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 , 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) 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 ' ¥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 ) 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 SX'dg Sin ( Q'^ Js-x'&n(6-) 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.