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Detection efficiency of plastic scintillators for elastically scattered positive pions Felawka, Larry Thomas 1973

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DETECTION EFFICIENCY OF PLASTIC SCINTILLATORS FOR ELASTICALLY SCATTERED POSITIVE PIONS by LARRY THOMAS FELAWKA B.Sc, U n i v e r s i t y of Manitoba, 1968 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 U n i v e r s i t y of B r i t i s h Columbia A p r i l , 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 at 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 granted by the Head of 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 of 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 e f f i c i e n c y for detecting p o s i t i v e pions which have been scattered e l a s t i c a l l y from a target has been calculated f o r low pion energies. Measurements of the e f f i c i e n c y at pion k i n e t i c energies 12 - 77 MeV were made for a 12" long x 5" diameter NE110 p l a s t i c s c i n -t i l l a t o r o p t i c a l l y coupled to a P h i l i p s XP1040 photomultiplier tube. The e f f i c i e n c i e s varied from 94% at 12 MeV to 79% at 77 MeV. i i TABLE OF CONTENTS Page CHAPTER I INTRODUCTION • ..... 1 CHAPTER II CALCULATION OF EFFICIENCY 3 1. Losses from I n e l a s t i c Reactions ... 3 2. Losses Due to F i n i t e Pulse Integration Time .. 7 3. Losses Due to Pion Decay beforeSStppping 14 4. Tot a l E f f i c i e n c y 15 CHAPTER I I I CORRECTIONS TO EFFICIENCY ARISING FROM EXPERIMENTAL GEOMETRY .19 1. Pion Beam Contamination by Decay Muons 19 2. E l a s t i c Scattering Out of S c i n t i l l a t o r 26 CHAPTER IV EFFICIENCY MEASUREMENTS ............ 31 1. Experimental Arrangement 31 2. Data Analysis 35 CHAPTER V CONCLUSIONS 48 BIBLIOGRAPHY -.49 APPENDIX A - Pion-Muon Decay Kinematics 50 APPENDIX B - Distance Between Scattering Centre and Boundary of Detector 54 i i i LIST OF TABLES Page I. P r o b a b i l i t y P^ of TT Stopping i n S c i n t i l l a t o r without Undergoing I n e l a s t i c Reactions 9 I I . Pion Stopping Time ( i n Rest Frame of Pion) vs. Pion K i n e t i c Energy 16 I I I . T h e o r e t i c a l E f f i c i e n c i e s 17 IV. Muon Contamination of Pion Beam • 25 V. E l a s t i c Scattering out of S c i n t i l l a t o r 30 VI. Experimental E f f i c i e n c i e s 3 9 i v LIST OF FIGURES Page 12 1. I n e l a s t i c Reaction Cross Sections f or u on C 8 2. Region of Integration of P r o b a b i l i t y Density Function 12 3. P r o b a b i l i t y P 2 vs. Gate Length 13 4. T o t a l E f f i c i e n c y 18 5. Geometry for Muon Contamination C a l c u l a t i o n 20 6. JGeometryvforEElasticSSeatteringCCalculation 27 7. Flow Diagram for E l a s t i c Scattering C a l c u l a t i o n 29 8. Experimental Setup ........ 32 9. Pion Flux vs. Energy 33 10. Experimental Logic 34 11. T y p i c a l Time Encoder Spectrum at .4.9 MeV 36 12. ADG Spectrum at 49 MeV before R e s t r i c t i o n s 40 13. ADC Spectrum at 49 MeV a f t e r R e s t r i c t i o n s 41 14. ADC Spectrum at 12 MeV 42 15. ADC Spectrum at 24 MeV 43 16. ADC Spectrum at 35 MeV • 44 17. ADC Spectrum at 56 MeV 45 18. ADC Spectrum at 67 MeV 46 19. ADC Spectrum at 77 MeV 47 v ACKNOWLEDGEMENT S I would l i k e to take t h i s opportunity to express my thanks to my research supervisor, Dr. David A. Axen, for his encouragement and assistance. I should also l i k e to thank Dr. C.H.Q. Ingram for h i s h e l p f u l observations and information. V i 1 CHAPTER I. INTRODUCTION In order to distinguish, between p o s i t i v e pions which have been scattered e l a s t i c a l l y from composite n u c l e i and those which have scattered i n e l a s t i c a l l y e i t h e r the momentum or the energy of both the incident and the scattered pions must be measured. The most convenient technique for determining the incident momentum i s magnetic analysis. E l a s t i c a l l y scattered pions can be i d e n t i f i e d as w e l l i f instead of measuring the momentum eit h e r the v e l o c i t y (time of f l i g h t ) or the energy deposited while passing through some material i s measured. The t l m e - o f - f l i g h t technique has l i m i t e d usefulness for low-energy pions, as the path length required to measure the v e l o c i t y accurately i s usually large compared to the mean decay length of the pions, r e s u l t i n g i n large p a r t i c l e l o s s . A second l i m i t a t i o n i s the reduction of the s o l i d angle of sc a t t e r i n g by the long f l i g h t path. The pion energy can be conveniently measured by stopping the pions i n a p l a s t i c s c i n t i l l a t o r and measuring the pulse height (or charge) produced by a photomultiplier tube o p t i c a l l y coupled to the s c i n t i l l a t o r . Better energy r e s o l u t i o n i s obtained by stopping the pions completely rather than allowing them to pass through a r e l a t i v e l y t h i n s c i n t i l l a t o r . P l a s t i c s c i n t i l l a t o r s are preferable to sodium iodide c r y s t a l s because the short r i s e time (y 5 nsec) of the pulses produced reduces background from random events during the pulse i n t e g r a t i o n and hence improves o v e r a l l r e s o l u t i o n . The small range of the pions i n p l a s t i c s c i n t i l l a t o r (about 2 10 cm for 50 MeV pions) allows, the f a b r i c a t i o n of r e l a t i v e l y small and inexpensive detectors. Monoenergetic pions which enter the stopping counter do not produce pulses a l l of the same height. The following processes w i l l lead to deviations from the mean pulse height: 1) i n e l a s t i c reactions i n the s c i n t i l l a t o r 2) f i n i t e pulse i n t e g r a t i o n time 3) pion decay Before stopping i n the s c i n t i l l a t o r . The e f f i c i e n c y for detecting e l a s t i c a l l y scattered monoener-ge t i c pions i s defined as counts i n e l a s t i c peak £ = £ /JL) t o t a l number of counts Calculations of e are made i n Chapter I I . Upper l i m i t s to small e f f e c t s such as beam contamination by decay muons and e l a s t i c s c a t t e r i n g of pions out of the s c i n t i l l a t o r are made i n Chapter I I I . In Chapter IV experimental measurements of the e f f i c i e n c y for energies 12 to 77 MeV are described. 3 CHAPTER II. CALCULATION OF EFFICIENCY Events are l o s t from the peak i n the pulse height spectrum p r i m a r i l y because of the following processes: 12 1) pions undergo i n e l a s t i c reactions with the C n u c l e i before stopping 2) events are l o s t because of the f i n i t e pulse i n t e g r a t i o n time 3) pions decay to muons before stopping. If P^, P^ and P^ are the respective p r o b a b i l i t i e s that pro-cesses 1-3 do not occur, the e f f i c i e n c y i s given by e - P 1 P 2 P 3 (2) 1. Losses from I n e l a s t i c Reactions The procedure used to ca l c u l a t e losses due to reactions i s (1 2) si m i l a r to that used by Measday and Richard-Serre ' to cal c u l a t e losses of protons i n various types of materials. The p r o b a b i l i t y of nuclear i n t e r a c t i o n i n a s l i c e of material of thickness ds, composed of n chemical elements i s , by d e f i n i t i o n where a. = t o t a l r e a c t i o n cross section for the i ^ I element n ^ = number of atoms of 1^ element per un i t area. Also, n. . - n .ds., where n . i s the number of atoms of the i * " * 1 element Ax v i v i per unit volume. The p r o b a b i l i t y of i n t e r a c t i o n dP at a distance s i n the s c i n t i l l a t o r i s the product of the p r o b a b i l i t y that no i n t e r a c t i o n has occurred i n the distance s and the p r o b a b i l i t y that the i n t e r a c t i o n occurs i n the i n t e r v a l ds. Thus, 1*1 The cross section has been written as a\(.s) to stress the fact that the cross section varies greatly.ywith distance. Integrating, .s where S = maximum penetration of IT R The p r o b a b i l i t y of no i n t e r a c t i o n P^ i s thus + P, = exp Changing the v a r i a b l e of i n t e g r a t i o n from s to k i n e t i c energy E where E^ = k i n e t i c energy of incident p a r t i c l e s • ° 2 nvi diE — ™ C3) dE The quantity for a material composed of a sing l e element ( 3 ) i s given by the formula of Bethe " where + 2 ~z e = z = n = v Z = mo E c = I = elementary charge (e.s.u.) charge number of incident p a r t i c l e number of atoms of element/unit volume atomic number of element electron mass speed of l i g h t i n a vacuum p a r t i c l e speed/c geometric mean i o n i z a t i o n p o t e n t i a l of the element Assuming complete a d d i t i v i t y , then for a material composed of n elements dE 4s Using the r e l a t i v i s t i c expression for k i n e t i c energy, (4) E = M Qc 2 [Cl-62) 1 / 2 - l ] (5) where = p a r t i c l e mass we may express g as I 2 -•aC2+q) (I+a) 2 where a M Qc Also, ii Pf i N Q v i M. where p = density of material f_^ = f r a c t i o n of l^1 element by weight NQ = Avogadro's number .th = gram atomic weight of i element Substituting i n (4). olE V~*r -I Using the values (4) (6) / onooQ m"10 3/2 1/2 -1 e = 4.80298 x 10 cm g s z(pions) = 1 z ( 1 2 c ) N Q = 6.02252 x 10 2 23 0.511006 MeV 6 M( 1 2C) = 12.01115 g C N I ( 1 2 C ) = 78 eV Z(H 2) = 2 C5) M(H 2) = 2.01594 g I(H 2) = 18.7 eV (5) -1 2 the constants a and b were calculated to be 0.153396 MeV g cm and 12 -1 2 1310.2.7, re s p e c t i v e l y , for C and Q.3Q4649 MeV g cm and 54653.1, re s p e c t i v e l y , for H^. Equation (6) does not hold for small energies; the c o n t r i b u t i o n to i n t e g r a l i n eq. (3) below 0.1 MeV i s small and was neglected. Thus, 2 *i(E)nvi<lE ©I a t + 12 The reaction cross section for rr on C as a function of pion k i n e t i c energy i n the range 0 - 100 MeV was obtained by f i t t i n g a poly-nomial to the available cross section d a t a 1 ^ . E x E x p e r i m e n t a l cross sections and the l i n e a r ^ f i t are shownhinr.Fig. -!^ At .theseeenergies + 12 there i s no i n e l a s t i c s c attering of IT from protons. The C cross 12 section included e x c i t a t i o n of the 4.4 MeV l e v e l of C, as the r e s o l u -t i o n was i n s u f f i c i e n t to separate t h i s contribution. -3 The density of NE110 p l a s t i c s c i n t i l l a t o r i s 1.032 g cm and the molecular formula i s CH ^ , The f r a c t i o n of 1 2 C by weight ( f ^ 1.104 i s therefore 0.9152 and the f r a c t i o n H 2 (f 2> i s 0.0848. Table I l i s t s the values of P^ calculated from equation (3). 2. Losses Due to F i n i t e Pulse Integration Time Pions decay i n the following manner: TC ,+ r\= 26 ns e 8 9 TABLE I P r o b a b i l i t y P^ of i r + Stopping i n S c i n t i l l a t o r Without Undergoing I n e l a s t i c Reactions Pion K i n e t i c Energy (MeV) V 10 .998 20 .991 30 .981 40 .967 50 .948 60 .920 70 .883 80 .832 90 .763 100 .673 110 .564 120 .440 130 .312 140 .194 150 .103 10 Pions, which come to re s t before decaying w i l l y i e l d mono-energetic muons of k i n e t i c energy 4.1 MeV (see Appendix A). The f u l l energy peak i n the analogue-to-digital converter (ADC) spectrum w i l l correspond to an energy equal to the sum of the i n i t i a l pion k i n e t i c energy and the decay muon k i n e t i c energy of 4.1 MeV. As the pulse i s integrated for a f i n i t e time, a c e r t a i n f r a c t i o n of the pions w i l l not have decayed during that time and a c e r t a i n f r a c t i o n w i l l have decayed with a subsequent muon decay. Both of these cases r e s u l t i n l o s s of events from the peak i n the ADC spectrum. The f r a c t i o n of events corresponding to a pion decay but no muon decay within the in t e g r a t i o n time T was calculated i n the following manner: Using set theory notation, the following sets were defined: A = {set of a l l events such that T"1  decays i n the time i n t e r v a l (t , t ^ d t ^ } B E {set of a l l events such that y + decays i n the time i n t e r v a l ( t 2 , t 2 +dt 2 )} From the d e f i n i t i o n of conditi o n a l p r o b a b i l i t y , P(AftB) = P(A)P(B|A) where P(AflB) E p r o b a b i l i t y that both A and B occur P(A) = p r o b a b i l i t y that A occurs P ( B | A ) E p r o b a b i l i t y that B occurs, given that A occurs Since the decay times t ^ and t 2 are both exponentially d i s t r i b u t e d , - U T , P(ft» - S 11 P (B|A) = 5 i L + + where t 2 » T 2 a r e t' i e m e a n l i f e t i m e s of the T and u r e s p e c t i v e l y . ,. K M U & » iP t - -£ % -The i n t e g r a t i o n was performed i n the shaded region indicated i n F i g . 2. In t h i s region the pion has decayed and the muon has not. ^ A e - ( t - t . y T , j t P„ was plo t t e d as a function of T (using T_ = 26.024 ns and T„ = 2199.4 2 1 / (9) ns ) i n F i g . 3. The highest e f f i c i e n c y i s obtained when P„ i s a maximum. dP 2 Setting = 0 and solving for T, one obtains d l •max r a - r , 1 2 1 3 Gate Length (ns.) FIG. 3. P r o b a b i l i t y P„ vs. Gate Length 14 3. Loss Due to Pion Decaybefore Stopping The p r o b a b i l i t y that the pion decays i n the time i n t e r v a l Ct'.t'+dt') i s given by -t'/x: a,* where T = mean l i f e t i m e of IT" . In t h i s expression t' represents the time i n the r e s t frame + + + of the T . If the stopping time of the T i n the r e s t frame of the IT + i s T^, then the p r o b a b i l i t y P^ that the T does not decay before stopping i s e at - s / x 3 - \ — — — - — - e ( 7 ) T' was calculated i n the following manner: s By d e f i n i t i o n , CA = at where t = time i n lab frame From Eq. (5), ->lz E Also, from r e l a t i v i s t i c time dilation.-^ :. dt' -15 Integrating, o M6cM J/3 2 The above in t e g r a t i o n was performed numerically with. M QC = dE 139.576 MeV. Eq. (6) was used to evaluate - j— . ^ ds The r e s u l t s of the c a l c u l a t i o n of P^ using eq. (7) are given i n Table I I . 4. Tot a l E f f i c i e n c y i n Table III and are plo t t e d (along with experimental e f f i c i e n c i e s ) i n F i g . 4. For th i s c a l c u l a t i o n , the assumption was made that the s c i n t i l -lator;, was large enough to contain pions which were scattered e l a s t i c a l l y 12 1 from the C and the H n u c l e i i n the s c i n t i l l a t o r . A c a l c u l a t i o n of th i s e f f e c t assuming a f i n i t e counter s i z e i s made i n Chapter I I I . In addition, the small losses r e s u l t i n g from e l a s t i c s c a t t e r i n g from 12 n u c l e i (which are not as massive as C n u c l e i and have a r e c o i l ) were neglected. It should be emphasized that these c a l c u l a t i o n s were done f or p o s i t i v e pions and are i n v a l i d for negative pions. The e f f i c i e n c y was calculated from eq. (2); r e s u l t s are given 16 Pion Stopping Times ir"1" K i n e t i c Energy (MeV) TABLE II (in Rest Frame of Pion) 1 Stopping Time (ns) vs. Pion K i n e t i c Energy P r o b a b i l i t y P^ of No Decay before Stopping 10 0.066 .997 20 0.159 .994 30 0.262 .990 40 0.368 .986 50 0.475 .982 60 0.581 .978 70 0.685 .974 80 0.787 .970 90 0.886 .967 100 0.983 .963 110 1.076 .959 120 1.167 .956 130 1.254 .953 140 1.339 .950 150 1.422 .947 17 TABLE I I I Theoretical E f f i c i e n c i e s Pion Energy (MeV) P l P2 P 3 e = P P P 10 .998 .948 .997 .944 20 .991 .948 .994 .934 30 .981 .948 .990 .921 40 .967 .948 .986 .904 50 .948 .94848 .982 .882 60 .920 .948 .978 .854 70 .883 .948 .974 .815 80 .832 .948 .970 .765 90 .763 .948 .967 .699 100 .673 .948 .963 .615 110 .564 .948 .959 .513 120 .440 .948 .956 .399 130 .312 .948 .953 .282 140 .194 .948 .950 .175 150 .103 .948 .947 .092 18 1.0, •rl I -41 5o + T K i n e t i c Energy (MeV) I. , , I, 100 FIG. 4. T o t a l E f f i c i e n c y 19 CHAPTER III CORRECTIONS TO EFFICIENCY ARISING FROM EXPERIMENTAL GEOMETRY 1. Pion Ream Contamination 'by Decay Muons: A Monte Carlo computer c a l c u l a t i o n of the f r a c t i o n of p a r t i -c l e s entering the counter which are due to pion decays i n f l i g h t was made using the following model: (see F i g . 5 ) 1. a pion t r a v e l l i n g p a r a l l e l to the counter axis was incident on a plane perpendicular to the d i r e c t i o n of motion of the pion within a c i r c l e of radius R^. The coordinates of the incident point were random va r i a b l e s which were generated by the computer, assuming even d i s t r i b u t i o n across the c i r c l e . 2. the pion t r a v e l l e d a distance before i t decayed into a muon and a neutrino. The muon decayed i s o t r o p i c a l l y i n the centre of mass frame with lab angles 0 and cf> (spherical polar coordinates) and continued to t r a v e l i n a s t r a i g h t l i n e . The angles 6 and <j> were random variables generated by the computer. 3. the i n t e r s e c t i o n of the t r a j e c t o r y of the muon and the plane Z = D (corresponding to the face of the stopping counter) was calcu l a t e d . I f the point of i n t e r s e c t i o n lay within a c i r c l e of radius RQ (stopping counter radius) then a " h i t " was scored. If PQ = (p cos u i + p s i n wj + Z g k ) i s the point of decay and c|f = eg (sin 6 cos (j) " i * + s i n 6 s i n <j>j + cos 6k) i s the muon v e l o c i t y , 20 FIG. 5. Geometry f o r Muon Contamination C a l c u l a t i o n 21 then the equation of the muon tr a j e c t p r y is. where t =? time parameter P = (/©cosu> -v- c^tsiviG&osqgYi + (xi^ihco + e AtVmO s,n<? )y + feo + opt cos 6) k The muon struck the Z - D plane when 2^+ Cm COS© = D If cgt < 0 then the decay was i n a backward d i r e c t i o n and the muon did not i n t e r s e c t the Z = D plane. I f c$t k 0 then the coordinates of the point of i n t e r s e c t i o n were x« - /Ocos«o + "taw 8cos© 2 2 2 If x + y < R_ then a " h i t " was scored, s J s 0 I t i s assumed that the pion decay i s i s o t r o p i c i n the centre-of-mass frame. I f 0 and <j) correspond to 0 and § i n the centre-of-mass frame then,from Appendix A, g c = v e l o c i t y of C. of M. T - ( 1 - ^ ) " 1 / 2 Since the pion decays v i a the 2-body process, 6 i s independent of angle and i s equal to 0.272 (see Appendix A). 22 The random va r i a b l e s p, 03, Z^, 0 and $ were generated using the following procedure: i f the d i s t r i b u t i o n function to be simulated were f(x) and x v a r i e d from a to b, then the cumulative d i s t r i b u t i o n function ^ P= \ "f"("xO i X was c a l c u l a t e d . Ja The r e s u l t i n g equation was solved for x. If the quantity P i s given random values ranging from 0 to 1, assuming even d i s t r i b u t i o n , then the r e s u l t i n g values of x have the desired d i s t r i b u t i o n . The random numbers P were generated using the CDC 6600 subroutine RANF. Case 1: Generation of D i s t r i b u t i o n s for p and cu Assuming that the p r o b a b i l i t y per unit area i s constant (and hence equal to —^y) the p r o b a b i l i t y that the incident p a r t i c l e s t r i k e s an TrR§ area pdpdco i s I The p r o b a b i l i t y density function (PDF) i n terms of the v a r i a b l e p may be found by in t e g r a t i n g with respect to u). Thus, the p r o b a b i l i t y that the incident p a r t i c l e l i e s between pnand p+dp i s The cumulative d i s t r i b u t i o n function P i s thus .«. /o = RWP 2 3 S i m i l a r l y , the PDF for u> i s obtained by integrating with respect to p. Thus, / C O and 60 Case 2: Generation of D i s t r i b u t i o n for ZQ Assuming that the pion decays according to the exponential d i s t r i b u t i o n , the PDF i s then defined by where Z ^ = mean decay length of pion Then, (varies with pion energy) = \ - e Case 3: Generation of D i s t r i b u t i o n for 0 and <J> Assuming that the p r o b a b i l i t y of decay per unit s o l i d angle i s constant (and hence equal to •^0 the PDF i s defined by Integrating out the v a r i a b l e cf> , 24 Integrating out 9 , Calculations of the r a t i o of muons s t r i k i n g the target to t o t a l p a r t i c l e s (pions and muons) s t r i k i n g the target were made for various pion energies and are shown i n Table IV. For these c a l c u l a t i o n s D = 400 cm (distance between bending magnet H2 and counter), 1 = 6 cm and R n = 4 cm. 25 TABLE IV Muon Contamination of Pion Beam Pion Energy Mean Decay (MeV) Distance (cm) Mu's/Total 10 301 .042 20 433 .040 30 539 .043 40 632 .040 50 718 .044 60 798 .042 70 875 .042 80 948 .041 26 2. E l a s t i c Scattering Out of S c i n t i l l a t o r Both Coulomb s c a t t e r i n g and nuclear e l a s t i c s c a t t e r i n g were considered. Using formulas f or multiple Coulomb sca t t e r i n g (see, for example, "Techniques of High Energy P h y i i c s " , edited by D.M. Ritson, pp. 7-11^*^) the root mean square l a t e r a l spread of a beam of 80 MeV -2 pions which pass through a slab of C ^ of thickness 17.7 gem was c a l --2 culated to be <l.i.cm. As the range of 80 MeV pions i n CR^ i - s 18.2 gem , a s c i n t i l l a t o r of 5" diameter i s s u f f i c i e n t l y large to contain a l l 80 MeV pions undergoing Coulomb s c a t t e r i n g . The e f f e c t of nuclear s c a t t e r i n g i s more s i g n i f i c a n t , however, as the scattering d i s t r i b u t i o n s are generally l e s s peaked i n the forward d i r e c t i o n than i n the case of Coulomb s c a t t e r i n g . The assumption was made that nuclear s c a t t e r i n g was i s o t r o p i c and a c a l c u l a t i o n of the nuclear e l a s t i c s c a t t e r i n g out of a c y l i n d r i c a l s c i n t i l l a t o r of radius RQ = 6.35 cm and length L = 30.5 cm was made using a Computerized Monte Carlo procedure; i n addtion, a f i n i t e incident beam of pions was assumed (see F i g . 6). The following model was used: 1. a pion t r a v e l l i n g along a l i n e p a r a l l e l to the counter axis struck the counter at a random point l y i n g within a c i r c l e of radius R^ = A cm. A uniform d i s t r i b u t i o n across the area was assumed. 2. the pion continued to t r a v e l through the s c i n t i l l a t o r u n t i l i t stopped or scattered e l a s t i c a l l y (nuclear scatter-ing only). A mean free path length of 60 cm was used to generate a random sca t t e r i n g distance; the mean free path was calculated using an average cross section of 300 mb for 12C and 50 mb for "hi. 27 28 3. i f scattering occurred, the coordinates of the s c a t t e r i n g centre PQ and the pion energy at PQ were calcu l a t e d . The random sca t t e r i n g angles 6 and ty (in s p h e r i c a l polar coordinates) were generated and the distance between PQ and the i n t e r s e c t i o n of the s t r a i g h t l i n e t r a j e c t o r y of the scattered pion with the boundary of the s c i n t i l l a t o r was calculated. 4. another random scattering distance was generated; i f sc a t t e r i n g occurred then step 3 was repeated; i f not, then the distance was compared to the pion range to determine whether the pion escaped the s c i n t i l l a t o r . A flow diagram for the computer programme which performed the above c a l c u l a t i o n i s shown i n F i g . 7. The methods of generating the various random variables are the same as those described e a r l i e r . The d i s t r i b u t i o n of the s c a t t e r i n g distance s was assumed to be exponential with the mean free path corresponding to the mean value of s. The c a l -c u l a t i o n of the distance to the s c i n t i l l a t o r boundary S c i s described i n Appendix B. The r e s u l t s of the c a l c u l a t i o n are shown i n Table V; they should be considered as upper l i m i t s only, as the mean free-path was under*-estimated. I n i t i a l i z e Pion Energy Generate Point of Entry into S c i n t i l l a t o r Set s = L c «#  Generate Distances t r a v e l l e d before s c a t t e r i n g . Set s equal to the range i f no sc a t t e r i n g has occurred yes Calculate coordinates of sca t t e r i n g centre PQ. Calculate pion energy at P . Generate sc a t t e r i n g angles 0 and (J) Calculate distance s to c boundary of s c i n t i l l a t o r FIG. 7. Flow Diagram for E l a s t i c Scattering C a l c u l a t i o n 30 TABLE V E l a s t i c Scattering out of S c i n t i l l a t o r Pion Range Pion Energy i n S c i n t i l l a t o r (MeV) (cm) Scattered Pions/Total 20 1.9 <.0001 30 4.0 .0005 40 6.7 .0054 50 11.2 .049 60 14.0 .086 70 18.5 .14 80 23.6 .20 31 CHAPTER IV EFFICIENCY MEASUREMENTS 1. Experimental Arrangement The arrangement i s depicted schematically i n F i g . 8. A poly-thene target placed i n the primary proton beam of the 184" cyclotron at the Lawrence Berkeley Laboratory was used to produce p o s i t i v e pions. The pion beam was passed through bending magnet HI to remove undesired p a r t i c l e s such as scattered protons, negative pions and electrons. Beam focussing was effected by 2 p a i r s of quadrupole magnets (Ql, Q2, Q3 and Q4) and a second bending magnet H2 selected pions of the desired momen-tum. A f i n a l quadrupole magnet Q5 provided a d d i t i o n a l focussing i n the h o r i z o n t a l plane. S c i n t i l l a t i o n counters B l , B2, B3 and B4, each made up of pieces of 15 cm x 10 cm NE102 p l a s t i c s c i n t i l l a t o r of 0.5 cm thickness, defined the pion beam. The time of f l i g h t between B l and B2 (path length approximately 6m) enabled i d e n t i f i c a t i o n of p a r t i c l e type. A hodoscope array of 12 NE102 p l a s t i c s c i n t i l l a t i o n counters, each of dimensions 20 cm x 1 cm x 0.2 cm thickness, defined the incident momentum of the p a r t i c l e s . With the pion beam defined i n t h i s manner, the pion f l u x was ^ 1 x 10^/sec for 50 MeV pions. The pion f l u x as a function of energy i s shown i n F i g . 9. An event, defined by a B1-B2-B3-B4-C1 co i n -cidence, gave an interrupt s i g n a l to a NOVA 1200 computer (12K memory). A data a c q u i s i t i o n programme developed at the U n i v e r s i t y of B r i t i s h C o l u m b i a w a s used to read the contents of CAMAC scalers which con-tained information from the spark chambers, hodoscope, stopping counter analogue-to-digital-converter (ADC) and B l -> B2 t i m e - o f - f l i g h t encoder. 32 33 * * » ' So loo + TT K i n e t i c Energy (MeV) FIG. 9. Pion Flux vs. Energy 9<iie Patter* . f&^s-1 E T I O D / N L CAM AC Syark Chamber/—i 20 HH* lwterrapt L S Z Z . gate to -P-AHpe>? W . ^ J EA1Q1/S1 I CAM AC — ' 1 I^esct rnmpuTud j ^ J delay l i n e l i s c r i m i n a t o r coincidence unit FIG. 10. Experiment Logic 35 This data was. then transferred from the computer memory to magnetic tape for o f f - l i n e a n a l y s i s . Each 2400>ft. r e e l of tape was able to hold information on 80,000 events. 2. Data Analysis Histograms of the ADC spectra at various pion energies were made using a general purpose computer programme developed by W. Westlund at U.B.C. In order to obtain unbiased e f f i c i e n c y measurements only those events s a t i s f y i n g the following c r i t e r i a were considered i n the a n a l y s i s : 1. The p a r t i c l e passed through the centre region of the hodoscope 2. the angle between the two p a r t i c l e t r a j e c t o r i e s defined by the 2 p a i r s of spark chambers was <0.1 radians 3. the time of f l i g h t between B l and B2 f e l l between narrow l i m i t s to ensure p o s i t i v e i d e n t i f i c a t i o n of the pion. Figure 11 shows a t y p i c a l time encoder spectrum at 49 MeV. The l i m i t s indicated by the v e r t i c a l l i n e s served to i d e n t i f y the pion. 4. the pion was incident upon the face of the stopping counter and within a concentric c i r c l e of radius 4 cm. R e s t r i c t i o n s 2 and 3 eliminated some of the decay muon contam-i n a t i o n , as a f r a c t i o n of the pion decays occurred downstream of bending magnet H2 would r e s u l t i n non-collinear t r a j e c t o r i e s and i n f l i g h t times s l i g h t l y d i f f e r e n t from that of the pions. Upper l i m i t s to the remaining 36 lOOOh + + 10 2.0 3 0 Channel Number 4o 5o FIG. 11. Typical- •TdiSe\.,Encodef-™Spec'1rfum' at- 49 '.'-HeV 37 muon contamination were given i n Chapter I I I . Examples of t y p i c a l ADC spectra associated with. 49 MeV pions before and aft e r a p p l i c a t i o n of the r e s t r i c t i o n s are shown i n Fig s . 12 and 13. The overlapping muon and pion peaks were c l e a r l y separated. Pion spectra a f t e r r e s t r i c t i o n s are shown for various pion energies i n Figs. 14 to 19. E f f i c i e n c i e s were obtained by integrating the peak and d i v i d -ing by the t o t a l counts i n the spectrum. The 10 lowest channels i n the spectrum corresponding to pulses below 80 mV threshhold of the d i s c r i m i n -ator connected to CI anode (see F i g . 10) were assumed to contain no bona f i d e pion events. The experimental e f f i c i e n c i e s are given i n Table VI and are plo t t e d i n F i g . 4. Peak Integration The following procedure was used: 1. the peak centroid was calculated using the formula X - v»» ' where x. = channel number x y. = counts i n channel i J x n = no. of channels Substituting x. = x.. + i - 1 , where x. i s the number of the x 1 x st channel, i l chchannel, edinatftenabove equation: 38 = <i - 1 + where Y = X Mi the standard deviation a of the peak was then calculated from A t -\a Y Substituting the expressions f or x^ and x n Z .2L Y the peak was assumed to be Gaussian and was summed from x - 2.5a to x + 2.5a. The r e s u l t was divided by 0.9876 (the area under the Gaussian curve from -2.5a to 2.5a) to obtain the r e s u l t . 39 TABLE VI Experimental E f f i c i e n c i e s Pion Momentum At Magnet H2 Error (MeV/c) Pion K i n e t i c Energy (MeV) E f f i c i e n c y . . '. . % 78 12.0 .939 0.7 96 24.3 .909 0.7 114 35.3 .909 0.5 133 49.1 .887 0.6 143 56.3 .870 0.7 157 66.5 .827 0.8 170 76.8 .792 0.7 Corrected for energy loss i n material placed i n beam l i n e 40 loool-+ 5oo + 11 * • * s 10 20 3o 40 Channel Number 5o FIG. 12. ADC Spectrum at 49 MeV before R e s t r i c t i o n s 41 1oooL 4J O 500 _ 1 . • « * I— ' . ' I U I I I I I ' * ' ^ 1 , io 20 3o 4o 50 Channel Number FIG. 13. ADC Spectrum at 49 MeV a f t e r R e s t r i c t i o n s 42 1oo-» - 1 ' - - - ' 10 zo 3 o 4 o 5 0 Channel Number FIG. 14. ADC Spectrum at 12 MeV 43 3 0 0 cn 4-1 C o u AOOL loo • • • * • 10 20 3o 4 o 5o Channel Number FIG. 15. ADC Spectrum at 24 MeV 44 5oo -10 2o 3o 40 50 Channel Number FIG. 16. ADC Spectrum at 35 MeV 45 to 2o 5o 40 50 Channel Number FIG. 17. ADG Spectrum at 56 MeV 46 300 zool loo to to 30 40 50 Channel Number FIG. 18. ADC Spectrum at 67 MeV 47 460 L 360L 10 20 30 4© 50 Channel Number FIG. 19. ADC Spectrum at 77 MeV 48 CHAPTER V CONCLUSIONS The processes described i n Chapter II account f o r the observed e f f i c i e n c y ; the energy dependence of the e f f i c i e n c y i s due l a r g e l y to the e f f e c t of i n e l a s t i c reactions i n the s c i n t i l l a t o r . The i n e l a s t i c reactions l i m i t the usefulness of stopping counters to pion energies of 100 MeV or l e s s . The e f f i c i e n c y i s r e l a t i v e l y i n s e n s i t i v e to the dura-t i o n of the gate length beyond 120 ns, as i s seen i n F i g . 3. The good agreement between theory and experiment at low energies, where the e f f e c t of i n e l a s t i c reactions i s not prominent, indicates that the time of f l i g h t and s p a t i a l r e s t r i c t i o n s imposed for the experimental determination of the e f f i c i e n c y served to reduce contamination of the pion beam by decay muons to n e g l i g i b l e amounts. The e f f e c t s of nuclear e l a s t i c s c a t t e r i n g out of the s c i n t i l l a t o r appear to be much smaller than the worst case^-estimates. 49 BIBLIOGRAPHY 1. D.F. Measday and C. Richard-Serre, CERN 69-17. 2. D.F. Measday and C. Richard-Serre, Nuclear Instruments and Methods 76 45 (1969). 3. H. Bethe, Annalen der Physlk _5 3 2 5 (1930). 4. CRC Handbook of Chemistry and Physics (49th E d i t i o n ) . 5. TRIUMF Kinematics Handbook. 6. D. Stork, Physical Review 93 868 (1954). 7. R.L. Martin, Physical Review 87 1052 (1952). 8. Nuclear Enterprises Catalogue (1970). 9. "Review of P a r t i c l e Properties", Reviews of Modern Physics, V o l . 43_ No. 2, Part I I , A p r i l 1971. 10. "Interscience Monographs and Texts i n Physics and Astronomy", edited by R. Marshak, V o l . V - "Techniques of High Energy Physics", edited by D. Ritson, Interscience Publishers Inc., 1961. 11. D. Lepatourel and R. Johnson, "A Data A c q u i s i t i o n System Based on CAMAC and Supported by BASIC and FORTRAN" (Submitted to the CAMAC B u l l e t i n ) . 50 APPENDIX A T:+ •*• u + DECAY KINEMATICS 1) Energy of Decay Muon Pions decay v i a the reac t i o n TT » y + v y A pion t r a v e l l i n g i n a c e r t a i n d i r e c t i o n with t o t a l energy E^ decays into a muon and neutrino. The muon f l i e s o f f at an angle 0 from the pion d i r e c t i o n and has t o t a l energy E^. *TC+  Using the 4-vector notation of Special R e l a t i v i t y , the conser-vatio n of momentum i s written as p = p + p where p = (P,iE). ? = ordinary 3 dimensional momentum E = t o t a l energy i 2 , - i 51 2 2 2 Rut E = R + m where m = mass, i n MeV Squaring, 1= 0-/S z} But Substituting i n above equations and solving for E : If the pion i s i n i t i a l l y at r e s t , y = 1. Thus The k i n e t i c energy T^ and the v e l o c i t y |3 (in units of c) are thus 1 Using M = 139.576 MeV and m - 105.6594 MeV, T = 4.16 MeV and 6 T  y y B* = 0.272. 52 2) Transformation of Muon Angle from Centre of Mass tp Lab Frame Let the z^axis be p a r a l l e l to the d i r e c t i o n of motion of the pion. g = v e l o c i t y (in units of c) of decay muon i n lab frame. The Lorentz transformation between a re s t frame and a frame moving along the z-axis with v e l o c i t y B c (in units of c) i n matrix notation i s ct 1 0 0 1 0 0 0 0 0 0 0 0 Y 6 CY Y Y ct I- J 2 "1/2 where y = (1-8 ) The a s t e r i s k s r e f e r to quantities i n the moving frame. The r e s u l t i n g 4cequations are 5 3 x 3 x T - y z *=• Y z +•Scyct A A ct 3 6 Y Z + Y C T J *• I ax *- G dit " foCcV^ S i m i l a r l y , and 0 « * + A c Using s p h e r i c a l polar coordinates, these are expressed as Dividing the 2 N D by the 1 S T D i v i d i n g the T2- by the 3 ( s i n ^ - sin<§ ) : /&* sine* 54 APPENDIX B DISTANCE S BETWEEN SCATTERING CENTRE P. AND BOUNDARY c 0 OF DETECTOR (see F i g . 6) The equation of the scattered pion i s P = P Q + Sn where p Q = Xal -V l| 0"t -V and n i s a unit vector i n the d i r e c t i o n of scatte r i n g ; i .e. n - sin&cos^T + VuiD si&iqTj + tos S = distance parameter The t r a j e c t o r y of the scattered pion i n t e r s e c t s e i t h e r a) theofr,bnthfa£eonb)fthe rearc'face^br "h s c ^ e t h e f c y l i n d r i c a l ' wall* c y l i n d r i c a l wail. P o s s i b i l i t y 1: E x i t through front face The s t r a i g h t l i n e given by eq. 8ntintersectsetlieofront face i f 55 If S < 0 then the s c a t t e r i n g i s i n the forward d i r e c t i o n and the pion cannot i n t e r s e c t the front face. If S > 0 then i n t e r s e c -t i o n with the front face i s p o s s i b l e . The coordinates of the point of i n t e r s e c t i o n are x = x„ + S sin6cosd) s 0 y = y_ + S sin9sin<t> J s J0 2 2 2 If x g + y g - RQ then the t r a j e c t o r y i n t e r s e c t s with front face and otherwise i n t e r s e c t i o n occurs with the s c i n t i l l a t o r w a l l . P o s s i b i l i t y "2: E x i t through rear face The straight l i n e from eq. (8) i n t e r s e c t s the rear face i f If S < 0 then i n t e r s e c t i o n occurs with the w a l l . If S - 0 then the procedure outlined i n p o s s i b i l i t y 1 determines whether i n t e r s e c t i o n , then with the rear face occurs. If so, P o s s i b i l i t y 3 : E x i t through side The point of i n t e r s e c t i o n i s such that Solving for S, 56 the p o s i t i v e s o l u t i o n i s chosen, so that 

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