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Non-radiative positive pion absorption by deuterons at 49 mev Duesdieker, Giles Anthony 1973

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NON-RADIATIVE. POSITIVE PION ABSORPTION BY DEUTERONS AT k9 MEV by -GILES ANTHONY DUESDIEKER B.S., C a l i f o r n i a I n s t i t u t e of Technology, 19T0 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the department of Physics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1973 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t c o p y i n g or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada ABSTRACT The r e a c t i o n IfD-~~-^* was in v e s t i g a t e d at a pion lab k i n e t i c energy of U9 MeV i n a companion study to the e l a s t i c r e a c t i o n ff>>&-+7J*D at the l 8 V c y c l o t r o n of the Lawrence Berkeley Lab-oratory. Although evidence f o r d- wave absorption had been found by other experimenters at energies above 150 MeV, l i t t l e evidence f o r t h i s process was in d i c a t e d i n our analysis of t h i s experiment. A least-squares f i t of the center of mass d i f f e r e n t i a l cross sections to the standard parameterization y i e l d e d the values Y^, = 0 . 7 ^ 5 + .017 and A = 0 . 2 8 6 + . 0 0 9 , i n -d i c a t i n g a t o t a l cross s e c t i o n of 5 . 8 0 +_ .16 m i l l i b a r n s . i i TABLE OF CONTENTS Page CHAPTER I INTRODUCTION .... ' 1 CHAPTER I I EXPERIMENTAL ARRANGEMENT 5 A. The Pion Beam 5 B. The Deuterium Target • 9 C. Electronics and Hardware 12 D. Scattering Geometry l 6 CHAPTER I I I THE INCIDENT BEAM IT A. Useful Beam IT B. Calculation of E f f i c i e n c i e s 22 C. Decay Muon Contamination 23 D. The Transmission Co e f f i c i e n t . . . . . . . . . . . 26 CHAPTER IV THE SCATTERED BEAM. 28 A. Cross Sections. General Discussion..... 28 B. Angular Binning........................ 30 C. Proton I d e n t i f i c a t i o n hO D. Background Subtraction.......... ........ U8 E. Losses i n the S c i n t i l l a t o r s 55 F. Transformation to the CM 58 CHAPTER V RESULTS AND DISCUSSION...... 59 A. Results 59 B. Treatment of S t a t i s t i c a l Errors........ 6T C. Systematic Errors TO BIBLIOGRAPHY T 3 APPENDIX A - Monte Carlo Estimation of Decay Muon Contamination at the Target T^ APPENDIX B - w*0-*~f>p Kinematics T9 APPENDIX C - S o l i d Angle Calculation 82 i i i LIST OF TABLES Page I. Variation of Total S o l i d Angle with X' and Y' 31 I I . S o l i d Angles for Various Grids on the Counter 38 I I I . Best-Fit Parameterization of Data 63 IV. A Compilation of Data at Low Energies 6k V. Decay Muon Contamination at Various Target Geometries 7 8 i v LIST OF FIGURES Page 1. Maximum P a r t i c l e Energies from a # D I n i t i a l State.. 3 2. Experimental Arrangement. 6 3. Pion Production i n Polythene..., 7 1+. The Deuterium Target. 10 5. Logic Diagram of E l e c t r o n i c s . . . 13 6 . 1+9 MeV Pulse Height Spectrum .... 18 7. Spectrum of F l i g h t Time of Incident Beam 19 8 . Monte Carlo Decay Muon Spectrum 2h 9. Angular Acceptance of Stopping Counter.. 32 10. I n t e r s e c t i o n of Various Single P a r t i c l e Angular Bins with Stopping Counter Face. 35 11. D i s t r i b u t i o n of S o l i d Angles 36 12. V a r i a t i o n of S o l i d Angle with X' 39 13. Proton Separation. dE/dX vs. E P l o t . . . . . . . . . . . . . 1+1 l i t . ADC Spectra with and without Proton Flag R e s t r i c t i o n ... 1+2 15. Proton ADC C a l i b r a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . 1+6 1 6 . Locus of Proton Generation 1+9 17. Z* as a V a l i d Event Indicator 50 y Page 1 8 . Q-Spectrum of Foreground and Background 5U 1 9 . Cross Sections i n CM vs . < 0 ^ . Log Plot 60 2 0 . Cross Sections i n CM YS. ^os2Q^) 62 2 1 . Total Cross Sections vs. Pion Lab Energy 65 2 2 . Logic of Monte Carlo Estimation 77 23. 7r4D-*-f>J2 Kinematics - 80 2k. Geometry of Target and Stopping Counter 83 v i ACKNOWLEDGEMENTS I vould l i k e to thank my research supervisor, Professor Garth Jones, for the opportunity to work at the Lawrence Berkeley Laboratory, and for his help i n pointing out the many p i t f a l l s inherent i n performing scattering experiments of t h i s kind. I would also l i k e to thank Dr. C. H. Quentin Ingram for h i s constant willingness to of f e r suggestions when they were re-quested, and more importantly, when they were not. v i i 1 CHAPTER 1 INTRODUCTION The reactions Tr+D-*-pp and i t s inverse are two of the most thoroughly-studied reactions i n p a r t i c l e physics. Their docu-mentation may be a t t r i b u t e d t o t h e i r e a r l y importance i n deter-mining the spin of the p o s i t i v e pion from detailed-balance arguments. With the spin of the negative pion now well-known from pi-mesic X-ray studies, t h e i r importance has now come f u l l c i r c l e as a check on d e t a i l e d balance i t s e l f . E a r l y i n v e s t i g a t o r s parameterized the center of mass cross sections of these reactions i n the form ( i - l ) da* v , A ^ 2 d ^ = Tr + C O S 0 } which i s v a l i d i f s- and p- waves predominate the absorption. * Odd powers of cosO are absent since the angular d i s t r i b u t i o n of e i t h e r of two I d e n t i c a l p a r t i c l e s i n a two-body f i n a l state must n e c e s s a r i l y be even. In the case o f pp—<—TT+D, odd powers would be i n d i c a t i v e of a p a r i t y - v i o l a t i n g process i n which one p a r t i c l e would be emitted p r e f e r e n t i a l l y at a given center of mass angle. In 1 9 6 9 , Measday and c o l l a b o r a t o r s ^ studied TT+D—*-pp at f i v e energies i n the range 1^0-275 MeV and found evidence f o r some d- wave absorption, t h e i r parameterization being ( 1 - 2 ) da* • ^ 2 * _ h * = K (A + cos 0 - Bcos 0 ) dp* TT An e x c e l l e n t opportunity f o r improving the accuracy to which and A were known, and f o r i n v e s t i g a t i n g the p o s s i b i l i t y o f a d- wave term at low energies was afforded by the construction of an apparatus t o study the e l a s t i c r e a c t i o n TT+D —*-ir +D by the U n i v e r s i t y of B r i t i s h Columbia group at the Lawrence Berkeley Laboratory.- A l l protons from the non-radiative absorption r e a c t i o n iT +D-*-pp were present i n the background of the els r e a c t i o n , and therefore a v a i l a b l e f o r a n a l y s i s . 2 At 1+9 MeV, the av a i l a b l e f i n a l state channels f o r the i n i t i a l state are: 1 . ) pp (non-radiative absorption) 2 . ) yPP ( r a d i a t i v e absorption) 3 . ) TM?p (charge exchange absorption) h.) -rr+pn ( i n e l a s t i c s c a t t e r i n g ) 5.) IT D ( e l a s t i c s c attering) and the r e l a t i v e importance of these processes i s i n d i c a t e d by the r e s u l t s at 85 MeV of Lederman and Rogers^ , who measured the t o t a l cross sections f o r each of these reactions i n a cloud chamber experiment: REACTION CROSS SECTION (MB) , Tr +D^.pp 7 Tf +D—vyPP • 1 + . 0 TT D—>"ir pp 1 2 +^  + IT D—*-TT pn 2 1 u +D—*-rr +D 1 7 Figure 1 i l l u s t r a t e s the maximum k i n e t i c energy of some p a r t i c l e s a r i s i n g from these r e a c t i o n s . The d i s t i n g u i s h i n g c h a r a c t e r i s t i c of protons from the absorption re a c t i o n i s obviously t h e i r r e l -a t i v e l y high k i n e t i c energy, the r e s u l t of the conversion i n t o energy of the IkO MeV r e s t mass of the inc i d e n t pion. Although i t i s kin e m a t i c a l l y p o s s i b l e that protons from r a d i a t i v e absorption can compare i n energy with protons from TT+D—-vpp i f the gamma i s emitted with low energy, t h i s case i s s t a t i s t i c a l l y u n l i k e l y i n a three-body f i n a l s t a t e . As a matter of f a c t , a study by Haddock J of r a d i a t i v e pion capture i n deuterium ( 1 - 3 ) TT D->>-YPN 3 120 100 Lab Angle(degrees) FIGURE 1. Maximum P a r t i c l e Energies from a I n i t i a l State TJJ. ar lj-9 MeV The symbols PJCP-^.-.PJJ) associated with each curve represent the f i n a l state p a r t i c l e Pj and the f i n a l state c o n f i g u r a t i o n (P2...PJJ) , r e s p e c t i v e l y -has shown that the gamma spectrum i s strongly peaked at the upper kinematic l i m i t rather than the lower. One would therefore expect gammas a r i s i n g from ( I - U ) TT+D -> p p y to have a s i m i l a r spectrum under charge independence, with an accompanying low-energy proton spectrum. This assumption, i n conjunction with the r e l a t i v e l y small size of the radiative pion absorption cross section, would seem to preclude t h i s reaction as a source of any sizeable background. CHAPTER I I EXPERIMENTAL ARRANGEMENT A. The Pion Beam. A layout of the pion beam and scattering apparatus i s given i n Figure 2. F u l l d e t a i l s of the experimental arrangement are given by Westlund'1 , and ve s h a l l present only the most sa l i e n t features. Pions were produced by bombarding a polythene target with the extracted beam of the LBL l 8 V cyclotron. The proton beam had a mean energy of 730 MeV with a f u l l width at half maximum of 28 MeV. The beam was extracted at beam s p i l l s varying from 5 to 10 milliseconds. The microstructure of the beam consisted of 10 nanosecond pulses spaced by 50 nanosecond i n t e r v a l s , so that the o v e r a l l duty cycle varied correspondingly from 5 to 10 per cent. Since polythene'is a hydrocarbon, pions are generated i n the production target by the reactions C11-1) f>+f> —^ vr^D . (11-2) f>+*C—>7Tf«C CII-3) ^ , 2 c — > a C These pions are momentum-selected by the f i r s t bending magnet HI, which c o l l e c t s them at 129 degrees, minimizing the beam con-tamination from forward-scattered protons. Varying the p o l a r i t i e s and magnitudes of the currents i n the magnet system allowed a determination of the possible fluxes of both p o s i t i v e and nega-t i v e pions which could be achieved i n the target region. The res u l t s of such a survey are shown i n Figure 3:. The v e r t i c a l l i n e through the f i r s t maximum i n the positive pion spectrum denotes the f l u x corresponding to the magnet settings used i n the actual scattering experiment. Since no corresponding maximum at t h i s mo-mentum occurs i n the negative pion spectrum, we i n f e r that most of these pions are generated by reaction ( i l - l ) . I t i s int e r e s t i n g to note that t h i s i n fact i s the inverse of the pion absorption reaction to be studied. PROTON BEAM i HI 4 f 50 PRODUCTION TARGET FIGURE 2. Experimental Arrangement (Not to Scale) FIGURE 5. PION PRODUCTION FROM POLYTHENE J ! I L 7T 7T (X|0) J 1 1 1 I J _JL \ \ 100 200 300 MOMENTUM (MeV/c) 8 With, t h i s arrangement, t o t a l pion fluxes on the order of 2 x luVsec were obtained. The average energy of these pions was determined by range measurements i n copper, and y i e l d e d the value 1+9 MeV, using the TRIUMF range-energy table s . We ascribe an e r r o r of ±2% to the c a l c u l a t i o n using t h i s method. The energy spread of t h i s pion beam was of the order of 7 MeV FWHM as c a l c u l a t e d by the o p t i c s system. The p i o n beam was d i r e c t e d toward the target i n an evac-uated channel of 11 meters l e n g t h by the magnet system H1-Q5. D e t a i l s of the pion channel are presented by Reeve , but es-s e n t i a l l y the magnets form an achromatic focussing system with a d i s p e r s i v e midplane focus. Since a knowledge of the pions' p o s i t i o n at midplane i s necessary f o r determination of t h e i r momenta, the hodoscope H c o n s i s t i n g of 12 NE102 f i n g e r counters was p l a c e d here. Each counter was 12 mm wide and a l l elements except the outermost overlapped t h e i r nearest neighbors by 1 mm. Each element acted as a v i r t u a l source of monochromatic pions of d i f f e r e n t momenta.- By d e f i n i t i o n , an achromatic f o c u s s i n g system would focus the image of a point production t a r g e t isfto a point at the target plane. In f a c t , f i v e hodoscope elements i n c l u d i n g the outermost ones had pion peaks centered w i t h i n 0 . 7 1 cm of each other at the target plane. This i n -d i c a t e d that the channel i s achromatic to within ± 0 . 3 6 cm, which, corresponds to a momentum spread of ±0.l6% of the nominal i n c i d e n t momentum^. B. The Deuterium Target. The target f l a s k (Figure h) was of LBL design and rested i n a s t e e l vacuum chamber equipped with p e r i p h e r a l Mylar windows. Deuterium was introduced i n t o the c e n t r a l chamber, of 2 cm x 10. cm x 20 cm dimensions. As the l a t e r a l walls of the deuterium chamber were only of 0.002" thickness, precautions had t o be taken to insure against t h e i r outward bulging. This was prevented by surrounding them with a gas b a l l a s t region. L i q u i d b o i l o f f escaping from the top of the target was l e d i n t o t h i s region, thereby minimizing the transmural pressure gradient. In t h i s manner, the l a t e r a l walls of the target were maintained i n a p a r a l l e l o r i e n t a t i o n and the target thickness was constant independent of p o s i t i o n . Excessive b o i l o f f was prevented by surrounding the gas b a l l a s t region with several l a y e r s of aluminized Mylar, which acted as. a superinsulating device. Deuterium was supplied from c y l i n d e r s of the com-pressed gas used i n the deuterium bubble chamber at the Lawrence Berkeley Laboratory Bevatron. The concentration of deuterium atom i n the c y l i n d e r s was assessed at 98.9% by spectrometric a n a l y s i s , the remaining constituent being a 2.2% concentration of gaseous HD. This mixture was l i q u e f i e d by a Model 1023 hydrogen condenser supplied by Cryogenic Technology. Gas entered a vacuum-jacketed feedthrough i n the top r e f r i g e r a t o r flange at ambient temperature and a s l i g h t l y p o s i t i v e pressure, a f t e r which o i t was precooled to about 77 K i n the f i r s t - s t a g e r e -f r i g e r a t o r c o i l . Subsequently, i t passed to the second-stage r e f r i g e r a t o r , where i t was f u r t h e r cooled to 20 K. At t h i s stage, i t l i q u e f i e d on the condensing surface of a l i q u i d r e s e r v o i r , and flowed by g r a v i t y to the base of the target flask„ B o i l o f f returned v i a the gas b a l l a s t 10 0 . 0 2 " Mylar Vacuum Window-S t e e l -£73 mm •32k mm 5 5 0 mm Figure Ij. The Deuterium Target. H o r i z o n t a l Cross Section 11 region t o the l i q u i d r e s e r v o i r . The rate of f i l l i n g could be monitored v i s u a l l y through a 1 cm v e r t i c a l s l i t cut i n the f r o n t and rear layers of the s u p e r i n s u l a t i o n , and when approximately 18 cubic f e e t of deuterium had entered the system, the target was f u l l and ready f o r operation. 12 C. Electronics and Hardware. Useful incident beam i s defined by NE102 s c i n t i l l a t o r s B l and B2 (Figure 5), each of 0.2 x 10 cm x 15 cm dimensions. Pulses from each are amplified by XPIO^O phototubes, passed through discriminators, and led to the coincidence unit B1.B2, which has a resolution time of 5.5 nanoseconds. Each signal i s delayed so that v i r t u a l l y only pions w i l l r e g i s t e r a B1.B2 coincidence. Any structure i n the t o t a l spectrum of events s a t i s f y i n g t h i s coincidence may be seen using the time encoder, which acts as a check on the time of f l i g h t of the incident p a r t i c l e s . ' Scattered beam i s s i m i l a r l y defined by a r e g i s t r a t i o n of a B3.BU.C1 coincidence. B3 and Bh are e s s e n t i a l l y i d e n t i c a l to Bl'and.B2, while CI i s a 5" diameter NE110 stopping counter of 12" length. The combined function of these three counters i s to act as a scattered p a r t i c l e telescope which defines a region of acceptance for the scattered beam. A scattered "event" was defined by the r e g i s t r a t i o n of a (B1.B2).(B3.BU.Cl) coincidence i n the EVENT coincidence u n i t , which had a resolution time of 15 nanoseconds. This coincidence triggered the spark chambers and generated an interrupt signal to the NOVA 1200 computer. I f the com-puter were not busy, i t then read the contents of the scalers of a CAMAC data a c q u i s i t i o n system1 , which contained information concerning: 1. ) The time of f l i g h t of the incident p a r t i c l e from B l to B2 2. ) The energy deposited i n Bh and CI by the scattered p a r t i c l e 3. ) The'locations of the sparks generated i n Sl-U h.) Various coincidence rates Bk, being a t h i n counter, gave l i g h t outputs propor-t i o n a l to the rate of energy l o s s of the scattered p a r t i c l e . CI, being a stopping counter, measured i t s f u l l energy. The charge pulses from the phototubes connected to these s c i n t i l l a t o r s were integrated for a period of l60 nano-seconds, and the r e s u l t s were encoded by two analogue-t o - d i g i t a l converters (ADC's). These numbers were then placed i n the CAMAC s c a l e r s . The u n i t s Sl-U were magnetostrictive wire spark chamber of 7" by 7" dimensions. The wire diameters i n the chambers were 0.0125 cm, with wire spacing being 0.05 cm. One nylon wire of 0.0125 cm diameter was placed between each p a i r of copper wires. Triggering,each spark chamber started two 20 MHz o s c i l l a t o r s , one of which was stopped with the a r r i v a l of the s i g n a l from the spark, and the other of which was stopped with the a r r i v a l of a f i d u c i a l s i g n a l generated at the f a r end of the chamber. The lengths of these two pulse t r a i n s were d i g i t i z e d i n the routing u n i t s , and subsequently recorded i n the CAMAC s c a l e r s . Incident and e x i t t r a j e c t o r i e s of incident and scattered p a r t i c l e s could then be determined from the ca l c u l a t e d p o s i t i o n s of the sparks. Angular r e s o l u t i o n of each p a i r of chambers was better than one degree, and p o s i t i o n at the target plane could be found to wi t h i n one centimeter. Each r e g i s t r a t i o n of a u s e f u l coincidence caused the appropriate CAMAC scaler to be incremented by one. Co-incidence rates could then be determined by d i v i d i n g the t o t a l number of coincidences by the elapsed time i n d i c a t e d by the number of pulses emitted by the O.k Hz clock. T y p i c a l rates of the .various coincidences were of the order: 1.) B1.B2 loVsec 2.) B3.B1+.C1 10/sec 3.) EVENT 0.25/sec <h.) B l lO^/sec 5.) B2 10 /sec 15 In addition t o these rates, one could also estimate the number of random coincidences caused "between p a r t i c l e s generated on succesive spikes i n the microstructure of the proton beam. This was accomplished by introducing a 50 nanosecond delay into one input of a properly-timed co-incidence. Typical randoms rates measured i n t h i s fashion are: 1. ) (EL.B2L, 100/sec t\ 2. ) (EL.B2).(B3.BU.C1)_ 005/sec K Analysis of a l l tape-recorded information was performed on the LBL CDC 7600 computer. 16 D. Scattering Geometry. Data for the scattering experiments were taken at nominal scattering angles of 30 to 150 degree increments. In an attempt to minimize the v a r i a t i o n of the energy losses of p a r t i c l e s i n the target, the target was rotated by an angle » nominal scattering angles 0 i n the range 30 - 105 degrees. At larger values of U , a target angle of would imply that scattered p a r t i c l e s would have to t r a v e l inordinately large distances i n the target material and a "brick w a l l " geometry was chosen i n which 9^  = In t h i s configuration, the incident angle of the pion and the e x i t angle of the scattered p a r t i c l e are approximately equal, and losses due to long f l i g h t paths i n the target are avoided. In the "brick w a l l " geometry, the incident spark chambers interfered with the placement of the e x i t spark chambers, and the l a t t e r were retracted from t h e i r usual p o s i t i o n . The angular acceptance of the stopping counter for large-angle scattering was therefore smaller than for small-angle scattering. A l i s t of target angles and component distances with respect to the target, c o l l e c t i v e l y known as the "scattering geometry" for the p a r t i c u l a r nominal scattering angle 9 i s presented below. 8(degrees) (degrees) d (ram) dS2(mm) dg3(mm) dsl|(mm) d c (mm) 30 . 1 8 . 8 . -1+16 -3ll+ 197 299 33l+ k5 2 7 . 8 -1+16 -311+ 197 299 331+ 6 0 3 6 . 6 -l+l6 -3ll+ 197 299 331+ 75 . 1+5.0 -1+16 . -311+ 197 299 331+ 90 5 2 . 5 -1+16 -311+ 197 299 331+ 105 5 9 . 8 -1+16 -311+ 197 299 331+ 1 2 0 -30 .0 - 6 8 0 - 5 7 8 197 299 331+ 135 - 2 2 . 5 . - 6 8 0 - 5 7 8 ; 297 399 1+31+ 150 - 1 5 . 0 - 1 0 9 2 - 8 9 0 1+51 553 588 IT CHAPTER I I I • THE INCIDENT BEAM A. U s e f u l Beam. An a l y s i s of the in c i d e n t beam i s done with two purposes i n mind. F i r s t , one must r e j e c t those incident p a r t i c l e s about which i n s u f f i c i e n t information i s given f o r a proper a n a l y s i s . Second, one must d i s c a r d from the remaining p a r t i c l e s those which f a i l t o s a t i s f y c e r t a i n c r i t e r i a demonstrating that the p a r t i c l e i n question i s a v a l i d p r o j e c t i l e . From the most b a s i c standpoint, the c r i t e r i a f o r d e f i n i n g a v a l i d i n c i d e n t event are: 1. ) The in c i d e n t p a r t i c l e i s a pion 2. ) The pion can s t r i k e the deuterium t a r g e t 3 . ) The energy of the pion i s known The i n c i d e n t beam i s composed p r i m a r i l y of electrons , protons, muons, and pions. Since the times of f l i g h t of ele c t r o n s and protons are s u f f i c i e n t l y d i f f e r e n t from those of p l o h s , they are e a s i l y discriminated against by the 5.5 nanosecond r e s o l u t i o n time o f the B1.B2 coincidence u n i t . Muons i n the beam are generated from pion decay p r i o r t o or downstream from HI, the f i r s t bending magnet. In the f i r s t case, they, are momentum-selected by the magnet system and f a l l i n t o a peak i n the energy spectrum (Figure 6). The r e s o l u t i o n time o f B1.B2 i s s u f f i c i e n t to discriminate against these momentum-selected muons, although Figure T shows a s l i g h t overlap i n the timing, so that a small f r a c t i o n of them may i n f a c t r e g i s t e r a coincidence. In the second case, the unions possess a broad energy spectrum. These "decay muons" can a r i s e upstream from B2, making time of f l i g h t d i s c r i m i n a t i o n u n l i k e l y , or they may be generated FIGLRE 7 1500 1000 500 MOMENTUM SELECTED MUONS hs-1 SPECTRUM OF FLIGHT TIME OF INCIDENT BEAM 1.5' n. s. PIONS DECAY • MUONS H 0 + 5 F L I G H T T I M E ( n s , a r b i t r a r y o f f s e t ) 20 downstream from B2, rendering i t completely impossible. The e l i m i n a t i o n o f r e s i d u a l decay muons w i l l be considered i n greater d e t a i l i n section III-C. At t h i s l e v e l , we consider the f i r s t c r i t e r i o n t o be s a t i s f i e d i f an incident p a r t i c l e r e g i s t e r s a B1.B2 coincidence. The second c r i t e r i o n i s s a t i s f i e d using the information given by the spark chambers. The two points generated by the chambers SI and S2 define the incident pion's l i n e a r t r a j e c t o r y , which can be extrapolated to the target plane. I f the i n t e r c e p t of t h i s l i n e and the target plane i s a point (X',Y') such th a t -80mm € X' * 80mm and -35mm ^ Y'- 55mm, the pion w i l l enter the target i f i t does not s c a t t e r from the Mylar vacuum window. The region of confidence i s defined asymmetrically as the target was placed 10 mm below Y' * 0 i n spark chamber space, which was the n a t u r a l choice of a r e -ference coordinate system. The t h i r d c r i t e r i o n i s necessary f o r an analysis of the center of mass kinematics as discussed i n Chapter IV. The beam o p t i c s give the momentum of an incoming pion as ( I H - I ) where h, ^ Nominal pion incident momentum(l27MeV/c) \l-h0 = Hodoscope element f i r e d minus c e n t r a l hodoscope element H o r i z o n t a l displacement at target plane and V^ and V^ are constants c h a r a c t e r i s t i c of the magnet system. 21 Su f f i c i e n t information about an incident p a r t i c l e i s therefore given only i f i t registers a B1.B2 coincidence, properly triggers the incident spark chambers, and gives a good si g n a l i n the hodoscope. One may formally describe the t o t a l number of incident beam p a r t i c l e s which s a t i s f y these three c r i t e r i a , hereafter referred to as the "useful beam" as ( i n - 2 ) Na= 9132 € HbV 66x1 where J?/£?-Number of B1.B2 coincidences - Hodoscope e f f i c i e n c y 22 B. C a l c u l a t i o n of E f f i c i e n c i e s . One must define the e f f i c i e n c i e s of the preceding section i n a consistent manner. The e s s e n t i a l c r i t e r i o n i s that a u s e f u l p a r t i c l e must pass a l l t e s t s subjected t o i t , so that candidates f o r u s e f u l beam are eliminated s e q u e n t i a l l y i n passing through the apparatus. A l o g i c a l d e f i n i t i o n would be (III-3) e H Q D = n(H n BlB2)/n(BlB2) where n(H/3 B1B2) s i g n i f i e s that number of p a r t i c l e s i n the f u l l i n c i d e n t beam r e g i s t e r i n g a B1.B2 coincidence and (0 ) g i v i n g a proper s i g n a l i n the hodoscope H." S i m i l a r l y , the i n c i d e n t spark chamber e f f i c i e n c y would be defined as ( I I I - M = n(Boxir) H H BlB2)/n(HO B1B2) I t should be mentioned at t h i s point that the approach chosen i s somewhat s i m p l i s t i c i n that a l l e f f i c i e n c i e s are dependent upon p a r t i c l e type, energy, and p o s i t i o n i n the apparatus under consideration. These complications s h a l l be ignored i n the a n a l y s i s of the incident beam. The s a l i e n t p o i n t i s t h a t the e f f i c i e n c y of any module R must be defined only f o r those p a r t i c l e s which have properly r e g i s t e r e d a l l modules p r i o r to i t . I f , f o r example, a module R were placed downstream from Box 1, i t s e f f i c i e n c y would be defined as . _ n(Rrt EH Boxl n B1B2) eR ~ n(Boxl n H ^B1B2) With these ideas i n mind, the e f f i c i e n c i e s i n (111-2) can be measured and u s e f u l i n c i d e n t beam defined from them. 23 C. Decay Muon Contamination. In keeping with the formalism of the preceding s e c t i o n , one would i d e a l l y l i k e to measure a beam p u r i t y Pp such that ( I I I 5) P = n ( n o feisty n Boxl (1 H 0 B1B2) n(Boxl O H P B1B2) This number can be estimated quite w e l l using the form of . the laboratory angular d i s t r i b u t i o n of the decay muons, which i s known t o be markedly peaked at the maximum decay angle . . MS* (Figure 8) I t can be shown that 9^ has the value (III-6) £"* = Co* 1 M^fAprj} — l 8 . 0 8 degrees f o r a ^9 MeV pion By assuming that every pion t r a v e l s the c e n t r a l incident beam a x i s , and that every decaying pion emits a muon at ofa , one a r r i v e s at a lower bound f o r the beam contamination C^, where C^ - 1 - P. . (m-7) H»{I-«I,[-2&?]}=</.DZ-and ~yo = decay length of a k9 MeV pion =. T O 8 . 6 cm jr = z average radius of target normal to the beam — 8.8 cm The value quoted i s f o r 30 degree geometry, and demonstrates that the decay muon contr i b u t i o n to the in c i d e n t beam i s quite small with respect t o the pion component. 2k 15000 10000 4-EVENTS 5000 4-FIGURE %. MONTE CARLO DECAY MUON SPECTRUM 39773 EVENTS T « k 9 MeV 9. max _• V 1&08 DEGREES DECAY ANGLE IN LABORATORY (DEGREES] 25 A more exact estimation of the contamination using a Monte Carlo technique appears i n Appendix A. The contamination c a l c u l a t e d there i s where the notation P(A | B) s i g n i f i e s the c o n d i t i o n a l p r o b a b i l i t y of an event A occurring given that the event B occurs. The c a l c u l a t i o n assumes a 5 " diameter homogeneously d i s t r i b u t e d pure pion beam s t a r t i n g at the downstream edge of H 2 , the second bending magnet. Each p a r t i c l e i n the beam i s assumed to t r a v e l p a r a l l e l to the beam ax i s , and have an energy o f 1+9 MeV. As the contamination c a l c u l a t e d by t h i s method was r e l a t i v e l y i n s e n s i t i v e to the beam diameter chosen^ no f o l d i n g c o r r e c t i o n f o r the s p a t i a l d i s t r i b u t i o n of the incident beam was introduced. Although the experimental data i t s e l f could have been used to i n f e r the target decay muon contamination, no con-ceivable method'entailed making none of the p r o b a b i l i s t i c assumptions c h a r a c t e r i s t i c of Monte Carlo methods themselves. In view of the f a c t that the contamination was small and d i d not have t o be c a l c u l a t e d t o any high degree of accuracy, no obvious b e n e f i t was seen i n using the data i t s e l f t o c a l c u l a t e the contamination and the Monte Carlo value was s u f f i c i e n t f o r our purposes. Having now defined P^ , , we may write the number of pions i n the u s e f u l incident beam as U Q* Pfcr Jec*3 0 82 0 Tarjeh JHpBlO Box fj 26 D. The Transmission C o e f f i c i e n t . One has now a r r i v e d at an estimation of the number of pions about which s u f f i c i e n t information has been given f o r a proper a n a l y s i s . I t remains to determine how many of these w i l l enter the ta r g e t . One may formally define the f r a c t i o n entering'the region of confidence i n the target as the transmission c o e f f i c i e n t K , where one i s p r i m a r i l y i n t e r e s t e d i n the number of p a r t i c l e s entering the t a r g e t i n a s c a t t e r i n run, which records only those events s a t i s f y i n g a (B1.B2).(B3.BU.Cl) coincidence. For t h i s reason i t i s not appropriate to extrapolate the t r a j e c t o r y of the i n c i d e n t pions recorded by the computer to the target plane and c a l c u l a t e the f r a c t i o n entering the region of confidence. Obviously, more p a r t i c l e s w i l l reach CI from the proximal side of the t a r g e t than the d i s t a l , and a homogeneous f l u x w i l l appear t o be weighted toward +X'. To derive a true p i c t u r e of the inc i d e n t f l u x , two s a t e l l i t e experiments were performed p r i o r to and fo l l o w i n g the s c a t t e r i n g runs. In these experiments, known as "0 degree runs as the stopping counter was rotated to a mean s c a t t e r i n g angle o f 0 degrees, the computer recorded events s a t i s f y i n g a B1.B2 coincidence only. In t h i s fashion, a sample d i s t r i b u t i o n o f the inc i d e n t f l u x before and a f t e r the s c a t t e r i n g run was obtained. The values of K were . • T de r i v e d f o r each s a t e l l i t e run, and a mean value was assumed to apply to the s c a t t e r i n g run i f the values were consistent with each other. (111-10) K. I t must be emphasized that i n measuring a cross section 27 With K^ , defined, the number of useful pion p r o j e c t i l e s i s then expressed as (iii-iD = em5eH%PT K° vhere "s" and "0" s i g n i f y numbers derived from the scattering and 0 degree runs, respectively. 2 8 CHAPTER IV THE SCATTERED BEAM A. Cross Sections. General Discussion. Before proceeding t o a discussion of the analysis of the absorption associated with the rea c t i o n i n question, i t would be worthwhile f o r the sake of c l a r i t y t o review b r i e f l y the concept of the d i f f e r e n t i a l cross s e c t i o n . We consider the d i f f e r e n t i a l cross section t o be defined as the e f f e c t i v e area per unit target nucleus normal to the d i r e c t i o n of an impinging p r o j e c t i l e , i n which a nuclear reaction w i l l take place with one p a r t i c u l a r reaction product being emitted i n t o the s o l i d angle dfi . Let ATr/V p r o j e c t i l e s be introduced-homogeneously over a target m a t e r i a l of normal area a^- a rg densityJ> , atomic weight A and thickness t . I f N puT r e a c t i o n products t a r g , are detected i n a s o l i d angle J\fl at a mean angle Q , one may writ e SCFfT r 7 ' ' ( i v - i ) PlScaff.riM^eJ^ 7^7 = a TffRG where #j M r=:The t o t a l e f f e c t i v e area presented f o r s c a t t e r i n g i n t o ASl , 0 I f HNU[, are the t o t a l number of n u c l e i i n the target,- then and A/o ~ Avogadro's number Combining r e s u l t s , • 29 I t would be advantageous f o r our purposes t o define the cross s e c t i o n f o r TT+D —*-pp i n exactly the same fashion as the + + cross s e c t i o n f o r TT D—t>n D. The fundamental d i f f e r e n c e between these two reactions i s that the f i n a l state p a r t i c l e s - from the l a t t e r are d i s t i n g u i s h a b l e , whereas i n the former they are not. It i s therefore d e s i r a b l e t o t r e a t the protons i n the former case as d i s t i n g u i s h a b l e i n some sense. Consider a Gedankenexperiment i n which one proton i s "tagged" as i t sc a t t e r s from the deuterium, and that one has a counter which responds only t o "tagged" protons. This would be i n exact analogy with the e l a s t i c s c a t t e r i n g case i n which one could d i s t i n g u i s h between deuterons ( i f they i n f a c t escaped from the t a r g e t ) and pions by t h e i r pulse heights i n the stopping counter. The a c t u a l apparatus can make no d i s t i n c t i o n between the two protons, of course. I f the "tagged" proton sca t t e r s at 0^, i t i s counted, and the other proton s c a t t e r s at the angleO^. But i f the "tagged" proton s c a t t e r s atG^, the "untagged" proton w i l l s c a t t e r at 0^ and be counted, which i s p r e c i s e l y what one wishes t o avoid. One can formally circumvent t h i s by n o t i c i n g that tagged protons s c a t t e r at 0^ j u s t as often as untagged ones, so that one can simply d i v i d e the t o t a l number of protons by two t o e l i -minate the "double counting" e f f e c t . (IV-3) <to_ _ 1 JD2 CUT dfi 2 ptN An N_„S 30 B. Angular Binning. Formula (IV-3) i s s t i l l not an appropriate d e s c r i p t i o n of absorption i n the apparatus f o r a number of reasons. F i r s t , the target i s l a r g e r than the e f f e c t i v e beam size and therefore not uniformly i l l u m i n a t e d . A second problem i s that the s o l i d angle of the stopping counter i s strongly dependent on the p o s i t i o n i n the target from which i t i s viewed (Table l ) . T h i r d , each incident p a r t i c l e sees a d i f f e r e n t t arget thickness due t o a 6 degree angular divergence of the incident beam. Aside from introducing un c e r t a i n t i e s as to the values of t,N J N , and ASl which should be placed i n equation ( l V - 3 ) , the large target s i z e and beam divergence act i n conjunction with the large s i z e of the stopping counter face to y i e l d a t o t a l angular acceptance of about 50 degrees (Figure 9). This i s c l e a r l y inappropriate f o r d e f i n i n g the cross section as a function of angle, and i s a strong argument f o r the formulation of a j u d i c i o u s method of angular binning. The method * chosen was to s e l e c t a random sample o f incident pion events from the 0 degree run. This sample represented the i n c i d e n t f l u x f o r the corresponding s c a t t e r i n g run. The s o l i d angle subtended by the stopping counter f o r each of a number of angular bins was c a l c u l a t e d f o r each pion i n the sample. These s o l i d angles were a function of each p a r t i c l e ' s i n c i d e n t X' amd Y' coordinates i n the t a r g e t , and i t s laboratory d i r e c t i o n cosines. By d i v i d i n g the sum of the s o l i d angles i n a p a r t i c u l a r angular b i n by the t o t a l number of p a r t i c l e s i n the sample, an average s o l i d angle f o r each angular b i n was obtained. This average accounted f o r inhomogeneities i n both s p a t i a l and angular d i s t r i b u t i o n s of the i n c i d e n t f l u x , 31 TABLE 1 V a r i a t i o n of T o t a l S o l i d Angle with X and Y. 30 Degree Geometry H. (ma) X (mm) 0 ±20 — ± 3 0 =£4o 100 .04382 .04359 .04290 .04178 80 .04495 .04471 .04399 .04283 60 .04560 .04535 .04462 .04345 40 .04575 .04550 .04477 .04360 20 .04540 .04516 .04444 .04329 0 .04459 .04435 .04367 .04256 -20 .04337 .04315 .04249 .04144 -40 .04180 .04159 .04098 • .03999 -60 .03995 .03976 .03920 .03820 -80 .03791 .03774 .03723 .03640 -100 .03575 .03559 .93513 .03438 Values i n table are given i n steradians. Target Angle: 18.8 Degrees Stopping Counter Distance: 334 mm. Stopping Counter Radius: 40 mm. 32 200 + F I G U R E 9 . ANGULAR A C C E P T A N C E OF S T O P P I N G COUNTER 90 D E G R E E GEOMETRY" 150 -F UJ UJ 100 4 50 1159 E V E N T S PROTONS ONLY i t 8 ^ 9 ^ ' Too iii" SCATTERING A N G L E ( D E G R E E S ) 33 S p e c i f i c a l l y , i f a pion enters the target at X', Y' with d i r e c t i o n cosines <X , then the p r o b a b i l i t y f o r s c a t t e r i n g i n t o an angular b i n k i s r 7P 2 L JK The 0 degree run cannot p r e d i c t the Z' coordinate of any p o s s i b l e s c a t t e r i n g , so an estimate of Z' must be made. It i s expected that the absorption of pions w i l l be uniformly d i s t r i b u t e d along the thickness of the t a r g e t , so that a l o g i c a l approximation would be to assign Z' = 0, the mean absorption coordinate, t o every s o l i d angle c a l c u l a t i o n . In t h i s approximation, the expected maximum erro r i n c a l c u l a t i n g any one s o l i d angle w i l l be of the order (IV-5) where 2lM*l R R = AZ * = Mean distance to stopping counter 33k mm Largest p o s s i b l e deviation of the Z' coordinate from the mean f o r a v a l i d absorption event 10 mm I f N s o l i d angle c a l c u l a t i o n s are performed, the average e r r o r i n estimating the mean f o r a uniform d i s t r i b u t i o n of Z' between ±10.mm. Since U00 s o l i d angles were c a l c u l a t e d f o r each angular b i n , the expected e r r o r In the mean s o l i d angle was approximately 0.2%. Therefore no s i g n i f i c a n t e r r o r was incorporated by assuming Z 1 = 0 f o r a l l absorption events. 3k I f a large number of p a r t i c l e s N enters the t a r g e t , then one expects that A/°"r w i l l be absorbed and N^T protons w i l l s c a t t e r i n t o the angular b i n k, where I and ^ l a b e l s the ^ 7 t h pion t o enter the t a r g e t . This expression may be rewritten i n terms of an average over the in c i d e n t f l u x : The average was estimated by histogramming the values of the s o l i d angle times the apparent target thickness corresponding to each member of the group of 400 incident pions s e l e c t e d from the 0 degree run; <^ZASl)K immediately followed. The advantages of t h i s method are that ^&7*&fl}g i s normalized t o both the s p a t i a l and angular d i s t r i b u t i o n of the i n c i d e n t beam, and the s c a t t e r i n g angle may be r e s t r i c t e d t o an a r b i t r a r i l y s m a l l value ( a l b e i t with l o s s of s t a t i s t i c a l accuracy) f o r improving the angular d e f i n i t i o n of the cross s e c t i o n . The a c t u a l evaluation of s o l i d angles subject to the con-s t r a i n t of angular binning e n t a i l e d d e f i n i n g an area of the stopping counter face i n t o which a given p a r t i c l e could s c a t t e r w i t h i n the l i m i t s of the angular b i n . The l o c i of these p o i n t s assumed four b a s i c forms, examples of which are i l l u s t r a t e d i n Figure 10. A great many in c i d e n t p a r t i c l e s cannot see the stopping counter at a l l i n c e r t a i n b i n s , which leads t o a pathologically-shaped d i s t r i b u t i o n function f o r the s o l i d angle (Figure l l ) . In c a l c u l a t i n g the mean, these zeroes must be included. 35 F I G U R E 10. I N T E R S E C T I O N OF V A R I O U S S I N G L E P A R T I C L E A N G U L A R Bl N S W I T H S T O P P I N G C O U N T E R F A C E . 97. 5*- 102-5° Incident p a r t i c l e s enter target at X'=r 0, Y'= 0 w i t h di rect io n cos ines 87.5- 92.5° 97.5* 82.5'-o7. 5° 36 0 5 10 15 20 SOLID ANGLE (MILLISTERADIANS) Figure 1 1 ' . D i s t r i b u t i o n of S o l i d Angles 9 0 Degree Geometry . Angular Bin: 8 7 . 5 - 9 2 . 5 Degrees 3T The solid, angle involved a double i n t e g r a l (Appendix C) which was evaluated numerically by d i v i d i n g the counter face i n t o many wedge-shaped sections. Due to the l i m i t a t i o n of computer time, a compromise between the number of these c a l c u l a t i o n s which could be performed and the accuracy to which each could be ca l c u l a t e d had to be made. Table 2 shows the values o f various s o l i d angles, c a l c u l a t e d with various numbers of angular and r a d i a l d i v i s i o n s of the counter face. With respect to the 100 x 100 g r i d , the 25 x 25 g r i d i s about 2% accurate f o r any one s o l i d angle c a l c u l a t i o n , and i t was t h i s g r i d which was used i n each 0 degree run. Since the mean of the d i s t r i b u t i o n follows a Gaussian d i s t r i b u t i o n f o r large values of N, we may estimate the erro r i n the average s o l i d angle as ' A (Hi) Q± 6~(A<(l) l (IV-7) Ja ~~ ~MT YN^T ~ 5<7° Although the s o l i d angles c a l c u l a t e d were r e l a t i v e l y i n s e n s i t i v e to Y', they were extremely s e n s i t i v e to X 1, as Figure 12 c l e a r l y demonstrates. The r a p i d r i s e and f a l l of the s o l i d angle corresponds to the approach and departure of the center of the angular b i n annulus from the center of the counter face as the p a r t i c l e crosses the surface of the target from l e f t t o r i g h t . 38 TABLE 2 S o l i d Angles f o r Various Grids on Counter Face G r i d lQxlO/Error 2 5 x 2 5/Error 5 0 x 5 0/Error 100xl00/Erro] • Angular Bin 7 7 . 5 - 8 2 . 5 0.0 0.0 0..0 0.0 0.0$ 0.0$ o.'o$ 0.0$ 79-5-84 .5 3 . 1 8 8 2 . 2 2 3 2 . 2 7 6 2 . 2 3 1 4 2 . 9 / 0 -0 . 4 $ 2 . 8 $ 0.0$ 8 1 . 5 - 8 6 . 5 8 . 5 5 ^ 8 . 1 5 0 8 . 4 1 3 8.410 1 . 7 $ 3.1$ 0.1$ 0.0$ 8 3 . 5 - 8 8 . 5 16 . 0 3 1 6 . 1 7 1 5 . 8 5 15 . 7 8 1 . 6 $ 2 . 5 $ • 0 . 4 $ 0.0$ 8 5 . 5 - 9 0 . 5 22.41 1 8 . 9 7 1 9 . 1 3 . 1 9 . 2 9 1 6 . 2 $ - 1 . 7 $ - 0 . 8 $ 0.0$ 8 7 . 5 - 9 2 . 5 18.26 1 9 . 9 9 2 0 . 5 9 2 0 . 3 2 -10.1$ - 1 . 6 $ 0 . 8 $ 0.0$ S o l i d angles are c a l c u l a t e d i n m i l l i s t e r a d i a n s f o r a p a r t i c l e which enters p a r a l l e l t o the beam axis and s t r i k e s the t a r g e t i n the center. S c a t t e r i n g Angle: 9 0 Degree geometry Target Angle: 5 2 . 5 Degrees Distance to Stopping Counter: 334 mm R e s t r i c t e d Radius of Stopping Counter: 40 mm E r r o r s appearing below each s o l i d angle are the deviations from the value c a l c u l a t e d using the 100 x 100 g r i d . 39 25-11 311 243441121 6534 1 l 20-^  12 5122321 11121 21112 6 l 1 2 21 1 24 3 1 1221 1 I I 1 2 1 1 1 1 1 2 11 11 12 1 21 1 1 1 1 2 1 1 21 1 I I I 1 1 1 1 1 1 1 1 § 1 5 - 11 1 1 1 1 a 1 1 12 1 1 1 bo CO 11 1 1 12 2 1 1 S 1 1 . 2 1 1 2 1 1 H 1 11 21 1 1 1112 1 1 1 10- 13 2 1 1 1 3 ' 12 1 1 1 Q 1 . 1 11 1 12 1 1 1 1 11 1 11 11 1 11 1 2 1 1 11 1 1 1 1 1 11 1 1 2 1 1 21 1 4 11 11 11 2 1 1 1 1 1 1 1 1 1 1 1 11 11114 1122 1 2 135^3234333  -100 , 0 100 X (millimeters ) y Figure 12. V a r i a t i o n of S o l i d Angle with X 90 Degree Geometry Angular Bin: 87.5-92.5 degrees 340 Incident P a r t i c l e s Analyzed C. Proton I d e n t i f i c a t i o n Having now treated problems i n the incident beam, we'now turn our attention t o the i d e n t i f i c a t i o n of protons from the non-radiative absorption r e a c t i o n . Protons i n general may e a s i l y be i d e n t i f i e d by an . examination of dot p l o t s of the Bh ADC vs. the C I A D C , which e s s e n t i a l l y y i e l d dE/dX vs. E p l o t s . Protons separate w e l l from pions and other l i g h t p a r t i c l e s at a l l s c a t t e r i n g angles, as t y p i f i e d by the r e s u l t shown i n Figure 13. A software "proton f l a g " i s defined such that p a r t i c l e s f a l l i n g above the s t r a i g h t l i n e through the region of se-paration are treated as protons i n subsequent a n a l y s i s . Figures l U a and l V b show a comparison of the C I ADC spectrum before and a f t e r a p p l i c a t i o n 'of the proton f l a g . The l a t t e r contains not only protons from non-radiative absorption, r a d i a t i v e absorption, and i n e l a s t i c s c a t t e r i n g , but a l s o protons and heavier p a r t i c l e s a r i s i n g from reactions with carbon i n the various Mylar windows and s c i n t i l l a t o r s . The main purpose of a l l subsequent analysis i s t o separate these background events from the foreground, protons from the r e a c t i o n TT+D -»• pp. To perform t h i s separation, one wishes to f i n d some energy-dependent parameter which w i l l map a l l non-radiative proton events i n t o a narrow peak, and transform a l l other events i n t o a broad spectrum. This e f f e c t i v e l y increases the r a t i o of the height of the foreground peak t o the back-ground, and f a c i l i t a t e s subtraction of the l a t t e r from the former. Figure l U b has already shown that the C I ADC spectrum i s not of maximum be n e f i t i n t h i s e f f o r t due to i t s poor r e s o l u t i o n . There, the proton f u l l energy peak i s broadened by the energy spread of both incident and scattered beams, 4 i 1 0 0 0 -9 0 0 -8 0 0 -700-1 1 1 2 2 2 5 3 8 JL 1 1 1 1 1 1 1 1 2 1 3 1 1 2 1 1 1 1 3 2 1 2 1 2 3 1 2 1 2 3 2 1 3 1 1 5 2 1 4 5 5 3 7 2 2 1 3 5 6 6 5 5 3 3 5 3 3 3 1 7 7 4 5 8 5 4 4 3 4 3 5 4 7 9 5 8 3 5 2 2 2 3A0A0A1 8 8 6 5 2 2 1 1 2 6 8A2A3 2 2 1 1 7 7 8A5A1 7A3 1 2 5 2 6 A I 8 1 1 2 2 9A1 8 9 A 1 8A2 9 A 0 9A2A4A1B0A1B0 8A1 k 4:5 pq 1 8 8 A 0 5 A 7 A 2 7 A 4 A 3 A 8 B 0 C 3 A 5 A 1 8 k h 6 7AOA5A6AOA6A9B2DICOC4BI 9 4 9 6A4BICOD8C4D4CIA5 7A0A1A8B4C9E2E0C9A9 4 5 7 7A9B9C3C7C6A1 5 2A3C0C4B2C0A1 2 5 3 4B5A6A1 9 Figure .13. Proton Separation. DE/DX V S . E P l o t . Line between p a r t i c l e groups defines the software proton f l a g . 300T Figure lUa. CI ADC Spectrum. 9 0 Degree Geometry Figure lUb. CI ADC Spectrum. 90 Degree Geometry Proton F l a g R e s t r i c t i o n hk and a good number of background events have the proper energy to f a l l under the foreground peak. A more u s e f u l parameter i s the quantity Q defined as 1 • dT-?) E/-z which should be i d e n t i c a l l y equal t o 0 f o r ir+D •> pp by the conservation of energy. Since there are i n t r i n s i c experimental errors associated with the measurement of a l l three p a r t i c l e energies, one expects the Q-spectrum to be a l i n e spectrum broadened by the e f f e c t s of the i n t r i n s i c r e s o l u t i o n of CI and inaccuracies i n the optics system. The t h e o r e t i c a l value of Q f o r a l l other reactions assumes a wide range of values, so that the background spectrum w i l l appear t o be a broad band under the foreground peak. Therefore, Q s a t i s f i e s the c r i t e r i a r e q u i s i t e of a foreground i d e n t i f i e r . Since E i s known from the beam optics and = m^ , a knowledge of determines E and E_ . This follows ° CM 7r D . simply v i a the r e l a t i o n (iv-8) fa ~ f i r / f a * n J * Thus, only E need be found to f i x the value of E , and P P . therefore Q, f o r each absorption event. This i s most con-v e n i e n t l y found by c a l i b r a t i n g the proton ADC i n MeV/channel. Gooding and Pugh^ have found that a s a t i s f a c t o r y empirical d e s c r i p t i o n of the l i g h t emitted by p l a s t i c s c i n t i l l a t o r s i s given by the r e l a t i o n (IV-9) where a = a constant f o r a l l p a r t i c l e s = 0.025 g MeV - 1 cm - 2 and gJE A H5 Since a c a l i b r a t i o n i s necessary only i n the region of the peak, terms of order a "(d^/dX 2 ) and above may be neglected, y i e l d i n g (TY-9a) or (!V-9b) i . e . , a simple l i n e a r dependence of energy vs. channel number i s assumed i n the region of the peak. The r a p i d angular dependence of proton energy may be used i n a bootstrap ADC c a l i b r a t i o n f o r each s c a t t e r i n g run. I f the k i n e t i c energy of each proton i s c a l c u l a t e d assuming tha t i t came from non-radiative absorption, and t h i s i s placed i n a dot p l o t vs. ADC channel number, then those protons which i n f a c t come from t h i s r e a c t i o n form a c l u s t e r about a s t r a i g h t l i n e (Figure 15). Background protons and heavier p a r t i c l e s are dispersed about t h i s region. A s t r a i g h t - l i n e f i t by least-squares through the points i n the c l u s t e r w i l l give a good estimate of the ADC response i n the region o f the peak i n most cases. An exception occurred i n the large angle runs, p a r t i c u l a r l y i n the 150 degree geometry. At 150 degrees, the small angular acceptance of the counter and r e l a t i v e independence of proton energy from s c a t t e r i n g angle (Figure l ) meant that the only spread i n proton energy could come from the spread i n the -energy of the i n c i d e n t pions. However, a 7 MeV s h i f t i n i n c i d e n t pion energy leads only to a 1.2 MeV s h i f t i n the energy of the s c a t t e r e d proton, so t h i s e f f e c t i s likewise small. The t o t a l energy spread i n the scattered protons was t h e r e f o r e o f the order of 2 MeV, much l e s s than the 46 100 ADC 1000 Channel Number 1T00 Figure 1 5 . Proton ADC C a l i b r a t i o n 75 Degree Geometry K : O .O652 MeV/Channel 34.77 MeV i n t r i n s i c resolution of the counter. K was consequently-estimated to be quite small at large scattering angles, as indicated by an anomalously narrow peak i n the Q-spectrum. Although absolute values of K and A were therefore not P P found, the small spread i n proton energy i n fact almost negated the need for an absolute ADC c a l i b r a t i o n , and the method remained v a l i d as a device f o r i d e n t i f y i n g foreground protons. 1+8 D. Background Subtraction. Figure l 6 shows the locus of points from which protons are received i n the 30 degree s c a t t e r i n g geometry. Although i t i s apparent that a r e s t r i c t i o n on X' and Y' would eliminate some background events, t h i s would be somewhat useless f o r most protons generated i n the f l a s k jacket, B2 and B3, the spark chambers, and the f l a s k windows. A more e f f e c t i v e parameter f o r d i s c r i m i n a t i n g against t h i s background would be found i n the Z'-coordinate, as Figure 17 demonstrates. Although the r e s o l u t i o n of Z' i s not great enough to d i s c r i m i -nate against protons a r i s i n g from the f l a s k windows themselves, i t i s s u f f i c i e n t to discriminate against a l l the other major-sources of background. In f a c t , a Z 1 r e s t r i c t i o n eliminates up to ninety per cent of a l l remaining proton background with v i r t u a l l y no los s i n v a l i d foreground events i f the l i m i t s of the r e s t r i c t i o n are set at ±75 mm. Residual background was t r e a t e d by repeating each s c a t -t e r i n g experiment with the target empty and examining the nature of the Q-spectrum subject to the same constraints as the s c a t t e r i n g runs. Aside from a normalization f a c t o r , the Q-spectra thus obtained were a representation of a l l protons generated i n anything other than l i q u i d deuterium i n the s c a t t e r i n g run. s S p e c i f i c a l l y , i f n protons are observed i n the s c a t t e r i n g run, where (iv-io) n, = t& [pDi ^ J e e S 0 x 2 € / M P and P = The p r o b a b i l i t y f o r s c a t t e r i n g from deuterium i n t o P = The p r o b a b i l i t y f o r s c a t t e r i n g o f f other objects i n t o AJ2 ^tSMP = Computer e f f i c i e n c y and the e f f i c i e n c i e s of B3 and Bh are assumed to be 100%, k9 X ( m i l l i m e t e r s ) Figure x6. Locus of Proton Generation 30 Degree Geometry Y' Suppressed 50 TARGET,TARGET WINDOWS Figure 1 7 . Z as V a l i d Event Indicator R e s t r i c t i o n s Placed as Shown ho mm per b i n 51 and one likewise observes n protons i n the background run, (IV-lOa) ^B^KIN^J€BOX2e€o^ then the a c t u a l numbers of proton events i n the scattered and background runs can be estimated to be ( I V - l l a ) M B - NB[P1 ^ 2? 6 00X2^ CP MP Notice that (IV-12) JjV.W B = A/' P ' ^ ^ H O P ^ L K 8 Therefore, ( i v - i s ) N f lI T = N w r - A / x ^ Care must be exercised i n de f i n i n g the e f f i c i e n c i e s of the e x i t spark chamber e f f i c i e n c i e s , as these perform a d i f f e r e n t function than the i n c i d e n t chambers. In the i n c i d e n t case, a l l pions g i v i n g bad sparks are simply r e j e c t e d from f u r t h e r consideration, l e a v i n g a number of pions which are t r e a t e d as u s e f u l beam. "Bad sparks" may be defined according t o any a r b i t r a r y set of c r i t e r i a ; the only e f f e c t i s to change the number of pions which are t r e a t e d as v a l i d i ncident p r o j e c t i l e s . In the e x i t case, however, one must know the exact number of protons generated by the absorption r e a c t i o n , and any a r b i t r a r y d i s m i s s a l of events w i l l lead to a low value of the cross s e c t i o n . 52 Only those scattered events which give good sparks i n the e x i t chambers can be placed i n t o angular b i n s . A number of v a l i d proton events are therefore l o s t from the analysis due to spark chamber i n e f f i c i e n c y . I t i s t h i s f r a c t i o n which must be estimated c a r e f u l l y , and which should be repre-sented by the " e x i t spark chamber i n e f f i c i e n c y " . T h e o r e t i c a l l y one would l i k e t o know ^gox,z ^**iy -^or protons a r i s i n g from TT+D -> pp. Up to t h i s point i n the a n a l y s i s , these protons have been d i s t i n g u i s h e d by the software proton f l a g , the X', Y' r e s t r i c t i o n from the i n c i -dent chambers, and the Z T r e s t r i c t i o n determined by a l l four chambers.. Obviously, the l a t t e r constraint cannot be used to measure as i t presupposes good sparks i n box 2. We derived our estimate of 6 ^ by f i n d i n g that set of protons A whose X' and Y' coordinates at the target plane were w i t h i n the region of confidence, and by f i n d i n g the subset B c o n s i s t i n g of those members i n A which gave good sparks. The e x i t spark chamber e f f i c i e n c i e s were then defined as (TV-Ik) Ggoxg = With the spark chamber e f f i c i e n c i e s thus defined, the formula f o r the cross section becomes (IV-15) 55 = '2 72S _ Bl82S€„ao€ggXI J?8 where SPK ~ ^eati^eotz 53 The background remaining a f t e r a p p l i c a t i o n of the Z' con s t r a i n t was t y p i c a l l y about 10% of the foreground. This was subtracted channel by channel from the Q-spectrum of the foreground. Figure 18 shows a t y p i c a l foreground and normalized background Q-spectrum. F u l l spectra with no angular binning c o n s t r a i n t s showed asymmetric peaks with low energy t a i l s and f u l l widths at h a l f maximum of the order o f 5 MeV, or 5% of the proton energy i n the center of mass. Although large-angle s c a t t e r i n g runs y i e l d e d anomalously low values of the energy r e s o l u t i o n , t h i s was c e r t a i n l y due to the uncertainty i n the ADC c a l i b r a t i o n as discussed p r e v i o u s l y . We assume that the best value i s given by intermediate s c a t t e r i n g angle runs, where the energy v a r i a t i o n with angle i s most r a p i d , and y i e l d s values on the order of 5%. 60 50 - -LO UJ > UJ 30 - -20 - -10 - -FIGURE 18. Q - S P E C T R U M OF SCATTERING A N D BACKGROUND RUNS s c a t t e r i ng backgrou nd 75 degree g e o m e t r y 67.5°- 72.5° b i n — I — I — ^ 4 = — i • 1 - f - r H— \ — 1 : j i i „ - i — h - - 4 — H - E H F-• i . : i \ 1 [ Q(MQV) 55 E. Losses i n the S c i n t i l l a t o r s . A f r a c t i o n of the protons t r a v e l l i n g toward the stopping counter may be l o s t from the scattered beam due to reactions with carbon n u c l e i i n the s c i n t i l l a t o r s B3 and Bh. This f r a c t i o n may be estimated as (iv - 1 6 ) f a m £ l-£xp[-»«c<>?(0] - 0.3% f o r a 100 MeV proton where n^jQ are the number of carbon n u c l e i per square cen-timeter i n the combined B3 - Bh telescope. I f n protons are observed i n CI, then one may i n f e r N = n-'F^g^ protons l e f t the t a r g e t , where ' . .. ( I V - 1 7 ) FB3BU = 1 / ( 1 "  fB3Bh ] In a d d i t i o n , not a l l protons which a c t u a l l y reach the stopping counter appear i n the f u l l energy peak. Some simply disappear i n t o the s c i n t i l l a t o r casing without having donated t h e i r f u l l energy, and others undergo various nuclear reactions i n the s c i n t i l l a t o r i t s e l f . By r e s t r i c t i n g the analysis to a c i r c l e of hO mm radius concentric with the axis of the counter, the f i r s t e f f e c t i s minimized. The second e f f e c t was the sum of three contributions: 1. ) E l a s t i c c o l l i s i o n s with hydrogen and carbon n u c l e i 2. ) E x c i t a t i o n of the h.h3 MeV l e v e l i n carbon 3. ) A l l other i n e l a s t i c nuclear c o l l i s i o n s with carbon n u c l e i 56 E l a s t i c c o l l i s i o n s with hydrogen n u c l e i conserve energy, and any los s i n pulse height w i l l "be due to nonlinear response of the s c i n t i l l a t o r m a t e r i a l . Since t h i s e f f e c t cannot "be detected with the energy r e s o l u t i o n of the counter, i t s h a l l be neglected. S i m i l a r l y , i n e l a s t i c c o l l i s i o n s with carbon n u c l e i , only a few percent w i l l s u f f e r reactions i n which more than 2% of the proton energy i s t r a n s f e r r e d to the nucleus; t h i s e f f e c t s h a l l likewise be neglected. The h.h-3 MeV l e v e l i n carbon decays by gamma-ray emission, but since the i n t e r a c t i o n length f o r a k.h-3 MeV gamma-ray i s about 35 cm, most gammas escape from the counter. However, considering the width of the spectra, i t i s doubtful that t h i s e f f e c t w i l l make a noticeable co n t r i b u t i o n and i t s h a l l also be neglected. When any other type of i n e l a s t i c c o l l i s i o n occurs with a carbon nucleus, the r e a c t i o n products are neutrons, deu-terons, alphas, and other heavy p a r t i c l e s , which contribute s u b s t a n t i a l l y l e s s l i g h t than protons. We assume that a l l protons which undergo i n e l a s t i c c o l l i s i o n s with carbon n u c l e i o are l o s t from the f u l l energy peak i n accordance with Measday and use h i s numbers f o r the f r a c t i o n f ^ of protons which undergo such r e a c t i o n s . Defining F ^ as l / ( l - f ^ ^ ) and l e t t i n g n be the number of protons observed i n the f u l l - e n e r g y peak, we i n f e r that N protons would have f a l l e n i n the peak f o r a pe r f e c t counter, where (rv-18) J\[ = P 57 The laboratory cross section then assumes i t s f i n a l form; ( r v _ 1 9 ) ^ L . _ _ l _ _o N-pTr ^ Np = p p ^ s V l ^ ^ e ^ ^ flflT em >u £ s pS nipper B o 8 o 58 F. Transformation to the Center of Mass. The laboratory cross sections are transformed to the 10 center of mass by the relations ^ djf — yz (IV - 1 8 ) g* _ fji j f_j^_5f^^<^w^7_ L fate where i f the expression i n brackets i n the l a s t equation i s less than 0, @* *$Ky • {EDS&) i m p ! i e s t n e formal average of /«s^ over the angular bin i n the lab i n which the cross section i s measured. 59 CHAPTER V RESULTS AND DISCUSSION A. Results. Nineteen s c a t t e r i n g runs were analyzed using angular b i n s ranging i n s i z e from two to f i v e degrees. Five angular bins were made f o r each run, the b i n s i z e s being chosen t o subtend approximately equal areas on the stopping counter face independent of the distance of the counter from the t a r g e t . The cross s e c t i o n i n the center of mass was assigned to that angle which was the center of mass transform of the ari t h m e t i c mean of the b i n l i m i t s i n the l a b . No great error i s encountered i n making t h i s assignment (rather than using the arithmetic mean of the center of mass s c a t t e r i n g angle f o r each event) since each s c a t t e r i n g angle i s known only t o the nearest degree, and the d i f f e r e n c e i n the two estimates i s a l s o of the order of one degree. The derived cross sections are presented l o g a r i t h m i c a l l y i n Figure 19. The s i z e of each v e r t i c a l error bar r e f l e c t s : the s i z e i n the f r a c t i o n a l error i n determining the cross s e c t i o n independent of p o s i t i o n on the graph. The s i z e of the e r r o r bars represents the quadrature sum of the s t a t i s t i c a l e r r o r s i n determining the hodoscope e f f i c i e n c y , incident spark chamber e f f i c i e n c y , transmission c o e f f i c i e n t , numher of protons i n the peak, and the e r r o r s i n determining the mean s o l i d angle, plus an estimation of the erro r i n F ^ Further d i s c u s s i o n of these e r r o r s appears i n section V^-E. The h o r i z o n t a l e r r o r bars are the formal transforms of the angular b i n l i m i t s i n the lab to the center of mass. 6o 6l The same r e s u l t s appear i n Figure 20, where the cross section i s p l o t t e d l i n e a r l y as a function of the average value of the square of the cosine i n the angular b i n . This p l o t demonstrates consistency of r e s u l t s between forward and back-ward angles. The continuous l i n e drawn through the cross sections i n both figures i s a standard least-squares f i t of to the data. A sizeable d- wave cont r i b u t i o n would manifest i t s e l f i n Figure 20 as a deviation from the l i n e at high 2 * values of cos 6 . E v i d e n t l y , no obvious d- wave absorption occurs. k * The necessity of including a cos 8 term i n the f i t i s further obviated by comparing the reduced chi-squared f o r f i t s to hoth • JftT^/^&s^l/ both of which appear i n Table 3. Both values of chi-squared are l a r g e , since systematic e r r o r s have not been treated at t h i s p o i n t , but i t i s seen that no s i g n i f i c a n t reduction i n chi-squared occurs with the a d d i t i o n of the d- wave term. One therefore concludes that the r e s u l t s are consistent with B = 0. Fe include our r e s u l t s i n a compilation by Measday of the work of other experimenters at low energy of the non-r a d i a t i v e pion absorption r e a c t i o n and i t s inverse. Numbers which have been derived using d e t a i l e d balance appear i n parentheses. A phenomenological theory proposed by Gell-Mann and W a t s o n s t a t e s the energy dependence of the t o t a l cross section i s given by the r e l a t i o n (v-i) ^=^rl + ^ 5 « wjiere ^ i&- the center of mass- momentum of the pion i n terms-of the pion r e s t mass: n w c The v a l u e s of t o t a l c r o ss s e c t i o n 62 (qui) uoxq.03g SSOJQ 6 3 TABLE 3 Best - F i t Parametrization of Data F i t A -B CT^mb) X 2 n 2 * Cos 6 .7^5 + .017 .286 + .009 5.80 + .16 1.62 Cos & .846 + .050 .246 + .015 .172 + .017 5-79 + .38 1.6l 61+ TABLE 4 A Compilation of Data at Low Energies E((lab) A -B <j(mb) (22.9) (26.0) 37.6 47.0 0.32+.05 0.29+.09 0.223+.035 0.26+.15 (4.41+.35) (4.09+.31) 5.41+.40 6.1+.6 (0.54+.06) (0.52+.08) 0.77+.08 0.82+.22 49.0 0.246+.015 0.17+.02 5.80+.38 O.85+.O5 60.7 (70) (75) (88) (91) 0.248+.032 0.172+.016 0.220+.022 0.24+.05 O.27+.09 0.1+.17 0.06+.23 0.2+.6 7.18+.44 (S.58+.47) (7.85+. 45) (8.6+0.9) (7-4+1.2) O.98+.O8 (1.41+2.39) (1.15+4.42) (1.28+3.85) (0.98+.22) 65 66 given by Table k are graphed vs. 7^ i n Figure 21, together "with a f i t to a l l data points except our own as given i n the a r t i c l e by Measday . I t i s seen that our value coin-cides quite n i c e l y with the previously available data. 6 7 B. Treatment of S t a t i s t i c a l E r r o r s . The formula f o r cross sections i n the lab i s given by r e l a t i o n (IV-19) ( I V - 1 9 ) ^ ~ ^ ^ ' < . Z ^ > ' ¥e assume that the f r a c t i o n a l error i n c a l c u l a t i n g the center of mass cross sections i s the same as the f r a c t i o n a l error i n c a l c u l a t i n g the l a b o r a t o r y cross sections. In t h i s case, i f the i n d i v i d u a l errors are uncorrelated. Errors i n ^AZ} and j? s h a l l be omitted from the present d i s c u s s i o n , which s h a l l concern i t s e l f with the remaining terms. Using ( i l l - l l ) , one may write , „ ...... -Afai {J IP'M'J l^HOP J it) s There i s no true " s t a t i s t i c a l " error i n B1B2 , and other e r r o r s i n t h i s term s h a l l be tr e a t e d l a t e r . The remaining four terms are the s t a t i s t i c a l errors i n estimating the p r o b a b i l i t y that a c e r t a i n event does or does not occur. For example, € H o 0 i s an estimate of the p r o b a b i l i t y that an i n c i d e n t p a r t i c l e w i l l r e g i s t e r the hodoscope. T h i s i s estimated by passing N pions through the hodoscope and obser-ving the number n which give a proper r e g i s t r a t i o n . The s t a t i s t i c a l error i n measuring'G^p ~ (%) i s then given by- the standard formula f o r the erro r i n estimating the p r o b a b i l i t y of the binomial d i s t r i b u t i o n 5 ~Hao 68 The e r r o r s i n est imating &goxi Kr and ^ are c a l c u l a t e d s i m i l a r l y and t y p i c a l e r r o r s i n these parameters are o .W 3.0% K° . . . 0 . 3 % p I f one rewrites NQ^JJ as p (V-5) N = F . F N V ? ; OUT B3BI+ C I then the s t a t i s t i c a l e rrors i n ^ggg^ and F C 1 m a y he estimated as (v - 6 ) -p = ~—~r * 5% ~ 0.01 A and CV-T) = _ A _ x Si U.5Z Fa l-fa f o r f i n the range 5 - 11%. The f i g u r e of 5% on the frac-OX t i o n a l e r r o r i n f ^ and fg^BU * s ^ a s e < 3 - o n a n estimate by Measday . Since hO to hOO counts were observed i n the v a r i o u s angular-binned Q.-spectra, the s t a t i s t i c a l error i n N i s merely CV-8) AN ^ 1 E r r o r s i n the s o l i d angle are of the order of 5% as discussed i n section IV-B. 69 S u b s t i t u t i n g the above r e s u l t s i n t o equation ( V - 2 ) , the t o t a l s t a t i s t i c a l f r a c t i o n a l error becomes (v-2) - J ^ L - 7.7-/707. I t i s apparent that the t o t a l s t a t i s t i c a l error i s dominated by the number N of the counts i n the Q-spectrum. TO C. Systematic E r r o r s . A number of systematic errors made i n the cross s e c t i o n measurement remain to be considered. Since an estimation of the magnitude of some of the more important of these errors i s contingent upon f u r t h e r experimentation, no systematic e r r o r which s h a l l be mentioned below has been incorporated i n t o the data a n a l y s i s presented e a r l i e r i n t h i s chapter. One major source of error may l i e i n the assumption that the target i s of uniform thickness. Although the gas b a l l a s t system i s designed to maintain the walls i n a p a r a l l e l o r i e n t a t i o n , t h i s has not i n f a c t been v e r i f i e d . Since even a 1 mm d e v i a t i o n of the target thickness leads to a 3% error i n the cross s e c t i o n , a survey of the target f i l l e d with f l u i d s of various d e n s i t i e s w i l l c e r t a i n l y be necessary i n ' the f u t u r e . The p h y s i c a l state of the deuterium i s also important. I t has already been remarked that the p u r i t y of the deuterium was of the order of 9Q.9%- In a d d i t i o n , we have assumed that the temperature of the deuterium corresponded to the temper-ature of the second-stage r e f r i g e r a t i o n s t a t i o n , implying an avera-e density of 0.165 g/cm . In f a c t , the temperature v a r i e d from run to run. Since 3^?r i s approximately 0.002 g/cm /°K, a misestimate of 2°K leads to a 2.5% change i n the cross s e c t i o n . Another source of error l i e s i n estimating the number of r e a l pion coincidences i n B1.B2. The random c o n t r i b u t i o n to B1.B2 has been measured to be of the order of 1% by delaying the s i g n a l s from B l by 50 nanoseconds over the time of f l i g h t of the incident pions, and p l a c i n g these i n c o i n -cidence with B2. Since 75% of the B2 rate c o n s i s t s of par-t i c l e s which pass through B l , t h i s measurement overestimates the " r e a l " randoms r a t e since t h i s c o r r e l a t e d B2 r a t e makes 71 a c o n t r i b u t i o n to the measured randoms r a t e . The r e a l randoms rate i s more c o r r e c t l y estimated as (v.?) H«mz & 81 %iez jV2 vhere i s the "uncorrelated" B 2 r a t e . Since the t o t a l rate i n B 2 i s approximately 4/3 the rate i n B 1 . B 2 , one has that 1 / 3 ( B 1 . B 2 ) . In t h i s approximation, the f r a c t i o n a l c o n t r i b u t i o n of the " r e a l " randoms i s estimated to be of the order Loss of protons i n B3 has already been considered. Protons may also be l o s t due to i n t e r a c t i o n s i n the .deuterium, the Mylar e x i t windows, and the spark chambers. An estimate of the f i r s t and l a r g e s t c o n t r i b u t i o n i s ( M i ) r ^ "5*1 ^ 0 ? % where t i s the nominal target thickness. This c o n t r i b u t i o n increases as the target angle increases, which leads to longer f l i g h t paths f o r protons leaving the target. Further l o s s of protons r e s u l t s from t h e i r m u l t i p l e s c a t t e r i n g i n B3, which improperly bins them, and also may have a small e f f e c t i n s c a t t e r i n g more protons i n t o the kO mm radius area on the counter face than out. Both e f f e c t s are probably small since the root mean square s c a t t e r i n g angle from B3 i s approximately 0.8 degrees, but the e f f e c t bears f u r t h e r i n v e s t i g a t i o n . Protons which reach the counter may be displaced out of the Q-spectrum peak due to random coincidences with back-72 ground events. The average rate f o r p a r t i c l e s of greater 3 than 2 MeV energies was measured to he about 5 x 10 /second, which corresponds to an instantaneous rate of 5 x 10 /second f o r a duty f a c t o r of 10%. An estimate of the magnitude of t h i s e f f e c t i s then Q-ST-. P o s i t i o n a l dependence of spark chamber e f f i c i e n c i e s was neglected i n c o r r e c t i n g f o r the number of protons l o s t due t o bad sparks i n the e x i t chambers. Such a dependence i n f a c t occurs, since s i g n a l s from the spark are attenuated as they t r a v e l along the magnetostrictive wire. This e f f e c t i s increased i n cases where the magnetization of the wire has decreased below i t s normal value. I f the attenuation i s s u f f i c i e n t l y l a r g e to c o n s i s t e n t l y decrease the s i z e of the pulses below the-discriminator threshold, then an apparent decrease i n e f f i c i e n c y of regions i n the chamber d i s t a l t o the pulse a m p l i f i e r should be observed. The o v e r a l l r e s u l t would be to introduce angular bin-dependent corrections t o the number of protons observed i n each Q-spectrum. 73 BIBLIOGRAPHY 1. D.F. Measday and C. Richard-Serre, Nuclear Physics B20,1*13 (1970). 2. L.M. Lederman and K.C. Rogers, Physical Review 105, ' 21+7(1957). 3. R.P. Haddock et. a l . , Physical Review 105, 2l+7(l957). 1+. W. Westlund, Ph.D. Thesis, University of B r i t i s h Columbia, (1973), unpublished. 5. TRIUMF Kinematics Handbook. ' . 6. P.A. Reeve et. a l . , TRIUMF report no. VP9-73-10 (1973). 7. D. LePatourel, M.Sc. Thesis, University of B r i t i s h Columbia,(1973), unpublished... 8. T.J. Gooding and H.G. Pugh, Nuclear Instruments and Methods 1, 189(1962). 9. D.F. Measday and C. Richard Serre, CERN 69-17(1969). 10. Rutherford High Energy Physics Handbook. 11. M. Gell-Mann and K.M. Watson, Annual Review of Nuclear Science k, 219(1951+). 0 Ik APPENDIX A Monte Carlo Estimation of Decay- Mu Contamination at Target We would l i k e to estimate the quantity Q = f^fj Jecys AB2AjU reacts Ujd \&>A A tif\Bl] I t i s assumed that no muon a r i s i n g from decay p r i o r to reaching the downstream, edge of H2 w i l l encounter the tar g e t , since these muons w i l l "be deflected by H2 from the incident "beam ax i s . A l l other p a r t i c l e s may reach B2 and the target i n three ways: 1. ) Pions may not decay, and may reach B2 and the target i n a l i n e a r t r a j e c t o r y . 2. ) Pions w i l l decay p r i o r to B2 at small angles and t h e i r daughter muons w i l l reach B2 and the t a r g e t . 3. ) Pions w i l l decay a f t e r passing through B2 and the muon daughter w i l l then reach the ta r g e t . We s h a l l estimate muon contamination as where, cases 1.) and 3.) w i l l predominate. The incident pion beam at H2 w i l l e s s e n t i a l l y be homo-geneously d i s t r i b u t e d over a c i r c l e of 5" diameter. One there-: fore chooses s t a r t i n g coordinates f o r the f i r s t p a r t i c l e : (A-2) X = #WffM [ t y ' J l and t'^'')'^^^^ s k a H represent random numbers between 75 0 and 1 i n t h i s and the following discussion. The p a r t i c l e now t r a v e l s p a r a l l e l to the heam axis u n t i l decay occurs, which w i l l happen at the distance • , where (A-10 S^^-X^LnM and =• CpnT/rily. , the decay length of a 1+9 MeV pion I f Sju. i s l e s s than the distance from H 2 to the ta r g e t , a muon has been generated. A s o l i d angle f o r decay i n the center of mass ( i n which decay i s i s o t r o p i c ) i s generated randomly. ( A - 5 ) <p* - 2?-n." These angles are then transformed to the laboratory frame. CA-7) d = <£* The parameters X, Y, ^  , ^ 5 , and @ define a decay p o s i t i o n and subsequent t r a j e c t o r y . The p o s i t i o n of the muon i n the planes of B 2 and the target are c a l c u l a t e d , and i f the points are within the boundaries of both, the muon i s counted. -If Eju i s greater than the distance -from H 2 to the t a r g e t , no muon i s generated and one need only ask whether X' and Y' are w i t h i n the boundaries of both. 76 The l o g i c of the c a l c u l a t i o n i s diagramatically shown i n Figure 22, and the r e s u l t s of the calculations for -various target geometries are presented i n Table 5. 78 TABLE 5 Decay Muon Contamination at Various Target Geometries Scattering Angle Target Angle N(«) N( H) Contamination (%) 30 18.8 4636 227 4.67 + .32 45 27.8 4581 257 5.31 + .34 60 36.6 4704 213 4.33 + .30 75 45.0 4o6i 178 4.20 + .32 90 52.5 3316 134 3-88 + .34 105 59.8 2578 149 5.46 + .46 120 -30.0 4741 191 3.87 + .29 135 -22.5 4750 236 4.73 + .32 150 -15.0 474l 212 4.28 + .30 79 APPENDIX B + n D~»-pp Kinematics C a l i b r a t i o n of the proton ADC requires knowledge of proton k i n e t i c energy T x vs. lab angled? and inc i d e n t pion energy E . Referring to Figure 23, conservation of 4-momentum requires (B - l ) P 5 t + P D = P 1 + P 2 or (B - 2 ) ( P j t . P ; L ) 2 = ( P 2 - P D ) 2 which implies (B - 3 ) m^2 - 2 ( E j l E x - p^p-jcos^ ) = mD - 2 m D E 2 Eg may be eliminated by the ..relation (B - 4 ) E 2 = E ^ + m D - E x so that one has (B-5 ) (m/+m D 2 +2m DE J t )+2p r t'^^m~2 c o s 0 = 2 ( E j t + m D ) E 1 Noticing that (B - 6 ) S H C P ^ + P D ) 2 = m^+mjjS+amjjEjj . and (B - 7 ) E j ( + mD = E l a b = E (B - 5 ) s i m p l i f i e s to (B - 8 ) s + 2p"|E,2-m 2 cos0 = 2EE, rnv 1 p 1 Squaring and rearranging, (B-9) 4 ( E 2 - p J t 2 c o s 2 0 )E 1 2-4sEE 1+(s 2 + 4 m p 2p n 2cos 2 ( 0 ) = 0 Figure 23- If D-^PP Kinematics 81 Which i s a simple quadratic i n E_^  with the s o l u t i o n -b ±1 b 2 - 4ac ( B - 1 0 ) E1 = ^ • where a = 4 ( E 2 - p ^ c o s 2 ^ ) . b = -4sE C = S 2 + ^ H l p ^ ^ C O S 2 ^ And the k i n e t i c energy of the proton i n the lab i s given by ( B - l l ) T± = E x -82 APPENDIX C S o l i d Angle C a l c u l a t i o n It i s desired to' estimate the quantity AZASI f o r an i n -cident p a r t i c l e with a given X' and Y' i n the target, and laboratory d i r e c t i o n cosines ois, o<2 , oCj , with the r e s t r i c t i o n that the p a r t i c l e must scatter within the angular b i n The general formula for c a l c u l a t i n g s o l i d angles i s f  R J. R z Referring to Figure 2k, ic-2) "-mm r and CO are allowed to assume the values C^T-I's 0 — (i)~£7T* For a p a r t i c u l a r t^ui) , one then asks whether the i n c i d e n t and e x i t vectors Xx an(^ R subtend an angle i n the angular b i n to 0^ • (c-M ^Hr1^ I f not, the p a r t i c l e i n question cannot p o s s i b l y s c a t t e r i n t o the d i f f e r e n t i a l area fdrda> chosen and no contr i b u t i o n from t h i s area i s made. I f so, then the s o l i d angle i s i n -cremented by the value (C-5) Jffi = , frdrJuj ~W f p K • v • 7 5 / 2 r> //"*•/- a * br cceui +Vr sen <oj 83 Figure 2H- Geometry of Target and Stopping Counter Qk vhere ^ = df! + x'*> SX'dg Sin ( Q'^ Js-x'&n(6-<l>) which i s the s o l i d angle subtended by the d i f f e r e n t i a l area The apparent target thickness £>l i s simply given by where CK^  i s the d i r e c t i o n cosine of the incident p a r t i c l e with Z 1 i n the target plane and od,^2 j 0^ 3- are the d i r e c -t i o n cosines of the p a r t i c l e with X and Z i n the laboratory-f raj^e. 

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