicated place. F r o m the point of view of physics, two forces of nature were quantitatively understood. Grav i ty had been described by Sir Isaac Newton three centuries earlier and its existence had long been put to use by mankind^. D u r i n g the nineteenth century, James Clerk M a x w e l l had unified the forces of electricity and magnetism^ wi th his concise description of the electromagnetic interaction. It goes without saying that this knowledge brought about a proli feration of applications around the t u r n of the century which have become irreplaceable i n modern society. However, it was the discovery of radioactivity i n 1896 by Becquerel that opened up a whole new "can of worms," spawning the entirely new field of "sub-atomic" physics. A l though this discipline has been actively studied for less than a century, it has already provided us w i th a new means of looking at the world. N o longer can we content ourselves w i th only two forces of nature. We have discovered a very symmetrical subcutaneous structure existing just below the skin of the "everyday" world around us. It is probably safe to say that the implications of this modern realm of physics are far from being fully felt. There already have been several major ^ Obvious uses of gravity are the power obtained from a water dam or sliding down a hill on a toboggan. This knowledge had also been used in scientific applications to provide a quantitative description of the motion of the planets, etc. ^Electricity and magnetism had been known for many centuries as two separate effects. applications^ resulting from work i n sub-atomic physics, despite the fact that a complete quantitative picture (of the symmetrical subcutaneous substructure) does not yet exist. One could draw an analogy wi th the compass, a device whose services were used long before a quantitative understanding of electromagnetic interactions was achieved. Today the world of science can be loosely portrayed as a network of several disciplines as depicted i n F i g . 43. The position of a discipline i n this network by no means reflects its relative importance. Instead the ensemble is intended to demonstrate that the science of one field can contain components which are fundamental to the science of another discipline. Ul t imate ly the l inks between each of these fields are of v i ta l importance; they allow knowledge from one discipline to be applied i n another. The work on which this thesis is based is associated w i t h the lower echelons of this network, and i n part icular w i th the field of intermediate energy physics. A . l Sub-Atomic Physics Primer The research which takes place at each level of sub-atomic physics pr imari ly focuses on the properties of the particles which exist at these levels, and the mechanisms by which they interact. A t the most fundamental level, particle physics, the constituent particles of matter are studied. Current understanding suggests that quarks and leptons listed i n Table X X fulf i l l this role. These particles are divided into three generations, according to the patterns i n which they appear i n nature. The ir interactions are governed by the exchange of another fundamental particle, called a boson, which carries momentum, energy, and other information between the two interacting particles. ^Obvious examples are nuclear power, medicines, imaging techniques, muon spin resonance (/xSR), etc. M e t a l l u r g y (Senlagy Zoology Mebxcxne J n n r g a n t r E^cxence îStnlngg Olljf m t s t r g A t n m t r p i jgs trs N u r l e a r }pl|gstrs 3 i n i e t m e b x n i e E n e r g y ^Ifystrs îStglj îEnrrgy ^I jystrs Figure 43: Network of scientific disciplines. 179 , , Table of the constituent particles of nature. Each co lumn cor-l a b l e A A : , , ,. responds to a generation. I II III charge leptons ( , ) ( ; ) ( ^ ) 1 quarks ( ^ ) ( ^ ) ( ^ ) _ ! mass Of part icular interest to this discussion is the strong interaction, which occurs between quarks. The strong force is mediated by the exchange of a boson called a gluon. A n interesting property of the quark interaction is that its strength seems to increase as the interacting particles get farther apart. A result of this property is that quarks or gluons have never been seen as free entities (a feature known as "confinement"). It is the strong interaction which allows the quarks to be bound together to form baryons, the " fundamental" particles of nuclear physics. Baryons are made up of sets of three quarks. A l though quark combinations can be made from a l l generations of Table X X , only the first generation exists i n the world around us (this generation has the least mass). The u and d quarks form neutrons and protons, as well as another important baryon, the delta ( A ) . These particles, l isted i n Table X X I , can be distinguished by their total spin (defined i n section 1.2), mass and charge (the A exists i n four charge states). The more massive A prefers to decay quickly (within 6 x 10"^^ seconds) into the lighter neutron or proton and hence is generally not directly seen i n experiments. The interactions between these baryons does not involve directly the exchange of gluons as a result of the "confinement" restriction. Instead, mesons (quark, anti-quark^ (qq) pairs) are exchanged, as i l lustrated i n F i g . 44. Most •'An anti-quark is the anti-particle of a quark. Figure 44: Schematic representation of the nucleon-nucleon interaction i n terms of quarks. common i n the nuclear environment are the pions, l isted i n Table X X I , as they make up the lightest qq pair , and are hence the lightest mediator of the nuclear force. A l so l isted are rho (/?) mesons which are pion-like i n quark content but have larger mass and a spin of 1^. One can imagine the nuclear environment (inside a nucleus) as a group of neutrons and protons (nucléons) surrounded by a sea of mesons which act as mediators of the nuclear binding force and thus hold the nucléons together. M a n y complicated processes are occurring w i t h i n this environment. Often, on very short time scales, mesons are produced or absorbed by the nucléons resulting i n "excited" baryons, including the A , which then decay by producing or absorbing other mesons. Due to the large numbers of particles and possible processes occurring w i t h i n the nucleus, most models describing the nuclear environment are only able to deal w i th bulk properties. However, a more elementary description of the nuclear environment is desirable. To describe what is happening in this "nuclear wor ld " , one would like to be able to use the fundamental particles of Table X X and their interactions to provide a general description of nuclear phenomena. Unfortunately, at energies relevant to nuclear physics, baryons and mesons appear as rubber balls. One can "dent" the surface without revealing the r ich quark and gluons structure hidden deep below the surface of the nuclear particles. In addit ion, our current understanding of quark interactions, which describes the world inside a single nucléon reasonably wel l , are not capable of describing the longer range interactions of the nuclear environment. Thus , the transit ion from the very short range interactions of quarks, to the longer range interactions of baryons is not well understood. Th i s transit ion is the l ink between the worlds of particle and nuclear physics, and thus there remains a significant "grey area" between these two widely studied branches of science. It is here that one may introduce the field of few-body intermediate energy physics^. Th i s field occupies an important niche between the fundamentals of particle physics and the more pragmatic world of nuclear physics. It is the goal of few-body physics to simplify a system to as few particles as possible i n order to understand the ult imate structure and interactions of those particles. W i t h a detailed understanding of few-body nuclear particle interactions one might, on one hand, obtain insight into the transit ion between quark and baryon interactions, and on the other hand , acquire greater understanding of the interactions of the particles i n the nuclear environment. ^"Intermediate energy" currently refers to a range of energy from lO's of MeV to a few GeV. In this range, individual particles of importance to nuclear physics can be created and studied. Table X X I : Properties of important particles from nuclear physics which are discussed throughout this thesis particle type charge spin n mass ( M e V ) sub-structure 7r+ boson +1 0 140 ud 7r~ boson -1 0 140 ûd boson 0 0 135 \{uû - dd) boson + 1 1 770 ud p boson -1 1 770 ûd p' boson 0 1 770 \{uu — dd) p fermion +1 1 ? 938 uud n fermion 0 i 939 udd A++ fermion +2 1232 uuu A + fermion +1 1232 uud A ° fermion 0 3 f. 1232 udd A - fermion -1 3 2 1232 ddd 


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