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Measurement of the total cross section for the π - 2π reaction p(π⁺, π⁺ π⁰)p near threshold Suen, Nelson 1993

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MEASUREMENT OF THE TOTAL CROSS SECTION FOR THE it-27r REACTION p( Π+ , Π+Π 0)p NEAR THRESHOLD  Nelson Suen B.A.Sc. University of British Columbia  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA August 1993 © Nelson Suen  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Department of Physics The University of British Columbia 6224 Agriculture Road Vancouver, Canada V6T 1W5  Date:  vr.44 ici13  o A  Abstract  A feasibility study for measuring the total cross section of the 71--27 reaction, 71- + p --> 7r ± 7r°p, was performed at incident pion kinetic energies of 195 and 201 MeV. It was not possible to measure the total cross section with the present apparatus. Modifications and improvements to the present apparatus are presented.  ii  Table of Contents Abstract List of Tables^  vi  List of Figures^  vii  Acknowledgement^  xi  Dedication^  xii 1  1. Introduction^  1.1 Motivation for the Experiment^  1  1.2 World Data on Reaction^  2  1.3 ...Survey of Chiral Symmetry ^  4  1.3.1 Weinberg and Schwinger^  4  1.3.2 Olsson and Turner^  5  1.3.3 Oset and Vicente-Vacas^  5  1.3.4 Current Theory^  5  1.4 Predictions of the Olsson-Turner Model^  6  1.5 Chiral Perturbation Theory^  8  1.5.1 Feynman Diagrams^  10  1.5.2 Sensitivity of the 7-7r (Pole) Contribution^ 15 2. Description of experiment^  23  2.1 Introduction 2.2 General Approach^  23  2.3 Details of Set-up^  24  iii  2.4 A Brief Tour of a Reaction Event^  31  2.5 Strategy for Eliminating Background Reaction Events^32 2.6 Conclusion^ 3. Modelling of experiment^  36 36  3.1 Introduction^  37  3.2 Phase Space^  37  3.3 Simulation^  42  3.4 Conclusion^  48 49  4. The Experiment^  4.1 Introduction^  49  4.2 Initial Set-up^  49  4.3 Running of the Experiment ^  51  4.4 Conclusion^  52  5. Experimental Results^  53  5.1 Introduction^  53  5.2 Analysis of Data^  53  5.2.1 Time-of-Flight Cut^  54  5.2.2 Events in the AE vs E Plane^  55  5.2.3 Normalization of Spectra^  58  5.3 Cross section^  60  5.4 The CH 2 -C Spectra^  61  5.5 The Subtraction Problem ^  63  iv  5.6 Conclusion^ 6. Redesign of experiment^  67 68  6.1 Introduction^  68  6.2 Hole in the S1-S2 Telescope^  68  6.3 Increasing Time-of-Flight Separation^  70  6.4 Triplet Lens^  71  6.4.1 The Triplet Arrangement^  72  6.4.2 The Triplet Simulation^  72  6.4.3 Results from the Simulation^  73  6.5 Conclusion^  78  7. Final Conclusions^  79  Bibliography^  80  Appendix A Electronics^  82  Appendix B Listing of Routines for Driving MOLLI^  86  Appendix C Detectors^  92  v  List of Tables  1.1^World data on the 7r ± p —> Ir + 71-°p channel 1.2^Scattering lengths predictions from different theoretical models  ^  8  1.3^Proportion of contributions to the total cross section for different diagrams for the 12 ir + 7rO channel^ 1.4^Proportion of contributions to the total cross section for different diagrams for the 12 71- +7r± channel^ 2.1 Maximum angles and n^35 C.1^Detector sizes^  vi  92  List of Figures 1.1^World data on the li- ± p —> ir + ir°p channel^  3  1.2^Feynman diagrams showing 11--ir scattering imbedded in 7-27 scattering ^3 1.3^Tree level diagram^  9  1.4^Loop level diagrams which contribute to scattering amplitudes ^9 1.5^Total cross section (ir + r° ) based on the Donoghue (ChPt) scattering amplitudes 10 1.6^Total cross section (7rtr + )^  11  1.7^Feynman diagrams for the Ir+iis ) channel^  11  1.8^Feynman diagrams for the  13  7 ± /r ±  channel^  1.9^Different contributions to the total cross section ^  14  1.10 Different contributions to the total cross section^  14  1.11 71- ± ir° channel, total cross section divided by phase space ^16 1.12  7r + ir +  channel, total cross section divided by phase space^17  1.13 Fractional change in the total cross section^  18  1.14 Fractional change in the total cross section (7 + 7°) compared against existing experimental error levels^ 19 1.15 Fractional change in the total cross section (r+7+) compared against existing experimental error levels^ 19 1.16 The ratio of E's for the two channels: E(ir + 7- + )/E(7 + 71-O), with same variation in the alpha parameters as previous defined (see Fig. 1.11) ^ 21 1.17 Changes in the E ratio due to the variation in the alpha parameters as well as a ±20% change in the A coupling constant^ 22 2.1^Confinement of protons to small cone angles ^  24  2.2^Experimental Set-up^  25  vii  2.3  A 'uniform' distribution of pions  29  2.4  A 'uniform' distribution of protons  29  2.5  The 'band' structure for different masses in the AE vs E plane  30  2.6  Typical 'pile-up' event  31  2.7  Typical event vetoed by the 'C' detector  33  2.8  Pile-up events are those that appear in the left window  34  3.1  (phase space of w + p --> w + w°p channel)  38  3.2  (phase space of ir + p -› ir + 15)p channel)  38  3.3  (phase space of w+p --> w -l- w'p channel)  38  3.4  (phase space of w + p --> ir + w°p channel)  38  3.5  (phase space of w + p -› w + p channel)  39  3.6  (phase space of w + p -> w + p channel)  39  3.7  (phase space of w + p -> ir + p channel)  39  3.8  (phase space of w + p -> w + -yp)  40  3.9  (phase space of w+p -› w+-yp)  40  3.10  (phase space of ir + p -› w + -yp)  41  3.11  (phase space of ir + p -> w + -yp)  41  3.12  (phase space of w + p -> w + -yp)  41  3.13  Reaction events as seen in the AE vs E plane  43  3.14  Surface plot of reaction events  43  3.15  Reaction events as seen in the .6,E vs E plane with C veto enabled  43  3.16  Surface plot of reaction events with C veto enabled  43  viii  3.17  Elastic scattering background  43  3.18  (A-±p —> 71- + Tp events as seen in the AE vs E plane)  45  3.19  Or -F p —> ir + -yp events as seen in the AE vs E plane)  45  3.20  (71- ± p —> 7- + T ± p events as seen in the 1E vs E plane)  46  3.21  (7r + p —> ir +,ep events as seen in the AE vs E plane)  46  3.22  (en —> irtip events as seen in the AE vs E plane)  47  3.23  (en ---> 7i- + ep events as seen in the AE vs E plane)  47  3.24  (ir'n —> ir ÷ irp events as seen in the AE vs E plane) C veto enabled  48  3.25  (en ---> 7r ± ir-p events as seen in the AE vs E plane) C veto enabled  48  5.1  Typical spectra from a CH2 target run  53  5.2  Detail of Time-of-Flight spectrum from a CH 2 target run  55  5.3  Typical Si vs S2 spectra  56  5.4  S1 vs S2 detail, raw spectrum.  57  5.5  Surface plot of Si vs S2, raw spectrum  57  5.6  Si vs S2 plot, with time-of-flight cut  57  5.7  Surface plot of S1 vs S2 spectrum with time-of-flight cut  57  5.8  H2 spectrum  in the S1 vs S2 plane, with no time-of-flight cut  61  5.9  H2 spectrum  in the S1 vs S2 plane, with optimal time-of-flight cut  62  5.10  Negative contours in the Si vs S2 plane  64  5.11  Si Signal Instability  65  5.12  Subtraction of two S1 spectra, Run 34 minus Run 29  65  5.13  Instability in S2 signal.  66 ix  5.14 Subtraction of two S2 spectra, Run 34 minus Run 29 6.1^Angle correlation between outgoing pion and proton 6.2^The triplet set-up  ^ ^  ^  6.5^Field gradient of Q2, as a function of z  69 71  6.3 Raytrace of monoenergetic protons with Tp =36 MeV 6.4^Field gradient of Q1 as a function of axial distance z  66  ^  ^  ^  74 75 75  6.6^Raytrace of outgoing protons from the reaction w + p —> 7r + ir°p, with divergence in the ^ x,y directions < 5 °, y direction (dcd plane) 76 6.7^Raytrace of outgoing protons from the reaction^ w+p —> 7r + ir°p, with divergence in the x,y directions < 5 °, x direction (cdc plane) 77 A.1 Block diagram of 'beam' logic. ^  82  A.2 Block diagram of 'detector' logic. ^  83  A.3 Block diagram of 'event' logic. ^  84  A.4 Various modules.^  85  C.1 Target geometry  ^  C.2 Detector geometry  ^  x  93 94  Acknowledgements  First and foremost, I would like to thank Dr. Richard Johnson for supervising and supporting this work. His insights, experience and humour have been invaluable in the completion of this work. I would also like to thank Dr. Ami Altman for his assistance in so many areas and his excellent explanations for so many of my questions. Special thanks must be given to Dr. Eli Friedman for his PH3J5 program for performing the Monte Carlo simulation and his readiness in sharing his knowledge; and also to Dr. David Axen for reading this thesis. Finally, I would like to thank my old friend Patrick for his good advice and encouragement.  xi  Dedication  To the Rose of Sharon  xi i  Chapter 1  Introduction  1.1 Motivation for the Experiment  The motivation for performing this experiment has its roots in the study of the strong interaction, in the low energy regime. Early theoretical work in the field have been based on the idea of chiral symmetry' breaking. Contemporary treatment of the subject in terms of quantum chromodynamics (QCD) still relies on the idea of chiral symmetry which is the only rigorous formalism of QCD at low energies.  Traditionally, the test of the validity of various theories that make use of chiral symmetry breaking takes place in trying to measure physical quantities associated with the reaction ira + 7re3 --> 71-7 + 7r° (7-7r scattering), perhaps the most fundamental of all hadronic processes'. Because of the short lifetimes of the pions, 7r-ir scattering cannot be observed directly. Instead, one resorts to indirect means: one such method involves the measurement of cross sections for the w-27 reactions. Cross sections for these reactions ' Chiral symmetry in this case refers to the symmetry that exists if quarks were massless. In terms of quark theory, the QCD Lagrangian in the chiral limit would consist of two separate parts: one for right-handed particles another for left-handed particle with no coupling between the two. Hence, in the massless quark limit, the left-handed states do not mix with the right-handed states. In the real world, chiral symmetry is not preserved since quarks are not massless. 2 'most fundamental of all hadronic process' because a-ir reactions involve the self-interaction of the lightest particle in the hadron spectrum of particles.  1  Chapter 1. Introduction  near threshold can be used to calculate w-w scattering quantities (Fig. 1.2). Early work [15,16] in the field suggested the behaviour of it-'r interactions is embodied in a single theoretical parameter' other physical quantities such as scattering lengths are given in terms of Contemporary theory based on QCD such as chiral perturbation theory makes somewhat different predictions from those of earlier theory. It is the intent of the current experiment to test the predictions of existing theory by measuring the cross section of a ir-2ir reaction. For a survey of measurements of different ir-2ir channels and the theoretical description of chiral symmetry breaking refer to [3,23].  1.2 World Data on Reaction  For the it-2w reaction, w + p —> w + w°p, very few measurements have been performed to date: indeed no measurements exists near the threshold energy (T,r+ = 164.75 MeV). T,r÷ (MeV)  Ottb)  Reference  230+13  18+1  [10] (1975)  275+15  48+1'5'  [10] (1975)  294+4  120+50  [11] (1972)  300+?  _ 110±40  [30] (1963)  Table 1.1 World data on the 7r + p ir + 7C O p channel (reproduced from [2]).  3  Known as the 'chiral symmetry breaking' parameter.  2  Chapter 1. Introduction  10  3  lllll  ^iiiiii^llllll^  10 -2 ^ 160^200^240^280 320 T Tr+ (MeV) .1^1111^ll 1 lll 111111^1111111^11111111^1111111^11111111^11  World data for the 7r + p —> ir + ir° p channel (reproduced from [2]). The Oset and VicenteVacas model was used to generate the curve for the total cross section.  Figure 1.1  p^/ ^p^ pole diagram  equivalent  ^  o ■  (rv+  7r-27  /  ^ N  ,  oA  N^  ,„..„......„.„...„......:  N  7T  x  x  zo  +'N , N,,,,, N.  ,  AL, z^-.^0  7 + 21^ ..,^ x^ z^7T-7T  N.  N  N.,  Feynman diagrams showing 7r-7r scattering imbedded in 7-27r scattering. Figure 1.2  3  Chapter 1. Introduction  It is not at all surprising the data is sparse since, as we shall see, the measurement of the cross section for this reaction is a challenging undertaking, due to the host of background reactions.  1.3 A Semi-Historical Survey of Chiral Symmetry  1.3.1 Weinberg and Schwinger  The first work on chiral symmetry was performed by Steven Weinberg during the early 1960's. Weinberg introduced chiral symmetry breaking to current algebra and the partial conserved axial current (PCAC) hypothesis', in order to calculate the ir-ir scattering lengths'. Based on this approach, Weinberg also developed a Lagrangian for the  7 7 -  interaction [13]. At about the same time, Schwinger, using a different approach arrived at a different ir-ir Lagrangian [14].  4 the notion of a partially conserved axial current originates from the idea allowing the quarks to have a small mass (a few MeV's) and therefore breaking the chiral symmetry. It can be shown that the preservation of chiral symmetry implies the conservation of the axial current. To slightly break the symmetry with small quark masses implies that the axial current is partially preserved.  5  the scattering is defined as  1 bin a =a 2 4n 1- e° where a is the scattering length and X is the wavelength associated with the incident particle.  4  Chapter 1. Introduction  1.3.2 Olsson and Turner: generalization of the 7-7 Lagrangian derived by Weinberg and  Schwinger.  During the late 1960's Olsson and Turner [15] constructed the most general form of the 7r  -  ir  Lagrangian, which can be considered as a family of Lagrangians because it contains a single free parameter E. According to this model, at low energies, E the chiral symmetry breaking parameter alone determines the strength of the  7r ir -  interaction at low energies.  Furthermore, the Weinberg and Schwinger Lagrangians are two specific cases of the OlssonTurner family of Lagrangians corresponding to the E values of 0 and 1, respectively.  13.3 Oset and Vicente Vacas -  The model constructed by Oset and Vicente-Vacas for 7-27r reactions, adds to the Olsson  and Turner model the effects of the intermediate isobar states of the N* and A [22].  1.3.4 Current Theory  One obvious short coming of the Olsson-Turner model is that it does not include  71 -7 -  -  rescattering effects. Modern theories that include rescattering effects make significantly different predictions on the scattering lengths. One such approach based on QCD is chiral perturbation theory (ChPT) [17,23,27]. Predictions made by ChPT for the 7r+p ---> ir + ir°p reaction will be discussed below.  5  Chapter 1. Introduction  1.4 Predictions of the Olsson Turner Model -  In the framework of chiral symmetry, the Olsson-Turner Model makes specific predictions that can be experimentally verified. Specifically, we wish to investigate the nature of chiral symmetry breaking by studying 7r-ir scattering amplitudes at zero relative momentum. Because of the short lifetimes of the 7r, measurement of the amplitudes must be done indirectly. One such method is to measure the cross sections of the 7rN .- 7r7rN (ir-2ir) reactions, near threshold [15].  According to Olsson and Turner's model, the total cross section at threshold for the 7r-27r reactions is given by [10,16,20]  a OW  —  Tr icIV)=a(iTnN) 2Q 2S x(phasespace)  For the reaction w + p --> 71-+ 7r° p, the total cross section becomes a =a(TE +Tr °p) 2 Q 2 X (phasespace)  where^Q  =  ^  (1.2)  momentum of incident ir + in the center of mass system  a(7 ° 71- ±p) =  the reaction amplitude at threshold, dimensionless in this notation  6  Chapter 1. Introduction  S = statistical factor accounting for pions in the final state; S = 1/2 if final  pions are identical; S = 1 otherwise.  The threshold amplitude is related to the chiral symmetry breaking parameter in the following way [15,19] 24a(Tc+TE°p)= 1.51 +0.6  ^  (1.3)  Furthermore, the s-wave^scattering lengths ° for isospin I =0 and I =2 are given by a2 _ +2  (1.4)  ao 5 _7  2  and 2a0 -5a2  3m , 7  -  (1.5)  where fr = the pion decay constant m,,. = pion mass  6 Because the spin (intrinsic angular momentum) of the pion is zero and at threshold the angular momentum 1= 0, only s-waves are present.  Symmetric wave function (boson symmetry) under the interchange of pions of a 7r-ir system further dictates that the isospin be even. Therefore for the ir-7r system at threshold, I =1 isospin components vanish leaving only 1=0,2 components [3]. In terms of the scattering lengths a, only ao and a 2 remains.  7  Chapter 1. Introduction  Combining (1.4) and (1.5) yields and using fir = 93.3 MeV ao = (0.156  0.0560 Om 7:  (1.6)  -  -  -  a2 = -(0.045 +0.0224 Om Tc1  (1.7)  -  In summary, according to the Olsson-Turner model, by measuring the total cross section  a,  the threshold amplitude a(7 ± 7- °p) can be determined using (1.2). It follows that by (1.3) allows is determined and by (1.6),(1.7) the scattering lengths are found. Table 1.2 gives the values for the scattering lengths for E = 0. ao  a2  Weinberg [18] (Olsson-Turner ^=0)  0.16  0.045  Gasser & Leutwyler [17] (Chiral Perturbation Theory)  0.20  -0.042  Table 1.2 Scattering lengths predictions from different theoretical models. a, are in units of (m„) -1 , where I is Isospin.  1.5 Chiral Perturbation Theory  Gasser and Leutwyler [24] have made predictions on the  7r-ir scattering  lengths using ChPt  (see Tab. 1.2). Based on Gasser and Leutwyler's work, Donoghue [28] has calculated the 7r- 2-  chiral scattering amplitudes. ChPt makes use of chiral effective Lagrangians [27], which  are classified according to expansion in terms of energy. The lowest order Lagrangian is of the order E 2 (energy squared). At this order, the 8  7--7r Feynman  diagrams are at 'tree-level'  Chapter 1. Introduction  n  7,  Figure 13 Tree level diagram.  (1)  ^  (2)  Figure 1.4 Loop level diagrams which contribute to scattering amplitudes.  (Fig. 1.3); that is, no rescattering effects are considered.  The predictions made at this order reproduces the scattering lengths proposed by Weinberg [1] (which corresponds to the Olsson-Turner model with E =0). The Lagrangian at order E 4 has been calculated by Gasser and Leutwyler [27]. At this order, calculations involve oneloop diagrams (Fig. 1.4). Imbedded in the Lagrangian to order E 4 are two 'free parameters' « i and ii2 which are to determined by experiment. In what follows, the 7-7 scattering amplitudes (see [28]) derived by Donoghue will be used in conjunction with the intermediate isobars  A  and N* portion of the Oset and Vacas-  Vicente model to generated cross sections for the different 7-27 channels. The following calculations will follow the approach by V. Sossi [29]. Figures 1.3 and 1.4 show the cross section generated by this approach. The solid curves for the total cross section are determined by setting the parameters of the Lagrangian to values derived in ref. [28] « i = -0.007 and a2 = + 0.013 9  Chapter 1. Introduction  Tr +p^71. -1- 71. Op it^  ili1,1111.11/^111.11  1o2 pole term + leading terms + pole term + leading terms  pole term  °  10 -  10 -1 150^200^250^300^350 T Tr+ ( MeV )  400  Figure 1.5 Total cross section based on the Donoghue (ChPt) scattering amplitudes along with isobars described by the Oset and Vacas-Vicente model. The solid curve is determined by setting = -0.007 and « 2 = + 0.013. And the vertical dash marks represent points calculated by varying ± 100% and « 2 +50%.  To determine the sensitivity of the cross section to the parameters a l and &2, calculations were performed by varying the parameters by 100% and 50% respectively (these points are shown as vertical dash marks in Fig. 1.3 and 1.4). The 7 + 7+ channel is included here for comparison as will be discussed.  1.5.1 Feynman Diagrams  Figure 1.7 depicts the Feynman diagrams for the 7r + p ---> -71- + -7r °p reaction. The diagram of primary interest is the pole (1), for studying  7 7 -  10  scattering (see Fig. 1.2). The set of 3-point  ^ ^ ^  Chapter 1. Introduction  Tr+p^-rr+Tr+n  10 -2  150^200^250^300^350 T (MeV)  Figure 1.6 Total cross section (see caption on Fig. 1.2).  ^ 7T 1  ^+1  +  /  \\ / , it I^ -N \ ins IT I^ 0.' J.1^ P^ P^P^ p  (3) /  rr  IT4. + — ■ — 4 — .- — ir°^ \  ^n  ^o  400  ,  \  ^  (4)^  (5)  \  / +^+ 1^\ Tr + / It o^ IT +//^0 / /1' / it^Tr1 , / IT \^/^TT / + \ /^\ 1 f^I  +  ^4^\  ^Tr  ^  \ 1^1 1/ I 1^ I \ I /^\ V^I^\^I^/  P^ P^p^p^n^p^p^P^P (7)^ (8) I^n I^I^+ 1^(61^I^ Tr+l1 t +I ^rr o ; Tr I n + I n + 1^rt.+ 1 it " I^I 71" I I.^ II^ A^ f 0I 01 ^ 1^I I^I^I^ I^I^I^ I^I^I^I^I^I^ I^I^I P^p^n^p^p^n^n^p^P^n^p p 0  ^0  i^  I^ ^ I^  Figure 1.7 Feynman diagrams for the 7r + ir° reaction channel. Note: 3-point diagrams involving isobars are not depicted. Diagrams (2)-(8) are called in this paper the 'leading diagrams' for this channel.  11  Chapter 1. Introduction  diagrams (6)-(8) in Fig. 1.7 are incomplete, as only nucleon states have been shown. There are 22 other 3-point diagrams containing a mix of (nucleon-0), (nucleon-0-0) and (nucleon0-N*) states (see ref. [3] for more diagrams). All diagrams contribute to the total cross section and are used to determine the total cross section. For this channel, the pole term accounts for 30% of the total cross section at T„ =180 MeV and less at higher energies (Table 1.3). Even at modest energies, isobar states begins to dominate causing the pole term contribution to diminish to 10% at 240 MeV (Fig. 1.9). 180  T, + (MeV)  240  % of total cross section^a pole diagram  30  10  leading diagrams (see Fig. 1.7)  48  44  isobars and nucleons diagrams  22  46  Table 13 Proportion of contributions to the total cross section for different diagrams for the eirO channel.  180  T„,(MeV)  240  % of total cross section^a pole diagram  88  45  leading diagrams (see Fig. 1.8)  1  37  isobars and nucleons diagrams  11  18  Table 1.4 Proportion of contributions to the total cross section for different diagrams for the el' channel.  Figure 1.8 depicts the Feynman diagrams for the ir + p 7 + 7 + n reaction. Again, the set of 3-point diagrams are incomplete, as only nucleon states have been included. But overall there are far less number of diagrams for this channel especially those that involve the 12  ^ ^ ^ ^  Chapter 1. Introduction  (2)  71  n^p  p  (0)^ + / \  71I  T  /  (4)  /IT +^+ \^+ /^+ / TV \ 7 /^7 /  / \ 1^ 1 l^A \/^/  /^V^ \/^/ n^n^P^p^n (5) +  ^I  /  ^I p^n^p^n  Figure 1.8 Feynman diagrams for the 7r + 7r + channel. Note: 3-point diagrams involving isobars are not depicted. Diagrams (2)-(5) are called in this paper the 'leading diagrams' for this channel.  isobar states. There are only 11 other 3-point diagrams containing a mix of (nucleon-0), (nucleon-0-0) and (nucleon-A-N`) states (see ref. [3] for more diagrams), in contrast to the 22 for the ir + ir° channel. Further, in sharp contrast, for this channel, the pole term accounts for 88% of the total cross section at T, r+ =180 MeV (Table 1.4). Isobar states play much less of a role for this channel whose contribution to the total cross section increases from 11% to 18% between 180 MeV and 240 MeV (Fig. 1.10).  13  Chapter 1. Introduction  +p^ir+7rop  40  1111111iIII^ i11.1!■  1[11.11111  pole term + leading terms + other terms  30  _o b  10 pole term + leading terms pole term _______i  0 170^190^210^230^250 T 7T+ (MeV)  Figure 1.9 Different contributions to the total cross section. Tr + p^Tr+Tr+n 40  30 -  11111111  111  111111111  111  11111^111  pole term + leading term +  pole term + leading term  10  pole term  180^200^220 T (MeV)  ^  240  ,  Figure 1.10 Different contributions to the total cross section.  14  Chapter 1. Introduction  1.5.2 Sensitivity of the 7 7 (Pole) Contribution -  As mentioned above, the pole diagram accounts for only a fraction of the total cross section; since the physics of 7-7 scattering is imbedded in the pole term the magnitude of its contribution to the total cross section is determined by the parameters « 1 and &2. As shown in Figs. 1.5 and 1.6, by varying the these parameters and looking at the changes in total cross section one can determine the sensitivity of the pole term. It will be shown below that the 7 + ir° channel is 'insensitive' to changes in « 1 and '& 2 in the region near threshold (say, from threshold up to 320 MeV). By 'insensitive', two things are implied: first, the variation in cross section is small relative to other channels such as the ± + 7r 7  and second, perhaps the most important reason, the variation is small relative to  existing experimental error levels'.  For Figures 1.11 and 1.12, the phase space dependence of the total cross section  a  has been  divided out. It is evident that for the 7r + 7r° channel that the variation in cross section is far smaller than errors on existing data points. For this reason, it is not possible to extract useful information about the « 1 and a2 parameters, without improving on the previous experimental error levels which are typically ±50% of a. To pin down the alpha parameters errors should be reduced to — ± 10% of a. In contrast, the variation for the is larger or comparable to errors on existing data points.  ' i.e., the size of the experimental errors on existing experimental data.  15  ir + 7r +  channel  Chapter 1. Introduction  7T  +  p -->  + 0 7T 7T p  Figure 1.11 7r + ir° channel, total cross section divided by phase space (dimensionless). The vertical dash marks  represent points calculated by varying -«, +100% and «2 +50% (from the base values of « 1 =-0.007 and «2 = +0.013), while the broken line outlines the region mapped out by this variation.  16  Chapter 1. Introduction  -rr +p -4 Tr +7 r +n -  6  1^1^1^1^11^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1  5-  Sevior et al. [2] Kravtsov et al. [26]  0  1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1  1^1^1^1^1^1^1^1^1^1^l^ll  150^200^250^300^350 T Tr+ (MeV)  400  Figure 1.12 ir+7r+ channel, total cross section divided by phase space (dimensionless). The vertical dash marks represent points calculated by varying ii i +100% and '&2 +50% (from the base values of Ei,= 0.007 and Ei2 = + 0 .013), while the broken line outlines the region mapped out by this variation. -  17  Chapter 1. Introduction  Recent measurements by Sevior et al. [2] near threshold contain errors which are small enough to constrain the alpha parameters. The errors for the Sevior experiment are typically — 20% of the total cross section, which are less stringent than the 10% requirement for the 71-± 7r° channel. Figure 1.13 show the fractional change in the total cross section as a result of varying the alpha parameters: it is immediately evident that the 7r ± ir° channel cross section is far less sensitive to such variation. Figures. 1.14 and 1.15 show the experimental error levels for existing data.  150^200^250^300^350^400 T ,T+ (MeV)  Figure 1.13 Fractional change in the total cross section as a result of  variation in the alpha parameters, for the irtir ° and 7rir + channels.  18  Chapter 1. Introduction  Tr +p^7T+Trop  1.0 = 0.8 0.6 -  experimental error level on existing data  b  0.4 = 0.2 — 0.0  150^200^250^300^350^400 T n+ (MeV)  Figure 1.14 Fractional change in the total cross section compared  against existing experimental error levels. rr+p —> Tr + Tr + n  1111^11111^1.111111111^11111^1111  1.0 0.8 — 0.6 0.4 — 0.2 0.0  experimental error level on existing data  150^200^250^300^350^400 T n+ (MeV)  Figure 1.15 Fractional change in the total cross section compared  against existing experimental error levels.  19  Chapter 1. Introduction  If one defines the quantity E = (rips, where a is the total cross section and ps is the phase space, then another approach is to consider the ratio of E's for the two channels: E(71-+7+)/E(irtir°). Figure 1.16 shows the plot of this ratio, again with same variation in the alpha parameters as previous defined (see Fig. 1.11). In contrast to Figures 1.11 and 1.12, which shows a small cross sectional variations at energies near threshold and then progressively larger variations at higher energies, the ratio displays a large changes near threshold with progressively small changes at higher energies. While constraining the alpha parameters still require the experimental errors to be smaller than 20% and 10% for the 7+ 7+  and 7r + ir° repectively, this approach offers a different perspective in analyzing the cross  section data. Figure 1.17 shows changes in the E ratio due to the variation in the alpha parameters as well as a ±20% change in the A coupling constant. It is clear that for the ratio, changes due to the alpha parameters dominates when compared to those arising from the A coupling constant.  20  Chapter 1. Introduction  3.5 _  1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1  „, variation in a, and a 2  3.0 =  1  ..-... i  1^i^  i^  : N,  •  N  i^N  i^ I^i I^  .\ ,  :^N  ___I _ — .:„.,...^I^i^ s..„  ...... :  1 ; .,^I ■,.  \ • 1  • -...„.  ....,^: ■:.. ....„^i ---. L  ''. ■.,..  0.5 0.0 ^ 150  ..----  -..... : ".... I1  ■....  ---,  •,.  I  ,......  ....... ---1 ---- --- •  - _J 1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1  200^250^300 T rr+ (MeV)  350  400  Figure 1.16 The ratio of E's for the two channels: E(ir + 7r ± )/E(ir + 7r°), with same variation in the alpha parameters as previous defined (see Fig. 1.11)  21  Chapter 1. Introduction  3.0 _  1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^I^1^1^1^1^1^1^1^1^1^I^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1  — — variation in a 1 and a 2 0 0 0 0 0 ±20% in A coupling  2.5 -----. Q 4_ 2.0 w 4-+ k w  boundary defining variation in a, and  02  boundary defining variation in A coupling  1.5 1.0 = 0.5  0.0  ,^1^1^1^1^1^1^1^1^,  150^200  1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1  250^300 T 7T+ (MeV)  ^  I^1^1  350  ^  ^  ^  1^1^1^1^1^1^1  1^1^1  1^1^1  400  Figure 1.17 Changes in the E ratio due to the variation in the alpha parameters as well as a ±20% change in the A coupling constant.  22  Chapter 2 Description of Experiment  2.1 Introduction  This chapter will address how one measures the cross section of reaction (2.1). The experimental apparatus and the technique for eliminating the background events will be introduced.  2.2 General Approach  The purpose of the present experiment is to measure the total cross section for the reactions nip -.7c++7r°-Fp  ^  (2.1)  near threshold energy of 7r° production (164.75 MeV). At a first glance, this measurement appears relatively simple. However upon closer inspection one finds that there are a number of other reactions occurring at the same time. Therefore, the difficulty in the measurement is to separate the events arising the principal reaction from those of the background reactions. The first category of background reactions arises from other 7r+p channels (reactions (2.2)-(2.4)).  I Henceforth in this paper, reaction (2.1) will also be referred to as the 'principal' reaction. Of course, this convention is only a necessary prejudice of this paper.  23  Chapter 2. Description of Experiment  The second category (reaction (2.5)) arises from  71-+  collisions with neutrons which are  present together with the protons in the solid the C and CH 2 targets. Ignoring the background reactions for a moment, the technique that can be used to measure the total cross section is to take advantage of the small angles 2 (Fig. 2.1) of the outgoing protons near threshold. For example, for beam pions with kinetic energy of T„ =220 MeV, the reaction protons are confined within a cone of  (0p)Mi,  =36 ° [3]. The angles are smaller  for lower T„, with (Op)max — 7 ° at threshold. Hence, if there were no competing reactions, one can account for all events by simply detecting the outgoing protons within the cone. Before going on to discuss how to distinguish background events from those of the principal reaction, it is necessary to introduce the experimental set-up.  2.3 Details of Set-up +  T=220 MeV  Targets  The targets used in the experiment are solid  outgoing protons confined within cone  carbon (C) and polyethylene (CH 2 ). Due to the low energy of the reaction protons and Figure 2.1 Confinement of protons to small cone  angles.  2  i.e., angle of the proton with the beam axis  24  Chapter 2. Description of Experiment  Figure 2.2 Experimental Set-up ep -Tc+ +p^  (2.2)  Tc +P ' 1E + + Y ÷P^  (2.3)  ep  en  -  -  e+ic++n^  (2.4)  TE++e f p^  (2.5)  - -  pions, it is necessary to use very thin targets' (- 0.2 g/cm2 ). The reason that 2 different 3 A CH, target with thickness of 0.5 g/cm 2 , will stop protons with kinetic energy up to 22 MeV and pions up to 9 MeV.  25  Chapter 2. Description of Experiment  targets are used are as follows. Since we are only interested in the events arising from 7+14 2 (i.e. w+p) collisions with the CH 2 target, some method must be introduced to remove the unwanted events from w+C collisions. The way to achieve this is to make 2 separate measurements: one with a CH 2 target and an other with a C target. The results from the 2 measurements are then subtracted (CH 2 events - C events). Subtracting the C events from the CH2 events effectively removes the events from scattering off carbon leaving only the events from scattering off the 2 protons. This method has already been successfully implemented in measurement of the total cross section of the single charge exchange (SCX) reaction /IT —> w'n [7,8]. A more detail treatment of this subtraction method is discussed in [5,6].  The Proton Absorber  An aborber made of polyethylene (CH 2 ) with thickness 1.9 g/cm 2 is placed in front of the beam window to remove protons that have 'leaked' through with the beam pions.  Beam Defining Counters  Four scintillator counters (NE102) V, Bi, B2 and B3 are used for beam definition. An incident beam pion accepted is defined' as the coincidence of B's anticoincidence with V. The purpose of the V or 'halo' counter is to eliminate any stray (outside the main beam)  4  i.e., in boolean algebra B1 AND B2 AND B3 AND (NOT V).  26  Chapter 2. Description of Experiment  particles that causes a coincidence in the B counters (see Appendix A and C).  The Event Detectors  Events are defined by a group of 4 detectors, also NE102 scintillators: C, S1, S2 and S3. The S1 and S2 detectors make up the 'telescope' array which is the heart of the experiment for looking at reaction events. The principle of how this telescope works will be discussed in detail below. The rest of the detectors C and S3 are used to reject unwanted background events. The C or 'cylindrical' detector is used to reject events from background reaction (2.5). The S3 veto detector is used to eliminate events from background reactions with high energy particles that manage to traverse the entire array of detectors and target, 'coming out the other side'. These particles mainly arise from elastic scattering and beam particles that did not interact' with the target. An 'event' is defined' as the coincidence of Si and S2, anticoincidence with C and anticoincidence with S3 (see Appendix A and C).  The S1 S2 Telescope Array -  A single scintillator detector by itself cannot identify what type of particle has traversed it. To see why this is the case, we first investigate what happens to a charged particle travelling through a medium. For a charged particle whose mass is much greater than the mass of the  5  i.e., the strong interaction.  6  In boolean form, an 'event' is (Si AND S2) AND (NOT C) AND (NOT S3).  27  Chapter 2. Description of Experiment  electron, energy loss in traversing a medium is due primarily to the interaction of the particle with the atomic electrons in the medium. The mean rate of energy loss due to the ionization of electrons is given by the Bethe-Bloch formula' is given by dE 47ENz 2e 4Z{ (2m e 13 2 c 2 ) _02 In ^ dx mep2c2^A1-02)  (2.6)  where^me = electron mass z = charge (in units of e) of traversing particle /3 = v/c v = velocity of traversing particle Z = atomic number of atoms in medium N = number density of atoms in medium x = path length in medium I = an effective ionization potential, averaged over all electrons — 10Z eV Suppose that we have a very thin' medium (Si) with width dx, then knowing all the medium parameters, one can extract the velocity of the traversing particle by measuring dE. There is no way to identify the mass of particle since (2.6) is independent of this mass. The way to identify the particle is to determine its total energy by adding a second thicker medium  A semi-classical derivation of this formula is given in [8]. This equation is the 'basic' Bethe-Bloch formula without any terms for shell corrections or density corrections at higher energies. 8 The 'thin' assumption is to simplify the argument making the velocity approximately constant through the medium. For a finite ox medium the velocity is through the medium is not constant but one can easily devise an algorithm to extract the velocity upon exit of the medium by dividing the entire medium into smaller pieces and accounting for the velocity difference upon entrance and exit of each little piece of material and proceed then through the entire medium.  28  Chapter 2. Description of Experiment  (S2) which stops the particle completely. The second medium records an energy deposition E. Hence, the total energy of the traversing particle is E+ dE. The mass M of the particle is given by  13 in (2.7) given by (2.6).  In practice, one does not calculate the mass explicitely to identify the particle. By plotting  Proton kinetic energy distribution  Tr" kinetic energy distribution^  20 ^  20  15 -  15 -  z'10  z ° 10 -  -  10  20^30 K (MeV)  40^50^  r  0^10^20^30^40^50^60^70 K (MeV)  Figure 23 A 'uniform' distribution of pions. Figure 2.4 A 'uniform' distribution of protons.  the S1 signal versus the S2 signal (i.e., AE vs E) one can readily identify a particle. Sending a host of particles with the same mass but different energies through the S1-S2 telescope will trace out a unique 'band' in the AE vs E plane. To show this effect, imagine sending a uniform distribution of 7 + with kinetic energy between 0 and 50 MeV (Fig. 2.3) and also 29  Chapter 2. Description of Experiment  a uniform distribution of protons between 0 and 70 MeV (Fig. 2.4). Figure 2.5 shows the band structure corresponding to each type of particle.  Simulated 31 vs S2 Data I  50  I,t 40 (  30  o 20 z  U Jf)-  10  0  0^10^20^.30^40  ^  50  S2 CHANNEL NO. (2 MeV PER CHANNEL)  Figure 2.5 The 'band' structure for different masses in the .6,E vs E plane. The top group is the proton band while the bottom is the pion band. Progressively larger massess would trace out bands higher up in the plane.  30  Chapter 2. Description of Experiment  C  S1^S2 S3 Figure 2.6 Typical 'pile-up' event. The two gamma  rays come from the decay of the 7e.  2.4 A Brief Tour of a Reaction Event  With the telescope, we are now ready to look at reaction events. As shown above, one can not only identify whether a pion or proton has struck the S1-S2 array but also the energy of the particle. A beam pion begins upstream and then through the aborber, through the V halo veto, gets accepted by B1 B2 B3 and then reacts with the target. Let us suppose that the principal reaction takes place then what is emitted from the target is a proton, a Irk, and a 7-° . The ir° will escape without being detected'. The proton is restricted by phase space to be within a certain angle: all that is needed is to make S 1-S2 large enough to cover the cone angle, catching all protons and some of the ir + with the detector. In this way, with the  9 In principle, one can detected the ir° via its decay to 2 gamma rays. In reality due to the high energy of the gamma's (interaction cross section falls off with increasing energy) from the 7r° of this reaction, a very large and expensive detector is needed.  31  Chapter 2. Description of Experiment  absence of background reactions, the total cross section of the principal reaction is determined.  2.5 Strategy for Eliminating Background Events  To eliminate the background reactions, we need to determine which background events will fall in the acceptance angle of the S1-S2 telescope. Since we can achieve mass and energy resolution with the telescope, if the background events have energies which are radically different than (2.1), then they will appear in a different region of the AE vs E plane. So, the general strategy in modelling the background is to send a distribution of particles that belong to the phase space of a particular background reaction and determine their signatures in the AE vs E plane.  The result of this analysis (see Chap. 3) is clear, to distinguish events from the reaction of interest and background reactions, one should consider a smaller group of outgoing protons that fall in the kinematical cone: those that strike the S1-S2 detector accompanied by a ir+, (Fig. 2.6) Thus we ignore those protons that hit the telescope without an accompanying pion. The reason for rejecting these protons is that they fall in a region AE vs E riddled with background events. However, the signatures of the smaller group of (proton & irk) or 'pileup' events are uniquely determined in the AE vs E plane, with the only exception of reaction (2.5). This reaction also possesses simultaneous events of the form of either 71-+p or 7r p, -  32  Chapter 2. Description of Experiment  C TV  /  t  /^  (<^ B3 Target  ............  /  S1^S2 S3 Figure 2.7 Typical event vetoed by the 'C' detector.  which appear in the same region of the AE vs E plane However, this competing reaction is accompanied by an additional charged pion. This difference is used to eliminate the overlap in events: a cylindrical veto counter (C) is introduced to reject simultaneous irp events coincident with another charged ir (Fig. 2.7).  As it will be discussed later (see Chapter 5), the use of time-of-flight will be very important in separating another type of background events from those of the principal reaction. These background events are different from those' above. In this case the background comes  '° i.e., reactions (2.2)-(2.5).  33  ^  Chapter 2. Description of Experiment  from the beam interacting with the 'B' and 'S' detectors. In a very real sense, these detectors are themselves targets, producing a host of events. The trick here is to isolate only those events that are coming from the target: this is where time-of-flight separation is crucial. To summarize, the elimination of the background events is a 2 stage process: first one  Angle Correlation Between 7-1 + and p 30  1^1 1^1^1^1^1^1^1^1^1^1^1  pile—up region o 0 0^0^ 0 25 -^ *g, ®° go. (0080,0 0 824 :0020,,s0.0 6 77 p -> 7i IT p 00 .°°01 00 it °°-°.%°° 000 '013: 4° 0,3010 20 . ":98° , , .o0„. o %^ : ,,,. ; ,,,,,v 00,„ 0 - 00^. 0. ----:^  °:vif°'"°:"4  .,^.0, „40tv'' oc, 0%0 :„ 00., At, 0 ,% *0  0  0  t 4141( SS0t,0° °,40;`01 :4„0° ° 0^ ° + 0 00 0^ ,t,sto:ref,„ s00°, 0 o ,2.0 0° ^ ^0 ° Pe :7, ,?,g ,t, 0 1, % Pbrgb° ° 0, - ;°*0 *^ °,,,^,, cr-_)^ 0 0 „„ At- i^ ° : .o°. i° ^ 7*, °,, ° °°.,$>: ° ^° ;^ oy8 0,.,.fgAsto°1 leo t^,, ..,9,,,,,,,4. .  -0 15  10  00  ,00 k o000 0  ° 'f°:. 0 0:8 g, .! : 0 °^00* g,,, 8 0: ego 0 °*.: 0 0 00:^*0:0 ito  0  *0 % 0 : °° 01 0 * 00°^0% ° ,,^o^  0  °; * ,,^. 0^0 5 7.° 0 f 4.0 :',"°% 0 * 0^" 0 0' : ^o,, $^9 ^ 0 0^o^04,^ .o  g  o^o^ *  o .  00 °  .  50  ^  0,  100^150^200 (deg.)  Figure 2.8 Pile-up events are those that appear in the left window.  eliminates the 'detector' background using time-of-flight; second having separated out the target events during the first stage, then one uses the 'pile-up' events to separate 'target' background events from those of the principal reaction.  34  Chapter 2. Description of Experiment  By restricting the accepted events to pile-up events, one can only measure a fraction of the total cross section of (2.1). This fraction is sometimes referred to as the 'efficiency' n . From phase space considerations alone, the proportion of pile-up events to total events yields n = 0.22, at an incident pion kinetic energy of 200 MeV. Table 2.1 shows different values of n at different pion kinetic energies, with an acceptance angle corresponding to (0 p)., for the first 3 energies and 30 ° for 200.0 MeV. T,,, (MeV)  (01).  (0.,+).  71  167.0  7.19°  40.96°  0.120  180.0  18.40°  175.61°  0.159  190.0  23.28°  176.55°  0.184  200.0  26.99°  176.86°  0.221  Table 2.1 Maximum angles and  different energies.  n (percent of total cross section measured) from phase space calculations at  Figure 2.8 shows the correlation between angles of the outgoing 71- ± 's and protons: the pileup events appear in the window on the left. Monte carlo simulations that account for nonlinearity of light output in the scintillor detectors and a 10% FWHM il photoelectron statistics give 71— 15%.  'Full Width at Half Maximum'. For an explanation of photoelectron statistics as it relatives to energy resolution of scintillator detectors see 'Glenn F. Knoll, Radiation Detection and Measurement, John Wiley & Sons (1979), Chapter 10. 11  35  Chapter 2. Description of Experiment  2.6 Conclusion  In this chapter, we have described the expermental apparatus; also, the 2 different types of background events were introduced: the first comes from the beam interacting with the detectors while the second comes from background reactions. Different methods for eliminating these background events were introduced. In the next chapter, we will look in detail the technique for eliminating events from background reactions developed from Monte Carlo simulation.  36  Chapter 3  Modelling of Experiment  3.1 Introduction  In the previous chapter, we discussed the use of 'pile-up' events for measuring the cross section of reaction (2.1). Much of this analysis is based on the Monte Carlo simulation of the experimental set-up. A 'standalone' routine 'PH3J5' was written by Eli Friedman of the Hebrew University Jerusalem to perform this analysis. We will demonstrate how events from reaction (2.1) can be isolated from those of reactions (2.2)-(2.5).  3.2 Phase Space  To gain a better understanding of the difficulty in separating the reaction events from background reaction events, let us consider the phase space of all the reactions.  All the calucations below are performed at a kinetic energy of T , =200.0 MeV, for the ir  incident 7r + .  For reaction (2.1) ir±p —> W + 7r°p, Figures 3.1-3.3 show the number distributions in the kinetic energy space for each of the particles on the right-hand side of the reaction. An important phase space feature of the reaction is shown in Fig. 3.4, where we see that the outgoing protons are confined to a relatively small angle, (0 p ).= 27 0 . 37  Chapter 3. Modelling of Experiment  Tr+ Kinetic Energy Distribution^ 250  Proton Kinetic Energy Distribution  I^I^1^I^I^  250 1  i^i + 7T p  +^+ 0 IT p -, IT IT p  200H  -,  ^0 7T 7T p  200H  150 -  150 -  z  z 100 -I  100  50H  -  50H  10  20^30^40 T (MeV)  Figure 3.1^  Figure 33^  10^20  50^60  30^40^5 T (MeV)  11?^ i 60^70  Figure 3.2  Figure 3.4  For elastic scattering (2.2) w+p —> ir+p, Figure 3.5 shows that the kinetic energy range of the outgoing w's do not overlap the range of outgoing r's of (2.1). However, Figure 3.6 shows  38  -  Chapter 3. Modelling of Experiment the kinetic energy range of the outgoing protons contains the range of the protons from the principal reaction (2.1). It is already evident by including the elasic channel in our analysis that some way must be used to separate the reaction events from background events.  Tr*  Proton Kinetic Energy Distribution  Kinetic Energy Distribution^  60 ^  60  rt+p  50 -  50 -  40 -  40 -  230-  z 30 -  20 -  20 -  10 -  10 -  50  -rrTp^rr*p  50^100^150 T (MeV)  100^150^200 T (MeV)  Figure 3.6  Figure 33^  Angle Correlation Between Tr* and p 100 80 -  (D•i, 60 S  cz ° 40-  20 -  0  0  100^150 19„, (deg.)  Figure 3.7  39  200  ^  200  Chapter 3. Modelling of Experiment For reaction (2.3) 7r±p --› 7±- yp, Figures 3.8-3.9 show a substantial overlap in the energy ranges of the outgoing 71-+ 'S and protons with those of the principal reaction.  For reactions (2.4) 7r+p .-->  7r+,-+n  and (2.5) w+n --> w+71--p, the phase space distributions are  almost identical to that of principal reaction, because the particle masses associated with these reactions are virtually identical to those of (2.1).  Tr* Kinetic Energy Distribution ^  Proton Kinetic Energy Distribution  50  80  40 -  60 -  30 z  z 40 20 20 -  10  -  50^100^150 T (MeV)  1--,-t  Thrl 50  200  100^1 5 0 T (MeV)  200  Figure 3.8 Figure 3.9  The conclusion that one comes to by considering the phase space distributions is the difficulty in measuring the reaction cross section of (2.1) since the energies of the outgoing particles are very simular. In the next section, a technique is proposed to measure the cross section of the principal reaction based on Monte Carlo simulation.  40  • •^  Chapter 3. Modelling of Experiment  Angle Correlation Between p and -y  Angle Correlation Between rr+ and 7 200  80 +p^Tr 7p  o^  150 —  ^.°°„!•*4  100 —  m 8-  °%:„:^  50 —  .9.  4  •  • „ s °°  °.°, ::et: As..i°!^ %°^ °°.•°°.° ° °^  ...t^° 99.11  o  ..°°  •„•.e:°> %.• ° toe °^°°° ••^ ••••,„•  :52,40 — ° • • .1.^ °°  •^ . 9 ° ^ %to^ ;te . °.• •  " °"„,98°.^'•90°,19.^  8;e: tier 444°-141/tize° :* ° .$  mn  °....  Figure 3.10  ^ 50^100 150 e n , (deg.)  0  200  ^  0  ••^.4°^  ^  50  +  and p  80  60 —  20 —  0  0  ^  100^150 (deg.)  Figure 3.12  41  ^  .•••^° is%%,!° .*°^o  ^  Figure 3.11  Angle Correlation Between rr  •  °..9•0•*8^ "8 ° 8*^:s  0  °  •••• • * °^•° :91°° • * 7;18'14 ;;999:;:•••^ •  20 —  , 1t° 0  o  60 — „40 .9t,^  a)  irr+p^/1. +7P o^  200  100  ^  B y (deg.)  150  ^  200  Chapter 3. Modelling of Experiment 3.3 Simulation  Monte Carlo simulation of reactions (2.1)-(2.5) yield the following results in the AE vs E plane. For the analysis below, the acceptance angle for the S1-S2 array has been set to 30 ° (see sections 2.3, 2.4) and the incident pion kinetic energy, to 200.0 MeV. The total number events in the AE vs E plane are typically — 5000, based on 20 000 events 'Monte Carloed'. It will be shown below that the pile-up events can be used to measure the total cross section of reaction (2.1). For reaction (2.1) w±p ---> wrfw °p, Figure 3.13 shows three distinct groups are present in the AE vs E plane. The lower band is created by  7  .+  particles while the higher band protons.  The events outlined by the polygon window are the 'pile-up' events created by simultaneous 7r+  and p hits of the S1-S2 detector (see section 2.4). The C veto (see section 2.2) has been  disabled, for this figure. For Figure 3.15, the C veto has been activated; one can see that the middle proton band on Fig. 3.13 has disappeared in this plot as a result of a scattered Ir +  triggering the C veto.  The question arises: where do the w's in the lower pion band come from? The only w+ that one should see are those that fall with the acceptance angle, and hence counted as a 'pile-up' event. The other w+'s that fall outside the acceptance angle should simply not be counted. The w's in the lower band  are  from 'pile-up' events. There are many pile-up  events where the protons are of very low energies and gets stopped in Si and hence one  42  Chapter 3. Modelling of Experiment  rr+p ->^+rr'p 8  7r  +p  ^•r+•°p  6  w4  2  0  0^20^40^60 E (MeV)  ^  80  ^  100  Figure 3.14 Surface plot of reaction events shown  Figure 3.13 Reaction events in the AE vs E plane.  in Fig. 3.13.  •1. +p  8  ^  1 +7 0 p  Tr'p -> -R+71- Op  6 .. •  w4  2 .6.??E815 00..  0  0^20^40^60 E (MeV)  ^  80  ^  100  Figure 3.15 Reaction events in the AE vs E plane, with C veto enabled.  Figure 3.16 Surface plot of events shown in Fig.  3.15. (note: this Figure and Fig. 3.14 do not have common scales 'out of the AE vs E plane').  43  Chapter 3. Modelling of Experiment only sees the 71- ± coming through both detectors. Such an event looks like a 'lone pion event'. There is no ambiguity associated with 'lone proton events' appearing in the proton band, since the phase space of the l.'s is not restricted to the acceptance angle (see Fig. 3.4).  8  7cl-fp —> -rr'p  6 a)  w4  2 0  0 0  I^I ^  20  40^60 E (MeV)  ^  80  ^  100  Figure 3.17 Elastic scattering background.  For the elastic channel (2.2) ir±p ir+p, Figure 3.17 shows that outgoing particles falling well away from, the 'pile-up window'. Hence, this background reaction is removed.  44  Chapter 3. Modelling of Experiment  For (2.3) 7r+p —> ir+-yp, Figures 3.18-3.19 show that events fall outside of the pile-up window.  ir+P 8  7T  +p . 7'-rp  2  0  1^1^1  0^20^40^60 E (MeV)  80  100  Figure 3.19  Figure 3.18  45  - 7  +7P  Chapter 3. Modelling of Experiment For the reaction (2.4) Ir+p —> -irtir+n, there is a pion band as well as a group of simultaneous i+71-+; both features lie outside the pile-up window: (there is a single count in the pile-up window, which is considered negligible).  Tr+p -> rr+Tr+n 8  Tr +p , Tr -Err +n  6 > a) w4 a 2  0  0^20^40^60 E (MeV)  ^  80  ^  100  Figure 3.20  Figure 3.21  The background reaction (2.5) T±n —> rte p, appears to cause the most difficulties. The phase space of this reaction is virtually identical to (2.1); as well, the reaction products on the right-hand side are almost the same. It is not suprising then that the signature of this reaction in the AE vs E plane (Fig. 3.22) looks very similar to that of the main reaction (Fig. 3.13). In particular, one sees that there is a significant number of events in the pile-up window, representing — 27% of the total cross section for this reaction. There is one important difference between this reaction and (2.1): both pions on the right-hand side are 46  Chapter 3. Modelling of Experiment charged. We will take advantage of this fact to eliminate this background. By using the cylindrical veto C most of the events from this reaction can be eliminated (Fig. 2.7). The no. of events now appearing in the pile-up window is approximately 0.7% of the total cross section. In principle, this method of removing the events from reaction (2.5) is redundant since the subtraction of CH 2 and C spectra (see sec. 2.3) should eliminate this background altogether. Nonetheless, we have chosen a more conservative route by eliminating events from this reaction before the subtraction. It is evident that the '0.7% of the total cross section' background mentioned above should be eliminated in the CH2 -C subtraction.  Tr +n --> Tr +Tr p  8  Tr +n -> Tr*Tr - p  6 > a) ._. w4  a  2  0  I^f^I  0^20^40^60 E (MeV)  80  100  Figure 3.23  Figure 3.22  47  Chapter 3. Modelling of Experiment  Tr +n —> 7r +7T  8  rr +n^rr 4- 71-  6  w 4  2  0  0^20^40^60 E (MeV)  Figure 3.24  ^  80  ^  100  Figure 3.25  3.4 Conclusion It is evident from the phase space of the different ir+p channels that it is very difficult to  isolate the events from the principal reaction (2.1). We conclude that to measure the total cross section, one possible strategy is to consider the 'pile-up events' in the AE vs E plane. This strategy successfully isolates a fraction of the events from the principal reaction.  48  Chapter 4  The Experiment  4.1 Introduction  A feasibility study of the apparatus based on chapter 2 was performed at TRIUMF, on the M11 beam line during July and August of 1992. This chapter is devoted to highlighting what occurred during the experiment.  4.2 Initial Set up: Additions and Modifications to Original Set up -  -  Even during the intial set-up, it was apparent that a large background signal was present due to the beam interaction with the S detectors. Therefore, several steps were taken to try and reduce this background and isolate the events coming from the target.  Absorber Introduced  It was discovered early on during set-up that protons that have leaked through along with the beam pions appeared in the S1 vs S2 plane. In order to eliminate this background, a 6 mm CH 2 absorber was introduced placed in front of the beam window (see Fig. 2.2).  The S2 Problem  Attached to the S2 detector, are 2 photomultiplier: S2-Left and S2-Right. During a trial run, it became apparent the signals from the S2-Left and S2-Right photomultipliers did not 49  Chapter 4. The Experiment  match up: when the signal (No. of counts versus channel number) from one side was superimposed on the other, one signal was skewed relative to the other. This problem, as it was discovered, was a result of a rate dependance of the photomultipliers caused by a high voltage setting ( —1800 V). The voltage was lowered and to compensate for this, an amplifier was used to boost the S2 signals.  Upon inspection of the S2-Left and S2-Right signals matching was achieved. However, while the matching problem was solved another problem with the S2 photomultipliers was transparent to the experimenters was discovered only during analysis of the data (discussed in Chapter 5).  S1 Threshold  Another technique was used to further lower the background events. The threshold of acceptance (discrimination level) was raised electronically on the S1 detector, to reject unwanted high energy events appearing away from the events in the 'pile-up' window.  S2 Detector Moved Downstream  To achieve better time-of-flight, the S2 and S3 detectors were moved 11 cm downstream'. It was not possible to further increase the time-of-flight because of the S2 detector size. ' The convention is that the beam starts 'upstream' from the beam pipe and proceeds 'downstream' through the target and detectors.  50  Chapter 4. The Experiment  4.3 Running of the Experiment  Calibration  It is necessary to calibrate the S1-S2 array detectors: that is, to assign actually energies to the channels 2 corresponding S1 and S2 signals. The calibration was performed by sending pions and protons known energies through the S1-S2 array. Beam pions of 196.4 MeV and 'leaked-through' beam protons of 29.3 MeV were used. Both energies are the kinetic energies just before the target.  Main Runs  After the initial set-up, the main runs for attempting to measure the cross section took place at 2 different energies' 195.2 MeV and 201.2 MeV. Each run is about 10 hours long, with a total number of beam events of 10 10 . A rough estimate of the cross section ( — 10 gb) suggests that the number of events from the principal reaction — 150 events at 195.2 MeV. The number of events at the higher energy is expected to be higher (perhaps a factor of two) since the cross section increases with energy (see Fig. 1.1).  2  Channels here refer to the channels of the analogue to digital converters (ADC) used to record the signals.  3  i.e., the kinetic energy of the incident 7,-+ at the center of the target.  51  Chapter 4. The Experiment  4.4 Conclusion  In spite of the difficulties encountered during the intial set-up of the experiment is was possible to collect data at 2 different energies. In the next chapter, we will describe the analysis of the data and also the results extracted from the data.  52  Chapter 5 Experimental Results 5.1 Introduction  In this chapter, the method for analyzing data will be outlined; as well, the results from the experiment will be presented.  == EXPERIMENT 655 RUN 28 18:15:50 1-SEP-92 14-11.47-1993 14:45 ^ S [AOC^  ^  /PLOT XHI  /PLOT 892  TOTAL COUNTS = 10507731.  521.ADC  TOTAL COUNTS = 10986264.  120000  50000 ^  100000 -  40000  80000 30000 60000 20000 40000 10000 -  20000 7  200  1000  S2RAOC  /PLOT XH3  200  400^600  1000  S2ADC  /PLOT 8114  TOTAL COUNTS - 10903096.  600  TOTAL COUNTS - 10946222.  40000 ^  25000 "  20000 -  30000 -  15000 20000 10000 /0000 -  5000 -  0  0 200  400  600  800  1000  0  500  1500^2000  Figure 5.1 Typical spectra from a CH, target run. Vertical scales represent the number events; Horizontal  scales represent energy in 'channel no.'.  5.2 Analysis of Data  The analysis of the results was performed using the TRIUMF software packages MOLL', NOVA, FIOWA and REPLAY (see Appendix B).  53  Chapter 5. Analysis of Results  The procedure is relatively simple: for a particular run, play back the data event by event and histogram the signals for the various detectors. Figure 5.1 shows the spectra from the Si and S2 detectors for a typical run.  5.2.1 Time of Flight Cut -  -  Figure 5.2 shows a detail of the time-of-flight spectrum between the B3 and S2. The spectrum identifies where events originate from along flight path across the experimental apparatus. The first two peaks describe events taking place in S2 and S1. The small bump of the right-hand side represents events coming from the target. However, it is clear that the 'tail' of the S1 and S2 peaks are superimposed on the target peak. The time-of-flight strategy is to introduced a limit of acceptance in time: e.g., only those events appearing with t > channel 350 are accepted. Such a 'cut' should eliminate most of the background events from 51 and S2. However, because the tails of S1 and S2 overlaps the target peak, there is no way, with the existing set-up to eliminate all the S1 and S2 events. In principal one can 'stretch' out the time-of-flight spectrum by increasing the distance between the target and each element (B3, Si, S2). But realistically one cannot get an arbitrarily large separation because of detector size limits and loss of event particles through the air. The limiting factor in the present feasibility study are the sizes of Si and S2: to preserve the cone angle (see Fig. 2.1) for capturing all the protons from the principal reaction. The timeof-flight problem will be discussed in more detail in Chapter 6.  54  Chapter 5. Analysis of Results  == EXPERIMENT 655 RUN 28 18:15:50 1—SEP-92 STIME  /PLOT %H5(X=100,499)  15—MAY-1993 13:47  TOTAL COUNTS = 11936656. I^ I S2  700000 600000 —  51 500000 — 400000 — 300000 — Target  200000 —  B.3 100000 —  0  '^I 100^200  300  400  500  Detail of Time-of-Flight spectrum from a CH, target run. The vertical scale represents the no. of events while the horizontal represents time in 'channel no.' with each channel denoting 50 ps (picoseconds). Figure 5.2  5.2.2 Events in the AE vs E Plane  Figure 5.3 shows the 2-diminensional spectra in the Si vs S2 plane (AE vs E). Starting from the top left plot and progressing clockwise, the first plot is the 'raw' spectrum without a software time-of-flight cut. There is however, a hardware cut for all spectra introduced in S1 (see sec. 4.2) and hence no events appear below channel 120 of S1. The second and third plots are the spectrum with a time-of-flight cuts at channel 350 (see Fig. 5.2) and with the same cut but the S2-left signal only, respectively. For comparison the last plot shows  55  ^  Chapter 5. Analysis of Results  -- EXPERIMENT 655 RUN 28 18:15:50 1-SEP -92 15-MAY-1993 14:23 ^ S 1VS2^  ^  /DENSITY 7: SI  /DENSTY 852  S1VS2_CNI  (q v)  747  1000  /000 ^  i  o  O.  800 -  800 -  a  600 -  600 V1  (.77  OJ  400 -  400 -  Li) 0)  200 -  200 -  II 0 0^500  1000  1500^2000  S2  C  S IVSZ_CN2  /DENSITY 7.53  500  0  0 0  1000 S2  1500  2000  S IVS2L_CN I  /DENSITY 854  0 1000 00  1000 ^ 0  800 -  800  (,) 0  600  600  ...  (71  0) 400  (7) 400 -.  (0  0 (0  200 -  ^0  0^500^1000^1500^2000  S2  C 0  200 -  ^ 0  0  200  400^600  800  1000  S2  Figure 53 Typical Si vs S2 spectra.  a spectrum with the time-of-flight cut at channel 375. Figures 5.4-5.6 show first and second spectrum in more detail. Figure 5.4 (time-of-flight cut at channel 350) will be the one used in the attempt in extracting events from the principal reaction. This time-of-flight cut represents the maximum possible channel for rejecting S1 and S2 events but without undesirably rejecting target events. B3 events cannot be eliminated since its signature in the time-of-flight spectrum is much too weak. 56  Chapter 5. Analysis of Results  ^ == EXPERIMENT 655 RUN 28 18:15:50 1-SEP-92 ^ S1VS2^ ^ /DENSITY 1S1 15-MAT-1993 14:11  == EXPERIMENT 655 RUN 28 16:15:50 1-SEP-92 ^ S1VS2^ 15-MAY-1993 14:13  /SURFACE %S1  1000 800 -  600 Li) 400 -  200 -  Figure 5.4 Si vs S2 detail, raw spectrum. ^Figure 5.5 Surface plot of Si vs S2, raw spectrum.  == EXPERIMENT 655 RUN 28 18:15:50 1-SEP-92  == EXPERIMENT 655 RUN 28 18:15:50 1-SEP-92 S1VS2_CN1  /DENSITY ZS2  /SURFACE %S2  17-M4Y-1993 15:51  ^ S1VS2_CN1  15-MAY-1993 15:13  0 _ 500  1000  2000  S2  Figure 5.6 Si vs S2 plot, with time-of-flight cut.^Figure 5.7 Surface plot of Si vs S2 spectrum with  time-of-flight cut.  57  Chapter 5. Analysis of Results  5.2.3 Normalization of Spectra  In order to subtract C target spectra from those of the CH 2 target. One must account for the number of beam events' arising from 2 separate runs for the 2 different targets as well as the different number density of each target (i.e., the different number of scatterers).  Let^I = no. of beam events 2 N^= number density of target particles in cm -3 Az^= thickness of target in cm n^= Nix , area density of target particles cm -2 AI^= no. of reaction events (or no. of particles scattered out of beam)  And, let the subscripts C, CH2, H2 denote the corresponding particles. Then, the normalized spectra (per unit beam per unit scatterer) for H2 is  Al ^AICH 2 _^2  n „,„ ''2 '`2^`-'''2 ''-'''2  In nu Irvir  A/c Ic n c  1  A beam event is defined as B1 AND B2 AND B3 AND (NOT V).  2  There is an implicit time interval associated with I in the 'event by event' scenario.  58  (5.1)  Chapter 5. Analysis of Results  But, 1H2 =ICH2  ^  and^nH2 =n CH2  Because the H2 molecules are bound to the C atom for the CH 2 molecule. For data analysis, we rewrite (5.1) in the following form ICH2nCH 2 =AICH2^/cnc Al II2^  mc  (5.2)  Equation (5.2) gives the no. of H2 events normalize with respect to a CH 2 run. n can be calculated from n=(N Ax)= co NA 8^  where^O^= pAx , the thickness of the target in g/cm 2 NA  = Avagadro's number  co^= molecular weight in g P^= the density of the target in g/cm 3  For CH 2 :^with S cH2 = 0.1480 g/cm 2^n cH2 = 6.352 x 1021 cm2 For C:^with S c = 0.1824 g/cm2^nc = 9.166 x 10 21 cm2 59  (5.3)  Chapter 5. Analysis of Results  5.3 Cross section  The total cross section a (in cm2 ) is given by the well known equation (5.4)  Where^I^= is the beam intensity in particles per unit time A/^=  change in intensity  A..1C^=  thickness of target in cm  N^= number density target particles in cm -3 n^= area density of target particles in cm 2  Hence, the total cross section for  H2  is essentially (5.1). The parameter n is introduced to  account for the fact that only a fraction of the total cross section in measured because we are only considering events in the pile-up window. AIH 2  n a H2 n I H2 H2 -  (5.5)  For a single proton, the cross section is  0H = a-2 u  2  60  (5.6)  Chapter 5. Analysis of Results  5.4 The CH 2 C Subtraction Spectra -  The C target produces a simular set of spectra. As discussed in section 2.2, to obtain the H2 cross section one must subtract the CH 2 spectra from C spectra. Figures 5.8 and 5.9  show the result of such a subtraction, with spectra normalized with respect to the CH 2 target run, eqn. (5.2). Figure 5.8 is a 'raw' subtraction without any time-of-flight cut introduced  Experiment 655 — 201 MeV — Tcut=0  ^  — Run 28/29  50 (I)  c 40 0 .........................  Q  30  20 0 o • 00.. o DO.  10 0 ^ 0  L9E8 " • 2 APAA000 0000  ,C7  •  0000  •^o 000000000  00 ^  COD MOO  10^20^30^40 S2 (2 MeV per channel)  ^  50  Figure 5.8 H 2 spectrum in the S1 vs S2 plane, with no time-of-flight cut. The total no. of events in  the plane is 2 879 241, while in the pile-up window the no. of events is 364 556.  61  Chapter 5. Analysis of Results  Experiment 655 — 201 MeV — Tcut=350 — Run 28/29 50 a)  E 40 - 30 Q (1)  20 co 10 U)  0 ^ 0  10^20^30^40 S2 (2 MeV per channel)  ^  50  Figure 5.9 H 2 spectrum in the S1 vs S2 plane, with optimal time-of-flight cut. The total no. of events in the plane is 241 790 while in the pile-up window it is 32 019.  while Figure 5.9 represents the maximum possible cut. The time-of-flight cut has decrease the total number of counts in the entire plane by a factor of 12. While in the pile-up window, the events have decreased by a factor of 11. Nevertheless, the number of counts in the pile-up window is 32 019, which remains much higher than the expected number of events from the principal reaction. 62  Chapter 5. Analysis of Results  Based on the number of beam events  IcTh = 8.082673  x 10 9 for the CH2 run, a total cross  section of ow= 100 and n = 0.15, a rough calculation using eqn. (5.5) shows that the number of events expected in the pile-up window is --150 events. Hence, the present background is some — 200 times larger than the signal.  The large background is the result of events originating from the different detectors, especially S2 detector which is very thick (11cm). The strategy for removing the 'detector' events had only limited success since there was a lack of time-of-flight resolution for separating out the target events, due to the limitations of the present apparatus.  5.5 The Subtraction Problem  A serious problem was discovered during the analysis of the data. Subtraction of the CH 2 and C spectra yielded negative numbers in the Si vs S2 plane. Figure 5.10 shows the negative contours in the 51 vs S2 plane for Fig. 5.9. The negative peak is at --100 counts. This peak does not appear to originate merely from random statistical fluctuations (i.e. noise) but rather from variation in the structure of the S signals. That is, the source of this problem originates from an instability of the Si and S2 signals during successive runs To illustrate this instability, let us consider the Si and S2 signals from 4 different runs of the same Carbon target at the same energy (201 MeV). If the S signals are stable, then subtraction of the normalized spectra with respect to the number of beam events should 63  Chapter 5. Analysis of Results  Experiment 655 — 201 MeV — Time Cut=350 — Run 28/29  35  E 30 E 0 C.)  eL  25  20 (0  15 (r)  10  0^2^4^6^8^10^12 S2 (2 MeV per channel)  ^  14  Figure 5.10 Negative contours in the Si vs S2 plane.  yield a 'zero' result with some accompanying noise. Figures 5.11-5.14 show that no such matching exists. Two reasons have been proposed to account for the 'drift' in the S signals. First, the problem may orignate from instability in the photomultipliers. Second, the energy of the beam is drifting; it is known that the uncertainty in kinetic energy of the beam pions is ±0.2 MeV, on the M11 beam line at TRIUMF. This subtraction problem remains unsolved and if the existing apparatus is used should be the subject of further study.  64  Chapter 5. Analysis of Results  S1 ADC from 4 Carbon Target Runs 120000 ^ 100000 80000 -  0 0  0000 -  40000 20000 0  0^100^150^200 50 S1 (Channel)  250  300  Figure 5.11 Si Signal Instability.  S1 Subtraction Run 34 minus Run 29  0 X  (I)  E' 0 0  —2  0  —3 -  I^I^1^ 800 200^400^600 S1 (Channel)  1000  Figure 5.12 Subtraction of two S1 spectra, Run 34 minus Run 29.  65  Chapter 5. Analysis of Results  S2 ADC from 4 Carbon Target Runs  0  200^400^600^800 S2 (Channel)  1000  Figure 5.13 Instability in S2 signal.  S2 Subtraction Run 34 minus Run 29 4000 3000 2000 V)  0 C)  1000 -  —1000 —2000 ^ 0^200^400^600 S2 (Channel)  800^1000  Figure 5.14 Subtraction of two S2 spectra, Run 34 minus Run 29.  66  Chapter 5. Analysis of Results  5.6 Conclusion  From the present feasibility study, it is apparent that several problems need to be overcome: first the background will have to be decreased by 2-3 orders of magnitude and second, the stability of the S1-S2 detectors will have to addressed, before meaningful cross section measurements can be obtained. In the next chapter, different methods will be explored to improve on the present apparatus.  67  Chapter 6 Redesign of Experiment 6.1 Introduction  In this chapter, alternative ways of measuring the cross section of the principal reaction will be considered. Modifications to the original apparatus will be introduced.  6.2 Hole in the S1 S2 Telescope -  As seen in the previous chapter the large number of reactions taking place in the S1-S2 detector has introduced a large background. One way to deal with this problem is to introduce a 'hole' in the center of both Si and S2 and hence allow the beam to pass through the array without the possibility of interaction. There is however, a disadvantage to this technique: some of the legitimate 'pile-up' events will be lost. From phase space calculations, at an incident pion kinetic energy of 200 MeV, about 6.5% of pile-up events will be lost as a result of a hole equivalent to a 5 ° cone angle. Therefore, the parameter n (representing the percentage of total cross section measured) will be further reduced by 1 percent to 14%. Figure 6.1 shows the events lost for a 5 ° hole. The total number of events in the entire plane is 5000. The total number of pile-up events is 1106 and the total number of events lost as a result of the hole is 72. Using Monte Carlo transport simulation, assuming the distance between the target and 'S' detectors is — lm, and a hole size of 5 °, about 2% of the beam interact with the 'S' detectors. Hence, we can estimate that the number of events in the pile-up window will decrease from 32 000 to 700 (using the sample run in sec. 5.1.5). 68  ^ ^ ^  Chapter 6. Redesign of Experiment  Angle Correlation Between  Tr+  and p  30 ^ pile—up region 25 -^°,17 69 iiia; oiii t e>p , 8°.° 7 o  2., rix,. 1 vs..  7V  p ->  + 0 7 lT p  00^.0 .00. ,,  —  cin 2 0 - '4 '1' 0; s/41, 8%lbo ir;;;;'4 : ;:: 411%, 0  8 15 -  ^0.;:0t?,  0 0  .. c 0.  ":  41- (0 0,. 0,^.. ^,. . .°808..8. . 47,2.tifityg.zire.0rop:..,:,,,,,i; ,, ,,}  ,  o  8t ^7  io —^  7 8 a.p. 0°801/4. o ° ° 8 ° b .* : ^0 0 0,0^ 99,!. 00 , 0,!,, $0: 0, :% 04.; ok. ,,,, 8 ,,°,0, 0 0 0 * 0: 0 0 00o  5 0  e 4.1  Oe e .. 0 O  .^00  ^  °o  i: 00  0 .  0  ,,,t,  0 0 o^0^0 ' ''^ 0. 0 0 t^...^ ^0^  t., .^ O. .0  s  o  -  0^.0 00  %^.  0 0^ 50^100 rejection region for 5 ° hole^0 ,1_ (deg.)  150  200  Figure 6.1 Angle correlation between outgoing pion and proton. The 'L' window in the lower left region represents events lost for a 5 ° hole.  There are two ways to introduce a 'hole' in the S detectors. One is to physically cut a hole. The difficulty in this technique is to assure that there will be proper light collection in the S detectors. The other is to introduce beam veto detectors much like B1, B2 and B3 downstream of the target and before the S1-S2 array. While the advantage to this technique is that one does not need to modify the S detectors, the disadvantage is that the veto detectors introduced will themselves have a background signature.  69  Chapter 6. Redesign of Experiment  6.3 Increasing Time of Flight Separation -  -  It is obvious that the larger the time-of-flight separation, the better events coming from the  target will isolated from those of the detectors, particularly S1 and S2. However, the distance between the target and the S detectors cannot be arbitrarily large, since a 'dimininshing returns' phenomenon starts to take effect because the outgoing protons from the principal reaction get absorbed by air. For instance, at an incident pion kinetic energy of T„ = 200 MeV, the mean kinetic energy of the protons is — 30 MeV. At a distance of 0.775 m, half the protons will be lost. A way to reduce the effect of this problem may be to use a 'bag' of helium to fill the distance between the target and S1-S2. To get an idea how much the distance between the target and the S detectors needs to be increased, let us suppose that the incident w+ kinetic energy is T„ =201 MeV. To separate the 'peak' arising the target events (see Fig. 5.2), we need to shift the peak to the right by about 100 channels (5 ns). For complete isolation of the target 'peak', we use the fastest possible particle coming from the target which is a 2-+ with kinetic energy of 201 MeV (elastic scattering) to determine the increase in distance needed to achieve this time-of-flight separation. The increase in distance needed is 1.37 m. Repeating the calculation with it's with a kinetic energy of 60 MeV (these are the maximum kinetic energy it's from the principal reaction), the increase in distance needed is 1.07 m. Clearly, it is not possible to let the outgoing particles from the reaction to travel through air since as shown above over half the pile-up events would be lost.  70  Chapter 6. Redesign of Experiment  L.^d  7*, p^D  ^ ^ ^ d L. L.  ^g1^g2^gl  01  ^  Q2  ^  Q3  Figure 6.2 The triplet set-up.  6.4 Triplet Lens  With increased time-of-flight separation, not only does one runs into the problem of particle loss through air but also the requirement of large S1-S2 detectors. At a distance of 2 m, to cover a 30 ° cone angle the S detectors would have to be 1.07 m in diameter. One interesting method has been suggested to overcome both of these problems. Essentially, the method proposes to use a magnetic quadrapole triplet as a lense to focus outgoing event particles, keeping them within a reasonable size envelope downstream from the target. And by housing the triplet with a vacuum pipe, one can achieve a huge time-of-flight separation (a distance of — 3 m between the target and S detectors can easily be achieved).  71  Chapter 6. Redesign of Experiment  6.4.1 The Triplet Arrangement  Figure 6.2 shows the triplet set-up used for the simulation below. Each quadrapole (Q1, Q2, Q3) has a bore diameter of D = 20.3 cm (8"), with a typical field gradient of g — 0.6 KG/cm and effective length of Le = 0.49 m. The parameters that one adjusts are 'd' , the distance between the quadrapoles and 'g' the field gradient. For simplicity, a symmetric triplet will be used, i.e., the distance between Q1 and Q2 equals that between Q2 and Q3; and the field gradient in Q1 equals Q3.  6.4.2 The Triplet Simulation  To model the effect of the triplet on outgoing particles from the principal reaction, two software routines from TRIUMF was used: RAY 1 RACE' and REVMOC. The first routine makes use of a field map of the quadrapole triplet and raytraces particles through the system; the optics (focii, focal length, etc) of a particular triplet arrangement can be determined. Unfortunately, one needs to use a second routine to raytrace particles from the principal reaction because RAY'11(ACE can only handle very small particle divergences. REVMOC is a monte carlo beam transport program that performs the final raytrace with events from the principal reaction.  Raytrace by Arthur Hayes, April 1980.  72  Chapter 6. Redesign of Experiment  To summarize and elaborate on the method used:  1.  Use RAYTRACE to find an optical set-up for a parallel stream of particles (e.g. protons) of fixed momentum. The optics of the system is then 'tuned' so that there is focusing in both transverse directions x and y. The focii for both transverse directions are made to coincide.  2.  Duplicating the set-up from RAYTRACE, use REVMOC to raytrace events from the principal reaction.  Note: All analysis will performed in vacuum, at an incident pion kinetic energy of 200 MeV.  6.4.3 Results from the Simulation  Protons from the principal reaction are used to see the effect of the triplet. It is evident that the range of the momentum (10-340 MeV) and the range of the divergence ( — 0-30 °) are too large for a realistic size triplet lense to handle. For example, for quadrapoles with bore diameter D = 20.3 cm, 94% of protons gets rejected': for bore diameter of D =30.5 cm, 88% gets rejected. To see the effect of a smaller range of divergence, we limit the x and y divergences to be less than 5 ° . For quadrapoles with bore diameter D =20.3 cm, 63% of protons gets rejected; for bore diameter of D = 30.5 cm, 41% gets rejected.  2  'rejected' implies that a particle has drifted outside a cylinder defined by the quadrapole bore diameter.  73  Chapter 6. Redesign of Experiment  Ray Envelope for Triplet 1 1 1 1 11 1 11 11 1 1 1 11111111111111111.11111111111111111 ,,,,, 1111111 1 1 1 11 1 I I 1111 II I I 1111 1111 1 11111111111111  8  ^lllllll 11111/1/111111111111111 lllllll 11111111 l^llll 11111111 lllllll 111111111111  0^1^2^3^4^5 z (m) Figure 63 Raytrace of monoenergetic protons with Tp = 36 MeV, zero divergence in the transverse (x,y)  directions.  Plots from analysis with constraint of < 5 ° divergence Triplet Tuning  Figure 6.3 shows the result of tuning the triplet set-up for a beam of monoenergetic protons with Tp = 36 MeV. This energy was chosen because it is the mean energy of outgoing protons at an incident 7 + energy of Tir , =200 MeV. For convenience and without loss of generality, the proton beam is chosen to have zero divergence in the transverse (x,y) directions. Note: 1.^The triplet occupies the space z= (0.00, 2.92)m for this part of the analysis. Subsequent analysis will shift the triplet to another location in z.  74  Chapter 6. Redesign of Experiment  2.  The beam envelope with the single focus (divergent-convergent-divergent plane) is chosen to be the y direction.  3.  The beam envelope with the double focus (convergent-divergent-convergent plane) is chosen to be the x direction.  4.^Rays from both envelope converge at a focus at z— 4.43 m.  Field Gradient of 01 and Q3  ^  Field Gradient of Q2  .30  .25  .25  F  .20  .20  E  .15  .15  0  cn  .10  .10  .05  .05 .00  Lt„..1111,111  0  20^40^60 z (cm)  .00  80  0  20^40^60^80 z (cm)  Figure 6.4 Field gradient of 01 as a function of ^Figure 6.5 Field gradient of 02, as a function of z. axial distance z. 03 has an identical field gradient.  The settings for focusing as shown in Figure 6.3 are d = 60.0 cm g1= 0.258 kG/cm g2 = 0.248 kG/cm.  Figures 6.4 and 6.5 show the field gradient of each quadrapole with these settings.  75  Chapter 6. Redesign of Experiment  Raytrace with protons from the principal reaction  Figures 6.6 and 6.7 show the result of the monte carlo simulation for a triplet with the above settings. The two vertical lines define the location of the triplet, z = (1.53,4.45) m, which is different from the previous location along z axis. It is apparent that even restricting the particles to divergences of less 5 °, the beam envelopes are still unrealistically large.  Raytrace of Triplet 50  1111111111^11^1111 1 1111111  ^  )111111m^)111 )11111^uuuuuuduul1 1 mil li^1u 11111^n 1111^nuuu1  40 30 20  E  10 0--10 —20 —30 —40 —50  0^1^2^3^4 z (m)  Figure 6.6 Raytrace of outgoing protons from the reaction ir + p^ir + ir°p, with  divergence in the x,y directions <^(dcd plane).  76  Chapter 6. Redesign of Experiment  Raytrace of Triplet 20  11111111111111111111111111111111111111 1111111111111111111 ^11111111,111111111111111111 1111111111111111t11111  15 10 5-  E (-)^0 —5 - 10 —15 —20  0^1^2^3^4  Figure 6.7 Raytrace of protons from the reaction 7r + p —> ir + 7r°p, with divergence in the transverse directions < 5 ° . This figure shows the x direction (cdc plane).  77  Chapter 6. Redesign of Experiment  6.5 Conclusion  In this chapter, several methods have been proposed to improve on the current set-up. Introducing a hole in the S detectors and increasing the time-of-flight separation remain two feasible methods in dealing with the background. Preliminary calculations show very impressive reductions in the background by implementing these methods. Nonetheless, these calculations only suggest that the background appears to be within the same order as the events of interest. It is difficult, if not impossible to test out these methods without further experimentation. It was also shown in this chapter that the triplet lens will not be useful in helping the gain more time-of-flight separation since the outgoing particle envelope remain unrealistically large.  78  Chapter 7  Final Conclusions  A feasibility study for measuring the total cross section for the 7r-27r reaction, 7 + p --> 7 + 7°p was performed. The data collected was not useful in extracting the total cross section. However, the data was useful in accessing the background events for the existing apparatus. The background signal is 2-3 orders of magnitude larger than the 'reaction' signal. Several ways were introduced to help reduce the background: however, it was not possible to state conclusively that these methods will reduce the background sufficiently to extract a cross section measurement without further experimentation. In chapter one, we also showed the motivation for performing this experiment in the context of chiral perturbation theory which suggests that in order for this reaction to be useful in extracting information about  r ir -  scattering, the experimental error for the total cross section must be less than — +10%. This constraint poses another challenge for measuring the total cross section for this reaction.  79  Bibliography  [1] Steven Weinberg, Phys. Rev. Lett. 17 (1966) 616. [2] Martin Sevior et al., Phys. Rev Lett. 66 (1991) 2569. [3] Neil Fazel, M.Sc. Thesis (1992) University of British Columbia, unpublished. [4] Eli Friedman, Triumf Research Proposal, Experiment 655 (1991), unpublished. [5] Eli Friedman et al., Phys. Lett. 231B (1988) 39. [6] Eli Friedman et al., Nucl. Phys. A in press. [7] Eli Friedman, Triumf Research Proposal, Experiment 598 (1990), unpublished. [8] Eli Friedman et al Phys. Lett. 302B (1993) 18. [9] J.D. Jackson, Classical Electrodynamics, 2nd ed., John Wiley, Chapter 13 (1975)  [10] Yu. A. Batusov et al., Sov. J. Nucl. Phys. 21 (1975) 162; Sov. J. Nucl. Phys. 1, (1965) 374. [11] M. Arman et al., Phys. Rev. Lett. 29 (1972) 962. [12] B.R. Martin, D. Morgan and G. Shaw, Pion-Pion Interactions in Particle Physics, Academic (1975). [13] Steven Weinberg, Phys. Rev. Lett. 18 (1967) 188. [14] J. Schwinger, Phys. Lett. 24B, (1967) 473. [15] M.G. Olsson and L. Turner, Phys. Rev. Lett. 20 (1968) 1127. [16] M.G. Olsson and L. Turner, Phys. Rev. 181 (1969) 2141. [17] J. Gasser and H. Leutwyler, Phys. Letters 125B (1982) 312. [18] A.N. Ivanov and N.I. Troitskaya, Soy. J. Nucl. Phys. 43 (1986) 260. [19] J. Lowe et al., Phys. Rev. C 44 (1991) 956. [20] M.G. Olsen et al., Phys. Rev. Lett. 38 (1977) 296.  80  [21] D. Mark Manley, Phys. Rev. D 30 (1984) 536. [22] E. Oset and M. Vicente-Vacas, Nucl. Phys. A 446 (1985) 584. [23] J. Gasser and H. Leutwyler, Phys. Rep. 87 (1982) 77. [24] J. Gasser and H. Leutwyler, Phys. Lett. 125B (1983) 325. [25] J. Gasser and H. Leutwyler, Phys. Lett. 125B (1983) 321. [26] A.V. Kravtsov et al., Nucl. Phys. B 134 (1978) 2622. [27] J. Gasser and H. Leutwyler, Ann. Phys. (N.Y.), 158 (1984) 142. [28] J.F. Donoghue et al., Phys. Rev. D 38 (1988) 2195. [29] V. Sossi, Phys. Lett. B 298 (1993) 287. [30] V. Barnes et al., CERN Report 63-27 (1963).  Appendix A Electronics  Figure A.1 Block diagram of 'beam' logic.  82  Appendix A. Electronics  Figure A.2 Block diagram of 'detector' logic.  83  Appendix A. Electronics  C212 BIT 0  t  C B S  OR LAM ^TDC START ^ GATE ADC TRIGGER CAMAC  B•S•C  (C212 STROBE)  B SAMPLE  'BLOCK EVENT HARDWARE'  START 222 GATE  NIM  ^ 'INHIBIT SCALERS'  OR VETO STOP 'COMPUTER'  END BUSY PULSE (OUT. REGISTER BIT 1)  MASTER 'COMPUTER BUSY' (OUT. REGISTER BIT 0)  Figure A.3 Block diagram of 'event' logic.  84  Appendix A. Electronics  ND021 OUTPUT REGISTER  START ...____OR LAM  S---• STOP 0 e__,.... MASTER  RF--• STOP CAP. PROBE wip  C212 COINCIDENCE BUFFER  STOP  TDC  0  BIT  B SAMPLE ^ BIT  C -0  OR LAM  oil  1 •--.— END BUSY  STARBURST 2.^ SCALER CLEAR  BIT  B•S•C  J11  STROBE  X  TERMINATE  Figure A.4 Various modules.  85  Appendix B Analysis Software  Listing of Routines for Driving MOLLIE'  There are 3 routines (define, dplot and scalers) in 3 source files (definel.for, dp2.for, and sca.for).  definel.for C this version is for multiple time cuts analysis C SUBROUTINE DEFINE C C CALL PTITLE1('EXPERIMENT 655') CALL PTITLE2('RUN NO. ') C C DETECTOR HISTOGRAMS C CALL TH1ST(1,'S1ADC$') CALL PHIST(1, 0.0, 2.0, 500, 0) C CALL THIST(2,'S2LADC$') CALL PHIST(2, 0.0, 2.0, 500, 0) C CALL THIST(3,'S2RADC$') CALL PHIST(3, 0.0, 2.0, 500, 0) C CALL THIST(4,'S2ADC$') CALL PHIST(4, 0.0, 2.0, 1000, 0) C c*********************************************************************** C S1 VERSUS S2 HISTOGRAMS C CALL TSCAT(1,'S1VS2 @S2gS1@$') CALL PSCAT(1, 0.0, 40.0, 50 , 0.0, 20.0, 50) C  MOLLI stands for 'Multi Offline Interactive Analysis' and is a software package for offline analysis of data. for more information see the documentation titled 'MOLLY by Anne W. Bennett (1983) and Corrie Kost (1985).  86  Appendix B. Listing of Routines for Driving MOLLI C CALL TSCAT(3,'S1VS2_CN2 @S2@S1@$') CALL PSCAT(3, 0.0, 40.0, 50 , 0.0, 20.0, 50) C CALL TSCAT(4,'S1VS2L_CN1 @S2@S1@$') CALL PSCAT(4, 0.0, 20.0, 50 , 0.0, 20.0, 50) C CALL TSCAT(5,'S1VS2R_CN1 @S2@S1@$') CALL PSCAT(5, 0.0, 20.0, 50 , 0.0, 20.0, 50) C CALL TSCAT(6,'S1VS2L_CN2 @S2@S1 @$') CALL PSCAT(6, 0.0, 20.0, 50 , 0.0, 20.0, 50) C CALL TSCAT(7,'S1VS2R_CN2 @S2@S1@$') CALL PSCAT(7, 0.0, 20.0, 50 , 0.0, 20.0, 50)  C*********************************************************************** C TIME OF FLIGHT HISTOGRAMS C CALL THIST(5,'STIME$') CALL PHIST(5, 0.0, 2.0, 1000, 0) CALL THIST(6,'TCAP$') CALL PHIST(6, 0.0, 2.0, 1000, 0) CALL THIST(7,'TRF$') CALL PHIST(7, 0.0, 2.0, 1000, 0) C RETURN END  Appendix B. Listing of Routines for Driving MOLLI dp2. for C this version is for multiple time cuts analysis C SUBROUTINE DPLOT C C C C SUBROUTINE DPLOT C C COMMON /EVENT/ RAW(50) C COMMON /IREC/ IRAW(50) C REAL*4 TCUT1 /350.0/ REAL*4 TCUT2 /360.0/ C^REAL*4 TCUT3 /375.0/ INTEGER*4 EMASK /1/ INTEGER*4 BMASK /2/ C REAL*4 S1ADC, S2LADC, S2RADC, S2ADC REAL*4 STIME, CAP PRB, RF INTEGER*4 BITS — INTEGER*4 EVENT, BSAMPLE C S1 ADC = RAW(2) S2LADC=RAW(5) S2RADC=RAW(4) S2ADC=S2LADC + S2RADC  C STIME = RAW(7) CAP_PRB=RAW(8) RF=RAW(9) C BITS=RAW(11) EVENT= (EMASK .AND. BITS) BSAMPLE= (BMASK .AND. BITS) C^WRITE(6,*) BITS,EVENT,BSAMPLE C IF (EVENT .EQ. EMASK) THEN CALL HIST(S1ADC, 1., 1) CALL HIST(S2LADC, 1., 2) CALL HIST(S2RADC, 1., 3) CALL HIST(S2ADC, 1., 4) CALL HIST(STIME, 1., 5) CALL SCAT(S2ADC, S1ADC, 1. , 1) CALL SCAT(S2LADC, S1 ADC, 1. ,4)  88  Appendix B. Listing of Routines for Driving MOLLI CALL SCAT(S2RADC, S1ADC, 1. ,5) C  C  IF (STIME .GT. TCUT1) THEN CALL SCAT(S2ADC, S1ADC, 1. , 2) END IF IF (STIME .GT. TCUT2) THEN CALL SCAT(S2ADC, S1ADC, 1. , 3) CALL SCAT(S2LADC, S1ADC, 1. ,6) CALL SCAT(S2RADC, S1ADC, 1. ,7) END IF END IF  C IF (BSAMPLE .EQ. BMASK) THEN C CALL HIST(CAP_PRB, 1., 6) CALL HIST(RF, 1. , 7) END IF RETURN END  89  ^  Appendix B. Listing of Routines for Driving MOLLI sca.for SUBROUTINE SCALERS (*,*,*) C C= = Suen version C= = To fill the scaler values C C IMPLICIT NONE include 'molli$DIR:scalers.inc' include 'molli$D1R:molli_units.inc' include 'molli$D1R:mflags1.inc' include 'molli$DIR:pointer.inc' include 'molli$DIR:irec.inc' INTEGER*2 INT2(2) INTEGER*2 MASK(6)/1,2,4,8,16,32/ INTEGER*4 KOVER/16777216/ C C default integer declaration in INTEGER*4 C INTEGER INT4, NWORD, IPOINT, IVAL, K, I, J, NVALUE EQUIVALENCE (INT2(1),INT4) C^ C= = Each scaler uses 2 INTEGER*2 words in IREC. C= = These are combined to a single INTEGER*4 word in SCALER C= = Ignore this Type SCALER event if it is the first event of a run C INT2(1)=IREC(KOUNT+4) INT2(2) =IREC(KOUNT+5) IF(INT4.EQ.1)RETURN NWORD=IREC(KOUNT+1)/2 NBLOCK= (NWORD-5)/14 IF(NBLOCK.LT.1)RETURN 1F(NBLOCK.GT.n_scal_m)THEN WRITE(prunits(1),50)NBLOCK,n_scal_m If(log)WRITE(prunits(2),50)NBLOCK,nscalm 50^FORMAT('OType "SCALER" Event with ',I4,' block;',/, *^' Array sizes in ANALYZE can handle only',i3,' blocks') RETURN1 END IF DO J =1,NBLOCK  90  Appendix B. Listing of Routines for Driving MOLLI IPOINT=KOUNT+8+14*(J-1) C nscale is 6 DO I =1,NSCALE INT2(1)=IREC(IPOINT) INT2(2)= IREC(IPOINT+ 1) SCBUF(I,J) = INT4 SCALER(I,J) =SCALER(1,J)+INT4 IVAL = I+ (J-1)*NSCALE !POINT = [POINT + 2 ENDDO ENDDO RETURN END  91  Appendix C  Detectors are made of NE102 plastic scintillator. B1  0 32 x 3.2 mm (0 1.3" x 1/8")  B2  0 32 x 3 2 mm (0 1.3" x 1/8")  B3  0 27 x 3.2 mm (0 1.1" x 1/8")  S1  0 2032 x 16 mm (0 8" x 1/16")  S2  0 355.6 x 101 6 mm (0 14" x 4")  S3  0 365.8 x 12.7 mm (0 14.4" x 1/2")  C  (outside dia.) 0 203.2 x 340 x 3 2 mm (0 8" x 13.4"^x 1/8")  Table C.1 Detector sizes.  92  Appendix C. Detectors  Target Locations: T1 for 220 MeV T2 for 195 MeV  c'3  Figure C.1 Target geometry.  93  Appendix C. Detectors  Not to scale; all dimensions in mm.  Figure C.2 Detector geometry.  94  

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