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The technical characteristics of a magnetic spectrograph used for (P,π ) studies at TRIUMF Ziegler, William Anthony 1983

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THE TECHNICAL CHARACTERISTICS OF A MAGNETIC SPECTROGRAPH USED FOR (P,7r) STUDIES AT TRIUMF . by WILLIAM ANTHONY ZIEGLER B.Sc, University Of Regina, 1980 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF 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 January 1983 © William Anthony Ziegler, 1983 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics The University of British Columbia 2075 Wesbrook Place Vancouver,•Canada V6T 1W5 Date: 18 January 1983 i i Abstract An experimental and theoretical analysis of a magnetic spectrograph is presented. The spectrograph was used for the analysis of pions produced in reactions of the type A(p\7T*)A+1 at the TRIUMF cyclotron. An experimental study of the response of the spectrograph to monoenergetic pions was carried out using pions produced by the reaction pp-»djT*. Both the instrumental lineshapes and effective solid angles were obtained for three pion energies, 50 MeV, 70 MeV, and 100 MeV. Theoretical studies of the spectrograph were performed by means of a Monte Carlo analysis simulation, which included both multiple scattering and pion decay. Good agreement was obtained between the experimentally determined lineshape and that predicted from the Monte Carlo analysis. The lineshapes were fitted analytically by a linear least squares method. Both the parameters resulting from the f i t and the effective solid angle were found to be dependent on pion energy. The one experimental feature which was significantly underestimated by the Monte Carlo analysis was that of multiple scattering, that i s , multiple scattering of the pions in the pole face structure of the magnet. Table of Contents Abstract i List of Tables i List of Figures I. INTRODUCTION II. THE "RESOLUTION" SPECTROGRAPH SYSTEM 2.1 General Description 2.2 Pion Production Target And Scattering Chamber 2.3 Magnet 2.4 Beam Monitoring 2.5 Particle Detection System 2.5.1 Associated Particle Detection 1 2.6 Electronics 1 III. EXPERIMENTAL DATA 1 IV. DATA TREATMENT 2 4.1 Counter Data 2 4.2 MWPC Data 2 4.2.1 Co-linearity 2 4.2.2 Y Target . .2 4.3 Focal Plane And Dispersion Relation V. MONTE CARLO ANALYSIS VI. REVMOC DATA TREATMENT 6.1 Data Conversion 6.2 Tests On The Data 6.3 Pole Face Scattering Correction ... 6.4 The Kinetic Energy Spectra VII. LINESHAPE 7.1 Analytic Fit 7.2 Energy Dependence VIII. THE EFFECTIVE SOLID ANGLE 8.1 Calculation 8.2 Energy Dependence IX. CONCLUSION BIBLIOGRAPHY APPENDIX A - THE "RESOLUTION" SYSTEM APPENDIX B - MWPC PREAMPLIFIER APPENDIX C - THE Y TARGET POSITION APPENDIX D - REVMOC6 iv List of Tables I. Data Array 16 II. Experimental Data 19 III. Pole Face Scattering Correction Factors 46 IV. Lineshape Fit Parameters 50 V. Effective Solid Angles 56 V L i s t o f F i g u r e s 1. The " R e s o l u t i o n " S p e c t r o g r a p h S y s t e m 4 2. M a g n e t i c f i e l d v e r s e s t h e C u r r e n t f o r t h e magnet S p e c t r o g r a p h 6 3. S c h e m a t i c o f P o l a r i m e t e r 7 4. Beam C u r r e n t C a l i b r a t i o n 8 5. P o l a r i m e t e r A n a l y z i n g Power 9 6. CE C o u n t e r 10 7. M u l t i w i r e P r o p o r t i o n a l Chamber 12 8. A s s o c i a t e d P a r t i c l e D e t e c t i o n H o r n 13 9. O v e r a l l E l e c t r o n i c s 15 1 0. Beam L i n e 1B 17 11. E n e r g y L o s s d i s t r i b u t i o n 21 12. T i m e - o f - F l i g h t d i s t r i b u t i o n 21 13. S t a n d a r d D e v i a t i o n d i s t r i b u t i o n s , X a n d Y 24 14. Y T a r g e t d i s t r i b u t i o n f o r T,r = 70 MeV 26 15. XFP d i s t r i b u t i o n f o r T T = 70 MeV 27 16. R a t i o o f 10 b i n s a t p e a k / t o t a l XFP d i s t r i b u t i o n v e r s e s Y T a r g e t C u t W i d t h 28 17. Momentum S p e c t r u m f o r Tjr=70 MeV 33 18. K i n e t i c E n e r g y S p e c t r u m f o r Tj-=70 MeV 33 19. REVMOC Momentum S p e c t r u m f o r T f f = 70 MeV 39 20 . REVMOC K i n e t i c E n e r g y S p e c t r u m f o r T x =70 MeV 40 2 1 . K i n e t i c E n e r g y S p e c t r a C o m p a r i s o n f o r Tff = 50 MeV 41 22. K i n e t i c E n e r g y S p e c t r a C o m p a r i s o n f o r T*r = 70 MeV 41 23. K i n e t i c E n e r g y S p e c t r a C o m p a r i s o n f o r T,r=100 MeV 42 24. Y T a r g e t C o m p a r i s o n f o r ^ = 5 0 MeV 44 25 . Y T a r g e t C o m p a r i s o n f o r T^=70 MeV 44 26. Y T a r g e t C o m p a r i s o n f o r T^ = 100 MeV 45 27. C o r r e c t e d K i n e t i c E n e r g y S p e c t r a C o m p a r i s o n f o r T, =50 MeV 47 28. C o r r e c t e d K i n e t i c E n e r g y S p e c t r a C o m p a r i s o n f o r T,, =70 MeV 48 29. C o r r e c t e d K i n e t i c E n e r g y S p e c t r a C o m p a r i s o n f o r T , r=l00 MeV 49 30. Y T a r g e t G e o m e t r y 64 1 I. INTRODUCTION The magnetic spectrograph discussed here (and named "Resolution" by the University of British Columbia pion production group) has been used to study the pion production reactions, A(p*,7T*)A+1 , where the resultant A+1 nucleus is in its ground state or a low-lying excited state. In order to be able to measure the differential cross sections for such reactions a detailed understanding of the spectrograph is required; in particular, features such as the acceptance and instrumental lineshape. An energy spectrum characterizing the pions from such reactions consists of a number of discrete peaks, each peak associated with the resultant nucleus l e f t in one of i t s low-lying excited states (including the ground state). In addition, there is a continuum pion distribution associated with the multi-body final state composed of a pion together with several nuclear particles (the "break-up" reaction). In order to resolve the various components of the resultant pion spectrum, the response of the spectrograph to a monoenergetic line is requi red. Therefore the main goal of this thesis is the determination of the effective solid angle and its dependence on pion energy, and the generation of the monoenergetic instrumental lineshape for the kinetic energy spectrum of the pions together with the energy dependence of this f i t . This was accomplished by ivestigating the pions produced by a reaction characterized by a single discrete line, the pp-*dn+ reaction. Using incoming 2 protons of 450 MeV, three different pion energies (50, 70, and 100 MeV) were investigated by setting the spectrograph at the appropriate angle (87°, 70°, and 51° lab, respectively). This energy range spans the region of interest for the pion production studies carried out for nuclear the targets. Since a CHA target was used for the pp-*drr+ studies, i t was impossible to detect only the monoenergetic pions from this reaction. The continuum arising from the carbon as well as the pions from the "break-up" reaction, pp-*pnrr +, contaminated the discrete line. For this reason, an "associated particle" technique was employed wherein another counter was used to detect the deuterons in coincidence with the pions detected by the spectrograph. By modelling the system with a Monte Carlo program and comparing this data with the experimental data, a good understanding of the system was developed. Thus both the solid angles and the lineshapes were determined as a function of energy. The thesis begins with a brief description of the system, the experimental data obtained, and the method used for handling the data. Next, the method used for modelling the system is discussed, and a comparison of the results of this analysis with data obtained from the experimental measurements is presented. Finally the energy dependence of the parameters describing the lineshapes and the effective solid angles are discussed; 3 II. THE "RESOLUTION" SPECTROGRAPH SYSTEM 2.1 General Description The basic instrument is a 65.0 cm Browne-Buechner magnetic spectrograph, (Browne 1956). The particle detection system consists of a counter telescope composed of three s c i n t i l l a t i o n counters together with three helically wound delay line multiwire proportional chambers (MWPC), (Lee 1974). The layout of the spectrograph system is shown in Figure 1. Geometrical details and demensions are listed in Appendix A. The three s c i n t i l l a t i o n counters provide the event definition as well as timing and energy loss (dE/dx) information. The MWPC's, on the other hand, provide position information for each particle from which the exit trajectories are determined. The intersection of the trajectories with the focal plane determines the particle's momentum. 2.2 Pion Production Target And Scattering Chamber The targets (in this case a 149 mg/cm2 CHj. and a 93 mg/cm2 C) were mounted on a target ladder (which was designed to accommodate up to six different targets) situated within an evacuated scattering chamber. The choice of target could be selected remotely. However, the target angle was defined by the physical orientation of the target ladder (and l i d of the scattering chamber) with respect to the chamber i t s e l f . This could be set to any desired angle by suitable manipulation but was not capable of remote control. The target ladder is constructed such that two of the targets could be mounted z C 2 — 4 g = • ! C l l—C=D : CD ' MWPC 3 1 * = 2 I He BOX 2 t 4 0 cm. t F i g u r e 1 - The " R e s o l u t i o n " S p e c t r o g r a p h S y s t e m 5 at 90° with respect to the other four, so that rotating the ladder was not normally necessary during a run. For the measurements described in this work, desired target angles were either 45° or 135° depending on whether the spectrograph was at a forward or backward scattering angle. With identical targets mounted at each of the two angular positions in the ladder, scattering measurements could be obtained at a l l angles without the need for local access to the target system. The target is positioned at the center of the target chamber, which was under beamline vacuum. The evacuated chamber was isolated from atmosphere by a 0.127 mm mylar window that allows measurements over an angular range of 25° to 155°. 2.3 Magnet The 65.0 cm Browne-Buechner magnet was obtained together with its vacuum box from the Chalk River Nuclear Laboratories in 1978. The vacuum box situated in the 32 mm gap between the poles of the magnet possessed a 23.8 mm radius circular mylar entrance window and a 640 mm x 203 mm rectangular exit mylar window. Since the magnet i t s e l f was complete, the particle detection system was the only part of the spectrograph that had to be developed locally. This spectrograph, since i t was designed i n i t i a l l y for low energy, high resolution nuclear studies, had some restrictive features as far as use as a pion spectrograph was concerned. The magnet was restricted to the angular range of 46° to 135° by the geometry of the magnet yoke. As well, with the existing power supplies available at TRIUMF, the coils of the magnet (basically high voltage, low current 6 conf igurat ion) l i m i t e d the maximum magnetic f i e l d strength to approximately 1 T e s l a . As a r e s u l t , the maximum pion k ine t i c energy that could be handled was approximately 100 MeV. The f i e l d of the magnet was monitored by a nuclear magnet resonance magnetometer (NMR), (Fremont 1978), pos i t ioned in the 32 mm gap. The pos i t ion i s shown in Figure 1. Figure 2 shows the dependence of the f i e l d on current when driven by the TRIUMF power supply (Canadian Dynamic, 120 V, 500 A ) . Figure 2 - Magnetic f i e l d verses the Current for the magnet Spectrograph 7 2.4 Beam Monitoring For monitoring both the intensity and polarization of the incident proton beam a four-arm polarimeter was used. Figure 3 is a schematic of the polarimeter's geometry together with i t s electronics. This polarimeter was based on a design developed Figure 3 - Schematic of Polarimeter by the University of Alberta at TRIUMF. It is based on monitoring pp elastic scattering with the target protons provided by the hydrogen in a thin CH^ target. Background events from the carbon are discriminated against by relying on detection of both the scattered proton and the recoil proton in a two-arm system situated at the angles appropriate to free pp scattering. A count corresponds to the coincident detection of a proton scattered at 17° to the right (left) with respect to 8 the beam direction and a recoil proton detected at 71.3° to the le f t (right). The sum of a l l left scattered (L) and right scattered (R) counts is essentially independent of the polarization of the beam, and therefore provides a measure of the beam current. A beam current calibration curve is given in Figure 4. The difference, L-R, though is a direct measure of the proton polarization (the polarization P is given by P=Ajj1 (L-R)/(L+R)). Figure 5 gives the polarimeter' s analyzing power, hy. This calibration data was supplied by the University of Alberta group, (Hutcheon 1979). 9 .50r < .30+ 100 200 300 T p (MeV) 400 500 Figure 5 - Polarimeter Analyzing Power 2.5 P a r t i c l e Detection System The detection system shown in Figure 1 consists of three s c i n t i l l a t i o n counters and three MWPC's. In Figure 1, i t i s clear that the CE counter i s d i f f e r e n t from counters C1 and C2. Both C1 and C2 have photomultiplier tubes (RCA 8575) mounted at each end. The time of t r a n s i t of the p a r t i c l e through the s c i n t i l l a t o r can therefore be determined (with sub-nanosecond resolution) using a 'meantimer' (LRS 624), e l e c t r i c a l l y connected to each photomultiplier p a i r . In the case of the CE counter, however, only one photomultiplier was employed, thus predicating against use of a meantimer. In t h i s case, the CE counter was constructed so that the time for the l i g h t to travel (via appropriate l i g h t guides) to the tube was nearly the same 10 In setting up the discrimination levels (the overall electronic schematic is shown in Figure 9) and high voltages for the photomultiplier tubes, the same procedure was carried out for a l l five tubes. First the discrimination levels were set at an appropriately low level so that the photomultiplier tubes would always operate in their linear region, but well above electronic noise level. Next the high voltage was increased keeping an eye on the amplitude distribution of the relevant signals as determined by CAMAC analogue-to-digital converters 11 (ADC). Section 2.6 describes obtain these distributions. The the discrimination level cut distribution, that is such that the electronic setup used to high voltage was set such that off was well below the f u l l cut ADC Value off The MWPC's were based on the chambers described by Lee,1974. A schematic representation of these chambers is shown in Figure 7. The preamplifiers used with these MWPC's were based on a Los Alamos Scientific Laboratory design, (Studebaker 1974). A schematic of the preamplifier is contained in Appendix B. The position of the three MWPC's is shown is Figure 1. The K1 signals were taken from the side of the chamber at the lower X position and the A1 signals from the lower Y position. 12 IAI I K I IK2 Figure 7 - Multiwire Proportional Chamber K indicates cathode output and A anode. The signal coding shown is for MWPC1. (Chambers 2 and 3 output would be referred to as 2K1, 3K1 respectively.) When applying the negative high voltage to the MWPC's, the dc leakage currents that resulted were the main concern. In normal use of a wire chamber the voltage is set so that the chambers are essentially 100% effecient. However, for our application, where the cross sections measured for pion production from nuclear targets are very small (~nb/sr), use of a large beam current was required in order to obtain satisfactory data rates. Background from the beam produced a significant dc leakage current. So in order to retain the high current (~15nA) and a reasonable dc leakage current (<2>uA), a compromise value for the high voltage was selected corresponding to operating efficiencies between 0.85 and 0.90. I f higher voltages were applied (and thence higher leakage currents), the chambers tended to breakdown by spontaneous discharges. 13 2.5.1 Associated Particle Detection An associated particle detection system was developed primarily to provide the capability for detecting the deuteron in the pp-»drr+ reaction as required for the lineshape studies described in Section 6.4. In addition, however, i t was used to obtain data for the pd«*tTr* reaction, and also preliminary data on reactions of the type p12C-»Xrr*d. The system was basically a horn-shaped chamber installed in the beam pipe immediately downstream of the target chamber, as shown in Figure 8. With this system, an Figure 8 - Associated Particle Detection Horn associated particle detector (s c i n t i l l a t i o n counter, ZC) could be mounted in a i r , outside the mylar window, and use to detect the associated particles over an angular range of 6 . 8 ° to 11.0°. 14 In the case of the pp+drr* reaction, the counter, ZC, detected the recoil deuteron. 2.6 Electronics The basic electronics for the system is illustrated in Figure 9. Parallel event-defining logic was utilized providing both true and random event definitions. This provides the ab i l i t y to do random subtractions. The timing for the three (four)-fold coincidence system was as follows: C I 10ns C 2 C E 10ns 10ns. 60ns Z C 10ns Thus, since the "period between successive cyclotron - RF pulses was 43 ns, the CE distribution included the random events. The start signal for the time-to-digital converters (TDC) was provided by the coincidence between the strobe and the mean time of Cl with the timing logic arranged so that the time of the output pulse was correlated with the time of Cl (rather than the strobe i t s e l f ) . The strobe is simply the event definition vetoed by the busy signal from the computer (imposed so that only one event was examined at one time). Figure 9 shows the basic setup of this veto, excluding some electronic logic C I L CIR •7V_ I /*T\ , C E , / - N C 2 L , ^ - N C 2 R ^ Z C S I G N CONFIGURATION ®Wi ©A <- »r T T 2) J l J TRUE RAND.1 EVENT EVENT' O R r [40ns IT) START ^y) STROBE n R COMPUTER BUSY M . W . P . C . L O G I C IKI I K 2 IAI I A 2 2 K I 2 K 2 2 A I 2 A 2 3 K I 3 K 2 3 A I 3 A 2 R F DISCRIMINATER L R S 621 T Y P E II COINCIDENCE W (2 FOLD) rn LRS 6 2 2 I MEAN TIMER LRS 6 2 4 J I L L COINCIDENCE W (4 FOLD) I T I L R S 4 6 5 DELAY LOGIC J LRS 4 2 9 6 CAMAC SCALER KINETICS 3615 y-L-CAMAC ADC LRS 2 2 4 9 PATTERN UNIT E G G C 2 I 2 6 CAMAC TDC LRS 2 2 2 8 F i g u r e 9 - O v e r a l l E l e c t r o n i c s 16 designed to veto the system during the time i t took the computer to service a CAMAC interrupt (typically a few hundred microseconds). As is clear from Figure 1, C1 is one of the last s c i n t i l l a t o r s to f i r e . Thus if C1 is used to in i t i a t e the timing, a l l the other signals must be delayed appropriately. The scalers and pattern unit are simply monitoring devices providing essential information for off-line analysis. The spin configuration of the protons was monitored by scalers with veto input as shown in Figure 9. Values of the ADC's and TDC's were read by the on-line computer system and recorded on magnetic tape for subsequent off-line analysis. The data array recorded on the magnetic tapes is shown in Table I. D(21 ) CE ADC • D(33) CE TDC D(22) C1L ADC D(34) C1L TDC D(23) C1R ADC D(35) C1R TDC D(24) C2L ADC D(36) C2L TDC D(25) C2R ADC D(37) C2R TDC D(26) ZC ADC D(38) ZC TDC D(40) RF TDC D(41 ) 1A1 TDC D(49) 1K1 TDC D(42) 1A2 TDC D(50) 1K2 TDC D(43) 2A1 TDC D(51 ) 2K1 TDC D(44) 2A2 TDC D(52) 2K2 TDC D(45) 3A1 TDC D(53) 3K1 TDC D(46) 3A2 TDC D(54) 3K2 TDC Table I - Data Array 17 III. EXPERIMENTAL DATA The "Resolution" Spectrograph system was situated on beam line 1B at TRIUMF. An illustration of beam line 1B is given in Figure 10. In order Figure 10 - Beam Line 1B to develop an understanding of the spectrograph system, pions from the pp-#drr* reaction were studied. Since a CHA target was used in this study, the pions detected at the exit of the spectrograph could arise from any of the following reactions: PP4dn* ( 3 - D pp+pniT (3-2) p , 2 O X » * (3-3) where X=11B+p+n,etc. In reactions (3-2) and (3-3), the final state consists of three 18 (or more) bodies. Thus, the pion momentum distribution for these cases is a continuum, on which the monoenergetic peak of reaction (3-1) is superimposed. In order to enable carbon subtraction (reaction (3-3)), both the CHa target and a carbon target were mounted on the remotely controlled target ladder. In order to reduce contamination from the "break-up" reaction (3-2), the associated particle detector described in Section 2.5.1 was used to detect the deuteron in coincidence with the pion detected by the spectrograph. But since the associated particle detector limited the solid angle of the system, the deuteron coincidence was only demanded in the off-line analysis. That i s , ZC was not included in the hardware event definition (refer to Figure 9). However, ADC and TDC information appropriate to ZC were recorded for these events characterized by a particle through ZC. Thus, in calculations of the solid angle of the spectrograph (Section 8.1), where the coincidence requirement would unfavourably affect the solid angle, the ZC coincidence demand was dropped. The ZC cut was particularly useful, however, in studies of the lineshape. Though the associated particle detector did eliminate most of the "break-up" contamination, there was always some present. The carbon continuum was subtracted using the carbon target runs, but the remanent pp^pnrt* contamination could not be eliminated so easily. In order to determine how much of this contamination remained, the ZC counter was t i l t e d in order to reduce the solid angle i t subtended at the production target. Thus knowing the change resulting in the pion spectrum as a result of the 19 reduction in solid angle, the number of "break-up" pions remaining could be estimated by extrapolating to zero solid angle. The following set of runs were recorded: Tp=450 MeV Tgt: C% and C Tff(MeV) &7T B(T) ZC t i l t * 100 51° .989 45° 90 57° .867 45° .895 45° .929 45° .970 45° 80 63° .863 0° 70 70° .792 0° 45° 85° 60 78° .726 0° 50 87° .649 0° * where 0° means maximum solid angle and 90° would be zero solid angle. Table II - Experimental Data In addition, one more data set was recorded, a cosmic ray run. This was a run for which only C1 and C2 (refer to Figure 1) were included in the event definition (with no beam striking the target). That is the cosmic rays passed through C2, Cl, MWPC3, MWPC2, and MWPC1. Since the cosmic rays extended over the whole physical active area of the MWPC's, the dimensions of the chambers in TDC bins could be readily inferred from such data and used to calculate the conversion from TDC bins to linear units (mm) and vice versa, (Section 6.1). 20 IV. DATA TREATMENT 4.1 Counter Data The f i r s t step of the "off-line" data treatment was to eliminate the background events. This was accomplished by employing a series of tests on the signals from the sci n t i l l a t i o n counters. The f i r s t of these was the "true event" test, which subtracts the random events from the total event distribution. The ADC's of C1, C2, CE, (and ZC where applicable) provided energy loss information; separation of pions from protons was proven possible at a l l pion energies of interest because protons were predominantly low-energy multiply-scattered protons of large dE/dx, while the pulse-height separation of pions and electron-like events improved slightly with decreasing pion energy. Figure 11 illustrates a typical dE/dx distribution. The protons are actually off the right-hand side of the page and the electrons are in the f i r s t peak. In addition the counters provide time-of-flight measurements. For optimum timing discrimination between particles of similar velocity (say v1 and v2), one wants, £T=s/v1-s/v2 = s(1/v1 -1/v2) to be large. Thus AT is proportional to s, the path length of the particles. Since the "start" was trigger by C1, timing to CE, C1, and C2 involves only a small path length. Thus 4T is small. Timing on the cyclotron RF involves essentially the measurement of the time interval from arrival of the beam at the production target until the created particle traverses C1. Thus AT is proportional to the distance from the production target to 21 9750 9 5 0 0 9 3 5 0 9 0 0 0 S750 8 5 0 0 8350 »0OO 7750 7500 7350 7000 6 7 5 0 6 5 0 0 6 2 5 0 6000 5750 5 5 0 0 5 2 5 0 5 0 0 0 4750 4500 4 2 5 0 4 0 0 0 3750 3500 3350 3000 2750 2 5 0 0 2350 2000 (750 1500 1350 1000 750 5O0 250 0 Pions Electrons > \ XXX8 XXXX x x x x x 3 X X X X X 5 XXXXXXX X X X X X X X9 777 ' 9XXX2 x x x x x 8XXXXX7 XXXXXXX 3 X X X X X X X 1 XKXXXXXXX x x x x x x x x x XXXXXXXXX8 1XXXXXXXXXX x x x x x x x x x x x X X X X X X X X X X X 9 x x x x x x x x x x x x 7 X X X X X X X X X X X X X X X X X X X X X X X X X 9 IX X X XX X X X X X X X X X X X X X X X X X X X X X X X X 2 1 X X X X X X X X X X X X X X X X x x x x x x x x x x x x x x x x x X X X X X X X X B 9 X X X X X X X X X X X X X X X X X 5 8 « X X X X X X X X X X X X X X X X X X X X X X X X X X X XXXXXXXXXXXXXKXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXKXXXXXXXXXXXXX9 9XXXXXXXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX9 8XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX4 1 X X X X X X X X X X X X X X X X X x x x x x x x x x x x x x x x x x XXXXXXXXXXXXXXKXXXXXXXXXXXXXXXXXXXXX 4XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX9 2 X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X K X X X X 9 1 1XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX52 124XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXB5521 1 3 5 7 8 9 X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X 8 8 7 8 7 7 Typical Cut 1 Protons \ 1 1121112234433263542344 6 6 6 6 6 6 7 7 7 7 6 7 8 8 9 9 X 9 9 9 X X X X X X X X X X X X X X X X X X X X X X X X K X X 1 11 111111 1 11 11 1 1 1 1 11 1 1111 1 111 1 1 1 1111111 12222322222222222232222223322222222232222 1 111222233334444555S666677778B8899990000111 1322233334444555566667777888899990000 111 122223333444455556666777788689999 0 2 5 7 0 3 5 7 0 2 5 7 0 2 5 7 0 2 5 7 0 2 5 7 0 2 5 7 0 2 5 7 0 2 5 7 0 2 5 7 0 2 5 7 0 3 5 7 0 3 5 7 0 3 5 7 0 2 5 7 0 2 5 7 0 2 5 7 0 2 5 7 0 2 5 7 0 2 5 7 0 3 5 7 0 2 5 7 0 2 5 7 0 2 5 7 0 3 5 7 0 2 5 7 0 2 5 7 0 2 5 7 0 3 5 7 0 2 5 7 05O5O505050505O50505050505050505050505050505050505O5050505O505050505O505O505CI505050505050505O505050505050505O5O5IJ5O5O5O5 TDC BINS Figure 11 - Energy Loss distribution 320CO 3 1200 304O0 29600 38800 28000 37300 36400 35600 34800 24000 33300 22400 31600 20800 20000 19200 18400 17600 16800 16000 15200 14400 13600 12800 12 OOO 1 1200 10400 9600 8 8 0 0 8000 7200 6 4 0 0 5600 4 8 0 0 4 OOO 3200 2400 1600 BOO O Protons i Electrons X66 XXXX1 XX XX 6XX XXX xxxx XXXX9 XXXXXX X X X X X 3 9 9 XXX XXX XXX XXX XXX XXX XXX3 xxxx xxxx xxxx xxxx xxxx 6XXXX xxxxx xxxxx xxxxx XXXXX9 XXXXXX XXXXXX XXXXXX 7XXXXXX5 xxxxxxxx XXXXXXXX82 4XXXXXXXXXX5 Pions Typical Cuts -Electrons -I-681 27XXX 8XXXXXXXXXXXXXXX6j6XXXXXXXXXXXX) |9XXXXX93 X X X X X X X X X X X X X X X X x J x x X X X X X X X X X X x 4 x X X X X X X X X e e 7 6 5 S 5 4 4 4 3 3 3 3 3 2 3 2 2 2 1 1 1 1 1 I U 1 1 1 1 1 1 1 1 1 1 1 1 1 H 1 1 1 1 1 H 1 l -I- -I-111111111 f t 111111111111 11232333322222223222222332233333333333333333333333334 112223344455666778BB990OO112223344455666778889900011222334445566677B8B9900011222334445566677888990001 1223334445566677 0 4 8 3 6 0 4 8 2 6 0 4 8 2 6 0 4 8 2 6 0 4 8 2 6 0 4 8 2 6 0 4 8 2 6 0 4 8 2 6 0 4 8 2 6 0 4 8 2 6 0 4 8 2 6 0 4 8 2 6 0 4 6 2 6 0 4 8 2 6 0 4 8 2 6 0 4 8 2 6 0 4 8 2 6 0 4 8 2 6 0 4 8 2 6 0 4 8 2 6 0 4 8 2 6 0 4 8 2 6 0 4 8 2 6 0 4 8 3 6 TOOOOOOOOOOOOOOC^ TDC BINS Figure 12 - Time-of-Flight distribution 22 C1. This path length is much larger than the other ones mentioned. Thus the RF TDC distribution (shown in Figure 12) showed the largest separation of pions from protons and elctron-like events. Because of the periodicity of the RF timing signal, these time difference are not unique, and "wrap-around" distributions can result (see, for example, the electron contribution at each end of the distribution shown is Figure 12) . 4.2 MWPC Data After this f i r s t level of background elimination, further selection of data was performed by using the information from the MWPC TDC's. There were two tests applied to the MWPC data. The f i r s t was a co-linearity test, which determined the extent to which the particle travelled a straight line through the three MWPC's. The second was the Y target test. This traced the Y position of the particle back to the pion production target. By insisting that the particles originated in the illuminated region of the target, the particles which were associated with pion decay or had been multiply-scattered somewhere on route could be distinguished and eliminated from the sample under consideration. 4.2.1 Co-1inearity Before the test for co-linearity can be implemented, position information from the three MWPC's must be intercalibrated. Even though the three electri c a l l y identical MWPC's were physically aligned, their electronic timings had to 23 be mutually calibrated. (In particular the cable propagation delays between the chambers and their respective TDC's were not identical.) An i n i t i a l calibration was performed using a very prominent peak on a spectrum such as that from the reaction pp->dK+. By selecting the value for the magnetic f i e l d of the magnet so that the pions leave the magnet with a vertical trajectory, one could insist that the pions strike the same position on each MWPC. The definition of the position on MWPC1 (in TDC channel units) was Yl- P W - ^ + YM (4-2) where from Section 2.6 Table I, D(49) and D(50) are the MWPC1 cathode signals (1K1 and 1K2, respectively). Likewise D(41) and D(42) are anode signals (1A1 and 1A2,respectively). The positions on MWPC2 and MWPC3 are defined similarly. Thus, i f the TDC time calibration (time increment/bin) is the same for each TDC channel, the calibration of Xj or Yj (in mm/timing bin) would be the same for each chamber, leaving just the constants XjO and YjO to be determined. By comparing the peak positions for these vertically travelling pions on each MWPC, the XjO and YjO were defined (to within an overall additive constant). The co-linearity test was performed by accumulating distributions of both the X and Y standard deviations, 24 2 3 4 0 2 2 8 0 2 3 2 0 2 1 6 0 2 l O O 2 0 4 0 I 9 6 0 1 9 2 0 I 8 6 0 teoo 1 7 4 0 1 6 8 0 1 6 2 0 1 5 6 0 1 5 0 0 1 4 4 0 1 3 8 0 1 3 2 0 1 2 6 0 1 2 0 0 1 1 4 0 1 0 8 0 1 0 2 0 9 6 0 9 0 0 8 4 0 7 8 0 7 2 0 6 6 0 6 O 0 5 4 0 4 8 0 4 2 0 3 6 0 3 0 0 3 4 0 1 8 0 1 2 0 6 0 8 3 6 XXX XXX XXX2 -xxxx X X X X 5 xxxxx XXXXX5 xxxxxx xxxxxx xxxxxx X X X X X X 8 xxxxxxx XXXXXXX 1 XXXXXXXX XKXXXXXX3 xxxxxxxxx XXXKXXXXX XXXXXXXXX7 xxxxxxxxxx X X X X X X X X X X 2 xxxxxxxxxxx xxxxxxxxxxx XXXXXXXXXXX4 xxxxxxxxxxxx X X X X X X X X X X X X 9 XXXXXXXXXXXXXB XXXXXXXXXXXXXX3 XXXXXXXXXXXXXXX4 XXXXXXXXXXXXXXXX8 XXXXXXXXXXXXXXXXX91 X X X X X X X X X X X X X X X X X X X 8 5 2 XXXXXXXXXXXXXXXXXXXX X X 6 6 6 3 4 2 XXXXXXXXXXXXXXXXXXXKXXXXXXXXX9 ' I I I Typical Cut X 8 9 9 9 7 6 6 7 5 7 4 5 7 6 4 5 6 4 6 4 5 4 5 4 3 5 4 5 6 5 3 4 4 4 5 5 5 4 5 4 4 3 4 4 4 4 4 5 4 5 3 4 3 4 3 5 4 5 4 3 4 4 5 4 4 4 4 4 ° 0 0 0 0 , ' ' '"">""33""'™™^^ 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 3 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 TDC BINS 2 4 6 8 0 2 4 6 B 0 2 4 6 8 0 2 4 6 8 3 1 2 0 3 0 * 0 2 9 6 0 2 8 8 0 2 8 0 0 2 7 2 0 2 6 4 0 2 5 6 0 2 4 8 0 2 4 0 0 2 3 2 0 2 2 4 0 2 1 6 0 2 0 8 0 2 O O O 1 9 2 0 1 8 4 0 1 7 6 0 1 6 8 0 8 4 X X 6 X X X X X X 6 xxxx X X X X 7 xxxxx X X X X X X X X X X 4 X X X X X X X X X X X X xxxxxx X X X X X X 1 1 6 C O - X X X X X X X 1 5 2 0 1 4 4 0 1 3 6 0 1 2 8 0 1 2 0 0 1 1 2 0 1 0 4 0 9 6 0 8 8 0 BOO 7 2 0 6 4 0 5 6 0 4 8 0 4 0 0 3 2 0 2 4 0 1 6 0 8 0 X X X X X X X X X X X X X X 2 X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X 9 X X X X X X X X X X X X X X X X X X 4 X X X X X X X X X X xxxxxxxxxxx xxxxxxxxxxxg xxxxxxxxxxxx xxxxxxxxxxxxx X X X X X X X X X X X X X 9 X X X X X X X X X X X X X X S 1 xxxxxxxxxxxxxxxxs I X X X X X X X X X X X X X X X X X X 5 5 I X X X X X X X X X X X X X X X X X X X X 9 8 6 6 7 4 l * 4 3 2 t 4 4 3 1 1 1 4 2 1 1 2 11 1 I X X X X X X X X X X X X X X X X X X X X X X X X X 4 f X X X X X X X X X 9 X X X X X X X X X 9 X X 8 X 9 X 9 9 9 X X 9 9 X X 9 8 6 8 6 7 7 8 6 6 6 7 6 6 5 6 5 4 4 4 4 4 5 5 5 4 3 4 4 3 3 3 2 2 2 3 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 O 0 0 1 1 3 2 2 3 3 4 4 4 5 5 6 6 6 7 7 8 8 6 9 9 0 0 0 1 1 2 2 2 3 3 4 4 4 5 5 6 6 6 7 7 8 8 6 9 9 0 0 0 1 1 2 2 2 3 3 4 4 4 5 5 6 6 6 7 7 B 8 8 9 9 0 0 0 1 1 2 2 2 3 3 4 4 4 S 5 6 6 6 7 7 8 B 8 9 9 O482604826O482604e2604826048260482604B2604B2604826O48260482604826O48260482604826O4B26O48260482604B26 TDC BINS Typical Cut Figure 13 - Standard Deviation distributions, X and Y 25 quantities determined from straight line least squares f i t s through the chambers on an event-by-event basis. Figure 13 illustrates the typical standard deviation distributions. If the standard deviation associated with a particular trajectory was large, the particle was eliminated since i t was probably either a muon arising from a pion decaying in the chamber system or a large angle multiply-scattered event. If the additive constants (XjO and YjO) were chosen correctly, the standard deviation distributions should be centred about zero. Thus there is a fine ajustment capability for these additive constants. 4.2.2 Y Target In order to determine the Y position at the target, the trajectory was extrapolated backwards from its final straight line path. In order to do this, the path length of the particle must f i r s t be evaluated. Since there is no focussing in the Y direction (except for small edge focussing effects),one can determine the Y position on the target. A simple extrapolation was employed, based on the "Y" slope and intercept obtained from f i t t i n g the Y position on each of the three MWPC's. Edge focussing effects were then investigated in some detail. Because of such focussing the "Y" slope of a particle trajectory changed somewhat at the edges of the magnet. For this investigation, the path was sp l i t into three parts (the two external and one internal regions) and the slightly changed slopes were used appropriately to determine the Y target positions. The overall effect of the edge focussing was small, 26 though the peak was somewhat sharper the increase in the number that were within the cuts was less than a few percent. Appendix C describes in detail how the Y position on the target was computed. A typical Y target distribution (including edge focussing) is shown in Figure 14. For convenience, the units on the abscissa are in TDC bins. Although the events in the central peak corresponed to "undisturbed" trajectories, whereas those in the t a i l s are associated with multiply-scattered or decay events, there was no clear separation between them. In order to determine in more detail the nature of the events in the various parts of the distribution a serious of studies were performed. For each of the three i t i O 1QRQ 1 0 5 0 1 0 7 0 P 9 0 oc,0 9 0 0 •?">0 r>AO R tQ 7 f l O 7 5 0 7 3 0 fiQ0 6 ^ 0 G 3 0 GOO ? ' 0 5 * 0 5 10 * n o * 5 0 a ? 0 ?90 3*?0 mo 3 0 0 2 7 0 2 1 0 7 10 I P O ' 5 0 1 7 0 9 0 CO ? 0 1 r w s x x x x x x x j 3BRYXXXXXXXXXX XXj 2 1 1 2 1 3 3 5 G 5 G 5 7 X 9 X X Y X X X X X X X X X Y X X X X X * X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X 3 1 X X X X Y X X X X X X f X X X X X X X X X X X X X X X X X Typical Cut Xfl 7 x x 5 x x t x 7 X X * 7 X 1 XXXXXXX XXVYXXX y y x t x » x p y . y y x x x y q J X Y X X X X X X X x x x x x x * * xx y x x x x x y x x y X X x x x x x x x x 3 x xyX x x x x xxx x x x x x x x x x x x XXXXXXXXXXX I RXXXYXXXX XXXX 9YXXXXXXXXXXXX xxx y x x x x xx y x x x ? x x x Y X X X x xx xxy xy j y y v x x x x x x x x x x x x x J x x x x x x x x x x x x y x x x x j X YXXXYXXXXXX YXXXX' x x x x x x x x x x x x x x x x x | x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x l x x x x x x x x x x x x x x x x x fXfiX XXXXXXXXXXXXXXXXX 'XXX IB 3 XX l rXXXXXXXXXXXXXXXXX fXXXXX? 1 3Xt XX 'XXXXXXXXXXXXXXXXX X x x x x x x-t ROXXXyX 'XXXXXXXXXXXXXXXXX l X X X X X X X X 7 « l l l x x x x x x x | f x x x x x x x x x x v x x x x x x | f x x x x x x x x y y x x x x x y x y y x x y x x 2 S ! i x x x x x x x x x x x x y x x x 2 X X X X X X X X X X X X X X X X X K X X X X X X X X X X X X X X X X P X X ? ^ x x Y x x x x x x x x x x x x x x x x x x v 5 n r > i 3 K 3 7 12 i 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 - - 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 7 7 2 7 2 2 7 3 7 2 2 ? ? ? 2 2 ? 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 09aRB77pG«;5.t<!3322 1 I C 0 Q Q R R 7 7 5 6 5 5 4 4 3 3 2 2 1 1 - 1 1 2 2 3 3 4 4 S 5 S 6 7 7 R R P 9 0 0 1 1 2 2 3 3 4 4 5 5 * * 7 7 R R P 9 0 O I 1 2 2 3 3 i -1 5 ? * ? 7 7PP9<»001 12233•»-155<5<;77PP99 0 5 0 5 0 5 0 S 0 5 0 5 0 5 0 9 0 5 0 * 0 5 O 5 0 3 0 R O 5 0 5 0 P O 5 0 5 ^ ^ TDC BINS Figure 14 - Y Target distribution for %=70 MeV 27 2960 2880 2800 2720 2640 2560 2480 240O 2320 2240 2 160 2080 2 OOO 1920 1840 1760 16B0 1600 1520 1440 1360 12B0 120O 1 120 1040 960 880 eoo -720 640 560 480 4 CO 3 2D 240 " 160 80 1 1 XX XX 1 XXX XXX XXX 4X XX XXXX •xxxx xxxx X X X X X X X X X X X X xxxx X K X X XXX X4 X X X X X XXXXX 6XXXXX XXXXXX XXXXXX xxxxxx X X X X X X 2 XXXXXXX XXXXXXX 7XXXXXXX6 1 X X X X X X X X X X 7 6 5 3 3 I 1 1 1 2 1 1 2 1 1 1 1 2 1 1 2 1 2 2 2 4 ^ 5 7 X X X X X X X X X X X X X X X X 7 8 6 7 9 4 6 5 4 5 3 3 3 4 3 3 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11111 1 1 1 1222222222233333333334444444444555555555566666666667777777777e8eBBft88S899999999990r>OOOCOCOO 12345678901234567690 1234567B9012 34567B90 1234567890 1234567890 123456789012345678901234567B901234567890123456789 0COC<trxXXXX>00C>CK)0un0000f >0TV)0C*X)0C*XX>0000< KJOOt*XX>000000000< >OC<KX>O<X>OOO<KMX>C*XX^^ 0O0CKX>0O0C<)CHXXX>O0C<XXX^ TDC BINS Figure 15 - XFP distribution for Tp- = 70 MeV pion energies of concern (50 MeV, 70 MeV, and 100 MeV) a series of runs were performed having cuts of varying width placed around the Y target peak. In particular, the effect of such a cut on the number of events in the ten central TDC bins at the peak of the XFP distribution was examined (the definition of the focal plane is described in Section 4.3). Figure 15 shows such an XFP distribution obtained using the ZC in the event definition. The information obtained from these Y target studies is illustrated in Figure 16, where the ratio of the number of events in the ten central bins to the total of the XFP distribution is plotted against the Y target cut width. It is clear that there is a point at which no further gain is achieved by tightening the cuts further. 28 O 100 MeV pions O 70 MeV pions 0 . 1 5 ' — 1 — • — • — • — • — • — • — • — • — • — • — • — • — • — • •—• 0 20 40 60 80 100 120 140 160 Width (bins) Figure 16 - Ratio of 10 bins at peak/total XFP distribution verses Y Target Cut Width That i s , just as many good events are lost as bad events i f the cut is less than about 80 to 110 bins (depending on pion energy). Therefore for the most efficient cuts, the widths were taken as 110 bins for 50 MeV pions, 90 bins for 70 MeV pions, and 80 bins for 100 MeV pions. 4.3 Focal Plane And Dispersion Relation After selecting the good events by means of the cuts described in Sections 4.1 and 4.2, the momentum of each remaining particle was determined from its position on the focal plane (XFP) (determined from the least squares f i t of the trajectories) using the dispersion relation for the spectrograph. The dispersion relation is the expression relating the pion momentum and XFP to the magnetic f i e l d of the magnet. 29 The determination of the position on the focal plane was straightforward. It is just the X component of the intercept of the focal plane (assumed to be a planar surface, Z=a-bX) and the straight line trajectory of the particle as defined by the MWPC position information. In order to empirically determine the position of the focal plane, the pp-tdrr* peak was used. From the straight line trajectories for each particle, X distributions of the pp-*d;r + peak were reconstructed at a variety of heights above the magnet. In this way, the height at which the peak was narrowest (that i s , optimally focussed) was obtained. Thus the focal plane was defined in terms of the X positions and heights of these focussed peaks. The height of the best focussed peak could only be obtained however to within ±25 mm since convergence of the rays was inadequate to • define the optimum focus more precisely. Thus the focal plane was defined as: Z=a-bX (4-3) where a=646±l3 mm b=0.94±0.03 mm/bin. The units of X is TDC bins and that of Z is mm, the vertical height measured from the middle of MWPC1, (see Figure 1). This focal plane though, was defined in terms of the "narrowest" peak measured at a given height, rather than in terms of the X-intercept of the trajectories with the focal plane. Thus, to complete the definition of the focal plane, the peak was reconstructed on planes parallel to that of Eqn. (4-3) (with slope of -0.94 mm/bin) but at different heights (Z-intercepts). It was found that a Z-intercept of 700±20 mm produced the 30 narrowest peaks. In this way, the value of a in Eqn. (4-3) was determined to be a=700±20 mm. The XFP value for a particular trajectory is simply the X-intercept on the focal plane of the particle trajectory. X=X0+XSLOPE*Z (4-4) where X0 is the X-intercept of the trajectory at the height of the middle of MWPC1 in bins and XSLOPE is the slope in bins/mm. Appropriate values for X0 and XSLOPE were obtained for each trajectory by the straight line f i t to the three positions on the MWPC's. Solving (4-3) and (4-4), the X position on the focal plane was obtained: y p p 2 - r— (4-5) where a=700±20 mm b=0.94±0.03 mm/bin. Similarly, the Y position on the focal plane (YFP) is simply the Y-intercept of the trajectory with the focal plane: YFP=Y0+YSLOPE*Z(focal plane) or: YFP=Y0+YSLOPE*(a-b*XFP) (4-6) where Y0 i s the Y-intercept in bins, YSLOPE is the slope in bin/mm of the Y trajectory. Now to determine the dispersion relation, the pp^dir* reaction was used again. From Browne-Buechner theory (Browne 1956), XRyiRf-R ) (4-7) X 2 31 where Rp is the effective radius of the magnet and R is the radius of curvature of the particle. Combining Eqn. (4-7) with the fact that P:fBR*0.3BR (4~8) where P is the momentum (in MeV/c), B is the f i e l d of the magnet (in T), and R is the radius of curvature (in mm), the dispersion relation i s : where R0 is in mm and X is in mm. Now, using the pp*>drr+ data for different magnetic fields, the resulting XFP values were f i t to a function of the same form as Eqn. (4-9), / - A*XFP t h a t i s : - * U 1 I -D'XFP Fitting the three constants yielded: i*J««--'»xff < 4 - , 0 ) where the units of XFP is TDC bins. By comparing (4-9) with (4-10) a calibration of XFP (mm/TDC bin) was inferred. For ^=650 mm, (4-9) yields, for vertical rays (X=0), P/B=195. From (4-10), P/B=195 yields XFP, =526, that is XFP for vertical rays. Therefore, with XFP=XFPtf+X(bins), (4-10) becomes: £ = 1X1.5 6 .8116 -3.15*10 «X(t>>*s) 32 or P_ (4-11) Now, by comparing (4-11) to (4-9),4.19*10-ftX(bins)=X(mm)/2R0. Therefore, X(mm)/X(bins)=0.545. Thus This is consistent with the direct measurements described in Section 6.1. Thus by using Eqn. (4-10) the momentum of each particle was determined and momentum spectra thereby obtained. The kinetic energy was obtained for each particle from the momentum defined by (4-10), using the kinematic relation, where 1 % is the mass of the pion, 139.57 MeV/c2. Therefore the two spectra are just the appropriate distributions of particles that satisfied the various tests and cuts. Figure 17 and 18 show the spectra for 70 MeV pions from the reaction pp-»dn +obtained with incoming protons of 450 MeV and the lab angle of the pions of 70° with the ZC coincidence. XFP(TDC bins)=X(mm)/0.545+526 (4-12) (4-13) CO C3 X X X X J c x x x x x x x x x x t x x x x x x x x x x ty, x x x x x > - tr u - tx t£ *- tr ir. - a i-. • O • r- C • r- C • ir a • u. a a a • <j a • ^ — • u C • u" C • m a. • ir. o. • ir. a • ir- tr • ir, t-• ir- tr If- IX ttf IT. IT. IT. — r. o> — r; o — r> a u a> 2 > CD o I'-ll 1* V-i O 6 u 4J o CD a wi 6 3 4-> C a) e o 2 OJ I-I 3 cn fc, c x x x x x x x x x x ( X X X X X X X X X X ( x x x x x x x x x x c x x x x x x x x x x -( x x x x x x x x x x . C X X X X X X X X X X I ( X X X X X X X X X X I ( X X X X X X X X X X ( X X X X X X X X X X c c x x x x x x x x U> 01 <t> at U> Cl IP CO 10 tC IP CD IC CD -IP cc > ? S - S S S S S S S S 2 0 2 9 ° £ 0 £ e o o o ° o o o o o o o o o o o o o o o o > (U 22 o I'-ll E* u O e 3 u 4-> O UJ a CO >1 cn u OJ c ta CJ • rt 4-) 0) c I ao u 3 cn fc, C COOOC-CCCCCC- OOOOOO CO *: r. r\ r. - — c O 0 n a f tx tr i/. tr «c r> r• c* o o c OOOO o o c o o o o o o e OOO c O^arst fC^i i '^ i io^f f*" . uc«otro.tf t> - C C o> on M - u ii f - t « r. r- n •• -34 V. MONTE CARLO ANALYSIS In order to model the "Resolution" system, a version of the Monte Carlo program, REVMOC (Kitching 1971, Stinson 1973, Reeve 1977, Mathie 1980), was used. REVMOC is a computer program that traces charged particles through various optical elements in a system. These elements can range from vacuum d r i f t lengths, absorbers, and collimators of many different shapes to magnets of dipole or quadrapole configuration. REVMOC allows the particles to suffer multiple scattering, stop in the material, or even decay with the trajectory of the final particle traced as well as that of the original. The sixth version of REVMOC called REVMOC6 was used. The special features of this version are described in Appendix D. To model the three energies of interest and to simulate the experimental conditions as closely as possible, pion production at the target was simulated using the pp«*djr* reaction kinematics for incoming protons of 450 MeV. A parallel beam of incident protons was assumed, distributed over a rectangular area on the target whose dimensions could be varied. REVMOC traced these pions through the spectrograph for each of the three spectrograph angles 87°, 70°, and 51°, corresponding to 50, 70, and 100 MeV pion kinetic energy, respectively. The trajectories of a l l pions that were created within a solid angle of 25.6 msr, centred about the angle of interest, were followed. This solid angle completely included the entrance window of the magnet. By starting with 100,000 pions into this solid angle, the effective solid angle of the system is determined by 35 Z\A e = lS.6msr1&- (5-1) / 00/000 where Np- i s the number of p i o n s t h a t s u c c e s s f u l l y t r a v e r s e the system, s a t i s f y i n g t he same k i n d of t e s t s and c u t s t h a t c h a r a c t e r i z e the e x p e r i m e n t a l d a t a . REVMOC g i v e s the p o s i t i o n , b o t h X and Y (see F i g u r e 1 ) , a t the end of ev e r y d i f f e r e n t i a l element u n t i l t he p i o n i s e i t h e r s t o pped or out of the g e o m e t r i c a l a c c e p t a n c e of the d e t e c t i n g e l e m e n t s . By o u t p u t t i n g the X and Y p o s i t i o n s a t each of the t h r e e MWPC's, on an eve n t - b y - e v e n t b a s i s , e x p e r i m e n t a l event d a t a were s i m u l a t e d , so t h a t the same d a t a t r e a t m e n t c o u l d be a p p l i e d f o r each c a s e . That i s the same a c c e p t a n c e t e s t s were a p p l i e d t o the REVMOC d a t a as were a p p l i e d t o the e x p e r i m e n t a l d a t a . 36 VI. REVMOC DATA TREATMENT 6.1 Data Conversion There were two essential differences between the REVMOC data and the actual data. The MWPC positions determined by REVMOC were the physical distances (in mm) between the trajectory intercepts and the centre of the MWPC's. For the actual data, on the other hand, the positions were recorded in the form of TDC bins, with the starting point (the zero) defined in an arbitrary fashion, being simply dependent on electronic cable delays. In order to u t i l i z e an identical data treatment, the two sets of data had to be converted to a common system of units. We chose to express the REVMOC positions in TDC bin units. The second difference was the fact that the REVMOC MWPC position data was exact whereas the actual MWPC position information was blurred somewhat by the fin i t e spatial resolution characteristics of the MWPC's themselves. Therefore, in order to effect r e a l i s t i c comparisons, i t was necessary to incorporate an appropriate positional uncertainty into the REVMOC data (a resolution factor). To obtain a conversion from millimeters to bins, the cosmic ray data (described in Chapter III) was used. Since the cosmic rays extended over the whole physical active area of the MWPC's, the dimensions of the chambers (in TDC bins) were obtained. Comparison of the dimensions in these units with the physical dimensions (in mm) would, in principle, provide the desired conversion factor. Extraction of these calibration factors was 37 hampered, however, by the lack of a sharp cut off in the distributions on the MWPC's. As a result, the MWPC dimensions (in TDC bins) could only be obtained with an error of 10%. The conversion factors so determined were: X direction: 0.54±0.05 mm/bin Y direction: 0.56±0.06 mm/bin. The centre of these distributions were: X direction: 500±50 bins Y direction: 0±20 bins. Thus, the conversion factors appropriate to the REVMOC data were: X(bins)=X(mm)/(0.54)+500 (6-1) Y(bins)=Y(mm)/(0.56) (6-2) where X(mm) and Y(mm) were the positions determined from REVMOC (in mm). By comparing the REVMOC data with the actual data, i t was possible to further improve the accuracy of the conversion relation for the X component. A series of runs had been obtained for the pp-»d7r+ reaction with the spectrograph at a fixed angle but for different magnetic f i e l d settings, (the measurements used to define the dispersion relation, Section 4.3). The same runs were modelled with REVMOC. By changing the magnetic f i e l d value in REVMOC, the pp-j»drr+ peak moved across the focal plane appropriately. By comparing the changes in the positions of the peak from one run to another, (for both the actual and REVMOC data sets), a more accurate calibration was obtained. In addition, for the same magnetic f i e l d the peaks 38 characterizing both the actual and REVMOC data should occur at the same place. Thus both the conversion factor and definition of the centre of the MWPC's were improved. The result of this comparison yielded: X(bins)=X(mm)/(0.50±0.02)+512±5. (6-3) In Section 4.3, a similar conversion relation for the XFP was obtained, Eqn. (4-12). Although Eqn. (6.3) is for the position on the MWPC's themselves and Eqn. (4-12) is for the position on the focal plane, the two equations are completely consistent. In order to include the fini t e resolution of the MWPC's, the REVMOC distributions were broadened by a Gaussian resolution function. A suitable random number was added to each of the positions determined from REVMOC on an event-by-event basis. Since a l l MWPC's were physically identical, the same spatial resolution function was applied to each of the three MWPC's. The "widths" for the Gaussian distributions were determined by matching the resulting standard deviation distributions (described in Section 4.2.1) obtained for the REVMOC data to those obtained for the real data. It was found that for the X direction the resolution was 1.3 bins (0.65 mm) and in the Y direction the resolution was 2.7 bins (1.5 mm). Since the wire spacing in the cathode is 1.5 mm and the anode is 3.0 mm, these resolutions are very reasonable, approximately one half a wire spacing in each case. 39 6.2 Tests On The Data As was the case for the actual data the f i r s t test applied to the REVMOC generated data was the "true event" test. The REVMOC data was checked to determine whether the pion traversed a l l three counters successfully. The next two tests applied to the actual data were not applied to the REVMOC data, since these corresponded to dE/dx and time-of-flight tests. Since the REVMOC model only generated pion events, such tests were unnecessary. Finally the co-linearity and Y target tests were applied using the same cuts as for the actual data. By this means, momentum and kinetic energy spectra were generated as illustrated in Figures 19 and 20. 1 140 1 1 10 ioeo 1050 1020 990 960 930 BOO 870 840 810 780 750 720 690 660 630 600 570 540 510 480 450 420 390 360 330 300 270 240 210 180 150 120 90 60 30 o XX XX XX6 XXX XXX XXX XXX 9XXX XXXX XXXX2 XXXXX XXXXX XXXXX XXXXX XXXXX XXXXX XXXXX 6XXXXX xxxxxx xxxxxx xxxxxx XXXXXXX XXXXXXX XXXXXXX XXXXXXX 5XXXXXXX XXXXXXXX XXXXXXXX1 xxxxxxxxx xxxxxxxxx BXXXXXXXXX 6 X X X X X X X X X X 9 15XXXXXXXXXXXX2 11 1 111 2 1 1 1 1 1 1 1 1 1 1 2 2 1 2 3 3 2 3 3 2 2 2 2 3 5 4 4 3 3 4 4 5 7 9 5 7 8 X X X X X X X X X X X X X X X X 6 2 2 3 3 1 1 2 2 3 2131 2 11 1 1 111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 3222222222333333333333333333334444444444444444444455S555555555555555556666666666666666666677777777777777777777B88B8BB688 5S66778B990011223344556677B899001 1223344556677B8990OI 12233445566778899001 1223344556677B899O0112233445566778899001 1223344 O50SO5O5O505O5OSO5O50505O50505O5O5O5O5O5O5O505O5O5O5O5OSO5O5O5O505O5O505050505O505O50505050505O505O5050505050505O5O5O505 (MeV/c) Figure 19 - REVMOC Momentum Spectrum for ^ =70 MeV 40 1560 1520 1480 1440 ( 4 0 0 1360 1320 1280 1240 1 160 1 120 XX 9XX 1 0 8 0 . x x x 1 0 4 0 | x x x B 1000 ' X X X XXX 1 8 8 0 1 C 8 4 0 XXXX xxxx xxxx xxxx 2 X X X X XXXXX XXXXX XXXXX XXXXX XXXXX XXXXX1 XXXXXX XXXXXX xxxxxx 7 XX XX XX XXXXXXX XXXXXXX3 5XXXXXXXX 6XXXXXXXXX1 <0 |1 1 1 1 12 21121 1 1 1 2 2 1 3 3 3 3 2 2 3 4 5 3 3 5 5 B 7 7 8 9 X X X X X X X X X X X 4 2 3 2 1 2 2 1 2 2 2 21 1 1 1 1 - I - I • • I 9 6 0 ' * * * N 9 2 0 I X X X X „ X X X X 0 B O O U 7 6 0 U 7 2 0 T 6 B 0 S 6 4 0 6 0 0 5 6 0 5 2 0 4 S 0 4 4 0 4 0 0 3 6 0 3 2 0 2 6 0 2 4 0 2 O 0 1 6 0 1 2 0 6 0 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 S 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 8 8 6 8 8 8 8 B 6 8 8 8 B B 8 8 8 8 9 9 9 9 9 9 9 9 9 9 5 5 6 6 7 7 B 8 9 9 O 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 B 9 9 O 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 B 9 9 O 0 1 1 2 3 3 3 4 4 5 5 6 6 7 7 B 8 9 9 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 B 9 9 O 0 1 1 2 2 3 3 4 4 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 T V (MeV) Figure 20 - REVMOC Kinetic Energy Spectrum for ^  = 70 MeV 6.3 Pole Face Scattering Correction In comparing the actual and REVMOC spectra, i t was found that for a l l three energies, the low energy t a i l of the REVMOC kinetic energy spectra was smaller in magnitude than that of the actual spectra, This is illustrated in Figures 21, 22, and 23, where the area of each distribution has been normalized to unity and the peak region reduced by a factor of 10. The actual data were obtained using the two-arm configuration, with the spectrograph detecting the pions from the CH^ target and the ZC counter detecting the associated deuterons, (described in Chapter III). In order to correct for quasi-free events orginating from the carbon, a pure carbon run was also taken and a normalized carbon subtraction performed on the final spectra. The difference spectra were s t i l l not purely monokinetic pions from pp*d/r+, 41 (x10 3) 8 0 0 . 0 F i g u r e 21 - K i n e t i c Energy S p e c t r a Comparison f o r Tjr=50 MeV ( x l C T M F i g u r e 22 - K i n e t i c Energy S p e c t r a Comparison f o r T y=70 MeV 42 ( x i c r 3 ) Figure 23 - Kinetic Energy Spectra Comparison for ^ =100 MeV however, but were contaminated by those pions from the pp^pnr* reaction for which the associated proton traversed the ZC counter. Even though the ZC counter was t i l t e d , in order to reduce the solid angle i t subtended at the production target, there was s t i l l some contamination. Even use of the TDC and ADC distributions for the ZC counter (that i s , the time-of-flight and dE/dx distributions of the deuterons and "breakup" protons) was insufficient to separate such protons from the deuterons, as the protons associated with "break-up" pions having a kinetic energy very close to that of the two-body peak, have essentially the same velocity as the deuterons. Since their electrical charges are the same, they are very d i f f i c u l t to identify 43 without employing energy or momentum determination as well. The energetics of this reaction is such that these "breakup" pion events also occur where the observed low energy t a i l i s . Such contamination does not, however, account for a l l the discrepancy in the magnitude of the t a i l s observed between REVMOC and experiment. The origin of the excess number of pions in this region is attributed to those degraded pions which multiple scatter in the material of the spectrograph pole faces. The magnet apparently possesses ridges on the pole faces of the magnet, as well as some baffles inside the gap, but the positions and geometry of such are unknown, since the magnet box was never dissembled at TRIUMF. With the existing REVMOC program, i t would have been impossible to model these ridges and baffles even i f the positions and geometry were known, since dipole magnets are treated within REVMOC as having smooth pole faces. Thus the model underestimated the amount of multiple scattering in the magnet. Further evidence for this interpretation is provided by the fact that those experimental spectra obtained for lower energy pions possessed comparatively larger t a i l s . As well, the ta i l s of the Y target distributions for the REVMOC data are smaller than those of the experimental data as illustrated in Figures 24, 25, and 26 where the area of each distribution has been normalized to unity. Such would be expected i f the pions far from the centre of the distributions arose from multiple scattering effects. In summary, the REVMOC model provides a very good description of the actual data except for the 44 Figure 24 - Y Target Comparison for Tff=50 MeV Figure 25 - Y Target Comparison for ^  = 70 MeV 45 o Experimental x REVMOC - 2 4 . 0 - 1 6 . 0 - 8 . 0 0 .0 8 . 0 16.0 2 4 . 0 3 2 0 ( x i c n TDC BINS Figure 26 - Y Target Comparison for TT=100 MeV magnitude of the low energy t a i l s , (although the shape of the ta i l s appears to be well described). The underestimate of the low energy t a i l by the REVMOC model is attributed to inadequate treatment of the pole face scattering. As a result of being unable to model the pole face scattering properly, a phenomenological correction to the REVMOC data had to be developed before i t could be compared to the experimental data in detail. A major convenience of the Monte Carlo model was the ab i l i t y to examine separately the nature of the individual spectral components of the pion spectrum. For example, the shape of a pure- monokinetic pion spectrum could be examined both with and without the contribution of pole face multiple scattering. From these, the spectral shape of the suffering 46 multiple scattering component could be isolated. It was assumed that the corresponding component of the experimental spectrum which arose from pole face scattering had the same shape as that generated by REVMOC but was larger in magnitude. However, since the experimental spectra also contained some pp*pnjr+ contamination in the t a i l s , the correction factor for this underestimated pole face scattering could not be simply obtained by comparing the magnitudes of the experimental and REVMOC spectra in the appropriate region. Instead, the correction factors were obtained by comparing the Y target distributions. Since the pions from the pp^ pnTT* contamination should be distributed evenly over the whole of the Y target distribution, rather than situated in one area like in the XFP distribution, the percentage of contamination is very small for any small region on the Y target distribution. Therefore the ratio of the number of pions in the experimental distribution in a region far from the centre to that of the REVMOC distribution, when the spectral peak components are normalized, is a measure of the extent to which the pole face scattering was underestimated by REVMOC. In this manner, correction factors were obtained for each of the pion energies considered with an accuracy of about 5%. The results are shown in Table III. TV 50 MeV 70 MeV 100 MeV Correction Factor 1 .64 1.14 1 .06 Table III - Pole Face Scattering Correction Factors 47 6.4 The Kinetic Energy Spectra After correcting for the inadequate degree of pole face scattering, the REVMOC kinetic energy spectra agreed well with the real spectra.- Figures 27, 28, and 29 show the kinetic energy spectra for both the REVMOC and experimental data. The total area has been normalized to unity in each case, with the peak region reduced by a factor of 10. Some differences are s t i l l apparent. The f i r s t is that the experimental peaks are sharper than those of the REVMOC model. The most likely origin for this discrepancy is the assumed energy spread of the proton beam. In the REVMOC calculation, i t was 2 4 . 0 -3 18.0 - H 1 2 . 0 -a o Experimental x REVMOC — Fit 2 4 0 . 0 3 2 0 . 0 i i i | i i i i i i i i i j i i i 4 0 0 . 0 4 8 0 . 0 5 6 0 . 0 6 4 0 . 0 7 2 0 . 0 8 0 0 . 0 CxIO-') TV (MeV) Figure 27 - Corrected Kinetic Energy Spectra Comparison for Tn =50 MeV 48 (x10 "M 2 4 . 0 -d 1 8 , 0 - H 1 2 . 0 -3 6 . 0 - H 0 . 0 o Experimental x REVMOC — Fit i i i i i | i i i i i i i i i | i i I i i i rrr 5 6 . 0 6 4 . 0 7 2 . 0 8 0 . 0 V C M e V ) 8 8 . 0 w i | i i i i i 9 6 . 0 Figure 28 - Corrected Kinetic Energy Spectra Comparison for TV =70 MeV Cx IQ M 1 9 . 2 -d 1 4 . 4 -H 3 . 6 -3. 4.8 -d o Eperimental x REVMOC - Fit - 0 . 0 j i i i i i i ITfTjTi i i i i i i i | 6 9 . 1 - - - 1 0 1 . 6 1 0 9 . 6 1 1 7 . 6 7 7 . 6 8 5 . 6 9 3 . 6 T V (MeV) 1 2 5 . 6 Figure 29 - Corrected Kinetic Energy Spectra Comparison for T„.= 100 MeV 49 assumed that the beam had a mean energy of 4 5 0 MeV with a 0.2 MeV standard deviation (normally distributed). It appears that the experimental data were obtained with a beam having a somewhat smaller spread N O . 17 MeV). But since the areas under the peaks of both the experimental and REVMOC data are the same, this is not a major problem. Since each reaction studied would have a different relative peak width depending on the kinematics, i t is more important to have a good description of the fraction of events contributing to the t a i l . Again, by inspection of Figures 2 7 , 2 8 , and 2 9 , i t is seen that the corrected REVMOC distributions s t i l l underestimate the magnitude of the low energy t a i l s but only in the region close to the prominent peak. This remaining discrepancy is consistent, however, with the "breakup" contamination of the real data from the pp*pn7r* reaction. The magnitude of the discrpancy in the low energy t a i l region was observed to be essentially proportional to the solid angle subtended by the ZC counter, thus substantiating the interpretation of this component in terms of the "break-up" reaction. We thus feel confident that the "corrected" REVMOC distributions are able to account for the correct "low energy t a i l " component of the observed lineshape to a 5% level. That, is the component of the spectrum that suffers pole face scattering is considered accurate to 5 % . 50 VII. LINESHAPE 7.1 Analytic F i t In order to apply these lineshape results to the analysis to experimental data of the A(j5,7r J)A+1 reactions, i t is necessary to have an analytic description of the lineshapes. After trying a number of different experssions, the one which was found most successful in giving a good f i t to the REVMOC kinetic energy lineshape was of the form The total number in the lineshape has been normalized to unity. The f i r s t term, a Gaussian type, characterizes the peak component of the spectrum, whereas the second term, the exponential decay (with a Fermi-type cut off at T^B) is used to describe the t a i l component. Using a linear least squares f i t t i n g procedure, Opdata (Kost 1 9 7 9 ) , a l l six parameters were fit t e d simultaneously. The best f i t values for the f i t s shown as solid lines in Figures 2 7 , 2 8 , and 2 9 are given in Table IV. Tjr A B C D F G 7#pt 50 MeV . 1 9 6 ( 4 ) 4 8 . 5 9 ( 1 ) 1 . 0 8 ( 3 ) . 0 3 4 ( 2 ) 3 . 0 6 ( 1 2 ) 1 . 0 6 ( 4 ) 1.30 70 MeV . 2 0 4 ( 3 ) 6 9 . 6 2 ( 1 ) 1 . 3 4 ( 3 ) . 0 1 2 4 ( 9 ) 5 . 1 0 ( 2 4 ) 1 . 5 1 ( 7 ) 1.04 100 MeV . 2 0 0 ( 3 ) 1 0 0 . 6 1 ( 1 ) 1 . 5 0 ( 3 ) . 0 0 7 6 ( 6 ) 7 . 1 2 ( 3 9 ) 2 . 0 4 ( 1 1 ) 0.98 Table IV - Lineshape Fit Parameters 51 7.2 Energy Dependence Since both the width and position of a peak are dependent on the kinematics of the reaction and the value chosen for the magnetic f i e l d , the exact numerical values for the parameters A, B, and C are relatively unimportant. The parameters of importance are those of the t a i l ; D, F, and G. By plotting a log-log graph of the pion energy verses D, i t was found that the D parameter could be expressed in the form: D=P/T^  (7-2) The pion energy, T^, was that appropriate to the centroid of the peak, that is parameter B in the lineshape f i t . A linear least squares f i t of (7-2) to the data of Table IV yielded the - value (with X2/pt.=1.81): P=70(3). The other two parameters, F and G, can be described simply in terms of a linear dependence on the pion energy. Appropriate least squares f i t s yielded: F=R+S*T %2/pt=.77 R=-.93(40) S=.083(7) G=U+V*T ^/pt=.37 U=.13(12) V=.019(2). The poor f i t of D forces the conclusion that other pion energies must be studied in order to tie down the energy dependence. 52 VIII. THE EFFECTIVE SOLID ANGLE 8.1 Calculat ion As stated earlier (Chapter V, Eqn. (5-1)) the effective solid angle is given by A£L**AZ6msr • ^  (5-D ^ 100,000 where Njr is the number of pions that successfully traverse the system, meaning that they satisfy a l l the tests imposed on the real events, while undergoing the possibility of decay, multiple scattering, and absorption. Therefore the effective solid angle, ASLg, is not just a geometrical solid angle, but includes a l l the effects of decay, multiple scattering, absorption, and as well the effects of the tests on the data. An effect not included, however, is the efficiency of the particle detection system which is assumed to be 100% for the purpose of the REVMOC calculations. Thus, when applied to real data, the data must f i r s t be corrected for detector inefficiency before the effective solid angle is considered. In principle, since the final lineshape for each energy is now known, determination of the effective solid angle should be an easy task. In fact, the situation is complicated by the fact that the geometry of the baffles and ridges in the magnet is not known exactly (a problem already encountered in the consideration of pole face scattering (Section 6.3)). For the present considerations, i t is not clear whether the baffles and 53 ridges effectively reduce the gap size below that of 32 mm as assumed in REVMOC, since the Afl^ot Eqn. (5-1).is appropriate only to a system which matches that modelled in the REVMOC calculations. The problem is there is no way of knowing whether the baffles and ridges, by multiple scattering, actually did increase the number of pions in the spectrum or whether they simply degraded the pions in the peak. Thus the use of Eqn. (5-1) became impossible. With the lineshapes known, however, the effective solid angles may be obtained from the experimental data using the known cross sections of the pp-j>djr+ reaction. That i s , using the data described in Chapter III (without the ZC coincidence), the lineshapes described in Section 7.1, and the known cross sections from the literature, the ,4Ag's were obtained. The definition of the differential cross section i s : der s ( 8 _ D where: Njr is the number of pions detected Hp is the number of protons that hit the target (nN„/GMW){ft) is the number of scattering centres in the target where: n is the number of scattering centres per molecule He is Avogadro's number GMW is the molecular weight of the target material 54 (ft) is the thickness of the target (in gm/mm2). £ is the efficiency of the detectors is the effective solid angle of the spectrograph (in msr). Therefore for the pp-»d7r+ reaction where: (nNp/GMW) is just and (ft) is 2/14(ft)v^, since the target is at a 45° angle with respect to the beam and only 2/14 of the the CH^ target consists of free protons, the effective solid angle is given by: ^ (8-2) 75. In this case, £ is the efficiency of the MWPC's. Assuming that the three chambers are independent, £ is defined as: or r _ CH'**'*) fa'K-Vi) (8-3) (Pi'**) W W where: (N^'Nj) is the number of events for which both MWPCi and MWPCj simultaneously detect the particle. (N^N^'Nj) is the number of events for which a l l chambers simultaneously detect the particle. The efficiency of each of the MWPC's, £ •, was typically between .85 and .90 (refer to Section 2.5). Such an experimental determination of Asig also suffered from an additional problem. The number of pions detected, NJJ., 55 could not be obtained simply from the experimental data. Even though the carbon subtraction could be performed, a large s t a t i s t i c a l uncertainty characterized the t a i l (non-peak) region due to the large carbon subtraction. In addition, the contamination from the pp^pnrr* "break-up" reaction could not easily be eliminated. Thus the number of pions in the t a i l regions were poorly defined in the experimental spectra. However, since the lineshape is known from Section 7.1, the number of pions in the t a i l s (NT) could be determined by the number of pions in the experimental data peak (Nfr), the number of pions within tight cuts placed around the pp-*d»r+ peak. In particular, by applying the same cuts on the theoretical lineshape as placed on the experimental peak, the ratio of pions not within these cuts to those within the cuts was determined. Thus the numbers of pions in the t a i l s were determined. Therefore, N^ .=NC.-N5+Nr (8-4) where: Ng. is the number of pions in the peak region of the experimental spectrum (including the carbon background). Ng is the carbon continuum background within the peak region, determined from the carbon target run. N r is the number of pions not in the peak region, as determined by the REVMOC lineshape. The number of beam protons, N^, was obtained from the polarimeter as described in Section 2.4. The known differential cross sections for the pp*djr + reaction, , were taken from JJL (Jones 1981). In this way, following effective solid angles 56 were obtained: 50 MeV 87° 25469 .771 1 . 469* 1 0 1 3 88.6 ^ b/sr 1 .40 (.1 2) msr 70 MeV 70° 19738 .754 1 . 237* 1 0 1 3 79.5^b/sr 1 . 47 (.1 3) msr 100 MeV 51° 2001 3 .736 9.236*1012 98.8/ub/sr 1.64(.14) msr Table V - Effective Solid Angles The error in % arises from the s t a t i s t i c a l errors of N^  and N^, together with the 5% uncertainty in NT reflecting the uncertainty in the basic lineshape (Section 6.3). The uncertainty in N^o was estimated by comparing N^ to Up for similar runs taken on various days. The error in N^ was estimated to be 5%. The absolute error in from the literature was estimated to be 7%, whereas the relative error as a function of angle (and thus pion energy) was only 2%, since the experimental data for a l l three pion energies were obtained from experimental data taken at a single proton energy. To illustrate how these errors were used to estimate the error in the effective solid angles, the calculation for the 50 MeV pions is shown ex p l i c i t l y . where: % = 23995 f (the normalization constant between the CH^ and C runs)=1.222 N5=3252 NT=5448 57 Therefore the absolute error in the effective solid angles, , is approximately 8.7%, whereas the relative error is approximately 5.5%. 8.2 Energy Dependence One might think that the solid angle of a spectrograph should not depend on the energy since solid angle is basically a geometrical concept. In our case, however, the effective solid angles include not just the geometrical constraints, but the effects due to multiple scattering, decays, and the data selection tests, which are a l l energy dependent. Thus, the effective solid angle does depend on the pion energy. Assuming for simplicity a linear dependence on energy, the effective solid angle can be expressed as: 4JO<=P+Q*T where: %2/pt.=.22, P=1.22±18*10"3, Q=3.9±2.5*10"6 using the relative errors ASLg. The units of A&e is sr and T^ is MeV. Due to the large uncertainties in ASL±, the detailed form of the energy dependence cannot be tested at this time. 58 IX. CONCLUSION The effective solid angles of the magnetic spectrograph and the lineshapes of the pion kinetic energy spectra were obtained at the three pion energies 50, 70, and 100 MeV. In the lineshape determination, the Monte Carlo program underestimated the low energy t a i l . Although the REVMOC t a i l could be normalized to provide a good f i t to the experimental data, an uncertainty of 5% in the number of pions in the t a i l s t i l l remained. The effective solid angles for pion detection were determined to be 1 .40, 1.47, and 1.64 msr for 50, 70, and'100 MeV pions, respectively. The absolute error of is 8.7%, whereas the relative error is 5.5%. Although estimates of the energy dependence of both the lineshape f i t parameters and the effective solid angles were obtained, the three energies investigated were insufficient to enable the exact energy dependence to be determined. Several more pion energies w i l l be investigated in the near future in order to attempt to determine such energy dependence more exactly. The only firm conclusion one can make about the energy dependence at this stage is that neither the lineshape parameters nor the effective solid angles are independent of energy. However, i t is felt that use of the energy dependence provided here are sufficient to enable interpolations of either the lineshape or effective solid angles to a precision compatible with the errors associated with experimental data. 59 BIBLIOGRAPHY 1. Browne, CP. and Buechner, W.W. 1959 Rev. Sci. Instr., 27, 899. 2. Fremont, G. 1978 "Magnetometer", model CERN 9298. 3. Hutcheon, D. 1979, private communication. 4. Jones, G. 1981 "Pion Production and Absorption in Nuclei-1981", ed. R.D. Bent, AIP Conf. Proc, 79,15. 5. Kitching, P. 1971 TRIUMF internal report TRI-71-2. 6. Kost, C. 1979, private communication. 7. Lee, D.M., Sobottka, S.E., and Thiessen, H.A. 1974 Nucl. Instr. Meth., 120, 153. 8. Livingood, J.J. 1961 "Cyclic Particle Accelerators", Van Nostrand. 9. Mathie, E.L. 1980 Ph.D. Thesis, University of British Columbia. 10. Reeve, P., Hsieh, W., and Kost, C. 1977 TRIUMF internal report TRI-DN-77-6. 11. Stinson, G. and Kitching, P. 1973 TRIUMF internal report TRI-DNA-73-4. 12. Studebaker, J.K. 1974 Los Alamos Scientific Laboratory internal report LA-5749-MS. 60 APPENDIX A - THE "RESOLUTION" SYSTEM A l l demensions are given in mm. Element Length Size Target Target Chamber Target Chamber Window Distance between Target Chamber Exit and Magnet Entrance Magnet Entrance Window Entrance Pipe -Cylindrical Part -Conical Part Magnet Exit Box Magnet Exit Window Distance between Exit Window and CE Counter CE Counter Distance between CE and MWPC1 MWPC 1 He Box Window He Box He Box Window MWPC 2 He Box Window He Box He Box Window MWPC 3 Distance between MWPC3 and C1 Cl Counter Distance between C1 and Al Sheet Al Sheet 2.23* 38x38 radius:151 .127 159 .127 25 315 240 .191 32 1 .02 1 07 77 .025 375 .025 77 .025 374 .025 77 100 6.35 222 3.18 angular range: 25°-155° radius:23.8 radius:23.8 Ent. radius:23.8 Exit radius:76.8 Gap:32 Eff. radius:650 Actual radius:618 800x220 640x203 610x102 982x406 900x400 900x400 900x400 982x406 900x400 900x400 900x400 982x406 600x200 900x400 Material CH^ Al(evacuated) Mylar Air Mylar Stainless Fe (evacuated) Stainless Fe (evacuated) Fe(evacuated) Stainless Fe (evacuated) Mylar Air NE 1 1 0 Scintillator Air see MWPC Mylar A l ( f i l l e d with He at atmospheric pressure) Mylar see MWPC Mylar A l ( f i l l e d with He at atmospheric pressure) Mylar see MWPC Air NE 1 1 0 Scintillator Air Al 61 Distance between Al Sheet and C2 C2 Counter 1 1 1 6.35 800x200 Air NE1 10 Scintillator MWPC Entrance Window Gases Exit Window Helical Cathode Wire Anode Wire .025 77 634x234 982x406 Mylar 82% Ar 17% Isobutane 1% Freon-Methylal Mylar Cu plated Al .025 634x234 diameter:.0762 spacing:1.5(pitch) diameter:.0203 spac ing:3.0 Active area of wire planes:621x201 Au plated W * The target thickness given in mm includes the due to the 45° rotation of the target. The thickness in gm/cm2 is 149. 62 APPENDIX B - MWPC PREAMPLIFIER o + l2 o OUT o - l 2 N O T E S : 1. RI4 IS USED FOR GAIN TAILORING. (NECESSARY) 2. CU IS FOR OVERSHOOT SUPPRESSION. (OPTIONAL) (Studebaker 1974) 63 APPENDIX C THE Y TARGET POSITION The calculation of the Y target position involves the calculation of three separate path lengths and three slopes. To make the calculation of the path lengths less complicated, the starting point of the calculation is the Y position and Y slope of the particle's trajectory as i t crosses a horizontal plane at a height of 650 mm (or R0) above the magnet exit. That is at where Z=231.5 mm is at 650 mm above the magnet since Z=0 is at the middle of MWPC1. Y0 and YSLOPE are determined from the particle's trajectory through the MWPC's. The calculation involves several approximations. a) Since the YSLOPE is small, the path lengths are approximated by the projection in the vertical plane. b) The effect of edge focussing on the path lengths is neglected. It was estimated that these small effects change the total length by less than one percent. c) A uniform magnetic f i e l d with "hard" edges was assumed. The three path lengths that must be calculated are: PATHEN, the length between the target and the magnet entrance. PATHMG, the length inside the magnet i t s e l f . PATHEX, the length from the magnet exit to a height R above the magnet exit. Figure 30 illustrates the geometry of the paths. First consider PATHEN, in Figure 30, ST. Looking at 40TS, one sees: YR0=Y0+YSLOPE*231.5 (ST) cosS-r R0to$y =3LR<, (1) and (2) Therefore from (2), (ST)' (3) Substituting (3) into (1): cost + Roto,T = £Ra SinS 6 4 Figure 30 - Y Target Geometry 65 therefore: f s s / t ' f a i * * ) - * ( 4 ) substituting (4) -into (3): or: ^ " JWi 05 7) = cos 8- co$[$ih'(isi»sJl) Therefore: PATHM* R0(ZtosS'-Jl- HsiSf) { 5 ) PATHMG is much simpler since i t is the arc of a ci r c l e , PATHMG=R<* Since P/R, =R/R0 where P© is the central ray momentum, Browne-Buechner theory enables R to be written in terms of Re>. Using the dispersion relation in Section 4.3, Eqns. (4-10) and (4-9), and thus, — = 227.5 / — (4-10) JL. a.**.  1 (4-9) 6 J I - X/JL*. At P0 , X=0 A = O.IR0 o ~%o ft ' y I - 3.H?>*lO'H'* XFP Therefore (7) 66 where (R/Rd) is given in terms of XFP in (6). For PATHEX, from Figure 30, therefore substituting (9) into (8), « » « - £ f c - ^ n £ < , 0 ) Now collecting ( 5 ) , ( 7 ) , and ( 1 0 ) , PATHS A/ * y? 0 (3Lt*a-J/-'/$/n*s'] ( 5 ) PAT//*** fa(/fa)oL ( 7 ) fAT/f£X ' - ( 1 0 ) the problem reduces to evaluating four angles: £ the vertial angle from the target octhe bend angle the angular position of the emerging ray at the edge of the pole face $ the angle off the vertical of the final ray. ©is the simplest to evaluate, since the trajectory through the MWPC's is known, & s t«hml(x$Lof>a) o n The other three angles, on the other hand, are more complicated. 67 Again using (12), = 0 Therefore solving for cosoc, (4) (12) (13) Starting again with dOTS, therefore using (4) f r 5 / V 7 ^ M » C ) - / Next looking at «4P0S and using the Law of Sines, 5/V»// ~ /'"fir) where (fi0)= R0 but (ft>) * - <y><;/> ^ o)*- .Z ^^$«> using (12): = + fi* - JLRR, SUxC*tf»S% or: (pot* fit+f£-HAKs'n* ( 1 4 ) Therefore using (14), (13) becomes fit*A*- HAKcS''"* -or: £o$ot~- \ - — — (15) 68 S t i l l looking at APOS, one sees Again using (12), j L - TT' 4" " If tj/»~'(Z*i»S) or: fi* V-* + (16) Now looking a t f l , one sees, Using (4) and (16), fl - f/Y'fa.'tf) - 2+TT - / zT - o * +J- S^''S)j or: t* s - S i * ' ( 1 7 ) Looking at the angles about U, i.Mv'zw-i-ifc-fj^^+t'n de) and or: &--f~Ll+ 0-71) Again using (12) and also (17), or: , <?C = X - 0 + J (19) Therefore C05<£s Co9(9-S'^) 69 or: (20) Now by equating (15) and (20), or: ffi+j\ + l*0 (21) Now since 0 and (R/R0) are known, one can solve for sin £ iteratively. Since<J<4.4° by geometry, one starts with the approximation that cos/=1 and solves the quadratic in sin<T for s i n r e . Next, let cosS=cos^0 and solve for s i n i ) . This procedure converges quite quickly with two iterations normally adequate. Once / is found, can be found by using either (19) or (15), and can be found by using (17) once is found. Therefore a l l three of the path lengths are determined. The next step involves determination of the three slopes. The Y component of the slope of the trajectory, at any point along the trajectory, is given by where Vy is the Y component of the momentum and P is the magnitude of the momentum. The primary effect of edge focussing is to alter the Y component of the slope of the particles at the magnet edges. since the magnetic f i e l d is in the Y direction, (Livingood 1961). B is the magnetic f i e l d , Y is the Y position measured from the centre of the gap, and X is the angle between the trajectory of the particle and a radial line from the centre of the magnet, see Figure 30. For PATHEX, the Py is simply the projection of the measured momentum. Therefore, m=Py /P (22) The change in P^  i s : (23) 70 Py = Pecs & * YSLOPE »(.5(>) < 24) where Pcos6» is the vertical component of the momentum (P 2), and YSLOPE*(.56) is the vertial projection of the slope of the trajector (Py/P^). The (.56) factor converts the Y position from TDC bins into mm, Section 6.1, Eqn. (6-2). Therefore from (22) and (24), the slope is Py/f>* XVOp£*co$6>*(.50 (25) Before the other two slopes can be calculated, the change in Py at the magnet edge must be determined. This requires the evaluation of the angle X from Eqn. (23) in terms of angles that are known. Examination of point U in Figure 30 shows that but from (18), LPlA-^- X-+ cj-ft (18) Thus X- - -J - 6- <fi (26) The same X describes the angle of the particle entering the magnet. This can be most easily seen by comparing the points U and S in Figure 30. Since ZPUW and ZPST are right angles, and ZPSO and ZPUO are both 7}, and since OQ and ON are straight lines, ZNUV and ZTSQ must be equal. Therefore using (26), Eqn. (23) becomes APy = .3BYi**(f-e-fi (27) where 4Py is in MeV/c, B is in T, and Y is in mm. The slope for PATHMG is then PyWP where and from (27): APy^.16* YEX* (.SL)« t«h (f - e- f) . where YEX is the position at the magnet exit defined by: 71 YBX * Y*o - $) pATH BX/(-») o r u s i n g ( 2 5 ) : Y£X - \Ro - Y$LOP£ * cas & * PAT/tEX ( 2 9 ) T h e s l o p e f o r P A T H E N i s t h e n ( P ^ + J P y M / P w h e r e A(t*.$6 + XB#*(.x)*tm*(£-a-4) <30) w h e r e Y E N i s t h e p o s i t i o n a t t h e m a g n e t e n t r a n c e d e f i n e d by: T h u s t h e Y p o s i t i o n o n t h e t a r g e t i s : T h e (.56) f a c t o r s a r i s e t o k e e p t h e u n i t s c o n s i s t e n t . 72 APPENDIX D - REVMOC6 The original REVMOC program (Kitching 1971) has been revised many times (Stinson 1973, Reeve 1977, Mathie 1980). The version used in this work was adopted from REVMOC6 (Mathie 1980). REVMOC6 has two additional subroutines compared to previous REVMOC versions. One is a two-body kinematic package and the other involves a circular pole face dipole magnet routine, BEND2. In the previous REVMOC versions the TARGET card served l i t t l e more than to define the physical size of the target, whereas in REVMOC6 i t is used to input kinematical data for the relevant two-body target reactions. The variables read in on the TARGET card are listed below: Zj- is the target thickness which must include any effective increase in thickness i f the target is t i l t e d with respect to the i n i t i a l beam. X7 is the target horizontal width. &r is the polar angle of the secondary beam with respect to the i n i t i a l beam. M, ,M^ ,My are the masses of the particles in the two-body kinematic reaction M,+M^ *M5+MV with particle 3 being traced by the Monte Carlo program. T)B is the incident particle central energy ATp is the spread of the Gaussian energy distribution. and the seventh input variable must be zero. It is used as a general flag directing the program to use the new dipole bend routine, BEND2. If the M,,M^ ,My variables are not zero a two-body reaction is assumed and t5y is randomly chosen and the momentum of Mj is calculated by: TARG Zf , Xy , M j , M^  , My, e?, where: A C ± 3L SjA^Mftf (1) where 0--73 and E, is the incident particle energy chosen from the Gaussian distribution of Hp ±ATp. The new dipole magnet routine, BEND2, is a subroutine which is called in both TEST and TRACE. TEST and TRACE are two other subroutines in REVMOC which trace the particle's trajectory through the system of elements. TEST is called f i r s t and only checks if the particle's trajectory stays inside the geometrical constraints with no nuclear scattering, coulomb multiple scattering, or decays allowed. TRACE on the other hand also traces the particle's trajectory but allows the particle to scatter and decay. BEND2 is the subroutine that simulates the passage of a charged particle through a uniform magnetic f i e l d using an effective edge model. The sign convention used refers to a positive particle deflected in the positive X direction by a negative B. The general form of the BEND2 data i s : BEND B2 * L,B,R/A^  ,R 0„ r , ,Posit, ,D,G TMIN,RHO,XL,XR,YU,YL where: B2 indicates that the new BEND2 routine must be used. * indicates that two more data lines associated with the element must be read. L is the nominal path length in meters. B is the magnetic f i e l d strength in kG. R iN and "ROUT are the radia in cm to the effective edge of the f i e l d at the entrance and exit respectively. Posit is the shift along the X axis of the magnet center in cm from the nominal beam axis (X=0). Posit>0 means the magnet center is shifted in the positive X direction. D is the length of the collimator element used in the pole face scattering calculation in cm. G is half the magnet gap width in cm. 74 The next two lines are used in the description of the collimator. The collimator has material with density RHO, dimensions (XL,XR,YU,YL), composed of elements with proportional composition P/ and atomic number Z/. TMIN is the internal step size for the collimator multiple scattering calculations. The details of the REVMOC collimator calculations are explained by Kitching, 1 971 . There is one more special feature. If R ^ <0 the multiple scattering is neglected. The reason BEND2 was added to REVMOC was to allow for the fact that the f i e l d is curved. Depending on the position of each particle at the end of the previous element, some particles might travel for a distance without any affect of the fi e l d , whereas others are affected immediately. Thus BEND2 tracks each particle from i t s position at the end of the previous element to the effective edge of the magnetic f i e l d . It then adjusts the particle's direction of motion as i t crosses the effective f i e l d edge to account for the edge focussing due to the fringe f i e l d . Then i t fina l l y traces the particle through the uniform magnetic f i e l d , with or without the scattering and decays, these latter depending on whether BEND2 was called by TEST or TRACE. Next i t adjusts the direction of motion of the particle due to edge focussing and traces i t s path to the next element. 

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