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Analyzing powers and differential cross sections for pn --> [pi]⁻pp(¹ ₀) at 403 and 440 MeV Duncan, Fraser Andrew 1993

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Analyzing Powers andDifferential Cross Sections for-*— 1pn PP(So)at 403 and 440 MeVFraser Andrew DuncanB.A.Sc. University of British Columbia, 1985M.A.Sc. University of British Columbia, 1988A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF PHYSICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIADecember 1993©Fraser Andrew Duncan, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)_______________________Department of PH’.S( c5The University of British ColumbiaVancouver, CanadaDate________________DE-6 (2/88)AbstractThere is considerable interest in the pri —f rpp reaction which can proceed bya nonresonant channel from the isospin 0 pn initial state (an N/ intermediate statecannot be formed). This thesis describes a measurement of analyzing powers and tripledifferential cross sections for a subset of this reaction, pn — irpp(’So) by isolating thequasifree process in pd —* irppp3. The experimental arrangement selects the relativeS-wave component of the outgoing “diproton”. The experiment was done on TRIUMFbeam line lB using a LD2 target; the pion was detected in a magnetic spectrometer, thetwo outgoing protons in a scintillator bar array. The spectator proton was undetected.Data were taken in August 1989 at 353, 403 and 440 MeV beam energies. Of thesethe 403 and 440 MeV data are analysed in this thesis and analyzing powers and tripledifferential cross sections as a function of pion scattering angle extracted at centre ofmass kinetic energies, TOM, of 55 and 70 MeV (corresponding to the 403 and 440 MeVbeam energies respectively).Partial wave analysis of the data shows that, while the isospin 0 channel dominatesthe reaction, contributing approximately 75% of the cross section at the energies studiedhere, there are significant contributions from the s and d-wave pion, isospin 1 channels.Of particular importance is the contribution from the s-wave pion, isospin 1, channelwhose interference with the isospin 0 channels produces the characteristic shapes of thecross sections and analyzing powers observed in the data. The d-wave pion, isospin 1channels, are also required to fully explain the observed analyzing power distributions,and are essential for the TOM = 70MeV data. Comparisons of the pion productiondata measured in this experiment with pion absorption measurements on 3He, wherethe absorption process is rpp(’So)—pn, show a shift in the shape of the differentialcross section which can be interpreted as due to differences in the wave functions forthe free and bound diproton.11ContentsList of TablesList of FiguresAcknowledgements1 Introduction1.1 NN—*TrNN1.2 Partial Wave Analysis and Observables.1.3 Previous Workviviiixi1510142 Apparatus and Procedure2.1 Experimental Procedure2.2 Kinematics of pd —* irppp3 .2.3 Beam Line lB2.4 Target and Scattering Chamber2.5 QQD Pion Spectrometer2.6 Scintillator Bar Array2.7 Coordinate Systems and Trajectory2.8 Electronics2.9 Data Acquisition and Analysis3 Calibration3.1 Beam3.1.13.1.23.1.33.1.4CalibrationBeam EnergyBeamspot on Target . .Beam CurrentBeam Polarization2323263031323739Notation4251525252535455Hi3.2 Spectrometer Calibration3.2.1 Wire Chamber Calibration3.2.2 Quadrupole Calibration and3.2.3 Alignment of the Apparatus3.2.4 Vertex Reconstruction3.2.5 Momentum Calibration3.2.6 QQD Energy Loss Corrections3.2.7 QQD Wire Chamber Efficiency3.2.8 QQD Acceptance3.3 Scintillator Bar Array Calibration .3.3.1 Bar TDC calibration3.3.2 Horizontal Bar Position Calibration3.3.3 Vertical Bar Position Calibration3.3.4 Bar TOF3.3.5 ADC Calibration3.3.6 Bar Energy Calculation3.4 Calibration Summary55566065757581828695969698100105113113116,• . . . 116116121• . . . 121123129131140141149154155163WC Calibration Checkusing Experimental Data4 Analysis4.1 Definitions4.1.1 Event Reconstruction4.1.2 Energy Conventions4.1.3 Cross Section and Analyzing Power Definitions4.1.4 Extracting Cross Sections4.1.5 Nuclear Shadowing4.2 pd —* dirri54.3 pd—irppp4.3.1 Background Contamination4.3.2 P-Wave Contamination4.3.3 Quasifree pn—+ 7rpp(1S0)Analyzing Powers4.3.4 Quasifree pri —* irpp( Differential Cross Sections4.3.5 Four Body Contamination5 Discussion5.1 Partial Wave Analysis5.2 Comparison With Other Experiments169170178iv5.3 Comparison with Theory.1836 Conclusions 190Appendices 197A Partial Wave Analysis of j5n —* Trpp(1S0) 197B Handler Parameterization of pn — rpp 203C Beam Line lB SEM Current Monitor 214D Beam Line lB Polarimeter 217E Spectator and Four Body Models 221E.1 Spectator Model 221E.2 Four Body Model 226F Monte Carlo Simulation 228F.1 Simulation of Beam, Target and Scintillator Array 231F.2 Simulation of the Spectrometer 232G Yields and Phase Space Integrals 238G.1 Predicting Reaction Yields 238G.1.1 pd irppp5 (Spectator Model) 239G.1.2 pd —* irppp (Four Body Model) 243G.1.3 pd — dirn (Spectator Model) 245G.2 Phase Space Integrals 245Bibliography 248VList of Tables1.2 Allowed transitions for pri — 7rpp(’So)1.3 Piasetzky’s Partial Wave Solutions for irpp(10— pri2.1 Dipole Field ParameterizationBeam EnergiesQ QD alignment model parametersSummary of measurementsFitted QQD Optics Coefficientspd —* data used to calibrate QQD acceptanceBar Vertical Position CalibrationTDIF calculated for pd — 7rppp3 and pd —* dirn.Range of TDIF for different reactionsGlauber Correction for quasifree pri —* irpp .Experimental and Monte Carlo TDIFQuasifree pn —* irpp(1S0)ANOQuasifree pn —* irpp(1S0)cross sectionsGoodness of Fit for PWAPartial Wave solutions for 55 MeVPartial Wave Solution for 70 MeV5.4 Relative partial wave strengths for Coupled Channels Model5.5 Analyzing Power Zero Crossing6.1 Relative Partial Wave StrengthsA.1 Partial Wave Amplitudes for pri — irpp(1S0)B.1 Allowed Transitions for rip —* irpp1.1 Isospin decomposition for NN —* 7rNN 615203.13.23.33.43.53.63.73.84.14.24.34.45.15.25.335536874799299105115130148157160171173174186188• . . . 194• . . 201205viC.1 SEM Calibration.216D.1 Polarimeter targets 220D.2 CH2 analyzing powers and expected polarirneter rates 220viiList of Figures1.1 Isopin cross sections for NN — irNN 71.2 Allowed transitions for pri — irpp(’So)1.3 Ponting ANO and Piasetzky PWA .2.1 242.2 282.3 292.4 332.5 352.6 382.7 402.8 412.9 432.10 472.11 492.12 501321Experimental LayoutAngular Range of DiprotonRelative yield of S and P-wave diprotonsThe QQD Pion SpectrometerSpectrometer dipole field shapeScintillator Bar ArrayCoordinate SystemsTrajectory NotationElectronicsWire chamber readout electronicsE460 hardware triggerE460 Trigger Timing3.1 Wire chamber position reconstruction 573.2 Transport Through Quadrupole 613.3 Empty Target Traceback with Quad on 643.4 Spectrometer Alignment Model 673.5 Comparison of data and QQD alignment model 693.6 Fit of vertical QQD alignment model to data 713.7 &— 6QQd reconstructed from triton distributions 733.8 pp ‘ di ANO(1ab) 743.9 QQD momentum calibration 803.10 Scatterplots of WC2 and WC3 853.11 QQD Solid Angle calculated by Monte Carlo simulation . . . 88viii3.12 Experimental and Monte Carlo QQD Yield as a function of 53.13 Average ratio of experimental and Monte Carlo yield3.14 Ratio of relative pd —* tir data yield to Monte Carlo solid angle3.15 Ratio of pd * data yield to Monte Carlo yield899094943.16 Triton distributions in barsTOF spectrumScintillator bar TOP calibration.Scintillator bar ADC calibrationPID plots for pd —* irppp3Tritons “falling out” of their PIDADC4UPPID cutTDIF plot for pd —* tirband97• 101• 104• . . • 108• • • . 109• • . . 110111• 112• 1143.173.183.193.203.213.223.233.244.14.24.34.44.54.64.74.84.94.104.114.124.134.144.154.164.174.184.194.204.214.22Naming convention for pd 7PPPs 117Centre of Mass system for irpp 122Reid Soft Core Deuteron Wave Function 129Quasifree pp —* dir TOM 132pd —* dirn. TDIF 133Simulation of background for pd — dirn 134Quasifree pp —* dir d/d2 vs TOM 137Q uasifree pp —p dir+ differential cross section 139Quasifree j5p — dir analyzing power 140pd —* irpJpS TDIF 143TDIF with cuts on PB counters 146Analyzing Power vs P 150Experimental and Handler P distributions 151Handler P distributions fitted to experimental data 152Fraction of S wave diprotons accepted 153Quasifree pn —* irpp(1S0)analyzing power 156Analyzing Powers for Hahn and Ponting 158Quasifree pri —* 7rpp(’So) TOM 159Quasifree pn —* 7rpp(1S0)triple differential cross sections 161403 MeV and 440 MeV du in the same energy bins 162pd“ lrppps Ps distributions 165Ps distribution for low spectrometer setting 168ix5.15.25.35.45.55.65.75.85.95.10datadataHandler’s Event definitionMonte Carlo simulation of Handler variablespri —* irpp total cross sectionsHandler’s differential cross sectionsHandler h2(P) function and3u/dQ.dMSimulation of pd — irppp event in GEANTQQD Monte Carlo Simulation, Plan ViewQQD Wire Chamber distributionsQQD Wire Chamber distributionsQ QD momentum reconstruction171172175176177178180Partial Wave Fits using s and p and d-wave pionsPartial wave fits without constraintQuasifree pn —* irpp(1S0)Partial Wave Analysis, TCM = 55MeV . .Quasifree pn — irpp(’So) Partial Wave Analysis, TCM = 70MeV . .Argand plot of partial wave amplitudesComparison of quasifree pri — irpp(1S0) ANO and Piasetzky’s PWAComparison of quasifree pn —* irpp(0)and Handler S waveComparison of quasifree pn — ir°p(’So) and3He(ir,pn)n differentialcross sectionsComparison of quasifree pn —* irpp(’S0)Comparison of quasifree pn —* irpp(’So)and theoryB.1B.2B.3B.4B.5and CCM modelD.1 TRIUMF beam line lB In Beam PolarimeterF.1F.2F.3F.4P.5182185187204208209210212218230232235236237xAcknowledgementsI wish to thank my thesis supervisors, Ed Auld and Pat Walden for suggestingthis experiment and for the guidance they have provided over the course of the analysisand interpretation of the data. I wish to thank Hartmut Hahn, the other graduatestudent who worked on the analysis of the data for many hours of fruitful discussionand shared anguish. I wish to thank Garth Jones and Dave Hutcheon for their assistance and advice during the analysis. I wish to thank Jouni Niskanen for the manydiscussions on the theoretical aspects of the experiment and for permission to includehis calculations in the discussion section of this thesis. And I wish to thank the otherphysicists who participated in this experiment: Chaim Aclander, Danny Ashery, DaveGill, Elie Korkmaz, Sharon Maytal-Beck, Murray Moinester, Dave Ottewell, Azriel Rahay, Shiomit Ram, Martin Sevior and Ruthie Weiss. I thank the staff and techniciansof TRIUMF without whose support and assistance the experiment would not have beenpossible. In particular I thank Walter Keliner, the technician in charge of the liquiddeuterium target for his assistance it really went above and beyond the call of duty.I thank my best friend and wife, Sandra, for the help, support, understandingand patience she has given me over the course of my studies. I thank my parents foran upbringing that taught me to strive for what I waiited in life and that nothing isimpossible. I dedicate this work to my father, Donald Hudson Duncan who could notsee its completion but is still a part of it.xiChapter 1IntroductionThe pion production process, NN irI\TI\T refers to the class of reactions where twonucleons (protons and or neutrons) interact producing two final state nucleons anda pion (ir+, 7O, rj. The final state nucleons may he free or bound in a deuteron.Even though it is the simplest pion production mechanism, a complete understandingof this process has proven elusive and much effort has been put into both experimentalobservations and theoretical calculations of the various NN —* TrNN reactions sincethe discovery of the pion[1] in 1947.Experimental efforts to study NN — irNN processes consist of measuring sufficient observables for each reaction to parameterize it with a partial wave analysis. Partial wave analysis allows the isolation of the individual transitions contributing to thereaction, which can then be compared to theoretical calculations. The measurementsof observables for the different reactions vary in complexity. Factors that determinethe difficulty of a measurement include: the number of final state particles (two if thefinal state nucleons form a deuteron, three if they are unbound), the charge statesof the particles and the spin states of the particles. The best studied NN—f irNNreaction, pp —> dir+, has its strongest partial wave amplitudes reasonably well known.The partial wave analysis of pp — dir+ shows that the dominant mechanism for pion1CHAPTER 1. INTRODUCTION 2production is through the formation of an intermediate NA state,NN —f NA —* irNN.This is the well known A resonance from pion-nucleon scattering (see [2]). Earlytheoretical effort has centred around describing pion production in terms of the Aresonance[3] but it has become apparent that while resonant production is domimam,it is not the only mechanism (see [4]).Studying the monresonant pion production mechanisms offers the possibility ofinvestigating the finer details of the nucleon-nucleon and the pion-nucleon interactions.Because these mechanisms are somewhat weaker than resonant production, they maybe more sensitive to short range interactions between the nucleons — perhaps even tothe extent that it is necessary to consider quark degrees of freedom in the nucleonsand mesons. Additionally, since the nonresonant channels are much weaker than theresonant channels, nonresonant production may be more sensitive to the initial and finalstate interactions of the nucleons. Thus a complete understanding of the NN —p irNNprocess must include the weaker nonresonant production in addition to the dominantresonant mechanisms.The strength of the A resonance makes it difficult to study nonresonant pionproduction. The cross section from the A is so large that it masks the weaker contributions from the nonresonant channels. Thus the strategy for studying nonresonantproduction in reactions such as pp — dir+ is to measure observables that enhance production amplitudes where A formation is forbidden. This often leads to complicatedexperiments involving difficult polarization measurements.Alternatively, one can study reactions where resonant production is suppressed(there are no NN —* 7r1\hI\T reactions where intermediate I\TA states are completelyCHAPTER 1. INTRODUCTION 3forbidden). One such reaction is a subset of the pn — irpp reaction,pn — irpp(’So), (1.1)where a proton and a neutron interact to form a negative pion and two protons in arelative ‘S state (which has isospin 1 and is referred to as a “diproton”). Reaction1.1 does not allow the dominant NL\ channels found in processes such as pp —* d7r+and is thus hoped to have a greater contribution from the nonresonant pion productionprocesses.Because of the neutron in the initial state, pion production from pn—f irpp(’So)necessitates either a neutron beam with a proton target or a proton beam with a boundneutron target. The advantage of using a proton beam and a bound neutron targetis that proton beams have a much higher intensity than the available neutron beamspermitting experiments to be performed iii a fraction of the time it would take withneutron beams. As well, there are well established technologies for preparing highintensity polarized proton beams making spin observable measurements relatively easyto accomplish. The cost of using a proton beam is the complication of dealing with thebound neutron. Being bound, the neutron is off its mass shell and has a momentumdue to the binding1. As the simplest nucleus with a neutron and with a small bindingenergy, deiiteriurn is the material of choice for a bound neutron target.The reaction pd — irppp can be used to extract information on the reactionpri —* 7rpp(’50)by designing an experimental apparatus to enhance the relative ‘S0state of the two final state protons from the pn —* irpp(’So) process and by assumingthe following two hypotheses:1The energy, E, momentum, F, and mass, rn, of a free particle obey the relation, E = i/m2 + P2.This is the relationship of a sphere in P space and is referred to as the “mass shell”. For a bound orvirtual particle, this relationship is no longer necessarily true and the particle is said to be “off its massshell”.CHAPTER 1. INTRODUCTION 41. The proton in deuterium behaves as a “spectator” to a collision between a beamproton and the bound target neutron, n. That is to say, the proton does notinteract and leaves the reaction with only the momentum it had due to the nuclearbinding. This momentum is called the Fermi momentum (or Fermi motion) ofthe proton[5]. The pion production from the deuteron is then referred to as,pd —* irppp3,where Ps is the spectator proton which has a spectator momentum, Ps. Withproper consideration of the off shell target neutron’s energy and momentum,cross sections for the quasifree reaction,pn —* irpp(’So),can be extracted from a measurement of the pd— IrPppS reaction.2. The transition matrix element for the quasifree reaction, MQ(S, t), is identical tothe matrix element for the free reaction, MF(S, t), i.e.MQ(S, t) MF(8, t),where s and t are the Mandlestam variables.With these hypotheses, the free pri —* irpp(1S0)cross section can be extracted fromthe pd _* PPPs process.This thesis describes TRIUMF experiment E460 (conducted in August and September of 1989) in which measurements of pd —* irppp3 were made at three proton beamenergies: 353, 403 and 440 IVIeV. From these data, analyzing powers and differentialcross sections for the free reaction pri —* 7rpp(So) were extracted. The measurementCHAPTER 1. INTRODUCTION 5of both cross sections and analyzing powers permits a partial wave analysis to be doneon the data. In this work, the experiment, its design, and calibration are presented.The extraction of cross sections and analyzing powers from the 403 and 440 MeV dataare discussed and conclusions drawn. The analyzing power of the 353 MeV data ispresented in a separate work (Hahn[6]) as well as an independent extraction of the 403MeV analyzing power.1.1 NN—rNNThe NN — irNN process refers to the collision of two nucleons producing a finalstate of a pion and two nucleons. The initial state nucleons can be either two protons, a proton and a neutron or two neutrons. In the final state, the pion can bepositively charged, neutral, or negatively charged. The final state nucleons are againeither protons, neutrons or one of each. If the final state nucleons are a proton and aneutron, they can be either bound in a deuteron or unbound. With the above reactionconstituents, there are ten possible NN — 7rNN reactions. These are listed in table1.1.In 1954, Gell-Mann and Watson [7] and Rosenfeld [8] postulated charge independence of the NN —* irNN reaction and described it in terms of cross sections dependingonly on the isospin of the initial and final state nucleons. Under this scheme, there arefour allowed cross sections and the total cross section for any NN —* 7rNN reactioncan be described as a weighted sum of either one or two of these. Table 1.1 lists thetotal cross sections for the various NN —÷ irNN reactions in terms of the isospin decomposition. For a given cross section, °i,p, I is the total isospin of the initial NNsystem and I’ the isospin of the final NN system. The cross section uio(d) is the crosssection to the bound state of the pn system, the deuteron. The cross section aio(pn)is for the unbound pn system. Notice that only the U and a10(d) cross sections canCHAPTER 1. INTRODUCTION 6pp +pfl g = g11 + uio(pn)pp_*qr+d u=uio(d)ppopp g=Jllpn —÷ rpp a (a01 + a11)pn—’ir°d u=uio(d)pn —* lr°pn a = (a01 + aio(pn))p ‘ = (u01 + a11)nfl —÷ irpn a aio(pn) + aunn —÷ 7rd o- = aio(d)flflK°flfl a=a11Table 1.1: Isospin decomposition for NN —* irNN. In the cross section notation cup, Iis the total isospin of the initial NN system and I’ the isospin of the final NN system.be measured directly. Ver West and Arndt[9] parameterized the isospin cross sectionsusing phenomenological functions fitted to the data available in 1982. The Ver Westparameterizations of the cross sections are plotted in figure 1.1 as a function of theexcess kinetic energy in the 7rNN CM system, TOM. This quantity is given by,TOM = W — (m7, + rn1 + m2),where W is the total CM energy, m the mass of the pion and m1 and m2 are the massesof the two nucleons. If the two nucleons form a deuteron, then TOM = W — (m7, + md).Of the reactions in table 1.1, the most studied has been pp —* dir+. The reason forthis is that the reaction is a two body process with all the particles in both the initialand final state carrying an electromagnetic charge making production and detection ofsuch a reaction relatively simple. In addition, the cross sections above threshold arebCHAPTER 1. INTRODUCTION 7151296300 500Figure 1.1: Isospiii cross sections for NN —* irNN. I is the the initial NN isospin I’ isthe final NN isospin. TCM is the excess kinetic energy of the irNN CM system. FromVer West[9].100 200 300 400TCM(MeV)CHAPTER 1. INTRODUCTION 8moderately large. There are extensive data for pp —* d7r+ in both the pion productionprocess and in the inverse pion absorption reaction, +d pp. Extensive study of theabsorption channel is made possible by the availability of high intensity pion beams atthe Meson Factories (TRIUMF, PSI, LAMPF). In addition to total and differential crosssections, there have been various measurements of spin observables in which both theinitial and final state particles are polarized. From these data, a reasonably completeset of partial wave amplitudes has been found for the reaction[1O]. The amplitudeanalysis shows that the reaction proceeds predominantly through an intermediate NAstate which decays to dir+,1pA +—nA f — drDescribing the state of the incident protons as2S+Lj,r where S is the total spin of thetwo protons, L the orbital angular momentum, J the total angular momentum and irthe parity, the dominant transitions for pp —÷ dTr are from the ‘D2+ and 3F- initialstates. For the 1D2+ transition, the intermediate NA has an angular momentum ofLNI = 0 and for 3F-, LN/ = 1.The A with a mass of 1232 MeV is familiar in irN scattering as the first of severalresonant irN states (some others being N*(1440), N*(1520) and N*(1535) where thenumbers are masses in MeV). It has a width of 115 MeV, corresponding to a lifetimeof 6 x 1024s, spin 3/2 and isospin 3/2. Pion production through an intermediate NAstate dominates the NN —* rNN reaction making it difficult to examine nonresonantproduction processes. In reactions where the A resonance is strong, such as pp —* dir+,nonresonant channels are examined by complex experiments measuring spin observables. The smaller nonresonant channels can contribute more significantly to the spinobservables via interference terms.CHAPTER 1. INTRODUCTION 9A different approach to studying nonresonant channels is to examine reactionswhere the L resonance is forbidden. The N\ state can form either an isospin 1 or 2state, thus it can be formed in the NN —* irNN reaction from the u,o(d), the aio(pn)and the a channels. The a01 channel, having an initial isospin of zero, cannot forman intermediate NL\ state. From table 1.1, the reactions including a01 are,pnpp a=(ao,+a,i), (1.2)a = (aoi + an) (1.3)andpa ‘ °pn a = (a0, + aio(pn)). (1.4)The and ir channels are identical except for Coulomb interactions. None of thesereactions proceed omiy by a01 and all have a neutron in the initial state which complicates experiments.Of reactions 1.2, 1.3 and 1.4 the simplest to study experimentally is pri —* irppsince the final state has no uncharged particles. This work studies a particular sub-channel of the pa —* irpp reaction, namely that in which the final state protons form arelative ‘S0 state (i.e. pa —* 7rpp(1S0)). This channel is of interest for several reasons:First of all, it is experimentally simpler to study than the full process (where it is necessary to have a 4ur detector in order to get the complete kinematics of the final statediproton). The S wave protons, on the other hand, do not have an angular dependencein their centre of mass so it is not necessary to measure their angular distribution.Secondly, by looking at the pa —* 7rpp(1S0)suhchannel of pa —* irpp, the strongtransitions from the 1D2- and ‘F3- initial states (observed in pp — dir+) are forbiddenwhich should strongly suppress the Li contributions to the a11 cross section. AndCHAPTER 1. INTRODUCTION 10finally, for pn —* irpp(1S0)there are two particles with spin one half in the initialstate (the nucleons) and two spin zero particles in the final state (the pion and thediproton). This requires only two spin independent transition amplitudes (and thusthree independent parameters) to describe the reaction2.With only two spin independent amplitudes contributing to the reaction, it is relatively easy to perform a partialwave analysis on pn —* irpp(1S0). For the full pn —b irpp reaction, the diproton isallowed to be in a P-wave and higher, greatly increasing the number of partial wavescontributing to the reaction. This makes both partial wave analysis and theoreticalcalculation more difficult.The greatest difficulty with studying pri —* irpp(1So) is isolating the S-wavediproton. The other allowed diproton angular momentum states can contaminate thedata. There is, however, a Final State Interaction (FSI) between the protons when theyare in a S-wave, that enhances the probability of their having a low relative momentum.By building an apparatus that selects protons with low relative momentum, the S-wavecan be enhanced. This was the technique employed in this experiment.1.2 Partial Wave Analysis and ObservablesIn order to compare the experimental data to the various theories, it is sometimes usefulto perform an Amplitude Analysis which decomposes the transition matrix elements,(fHj)for the process into a specific basis. Observables can be calculated from the amplitudesor, alternatively, amplitudes can be fitted to the observables. By fitting the amplitudessimultaneously to many observables, an averaging occurs that smooths over errors in2This can be compared to reactions such as pp —* dr+ which has two spin one half nucleons in theinitial state and a spin one deuteron in the final state requiring four spin independent amplitudes andseven independent parameters to describe it.CHAPTER 1. INTRODUCTION 11any one particular measurement. Theories assuming specific transition operators, T,can be used to generate amplitudes which can then be compared to the amplitudesfitted to the data3.One possible basis for amplitude analysis is a Partial Wave Analysis also calledthe LS decomposition. The amplitudes are written out in terms of the orbital andspin angular momentum of the initial and final states. The matrix element, (fiTli),describing the transition between a given initial and final state can be decomposed intoan expansion in spherical harmonics such as(fTIi) = CYj(6,q5)a (1.5)In this notation, j refers to the quantum numbers of the initial and final states. Theterm is a numerical factor containing Clebsch Gordan coefficients for the couplingof spin and orbital angular momentum and Yj(6, ) is a spherical harmonic4 whichdescribes the scattering angle distribution between the initial and final state particles(0 is the angle between the production particle and beam direction, is the anglethe scattering plane makes with the horizontal plane in the laboratory), and a1 is anamplitude.The partial wave expansion used to fit the observables (differential cross sectionsand analyzing powers) measured in this experiment is discussed in appendix A. Itshould be noted that these amplitudes cannot he directly compared to other analysesfor example Piasetzky[12] and Niskanen[13]. Different expansions differ by conventionsused for numerical factors and the relative phases of the amplitudes. It is thus necessaryto use caution when comparing the results from different works.3See Blankleider[11] and Garcilazo and Mizutani[4] for a discussion of NN—, KNN theories.4A spherical harmonic is correctly written as Y’(9, 1). In the notation Y(9, ), j is being used torepresent L and M (along with any other quantum numbers).CHAPTER 1. INTRODUCTION 12Regardless of the partial wave expansion, some general comments can be madeabout the relationship between the amplitudes and observables. An observable, (9, iscomposed of the real and imaginary parts of bilinear combinations of matrix elements(amplitudes) which can be written as0 [CRRe{a3a}+ CiIm{aa}J,j,kwhere j and k are summations over amplitudes and CR and C1 are the spherical harmonics and numerical factors. Dropping the C1 and CR terms, the spin averaged crosssection is written asdg a)2 + Re{a2a},i jkwhich is a summation over the absolute squares of the amplitudes, IaI2, and a summation over the real parts of the interference terms, Because of the square terms,the presence of a few large amplitudes can mask contributions from smaller amplitudes.Spin Observables on the other hand are a summation of interference terms such as,ZkIm{aak}spin duBecause the square terms enter here only as a multiplicative factor times some interference terms, spin observables are more sensitive to smaller amplitudes. Theseamplitudes manifest themselves as an interference with a large amplitude. Thus themeasurement of various spin observables can improve the accuracy and quality of thepartial wave analysis.Figure 1.2 shows the first few allowed transitions for pn —* Trpp(’So). Thenotation used is that of Rosenfeld[8]. The NN state is defined as (2S+1)L where S isthe total spin, L the orbital angular momentum, J the total angular momentum andCHAPTER 1. INTRODUCTION 13Figure 1.2: The allowed transitions for pri— 7rpp(’So) for J 2. The pn) stateis described by(29+1)L where S is the total spin of the two nucleons, L the orbitalangular momentum, J the total angular momentum and ir the parity. The 17r119) stateis described by (2s+1)Ll where S is the total spin of th two nucleons, L and j are theorbital and total angular momentum of the two nucleons respectively, 1 is the orbitalangular momentum of the pion, J the total angular momentum of the system, and irthe parity.PH>0 1 0 13s 3p0 ,3p2 3D ,3D 3F2pp) 1S0s ‘SOP1+INN 1 1CHAPTER 1. INTRODUCTION 14ir the parity of the NN system. The irNN system is denoted by (2S+l)Ll whereS, L and j are the spin, orbital angular momentum and total momentum of the NNsubsystem. The pion orbital momentum is 1 and the total angular momentum andparity of the irNN system is J and ir respectively. The selection rules are:• the initial total angular momentum equals the final total angular momentum;• conservation of parity;• the NN wave function is antisymmetric over spin, orbital angular momentumand isospin.As can be seen in table 1.1, the final state pp system has isospin = 1 while theinitial pn system can have = 0 or I = 1.Table 1.2 shows the five allowed transitions for which J 2. The five amplitudescan be described by nine parameters5. If the contributing channels for the pn —*7rpp(1So) process are either pure o-01 or pure o• then the differential cross sectionwill be symmetric about 900 and the analyzing power will be antisymmetric[13]. Thusa nonsymmetric cross section or a nonantisymmetric analyzing power indicates thepresence of both a01 and U. Table 1.2 also shows the allowed intermediate NN andN/ states for each transition. Unlike in pp —* dir+where an S wave N/ can be formed,the lowest permitted angular momentum for a N/ state is LN, = 1.1.3 Previous WorkWhile pn ‘ pp is experimentally the simplest pion production channel for the studyof aol, there are relatively few data available on pn —* irpp and even fewer dataavailable for the pn —* irpp(1S0) subchannel studied in this work. There have been5There are 5 complex numbers but one phase is arbitrary.CHAPTER 1. INTRODUCTION 15Initial State Intermediate States Final StateI NN NN N/ irNNj I Amplitude1 3D 3D 3D 1C 11. .1 Q— 1 Ps0 3S1+ 3D1+ ‘Sopi+ 1 asp3S1+0 3D+ 3S1+ ‘S0p1+ 1 aD31-iLJ1+1 3P2_ 3F— 3P2_ 1S0d2— 1 apd3D 5D‘2— 123’;,25’;,-‘ 21 3F2— 3P2_ 1S0d2_ 1 aFd3’;’ 5n‘2 123’;’‘ 25’;’‘ 2Table 1.2: Allowed transitions and intermediate J\TN and N states forpri— irpp(’So) for 1,, <2. I, is the initial NN isospin, Ij the final state NN isospin.CHAPTER 1. INTRODUCTION 16three approaches to pn —÷ irpp experiments. These are,rip —* irpp, (1.6)pd — irppp5, (1.7)andir3He“ Ps, (1.8)where reaction 1.8 is used to extract information on negative pion absorption on adiproton, irpp(’So) —* pri. All three of these methods of extracting information onpn —f irpp have been exploited and all have their own strengths and weaknesses.Since there are no free neutron targets, the most straight forward reaction to studyis rip —* irpp where a neutron beam strikes a hydrogen target. These experimentsare an unambiguous measurement of the pri —* irpp process but are complicated bythe need to produce a neutron beam of known energy and adequate intensity. In1955 Yodh[14j measured rip —f irX and extracted rip —* irNN differential crosssections using a liquid hydrogen target and photographic emulsions as counters at anincident neutron kinetic energy of 409 MeV. Yodh found a significant asymmetry for thedifferential cross section indicating the presence of both u and J contributions to thereaction. Handler[15] performed a liquid hydrogen bubble chamber experiment usinga neutron beam with a kinetic energy distribution varying from the pion productionthreshold (approximately 287 MeV) to 440 MeV and peaked at approximately 409MeV. The data from this experiment were fitted with a phenomenological model ofthe pri —* irpp reaction allowing final states with l = 0 or 1 and L7, = 0 or 1 (seeappendix B for a discussion of this model). The parameterization showed a differentialcross section with a similar asymmetry at 409 MeV to that measured by Yodh (althoughCHAPTER 1. INTRODUCTION 17with a lower absolute normalization).Other rip —* irpp experiments include: Dzhelepov et al[16] (T=600 MeV);Thomas e al(17] (T=790 MeV) and Terrien e al[18] (T = 572, 784, 1012MeV). Mostof these measurements are differential and total cross sections. Terrien however measured analyzing powers for np —+ irpp over a pion centre of mass angular range of 20°to 160°. Terrien selected an energy range for the diproton where it is expected to bedominantly in a relative P-wave. Dzhelepov et al found evidence for a small contribution to the total cross section from oçj (20%) at T=500 MeV while Thomas et alestimated that a01 contributed less than 4% at T=790 MeV.Kleinschmidt et. al[19] measured differential cross sections for the companionreaction to rip —* lrpp, namely rip ,‘ +riri (which is seen from table 1.1 to havean identical total cross section to pri — irpp). Data were taken at neutron beamenergies ranging from T = 480MeV to 578 MeV for centre of mass pion angles lessthan 35°. They were unable to fit cos & terms to the limited angular distributions ofthe data which could be interpeted as a lack of contribution from However, a morerecent measurement by the same group (Bannwarth et al. [20]) measuring the ratio ofrip —* irpp to rip _+ n+riri differential cross sections at 8 = 166° indicates a significanta01 contribution in a similar energy range (T = 460 to 560 MeV).As discussed on page 3, using the pd —f irppp3 reaction to extract informationon pri —* -irpp requires two assumptions. The first assumption is that the protonin the deuteron is a spectator to the reaction. The second assumption is that theresulting quasifree reaction on the bound neutron target, pri* —* irpp, is the same asthe free reaction. The Fermi motion complicates the analysis of the data by varyingthe centre of mass energy of the pfl* collision. The advantage of quasifree productionover neutron beam experiments is the relative ease of preparing high intensity, proton(or deuteron) beams. Polarized proton beams are also straight forward to prepare forCHAPTER 1. INTRODUCTION 18spin dependent experiments. Experiments of this form include: Brunt et al[21] (P=1.825, 2.110 GeV/c); Cohn et al[22] (P= 3.7 GeV/c); Dakhno et al[23] (T= 528 978MeV); and Tsuboyama et al[24] who used a deuteron beam, measuring dp —* irppp(P=1.04 1.89 GeV/c). Tsuboyama et al compared their extracted pri —p ‘irpp totalcross sections to pp —* ir°pp data (which is pure an) and found no contribution fromaol in the range TOM = 80 MeV to 380 MeV (there was some asymmetry appearingat 380 MeV which Tsuboyama interpreted as the first appearance of ao,). This is incontrast with Handler[15] and Yodh[14] (also measuring the pri —> irpp total crosssections) who found significant a01 at TOM = 55MeV and with Kleinshmidt[19] whofound significant a01 in the energy range TOM = 85MeV to 185 MeV.Pion absorption on 3He, reaction 1.8, is used to extract information on pionabsorption on the ‘S0 diproton, irpp(’So) — pn using the same assumptions made forpion production from deuterium — namely that the neutron in the 3He is a spectator tothe process and that the absorption on the bound diproton (which is in a ‘S0 state[25]),ir_{pp(1S0)}*—* pn is the same as absorption on a free diproton, irpp(’So) —* pn. Theprinciple of detailed balance can then be used to relate the pion absorption reactionto the production reaction, pn —* irpp(1S0). Relating pion absorption onto a bounddiproton to free pion production through the principle of detailed balance is opento question since the radial parts of the free and bound diproton wave functions aredifferent. However, lacking evidence to the contrary, the validity of the procedure hasbeen assumed.The advantage of pion absorption experiments is the restriction of the processto a two body final state. However, these experiments are complicated by the need todetect a neutron and by the 3He target Fermi motion. Furthermore, spin dependentmeasurements (i.e. measuring the polarization of the final state nucleons) are almostan impossibility due to the low efficiencies of polarimeters, the low efficiency of neutronCHAPTER 1. INTRODUCTION 19detection and the low intensity of available negative pion beams (and of course, thelow reaction cross sections).Absorption experiments have measured differential cross section distributions atvarious pion beam energies. The apparatus typically consists of a liquid 3He target andcounters which detect the final state proton and neutron in coincidence. The spectatorneutron remains undetected. Hahn[6] measured differential cross sections at T,-=37MeV, Aniol et al[26] at T= 62.5 and 82.8 MeV, Weber et al[27] at T7r= 64, 119, 162and 206 MeV and Mukhopodhyay et al[28] at 165 MeV. All these measurements founddefinite asymmetries in the differential cross sections about 900 requiring the presenceof both the a01 and U. The total cross sections extrapolated from the differential crosssection data show little energy dependence.Piasetzky et al[12j did a partial wave decomposition of irpp(’So) — pn for lv,. 2.They fitted these amplitudes to the 62.5 MeV pion absorption differential cross sectiondata[26j. However, because of the number of degrees of freedom, further restrictionshad to be made on the amplitudes. Piasetzky used Watson’s Theorem[29j to relate therelative phases of the amplitudes to (experimentally measured) pn elastic scatteringphase shifts6 leaving only real parameters. The fit resulted in five solutions for thepartial wave analysis. However, Gal noted an ambiguity in the Piasetzky partial waveanalysis and pointed out the equivalence of some of the solutions[30j. Piasetzky et alargued that at the low pion energy considered, 1,, = 2 should not play an importantrole in pion absorption and rejected three solutions which had significant pion d-wave,leaving two solutions. These solutions are shown in table 1.3 as the fractions of the totalcross section contributed by each amplitude and are labeled “S” and “D” depending onwhich of the S or D wave channels (amplitudes, asp and a from table 1.2) is stronger7.6Watson’s theorem notes that elastic NN scattering is much stronger than the inelastic process NN7rNN. Invoking unitarity for pn elastic scattering and treating pn irpp(’So) as a perturbation, thephases of the pion absorption channels can be related to the phase shifts of elastic pn scattering.7Actually the Partial Wave Solutions showil in table 1.3 are not the ones from the original paper byCHAPTER 1. INTRODUCTION 20PWA solution Theoreticaltransition amp “S” “D” quark meson—* ap 0.07 0.07 0.02 0.00‘Sopi+ —> as 0.52 0.09 0.87 0.02—* 3D1+ aD 0.41 0.84 0.11 0.98Table 1.3: Piasetzky’s Partial Wave Solutions for irpp(’So) —* pn. The strengths arethe fraction of the total cross section contributed by each amplitude. From reference[331.Also shown in table 1.3 are the amplitudes for two models that were proposedfor pion absorption on the diproton in 3He. Both models assume that the neutron in3He is a spectator and that the diproton is in a ‘S0 state at rest in the lab frame.The Meson Exchange Model by Maxwell and Cheung[31] had the pion absorbed ontoa NN or N intermediate state that exchanged a virtual meson. It predicted thatthe pion absorption would be almost entirely through the aD channel. The QuarkCluster Model by Miller and Gal[32] presumed that the diproton could be treated as acluster of six quarks and that the pion was absorbed onto a single quark. It predictedthat the dominant absorption amplitude was asp. Neither model predicted significantabsorption through the ap amplitude which is in agreement with Piasetzky’s partialwave analysis. It is seen from table 1.3 that Piasetzky’s solution “D” has similarstrengths to the meson exchange model while solution “S” agreed with neither. Thepartial wave solutions could be distinguished by the polarization of the proton in thepn system.As noted above, polarization measurements of the final state nucleons from pionabsorption on 311e are particularly difficult. However an equivalent measurement isthe analyzing power in the production channel, j5 —* irpp(’So). In an attempt toPiasetzky (reference [12]). Instead they are solutions from reference [33] which used a more recent pnphase shift analysis than Piasetzky.CHAPTER 1. INTRODUCTION 211.000.750.500.250.00—0.25—0.50—0,75—1.00120 150 180Figure 1.3: AND measured by Ponting{33](points) and Piasetzky partial wave analysissolutions “S” and “D” (lines).determine which of the Piasetzky solutions was correct, Ponting et. al.[33] (see also [34])measured the analyzing power for j5 — irpp(1S0) using j5i— Irppps. The kinematicsof the reaction were selected to enhance the 1S0 state of the diproton. A proton beamenergy of 400 MeV was chosen to correspond to the 62.5 MeV pion absorption data8.AND data were taken over an angular range of 58° — 81°. The analyzing power dataand the Piasetzky solutions are shown in figure 1.3. While the data tended to favoursolution “S”, it is seen that neither solution is able to reproduce the zero crossingobserved in the data. Solution “S” shows almost equal contributions from asp andNoting from table 1.3 that the meson model proceeded almost completely via aDwhile the quark cluster model preceded mostly through asp, Ponting suggested that8The irpp centre of mass energy of the quasifree pion production experiment is approximately 55MeV. It has been observed that because of the binding energy of the 3He nucleus and the Fermi motionof the spectator neutron, the rpp CM energy of the pion absorption experiment is in fact much lowerthan that of the production experiment, being approximately 46 MeV.0 30 60 900(deg)CHAPTER 1. INTRODUCTION 22the favoring of solution “S” was an indication that the reaction proceeded by both themeson and the quark mechanisms. Ponting concluded that pn —> 7rpp(’So) containssignificant contributions from short range interactions.Subsequent to the Ponting measurement, Niskanen performed a Coupled Channels Model (C CM) calculation for pion absorption on 3He [13] which was able to producea polarization similar to that of the Piasetzky solution S using only mesonic degreesof freedom. It predicted that the dominant transition was aD. However, as withthe partial wave analysis, the zero crossing of the CCM polarization calculation wasat a significantly higher angle than the Ponting measurement. More recent work byNiskanen has incorporated a Heavy Meson Exchange into the Coupled Channels Modelwhich has shifted the zero crossing to a more forward angle in better agreement withthe Ponting data[35].Another calculation by Bachman, Riley and Hollas [36] based on the DKS (Dubach,Kloet and Silbar) Model [37] was also able to produce an analyzing power for pnirpp(So) which had the basic “negative going to positive” shape of the Ponting data.However the model had a zero crossing near 900 compared to the 72.7° of the PontingData. The authors of [36] did not discuss the amplitudes which contributed to thecalculated analyzing power.In light of the considerable theoretical interest and the far from satisfactory agreement between the Piasetzky partial wave analysis and the Ponting analyzing powerdata, it was decided to extend the measurement of Pouting, covering a larger angularrange and taking measurements at beam energies of 353, 403, and 440 MeV. Both ANOand differential cross section data were taken, allowing a complete data set for partialwave analysis of the pri—irpp(’S0) reaction. This has the advantage of eliminatingquestions about the validity of assuming the equivalence of pion absorption and pionproduction data.Chapter 2Apparatus and ProcedureThis chapter discusses the experimental procedure used in this work, mentioning thevarious types of data collected and their purpose. The apparatus for this experimentis shown in figure 2.1. The arrangement consisted of a proton beam striking a liquiddeuterium target with a magnetic spectrometer to detect pions and a scintillator bararray to detect protons, deuterons or tritons depending on the reaction being investigated. The magnetic spectrometer could be rotated about the target where as thescintillator bar array was fixed. This apparatus differs from that used by Ponting[33,34]in two respects: a different detector arrangement was used for proton detection whichallowed higher event rates and a modified scattering chamber was used which alloweda larger detection range for the pion spectrometer. The features of the pd —f irppp3reaction that lead to the design of the apparatus shown in figure 2.1 are described andvarious aspects of the apparatus are discussed. The data acquisition electronics, theevent triggers for each data type collected and acquisition software are also described.2.1 Experimental ProcedureThe procedure for the experiment consisted of: a design phase, during which MonteCarlo simulations of the experiment based on the parameterization of the reaction byHandler[15] were used to determine the optimal configuration of the apparatus and23CHAPTER 2. APPARATUS AND PROCEDUREwire chambers24Figure 2.1: Experimental Layout. A polarized proton beam strikes a liquid deuteriumtarget producing a pion and two protons. The pion is detected to the left of the beamin the QQD spectrometer which makes an angle of 0QQD with respect to the beampipe. The two protons are detected to the right of the beam in a plastic scintillatorbar array located approximately 4m downstream of the target which is in coincidencewith a counter at the exit of the scattering chamber. 6QQD is the angle the QQD makeswith the beam pipe.t veto countersbeam dumpscintillatorbar arraybeam pipedipolescintillatorsproton barrelchambermagnetic spectrometer4Zwire chambersiTquadrupolescintillators scintillatorIbeamCHAPTER 2. APPARATUS AND PROCEDURE 25expected data rates; a setup and timing phase, in which the apparatus was assembledand the electronic’s timing adjusted; a data taking phase in which the products fromthe three reactions,pd —f tir, (2.1)pd—d7rn3 (2.2)andpd PPPs (2.3)where taken; and, finally, an analysis phase. The analysis of the experiment involvedcalibrating the apparatus, removal of background from the data and extraction of theanalyzing powers and cross sections. The cross section extraction required extensiveMonte Carlo calculation of the acceptance of the apparatus.The procedure used to take the three different data types (equations 2.1, 2.2 and2.3) and the purpose each data type was used for is described below:pd tir+ The pion and triton from this reaction were detected in coincidence thepion in the spectrometer, the triton in the scintillator bar array. The two bodykinematics of this reaction were exploited to calibrate the spectrometer’s pionmomentum reconstruction. The pions from this reaction were also used to determine the spectrometer’s acceptance. The triton’s were used to calibrate thescintillator bar energy reconstruction and to measure the spectrometer angle viatwo body kinematics. Many runs using different beam energies, spectrometersettings and scintillator bar array positions were taken.pd dir+n3 Measurements of cross sections and analyzing powers of the quasifree j5 —*CHAPTER 2. APPARATUS AND PROCEDURE 26dir+ reaction using the bound proton in deuterium were made. The purpose ofthese measurements was to check the apparatus calibration and to access theviability of the proposed procedure for extracting the free pn —* 7rpp(’So) crosssections and analyzing powers from the quasifree reaction. Data were taken atthree angles at a beam energy of 353 MeV and at one angle at 403 MeV. Thepion was detected in the QQD spectrometer, the deuteron in the scintillator bararray. The spectator neutron from the deuterium target, r, was undetected.pd —* IrPPPS These measurements were the intent of this experiment. The pion wasdetected in the spectrometer, the two protons from the quasifree pri —* irpp(1S0)reaction in the scintillator bar array. The spectator proton from the deuteriumtarget, p3, was undetected. Cross section and analyzing power data were takenat beam energies of 353, 403 and 440 MeV with the pion spectrometer at anglesranging from 28° to 93°.Additionally, there were empty target data taken for which the particles detected inthe QQD originated from the target cell walls. These runs were used for backgrounddetermination during the pd —* IrPPPS data taking, calibration of the spectrometer’swire chambers, and alignment of the spectrometer with respect to the target cell.2.2 Kinematics of pd—7rpppTo understand the detector geometry used in this experiment (figure 2.1), it is necessaryto examine the kinematics of the pd —* irppp3 reaction. Since the goal was to use thequasifree reaction to reconstruct observables for the free pn —* irpp(1S0)process itwas necessary to:• Minimize the participation of the spectator proton in the reaction.• Maximize the detection of the S-wave diproton state.CHAPTER 2. APPARATUS AND PROCED URE 27To this end, the apparatus was configured as if to detect a two body process,pn—*That is, a proton strikes a virtual neutron and produces a pion and a diproton (the twoprotons) in a ‘$0 state with mass The diproton can be described by the invariantmass, of the two proton system which can be written as, = 2m + whereis the kinetic energy of the two protons in their centre of mass system. can bethought of as an excitation energy of the diproton and can in general vary from zero tothe total centre of mass kinetic energy of the irpp system. However, the apparatus usedin this experiment only accepts diprotons with less than or equal to approximately10 MeV. This range of is adequate because while the two protons in a ‘So state arenot bound, they experience an attraction that causes them to favour < 10 MeV.The detection apparatus consisted of a small solid angle spectrometer located tothe left of the beam to detect the pion over an angular range of 300 to 1000 and a largesolid angle scintillator bar array to detect the two protons, placed at a small angleto the beam on the right of the beam pipe. The positioning of the diproton detectorwas determined by considering the kinematics of the quasifree two body process. Thevirtual target neutron is lighter than a free neutron due to the binding energy of thedeuteron and the Fermi motion of the spectator proton. If the bound neutron wasat rest, it would have a mass of 937.4MeV/c2 (2.2 MeV lighter than a free neutron).The spectator proton has an average Fermi momentum of approximately 45 MeV/c.This further reduces the mass of the target neutron to 935.2 MeV (4.4 MeV lighterthan a free neutron). The total energy of the collision between the beam proton andthe target neutron depends on the magnitude and direction of the target neutron’sFermi momentum. The total laboratory energy of the interaction can vary by plus orminus 50 MeV or more, depending on whether the neutron is moving towards or awayCHAPTER 2. APPARATUS AND PROCEDURE 2825- —20 -- /0 30 60 90 120 150 1608(deg)Figure 2.2: Angular Range of Diproton. Solid line angle for a diproton of mass= 2m at beam energy 400 MeV. Dashed lines maximum range of single protonsfor differing combinations of beam energy (350 MeV to 440 MeV) and (0 to 10 MeV).from the beam proton. The solid line in figure 2.2 shows the variation of the diprotonscattering angle as a function of the pion angle for an incident proton energy of 400MeV and a diproton with zero excitation energy. With = 0, the two protons willtravel together. When the diproton has an excitation energy, the protons diverge. Thedashed lines in figure 2.2 show the extreme range of proton angles for combinations of= 10MeV and changes in effective lab energy (ranging from -50 MeV to +50 MeV).It is seen that over the angular range of the pion used in this experiment (approximately30° to 100°) the protons can vary in angle from —10° to +20°. The proton bar arraywas placed approximately at the centre of this range, spanning from 2° to 16°.The quasifree pn—f 7rpp events detected will not have the protons entirelyin a relative ‘S0 state. In fact, the parameterization of the reaction by Handler[15]indicates that at the energies studied in this experiment, the dominant contribution tothe cross section is from P-wave protons. The relative yields of deecable S and P-waveCHAPTER 2. APPARATUS AND PROCEDURE 29Figure 2.3: Relative yield of S and P-wave diprotons as a function of P for:a)Acceptance determined only by pion spectrometer. b)Acceptance determined byspectrometer and proton scintillator bar array. The solid histogram is the total S andP-wave distribution. The dashed histogram is the P-wave.diprotons expected in this experiment were modeled using Monte Carlo simulations ofthe apparatus which incorporated the Handler parameterization of pri —+ irpp (seeappendices F and B) and the Spectator Model for the pd — irpppS reaction (appendixE). The simulations indicate that a kinematic quantity which is useful in differentiatingthe S and P-wave diprotons is P, the relative momentum of the two protons in the irppCM system,P -Pj and P are the momenta of the two protons (labeled 1 and 2). Figure 2.3a showsa Monte Carlo simulation of the P distribution for a case where the limiting apertureis the pion spectrometer and all protons are accepted. It is seen that the dominantcontribution is from P-wave diprotons. Figure 2.3b shows P for the actual detectorconfiguration (including the scintillator bar array which restricts the relative momentaof the two protons). The diprotons in a relative P-wave are greatly suppressed. Byputting a software cut on the data at a low value of P, the diproton can be selected tob)200P(MeV/c)200P(MeV/c)CHAPTER 2. APPARATUS AND PROCEDURE 30be almost completely in a 1S0 state.The enhancement of the S-wave diproton, which is due to a Final State Interaction(FSI) between the protons in the relative S-wave, increases the cross section at lowrelative momentum. The relatively small solid angle of the diproton detector forces theprotons to have a low relative momentum and while this restriction has little effect onthe S-wave diprotons, it greatly cuts the yield of P wave events.2.3 Beam Line lBThe experiment was performed on TRIUMF beam line lB [38] using a proton beampolarized vertically in the lab frame. The beam spin was cycled between up, down andoff— typically 3 mm up, 3 mm down and 0.5 mm off. Beam currents varied from 0.2 nAto 1 nA depending on the QQD spectrometer angle; the main limit on beam currentbeing the singles rates in the QQD front end chambers, especially at forward QQDangles. Beam polarization was measured with an In Beam Polarimeter (IBP) locatedsome distance upstream of the target before the final lB bender magnet. Beam currentwas measured with a Secondary Emission Monitor (SEM) located approximately 5m downstream of the target, which was subsequently calibrated against a FaradayCup. The beam profile was determined with two monitors, one located approximately1.5 m upstream of the target, the other downstream just before the SEM. The beamtarget position was determined by moving a scintillator screen in front of the targetand observing the beam spot with a television camera. This scintillator screen wasretracted during data taking. A 0.001 in. stainless steel vacuum window1 was located1.5 m upstream of the target and just downstream of the beam monitor. The purpose ofthis window was to prevent any gas escaping through a target rupture from damagingthe beam line. Unfortunately it also contributed a significant amount of background‘Since engineering is still done in the English system, dimeisions will be reported in English units,where appropriate, to avoid roundoff errors.CHAPTER 2. APPARATUS AND PROCEDURE 31from scattered beam protons. Most of this background was eliminated by placing acollimator inside the beam pipe and constructing a 20 cm thick lead wall around thebeam pipe to shield the scintillator bar array.2.4 Target and Scattering ChamberThe liquid deuterium target used in this experiment was supplied and maintained bythe TRIUMF cryogenic targets group. The aluminum target flask was a cylinder with2.000 in. outside diameter and 0.004 in. thick walls. The flask was oriented with itsaxis vertical in the lab. There were 5 layers of mylar “superinsulation” (each 0.00025in. thick, for a total of 0.00125 in.) wrapped around the flask except where the beampassed. The LD2 was pressurized at 16.5 to 17 PSI corresponding to a temperature of24.1 K and a density of 0.162+0.001 gm/cm3. The flask was mounted in a rectangular scattering chamber with a 0.005 in. thick Kapton exit window to the left of thebeam for the pions to exit towards the QQD spectrometer (figure 2.1). Backgroundmeasurements on the target flask were made by forcing the liquid deuterium out of thetarget cell using deuterium gas. The “empty” target thus contained cold deuterium gaspressurized at 17 PSI with a density of approximately 1.5% that of the liquid deuterium.Downstream from the target, the scattering chamber was attached to a largecylindrical extension referred to as the “proton barrel” which provides a large windowfor the protons to exit the scattering chamber towards the bars. The beam exits theproton barrel through a 4 in. beam pipe to the SEM and beam dump. In the endof the proton barrel, to the right of where the beam pipe was attached, is a 0.050in. thick, approximately rectangular, stainless steel window to allow the protons toexit towards the scintillator bar array, while suffering as little multiple scattering aspossible. Mounted against the exit of the proton barrel window were two 1/16 in.thick rectangular scintillators. The scintillators split the aperture horizontally at beamCHAPTER 2. APPARATUS AND PROCEDURE 32height and are referred to as PBU (Proton Barrel Up) and PBD (Proton Barrel Down).The proton barrel counters are used to identify particles originating from the targetand to reduce triggering from randomly scattered beam protons.The target and scattering chamber differed in two respects from that used byPonting et al[33]. First, an aluminum target flask was used instead of Kapton. This wasnecessary for the higher beam currents used in this experiment. Secondly, the scatteringchamber was redesigned with a bigger pion exit window to allow the spectrometer to bemoved over a larger angular range. The original scattering chamber allowed an angularrange from 45° to 82° while the modified scattering chamber allowed a range from 28.8°to 110°.2.5 QQD Pion SpectrometerThe spectrometer was assembled using the dipole and one of the quadrupoles of the (socalled) TRIUMF QQD [39,40,41,42,43,44]. The spectrometer was used to reconstructthe momentum and trajectory of the pion from reactions 2.1, 2.2 and 2.3. The reconstructed pion trajectory could then be traced back to the known beam position on thetarget to determine the event vertex in the LD2 target cell.The spectrometer (figure 2.4) has four proportional wire chambers labeled WC2,WC3, WC4 and WC5 (following a numbering convention used when there was a fifthwire chamber). WC2 is mounted before the quadrupole, WC3 is located betweenthe quadrupole and dipole magnets. WC4 and WC5 are located after the dipole atapproximately the focal plane. A 1/8 in. thick scintillator referred to as ST (STart) ismounted before WC2. After WC5, are two banks of scintillators, Si and S2 forming onebank, S3 and S4 forming the second. Each of the five scintillators on the spectrometerhave one photomultiplier tube connected to it.A pion traversing the QQD passes through WC2, the quadrupole magnet, WC3,t46.6540.0Figure 2.4: The QQD Pion Spectrometer consists of a quadrupole magnet and a 700bend dipole magnet instrumented with four wire chambers and five scintillation counters. The ST scintillator and wire chamber 2 (WC2) are located before the quadrupolemagnet. Between the quadrupole and the dipole is WC3. downstream of the dipoleare WC4 and WC5 followed by the two scintillator banks S1&S2 and S3&S4. (Alldimensions in cm.)CHAPTER 2. APPARATUS AND PROCEDURE 33/70 degS4/ /S2WC5 /WC4/dipole magentWC3 139.065quadrupole magnetWC2STpivot +CHAPTER 2. APPARATUS AND PROCEDURE 34the dipole, WC4, WC5, and the back end scintillator array. The five scintillators onthe spectrometer are used to trigger an event. The ST counter covers the aperture ofWC2 while each pair of scintillators at the back end of the spectrometer (S1& S2 orS3& S4) span the exit apertures of WC4 and WC5. A coincidence between ST and Sior S2 and S3 or S4 ensures that the pion has both entered and exited the spectrometer.The pion positions in chambers 2 and 3 and knowledge of the quadrupole’s opticsare used to reconstruct the initial trajectory of the pion. The dipole magnet is themomentum analyser and the position of the pion in either WC4 or WC5 along withthe initial trajectory is used to reconstruct the pion momentum. Using WC4 and WC5to make separate momentum measurements provides a redundancy which is used toreject muons from pion decay inside the spectrometer. The event vertex in the targetis found by tracing the initial ray back to its intersection with the beam. Because ofthe materials the pion must travel through, an energy loss correction is made to thepion energy as measured by the QQD when calculating the vertex energy of the pion.The spectrometer optics consisted of a quadrupole magnet (defocusing horizontally, focussing vertically) followed by a dipole magnet which bent the central ray 700to the left in the horizontal plane. The quadrupole magnet was 25.05 cm long with a 13cm diameter aperture which held a 5” (12.7cm) beam pipe. The dipole was a uniformgap, circular magnet with a 29.7cm radius and a 12.1 cm gap. A plot of the dipolefield shape (figure 2.5) measured by Whittal[45] shows the fringe field extending some30 cm beyond the edge of the dipole. The fringe field can be parameterized using theEnge function [46],B=B0, (2.4)1 + eswhere, B0 is the internal magnetic field. S is the parameterization,0 2 3 43 = C0 + C18 + CS + C3S + C48CHAPTER 2. APPARATUS AND PROCEDURE 35640Figure 2.5: Spectrometer dipole field shape. Points measured field shape. Curvefitted field shape.and s is the distance from the pole edge (z) measured in units of the dipole gap, D (i.e.s= z/D). Table 2.1 has the parameters for the Enge function fitted to the dipole. Thisparameterization of the field is used in the Monte Carlo simulation of the spectrometer.The equivalent Sharp Cutoff magnet radius (assuming a uniform field up to the cut offradius) is 38.46 cm.All four wire chambers on the spectrometer are delay line proportional chambers.Co = -2.461225Cl = 6.352449C2 = -3.947711C3 = 1.260056C4 = -0.117003Dipole Field Parameterization1210820 10 20 30 40 50 60 70 60r(cm)Table 2.1:CHAPTER 2. APPARATUS AND PROCEDURE 36Wire chambers 2 and 3 are of identical design and consist of 3 wire planes: a positivehigh voltage anode plane (with the wires running horizontally) sandwiched between twocathode planes at ground potential. One of the planes has its wires oriented vertically(the x plane) and the other has its wires oriented horizontally (the y plane). The wirespacing on each plane is 2 mm and the spacing between wire planes is 4.76 mm [42].A particle passing through a chamber leaves a trail of electrons knocked free from thechamber gas molecules. These electrons are attracted to the nearest anode wire. Nearthe anode, where the electric field is strong, the electrons undergo an “avalanche” 2resulting in a large number of electrons (on the order of 1O) reaching the cathode wire.The pulse arriving at the cathode wire induces image charges on the x and y planewires. The induced pulses in the x and y plane wires travels to the end of the wireswhich are attached at regular intervals to a delay line. A pulse reaching the delay linesplits and travels to each end of the delay line. By measuring the relative arrival timeof the signals at each end of the delay line, the position of the event in the x and ycoordinates is determined.A feature of this wire chamber configuration is that because the y plane wiresare parallel to the cathode wires, the signals occur at discrete positions. Provided thetiming resolution of the signals in the delay line is sufficiently fine, a histogram of they position of the events in the chamber will occur at discrete positions each peakcorresponding to the location of a cathode wire. Such a histogram (called a “picketfence”) is routinely seen for WC2 and WC3 and is used to calibrate the distance scalefor the wire chambers. The resolution of the y position measurement is one half of awire spacing. On the other hand, the x plane wires are perpendicular to the cathodewires. The x plane wires couple to a continuously variable positional source giving a2An avalanche is when the electrons are accelerated to a high enough energy to knock other electronsfree from the gas molecules. These electrons in turn are accelerated and knock other electrons free. Thisexponential growth process continues and thus the few free electrons left by the passing charged particlecan result in a pulse of on the order of iO electrons reaching the anode wire.CHAP TER 2. APPARATUS AND PROCEDURE 37position resolution finer than the cathode wire spacing. As a consequence of this, thehistogram of event position for x shows a continuous distribution, not the picket fenceof the y plane.Chambers 4 and 5 are also of the cathode plane, anode plane, cathode planedesign, except that the anode plane is at ground and the cathode planes are at negativehigh voltage. The wire spacings are 2 mm and the wire plane spacings are 6 mm[47]. The timing resolution of WC4 and WC5 is not adequate to see a picket fenceso the distance calibration for these chambers is done by directly measuring the signalpropagation speed of the delay lines.2.6 Scintillator Bar ArrayThe scintillator bar array consisted of two banks of eight scintillators located approximately 4 m downstream from the target and approximately 15 cm to the right of thebeam line (figure 2.6). Each bar was a lm long by 12.5cm wide by 10cm thick block ofNE11O plastic scintillator and was instrumented with two photomultiplier tubes; oneat at each end. The “front bars” were labeled in ascending order from the beam pipe,BAR1, BAR2, ..., BAR8. The “back bars” were labeled BAR11, BAR12, ..., BAR18again in ascending order away from the beam pipe. The photomultipliers on BAR1were referred to as 1U (top) and 1D (bottom), on BAR2, 2U and 2D, etc. Behind theback bars was a bank of scintillators used to detect and veto elastically scattered beamprotons which, because of the close proximity of the bars to the beam line, would haveotherwise swamped the data acquisition system. There were eight veto scintillators,four covered the bars above the beam height and four below. Veto V1U and V1D werelocated behind BAR11. V23U and V23D were located behind bars 12 and 13. V45Uand V45D covered bars 14 and 15. V678U arid V678D covered bars 16, 17 and 18. Theveto counters nearer the beam line were made narrower to reduce the counting rateCHAPTER 2. APPARATUS AND PROCEDUREbeamdumpbeam pipe38photornultipliers‘UP’ vetocountersBARS/‘DOWN” vetocountersV678U & V678DV45U & V45V23U &V1U & V1Dphotomultipliersfloortarget/Figure 2.6: Scintillator Bar Array, plan view. Insert: elevation view.CHAPTER 2. APPARATUS AND PR OCED URE 39(the bars closer to the beam pipe experienced a larger background flux). The protonenergy was determined by the Time of Flight (TOF) of the protons travelling from thetarget to the bars (with the timing started by the pion firing the ST counter on thespectrometer). Horizontal proton position was determined by bar number, vertical proton position by the travel time of the light propagating from the proton strike to eachend of the bar. The minimum kinetic energy required for a proton to reach the frontbars was 44 MeV. The maximum energy of a proton stopped in a front bar was 126MeV. The maximum energy of a proton stopped in a back bar was 180 MeV. Protonswith energy greater than 180 MeV would exit the back bars and enter a veto counterwhich would then fire, rejecting the event.This proton detector differed from that used by Ponting el. al. [33] where theproton energies were determined in a NaT scintillator bar array located at the exit ofthe proton barrel. The reason for going to a plastic scintillator/TOF measurement wasto allow an increased counting rate (plastic scintillators are capable of higher rates thanNaT).2.7 Coordinate Systems and Trajectory NotationFigure 2.7 shows the various coordinate systems used in this experiment — all wereright handed. The origin of the laboratory system, (XL, YL, zL) was at the centre ofthe LD2 target cell. The ZL axis pointed along the beam pipe (downstream). The YLaxis pointed vertically up in the laboratory. The beam was typically offset from thezL axis by (XB,yB). Each of the spectrometer’s wire chambers had it’s own coordinatesystem centred on the optical axis. The z axis of each wire chamber coordinate systempointed downstream through the spectrometer. The y axis pointed vertically up in thelaboratory. In addition, there was the QQD frame (x0, Yo, zo) which was located at thepivot of the spectrometer, 46.65 cm before WC2. The z0 axis was located on the opticalCHAPTER 2. APPARATUS AND PROCEDUREdipole40WC5target/Figure 2.7: Coordinate systems used in the analysis of the experiment. There are fourcoordinate systems associated with the QQD wire chambers, The QQD system locatedat the pivot of the spectrometer and the lab system located at the centre of the target.quadpivot(xQQD,yQQD,zQQD)/beam(xB,yB)CHAPTER 2. APPARATUS AND PROCEDURE 41xzFigure 2.8: The trajectory of a particle is specified by projections onto the xz planeand the yz plane. The angle made by the projection onto the xz plane and the z axisis &. The angle made by the projection onto the yz plane and the z axis is .axis, pointing downstream. The Yo axis pointed vertically up in the laboratory frame.The x0y plane is sometimes referred to as the iarget plane. The pivot was supposedto be centred at the origin of the lab frame, but was found to be displaced slightly. It’sposition in the lab frame is given by (XQQD, YQQD, zQQD) see section 3.2.3.The trajectory of a particle passing through the spectrometer is usually describedin terms of the position and angle the trajectory makes at a plane. For example at thetarget plane (zo = 0), the trajectory of a pion is described by,(x0, o, Yo, q5o).Figure 2.8 shows that the angle between the projection of the trajectory onto thex0z plane and the z0 axis is Similarly, the angle between the projection of thetrajectory onto the y0z plane and the z0 axis is This notation is used frequentlyin the discussion of the spectrometer in chapter 3.CHAPTER 2. APPARATUS AND PROCEDURE 422.8 ElectronicsThe data acquisition electronics for the experiment had two aspects. First there wasthe event trigger. This consisted of logic circuits which determine if certain of thedetectors (for the most part, scintillators) have fired in coincidence. If the correctconfiguration of detectors fired, the trigger circuitry started the recording process andrequested the computer to service the event. The necessary trigger condition dependedon which of the three reactions, 2.1, 2.2 or 2.3 was being detected. The second aspectwas the electronics to digitize timing and pulse sizes from the various scintillators andwire chambers of the apparatus. These digitized signals were transferred to the dataacquisition computer when a valid event trigger was generated.The event trigger had two levels. At the first level, the signals necessary for atrigger were brought together in the Master Coincidence (MC). If there was a MasterCoincidence and the computer was not already busy reading out an event, the LAM(“Look At Me”) coincidence was valid. It was the LAM coincidence that generatedthe request for the computer to read the event. Sometimes there would be a MasterCoincidence and the computer was all ready reading the event. In that case the newevent would be lost. When calculating cross sections, it is necessary to know what fraction of the events are lost, or conversely, what fraction of the time the data acquisitionsystem is able to read events. This is referred to as the computer Live Time, LT, andis calculated as LT = (LAMS/MC).The electronics are shown in figure 2.9. Each scintillator on the QQD and theProton Barrel had one photomultiplier tube, the signal from which was sent to anAnalog to Digital Converter (ADC) and a Time to Digital Converter (TDC). Theevent logic was formed and signal shaping was done with NIM modules while the AD Cs,TDCs and computer interface electronics were CAMAC. The CAMAC data were readby a J11 preprocessor which passed the data to the acquisition system running on aCHAPTER 2. APPARATUS AND PROCEDURE 43BAR 11V 1UPBUPBDFigure 2.9: Trigger electronics for the experiment (not including the Master Coincidence, LAM Coincidence and the Gate Coincidence — see figure 2.11). Shown arethe electronics for BAR1, BAR11, the QQD electronics and the proton barrel counterelectronics. BAR2 through BAR8 have the same electronics as BAR1. BAR12 throughBAR18 have the same electronics as BAR11.ADCTDCS4MC ( BARS)MC(WCH)MC(QQD)GC(QQDWIDE)LAM(ST)CHAPTER 2. APPARATUS AND PROCEDURE 44VAX 750. The data were recorded on 6250 bpi tapes.The scintillator bar array consisted of two banks of eight bars with veto countersbehind the second bank. The front bank of scintillators and the veto counters wereused in the event trigger while the back bars were only read into the data acquisitionsystem. The electronics for BAR1 and for BAR11 are shown at the top of figure 2.9.BAR2 through BARS had identical electronics to BAR1 while BAR12 through BAR18had identical electronics to BAR11. For both the front and back bars the electronicsstarted out the same. The signals from the photomultipliers on each end of a bar werefed into a passive splitter. One output of the splitter was attenuated and went to anADC. The other output went to a Constant Fraction Discriminator (CFD) which inturn went to a TDC. While this was the full circuity for the back bars, the front barshad additional electronics for use in the event trigger.To use a front bar in the event trigger it was necessary to realize that there was atime delay between the particle striking the bar and the resulting light pulse reaching aphotomultiplier tube. This delay depended on the distance of the particle hit from thephotomultiplier. While this variable time delay was useful for determining the positionof the hit in the bar (by taking the difference in firing times of the top and bottomphotomultipliers) it was undesirable for triggering. A signal whose time delay from theparticle hit was a constant could be formed using a meantimer. A meantimer generatesa signal which is the mean of the two input signals— a time that is independent ofwhere the particle strikes the bar. Figure 2.9 shows the signals from the 1U and 1Dconstant fraction discriminators being fed into a meantimer. Since both signals mustbe present in the meantimer for it to generate an output, it is equivalent to a logicalAND. The output of the meantimer went to a coincidence unit where it could be vetoedby either the V1U or V1D scintillators firing (which meant that the particle did notstop in the bars and was probably a beam proton). The signal from this coincidenceCHAPTER 2. APPARATUS AND PROCEDURE 45unit was called BAR1 and could be written in terms of the bar and veto signals as,BAR1 = 1U. 1D (V1U + V1D).(In the notation used here, “A• B” means “A AND B”, “A + B” means “A OR B”and “A” means “NOT A”.) BAR2 through BAR8 had similar circuits except that theveto scintillators covered more than one bar (BAR2 and BAR3 were covered by oneset of scintillators, BAR4 and BAR5 by another set , and BAR6, BAR7 and BAR8 bya third set). Thus the logical functions for BAR2 through BAR8 were written as,BAR2 2U 2D . (V23U + V23D),BAR3 = 3U . 3D . (V23U + V23D),BAR4 = 4U . 4D . (V45U + V45D),BARS = 5U SD. (V4SU + V4SD),BAR6 = 6U 6D . (V678U + V678D),BAR7 = 7U . 7D. (V678U + V678D),andBAR8 = 8U . 8D. (V678U + V678D).The BAR1 through BAR8 signals were fed to a Majority Logic Unit (MLU) whichgenerated an output provided a minimum number of inputs (or more) were present.This minimum was selectable for the different trigger conditions:1. For the pd—f irppp3 reaction, the minimum number of bars firing was two.2. For the pd — dir+n3 reaction, the minimum number of bars firing was one.CHAPTER 2. APPARATUS AND PROCEDURE 463. For the pd — tlr+ reaction, the minimum number of bars firing was one.The output of the MLU was a 100 ns wide pulse labeled BARS which was used in theMaster Coincidence (MC).On the QQD, the scintillator pairs Si, S2 and S3, S4 formed planes (see figure2.4). A valid QQD trigger was when a pion had passed through the ST counter andeach of the two scintillator planes. To this end, the Si and S2 signals were shapedwith discriminators and then combined together in a logical OR unit (the pion havingpassed through either Si or S2). The same was done for S3 and S4. The signals fromthe logical OR’s were put in coincidence with the shaped ST signal to form the 20 nswide QQD signal,QQD = ST• (Si + S2) . (S3 + S4),used in the Master Coincidence. The ST signal was also shaped with a discriminatorto form a 5ns wide pulse which was used in the LAM coincidence. All the TDC’s arestarted from this ST signal.Each delay line proportional wire chamber had two wire planes (x and y). Thesignals from the ends of the delay lines for each wire plane were shaped with discriminators and connected to TDC’s (figure 2.10). The WC2 x plane signals are called X2Pand X2M where “P” means the positive x end of the delay line and “M” means thenegative x end of the delay line. The y plane signals are called Y2P and Y2M. Thesame naming convention is used for the delay line singles from WC3, WC4 and WC5.In addition to the delay line signals, WC2 and WC3 had fast anode signals which couldbe used in the trigger. These anode signals were shaped with discriminators and thenfed into a coincidence, the output of which was referred to as WCH,WCH = WC2• WC3,CHAPTER 2. APPARATUS AND PR OCED URE 47cathode wiresFigure 2.10: Readout electronics for wire chamber 2’s x plane. The other wire chamberplanes are instrumented similarly.which could be optionally placed in the Master Coincidence.The proton barrel scintillation counters, PBU and PBD were discriminated andused to form a logical OR with with an output width of lOOns called BARREL,BARREL = FBU + PBD,which was used in the Master Coincidence.The trigger circuit is shown in figure 2.11. The BARS, QQD and BARRELsignals are brought together to form the Master Coincidence,MC = BARS. QQD . BARREL{.WCHJ,where the term {.WCH} is the optionally included coincidence with the signals fromWC2 and WC3. The LAM coincidence was between the MC and a narrower ST signalto ensure that the timing was with respect to the ST counter. In addition to generatingdelay linex2X2MTDC TDCCHAPTER 2. APPARATUS AND PROCEDURE 48the LAM request to the acquisition computer, the LAM module generated the startsignal for the TDC’s and the gate for the QQD ADC’s. The LAM signal consisted ofa coincidence between the MC and the ST and BUSY, where BUSY is a latch whichis set by the LAM and reset by the computer when the event has been read into thecomputer. The BUSY latch prevents the start of processing for a new event before thecurrent event has been read. Thus the LAM signal isLAM = MC . ST. BUSY.The timings of the MC and LAM coincidence are shown in figure 2.12a and b. Thetiming was arranged so that the LAM was triggered by the ST counter. In additionto signaling the data acquisition system, the LAM started the TDC’s and the QQDADCs. Because the LAM was triggered from the ST, this meant that all TDC timingwas relative to the ST (pion) signal.Because the arrival time of protons at the bars could be spread out over a largetime, it was necessary to have a separate coincidence to open the gates for the scintillator bar ADC’s. This coincidence was called the Gate Coincidence (GC) and consistedof a triple coincidence between the BARSNARROW, QQDWIDE and BARREL,GC = QQD WIDE BARSNARROW. BARREL.where QQDWIDE was a 100 ns wide QQD pulse and BARSNARROW was a 10 ns wideBARS pulse. As can be seen in the timing diagram (figure 2.12c), the Gate Coincidencewas timed off the BARSNARROW rather than the QQD (as in the Master Coincidence)or ST (as in the LAM Coincidence).CHAP TER 2. APPARATUS AND PR OCED URE 492OnsWCH(optional)lOOns lOOnsFigure 2.11: E460 hardware trigger. Shown are the Master Coincidence (MC), theLAM (“Look At Me”) Coincidence, and the Gate Coincidence (GC).QQD EARS BARRELQQD BARS BARRELLONG NARROWiOnsSTLAM6OnsIIADC GATESTDCSTARTCPUENDBUSYCHAPTER 2. APPARATUS AND PROCEDURE 50a) Master CoincidenceBARS________lOOns1<BARREL lOOnsQQD2ons1b) LAM CoincidenceMC_____________ ______________2Ons____ __ _ __ ____ ____ __ _ __ST__c) Gate CoincidenceQQDWIDEIBARREL lOOnsBARSNARROWFigure 2.12: E460 Trigger Timing. a) Master Coincidence b) LAM Coincidence c) BarADC Gate CoincidenceCHAPTER 2. APPARATUS AND PROCEDURE 512.9 Data Acquisition and AnalysisThe data acquisition system consisted of a VAX 750 running the TRIUMF VDACS dataacquisition program [48]. VDACS used a Jil. preprocessor loaded with a user writtenprogram which allowed preprocessing of the data before it was recorded to tape. Thepreprocessing for this experiment consisted of optionally rejecting the event if one ormore of the wire chambers on the QQD failed to fired. For a wire chamber efficiency ofapproximately 50%, this lead to a 50% reduction in events written to tape. However, inorder to determine the efficiency of the wire chambers, a sample of events were writtento tape, regardless of whether or not the wire chambers had fired. Typically, half theevents were accepted regardless of the wire chamber condition.The on line analysis was performed with LISA[49j, an analysis package (writtenin Fortran) in common use at TRIUMF. LISA runs parasitically off VDACS, analysingevents when the computer is not busy writing data to tape. While the on line dataanalysis fully reconstructed each event, the calibrations used for the apparatus wererelatively crude compared to the off line calibrations.Off line, LISA was used to analyse the raw events, make cuts on the data andcreate skim files containing the calibrated particle 4 vectors. The skim files were thenanalysed by AN460 (ANalysis E460). AN460 was a C program written by the authorspecifically for the analysis of this experiment. It combined separate data files for agiven detector configuration, sub-binned the data in smaller angle bins for each QQDangle setting, and then calculated the analyzing powers and cross sections. The off lineanalysis was done mostly on a Vaxstation 3100 M76 running VMS. However, MonteCarlo acceptance calculations were performed on Decstations running UNIX.Chapter 3CalibrationThe degree to which the quasifree pri —* rrpp(’So) reaction can be resolved fromthe pd —* lrpPps data and the ability to remove background contamination from thedata depends on the quality of calibration of the apparatus. The calibration can beconsidered in three parts: the proton beam, the pion spectrometer and the scintillatorbar array. Beam calibration includes determination of the beam energy and beamposition at the target, the beam polarization and the beam current. Spectrometercalibration consists of the determination of the pion trajectory, the pion momentum,and the spectrometer’s solid angle. The scintillator bar array calibration consists ofdetermination of the particle trajectory, particle time of flight, and particle energydeposit.3.1 Beam Calibration3.1.1 Beam EnergyThe beam line lB proton beam energy is selected by positioning the Xl stripper insidethe cyclotron at a radius that corresponds to the desired beam energy. If the stripperis positioned incorrectly, the beam energy will not he what is expected. Studies haveindicated an inconsistency of up to 1.0 in. in positioning between the stripper locationand flags used to intercept the beam circulating in the machine[50]. These studies52CHAPTER 3. CALIBRATION 53TB(MeV) Tcentre(MeV)300 298.310353 351.500403 401.598440 438.656Table 3.1: Proton Beam energy extracted from Cyclotron, TB, and at the centre of thetarget cell, Tcentre.do not indicate whether the stripper or the flags (or both) are at fault. If the erroris due entirely to the stripper, a 1.0 in. position error can lead to a 4.7 MeV shift inproton beam energy at 440 MeV. While this experiment was not intended to accuratelymeasure the beam energy, it did have data used to calibrate the QQD momentum whichmay indicate a 1.4 MeV inconsistency between the 300 MeV and the 403 MeV beamenergies (this is discussed in section 3.2.5). Based on these data, an uncertainty of 1.4MeV is assigned to the beam energy.In addition to the systematic uncertainty, there was a spread in beam energies dueto the width of the stripping foil used in the beam extraction. For lB this was typically0.15% in momentum. Shown in table 3.1 are the beam energies at the target centre forthe nominal beam energies used during this experiment. The corrected energy at thetarget centre was due to the energy loss in the stainless steel window upstream of thetarget, the target cell wall and the LD2.3.1.2 Beamspot oi TargetThe beam profile was determined using two profile monitors, number 6 located 213 cmupstream of the target and number 7 located 457 cm downstream. The beam positionat the target was measured with a retractable scintillator screen located 6 cm upstreamof the target centre. During normal data taking, both the monitors and the scintillatorscreen were retracted from the beam. Part way through the experiment, monitor 7CHAPTER 3. CALIBRATION 54failed but the target position and beam profile could still he determined from monitor6 and the scintillator screen. The consistency between the alignment of the monitorsand the scintillator screen was better than 0.16 cm in both x and y. The angle thebeam makes with respect to the axis of the beam pipe was less than 0.17°. The angulardivergence of the beam was less than 0.08°. The full width of the target spot wasapproximately 0.8 cm for the 403 MeV beam tune, but less than 0.4 cm for the 353and 440 MeV tunes.In general, the beam did not strike the target along the z axis in the lab frame(as defined in section 2.7). The actual location of the beam on the target depended onthe beam tune and could be as much as 0.4 cm offset from XL = 0 = YL. The positionof the beam spot for each beam tune was specified by XB and YB, the offset in the labframe (see figure 2.7).3.1.3 Beam CurrentThe principle method for measuring beam current was a Secondary Emission Monitor (SEM) located 5 m downstream of the target (see Appendix C). After the datataking, the SEM was calibrated against a Faraday Cup. Unfortunately, during theSEM calibration, there were no scans to measure the SEM response as a function ofbeam position. The importance of such a scan was discovered several months after theexperiment by another group[51j. Their scan revealed a 4% drop in the SEM responseat the nominal beam position. This was probably due to poor vacuum in the SEMallowing the build up of a surface deposit on the foils. The deposit in turn changes theemission properties of the foil and thus the SEM calibration. As discussed in the previous section, the profile monitor located just before the SEM failed part way throughthe experiment and the beam position with respect to the region of poor SEM responsecould not be determined for much of the experiment. Thus it is necessary to quote aCHAPTER 3. CALIBRATION 554% systematic error in beam current.The In Beam Polarimeter (IBP) could also be used to measure the beam current.However it was less desirable than the SEM for two reasons. First it was located somedistance upstream of the target past a bending magnet. A current measured at thepolarimeter was not necessarily the current at the target; beam could spill out of thebeam line between the polarimeter and the target. Secondly, the polarimeter’s currentcalibration depended on the areal density of hydrogen in the polarimeter target (aCH2 foil) which could vary with radiation damage or warping due to beam heating.In fact the polarimeter estimated the beam current to be 30% higher than the SEMthroughout the experiment. Considering the above uncertainties, only the SEM beamcurrent measurement was used for the data analysis.3.1.4 Beam PolarizationThe lB proton beam polarization was continuously monitored during the experimentusing the In Beam Polarimeter (IBP). As discussed in appendix D, the IBP determinesthe beam polarization by measuring the asymmetry of proton proton elastic scattering(for which the analyzing power is known) using a CH2 target. The beam polarizationwas typically 70% and was measured to a statistical accuracy of typically 0.3%. Thesystematic error in beam polarization is estimated at 1% due to the uncertainty in theproton-proton elastic scattering analyzing power.3.2 Spectrometer CalibrationThe spectrometer is used in this experiment to determine the momentum and trajectoryof the pion. The reconstructed pion trajectory is extrapolated back into the target cellto determine the event vertex. To determine cross sections, it is necessary to know theacceptance of the spectrometer. Calibration of the spectrometer consists of:CHAPTER 3. CALIBRATION 56• Determination of the positions of particles in each wire chamber.• Measurement of the quadrupole optics parameters to allow reconstruction of theinitial trajectory of a particle entering the spectrometer.• Measurement of the location and orientation of the spectrometer with respect tothe target cell.• Measurement of the transport parameters of the full spectrometer which are thenused to determine the momentum of the pion.• Measurement of the spectrometer acceptance.3.2.1 Wire Chamber CalibrationThe wire chambers need to be calibrated for them to be used to reconstruct the positionsof pions passing through them. When calibrated, the chambers report the position of aparticle in a coordinate system whose origin lies on the optical axis of the spectrometerwere it passes through the chamber. The two coordinates are x, which lies in the bendplane of the dipole magnet, and y, which lies in the nonbend plane (see figure 2.7). Allfour chambers are of the proportional delay line type with two orthogonal wire planes.Each wire plane requires a scale and an offset calibration making a total of sixteencalibration constants for the four chambers.As described in section 2.5, a particle passing though a chamber knocks freeelectrons which induces a voltage on the nearest wire in a wire plane. The voltagepulse travels to the end of the wire which is attached a delay line which runs the lengthof the wire plane. The voltage pulse splits and travels to each end of the delay linewhere the signals are amplified, discriminated and sent to TDC’s (see figure 3.1). Theends of the delay line are label positive and negative, corresponding to the posi.tive andnegative directions of the wire chamber coordinate system. The TDC at the positiveCHAPTER 3. CALIBRATION 57TDCM TDCPFigure 3.1: Wire chamber position reconstruction. The delay line has length, D, thestruck wire is a distance, d, from the negative end of the delay line. For a propagationvelocity in the delay line of v, the travel time of the signal from the wire to the negativeend of the delay line is d/v and to the positive end, (D— d)/v. From the negative endof the delay line there is a further delay of t before the negative TDC fires. From thepositive there is a delay of t2 before the positive TDC fires.struckwiredti t2end of the delay line registers a time referred to as TDCP (TDC Positive) and theTDC at the negative end of the delay line registers a time referred to as TDCM (TDCMinus). If the delay line has a length D and the wire is attached at a distance d fromfrom the negative end, then the pulse travelling to the negative end of the delay linewill take a time d/v (where v is the speed of propagation in the delay line) to reachthe end. The pulse travelling to the positive end of the delay line will take a time(D — d)/v. There will be a constant time delay, ti, between the signal reaching thenegative end of the delay line and the TDCM firing. Similarly there is a time delay, t2,before TDCP fires. Taking the difference of TDCM-TDCP then gives,d ID—d \TDCM - TDCP =- + ti - +t2),V JCHAPTER 3. CALIBRATION 582d D= —+tl—t—,V V2d= —+C,Vwhere C = 11 — t2 — D/v is a constant. Defining the scale, SCL = v/2, and offset,OFF = —C/SCL, and solving for the position of the wire, d, one gets the relationship,d = SCL x (TDCM - TDCP) + OFF. (3.1)Equation 3.1 assumes that TDCP and TDCM measure time in the same units.The calibrations of all the TDC’s in the experiment were checked with a quartz crystalpulser which found deviations of up to 2% from the expected calibrations, so in the offline analysis, the TDC signals were corrected to read 250 Ps per channel before beingused to calculate the wire chamber positions.The scales for the y planes of chambers 2 and 3 were determined from the histograms of the y distributions. As discussed in section 2.5, the y signals come fromdiscrete locations and a histogram of y forms a series of discrete peaks, each at thelocation of an anode wire (the so called “picket fence”). The y SCL was adjustedso that the peaks had the known anode wire separation (2mm). This procedure doesnot work for the x planes of chambers 2 and 3 since the signal source is continuouslyvariable. Instead, because the x and y wire planes and delay lines in each chamberare of identical fabrication, it is assumed that the propagation velocity (and thus theSCL) is the same for both x and y.WC4 and WC5 were of different construction from chambers 2 and 3 and thetiming resolution for y was not sufficient to observe the picket fence and determinethe delay line velocity. Instead, the SCL’s were determined for these chambers from adirect measurement of the delay line velocity [47].CHAPTER 3. CALIBRATION 59The exact location of a particle passing through WC2 and WC3 is needed toreconstruct the initial trajectory of the particle. The calibration offsets, OFF, forwire chambers 2 and 3 were chosen to make the zero position in each chamber lie onthe optical axis a line that passes through the centre of the quadrupole magnet(see figure 2.7). The offsets for chamber 2 were determined by placing a collimator(consisting of a 0.75” thick brass plate with a 0.25” hole centred on the optic axis) infront of the QQD aperture. Histograms of x2 and Y2 showed peaks at the optical axisand the offsets were adjusted accordingly.For chamber 3, the collimator could not be used since it was located too far fromthe chamber. Instead, data were taken with only WC3 in the trigger. A scatterplot ofx3 vs y3 showed a circular region of events that fell inside the 12.7 cm diameter beampipe. It was assumed that the beam pipe passed through the centre of the quadrupolemagnet and a 12.7 cm circle was superimposed on the scatterplot and the wire chamberoffsets adjusted to fit this circle over the data.The resulting error for the location of an event in WC2 and WC3 is a randomerror due to the wire plane resolution of 1mm (one half of the anode wire spacing of 2mm)1 Systematic errors, due to the determination of the offsets of the chambers, areestimated as 1 mm for WC2 and 2 mm for WC3 (for which the offset was more difficultto determine).The offsets for chambers 4 and 5 should also, in principle, he defined with respectto the optical axis. However, this was not critical for the momentum calibration whichincluded a fitted offset to correct for any error in the wire chamber calibration (seesection 3.2.5). A rough offset for the x plane of each chamber was determined byplotting x vs the momentum of pions traversing the chamber. Particles travelling1Actually this is an over estimate of the x random error. Because the x signal source is continuousit is possible to achieve better than one half a wire spacing resolution. However, since the x resolutionwas not explicitly measured, the error is overestimated at 1 mm.CHAPTER 3. CALIBRATION 60at the central momentum of the spectrometer should intersect the wire chambers atx = 0. By fitting polynomials to the momentum vs x distributions, the zero positionsand offsets were determined. Because the y coordinates in WC4 and WC5 are relativelyunimportant, the y offsets for chambers 4 and 5 were not determined and set to zero.The random positional errors in WC4 and WC5 are estimated at 2 mm.3.2.2 Quadrupole Calibration and WC Calibration CheckTo calculate the momentum of a pion detected in the QQD and the vertex of an eventin the target, it is necessary to reconstruct the pion’s initial trajectory. The initialtrajectory of the pion is reconstructed at the target plane (the xo— Yo plane shownin figure 2.7) using the measured positions of the particle in WC2 (located beforethe quadrupole) and WC3 (located after the quadrupole) and using knowledge of thequadrupole’s optics which, to first order, are described by two parameters, the effectivelength of the quadrupole magnet, L, and the strength of the quadrupole, k. Becausethe optical parameters of the QQD quadrupole magnet were not known prior to theexperiment, it was necessary to measure them.The trajectory of a particle passing through the spectrometer is written in termsof position and angles the trajectory makes at a specified plane. The trajectory atWC2 is(x,&yq),at WC3 the trajectory is (x3,8yq),and at the target plane,the trajectory is (x0,6o, yo, o) where the angles are defined in figure 2.8. Using a firstorder optics model for the quadrupole (see reference [52]), the transport of the ray fromWC2 to WC3 is represented by a drift from WC2 to the quad, a first order transportmatrix through the quad and a drift to WC3 (see figure 3.2). This combined transportmatrix can be written as,— 1 q cosh’ sinh& 1 p x2 32— 0 1 k sinh b cosh b 0 1 02 ‘CHAPTER 3. CALIBRATIONxcyd1 L61Figure 3.2: The quadrupole magnet is focussing in the y direction and defocusing inthe x direction. At the pivot, the trajectory of the ray is specified by (x0, andby (yo, At WC2 the ray is specified by (x2, 62) and (y2, 2) where 82 = 6 and= qo (since the ray is not deflected between the target and WC2). At WC3 the rayis specified by (x3, 83) and (ye, 53).xzCHAPTER 3. CALIBRATION 62for the x or horizontal plane and,— 1 q COS? sinb 1 p Y2 33— 0 1 —ksinb cos’/ 0 1 c’2for the y or vertical plane, where p is the distance from WC2 to the quad and q thedistance from the quad to WC3. The trajectory is represented by the two componentmatrices [x, ] and [y, q], x and y give the trajectory’s x and y position at a specifiedz, and & and q5 are the trajectory slopes dx/dz and dy/dz respectively. Indices 2 and 3refer to the trajectory slope and position at WC2 and WC3 respectively. The centralmatrix in both equations 3.2 and 3.3 represents the transport of the trajectory throughthe quad. As written, the quad is defocusing in the x plane and focussing in the yplane. For the quad, ‘ = kL, where L is the effective length of the quadrupole and kis strength of the quadrupole. The quadrupole strength is given by the equation,k2 = , (3.4)where q is the charge of the particle, P is the momentum measured in MeV/c, and Gis the field gradient in gauss/cm,G==ay 8xIf the first order optics model is an adequate description of the quadrupole magnet,only two parameters are required to describe the transport matrix: the strength, k andthe effective length L.From figure 3.2 it is seen that the pion trajectory at WC2 can be written in termsCHAPTER 3. CALIBRATION 63of the initial trajectory as,— 1 d1 x0— 0 1 ‘ (3.5)andY2 — 1d Yo 3672 — 0 1where d1 is the distance of WC2 from the target plane and the index “0” refers tothe trajectory parameters at the target plane. Noting from equations 3.5 and 3.6 that= &o and 2 = qo, equations 3.5, 3.6, 3.2 and 3.3 can be combined to solve for thethe initial target trajectory in terms of the particle positions in WC2 and WC3. Thissolution is,= — d180, (3.7)— x3—(cosl&+qksinh)x2 (38)— pcoshb + sinh + q(pksinh + coshb)’Yo Y2 — d1q0, (3.9)and— i13 —(cos’b —qksin’)y21• . .p cos b + sin + q(—pk sin ib + cosBecause the optical properties of the quadrupole were not known, it was necessary todetermine k and L from the experimental data. The data used were the empty targetruns where the particles originate from the two points where the beam passes throughthe aluminum target wall. Each run gave two known positions in x0 and one knownposition in yo. By using empty target runs that were done at different spectrometerangles and beam tunes, a variety of known x0 and Yo positions were available to fit theCHAPTER 3. CALIBRATION 64a) b)zFigure 3.3: Empty Target Traceback with quadrupole magnet on, spectrometer at 900.a) x0, b) y.quadrupole optics parameters. A global fit to all the empty target data gave,k = 0.031 + 0.02cm1,andL = 25.2 + 0.4cm.It should be noted that the first order optics model could not completely fit the data,indicating an inadequacy of the first order model. This is partly because the momentumdependence of k (as seen in equation 3.4) is ignored when reconstructing the trajectory.This is because the momentum is determined after the trajectory is measured.Figure 3.3 shows the reconstructed empty target for the QQD at 90°. The errorin the reconstructed target position is approximately 0.5 cm for both x0 and Yo• Theerrors in 8 and qo are approximately 0.7°. Since x0 is calculated from via equation3.7, the error in x0 is strongly correlated to the error in 8. Similarly, the error in Yois correlated to the error in g5o. Approximately half the random errors were due toCHAPTER 3. CALIBRATION 65multiple scattering of the pion as it traveled from the target to the QQD mostlyin the ST counter just before WC2. The remaining uncertainty was due to the wirechamber resolution.A test of the wire chamber calibration was to see if the separation of reconstructedempty target peaks matched the known target width. For the QQD set at 90°, pionsdetected from the target walls in the empty target runs should have had a measuredseparation of 5.02 cm . However, measurement of the reconstructed target width withthe quadrupole magnet turned on (see figure 3.3a) gave a target width of 5.4 cm. Ameasurement was also made with the quadrupole magnet turned off, thus removingany uncertainty due to the quadrupole calibration, and this also gave a target widthof 5.4 cm. This indicated that the problem may be with the calibration of the wirechambers. The delay line velocities of x2 and x3 could not be measured because of thelack of “picket fences” in their spectra. Since they were of identical design, the x and yplanes of each chamber were assumed to have the same delay line velocities. A possiblereason that the target reconstruction was wide was if the delay line velocity of x2 wasoverestimated by 8%. By reducing the delay line velocity by this amount, the anomalously large target width could be explained. However, since no direct measurement ofthe velocity was made, it was decided not to alter the x2 calibration.3.2.3 Alignment of the Apparatus using Experimental DataWhen the experimental apparatus was set up for this experiment, the QQD was alignedso that its pivot would be at the centre of the target cell and the angle readout on thespectrometer track,6QQD, would indicate the angle the spectrometer’s optical axis madewith respect to the beam pipe. The other detector in this experiment, the scintillatorbar array located downstream of the target, was located with distance measurementswith respect to the beam pipe and the target cell. However, analysis of the experimentalCHAPTER 3. CALIBRATION 66data showed significant misalignments of both detectors. This section discusses howthe experimental data was used to improve the alignment of both the spectrometer andthe scintillator bar array. The alignment of the bar array is discussed in this sectionrather than in the scintillator bar array calibration section (section 3.3) because thearray alignment is closely linked to the spectrometer alignment.Knowing the alignment of the spectrometer is very important when calibratingthe apparatus and the most important parameter for the alignment is the true angle thespectrometer makes with the beam pipe. The spectrometer angle was nominally readoff an indicator on the spectrometer as 0QQD. If the spectrometer readout is misalignedby some angle &off, the reconstructed pion angle, 9-, is given by,= O + 6QQD + off, (3.11)where 6 is the trajectory reconstruction angle as discussed in section 3.2.2. The pionscattering angle is used to determine the energies of the pions used in the QQD momentum calibration. It is also needed in the Tel Aviv Bar TOF calibration, and, ofcourse, it affected the final angular distributions calculated for this experiment. Thusthe measurement of 0off is important.That the spectrometer was misaligned can be seen in figure 3.3a which shows thereconstructed target cell walls when the spectrometer was oriented at 0QQD = 900. Ifthe spectrometer was properly aligned with the target cell, then the cell walls wouldbe located at x0 = +2.51cm. In fact, the cell walls were located at approximately -2and +3 cm, indicating that the optical axis of the spectrometer passed through the celldownstream of the target centre. An indication from a previous experiment thatwas not exactly 0, was given by Rozen[43] who reported &off = —2° (no error bar).This result could only be considered an indication of a problem since the exact valueof &of f can depend on the alignment of the spectrometer when it is set up for eachCHAPTER 3. CALIBRATION 67optical axis/Figure 3.4: Spectrometer Alignment Model. Horizoitally, the pivot of the spectrometer is offset from the centre of the target by (XQQD,ZQQD). The optical axis of thespectrometer points towards the pivot, but has an angular offset of from0QQD, theangle read off the spectrometer track. Vertically, the optical axis is shifted a distanceYQQD from the YL = 0 plane.experiment. The reconstructed vertical position of the beam shown in figure 3.3b wasalso found to be in error, indicating that the optical axis was displaced from the YL = 0plane.To explain the empty target cell data, a model of the spectrometer’s misalignmentwas devised. Specifying coordinates in the lab frame, this model (figure 3.4) assumedthat the spectrometer pivot was located at a position (XQQD, zQQD). The model hadthe spectrometer optical axis passing through the pivot, but with the offset,between the angle indicated in the QQD angle readout, &QQD, and the optical axis.The alignment model also allowed a vertical displacement, YQQD, of the optical axisfrom the YL = 0 plane.The horizontal parameters of the alignment model, XQQD, zQQD, and werefitted to the empty target runs taken during the experiment at various spectrometerangles. The target wall peaks from the x0 plots were compiled for these runs and two0QQD pivot(xQQD,yQQD,zQQD)XLZLtarget cellCHAPTER 3. CALIBRATION 68XQQD 0.46 + 0.03cmYQQD —0.14 + 0.01cmZQQD 0.69 + 0.02cm0off —1.14+1°Table 3.2: QQD alignment model parametersquantities calculated, the width of the projected target cell,X = XR — XL,and the centre of the projected target cell,XR+XLXc = 2where XL and XR are the left and right peaks in the x0 histogram (see figure 3.3a).The (XQQD, zQQD,8°ff) model was fitted to the empty target data using the CERNfitting program, MINUIT[53]. As discussed in section 3.1.2, the different proton beamtunes used in this experiment had the beam striking the target at different locations,given by XB and YB (specified in the lab frame). The data used to fit the horizontalspectrometer alignment had three different XB values, 0.0, 0.2 and 0.4cm. Each beamspot resulted in a different apparent width of the target walls when reconstructed.This difference was accounted for when fitting the alignment model. Figure 3.5 showsa comparison between the empty target data and the fitted model for each of the threebeam spots. Shown are the fits to X (left column) and to X (right column). Themodel parameters are listed in table 3.2. The error of +1° for quoted in table3.2 is the extreme variation of that could he achieved by varying the calibrationparameters of the wire chambers and the quadrupole optics. The model had little orno success at fitting the apparent width of the target cell, X near 6QQD = 90° butCHAPTER 3. CALIBRATION 6920 30 40 50 60 70 80 90 100oQQ(deg)Figure 3.5: Comparison of empty target data (points) and QQD alignment model(lines) for each of the three beam spots. The left column is the apparent width of thetarget, x, and the right column is the target centre, x. The top row shows the datafor XB = 0.0cm, the centre row for XB = 0.2cm and the bottom row for XB = 0.4cm.654032I I I I I00— I I I I I I I —X 0cmI I I I I —20 30 40 50 60 70 80 90 10OQQfl(deg)I I I I I I —=0 2cm210—1210—165403210x =0.2cm20 30 40 50 60 70 80 90oQQ(deg)‘C 0 20 30 40 50 60 70 80 90 1(UQQ(deg)010X B° .4cmCHAPTER 3. CALIBRATION 70it was reasonably successful at fitting the centre of the cell, x. The inability of themodel to fit the width of the target cell was due to the error discussed in section 3.2.2which causes the apparent width of the target cell to be too wide. This causes a 2mmsystematic error to be present in the reconstructed x0.The vertical offset, YQQD, was fitted separately from the horizontal alignmentparameters using empty target data with three different beam positions, YB = 0.0cm,YB = —0.3cm and YB = —0.4cm. The fitted YQQD is listed in table 3.2 and the figure3.6 shows the fits of the vertical offset to the data. It is seen that a simple offset modeldid not explain the Yo data. However, since the experiment was relatively insensitiveto an error in the vertical position of the target vertex, it was not considered worthpursuing. Thus there is a 6 mm systematic error present in i/o.An alternate method of determining the QQD offset angle was to exploit the twobody kinematics of the pd /i+ reaction. Various pd + calibration runs weretaken with the spectrometer set at different angles. By measuring the distribution oftritons detected in the scintillator bar array to the right of the beam, the correspondingpion angle to the left of the beam could be predicted. Measuring the triton angle wascomplicated by the large angular width of each bar (1.9°) and the multiple scatteringof the tritons on the way to the bars (in the target cell and proton barrel exit window).The scattering was sufficient to deflect tritons into bars that should have been kinematically impossible to reach. In the analysis it was assumed that the triton multiplescattering was symmetric and that the distribution in the bars could be approximatedby a gaussian centred at the mean triton scattering angle. Only the three bars centredabout the expected triton position were in the hardware trigger during the calibrationruns. The data were analysed with cuts of +3° on the wire chamber trajectory angles&o and q to ensured that the triton distribution corresponded to pions centred aboutthe optical axis of the spectrometer.CHAPTER 3. CALIBRATION0.80.6-04-0.2-0.0-—0.2-—0.4 -—0.6-—0.820.80.6-0.4-0.2-0.0-—0.2 -—0.4 -—0.6-—0.820.80.6-0.4 -0.2-0.0-—0.2-—0.4 -—0.6 -—0.8 -2071Figure 3.6: Fit of vertical QQD alignment model(line) to data(points). a)yB = 0.0cm,b)yB = —0.4cm, c)YB —0.3cm,I. II I I I I I —) 30 40 50 60 70 80 90 1(b° UQQD1e— I I I I I I —IU • UII— I I I I I I —0 30 40 50 60 70 80 90 10b=°4 OQQ(deg)I I I I I IUU -I I I I I30 40 50 60 70 80 90 100b°3 6QQ(deg)CHAPTER 3. CALIBRATIOI\T 72The data consisted of a series of pd —* runs at 403 MeV beam energy anddiffering QQD angles ranging from 62° to 1100. A gaussian distribution was fitted tothe triton distribution in the bars for each run. The mean triton angle for each runwas determined from the fit and from this angle the corresponding pion angle, , wascalculated. The difference between & and 6QQD is the QQD angle offset. Figure3.7a shows a plot of ir — 8QQD as a function of 6QQD reconstructed from the tritondistributions with the bars assumed to be at their measured location, 13.4 cm from thecentre of the beam pipe. The QQD angle offset shows a strong dependence on &QQD.If the bar position was correct, figure 3.7a should show &off independent of8QQD. Infact, with the exception of two points (shown as open boxes) the data shows a strongangular dependence. This indicates that the bar array position was not the measured13.4 cm from the beam pipe centre. Figure 3.7b shows &off as a function of OQQDassuming that the bars are shifted 1.25 cm further away from the beam pipe. Withthis shift in bar position the data’s angular dependence was removed and a fit to thedata (except for the two open boxed points which are inconsistent) gave an angle offsetof &off = 1.2° ± 0.3°.The QQD angle offset was also checked using the quasifree i5 —* dir+ analyzingpower2. Figure 3.8 shows a parameterization based on the world data set[72] of thefree p —* dir+ analyzing power (solid line) as a function of laboratory pion scatteringangle. The measurement of the quasifree ANO from this experiment is shown as thepoints. The experimental data were fitted to the free ANO with an angular shift. Theresult of the fit was a = —0.6° ± 1.9°. The large error bar associated with thismeasurement lends only qualitative support to the previous measurements.Table 3.3 summarizes the various attempts to determine the QQD angle offset.The triton distribution and ANO data confirm the QQD offset determined from the2the analysis of the j5p —+ dir data is discussed fully in chapter 4.CHAPTER 3. CALIBRATIONa)b)543q-)00—1nominal bar positions73Figure 3.7: &lr 8QQD reconstructed from triton distributions, a) Nominal bar positions.b) Data reconstructed assuming the bars are moved 1.25 cm further away from the beampipe. The two open points were not used to fit the meanshifted bars21— 00’—4CHAPTER 3. CALIBRATION 74Cz0.20.0—0.2—0.4—0.6—0.8Figure 3.8: j5p —* dir ANO(&lab). Solid line free analyzing power. Solid points —experimental quasifree analyzing power from this experiment, shifted to fit the free ANOmethod 6offalignment model —1.14° + 1°triton distributions —1.2° ± 0.3°j5p —> d7r ANO —0.6° ± 1.9°90O1(deg)Table 3.3: Summary of doff measurements.CHAPTER 3. CALIBRATION 75alignment model. While the triton data gives a more accurate value of 8H it wasdecided to use the value from the alignment model. The value from the alignmentmodel had already been incorporated into the analysis and since this value is withinthe error bars of the value from the triton distributions, it did not seem worth the effortto redo the analysis at the triton value.3.2.4 Vertex ReconstructionThe event vertex in the target cell was reconstructed by assuming that the lab positionof the event was at the centre of the beam, XL XB. The trajectory of the pion in theQ QD frame, (x0,6, Yo, o) was projected back into the target until it intersected theXL = XB plane. The misalignment of the QQD was accounted for in the projection.3.2.5 Momentum CalibrationThe purpose of the QQD spectrometer is to determine the momentum of the pion fromthe reaction pd PPPs A magnetic spectrometer does this by passing the particlethrough a dipole magnet where it is deflected through an angle, 6, which depends onthe magnetic field strength, B, and the particle’s momentum, F, and the integratedlength of the magnetic field. The specific equation is8=fBdl, (3.12)where q is the charge of the particle and f Bdl is the integral of the magnetic field ofover the path length of the particle.There are various algorithms for determining the momentum of a particle in theQQD (see [40,41,42,43]). For this experiment, the momentum of the pion is determinedfrom the second order transport equations for the spectrometer which describe thetransport of the particle from the target plane to either WC4 or WC5. The transportCHAPTER 3. CALIBRATION 76equations give the position in WC4 or WC5 as a second order polynomial function ofthe initial trajectory and momentum of the particle. The momentum of the particleis found taking the observed initial trajectory and the position in WC4 or WC5 andsolving for the momentum. Calibration of the spectrometer consists of determiningthe coefficients of the polynomial. This is done using pd _* data for which thepion momentum can be determined by two body kinematics and the measured pionscattering angle (the calibration of which was determined in sections 3.2.2 and 3.2.3).The transport polynomial is fitted to these data and the coefficients extracted.In a spectrometer, the momentum of a particle, F, is usually described in termsof the relaiive rnomenum,6 = x 100%, (3.13)where F0 is called the central momentum of the spectrometer. For a given magneticfield setting of a spectrometer, B, P0 is the momentum that a particle must have totravel along the optical axis of the spectrometer. For the QQD this means a particlethat is deflected through an angle of 700 by the dipole. From [54], when the QQDdipole is set at a field of B = 11274.4G, the central momentum of the spectrometeris P0 = 171.36MeV/c. From these data, the central momentum for any dipole fieldsetting is given by= (171.36MeV/cB. (3.14)\ 11274.4G )It should be noted that the specification of the ratio ofP0/B is not critical to momentumreconstruction. This is because the spectrometer is calibrated against pions of knownmomentum. If P0/B is not correctly specified, the calculated 6 will be effected but notthe reconstructed particle momentum, P.The initial trajectory in the QQD frame, (x0, 00, Yo, is reconstructed from theCHAPTER 3. CALIBRATION 77position measured in wire chambers WC2 and WC3 as discussed in section 3.2.2. Theposition of the ray at one of the wire chambers after the dipole, Xf (which can be eitherx4 or x5) is described to second order by the functionXf =R11x0+R266+Ti11x+T112x06+ T122& +T116x06+T126&06+ T16662 + XFOFF(3.15)where the coefficients and Tjk depend on the optics of the spectrometer. The termXFOFF is an offset to account for the fact that the Xf = 0 may not lie exactly on theoptical axis of the spectrometer (see section 3.2.1).The unknown is written as a quadratic equation consisting of optics coefficientsand the known initial trajectory,A52 + B6 + C = 0, (3.16)where the coefficients, A, B and C areA = T166, (3.17)B =R16+T16x0-]-T28 (3.18)andC R11x0+ R12&0 + T111x + T1226,+T112x0&o + XFOFF — xj. (3.19)CHAPTER 3. CALIBRATION 78This quadratic, solved for 6, gives-B+VB24AC2ANote that there appear to be two solutions. In fact only one of them is valid, (—B —V’B2 — 4AC)/2A, which was found by inspection. The second arises from the truncation (to second order) of the infinite series necessary to describe the optics transferfunction through the spectrometer.The pd —* data used to fit the transfer coefficients in equation 3.15 was referred to as the Fixed B Field Data. These data consist of a series of runs at a beamenergy of 403 MeV spanning the range of relative momenta 6 = —12% to 6 = +20%.The spectrometer was kept at a constant magnetic field setting (keeping Po constant).The momenta of the detected pions were varied by moving the spectrometer to different scattering angles. For each pion detected, the trajectory was reconstructed and themomentum at the target vertex calculated from two body kinematics. Before calculating the relative momentum in the spectrometer, 6, for the event, it was necessary tocorrect for the energy loss of the pion travelling from the target to the spectrometerentrance. This is discussed in section 3.2.6. Once the pion momentum at the entranceto the spectrometer was known, 6 could be calculated using equations 3.14 and 3.13.The coefficients of the second order optics model (equation 3.15) were fitted to allthe events of the Fixed B field data using the MINUIT[53] fitting program. Separatecoefficients were fitted for the x4 and x5 transfer functions. This gave two independentdeterminations of the relative pion momentum, 64 and 65. The optics coefficients aregiven in table 3.4.Figures 3.9a and b show the difference between the reconstructed and the truemomentum for the Fixed B field data for 64 and 65 as functions of the “true” pionCHAPTER 3. CALIBRATION 79coef WC4 WC5XFOFF 0.260095 2.00584Ru 1.07403 8.870123x102R12 0.113963 3.821447x10R16 -0.638168 -0.93088Till -9.759337x102 -0.137074T112 -l.987402x10 -2.719351 x 10—2T122 -8.4856l0x10 -1.158215x 1OT116 4.908166x102 7.422223x10T126 4.208245x103 6.346492x10T166 3.3l4430x10 5.317017x10Table 3.4: Fitted QQD Optics Coefficients. The units are for x in cm, in milliradiansand 6 in percent.relative momentum (as calculated from two body kinematics). Each data point showsthe average difference for a complete data run. The Fixed B field data shows a deviationin the mean difference between the true and reconstructed momenta on the order of0.3%. The vertical “error bars” in the figures are actually the standard deviations ofthe difference distributions for each run, not the error in the mean for each run. Thatis, the bar shows the error in determining the relative momentum for a single event. Itis seen to vary from about 1% at 6 = —10% to about 1.5% for 6 +20% for 64. For65, the error is larger, ranging up to about 2% at 6 = +20%.Figures 3.9c and d show the error in reconstructed momenta for a different set ofpd —* tir runs using the calibration coefficients fitted to the Fixed B field data. Thisdata set is referred to as the Fixed Angle Data and consists of a series of runs taken at abeam energy of 300 MeV with the spectrometer kept at a constant angle. Each run hadthe spectrometer set at a different magnetic field, thus changing the relative momentaof the pions. The Fixed Angle data spans a relative momentum range of 6 = —13%to 6 = +32%. In addition to the data points being scattered over approximately 0.6%and having larger standard deviations for each run than the Fixed B field data, thereCHAPTER 3. CALIBRATION2‘c 0—1—280Figure 3.9: A comparison of the reconstructed pion momentum and the momentumcalculated from two body kinematics for: a) WC4 using the Fixed B field data, b)WC5 using the Fixed B field data, c) WC4 using the Fixed Angle data, d) WC5 usingthe Fixed Angle data. The horizontal axis shows the relative pion momentum, (5,calculated from two body kinematics for the reaction pd —* tir+. The vertical axisshows the difference between the relative momentum calculated from the wire chamber(either 64 or 65) and 6. Each point shown on a graph is the average over a completedata run. The horizontal error bar is the standard deviation for the distribution of 6for each run. Each vertical error bar is the standard deviation for the distribution of64 — 6,, (or 65 — 6,,), moi the error in the mean difference.a)3b)llil.Ili JIlIlLII ii 111•1•l—33—10 6 ib 20 30 4I I I Ic)‘0‘0a):1111 111 1 ‘0‘0—q—20 —10 0 10 20 30 40 —20 —10 0 10 20 30 4J6(z) 6(z)CHAPTER 3. CALIBRATION 81is a 0.4% constant offset for the reconstructed 6. A possible explanation for this offsetis that there is a discrepancy of 1.4 MeV between the 300 beam energy at which theFixed Angle data were taken and the 403 MeV beam energy at which the Fixed B fielddata were taken. A discrepancy in energy of this order of magnitude is consistent withthe accuracy to which the beam energy is known (section 3.1.1).To summarize, the spectrometer pion momentum reconstruction was calibratedusing pions with known momentum from the reaction pd tir. Two different datasets show an inconsistency in momentum calibration on the order of 0.4% which willbe considered as an estimate of the systematic error. The calibration spans the range6 = —13% to 6 = +32%. The random error in the reconstruction of the momentum ofa pion is estimated at 2%. Combining the systematic and random errors linearly, theerror in the reconstructed pion momentum is estimated as 2.4%.3.2.6 QQD Energy Loss CorrectionsThe pion loses energy travelling from the centre of the target to the beginning of theQQD. The QQD momentum is calibrated from the beginning of the QQD (after theST). Thus to measure the energy of the pion at the event vertex in the target, acorrection must be made to the energy calculated from the momentum reconstructedfrom the QQD data. In addition to correcting for the energy loss of the pion, the timeof flight (TOF) of the pion from the target to the ST counter on the QQD has to becalculated. This is because all the scintillator timing in the experiment is started froman event in the ST counter. The flight time of a proton arriving in the scintillator bararray is really the difference between the TOF a proton travelling from the target tothe bars and the TOF of a pion travelling from the target to the ST counter. To getthe true flight time of a proton, it is necessary to add the pion TOF to the proton’smeasured time of flight. For the pion energies used in this experiment, the pion TOFCHAPTER 3. CALIBRATION 82was approximately 2 ns.Energy loss and TOF of the pion are calculated using the Bethe Bloch formula{55]for a set of pions with initial kinetic energies ranging from 20 MeV to 250 MeV. Apolynomial was then fitted to this data to describe the initial kinetic energy of the pionin terms of the pion kinetic energy at the entrance to the spectrometer. Similarly, apolynomial was fitted to the data to describe the TOE of the pion as a function of thekinetic energy at the spectrometer. Determination of the pion’s vertex energy and TOFconsisted of first calculating it’s momentum in the spectrometer using the momentumreconstruction discussed in section 3.2.5 and then applying the polynomials. The useof the fitted polynomials to calculate energy loss and TOF was more efficient in theuse of computer time than using the Bethe Bloch formula on an event by event basis.The model used for the energy loss corrections assumed that the pion originatesat the centre of the target and that the QQD is oriented at 90 deg. The pion travelsthrough 2.54 cm of LD2, the target cell wall, the scattering chamber window, air andthe ST counter. Variations in energy and TOE due to QQD angle are negligible. TheTOE variation is on the order of 0.OOl6ns and the energy loss variation is around0.02MeV. The assumption that the pion originates at the center of the target is alsovalid. Variations in the beam position in the target and variations in the position ofevent origins results in only a small change in TOE and energy loss.3.2.7 QQD Wire Chamber EfficiencyTo calculate cross sections, it is necessary to know the efficiencies of the detectors.While the scintillators could be assumed to have 100% efficiencies (within the accuraciesof this experiment), it was necessary to measure the efficiencies for the QQD wirechambers. Since the efficiency of a wire chamber could vary depending on the flux ofparticles hitting it, it was necessary to measure the efficiency continuously throughoutCHAPTER 3. CALIBRATION 83the experiment.The efficiency of an individual wire chamber was calculated by comparing thefiring of one of the four wire chambers against the other three. An event in which threechambers fire was very likely to be caused by a particle traversing the spectrometerand thus the fourth chamber should also fire. The ratio of the number of events forwhich all four chambers fire to the number of events for which all but the chamber inquestion fire is the efficiency of that chamber. For example the efficiency of WC2, e2,is given by,= jV2345 (3.20)N345where N2345 is the number of events for which all four chambers fired and N345 is thenumber of events for which only chambers 3, 4 and 5 fired. The total wire chamberefficiency, is(3.21)This assumes that there are no correlations between the efficiencies of the individualwire chambers.The data used to calculate chamber efficiencies consisted of events written totape by the data acquisition system. The trigger for these events was a coincidencebetween one or more bar scintillators (depending on the reaction being studied) and thespectrometer scintillators. The data acquisition system utilized an event preselectionwhich did not write events to tape if one of the wire chambers failed to fire (see section2.9). To calculate the wire chamber efficiencies, every second event was written to tapewithout checking the wire chambers. While this procedure did save tape, it meantthat only half the number of events possible were available to calculate wire chamberefficiencies. To reduce the statistical error, it was assumed that the wire chamberefficiencies were independent of beam polarization. Thus the average efficiencies ofCHAPTER 3. CALIBRATION 84the chambers over all spin states were used in the analysis rather than calculating theefficiency separately for each beam polarization. In software, the events used in theefficiency calculation only had cuts to determine if the chambers had fired. A chamberwas said to have fired if all four of the TDC signals (two x, two y) for the chamberwere valid.During the pd —* r ppp8 data taking, the average total wire chamber efficiencywas 59% with a statistical error of 3%. Chambers 4 and 5 were newer and well shieldedfrom the background radiation flux from the target and tended to have efficienciesaround 90%. Chambers 2 and 3 were older and exposed to a higher background fluxand tended to have efficiencies of about 80%. The efficiencies of chambers 2 and 3tended to be higher at back angles where the background flux of particles on thechambers was lower.During the pd data taking phase of the experiment, wire chambers 2 and3 failed. This occurred because they were operated in too high a particle flux caused bythe QQD being oriented at forward angles with large beam intensities. The high fluxon the chambers caused the chamber gas to break down and build up a residue on thewires. The gradual build up of this residue caused the efficiency of the chambers to dropuntil they became inoperable (total chamber efficiency dropped to 16%). Chambers 2and 3 were removed and cleaned. Subsequently, the beam current was reduced whenthe QQD was at forward angles in order to reduce the particle flux in the chambers.Figure 3.10 shows plots of x2 vs Y2 and x3 vs y3 taken from a run just before thewire chambers failed. These plots are generated for events where all four wire chambersfire. The oval curve in each plot shows the extent of the beam pipe between chambers2 and 3. The dense region is the image of the quad’s limiting aperture projected ontoWC2 and WC3. The halo is caused by particles penetrating the beam pipe. The pionflux across the chambers is expected to be approximately uniform. However large areas)-‘iCDCDCD-CDrJD-•CDCDCD-CD—.c-+iCDHi-_.U)CDU)0_.÷O-tC)CD°CDCDCDC-hCU)C)ct-CD‘N)CDCD:DCDC’.CDCDr-j•j.0jO•(ThCDCDCD—i—U)c-.i 0)CDC)CD—I—.I—i-CCD—.0roo--U)0—c--CDOCDCDo0)C)C)U)U)—.I---C)c-t-c-C)oCDCD 0c-t--CCDCD C)C)C)00U)O0CDCDCD)CMCDc-4--t-CDCDU)y3(O.O1cm)000000000,0000.-.-,000000000000000000000000000000000000o000000000000000000000000000000000000000000000000000000000000000..•-00000000000000000000.‘oOOOOOOOOOOOOOoOoOooV00000000000000000000000‘O))OaOQOoOOOOOOOOOQQOQOOOoO°.Q000D00000000000000000000000.o000)0a00000000o000000000000•00000g00000000000000000000000o0000D00000000000000000000000000)0000000000000000000o00000oo•.•000000000000000000000000000000-00000oOOoo0’0O0oOODJ0O0D0•0000000000000000000000000000.0000c0a00000ooo00oo000000000•00000000oo0ooo0o000000000000-oO[0000000oo000000000000000o000000oooco0000000000000000.--0000000ooo000o000000o000000o0000000oocccooa.0)000000•00000000000000000000000o0O0O00uS000o000000000°0flQQflQflflo.0000000000.000000000...0000000000000.000000000000000.0oo000000000000000000.000000000000000W00000000..oc000000000000000000000o..0o0oo000fl00DW00000oo•.oooooanoooj1Juaoo...0000000000000000JI000o.•..o)fl)o.oo.00000.000000...00000.000...00000.-..000000000.-.000000-0000D)o..•0000....0y2(O.Olcm)COC)COC)COC)COC)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)IIIIHI 0C)-CO 0-0 F CO C) 0 0CD0 CO 0-C) F CO C)- C) 0C)C)Co COCHAPTER 3. CALIBRATION 86of the chambers have become dead in this region just prior to complete failure.3.2.8 QQD AcceptanceA magnetic spectrometer is used to measure the momentum of particles that successfully pass through it. Whether or not a particle will pass through (or be accepted by) thespectrometer depends on the particle’s initial trajectory and momentum. As discussedin section 4.1.4, in order to calculate the differential cross sections for pn —* irpp(’S0)from the observed yield in the detectors, it is necessary to determine what fractionof the possible pion trajectories will be accepted by the QQD spectrometer. In thissection a numerical model of the spectrometer acceptance is used to calculate the expected yields for two different reactions: quasifree pp —* d7r+ and pd —* tlr+. Theseexpected yields are compared against experimental data to assess the accuracy of thespectrometer acceptance calibration.Specifying the initial trajectory of a particle entering the spectrometer in theQ QD frame as = (xo, 8, yo, qo, 6) (see section 2.7) an acceptance function, A(i), forthe spectrometer can be defined as,A1 — A” — ‘ 1 if ray passes through spectrometer,rO) x0, o,Yo,Yo,— j 0 otherwise.A straightforward way to calculate A(r) is to numerically model the spectrometer usinga computer. To determine if a particle will successfully traverse the spectrometer, thecomputer raytraces the trajectory through the spectrometer optics and apertures. Suchan approach is taken for the calculation of the QQD’s acceptance function. Details ofthe numerical model (a Monte Carlo simulation) of the QQD are found in appendix F.Since the acceptance function is a five dimensional volume in the coordinate spaceof (x0, Oo, yo, o, 6), it can be difficult to assess the quality of the numerical model ofCHAPTER 3. CALIBRATION 87the spectrometer. A simplification can be made by considering the spectrometer’s solidangle, , which is the portion of a spherical surface around the target plane subtendedby the spectrometer. Solid angles for spectrometers are usually give in millisteradians.Integrating the acceptance over the pion solid angle, , and averaging over the positionin the target plane, x0 and Yo gives the solid angle as a function of 6,Z12QQD(S) = J dxo f dyo f dA(xo, 60, yo, qo, 6) (3.22)The solid angle is expected to be greatest near the central momentum of the spectrometer (6 = 0) and then fall off at large positive or negative 6.The details of the simulation of the QQD are discussed in appendix F. A comparison of the various wire chamber distributions and calculated quantities (such as thetarget trace back) show a general agreement between the Monte Carlo simulation andthe experimental data, but there are small differences. Because of these differences,cuts were made on the data to restrict the acceptance in WC2 and WC3 to the centralregions (1x2 2.5cm, Y21 4cm, x3 4cm, y 4cm) where there was agreement between experiment and simulation. In addition, cuts are placed on 6 and o,restricting them to less than 3°. These cuts help reject decay muons which tend to havelarge reconstructed target angles. Figure 3.11 shows the QQD solid angle, QQD(6),as calculated by the Monte Carlo simulation. The solid line shows the full acceptanceof the QQD with no cuts on WC2, WC3, & or.The dashed line shows the muchreduced solid angle of the QQD when the cuts are made on WC2, WC3, 6 and o.3.2.8.1 Comparison with pd — dirn dataThe accuracy of the Monte Carlo simulation of the spectrometer was checked againstpd —* d7r+ri5 data taken in this experiment. The analysis of the pd —* d7r+n8 dataCHAPTER 3. CALIBRATION 882520Cf)Sc 1050Figure 3.11: QQD Solid Angle calculated by Monte Carlo simulation. Solid line fullacceptance. Dashed line restricted acceptance used for cross section calculations.is based on the assumption of the Spectator Model (see section 4.1.4) in which thecross section for pd —* dir+n8 is a function of the cross section for the free pp —* dir+reaction. This model for the pd —* dir+n3 cross section was used as the event generatorfor a Monte Carlo simulation of the experimental arrangement used to take actualpd —* d7r+n.9 data. The quality of the spectrometer simulation is accessed by comparingthe predicted yields of the Monte Carlo simulation against the experimentally observedyields.Figure 3.12a shows the a comparison between the experimentally observed andMonte Carlo predicted yields in the spectrometer as a function of the relative pion momentum, 8, for a typical pd dir+n8 run. The general shapes of the distributions agreealthough they are peaked at slightly different momenta. A more accurate comparisonof the experimental and Monte Carlo data is shown in figure 3.12b where the experimental yield is divided by the Monte Carlo yield. The experimental data and MonteCarlo are in near agreement in the range -10% to +30%. At high 6 the Monte Carlooverestimated the experimental yield. At low 6 the Monte Carlo underestimated it.—50 —40 —30 —20 —10 0 10 20 30 40 5(ô()CHAPTER 3. CALIBRATION 892.01.81.61.41.21.00.80.60.40.20.0I I I Ia)40003500-3000-2500-2000-1500-1000500 -0--——40b)—30 —20 —10I I0 10 20 30 40 50 6ä4(%)L)Figure 3.12: Comparison between experimental and Monte Carlo QQD yield as a function of relative momentum, 6, for a typical pd dir+n8 data run. a) Solid histogramexperimental data; dashed histogram Monte Carlo yield. b) Ratio of experimentalyield to Monte Carlo yield.1064(%)CHAPTER 3. CALIBRATION 902.0-I I I I I I I1.8 -1.6 -r 1.21,0 -__________________________________________-0.6-0.4- -0.2- -0.0- I I I I I -—40 —30 —20 —10 0 10 20 30 40 50 60Figure 3.13: Ratio of experimental and Monte Carlo pd —* dirn8 yield between6 = —10% and 6 = 20% averaged over all the pd —* dirri3 data runs.These discrepancies were believed due to the imperfect knowledge of the QQD optics.Particles with momenta at the edges of the spectrometer’s momentum acceptance willtravel through the regions where the knowledge of the magnetic fields is poor.While figure 3.12b shows fair agreement between the experimental data and theMonte Carlo simulation in the region of relative momentum —10% < 6 +30%, forthe reasons discussed in the next section, it was decided to restrict the useful data fromthe spectrometer to the region,—10% <6 <20%.Figure 3.13 shows the ratio of experimental yield to Monte Carlo yield averaged overall the pd —* dirn3 runs for the restricted range of 6[—10%, 20%] (different from figure3.12b in which data from only a single run were shown). The ratio of experimentalCHAPTER 3. CALIBRATION 91to Monte Carlo yield is found to be almost flat. On average, the Monte Carlo underestimates the experimental yield by 2%.3.2.8.2 Comparison with pd —* tir dataAn alternative to using the pd — dir+ri3 data to check the Monte Carlo simulation ofthe spectrometer is to use pd data. The pd tir+ data has the advantage of asimpler final state and not requiring the Spectator Model to interpret it. However, theworld data base for pd ‘ is sparse. The experiment by Lobs ei al[56] measuredcross sections at 305 and 400 MeV. These measurements are reasonably close to the298.3 and 401.6 MeV data taken in this experiment for the purposes of calibration, butan overall normalization uncertainty of 20 to 30% was estimated for the Lobs data[57].An additional complication is that the pd ,‘ data were taken just before the failureof WC2 and WC3 discussed in section 3.2.7. These difficulties mean that the pd —*data can only be used for a qualitative check on the acceptance calibration.None of the experimental data in the current work lie exactly at measured pointsfrom [56]. However, there is little energy dependence for the pd —* tir data in theregion of interest and there is little angular dependence at back angles. A comparisonbetween the Lobs data and the runs from this experiment is shown in table 3.5.To compare the Monte Carlo cross section to an experimental yield, Y, the apparent spectrometer solid angle is calculated from the experimental data without correctingfor the loss of pions due to decay in the spectrometer. This solid angle is given by,L x xNx--x’(3.23)T Ewc B A dO* dcwhere LT is the detector live time defined as the ratio of LAMS to master coincidences,LT = LAMS/MC, is the wire chamber efficiency, NB the number of beam protons,CHAPTER 3. CALIBRATION 92Lobs e al{56] This ExperimentTB(MeV) &lab(deg) da/d2*(ib/sr) TB(MeV) &lab(deg) 6(%)305 91.41 0.57+0.03 298.3 91.9 -1.7298.3 91.9 29.9400 106.1 0.71+0.04 401.6 102.5 1.3401.6 102.5 -8.7401.6 107.5 -11.4Table 3.5: pd > data from Lolos[56j used to calibrate QQD acceptance. Theposition of the pion peak at the focal plane is given by 6, the relative momentum inthe spectrometer.number of target deuterons per unit area, the centre of mass differential crosssection of the reaction and the Jacobian to convert from the centre of mass tolab solid angle. This apparent solid angle is then compared against the Monte Carlosimulation of the same experimental arrangement. The Monte Carlo simulation includespion decay so it should calculate an apparent solid angle which is equivalent to the solidangle derived from equation 3.23.The pion decay has to be corrected in the Monte Carlo, rather than applying acorrection to the data, because the decay correction needs to be applied on an eventby event basis. Some fraction of the muons from the decay of pions will traverse thespectrometer and be included in the experimental yield. Thus it is necessary to fullysimulate the decay process in the Monte Carlo simulation to accurately assess thenumber of decay muons accepted. A simplistic estimate of the pion survival fraction,D- can be calculated from the distance the pion travels, 1 and the pion mean life time,D, =where 3 is the relativistic velocity of the pion, y the Lorentz contraction for the pionand c the speed of light. This formula overestimates the loss of pions since it assumesCHAPTER 3. CALIBRATION 93that any pion that decays in the spectrometer is lost from the experimental yield.If the pd — t7r+ cross section is not known, the absolute spectrometer solid anglecan not be calculated from equation 3.23. However the shape of the LQqD(6)functioncan be determined from pd —* data without knowledge of the cross sections. Thiswas done by positioning the spectrometer at a fixed angle (beam energy 300 MeV,&QQD = 78°), meaning that the pions entering the QQD had a constant energy. Themagnetic field of the spectrometer was varied for a series of runs, thus changing therelative momentum of the pions. Dropping the cross section and Jacobian from equation3.23 gives a relative yield, Y’ which is proportional to the solid angle. Thus equation3.23 becomesYl-Y— LT x c x NB XThe ratio of the relative yield to AQMC, the QQD solid angle as measured by theMonte Carlo, should be a constant independent of 6 and is shown in figure 3.14. Thethree open points are data runs taken immediately before the complete failure of thewire chambers as discussed in section 3.2.7 and should be considered unreliable. Itis seen that the ratio is approximately constant in the region 6 = —7% to 6 = 20%.Unlike the pd —* data in the previous section which showed good agreement withthe Monte Carlo up to 6 = 30%, the three right most points in figure 3.14 show anincreasing deviation between the Monte Carlo simulation and the pd —* tlr+ data above6 = 20%. This discrepancy above 6 = 20% is very likely due to the problems with thewire chambers discussed in section 3.2.7. However, it was decided to be conservativeand only use data from the spectrometer in the range 6 = —10% to 6 = 20%.The absolute solid angle for the spectrometer can be calculated from the five datapoints shown in table 3.5. The ratios of /2E/L\MC for these runs are shown in figure3.15. The 403 MeV runs show a variation in the solid angle normalization ranging fromCHAPTER 3. CALIBRATION0.0—501.2 -1.0Cc0.4 -0.2-940.0-—50 —40 —30 —20 —10 0 10 20 30 40 50Figure 3.15: Ratio of zQ calculated from the pd —* tir data and from the Monte CarloSimulation for 300 MeV beam energy (open points) and 403 MeV (solid points).I I I I I I1.01080.6-4(tCiCT—40 —30 —20 —10 0 10 20 30 40 50Figure 3.14: Ratio of the relative pd + data yield, Y’, to Monte Carlo calculationof solid angle, Ac2MC (arbitrary units). The line is drawn to guide the eye. The threeopen points were runs taken just before WC2 and WC3 failed.1.4— I I ID 300 MeV• 403 MeVCHAPTER 3. CALIBRATION 950.83 to 1.15 compared to the Monte Carlo calculation in the range of 6 = —12% to6 = 0%. The 300 MeV data show an experimental solid angle 0.7 of the Monte Carloat 0% and 0.35 at 6 = 30%. The latter point is consistent with the data in figure 3.14which also predicted a low ratio at high momentum.Considering the uncertainty in the normalization of the pd —* tlr+ cross sectionsand the problems with the wire chambers during these runs, it was decided that thepd —* dirn data were a more reliable indication of the quality of the Monte Carlosimulation of the spectrometer. However, to be conservative, the allowed momentumacceptance of the spectrometer was restricted to the range 6 = [—10%, 20%] in case thelow discrepancy between the Monte Carlo and the pd —* tlr+ data in the range 6 = 20%to 30% was real.3.3 Scintillator Bar Array CalibrationThe scintillator bar array was used in this experiment to detect the particle or particlesassociated with the production of pions from the proton beam interaction with thedeuteron target. Specifically, two protons were detected (each in a separate bar) forthe reaction pd —* IrPPPS, deuterons were detected for the reaction pd — dir+n8 andtritons for the reaction pd —* tir. The horizontal position of a particle detected in thebar array was determined by which bar fired. The vertical position was determined bythe difference in propagation delay of the light signal generated in the bar travelling tothe photomultiplier tubes at each end. The position of the event in the bar array wasused with the event vertex reconstructed from the pion trajectory (as reconstructedby the spectrometer) to determine the trajectory of the particle. The energy of theparticle was determined by the time of flight of the particle from the target to the bararray.This part of the thesis discusses the calibration process for the scintillator barCHAPTER 3. CALIBRATION 96array. The following sections deal with: calibration of the bar time to digital converters; the horizontal and vertical position calibration of the bar array; the bar time offlight calibration; energy deposit and analog to digital converter calibration and finallyreconstruction of the particle energy from the measured time of flight.3.3.1 Bar TDC calibrationAs discussed in section 3.2.1 for the QQD wire chamber TDC’s (time to digital converters), the TDC’s on the scintillator bars had to be calibrated against a quartz crystalpulser. This was done and the the individual TDC signals were scaled in software toread 50 Ps per channel.3.3.2 Horizontal Bar Position CalibrationIn the data analysis, the horizontal position of a particle striking the bars could only beresolved to the width of one bar (12.5cm). Hence in the reconstruction of momentumvectors of particles striking the bars, the particle’s horizontal position was placed at thecentre of the bar. This coarse granularity did not greatly degrade the resolution of theexperiment. In fact initial Monte Carlo studies showed that no significant improvementwould be gained by better positional measurements with wire chambers.While the scintillator bar array was positioned at several locations in the experimental area during the initial calibration phase of the experiment, most of the theexperimental data were taken at one location. It was necessary to know this scintillatorarray location for the proper reconstruction of the particle trajectories. To this end aninitial measurement of the array position with respect to the beam pipe and target wasdone during the experiment. The bars were located approximately 4 m downstream ofthe target cell with BAR1 a distance of 13.4 cm from the centre of the beam pipe. Theaccuracy of the position measurements was estimated as 1 cm. While a 1 cm accuracywas adequate for the position of the bars along the beam direction (ZL coordinate in theCHAPTER 3. CALIBRATION 97a) nominal bar position b) shifted bar position500 I I I I 5rj I I I400 400cr300 cr3000 0a a0200100 1000 I I I 0 i I I I—1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9bar barFigure 3.16: Triton distributions in bars for a typical pd data run. a) The solidhistogram is the experimentally observed triton distribution. The dashed histogramis the Monte Carlo prediction of the triton distribution assuming the bars are at theposition measured during the experiment. BAR1 is closest to the beam pipe, BAR8furthest away. b) The solid curve is the same experimental distribution as in (a), butthe dashed curve is the Monte Carlo predicted triton distribution with the bars moved1.25 cm further away from the beam pipe.laboratory reference frame), it was desired to improve the measurement perpendicularto the beam pipe (XL coordinate in lab frame).An improved measurement of the bar array distance from the beam pipe wasachieved using the pd —* data. Even with the coarse granularity discussed above,the horizontal position of the bars could be checked quite accurately using experimentaldata. This procedure was discussed somewhat at length in section 3.2.3. It was shownthat by studying the horizontal distribution of tritons in the bars from pd —*the two body angular correlation could be used to fix the position of the bars. Theanalysis in section 3.2.3 found that to explain the pd —* tir data the bars had to belocated 1.25 + 0.25 cm further from the beam pipe than was found from the physicalmeasurement. Figure 3.16a shows the experimentally observed distribution of tritons inthe bars for a typical pd _* run compared to a distribution predicted from a MonteCarlo simulation of pd —* tlr+. The experimental triton distribution shows an almostequal number of events in BAR6 and BAR7 while the Monte Carlo simulation predictsCHAPTER 3. CALIBRATION 98distribution is more heavily weighted to BAR7. When the bars in the simulation areshifted 1.25 cm further away from the beam pipe (figure 3.16b), the Monte Carlopredictions are seen to be in much better agreement with the experimental data. Thus,in the data analysis, the bars were assumed to be at the position determined by thisstudy (14.65 cm from the beam pipe).3.3.3 Vertical Bar Position CalibrationIn contrast to the horizontal position, a particle’s vertical position in the bars could bedetermined continuously by the difference in TOF (Time Of Flight) of the light signalto each end of the bar. This is the same principle used to determine the position in adelay line proportional wire chamber described in section 3.2.1. The formula for thevertical position in a scintillator bar isy = SY x (td — t) + OY, (3.24)where td and t, are the (calibrated) TDC signals from the bottom and top of the bar.SY is a scale factor to convert from time difference to position and OY is an offset toset y = 0 at the centre of the bar. Measuring SY and OY is the vertical bar calibration.To determine SY, a 106Ru source was placed on BAR2 at various locations andthe ratio of difference in positions, Ay, to difference in time difference, LStd — t) wasused to determine SY,SY Ly— A(td — t)which was found to be 6.9 + 0.lcm/n.s. This value was confirmed by measurements ofthe distributions of events in the bars when they are fully illuminated. The height ofthe bars determined by the spread of event position in the analysis equaled the physicalheight of the bars using the SY = 6.9cm/ns calibration. From this full illumination,CHAPTER 3. CALIBRATION 99c5/2This Work 6.9 + 0.1E561 [58] 8.34MDAlister [59] 7.54Table 3.6: Bar Vertical Position Calibration estimated from the speed of light andexperimentally measured by various groups. c5 is the speed of light in the scintillatormaterial. SY should be 1/2 of this value.the offset for each bar could be determined to an accuracy of +2cm. The error inthe reconstructed position of an individual event was estimated from the width of thereconstructed peaks using the ‘°6Rti source (which was essentially a point source) andwas found to be +6cm.The value of SY found in this calibration was unexpected. SY is expected to beequal to half the effective speed of light in the scintillator, Cs. The bars are made ofNE11O plastic scintillator which has an index of refraction of 1.58 giving an effectivespeed of light in the bar ofc 2.9979 x 10’°cm/s= = = 18.974cm/us.n 1.58SY would then be expected to be 9.49 cm/ns. Other experimenters using the samescintillator bars found values of SY lower than c5/2, but still larger than the valuefound in this work. A summary of the other SY measurements are given in table 3.6The differences in SY are unexplained and the value measured during this experimentis used for the current analysis.Measurement Reference SY(cm/ns)9.49CHAPTER 3. CALIBRATION 1003.3.4 Bar TOFThe Time Of Flight (TOF) of a particle travelling from the target to the bars wasdetermined from the calibrated TDC signals from the top of the bar, t, and the bottomof the bar, td by,TOF= (t + td)+ OTOFF + t (3.25)where OTOFF is a calibration offset. Because the timing was started by the pionstriking the ST counter, the flight time of the pion to the ST (tv,.) must be added to thetotal time of flight (this is between 1.5 and 2 ns). Flight times for protons, deuterons,and tritons travelling to the bars ranged from 17 ns to over 60 ns. t7, was calculatedby the QQD routines after the momentum of the pion was determined. The analysisroutines used a polynomial to calculate the TOF of the pion (assumed to start at thecentre of the target) using the pion energy calculated from the QQD data (see section3.2.6).A typical TOF spectra of BAR2 (near the beam pipe) for a pd PPPs run atTB = 403MeV (figure 3.17) shows three distinct features — a narrow peak at approximately 19 ns, a broad peak near 35 ns and another narrow peak at approximately 62ns.The broad peak at 35ns is due to the protons from quasifree pn —÷ irpp, and its widthreflects the variation in flight times that these protons can have. The l9ns positionof the first narrow peak corresponds to protons with kinetic energies of approximately400 MeV indicating that these are beam protons. These beam protons are elasticallyscattered from the target via pd —f pd or they undergo scattering off the proton orneutron in the deuteron or they can scatter off the beam pipe or scattering chamber.Most of these “elastic” protons do not stop in the scintillator bars3 and fire the vetocounters which prevents them from being used to form a trigger for a valid quasifree3Some elastic protons do undergo nuclear interactions and are effectively stopped in a bar.CHAPTER 3. CALIBRATION 10112010080U)40200Figure 3.17: TOF spectrum for a pd —* irppp3 run showing protons from quasifreepri —* irpp, elastic beam protons and accidental beam protons.pn — irpp event (the trigger is two protons stopping in two separate bars). However,if a random beam proton happens to fire a bar when a valid quasifree pri —* irpp eventoccurs, the beam proton fired bar will be read into the computer along with the restof the event. The number of elastic protons detected depends on the singles rate forthe bar. The bars closest to the beam pipe had singles rates on the order of 5 MHz.The duty cycle of the cyclotron is 23 MHz meaning that the beam arrives in bunches(called “beam buckets”) every 43.4 ns. The probability that a bar is firing in a givenbeam bucket is,5MHzO223MHzThis implies that the elastic peak will be about 1/5 the size of the pri —* irpp peak infigure 3.17 which is what is observed. It should be noted that for bars further from thebeam pipe, the singles rates decrease and the elastic peaks are smaller.The narrow peak at 63 ns is caused by the same processes that produce theelastic peak. A quasifree pn —* rpp event from a given beam bucket triggers the data40 60TOF(ns)CHAPTER 3. CALIBRATION 102acquisition system and the computer starts to read the event. Meanwhile another beambucket arrives and elastic protons from this bucket can hit the bars, generating a signal43.4 ns after the elastic protons. Such an event is called an “accidental” coincidenceand the resulting peak in the TOP spectra is called the accidental peak. Since theaccidental and elastic peaks should be separated by the 43.4 ns spacing of the beambucket, they can be used to check the calibration of the TDC scales. A measurementof the spacing for several runs gives 43.3+0.1 ns which is in good agreement with theexpected result.In equation 3.25, OTOFF is a constant of the electronics. It was measured bymoving the ST counter with its cables from the QQD to directly in front of the bars(two at a time) and triggering on elastically scattered protons passing through boththe ST and a bar. This procedure gives a precise measurement of OTOFF and alsogives the timing resolution of the bars (typically about 0.4 or 0.5ns) from the width ofthe resulting TOF peak.OTOFF measured by the above procedure was checked using the TOF of tritonsgenerated by the pd — tlr+ reaction. The energy of each good triton event was determined by measuring the angle of the associated pion and using two body kinematics.Using the known energy of the triton, the TOP of the triton from the target vertex tothe position where it strikes a scintillator bar can be calculated, including the effectsof energy loss in the materials between the target and the bars. This calculated TOFcan be compared to the actual TOF measured from the bars. If the calibrations arecorrect, the data should lie on a line of unit slope passing through the origin. However,it was found that an offset, a0 had to be added to the measured TOF, tmeas, for it toagree with the TOP calculated from the kinematics, tkin,kin trneas + a0.CHAPTER 3. CALIBRATION 103Depending on the bar, the size of the offset was up to 1 ns. Figure 3.18 shows plots oftkjn VS tmeas + a0 for all eight bars. Each point in figure 3.18 is the averaged over allthe events for a data run.While it is not clear why there is a disagreement between the two methods forcalculating the bar TOF offsets, there is a test to see which is more consistent with theother calibrations of the apparatus. TDIF is defined as the initial energy of the systemminus the final energy. For pd ,‘ this is,TDIF = EB + ?nd — (E + E’), (3.26)where EB is the total beam energy and md is the mass of the deuteron. E is the energyof the triton calculated from the bar TOF and E7, is the energy of the pion calculatedfrom the spectrometer reconstruction of the pion momentum. If the calibrations forthe beam energy, pion momentum reconstruction and triton energy measurement arecorrect, TDIF should equal zero. In fact what one gets is a peak with a width whichis a measure of the random errors in the energy reconstruction and whose position isa measure of systematic errors in the energy reconstruction. For a series of runs at aconstant beam energy and various QQD angles, the TDIF peak was found to vary from-0.11 MeV to -1.44 MeV using the TOF offsets determined from triton data comparedto a value of about -3 MeV when using the TOF offsets determined from positioningthe ST counter against the bars.The TDIF spectra could also be checked for the pd—irppp3 and pd —* dirn3data. Table 3.7 shows position and width of the TDIF spectra for runs from eachdata type using the two different calibration methods. Neither the ST nor the tritonoffset calibration give a TDIF centred at zero. However, the triton calibration producesTDIF’s that are closer to zero and have a narrower width. Because of the better TDIFfor all three reactions, it was decided to use the triton calibration for the TOP offsets.CHAPTER 3. CALIBRATION 104Figure 3.18: Scintillator bar TOF calibration using pd —* tir. The TOF calculatedfrom two body kinematics, tkjn is plotted against the measured TOF, tmeas plus anoffset, a0. The line has unit slope passing through the origin.bar 1I Ibar 255(I)—‘4540bar 3/35 40 45 50 55 6t+a(ns)bar 4I I Ibar 5 bar 65550+‘ 454035.36055.50a.‘-‘ 45.4075 5 60t+a0(ns))t+a0(ns)bar 77/60bar 86035y_35 4b 45 5b 55t+a0(ns)45 50 55t+a0(ns)60CHAPTER 3. CALIBRATION 105offset pd—÷irppp pd—*dirn3TDIF(MeV) FWHM(MeV) TDIF(MeV) FWHM(MeV)ST -3.7 13.995 -5.01 13.839triton 2.61 12.672 2.94 12.757Table 3.7: TDIF calculated using the ST and triton offsets.However, it should be noted that using tritons to calibrate the scintillator bars tiesthe bar calibration to the QQD angle calibration, because the pion angle reconstructedfrom the QQD data is used to determine the energy and TOE of the triton.3.3.5 ADC CalibrationThe intent of the ADC calibration was to use the TOE of a particle to the bars andthe energy deposited in the bars as a means of identifying the particles. The ADCsignal was assumed to be linearly related to the number of photons reaching the photomultiplier tube. The number of photons propagating through the bar from wherethe particle strikes is assumed to decay exponentially. Thus the ADC signal from theupper photomultiplier is given by,= GuLe + b (3.27)where L is the number of photons generated in the bar by the energy deposited, cv theattenuation constant of the signal in the bar, x the distance from top of the bar, and 1the length of the bar. The offset b is the zero offset of the ADC. Similarly for down,Ad = GdLe(L7) + bd (3.28)While in plastic scintillators, the number of photons generated as a function of energy deposited, Ed, is a nonlinear function, for sufficiently large energies, it can beCHAPTER 3. CALIBRATION 106approximated by a linear function[55],L alEd + a0. (3.29)The slope, a1, is independent of the particle type depositing the energy but the offseta0 varies for different particles types. Combining equations 3.27 and 3.28 together,substituting in equation 3.29, and solving for Ed gives,Ed =— b)(Ad— bd)eo —. (3.30)a1 GGd a1By combining constants, equation 3.30 can be simplified toEd = ciJ(A — b)(Ad — bd) — co, (3.31)which gives a four parameter model that is basically the geometric mean of the twoADC signals. The procedure for fitting the model was similar to that used to fit theTOF offsets. Using pd _* data, the reconstructed pion angle was used to determinethe energy of the associated triton. The energy loss of the triton from the event vertexthrough to the bar was calculated and the remaining energy gave the energy depositin the bar. This was done for each event. Equation 3.31 was then fitted to all the datausing the program MINUIT.Two difficulties arose. First, MINUIT was unable to give reasonable fits usingequation 3.31. This is most likely due to parameters b, bd and c0 being basically offsetsand competing with each other. When the parameter c0 is fixed at zero, the model forphoton production (i.e. equation 3.29) reduces toL = alEdCHAPTER 3. CALIBRATION 107and MINUIT is able to fit the data. The second difficulty arose trying to use theparameters from the triton data to reproduce the proton and deuterium data. Themodel did not work, indicating that even the three parameter model (co fixed) has toomany degrees of freedom. The triton data for Ed were too restricted in their range toget a good fit to the proton and deuteron data which had much lower energy deposits.To improve the fit, elastic proton data were included in fitting process. The energy ofthe elastics were calculated assuming the kinematics of elastic pd scattering from thetarget. Because these elastic protons had a much smaller energy deposit, the model wasforced to a more general solution. Figure 3.19 shows the plots of the energy depositin the bars calculated from the reaction kinematics, Ed(kirl), vs the energy depositmeasured from the ADC’s and fitted model, Ed(meas).A plot of the energy deposit in a bar vs the TOF to the bar can he used toidentify different particle types. Each species of particle will lie in a distinct bandin such a plot. Figure 3.20 shows (so called) Particle Identification (PID) plots for apd•‘ “PPPs run. The chevron shape of the PID band can be explained as follows.Events on the large TOF side of the chevron are particles stopped in the bar. As theenergy of the particle increases, the TOF decreases and the Ed increases. At the tip ofthe chevron, the particle has enough energy to traverse the bar. As the energy of theparticle increases beyond this point, the energy deposit becomes,dEdxwhere dE/dx is the energy deposited per unit length of scintillator, and t is the thicknessof the scintillator. With increasing energy, dE/dx decreases, and since t is constant,the amount of energy deposited in the scintillator decreases. Hence, at this point, asthe energy continues to increase, both the TOF and the Ed decrease.While the PID plots in figure 3.20 show a reasonable calibration, it was found—440V.SiooVtcal 60—140V.6 100V-dlal 60-CHAPTER 3. CALIBRATION 108bar 1 bar 2 bar 3bar 4180bar 5180bar 6140 -20100 -/60 -180bar? barB2020 60 100 140 180Edep meas (MeV)20/_I20 60 100 140 180Edap meas (14eV)Figure 3.19: Scintillator bar ADC calibration using pd —, tir. The x axis in eachplot is the Ed calculated from the ADC data using the model. The y axis is the Edreconstructed from the pd tir+ kinematics. Each point is the mean for a data run.CHAPTER 3. CALIBRATION 109Figure 3.20: Particle Identification (PID) plots for run 169. A plot of energy depositedin the bar vs TOF to the bar gives a characteristic band for each particle species. Thescatter plot shows a band of protons from pd— ITPPPS. The spots at approximately20 and 60 ns are elastic and accidental protons respectively. The three curves shownare theoretical PID curves for protons (p), deuterons (d), and tritons (t).ban bar2 bar3( I I I I I- 200- 180- 160- 140- 120100--e 80- 60- 4020- 03bar 4CI I I I I10 20 30 40 50 60 70 8TOF(ns)bar 5 bar 6200180160140120100806040200200180160140120b00806040200200180160140120 -100 -80-60 -40 -20 -0-bar7 bar8200• 18016014012010080• 6040200 ib 2b 3b 40 50 6b 70 80 0 10 20 30 40 5b 60 70 80TOF(ns) TOF(ns)CHAPTER 3. CALIBRATION 1102000- I I I I I -1500- -1000500- -U- I I I I0 200 400 600 600 1000120014001600TOF(0.05 ns)Figure 3.21: Tritons “falling out” of their PID band.that as the experiment progressed, the calibration of the AD C’s on the bars degraded.This change in calibration was greatest for the bars closest to the beam pipe. By theend of the experiment, some of the calculated energy deposits were in error by a factorof two.Another problem experienced with the ADC’s can be seen in figure 3.21. Thisfigure is a PID plot for a pd run showing tritons. The plot is made with tightcuts on the pion to cleanly identify the triton peak. However, there is seen a long tailof events “falling out” of the triton spot in the PID plot. These are events for whichthe measured energy deposit in the bars is too small. The problem can be traced totoo small of an ADC signal as shown in figure 3.22. For the triton runs, the eventsare well located at the centre of the bars so attenuation of the light signal to thephotomultipliers is constant. Thus the ADC UP signal shown in figure 3.22 should bea well defined peak. A peak is seen in figure 3.22, but there is also present a long tailCHAPTER 3. CALIBRATION 1115040304)V20100Figure 3.22: A plot of ADC4UP (the ADC from the upper end of BAR4) shows theexpected triton peak at approximately 160. Also present is a long tail and pedestal.and a pedestal. This “fall out” is present for all the ADC signals from all the bars.Fall out from the PID bands is seen for pri —* irpp, pp —p dir+, and pd —* runs.The cause of this fall out is not known.The purpose of the ADC calibration was for particle identification in the barsto remove background. To that end, a cut in the LISA analysis was developed thatselects events only in the desired particle band in the PID curves. Figure 3.23 showsthe PID cut selecting protons for a pd —* irppp3 run. The box around the particleband is centred on the theoretical PID curve and is adjusted to encompass the particleband but reject the background. However, because of the problem with events fallingout of their PID bands, this cut could only be used on data extracted to calculateanalyzing powers where absolute normalization is not important. Where normalizationi important, such as for cross section measurements, the PID cut could not be used.Because the ADC calibration changed during the experiment, it was necessary to scaleADCRUM 169 ZC OIOOI 54.00)I I I_______Figure 3.23: PID cut for protons in BARS. The horizontal axis is the TOF to the barin 50 ps units; the vertical axis is the energy deposit in the bar in 0.1 MeV units. Thedashed curve around the particle band shows the PID cut.1600’1400’1200C1000’N800’600400200’CHAPTER 3. CALIBRATION2000’1800’112I . I30—AUG—93 13!07!02I200 400 500 800 1000 1200 1400 1600SPE 6=PID8 St 3.834E-’-03 K :181 TBCHAPTER 3. CALIBRATION 113the PID box to fit over the experimental band for each angle and energy.3.3.6 Bar Energy CalculationThe energy of a particle starting at the centre of the target is calculated from it’s TOFusing a sixth order polynomial. Separate polynomials for the TOF to each bar werecreated for protons, deuterons and tritons by numerically integrating the energy lossfor particles with initial kinetic energies ranging up to 200 MeV. Because the particlesdo not originate at the centre of the target a correction to the calculated initial energywas made by estimating the actual energy loss in the target by numerically integratingdE/dx for the particle through the target given the vertex from the QQD data. Thedifference between this energy loss and the one assumed by the polynomial is thecorrection. This correction was calculated for each bar particle detected in the event.A rough estimate of the error in energy reconstruction of the bars is +3 MeV for aproton with an initial kinetic energy of 100 MeV.3.4 Calibration SummaryThe quality of the overall calibration of the apparatus is indicated by the quantity,TDIF, defined as the difference between the energies of the initial and final statesof the reaction being studied. Plotting a TDIF histogram (figure 3.24) shows a peakwith a certain width and mean value. The width of the peak indicates the energyresolution of the apparatus while it’s position indicates systematic errors in the energyreconstruction. For pd —* tir, TDIF is given by (equation 3.26),TDIF = EB + md - (E + E)and is thus a measure of the combined calibration for the beam energy, the QQDreconstruction of the pion momentum and the scintillator bar TOF measurement ofCHAPTER 3. CALIBRATION ]J4350300250150100500—100 —75 —50 —25 0 25TDIF(MeV)Figure 3.24: TDIF plot for pd — tir. The position of the peak shows the systematicerrors in the energy reconstruction for an event while the width of the peak indicatesthe energy resolution of the apparatus.the triton energy. Because the QQD momentum reconstruction and the bar TOFcalibrations are based on the two body kinematics of the pd + reaction and onthe pion scattering angle, TDIF is also a measure of the QQD angle alignment (whichis related to the bar position calibration) and pion trajectory reconstruction.As discussed in section 3.3.4, the TDIF peak for pd —* tlr+ is noi centred at zeroand in fact shows a dependence on spectrometer angle, ranging from -0.11 to -1.44MeV. A similar angular dependance is found for the TDIF for pd —* irppp3 (table4.2). The pd —f d7rn9 data also shows a shift of TDIF from zero. The range of theTDIF peak for the different reactions is shown in table 3.8.Various attempts were made to find the cause of the error in energy reconstruction. The beam energy was varied, the spectrometer angle calibration changed, scmtillator bar array position moved, etc. However, no single effect could be found toCHAPTER 3. CALIBRATION 115TDIFreaction min(MeV) max(MeV)pd -1.4 -0.1pd —* dirn8 -2.2 -0.3pd,‘ PPPs 1.4 4.8Table 3.8: Range of the TDIF peak for the different reactions studied in this experiment.account for the observed TDIF behaviour. While not perfect, the calibrations presented in this chapter are considered to have the best overall consistency for the threereactions studied in this experiment.Chapter 4AnalysisThis chapter discusses the analysis of both the pd — rppp3 data which was the goalof the experiment and the pd—÷ dirri8 data which was used to check the calibrationof the apparatus and the experimental procedure. First some definitions and notationused in the analysis of the data are described followed by the procedure used to extractthe quasifree differential cross sections. Then there are discussions of the pd —* dlr+nsanalysis and the pd —* 7rPPPS analysis.4.1 Definitions4.1.1 Event ReconstructionThe experiment consisted of a polarized proton beam striking a neutron in a liquiddeuterium target producing a pion and two protons in the process shown in figure 4.1.The 4-momentum conservation equation for this reaction isPB +Pd Plr+Pl+P2+PS, (4.1)wherePB = (EB, ) is the 4-momentum of the beam proton,Pd = (Ed, Pd) is the 4-momentum of the deuteron target,116CHAPTER 4. ANALYSIS 117spectrometerscintillatorbar arrayFigure 4.1: Naming convention for pd— IrPPPS. A proton with momentum PB strikesthe quasi neutron P* in deuterium producing a pion (Pm-) and two protons (F1, F2)which are detected. The spectator proton from the deuteron (F8) remains undetected.Ps = (E8,P5)P =(E,15)Pi =(E1,1)andP2 =(E,1)is the 4-momentum of the spectator proton,is the 4-momentum of the final state pion,is the 4-momentum of proton 1,is the 4-momentum of proton 2,where E1 denotes total energy and P, denotes 3-momentum. The magnitudes of 3-momenta are denoted by P. Assuming the proton spectator exits unimpeded from thenucleus, Fd can be equated to,P1deuteron P2Pd = Pn* + PS, (4.2)CHAPTER 4. ANALYSIS 118where P* = P) is the 4-momentum of the bound neutron and P. =—Ps. Substituting equation 4.2 into equation 4.1 gives the 4-momentum conservation equation,(4.3)for the quasifree pfl* 7rpp process.The trajectory and momentum of the pion is measured in the QQD spectrometerfrom which the pion 4-momentum is reconstructed. The pion trajectory is traced backto the intersection with the beam to calculate the event vertex. The two protons aredetected in the scintillator bar array. The horizontal position is determined by the barnumber. The vertical position of the proton is determined by TOP of the light signalto each end of the bar. Using the bar position of the proton and the event vertex,each proton’s trajectory is calculated. The proton energy is determined by the TOPto the bar with corrections for energy loss through the materials to the bars. Theproton 4-momenta can then be reconstructed. The spectator proton from the deuteronis assumed not to interact in the reaction and carries only the Fermi motion it had inthe deuteron.The four final state particles have sixteen kinematic parameters. Forcing theparticles to be on their mass shells gives four constraints. The conservation of total energy and momentum give another four constraints leaving eight parameters tobe determined. Measuring the momentum and trajectory of three of the four finalstate particles gives nine measured parameters meaning that the kinematics are overconstrained. This over-constraint allows the determination of the spectator momentum,Ps, which is given by,(4.4)CHAPTER 4. ANALYSIS 119(note that 1 = 0 and does not have to be included in 4.4) and the missing energyof the system, TDIF, which is defined as the difference between the initial and finalenergy of the pd —* irppp3 system,TDIF=E-Ef. (4.5)The initial energy of the system, E, is,E=EB+md, (4.6)and the final energy of the system, E, isEf=E+El+E2+ s, (4.7)where Es = + is the energy of the spectator proton. Combining equations 4.5,4.6 and 4.7, TDIF can be written as,TDIF=EB+md—(E+El+E2+ s) (4.8)The total CM energy of the rpp system is calculated from the laboratory 4-momentaof the detected final state particles,w2 = (2+P1+P)= (E + E1 + E2)- ( + + )2 (4.9)In the -lrpp CM system, the following kinematical quantities can be defined,= (E, 1) the 4 momentum of the beam proton,= (E, P) the 4 momentum of the quasi neutron in the LD2 target,CHAPTER 4. ANALYSIS 120= (E, 1-f;) the 4 momentum of the final state pion,= (Er, 1) the 4 momentum of proton 1and= (E, P) the 4 momentum of proton 2.The CM pion scattering angle, 8. is defined by,cos(6) = (4.10)A useful kinematic quantity is the relative momentum of the two protons in the CMsystem,(4.11)which is a measure of the “excitation” of the diproton. A more direct measure of theenergy of the diproton is it’s invariant mass, defined by,= (P + 22)= (E1 + E2)—(P1 + P2)2, (4.12)or the excitation energy, defined as,= — 2m (4.13)where m is the rest mass of a proton.CHAPTER 4. ANALYSIS 1214.1.2 Energy ConventionsIn general, the best specification of the energy of an interaction is the total CM energy,W, or equivalently, the Mandlestam variable, s = W2. However, this can be a cumbersome number to use and experimentalists tend to prefer the beam kinetic energy, TB.TB is an adequate variable for fixed target experiments where the relationship betweenTB and W is clear. But for the current work, where the target is a virtual neutron,there is no longer a one to one correspondence between the beam energy and the totalCM energy. The energy unit chosen for this work is the Centre of Mass Kinetic Energy,defined as the total CM energy, W, minus the rest masses of the NN system, whichis,TOM = W — (m’, + m1 + rn2). (4.14)If the 7rNN system consists of a pion and a deuteron, TOM is as defined,TOM = W — (mg, + 7fld).Another quantity sometimes used to specify the energy of the system is j, defined asthe maximum CM pion momentum divided by the pion mass,= P(max)flr—1 (W2 — (m + (mi + m.2))(W — (7n — (m1 + rn2)) 15)m 2W4.1.3 Cross Section and Analyzing Power DefinitionsAngles and spin observables are defined according to the Madison Convention[60]. Inthe CM system (figure 4.2), the z axis is in the beam direction pointing downstream, ypoints vertically up in the laboratory. The x axis completes a right handed coordinateCHAPTER 4. ANALYSIS 1221Tp nFigure 4.2: Centre of Mass system for irpp. The coordinate system is located at thereaction vertex with the z direction pointing in the same direction as the beam proton.The y axis is out of the scattering plane and the x axis completes a right handedcoordinate system.system, pointing to the left of the beam when looking downstream. The azimuthalscattering angle is the angle with respect to the z axis in the z — x plane. A positiveangle is counter clockwise about the y axis. For a normally polarized proton beam, thespins of the protons point along the y axis. Spin up is in the positive y direction, spindown in the negative direction.According to the Madison convention, the experimentally measured cross sectionsfor the beam polarized spin up, o, and for the beam polarized spin down, o, can bewritten as,= oo+Pai, (4.16)andIT=U0 — (4.17)CHAPTER 4. ANALYSIS 123where P+/P_ is the polarization of the beam for spin up/down. Rearranging equations4.16 and 4.17, the cross sections a0 and a1 can be written as— Pa++PoJo—(4.18)andcT+ —a1=(4.19)The analyzing power for a normally polarized beam and unpolarized target is definedasA °•____________tiNO==ao Po++P+o4.1.4 Extracting Cross SectionsThe goal of the experiment is to extract cross sections and analyzing powers for thepri —* irpp(1S0)reaction from the pd —* irppp3 data. This extraction is achieved byassuming the validity of the Spectator Model for pion production from the deuteron(appendix E). The Spectator Model assumes that pion production proceeds solelyby a collision between the beam proton and the bound neutron. While the spectatorproton is assumed not to participate in the reaction, it does have an influence. Thebound nucleons in the deuteron have a Fermi motion (described by the deuteron wavefunction). The spectator proton is treated as an unbound particle with a momentumdistribution given by the deuteron wave function. The consequence of this is thatthe target neutron is off it’s mass shell (due to the binding energy of the deuteronand momentum of the spectator), and has an energy and momentum that dependson the momentum of the spectator. The pd—+ 7rPPPS reaction can be thought of asa collision between a beam proton with a well defined energy and momentum and aCHAPTER 4. ANALYSIS 124nonstationary target neutron which has a varying energy and momentum. The effectis that the total energy of the quasifree pri —* irpp reaction is smeared out. Whenrelating the pd 7rPPps cross sections to the quasifree pn —* rpp cross sections, itis necessary to consider the range of energies involved. As well, there are kinematicalfactors arising from the apparent change in the beam flux due to the movement of thetarget neutron.The pn — irpp process has twelve kinematical parameters for it’s three finalstate particles (three energies, nine momentum components). Consideration of theseven constraint equations (all three particles are constrained by E2 = m2 + F2, totalenergy is conserved and all three total momentum components are conserved) leavesfive kinematical parameters necessary to describe the reaction. The five parameterschosen in this analysis are:Q The solid angle of the pion in the irpp CM system (two variables);The square of the invariant mass of the two protons defined by equation 4.12;andThe solid angle of proton 1 in the pp CM system (two variables).In this notation, any variable with (*) denotes the irpp CM system and (‘) denotes thepp CM system. The five fold differential cross section for pri —* irpp is then writtenas,d5Jpnr—ppd2;dMdThe reaction pd —* rPPPS on the other hand, has the additional final state spectatorproton giving it sixteen parameters and eight constraints. Thus pd —* -irppp3 requireseight kinematical variables to describe the reaction. Five of the eight parameters chosenCHAPTER 4. ANALYSIS 125are those for pn —÷ irpp. The other three are the momentum components of thespectator proton, Ps, in the laboratory reference frame. Thus the pd — rppp5 eightfold differential cross section isd80pd—ir — pppdc;dM2dcdfl5Using the spectator model discussed in appendix E, the pd —* 7rppp differential crosssection is derived from the free pn —p irpp cross section by the relationship (equationE.15),i8 15 TTTD* ..L2fij(0pd—irppp LL pnirpp VY ‘B Y US— dc2dMd EThPB (2ir)3That is, the quasifree differential cross section is equal to the free cross section multiplied by two factors. The term q52(Ps)/(2ir)3 is the momentum distribution of thespectator proton, or equivalently the off shell target neutron and is normalized toI P k’S) 3JFS (2r)Note that because the spectator proton is assumed not to interact, all the offshell behaviour is forced onto the target neutron. The other term, (WP)/(EPB), is calledthe Flux Facior and it arises because the target neutron is no longer stationary. Sometimes the neutron is moving towards the beam proton. This is equivalent to increasingthe flux of beam protons on a stationary target which increases the probability of aninteraction. Alternatively, the neutron can be moving away from the beam protonwhich decreases the beam flux and the probability.The desirable observable to measure for the pn—irpp(’S0) system would beCHAPTER 4. ANALYSIS 126the differential cross section,(4.22)To extract this cross section it is necessary to integrate the measured five fold differentialcross section over the entire diproton phase space. This operation is given bydci; = I dcdMdcdM (4.23)To evaluate equation 4.23 a problem arises due to the limited part of phase spaceaccepted by the apparatus in this experiment. While it is reasonable to assume thatthe cross section is isotropic in (since the diproton is in an S wave), the real problemcomes in integrating equation 4.23 over It is not clear, having measured d5a overa small (but most important) slice of how to analytically extrapolate d5a over allfor the diproton in a 1S0 state. To avoid such questions in the interest of obtaininguseful cross sections, it was decided to compute the triple differential cross section,d3udQ*dM2 (O) , (4.24)PP O<T <Tm’— pp_ ppaveraged over some interval in M. For convenience of writing, this interval is writtenin terms of (the excitation energy of the diproton defined by equation 4.13) but theaveraging is still over the range0 <M(max).1n fact, because the solid angle, , represents two angle variables, the cross section is doublydifferential and should be writtend2odc2;However, it is conventional to treat it as a single differential cross section and write it as in equation4.22.CHAPTER 4. ANALYSIS 127The value of T is selected to exclude the P wave diproton contributions to the crosssection (see section 2.2). As a short hand for writing equation 4.24, the averaging overthe interval in will be implicit and not written down. Equation 4.24 will then bewritten,d3u (4.25)Assuming the cross section is independent of c,, the triple differential cross section isobtained from the measured d5a using,— 1 d3u (4 26)df2.dMdQ’7 — 27rd$TdMThe factor 1/2ir comes from the averaging over d2,. The integration is only over2ir because of the indistinguishability of the two final state protons. Including wirechamber efficiency,,and detector live time, LT, the detector yield, Y, isINAPtY = LTeWCJ A NBaUNAPt WP df2’ fl= NB A LT€WCJ EPB ddMdQ (2ir)3 s= NBNPtLTEWC f J°J ddMdd15, (4.27)where NB is the number of beam protons, NA is Avogadro’s number, p is the targetdensity, t is the target thickness and A is the molar mass of the target. The integralin equation 4.27 includes the effect of pion decay implicitly in the limits of integration.Averagingd3o/dQdM% over the detector acceptance by using the relationship,V (f) = (4.28)CHAPTER 4. ANALYSIS 128the yield is given by,Y = LTeWCNBPI_Kd2) f (4.29)For ease of writing the substitution,I d3u \ d3o\ dZ;dMj “ dc2dM’is made and it will be understood that the cross sections extracted from this experimentare averages over the relevant regions of phase space. Defining the Phase Space Integral(PSI), Q, asQ = (4.30)the differential cross section is then calculated from the yield by the equation,d3u YddM = (4.31)where g(TcM) is a factor to correct for the shadowing of the target neutron by thespectator proton (see section 4.1.5). The phase space integral is the effective solid angleof the apparatus including all the detectors, multiple scattering, beam profile etc. It iscalculated by Monte Carlo (see appendix G) for each set of cuts the data is binned by.The Monte Carlo simulation includes pion decay and the tracking of the decay muon.Thus the Monte Carlo calculation of Q includes the pion decay correction.The Reid Soft Core (RSC) deuteron wave function{61] shown in figure 4.3 isused for the spectator proton momentum distribution. This probability distributioncombines both the S wave and D wave components of the deuteron and is normalized.012.010.008.004.002.000400129CHAPTER 4. ANALYSIS.006c:5ct0 50 100 150 200 250 300 350P(MeV/c)Deuteron(RSC) wave function[61].Figure 4.3: Reid Soft Coreby equation 4.21. The RSC wave function is almost identical to the Hulthén wavefunction[62] used by some authors.4.1.5 Nuclear ShadowingIn addition to the effects of the Fermi motion on the cross section, it is necessary toconsider the shadowing of the target neutron in the deuteron by the spectator proton.First pointed out by Glauber[63], the shadowing effect is due to the beam protoninteracting with the spectator proton rather than the target neutron, thus reducing theavailable beam flux for pion production from the neutron. For quasifree pri —* irpp,the shadowing effect reduces the observed cross section by a factor, f(TcM), given by,f(TcM)=4r2t(T0M) (4.32)CHAPTER 4. ANALYSIS 130TCM f(TcM)55 MeV 5.5 + 3%70 MeV 5.8 + 3%Table 4.1: Glauber Correction for quasifree pri —* irpp at the centre of mass energiesstudied in this experiment.where r is the average radius of the deuteron and is taken from Ericson and Weise[2]to be 1.963 ± 0.OO4fm. The term o(TCM) is the total pp cross section at the relevantCM energy and is taken from the SAID2 data base[64]. The cross sections extractedfrom an experimental yield must then be multiplied by a term,1gpn=1 £ f1— Jpn’. CMto correct for the shadowing.For TCM = 55MeV, the shadowing factor is 5.5% and for TOM = 70MeV it is5.8%. The largest error in the correction is the uncertainty over which value of r to usefor the correction. Various values have been used in the literature and these vary by asmuch as 30% [24]. This gives an uncertainty of ±3% to the shadowing correction usedin this experiment (table 4.1). Tsuboyama{24] notes that the protons scattered fromthe spectator proton in the deuteron can still interact with the target neutron. Thismakes equation 4.32 an overestimate of the shadowing effect. However, consideringthe relatively large overall normalization uncertainties in this experiment, it was notdeemed necessary to correct for this effect.2SAID (Scattering Analysis Interactive Dial-in) is an interactive data base for nucleon-nucleon scattering data developed by RA. Arndt and L.D. Roper.CHAPTER 4. ANALYSIS 1314.2 pd—dirnThe purpose of examining the quasifree pp —* dir data is to demonstrate the validityof using the Spectator Model to extract information about the free pn —* irpp reactionfrom pd —* irppp data. While pp —* dir has only a two body final state comparedto the three body final state for pn —* rpp, the reaction kinematics are similar andpresumably the role of the spectator nucleon in deuterium is similar for the two reactions. By comparing the observed yields of pp —÷ dir against the predictions of aMonte Carlo simulation based on a parameterization of the free pp —* dir+ reaction byRitchie[65], the validity of the Spectator Model can be examined. Further, the MonteCarlo simulation can be compared with the experimental distributions to assess thedetector calibrations in particular, the QQD acceptance (this is discussed in section3.2.8).Eight pd — dlr+ns data runs were taken during this experiment. The data consistof a single run at a beam energy of 403 MeV (ird CM kinetic energy, TOM 52.5MeV)and seven runs at three angles at a beam energy of 353 MeV (TOM = 30MeV). Attwo of the three 353 MeV angles, data were taken at different spectrometer momentumsettings. Figure 4.4 shows the reconstructed TOM for a typical quasifree pp —* dir+ run.The target proton Fermi motion causes the centre of mass energy to be smeared outover more than 20 MeV.A measure of how effectively the quasifree pp —f dir+ process is isolated in thedata is found in the plot of TDIF, defined asTDIF=EB+md-(E+Ed+Es).Figure 4.5 shows a TDIF spectra for a typical pp —* d7r run. The peak is the pd —*dir+n8 events. The asymmetric background is due mostly to quasifree pp —* lr+pn fromCHAPTER 4. ANALYSIS 13230002000(I)4)1000020 30TCM(MeV)Figure 4.4: The ird CM kinetic energy, TOM, for the quasifree J5p — dr reaction at abeam energy of 353 MeV. The CM energy is smeared out due to the Fermi motion ofthe bound target proton.which the proton is mistakenly reconstructed as a deuteron (because there is no cut onthe bar data to identify the deuteron, protons can be accepted accidentally). Figure4.6 shows a Monte Carlo simulation of the pp * +pfl background. The simulation isable to reproduce the observed behaviour of the data.The extraction of free pp — dir+ differential cross sections from the pd — dr+n3data using the Spectator Model is similar to the procedure discussed in section 4.1.4for extracting the pri — irpp(’So) cross sections from the pd —* irppp3 data. Forpp —* dir, there are two final state particles with eight kinematical variables. Sixconstraint equations (total energy, total momentum, each particle on its mass shell)gives two free parameters to describe the reaction. Selecting the pion solid angle in theCHAPTER 4. ANALYSIS 133300025002000rn15001000500Figure 4.5: TDIP spectra for pd —* dirn3. The histogram shows the data consisting ofa peak of good events centred near TDIF=0 and an underlying asymmetric backgroundextending from TDIF=50 MeV to below TDIF=-100 MeV. Shown also are a gaussianfit to the peak and a fourth order polynomial fit to the background.0TDIF(MeV)CHAPTER 4. ANALYSIS 134500400300Q)2001000—100Figure 4.6: Simulation of background for pd —* dirn5. The solid histogram is theexperimental data. The solid curves are fits to the peak and background data as infigure 4.5. The dashed histogram is a Monte Carlo simulation of a quasifree pbackground (the amount of simulated background was scaled to compare shapes withthe experimental background).100 —50 0 50TDIF(MeV)CHAPTER 4. ANALYSIS 135ird centre of mass, the double differential cross section is,du(rndir ),10*For pd —÷ dir+n8,with the spectator neutron the differential cross section is written,d5 Updd7r+ flddP5which using the spectator model discussed in appendix E can be written as (equationE.16),d5Jdd+ WPq2(Ps) dcT +dcdfl = EPFB (2ir)3 -j—(PP— dir ), (4.33)and the experimental yield isY = LT€WCNBNAp f ‘S) (pp - dirjdQd]= LTeWCNBNApt— dir) J r;B (4.34)where the cross section outside the integration symbol in equation 4.34 is taken as anaverage over the detector acceptance and the other terms are defined in section 4.1.4.The phase space integral, Q, is defined as,2 p= I EP (2ir) (4.35)and is calculated by Monte Carlo integration as discussed in appendix G. The averageCHAPTER 4. ANALYSIS 136differential cross section calculated from the detector yield is(pp d)=(4.36)where1gpp= 1— fpp(TCM)(4.37)is the Glauber shadowing correction for quasifree pp —÷ dir+ (see section 4.1.5). Theterm f. is given by,=4r2TG (4.38)where r is the deuteron radius and o(TcM) is the total pn interaction cross sectionat the relevant energy.Figure 4.7 shows the reconstructed differential cross sections for three of the eightquasifree pp — dr’ runs, binned by TGM in 5 MeV wide bins. Also shown in figure4.7 is the Monte Carlo prediction of the quasifree cross section based on the parameterized world data for pp — dir+ and the spectator model. The agreement between theexperimentally measured and Monte Carlo predicted quasifree cross sections are consistent with the statistical errors. One can use this successful comparison to confidentlycalculate the differential cross section for pp —f dir+.The size of the bin width to be used to extract the free pp —* dir+ cross sectionfrom the quasifree data depends on two considerations. First, it is desirable to choose abin wide enough to have a reasonably small statistical error. Second, it is necessary tochoose a bin width small enough so that the measured cross section corresponds to thefree cross section at the central energy of the bin. This can be understood by lookingat figure 4.4 which shows the detector yield as a function of TCM. The detector yieldis an integration over the cross sections as a function of energy. If the energy bin isCHAPTER 4. ANALYSIS 13750T=353 MeV eQQ34.2deg40 oC-30C20b1000 10 20 30 40 5TCM(MeV)TB=353 MeV &QQ5Odeg403020- CbI I I10 Q C00 10 20 30 40 50TCM(MeV)Du I ITB=353 MeV QQ5?2deg4030C-CC.10 I00 10 20 30 40 50TCM(MeV)Figure 4.7: Quasifree pp —* dir da/d!2 vs TCM for three pd -4 drn8 data runs. Solidpoints — experimental data. Open points Monte Carlo predictions. The error barson the experimental points are statistical uncertainties.CHAPTER 4. ANALYSIS 138too wide, the contributions to the yield will not be symmetric about the central energyof the bin. The smaller the energy bin, the more closely the measured cross sectioncorresponds to the cross section at the central energy of the bin.The choice of bin size was made by using Monte Carlo simulation to predictthe detector yield based on the parameterized free cross section and the SpectatorModel (see appendix G). The Monte Carlo simulated yield was then put into equation4.36 and the quasifree differential cross section calculated. This quasifree cross sectionwas compared to the free cross section to see how large the error would be whenreconstructing the differential cross section. The results of these studies indicated thata TCM bin centred at 30 MeV with a width of +10MeV would give minimal distortionof the reconstructed cross section (2.8%) and still have reasonable statistics.Figure 4.8 shows the reconstructed quasifree differential cross sections for three ofthe seven data runs taken at TB = 353 MeV, binned at TOM = 30 + 10MeV comparedto the free pp — dir differential cross section from the parameterization by Ritchie[65].The Glauber correction for the quasifree data was f(30MeV) = 7 + 3%. The errorbars shown are statistical.The three runs shown in figure 4.8 were chosen because the Monte Carlo simulations of the quasifree yield predicted that they would have the least distortion ofthe cross section due to the energy binning (other runs were expected to have higherdistortions due to the detector setting used for those runs). All seven of the TB = 353MeV runs were used, however, to estimate the systematic error in quasifree pp —* dir+cross section normalization. The average error between the measured and the MonteCarlo predicted quasifree cross sections in the TOM = 30 + 10MeV bin was 5%. Thelargest difference between a measured and simulated quasifree cross section was 13%.To be confident of the upper bound on the systematic error of the quasifree pp —* dir+differential cross section measurement, a systematic error of 15% is quoted.CHAPTER 4. ANALYSIS 139403530-Qc15b1050Figure 4.8: Quasifree pp —* dir+ differential cross section (points) compared to the freepp —* dir+ differential cross section (line) at TOM = 30MeV. The error bars on thequasifree points are statistical errors.Analyzing powers were also extracted from the pd —* dir+ri3 data. Figure 4.9shows the quasifree pp —* dir+ analyzing power measured in this experiment comparedwith a parameterization of the free ANO at T01 = 30MeV taken from Walden[66].The data is subdivided into three angular bins for each QQD setting. The quasifreemeasurement shows good agreement with the free analyzing power. However, it shouldbe noted that the 403 MeV run does not have such good agreement. At a CM angleof approximately 60°, an analyzing power of -0.2 is expected. The measured analyzingpower is -0.35 three standard deviations from the expected result. Unfortunately,there is only one data run at 403 MeV so it is not known if this is an isolated effect orif it is common to all 403 MeV data. The cross section for this data is in reasonablyagreement with the predictions of the spectator model. Because the 353 MeV data isin good agreement with the free pp — dir it is felt that the discrepancy with the single90O(deg)CHAPTER 4. ANALYSISCz0.20.0—0.2—0.4—0.6—0.8140Figure 4.9: Quasifree pp —* d-ir AND at TOM = 30MeV (points) compared to a parameterization of the free pp —* dr AND at TOM = 30MeV by Walden[66].403 MeV analyzing power measurement could be ignored.4.3 pd—rpppThe pd — irppp3 data were taken at three beam energies in this experiment: 353,403 and 440 MeV. Data were taken at six spectrometer setting for the 403 MeV data,and at five settings for both the 353 and 440 MeV data. The 403 and 440 MeV dataare analysed in this thesis while the 353 MeV data is analysed independently by Hahn[6] who also performed an independent analysis of the 403 MeV data. Analysis of thepd 4 7r PPPs data consisted of:• Identification of quasifree pn —* irpp events and isolation from the background.• Selecting cuts on the data to minimize the contamination of P wave diprotons.• Extraction of the analyzing powers and differential cross sections.CHAPTER 4. ANALYSIS 141In addition, the data were examined for evidence of contamination from four bodyprocesses.4.3.1 Background ContaminationThe first requirement for the analysis of the pd — irppp data was to identify validevents and exclude background processes. Because of the multiparticle final state, theonly way to ascertain if an event was due to pd —f irppp was to reconstruct the missingenergy of the system, TDIF, defined by equation 4.8. A spectra of TDIF shows avalid event in a peak centred near zero, the exact position and width of which dependson the calibration of the apparatus and the experimental resolution. On the otherhand, background events will show a much broader range of TDIF, the exact natureof which depends on the type of background present.Types of background expected in this experiment are varied. Muons from piondecay will contaminate the spectrometer. Beam protons, instead of protons from thequasifree pn —* irpp reaction will be detected in the scintillator bars. It is possible fora single proton to strike a scintillator bar and then multiple scattering into an adjacentbar, triggering both. Such an event could be mistaken as two separate proton signals.Exclusion of these background processes is done by making cuts on the data.For the purposes of cross section calculations, the pri —* 7rpp data had relativelyloose cuts. The QQD data had the tight cuts on t9 and o of +3°. Wire chamber 2 had2.5cm, Y2 <4cm and wire chamber 3 had both x3 and y3 less than or equal to4cm.A software cut rejecting events outside a window of 25 to 50 ns is placed on theTOF of particles to the scintillator bars to eliminate the elastic and accidental beamprotons. A valid diproton event is defined as two bars firing within the TOE cut. Ifthe bars are adjacent, the vertical separation of the events must be greater than 10CHAPTER 4. ANALYSIS 142cm. This cut is to prevent a false signal from a proton striking a bar and multiplescattering into an adjacent bar, generating an additional signal. It would have beendesirable to include an additional cut on the bar TOF vs energy deposit spectrum (theso called PID spectrum discussed in section 3.3.5) to positively identify the protons butthe problem with events “falling out” of the particle bands (see page 110) preventedthis.It should be noted that for the 403 MeV 60.1°, 74° and 90° runs, BAR3 had afaulty connection making it useless for data taking. For the runs at these angles, BAR3was taken out of the analysis. BAR3 was also disabled in the Monte Carlo phase spaceintegral calculations.The quality of the isolation of quasifree pn —* irpp events is assessed with theTDIF plot. Figure 4.lOa shows the TDIF spectra using the cuts on the data intendedfor cross sections. There are three basic features to the spectra: The gaussian peakcentred approximately at TDIF=0; the “bump” centred approximately 27 MeV abovethe peak, and the “hump” — a background stretching from TDIF < —100MeV toapproximately TDIF = 60MeV.Figure 4.lOb shows the TDIF spectra for an analysis using four additional cutson the data. The quantity DDIF 64 — 65 is the difference in pion momentumreconstructed from WC4 and V41C5. A value of DDIF far from zero is indicative ofa muon from pion decay in the spectrometer after wire chamber 3. A muon from adecay after WC3 will have a different trajectory through the spectrometer and thepositions in WC4 and WC5 will not be consistent with the same initial momentum.The quantity Y4M5= — y5 is the difference in the nonbend plane positions of thepion at WC4 and WC5. Having passed through the spectrometer, a pion is constrainedto a small value of Y4M5. A large positive or negative value is again indicative of amuon from pion decay after the dipole magnet. The quantity QTOF is the time ofrjq)(UQ)Figure 4.10: Missing Energy, TDIF, forpd —* irppp3. a) The minimal cuts on the dataused for cross section analysis. b) The strong cuts on the data used for the analyzingpower analysis. The histogram is the experimental data. The solid curve is a fit to thehistogram. The model fitted to the data has two gaussian curves to describe the peakat TDIF = 0 and the bump bump at TDIF = 27MeV and a fourth order polynomialto fit the underlying background. The dashed curves show the individual componentsof the model.143CHAPTER 4. ANALYSISa) no cuts250200150100500b)140TDIF(MeV)ANO cuts120100806040200TDIF(MeV)CHAPTER 4. ANALYSIS 144flight of the particle travelling from the ST counter before WC2 on the spectrometer toeither Si or S2 which are located after WC5. QTOF can be used to distinguish betweenpions and electrons travelling through the spectrometer by their different TOF’s. TheQQD has cuts on DDIF <15%, Y4M5j 2cm and a tight cut on QTOF aroundthe pion peak. As well, the bars have a PID cut to strongly identify the protons.These additional cuts greatly reduce the hump background (but leave the bump intact).Unfortunately, because the calibrations of these quantities are not well understood, thedetector acceptance cannot be calibrated with these cuts and they cannot be used forcross section calculations. They are, however, useful for analyzing power measurementswhere the absolute acceptance is not needed.According to the Monte Carlo simulation and depending on the beam energy andspectrometer angle, between 1/4 and 1/2 of the pions decay before reaching the S3 andS4 scintillators at the back end of the spectrometer. Of the muons from these decaysbetween 16% and 57% are accepted by the cross section cuts (not including the PIDcut, DDIF, Y4M5 and QTOF). Most of the muons that are accepted by the apparatusare from pion decays far back in the spectrometer. The high percentage of muons thatare accepted by the apparatus illustrates the need to calculate the effective detectoracceptance by Monte Carlo simulation.The peaks in figure 4.lOa is comprised of good quasifree pn —* irpp events theevents desired. The hump background consists of random coincidences in the detectorsand are a “true” background unrelated to pn —* irpp events. As was discussed above,the hump background can be decreased by using tight cuts on the data to cleanlyidentify events.The events creating the bump at 27 MeV in the TDIF spectra have a morecomplicated origin. From the PID spectra, the particles striking the bars and creatingthe bump at TDIF 27 are protons. There is no obvious difference in the PID spectraCHAPTER 4. ANALYSIS 145between the bump events and the good peak events. The bump is caused by pri —* irppevents where one of the protons has not left the proton barrel through the exit window.The proton barrel counters are placed at the exit of the proton barrel to identify validprotons and help reject beam protons that would otherwise swamp the detector. Fora bump event, one of the pri — irpp protons has left the proton barrel through theexit window, firing the PB counters. Another proton, either a pri —* irpp or beamproton, leaves the proton barrel, passing through the 3/8” thick stainless steel barrelend. These protons can loose between 20 and 60 MeV more energy than if they passthrough the thin proton barrel window. The geometry of the proton barrel windowis large enough to cover the bars, however, the protons missing the window can stillmultiple scatter into the bars firing the detectors as a valid event.The hypothesis that the bump events are caused by the detection of a proton notexiting through the proton barrel exit window can be tested. There are in fact twoproton barrel counters (see figure 2.1), PBU and PBD, splitting the aperture aboutthe reaction plane. While valid events can occur if only one barrel counter fires (bothprotons pass through the same counter) most of the invalid events should occur whenonly one counter fires. Figure 4.lla shows TDIF with no cut on PBU or PBD. Infigure 4.llb TDIF is plotted for events where both PB counters have fired, implyingthat both protons have left the barrel by the window. The relative amount of bumpto peak has dropped substantially. Figure 4.llc shows TDIF for events where onlyone barrel counter fires. Almost all of the bump events occur when only one barrelcounter fires. The conclusion is that the bump events are caused by protons not passingthrough the barrel counters. The small number of bump events for the two countercondition can be explained by the counters being fired by a pn—7rpp proton and abeam proton. The other pn proton passes around the barrel counter.Is the width of the TDIF peak consistent with the estimated errors in the event—75 —50 —25 0 25 50TDIF(MeV)Figure 4.11: TDIF for a) either one or both Proton Barrel counters fire. b) Both ProtonBarrel counters fire. c) Only one Proton Barrel counter fires.146CHAPTER 4. ANALYSISa) No cut on PB counters-I-)a)b) Both counters firec)120one counter fires100•80200 -—100 100CHAPTER 4. ANALYSIS 147reconstruction? A simplistic calculation of the expected width of the TDIF peak is asfollows. A typical event consists of a pion with kinetic energy of 100 MeV and twoprotons with, say, kinetic energies of 100 MeV each. The momentum resolution ofthe pion spectrometer is, from section 3.2.5, 2% which corresponds to an error in thepion energy of 3 MeV. From section 3.3.6, the errors in the proton energies are alsoapproximately 3 MeV. The error in TDIF, TDIF, is given by,TDIF = + L + T2 + (4.39)where T1, -T2 and A15 are the errors in the pion, proton 1, proton 2 andspectator proton kinetic energies respectively. Because the spectator energy is small(less than 5 MeV) it contributes a negligible error to LTDIF which is thus given by,/.TDIF +32 +32 5.2MeVFigure 4.10 shows a TDIF peak with width, a = 5.6MeV. However, this simpleestimate ignores the error in beam energy, the variation of pion and proton energies,correlations between errors etc. A more accurate error estimate can be derived fromthe Monte Carlo simulation . Table 4.2 lists the positions and widths of the TDIFpeaks from the experimental and Monte Carlo TDIF spectra for all the pd —* irppp5runs. The Experimental data shows an increase in the width of the TDIF peak withincreasing QQD angle. The Monte Carlo simulation also predicts an increase in thewidth, but tends to be about 1 MeV narrower. Table 4.2 also shows that the meanvalue of TDIF shifts with angle from negative values to positive values. For the 403MeV data, this is on the order of a 3 MeV shift. For the 440 MeV data, it is a 6 MeVshift. This is indicative of an error in the calibration of the apparatus but it is notclear if the problem is with the QQD, the scintillator bars or both. The Monte CarloCHAPTER 4. ANALYSIS 148TB OQQD Expt MC(MeV) (deg) mean(MeV) a(MeV) mean(MeV) o(MeV)403 28.8 -1.4 4.1 1.4 4.237.2 -0.8 4.6 1.2 4.048.1 -0.6 4.6 1.0 4.160.1 -0.5 5.6 1.6 4.474.0 0.9 5.8 1.7 4.890.3 1.5 5.9 1.4 5.3440 41.0 -1.0 5.1 1.5 4.350.4 1.0 5.1 0.9 4.962.6 2.0 5.7 1.0 4.876.3 3.1 7.0 1.5 5.993.2 4.8 7.6 1.5 6.0Table 4.2: The mean position and widths of the experimental and Monte Carlo TDIFspectra for pd — PPPsalso has TDIF offset from zero but it is approximately constant at about 1.5 MeV.Contributions to the event rate from the target cell were measured by takingdata with the target cell empty. “Empty” meant that the liquid deuterium in thetarget cell was replaced with deuterium gas at 1.5% of the liquid’s density. The ratesfrom the empty target cell were in fact 1.5% of the full target cell rates, indicatingthat there was no significant background from the aluminum target cell walls. It mayseem surprising that there was no background from the aluminum since it had an arealdensity of neutrons 7% that of the LD2 — over four times larger that the deuteriumgas. However, the binding energy of 27A1 is much larger than deuterium resulting in amuch larger Fermi motion. The combination of large Fermi motion and large bindingenergy makes it kinematically unfavourable for the detectors (which were tuned to aquasifree reaction on deuterium) to detect events from aluminum.CHAPTER 4. ANALYSIS 494.3.2 P-Wave ContaminationAs discussed in chapter 2, the Monte Carlo simulation, based on the Handler parameterization of pri —p irpp[l5], predicts that as the relative CM momentum, F, of thetwo protons increases, the diproton state becomes dominantly P-wave. Figure 4.12bshows the analyzing power plotted as a function of F for beam energy 403 MeV andpion angle 6QQD = 75o The Monte Carlo simulation predicts that this run has thegreatest P-wave contamination at 403 MeV. It is seen that ANO is near unity at lowF but then drops off dramatically at around 75 MeV/c. This shift in the analyzingpower is interpreted as more and more P-wave diproton entering the data sample. TheP dependence of the experimental ANO is the most dramatic at this energy. For moreforward or backward angles, the F dependence of ANO is either small or negligible. The440 MeV data behaves similarly.The P-wave content of the data sample was reduced by placing a cut on the dataof F 75MeV/c. The analyzing power distributions were found to be stable for valuesof P less than this. This cut of 75 MeV/c on P is equivalent to 1.5MeV (whereis the excitation energy of the diproton). All the analyzing power and cross sectiondata are analysed with this cut.An estimate of the P-wave contamination using the P 75MeV/c cut can beestimated by comparing the experimentally observed F distributions to distributionsgenerated using the Monte Carlo simulation based on the Handler parameterization ofthe reaction. Figure 4.13 shows the experimental and Monte Carlo P distributions. Forthe 403 MeV data, the Handler P distributions agree with the experimental distributions at large and small angles but not in the range 48 to 74°. Similarly, the 440 MeVdistributions do not agree at similar angles. A possible cause for this disagreement maybe an overestimate of the P-wave contribution in Handler’s parameterization. Figure4.14 shows the S-wave and P-wave Monte Carlo P distributions fitted to the experCHAPTER 4. ANALYSIS 150a)250200150100500b)1.000.750.500.250,00—0.25—0.50—0.75—1.00Figure 4.12: a)function of P.P(MeV/c)P(MeV/c)P distribution at 403 MeV OQQD = 48.1°. b) Analyzing Power as aCHAPTER 4. ANALYSIS 151.014403MeV 37deg100 200 300 4(P(MeV/c).012• 440MeV 63deg.010 -E/nfl”Figure 4.13: Experimental(points) and Handler(solid histogram) P distributions. Alsoshown are Handler’s predicted P-wave contribution to the P distribution (dashed histogram). The experimental and total Handler distributions are normalized to unit area.>0440MeV 50deg.012.010.008.006004.002.000ido 200 300P(MeV/c)4 )0 ide 200 300P(MeV/c)400100 200 300P(MeV/c)100 200 300P(MeV/c)CHAPTER 4. ANALYSIS.012.010.008.006.004.002.0000152Figure 4.14: Handler P distributions fitted to experimental data by varying the relativefractions of the S-wave and P-wave contributions. The experimental data are thepoints, the total Handler distribution is the solid histogram and the Handler P-wavecomponent the dashed histogram. The experimental and total Handler distributionsare normalized to unit area.-— 440MeV 50deg.010.008.006.004.002P(MeV/c)uuu100 200 300 410P(1{eV/c)0100MeV 93deglao 200 360P(MeV/c)4U0CHAPTER 4. ANALYSIS 153b) 440 MeV20 30 40 50 60 70 80 90 100oQQ(aeg)Figure 4.15: Fraction of S wave diprotons accepted by the apparatus (without theP 75MeV/c cut) as a function of QQD angle. Dashed line— nominal Handlerparameterization. Solid line Handler parameterization fitted to experimental Pdistributions, a) 403 MeV data; b) 440 MeV data.imental P distributions. The fitting procedure consisted of varying the fraction of Swave diprotons (fs) to fit the experimental data. The form of the fit is= fsNs(P) + (1 —f8)Np(P),where Nexpt, Ns and N are the experimental P distribution, the Handler S-wave Pdistribution and the Handler P-wave P distribution respectively, all normalized to unitarea. The resulting fits are seen to be able to fit the experimental data quite well atboth energies and all angles. This seems indicative that while Handler kinematicaldistributions are correct, the relative fractions of S and P-wave contributions to thereaction are not.Figure 4.15 shows the fraction of S wave diprotons accepted by the experimentalapparatus without the P 75MeV/c cut. Shown in the figure are the estimated S wavefraction for the nominal Handler parameterization and for the Handler parameterizationfitted to the experimental P distributions. The nominal parameterization estimates upa) 403 MeVN0.60.50,4Cl) 0.3C.) 0.20.10.0 I I20 30 40 50 60 70 80 90 100OQQjdeg)00C)ClClCl)C.)CHAPTER 4. ANALYSIS 154to a 40 % P-wave contamination of the data sample for the 403 MeV data and 70% forthe 440 MeV data without the P < 75MeV/c cut. With the P < 75MeV/c cut, thenominal Handler parameterization estimates a 9% and 10% P-wave contamination forthe 403 and 440 MeV data respectively. The fitted Handler parameterization estimates,for both energies, a maximum P-wave contamination of approximately 18% without and4% with the P cut.4.3.3 Quasifree pn —* irpp(’So) Analyzing PowersSince absolute acceptance normalizations are not needed for analyzing power calculations, the data had the strong cuts (discussed in section 4.3.1 placed on them to reducethe background that resulted in the reduced background seen in figure 4.lOb. Theseincluded cuts on DDIF and Y4M5 to reduce the acceptance of muons from pion decayand a cut on the PID curves to identify the protons reaching the scintillator bars. Background is further reduced with a cut on the spectator momentum of Ps 100MeV/c.The bump events (see section 4.3.1) are rejected by a cut of +10MeV centred on theTDIF peak (see figure 4.10).However, unlike the differential cross section analysis which had tight cuts onthe angle of pions entering the QQD, the analyzing power analysis allowed the fullangular acceptance of the spectrometer (approximately +5°). For the cross sectionanalysis, the QQD acceptance was restricted to +3° to help reject muons, but this isnot necessary for the analyzing power analysis where the DDIF and Y4M5 cuts rejectthe muons. The P 75MeV/c cut is used to isolate the S wave diproton. Because theQ QD acceptance is so wide, it can be subdivided into 3 angular bins. These bins areautomatically selected in the analysis to have approximately equal numbers of events.Figure 4.16 shows the resulting analyzing power curves for the TB = 403 and 440MeV data (corresponding to TCM = 55 and 70 MeV respectively). The data are listedCHAPTER 4. ANALYSIS 155in table 4.3. The 6 = 50.5° data point was actually calculated to be —1.016 + 0.072which is within it’s error of —1. It was shifted to —1.000 ± 0.056. The 403 MeV and440 MeV data have similar zero crossings at 67.8° and 68.2° respectively. The 440 MeVdata show a second zero crossing at approximately 117°.These analyzing power curves do not have cuts on the CM energy of the irppsystem. The reason being that the analyzing powers change slowly with energy. Anindependent analysis of the 403 MeV beam energy analyzing power was performed byHahn [6] and is compared with this analysis in figure 4.17 The two separate analysesare in good agreement. Also shown in figure 4.17 is the 400 MeV data of Ponting. It isseen that the zero crossing of the Ponting data is shifted to an angle 4.9° higher thenthat found in this work. This difference is attributed to an error in Ponting’s anglecalibration.4.3.4 Quasifree pri —* 7rpp(’So) Differential Cross SectionsIn section 4.1.4 the procedure for extracting free pn —f irpp(’So) differential crosssections from the pd —* 7rPPPs data was discussed. Because this experiment oniysamples a small fraction of the diproton mass distribution, only a triple differentialcross section can be extracted from the data. This cross section is written asd3o-ddM”where it is implied that it is an average over the detector acceptance and a range of thediproton mass squared, The range of averaged over is expressed as a rangeof diproton excitation energy, (defined by equation 4.13),CHAPTER 4. ANALYSIS 156Figure 4.16: Quasifree pri —* irpp(1S0)analyzing powers, ANO, at a) TOM = 55MeV(TB = 403MeV) b) TOM = 70MeV (TB = 440MeV) beam energies, as a function ofpion centre of mass angle, .TCMSS MeVI I ia)1.00.50.0—0.5—1.0b)1.00.50.0—0.5—1.0=— I I FIllIllIllIll0 30 60 90 120 150 1806 (deg)TCM=?O MeVI I I I I I I I I=0 30 60 90 120 150 1800 (deg)CHAPTER 4. ANALYSIS 157TCM(MeV) (deg) ANO55 43.9 +0.1 -0.835 + 0.06647.1 +0.1 -0.880 + 0.06750.5 +0.1 -1.000 + 0.05656.2 +0.1 -0.820 + 0.06359.8 +0.1 -0.784 + 0.06363.8 +0.1 -0.544 + 0.06073.6 +0.1 0.648 + 0.05877.6 +0.1 0.980 + 0.06481.7 +0.1 0.863 + 0.06289.4 +0.1 0.748 + 0.06493.6 +0.1 0.653 + 0.06397.6 +0.1 0.512 + 0.062105.6 +0.1 0.469 + 0.046109.8 +0.1 0.377 + 0.045113.6 +0.1 0.472 + 0.048121.9 +0.2 0.281 + 0.076125.7 +0.1 0.226 + 0.075128.8 +0.1 0.238 + 0.07870 61.8 +0.1 -0.488 + 0.07465.4 +0.1 -0.245 + 0.07069.3 +0.1 0.132 + 0.06976.1 +0.1 0.623 + 0.07480.0 +0.1 0.930 + 0.08283.8 +0.1 0.757 + 0.08191.9 +0.1 0.480 + 0.06995.7 +0.1 0.428 + 0.06899.5 +0.1 0.346 + 0.068107.0 +0.1 0.097 + 0.059110.9 +0.1 0.059 + 0.059114.7 +0.1 0.150 + 0.060123.1 +0.1 -0.057 + 0.054126.6 +0.1 -0.129 + 0.054129.8 +0.1 -0.133 + 0.056Table 4.3: Quasifree pn —÷ irpp(’So) analyzing power, ANO, as a function of the centreof mass production angle of the r, 8, at TCM = 55MeV (TB = 403MeV) andTCM = 70MeV (TB = 440MeV).CHAPTER 4. ANALYSIS 158z1.00.50.0—0.5—1.0TcM’SS MeVFigure 4.17: pn —* irpp(1S0) analyzing powers, ANO, for this experiment analysed in this work (solid points) and by Hahn[6] (open points) at TB = 403MeV(TCM = 55MeV). The crosses are the measurement by Ponting [33] at TB = 400MeV.The zero crossing of the analyzing power measured in this work is at an angle of 67.8°while that measured by Ponting is at 72.7° 4.9° higher.90O(deg)CHAPTER 4. ANALYSIS 159.05.04Cl).03CdCd.01.00160Figure 4.18: Quasifree pn —* irpp(’So) centre of mass kinetic energy, TOM, for runs atbeam energies 403 and 440 MeV.The maximum value, T’ is related to the value of the P cut used to isolate the S-wavediproton (see section 4.3.2). The P cut at 75 MeV/c corresponds to = 1.5MeVso the triple differential cross sections are averaged over M% in the range, 0 T1.5MeV.As for the pd — dir+n8 data, the Fermi motion of the target particle smearsout the energy of the irpp CM for pd — irppp3. Figure 4.18 shows the reconstructedirpp CM kinetic energy, TOM, distributions for typical runs at 403 and 440 MeV beamenergies. The target neutron Fermi motion can be seen to spread out the CM energyby over 40 MeV. The average CM kinetic energy for the TB = 403MeV data is TOM =55MeV and for the TB = 440MeV data, TOM = 70MeV.Following the procedure for the extraction of the quasifree pp —* dir+ cross sections in section 4.2, the quasifree pri —÷ 7rpp(’So) data is binned at the central irpp CMkinetic energy with a bin width of +10MeV. For the 403 MeV beam energy data the0 20 40 60 80 100 120 140TCM(MeV)CHAPTER 4. ANALYSIS 160* j3 / 2CM ,. U0/U u(MeV) (deg) (pb/srMeV255 47.2 + 0.2 12.0 ± 1.260.6 + 0.1 4.7 + 0.277.7 + 0.1 6.1 + 0.393.6 + 0.1 15.0 + 1.0109.7 + 0.1 32.2 + 2.5125.3 + 0.1 42.9 + 3.170 66.3 + 0.1 7.5 + 0.680.5+0.1 7.6+0.595.9+0.1 19.9+1.1111.4 + 0.1 40.3 + 2.1126.6 + 0.1 58.3 ± 2.2Table 4.4: Quasifree pri —* irpp(1S0) centre of mass triple differential cross sectionsaveraged over the diproton excitation energy range, 0 1.5MeV.corresponding bin is TOM = 55 ± 10MeV and for the 440 MeV beam energy, the binis TOM = 70 + 10MeV. The resulting centre of mass triple differential cross sectionsare shown in figure 4.19 and are tabulated in table 4.4. Also shown in figure 4.19 aresecond and third order Legendre Polynomial fits to the data. It can be seen that forboth energies, the second order Legendre polynomial is not able to fit the data preciselyand that a third order fit is required.A test of the energy calibration of the apparatus is to compare the 403 MeV dataagainst the 440 MeV data by binning them in the same energy bin. Thus the 403 MeVdata are binned at the TOM energy used for the 440 MeV data and vise versa. Figure4.20 shows both data sets binned at TOM = 55MeV and 70 MeV. The data are seen toagree in the 70 MeV energy bin. However, the 440 MeV data are high at back anglesfor the 55 MeV bin. The cause of this difference is unknown and can be speculated tobe a manifestation of the incomplete understanding of the calibration of the apparatus.The errors in the cross sections shown in figure 4.19 are the statistical countingCHAPTER 4. ANALYSIS8070ci)50—‘ 402010080,_ 70>ci)5030C-db-O10060TCM=55 MeV180180161Figure 4.19: Quasifree pn —* irpp(’S0) centre of mass triple differential cross sectionsat TCM = 55MeV and TCM = 70MeV as a function of centre of mass pion angle, 6,. Forboth energies, the cross sections are averaged over in the range 0 < 1.5MeV.Points experimental data. Dashed line — second order Legendre polynomial fit todata. Solid line — third order Legendre polynomial fit.0 30 60 90 120 150Q(deg)TCMr=70 MeV0 30 60 90 120 1509(deg)CHAPTER 4. ANALYSIS 162TCM=SS±1O MeV TCM=70±tO MeVI I I I I• 403 MeV data • 403 MeV data0 440 MeV data ,_70 0 440 MeV data60 6050 5030 3020 201ol00 30 60 90 120 150 180 0 30 60 90 120 150 180O,(deg) Q,(deg)Figure 4.20: Both the 403 MeV data (solid points) and the 440 MeV data (open points)binned at TOM = 55MeV and TOM = 70MeV. Two of the 403 MeV data points areabsent from the 70 MeV binning because the yields were too low to calculate reliablecross sections.errors added in quadrature to the statistical errors in the phase space integrals calculated by the Monte Carlo simulation. The phase space integrals have statistical errorsof less than 2.5%.There are various possible sources of systematic error in the cross section calculation. The beam current has a 4% systematic error due to the change in SEMefficiency with beam position (section 3.1.3). The QQD acceptance calculation has a2% systematic error (see section 3.2.8 and figure 3.13). In addition, there could besystematic errors in the scintillator bar positions and the QQD angle offset. Errorsin these parameters will effect the phase space integral calculation in the Monte Carlosimulation. Varying the bar positions by +1cm showed a change in the phase spaceintegral of 2.7% when averaged for several runs. Similarly, a +10 shift in the QQDoffset angle caused an average change of 2.5% in the phase space integral. The errorin estimating the areas of the peaks in the TDIF spectra is 4%. Adding these errorsCHAPTER 4. ANALYSIS 163linearly gives an estimate of the systematic error of= 15.2%,or, adding them in quadrature gives,Equad = 7%.These estimates of e compare reasonably to the observed variations between thereconstructed quasifree pp —p d7r+ differential cross sections and the free pp —* dir+differential cross sections discussed in section 4.2. These data were found to have amaximum deviation of less than 15%. To be conservative, a value of 20% is assignedto the systematic error for the reconstructed quasifree pn — irpp(1S0)cross sections.4.3.5 Four Body ContaminationIn section 4.1.4 the procedure for extracting the quasifree pn —p irpp process frompd — 7rppp is described. The assumption is that the proton in the deiiteron actsas a spectator, leaving the reaction with only the Fermi motion it had inside thebound deuteron. The analysis of this experiment uses the Reid Soft Core (RSC) wavefunction shown in figure 4.3. The RSC wave function peaks at 45 MeV/c and has atail extending beyond 400 MeV/c. In this section the validity of the Spectator Modelfor the interaction is assessed against an alternative Four Body Model in which all fourfinal state particles uniformly fill the entire four body phase space.In a 4ir experiment, the experimentally observed spectator momentum (i.e. Ps)distribution can be compared directly against the expected theoretical deuteron wavefunction. Tsuboyama[24j made such a comparison at TCM = 380MeV and foundthat the experimental Ps distribution agreed well with the Hulthén wave functionCHAPTER 4. ANALYSIS 164(which is equivalent to the RSC wave function) up to a spectator momentum of 300MeV/c. Beyond 300 MeV/c, Tsuboyama observed a significant enhancement of theexperimental distribution which was attributed to the “spectator” proton participatingin the interaction. The size of the enhancement was approximately 9% of the totaldistribution.Unlike Tsuboyama’s 4ir experiment, the limited acceptance of the apparatus usedin this experiment restricts the range of spectator momentum to approximately Ps150MeV/c and to compare the experimental and theoretical Ps distributions, it isnecessary to calculate the expected Ps distribution by Monte Carlo simulation. Figure4.21 shows the experimental Ps distributions for all eleven data runs for the 403 MeVand 440 MeV beam energies compared against the Spectator Model. The experimentaldata are seen to agree well with the spectator model for most of the runs. For the 403MeV 48° and 60° data, the spectator model is shifted approximately four to eight MeV/chigher than the experimental data. This disagreement can be explained if the MonteCarlo does not accurately simulate the experimental resolution. If the experimentalresolution in the real apparatus is better than in the simulation, the Monte Carlospectator momentum distribution will be peaked at a higher value.One can test whether the spectator hypothesis for the pd —* irppp reaction isvalid by considering an alternate Four Body Model (FBM) in which the beam protonstrikes the deuteron, forming a four body final state of a pion and three protons (one ofwhich is the proton from the deuteron). Under this hypothesis, the “spectator proton”shares with equal probability the energy of the final state with the pion and the othertwo protons. The resulting FBM prediction of the Ps momentum distribution can becompared to the experimentally observed Ps distribution which is expected to matchthe Reid Soft Core Distribution if the Spectator Model is valid.Appendix E derives an expression for the Four Body Model differential cross.0200015-010.005.000.025403MeV 37de20ido idoP(MeV/c)Figure 4.21: pd —* irppp3 P8 distributions for all QQD angles. The points are theexperimental data. The solid histograms are the Spectator Model calculations. Thedashed histograms are the Four Body Model calculations. The distributions are normalized to unit area.CHAPTER 4. ANALYSIS 1650I.025.020.015.010.005.000403MeV 48deg0 50 100 150 2(P(MeV/c).025440MeV 76deg100 150P(MeV/c)[.:LII0I”0 50 100 150P(MeV/c)200CHAPTER 4. ANALYSIS 166section as a function of the irppp CM pion solid angle, Q, the square of the invariantmass of the three protons (S, 1 and 2), M12 the solid angle of the “spectator” protonin the ppp CM frame, Q’, the invariant mass of protons 1 and 2, M12, and the solidangle of proton 1 in the 12 CM frame, 2’1’. The cross section is given by equation E.19,18 n* ri’ ni”U FB 2____________________ddM12Q’d ?2’’ = FB 256(27r)8PWMs1 i’where quantities with a (*) are in the irppp CM frame, quantities with (‘) are in theS12 CM frame and quantities with (“) are in the 12 CM frame. W is the total energyof the final state in the irppp CM frame and P is the momentum of the beam protonin the irppp CM frame. The quantity MFB is the (unknown) relativistic invariantmatrix element. For the purpose of calculating kinematical distributions for the FourBody Model, the matrix element is set to 1.The predicted distribution of P8 from the FBM is determined by integrating thefour body differential cross section to determine the yield,FB NAPt= LTfWCNB Ax J ddM12ddM2dQ’ (4.41)and binning it as a function of Ps. The integration of equation 4.41 is done by a MonteCarlo simulation of the reaction with each event weighted by equation E.19 (see appendix G). The momentum of the pion and two of the three protons are reconstructedfrom the spectrometer and scintillator bar array. The momentum of the third (undetected) proton is then reconstructed. The momentum distribution of this reconstructedthird proton, Y(P5), is the FBM Ps distribution.Looking at the Monte Carlo simulations of the Four Body Model spectator moCHAPTER 4. ANALYSIS 167mentum distributions in figure 4.21, it is seen that there is no agreement with theexperimental data. At small angles the Four Body Ps distributions look similar inshape to the experimental data but are shifted approximately 25 MeV/c higher. Atback angles for both the 403 and 440 MeV data, the Four Body distributions are muchbroader that the experimental distributions and are peaked approximately 40 MeV/chigher.An estimate of the expected event rates due to the 4 body process can be madeby assuming that the totalpd —4 irppp cross section is due entirely to 4 body processes.This estimate was made by extrapolating Tsuboyama’s measurement of the total dp —*irppp cross sections to the energies investigated in this work and calculating the fourbody Monte Carlo estimate of the experimental acceptance. The resulting event rateturned out to be 11 to 60 times smaller that the event rate observed in this experiment.This means that between 90 and 98% of observed yield could not be explained by the4 body hypothesis.The consistency of the spectator model was also investigated by looking away fromthe region where the spectator momentum is near zero. One way this was investigatedwas to set the QQD spectrometer to select pions of lower energy than that expectedfor a zero momentum spectator. In figure 4.22 a comparison is shown between Psdistributions obtained at a nominal setting and the same setting with the QQD set ata 30% lower B-field. In both cases the spectator model does a better job of explainingthe observed Ps distribution than the Four Body Model. The conclusion is that thereis no evidence of processes other than the quasifree pri —* irpp involved in pd —* irpppfor the data sample obtained in this experiment.To further the case that only the quasifree pri —4 irpp process is seen in thisexperiment, the cross sections calculated from the runs for the nominal and low QQDsettings can also be compared. The CM energies of the events from the nominal andCHAPTER 4. ANALYSIS 168.025.0200-I-).015.005.000Figure 4.22: Ps distributions for a) QQD at nominal setting; b) QQD set to selectlow energy pions. The points are the experimental distribution. The solid histogramsare the Spectator Model calculations. The dashed histogram are the Four Body Modelcalculations. The distributions are normalized to unit area.low QQD setting overlap at TOM = 40MeV. The differential cross section calculatedin the range TOM = 40 + 5MeV is,dQdM2 = 8.9 + 0.8(pb/sr . MeV),for the nominal QQD setting and 10 + 2pb/sr . MeV for the low QQD setting. Thesecross sections agree within error. This study is equivalent to binning the 403 MeV dataat 440 MeV energies and vise versa. Such a comparison has already been presented infigure 4.20. The data binned both ways are consistent as was discussed in section 4.3.4.In summary, the alternative hypotheses of spectator and four body processes havebeen applied to the analysis of the experimental data. The data support, in all cases,Ps distributions consistent with the Spectator Model. The observed distributions arenot consistent with the Four Body Model. The conclusion has to be that no evidencefor four body processes is seen in the energy regions examined in this experiment.P(MeV/c) P(MeV/c)Chapter 5DiscussionThe comparison of the results from this experiment to other experiments and to theavailable theoretical calculations is facilitated by a partial wave analysis of the data fromwhich the amplitudes and strengths of the different reaction channels are extracted andcompared to theory and other experiments. Comparisons of the triple differential crosssections extracted from the data of this experiment to other experimental measurementsand theoretical calculations are complicated by the fact that, with the exception of themeasurement by Handler{15], all the measurements and calculations are of the singledifferential cross sections with respect to the pion solid angle, du/d. Thus to comparecross section distributions from this work with other experiments and theory, it isassumed that the shapes of the cross section distributions do not change significantlyover the diproton mass distribution. The partial wave analysis discussed in section 5.1is, strictly speaking, only valid for the small part of the diproton mass distributioninvestigated in this experiment. However, it is to he noted that a significant fractionof all diproton events lie within this small cut in the diproton mass spectrum due tothe final state interaction between the two final state protons.169CHAPTER 5. DISCUSSION 1705.1 Partial Wave AnalysisAs it was discussed in section 1.2, allowing up to d-wave pions, gives rise to five partialwave amplitudes with nine free parameters necessary to fit the data. These amplitudesand their associated transitions are listed in table 1.2. The redundancy of one ofthe phases of the five amplitudes was removed by fixing ap3 to be real. Using theexpressions for cross section and analyzing power developed in appendix A, a partialwave amplitude fit was done to the differential cross section and analyzing power datafor both the 55 MeV and 70 MeV data. Figure 5.1 shows the partial wave fits usingeither s and p-wave pious or s, p and d-wave pions. It can be seen that using only sand p-wave pious is insufficient to fit either the TCM = 55MeV or the TCM = 70MeVdata. This is particularly true for the 70 MeV data where d-wave pion contribution isnecessary to fit the second zero crossing of the analyzing power at 116.7°.When fitting with s, p and d-wave pions, it was necessary to add an artificialconstraint to the fit because the angular range of the data is not sufficient to constrainthe fit at forward and backward angles. If fitted without a constraint, the fitted crosssections tended to suddenly drop at small and large angles as can be seen for the 70 MeVdata in figure 5.2. In general, the differential cross sections for NN —* NN processesare not observed to have such complicated shapes. Thus making the assumption thatthe cross sections should vary smoothly, without sudden wiggles, a constraint wasplaced on the fitting process to prevent the slope of the fitted cross sections from beingnegative at back angles. This constraint was applied to both the 55 and 70 MeV data.The effect on the 55 MeV fit was minimal, but the 70 MeV fitted cross section becamemore reasonable. Table 5.1 shows the x2 per degree of freedom, i’, for the s,p andd-wave amplitude fits with and without the constraint. It is seen that the constrainthad little effect on the x2/v.When fitting the 55 MeV data, three different solutions with identical x2 per100908070604030b 20100TGM(MeV)(MeV)557010090807060504030b100x2/v(spd) x2/v(spd)no constraint constrained1.343 1.3470.876 0.972Table 5.1: Goodness of Fit for PWA. The s and p-wave fits are incapable of fitting thedata. The constrained s, p and d-wave fits are only slightly worse that the unconstrainedfits.CHAPTER 5. DISCUSSIONTCM=55 MeV171TCM=7O MeV1.00.51.00.00.5z 0.0—1.0—0.590 120o (deg)—1.090 120 150o, (deg)Figure 5.1: Partial Wave Fits using s and p-wave pions (dashed line) or s, p and d-wavepions (solid line). It is seen that d-wave pions are needed to fit both the 55 and 70MeV data.x2/u(sp)13.73034.636CHAPTER 5. DISCUSSION 1721009080706050403020100TCM=SS MeV TCM=7O MeV1009080706050403020100TCM=7O MeV1.0 1.00.5 0.50.0 0.0—0.5 —0.5—1.0 —1.0UFigure 5.2: Partial Wave fits without constraint.is seen to decrease at forward and back angles.730 60 90 120 150 1808 (deg)The 70 MeV differential cross sectionCHAPTER 5. DISCUSSION 173soil sol2 soi3ap ( 17.9±3.1) ( 15.24±0.63) ( 16.4±3.2)asp ( 9.l±l.9)+i( 1.1±2.7) ( 11.98±0.93) +i( 7.71±0.50) ( 8.0±l.6)+( 1.7±1.2)aD (-12.0±3.0)+i( 6.8±2.9) (-10.34±0.68) +i( 0.38±0.42) ( -6.9±l.4)+i( 2.9±2.7)apd ( 4.334±0.003)-fi(-l.3±3.9) ( -1.64±0.58) +i( 1.24±0.63) ( l.5±l.3)+i(-2.0±l.0)aFd ( 0.9± 2.5)+i( 1.2±1.9) ( -3.5766±0.0004)+i(-0.87±O.36) ( 8.4±l.l)+i(-6.8±l.0)fp3 0.26±0.11 0.18±0.02 0.21±0.09f 0.20±0.11 0.49±0.07 0.16±0.07fDp 0.45±0.22 0.26±0.04 0.13±0.06fa 0.08±0.04 0.02±0.01 0.03±0.03fFd 0.0 1±0.03 0.05±0.004 0.47±0.12f 0.65±0.13 0.74±0.03 0.3±0.1314±74 pb/MeV2 313±22 pb/MeV 314±48 pb/MeV2Table 5.2: Partial Wave solutions for 55 MeV. The a’s are complex amplitudes withunits [pb/(sr . MeV2)j’/ The f’s are the fractions of the total cross section due toeach amplitude except for foi which is the fraction of the total cross section due toall the u01 amplitudes.degree of freedom were found. The partial wave amplitudes, the relative contributionsof the amplitudes to the total cross section and the integrated total cross sectionsfor these solutions are shown in table 5.2. The fraction of the integrated total crosssection due to a01 (foi in table 5.2) shows that both solutions 1 and 2 predict that themajority of the total cross section is due to a01 (the asp and amplitudes) and thatthe contributions from the d-wave pion amplitudes (apd and aFd) are small. Solution 3on the other hand predicts that the d-wave amplitude, aFd is dominant and that 70%of the total cross section is due to o — the opposite of solutions 1 and 2. All threesolutions predict a significant ap8, which is necessary to produce the zero crossing atapproximately 700.The partial wave fit to the 70 MeV data only had the one solution shown intable 5.3. This probably comes from the fact that the complex shape of the 70 MeVanalyzing power allows less ambiguity. Again, as was seen in all the 55 MeV solutions,ap5 is seen to be significant. However, asp is found to be stronger than a, which issimilar to solution 2 for TOM = 55MeV. The integrated cross section is 78+11% aoi.CHAPTER 5. DISCUSSION 174amplitude valueap8 (14.9±4.5)as (14.0±1.6) +i(12.1±4.7)aD (-8.76±O.76)+i(-5.1±5.2)apd (-3.6±2.0) --i( 1.4±1.3)aFd (-4.0±1.6) +i(-0.5±2.6)f3 0.13±0.09fs 0.60±0.26fDp 0.18±0.11fPd 0.04±0.04fFd 0.05±0.04foi 0.78±0.11430± 110 pb/MeV2Table 5.3: Partial Wave Solution for 70 MeV. The a’s are the complex amplitudes withunits [pb/(sr MeV2)]112. The f’s are the fractions of the total cross section due to aparticular amplitude — except for foi which is the fraction of the total cross sectiondue to the all u01 amplitudes.Using the argument that the partial wave amplitudes should not change dramaticallywith increasing energy, the 55 MeV solution 2 (having the strongest resemblance tothe 70 MeV solution) can be picked as the most probable solution for the partial waveamplitudes for that energy. It should be noted that this solution predicts a 74 + 3%contribution to the total cross section from the 001 channel which is in agreement withan estimate by Stanislaus e al., who also estimate a 74% contribution from o atapproximately the same energy [67]. The resulting partial wave fits to the data areshown in figures 5.3 and 5.4. Shown also are the observables generated only from thea01 and o amplitudes. Figure 5.5 shows an Argand plot[68] of the imaginary vs realparts of the amplitudes from the TOM = 55MeV (solution 2) and 70 MeV partial wavefits. It can be seen from figure 5.5 that the 55 MeV partial wave amplitudes havesmaller error bars than the 70 MeV amplitudes.Figure 5.6 shows a comparison between the 55 MeV analyzing power data fromthis experiment and the two partial wave solutions of Piasetzky[12]. It is seen thatneither of Piasetzky’s solutions are able to reproduce the zero crossing of the analyzingC’?>ci)-cC’:?Figure 5.3: Quasifree pri —* irpp(’So) Partial Wave Analysis, TOM = 55MeV. Thesolid curve is the total fit to the data. The dashed curves are the observables usingonly the ooi amplitudes and the dotted curve using only the J amplitudes.CHAPTER 5. DISCUSSION 175TCM=55 MeV10090607060504030201001.00.50.0—0.5—1.06CM ( d e g)TCMrrSS MeV0 30 60 90 120 1500 (deg)180Figure 5.4: Quasifree pn —* irpp(’So) Partial Wave Analysis, TCM 70MeV. Thesolid curve is the total fit to the data. The dashed curves are the observables usingonly the amplitudes and the dotted curve using only the J amplitudes.CHAPTER 5. DISCUSSIONTCMZr?O MeV1761009080706050403020100Q)-DcCz0 30 60 90 120 1508CM ( d e g)TcM=?O MeV1801.00.50.0—0.5—1.090O (deg)Figure 5.5: Argand plot of the imaginary vs real parts of the partial wave amplitudesfor TOM = 55 (solution 2) and 70 MeV.CHAPTER 5. DISCUSSION 177Im[a]20aPs—ii Re[a]20Im[a]12Re[a]12—20Im[a]20aSpaDp—12—20Im[a]6Re[a]20Re[a]6—12aPd705Im[a]6—6aFd—6 —6CHAPTER 5. DISCUSSION 1781.00.50.0—0.5—1.0Figure 5.6: Comparison of quasifree pn — rpp(’So) analyzing power from this experiment and Piasetzky’s PWA at TCM = 55MeV.power. Both of the Piasetzky solutions predict a small contribution from ap3. Theabove analysis shows that a much larger contribution is needed to reproduce the smallangle of the zero crossing. Neither solution has analyzing powers approaching unity asis observed for the experimental data.5.2 Comparison With Other ExperimentsIt is possible to compare the differential cross sections measured in this experiment tocross sections measured by Handler[15] for np —* 7rpp(1S0)and to differential crosssections measured for3He(ir, pri)n experiments. However, because this experimentmeasures triple differential cross sections, the only direct comparison is with Handlerwhere triple differential cross sections can also be extracted. To compare the shapesof the cross sections from this experiment with the single differential cross sectionsmeasured for the pion absorption experiments, it is necessary to scale the data.90O(deg)CHAPTER 5. DISCUSSION 179The parameterization of Handler[15] (see appendix B) fitted the pn —* irpp crosssection to a data distribution which ranged from TOM = 0 (threshold) to 70 MeV withthe peak concentration of events at TOM = 56MeV. The comparison of Handler’sparameterization with the triple differential cross section measured in this experimentis done by integrating Handler’s five fold differential cross section over the solid angleof one of the protons, Q, and over the same range of diproton excitation energy, T,,,,, asin the measurement presented here, namely 0 1.5MeV. Figure 5.7 shows thecomparison between this experiment and Handler’s pn —f irpp(1S) cross section forTOM = 55 and 70 MeV. Handler’sd3a/dQdM7,(solid curve) is high compared to thiswork. However, by scaling the Handler cross section to fit the data (dashed curve), itis seen that the shapes of the Handler cross sections are similar to the 55 and 70 MeVdata of this experiment with the minima of the cross sections at the same angle as thedata. The 55 MeV Handler cross section has better agreement with the data than the70 MeV cross section (which has too low a minimum). This is not surprising since thebulk of the data that were fitted by Handler were near 55 MeV.The scaling factors used to fit the Handler cross sections to the data from thisexperiment show that Handler overestimates the differential cross sections by 1.8 +0.2 for the TOM = 55MeV data and 2.0 + 0.2 for the TOM = 70MeV data. ThatHandler overestimates the triple differential cross section compared to the data fromthis work is not surprising. In appendix B it was observed that Handler overestimatesthe pn —* irpp total cross section also by a factor of approximately 1.8 in the energyregion studied in this experiment. As well, appendix B shows that extracting tripledifferential cross sections from the Handler parameterization of pn —* irpp(’ S) isuncertain because of the sensitivity of the parameterization to the form of the functionused to describe the final state interaction of the two protons. Changing the form of thisfunction resulted in a 26% change in magnitude of the triple differential cross sections.CHAPTER 5. DISCUSSIONa)b)8070a)C,)%% 50-o30-dob-d 10080,—70C’)a)U)-D300-dbrOb06060MeV180Figure 5.7: Comparison of quasifree pri —* irpp(1S0)triple differential cross sectionwith the Handler parameterization for a) T01117 = 55MeV and b) TCM = 70MeV. Thepoints are3u/dQ.dM% measured in this experiment. The solid line isd3u/dQdMfrom Handler, averaged over M% in the same interval of 0 1.5MeV as theexperimental cross section. The dashed line is the Handlerd3a/dQdM% scaled to fitthe experimental data.90O(deg)CHAPTER 5. DISCUSSION 181The absolute magnitude of triple differential cross sections extracted from Handler’sparameterization should thus be considered unreliable. It should also be noted that insection 4.3.2 it was found that the Handler parameterization over estimated the P-wavecontribution to the event yield observed in this experiment.Thed3u/ddM measurement from this experiment can also be compared toto the du/dQ measurements from the pion absorption experiments. However, it isnecessary to assume that the shape of the differential cross section does not changesignificantly when integrated over the square of the diproton mass distribution, M.Then the cross sections can be compared by scaling the absorption cross sections tofit the production cross sections as is done in figure 5.8 where the TCM = 55MeVproduction cross section is compared to the scaled T, = 82.8MeV pion absorption crosssection; which when corrections are made for the binding energy of the 3He nucleusand for the Fermi motion of the spectator neutron has a centre of mass kinetic energyof 61.0 MeV. It is seen from figure 5.8 that the pion production cross section from thismeasurement has a minimum at a significantly smaller angle than the absorption data.The minimum of the production cross section is located at 67.3° while the minima ofthe absorption cross section is at 80.9°, a difference of 13.6°.The higher angle for the pion absorption minimum is also inconsistent with theanalyzing powers measured for the production channel in this experiment. That is tosay, the amplitudes required to fit the zero crossing angle of the production analyzingpowers are incompatible with the amplitudes required to fit the angle of the minimumin the absorption data. Fitting the absorption data requires a smaller contributionto ap3 than is found in the production data. This results in a larger fraction of Oncontributing to the cross section as determined by the Piasetzky analysis (93%) thanis found from the partial wave analysis done in this work on the production data(74 + 3%). This difference in shape between the absorption and production crossCHAPTER 5. DISCUSSION 18280Q)Cl)50-D4030-dC-db-ilo0180Figure 5.8: Comparison of quasifree pn pp(’So)d3/dQdM and3He(,pn)nda/dQ as a function of the pion centre of mass angle, 6. The TCM = 55MeV production data is compared to the T = 82.8MeV (TGM = 61.0MeV) absorption datafrom [26]. The pion absorption cross section has been scaled to fit the production data.The lines are second and third order Legendre Polynomial fits to the absorption data(dashed) and production data (solid) respectively.T ‘55 MeV0 30 60 90 120 150O(deg)CHAPTER 5. DISCUSSION 183sections may invalidate the assumption that the shape of the production cross sectiondoes not change significantly over the diproton mass distribution. Alternatively, it maybe evidence that the pion absorption process on the diproton in 3He is significantlyaltered from the unbound process. A plausible explanation for this difference betweenthe absorption and production processes is given by a recent theoretical calculation byNiskanen[35]. This calculation shows (see section 5.3) that the position of the crosssection minimum is sensitive to the shape of the diproton radial wave function whichis different for the case of the unbound diproton in pion production compared to thecase of the bound diproton in pion absorption.5.3 Comparison with TheoryMost of the theoretical calculations for the pri —* irpp(’ S0) system are for the absorption process onto a 3He nucleus. All the calculations consider differential cross sections or analyzing powers averaged over the square of the diproton mass distribution,M%, making direct comparison with data from this experiment impossible. Thus thed3u/ddM% differential cross sections measured in this experiment are compared tothe theoretical calculations for da/dQ. by scaling the theoretical angular distributionsto fit the experimental ones. This allows a comparison of the shapes of the distributionsprovided the assumption that the shapes of the distributions are independent of isvalid. Similarly, it is assumed that the analyzing power is independent of the diprotonmass.The Meson Exchange Model (MEM) of Maxwell and Cheung[31] was used tocalculate differential cross sections for pion absorption by the diproton in 3He. Thedashed line in figure 5.9a shows a comparison between the Meson Exchange Modelcross section and the data from this experiment at TOM = 55MeV. The model doesnot reproduce the strong dip in the cross section nor does it have a minimum as farCHAPTER 5. DISCUSSION 184forward as the data. Also, the Meson Exchange model predicts that the cross sectionis forward peaked while the data is peaked at back angles. From table 1.3 it is seenthat almost all of the strength for this model is in a,, while the partial wave analysisfinds significant contributions from ap3, asp and a (table 5.2).An analyzing power was calculated by Bachman, Riley and Hollas[36] using theDKS (Dubach, Kloet and Silbar) model[37] for the pion production reaction pn —*irpp(1S0)at an energy comparable to the TCM = 55MeV data. It is seen in figure5.9b that the DKS model has the correct basic shape to the asymmetry but it’s zerocrossing is close to 900, compared to the 67.8° for the data. The amplitudes from theBachman calculation were not reported.Both cross sections and analyzing powers were calculated for the Quark ClusterModel (QCM) by Miller and Gal[32] and are compared to the 55 MeV data in figure5.9. Like the DKS model the basic asymmetry of the experimental analyzing powersare reproduced. However the zero crossing is much further forward, in better agreementwith the data. Unlike the Meson Exchange Model, the Quark Cluster Model has itscross section peaked at back angles. Still, the minimum in the cross section is muchshallower than the data and at a larger angle. This model has most of its strength inasp with a small contribution from aD (table 1.3).By far the best agreement with both the cross sections and analyzing powers at55 MeV is found for the Coupled Channels Model using the heavy meson exchangemechanism (CCM(hme)) calculated by Niskanen[35J. While the calculation was originally performed for pion absorption onto the diproton pair in 3He, Niskanen modified itto examine pion production by changing the diproton wave function used from that fora bound diproton to that for a free diproton and approximating the kinematics for thepion production process. This is not a completely satisfactory procedure for studyingpion production since the integrations over the free diproton wave functions are moreCHAPTER 5. DISCUSSION 18570ci)Cl) 50-D— 40-d 10080MeVMeVN1.0 -0.5 -0.0—0.5—1.0Q, (deg)Figure 5.9: Comparison of quasifree pn. —* irpp(1S0) data and theory atTOM = 55MeV. a) The experimentald3/ddM compared to the Coupled ChannelsModel with heavy meson exchange (CCM(hme)), the Meson Exchange Model (MEM)and the Quark Cluster Model (QCM). All the theoretical curves have been scaled to fitthe experimental data. b) The experimental ANO compared to the Coupled Channelsmodel with heavy meson exchange, Quark Cluster Model and the DKS model.0 30this exptCCM(hme)QCM- - -- DKS60 90 120 150 180CHAPTER 5. DISCUSSION 186amp TOM = 55MeV TOM = 70MeVf8 0.22 0.17fs 0.52 0.53fDp 0.25 0.29fPd 0.0001 0.0001fFd 0.01 0.01Table 5.4: Relative partial wave strengths for Coupled Channels Model with heavymeson exchange (CCM(hme)).complicated than the integrations over a bound wave function. However, a proper calculation of the pion production process is in progress and the results are expected soon[69]. Shown as the solid lines in figure 5.9, the Coupled Channels Model is seen tohave very good agreement with the shape of the cross sections and is able to reproducethe zero crossing of the analyzing power. The only significant difference between theCCM model and the data is that for the analyzing power the data has a more dramaticswing in the asymmetry from negative to positive values. The contributions to thetotal cross section from the different amplitudes for the Coupled Channels Model withheavy meson exchange are shown in table 5.4 for TOM = 55 and 70 MeV. It is seen thatthe relative strengths of the CCM amplitudes ap3, as and aD at TOM 55MeV aresimilar to the strengths from partial wave solution 2 (the favoured solution) from table5.2. The Coupled Channels Model predicts much smaller amplitudes for apd and aFdhowever.The apparent success of the Coupled Channels Model at TOM = 55MeV is dueto the incorporation of the heavy meson exchange mechanism for calculating the ap3amplitude. The first CCM calculations [13] used an S wave rescattering mechanism forthe calculation of ap8 and had both the zero crossing of the ANO and the minimum ofthe cross section at higher angles. This is shown in figure 5.10 for the three energies,TOM = 33, 55 and 70 MeV.CHAPTER 5. DISCUSSION 18780N>60-50403020be 100N6050403020b100- —— CCM swr0 30 60 90 120 150 180O, (deg)Figure 5.10: Comparison of quasifree pri — 7rpp(’So) data and CCM model atTOM = 33MeV (top row), 55 MeV (middle row) and 70 MeV (bottom row). Thesolid curve is the CCM calculation using heavy meson exchange (CCM hme) and thedashed curve is the CCM calculation using S wave rescattering (CCM swr). The CCMcross sections are scaled to fit the experimental data. The 33 MeV analyzing powerdata is from Hahn[6].1.00.50.0—0.5—1.0TCM=55 MeV1.00.50.0—0.580 1.00.50.0—0.590 1206(deg)—1.090 120 150o (deg)CHAPTER 5. DISCUSSION 188TGM Expt CCM(hme)33 658° 603°55 678° 685°70 68.2° 71.3°Table 5.5: Analyzing Power Zero Crossing. Experiment and Coupled Channels Modelusing Heavy Meson Exchange.While the agreement with the experimental ANO zero crossing is impressive at 55MeV, figure 5.10 shows that the CCM(hme) zero crossing has an energy dependence,shifting it to larger angles with increasing energy which is not seen in the experimentaldata. Table 5.5 shows that the experimental zero crossing shifts only 2.4° over therange of energies measured in this experiment, where as the CCM(hme) predicts thatthe zero crossing shifts 11.0° over this same energy range. In addition, the second zerocrossing at TaM = 70MeV is not reproduced. The failure to reproduce the secondzero crossing is likely due to the CCM(hme) predicting small contributions to the crosssection from ap and aFd (see table 5.4). The partial wave analysis in table 5.3 showsthat a significant contribution from these amplitudes is required to produce the secondzero crossing.Another thing to note about the CCM result is that while the heavy meson exchange mechanism is necessary to reproduce the shape of the differential cross sectionand the zero crossing of the analyzing power (at TaM = 55MeV), the result is alsosensitive to the form of the wave function used for the diproton. As stated above, theCCM(hme) results presented here are based on a pion absorption code that approximates the kinematics for pion production and uses an estimate of the diproton wavefunction expected for pion production. By using a different wave function that is expected to more closely resemble that of the bound diproton in 3He, Niskanen was ableto reproduce the shape of the pion absorption differential cross section[35] (which hasCHAPTER 5. DISCUSSION 189a minimum at an angle 13.6° higher than the production data). So far, the CCM(hme)pion production calculations are all based on the code for pion absorption. It will beinteresting to see results of the Coupled Channels Model applied specifically to the pionproduction reaction. This is a calculation which is currently in progress [69].Chapter 6ConclusionsThis experiment measured triple differential cross sections and analyzing powers forthe reaction n —* irpp(1S0)by isolating the quasifree reaction from d —* irppp8.This was an extension of the analyzing power measurement by Ponting{33] at a protonbeam kinetic energy of 400 MeV, and data were taken at beam energies of 353, 403and 440 MeV (which correspond to centre of mass kinetic energies of TCM = 33, 55and 70 MeV). The observed shapes of the spectator proton momentum distributionsfrom the data were consistent with the Spectator Model of pion production from thedeuteron target as were the experimental yields. There was no evidence for four bodycontamination. The diproton ‘S0 state was isolated by a cut on the relative momentumof the two final state protons. Comparisons with the relative momentum distributionsof Handler[15] indicate that there is at most a 4% P-wave diproton contamination ofthe data sample.Studies of quasifree pp — dir indicated that while the calibrations were adequateto extract cross sections and analyzing powers at the mean centre of mass energy foreach incident beam energy, the calibrations were not adequate to subdivide the datainto smaller energy bins. Hence, for the quasifree pri —f irpp(1S0)reaction, the sameenergy binning was used. The maximum observed deviation of the extracted quasifreepp — dir+ cross section measurements from the expected free cross sections was 15%.190CHAPTER 6. CONCLUSIONS 191This number was used as an estimate of the systematic error that could be present inthe pn —* 7rpp(1S0)cross sections presented in this thesis. To be more conservative instating the estimated maximum systematic error that could he present, the 15% hasbeen rounded up to 20%. Thus it is expected, to a high degree of certainty, that thenormalization of the pn —b rpp(’S0) cross sections presented here cannot be morethan 20% in error.The basic shape of the analyzing powers were similar at all three energies, thedominant feature being a rapid cross over in the asymmetry from approximately -1to +1 with an almost constant angle for the zero crossing. The asymmetric shape ofboth the analyzing powers and differential cross sections about 90° clearly indicate thepresence of both a01 and a channels in the reaction. The TCM = 70MeV (TB =440MeV) analyzing powers have a second zero crossing at a centre of mass pion angleof 116.7° which requires a significant contribution of d-wave pions in the reaction.The TCM = 55MeV (TB = 403MeV) analyzing powers were similar to the Pontingmeasurement with one notable difference. The Ponting analyzing power has a zerocrossing at a centre of mass angle 4.9° higher than this measurement. Because thespectrometer angle was independently measured as part of the calibration procedurefor this experiment, the different zero crossing angle found by Ponting is attributed toa calibration error in the Ponting experiment.The clear signal of both the isospin 0 and 1 channels being present in this reactionis in qualitative agreement with the results of Handler[15] and Yodh[14] at similarenergies (TCM 70MeV) and of Bannwarth[20] (who measured ratios of npto np —* irnn differential cross sections) in the range TCM = 85MeV to 185 MeV.However, the analysis of Tsuboyama[24j saw no appreciable a01 for the full pri —* rppreaction for energies above TCM = 80MeV until a TCM of 380 MeV was reached. Theapparent importance of a01 at lower energies and its disappearance at higher energiesCHAPTER 6. CONCLUSIONS 192can probably be attributed to the channel pri —* 7rpp(’So) which has a significantisospin 0 contribution, becoming less important at higher energies with respect to thechannel pn —÷ 7rpp(P — wave) which is mostly U.The angles of the minima of the triple differential cross sections measured byHandler are similar to those found from the analysis of this experiment (approximately700) for both the 55 and 70 MeV data. The shape of the Handler differential crosssection is similar to that measured in this experiment at 55 MeV but shows a lowerminimum than the 70 MeV data. The Handler parameterization overestimated themagnitude of the triple differential cross sections by a factor of two compared to themeasurement of this experiment. This is similar to Handler’s overestimate of the pn —irpp total cross section by a factor of approximately 1.8 in the energy range examinedin this experiment (see appendix B). It is noted that the model dependent nature ofthe Handler parameterization of the pn — irpp(1S0)cross sections makes it unreliablefor the extraction of triple differential cross sections. It was also observed in section4.3.2 that Handler overestimates the P-wave diproton contributions to the reaction.Since this experiment measures triple differential cross sections and the pion absorption experiments measure single differential cross sections, it is necessary to assumethat the shape of the differential cross section does not change significantly over thediproton mass distribution in order to compare the shapes. Such a comparison between the TOM = 55MeV production data and the TOM = 61.0MeV (T7 = 82.8MeV)absorption data found that the pion production cross section had a minimum at anangle 13.6° lower than the absorption cross section. This may be an indication that theassumption that the shape of the cross section does not change over the diproton massis invalid or it may be an indication that the neutron in the 3He is not a spectator in theprocess, but rather an active participant. Alternatively this shift may be an indicationthat the matrix element is dependent on the difference in the radial wave functions forCHAPTER 6. CONCLUSIONS 193a hound and unbound diproton. This last hypothesis is supported by recent theoreticalcalculations by Niskanen[35].The previous partial wave analysis of the pion absorption channel by Piasetzkyconcluded the importance of the a5 or a amplitudes with almost no apt. Thatanalysis predicted that 93% of the total cross section was due to O.j. However, thePiasetzky partial wave solutions failed to reproduce the zero crossing of either thePonting analyzing powers or those measured in this experiment. The reason for thisfailure is the insufficient ap3 amplitude in the Piasetzky analysis.The partial wave analysis done for this experiment found it necessary to includet7- < 2 and found three solutions for the 55 MeV data and one solution for the 70 MeVdata. While solution 2 of the 55 MeV fits (shown in figure 5.3) is preferred (becauseof its similarity to the 70 MeV solution), all of the solutions show the importance ofall three of the i, < 1 amplitudes ap5, asp and aD. The ape, I = 1, amplitude isparticularly important in that its interference with the asp and aD, I = 0 amplitudesproduces the observed low angle zero crossing of the analyzing power. The necessityof including d-wave pions in the partial wave analysis is most apparent for the 70 MeVanalyzing power data which has a second zero crossing at 116.7°. However, even the 55MeV partial wave fits are significantly improved by the inclusion of d-wave pions. Thisis the first clear evidence for the participation of d-wave pions in the pn —* irpp(’S0)reaction at these energies. The previous analysis by Piasetzky assumed that the d-wavecontributions were negligible.Solution 2 from the 55 MeV partial wave analysis and the single solution from the70 MeV partial wave analysis are shown in table 6.1 along with the Piasetzky partialwave solutions and various theoretical calculations. It is seen from table 6.1 that twoof the earlier models for pn —* irpp(’So), the Meson Exchange Model[31] (MEM) andthe Quark Cluster Model[32J (QCM) showed almost all of the reaction strength inCHAPTER 6. CONCLUSIONS 194TCM solution fs fD fPd fFd55 PWA 0.18 + 0.02 0.49 + 0.07 0.26 ± 0.04 0.02 + 0.01 0.05 ± 0.004Pia “S” 0 07 0.52 0 41Pia “D” 0.07 0.09 0.84MEM 0.00 0.02 0.98QCM 0.02 0.87 0.11CCM(hme) 0.22 0.52 0.25 0.0001 0.0170 PWA 0.13 + 0.09 0.60 + 0.26 0.18 + 0.11 0.04 + 0.04 0.05 + 0.04CCM(hme) 0.17 0.53 0.29 0.0001 0.01Table 6.1: The fraction of the integrated pri —4 irpp(’So) cross sections, f, contributedto by the different partial wave amplitudes. The PWA solutions are the partial wave solutions from this work (the 55 MeV PWA is solution 2); Pia “S” and Pia “D” are the Piasetzky solutions[12]; MEM is the Meson Exchange Model of Maxwell and Cheung[31];QCM is the Quark Cluster Model of Miller and Gal[32] and CCM(hme) is the CoupledChannels Model with heavy meson exchange by Niskanen[35].either the a channel (MEM) or the a8 channel (QCM) with little or no strengthin the ap5, I = 1, amplitude. The Pouting analyzing power measurement tended tofavour Piasetzky’s solution “S” where the reaction proceeded approximately equallythrough both of these channels. This led Pouting to speculate that both the MesonExchange and Quark Cluster reaction mechanisms were present. However, the recentresults of the Coupled Channels Model using heavy meson exchange (CCM(hme)) byNiskanen[35] show a contribution from ap comparable to the results of the partialwave analysis on the data from this experiment and show reasonable agreement withthe data without invoking nonbaryonic degrees of freedom.In light of these recent theoretical results from Niskanen using the Coupled Channels Model with heavy meson exchange, future experiments on the pn — irpp(’So)system will clearly benefit from theoretical input. In particular a study of the observables (cross section and spin) that distinguish between the different possible reactionmechanisms (heavy meson exchange or s-wave rescattering) will be needed in planningfuture experiments. Also a study of the observables which shed light on the apparentCHAPTER 6. CONCLUSIONS 195differences between the pion absorption onto the bound diproton in 3He and the pionproduction with a free diproton will he useful.From a purely experimental point of view, it is desirable to constrain the solutionsfor the partial wave analysis more tightly. This can be done by both extending the rangeof the cross section and analyzing power measurements to smaller and larger centre ofmass angles and by measuring other spin observables such as ALT or ALL1. It is alsodesirable to further study the energy dependence of the observables for R —* irpp(1S0).By going to lower energies where the s-wave pion amplitude is expected to dominate,it may be possible to investigate the heavy meson exchange or s-wave rescatteringmechanisms. By extending the measurements to higher energies, through the L massrange (TCM 154MeV), possible resonant behaviour of the o channels may bedetectable. Finally it is desirable to measure the pn —* irppQSo) differential crosssection integrated over the diproton mass distribution (i.e. da/d). However, such ameasurement necessitates measuring the P-wave diproton contributions to the pn —*rpp reaction as well. Full measurements of the rip —* irpp reaction including crosssections and spin observables are being performed[70] but are much more complicatedthan the experiment discussed in this thesis — both in the implementation of themeasurements and in the analysis. It will be interesting to compare the measurementspresented here to the results of these new experiments when they become available.In conclusion, by measuring both differential cross sections and analyzing powers,this experiment has shown the importance of the 3P0— — 1S0s— transition amplitude,apt, with it’s significant Oii contribution and the necessity for the inclusion of d-wavepions. The shape of the differential cross section curve measured in this experimentat TOM = 55MeV is found to be different from the shape of a pion absorption crosssection measured at a similar energy, indicating a difference between the production1The optimum observables to measure would have to be determined by studying the sensitivity ofthe solutions to the various observahies.CHAPTER 6. CONCLUSIONS 196and absorption mechanisms. Although the a01 contribution estimated from this analysisis not as large as the 93% estimated from the absorption data by Piasetzky, it doesshow that C is greatly suppressed compared to other NN —* irNN processes. Thedominance of the a01 channel and the resulting suppression of the Li resonance clearlymakes pn —* irpp(’So) a good reaction for the study of nonresonant pion production.Appendix APartial Wave Analysis of pn —* irpp(’So)The intent of Partial Wave Analysis is to express observables for a reaction in a formthat is independent of the angles and spin states of the particles involved. This isdone by writing the matrix elements as products of transition amplitudes, aif, fromthe initial to final states and functions describing the angular dependency of each finalstate. The basis used to expand the matrix elements is called the LS basis (orbitalangular momentum, L, and spin, 5).The particular observables studied in this experiment are the spin averaged differential cross section,da = + da_], (A.1)and the analyzing power,du — d_ANO= , A.2du + du_where the cross sections du+ and du_ are the cross sections for the beam proton polarized spin up and down respectively striking an unpolarized neutron target. To derivethe expressions for the polarized cross sections, d+ and du_, a coordinate system ischosen where z is along the beam direction, pointing downstream, y is vertically upand x is horizontal (completing a right handed coordinate system). The axis of quantization is the z axis (along the beam). The spin state of a proton polarized verticallyis then written1 ip+) = +) +The cross section is given by the sum over the square of the matrix elements divided197APPENDIX A. PARTIAL WAVE ANALYSIS OF FN - PP(1S0) 198by the product of the spins of the initial states, s,dcr = (fTji)I2. (A.3)The beam polarized cross section has only two allowed initial states, neutron polarizedup, n+), and down, n—), giving s = 2. Combining the neutron spin states with thepolarized beam proton give the allowed initial states,i)= { 0 fl+) + +) \/ — (A.4)p )Øn—)_+—) ——)and the polarized cross section for beam spin up or down, du±, becomes,du = ++) + - +)}+ (fT{ + -) +-_>}]= + +) + i(fT - + j(fT + -) + i(fTj - _)j2j. (A.5)The matrix elements, (fTi), are written as expansions over the initial and final statesin terms of the complex amplitude, Cif,(f{Ti) = (f6(Jf, J)6(Mf, M)6(Pf, P)af Ii), (A.6)ifwhere the delta functions conserve total angular momentum, J, its projection onto thez axis, M, and the parity of the initial and final states, P. The summation is over theinitial and final state quantum numbers.The initial state can be decomposed into a spacial part, qj(x), and a spin part,(1, 2),Ii) = (x) 0 (1, 2),The indices 1 and 2 refer to the spins of the two initial state nucleons. The spacial partis a plane wave describing the incident particle, travelling with momentum, k, parallelto the z axis, and can be decomposed into orbital angular momentum partial waves,L, Mi). Expanding the plane wave givesqS(x)APPENDIX A. PARTIAL WAVE ANALYSIS OF PN ,‘ irPP(’So) 199= jLt2L + ljL(kZ)Y(&, i). (A.7)where 0,== 0 is the angle of the incident particle with respect to the z axis. TheBessel function, jL(kz), the spherical harmonic, 1’(0, i) and the term areabsorbed into the amplitude and are dropped from the description of the initial stategiving,j(x) = + 1 IL, 0) (A.8)L1 =0Note that the projection of the initial state orbital angular momentum onto the z axisis always zero. The spin part of the initial state wave function, described by the totalspin of two nucleons, S, and the projection of the total spin, ms1, can be decomposedinto a sum of the individual spins of the nucleons,(A.9)where the summation is over S, the total spin of the two nucleons and mS, the projection of S. For a given initial state, the spin projections of the individual nucleons arespecified, eg. for 1+—), ms1 = 1/2 and m52 = —1/2. The term (Si, ins,; 52, ms2 S, ms1)is a Clebsch Gordan coefficient weighting the mixture of S and S2. Combining thespacial and spin parts, the initial state is,i) = +L, 0; S, ms1; J, ]14) (A.10)where the summation is over J, the initial state total angular momentum, M, theprojection of J, L, S and rns. The term (Li, 0; S, ms lJ, M) is a Clebsch Gordan coefficient weighting the combination of L and S which form the total angularmomentum J.The final state, (f is treated in a simpler fashion than the initial state. Thespacial part of the final state is written,f(X) =ymLf(9) (A.11)Lf =0APPENDIX A. PARTIAL WAVE ANALYSIS OF FN — PP(’S0) 200and the complete final state is written,(f I =ymLf( )(Lf, mLf; Sf, ms1; Jf, Mf I, (A.12)where any numerical factors are absorbed into the amplitude. The spherical harmonics,mLfY (0, ) describe the angular dependence of the final state.Inserting equations A.10 and A.12 into A.6, the matrix element becomes,(fITIi) = (Lf,mLf;Sf,msf;Jf,MfY’(0,)x 6(Jf,J)6(Mf,M)8(Pf,P)afx i/2Lj+1(Si,ms;, sISi,ms)(Lj,0,Sj,msJJi,Mj)x (A.13)where the summation is over J, L, S, m8, Jf, Mf, Lf, rnLf, Sj and ms1. Invokingthe delta functions conserving the angular momentum, J = Jf and is replaced with J.Similarly, ms = M = Mf and is replaced with M. Because it equals rns, M can onlyhave values 0 and +1. For pn — irpp(’So), the final state nucleons have a combinedspin of zero so Sf = m81 = 0 and mLf = M. The final state angular momentum, Lf isthe angular momentum of the pion, l7 (and is equal to J but l. will be kept for clarity).The initial state nucleons are spin 1/2 particles which couple to form a total angularmomentum of zero or one. However, only the 5, = 1 state couples to the final state so(Si,ms1;S2,msIS,m ) becomes, (l/2,ins1;1/2,rnsj1,rns) and the summation overS is dropped. Then equation A.13 becomes,(fITIi) = (lw, M; 0,0; J, MM(6, g)x 6(Pf, Pj)af2Lj + 1(1/2, ms1; 1/2, ins2 1, M)(L, 0,1, MIJ,M)x IL,0;1,M;J,M), (A.14)where the summation is over J(= lv), M and L. The angle 0 is the polar angle of thescattered pion and is the angle of the scattering plane with respect to the xz plane.The allowed transitions and the amplitude notation for pn —+ 7rpp(’So) up to J = 2are given in table A.1.The first few terms of the matrix elements in equation A.5 can he written outexplicitly. The spacial component of the initial state wave function is,(x) = S+P+vD+vF+..., (A.15)APPENDIX A. PARTIAL WAVE AJ\TALYSIS OF PN -+ irPP(1So) 201Ipn) (irppf3P0- — ap8 —* 1S0s-3S1+ — asp —÷ ‘S0p1+3D1+ <— a, —* 1Sop+3P2-—apJ —4 ‘S0d2-3F2— aFd —* 1S0d2_Table A.1: Partial Wave Amplitudes for pn —* irpp(1S0)using the spectroscopic notation for the orbital angular momentum states. The spincomponent of the wave function is given by,= 11)o(1,2)= == Il-i)As noted above, only the spin triplet couples to the final state so the cv(+—) andc(—+) terms are written without 00) components. The initial states that couple tothe irpp(’So) final state for J 2 are then+ +) = ® 11) = 3s1+ + 3p2 + D1+3F2_, (A.16)+—) = 0 10) = + 3P2- — 3p0_ — 3D1+ —F2-, (A.17)I — +) = ® j1o) = + 3P2- — 3p0_— 3D1+ —F2-, (A.18)and——) = o 1 — 1) = 3s1+ + 3p2_+ D1+3F2-. (A.19)The final state is written (J < 2),(f I = o(lSos0)+ p+)+}M(1Sop_), (A.20)APPENDIX A. PARTIAL WAVE ANALYSIS OF PN -+ PP(’S0) 202where again M = 0, ±1. Substituting equations A.16 through A.19 and A.20 intoequation A.6 and only allowing the transitions in table A.1, the matrix elements up toJ 2 are,(fT++) = (as+ aD)Yt’ +(apd+aF)Y’, (A.21)(fTI +—) = —apY0°+ (as — aD)Y° + (apd — aFd)Y2°, (A.22)(fIT — +) = + (asp—aD)Yl° + (apd — aFd)Y2°, (A.23)(A.24)and(fT— —) = (asp + aD)1+ (apd + aFd)Y21. (A.25)Appendix BHandler Parameterizat ion of pn —* rppHandler[15] measured the np —* irpp reaction using a hydrogen bubble chamber anda neutron beam with a kinetic energy distribution ranging from 287 MeV (the pionproduction threshold) to 440 MeV. The beam energy distribution was peaked at 409MeV. The data were fitted with a Gell-Mann and Watson parameterization[7j whichcan be used to generate differential cross sections for the pn —* irpp as part of a MonteCarlo simulation.Figure B.1 shows the momentum vectors of the reaction products in the irpp centreof mass system. The incident neutron is labeled N, the two final state protons andq’, and the pion Q. Defining some kinematical variables, P is the relative momentumof the two protons in the rpp CM system,(B.1)p is the cosine of the angle between the beam neutron and P,Up = cos(P, N), (B.2)UQ is the cosine of the angle between the beam neutron and the pion vector, Q,= cos(,]), (B.3)u is the cosine of the angle between 1 and ,(B.4)and r is the ratio of the pion momentum to its maximum allowed momentum (where203APPENDIX B. HANDLER PARAMETERIZATION OF PN - -PPQq204Figure B.1: Momentum vectors of a rip — irpp event in the irpp CM system.the two protons have no relative momentum, i.e. P = 0),marMost of the momentum and energy variables are expressed in units of the pion mass,m. It is important to note that the angles are defined with respect to the beamneutron unlike in this experiment where the angles are defined with respect to thebeam proton.Handler considers the allowed transitions for the rip — irpp reaction with thefollowing assumptions:1. the final state diproton can only have angular momentum 0 or 1 (S or P-wavediproton),2. the final state pion can only have an angular momentum of 0 or 1 (s or p-wavepion),3. except for final state interactions, the momentum dependence of the final statepartial waves is determined by phase space,NPAPPENDIX B. HANDLER PARAMETERIZATION OF PN -* irPP 205Initial Final TransitionType state Final I P amplitudeSs 3P0 ‘S0. 1 - ah(P)Sp S0p 0 + b1oh(P)Q3D1 13OPi 0 + 2oh(P)Q1S0 3P0s 1 + c1P1D2 3P2s 1 + c2PP. 3S 3P1s 0 + c1oP3D1 3Ps1 0 + c2oP3D2 3P2s 0 + c3oP3P0 3P1p0 1 - d10PQ3P1 3P0p1 1 - d01PQ3P1 3P1p 1 - d1PQ3P1 3P2p1 1 - d21PQPp 3P2 3P1p2 1 - d12PQ3P2 3P2p 1 - d22PQ‘P1 3Pop 0 - dooPQ‘P, 3P,p1 0 - d10PQ‘P, 3P2p, 0 - d20PQTable B.1: Allowed transitions and energy dependence of the transitions for the Handlerparameterization of np —* 7rpp.4. only when the final state protons are in a relative S wave is there a significantfinal state interaction.Based on these selection rules, Handler lists the allowed transitions in table B.1. Thetransition amplitudes are expressed as functions of the kinematic variables P, Q, .t,,up and ,u. h(P), shown in figure B.5a, is a function which describes the final stateinteraction between the protons when they are in a relative S state. The total transitionprobability is expressed as a sum of transitions to different irpp states,f = fpp + fps + fsp + fss + + fss5, (B.5)APPENDIX B. HANDLER PARAMETERIZATION OF PN- PP 206where the expressions for the individual transition probabilities fitted to the bubblechamber data are (expressing the momentum variables and masses in units of m):f1 = P2Q[(l.32 + 0.13) + (0.92 + 0.21)it+(—0.05 + 0.21)i4 + (—0.13 + 0.19)112+(0.76 + 0.48)ipf1Q], (B.6)fp8 = 0, (B.7)fs = h2(P)Q[(—0.11 + 0.045) + (1.32 + 0.08)4], (B.8)fss h2(P)[(0.25 + 0.03)], (B.9)P2Q[(0.17 + 0.06) + (0.37 + 0.16)zQ+H0.22 + 0.10)upJ, (B.10)(B.11)andfss = h2(P)Q[(O.73 + O.04)1uQJ. (B.12)Handler expresses the five fold differential cross section in terms of the solid angleof the pion in the irpp centre of mass, r and,the solid angle of one of the protonsin the irpp centre of mass,ddrdQ = 16ir2As pmax12(T7 - E), (B.13)where a = 53.3 + 4.8ub is the total cross section averaged over all incident neutronenergies, A8 = 35.25 is a normalization factor, /3 is the relative velocity of the incidentneutron in the CM frame, E is the pion CM energy, W is the total CM energy and fis the sum of the Handler transition probabilities, defined in equation B.5.For the purposes of Monte Carlo event generation, it is more convenient to expressthe differential cross section in terms of the variables (Q, JVfQ, Q) where J/I7, is theinvariant mass of the diproton and is the solid angle of one of the protons in the ppCM system. The differential cross sections are related by,,15‘—‘x J B 14— ddrd ‘APPENDIX B. HANDLER PARAMETERIZATION OF PN -+ 7rPP 207where J is a Jacobian transformation,Or___________________.(B.15)OM 2WP ‘ j ( 1:?lrmaxt-’lr I )The term /3 is the velocity of the pp CM in the irpp CM system, is the Lorentzcontraction and = cos(P’, where F’ is the relative momentum of the two protonsin the pp centre of mass frame. Because Handler parameterizes the reaction with respectto the neufron, it in necessary to make an angular change to compare with pn —÷ irpp,namely,=—To test the Handler parameterization, a Monte Carlo simulation of Handler’sexperiment (a 4r detector and a neutron beam with the proper energy distribution) wasperformed. The Monte Carlo simulation was used to successfully reproduce kinematicaldistributions from Handler’s parameterization of his experimental data (figure B.2).Handler quotes a total cross section of 90 + 81ib at TB = 409MeV, (TOM = 56MeV).The Monte Carlo calculates a cross section of 89.8ib which is in good agreement withHandler model.Figure B.3 shows Handler’s total pri — irpp cross section with the dashed linesextrapolating the fit beyond 70 MeV. The upper and lower lines represent an error of9% to the fit’ Comparing Handler to other data, it is seen that, with the exception ofthe old 1955 measurement by Yodh[14], the Handler parameterization tends to be high.In the range TOM = 55MeV to 70 MeV, the Handler pn —* rpp total cross section isapproximately 1.8 times higher than the Ver West[9] parameterization shown in figureB.3.Figure B.4 shows Handler’s differential cross sections for various energies. Asignificant asymmetry of the differential cross section about 90° is seen at all energies indicating significant contributions from both the 00, and reaction channels.It is seen that at TOM = 20MeV, the S wave cross section dominates but becomesincreasingly less important as the energy increases.It is desired to compare Handler’s model for pu —* 7rpp(’So) cross sections to1llandler only quotes an error of 9% for the cross section at TCM = 56MeV. The same error isassumed for all energies and for the pn — irpp(’So) cross section as well.1.25:2 1.000.751t:20.500.250.000.0 0.2 0.4 0.6 0.8 1.0ILFigure B.2: Monte Carlo Simulation of the Handler kinematic variables. The variablesjig, /JQ, t and r are defined in the text. The variable b is the angle between thetwo protons in the irpp CM system. The curve is the Handler parameterization. Thehistogram is the Monte Carlo simulation based around the Handler parameterization.APPENDIX B. HANDLER PARAMETERIZATION OF PN -÷ PP 2081.50/LQ1.4 I I0.8. 0.6-4-)0.40.20.0• I0.0 0.2 0.4 0.6 0.8Co.4—).4-)CeI-C1.0rcos4APPENDIX B. HANDLER PARAMETERIZATION OF PN FP 209120010008006004002000 40 80 120 160TCM(MeV)Figure B.3: fl — rpp total cross sections from Handler[15] (solid lines), Ver West{9}(dotted line), Tsuboyama[24] (open squares), Dakhno[23] (crosses), Yodh[14] (solidtriangles). The dashed lines extend the Handler cross section beyond the valid rangeof the parameterization. The upper and lower lines for the Handler parameterizationpn-ipp total cross sections0show 9% error bounds.APPENDIX B. HANDLER PARAMETERIZATION OF PN -+ irPP 210Figure B.4: Handler’s differential cross sections for pri —* irpp at TCM = 20, 30,40, 50, 60 and 70 MeV. The solid line is the total differential cross section includingcontributions from both S and P wave diproton states. The dashed curve shows justthe S wave diproton cross section.I 1jeV, T=3DMeV10863-Cb2-0108108I:150 11-30 60 90 120 150&(deg)TMeV00 30 60 90 120O(deg)TCM=SOMeV3-b625201530 60 90 120&(deg)TCM=6OMeV10 11 0 0 30 6b 90 120 150O(deg)T=70MeV60 90 120e(deg)60 90 120O(deg)APPENDIX B. HANDLER PARAMETERIZATION OF PN - rPP 211the triple differential cross sections,d3udQdM’averaged over the diproton mass squared, in an interval of the diproton excitationenergy, = — (2m), given by,0 <Tfl. < 1.5MeVmeasured in this work (see chapter 4). This interval of corresponds to the cut ofP <75MeV/c used to isolate the S-wave diproton (section 4.3.2). However, this shouldbe done with caution. Handler fitted 4079 events with a model dependent descriptionof the S-wave proton-proton interaction over a centre of mass energy range of 70 MeV.As is described in the following, the magnitude of the triple differential cross sectioncalculated from the Handler model is sensitive to the form of the function used to modelthe proton-proton interaction.The exact shape of the function, h2(P) used to describe the proton-proton finalstate interaction (PSI) is uncertain. Handler uses a square well potential fitted toproton-proton scattering data (figure B.5). An alternate calculation of the final stateinteraction by Niskanen[75] based on a prescription by Watson[76] is also shown infigure B.5a. It is seen that the Niskanen h2(P) curve is peaked at a slightly highervalue of P than the Handler curve and has a lower tail at high P.The corresponding Handler parameterization of the pn —* 7rpp(’So)d3u/ddMat TCM = 55MeV and & = 750 as a function of is shown in figure B.5b using boththe Handler and Niskanen final state interaction calculations. The cross section for theNiskanen PSI was calculated by substituting the Niskanen h2(P) distribution into theHandler cross section formula and normalizing the total pn —f irpp cross section toHandler’s this procedure should be considered approximate only2. The peaks in theHandler and Niskanen triple differential cross sections near = 1MeV correspondto the peaks in h2(P) near P = 30MeV. The vertical lines in figure B.5 correspondto the cut on P 75MeV/c. It is seen that the cross section using the Niskanen FSIis enhanced in the region 0 1.5MeV. This results in approximately a 26%increase in3/dcdM averaged over 1’VI2 in this region.2This cross section calculated from the Niskanen FSI should not he confused with the CoupledChannels Model cross sections calculated by Niskaneii and discussed elsewhere in this thesis.APPENDIX B. HANDLER PARAMETERIZATION OF PN -* iCPP 212a)30706050403020100600b)cvU]-c020’-dC-db-dFigure B.5: a) h2(P) calculated by Handler (solid line) and by Niskanen{75] (dashedline). b)d3u/dQdM for pn —* irppQSo) as a function of diproton excitation energy,= — (2m) for irpp CM kinetic energy TCM = 55MeV and irpp CM pionscattering angle O. = 750 using the Handlerh2(F) function (solid line) and the Niskanenh2(P) function (dashed line), the vertical dash-dot line in both figures shows theP 75MeV/c (or equivalently the 1.5MeV) cut used to isolate the S-wavediproton.0 100 200 300 400 500P(MeV/e)100 5 10 15 20 25 30 35 40 45 50 55T (MeV)ppAPPENDIX B. HANDLER PARAMETERIZATION OF PN -+ irPP 213Because the Handler triple differential cross sections are sensitive to the exactshape of the h2(P) function used to describe the proton-proton final state interaction and the shape of this function is not certain, Handler’s parameterization of thepn —* irpp(1S0)cross sections should be considered unreliable when used to calculatedtriple differential cross sections in small bins. The region of validity of the parameterization is considered to be total cross sections, t0 and single differential crosssections, du/d.Appendix CBeam Line lB SEM Current MonitorA Secondary Emission Monitor (SEM) is a beam current monitor placed in the beamafter the target and before the beam dump. It consists of a series of thin aluminumplates in a vacuum chamber, alternating between collector plates and emission plates.The emission plates are set at a negative potential of several hundred volts. Thecollector plates are connected through a device called a current integrator, to ground.The positive proton beam passes through the SEM, depositing an amount of energy,E, in the emission plates,= (C.1)where /x is the thickness of the emission plates and dE/dx is the energy loss for aproton in aluminum. A fraction of the deposited energy knocks electrons free fromthe surface of the emission plates. These electrons are attracted to the collector plateswhere the charge accumulates. When the built up charge on the collector plates reachesa certain level, the current integrator discharges the collector plates to ground and emitsa pulse. The discharge level is adjustable on the integrator and is usually set at b_bC/pulse (Coulombs per pulse). The total charge of electrons collected, Qe, is given by,Qe = iiAxQ, (C.2)where Q, is the total charge of the protons which passed through the SEM. The efficiency with which secondary electrons are produced, ?], is assumed independent of theproton beam energy and the potential of the emission plates. The SEM signal is givenby,SEM_Qe_AXdE( C3CF CF dx’214APPENDIX C. BEAM LINE lB SEM CURRENT MONITOR 215where CF is the charge/per pulse setting of the SEM current integrator.The efficiency for producing secondary electrons, i, is dependent on the conditionof the surface of the emission plates, and not directly calculable. Thus, it is necessaryto calibrate the SEM against an absolute current measurement. In principle this needonly be done once after the SEM is assembled and installed in the beam line. However,if the SEM is kept under a poor vacuum, surface films can build up on the plates,affecting the calibration. It was decided to check the SEM calibration at the end ofthis experiment against a Faraday Cup rather than rely on the original calibrations forthe device.A Faraday Cup consists of a large copper block supported in vacuum in whichthe beam protons are stopped. The copper block is encased in (but insulated from) aconducting container held at a positive potential of several hundred volts with respectto ground. The block is connected via a current integrator to ground. Beam protons arestopped in the block and the surrounding positive potential keeps them trapped there.The accumulated positive proton charge is then measured with the current integrator,FCUP = , (C.4)and the ratio of SEM to FCUP gives,(SEM=i1Ax, (C.5)which is a constant.The SEM was calibrated against the Faraday Cup at the three energies pd —*PPPs data were taken at: 353, 403 and 440 MeV. However, some data were alsotaken at 300 MeV and at 450 MeV. For these energies, the nearest measured values of() 1 were used. The calibrations are listed in table C.1. The dE/dx’s used intable C.1 are interpolated from Janni[71]. The systematic error is due to a 4% positiondependent change in the SEM response (see section 3.1.1).APPENDIX C. BEAM LINE lB SEM CURRENT MONITOR 216T dE/dx SEM nC/count Systematic(MeV) (IVIeVcm2/gm) Error300 2.7951 0.2693 7.527 0.1329 0.007353 2.5666 0.2693+0.0002 6.912+0.002 0.1447+0.0001 0.007403 2.4054 0.2710+0.0008 6.519+0.008 0.1534+0.0005 0.008440 2.3115 0.274 +0.005 6.33 +0.05 0.158 +0.003 0.008450 2.2867 0.274 6.27 0.160 0.008Table C.1: SEM Calibration for this experiment.Appendix DBeam Line lB PolarimeterThe TRIUMF beam line lB In Beam Polarimeter (IBP) measures the beam protonpolarization by comparing the asymmetry in elastic pp scattering to the left and rightof the beam direction. The target protons are contained in a thin CR2 target mountedon a target wheel that allowed the selection of four different targets. The scatteredprotons are detected in scintillator arms set at 17° to the left and right of the beam(figure D.1). The recoil protons are detected at the conjugate angle, 700. An event inthe left arm of the polarimeter is triggered by a triple coincidence between scintillatorsP1, P2 and P3. An event in the right arm by a triple coincidence of P4, P5 and P6.The beam polarization is determined from the asymmetry, ECH2, of events in theleft and right arms of the polarimeter,NL — NRECH2 = PAGH2 = (D.1).LVL — JVRwhere P is the beam polarization and ACH2 the analyzing power of the CH2 polarimeter target. NL and NR are the number of events in the left and right arms of thepolarimeter. Solving equation D.1 for beam polarization gives,= CH2 (D.2)ACH2If the polarimeter target was pure hydrogen, then the analyzing power of the targetcould be determined directly from the well measured analyzing powers for pp scattering(the world data for pp scattering is available from the SAID data hase[64]). However,since the target is CR2 there is quasifree pp scattering from the carbon as well as thehydrogen which complicates the analysis.217APPENDIX D. BEAM LINE lB POLARIMETERtargetbeamP6/P5P4scattered proton218andFigure D.1: TRIUMF beam line lB In Beam Polarimeter.The numbers of counts in left and right arms of the polarimeter are given by,NL = N+NL’,NR = N+NN,(D.3)(D.4)where N and Nj’ are the numbers of counts in the left arm of the polarimeter due tocarbon and hydrogen respectively. Similarly, N and Nj1 are the numbers of countsin the right arm of the polarimeter. The number of counts, N, due to a given process,depends on the cross section, , the solid angle of the polarimeter arm, the arealdensity of the target, ricH2, and the number of beam protons, NB,N = NBriCH2O-/ (D.5)Following the Madison convention[60], the cross section for a scattering to the left orright of the beam is given by,P1P2/P3.recoil proton= ao(1+PA) (D.6)APPENDIX D. BEAM LINE lB POLARIMETER 219and= uo(l—PA), (D.7)where A is the analyzing power for the process. The number of events detected in theleft and right arms of the polarimeter can then be written as,NL = NBCH2 [o(C) + 2uo(H) + P(uo(C)Ac + 2c7O(H)AH)] (D.8)andNR = NBOH2 [o(C) + 2a0(H) + P(ao(C)Ao + 2o-o(H)AH)]. (D.9)Substituting equations D.8 and D.9 into equation D.1, the asymmetry of the polarimeter is written— oo(C)Ac +2a0(H)AH DCH2— uo(C) + 2u0(H) ( .10)= PACH2, (D.11)where ACH2 is,A— uo(C)Ac +2u0(H)AHOH2 —ao(C) + 2o-0(H)There is an energy dependent parameterization of AOH2 in terms of the pp analyzingpower, AH, available[72],ACH2= 1.0830— () [0.07486 — () 0.00823] , (D.12)where AH is taken from the the SAID data base and has an estimated uncertainty of1%. Assuming the uncertainty in formula D.12 is negligible, the 1% uncertainty in AHleads to a 1% systematic uncertainty in the measured beam polarization. Reference[72] has a similar formula for calculating the event rates, R, in the polarimeter,R = pt {44.17— () [1.171 — () 1.250] }, (D.13)APPENDIX D. BEAM LINE lB POLARIMETER 220target material thickness(mg/cm2)1 thick CR2 67.84572 thin CR2 5.3623 empty4 carbon 25.4318Table D.1: Polarimeter targets. Target 2 was used throughout the experiment.Tp(MeV) 0CM AH AGH2/A ACH2 R(cnts/nC)300 36.45 0.4054 0.93249 0.3780 3522353 36.861 0.4414 0.921297 0.4067 3773403 37.244 0.4663 0.9149768 0.4267 4054440 37.524 0.4820 0.9129488 0.4400 4289450 37.599 0.4861 0.9127875 0.4437 4357Table D.2: GIl2 analyzing powers and expected polarimeter rates for target 2.where R = RL + RR is the total event rate in the polarimeter, expressed in counts pernanocoulomb of beam (cuts/nC), p is the density of the target in gm/cm3 and t is thethickness of the target in cm.The four selectable targets of the polarimeter are listed in table D.1. Except forthe calibration runs at the end of the experiment, target 2 was used throughout theexperiment. The CH2 analyzing powers, ACH2, and event rate, R, for target 2 aregiven in table D.2. In fact, the observed rates were approximately 1.3 times larger thanthose reported in table D.2. This discrepancy is unexplained and the SEM (SecondaryEmission Monitor) is used to measure the beam current throughout the experiment.Appendix ESpectator and Four Body ModelsThis experiment extracts free pu —+ irpp cross sections from measurements of pd —*rppp data. To do this it is assumed that pion production from the deuteron is entirelyfrom the bound neutron with the proton in the deuteron being only a spectator. Themodel developed to extract the free cross sections is called the Speciator Model andis explained in this appendix. An alternate hypothesis for pion production from thedeuteron assumes that the proton in the deuteron is an active participant. A modelthat describes the kinematic distributions of the final state particles from pd — rpppusing this hypothesis, called the Four Body Model, is also described in this appendix.E.1 Spectator ModelThe Spectator Model[73J makes two assumptions to extract cross sections for the priirpp reaction from the pd —* irppp reaction. Consider a reaction,p+J\T_f, (E.1)where a proton, p, strikes a nucleon, N, undergoes an interaction and forms a finalstate, f, with some number of particles, Nf. This process, referred to as the freereaction, is specified by its differential cross section which is a function of the energyof the interaction (usually in the pN Centre of Mass) and the kinematical parametersof the final state particles. Now consider a nucleus, A, which is comprised of the sametype of nucleon, N, as in the free reaction plus other nucleons which will be collectivelyreferred to as S, i.e. A = N + S. Suppose a proton strikes this nucleus in a reaction(which will be referred to as the bound reaction),p+A f+ S,221APPENDIX E. SPECTATOR AND FOUR BODY MODELS 222producing a final state that consists of the same final state particles as for the freereaction, f, and an additional (single) particle S which is comprised of the “spare”nucleons from A. The first assumption of the spectator model is that the bound reactioncan be thought of as,p+N*+Sf+S,where the proton strikes the bound nucleon (designated by N* to indicate that it isbound) without interacting with the rest of the nucleus, 5, which is referred to as aspectator. In the final state, this spectator particle is, of course, a real particle and ithas the same Fermi momentum it had as constituent of the nucleus, denoted by Ps.The fundamental interaction, p + N* f, is referred to as the quasifree reactionand differs from the free reaction in two ways. First, the target nucleon is off its massshell due to the binding energy of the nucleus. Second, the target nucleon is movingwith an equal and opposite momentum to the spectator particle in the lab frame (therest frame of the nucleus, A). A relationship between the cross sections for the quasifreereaction with the moving offshell target nucleon and the bound reaction can be derived.The second assumption of the Spectator Model is that for a given centre of massenergy, the matrix element for the quasifree reaction, MQ = (fIMQIp, N*), and thefree reaction, M = (fMp,N), are equal,(fIMQIp,N*) = (fMIp,N). (E.2)That is to say the processes are the same even though the quasifree reaction involvesa nucleon that is off its mass shell and in a nuclear environment. The differentialcross section for the quasifree process is thus equal to the differential cross section forthe free process. Combining this with the first assumption of the Spectator Model, arelationship can be derived between the bound and free reactions.To derive the relationship between the free and bound cross sections, one startswith the Relativistic Fermi’s Golden Rule[73] which gives the cross section in terms ofthe Lorentz invariant phase space,2EB2ETIvB — vTda = M2(2)44PF—?‘) }, (E.3)where EB and ET are the energies of the beam and target particle, yB— vTI is therelative velocity of the beam and target particle and du is the differential cross sectionfor the reaction. M is the matrix element for the process and is a relativistic invariant.APPENDIX E. SPECTATOR AND FOUR BODY MODELS 223P and Pf are the total four momenta of the initial and final states respectively andthe terms,1 dSP,2Ej(2ir)3’are relativistic invariant phase space elements for each of the final state particles. Eachside of equation E.3 is a relativistic invariant, meaning that it will give the same resultevaluated in any reference frame.Evaluating the golden rule for the free reaction, p + N— f, in the pN centre ofmass frame gives,2E2Ev- vdJF = M2(2)46PF- P) fl }, (E.4)where (*) denotes pN CM quantities and the product of phase space elements is overthe number of final state particles, Nf. The term v— v7 can be rewritten,P* P* P* P* P** * — B N_ B B_ BV — VTT — —— -I- —E* * * * * *B N B N NBwhere, in the pN centre of mass, P7 = —P and W (E + E) is the total CMenergy. Substituting this result into equation E.4 gives,4PWdUF = M 2(9)44(p - P1) { (2)3 (E.6)Now evaluating the bound reaction, p + A— f + S, in the laboratory frame,2EBAV &B =T2(27r)46PF—1:,I) ii } 2ir)3’ (E.7)where because the target nucleus is at rest, its total energy, EA is equal to it’s mass,MA and yB — vT is B• Equation E.7 has the same final state phase space productsas for the free reaction but also has an additional term for the spectator particle. Thematrix element for the bound reaction, T, is related to the matrix element for thequasifree reaction, MQ, by,= (S,fMQp,A)2APPENDIX E. SPECTATOR AND FOUR BODY MODELS 224= I(S(fMQIP)A)2. (E.8)The interaction amplitude is a relativistic invariant and requires for each particle’s wavefunction, a normalization (b) = 2E, where E is the total energy of the particle. Thetarget nucleus for the bound reaction, A, is at rest and so has a normalization,(AA)=2MAThe target nucleus wave function is rewritten in terms of the target nucleon and thespectator particle,*_A) = IN )IS)(Ps), (E.9)where (P5) is the wave function in momentum space of the spectator particle and hasthe normalization,f 2(p)S = 1.For the case of a spectator proton in deuterium, g!(Ps) is the deuteron wave function. The wave functions for the target nucleon and spectator have normalizations,(N*IN*) = 2E and (SIS) = 2E5 respectively. Thus the total normalization for IA) is(AlA) = (N*IN*)(SlS)2MEf2(p)S2MA= 2EN* X 2E x X 11N*= 2MAas expected. Substituting equation E.9 into equation E.8 givesIT2 = (SKfIMqlp) IN)IS)(Ps)= (fIMQIP,N*) (E.1O)The second assumption of the spectator model equates the free and quasifree processesas shown in equation E.2. As well, the energy of the bound nucleon, EN* = MA —APPENDIX E. SPECTATOR AND FOUR BODY MODELS 225is replaced with EN = + P — the energy of the free nucleon’. The square ofthe matrix element for the bound process is then= (fMp,N)2MA2ES() (E.11)Substituting equation E.11 into equation E.7 gives,2EB2EA(VB)daB =2MAES(P)IMI464P- Pi) } (2r)’(E.12)and substituting in equation E.6 gives,2M4E 1 d3P2EB2EA(VB)dO-B2EN (Ps)4WdJE23. (E.13)Noting that VB = PB/EB and EA = MA (the target nucleus is at rest) and simplifyingone gets the relationshipWPB 2 d3PdJB= ENFB(Ps)(3 (E.14)The term (WP)/(ENPB) is referred to as the Flux Factor. It arises because thenucleon is no longer stationary. Sometimes the nucleon is moving towards the beamparticle. This is equivalent to increasing the flux of the beam on a stationary targetand increases the yield from the reaction. Alternatively, the nucleon can be movingaway from the beam particle which decreases the beam flux and the yield. The P usedin the flux factor is the momentum of the beam particle in the centre of mass of thefree reaction with CM energy W. The energy of the target nucleon in the lab frame,EN, is defined as,1The difference in the two normalizations for the energy of the target nucleon is small. If the nucleusis deuterium and the target particle is a neutron, the average spectator momentum is approximately 45MeV/c. This gives a ratio,1.005.ENAPPENDIX E. SPECTATOR AND FOUR BODY MODELS 226since Iv = —Ps. The mass of the target nucleon, inN, is its free mass. That is, thetarget nucleon is defined to be on its mass shell with momentum—Ps.One can write out the explicit relationships between cross sections for the freeand bound reactions of interest in this experiment. For pd —÷ irppp3, the spectatormodel gives the relationship,d8 T,T7 *,2( ‘ ,j5pd—irppp— B Y ‘S Sj pnrpp— EPB (2ir)3 dQdMdc2’’where the pn —* 7rpp five fold differential cross section is written with respect to thevariables, Q, the pion solid angle in the irpp CM system, the square of theinvariant mass of the two final state protons and , the solid angle of one of the twoprotons in the pp CM system. For the pd dirn reaction, the relationship betweenbound and free cross sections are,d5ud_+d. — 1/VP 2(Ps)dad+ E16dd1 — EPPB (2K)3 dQ ‘ . )where Q is the pion solid angle in the dir CM system and the spectator is now aneutron.E.2 Four Body ModelAn alternative to the Spectator Model for the description of the pd —* irppp reactionis the Four Body Model. The Four Body Model assumes that the proton strikes thedeuteron, an interaction occurs and a final state is produced in which all four of thefinal state particles, the pion and three protons, have equal probability of sharingthe energy of the interaction. While in this model all three protons have the samebehaviour, only two of them will be detected in the experimental apparatus along withthe pion. The momentum and trajectory of the third proton will be reconstructed fromthe observed particles and will be called the “spectator”, even though there is nothingthat distinguishes it from the other protons.To determine the expected momentum distribution for this reconstructed spectator proton in the Four Body Model, Fermi’s Golden Rule can be developed for thefour body final state. Designating the final state pion by ir and the three final stateAPPENDIX E. SPECTATOR AND FOUR BODY MODELS 227protons by 5, 1 and 2, the golden rule is written as,2E2EIv — VJdUFB =IMFB2 {(2)44(PF— } ,(E.17)where the left side of equation E.17 is evaluated in the centre of mass of the pd system(equals the centre of mass of the irSl2 system), denoted by (*). E is the energyof the beam in the pd CM, E is the energy of the deuteron target in the pd CM,—is the relative velocity of the beam proton and deuteron in the pd CM. Theterm MFB is the relativistic matrix element for the four body process and the termin braces is the differential phase space, dQFB. As discussed in the previous section,— = (PW)/(EE) where P is the beam proton momentum in the pd CMand W is the total energy in the pd CM. Equation E.17 can be written,4P,WdUFB = M2dQFB, (E.18)and the differential phase space can be rewritten as,dQ4= 64( 2 dM12!2dM2!2’1’,where (‘) refers to the CM12 system and (“) to the CM12 system. Ms12 is the invariantmass of protons S, 1 and 2. M12 is the invariant mass of protons 1 and 2. The FourBody differential cross section is then written as,* I IIUt7FB 2 ir Si ElddM12c2d ?Q’ = JVL FB 256(2ir)8PW2M12Appendix FMonte Carlo SimulationMonte Carlo simulations were used extensively at all stages of this experiments. Priorto the experiment, Monte Carlo simulation was used in experimental design to see ifsuch an experiment was feasible. Important quantities such as,• the expected data rates,• P-wave contamination of the data,• and detector resolutionneeded to be determined by Monte Carlo calculation before such an experiment asdescribed in this thesis could be contemplated. However the most important use of theMonte Carlo simulation came during the off-line analysis where it was used to calculate:• the phase space integrals used in the calculation of differential cross sections,• the expected yields from the reaction pp — dir+ to check the absolute normalization of the apparatus,• and the four body contamination of the data.The Monte Carlo simulation of E460 (this experiment) has gone through several incarnations. The original simulation was written by Ponting[34] in Fortran for the analysisof the first run of E460 in 1987. This simulation was expanded and modified to includethe current detector geometry by the Tel Aviv group. It was used to estimate the datarates and P-wave contamination prior to the second E460 run. In parallel with the TelAviv Monte Carlo, this author developed an independent code written in C. After theexperiment, the C code was converted into a GEANT code called G460 which is theMonte Carlo referred to throughout this thesis and described in this appendix.228APPENDIX F. MONTE CARLO SIMULATION 229G460 is a GEANT Monte Carlo simulation of particle interaction events trackedthrough the E460 apparatus. GEANT[74} is a large Fortran program package developedat CERN for the Monte Carlo simulation of subatomic experiments. The user writesroutines to describe the detector geometry, to generate particle event kinematics andto place experimental cuts on successful events. The GEANT routines then track theparticles which undergo interactions, energy loss, multiple scattering and decay. Thedetector description and treatment of particles is fully 3 dimensional. Particle decaysand interactions are based on the current world data and are handled relativistically.Although much of GEANT’s power was unused, it was nevertheless necessary for theE460 Monte Carlo simulation because of its 3 dimensional treatment of the apparatusand its energy loss and multiple scattering routines. Figure F.1 shows the simulationof a pd —* irppp event in GEANT. The pion is seen leaving the scattering chamber tothe top of the figure and is detected in the QQD spectrometer. The two protons fromthe quasifree pri —* irpp reaction leave the proton barrel to the right of the figure andare detected in the scintillator bar array. The spectator proton from the deuteriumtarget is not tracked through the apparatus.The heart of the Monte Carlo is the event generator. While the GEANT routineshandle the tracking of particles through the detectors, it is the users’s responsibilityto specify the momentum and trajectories of the final state particle from a collisionbetween the beam particle and the target. What makes the event generator for thisexperiment somewhat unusual is the fact that the target is not considered to be asimple stationary nucleus, but is a composite consisting of a virtual target and a realspectator. The spectator particle 4-vector is not influenced by the event kinematics,but is chosen such that the spectator momentum distribution is equal to the expectedFermi momentum distribution of the stationary nucleus. The virtual target is given amomentum equal and opposite to that of the spectator. The event generation consistsof the following steps:1. Generation of the momentum and energy of the spectator nucleon. For quasifreepn—f irpp or pp — dir+, the reaction actually simulated is pd —* 7rPPPS andpd ir+dn.. respectively. The spectator nucleon is given the Fermi motion ofthe deuteron. For pd * the target particle is stationary and there is nospectator.2. Calculation of the total momentum and centre of mass energy of the beam andvirtual target particle.APPENDIX F. MONTE CARLO SIMULATION 230Figure F.1: Simulation of pd—irppp event in GEANT. The pion passes throughthe QQD at the top of the diagram. The two protons from the quasifree pn —* -irppreaction are stopped in the scintillator bar array.APPENDIX F. MONTE CARLO SIMULATION 2313. Calculation of the momentums and energies of the final state particles in thecentre of mass of the beam and target particles. To improve the efficiency of theMonte Carlo, the pion is aimed into a small cone centred on the spectrometer’saperture.4. Calculation of the statistical weight of the event. This can be based on phasespace, for example, or differential cross sections for the reaction which have beenspecified as part of the input.5. Conversion of the reaction product 4-vectors to the lab frame.6. After the event generation, GEANT tracks each resultant 4-vector through theapparatus. It is then determined which detectors would have fired and if thetrigger conditions are met, the event is analysed.F.1 Simulation of Beam, Target and Scintillator ArrayThe representation of the E460 experimental apparatus necessarily includes variousapproximations. The lB experimental area is represented by an air volume large enoughto contain all the detectors. The scattering chamber and proton barrel are representedas vacuum spaces of the appropriate dimensions but the actual steel vessels are notincluded. The scattering chamber window (towards the QQD) and the proton barrelwindow are included in the simulation. The target cell consists of the aluminum vesseland the LD2. The mylar insulation around the target cell has negligible effect on theparticles and is not simulated. Rather than simulate the wrappings of the proton barrelexit counters, the scintillator material is thickened slightly to account for the energyloss in the wrapping. The Tel Aviv scintillator bar array is represented by volumes ofmaterial with the appropriate dimensions and properties. The veto counters are eachrepresented by separate volumes.The beam characteristics (position, width, mean direction) are known at thescintillator screen position 6 cm upstream of the target. The G460 simulation createsparticles at this position with gaussian distributions for position, angle and momentum using the parameters determined (from the beam monitor and target scintillatorscreen data) for each beam tune. The scintillators (and wire chambers on the QQD)are represented only as volumes with appropriate energy loss and multiple scatteringAPPENDIX F. MONTE CARLO SIMULATION/232Figure P.2: QQD IVionte Carlo Simulation, Plan View. The various thin boxes representeither wire chambers, scintillators or apertures that the particles must pass through.The solid curves are pion trajectories, the dashed curves are muon trajectories and thedot-dashed lines are neutrino trajectories.characteristics. A particle striking a scintillator is assumed to generate a signal with100% efficiency. No attempt is made to model the light propagation or the phototuberesponse of the scintillators.F.2 Simulation of the SpectrometerThe complexity of a four body final state has necessitated an accurate knowledge of theQ QD acceptance. For this purpose, a detailed Monte Carlo model of the spectrometerhas been created. The Monte Carlo simulation of the spectrometer is comprised ofthree parts: the optics, the apertures and the detectors. Figure P.2 shows the GEANTsimulation of the QQD as well as several pions (solid lines) traced through the spectrometer. Two of the pions have decayed into muons (long dashed lines) and neutrinos/—1• /10cmAPPENDIX F. MONTE CARLO SIMULATION 233(long-short dashed lines). One of the pions decayed after leaving the spectrometer butthe other has decayed inside the QQD. The muon from this decay has successfullypassed through the spectrometer and would be counted as a good event. Because it’sposition in the final wire chambers is quite different from that of a pion, the momentumof the original pion will be erroneously reconstructed.The QQD optics consist of a quadrupole magnet and a dipole magnet. Thedipole is modelled as a magnetic field region located between two iron pole pieces.The magnetic field points vertically in the lab frame and in the mid plane of thedipole, varies radially from the centre of the dipole (following the field profile of figure2.5). Off the median plane the field strength is the same as on the median plane.The quadrupole is modelled as a thick lens with the optics parameters fitted to theQ QD wire chamber data in section 3.2.2. When a particle reaches the centre of thequadrupole, it’s trajectory and position is changed such that it will correspond to thethick lens optics fitted to the experimental data. Aperture checks are made on theparticle at the beginning, centre and end of the quadrupole.The wire chambers are represented as thin air volumes at the appropriate locations in the spectrometer. The plastic chamber windows are not modelled. The actualwire planes are not simulated. Instead, the position in the chamber is reported witha random uniform smearing of plus or minus one half wire spacing. The scintillatorsare modelled as volumes of scintillator material. The pair of scintillators Si and S2are modelled as a single scintillator S12. Similarly S3 and S4 are modelled as a singlescintillator, S34.In addition to the magnetic fields and the detectors on the QQD, there are variousapertures that the particles must traverse. These correspond to restrictions in theQ QD vessel and apertures of the magnets and wire chambers. They are modelledas thin volumes. The simulation notes whether or not the particle passes throughthese aperture volumes. If one of the volumes is missed, the event is rejected. Thismethod for simulating the apertures does not allow for such effects as slit scatteringand penetration of material which are possible in the real spectrometer. The QQDmodel also incorporates the misalignment of the spectrometer discussed in chapter 3.The purpose of the detailed simulation of the QQD is to allow a precise estimate ofthe spectrometer’s acceptance and the muon contamination from pion decay. Thus thetrajectory and momentum of a simulated event in the Monte Carlo are reconstructedfrom the wire chamber data using the same algorithm and parameters as a real dataevent. Because the optics of the Monte Carlo model are not identical to the optics of theAPPENDIX F. MONTE CARLO SIMULATION 234real spectrometer, it is to he expected that there will be differences in the reconstructedpion trajectories and momenta. The quality of the Monte Carlo simulation of the QQDcan be judged by comparing distributions of various quantities reconstructed using realdata and using Monte Carlo data. For these tests, pd —* thr+n5 data are used.Figure F.3 shows reconstruction of the wire chamber x and y distributions. Thesolid histograms are the experimental data and the dashed histograms are the MonteCarlo. The histograms are normalized to the same area. The plots of wire chambers2 and 3 show agreement with the general shapes but tend to be shifted slightly. Thiscould be due to the experimental offsets of wire chambers being not right or it could bedue to inaccurately placed apertures or incorrect optics in the QQD model. x4 and x5are in agreement but the Monte Carlo distributions for y and y5 tend to be narrowerthan the experimental distributions. This probably indicates that the nonbend planeoptics of the QQD model are not correct. This is not surprising since, because thereis no data on the magnetic field off the median plane of the dipole, the model assumesa constant field with value equal to that on the median plane. This is a significantsimplification of the dipole fringe fields.Figure F.4 shows the experimental and Monte Carlo distributions for x0, 00, Yoand ç. The x0 and 8o trace back distributions look in good agreement. However, thisis not surprising since the simulation uses the model that was fitted to the tracebackdata in section 3.2.2. The agreement with the Yo and j distributions are not as good.Again this is not surprising since the QQD model was not able to fit the y trace backdata well.Also shown in figure F.4 is the quantity Y4M5= y—y5. The shapes of the experimental and Monte Carlo distributions are similar but not the positions. This is notunexpected because the QQD y and y5 offsets were never calibrated during the experiment, so neither y4 nor y5 are necessarily centred correctly in the experimental data.Similarly, the difference, Y4M5, will also not be centred correctly in the experimentaldata.Figure F.5 shows the experimental and Monte Carlo 6 distributions. Again thehistograms are normalized to unit area. There are slight differences between the experimental and Monte Carlo data but the general shapes agree. Also shown in figureF.5 is DDIF=64—65, the difference between the two reconstructed momenta. Thetail on the high DDIF side of the peak is due to pions decaying in the spectrometerand resulting muon triggering the detectors. The low DDIF tail is due to the breakdown of the momentum calibration for extreme rays.30000a)20000Figure F.3: QQD Wire Chamber distributions. Solid line — experimental pd —* d7rndata. Dashed line — Monte Carlo Simulation data. The histograms are normalized tothe same area.APPENDIX F. MONTE CARLO SIMULATION 2351600014000120001000080006000400020000500040000 —40 —20 0 20 40 6y(cm)10000a)14000120001000080006000400020000——60 —40 —20 6 20 40 60y5(cm)APPENDIX F. MONTE CARLO SIMULATION600040000200050004000300001 20001000236Figure F.4: QQD Wire Chamber distributions. Solid line — experimental pd — dirn8data. Dashed line Monte Carlo Simulation data.A600000 — o0(deg)03 —4 —2 0y0(cm)2 4 (APPENDIX F. MONTE CARLO SIMULATION 237riQ)Figure F.5: QQD momentum reconstruction, a) 6; b) 85 c) DDIF = 64 — 65. Solidline — experimental pd — dir+n3. Dashed line Monte Carlo Simulation data.12000100006000600040002000Appendix GYields and Phase Space IntegralsThe Monte Carlo simulation was used to calculate the expected yields for different configurations of the experimental apparatus and the Phase Space Integrals (a measure ofthe acceptance of the apparatus) used in the calculation of cross sections. Both theyield calculations and the phase space integrals require calculating multidimensional integrals with complex limits determined by the properties of the experimental apparatusand the software cuts placed on the data. The Monte Carlo method is an effective wayof performing such integrations and allows the inclusion of such effects as experimentalresolution and pion decay in a simple manor.G.1 Predicting Reaction YieldsIn this experiment there are three processes for which the experimental yield is calculated,pd —* irppp(Spectator Model), (G.1)pd — irppp(Four Body Model) (G.2)andpd —* dirn8(Spectator Model). (G.3)Consider a reaction with N final state parameters and an N fold differential crosssection given by, da/dQ, where Q represents the final state variables and is referredto as the Pha&e Space of the reaction. Then if NB beam particles strike a target with238APPENDIX 0. YIELDS AND PHASE SPACE INTEGRALS 239areal density of target particles, j, the yield, Y, of events in a detector array isY=NB77J (G.4)Acceptance dQwhere the integration is over the phase space of the final state particles. The limitsof integration are determined by the acceptance of the detector system. Instead ofperforming an integration with complicated limits, an integration is performed overthe entire phase space, Q, with the integrand weighted by an Acceptance Function,A(Q),Y = NBi f ZA(Q), (G.5)where A(Q) is defined as,A1 f 1 if event accepted by detectors—0 otherwiseand depends on the configuration of the experimental apparatus and any software cutson the data. A(Q) is determined numerically by the Monte Carlo Simulation, i.e.the trajectories of the particles are tracked through the apparatus and if they are obstructed, then the acceptance function is assigned a value of zero. If the particles areaccepted by the detectors and the reconstructed kinematical quantities of the reactionare within the software cuts the acceptance function is one. Because all of the reactions studied in this experiment involve a pion, the Monte Carlo simulation includespion decay on an event by event basis when determining the acceptance of the event.Experimental resolution is incorporated into the integration by simulating the randomerrors in the reconstruction of the event.G.1.1 pd PPPs (Spectator Model)The reaction pd —f PPPs is the production of a negative pion from deuterium viathe quasifree reaction pn—f irpp. The final state has four particles, requiring eightparameters to describe it. As discussed in section4.1.4, the parameters used arethe solid angle of the pion in the irpp CM system, M, the square of the invariantmass of the two protons produced from the quasifree pn—p irpp reaction, 2,, the solidangle of one of the two protons in the pp CM system and Ps is the momentum of theAPPENDIX G. YIELDS AND PHASE SPACE INTEGRALS 240spectator proton. The eight fold differential cross section is written,d8 SMpd—*inppp _ (G.6)dQdMdQ1,dPsSubstituting G.6 into equation 0.5, the yield for pd —* iPPPS 5,d8 SM= NB J ddMdd1A(, , (G.7)Here A(, , fls) is the acceptance function for the eight final state parameters.As discussed in section 4.1.4, under the assumptions of the Spectator Model the eightfold differential cross section can be written in terms of the five fold differential crosssection for pn —* irpp, i.e. (equation E.15)d8 SM TTD* .2(npd—rppp VY B Y 5) U Upn_*pp— EPB (2ir)3 dQdMd’where (P5) is the wave function of the deuteron in momentum space with the normalization,J 2(p)S = (9’)3 J2(P5)Pdd=1, (0.9)and the term, (WP)/(EPB) is the flux factor. The five fold differential cross sectionfor the free pri — rpp reaction,Upn_rppdc;dJV1,dc’is taken from a parameterization of the pn —* irpp reaction by Handler[15} as discussedin appendix B. Substituting equation 0.8 into equation 0.7 gives the expected yieldfrom pd —* irppp in as a function of the free pn —* irpp cross section and the spectatormodel,7SM —1pd—*irppp —NB J EP M Q,,APPENDIX G. YIELDS AND PHASE SPA CE INTEGRALS 241where it should be remembered that the integration is over all allowed phase space andthe acceptance function, A(Q), determines if the event is accepted by the apparatus.The eight dimensional integral, equation G.1O, is quite complicated and the technique used to evaluate it is called Monte Carlo integration. In its simplest form, MonteCarlo integration notes that for large N, the integral of a function f(x) between thelimits a and b is [73],pb zxNJ f(x)dx = lim —f(x), (G.11)a Nwhere x is a uniformly distributed random number in the range [a, b] and /.x = (b—a).For multi-variable functions with independent variables, for example f(x, y, z), theMonte Carlo integral becomesd f zxAAzNj dxf dyf dzf(x,y,z)=lirn (G.12)where /.y = (d — c) and /z= (f — e) and three uniform random numbers, a x, <b,c < y < d and e < z < f are chosen. If on the other hand, one of the variables is afunction of the other variables, e.g. z = z(x, y), the integral becomesb d h(x,y)f dxf dyf dzf(x,y,z) = lim (G.13)a c g(x,y)where L\z = [h(x,y) — g(x,y)j is evaluated for each pair of x and y, chosen. Thisprocedure can be extended to the eight variable integration of equation G.1O,= NBNAptzXMddMdQ c, ]).(G.14)The subscript, i, on the variables j and Ps indicate that these are therandom variables selected for each event. N is the total number of events generated.is the solid angle the pion is scattered into in the total CM system. is the solidangle Pi is scattered into in the diproton CM frame. is the range allowed for thewhich varies for each event depending on the total CM energy, W, which in turndepends on the spectator momentum. The term q52(Ps)/(27r)3is absent from equationG.14 because it is absorbed into the selection of Psi. The procedure for generating aMonte Carlo event is:APPENDIX G. YIELDS AND PHASE SPACE INTEGRALS 2421. The spectator momentum, Is, is randomly selected such that its trajectory isisotropic in space and it satisfies equation G.9. This is done by forming thequantity, i,1 Psu = h(Ps)= (2)3 J 2(P)PdPd8.which gives,4 “Ps‘U = h(Ps)= J q2(P)PdP(2ir) osince the distribution of Ps is isotropic over The variable u, is selected fromthe uniform distribution,0 ‘u 1,and then Ps = h’c). The bound target neutron is given a momentum oppositeto the spectator, Pr-,= Ps.2. The centre of mass energy, W, of the collision between the beam proton and thebound target neutron is calculated. The Lorentz transformations between the labframe and the irpp CM frame are calculated.3. In the irpp CM frame, the square of the invariant mass of the diproton is randomlyselected in the range,(2mp)2 M (W — m)2.Because the centre of mass energy, W, depends on the spectator momentum, 1-vs,the allowed range of is different for each event,= {W—(rn. + 2rn)]4. is chosen by isotropically selecting the trajectory of the pion from the coneA in the irpp CM frame. This scattering cone is centred on the spectrometeraperture and its size is chosen to fully illuminate the aperture.5. is chosen by isotropically selecting the trajectory of proton 1 in the pp CMframe. The solid angle of the cone that the trajectory is selected from is = 2Kbecause of the indistinguishability of the two final state protons.APPENDIX G. YIELDS AND PHASE SPACE INTEGRALS 2436. The 4-vectors of protons 1 and 2 are transformed from the diproton CM systemto the irpp CM system.7. The pn —* irpp differential cross section is calculated from the Handler parameterization (see appendix B).8. The 4-vectors of the pion and protons 1 and 2 are transformed from the irppCM system to the lab system.G.1.2 pd—+ irppp (Four Body Model)To calculate the reaction rates for Four Body pion production from. deuterium, it isconvenient to use a different set of final state variables for the differential cross sectionsthan is used for the Spectator Model. The Four Body Model assumes that the threefinal state protons and the pion share with equal probability the energy of the finalstate. The protons are numbered S, 1 and 2. The rppp system is designated by (*),the ppp CM system by (‘), the CM system of protons 1 and 2 by (“). The variablesused to describe the final state phase space are: c, the solid angle of the pion in the-irppp system; M12, the square of the invariant mass of protons 5, 1 and 2; c, thesolid angle of proton S in the ppp CM system; M12, the square of the invariant massof protons 1 and 2; and c, the solid angle of proton 1 in the CM system of protons 1and 2. Then, as described in section 4.1.4, the Four Body differential cross section forpd —* irppp is written as,,8 FB D* D! DII______________________— M 2 7rS1 1 G 1dcdM12d ?2’ FB 256(2ir)8PTiVMsi’where MFB is the matrix element for the reaction, P the pion momentum in the rpppCM system, P the momentum of proton S in the ppp CM system, and P the momentum of proton 1 in the proton 1 and 2 CM system. Following the same procedure as forthe pion production using the spectator model, the Four Body differential cross sectionis inserted into the yield equation (G.5) and converted to a Monte Carlo integration,7FB S1 MFBpd—irppp _ NB71 N 256(2ir)8N / p*p, p,1‘V 1ir S 1 zM2 zM2i D*LTT2,f S12i 12ii=1 \1 B i” L11512L1112APPENDIX G. YIELDS AND PHASE SPA CE INTEGRALS 244xA(c,‘S12i’ 12i’ ‘)) , (G.16)where z\ is the cone the pion is scattered into in the irppp CM frame, z2 is thecone proton S is scattered into in the ppp CM frame and A is the cone proton 1 isscattered into in the 12 CM frame. The range of the square of the invariant mass ofprotons S, 1 and 2, AM12 varies for each event, as does the range of M12, LMf2.The procedure for generating an event is:1. The centre of mass energy, W, of the beam proton and the target deuteron (theFour Body Model assumes a collision between a proton and a deuteron, not abound neutron) is calculated. Lorentz transformations between the lab frameand the irppp CM systems are calculated.2. In the irppp centre of mass system the invariant mass squared of particles S, 1, 2is selected in the range,< M12 < (W —and /M12 = (W — 7n,.)2 — (3m)2.3. The trajectory of the pion is isotropically selected in a cone of solid angle L\Qin the total CM system. is centred on the QQD aperture and is made largeenough to fully illuminate the aperture.4. In the S12 CM system, the invariant mass squared of particles 1 and 2 is selected,(2rn)“12i (M812 — m)2,and ZMj2 = (Ms12 — m)2 — (2m).5. The trajectory of proton S is isotropically selected in a cone, z\c2 = 4irsr.6. In the 12 CM system, proton 1 is scattered into a cone, AQ1 = 4irsr.7. The 4-vectors of protons 1 and 2 are transformed from the 12 CM system to theS12 CM system.8. The 4-vectors of protons S, 1 and 2 are transformed from the S12 CM system tothe irppp system.9. The four vectors of the pion and three protons are transformed from the irpppCM system to the lab system.APPENDIX G. YIELDS AND PHASE SPACE INTEGRALS 245G.1.3 pd —* dirn8 (Spectator Model)For the pd —* dir+n3 reaction, the final state has three particles with five parameters.Using the Spectator Model, the pd —* dirn differential cross section is written interms of the free pp —+ dir+ cross section, -(pp —* dir+),= WFB 2(ps)4(pp-* dirj, (G.17)where the flux factor and spectator momentum terms are the same as for equation G.8.Analogous with the procedure in section G.1.1 the Monte Carlo yield integral is givenby the equation,= NBA (pp .‘ (G.18)where the free pp —* dir+ differential cross section taken from an energy dependentparameterized by Ritchie[65] and A(Q, P8) is the Monte Carlo Acceptance function.The procedure for generating an event is:1. The spectator momentum, Ps is generated as in section G.1.1, except that thespectator is now a neutron instead of a proton.2. The dir CM energy, W is calculated and the Lorentz transformation between theCM and lab calculated.3. In the dir CM system, the trajectory of the pion is isotropically selected from acone of solid angle LQ which is centred on the spectrometer aperture.4. The CM momenta of the pion and deuteron are calculated.5. The 4-vectors of the pion and deuteron are transformed to the lab frame.G.2 Phase Space IntegralsThe phase space integral is used in the calculation of the differential cross section fromthe measured experimental yield. The yield integral, equation G.5, can he rewritten,Y = NB1 JA(Q)dQ, (G.19)APPENDIX G. YIELDS AND PHASE SPACE INTEGRALS 246were da/dQ becomes the average differential cross section over the integral which iscalled the phase space integral,Q = fA(Q)dQ. (0.20)Solving equation 0.19 for the differential cross section gives,du YGdQNBjQ .21)For the Spectator Model, the evaluation of the phase space integral is almost identicalto the evaluation of the yield except that the term NB is dropped from in front of theintegral and the differential cross section for the free reaction is dropped from insidethe integral. The phase space integral for pd — dirri3 is then47* 2(P ) —Qdr+n. (2 (0.22)= B(ç*.]5) (0.23)N EPPBwhere equation 0.23 is the Monte Carlo integral. The notation used in equation 0.23and the procedure for selecting the Monte Carlo event is the same as in section G.1.3.For calculating the phase space integral for pd —* irppp3, there is a slight modification to the above procedure. As discussed in section 4.1.4, the pn —÷ irpp(1S0)differential cross section is assumed to be independent of and five fold differentialcross section becomes,d5cr 1 dodQdMdQ = 2irddM ( .24)and da/ddM is the differential cross section extracted from the data. The phasespace integral then becomes,= J —A(c, , (G.25)= AM -A(1, , ]), (G.26)APPENDIX 0. YIELDS AND PHASE SPACE INTEGRALS 247where the notation used in equation G.26 and the procedure for generating the MonteCarlo event are the same as in section G.1.1.Bibliography[1] C.M.G. Lattes et. aL, “Processes involving charged mesons”, Nature 159, 694(1947)[2] T.E.O Ericson and W. Weise, Pions and Nuclei, Clarendon Press, Oxford (1988)[3] L.C.L Yuan and S.J. Lindenbaum “Pion Production in H and Be by 1.0- and 2.3-BeV Protons” Phys. Rev. 103, 404 (1956);S.J. Lindenbaum and R.M. Sternheimer “Isobaric Nucleon Model for Pion Production in Nucleon-Nucleon Collisions” Phys. Rev. 105, 1874 (1956);S. Mandeistam “A Resonance Model for Pion Production in Nucleon-Nucleon Collisions at Fairly Low Energies”, Proc. Roy. Soc. A244, 491 (1958)[4] H. Garcilazo and T. Mizutani, irNN Systems, World Scientific (1990)[5] E. Fermi et al Nuclear Physics University of Chicago Press, Chicago, pg 159 (1950);J. M. 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