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Determination of the cross section and matrix element for H([pi]⁺,[pi]⁺[̧pi]⁺)N near threshold : a preliminary… Lange, J. Brandon 1992

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D E T E R M I N A T I O N OF T H E CROSS SECTION A N D MATRIX E L E M E N T FOR H(ir+ ,ir+7r+)N N E A R THRESHOLD: A P R E L I M I N A R Y ANALYSIS By J. Brandon Lange B.Sc.(Hon), McMaster University, 1990  A THESIS SUBMITTED IN PARTIAL FULFILLMENT O F THE REQUIREMENTS FOR THE DEGREE OF MASTER OF 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  October 1992 © J. Brandon Lange, 1992  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at t h e University of British Columbia, I agree t h a t t h e Library shall make it freely available for reference and study. I further agree t h a t 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 T h e University of British Columbia 6224 Agricultural Road Vancouver, Canada V 6 T 1W5  Date:  Abstract The total cross section for the reaction ir+p —»• 7r + 7r + n was measured at incident -K kinetic energies of 184 and 200 MeV.  The threshold value for the matrix element  |a(7r + 7r + )| was also determined through a preliminary analysis. T h e results are thus far in disagreement with the world's d a t a set and suggestions for further work have been made.  n  Table of C o n t e n t s  Abstract  ii  List of Tables  vi  List of F i g u r e s  viii  Acknowledgments  x  1  Introduction  1  2  P r e d i c t i v e Chiral S y m m e t r y  4  2.1  Chiral Symmetry  4  2.1.1  6  2.2  The Partially Conserved Axial Current Hypothesis  6  2.3  The Model of Olsson and Turner  8  2.4 3  Goldstone Bosons  2.3.1  Scattering Length Predictions  10  2.3.2  N* Contributions At Threshold  11  Summary  12  The Experiment  14  3.1  Apparatus  14  3.1.1  The Active Target  14  3.1.2  The Deflecting Magnet  16  3.1.3  The Neutron Bars  17  iii  3.2  4  3.2.1  Detection of 7r+ —• y^v^  3.2.2  Yield Determination  18 Decays  20 24  Neutron Detection  26  4.1  Introduction  26  4.2  T D C Calibration  27  4.2.1  Time of Flight  27  4.2.2  Position Reconstruction  30  4.3  ADC Calibration  31  4.4  Energy Determination  34  4.4.1  36  Background  4.5  Monte Carlo Corrections  39  4.6  Neutron Detection Efficiency  41  4.6.1  C P U Livetime (A)  41  4.6.2  Position Acceptance ( r )  42  4.6.3  Stopped TT'S  42  4.6.4  T h e Branching Ratio (Pf)  43  4.6.5  Solid Angle Acceptance (ft)  43  4.7 5  Procedure  Summary  43  T h e Active Target  47  5.1  Composition of the Target  47  5.2  Energy Calibration  47  5.3  T h e CCD D a t a  49  5.3.1  Modeling The CCD Signals  50  5.3.2  T h e CCD Pulse Heights  52 iv  5.3.3 5.4 6  7  8  \x Decay Times  54  Stopped -K Detection Efficiency  56  T h e Incident B e a m  61  6.1  7r-Fraction  62  6.2  Acceptable Beam Burst Pattern  62  6.3  Multiple Beam Particles  64  The Results  68  7.1  The Acceptance  68  7.2  Determination of the Yields  70  7.3  T h e C P U Livetime  74  7.4  The Results  74  7.5  Further Work  78  Conclusions  81  Bibliography  82  v  List of Tables  2.1  Theoretical predictions for the isospin 0 and 2 s-wave scattering lengths for the 7r — 7r interaction  10  3.2  Deflecting angles and neutron bar clearances  18  3.3  Expected values of Tsum  19  4.4  Monte Carlo predictions for the neutron detection efficiency  34  4.5  CPU livetimes for the neutron detection efficiency calibration runs . . .  42  4.6  Stopped 7r d a t a for the calibration runs  43  4.7  Neutron Detection Efficiency parameters for 8.9 MeV neutrons  44  4.8  Neutron Detection Efficiency parameters for 68 MeV neutrons  46  4.9  Comparison of the neutron detection efficiency with the Kent State Monte Carlo code predictions  46  5.10 Number of target nuclei in the active target 6.11 Poisson probability of only one particle occurring in a beam bucket.  47 . .  67  6.12 Number of useful beam 7r incident upon the target  67  7.13 Number of simulated 7rp —• rnrp reactions per 1 x 10 12 incident 7r's. . . .  68  7.14 Experimental acceptances  70  7.15 Kinematic ranges for the final state products allowed to contribute to the reaction  71  7.16 The experimental yields  74  VI  7.17 C P U livetimes for Theam = 200 and 184 MeV  74  7.18 Preliminary cross sections  75  7.19 Kinematic parameters for 7r's of various beam energies  78  7.20 Preliminary reaction amplitudes  78  vn  List o f F i g u r e s  2.1  Theoretical predictions for isospin-0 and isospin-2 IT — ir scattering lengths. 11  2.2  Feynman diagrams believed to contribute to 7r-production at threshold.  12  3.3  Layout of the apparatus  15  3.4  T C A P spectrum for 7 ; = 1 8 4 MeV  17  3.5  First level trigger hardware circuit for target section A  21  3.6  Spectrum of the difference in detected energy between the long and short ADC's for target section C  22  3.7  Digitized long ADC signals target section A  23  3.8  Digitized long ADC signals for each section of the active target  25  4.9  Logic circuitry for the neutron bars  28  4.10 An individual neutron bar T D C spectrum  30  4.11 Position reconstruction for a front neutron bar  31  4.12 Typical neutron bar ADC output and comparison to the Kent State Monte Carlo code's predictions  33  4.13 Reconstructed kinetic energies for the 8.9 and 68 MeV neutron spectra.  35  4.14 Reconstructed kinetic energies for the expanded 8.9 and 68 MeV neutron spectra  38  4.15 Monte Carlo simulation of the (7r~p —• ynn) background contribution to the kinetic energy spectrum  39  4.16 Neutron yields for the 8.9 and 68 MeV calibration runs  vm  40  4.17 Neutron detection efficiency predicted by the Kent State Monte Carlo code for b o t h the front and back neutron bars as a function of kinetic energy  45  5.18 Long ADC spectrum for 200 MeV passing 7r's in target section E  49  5.19 Calibrated pulse height versus calibrated short ADC signal for passing 7r in target section A  53  5.20 Calibrated /.i pulse height spectrum  54  5.21 A comparison between the Second and Third Level Triggers  55  5.22 Aggregate time between fit pulses for 200 MeV incident 7r's for all target sections  56  5.23 Dot plot of calibrated Long ADC values versus calibrated Short ADC values for an incident 200 MeV 7r beam stopped in target section D. . .  57  5.24 Stopped IT detection efficiency as a function of long ADC value for target sections A through E  60  6.25 Hardware circuit to determine the beam burst p a t t e r n  63  6.26 S3 Prime spectra for events (a), and beam samples (b)  65  7.27 T h e effect of kinematic cuts in determining the yield  71  7.28 T h e experimental yields  73  7.29 Preliminary total cross sections  76  7.30 Preliminary reaction amplitudes  77  7.31 Neutron ADC output for any bar  79  IX  Acknowledgments  'Before I started university, ogy, psychology  I wish I knew that philosophy  is really psychol-  is really biology, biology is really chemistry,  really physics, physics is really math, and that math is really -  chemistry  is  philosophy.'  anonymous  Many thanks are due in the completion of this thesis, none more so than to my supervisor, Dr. Martin Sevior, who spent many patient hours 'getting me up to speed' and guiding my work. I would also like to express my gratitude to Kel Raywood who spent many hours helping m e win my battle against our software, extracting the -K detection efficiencies and instructing me about the A.F.L. To them I a m very grateful. I would also like to thank the many people who worked on E561(1991). Colleagues play an important part in making it through graduate school with one's sanity intact. In particular, I would like to thank Giovanni Pari for providing me with my inspiration [1]. Also deserving recognition are Andrew, Eraser, Gertjan and Jim for providing 'the wisdom of those who have gone before' and to G a r t h , Henry and Irene who La Proofread I^TgXt. I would be remiss without mentioning my parents, who encouraged my education from the start. None of this would be possible without them. And finally, I would like to acknowledge Mr. Robb Taylor, my high school computer science and calculus teacher who re-enforced my love of science, and Mr. Ron Wessner, m y high school physics teacher who gave me a love of physics, both of whom to which this thesis is thoughtfully dedicated.  x  Chapter 1  Introduction  Q u a n t u m Chromodynamics (QCD) has been quite successful in describing the strong interaction at energies above 5 GeV, but is difficult to apply at lower energies. Before QCD can be definitively accepted as the correct theory of the strong interaction, it must be understood at all energies. One of the simplest strongly interacting systems is the 7r — TT system, which involves only two of the lightest mesons. Weinberg [2] described the system by a Lagrangian which incorporated chiral symmetry (c./.  chapter 2.)  Olsson and Turner [3] then  showed that Weinberg's Lagrangian is a member of a family of Lagrangians which are described by a theoretical parameter, £, which is varied to alter the strength of the 7r — 7r interaction. A number of different theories have been developed to describe the higher order corrections to Olsson and Turner's family of Lagrangians, but none have been confirmed by experiment. Olsson and Turner made direct predictions of the scattering length for the TT — TT system. T h e scattering length, a, is defined in terms of the limit of the interaction cross-section as the wavelength of the incoming particle goes to infinity:  where a is the scattering length.  — lim a = a2  (1.1)  4-7T A--00  V  '  The scattering length of an interaction is of par-  ticular interest since it characterizes the low energy behaviour of the interaction. An experimental determination of the scattering lengths for the it — ir system would hence provide an interesting test of the symmetries inherent to the strong interaction. 1  Chapter 1.  2  Introduction  T h e best way to measure the TT—TT scattering length would be a direct 7r—-K scattering experiment. However, the mean lifetime of the charged TT is only 26 nanoseconds; the lifetime of the TT° is much less than that. Indirect methods must therefore be used. T h e reactions that have been the most extensively investigated are K decays and pion induced pion production reactions such as (irp —» inrn). T h e best K decay experiment performed to date was performed by Rosselet et al. [4] who studied the reaction K+ —> ir+ir~e+ve.  The advantage of such a reaction is that  the only particles that feel the strong nuclear force are the two 7r's in the final state. Rosselet's collaboration undertook to evaluate the phase shift difference, 8, between the isospin-0 and isospin-1 dipion states. They then used the model of Basdevant et al. [5] to evaluate the isospin-0 scattering length, ao- Unfortunately, a subtle analysis of the angular and energy distribution of the final state particles, which requires very high statistics, is needed to obtain an accurate measurement of CLQ. Rosselet et al. ran for 4 months to accumulate 30,000 events, resulting in a value of 0.28±0.05 for a 0 , a 22% error. The second class of reactions are difficult to do well since there is a large physics background. T h e itp —> -Kirn reactions have three strongly interacting particles in the final state and may proceed through a variety of Feynman diagrams [7]. These background processes may be substantially reduced by determining the cross-section of such reactions at energies close to threshold. Pion induced pion production reactions, which are also referred to as (TT, 2rc) reactions, are purely strong interactions and thus reasonable statistics may be reached on the order of a few days. In 1989, Sevior et. al. [8] reported results for the 7r+p —• 7r + 7r + n reaction. While their results were consistent with an isospin analysis of the world's d a t a set [9] they were in substantial disagreement with an experiment by the OMICRON collaboration [10]. Sevior et al. [11] repeated their experiment in 1991 for both incident 7r+ and 7r~  Chapter 1.  Introduction  3  beams with an improved experimental arrangement. This thesis will detail the d a t a analysis of 7r+p—> 7r + 7r + n.  These d a t a can then be used to determine the isospin-2  scattering length of the TT — -n interaction and hence make an interesting test of the strong interaction at low energies.  Chapter 2  P r e d i c t i v e Chiral S y m m e t r y  2.1  Chiral S y m m e t r y  Q u a n t u m Chromodynamics (QCD) was developed in order to describe the strong interaction, and has been generally accepted. However, QCD has proven to be difficult to apply at low energies. At high energies, QCD may be applied using perturbative methods. However, these methods break down at intermediate energies, and one is forced to use a variety of models 1 , none of which have been established as entirely correct. At low energies, (E < 500 MeV), one is again able to make predictions about strong interactions with confidence. These low energy predictions are based on chiral symmetry. Quarks have an internal angular momentum, and it is this 'spin' that gives rise to chiral symmetry. Let us first consider a system of NL quarks with left-handed helicity and NR right-handed quarks. Further, we can allow each system to have /  flavour  states available. Now if we take the quarks to have zero mass, the two systems become discrete. This is because, in QCD, left-handed particles do not interact with righthanded particles, and it is therefore not possible to boost to a frame where some of the particles appear to change helicity since the quarks have zero mass. Consequently, the left- and right-handed quarks comprise completely separate systems and do not affect each other. This is known as an SU(/)L  <8> SU(J)R  to be chirally symmetric. ^ o l e models and quark models are two examples.  4  invariance, and the system is said  Chapter 2. Predictive  Chiral  Symmetry  5  Now let us consider a system where the quarks each have a small mass. In this case, it is possible to boost to a frame where some quarks appear to change their helicity state, and t h e two sub-systems become kinematically linked, at the very least [23]. In this case, we must look at t h e wavefunction for t h e entire system, tfr. Since any given quark can have either positive or negative helicity, we must use a projection operator to separate t h e left- and right-handed helicity states.  V>L = TLip  where T satisfies:  rL + rR = I r i = v2R = I Now the Lagrangian of the total system is:  C = tf(i>y»d„ - m)r/>  (2.2)  where ifi = ipL + ij>R- Expanding equation 2.2 gives £ = ipLi^d^L  + ij)RiYd^R  + ipLmipR + 4>Rm^L  (2.3)  In this case, t h e chiral symmetry is explicitly broken by t h e presence of the quark mass because the Lagrangian now contains terms involving both ipL and ipR. In the limit of a zero quark mass, the left- and right-handed wavefunctions do not interact with each other, the chiral limit is reached and chiral symmetry is achieved. Spontaneous symmetry breaking is more subtle in that it occurs only when the equations of state (i.e. the Hamiltonian or the Lagrangian) show some symmetry, but the ground state is degenerate [22]. In the case of massless quarks, we may imagine  Chapter 2. Predictive  Chiral  Symmetry  6  our original system permeated with a plethora of virtual quark-antiquark pairs, but being virtual, they do not add to the total energy of the system. T h e total system may be composed of any combination of right- and left-handed particles, since we have put no restrictions on Ni nor on NR.  Thus, there is a degeneracy of states in  overall handedness, each being a phase space rotation of the other states. The system must select one of these degenerate states to be its ground state and in so doing, spontaneously breaks the ground state symmetry.  It is this spontaneous breaking  of symmetry that is referred to as Chiral Symmetry Breaking, and not the explicit symmetry breaking brought on by non-zero quark masses [21].  2.1.1  Goldstone Bosons  T h e degenerate ground state is also responsible for the generation of Goldstone bosons. If we apply the energy operator to each of the 'ground' states, they all return a zero energy. However, quantum mechanics requires that each state not describing the vacuum must be represented by a particle, and since these particles have zero energy, they must therefore have zero mass. In QCD, these particles take on the role of force mediators for the strong interaction. In the real world, quarks have mass, and so do the Goldstone bosons. For the strong interaction, the meson octet, composed of 7r's, K ' S and 77's, has been identified as the Goldstone bosons.  2.2  T h e Partially C o n s e r v e d A x i a l C u r r e n t H y p o t h e s i s  Noether's theorem [21] states that any Lagrangian that is symmetric under some transformation generates a conserved quantity or 'charge'. 2 Further, a conserved quantity implies a conserved 4-current. Lee [13] has shown that the Lagrangian expressed in 2  This charge may be electric charge, mass or something more exotic.  Chapter 2. Predictive  Chiral  Symmetry  7  equation 2.2 generates eight vector currents and eight axial-vector currents under chiral symmetry: vx  =  iq'(x)j47\-rq(x)  (2.4)  ax  =  iqHx)y4Jxj5^Tq(x)  (2.5)  where r are the Pauli spin matrices and the quark field is:  * 0 = [ MX) )  (2-6)  The eight particles of the meson octet correspond to the eight axial-vector currents generated by the Lagrangian. For zero mass quarks, all sixteen currents are conserved. It should be noted that the Lagrangian used to construct the currents in equations 2.4 and 2.5 has a zero mass term. If we allow the quarks to take on a small mass (on the order of 5 MeV, say) then we may take chiral symmetry to be an approximate symmetry. One result of this approximation is that the axial currents are only partially conserved, leading to the notion of a partially conserved axial current (PCAC). Lee goes on to show that the conservation law for the axial currents becomes:  f^ = Umlfax) where fv = 93.3MeV  (2.7)  is the pion decay constant, m^ is the pion mass and the pion  field, (ftv(x) is given by: ^  =  i  'T-^T5'747s^  (2-8)  Equation 2.7 is the basic equation of the P C A C hypothesis, which is a useful tool for making predictions regarding the ir — -IT interaction, especially ir — IT scattering amplitudes.  Chapter 2. Predictive  2.3  Chiral  Symmetry  8  T h e M o d e l of Olsson and Turner  The formalism to break chiral symmetry was developed by Weinberg [2] in the 1960's. He was able to combine the P C A C hypothesis with current algebra to develop his leading-order "tree-level" Lagrangian for 7r — n scattering in parallel with Schwinger [15]. T h e Lagrangians developed by Weinberg and Schwinger are considered leading order in t h a t they contain terms up to order q2 only, where q is the m o m e n t u m of the TT'S. 3  Olsson and Turner [3] have shown that if one requires the Lagrangian describing the 7r — 7r interaction to be as general as possible, but still chirally symmetric, this can be accomplished by a family of Lagrangians. In generalizing the work of Weinberg and of Schwinger, Olsson and Turner took P C A C to break chiral symmetry by equation 2.9:  f^  = Um2J^x){l  + h0cf>2 + •••)  (2.9)  rather than by equation 2.7. By defining a chiral symmetry breaking parameter, £, as e = l + 4/>0  (2.10)  they were also able to demonstrate that £ was the sole factor in determining the strength of the 7r — 7r interaction. Indeed it is the value of £ t h a t selects which member of Olsson and Turner's family of Lagrangians is under study. In particular, Weinberg's results corresponds to £ = 0 whereas a value of £ = 1 reproduces the Schwinger calculation. Olsson and Turner have shown that £ is also the only adjustable parameter in the ratio between the isospin-0 and isospin-2 scattering lengths: a0  a2 3  2o£ '  ~ 7  £+ 2  The momentum terms come from derivatives of the •n field in the Lagrangian.  (2.11)  Chapter 2. Predictive  Chiral  Symmetry  9  Using isospin arguments, they also showed that the scattering lengths are further constrained by the relation 2a0-5a2 = - ^ -  (2.12)  where /j, is the reduced mass of the TT — TT system. Combining equations 2.11 and 2.12 results in a0  =  a2  =  (0.156-0.0560 O ^  1  (2-13)  -(0.045 +0.0224 0 m ; 1  (2.14)  where /„. = 93.3 MeV has been used. In later work, Olsson and Turner [16] predicted the reaction amplitudes for three (TT, 2TT) processes at threshold in terms of £: a(7r-7r+)  =  (122.7) 2 (-1.36 + 0 . 6 O m~2  (2.15)  a(7r°7r°)  =  (122.7) 2 (2.11 - 0.3 0 m~2  (2.16)  a(7r+7r+)  =  (122.7) 2 (1.51 + 0.6 0  (2-17)  m  ?  They were able to further relate the reaction cross-sections to the threshold amplitudes by: u = a(7T7r) 2 P cm $ nb  (2.18)  where Pcm is the centre-of-mass m o m e n t u m of the incident n in MeV and $ is the phase space available to the three body reaction. Turner, in his Doctoral thesis [6], showed that the phase space was dependent upon the energy of the incident TT in the following manner: $ = 1.28 x 1 ( T 5 T ^ MeV  (2.19)  where T*m is the energy above threshold in the centre-of-mass frame and the 1.28 x 1 0 - 5 is a conversion factor which also sets the units of the phase space factor to MeV. Therefore, by measuring the cross-sections of ir+p —> 7r + 7r + n and ir~p —> 7r _ 7r + n, one can determine the reaction amplitudes and therefore the scattering length for the isospin-0  Chapter 2. Predictive  Chiral Symmetry  10  doim'1) 0.16 0.20 ± 0 . 0 1 0.20 0.31 0.20  Theory Weinberg [2] Gasser and Leutwyler [17] Jacob and Scadron [18] Lohse et. al. [19] Ivanov and Troitskaya [20]  a 2 (m; 1 ) -0.045 -0.042 ±0.02 -0.028 -0.027 -0.060  Table 2.1: Theoretical predictions for the isospin 0 and 2 s-wave scattering lengths for the 7r — 7r interaction. and isospin-2 processes.  2.3.1  Scattering Length Predictions  A number of groups have made predictions for the s-wave scattering lengths using a variety of methods. Weinberg's [2] first order (£ = 0) calculation gave a 0 = 0.16m~ a and a2 = —0.045m" 1 . It should be noted here that only £ = 0 is consistent with QCD, and hence there is a strong theoretical prejudice towards £ = 0. However, Weinberg's first-order  prediction does not take into account the notion of 7r's re-interacting, and a  number of people have found that these re-interactions provide substantial corrections to the leading order calculation. Gasser and Leutwyler [17] have used chiral perturbation theory, (xPT), the next-to-leading-order Lagrangian to order q4. Weinberg's work for the f0 resonance. on different models. Lohse et.  to construct  Jacob and Scadron [18] corrected  Other groups have based their calculations  al [19] used a meson exchange model while Ivanov  and Troitskaya [20] made their determination based on the Quark Loop Anomalies Dominance (QLAD) model. These results are summarized in table 2.1, and shown in figure 2.1.  Chapter 2. Predictive  Chiral  Symmetry  11  o.oo-0.02  Meson Ex.  Jacob a n d S c a d r o n -0.04  Weinberg^  ^ A.  ei - 0 . 0 6  QLAD  cti  -0.08  -0.10  Olsson a n d T u r n e r (£) .00  .30  .35  a 0 (m* l) Figure 2.1: Isospin-2 versus Isospin-0. The various predictions listed in table 2.1 are shown. The straight line corresponds to the model of Olsson and Turner [3]. 2.3.2  N* C o n t r i b u t i o n s A t T h r e s h o l d  There are only four processes believed to contribute to pion production at threshold, and these are shown in figure 2.2; all other processes are discriminated against due to insufficient energy or because of the s-wave nature of the interaction near threshold. Fazel [7] has shown that isospin considerations prohibit the N* diagrams from contributing to the H(7r+ —> 7r + 7r + )n reaction. He has further shown that inclusion of the N* mechanism alters the threshold reaction amplitudes in the following manner: a(n  •K  ^ =  ,82,  ( T ) 2 ( - l - 3 6 + 0.6O + C  (2.20) (2.21)  7r°)  =  (^)2(2.11 - 0.30 -  a(7r+7r+)  =  ( ^ ) 2 ( l . 5 1 + 0.6O  O(7r  0  +  U  ±=C  V2  (2.22)  Chapter 2. Predictive  Chiral  n  Symmetry  TX  rn  12  n  n  N  Figure 2.2: The four diagrams thought to contribute to 7r-production at threshold are: (a) the 7r-pole term, (b) the contact term, (c) and (d) the N* mechanism. N . B . There is some uncertainty as to the nature of the mediating particle in the N* mechanism. It is shown here as an e, but has also been postulated to be a a or an /o[12]. where C is the contribution due to the N* diagrams.  2.4  Summary  Theoretically, there is substantial disagreement between the predictions for the s-wave scattering lengths. As can be seen in figure 2.1, no single value of £ can reproduce the xPT  prediction. QCD requires £ = 0 for all scattering lengths. However, higher order  models suggest that this is not the case. In fact, there is no reason for £ not to be  Chapter 2. Predictive  Chiral  Symmetry  13  reaction dependent; that is, the value of £ applicable for TT+ — 7r+ scattering may not be the same as for 7r+ — 7t~ scattering. The need for precise experimental d a t a for all reaction channels is therefore clear.  Chapter 3  The Experiment  The experiment was performed on the M - l l beamline at T R I U M F . This beamline delivers a high flux of monoenergetic 7r's whose kinematics are well known [24]. Incident 7r kinetic energies of 172, 184 and 200 MeV were used. Since the reaction threshold was 172.3 MeV, the d a t a from the 172 MeV beam supplied an accurate estimation of the background level. T h e experiment utilized an active scintillator target thick enough to stop the low energy product 7r's from the ir+p —> 7r+7T+n reaction. These stopped 7r's were tagged by the decay process 7r+ —• yi^v^.  3.1  Apparatus  The apparatus was assembled as shown in figure 3.3, and was composed of three major sections: an active target, a deflecting magnet and a neutron detection assembly. These sections are treated individually, below.  3.1.1  T h e Active Target  The active target and beam defining scintillators are shown in the enlarged oval in figure 3.3. A beam degrader, which consisted of blocks of Al with various thicknesses, was used in calibration runs to stop the incident beam particles in the individual sections of the active target; it was removed during d a t a taking runs. The four 'in-beam' scintillators, 5*1, S2, S3 and S4, were composed of PILOT-U scintillation plastic (Cf/i.i) and were used to define the beam. Three scintillators (Si, S2 14  Chapter 3. The  Experiment  15  7r-Tracking Section  PACMAN Deflecting Magnet  Veto Scintillators Front Bars  Neutron Bar Array Figure 3.3: Layout of the apparatus.  Back Bars  Chapter 3.  The  Experiment  16  and S4) were 80 x 80 x 2 mm3 while the other one (S3) was of size 20 x 20 x 2  mm3.  Each scintillator was connected to a time-to-digital-converter (TDC) as well as two analog-to-digital-converters (ADC) which gave information about the time of interaction and about the energy deposited in each scintillator by passing charged particles, respectively. For each scintillator, one ADC was gated for 15 ns and used to determine the average pulse shape (c.f. section 5.3.1) delivered by a charged particle, while the other ADC was gated for 80 ns and used to determine the energy deposited in each scintillator. Within the M - l l beamline, there is a capacitive probe.  This probe detects the  passage of a proton burst from the main cyclotron, once every 43 ns. T h e timing of the in-beam scintillator T D C ' s was initiated by a pulse detected with S3. Thus the T C A P timing spectra (c.f. figure 3.4) was denned as the difference in time between a proton detection by the capacitive probe and a charged particle detection at S3. The T C A P spectra, being proportional to the time-of-flight ( T O F ) of particles down the M - l l beampipe 1 was used to discriminate 7r's from positrons. The active target was composed of five sections of PILOT-U scintillator material labeled A through E. Each section was of dimension 40 x 40 x 6 mm3.  The thickness  of the target sections was selected to distinguish passing 7r's (which deposit 1.26 MeV of electron equivalent light in each segment) from /u's observed from 7r+ —> /U+z^ decays. Three methods were used to detect the decay /w's, each of which are detailed in section 3.2.1, below.  3.1.2  T h e Deflecting Magnet  T h e PACMAN magnet was used to deflect any charged particles leaving the 7r-tracking section away from the neutron bar array, located 1.70 m downstream. The pole gap Actually, the TCAP value was proportional to (TOF mod 43 ns)  Chapter 3.  The  Experiment  17  2000  1500  d 1000-  o o 500  0  100  200  300  400  500  TCAP Figure 3.4: T C A P spectrum for T^=184 MeV. was opened to a separation of 30.2 cm. The deflection angles and beam clearances, the distances by which the bent beam missed the front of the neutron bar detection assembly, are shown in table 3.2. The deflecting angles were calculated from 6{radians)  0.3qBL  (3.23)  P where q is the charge of the passing particle in units of elementary charge, B is the magnetic field strength in Tesla, L is the length of the magnetic field region in metres and p is the m o m e n t u m of the particle in G e V / c [25]. T h e clearances were determined using the neutron bar width of 105 cm.  3.1.3  The Neutron Bars  The reaction neutrons were detected using a neutron bar array. This assembly was composed of fourteen bars in two vertical columns of seven bars each. Each bar was of  Chapter 3.  The  Experiment  T„(MeV)  18  p^MeV/c)  200 184 172  310 292 279  0;af)(deg) Clearance (cm) 18.6 20.7 22.6  68.3 76.6 84.4  Table 3.2: Deflecting angles and neutron bar clearances. dimension 15 x 15 x 105 cm3. At the end of each neutron bar, there was a phototube with a T D C and an ADC. Neutrons were detected through collisions with the nuclei of the scintillating material which released charged particles causing light emission in the scintillators. The resultant photons were detected at b o t h ends of the bar. T h e T D C ' s supplied the time at which these photons were detected and the ADC's indicated the energy deposited in the scintillators by the neutrons. The T D C d a t a were used in evaluating the kinetic energy of the product neutrons while the ADC d a t a were utilized in determining the neutron detection efficiency,  (c.f.  section 4.6)  3.2  Procedure  T h e cross-section for a reaction is defined by the following formula:  Yield  (3.24)  a = NiNi tgt where iV,- denotes the number of incident 7r's with an acceptable beam p a t t e r n  (c.f.  chapter 6) and Ntgt is the number of target hydrogen nuclei per unit area in the target. T h e Yield is given by  Yield =  Nrj U(r]n;r]n)  A  (3.25)  Chapter 3.  The Experiment  19  T„(MeV) 200.0 184.0  Tsum(MeV) 59.1 43.1  Table 3.3: Expected values of Tsum. where II is the total experimental acceptance and rjn and r/n are the detection efficiencies of the neutrons and 7r's, respectively and A is the C P U livetime. T h e detection efficiencies will be discussed in their own chapters, below, and the experimental acceptance will be examined in chapter 7. The CPU livetime will be examined in section 4.6.1. It should be noted that A only affected the Yield since Ni was determined using scalars, which were not inhibited from counting while the C P U processed data, and Ntgt was, for practical purposes, constant. The number of coincidental detections of a single neutron and at least one ir+ —-> y+u^ decay, Ncoinc,, was evaluated using the total kinetic energy of all reaction products. TSum — Tv + Tn  (3.26)  where Tn is the kinetic energy of the neutron and Tn is the total energy deposited in the active target by the charged particles. By including the energy deposited in the target before the reaction took place, the loss of beam energy due to the initial 7r travelling through the target was taken into account. For a (TT, 2TT) event, the total kinetic energy is given by: -'•sum  =  -'•beam  f^-K  yl^n  fTip)  yo.Zl)  The Tsum values expected in this experiment are given in table 3.3. T h e neutron kinetic energies were determined by evaluating their time-of-flight ( T O F ) to the neutron bars. T h e 7r kinetic energies were determined from the light released in the active target.  Chapter 3.  3.2.1  The  Experiment  20  D e t e c t i o n o f TT+ —* n+u^ D e c a y s  In order to determine whether or not a (7r, 2n) reaction had taken place, the observation of a single neutron and a decay muon, the signature of a generated IT was required. While the detection of a neutron required the observation of a photon at each end of a neutron bar, three triggers were used to determine whether a /J, was observed. These triggers are discussed here and are examined in more detail in chapter 5, below. The first level trigger utilized a hardware circuit (see figure 3.5) to examine the scintillator signals from the active target. T h e raw signal was fanned out to two discrimination units. The first unit allowed only the initial narrow pulse (due to the beam 7r) to pass, widening it to 80 ns. This widened pulse was then put into coincidence with the original pulse, which was timed such t h a t the narrow TT pulse was out of time with the wide pulse. However, a second pulse coming from a decay \x would be in time with the wide gate, signalling its presence. T h e second discrimination unit acted as a saturation veto. If the scintillator ADC from any section of the active target became saturated, the event could not be accepted because it would not be possible to determine the total energy deposited in the target. As such, the discriminated signals from each section were fanned together. A saturation of any one section was enough to reject an event. The second level trigger used was the comparison of two ADC units, one gated for 15 ns and the other gated for 80 ns. For a single target section, the narrow gated, or 'short', ADC would observe only the prompt -K pulse, whereas the wide gated, or 'long', ADC would see both the prompt pulse and any secondary pulse present. Thus, by taking the difference between the normalized outputs of the long and short ADC's, a non-zero result would indicate the presence of a second pulse with the additional information of the energy of the second pulse. A sample distribution of the difference  Chapter 3.  The  Experiment  21  ir <  A-  V High High High High  l 2 3 4 5  B C D E  LeCroy LeCroy BELIAL LeCroy LeCroy  IT  E>  1st Level Trigger  Event Coincidence •-Scalar 428F F a n - i n / F a n - o u t 821Z Quad. Discriminator Discrimination Unit 429A F a n - i n / F a n - o u t 465 Coincidence Unit  Figure 3.5: First level trigger hardware circuit for target section A  Chapter 3.  The  22  Experiment  50000 -r  40000 Ul .+J 30000 Pi  o (j  20000 -  10000 -  2  0  2  4  6  8  ADC Energy Difference (MeV) Figure 3.6: Spectrum of the difference in detected energy between the long and short ADC's for target section C. between the long and short ADC d a t a for target section C is shown in figure 3.6. The small peak at 3.6 MeV indicates the detection of a decay \i. The third and final trigger was the most powerful. The phototube signals from the active target and the first three in-beam scintillators were fed into a 500 MHz transient digitizing sampler which recorded the raw signal into 2 ns bins. A typical ADC is shown in figure 3.7. The raw signal is illustrated in part (a). Figure 3.7 (b) was generated by fitting a baseline to the signal shown in (a) from which the signal was subtracted; this process was denoted as baseline subtraction. A sample ADC output from the active target, as shown in figure 3.8, clearly showed the presence of a second pulse, the pulse heights of the observed peaks and the relative time between the two pulses. In figure 3.8, the negative going pulses have been baseline  Chapter 3.  The  23  Experiment  """-  m •£j 600-  Q  50  °-  /  <! £  f^—  400-  iT c  i  10  SO  30  40  5  Time (ns) (a) 350  Time (ns) (b) Figure 3.7: A sample of raw output from the long ADC in target section A is shown in (a). A baseline has been fitted to the data, from which the signal in (a) was subtracted, (b). The presence of a /J, is also indicated.  Chapter 3.  The  Experiment  24  subtracted resulting in positive going signals.  3.2.2  Yield D e t e r m i n a t i o n  As stated above, the yield was evaluated from the total kinetic energies of the reaction products. In computing these sums, at least one -K decay had to be observed, but sometimes b o t h were seen. This led to two Tsum spectra; those computed with one observed decay //, and the other where two ^u's were detected. The spectra comprised of a single decay /i signal had a large background from 7r + C —> 7r + nX, where X may be anything allowed by kinematic constraints. This background was reduced by applying kinematic cuts to the energy spectra of the individual product energy spectra t h a t were consistent with the 7r+j) —•>• 7r + 7r + n reaction.  The  extra requirement of observing two decay fi's significantly reduced the background. The remaining background was estimated by fitting the lower energy spectra to the background regions. T h a t is, the d a t a from incident TT energies of 172 and 184 MeV were used as background for the 200 MeV runs, while the background for the 184 MeV d a t a was determined from the 172 MeV data. The results are discussed in chapter 7, below.  Chapter 3.  The  Experiment  25  Time (ns) HI 360  Time (ns) O  360  Time (ns)  Time (ns) H  360  Time (ns) Figure 3.8: Digitized ADC signals for each target section. A beam TT is tracked through each section, and the presence of a decay /i is clearly seen in target section E. The vertical scales are in raw ADC units.  Chapter 4  Neutron Detection  4.1  Introduction  T h e detection of the reaction neutron was required to record an event. Since the kinetic energy of the neutrons was used in determining the experimental yield, an accurate energy calibration was required. This was obtained by calibrating the time-of-flight to the neutron bars using two monoenergetic neutron sources of 8.9 and 68 MeV. These energies span the range of the neutron energies from the ir+p —> 7r + 7r + n reaction. T h e lower energy source was obtained with a liquid hydrogen (LH2) target and a 7T~ beam with an initial kinetic energy of 30 MeV. The ir beam was degraded to stop in the target, initiating the reaction ir~p —> jn.  In this reaction, a 'pionic' atom  was created as the stopped TT began to orbit a target proton. The ix~ and the proton interact, resulting in, among other things, an isotropic monoenergetic 8.9 MeV neutron source. The upper energy calibration was performed using the reaction TT~d —> nn. This mechanism also produced an isotropic monoenergetic distribution of neutrons, but of 68 MeV. Since this is also a two body reaction resulting from a stationary source, the neutrons were emitted back-to-back. The calibration was twofold. First, the T D C information was calibrated by successfully reconstructing the kinetic energy of the monoenergetic neutrons. Second, the detection efficiency was determined by the calibration of the ADC data. A group at Kent State University [26] have developed a Monte Carlo code that gives the neutron  26  Chapter 4. Neutron  Detection  27  detection efficiency for neutron detectors composed of Ne-102 scintillation plastic as a function of the detection threshold. It was the aim of the ADC d a t a to verify that the Kent State code was valid at 8.9 and 68 MeV. A schematic diagram of the electronic logic circuits for the neutron bars appears in figure 4.9.  4.2  T D C Calibration  The neutron bar T D C ' s provided both timing and position information. From these spectra, the neutron kinetic energy was calculated by reconstructing its trajectory and evaluating its time-of-flight from the active target. T h e analyses of the timing and position information are discussed in separate sections, below. T h e procedures outlined in these sections were performed for each of the fourteen neutron bars individually, but are described for a single bar.  4.2.1  T i m e of Flight  To calibrate the time-of-flight, the d a t a from the T D C ' s at each end of the neutron bar h a d to be b o t h scaled and offset. T h e scaling was done by examining the signal produced by the T D C ' s based on a 25 MHz sampler as a source. Four sharp peaks were seen for each T D C spectrum. The means of these four peaks were then scaled to a separation of 40 ns by a least squares fit. For the 8.9 MeV neutrons (TT~P —>• jn),  the offset was determined by positioning  the neutron peak. T h e neutron peak was selected as it was easily identified for each T D C , unlike the peak attributed to the photons. The determination of the T D C offset was complicated by the fact that the neutrons were isotropically distributed and that the detector was planar. This mismatch led to an observed neutron time-of-flight spectrum with a shape not unlike a skewed Gaussian.  Veto B3 Veto B2 Veto Bl  Veto Tl Veto T2 Veto T3  _Jr LeCroy 428F F a n - i n / F a n - o u t LeCroy 2249A ADC LeCroy TC455 Quad. C.F.D. LeCroy 2228A TDC  1 2 3 4  8  7  6  5  EXK  ^ Scalar  S1-S2-S3-S4  Sl'+S2'+S3*  (oi. Figure 9.6)  High Scintillators  LeCroy 465 Coincidence Unit  LeCroy 821Z Quad. Distriminator  LeCroy 429A Logic Fan—in/Fan—out  LeCroy 624 Octal Meantimer  -*• Scalar  Event Coincidence  O 13  »—.  CD O ^+-  rt>  §  c  ft)  erf-  8  Chapter 4. Neutron  29  Detection  It was not readily apparent as to where the peak should be placed, as there was no way to know which part of the distribution corresponded to the center of the bar in question. (The time-of-flight calculations were made to the center of each bar.) Two methods were utilized to rectify this problem. First, an analytic fit was performed to the d a t a using a skewed Gaussian.  This  method did not enjoy great success as specifying a suitable background region was rather difficult. This method was discarded in favor of a Monte Carlo analysis. The Monte Carlo program N M O N T E was written to simulate the neutron distribution. It should be noted that the LH2 target was 10 cm thick. However, since all the neutrons under study must have h a d their originating ir travel through S3, the target was truncated to a rectangular box of dimensions 2 x 2 x 10 cm3.  The program was  run for one million events, and the results scaled to the peak height of the data. T h e simulation was in excellent agreement with the d a t a in shape as shown in figure 4.10, and the offsets were easily determined. To evaluate the T O F spectra, the T D C values at either end of the bars were averaged. (It should be noted that it was required for a signal to be received at both ends of the neutron bars to consider a neutron as 'detected'.) This resulted in the time-offlight plus the time for a photon to propagate along half the length of a neutron bar, a constant value which was subtracted off. As a check, the positions of the 7 peaks were verified to occur at the correct time, and the difference in time between the neutron and photon peaks was in agreement with the relativistic calculations. For the 68 MeV neutrons (7T-G?—> nn), the offset factors obtained from the 8.9 MeV runs were verified through time-of-flight calculations and a Monte Carlo analysis.  Chapter 4. Neutron  Detection  30  2000  2000  1500 -  1500  1000  -1000  500 -  70  500  75 80 85 Time (ns) (a)  90  70  75 80 85 Time (ns)  90  (b)  Figure 4.10: Neutron bar T D C spectra: the Monte Carlo spectrum as generated by N M O N T E (a), and the spectrum generated by 8.9 MeV neutrons in the left T D C in the front centre bar (b). The data has been background subtracted and scaled to the same count height as the computer simulation for ease of comparison. 4.2.2  Position Reconstruction  T h e time-of-flight calibrations for a bar's two T D C ' s were determined by the average of the time of observation at each end of the bar. T h e lateral position of the interaction along the bar was determined by taking the difference between the T D C ' s at each end and scaling the resultant distribution so that the full width at half maximum was equal to the full width of the neutron bar. In this way, the position histograms were a fine adjustment to the T D C offsets. A sample position reconstruction is shown in figure 4.11. The spread of the data at the edge of the bar was due to the finite timing resolution and from photons undergoing internal reflection as they traveled along the neutron bar. The thickness of the spread was defined as the difference in position between the distribution at 90% of maximum and at 10% of maximum. For the spectrum shown in figure 4.11, the F W H M resolution  Chapter 4. Neutron  Detection  -0.5 0.0 0.5 L a t e r a l Position (m)  31  1.0  Figure 4.11: Position reconstruction for a front neutron bar. was 22.6 cm, or 2 1 % of the bar width. This position resolution corresponds to a timing resolution of 1.2 ns.  4.3  A D C Calibration  T h e ADC's measured the energy deposited by the neutrons in the scintillator. By determining the threshold detection energy, the neutron detection efficiency was evaluated by the percentage of neutrons having an energy above the detection threshold. As with the T D C ' s , the ADC's at each end needed to be offset and scaled. For both calibrating reactions, only one neutron was available for detection. In determining the offsets, the fact that only one bar could detect the neutron was exploited. Histogramming all ADC outputs for every neutron detected resulted in a sharp peak indicating the ADC pedestal value and a small distribution at higher energies. The peak was  Chapter 4. Neutron  Detection  32  generated by neutrons detected in any bar but the one under consideration; the small distribution came from neutrons detected in the current bar. The offsets were then obtained by positioning the mean of the peak at 0 MeV. The scaling factors were determined by examining the ADC spectra generated by the 8.9 MeV neutron source only for neutrons detected in the bar of interest in order to calibrate the ADC d a t a in MeV of electron-equivalent-light. The Kent State Monte Carlo code was invoked to give this calibration. The Kent State code followed neutrons into a detector composed of scintillation plastic. In determining whether or not a neutron had interacted with the detector, the code used cross section d a t a for n-p elastic scattering and n-Carbon nuclear interactions that produce 7 rays a n d / o r charged particles such as protons and a particles. After having determined that an interaction had taken place, the code simulated the photons generated by scintillation and compared their energy with an energy detection threshold, specified by the user. A distribution of the number of neutrons detected as a function of the energy deposited by the neutrons in the detector was then output. T h e computer simulation was run for single photon threshold detection levels ranging from 0.01 MeV u p to 0.24 MeV. It was found that a photon acceptance level of 0.13 MeV fit the leading edges the most consistently. T h e leading edges of the neutron ADC spectra were then scaled such that the half maximum of their leading edge occurred at 4.1 MeV, in agreement with the Monte Carlo simulation. A sample of a neutron ADC distribution is shown in figure 4.12 with the Kent State Monte Carlo prediction, scaled in counts to the data. The Monte Carlo prediction for the neutron detection efficiency was also determined from the family of figures of which figure 4.12 is a member. T h e neutron detection threshold was determined for each bar by noting the lowest electron-equivalent-energy at which the real and simulated d a t a agreed. Since this threshold should depend only  Chapter 4. Neutron Detection  33  _L  300 *  _L  x  Software Cut Monte Carlo Data  W 200  O ^ 100  0 0  1  2  3  4  5  ADC Output (MeV) Figure 4.12: A comparison of data (solid line) and Monte Carlo simulation (heavy dashed line) for a particular front neutron bar. The hatched area denotes the data passing the ADC software cut which is indicated by the light dashed line.  Chapter 4. Neutron  Detection  (MeV) 8.9 68  34  Neutron Bars Front Back Front Back  Monte Carlo NDE Predictions (%) 29.59 7.33 20.20 15.04  Table 4.4: Monte Carlo predictions for the neutron detection efficiency. upon the geometry of the detectors, and since each individual neutron bar was identical, the greatest detection threshold, 1.9 MeV, was taken to be true for the entire assembly and applied to each neutron bar. The establishment of a neutron detection threshold had two implications. First, the application of a threshold reduced the event background. Second, the value of the threshold determined the neutron detection efficiency predictions. T h e predicted values are shown in table 4.4.  4.4  Energy Determination  T h e kinetic energies of the neutrons were determined from b o t h the time-of-flight information and the reconstructed trajectory. While a neutron could interact with the scintillator anywhere along its length, there was no way to determine the depth of penetration at the point of interaction. Hence, when calculating the length of the neutron's p a t h , it was assumed that the position of interaction was along the central axis of the bar. The kinetic energy was then calculated by Tn = m „ ( 7 - l)c 2 where  (4.28)  Chapter 4. Neutron  Detection  35  500  c 400 -  500  (b) T n =68 MeV  -400  0  300 -  u  300  200 -  N  200  100  T  100  S  0  5  7  9  11  13  15  Neutron Energy (MeV)  50  60  70  80  N e u t r o n Energy (MeV)  Figure 4.13: Reconstructed kinetic energies for the 8.9 (a) and 68 MeV (b) neutron spectra. Dn  fi = TOFxc  (4.30)  where Dn is the distance from the center of the target to the point of interaction, TOF is the calculated time of flight and c is the speed of light. A sample energy spectrum for both the 8.9 MeV neutrons (a) and the 68 MeV (b) d a t a is shown in figure 4.13. It is the sum of all counts in these histograms, after all software cuts have been applied, that gives the number of observed neutrons, n^ttcttdAll of the d a t a events included in figure 4.13 were required to have a well defined beam rr (c.f. section 4.5) and an identified neutron whose time-of-flight was consistent with that of a neutron whose reconstructed position fell within the dimensions of the neutron bar array.  Chapter 4. Neutron  4.4.1  36  Detection  Background  T h e neutron kinetic energy spectra shown in figure 4.13 are composed of events that have passed the ADC software cut discussed above. While the background has been reduced by the application of the ADC threshold, it has been by no means eliminated. In order to obtain an accurate yield, the background, as a function of neutron energy, had to be modeled as accurately as possible. Three methods were undertaken to describe the background, each of which is described, below. The first method made a two-point linear fit to the background. For each of the two background regions (the hatched areas in figure 4.13) the average number of counts was determined and placed at the mean of the region. These two points were then fit to a straight line, and the yield was taken as the area between the background regions, above the fitted background. Similar to the first method, the second method made a linear least squares fit to the d a t a in the higher energy background region. T h e yield for this method was calculated in exactly the same manner as for method one. The resulting yields for both methods were then averaged to evaluate ridetectedHowever, neither method was considered as very successful for two reasons. First, the pre-ADC corrected d a t a for the 8.9 MeV d a t a indicated a non-linear background which was not taken into account.  Second, the 68 MeV d a t a had a background  (TT~P —> 7?m) process which was responsible for the 'lip' on the energy distribution in figure 4.13 that was not fit very well by either method one or method two. T h e third and most robust method to model the background was essentially to compare energy distributions. As stated in section 4.4, above, a rather tight gate was placed on the neutron time of flights. These gates were then expanded to included as much of the time of flight spectra as possible while still excluding the 7 peaks. This  Chapter 4. Neutron  37  Detection  resulted in energy spectra from 5 to 80 MeV. The 8.9 MeV and 68 MeV spectra were used to obtain the shape of the background under each other's neutron peak. In the case of the 68 MeV yield a further correction for the (TT~P —>• jnn)  process yield had to be made. T h e center of the background region  (see figure 4.14) was selected as 41.1 MeV. For a given bar, the two energy distributions were scaled in counts (since an unequal amount of d a t a was taken for the two calibrating reactions) to agree with the other spectrum at the center of the background region, and then subtracted as background. The yield was then taken as the sum of the remaining counts. The 8.9 MeV d a t a was used as a background estimation for the 68 MeV d a t a in a similar fashion. To eliminate t h e (TT~P —» jnn)  'lip', which extended under the main energy distri-  bution, a Monte Carlo program simulated a three body decay for 47000 events. The results are plotted in figure 4.15. A straight line was then fit to the neutron spectrum. T h e distribution was then scaled to the center of the 'lip' for each bar and subtracted from the ADC corrected 68 MeV data. As with the 8.9 MeV data, all positive counts were included in determining the number of neutrons detected for each neutron bar. The neutron yields are shown in figure 4.16 for both energy calibrations. The count rate drops severely for the most vertically displaced bars (numbers one and seven) since they were shielded by the PACMAN magnet. As a result, only the ten central bars were used in computing the reaction yields. Also included in the figure are the ratios of the yields for the front to the back bars. The ratios are consistent with the Kent State Monte Carlo predictions.  Chapter 4. Neutron  Detection  38  500  (a) 400-  W -t-J  O  o  300200-  100-  I  Background Region Ur.l»,l<UvaV»J^«ul,.luw<.«^»>H'l»J^i-»J^..r„..i~  40  60  80  Tn (MeV) 1500  (b) 1200W •+->  PI  o o  900600 300  Background Region rH^«>^H*^V>W><>>%vJ«*t*^<^>SMW^V  40  Tn (MeV) Figure 4.14: Reconstructed kinetic energies for the expanded 8.9 and 68 MeV neutron spectra.  Chapter 4. Neutron  Detection  39  250 nny IConte Carlo  200  w -4-> 150 H  a O 100 -  o 50 H ^4A/lr. 50  55  60  65  70  80  Tn (MeV) Figure 4.15: Monte Carlo simulation of the (ir p —> jnn) the kinetic energy spectrum. 4.5  background contribution to  M o n t e Carlo C o r r e c t i o n s  As stated above, a well defined beam ir was required to accept an event. The definition of a 'well defined b e a m 7r' was as follows. T h e particle was required to pass through the first three in-beam scintillators, but not the fourth. Further, the particle had to deposit energy in S2 and S3 and have a T C A P value consistent with that of a ir. While this is the best achieved test to determine a beam ir that stopped in the target, it could not eliminate a variety of other processes. Two examples of such a 'contamination' include ir that entered the target and were elastically scattered out of the target, and ir t h a t decayed between S3 and the target. In order to correct for this background, the CERN Monte Carlo package, G E A N T , was invoked. The simulation determined that 3.6% of the 8.9 MeV and 3.8% of the 68 MeV well defined beam ir were falsely labeled as such. The correction was applied by dividing the values obtained from the d a t a by (1 — T ) , where T is the fraction of falsely identified well defined beam  Chapter 4. Neutron  Detection  40  Tn=8.9MeV  T n =68MeV  c 0  u N T S BAR • — • Front Bars  BAR O—O Back Bars  BAR  Figure 4.16: Neutron yields for the 8.9 (a) and 68 MeV (b) calibration runs. The ratio of the yields for the front bars to the back bars are shown below (c) and (d).  Chapter 4. Neutron  Detection  41  7r's.  4.6  N e u t r o n D e t e c t i o n Efficiency  T h e first parameter in the denominator of equation 3.25, r]n, is the neutron detection efficiency (NDE) which is defined as AT 7""i TP  i v JJIL/  '''detected  =  . . >.,.—,,. {nincident){A-){r)  4.31)  where A is t h e livetime of t h e software, a n d V is t h e correction d u e to acceptable position reconstruction, both of which are described in their own subsection below. T h e number of neutrons incident upon the detector array is given by ^incident  —  \^stopped  mm  (4-32)  where nstopped indicates the number of 7r's stopped in the target, Pf is the appropriate branching ratio, and 0 is the solid angle taken u p by the neutron bars. Each of these components of the NDE are treated separately, below. 4.6.1  C P U L i v e t i m e (A)  T h e hardware was not able t o continually accept d a t a due t o time spent processing information. The reaction yields were corrected by dividing the observed yield by the C P U livetime, which is the fraction of time that the hardware was available t o accept new data. The complement of the livetime, the C P U deadtime, is defined as 1 — A. T h e C P U livetime was determined by taking t h e ratio of d a t a recorded t o t h e amount of data presented to the hardware for consideration. This was done with scalar data. T h e amount of d a t a recorded was determined by taking the average of the number of LAM's (Look At Me's) and the number of C P U Busy signals. For each run, these two quantities differed by no more than 0.03 %. The number of times that d a t a was  Chapter 4. Neutron  Tn (MeV) 8.9 68  Detection  42  DATA 'EVENT + SAMPLE'S 2 502 574 2 359 330 1 204 370 1 126 478  CPU Livetime (%) 94.3 95.5  Table 4.5: CPU livetimes for the neutron detection efficiency calibration runs presented to the C P U was given by the ' E V E N T + S A M P L E ' scalar. The results of the CPU livetime calculations are shown in table 4.5.  4.6.2  Position Acceptance (r)  In considering neutron events, the lateral position of interaction was determined and histogrammed, as in figure 4.11 (c.f.  section 4.2.2). A gate was placed on this d a t a  to reflect the physical dimensions of the neutron bars. Only those events whose reconstructed position fell within this gate were accepted. Thus, a fraction of good events were eliminated from consideration.  4.6.3  S t o p p e d 7r's  T h e number of 7r's stopped in the target was determined by two methods using the in-beam scintillators and T C A P spectra. Four tests were applied to the T C A P spectra: a beam sample, a beam sample 7r that stopped in the target, a beam sample that was stopped in the target, and a beam sample ir that was not seen in S4. T h e first method took the ratio of the T C A P spectra gated on beam sample TT to the T C A P spectra gated with just b e a m samples and multiplied it by the total number of b e a m samples counted by a scalar. T h e second method took the ratio of the T C A P spectra gated on the fourth test to that gated by the third test, multiplied by the scalar value which counted (5*1 • S2 • S3 • S4).  The first method gave the number of  b e a m samples that were a ir stopping in the target, whereas the second method yielded  Chapter 4. Neutron  T 8.9 68  Detection  43  Method 1 (xlO 6 ) 55.814 36.615  Method 2 (xlO 6 ) 56.179 36.935  Average (xlO 6 ) 56.00±0.18 36.78±0.16  Table 4.6: Stopped TT d a t a for the calibration runs. the number of particles t h a t stopped in the target t h a t were beam sample 7r's. T h e results of b o t h methods are shown in table 4.6.  4.6.4  The Branching Ratio  (Pf)  T h e branching ratio for n~p —> •yn is known as the Panofsky ratio. Included in the ratio was a phase space factor, ( P S F ) . T h e Panofsky ratio has been measured to be 39.1% by Spuller et al. [27]; the phase space factor is one. T h e corresponding branching ratio for TT~d^  nn was determined to be 73.7% by Highland et al. [28]; the phase space  factor is two in order to account for the second neutron released by the reaction.  4.6.5  Solid A n g l e A c c e p t a n c e (fi)  T h e final correction in equation 4.32 is the solid angle acceptance of the detectors. It has been already pointed out t h a t the neutrons from b o t h reactions were isotropically distributed. Such being the case, the neutron bars took u p only a small fraction of the total solid angle available to the reaction. The solid angle taken u p by each neutron bar was calculated numerically; the results listed in tables 4.7 and 4.8.  4.7  Summary  T h e neutron detection efficiency has been determined at two energies; at 8.9 MeV using a liquid hydrogen target, and at 68 MeV using a liquid deuterium target. The kinetic  Chapter 4. Neutron  Detection  Neutron Bar Fl F2 F3 F4 F5 F6 F7 Bl B2 B3 B4 B5 B6 B7  44  '"•detected  r (%)  1790 6127 6506 6534 6490 5813 1512 956 1373 1335 1407 1413 1351 722  86.5 89.3 89.9 90.1 88.5 88.5 85.8 85.9 83.8 88.7 86.8 83.2 87.8 84.5  N.D.E. 3  (xlO- ) 1.49 1.52 1.53 1.54 1.53 1.52 1.49 1.37 1.39 1.40 1.40 1.40 1.39 1.37  (%)  8.78 28.65 29.93 29.86 30.32 30.53 7.48 5.16 7.50 6.81 7.32 7.69 7.03 3.96  Table 4.7: Neutron Detection Efficiency parameters for 8.9 MeV neutrons. energy of the detected neutrons has been accurately reconstructed by determining the position of detection along the neutron bar array and reconstructing b o t h the trajectory and time of flight. T h e ADC d a t a for the 8.9 MeV determined a neutron detection threshold of 1.9 MeV in electron equivalent light. The Monte Carlo predictions as a function of neutron kinetic energy are shown in figure 4.17. The average neutron detection efficiency for the middle ten bars are compared to the Kent State Monte Carlo predictions in table 4.9. Due to the uncertainties of the cross-section d a t a used in the code, an uncertainty of 10% was applied to the predictions. Since the percent difference between the measured and simulated neutron detection efficiencies is less than 10%, it was felt that the Kent State code acceptably modeled the detection efficiency for our neutron bars.  Chapter 4. Neutron  45  Detection  I  70  I  Total  6  g° O  Front Bars Back Bars  50  CD O  40  30  O O CD  20 -  0 0  20  30  40  50  70  Tn (MeV) Figure 4.17: Neutron detection efficiency predicted by the Kent State Monte Carlo code for both the front and back neutron bars as a function of kinetic energy.  Chapter 4. Neutron  Detection  Neutron Bar Fl F2 F3 F4 F5 F6 F7 Bl B2 B3 B4 B5 B6 B7  46  ^detected  r (%)  2494 11167 11368 11525 11078 10482 1202 3275 7217 7586 7352 7423 7208 1779  95.6 96.2 96.3 96.4 96.3 95.9 94.0 95.8 95.9 97.1 96.3 95.6 96.2 94.7  N.D.E. 3  (xlO- ) 1.49 1.52 1.53 1.54 1.53 1.52 1.49 1.37 1.39 1.40 1.40 1.40 1.39 1.37  (%)  4.48 19.64 19.79 19.98 19.28 18.51 2.20 6.42 13.95 14.34 13.98 14.25 13.88 3.53  Table 4.8: Neutron Detection Efficiency parameters for 68 MeV neutrons.  -Ln  (MeV) 8.9 68  Neutron Bars Front Back Front Back  Kent State Predictions (%) 29.6 ± 3.0 7.3 ± 0.7 20.2 ± 2.0 15.0 ± 1.5  Measured NDE(%) 29.9 7.3 19.4 14.1  Percent Difference +0.9 -0.8 +3.8 +6.4  Table 4.9: Comparison of the neutron detection efficiency with the Kent State Monte Carlo code predictions.  Chapter 5  T h e Active Target  5.1  C o m p o s i t i o n of t h e Target  T h e active target was composed of five sections of Pilot-U plastic scintillator. Each section was 0.237 inches thick, and of area 1.741 X 1.617 inches. In order to determine the cross-section, the number of target nuclei Ntgt had to be known and was calculated from: Ntgt  =  (Target  lmole \ molar mass  Density)(Thickness)  I#molecules\ mole  (#H  nuclei^  J \ molecule  (5.33)  J  Pilot-U scintillation plastic has a density of 1.032 g/cm3,  and a Hydrogen to Carbon  ratio of 1.1. The total number of target nuclei is given in table 5.10.  target section  iV^(xl023cm-2) 1.570 7 0.314 1  Table 5.10: Number of target nuclei in the active target.  5.2  Energy Calibration  T h e total kinetic energy of the final state 7r's, TV, was determined by the total energy deposited in the active target by the incoming beam IT and any decay products. As 47  Chapter 5.  The Active  48  Target  the Tv distribution was used in determining the experimental yield, an accurate energy calibration of the target ADC's was required.  An immediate complication to this  calibration was the observation of a baseline fluctuation with a frequency on the order of 50 Hz. This fluctuation was removed using two methods. First, for beam samples, the ADC signals in the first three in-beam scintillators were averaged. For 200 MeV 7r's, 0.44 MeV was deposited in the 2 m m thick scintillators; the baseline was then taken as the average ADC signal less 0.44 MeV. The second method, used for events, took the ADC signal from S4 as the baseline, since the definition of an event included a null signal in S4. The baseline was then subtracted from the ADC values for the target scintillators. As with the T D C ' s and ADC's connected to the neutron bars, the target ADC's needed to be b o t h offset and scaled. The offsets were determined using 'no-beam samples' while the scaling factors were evaluated using 200 MeV 'passing 7r's'. A 'no-beam sample' was defined as a sample of coincidence between the in-beam detectors with no signal in any scintillator. This resulted in a pedestal peak corresponding to zero MeV, allowing for a clear evaluation of the individual ADC offsets. A 'passing 7r' was defined as a TT that interacted with S4. As such, the calibration was made with beam samples taken from 200 MeV 7r's. It was determined that such particles deposited 1.32 MeV in each of the target sections. Since all of the 7r's passed through the target without decaying, the total energy deposited was seen by both the long and short ADC's. T h e prescription that follows was performed on all ten 1 target ADC's, but is described for a single ADC. T h e spectrum in figure 5.18(a) includes d a t a from beam buckets containing both single or multiple 7r's. These d a t a were separated from each other by placing gates about the peak and tail for each target section. A single -K in any given section was Recall that each target section has two ADC's.  Chapter 5.  The Active  Target  49  120  3000 Peak  (a)  2500 H  J  100  Tail CQ  L  w  r  80  <E>=2.65 MeV  § 60 H 1  o O  40 20-  0  n 8  E n e r g y (MeV)  10  -MJIT  0  2  /  I ^v„ 4  6  8  10  E n e r g y (MeV)  Figure 5.18: Long ADC spectrum for 200 MeV passing 7r's in target section E. T h e hatched area in (a) represents single TT beam buckets while the multiple TT distribution is shown in (b). then defined as any sample t h a t passed the peak cut in all of the other four sections. Similarly, a multiple TT bucket was defined as a beam sample that passed the tail cut in all of the other four target sections.  The d a t a from single -K buckets are shown  in figure 5.18(a) as the hatched area; the d a t a from multiple TC buckets are shown in figure 5.18(b). There is a clear peak at 2.65 MeV in figure 5.18(b), corresponding to a beam bucket containing two 7r's. T h a t this peak occurred at twice the calibration energy for a single passing n clearly indicated that the calibration was correctly implemented.  5.3  The CCD Data  Beam 7r's that passed the first two levels of triggering (c.f.  section 3.2.1) had each  of their target and in-beam scintillator long ADC data presented to CCD transient digitizers which digitized the signals into 2 ns bins. In order to identify the product  Chapter 5.  The Active  Target  50  H with any confidence, it was necessary to have an accurate calibration of the pulse height generated by a /j,. This calibration was achieved by the placement of Al b e a m degrading blocks of various thicknesses upstream of the first in-beam scintillator. Doing so made it possible to stop a -K beam with an initial kinetic energy of 30 MeV in each target section, thereby placing the 1: —> \iv^ decay in a known location, and allowed the accumulation of sufficient statistics in a reasonable amount of time.  5.3.1  M o d e l i n g T h e C C D Signals  The CCD signals were modeled by fitting an average pulse shape, as determined from beam samples, to their output. The average pedestal background of each CCD d a t a was obtained from the 'no-beam samples'. The average pulse shape in each target section was then determined by subtracting d a t a that had only one peak from this average background pedestal. Such d a t a were easily obtained by requiring the signals to be generated by passing 7r's. Since the CCD's observed negative pulses, this subtraction resulted in positive pulses, as seen in figure 3.8. Before the fitting was performed, the digitized signal from each CCD was scanned for local maxima (with a limit of five). The first local maximum was identified with the incoming beam ir. Of the remaining local maxima, the one with the greatest pulse height was assigned the role of a /j, candidate.  The heights and times of these two  peaks were then recorded and presented to the fitting routine as initial values for the parameters of the function: TV  f(t) = B + ^2At-p(t-tl)  (5.34)  i  where p(t) is the average pulse height shape, Ai and t; are the height and time of the ith peak, respectively, and B is the background. The non-linear least squares fitting routine MRQMIN [32] was used to fit equation 5.34 to the CCD output and to calculate  Chapter 5.  The Active  Target  51  the x2 value of fit for N equal to one (corresponding to a one peak fit) and for N equal to two (corresponding to a two peak fit). The minimum of the %2's for the two fits was used to determine the number of observed peaks and thus the best set of fitting parameters. MRQMIN is a widely used non-linear fitting routine that invokes the LevenbergMarquardt method. This robust algorithm is desirable since it performs it's minimization routine by a 'weighted combination' of two fitting routines: the 'steepest descent' method and the 'inverse-Hessian' method.  The 'steepest gradient' method alters a  functions parameters, a;, by the magnitude of the gradient of the function, scaled by some factor A: Sat = A • Pi  (5.35)  where  A  = -5 ir  (5 36)  -  By defining 1 d2X2 2 dakdai  (537)  and applying a dimensional analysis, one finds that equation 5.35 can be expressed as XauSai = pi  (5.38)  This method is useful when the choice of initial parameters results in a x2 value that is far from the minimum. In the 'inverse-Hessian method', a matrix of equations involving the second derivatives of the x2 function are iterativly solved until a minimum is found: ^2aki8ai  = (3k  (5.39)  This second algorithm is useful when the choice of parameter values is near the best solution and the x2 may be approximated as a quadratic vector function.  Chapter 5.  The Active  Target  52  By defining the parameter matrix as <*'„ =  o ^ ( l + A)  «;*  «i*  =  (5.40) 0'#*)  (5.41)  the relative weighting of each method is given by A, resulting in a final form of: J2a'kl6ai  = [3k  (5.42)  For small values of A, the matrix a' is nearly equivalent to the normal Hessian matrix; large values of A create a diagonally dominant a' matrix which is best solved using the steepest gradient method. Therefore, M R Q M I N utilizes a combination of two methods, weighted according to their effectiveness over the domain of the parameter space. This weighting decreases considerably the time required to achieve a good fit to the data. The CERN fitting package, MINUIT, also uses MRQMIN, but it was not employed in our analysis due the overly general nature of the package.  Since MINUIT was  designed to be applicable to a great variety of problems, it uses extensive CPU intensive matrix inversions to ensure that a determined minimum is global and not local. Indeed, the designers of MINUIT have admitted that the package is not readily applicable to online analysis, but rather is more appropriate for a 'one-shot' problem such as finding optimum operating parameters [33]. The approach used in this work was found to be some 30 times faster t h a n MINUIT.  5.3.2  The C C D Pulse Heights  T h e amplitudes of the reconstructed CCD d a t a were calibrated with 200 MeV passing 7r's. By using passing 7r's, no decays were present in either the short or the long ADC scintillator signals from the active target. Hence, the short ADC's saw only the energy due to the first pulse height. The height of the prompt pulse was therefore plotted  Chapter 5.  The Active  s^  4 -.  >  -  CD  S  '—'  Target  53  CD 3 -  a o  -  CD  m  i—i  2 PH «4-H  o  a 1 -  -t^>  Xi  &0 CD  • i-H  -  ffi n 0  1  2  3  4  Short ADC (MeV) Figure 5.19: Calibrated pulse height versus calibrated short ADC signal for passing 7r in target section A. A line of unit slope and zero intercept has been included for reference. against the short ADC, and the resulting line forced to have a unit slope with null intercept by scaling the pulse height to the same value as that of the short ADC. A typical dot plot of this type is given in figure 5.19. The parameters determined by the linearization of the prompt TV pulse were then applied to the pulse of the /j, candidate.  T h e resulting spectrum displayed a peak  centered about 3.9 MeV, as shown in figure 5.20. A /j, decay was therefore identified from events having a sufficient difference between the long and short ADC's observing the second peak, and whose second peak's pulse height was within the appropriate range, defined by the pulse height gates, (see figure 5.20) Figure 5.21 compares the second level trigger with the third level trigger.  The  spectrum represents a typical spectrum of the difference between the long and short ADC's of target section E. The hatched region is the distribution of the difference in  Chapter 5.  The Active  Target  54  1600  1200  800 -  O O 400 -  Second P u l s e Height (MeV) Figure 5.20: Calibrated second pulse height attributed to ^ ' s in target section E. The pulse height gate used in the final analysis has been included. ADC values that satisfy the tight pulse height gate shown in figure 5.20. The edge of the overlayed distribution was used to determine the location of an appropriate cut on the second level trigger, which is represented in figure 5.21 by the vertical line.  5.3.3  /J D e c a y T i m e s  A compilation of the delay time between the first and second pulses resulted in a halflife of 26.4 ± 0.5 ns, in agreement with the IT half-life of 26.030 ns. As is apparent in figure 5.22, there was a minimum reconstructed delay between the fitted pulses. This came about by two different reasons. First, the second level trigger could not identify such events since essentially all of the energy deposited in each target section was seen equally in the short and long ADC's. Second, the pulse height fitting routines were unreliable for peak separations of 10 ns or less. As such, a cut on the order of 16 ns was placed on all d a t a events. (A different cut value was used for each target section.) Also indicated by figure 5.22, there was a later time beyond which the counts fell below the  Chapter 5.  The Active  Target  55  30000  20000-  O  o  10000  ADC Difference  (MeV)  Figure 5.21: T h e distribution of the difference between the Long ADC and Short ADC for target section E for 200 MeV incident 7r's represent those events which passed the second level trigger. T h e hatched area is due to those events which pass the pulse height gate. T h e Second Level Trigger software cut has also been included.  Chapter 5.  The Active  Target  56  3000 Timing Cut  2500m 2000 -  T = 26.4±0.5 n s  § 1500H  o  \  O 1000H 500 0  0  20  40  60  80  100  Decay Time (ns) Figure 5.22: Aggregate time between fit pulses for 200 MeV incident 7r's for all target sections. exponential decay curve. This was due to the inability of the CCD Modeling routine to fit pulses which were not entirely digitized. As such, an additional upper cut on the order of 64 ns was placed on the data. The definition of a good fj, was altered to include these timing cuts.  5.4  S t o p p e d 7r D e t e c t i o n Efficiency  The second parameter in the denominator of equation 3.25 is the stopped TT detection efficiency, rjv. As stated in section 3.2.1, above, 7r's were identified by decay /x's. The main stage of this identification was the second level trigger, the ADC difference trigger. T h e second level of triggering was necessary in order to reject the vast number of events that consisted of only a passing IT, thereby reducing the dead time of of the data acquisition system to a manageable level. The rejection criteria, and thus the detection efficiency, of the second level trigger depended upon the contents of a Memory Lookup  Chapter 5.  The Active  Target  57  S h o r t ADC (MeV) Figure 5.23: Dot plot of calibrated Long ADC values versus calibrated Short ADC values for an incident 200 MeV TT beam stopped in target section D. Unit (MLU) within the trigger circuitry. Unfortunately, the presence of two bugs in the online software caused non-optimal values to be stored in the MLU, thus reducing r\v. This fact was discovered during the offline analysis. In order to calculate the stopped TT detection efficiency, it was necessary to determine the number of 7r's stopped in the active target. This was done using beam sample data since they were not subject to the second level trigger. For each target segment, a dot plot of long ADC d a t a versus short ADC d a t a was constructed; a sample of which is shown in figure 5.23. The main band of points corresponded to passing 7r's whose long ADC and short ADC values were very similar. ( N . B . A straight line on a dot plot of long ADC versus short ADC corresponds to a constant value in a 'long minus short  Chapter 5.  The Active  Target  58  A D C spectrum.) By fitting a straight line to this band and shifting it upwards in long value by a constant amount, an effective cut was defined, below which data was rejected. When this fit was performed online, a dot plot of the raw (i.e. uncalibrated) long ADC versus the raw short ADC displayed a 'kink' about which the d a t a was best modeled by two lines of slightly different slopes. The first bug came about in terms of parameterization of the data. The dot plot package used online did not give reliable straight line fits to the data. As a result, the parameters supplied by the plotting package were not very appropriate for the d a t a with a small short ADC value, but not too bad for d a t a with a high short ADC value. T h e reason this error was not detected during the online analysis was due to the second bug. When displaying the data, the decision as to which set of parameters to use in calibrating the d a t a was based upon a variable that had not been defined, rather than on the raw short ADC d a t a variable. Being undefined, the fitted variable had a zero value and so the parameters for the small short ADC values were used in the second level trigger for the entire experiment. While this bug did not affect the d a t a below the kink, it mis-calibrated the d a t a whose short ADC output was in excess of the kink value, resulting in a lower 'long minus short A D C cut value than was appropriate, needlessly rejecting otherwise acceptable events. As a result, histograms of 'long minus short A D C values were of the expected form (c.f. figure 5.21), So the compound error was not detected until the more in-depth offline analysis was made. However, it should be noted that this error was not catastrophic since approximately 90% of the d a t a occurred below the kink values. This compound error was rectified through a complicated analysis that expressed ?/„• as a function of the long ADC value [34]. The analysis began by carefully defining a 7r that stopped in a target section and whose decay fj, was detected within the same target section. A potential problem with this technique is t h a t the //'s from 7r's stopped near  Chapter 5.  The Active  Target  59  the edge of a target segment may travel into a neighbouring scintillator before being detected. (The range of a 3.9 MeV /J, in PILOT-U scintillation plastic is 1.4 mm.) To eliminate this 'dribbling' of //'s across scintillator boundaries, two short gates were defined: a wide short gate and a strict-short gate that have a higher threshold. Events involving dribbling /u's were thus eliminated from the efficiency calculation by requiring the incident 7r to pass the wider short ADC gates in the target section upstream of the one under consideration, pass the strict-short gate in the target section of interest, and fail to give a signal in the first target section downstream. This, in effect, accepted only those 7r's which decayed towards the back of the target section under study. The problem of the application of this method to the last target sections was alleviated by requiring no signal in the S4 scintillator. The total number of stopped 7r's was set as the number of 7r's that passed all such test combinations. Once the number of 7r's that passed these tests was determined, those samples were subjected to the gates applied to the pulse heights and times of the second pulses. The beam sample 7r's t h a t passed this second level of testing were then used to construct 'long minus short A D C spectra. By way of contrast, the actual d a t a analysis  (c.f.  chapter 7) computed the 'long minus short A D C spectra first and applied a cut to those spectra before applying the CCD pulse cuts. The rjv was therefore determined as a function of long ADC value by counting the number of events having a 'long minus short A D C value in excess of the second level trigger cut value. As shown in figure 5.21, the software cut has been set at 1.4 MeV. Graphs of r\v as a function of the long ADC value for the five target sections are shown in figure 5.24. T h e -K detection efficiency for each target section falls off according to how poor the online parameter fit was to the data. While the d a t a in sections A and E were modeled rather well, the parameters determined for sections B, C, and D were not so good.  Chapter 5.  The Active  Target  =  0.4 0.2 0.0  i^ o Pi  0.4  CD  • r—1  o  0.2  • rH «t-H «f-l  H  r, o  0.0 0.4  • f—1  -f->  o  CD  0.2  -i->  CD  Q  0.0  £  0.4  T> CD  0.2  OH  a o  0.0  +J  CO 0.4 0.2 0.0  0  10  20  30  40  Long ADC (MeV) Figure 5.24: Stopped -K detection efficiency as a function of long ADC value for t sections A through E.  Chapter 6  T h e Incident B e a m  T h e M-11 beamline delivered a high quality [24] positive TT beam to the experimental area. The 7r's were generated by colliding proton bursts from the main cyclotron with the T-1 target every 43 ns. T h e particle 'buckets' generated from these proton bursts were then focused into a m o m e n t u m envelope with a width of less than 2%. Due to the n a t u r e of the experimental set-up, a specific beam configuration was required. Specifically, an acceptable event required a single TT, generated by one proton burst, to be preceded and followed by empty beam buckets. T h e requirement for empty beam buckets prevented a following beam burst from simulating a TT —> fxv^ decay in the target. During the experiment, random samples of the beam were made and tested for their qualities. The definition of a beam 'sample' was taken as a coincidence between the first three in-beam scintillators, ( 5 1 • 5 2 • 53).  The determination of the useful  beam fraction was performed in three parts. First, the 7r-fraction of the beam had to be determined. Second, the fraction of beam bursts that had the pattern <full>,  <empty>  <empty>,  was evaluated. Third, Poisson statistics were invoked to calculate  the fraction of non-empty beam bursts that contained only one particle. These steps in the calculation are discussed separately, below.  61  Chapter 6. The Incident  6.1  Beam  62  7r-Fraction  T h e beam particles included 7r's, / / ' S and positrons. In order to distinguish between these particles, the difference in their time of flight to the active target due to their different masses was used. The T C A P spectrum shown in figure 3.4 was typical of the beam sample's time of flight down the M - l l beampipe. T h e placement of a gate about the 7r peak as shown in the figure, ably eliminated the positron fraction of the beam. Previous studies [35] have determined that the M - l l channel has a 1.2% \x contamination, which was corrected for in determining the total amount of useful beam. T h e 7r-fraction was evaluated for and applied to each run individually.  The  beam was found to be composed of 98% 7r's on average.  6.2  Acceptable B e a m Burst Pattern  The nature of the beam burst p a t t e r n was determined by hardware circuitry as shown in figure 6.25(a). Each of the first three in-beam as well as the target scintillators were connected to such a circuit whose output was denoted as the 'prime' of the originating signal. All of the scintillators except S3 were timed such that their ADC signals were similar to figure 6.25(b); S3's ADC timing is shown in figure 6.25(c). The original signal was fanned out through a discriminator. One of these signals was then put through a second level of discrimination to widen the pulse from the current beam bucket to 80 ns, just less than the time between three beam bursts. This wide gate was then p u t into coincidence with the original signal. For all of the scintillators except S3, the timing at the coincidence was such that the original signal arrived before the wide gate. However, the presence of any particles due to the following cyclotron burst were in time with the wide gate, signaling their presence. The timing of the small in-beam scintillator, S3, was delayed by 50 ns in order  Chapter  (a)  6. The Incident  Beam  63  L.  S3 — 1 1  LeCroy TC455 Quad. C.F.D.  2  LeCroy 821Z Quad. D i s c r i m i n a t o r  3  LeCroy 622 Coincidence Unit  Double Pulse  (b) i  .  LJ  (c)  i.  i  LJ Figure 6.25: Hardware circuit to determine the beam burst p a t t e r n (a). The timing of the ADC signals into this circuit for S i , S2 and the sections of the active target is shown in (b) whereas (c) shows the timing of the S3 data. The dashed pulses in (b) and (c) indicate where a pulse from a neighbouring beam bucket would be observed.  Chapter 6. The Incident  Beam  64  to determine the fraction of sample beam bursts which followed a bucket containing a particle. W i t h the timing shown in figure 6.25(b), if there was a particle in the previous beam bucket, it would produce the wide gate, and the particle in the current bucket would satisfy the coincidence. However, if the previous beam bucket was empty, then the current pulse would be the only one present in the signal, giving a null result. In order to verify the requirement of an empty beam bucket on either side of a bucket producing an event, the primed signals were histogrammed for both events and beam samples. An example of both histograms for the S3 scintillator is shown in figure 6.26. There are four distinct peaks in the S3 prime spectrum and they are labeled by letters in figure 6.26(b). As stated above, an event required its preceding and following beam bucket to be empty, and this was indicated in the event d a t a (figure 6.26(a)) by the disappearance of peaks A and B. T h e fraction of proper bucket patterns was determined by counting the number of events in the 'event disallowed' peaks in the beam histograms, dividing by the total number of counts in the histograms, and subtracting the value from one. This result was defined as 1 — T, where T was the fraction of beam samples not having the required configuration. As with the 7r-fraction, this was determined and applied for each run separately. It was found that 90 % of the beam had the necessary pattern.  6.3  Multiple B e a m Particles  The fraction of cyclotron proton bursts that resulted in two or more particles in a beam bucket were modeled by Poisson statistics. A Poisson distribution gives the probability of N events occurring in a time interval t as  p(N(t)) =  6  —JP-  (6.43)  Chapter 6. The Incident Beam  65  800  600 -f->  400 -  o o  200 -  500  1000  1500  2000  1500  2000  S3 Prime 1600 1200 800 -  o 400 -  0  500  1000  S3 Prime Figure 6.26: S3 Prime spectra for events (a), and beam samples (b).  Chapter 6.  The Incident  Beam  66  where A is the frequency of the events. For the M - l l beamline, the quantity \t is given by: A* = ^ ^  (6.44)  VRF  where Vbeam is the observed beam rate in the experimental area and URF is the R F frequency of the cyclotron, 23 MHz 1 . While the R F frequency of the cyclotron was constant, the observed beam rate was not. As such, a weighted average was calculated for each energy. T h e b e a m r a t e was calculated by taking the ratio of the SI scalar to the CLOCK scalar, converted to MHz. T h e SI scalar was used instead of the S A M P L E scalar for two reasons, b o t h stemming from the definition of a beam sample as (SI • S2 • S3).  First,  any particles interacting with the S4 scintillator were disallowed from consideration, reducing the number of counted beam samples. Second, since the S3 scalar was only one-sixteenth the size of the other in-beam scintillators, the requirement of interaction with S3 further reduced the number of beam samples from the number of particles delivered to the experimental area. T h e CLOCK scalar incremented while beam flux was incident upon the apparatus and d a t a was being written to tape. It was determined that one C L O C K unit corresponded to 6.9 ms. T h e b e a m rate was evaluated for each r u n individually, and weighted by the number of accepted events written to tape. Once the weighted beam frequency was determined, equation 6.43 was used to calculate the Poisson probability of a beam bucket containing one particle, two particles, three particles and so on. The final probability of one particle occurring in a beam bucket was evaluated by: P(1) =  ™-W  A frequency of 23 MHz corresponds to a period of 43 ns.  (6.45)  Chapter 6. The Incident  Beam  67  -* beam  b'beam  (MeV)  (MHz)  200 184  2.23 2.25  \t 0.0966 0.0980  P(l) (%) 95.25 95.18  Table 6.11: Poisson probability of only one particle occurring in a beam bucket.  (MeV)  Beam Scalar (xlO 9 )  Ni (xlO 9 )  Ratio (%)  200 184  169.093 308 302.907 827  142.047 423 251.021 792  84.00 82.87  -'-beam  Table 6.12: Number of useful beam TT incident upon the target. where Prob(i)  is the Poisson probability for having i particles in a beam bucket, given  by equation 6.43. T h e results are given in table 6.11. T h e results of determining the number of incident useful TT on the active target is shown in table 6.12. T h e ratio of useful b e a m particles to the total beam incident upon the target is also given.  Chapter 7  The Results  7.1  The Acceptance  T h e experimental acceptance was determined by a Monte Carlo simulation. Since the 7t+p —»• 7r + 7r + n reaction was examined near threshold, where s-wave processes dominate, the final state was taken to follow three body phase space. T h e Monte Carlo code followed TT'S into the active target, simulated the reaction of interest, followed the final state particles until they were stopped and then determined whether all of the detection requirements were met. A total of 1 X 10 12 incident 7r's were simulated, and the number of Trp —>• inrp reactions found to satisfy the detection requirements are given in table 7.13. In simulating the occurrence of a (71", 2n) reaction it was necessary to consider the energy degradation of the incident beam.  Since the cross section goes like PcmT*^  (c.f. equations 2.18 and 2.19) where Pcm is the centre of mass m o m e n t u m and T*m is the centre of mass kinetic energy of the incident 7r, the cross section is very sensitive to any energy loss the incoming TT beam experiences from traveling through the active  J-beam  (MeV) 200 184  # Detections per 1012 Events 123 467 87 756  Table 7.13: Number of simulated irp —> mrp reactions per 1 x 10 12 incident 7r's. 68  Chapter  7. The  Results  69  target. This energy dependence is particularly acute when near threshold. The starting locations of the final state products were, therefore, weighted by the dependence of the cross section upon the depth of penetration into the active target by the incident TT. The 7r's were tracked through the active target in variable step sizes such that the particle lost no more than 10% of its energy over one step. T h e light deposited in each scintillator was evaluated after each step and corrected for quenching effects. Once a 7r's kinetic energy dropped below 0.1 MeV, it was considered to have stopped and allowed to decay to yu's with the characteristic TT lifetime of 26.030 ns. T h e //'s were emitted randomly over a 4TT phase space and tracked until they were stopped. In determining whether or not the \i was detected as it would be in the experiment, the stopped -K detection efficiency for each scintillator, ?/„-, was 'folded' into the code. It was for this reason that the acceptance was written as a function of the detection efficiencies in the denominator of equation 3.25. T h e neutrons were followed to the face of the neutron bars and to their point of terminus. As with the /u's in the active target, the neutron detection efficiency was folded into the Monte Carlo Acceptance code. Only those neutrons that were found to be detected within the experimental constraints and within the physical dimension of the ten most central neutron bars were accepted. Requiring the detection of both the neutron and at least one decay //, the Monte Carlo acceptances are given in table 7.14 for the detection of at least one // and for both /!/'s. The acceptances decreased as the initial beam energy increased since the product phase space increases with energy while the detection phase space remains constant. As will be discussed in section 7.2, below, the Monte Carlo code was also utilized in determining the kinematic constraints on the neutrons and on the total energy deposited in the active target by the 7r's and //'s.  Chapter  7. The Results  70  Tbeam (MeV) One fi (%) 200 4.0 184 13.6  Two ^ (%) 0.503 1.48  Table 7.14: Experimental acceptances. 7.2  D e t e r m i n a t i o n of the Yields  In order to determine the experimental yield, an accurate determination of the background had to be made. As such, the d a t a from the 172 MeV incident 7r runs were taken as the background for b o t h the 184 and 200 MeV data. Since this energy was below the reaction threshold of 172.3 MeV, it modeled the substantial physics background from 7r + C —> 7r+nX processes, where X could be anything. Combining just the timing and ADC energy cuts for the neutrons with the pulse cuts from the active target resulted in spectra as shown in figure 7.27. It was readily apparent that the large background would reduce the significance of the signal since the signal would be on the same order of magnitude as the statistical fluctuations of the background. By placing a gate about the kinematic sum of the products  (c.f.  table 3.3), as shown in figure 7.27, a kinematic cut was put on the data. However, it was decided that this cut was too conservative in that the appreciable background under the peak broadened the resulting kinematic ranges. Therefore, the Monte Carlo acceptance code was allowed to determine the ranges of the neutron and charged particle kinematic energies that contributed to the (ir, 2ir) process. The ranges used in the final results are shown in table 7.15. For completeness, the ranges determined from the d a t a have been included. The d a t a meeting the Monte Carlo kinematic cuts for 200 and 184 MeV incident n beam are shown in figure 7.28; both methods of analysis (the requirement of observing  Chapter  7. The Results  71  500  200 sum  (MeV)  Figure 7.27: T h e large spectrum is Tsum for 200 MeV 7r's with no kinematic cuts applied to the data. The kinematic gate based on the d a t a is included as is the spectra of Tsum based on these kinematic cuts. The Monte Carlo cuts are slightly more severe.  Cut Source Data Monte Carlo  Tbeam (MeV) Tn (MeV) Tn (MeV) 200 4-49 12- 61 184 4 - 42 8-47 200 20-56 12-40 184 1 3 - 39 8 - 30  Table 7.15: Kinematic ranges for the final state products allowed to contribute to the reaction as determined from the d a t a and from the Monte Carlo acceptance code. Here, Tv refers to the total energy deposited in the active target by the charged final state particles.  Chapter  7. The Results  72  at least one decay /.i; and the observation of two yw's) are presented. T h e hatched regions are the estimated background determined from the 172 MeV d a t a meeting the same kinematic cuts as applied to the d a t a spectra. In attempting t o model the background as accurately as possible for t h e one-/j, method, the emphasis was on matching the leading edges of the d a t a and background spectra. This emphasis reflected the fact that Tsum represents the maximum energy available for detection. While some d a t a may have smaller reconstructed energies, none could have measured energies in excess of Tsum (within resolution). However, since the d a t a appears to sit on the leading edge, it was feared that fitting only the 'high side' would imprecisely reflect the background under the peak. Therefore, to determine the yields, the backgrounds were scaled over a continuum of distribution magnitudes and the limits of a reasonable fit were noted. T h e best fits (shown in figure 7.28) were taken as the mid-point of the magnitude scaling range. This method of fitting the background was preferred over a \  2n  * since the parameters determined using the x2 did not model  the d a t a very well, leaving a broad background under the d a t a peak. In estimating the background for the two-// method, t h e same method of fitting a continuum of backgrounds to the d a t a was applied. In determining the best fit, a greater emphasis was placed on equating the background with the extreme edges of the signal distribution in total counts t h a n on fitting the leading edge. The uncertainty in the yield due to background fitting was set at one half of the difference in yield given by the limiting conditions. This error was in addition to the standard error in the yield. T h e final yields are shown in table 7.16 along with both the background and standard errors.  Chapter 7. The Results  73  200  200  150-  150  w  GO  -p  100-  100-  o  o  o  o  20  40 sum  60  80  100  (MeV)  40 sum  100  (MeV)  20  40  60  80  100  (MeV)  60  80  (MeV)  100  sum sum Figure 7.28: The Tsum spectra for Tbeam = 200 and 184 MeV satisfying the kinematic cuts. The cross-hatched regions reflect the background from the TT+C —> ir+nX processes, where X can be anything, which also meet the same requirements.  Chapter  7. The  74  Results  One n Two JJ, Tbeam (MeV) 200 180 ± 40 ± 13 31 ± 2 ± 6 184 325 ± 44 ± 18 41 ± 5 ± 6 Table 7.16: T h e experimental yields. T h e errors quoted are for the background fitting error and standard error, respectively.  DATA Tbeam (MeV) 200 117 641 197 184 187 388 762  'EVENT + SAMPLE'S 138 476 215 218 998 145  CPU Livetime (%) 85.0 85.6  Table 7.17: C P U livetimes for Tbeam = 200 and 184 MeV. 7.3  The C P U Livetime  As with the detection efficiencies, it was necessary to evaluate the amount of time spent by the C P U processing d a t a because the electronics were unable to accept d a t a for some fraction of time. As detailed in section 4.6.1, above, the livetime of the data acquisition system was taken as the average of the LAM's and CPU busy signals, divided by the total number of beam bursts passing either the event or beam sample definition electronics, thereby warranting being written to tape. The results are given in table 7.17.  7.4  The Results  The cross sections were determined from equation 7.46: N  1 v  • come.  (7.46)  Chapter  7. The Results  75  -*• beam  (MeV) 200 184  One \x  Average a (fib) (lib) 0.32 ± 0.07 0.43 ± 0.08 0.37 ± 0.06 0.14 ± 0.02 0.16 ± 0.03 0.15 ± 0.02 T w o fj,  Table 7.18: Preliminary cross sections. Comparison with equation 3.24 shows that the number of target nuclei has been folded into the experimental acceptance along with the detection efficiencies. This was required to determine the weighting factor for the position distribution of the reaction for the acceptance determination. Here, the experimental yields, given in table 7.16, have been listed as iVco,-nc. for consistency with equation 3.25. The computed cross sections appear in table 7.18 and are plotted in figure 7.29. Coulomb corrections, as prescribed by Bjork et al. [36], increase the cross sections by 5% at 200 MeV and by 8% at 184 MeV. These cross sections have been evaluated using the d a t a in tables 7.16, 6.12, 7.17, 7.13 and 7.14. It is readily apparent from figure 7.28 that they do not agree with the world's d a t a set. Indeed, the total cross section at 184 MeV is just half t h a t measured by Sevior et al. [8]. T h e situation is much worse at 200 MeV where the measured cross section is only one third that of the previous data. T h e reaction amplitudes were determined from equation 7.47: |a(7r+7r + )| = . / ° g„ ^ 1 V n V 1.28 x 1 0 - 5 P c m T c ^  (7.47)  V  ;  The kinematic parameters used in evaluating the amplitudes are included in table 7.19. The computed amplitudes appear in table 7.20 and are plotted in figure 7.30.  As  with the cross sections, it is plain that the results of this experiment are in severe disagreement with the world's data. The errors quoted thus far for the cross sections and amplitudes are statistical only, coming from the uncertainty in the experimental  Chapter  7. The Results  76  10 r ^  ^  o  10 3  o CD W  w w  —  0  10 -  Oset a n d Vicente ( *= - 0 . 2 ) Kravtsov et al.  o u o  OMICRON  d 10  -1 B u r k h a r d t a n d Lowe  -p  o  Sevior et al.  This Work. -2  10  160  200  240  280  320  360  I n c i d e n t p i o n e n e r g y (MeV) Figure 7.29: A comparison of total cross section for 7r+p —» 7r + 7r + p as a function of incident TT energy. Some of the world's d a t a set has been included for comparison.  Chapter  7. The  Results  77  3.0 J  2.5-  This Work. Burkhardt and Lowe  2.0-  I 0.5-  £5 Sevior et al. — - Oset and Vicente  I  £  Kravtsov et al.  §  OMICRON  0.0  0  30  60 Tcm  90  T  120  150  (MeV)  Figure 7.30: A comparison of reaction amplitude for tr+p —> TT+TT+P as a function of incident w energy above threshold in the center of mass frame. Some of the world's d a t a set has been included for comparison.  Chapter  7. The Results  78  Tbeam (MeV) Pcm (MeV) 200 234.277 184 223.117 214.732 172.3  Tcm (MeV) 161.937 149.769 140.794  TL(MeV) 21.142 8.975 0.00  Table 7.19: Kinematic parameters for 7r's of various beam energies. The values at threshold have been included for comparison.  •L beam  One //  (MeV) 200 184  0.48 ± 0.08 0.77 ± 0.08  Two //  |a(7T+7T+)|  0.57 ± 0.08 0.52 ± 0.06 0.83 ± 0.11 0.80 ± 0.07  Table 7.20: Preliminary reaction amplitudes. yields. The errors quoted in the average results are computed by the quadratic sums of the individual methods. Up to this point, only a preliminary analysis of the stopped TT detection efficiency has been completed. It should be noted that both the One-// and Two-// methods agree with each other. This agreement implies that the work thus far is valid. However, since the d a t a disagrees with the world's d a t a a more complete analysis of the detection requirements is in order.  7.5  Further Work  There are a number of areas in which to progress with the analysis. First and foremost is a concerted effort to determine 7/ff. Since the stopped IT detection efficiency is vital to the Monte Carlo acceptance code, this should be the main focus of effort. A second area of the analysis that should be re-examined is the application of cuts to the time of the delayed // pulse in the active target and to the neutron bar ADC  Chapter  7. The  Results  79  — --  Existing threshold  15000-  Proposed threshold 10000-  o 5000-  ! \  0 "T  0  1  5  i  i  10  15  i n. 20  i  25  H n  30  N e u t r o n ADC Output (MeV) Figure 7.31: The spectra shows the aggregate neutron bar ADC spectrum for the 200 MeV data. T h e only requirement placed on the d a t a was that the beam -K be recognizable. T h e existing cut and proposed threshold cuts are also indicated. spectra. While the cuts to the /J, timing spectra eliminated a small fraction of the total number of events, the reaction yields were seen to decrease by about a factor of two. T h e elimination of these cuts would bring the d a t a into much better agreement with the world's data. For the neutron bar ADC data, two approaches should be examined. First, the detection threshold has been set at 1.9 MeV. This value was chosen to be as small as possible in order to not eliminate events too early in the analysis. However, an examination of the ADC spectrum for the total neutron bar assembly (c.f. figure 7.31) indicates that a larger threshold, say at 4 MeV, would be more appropriate in eliminating background processes. The second change to the neutron ADC d a t a is an amendment to the requirement that only one neutron bar detect a particle during an event. It is quite possible that  Chapter  7. The  Results  80  during t h e course of the six week r u n time that the bars became activated themselves. T h a t being the case, any low energy 7 ray in one of the extreme bars would be enough to veto a neutron signal from one of the central bars. It would be well worth examining the method of applying the thresholds to the neutron bar ADC's before applying the requirement of only one neutron bar signaling the detection of a particle. While this is not expected to increase the yield by much, it could be a meaningful improvement on the analysis.  Chapter 8  Conclusions  Using an active target to detect the charged final state particles, the total cross section for the 7T+p —* 7r + 7r + n reaction was measured to be 0.15 ± 0.02 /J,h and 0.37 ± 0.6 /ib at incident IT kinetic energies of 184 MeV and 200 MeV, respectively. In addition, the magnitude of the matrix element of the reaction was determined to be 0.80 ± 0.07 m " 1 and 0.52 ± 0.6 m~ a at these same respective energies. T h e quoted uncertainties are statistical in nature only. A full estimation of the systematic errors is yet to be done due to the great uncertainty in the understanding of the stopped 7r detection efficiency. By contrast, the neutron detection efficiency has been assigned a 10% relative uncertainty, contributing a 3% error to the cross sectional data. It is expected that the overall statistical error will be on the order of 15%. The preliminary analysis yields results in significant disagreement with the world's d a t a set. In fact, the results are a factor of 2 or 3 smaller. Suggestions for further work in the analysis of the d a t a have been made in an attempt to understand the reasons for this discrepancy.  81  Bibliography  [1] J. Stoney, New Scientist  2 0 / 2 7 (1984) 54  [2] S. Weinberg, Phys. Rev. Letters  18 (1967) 188  [3] M.G. Olsson and L. Turner, Phys. Rev. Letters  20 (1968) 1127  [4] L. Rosselet et a l , Phys. Rev. D 1 5 (1977) 574 [5] J.L. Basdevant, C D . Froggatt, adn J.L. Peterson, Nucl. Phys. B 7 2 (1974) 413 [6] L. Turner, Ph.D. Thesis (1992), University of Wisconsin, unpublished [7] N. Fazel, M.Sc. Thesis (1992), University of British Columbia, unpublished [8] M.E. Sevior et al., Phys. Rev. Letters 6 6 (1991) 2569 [9] A.V. Kravtsov et al., Nucl. Phys. B 1 3 4 (1978) 413 [10] G. Kernel et al., Phys. Lett. B 2 2 5 (1989) 198 [11] M. Sevior, TRIUMF  Research Proposal, E561 (1990), unpublished  [12] N. Fazel, Private Communication. [13] T.D. Lee, P a r t i c l e P h y s i c s a n d I n t r o d u c t i o n t o Field T h e o r y , Harwood Academic Publishers, Chapter 24, (1981) [14] S. Weinberg, Physica A 9 6 (1979) 327 [15] J. Schwinger, Phys. Letters  2 4 B (1967) 473  [16] M.G. Olsson and L. Turner, Phys. Rev. 181 (1969) 2141 [17] J. Gasser and H. Leutwyler, Phys. Lett. 1 2 5 B (1982) 312 [18] R.J. Jacob and M.D. Scadron, Phys, Rev, D 2 5 (1982) 3073 [19] Lohse et al., Nucl. Phys. A 5 1 6 (1990) 513 [20] A.N. Ivanov and N.I.Troitskaya, Sov. J. Nucl. Phys. 4 3 (1986) 260 [21] L.H. Ryder, Quantum  Field Theory, Cambridge University Press (1985) 82  Bibliography  [22] S. McFarland, Chiral Symmetry unpublished  83  and the CHAOS Experimental  Program  (1991),  [23] J . F . Donoghue, Chiral Symmetry as an Experimental Science. Lectures presented at the International School of Low-Energy Antiprotons in Erice. CERN-TH.5667 No. 1696 (1990) [24] T R I U M F U s e r ' s H a n d b o o k [25] T R I U M F K i n e m a t i c H a n d b o o k [26] N.R. Stanton, Ohio State University  Report,  COO 1545- 92 (1971)  [27] J. Spuller et al., Physics Letters 6 7 B (1977) 479 [28] V.I. Highland et al., Nucl. Phys. A 3 6 5 (1981) 333 [29] V. Paticchio et al., Nucl. Inst, and Methods  A 3 0 5 (1991) 150  [30] A.C. Phillips and F . Roig, Nucl. Phys. A 2 3 4 (1974) 378 [31] M. Kermani, B.Sc. Thesis (1991), University of British Columbia, unpublished [32] W.H. Press et al., N u m e r i c a l R e c i p e s : T h e A r t o f Scientific Cambridge University Press, Chapter 14, (1988)  Computing,  [33] F . James and M. Roos, M I N U I T : F u n c t i o n M i n i m i z a t i o n a n d E r r o r A n a l y s i s (1989) [34] K. Raywood, Private Communication. [35] G. Smith et al., Phys. Rev. C 3 8 (1988) 240 [36] C.W. Bjork et al., Phys. Rev. Lett. 4 4 (1980) 62  

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