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The efficiency of scintillation counters for gamma ray detection Leigh, John Laurence 1964

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THE EFFICIENCY OF SCINTILLATION COUNTERS FOR GAMMA. RAY DETECTION b7 JOHN LAURENCE LEIGH B . S c . , Univers i ty of B r i t i s h Columbia, 1962 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 thes is as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October., 1964 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 of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that 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 reference and study, I f u r t h e r agree that per-m i s s i o n f o r extensive copying of 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 that copying 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 gain s h a l l not be allowed without my w r i t t e n permission* Department of Testes The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada Date i i ABSTRACT The ef f ic iency of Nal(Tl) s c i n t i l l a t i o n counters for the detection of gamma rays i s discussed and experimentally determined,, Experimental e f f i c i enc i e s , based upon the number of counts above a bias equal to one-half the gamma ray energy and a s o l i d angle defined to the c r y s t a l effect ive centre, for a three-inch-diameter by three-inch long (3 x 3) c r y s t a l are given for gamma ray energies of 0.51, 1»275 and 6B3J+ MeV. Gamma ray spectra at 1B275 and 6„14 MeV are extrapolated to zero energy i n order to compare t heo re t i c a l e f f i c i enc i e s , based upon the integrated primary absorption, wi th the experimental results,, These resul ts show that , wi th experience, one can expect accuracy to bet ter than 5$ at these energies. Tables are given of the t heo re t i c a l e f f ic iency for 3 x 3 , 2% x Ui» and 5 x 4 c rys ta l s for several gamma ray energies and source-to-counter distances of from 0<,1 cm to 1 m0 v i i i ACKNOWLEDGMENTS I wish t o express my grati tude to Dr. G. M. G r i f f i t h s for h i s super-v i s i o n of the work which comprises t h i s t h e s i s 0 I am also indebted to Dr, J . B. Warren, Dr . G. Jones and Dr. B. L . White for many he lpfu l suggestions and discussions. In p a r t i c u l a r I acknowledge the help of my fel low graduate students of the Nuclear Physics Group - notably Mr. J . MacDonald, Mr, W. Falk, Mr. L . Monier and Mr. M. Relmann. To my mother and father, and to my wi fe , I owe sincere thanks for much patience and encouragement throughout my years of study. i i i TABLE OF CONTENTS page Chapter I - INTRODUCTION . . . . . . . . . . . . . . . . . . . . 1 Chapter I I - EMPIRICAL DEFINITION OF SCINTILLATION COUNTER EFFICIENCY 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 A. The Empi r ica l E f f i c i ency D e f i n i t i o n . . . . . . . . . . . 6 B„ Experimental Measurement of the Hal f Energy Bias Ef f xc iency . . . 0 0 . 0 0 . . . . . . . . 7 1. The S c i n t i l l a t i o n Counter . . . . . . . . . . . . . . 7 2. The Ef f i c i ency at 0.511 and 1.28 MeV . . . . . . . . 9 3. The E f f i c 4. The Rat io of E f f i c i enc i e s at 4.43 and 11,68 MeV . . . 19 5. Summary of 3 x 3 C r y s t a l Hal f Energy Bias E f f i c iency R6S*ul"bs 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 21 Chapter I I I - THEORETICAL EFFICIENCIES . . . . . . . . . . . . . . 22 A. Theore t ica l E f f i c i ency . . . . . . . . . . . . . . . . . 22 B . Experimental Studies of the Theore t ica l Ef f i c i ency . . . 23 1. Theore t ica l E f f i c i enc i e s at 6.14 MeV . . . . . . . . 24 2. Theore t ica l E f f i c i enc i e s at 1.28 MeV . . . . . . . . 30 3 . Rat io of Theoret ica l E f f i c i enc i e s at 4.43 and H o 6S M©V 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 32 C. Ef fec t ive Centres from Theoret ica l Ef f i c i ency . . . . . . Csilcizlcit ions 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 * 0 0 0 0 32 Chapter IV - STUDIES OF A Z% x k\ INCH Nal (Tl ) CRYSTAL . . . . . 34 A. The E f f i c i e n c i e s at 6.14 MeV . . . . . . . . . . . . . . 34 B. The Ratio of the E f f i c i enc i e s at 4.43 and 11.68 MeV . . . 35 Appendix A - REPRESENTATIVE SPECTRA FOR THE 3 x 3 CRYSTAL . . . . 35 i v page Appendix B - THEORETICAL EFFICIENCY OF Nal CRYSTALS . 38 A. Der ivat ion of the Ef f i c i ency Equation . . . . . . . . . . 38 B . Numerical Integration of the Ef f ic iency Integral . . . . . 40 C. Tables of Theoret ical E f f i c i enc i e s . . . . . . . 41 D» The Computer Program . . . . . . . . . . . . . . . . . . 45 Bibliography 0 0 . . . . . . . o . . . . . . . . . . . . . . . . 47 V LIST OF TABLES page I Ma te r i a l Between C rys t a l Face and Container Face . o * e . . . 8 I I Resolution of the S c i n t i l l a t i o n Detector . . . . . . . . . . 9 I I I Properties of the Alpha Detector . o o o o . o . . . . . . . 14 IV Alpha Detector P reanp l i f i e r Performance . . . . . . . . . . . 15 V Alpha Counter S o l i d Angle . » . « o » • . • . 16 V I Hal f Energy Bias Ef f i c i ency at 6.14 MeV . . o 18 V I I Half Energy Bias E f f i c i e n c i e s - 3 x 3 C ry s t a l . . . . . . . . 21 V I I I Theore t ica l Ef f ic iency Extrapolations at 6.14 MeV . . . . . . 29 IX Photofraction at 1.275 MeV . . . . . . . . . . a . . . . . . . 31 X Comparison of Analyt ic and Experimental Ef fec t ive C ent res X I E f f i c i e n c i e s at 6014 MeV - 2f x 4 i C ry s t a l . . . o . . . . . . 35 X I I C r y s t a l , Col l imator and Absorber Path Lengths . . . . . . . . 40 X I I I Theore t ica l E f f i c i enc i e s - 3 x 3 C rys t a l . . . . . . . . . . . 42 XIV Theore t ica l E f f i c i enc i e s - 2^ x kh Inch Cry s t a l . . . . . . . 43 XV Theore t ica l E f f i c i e n c i e s - 5 x 4 C rys t a l . . . . . . . . . . . 44 v i LIST OF FIGURES to follow page 1. S c i n t i l l a t i o n Counter Preamplif ier , • . 8 2. Caesium-137 Gamma Spectrum • , . , , . , » . . , . . . . 8 3. Resolution Function of the 3-Inch-Diameter by 3-inch-Long Nal(Tl) Sointillation Counter * * . » » . . » * . . 9 4. Cobalt-60 Decay Scheme . * » • • « « • • » • * « « • in text p.10 5. Sodium-22 Decay Scheme . . , , . . . . . . • . , . * In text p»12 6. F^(pf°( » y . . , . • • • « , . « , « « « , « , i n te:xt p»13 7. Fluorine-19 Target Chamber . , . , . . » , . , . . , , . 14 8. S o l i d State Detector Preamplif ier . . . . . . . . . . . 15 9. 5.3 MeV P o 2 1 0 Alpha P a r t i c l e Spectrum . . . . . . . . . 15 10* Alpha P a r t i c l e Window . , » , , , . , , . , , . , . . . 15 11. F^ (p ,o< , )0"^  E x c i t a t i o n Function . 16 12. 1.85 MeV Alpha Spectrum 16 13. 6.14 MeV Gamma Ray Spectrum . . . . . . . . . . . . . . 17 l4» "AAtt vs d - 6,14 MeV 18 V N 6 — 15. Alpha Detector Background . . . . . . . . . . . . . . . 18 16. Gamma Ray Spectrum from B ^ p , i r ) C l 2 - Ep - 300 kev. . * . 1 9 17. B ^ ( p , 2T ) C 1 2 . . . . . . . . . . . . . . . . . . . i n text p . 19 18. Hal f Energy Bias E f f i c i enc i e s (3x3) . . . . . . » # . 21 19. Comparison of 6 MeV Monte Carlo Spectrum and 6.I4 MeV Experimental Spectrum . . . . . . . . . . . . . . . . . 25 20. Theore t ica l E f f i c i ency Extrapolations at 6*14 MeV . . . 26 21. Zero Intercept Parameter JsJb ^ « . 28 22. Effect o f Lead Shie ld ing on Cobalt-60 Spectrum . . . . . 29 23. Effect of Lead Shie ld ing on the 6 .I4 MeV Spectrum » . * 29 v i i to follow page 24. Comparison of 1.275 MeV Monte Carlo Spectrum and Experimental Spectrum . . . . . . . . . . . . . . . . . . 30 25. Inverse Square Plot of Theoret ica l E f f i c i enc i e s (3 x 3) • 32 26. Theoret ica l E f f i c i ency Extrapolat ion at 6.14 MeV for the 2$ x 4^ Inch Crys t a l . « . . « . . . . . . . . . 34 27. Caesium—137 Gramma Ray . . . . . . . . . . . . . . . . . . 37 28o Sodium—22 Gamma Rays . . . . . . . . . . . . . . . . . . 37 29. Cobalt-60 Gamma Rays . . . . . . . . . . . . . . . . . . 37 30. Radio Thorium Gamma Rays . . . . . . . . . . . . . . . . 37 31. B 1 : i -(p, 2T)C 1 2 , Ep = 300 kev, Gamma Rays . . . . . . . . . 37 32. P ^ p , * * , f)01^, Ep = 340 kev, Gamma Rays . . . . . . . . 37 33. Geometry of C r y s t a l , Col l imator , and Absorber for t h e o r e t i c a l E f f i c i ency Calculat ions . . . . . . . . . . . 38 CHAPTER I INTRODUCTION A large part of the work of the Univers i ty of B r i t i s h Columbia Van de Graaff group has been concerned wi th measuring the gamma rays produced by capture reactions and the inverse of t h i s , photodisintegrat ion. To obtain quanti tat ive data on both processes i t i s necessary t o measure gamma ray f luxes . More than 60 years af ter the discovery of gamma rays t h i s i s s t i l l a d i f f i c u l t th ing to do wi th high accuracy over much of the energy range of nuclear gamma rays. The discovery of gamma rays can be a t t r ibuted t o V i l l a r d who, i n 1900, drew at tent ion to the fact that radium gave out very penetrating radiat ions which were detectable photographically and were non-deviable by a magnetic f i e l d . These resul ts were confirmed i n the same year by Becquerel and again very c l e a r l y demonstrated i n 1903 by Mme. Curie i n a c l a s s i c experiment reported i n her thesis t o the Faculty of Science, P a r i s . Rutherford studied the absorption of these radiat ions i n several materials by observing the ion iza t ion produced i n a i r with an electroscope. Although given the name gamma rays, the nature of these penetrating radiat ions was i n doubt i n 1904 when Rutherford*s book "Rad io -Ac t iv i t y " was published. Some favoured the hypothesis that they were Roentgen-like ether waves whi le others believed them t o be cathode rays moving at a v e l o c i t y close to that of l i g h t . Many subsequent experiments delineated the exponential absorption of gamma rays i n d i s t i n c t i o n to the de f in i t e range for alpha and beta rays i n matter. With increasing refinement, p a r t i c u l a r l y after the development o f the Wilson cloud chamber, these experiments showed that gamma rays d id not lose energy continuously i n t h e i r passage through matter but only when they transferred t h e i r energy to electrons, e i ther by the photoelect r ic effect , when a l l the energy i s t ransferred, or by a sca t ter ing process where only part of the energy i s t ransferred. Both processes at f i r s t suggested the pa r t i cu la te nature of the - 2 -gamma rays. However, l a t e r more deta i led studies of the scat ter ing showed a c l ea r analogy between gamma ray sca t ter ing and the Compton scat ter ing of X-rays . The l a t t e r were known to be electromagnetic waves because of t h e i r interference and po la r i za t i on cha rac t e r i s t i c s . The study of these processes helped to elucidate the d u a l i t y of quantum and wave-l ike properties of electromagnetic rad ia t ion which were incorporated in to quantum mechanics af ter 1926. With the development of quantum mechanics and the Dirac r e l a t i v i s t i c electron equation i t was possible to describe i n very sa t i s fac tory d e t a i l the angular proper t ies , the energetics and the p robab i l i t y of Compton scat ter ing by means of the K l e i n -Nishina formula. In addi t ion the photoelect r ic absorption was w e l l understood i n p r i n c i p l e , although accurate quant i ta t ive estimates of the cross sect ion required a more de ta i led descr ip t ion of the bound atomic electrons than was ava i l ab le at the t ime. The t h i r d main process by which gamma rays interact wi th matter, namely the production of posi t ron-elect ron pa i rs i n the f i e l d of the nucleus was not discovered u n t i l af ter D i r a c ? s p red ic t ion of negative energy electrons (positrons) and t h e i r discovery by Anderson i n 1932. In sp i te of de ta i led experimental confirmation of the accuracy of a very complete theory for the main primary absorption processes for gamma rays i n matter over a wide range of energies (Davisson and Evans, 195?) the problem of determining gamma ray . fluxes accurately has remained a vexed one for several reasons. By whatever method detected, gamma rays are not observed d i r e c t l y but only through the secondary electrons they eject from matter. Each of the three primary absorption processes produces a different primary electron spectrum. The r e su l t ing t o t a l spectrum i s i n general complex and d i f f i c u l t to interpret quan t i t a t i ve ly . Further, the shape of the electron spectrum, the angular d i s t r i b u t i o n of the electrons and the cross sect ion are a l l continuous functions of the gamma ray energy. Because of the range of the electrons i n matter and - 3 -subsequent absorption or escape of scattered quanta the spectrum ac tua l ly observed i s not the primary spectrum but one modified by secondary processes* This i s p a r t i c u l a r l y complicated for high energies where the electrons lose energy not only by ion iza t ion but also by the emission of bremsstrahlung rad ia t ion re su l t ing i n a continuous secondary photon spectrum. The ea r l i e s t attempts to obtain.other than r e l a t i v e i n t e n s i t i e s of the gamma rays emitted by radioact ive sources were ca lor imet r ic ( E l l i s and Wooster, 1925). The t o t a l energy due to f -rays was measured i n a d i f f e r e n t i a l c a l o r i -meter and compared wi th the t o t a l energy due to and p - rays . These measure-ments gave no information about the energy spectrum or the number of photons. Af ter the recognit ion that gamma rays followed p a r t i c l e emission several attempts were p a r t l y successful i n obtaining absolute gamma ray fluxes by cor re la t ing them wi th the number of pa r t i c l e s observed. Other early experiments involved the magnetic analysis of photoelectrons ejected from radiators exposed to the gamma rays using photographic detection of the electrons and l a t e r detection by means of geiger counters. These d id give some information about the energy spectrum and in tens i ty of the gamma rays, t h i s information becoming more accurate as photoelect r ic cross sections became bet ter determined. The e a r l i e r work determined gamma ray energies by f inding the absorption coeff ic ient of the rad ia t ion or by measuring the energy of the secondary electrons i n a cloud chamber placed i n a magnetic f i e l d . Later work up to 1948 tended to concentrate on (3 -ray spectrometers for measuring the energy of the secondary electrons or on measuring absorption curves of the secondary electrons wi th t h i n window geiger counters. The absolute y i e l d of the X~ -rays could be measured e i ther wi th a t h i c k walled ion chamber (Gray, 1936) or wi th a th ick walled geiger counter (Bradtet a l , 1946; Bleu ler and Zun t i , 1946). Both methods depended on a knowledge - 4 -of the range of a l l the secondary electrons i n the wal l s of the chambers. They were not very useful for complex spectra as they gave no information on the energy of the gamma rays producing the i o n i z a t i o n . A useful summary of t h i s pre-s c i n t i l l a t i o n counter work i s given by Fowler, Lauri tsen and Lauri tsen (1948). The advent of the s c i n t i l l a t i o n counter produced a great advance i n the technique of measuring energies, i n t e n s i t i e s , and l i fe t imes of gamma ray t r a n s i t i o n s . This detector received i t s i n i t i a l impetus from the work of Kallman (1947) who discovered that naphthalene was transparent to l i g h t generated i n i t by fast moving charged p a r t i c l e s , and that the l i g h t could be detected and amplif ied many times using photomult ipl ier tubes developed for t e l e v i s i o n . Following t h i s , organic phosphors such as anthracene, s t i lbene , and several p l a s t i c s and l i q u i d s were developed wi th large volumes capable of very short time re so lu t ion . Several attempts were made to develop inorganic phosphors wi th components of higher atomic number for greater gamma ray absorption which were transparent t o t h e i r own fluorescent r ad ia t ion . In 1948 Hofstadter introduced sodium iodide wi th a t ha l l i um ac t iva to r . Although t h i s has a slower decay time than the organic phosphors i t has a large l i g h t output and a large gamma ray absorption and has become the standard detector used i n p r a c t i c a l l y a l l gamma ray work s ince , except for that requir ing very high energy reso lu t ion . As a resul t i t i s important to have accurate information on the gamma ray detect ion e f f ic iency of sodium iodide s c i n t i l l a t i o n counters. A great deal of work has been done i n t h i s and other laborator ies to es tab l i sh the e f f ic iency of s c i n t i l l a t i o n counters for gamma rays. Accurate absprption coeff ic ients have been tabulated as a function of energy (Grodstein, 1957). A method su i tab le for machine ca l cu l a t i on has been developed for in tegrat ing the primary absorption over the c r y s t a l geometry (Rose, 1953) and severa l Monte Carlo type ca lcula t ions have been performed to t r y and predict - 5 -the spectrum shape to be expected as a function of gamma ray energy and c r y s t a l s i ze ( M i l l e r and Snow, I960; Zerby and Moran, 196l). However, p a r t i c u l a r l y for higher energy gamma rays i t remains d i f f i c u l t t o check the accuracy of these ca l cu la t ions . The present work involves the measurements of s c i n t i l l a t i o n counter e f f i -c iency; f i r s t l y according t o an arb i t ra ry d e f i n i t i o n of e f f ic iency and secondly according to the integrated primary absorption i n the c r y s t a l . - 6 -CHAPTER I I EMPIRICAL DEFINITION OF SCINTILLATION COUNTER EFFICIENCY A. THE EMPIRICAL EFFICIENCY DEFINITION Inspection of gamma ray spectra from sodium iodide ( tha l l ium activated) c rys ta l s shows one that i t i s imprac t ica l t o count a l l the pulses i n the spectrum. In the low energy region, room background and scat ter ing from shields and other equipment tends to d i s to r t the spectrum. Also i t i s often necessary t o eliminate the lower energy pulses i n order to reduce e lect ronic dead time losses . Therefore i t i s best to count only those pulses which appear above a convenient bias po in t . For most appl icat ions i t has been found that a bias of one h a l f the gamma ray energy i s convenient. The spectra are generally low and f l a t i n t h i s region and small gain sh i f t s i n the e lectronic system w i l l have l i t t l e effect upon the resul ts obtained. Also for energies at which p a i r production i s s ign i f i can t the p a i r peaks are w e l l above t h i s b i a s . Thus an empir ica l d e f i n i t i o n of e f f ic iency i s conveniently based upon the h a l f energy b i a s . I f one measures source-to-counter distances to a point w i t h i n the c r y s t a l , c a l l e d the effect ive centre, i t i s found that the count rate var ies as the inverse square of the distance. One can then define a h a l f energy bias e f f ic iency , , that i s distance independent provided any res idua l distance dependence i s included i n the effect ive centre p o s i t i o n . The number of counts expected above the h a l f energy b i a s , N c , for a source-to-counter face distance d i s then N - N !±l* wheres . •, A i s the s o l i d angle defined to the effect ive centre, W j f (d + e ) 2 -7-A i s the cross sec t iona l area of the c r y s t a l , and N^is the number of gamma rays emitted into the sphere. It i s assumed that for d greater than the c r y s t a l length the effect ive centre, e, i s distance independent. I t should be emphasized that the above d e f i n i t i o n i s s t r i c t l y empir ica l and i s dependent upon the source-counter geometry and sh i e ld ing . Algebraic manipulation of the preceeding equation leads to the fol lowing expression; This resul t i s very convenient for experimental inves t iga t ion for i t represents intercept -e . An experimentally determined p lo t of d vs ~\ / enables one to — V No ca lcu la te the h a l f energy bias ef f ic iency and the effect ive centre distance, B. EXPERIMENTAL MEASUREMENT OF THE HALF ENERGY BIAS EFFICIENCY 1, The S c i n t i l l a t i o n Counter The gamma ray detector employed was a standard three-inch-diameter by three-inch- long ( 3 x 3 ) cy l inder of Nal(Tl) mounted on a selected Dumont 6363 photo-m u l t i p l i e r . The u n i t , Harshaw 12S12/E, obtained from the Harshaw Chemical Company incorporates an external magnetic s h i e l d . The c r y s t a l i s mounted 0.1375 inches back from the container face. Table I l i s t s the mater ia l which makes up t h i s distance. and source-to-counter face distance d - 8 -TABLE I MATERIAL BETWEEN CRYSTAL FACE AND CONTAINER FACE ' MATERIAL Packed aluminum oxide Polyethylene disc Neoprene sponge rubber Aluminum container The assembly i s coupled to a t r ans i s to r i zed head ampl i f ie r the c i r c u i t diagram of which i s reproduced i n Figure 1„ The Harshaw Chemical Company has guaranteed a resolut ion of 8% or bet ter for the 0.661 MeV gamma ray of caesium-137 for t h i s crystal-phototube combination,, Subsequent measurement has confirmed t h i s wi th a resul t of 8„01 i 0.24$. (See Figure 2) The reso lu t ion over the energy range 0,511 MeV to 6,14 MeV has been measured and the resu l t s are included as Table I I below. Figure 3 shows a p lo t of reso lu t ion A E ^ E y where E r i s the gamma ray energy i n MeV, Also included for comparison i s the resolu t ion function for a 3 X 3 c r y s t a l wi th 7.8$ reso lu t ion at 0,661 MeV taken from Yuan and Wu (1963), page 636. DENSITY THICKNESS 67 mg/cm2 .0625 " 13 mg/cm2 .006" 43 mg/cm2 .050" 129.9 mg/cm2 .019" Dumonl 6363 FIGURE 1 S c i n t i l l a t i o n Counter Preamplifier 100 r* 100 ft 100 Ft 7E=> + 8-^  Volts 0C i c Q : 4 f S-20pf A H 100 K 6 8 X 220K< T Out < /OOJl OOIy*f 100 M .£ 2.N 38tj-"'-'''' 2.N38* + MOO. Volts /zooo /GOOD 14-000 IZ0O0 •P IOOOO c 3 o O 8000 cooo 4ooo 2000 QUI MeV 137 FIGURE 2 Cs Gamma Spectrum o- 18 5 (v-|eV < 8 - 0 / % 10 2.0 30 ¥0 Channe l Number 50 (oO - 9 -TABLE I I RESOLUTION OF THE SCINTILLATION DETECTOR RESOLUTION GAMMA ENERGY FOR 3 X 3 CRYSTAL FROM YUAN AND WU 0.511 MeV 8.9 * 0.3 % 8.7% 0.661 8,01 ± 0.24 7.8 1.27 6.18 * 0.17 6.2 1.33 6.08 * 0.14 6.1 2.62 4.94 * 0.12 4.9 6.14 3.86 t o . l 6 The h a l f energy bias ef f ic iency for t h i s c r y s t a l has been measured at 0.511, 1.28, and 6.14 MeV. In addi t ion an attempt to measure the r a t i o of the e f f ic iency at 4.43 MeV t o that at 11.68 MeV has been made. 2. The Ef f i c i ency at 0.511 and 1.28 MeV Cobalt-60 decays by (1~ emission t o the 2.505 MeV exci ted state of n icke l -60 which subsequently decays to the ground state by gamma ray cascade through the 1.332 MeV l e v e l (Nuclear Data Tables, I960 - see Figure 4 ) . The two Cobalt-60 gamma rays wi th energies of 1.332 and 1.173 MeV are i s o t r o p i c a l l y d i s t r i bu t ed . 10 8 4 2 0 1 I . I I I 1 I 1 i i i i i i i i -^> ^> f rom Yuan and Wu UBC c r y s t a l - Enerev (MeV) i i i i i i i 6-0 4 0 20 i i i i i i i i i i i i t i i i 0 •2 4 •6 -8 1-0 (Gamma Ray Ene rgy ) 1-2 (MeV)~^ 14 1-6 FIGURE 3 Resolution Function of the 3-Inch-Diameter by 3-Inch-Long Nal(Tl) S c i n t i l l a t i o n Counter -10-FIGURE 4 COBALT-60 DECAY SCHEME 2-505 I 1 1 o N i 6 0 Singh (1959) has measured the f l ux of gamma rays emitted by a cobalt-60 source (Co-60 #1) i n t h i s laboratory by a standard coincidence technique. He obtained a source strength value of 0.0237 * 0.0006 m i l l i c u r i e s on A p r i l 22, 1958. By counting a l l pulses above the average h a l f energy bias i n the Co-60 #1 spectrum the detector ef f ic iency at the average gamma ray energy 1.253 MeV was obtained. A plot of ~\ I^Ji vs d gave the effect ive centre at t h i s same energy. V A / C The resul t i s £^(1.25) = ( 6 0 ' 5 5 * 1.61)$ e ( l . 2 5 ) = 3.50 ± 0.40 cm The error involved i s mainly due to the o r i g i n a l uncertainty i n the source strength, wi th add i t iona l small errors from the counting s t a t i s t i c s , distance measurement, and the add i t iona l uncertainty i n the source strength at the time of the present ef f ic iency measurements. I f the o r i g i n a l source strength were I Q then, af ter a time t had elapsed, the source strength would be I J I 0 exp(- t ln2) H wheres t i i s the h a l f l i f e of the nucl ide , i n t h i s case - 1 1 -5.27 ± 0.05 years (Nuclear Data Tables, I960), The uncertainty A I i n I i s given by 4 1 = I 4 1 L / A [ e x p ( - - L ^ 2 ) , exp ( - -j- J^Z) \ [A LI 2 L i t JUZ A t ' / * (UZY where A t ^ i s the uncertainty i n the h a l f l i f e . 2 The time t between the o r i g i n a l source strength measurement ( A p r i l 22, 1958) and the present ef f ic iency measurements (August 3 , 1962) i s 4«28 years . Put t ing the numbers in to the above equation y i e l d s an uncertainty A i r 0.02591 for the source strength. The t o t a l error i n the ef f ic iency measurement i s the root mean square of the fol lowing errorss i . Gamma ray source strength 2.59$ i i . Counting s t a t i s t i c s 0.75 i i i . Distance measurements 0.90 To ta l 2,66$ The counting and distance errors quoted are for the worst case. In order to determine the h a l f energy bias e f f ic iency at 0.511 MeV i t was assumed that the s c i n t i l l a t i o n detector ef f ic iency for the 1.275 MeV gamma ray from sodium-22 was equal to that measured for 1.253 MeV using the ca l ibra ted cobalt-60 source. Sodium-22 decays to the 1.275 MeV l e v e l i n neon-22 by the fol lowing scheme (Nuclear Data Tables, I960). -12-F3GURE 5 INa 2 22 SODIUM-22 DECAY •10 K Capture 1-275 SCHEME o Kreger (1954) has shown that 10$ of the dis integrat ions to the 1.275 MeV exci ted state of neon-22 take place by electron capture and 90$ by posi t ron emission. Thus, for a source enclosed i n a container one can expect, for every 1.275 MeV gamma ray, 1.8 0.511 MeV gamma rays due to posi t ron ann ih i l a t ion w i t h i n the source and the container w a l l s . By counting the number of pulses above the 1.275 MeV h a l f energy b i a s , the known ef f ic iency was used t o determine the f lux of 0.51 MeV gamma rays emanating from the sodium-22 source. A p lo t of -\ /tlr vs d was then used to determine the effect ive centre and the h a l f energy b ias e f f ic iency at 0.51 MeV. The resu l t s were2 In addi t ion to the error due to the ef f ic iency uncertainty at 1.28 MeV . there are add i t iona l errors introduced. The largest i s that due to uncertaint ies introduced by the subtraction of the 1.28 MeV gamma ray t a i l from the 0.51 MeV spectrum. Distance measurement contributes errors of 1% or l e s s . The t o t a l error i s given by the root mean square of the fol lowing errors l £,(0.51) = (75.1 ± 2.4)$ e ( 0 . 5 l ) = (2.96 i 0.50)cm -13-i . Er ror i n 1.28 MeV eff ic iency 2.66$ i i . S t a t i s t i c a l errors 1.5 $ i i i . Measurement errors 1.0 % T o t a l 3.21$ 3. The Ef f i c i ency at 6.14 MeV The h a l f energy bias e f f ic iency at 6.14 MeV was determined by s imul -taneous counting of the alpha p a r t i c l e s and gamma rays from the w e l l known has been used and reported previously (Van A l l e n and Smith, 1941} G r i f f i t h s , 1958; G r i f f i t h s et a l , 1962). At 340 kev bombarding energy, protons are resonantly captured by f luor ine-19, forming neon-20 i n a h ighly exci ted s tate . The neon-20 decays by alpha p a r t i c l e emission to any one of three states of oxygen-16 (See Figure 6) , i e to the 6.15 MeV l e v e l , o(z t o the 6.91 MeV l e v e l , and eXj to the 7.12 MeV l e v e l . react ion F^(pt«,TC)0- at a proton bombarding energy of 340 kev. This method FIGURE 6 F 1 9 ( p , ^ , ^ ) 0 : ,16 /32(3 Curt Nt 1 0 Ground. Stite) -14-Alpha groups ©<o t o the ground state and <*tt t o the pa i r emitt ing state at 6.06 MeV i n oxygen-16 are not seen at t h i s bombarding energy. The 6014» 6.91, and 7»12 MeV leve ls of oxygen-16 subsequently decay by gamma ray emission to the ground state (Ajzenberg-Selove and Laur i t sen , 1959)» The in tens i ty r a t i o ?t * - 0.024 (Freeman, 1950) i n good agreement with the gamma ray r a t i o y» * *» = 0.023 t .002 (Dosso, 1957). The work of Devons and Hine (1949) and Chao (1950) has shown that the gamma rays and alpha p a r t i c l e s are separately i so t rop ic at the 340 kev resonance. The target chamber as previously used by Larson (1957) was employed for the measurements but was modified to incorporate a s o l i d state nuclear p a r t i c l e detector (RCA diffused junct ion detector type C-3-25-2.0) for counting the alpha p a r t i c l e s rather than the propor t ional counter previously used. The chamber i s i l l u s t r a t e d i n Figure 7. The RCA C-3-25-2.0 unit was chosen simply because of i t s low cost and r e l a t i v e l y large sens i t ive area. The most important propert ies of the detector are given i n Table I I I . TABLE I I I PROPERTIES OF ALPHA DETECTOR Mat e r i a l phosphorus diffused in to n-type s i l i c o n R e s i s t i v i t y 1000 ohm-cm Window Thickness 2 microns Sens i t ive area 20 mm' Operating voltage 25 vo l t s Leakage Current (measured) .02yaamps Resolution (5.3 MeV alphas) 50 kev A semiconductor preamplif ier was b u i l t as i l l u s t r a t e d i n Figure 8. This preamplif ier i s extremely easy to b u i l d , i s very compact, and i s i d e a l for appl icat ions not requir ing low noise l eve l s or exceptional r e so lu t ion . The noise figures for the preampli f ier wi th no capaci t ive loading are given i n Table IV. The spectrum for the 5.3 MeV polonium-210 alpha p a r t i c l e s i s shown i n Figure 9. The preamplif ier incorporated the 2N930 t r ans i s to r s when the spectrum was taken. TABLE IV ALPHA DETECTOR PREAMPLIFIER PERFORMANCE TRANSISTOR TYPE NOISE EQUIVALENT FIGURE * FWHM * COST* P h i l l i p s AFZ-11 PNP Motorola 2N2218 NPN Texas Instruments 2N930 NPN .0060 .0070 .0025 84.6 kev 98.7 35.2 $ 4.22 $ 5.80 $18.00 * Pulses t o st imulate 5.3 MeV alpha p a r t i c l e s ** Approximate as of Ju ly , 1964 The alpha p a r t i c l e s were counted i n an accurately defined geometry at 135° t o the Van de Graaff generator proton beam d i r ec t i on and at 90° to the target face. The s o l i d angle subtended by the alpha detector was defined by a s ta in less s t e e l window wi th an accurately measured diameter (Figure 10) which incorporated a 25/* inch sheet o f Grade C n i c k e l f o i l obtained from The Chromium Corporation of America which was required to reduce the scattered proton f l u x reaching the detector. The measurements required to determine the alpha FIGURE 8 S o l i d State Detector Preamplifier o Out 2- 2NfSO COO 500 1*00. +-> § 300 O O ZOO IO0 FIGURE <? 5 3 MeV P o 2 1 0 Alpha P a r t i c l e Spectrum FWHM 2 . 3 8 % ( 1 3 4 kev ) 150 i(,o iro Channe l Number lao 8' I n c h e s 5* ^ /6 2-56 -2/t /6 1 2 5 / * i n N i c k e l /6 T 32 2a = 3 . I 960 ± . 000Z,. mm 11 lb & Z7" 32' FIGURE 10 Alpha P a r t i c l e Window -16-detector s o l i d angle were made as indicated i n Table V . Refer to Figure 7 for an explanation of the symbols used. TABLE V ALPHA COUNTER SOLID ANGLE DISTANCE MEASURED Window diameter (2r) Distance A Distance B Target backing Detector window to target face R = A-B- t S o l i d Angle RESULT 3.1960 * .0004 mm 4.227 * .001 i n 0.9533 * .001 i n 0.014 * .0005 i n 3.2597 * .0025 i n (1.170 * .002) x 10" 3 steradians HOW MEASURED Trave l l i ng microscope over 15 diameters P rec i s ion ca l ipe r depth gauge Prec i s ion depth gauge Micrometer The errors quoted are a l l root mean square e r rors . The gamma rays were counted at 180° t o the alpha p a r t i c l e detector and correct ions t o the number of counts obtained were made i n order t o account for time dependent background, 6.91 and 7.12 MeV gamma rays and a lso for the absorption i n the 0.014 inch copper target backing and the 1/32 inch aluminum gamma ray window. The f luor ine targets were prepared by evaporation of powdered calcium f luor ide under vacuum ( « 5 x 10""'' mm Hg) onto the highly polished copper backings 0.014 inches t h i c k . E i g h t y - s i x amperes of current passed through a tantalum boat containing ^ 0 . 0 3 grams of CaF2 formed a target 9 kev th i ck to 340 kev protons at a distance of 25 cm from the boat. A l l targets were found to be 316 3Z¥ 332 3H0 3¥Q G e n e r a t i n g V o l t m e t e r R e a d i n g ( U n c o r r e c t e d ) (kev ) . 85 MeV FIGURE 12 1.85 MeV Alpha Spectrum /zoo /ooo\ at •P C 3 O O 4001 ¥oo V Noi se FWHM \tfo zoo\ \ b to : * ZO 30 Channe l Number ¥0 -17-v i s u a l l y uniform over t h e i r whole surface and exc i t a t ion functions (Figure 11) taken at various pos i t ions and times during the runs confirmed the resu l t s of the v i s u a l inspect ion . Measurements of the alpha p a r t i c l e and gamma ray fluxes were made by bombarding the targets wi th protons of <~ 340 kev energy from the UBC Van de Graaff accelera tor . I t was necessary to keep the beam current down to approxi-mately one microampere i n order to prevent- too rapid target de te r io ra t ion . When the alpha count rate indicated that the target was deter iora t ing badly the target holder was raised or lowered i n order that a new part of the target be exposed to the beam. A t y p i c a l alpha spectrum i s shown i n Figure 12 while a gamma ray spectrum appears as Figure 13. The proton beam s t r i k i n g the target was confined to a beam spot o f 2 mm diameter by the co l i imat ing arrangement shown i n Figure 7» This ensured that corrections to the alpha counter s o l i d angle due to f luctuat ions i n the beam spot pos i t i on and due to the f i n i t e beam spot s i ze were n e g l i g i b l e . Pulses from the alpha counter were fed in to 128 channels of a Nuclear Data Type 101 256-ehannel k i ckso r t e r whi le those from the gamma counter went to 256 channels o f a Nuclear Data Type 120 512-chanhel k i ckso r t e r and a lso to a type UBC NP-11 sca ler biased so as to count only those pulses appearing above the h a l f energy b i a s . A p lo t of ~\ / — vs d was made for f ive different source-to-counter face distances d. Here N ^ i s the number of alpha p a r t i c l e s detected whi le N c i s the number of gamma counts above the h a l f energy b i a s . The s traight l i n e obtained (Figure 14) gives the ef fec t ive centre distance for the 6.I4 MeV gamma ray. The h a l f energy e f f ic iency i s given by £ = NCL7°< O 2 0 ¥0 6 0 8 0 IOO I20 / V O I60 IQO Channe l Number FIGURE 13 6.14 MeV Gamma Ray Spectrum -18-wheres , N c are as previously defined C /^* • tJy a r e the alpha and gamma detector s o l i d angles. The gamma detector s o l i d angle i s defined to the effect ive centre, e(6.14) = 3.75 ± 0.25 cm, and i s given by (j - A (d • • 2 where A i s the c r y s t a l cross sec t iona l area = 45.606 cm . The resul ts for the f ive dif ferent t r i a l s are given i n Table V I . TABLE VI HALF ENERGY BIAS EFFICIENCY AT 6.14 MEV d d + e £ ^ PROBABLE ERROR 32.85 * .20 cm 36.60 *• .45 cm 56.03$ ± 1.15$ 56.25 ± .20 60.00 ± .45 58.44 ± 0.88 74.40 ± .20 78.15 ± .45 58.33 ± 0.86 106.00 ± .20 109.75 ± .45 57.83 ± 0.82 130.00 ± .30 133.75 ± .55 57.76 ± 1.00 Weighted average 57.84 ± 0.94 The errors quoted include contr ibutions from distance measurement, alpha counter s o l i d angle and counting s t a t i s t i c s . The accuracy of the counting s t a t i s t i c s was reduced by the subtract ion of appropriate background correc t ions . Time dependent background i n the alpha counter was neg l ig ib l e at the p o s i t i o n of the alpha peak but beam dependent background emanating from the target 200 1 . 8 5 MeV o , Beam Dependent Time Dependent • • 160 1 MeV 120 > > i | i i 1 1 , o o 5.3 MeV ( P o 2 K ? ) Cour 80 , / I 0 >c lo /  l 1 l i ! X 2 ? 1 1 1 \ \ -/ «/ o " n » n t X « » /o" 1 0 / 1 1 1 o°o / 0 o o \ \ o \ o \° \ \ o \ \ 0 I / / p / ° i \ 0 I \ 0 \ \ °°^o°° 0-°°-^ o * J i " ir . • * 1 H II , » « " T « » \o \ o \ 0 o " x * x X * X o j»o o o / ^ » y ° " K 1 l / 0 10 y - " w . T H M i * « « * 20 30 50 60 70 80 90 Channel Number FIGURE 15 Alpha Detector Background 1 -19-amounted to approximately 2% of the counts i n the alpha peak. This background was uniformly spread over a l l channels up to approximately 4 MeV where a very broad peak was evident (Figure 15)• Runs o f f the resonance gave.the same resul ts for the background. 4. The Rat io of E f f i c i e n c i e s at 4.43 and 11.68 MeV 11 12 The react ion B (p, t )C has a w e l l known low l y i n g resonance at a proton bombarding energy of 163 kev. (Ajzenberg-Selove and Laur i t sen , 1959)# This populates the 16.11 MeV l e v e l of carbon-12 which decays by gamma emission ei ther d i r e c t l y to the ground state or by cascade through the intermediate low l y i n g 4.43 MeV l e v e l (Ajzenberg-Selove and Laur i t sen , 1959* Figure 16). lilt Q"+ f (fy* t(>3 Key) FIGURE 17 B n ( p , r)G12 Uf3 There i s a one to one r e l a t i o n between the t o t a l number o f 4»43 MeV gamma rays and the t o t a l number of 11.68 MeV gamma rays given o f f . The angular d i s t r i b u t i o n of the 11.68 MeV gamma ray has been found to be proport ional to 1 + (0.26 -0.0l)cos*6 whi le that of the 4.43 MeV gamma ray i s propor t ional to 1 + (0.16 ± 0.02)cos 9 (Grant et a l , 1954) i n good agreement wi th the t h e o r e t i c a l p red ic t ions . - 2 0 -The number of counts expected above the h a l f energy b i a s , Nq, i s N Q = £ ^ u, and so the r a t i o of the 4.43 MeV to 11.68 MeV eff ic iency i s 64(4.43) = N c (4.43) N/11 .68) u^(4.43) ££(11 .68) N ,(4.43) Mc( 11.68) ty.(11.68) where the s o l i d angles are defined to the appropriate effect ive centres. I t i s t o be expected that t h i s r a t i o i s less than one since the t o t a l absorption coeff ic ient of Nal for 11.68 MeV gamma rays i s larger than that for 4.43 MeV gamma rays due t o the rapid increase i n the p a i r production cross sect ion i n t h i s energy range (Grodstein, 1957). Measurements at 118° ± 2° to the proton beam were made by bombarding a th i ck (550 / /«g/cm^) target of boron-11 obtained from Harwell , England, wi th protons of 280-300 kev energy. A representative spectrum i s shown i n Figure 17 wi th the background subtracted. The spectra were analyzed by subtracting background and an average c o n t r i -bution from the 16.11 MeV gamma ray t a i l which was assumed to be f l a t through t o zero energy to obtain the composite 4.43 + 11.68 gamma ray spectrum. An average 11.68 MeV t a i l was then subtracted from the 4*43 MeV por t ion of the spectrum t o obtain a completely separated 4.43 MeV gamma ray spectrum. The numbers of counts above the respective h a l f energy biases were taken and corrected for the d i f f e r i n g angular d i s t r i b u t i o n s . The effect ive centre of the c r y s t a l at 4.43 MeV was taken to be equal t o that at 11.68 MeV. Invest igat ion of t heo re t i c a l e f f ic iency p lo t s showed that t h i s introduced an error of less than 1% (see Chapter 3 ) . Thus Ut{4.43) = ( j / H . 6 8 ) . The r a t i o obtained was 1.055 * .029 which i s much higher than expected. From the r a t io s of the absorption coeff ic ients and t h e o r e t i c a l e f f i c i enc ies at 31.5 cm one expects a resul t c lose to 0.95. -21-Because of the resul t obtained i t was decided to repeat the measurement for a 2-3/4 x 4-1/2 inch c r y s t a l that had been previously invest igated. Here again the r a t i o was found t o be greater than one and more measurements, using a different target and bet ter beam co l l ima t ion , were made. De ta i l s of the resu l t s are presented i n Chapter 4. Suff ice i t to say at t h i s point that the inconsistency of the resul ts i s very d i s turb ing . 5. Summary of 3 x 3 C rys t a l Ha l f Energy Ef f i c i ency Results The resu l t s and probable errors are tabulated i n Table V I together with the measured effect ive centres. A l l effect ive centres are for distances large compared wi th the c r y s t a l length . Figure 18 i s a p lo t of the r e s u l t s . TABLE V I I HALF ENERGY BIAS EFFICIENCIES - 3 x 3 CRYSTAL ENERGY HALF ENERGY EFFICIENCY PROBABLE ERROR EFFECTIVE CENTRE (cm) 0.511 75.1 ± 2.40 2.96 ± 0.50 1.253 60.55 ± 1.61 3.50 ± 0.40 6.14 57.34 * 0.94 3.75 * 0.25 0 I I I I I I I I I I l _ _ L O I Z 3 k S G 7 & 1 10 II Gamma Ray Energy (MeV) -22-CHAPTER I I I THEORETICAL EFFIGIENGIES A. THEORETICAL EFFICIENCY Gamma rays of i n i t i a l in tens i ty I 0 , af ter t ravers ing a thickness of m a t e r i a l , ^ , are found to have an in tens i ty T0S/^\ Thus a f rac t ion / - e-^^ of the gamma rays have undergone at least a s ingle primary in te rac t ion , t rans-f e r r ing an amount of energy to the mater ia l depending upon the type of i n t e r -ac t ion , whether Compton sca t ter ing , photoelectr ic absorption or p a i r production. I f the mater ia l i s a Nal(Tl) s c i n t i l l a t i o n detector then each gamma ray undergoing a primary in te rac t ion w i l l contribute a pulse to the spectrum. The p o s i t i o n of the pulse i n the spectrum w i l l depend upon the energy transferred t o an electron i n the primary interact ions plus subsequent secondary interact ions and losses that take p lace . Since the h a l f energy bias e f f ic iency previously discussed depends upon the secondary in te rac t ions , i t i s not eas i ly ca lculable and therefore i t i s not poss ible t o achieve accuracies better than 15 or 20$ by use of theory at the present t ime. The author feels tha t , at t h i s t ime, i t should be pointed out that i t i s erroneous to compare measured h a l f energy bias e f f i c i enc ies defined to the ef fec t ive centre distances d i r e c t l y wi th the semi-empir ica l theory of P. P. Singh (1959) as was done i n h i s and other theses. The best that one can do i s to determine i f the ef f ic iency defined i n terms of a s o l i d angle subtended by the front face of the c r y s t a l approaches Singh*s value as the source- to-crys ta l face distance increases t o several c r y s t a l lengths. The t o t a l number of counts w i t h i n the spectrum i s not dependent upon the secondary interact ions taking place but depends only upon the number of primary in te rac t ions . I t i s r e l a t i v e l y easy to ca lcu la te the number of counts expected -23-i n the complete gamma ray spectrum and one can base an ef f ic iency d e f i n i t i o n on the t o t a l spectrum i f one makes some assumptions about the spectrum shape i n those regions that cannot be d i r e c t l y observed. We define the t heo re t i c a l e f f ic iency as the t o t a l p robab i l i t y that a gamma ray w i l l in teract wi th the c r y s t a l . For a broad p a r a l l e l beam impinging perpendicular t o the c r y s t a l face the t heo re t i ca l e f f ic iency i s 1 _ e - / ^ £ where! yiA i s the t o t a l absorption coeff ic ient for Nal at a gamma ray energy E y . i s the c r y s t a l length. For an i so t rop ic point source s i tuated on the c r y s t a l ax i s the t h e o r e t i c a l e f f ic iency i s given by o where! r i s the c r y s t a l radius . d i s the source- to-crys ta l face distance. £(<p) i s the gamma ray path length w i t h i n the c r y s t a l . i s the angle the gamma ray makes wi th the c r y s t a l a x i s . This expression i s derived i n Appendix B which includes a discussion of the computer program wr i t t en to carry out the in tegra t ion and tables of resu l t s obtained for three different c r y s t a l s izes and several gamma ray energies. B . EXPERIMENTAL STUDIES OF THE THEORETICAL EFFICIENCY As mentioned i n Chapter 2 i t i s not usual ly poss ib le t o observe a gamma ray spectrum accurately down to zero energy. Thus t o determine the t o t a l number of counts i n a spectrum i t i s necessary t o extrapolate the observed spectrum. The -24-next several pages are concerned wi th the accuracy which one might expect by such an extrapolation,, 1. Theoret ica l E f f i c i enc i e s at 6.14 MeV Since the gamma ray f l ux i s known most accurately at 6.14 MeV t h i s case s h a l l be discussed f i r s t . As wi th most gamma ray spectra, the 6,14 MeV spectrum from exhib i t s a reasonably f l a t t a i l over much of the lower part of the spectrum. At energies from 0 t o 500 kev however, backscattering from lead sh ie ld ing , room wa l l s and equipment d i s to r t s the spectrum. One usual ly sees a charac te r i s t i c backscattering peak at an energy of about 250 kev or l e s s , the height depending upon the geometry of the sh ie ld ing and other experimental d e t a i l s . Also p a i r production i n the lead sh ie ld gives r i s e to ann ih i l a t i on gamma rays of 0.51 MeV at higher incident gamma ray energies which contribute to the experimentally viewed spectrum. The f i r s t extrapolat ion that comes t o mind i s t o extend the f l a t t a i l through to zero energy; the next i s t o draw a l i n e to the o r i g i n of the axes from some convenient pos i t i on on the f l a t t a i l . I t has been noted that many invest igators ( for example Mainsbridge, I960) have done just t h i s and assumed that the t rue number of counts l i e s somewhere between these extrapolat ions. In the present study extrapolations of these two types give resul ts for e f f i c i enc ie s that are much too high and were thus deemed unsuitable. Monte Carlo ca lcula t ions of spectrum shape by M i l l e r and Snow (i960) and Zerby and Moran (1961) i n t h i s energy range have been i n good agreement wi th experimental resu l t s for that part of the spectrum comprising the three p a i r peaks. At energies below these peaks the ca lcu la t ions are much lower than experimental r e s u l t s . The t o t a l number of counts i n the Monte Carlo spectra i s -25-i n agreement wi th the t h e o r e t i c a l e f f ic iency as defined above. I t seems evident then that i n an experimental s i t ua t i on there i s much unexplained scat ter ing adding counts to the lower end of the spectrum and i f any extrapolat ion i s to agree wi th the t heo re t i c a l e f f i c iency i t must be such as t o remove these extra counts. I t was hoped that the Monte Carlo spectrum of Zerby and Moran for a 3 x 3 c r y s t a l at 6 MeV would give a guide by which the experimental curve at 6.14 MeV could be extrapolated to zero energy. Figure 19 shows a comparison between t h e i r spectrum for a c r y s t a l wi th 0.661 MeV reso lu t ion of 7*64$ and an exper i -mental spectrum taken wi th the UBC 3 x 3 c r y s t a l ( reso lu t ion at .661 MeV of 8.01$). It i s in te res t ing to note that the reso lu t ion of the UBC c r y s t a l at 6.14 MeV (3»86$) i s bet ter than that for the Zerby and! Moran c r y s t a l at 6.14 (4.3$)• The gross difference between the spectra cannot be explained by reso-l u t i o n alone, for the experimental spectrum shows far fewer counts i n the p a i r peaks than the Monte Carlo spectrum. I t would then seem j u s t i f i e d to have an extrapolated t a i l containing a la rger percentage of the counts than would be indicated by the Monte Carlo spectrum. Figure 20 shows the 6.14 MeV gamma ray spectrum wi th f i ve different ext ra-polat ions to zero energy; I . Extension of the f l a t t a i l t o zero energy. I I . A s t ra ight l i n e extrapolat ion from the o r i g i n tangent to the experimental spectrum. I I I . - V . Curved l i n e s drawn smoothly in to the p a i r peak and passing through an approximate analy t ic zero intercept . The zero intercept can be determined i n the fol lowing manner. I f one considers a l l in teract ions w i t h i n the c r y s t a l , i t i s evident that Compton scat ter ing i s the only in te rac t ion that can contribute pulses to the T Energy (MeV) Figure 19 Comparison of 6 MeV Monte Carlo Spectrum and 6.14 MeV Experimental Spectrum Note adjustment of energy scales so that photopeaks coincide at 6 MeV. -26-f i r s t channel of the k i ckso r t e r spectrum and then only i n the l i m i t of a s ingle s t ra ight ahead scattering,, I t i s necessary to determine the p robab i l i t y that an incident gamma ray w i l l undergo one s traight ahead Compton sca t ter ing and no other in te rac t ion i n t ravers ing the c r y s t a l . I f we denote ( d [ ) as the absorption coef f ic ien t per unit energy i n t e r v a l for ^ dE o Compton Forward Scat ter ing (CFS) then the gamma ray beam in tens i ty change A I due to t h i s type of scat ter i n a distance A x i s A I = dE I A x where I i s the o r i g i n a l beam i n t e n s i t y . Thus the p r o b a b i l i t y per uni t energy i n t e r v a l of the photon undergoing a CFS i n a distance A x i s given by [ A I \dL\ I and A x represents the p r o b a b i l i t y per uni t energy per uni t path length of the gamma ray undergoing a CFS. I f the t o t a l path length i s then the p r o b a b i l i t y of obtaining only a s ingle CFS event i n i s equal t o the p robab i l i t y of obtaining a CFS times the p r o b a b i l i t y of obtaining no other absorption processes. Thus the p robab i l i t y per MeV of a s ingle CFS i n a distance £ i s diE lo X We can obtain cLE by taking the l i m i t of the Kle in -Nish ina equation i n the ca3e of forward scat ter . The number-energy d i s t r i b u t i o n for Compton electrons per target electron i s given by \ z 2 , d(T\ 2.TC dE cLSl ^ 2 mc 2 0 - cK ( £ + * ) C O S 2 ( p FIGURE 20 Theoretical E f f i c i e n c y Extrapolations at 6.14 MeV -27-wheres (ft i s the e lectron scat ter ing anglec ok = mc = ,5108 MeV, the electron rest mass, and dcr i + o<(/ - cose)]2 / +- COS Z0 + o<2 (i - cosef i -f- <^(i - cose) i s the niuiiber-angle d i s t r i b u t i o n of Compton electrons, 9 i s the photon sca t ter ing angle r c i s the c l a s s i c a l electron radius . See, for example, Davisson and Evans (1952); Evans (1955)<> For the Compton forward scat ter 6 = 0° and (p i s found from the r e l a t i on cot (p = (/ + °<)tanf which gives (f = 90°. : Thus / <jer dSl CFS and dE CFS 2 2 This i s the cross sect ion per MeV per e lectron for CFS, Mul t i p ly ing by the electron density N gives Thus dU. dl dE ZTTr.zNmcz for Nal -28-It follows then that the p robab i l i t y of an event appearing i n the f i r s t channel of the k i ckso r t e r i s given by the average value of ^J=± ^tjtT^"^ m u l t i p l i e d by the channel width i n energy units and the f rac t ion of events appearing i n the first^ channel i s the above divided by the p r o b a b i l i t y that a gamma ray entering the c r y s t a l w i l l undergo any scat ter ing process. For 100$ accuracy i t would be necessary to do a numerical in tegrat ion over a l l path lengths i n the c r y s t a l . However, since Jsj/^^te not a rap id ly varying function over the range of interest (Figure 21) and since the extrapolation shapes only approximate the t rue shapes of the Compton t a i l , a rough estimate of the zero intercept was made. The number of counts i n the spectrum was taken as that defined by l i n e I I , Figure 20 and the f rac t ion of counts i n the f i r s t channel was taken to be where •£> i s the c rys ta l , length . This i s the t rue f rac t ion for a p a r a l l e l beam incident at r ight angles to the c r y s t a l face. A l l extrapolations were made to the zero intercept calculated by t h i s method. The intercept i s probably correct to 15$ for the distances used i n t h i s study. The experimental resu l t s obtained for the t o t a l e f f i c i enc i e s , based upon the extrapolations of Figure 20, for the f ive distances used to determine the h a l f energy b ias e f f i c i ency , are given i n Table V I I I . Absorption i n the c r y s t a l container i s less than 1$ and has been neglected. Theoret ica l resu l t s obtained by numerical in tegra t ion of the t o t a l absorption are also shown, as w e l l as deviations from the t h e o r e t i c a l for the "best" extrapolat ion. It i s evident that a wide v a r i a t i o n i n resu l t s i s poss ible due to the a rb i t ra r iness of T Pa th Length Z (cm) -29-extrapola t ion . As was expected the l i n e g iv ing best resul ts shows a larger percentage of the counts i n the Compton t a i l than would be expected from the spectrum of Zerby and Moran. TABLE V I I I THEORETICAL EFFICIENCY EXTRAPOLATIONS AT 6.14 MEV EXTRAPOLATION & t h ( x l O - 3 ) AT DISTANCE d (cm) d 32.85 56.25. 74.40 106.0 130.0 I 21.22 8.24 4.85 2.44 1.64 I I 17.8 6.93 3.99 2.05 1.38 I I I 16.6 6.45 3.79 1.91 1.28 IV 14.7 5.70 3.36 1.69 1.13 V — Very poor — ANALYTIC 17.38 6.37 3.74 1.89 1.27 ERROR IN "BEST" CURVE ( I I I ) . -4.49$ 1.26$ 1.34$ 1.05$ 0.79$ In order to determine i f sca t ter ing from the lead sh ie ld ing surrounding the c r y s t a l - p h o t © m u l t i p l i e r unit contributed great ly t o the spectrum, runs were made for both the 6.14 MeV radia t ion and for cobalt-60 r ad ia t ion wi th the counter i n and out o f the sh i e ld ing . The spectra obtained are p lo t t ed i n Figures 22 and 23. I t i s evident that at 6.14 MeV the lead shie ld ing does not contribute grea t ly t o the number of counts above 1 MeV and consequently the h a l f energy b ias e f f ic iency at t h i s energy i s not great ly dependent upon the lead cas t le sh i e ld ing . At gamma ray energies of 1.17 MeV and 1.33 MeV the sca t ter ing from the lead i s important i n contr ibut ing counts t o the whole spectrum and i f one i s to use the CO c zs o o 5000 t-000 3000 1000 IOOOX FIGURE 22 E f f e c t of Lead Shielding on Cobalt-60 Spectrum I 17 Ift Pb •5" /'0 Energy (MeV) -p c 3 O o 5"ooo L 2000 — — fo fo 6(83 0-5/ MfeV ' /oooU FIGURE 23 E f f e c t of Lead Shielding on the 6.14 MeV Spectrum 20 JO v o Energy (MeV) - 3 0 -h a l f energy bias e f f i c iency at these energies i t i s imperative that the sh ie ld ing be comparable to that used i n these measurements,. Since removal of sh ie ld ing does not reduce the 6.14 MeV t a i l , the great ly increased number of counts above that expected from the t h e o r e t i c a l e f f ic iency must be due to other scat ter ing i n the room and i n the c r y s t a l container i t s e l f . 2. Theore t ica l E f f i c i enc i e s at 1.28 MeV An attempt was made to f i t curves su i tab le for t h e o r e t i c a l efficiency-comparisons to the spectrum of the 1.28 MeV gamma ray from sodium-22. The gamma f l u x was determined by using the measured h a l f energy bias e f f ic iency at the same energy (Chapter 2) and the extrapolations were made using the approximate zero in tercept . The resul ts are essen t i a l ly the same as for the 6.14 MeV rad ia t ion , that i s , i t i s poss ib le to get w i t h i n 2$ for a su i t ab ly drawn curve but the range of curves gives a range of resu l t s which vary about the calcula ted t h e o r e t i c a l e f f ic iency by 12$. The Monte Carlo spectrum of Zerby and Moran at an energy of 1.275 MeV has a photopeak reso lu t ion of 6.25$ which i s equal to that of the UBC 3 x 3 c r y s t a l at the same energy (6.18 £ .12$). Comparison of the two spectra (Figure 24) i l l u s t r a t e s t h i s fact very w e l l . It would seem then that the extrapolations used to compare wi th the t heo re t i ca l e f f i c i enc ies should follow the Monte Carlo spectrum very c lose ly and indeed such an extrapolat ion gives good r e s u l t s . Because of the obvious prominence of the photopeak i t was f e l t that i t would be easier to use the photofraction f - Counts i n photopeak - N_ p To ta l counts Nq as the c r i t e r i o n for t heo re t i c a l e f f ic iency ca lcu la t ions . The number of counts i n the photopeak was found, and from the calculated absolute e f f ic iency and the ~i : n n ; n h275 Energy (MeV) EIGURE 24 Comparison of 1.275 MeV Monte Carlo Spectrum and Experimental Spectrum -31-known gamma ray f l ux I 0 , the photofraction was calculated so as to give agree-ment between the t o t a l counts i n the spectrum given by and that expected from the t heo re t i c a l e f f ic iency N c = * o £ t h The resul ts for four dif ferent distances are shown i n Table IX „ There was no evidence of any change i n the spectrum shape over t h i s range of distance,. TABLE IX PHOTOFRACTION AT 1.275 MEV d £ t h PHOTOFRACTION 50 .0 cm .961 1 0 - 3 .377 * .011 75.0 .400 1 0 - 3 .412 ± .012 125.0 .169 10" 3 .381 ± .012 150.0 .119 10" 3 .399 * .014 Weighted average .391 * .012 Zerby and Moran .404 * .012 The good agreement between experimental and Monte Carlo resu l t s indicates that one can use the photofraction 0.391 *.012 t o obtain the t o t a l number of counts i n a 1.28 MeV gamma ray spectrum to about 3.5$ accuracy providing s t a t i s t i c s are good. I t i s necessary t o point out that the photofraction l i s t e d above for the -32-Monte Carlo spectrum i s not that l i s t e d i n Table 1, page 13, of the report by Zerby and Moran (1961) but represents the r a t i o o f the area under the photopeak of Figure 22, page 37, of the same a r t i c l e to the t o t a l area i n the spectrum. 3 . Rat io of Theoret ical E f f i c i e n c i e s at 4.43 and 11.68 MeV Because of the wide v a r i a t i o n of resul ts obtained for the r a t i o of —^ - i t was f e l t that any attempt to extrapolate these curves to zero energy i n order to obtain —— '•— would be meaningless, (see Chapter 4 ) . £tk0/-68) C. EFFECTIVE CENTRES FROM THEORETICAL EFFICIENCY CALCULATIONS I f one p l o t s - a / — v s d for large distances d ( y 25 cm) one obtains a V Otk s t ra ight l i n e which crosses the negative d axis at the effect ive centre of the c r y s t a l . For smaller distances the graph deviates from the s traight l i n e and consequently the ef fec t ive centre changes. At close distances, d Q say, the ef fec t ive centre i s that point at which the s t raight l i n e tangent t o the ^/-pr vs d curve at d = d 0 , crosses the negative d a x i s . I t i s expected that the ef fec t ive centre w i l l move c loser t o the c r y s t a l face as the source moves c loser t o the face, for more scat ter ing events w i l l take place i n the front por t ion of the c r y s t a l . Figure 25 shows a p lo t o f vs d for the 3 x 3 c r y s t a l V <StK~ for energies of 6 . I4 , 1.28 and .50 MeV. For large distances the l i n e s are remarkably straight and indeed, on the smaller sca le , no devia t ion from the s t ra ight l i n e i s evident over the whole range of distances. The insert i n Figure 25 shows the deviat ion from the s t ra ight l i n e for small values of d for a gamma ray energy of 6.14 MeV. I f one i s to use the measured h a l f energy bias e f f i c i enc ies for a s c i n t i l l a t i o n detector at short source-to-counter face distances i t i s important 0 10 20 30 40 50 60 70 80 90 100 S o u r c e DCstance (cm) -33-that the change i n effect ive centre be taken in to account. I t should be pointed out tha t , when one assumes that effect ive centres as calculated from absolute e f f ic iency data may be applied d i r e c t l y to h a l f energy bias data, i t i s automatically assumed that the gamma ray spectra shape does not change wi th distance. This should not be so, for edge losses and secondary effects should change as source distances change. However, i n t h i s study, no spectrum shape change for any gamma ray was observed over a range of distances from 5 cm to 2.5 m. Thus the assumption i s probably a good one to w i th in 1$. Table X shows a comparison between the effect ive centres measured i n the laboratory, corrected for the 0.1375 inches of mater ia l between the c r y s t a l face and container face, and the analy t ic ef fect ive centres for large distances. The resu l t s agree to w i th in the rather large experimental e r rors . TABLE X COMPARISON OF ANALYTIC AND EXPERIMENTAL EFFECTIVE CENTRES (3, x 3 CRYSTAL) EFFECTIVE CENTRE E ^ EXPERIMENTAL ANALYTIC 0.511 2.61 ±0.50 cm 2.16 cm 1.28 3 .15* 0.40 3.13 6.14 3 . 4 0 * 0 . 2 5 3.15 -34-CHAPTER IV STUDIES OF A 2-3/4 x 4-1/2 INCH Nal(Tl) CRYSTAL Many researchers i n the Nuclear Physics Group at UBC have employed a 2-3/4 x 4-1/2 inch Nal (Tl ) c r y s t a l for the study of capture gamma rays and p h o t o d i s i n t e g r a t i o n » The h a l f energy bias ef f ic iency for t h i s detector has been previously measured at 0.51> 1.28 and 6.14 MeV by the method outl ined i n Chapter 2, ( G r i f f i t h s , 1958; Singh, 1959; G r i f f i t h s et a l , 1962). A. THE EFFICIENCIES AT 6.14 MEV Results of recent important experiments on the photodisintegrat ion of helium-3 depend grea t ly upon a knowledge of the ef f ic iency of t h i s counter and i t was decided to check the previous measurements at 6.14 MeV and t o see i f t h e o r e t i c a l e f f ic iency extrapolations were consistent wi th those of the 3 x 3 c r y s t a l . The h a l f energy bias ef f ic iency was remeasured by the method f u l l y de ta i led i n Chapter 2. The resul t obtained i s £ | ( 6 . 1 4 ) =(76.7 * 1.8)% which compares extremely w e l l wi th the previous resul t of ( 7 6 . 1 - 1 . 1 ) $ . Figure 26 shows a t y p i c a l 6.14 MeV spectrum wi th a s ingle extrapolat ion to the approximate ana ly t ic zero intercept drawn i n . This was the only extrapo-l a t i o n drawn for t h i s c r y s t a l and the r e s u l t s , tabulated i n Table X I , show that experience i s very important i n obtaining accurate r e s u l t s . +000 3Z00 ZHOO I bOO 800 _ 6 . 1 4 MeV-<40 SO 60 Channel Number 0 I 2 Energy (MeV) FIGURE 26 Theoretical E f f i c i e n c y Extrapolation at 6.14 MeV f o r the 2-3/4 x 4-1/2 Inch C r y s t a l -35-TABLE X I THEORETICAL EFFICIENCIES AT 6.14 MEV 2-3/4 x 4-1/2 CRYSTAL £ t h PROBABLE ERROR DIFFERENCE BETWEEN EXPERIMENTAL ANALYTIC IN EXPERIMENTAL £«.(exp) & 6«.(anal.) 29.25 cm 20.28 1 0 " 4 20.85 10~ 4 ±3.00$ 53.90 7.18 10~ 4 6.91 1 0 - 4 ±2.05$ 76.30 3.66 10-*+ 3.62 1 0 - 4 ±2.20$ 2.73$ 3.90$ 1.10$ The estimated error i n the experimental resu l t s i s based upon the assumption that the extrapolat ion i s the correct one and only s t a t i s t i c a l and measurement errors are included. S t a t i s t i c a l errors include both gamma ray and alpha p a r t i c l e s t a t i s t i c s since the gamma ray f l ux used t o ca lcu la te c?th ^ s based upon the number of alpha pa r t i c l e s counted. Absorption i n the c r y s t a l container i s less than 1$ and i s neglected. B . THE RATIO OF THE EFFICIENCIES AT 4.43 AND 11.68 MEV Several attempts were made t o determine the r a t i o of the h a l f energy bias e f f i c i enc i e s at 4.43 and 11.68 MeV. On three separate runs, comprising at least two different t r i a l s at different distances, resul ts were consistent w i t h i n themselves but deviated great ly from resu l t s of the other runs and from the resu l t s expected by comparison of the ca lcula ted t heo re t i c a l e f f i c i enc ies and by comparison of the respective absorption coef f ic ien ts i n sodium iodide; (11.68) «, .139 cm~ 1 , / ^(4.43) - .128 cm" 1 . Two different targets of boron-11, not previously used, of thicknesses -36-550/4 g/cm and 380 /^g/cm , were bombarded wi th protons of energies from 280 to 300 kev. Measurements were made at 90° and 45° t o the proton beam. I t i s evident that a de ta i l ed study, including angular d i s t r i b u t i o n s , must be made i n order t o determine the cause of the discrepancies. -37-APPENDLX A REPRESENTATIVE SPECTRA FOR THE 3 x 3 CRYSTAL The fol lowing pages contain representative spectra taken wi th the 3 x 3 Nal(Tl ) c r y s t a l for the gamma rays l i s t e d below. The ordinate i s given i n r e l a t i v e counts per MeV which represents the number of counts at energy E i n a unit channel at t h i s energy. Each spectrum has been normalized to 10 at a convenient peak. FIGURE GAMMA RAY SOURCE ENERGY OF GAMMA RAYS 27 Caesium-137 0.661 MeV 28 Sodium-22 0.511, 1.275 29 Cobalt-60 1.173, 1.332 30 Radio Thorium 2.615, .860, .729 .511, .830, .583 .239, .040 32 31 B 1 J - (p , a O C ^ , Ep=300 kev F 1 9 ( p , 0 < , ^ ) o l 6 , Ep=340 k ev 4.43, 11.68 6.14 R e l a t i v e C o u n t s Per MeV O • > V i -J> Cn Co o R e l a t i v e C o u n t s Per MeV O f\j - 4 > On OO O 0 -5 , v  1-0 Energy (MeV) I I I I i i i I i I 03 ^\<, O O . — CO . " ^ 2 o ( J ft ' O O 0 o o o 'O o o o o o o 0 o o o o o o 0 o o o ~o° C O » T l o O M o H * O > 3 ErJ 1 > t o t o V t o c o > - 1 : 1 1 1 1 1 1 1 1 1 1 CD -i DP t< CD do i i i i i i i i i i o o 0 o o CD o C pj — £ o -< V * o - 0 flj o <• CD -s O O o o o o o o _ o o o o O . ' . o o _ o o o° O fy^^^^^ . ^ ^ ^ 0 -5 , v  1-0 Energy (MeV) 1 1 FIGURE 27 Caesiutn-137 1 OO. 1 10 >a 8 2: 1. <0 Q. CO 6 £ 3 O C_) <0 Li > 1 •_J +-> (0 <—< <o " 2 0 1 • 1 1 •! . 1 1 ¥•¥•3 J FIGURE 31 . | B n ( p , / ) C 1 2 0 * • / \ V •. . . . ..^i^i \ -0 * 6 " a 4^ 0 e 0 • V 1 1 1 1 1 N 2 . <i 6 8 10 12 E n e r g y (MeV) 10 >8 « Z s. (1) - 6 «-> c 3 O O > •P (0 r - l CO 0 I 1 ; 1 1 .1 FIGURE 32 F 1 9 ( p , ^ , j r ) 0 1 6 M 1 1 0 • 1 "~ e 0 Q — 0 0 • " * I • . . • • < , • • • . • • „ . • . ' \ 0 0 „ I I 1 1 1 2 3 • b 5 6 Energy (MeV) -38-APPENDLX B THEORETICAL EFFICIENCY OF Nal CRYSTALS A. DERIVATION OF THE EFFICIENCY EQUATION The t heo re t i c a l e f f ic iency of Nal c rys ta l s for gamma ray detection i s defined as the t o t a l p robab i l i t y of a gamma ray emitted by the source lo s ing energy i n the c r y s t a l by Compton sca t te r ing , the photoelect r ic effect or by pa i r production. For a point source si tuated on the axis of the col l imator-absorber-crys ta l system of Figure 33 the gamma ray path lengths i n each mater ia l are functions of the angle of emission cf . The f lux of a beam of gamma rays, emitted at an angle (P , s t r i k i n g the c r y s t a l i s where the subscripts a and c refer to the absorber and co l l imator respec t ive ly , I 0 i s the i n i t i a l i n t ens i ty of the source assuming isotropy i n kTt geometry, and dSl i s the element of s o l i d angle subtended by an element o f c r y s t a l . The gamma ray in tens i ty leaving the c r y s t a l w i l l be where the unsubscripted l e t t e r s refer to the c r y s t a l . Thus the number of counts expected i n the c r y s t a l i s The t o t a l number of counts i s obtained by summing a l l contr ibutions from such elementary beams and i s given by C o l 1 i m a t o r h—TCOL^ H TABS H h TCRY ZCOL H -ZCRY H FIGURE 33 Geometry of C r y s t a l , Collimator, and Absorber f o r T h e o r e t i c a l E f f i c i e n c y Calculations -39-/\/t. = rjs.ll (, dSl HTTr and the p robab i l i t y that a gamma ray leave a pulse i n the spectrum i s The in teg ra l over © may be done immediately since the path lengths do not depend upon t h i s var iab le due to the symmetry of the geometry. Thus - (JJ. A 4- JI. /..) . .. t <d(p where ft = t a n " 1 £ where r i s the c r y s t a l radius and z i s the source-to-counter face distance. The path lengths i n the c r y s t a l ( • / ) , co l l imator (-/ c) and absorber (-/ a) are given as functions of (f i n Table X I I . In t h i s tab le r , z , t represent the c r y s t a l radius, source- tb-crys ta l distance and c r y s t a l length respect ively whi le sub-sc r ip t s c and a refer t o the co l l imator and the absorber. -AC-TABLE X I I CRYSTAL, COLLIMATOR AND ABSORBER PATH LENGTHS A. CRYSTAL CP > t a n - 1 _ L _ 2 - v C3C(P + z sec (/> z + t (D i t a n " 1 r ^ = t sec (f z + t B . COLLIMATOR > t a n " 1 r c ^ _ t c s e c # z c 0 ^ 4 t a n ~ 1 _ J c £c= 0 zc + t c t a n - 1 £S_ t a n " 1 r c a _ ( t p + z J t a n <P - r, z c + t C. ABSORBER s i n (/> c for a l l (/> ^ - t a s e c </> B . NUMERICAL INTEGRATION OF THE EFFICIENCY INTEGRAL The t h e o r e t i c a l e f f ic iency i n t eg ra l was programmed for numerical in tegra t ion be Simpson*s ru le for 100 in te rva ls of dx = cos <p by the IBM 1620 Computer i n the Fortran language. The program has been subsequently al tered so as t o be compatible wi th the requirements for running i t on the UBG IBM 7040 Computer. Data input i s i n the form of a s ingle card for each energy and each geometry for which the t h e o r e t i c a l e f f ic iency i s desired. I t was f e l t that - 4 i -t h i s method would give the greatest v e r s a t i l i t y of input although i t i s not best suited for table p r in tou t s . The var iables i n the program are named as fol lows; r 3 RCRY r c = RCOL z = ZCRY z c = ZCOL t 5 TCRY t c = TOOL t a = TABS = UCRY / L c 5 UCOL ju* = UABS E = ENERGY and data i s punched i n F7.0 format on a s ingle card i n the fol lowing order: Card Columns 1-7 8-14 15-21 22-28 29-35 36-42 Variab le TCRY TCOL TABS RCRY RCOL ZCRY Card Columns 43-49 50-56 57-63 64-70 71-77 78-80 Var iab le ZCOL UCRY UCOL UABS ENERGY blank Distances should be i n centimetres, absorption coeff ic ien ts i n inverse centimetres and gamma ray energy i n MeV. Resul ts .are pr in ted i n the form of a table for each card of input data. The table l i s t s the geometry of the c r y s t a l , co l l ima tor , and absorber together wi th the energy and calculated e f f i c i ency . The time taken on the IBM 7040 i s approximately 54 seconds for compilation plus 0 .9 seconds for each data card, A l i s t i n g of the ac tual program appears at the end of t h i s appendix. -42-C. TABLES OF THEORETICAL EFFICIENCIES The following tables include t heo re t i c a l e f f i c i enc ies for Nal c rys ta l s wi th no gamma ray co l l imator or absorber. TABLE X I I I THEORETICAL EFFICIENCIES - 3 x 3 CRYSTAL SOURCE GAMMA RAY ENERGY DISTANCE (cm) 0.50 1.28 4.43 6.14. 11.68 0.50 297.0 217.0 169.1 168.7 179.9 1.00 235.2 170.7 132.7 132.4 141.2 2.00 155.8 113.0 87.91 87.66 93.49 5.00 60.32 44.64 35.10 35.00 37.24 10.00 21.79 16.63 13.25 13.21 14.02 20.00 6.680 5.238 4.226 4.216 4.461 30.00 3.189 2.532 2.055 2.050 2.166 60.00 .8583 .6921 .5653 .5640 .5952 100.00 .3185 .2586 .2119 .2114 .2229 Notes A l l values i n the above tab le must be mul t ip l i ed by 10~° t o obtain the t h e o r e t i c a l e f f i c i ency . TABLE XIV THEORETICAL EFFICIENCIES - 2-3/4 x 4-1/2 Inch Crystal SOURCE GAMMA RAY ENERGY - MeV DISTANCE (cm) 0.50 1.28 4.43 6.14 8.06 9.17 11.68 0.10 356.1 262.9 206.2 205.7 209.5 212.2 219.0 0.50 281.4 205.5 160.7 160.2 163.3 165.4 170.7 1.00 219.0 159.4 124.7 124.3 126.7 128.3 132.4 2.00 , 141.4 103.5 81. 39 81.17 82.66 83.70 86.34 5.00 52.92 40.20 32.25 32.17 32.72 33.10 34.06 10.00 18.98 15.09 12.39 12.36 12.55 12.69 13.02 20.00 5.835 4.849 4.072 4.063 4.120 4.160 4.259 30.00 2.797 2.374 2.014 2.010 2.037 2.056 2.102 55.00 .8955 .7777 .6663 .6748 .6806 .6950 75.00 .4935 .4322 .3719 .3765 .3797- .3875 100.00 .2824 .2489 .2152 .2148 .2174 .2192 .2237 Note: All values in the above table —3 must be multiplied by 10 to obtain the theoretical efficiencies. TABLE XV THEORETICAL EFFICIENCIES - 5 x 4 CRYSTAL SOURCE GAMMA RAY ENERGY - MeV DISTANCE (cm) 0.50 1.28 4.43 6.14 11.68 2.00. 261.9 205.5 166.8 166.4 175.8 4.00 163.3 127.6 103.4 103.2 109.0 8.00 76.02 60.17 49.16 49.04 51.72 12.00 42.54 34.21 28.16 28.1G 29.59 20.00 18.47 15.19 12.64 12.61 13.25 80.00 1.421 1.221 1.037 1.Q35 1,082 Notes A l l values i n the above tab le must be mu l t i p l i ed by 10~J to obtain the t h e o r e t i c a l e f f i c i e n c i e s . The t o t a l absorption coeff ic ien ts used i n the ca lcula t ions were taken from a graph of the values given i n the tabula t ion by Grodstein (1957) and are l i s t e d below. ENERGY 0.50 1.28 4.43 6.14 8.06 9.17 11.68 ^ . ( c n r 1 ) .331 .185 .128 .127 .130 .133 .139 -45-D. THE COMPUTER PROGRAM $D) 0923 J . LEIGH ^FORTRAN C THIS PROGRAM CALCULATES GAMMA RAY DETECTION EFFICIENCIES FOR C SCINTILLATION CRYSTALS. PRINT 500 500 FORMAT (/44H CALCULATED EFFICIENCIES FOR GAMMA DETECTION) PRINT 700 700 FORMAT (//27X,7HCRYSTAL.7X,10HCOLLIMATOR,4X,8HABSORBER) DIMENSION Y(101), Z( lOl ) C GEOMETRY DATA INPUT 1000. READ 1,TCRY,TCOL,TABS,RCRY,RCOL,ZCRY,ZCOL,UCRY,UCOL,UABS,ENERGY 1 FORMAT (11P7.0) PRINT 401,RCRY,RC0L 401 FORMAT (/10HRADIUS(CM),16X,F10.5,4X,F10.5) PRINT 400, TCRY,TCOL,TABS 400 FORMAT (13HTHICKNESS(CM),13X,F10.4,4X,F10.5,4X,F10.5) PRINT 402, ZCRY,ZCOL 402 FORMAT (19HS0URCE DISTANCE(CM),7X,F10.5,4X,F10.5) PRINT 403,UCRY,UCOL,UABS 403 FORMAT(22HABS0RPTI0N COEF.(CM-l),4X ,F10 .5,4X ,F10 .5,4X ,F10.5) PHIMAX = ATAN(RCRY/ZCRY) PHIA o ATAN(RCRY/(ZCRY + TCRY)) CMAX = COS(PHIMAX) H = (1 .0 - CMAX)*.01 CA = COS(PHIA) C IF TCOLL IS ZERO THE NEXT STEPS ARE SKIPPED IF (TCOL)1000,301,302 302 PHIB = ATAN(RCOL/ZCOL) PHIBB - ATAN(RCOL/(ZCOL + TCOL)) GB . COS(PHIB) CBB . COS(PHIBB) C . CALCULATION OF INTEGRAND VALUES WILL PROCEED. 301 DO 320 I , 1, 101 AI = I COPHI - CMAX + (AI - 1.0>H SIPHI = SQRT(1.0 - C0PHI*»2) IF(SIPHI) 303,304,303 303 SIIN . 1.0/SIPHI 304 COIN = 1.0/COPHI C IT IS NOW NECESSARY TO FIND THE GAMMA PATH LENGTHS IN EACH C ABSORBER, THE PATH LENGTH IN THE CRYSTAL IS DETERMINED FIRST. IF (COPHI - CA) 3,4,4 3 YCRY = 1 .0 - EXP(-UCRY*(RCRY*SIIN - ZCRY»COIN)) GO TO 51 4 YCRY . 1.0 - EXP(-UCRY*TCRY*COIN) C IF NO COLLIMATOR THE FOLLOWING CALCULATIONS ARE SKIPPED. 51 IF (TCOL ) 1000,9,- 5 5 IF (COPHI - CBB) 6,9,9 6 • IF (COPHI -CB) 7, 7, 8 7 YCOL . EXP(-UCOL*TCOL*COIN) -46-GO TO 10 8 YCOL = EXP(((TCOL + ZC0L)*SIPHI*C0IN - RC0L)*SIEJ*(-UC0L)) GO TO 10 9 ICOL = 1 .0 C IF THERE IS NO ABSORBER THE EXPONENTIAL CALCULATION IS SKIPPED 10 IF (TABS) 1000,41, 40 40 YABS = EXP(-UABS*TABS»COIN) GO TO 320 41 YABS . 1 . 0 C INTEGRAND IS NOW FORMED 320 Y( I ) = YABS* YCOL* YCRY C INTEGRATION BY SIMPSON,S RULE STARTS SUMI = 0.0 DO 11 J • 1, 49 N . 2*J M - N + 1 11 SUMI . SUMI + 4.0*Y(N) - 2.0*Y(M) HI = H*.33333333 AREA1 = H1*(Y(1) + SUMI + 4»0*Y(lOO) + Y ( l O l ) ) EFFIC = . 5*AREA1 PRINT 12,ENERGY,EFFIC 12 FORMAT(13HEFFICIENCY AT,F9.4,7H MEV IS,E14 .8) GO TO 1000 END $DATA 7.62 0.0 0.0 3 .81 0.0 10 .0 0.0 . 212 0.0 0. 1.0 -47-BBLIOGRAPHT Ajzeriberg-Selove, F. A. and Laur i t sen , T. (1959), Nuclear Physics 11, 1. Bleu le r , E . and Zunt i , W. ( 1946) , Helv. Phys. Acta 19, 375. &-adt» H . , Gugelot, P. C , Huber, 0 , , Medicus, H . , Preiswerk, P. and Scherre, P. ' ( 1 9 4 6 ) , Helv. Phys. Acta l j , 77. Chao, C. Y . ( 1 9 5 0 ) , Phys. Rev. 80, 1035. Davis son, C. M. and Evans, R. D. ( 1952) , Rev. Mod. Phys. 2Zj., 79. Devons, S. and Hine, M. G. N . (1949), Proc. Roy. Soc. (London) 199A, 56. Dosso, H. H. (1957) , M. A. Thesis, Univers i ty of B r i t . C o l . E l l i s , C. D . , Wooster, W. A. and Dodds, J . M. ( 1952) , P h i l , Mag. j>0, 521. Evans, R. D. (1955), The Atomic Nucleus, McGraw-Hill Book C o . , I nc . , New York. Fowler, W. A . , Laur i t sen , C. C , and Laur i t sen , T. ( 1 9 4 8 ) , Rev. Mod. Phys. 2 0 , 236. Freeman, J . M. ( 1 9 5 0 ) , P h i l . Mag. 4_1, 1225. Grant, P. <J., Flack, F. C , Rutherglen, J . G. and Deuchars, W. M. ( 1954) , Proc. Phys. Soc. A67, 751. Gray, L . H. ( 1936) , Proc. Roy. Soc. A15_6, 578. G r i f f i t h s , G. M. (1958), Proc. Phys. Soc. 72, 337. G r i f f i t h s , G. M . , Larson, E . A. and Robertson, L . P. ( 1 9 6 2 ) , Can. J . Phys. 46, 402. Grodstein, G. W. ( 1957) , Nat»l Bureau of Standards C i r c . 583. Hofstadter, R. ( 1 9 4 8 ) , Phys. Rev. 7Jt, 100. Kallman, H. ( 1947) , Natur u Technik. Kreger, W. E . ( 1 9 5 4 ) , Phys. Rev. 96, 1554. Larson, A 0 G. L . ( 1957) , M. A. Thesis , Univers i ty of B r i t . C o l . Main3bridge, B. ( i 9 6 0 ) , Austr . J . Phys. 13, 204. Mainsbridge, E . ( i 9 6 0 ) , Nuclear Physics 2 1 , 1. M i l l e r , W. F. and Snow, W. J , ( i 9 6 0 ) , Rev. S c i . Inst . 3_1, 39. - 4 8 -Rose, M. E . (1953), Phys. Rev. 91, 610" Singh, P. P. (1959), Ph. D. Thesis , Univers i ty o f B r i t . C o l . Van A l l e n , J . A. and Smith, H. M. (1941), Phys. Rev. 69, 501. V i l l a r d , P. (1900), Comptes Rendue 130. 1178. Yuan, L . C. L . and Wu, C. (1963) E d . , Methods of Experimental Physics, V o l . 5, Nuclear Physics , Pb. A. Zerby, C. D. and Moran, H. S. (1961), Oak Ridge National Laboratory Report 3169 <-

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