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Some studies in Beta-Gamma and Gamma-Gamma-Angular correlation Darby, Edsel Kenneth 1952

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T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A F A C U L T Y OF G R A D U A T E STUDIES P R O G R A M M E OF T H E F I N A L O R A L E X A M I N A T I O N F O R T H E D E G R E E OF D O C T O R OF P H I L O S O P H Y of E D S E L K E N N E T H D A R B Y B.Sc. (University of Saskatchewan) 1946 M.Sc. (University of Saskatchewan) 1948 T H U R S D A Y , APRIL 24th, 1952, at 3:00 P.M. IN R O O M 303 PHYSICS BUILDING C O M M I T T E E I N CH A R G E : Dean H . F. Angus, Chairman Professor W. Opechowski Professor M. Kirsch Professor G. M . Shrum Dean Blythe Eagles Professor J. B. Warren Professor F. Noakes Professor C. A: Barnes Professor A. Earle Birney LIST OF PUBLICATIONS Negative Feedback. Dosage Rate Meter using a very small Ionization Chamber. H . E. Johns, E. K. Darby, J. J. S. Hamilton. American J. Roent-genology and Radium Therapy, 61, 550 (1949). Depth Dose Data and Isodose Distributions tor Radiation from a 22 Mev Betatron. H . E. Johns, E . K. Darby, R. N. H . Hawlour, L . Katz and E. L. Harrington, American J. Roentgenology and Radium Therapy, 62, 257 (1949). Dosage Distributions Obtainable with 400 KVP X-rays and 22 Mev X-rays. H . E . Johns, E . K. Darby, I. A. Watson, C. C. Barkell. Br. J. Radiology, 23,290 (1950). Depth Dose Distributions near Edge of X-ray Beam. H . E. Johns, E . K. Darby. Br. J. Radiology, 23, 193 (1950). A Radon Measuring Device. E . K. Darby, H . E. Johns. Am. J. Roentgenology and Radium Therapy, 64, 472 (1951). The Betatron in Cancer Therapy. H . E. Johns, E. K. Darby et al. Presented by H . E. Johns at 6th International Congress of Radiology, London, July 23 - 26, 1950. Ultra-Violet Photon Counting with Electron Multiplier. G. W. Williams, A. H . Morrish and E. K. Darby. R.S.I.,'21, 884 (1950). Energy Dependence of the Beta-Gamma Angular Correlation in Sb 1 2 4. E. K. Darby and W. Opechowski. Phys. Rev. 83, 676 (1951). Some Studies in Angular Correlation. E. K. Darby. Can. J. of Physics 29, 569 (1951) . T H E S I S SOME STUDIES^IN B E T A - G A M M A A N D G A M M A - G A M M A A N G U L A R C O R R E L A T I O N Abstract: The beta-gamma angular correlation for Sb 1 2' has been measured as a function of the beta particle energy in the range from 1.0 Mev to the end of the beta particle spectrum (2.4 Mev.) . As a beta particle spectrometer, use was made of a twelve channel kicksorter and a thick crystal beta particle scintillation counter. This was connected in coincidence with a gamma ray scintillation counter. Accordingly, the beta gamma coincidence counting rate W (Q, E), as a function of the angle 6 between the counters, and the energy E of the beta particles, was observed. The differential angular correlation coefficient W (180°, E) —W (90°, E) a (E) = W (90°, E) was found to vary smoothly from —0.17 at 1.0 Mev to —0.44 at the end of the beta particle spectrum. When a (E) is integrated, numerically, over all beta particle energies greater than 1 Mev, the value of the integrated angular correla-tion coefficient a = —0.24 + 0.02 so obtained, agrees with the value a = —0.23+0.01 which was measured directly. An attempt has been made to interpret these results in terms of the angular momenta of the particles emitted, using the theory developed by Falkoff and Uhlenbeck. Experiments on the gamma-gamma angular correlation of Co 0 0 and Sc4" performed with the same apparatus are in agreement with the previous results of other workers. G R A D U A T E STUDIES Field of Study: Physics Thermodynamics—Prof. C. A. McKay Spectroscopy—Prof. W. Petrie Electromagnetic Theory—Prof. L . Katz Nuclear Physics—Prof. H . E. Johns Quantum Mechanics—Prof. G. M. Volkoff Theory of Measurements—Prof. A. M. Crooker Radiation Theory—P,rof. F. A. Kaempffer Electronics—Prof. A. Van der Ziel Special Relativity—Prof. W. Opechowski j Other Studies: Advanced Differential Equations—Prof. T . E. Hull Radiochemistry—Prof. M . Kirsch and Dr. K. Starke SOME STUDIES IN BETA-GAMMA- AND GAMMA-GAMMA- ANGULAR CORRELATION by EDSEL KENNETH DARBY A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n PHYSICS We accept t h i s t h e s i s as conforming to the standard required from candidates f o r the degree of DOCTOR OF PHILOSOPHY. Members of the Department of Physics. THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1952. ABSTRACT The beta-gamma angular co r r e l a t i o n f o r Sb-^4 has been measured as a function of the beta p a r t i c l e energy i n the range from O.B'2 Mev to the end of the beta p a r t i c l e spectrum (2.4 Mev). As a beta p a r t i c l e spectrometer, use was made of a twelve channel kicksorter and a t h i c k c r y s t a l beta p a r t i c l e s c i n t i l l a t i o n counter. This was connected i n coincidence with a gamma ray s c i n t i l l a t i o n counter. Accordingly, the beta gamma coincidence counting rate W(0,E), as a function of.the angle 0 between the counters, and the energy E of the beta p a r t i c l e s , was observed. The d i f f e r e n t i a l angular c o r r e l a t i o n c o e f f i c i e n t : a(E).Jw(l80 O,E) - W(90 o,En L W(90°,E) J was found to vary smoothly from -0.17 at 1.0 Mev to -0.44 at the end of the beta p a r t i c l e spectrum. When a(E) i s integrated, numerically, over a l l beta p a r t i c l e ener-gies greater than 0.82- Mev., the value of the integrated angular co r r e l a t i o n c o e f f i c i e n t a z -0.24 — 0.02 was found. Direct measurements of the value of the integrated angular c o r r e l a t i o n c o e f f i c i e n t were also performed, and the r e l a t i o n to the above value &£ a considered. An attempt has been made to interpret .these r e s u l t s i n terms of the angular momenta of the p a r t i c l e s emitted, using the theory developed by Falkoff and Uhlenbeck. Experiments on the gamma-gamma angular correlation of Co^° and S c ^ performed with the same apparatus are i n agreement with the previous r e s u l t s of other workers. TABLE OF 'CONTENTS Page I Introduction. 1 II, Theory of Gamma-Gamma-Angular Correlation 7 III Gamma-Gamma Angular Correlation f o r Co^° and Sc^ b 11 IV Theory of Beta-Gamma-Angular Correlation 16 V Apparatus f o r Measuring the Integrated Beta-Gamma-Angular Correlation 19 VI Integrated Beta-Gamma-Angular Correlation f o r S b 1 2 ^ 21 VII Beta Gamma Angular Correlation as a Function of Energy f o r S b 1 2 * 24 VIII Results and Conclusions Regarding the D i f f e r e n t i a l Beta-Gamma-Angular Correlation f o r Sb12**- 28 TABLE OF APPENDICES Page Appendix I Appendix II Appendix I I I Appendix IV Appendix V Appendix VI Appendix VII Appendix VIII Appendix IX Appendix X Appendix XI Coincidence Counting with Direc-t i o n a l Correlation Between the Emitted P a r t i c l e s . 21 Procedure to Evaluate the True Coincidence Rate 34 Compton Scattering of Gamma Rays Between Coincidence Counters 37 Standard Deviation i n W(0) 39 C a l i b r a t i o n of the Kicksorter Channels i n Terms of Beta Energy 40 Calculation of the Beta-Gamma-Angular Correlation C o e f f i c i e n t . 41 Standard Deviation i n a(E) 44 Calculation of a From a(E) 45 Evaluation of Theoretical Beta-Gamma-Angular Correlation C o e f f i c i e n t s 47 E l e c t r o n i c C i r c u i t s 49 On the Influence of Absorption on the 50 Measured Value of a. Table I Table I I Table I I I Table IV Table V Table VI Tables VII Table VIII TABLES The Gamma-gamma-angular co r r e l a t i o n function f o r C o 6 0 The gamma-gamma-angular c o r r e l a t i o n function f o r Se^° The beta-gamma-angular c o r r e l a t i o n function f o r Sb* 2^ integrated over beta energies 0 . 9 - 2 .4 Mev. Averaging of d i f f e r e n t i a l beta-gamma-angular c o r r e l a t i o n c o e f f i c i e n t a(E) fo r S b 1 2 4 (Runs 1 to 4) Calculation of integrated angular • cor r e l a t i o n c o e f f i c i e n t a from the re s u l t s of a(E). Values of a(E 0) f o r possible t r a n s i t i o n s . Fermi Plot of Beta Spectrum a(E) f o r S b 1 2 4 (Run 4) Page 14 14 22 26 27 29 40 42 FIGURES Page F i g . 1 Disintegration scheme f o r C o D U 1 Fi g . 2 Coincidence counting 1 F i g . 3 General notation f o r successive nuclear t r a n s i t i o n s 7 Fig. 4 Counter arrangement f o r gamma-gamma-angular c o r r e l a t i o n 11 Fi g . 5 Coincidence counting c i r c u i t block diagram 11 F i g . 6 Lead Shielding to prevent sc a t t e r i n g 12 F i g . 7 E f f e c t of an n i h i l a t i o n r a d i a t i o n on W(0) 13 F i g . 8 Gamma-Gamma-Angular Correlation f o r C o 6 0 14 F i g . 9 Gamma^Gamma-Angular Correlation f o r Sc4o F i g . 10 Apparatus f o r Beta-Gamma Angular Correlation 19 F i g . 11 Disintegration Scheme f o r S b 1 2 ^ 21 F i g . 12 Integrated beta-gamma angular c o r r e l a t i o n f o r Sb 1 24 22 Fi g . 13 Apparatus f o r D i f f e r e n t i a l Beta-gamma-angular c o r r e l a t i o n 24 F i g . 14 The Four Runs f o r a(E) f o r Sb 1 2 / f 26 Fig . 15 The Averaged Results f o r a(E) f o r Sb12/<- 26 Fi g . 16 Counter Geometry 31 Fig . 17 Fermi Plot of the Beta Spectrum 40 ACKNOWLEDGEMENTS I am indebted to Professor ¥. Opechowski f o r valuable discussions i n the course of these experiments, and f o r help i n in t e r p r e t i n g the experimental r e s u l t s . I am also indebted to Dr. A. H. Morrish f o r his suggestions i n the early stages of the gamma-gamma co r r e l a t i o n experiments and to Mr. G. W. Williams f o r h i s assistance i n design and construction of the apparatus. I also wish to acknowledge the kind assistance of other s t a f f members of the Physics Department and the excellent laboratory f a c i l i t i e s supplied by the University of B r i t i s h Columbia Physics Department. This research was made possible by a grant from the National Research Council of Canada, and the author was aided by a Fellowship from the National Research Council of Canada. INTRODUCTION. The emission of a beta p a r t i c l e from a radioactive nucleus often leaves the product nucleus i n an excited state. The excited nucleus may then decay, by the emission of one or more gamma rays i n cascade, to the ground state. The di s i n t e g r a t i o n fit) scheme of Co (Fig. 1A) i s a simple example of two gamma rays i n cascade. The l i f e t i m e of the intermediate excited states i s i n -12 general very small (10 sec.) so the two gamma rays may be regarded as simultaneous for most purposes. I f we arrange two gamma ray counters near the Co source (Fig. 2) and connect them to a c i r c u i t designed to r e g i s t e r two simultaneous pulses (coin-cidence c i r c u i t ) , we w i l l be able to obtain a double coincidence  counting rate. This coincidence rate i s much smaller than the single counting rate i n either counter since the p r o b a b i l i t y of counting both gamma rays simultaneously i s small. I f , instead of having two gamma rays i n cascade, we have only one gamma ray following the emitted beta p a r t i c l e (Fig. IB),- we can obtain coincidence counts by using one beta p a r t i c l e counter and one gamma counter. Now the atoms i n the radioactive source* are i n t h i s case randomly oriented so i n the double-gamma cascade the f i r s t gamma-ray i n the cascade i s emitted i n an e n t i r e l y random The l i f e t i m e of the i n i t i a l state must be long enough f o r the angular momentum vectors of the nuclei to become randomly orien-ted, either by precession i n a magnetic f i e l d due to o r b i t a l electrons or by thermal agitation. C o 6 0 Z N . 6 0 i z 1.17 Mev. G a n n n a Rcny ' • •• < 1.33 Mev. > F I G . 1 A B D i s i n t e g r a t i o n S c h e m e s G a m m a Rays • Z P h o t o - C o 6 S o u r c e M u l t i p l i e r a n d Crysfa Co i n c i o l e n c e M i x e r S e a e r F I G - 2 C o i n c i d e n c e C o u r f t i n g 2. di r e c t i o n , however the second gamma-ray i n some cases may have i t s d i r e c t i o n of emission correlated with respect to the d i r e c -t i o n of emission of the f i r s t gamma-ray. This e f f e c t i s c a l l e d the angular c o r r e l a t i o n between the di r e c t i o n s of the two gamma rays, or more b r i e f l y , gamma-gamma angular c o r r e l a t i o n . We can measure t h i s gamma-gamma angular correlation-by observing the gamma-gamma coincidence rate as a function of 0, the angle between the counters. The gamma-gamma coincidence rate as a function of 0(after a small correction f o r f i n i t e counter size) i s propor-t i o n a l to the gamma-gamma angular c o r r e l a t i o n function, W(9), which i s the pro b a b i l i t y of the emission of the second gamma ray i n d i r e c t i o n 0 with respect to the f i r s t gamma-ray's d i r e c t i o n of emission. In a similar manner i f we observe the beta-gamma coincidence rate we sometimes observe a beta-gamma-angular  co r r e l a t i o n . The researches described i n t h i s t h e s i s concern the beta-gamma- and gamma-gamma- angular c o r r e l a t i o n observed i n several radioactive n u c l e i . As an example of the information which i s availa b l e from these experiments, l e t us consider the case of the gamma-gamma angular c o r r e l a t i o n i n more d e t a i l . Gamma ray t r a n s i t i o n s are c l a s s i f i e d on the basis of the angular momentum ca r r i e d away by the gamma ray. The multipole order of the t r a n s i t i o n i s 2^ where L i s the z-component of the angular momentum ca r r i e d away by the gamma ray. Thus i f the gamma ray has angular momentum L = l (in units of "ft), the t r a n s i t i o n i s c l a s s i f i e d as dipole. The t r a n s i t i o n s are c l a s s i f i e d further according to whether or not there i s a change of pa r i t y , that i s whether or not there i s 3. a change i n the symetry properties of the wave-functions of the two l e v e l s . In the case of dipole r a d i a t i o n i f the p a r i t y changes the t r a n s i t i o n i s c a l l e d e l e c t r i c dipole, i f i t does not change, i t i s c a l l e d magnetic dipole. I f we consider only the case of "pure" multipole t r a n s i t i o n s (that i s only one angular momentum associated with a given t r a n s i t i o n ) then the theory indicates that the form of the angular-correlation function i s as follows: X \ I whereZ . A i s the lowest multipole order of the two t r a n s i t i o n s . The c o e f f i c i e n t s d £ are functions of the multipole orders of the t r a n s i t i o n s , and of the angular momenta of the nuclear l e v e l s involved i n the t r a n s i t i o n s . For many cases tables ( 2 ) have been prepared which give numerical values for the d j . Experi-mentally, we can determine H Ck'L quite accurately, and.the values of the i n d i v i d u a l d{ approximately. . I f we have available some other information concerning either the angular momentum of the le v e l s or the multipole -order of the t r a n s i t i o n s , we can compare the experimental data with the tables, and usually obtain a d e f i n i t e assignment of angular momentum and multipole orders to the l e v e l s and tr a n s i t i o n s involved. As an example of the" procedure i n in t e r p r e t a t i o n of the data l e t us consider the case of Co^. The measurements f o r C o o Q give the c o r r e l a t i o n c o e f f i c i e n t d; - 0.1 5 7 - 0.01 Moreover from the graph (Fig. 8) of the c o r r e l a t i o n function we see that the function i s probably of the form .' 2 4" Although the accuracy of the r e s u l t s does not permit us to 4 . determine d, ^  ^ s e p a r a t e l y , the f a c t that there i s a term i n Cos^Q means that both t r a n s i t i o n s are of at le a s t quadrupole order. We have one other piece of information, namely that the angular momentum of the ground state i s probably 0 } since the product nucleus has an even number of both protons and neutrons. The tables may now be searched quite quickly and i t i s seen that the c o r r e l a t i o n function f o r the double t r a n s i t i o n * 2 2 4 — > 2 — * 0 i s W(0) = 1 -h 0.125 c o s 2 9 - + 0 . 0 4 0 eos^Q. This gives ^ s @t * &z ~ + 0.165 i n quite close agree-ment with the experimental value 0.157 * 0 . 0 1 0 . We conclude therefore that the angular momenta and multipole orders f o r the t r a n s i t i o n involved are very l i k e l y to be given by the following 2 -2. scheme 4 — ^ 2 — » 0 . The r e s u l t s f o r beta-gamma angular c o r r e l a t i o n may be interpreted i n a s i m i l a r fashion, leading to information *The angular momentum of a nuclear state i s often re f e r r e d to as the "spin". Where no confusion should a r i s e the term spin w i l l be used f o r the sake of brevity. The t r a n s i t i o n scheme 2. a 4 — ^ 2 —-9 0 . i s to be interpreted as follows: the i n i t i a l state has a spin of 4 ( i n u n i t s of -tl), and a t r a n s i t i o n takes place to an intermediate state having spin 2. This t r a n s i t i o n i s accompanied by a gamma ray having angular momentum 2 so i t i s c l a s s i f i e d as quadrupole. ;A t r a n s i t i o n then takes place from the intermediate state t o the ground state with spin 0 , t h i s t r a n s i t i o n also being c l a s s i f i e d as quadrupole. regarding the spins of the l e v e l s involved, the multipole orders of the gamma t r a n s i t i o n . The o r i g i n a l suggestion that gamma-gamma-angular co r r e l a t i o n might exi s t was due to Dunworth (1); Following t h i s suggestion, Hamilton (2) i n 1940 presented the theory of the e f f e c t giving the r e s u l t s that have just been outlined. Attempts (3) (4) to detect an angular c o r r e l a t i o n f a i l e d and i t was not u n t i l 1948 when Brady and Deutsch (5) reported the successful detection of a gamma-gamma angular c o r r e l a t i o n i n four n u c l e i . Their success was due l a r g e l y to the development of highly e f f i c i e n t gamma counters using c r y s t a l phosphors and photo-multipliers. During 1949 and 1950 numerous observers (6)(7)(8) reported confirmation of these r e s u l t s and new r e s u l t s , f o r other n u c l e i . Further d e t a i l s of the theory had also been given by Goertzel (9) (effect of an external magnetic f i e l d ) , and Ling and Falkoff (10) (effect of "mixed" multipole transitions).* Although many nuclei have exhibited a gamma-gamma-angular c o r r e l a t i o n , numerous attempts to detect a beta-gamma-angular c o r r e l a t i o n resulted i n f a i l u r e , (11)(hi)(13)(14)• By 1951 only two nu c l e i (Sb 1 24 j Rb^6) had yielded a d e f i n i t e beta-gamma-angular c o r r e l a t i o n (11)(12)(13), moreover the r e s u l t s reported were not consistent. S u f f i c i e n t theory to allow a reasonable inte r p r e t a t i o n of re s u l t s i n most cases had been developed i n 1950 by Falkoff and Uhlenbeck (15). Further inves-t i g a t i o n into the beta-gamma-angular c o r r e l a t i o n e f f e c t was required. The main o r i g i n a l contributions of the researches described i n t h i s thesis are the experiments on the beta-gamma-angular c o r r e l a t i o n i n Sb-^2^. In p a r t i c u l a r since the beta 1 p a r t i c l e s have a continuous energy d i s t r i b u t i o n , the dependence of the angular c o r r e l a t i o n on the energy of the beta p a r t i c l e s was the object of t h i s investigation. Such an energy dependence was predicted t h e o r e t i c a l l y by Falkoff and Uhlenbeck (15) but had not been observed previous to t h i s i n a quantitative way. The re s u l t s of t h i s in v e s t i g a t i o n have been .published, f i r s t i n a preliminary report (16) and l a t e r i n a : f u l l paper (17). Similar r e s u l t s have l a t e r been published by Stevenson and Deutsch (18) who used an e n t i r e l y d i f f e r e n t experimental method, the agreement with the present work being quite good. The gamma-gamma co r r e l a t i o n experiments were not new, Brady and Deutsch (5) having observed t h i s previously. However the accuracy obtained i n the present work i s somewhat better than i n the previous work. Moreover, the same apparatus was employed l a t e r i n the beta-gamma co r r e l a t i o n experiments, so the consistency of the gamma-gamma co r r e l a t i o n experiments lends support to the subsequent beta-gamma c o r r e l a t i o n r e s u l t s . 7. II . THEORY OF GAMMA-.GAMMA-ANGULAR CORRELATION. The theory of d i r e c t i o n a l c o r r e l a t i o n between successively emitted nuclear p a r t i c l e s has been developed i n considerable d e t a i l i n several papers, (2)(9)(10). The problem involves the determination of W(0), the r e l a t i v e p r o b a b i l i t y of the emission of p a r t i c l e 2 i n di r e c t i o n k 2 following the emission of p a r t i c l e 1 i n d i r e c t i o n k^ _5 the angle between k^ and "k^  being 9 and the t r a n s i t i o n taking place from A — ? B, B — ? C as shown i n F i g . 3. Hamilton (2) has applied second order time dependent perturbation theory to the gamma-gamma problem and obtained W(9) i n terms of the spins of the nuclear states involved and angular momenta of the emitted photons. Hamiltons theory involves two assumptions, namely: 1. That the l i f e t i m e of the intermediate state i s short compared with the nuclear precessional period. 2. That the t r a n s i t i o n s are "pure" multipole t r a n s i t i o n s and not due to a mixture of multipoles. Goertzel (9) has extended Hamilton's theory to cover the case i n which there i s a strong magnetic f i e l d present so that the f i r s t assumption above i s not f u l f i l l e d . Ling and Falkoff (10) have extended the theory to take into account mixtures of multipoles. Falkoff and Uhlenbeck (15) have generalized Hamilton's theory ( s t i l l retaining h i s assumptions however) and obtained W(0) i n parametric forms f o r a r b i t r a r y emitted p a r t i c l e s . S p e c i f i c a t i o n of the types of emitted p a r t i c l e s determines the parameters and gives W(Q) f o r the required type of co r r e l a t i o n . Choosing k^ as the axis of quantization the general r e s u l t of the theory (2) (15) il.s:, X Z [ 5 2 l ( B j H ^ ) | C p ) r j where IfS. I H. Lwl L p J denotes .the matrix element of the Hamiltonian describing the i n t e r a c t i o n of the second emitted p a r t i c l e and the nucleus f o r the t r a n s i t i o n from the substate m of the (nuclear) state B (a state as characterized by a d e f i n i t e value of the t o t a l angular momentum quantum number) to the substate p of the state C. The meaning of ^ /\^ > j |~J|(d)| ^>yy\ i s of course similar. S-^  represents the average o v e r a l l d i r e c t i o n a l information (spins, polarization) f o r the f i r s t t r a n s i t i o n except of course the d i r e c t i o n k-^  with a sim i l a r meaning f o r S£. The quantity .' S , K A J H l ( W l B j r = , < 2 ) i s the pro b a b i l i t y of the emission of the f i r s t p a r t i c l e along the axis of quantization f o r the t r a n s i t i o n between sublevels Aj^ B m • Pkv^ p &) . i s defined s i m i l a r l y so that ¥(0) i s given by ' In general considering a t r a n s i t i o n between states with t o t a l and Z- component angular momenta ~J}yn and ~$ Yy\ respectively and the emission of a p a r t i c l e having angular momen-tum quantum numbers L and M group t h e o r e t i c a l methods show that, 9. can be written , J L T _ n (4) (5) with • W\ + M a nol j ' * T + L . ^ J L J ' Where the C-y , when evaluated y i e l d numerical values depending only upon these angular momenta, and not upon the kind of p a r t i c l e emitted and the [ are the angular d i s t r i b u t i o n s which depend on the type of p a r t i c l e emitted. I t turns out that i f a l l the i n i t i a l sublevels are equally populated, and the d i r e c t i o n of emission of the p a r t i c l e i s specified, then the intermediate sublevels w i l l i n general not be equally populated. Since the intermediate sublevels are not equally populated, the second p a r t i c l e w i l l not l i k e l y have an i s o t r o p i c d i s t r i b u t i o n , thus exhibiting an angular c o r r e l a t i o n . The J- ^  may be expressed i n suitable parametric forms, i n t h i s case powers of cos 0 . The maximum number of independent parameters i s L. Falkoff and Uhlenbeck (15) obtain several convenient parametric forms. The parameters are specified according to the type of p a r t i c l e emitted. It i s then ~ J L J ' only necessary to evaluate the (j- ^  , which are independ-ent of the type of p a r t i c l e emitted, and then to obtain the ¥(0) as i n formula (3). The required sums have been evaluated by Hamilton (2) and are l i s t e d by Falkoff and Uhlenbeck (15) f o r L — 1 or 2. For higher values of angular momentum quantum numbers (of the nuclear states and the emitted p a r t i c l e s ) the computations of these sums become very d i f f i c u l t , and they have 10. been evaluated only f o r a ce r t a i n number of special cases. ' For example Lloyd (19) and Hess ( 2 0 ) . The gamma-gamma angular c o r r e l a t i o n functions have been tabulated i n a "canonical form" by Hamilton ( 2 ) . That i s , he writes the c o r r e l a t i o n function i n the form: and gives tables of R, Q and S as functions of the angular momenta involved. The same r e s u l t s may be obtained r e a d i l y from the tables of Falkoff and Uhlenbeck (15) by i n s e r t i n g the proper parameters for the gamma-gamma t r a n s i t i o n s . The following general r e s u l t s of the theory f o r gamma-gamma co r r e l a t i o n may be l i s t e d : j_ . 1. The co r r e l a t i o n function i s of the form ; W ( £ ) = l + 2 where L i s the minimum of L-^ , l>2-2. The c o e f f i c i e n t s 01 1 depend on the angular momenta of the three l e v e l s and on the multipole order of the t r a n s i t i o n but not on parity change. 2 3 . The highest power of cos 9 cannot exceed 2-J so there i s no cor r e l a t i o n i f the intermediate state has a spin of 0 , or £. . ' 11. I I I . GAMMA-GAMMA-ANGULAR CORRELATION IN C o 6 0 AND S c 4 6 . The d i s i n t e g r a t i o n scheme of Co^O which consists of a 0.31 Mev beta group followed by cascade gamma rays of energies 1.17 Mev and 1.33 Mev i s given i n Fig . 1 (21). Sc^ 6 has a similar decay scheme, (21), with a maximum beta p a r t i c l e energy of O.36 Mev, and cascade gamma rays with energies of 0.88 Mev and 1.12 Mev respectively. The Co source consisted of a wire of cobalt metal about '0.1 mm i n diameter and 4 mm i n length which had been i r r a d i a -ted i n the Chalk River p i l e to an a c t i v i t y of about 30,0 yu curies. The source:- was supported v e r t i c a l l y and enclosed i n a s u f f i c i e n t thickness of aluminium to stop a l l Co^O beta p a r t i c l e s . The Sc^ source consisted of ScO i n an aluminium capsule. The gamma counters employed 1 x 1 x i i n . anthracene c r y s t a l s cemented to RCA-5819 photomultipliers with Canada balsam. The c r y s t a l s were covered with a 0.001 i n . thick aluminium f o i l as a l i g h t r e f l e c t o r . The counters were mounted on a spectrometer table, the general arrangement being as shown i n F i g . 4« The source was r i g i d l y supported and accurately centred on a t h i n aluminium rod machined to f i t the hole i n the centre of the spectrometer table. The ele c t r o n i c s arrangement i s shown i n Fig . 5 with the derailed c i r c u i t s being given i n Appendix IX. The resolving time of the coincidence mixer i s 0.18 p-sec. The angular r e s o l u t i o n of the counters was measured using a n n i h i l a t i o n radiation from a Cu^ 4 source. An angle of 20° subtended by the counter at the source was selected as giving a sufficient,..' counting rate, without requiring too large a correction A r r a n g e menf A mi D ISC 5819 | Source • p.n ScalerMo) IZOO.V Sto bil i zedl Scqler(£>40) Amplifier Coi rtC\olencj Mixer 9ca lev (10) D j sc r i m m a t o r FIG 5 Coi incidence C i r c u i t f o r angular resolution. The true angular c o r r e l a t i o n between the gamma rays may be masked by three spurious e f f e c t s . These are as follows: 1. Compton scattering i n one counter produces a lower energy gamma ray which may be counted i n the second counter. Since the Compton scattering produces a count i n the f i r s t counter, the above process r e s u l t s i n a coincidence count. For the case of two 1 Mev gamma rays i n cascade, a c a l c u l a t i o n (See Appendix III) shows that the coincidence counts so produced, which we c a l l the "scattered coincidence rate", w i l l be about $0% of the true coincidence rate, i f we assume equal counting e f f i c i e n c i e s f o r both the 1 Mev gamma ray and the gamma ray scattered through 180°. With the foregoing assumptions the r a t i o of scattered to true coincidence rate when the counters are placed at 90 wil l . i n c r e a s e , because the s o l i d angle between the counters i s increased, although the number of scattered gammas per s o l i d angle i s s l i g h t l y decreased. This scattering from one counter to the other may be prevented by means of lead shielding such as i s shown i n Fig. 6. The source i s placed i n front of the aperture i n the lead shield and the scattering i s reduced d i r e c t l y i n the proportion of the area of the aperture to the area of the counter c r y s t a l . In t h i s way i t was possible to reduce the scattered coincidence rate to Jqq of i t s value without a shield. This method i s p a r t i c u l a r l y suited to low energy gamma rays, i . e . 500 Kev or l e s s , i n which case.the scattered gamma ray i s not appreciably lower i n energy. For gamma rays of the order of 1 Mev, the backward scattered gamma ray i s about 200 Kev, so i n t h i s case i t i s simpler to merely discriminate against the 200 Kev pulses, the 1 Mev gamma rays P M F I G . 6 - L e a d S h i e l d i n g 13. being counted with only s l i g h t l y reduced e f f i c i e n c y . 2. Compton and photo electrons may be ejected from the lead shielding i f i t i s used. Owing to the r e l a t i v e l y high efficiency, of the counters f o r counting beta p a r t i c l e s , the counting rate due to t h i s scattering may be large compared with the gamma counting ::• rate. This effect r e s u l t s mainly i n changing the e f f e c t i v e s o l i d angle of the counters so i t s presence i s undesirable. It may e a s i l y be eliminated by placing s u f f i c i e n t aluminium i n front of the counters to stop photo electrons from the gamma rays i n question. 3 . For gamma rays of energy greater than 1.02 Mev, lead shielding cannot be used with confidence since p a i r production i s then possible and the r e s u l t i n g a n n i h i l a t i o n r a d i a t i o n i s strongly-correlated.' For example with the shielding arrangement shown i n 60 Fig. 6 and a Co source (gamma energies: 1.17 and 1.33 Mev), the cor r e l a t i o n curve^shown i n Fig . 7 was obtained. With the bias of the discriminators set to eliminate pulses corresponding to gamma rays below 250 Kev and no lead shielding i n place, the curve^shown i n F i g . 7 was obtained. The curves coincide up to about 160° and the sharp r i s e i n curve B from 160° to 180° i s c h a r a c t e r i s t i c of that produced by a n n i h i l a t i o n r a d i a t i o n with the counters.set to subtend an included angle of 20°. It was concluded from the preceding i n v e s t i g a t i o n that the most s a t i s f a c t o r y method of avoiding spurious e f f e c t s i n meas-uring the angular c o r r e l a t i o n i n Co^° and S c ^ was to discriminate against Compton scattered gamma ray pulses. Accordingly the discriminators were calibr a t e d using the C o D U 1.3 Mev gamma pulses, and set to eliminate pulses corresponding to energies of le s s than 250 Kev. The only shielding employed was - " of aluminium 16 immediately i n front of each counter to stop beta p a r t i c l e s . The following procedure has been adopted i n the evaluation of the experimental data: 1. The data recorded consisted of the time T (hours), the t o t a l coincidence counts C(0), and the t o t a l single channel counts (scaled by 640) A and B. 2. Since the counter e f f i c i e n c i e s may vary s l i g h t l y (due to H.T. variations or moving one counter), the coincidence rate was divided by the i n d i v i d u a l channel counting rates to correct f o r these s l i g h t variations. Upon subtracting the accidental coincidence rate, calculated i n the usual manner, we obtain the quantity: C T(Q) = ( - 2TJX I0 7 where 2 T - 4 - 0 7 . AS are 3. The uncorrected c o r r e l a t i o n function, The d e t a i l s given i n Appendices I and I I . was next evaluated. was corrected f o r angular resolution by the method outlined i n Appendix II to obtain the true angular cor r e l a t i o n function, W(9). '5. The standard deviation was calculated s t a t i s t i c a l l y on the basis of the number of counts obtained, (see Appendix IV). 6. A comparison was made with the t h e o r e t i c a l values f o r W(9) calculated from Hamilton Ts tables (2). The evaluation of the experimental data i s shown i n Tables I and II f o r Co°° and S c ^ respectively. The corrected points (W(9)) are plotted i n Figs. & and 9 for Co D^ TABLE I Gamma-Gamma Angular Correlation i n Co 9 C T ( 9 ) W(9) -1 W(9) -1 S.D. W(9) (Theoretical) 90° 967 0.000 0.000 0.000 •1.000 100° 967 0.000 0.000 0.010 1.003 110° 973 0.011 0.012 0.012 1.014 120° 985 0.032 0.034 0.011 1.035 130° 1000 0.059 0.062 0.011 1.062 140° 1012 0.081 0.085 0.012 1.089 1 5 0 ° 1028 0 .109 0.114 0 .010 1.117 1 6 0 ° 1040 0 .130 0 .136 0 .010 1.142 1 7 0 ° 1053 0.153 0 .159 0 .012 1.158 ISO 0 1049 0 .147 0.153 0 .010 I . I 6 5 TABLE II Gamma-Gamma Angular Correlation i n Sc^ 6 w(e) 0 C T(9) W(0)-1 W(9)-l S.D. Theoreti-c a l . 90° 623 0.000 0.000 0.000 1.000 100° 627 0.006 0.007 0.012 1.003 110° 635 0.019 0.020 0.010 1.014 120° 641 0.029 0.031 0.010 1.035 130° 652 0.047 0.049 0.011 1 .062 140° 666 0.069 0.073 0.011 1.089 150° 682 0.095 0.099 0.008 1.117 160° 696 0 .121 0.127 0.010 1.142 170° 710 0.140 0.146 0.008 1.158 175° 713 0.145 0.151 0.0.10 I . I 6 3 180° 718 0.152 0.15$ 0.009 1.165 and Scr*" respectively. The s o l i d curve i n Figs. & and 9 represents the function: 1+ 0.\Z5CQS& +0.0+0 COS 4-which i s the t h e o r e t i c a l function W(9) corresponding to the t r a n s i t i o n 4~*2 ~*> 0 as given by Hamilton ( 2 ) . These r e s u l t s are in. good agreement with those of Brady and Deutsch ( 5 ) . It i s worth rioting that there i s a small systematic deviation of the experimental curves from the t h e o r e t i c a l values i n both Go^° and S c ^ . The cor r e l a t i o n i s s l i g h t l y l e s s than predicted i n both cases. These deviations are possibly explain-, ed by the precession of the nucleus duriiig the l i f e t i m e of the intermediate state, which would be expected to reduce the angular co r r e l a t i o n , ( 9 ) . Such a precession might a r i s e due to a mag-netic f i e l d at the nucleus from an excited e l e c t r o n i c state. The product nucleus may be i n an excited e l e c t r o n i c state f o r a short time, following the beta decay process. Frauenfelder (22) has performed an experiment on In"*""^ i n an attempt to show t h i s e f f e c t . The r e s u l t s indicate that the effect i s l i k e l y small i f the source i s m e t a l l i c , but may be appreciable f o r c r y s t a l l i n e sources, e s p e c i a l l y when very t h i n . The small deviations of the Co^O and S c ^ corr e l a t i o n curves (Figs. 8 and 9) from the t h e o r e t i -ca l values.are i n q u a l i t a t i v e agreement with the r e s u l t s of Frauen-fe l d e r , since the Co°° source was me t a l l i c and the Sc^6 source was i n the form of SeO^. Further investigation of t h i s effect would require considerable refinement of experimental technique. 16. IV. THEORY OF BETA-GAMMA ANGULAR CORRELATION Falkof f and Uhlenbeck (23) have also applied t h e i r general theory on angular c o r r e l a t i o n (15) to the p a r t i c u l a r case of the beta-gamma angular c o r r e l a t i o n . In order to apply the re s u l t s of the f i r s t paper to beta-gamma-angular correlations i t M , x i s only necessary to obtain the angular d i s t r i b u t i o n s F-^  (9) f o r the various possible beta-neutrino interactions. However the interactions involve the beta energy, W, so the angular d i s t r i b u -M tions are F^ (9 W), and are c a l l e d d i f f e r e n t i a l angular d i s t r i b u -tions. I f the d i s t r i b u t i o n s are integrated over a l l beta energies, so c a l l e d integrated angular.distributions are obtained: Correspondingly, we have two kinds of beta-gamma angular c o r r e l a -t i o n functions: the d i f f e r e n t i a l c o r r e l a t i o n functions ; VV ("0^  £•) (E i s the beta p a r t i c l e K.E.) and W(•£-, E 0 ) the integrated co r r e l a t i o n function. (E i s the maximum beta p a r t i c l e K.E.). o Falkoff and Uhlenbeck (23) have derived the necessary d i s t r i b u t i o n functions F ^ (9,W) f o r the f i v e possible beta-neutrino i n t e r -actions, f o r the f i r s t and second forbidden t r a n s i t i o n s . It i s possible therefore to compute the angular c o r r e l a t i o n functions from the r e s u l t s of t h e i r previous paper ( 1 5 ) . The following r e s u l t s of the theory are useful i n the inte r p r e t a t i o n of angular c o r r e l a t i o n data: (See Falkoff and Uhlenbeck (23) ) . 1. There i s no angular c o r r e l a t i o n for allowed beta t r a n s i t i o n s . 17. 2. In the case of a forbidden beta t r a n s i t i o n , the beta spectrin must have a "forbidden shape" f o r angular correla-t i o n to e x i s t . 3. For f i r s t forbidden beta-transitions, W(9) has the form: p 1 -T a cos Q. For second forbidden beta-transitions a term i n cos^Q may occur as well. 4. The d i f f e r e n t i a l angular co r r e l a t i o n i s greatest f o r beta p a r t i c l e s of maximum energy i n the beta-spectrum, and i s almost i s o t r o p i c f o r beta-particles near zero K.E. 5. The above theory assumes that a beta p a r t i c l e leaving the nucleus can be represented by a plane wave, which amounts to assuming that the nuclear charge z i s zero. Consequently the theory gives only approximate r e s u l t s . Fuchs and Lennox (24) have made some calculations taking into account the z-dependence, and have shown that f o r low values of z and certain types of in t e r a c t i o n (in p a r t i c u l a r i n the case of the int e r a c t i o n which gives r i s e to the "Bij-matrix element") the res u l t s of the simple theory (z - 0) represent a very good approximat ion. Since, i n general, the form of the integrated beta-gamma angular c o r r e l a t i o n function i s : W W = I + £ <*£ COS -9-, i t i s convenient to introduce the integrated angular c o r r e l a t i o n c o e f f i c i e n t , a, defined by: _ W(l*0*) , S i m i l a r l y the d i f f e r e n t i a l beta gamma correlation function W(0 E) i s of the form: w ( # E ) = ] + 21 01- (E) COS*^, 18. and the d i f f e r e n t i a l c o r r e l a t i o n c o e f f i c i e n t , a(E) i s defined as: a(E)-H ak(E)= »MZL _, W(n°E) 1 APPARATUS FOR MEASURING THE INTEGRATED  BETA-GAMMA-ANGULAR CORRELATION The gamma counter as used i n the gamma-gamma c o r r e l a -t i o n experiments i s quite suitable f o r t h i s case also. However, since we wish to measure the beta-gamma coincidence rate i t i s desirable to have a beta counter with a low e f f i c i e n c y f o r gamma counting, so the background gamma-gamma coincidence rate w i l l be as small as possible. Accordingly a beta counter consisting of anthracene f l a k e s of a thickness of 30 mg/cm connected to a 5819 photo-multiplier was used. The source was made as t h i n as possible i n order to . prevent scattering of the beta p a r t i c l e s i n the source. The sources were deposited from solution on a zapon f i l m using insulinc to obtain a uniform thickness. The weight was l e s s than 1 mg/cm^. The source was then enclosed i n a 5 i n . diameter p l a s t i c vacuum chamber to prevent scattering of the beta p a r t i c l e s i n the a i r . The beta counter was placed immediately outside the vacuum chamber In front of an aluminium window of 7 mg/cm to allow the beta p a r t i c l e s to enter the anthracene. The general arrangement i s shown i n Fig. 10. The absence of severe scattering e f f e c t s may be inferred from an experiment performed on the beta-gamma-correla-t i o n i n Sc^ . The beta-gamma coincidence rate i s is o t r o p i c i n th i s case as Novey. (14) has reported. With the set up as des-cribed i n t h i s section, the beta-gamma coincidence rate was found to be the same i n the 90° and 180° positions, within the experimen-x I am indebted to Dr. M. Kirsch and Dr. K. Starke f o r advice on the preparation of the sources. 20. t a l accuracy of about 1%. It was concluded therefore that the technique did not introduce any serious scattering e f f e c t s . L i n e Source M o u n t e d on Za^on A ToVacuum S y s t e m Al. W indow Beta Counter A nthra cene es " s s Movable G a m m o i C o u nte r F I G . 10 0 - / - A n g u l a r C o r r e l o t , o n E n e r g ,7 0 ^ 0 F I G . Disintegration Sc/. erne fo r VI INTEGRATED BETA-GAMMA-ANGULAR CORRELATION IN S b 1 2 4 . Many nuc l e i -have been investigated f o r beta-gamma angular c o r r e l a t i o n , but i n most cases the d i s t r i b u t i o n was found to be i s o t r o p i c . At the time t h i s research was undertaken only three cases had been reported i n which the c o r r e l a t i o n was not iso t r o p i c , namely Sb12/«-, (11) (12) Tni 1? 0, (14) a n d Rb^ 6 (13). There appeared to be considerable doubt as to the v a l i d i t y of the r e s u l t s f o r Tin 1? 0 and R b g 6 , K while i n the case of S b 1 2 4 the reported r e s u l t s were contradictory, but pointed to the d e f i n i t e existence of an angular c o r r e l a t i o n e f f e c t . Ridgway (11) reported the integrated c o r r e l a t i o n c o e f f i c i e n t f o r Sb"*"2 "^ as a : - 0 . 1 7 while Beyster and Wiedenbeck (12) gave a - - 0 . 2 6 . It was hoped that i f the integrated angular c o r r e l a t i o n c o e f f i c i e n t had a value )a|> 0 . 2 0 i t would be possible to make a measurement of 1 2.L. the d i f f e r e n t i a l angular c o r r e l a t i o n c o e f f i c i e n t a(E). An Sb source was therefore obtained from Chalk River and an experiment performed to obtain the integrated angular c o r r e l a t i o n c o e f f i c i e n t . The d i s i n t e g r a t i o n scheme of Sb^ 2^ (21) i s rather complicated as shown i n Fig. 11. The beta-gamma-angular co r r e l a -t i o n involves the highest energy beta group with 2 .37 Mev endpoint and the following .600 Kev gamma ray. I f we absorb the beta p a r t i c l e with energy l e s s than about 1 Mev, the beta-gamma coincidence rate The presence of a beta-gamma-angular c o r r e l a t i o n i n Rb o u has since been confirmed by Stevenson and Deutsch (18). w i l l involve mainly the 2.37 beta group and the 600 Kev gamma ray, with a few coincidences due to the 1.6 Mev beta group. A t h i n beta counter as described i n the preceding section was used. Of course the large number of gamma rays gives a r e l a t i v e l y high counting rate i n the gamma counter, which contributes to the accidental coincidence rate. ~ The r a t i o , D, of gamma-gamma to beta-gamma-coincidences was 0.115. The gamma-gamma coincidence rate shows l i t t l e i f any angular c o r r e l a t i o n i n t h i s case, being is o t r o p i c within the experimental accuracy of about 5%. The source was prepared by depositing sodium antimonate from.solution on to a zapon f i l m using i n s u l i n s to obtain a uniform thickness. An aluminium beta absorber weighing :.2-70 mg/cm was used i n front of the beta counter. This absorber removes a l l beta p a r t i c l e s of l e s s than about 0 . 9 Mev energy, so we obtain the angular c o r r e l a t i o n function Integrated over a l l beta energies greater than 0 . 9 Mev. The value 0 . 9 Mev was estimated from Feather's energy-range rela t i o n s h i p : R =-0.542E - 0.133 where R i s the range i n gms/cm2 and E i s the energy i n Mev. The t o t a l range of an electron which produced a count was about 310 mg/cm . The function ¥(0) as determined f o r Sb^ 2^ i s given i n Table III and F i g . 12. The evaluation of the data was carr i e d through as developed i n Appendices II and VI and consisted of the following consecutive steps: 1. The recorded data consisted of C ( 0 ) , the number of coin-cidences i n time T, the single channel counting rates (scaled by 640), A,&B, and D* the r a t i o of gamma-gamma to beta-gamma coincid-ence rate. (D was determined by absorbing the beta p a r t i c l e s completely i n aluminium). TABLE I I I  Integrated Angular Correlation i n So 0 Crn(Q) W(0)-1 W(Q) -1 W(6) S.D. 90° 364 - 0 . 0 0 0 - 0 . 0 0 0 1 .000 0 . 0 0 0 100° 364 - 0 . 0 0 0 - 0 . 0 0 0 1 .000 0.012 1 1 0 ° 358 - 0 . 0 1 7 - 0 . 0 1 9 0.980 0.003 1 2 0 ° 350 -0.038 - 0 . 0 4 3 0.955 0.003 130° 336 - 0 . 0 7 7 -0.086 0.910 0.003 1 4 0 ° 320 - 0 . 1 2 0 - 0 . 1 3 4 0.860 0.003 1 5 0 ° 311 - 0 . 1 4 6 - 0 . 1 6 3 0.330 0.010 160° .301 -0.173 - 0 . 1 9 3 0 . 7 9 9 0.003 1 7 0 ° 296 -0.189 - 0 . 2 1 2 0.773 0 . 0 1 1 180° 293 - 0 . 1 9 7 - 0 . 2 2 0 0.770 0 . 0 0 7 9 0 ° 1 2 0 ° 1 5 0 ° 1 8 0 ° -e-2 3 . 2. The accidental coincidences were subtracted from C(0) and the r e s u l t divided by the product of the single channel counting rates giving: C T ( * ) - ( cc*> r 3 . The uncorrected c o r r e l a t i o n f u n c t i o n : Wfc) - — T CT(<?0°) was computed. ' 4 . The correction f o r the gamma-gamma background was applied to ¥ ( 0 ) , thus obtaining .' W(^) -• W(&) ' T^t> *~ * T^D * (Appendix VI). 5. The correction f o r angular resolution was then applied i n order to obtain W(Q), the true c o r r e l a t i o n function. The plotted r e s u l t s (Fig. 12) can be f i t t e d c l o s e l y by 2 a curve ( s o l i d l i n e i n F i g . 12) .of the form: W(Q) - 1. + a cos 0 where a= - 0 . 2 3 - 0 . 0 1 . Unfortunately i t i s d i f f i c u l t to compare t h i s r e s u l t d i r e c t l y with the theory so the discussion w i l l be l e f t u n t i l the next section on the d i f f e r e n t i a l c o r r e l a t i o n c o e f f i c i e n t i n Sb . It may be remarked however that t h i s r e s u l t i s i n f a i r l y good agreement with that of Beyster and Wiedenbeck ( 1 2 ) , p a r t i c u l a r l y since the exact range of beta ray energies counted i s somewhat uncertain i n both cases. 711 24. BETA-GAMMA ANGULAR CORRELATION AS A FUNCTION OF ENERGY FOR Sb12l+. The theory of the beta-gamma-angular c o r r e l a t i o n predicts that the effect w i l l increase i n magnitude f o r higher energy beta p a r t i c l e s of the spectrum. In the case of Sb''"2 '^ the integrated angular c o r r e l a t i o n c o e f f i c i e n t was found to have . a value: a c . - 0 . 2 3 , only beta p a r t i c l e s of energy higher than 0 . 9 Mev ( i . e . up to.2 . 4 Mev) contributing to t h i s value. The effe c t should therefore be of such a magnitude near the end of the beta spectrum as to be e a s i l y observed, despite the small number of beta p a r t i c l e s . An experiment was therefore devised to measure the d i f f e r e n t i a l beta-gamma-angular c o r r e l a t i o n coeff-i c i e n t a(B), defined previously (p. 17 ) • In order to measure a(E) i t i s necessary to employ some type of spectrometer as the beta detector. I t has been shown by Hopkins (25) that the pulses produced i n a thick anthra-cene c r y s t a l s c i n t i l l a t i o n counter by beta p a r t i c l e s are propor-t i o n a l to the beta K.E.. energy, at least i n the energy range 0 . 1 - 3 . 0 Mev. It was therefore proposed to use a s c i n t i l l a t i o n spectrometer as the beta detector with a twelve channel kicksorter of Chalk River design as the pulse amplitude discriminator. The apparatus was arranged as shown i n Fig. 13 and functions as follows. The kicksorter and s c i n t i l l a t i o n counter c i r c u i t i s calibrated by determining the endpoint of the Sb-1-2^ 2.37 Mev beta group pulse spectrum. The twelve kicksorter channels are then,adjusted to cover equal energy i n t e r v a l s of about 0 . 1 Mev over the beta spectrum from 1 .0 Mev to 2 .37 Kev. Details of t h i s c a l i b r a t i o n are given i n Appendix V. The coincidence mixer Gamma Counter G a m m a Channel S ea ler V a c u u m C h a m b e r Be ta Counter J >ou rce Discriminators . And Co i nci'dence M i x e r Be ta -C h annei Scc( l e r Beta Pulse Amplifier G a t ( D e l a y 02/* -sec yTo 12. C h a n n e l K i c K s o r t e r .FIG-13 Differential B- ^-Angular Correlation 25. has the discriminator i n the beta channel set at about 0.9 Mev, so that only those beta-gamma coincidences are detected f o r which the beta energies are greater than t h i s value. Each coincidence count operates the gate c i r c u i t which opens to permit the beta pulse to reach the kicksorter. In t h i s manner the energies of each beta p a r t i c l e which has been counted i n coincidence with a gamma ray i s recorded. By measuring the' coincidence rates at the 90° and 180°positions of the two counters i t i s possible to determine the angular c o r r e l a t i o n c o e f f i c i e n t as a function of beta energy, that i s a(E). Since the beta counter now employs a thi c k c r y s t a l , i t i s sensitive to gamma rays as well. The ra t i o of gamma-gamma to beta-gamma coincidences D may be estimated making reference to Fi g . 11. The beta counter i s biased to about 1 Mev so only the 1.7 Mev gamma ray which occurs i n 60% of the dis i n t e g r a t i o n s , w i l l contribute to the gamma-gamma coincidence rate. The counting e f f i c i e n c y i s estimated at about .02 for t h i s gamma ray. The 2.4 Mev beta group comprises 20% of the di s i n t e g r a t i o n s and roughly a h a l f of the beta p a r t i c l e s are counted. On the average there-fore we would expect to f i n d a value of : 0.OZ0.6 = 0 | Z 0.2 0.5 The values measured experimentally f o r D ranged from 0.15 to 0.20 (see Appendix VI). Coincidence measurements were taken a l t e r n a t e l y i n the 90° and 180° counter positions f o r periods of about 12 hours. This procedure tends to average out the eff e c t s of small variations i n amplifier gain and photo m u l t i p l i e r high tension. The r e s u l t s of f i v e or six days counting i n t h i s 26. fashion were averaged and the d i f f e r e n t i a l c o r r e l a t i o n c o e f f i c i e n t a(E) computed. A group of readings such as t h i s w i l l be referred to as a run. The same computation procedure as was described on page 2.2, i n connection with the integrated c o r r e l a t i o n c o e f f i c i e n t i s applied i n t h i s case to each kicksorter channel, with two minor changes, as follows: 1. Since the source decays appreciably i n six or seven days (half l i f e equals 60 days) a correction must be applied to Cp(Q) as i l l u s t r a t e d inAppendix I I . 2. In order to obtain the accidental coincidence rate f o r each kicksorter channel, f i r s t of a l l , the beta counter pulse spectrum N(E) i s obtained on the kicksorter by "Qpening" the "gate" c i r c u i t . Then the t o t a l accidental coincidence rate i s computed i n the usual manner and the accidental rates f o r each kicksorter channel are computed as the product of the t o t a l accidental rate,and N(E). ( N(E) i s expressed as a f r a c t i o n of the t o t a l beta counter counting r a t e ) . The computational procedure used i s i l l u s t r a t e d f o r one run, i n Appendix VI. Three other s i m i l a r runs were made, however the computations are not tabulated here in d e t a i l . Fig. 14 gives the values f o r a(E) for a l l four runs. The values were combined as shown i n Table IV and F i g . 1$ to give a single curve. Also plotted i n Fig. 15 are two possible t h e o r e t i c a l curves (dashed) which w i l l be discussed i n the next section. It i s possible to calculate the integrated angular c o r r e l a t i o n c o e f f i c i e n t from the d i f f e r e n t i a l c o r r e l a t i o n TABLE IV The Beta-Camma-Angular Correlation C o e f f i c i e n t a(E) f o r S b 1 2 4 Run Energy 1.06 1.16 1.26 .1.36 I . 4 6 I . 5 6 1.66 I .76 1 .86 I . 9 6 2.07 2.17 a(E) 0.188 0 . 2 0 $ 0.219 0.2ft$ 0.273 0.298 0 .327 O.368 0.380 0.410 0.414 0 . 4 4 8 1 S.D. 0.013 0.015 0.016 O.017 0 . 0 1 8 0 . 0 2 0 0.023 0 . 0 2 6 0 . 0 3 4 0.045 0.053 0.070 2 a(E) 0.211 0 . 2 2 0 0.237 0.272 0 . 2 8 8 0.315 0 . 3 2 8 0.346 0.370 0 .388 0.431 0.445 S.D. 0 .014 0.015 0.015 0.017 0.018 0 . 0 2 0 0.023 0.025 O.029 0.033 0.047 O.O65 q a(E) 0.196 0.224 0 .254 0 . 2 8 0 0.308 0.334 0 . 3 6 0 0.382 0.400 0.412 S.D. 0.019 0 . 0 2 0 0.021 0 . 0 2 1 0 . 0 2 1 6 .022 0.025 0.025 0 . 0 3 0 0.030 . a(E) 0.188 0.218 0.250 0 . 2 8 0 0.314 • 0 . 3 4 6 0.380 0 . 4 0 8 0 . 4 2 0 0.436 0.440 O.444 S.D. 0.016 0.017 0.017 0.017 0 .019 0 . 0 2 1 0 . 0 2 4 0 . 0 2 6 0.023 0.028 O.O38 0 . 0 4 0 Aver- a(E) age S.D. o.d)95 0.218 0.240 0.270 0.296 0.323 0.349 0.376 0.393 0.411 0.428 0.446 0.008 0.009 0.009 0.009 0.010 0.010 0.012 0.013 0.014 0.018 0.026 0.040 11 \ v " - • o C o r r e l a t i o n Coeff ic ient - g(E) i i i i o O <o <o I 27. c o e f f i c i e n t and-the beta, spectrum N(E).. The d e t a i l s of the method are given i n Appendix VIII where i t i s shown that: where N(E) i s the beta cspec t runPas*measured on the kicksorter. The integration i s performed numerically i n Table V, f o r the energy range from E c r 0.8g£Mev. to E Q ^ 2.37 Mev. i n which we have measured a(E). The value so obtained i s a ^ -0 .24 %• 0.02. The problem of comparing t h i s value with the d i r e c t l y measured integrated c o r r e l a t i o n c o e f f i c i e n t a i s complicated by the. f a c t that the true beta spectrum N(E) i s d i s t o r t e d by the alum-inum absorber used to remove the lower energy beta p a r t i c l e s i n the case of the d i r e c t measurement of a. It i s shown i n Appendix XI that the value of a which we would expect to measure d i r e c t l y , may be estimated by making some assumption for.the amount of d i s t o r t i o n of N(E) and integrating a(E) i n a manner s i m i l a r to Table V. The r e s u l t of t h i s estimate i s that a should l i e between the values a r -0.24 £ 0.02 to -0.27 - 0 .02, depending on the amount of absorption of N(E).. The deviation of t h i s estimate from the measured value, a r -0.23 - 0.01 i s about-0.02 to t 0 .03. This small deviation may be interpreted as t h e . l i m i t of the systematic error. This systematic error i s due' to small d r i f t s i n amp l i f i e r gain and counter high tension. The influence of these d r i f t s on the determination of a(E) i s i l l u s t r a t e d i n F i g . 14, where the r e s u l t s of four separate determinations of a(E) are given. To conclude, i t should be emphasized that i t i s the value of a as obtained i n Table V, i . e . a = -0.24 1 0;02, which has an unambiguous t h e o r e t i c a l meaning. TABLE V Integration of a(E) to obtain a Beta Energy a(E) Spectrum N(E) "N/(E) 3 + a IE) a(E) ' N(E] 3 + aTE] 0 . 8 6 Mev - 0 . 1 3 0 25000 3700 -1130 0 . 9 6 - 0 . 1 6 0 23000 8100 -1296 1.06 - 0 . 1 9 0 21000 . 7470 -1419 1.16 -0.213 19200 6900 -1504 1.26 - 0 . 2 4 0 13000 6520 -1564 1.36 - 0 . 2 7 0 14970 5430 -1430 1.46 - 0 . 2 9 6 13100 4340 -1433 1.56 - 0 . 3 2 3 10930 4100 -1324 1.66 - 0 . 3 4 9 7690 2900 -1012 1.76 - 0 . 3 7 6 6550 2500 - 9 4 0 1.86 - 0 . 3 9 3 5070 1940 -762 1.96 - 0 . 4 1 1 3139 1230 -506 2.06 -0.423 1751 631 - 2 9 1 2.16 -0.446 349 323 -143 Totals: 61684 -14769 a = -14769 61634 = - 0 . 2 4 28 YIII RESULTS "AND CONCLUSIONS REGARDING THE DIFFERENTIAL  BETA-GAMMA-ANGULAR CORRELATION FOR Sb 1 2 /* From Fi g . 15 the value of a(E) near the end of the 2 .37 Mev beta-spectrum i s seen to approach a value i n the neigh-bourhood of a(E Q) = - 0 . 4 4 - 0 . 0 5 . In order to compare t h i s r e s u l t with theory the following a d d i t i o n a l information i s available. According to Langer et a l (26) the 2 .37 Mev beta spectrum i s very l i k e l y of the "alpha-type", corresponding to the f i r s t forbidden matrix element^Bjj-'. This implies a change of p a r i t y and the following possible changes of the angular momentum quantum number: A J - 0> - 1,-2. Moreover, the product nucleus Te 1 2^" has a spin i n the ground state of 0 since i t i s an even-even nucleus. I f the matrix element B{.y i s dominant, then the beta p a r t i c l e c a r r i e s away angular momentum L-^  — 2. From these considerations we get f o r e l e c t r i c dipole gamma radiation, that i s L 2 - 1, the following possible spin changes: 1 - > 1- -— > 0 0 ^ 1 3 — ? l -—> 0 and f o r e l e c t r i c quadrupole r a d i a t i o n (L ? = 2) 0 -—> 2 - »• 0 T 0 0 1 <s O 0 0 c 3 - =3> 2 -—>• •0 2 9 . The tables and formulae given by Falkoff and Uhlen-beck (23) have been used to calculate the value of the c o r r e l a -t i o n c o e f f i c i e n t for the maximum beta energy Eo = 2.37 Mev f o r each of the above cases. Further d e t a i l s of the procedure involved are given in Appendix IX. The values for a(E Q) so obtained are l i s t e d , Table VI. The values of a(E 0) f o r the t r a n s i t i o n schemes 1—>• 1 — * - 0 , 3 — > 2 — a r e r e l a t i v e l y close to the experimental value, so the complete functions a(E) were evaluated i n these two cases. The curves so obtained are shown i n Fig. 15. In interpreting these r e s u l t s , i t must be remembered that i n the energy range 1 .0 Mev- 1.6 Mev two beta groups are superimposed. The 1.6 Mev group ..''.. presumably has no beta-gamma-angular c o r r e l a t i o n , since the magnitude of the co r r e l a t i o n c o e f f i -cient decreases r a p i d l y below 1.6 Mev. With t h i s i n mind the re s u l t s favour the t r a n s i t i o n scheme 1 > 1 > 0 since any error would presumably tend to make the corr e l a t i o n more i s o t r o p i c . There i s , however, a serious d i f f i c u l t y i n accepting t h i s assign-ment, because the beta t r a n s i t i o n d i r e c t l y to the Te^2^" ground state would then be allowed, or at the most, f i r s t forbidden. It i s then d i f f i c u l t to account for the sc a r c i t y of these t r a n s i t i o n s . Very recently Nakamura etaal (27) have attempted to provide an al t e r n a t i v e i n t e r p r e t a t i o n of the r e s u l t s of Langer et a l ( 2 6 ) . Nakamura has considered, f t values and the nuclear s h e l l model i n h i s interpretation, but has given no consideration to angular c o r r e l a t i o n data a t c a l l . His t h e o r e t i c a l conclusions seem to be incompatible with our experimental r e s u l t s . However, further consideration i s being given to the problem i n an attempt TABLE VI Tran s i t i o n a(E Q) 1 — > 1 0 -0.44 2 $ 1 — * 0 +0.64 3 * 1 — ? 0 -0.14 +0.73 +0.86 -0.15 -0.37 +0.23 Experimental Valued -0.44 + 0.04 0 > 2 =50 1 » 2 3>0 2 > 2 — 9 0 3 * 2 —3> 0 4 > 2 — 0 to obtain a sa t i s f a c t o r y t h e o r e t i c a l i n t e r p r e t a t i o n of a l l experimental data which i s now available concerning Sb"*"2^. A few months after'the publication of our prelimin-ary report ( 1 6 ) , Stevenson and Deutsch (18) have published the re s u l t s of t h e i r measurement of the d i f f e r e n t i a l beta-gamma angular c o r r e l a t i o n a(E) f o r Sb 1 2^ i n which they made use of a magnetic beta ray lens spectrometer. There i s excellent agree-ment between t h e i r r e s u l t s and ours, though the two methods are quite d i f f e r e n t . It i s worthwhile noting the present state of beta-gamma-angular c o r r e l a t i o n experimentation. Angular c o r r e l a t i o n e f f e c t s have been reported'and the integrated beta-gamma-angular co r r e l a t i o n c o e f f i c i e n t measured for the following n u c l e i : Sb 1 2 /* ( 1 2 ) , Rb 8 6 ( 1 3 ) , I 1 2 6 (13), Tni 1? 0 (11) and K 4 2 (28). Only f o r Sb 1 2 4( 1 7)(13) and Rb 8 6(18) has the d i f f e r e n t i a l angular-c o r r e l a t i o n - c o e f f i c i e n t been reported. No angular c o r r e l a t i o n has been found f o r most commonly obtainable radioactive n u c l e i , including: Co 6 0, Na 2 2, Na22*-, C o ^ , S c 4 6 - and I 1 2 Z |-. APPENDIX I. 31. COINCIDENCE COUNTING WITH DIRECTIONAL CORRELATION  BETWEEN THE EMITTED PARTICLES. A simple cascade d i s i n t e g r a t i o n scheme i s i l l u s t r a t e d i n Fig. 1 B. Let us assume that counter 1 counts only the f i r s t p a r t i c l e emitted, and the second counter counts only the second p a r t i c l e . Let us denote the p r o b a b i l i t y of the counter 1 f o r detecting p a r t i c l e 1 whose d i r e c t i o n f a l l s within an i n f i n i t e s -i m a l s o l i d angle dw, around a d i r e c t i o n characterized by the unit vector CO, as: c ; (10,; "^.-ff ' T n e n the single counting rate i n counter 1 i s Nn = N £, , where N i s the 1 o l o source strength (disintegrations per sec.) and the integration being over the whole sensitive area of the counter. Introducing a sim i l a r notation f o r counter 2, we get, r e f e r r i n g to Fig. 16, f o r the d i f f e r e n t i a l coincidence counting rate: where W ( ^ i ^2) : V (^) i ^ L l 1 1 6 a n g u l a r ' . 00, - ^ c o r r e l a t i o n function, 0 being defined by: cos 9 -z • "— M we integrate between the l i m i t s indicated i n F i g . 16, making . the assumption that the counters 1 and 2 both subtend the same angles at the source we obtain the t o t a l coincidence rate: la w ° The subscripts 1 and 2 r e f e r to counters 1 and 2, while 01,02 re f e r to the center p o s i t i o n of counters 1 and 2 respectively. -s> — - 7 — • > I f we assume £, and € ^  are constant over d i r e c t i o n s u)t and i^i/l , and introduce the variables: flf, - (fl- (ft <• ' l i Of and C(z-- V>Z-(S>0Z expression f o r A/c c (47T) 1 , we obtain from the previous +4 +4 From the fac t p- i s small (10°) we now introduce the approxima-tions that cos 1 and 1?- can be neglected i n comparison with 0, that i s : 0 - Q0 +• C{z -fl I f we assume that Wi(0) has the simplest form consistent with the theory: elementary integration gives us: 1 + a , i - i ^ 2 4TT 1 4ir " AsO^, ^  become 'very small we obtain i d e a l r e s o l u t i o n , f o r which case the coincidence rate N ^  (Cl0o) i s : We have assumed that counter 1 has an e f f i c i e n c y of zero f o r the detection of the second p a r t i c l e , and s i m i l a r l y that counter 2 has zero e f f i c i e n c y f o r the detection of the f i r s t p a r t i c l e emitted. This condition can be obtained, p r a c t i -c a l l y , f o r beta-gamma coincidences. However i f these e f f i c i e n c i e s are not zero, the coincidence rate as expressed i n the previous formulae must include a second term, similar to the f i r s t one, but involving these e f f i c i e n c i e s . The case i n which both gamma rays are of abeiit the same energy (nearly r e a l i z e d for Co ) i s p a r t i c u l a r l y simple, since the coincidence rates are then given by the preceding formula with an additional f a c t o r of two. 34. APPENDIX I I . PROCEDURE TO EVALUATE THE TRUE COINCIDENCE RATE. The angular resolution of the c ounters was chosen So that 2fl(=2^ = 20 o = 0.349 radians (Fig. 16). I f we assume that W(0) = l + o | cos 0, which i s the simplest form consistent with the theory,.the coincidence counting rate f o r i d e a l resolu-t i o n i s : IVC &) * A/„ 6, 6 Z [/ + 0LC0SZ&] . (See Appendix I), Upon substitution of the above numerical values f o r the subtended counter angles we obtain f o r the observed coincidence rate: WUfl) Vo [• + (^0.02/ f 0.<?r* Co A)] tWe have assumed that: At Let us now define the experimentally observed angular c o r r e l a t i o n function as follows: Nc (Wj } and deter-mine the correction required i n order to obtain W(0) from i t . Applying the formula f o r |\Zc[-8"), we obtain: K 1 i +• 0.01/ a and since 0| i s generally small (about 0.2) t h i s may be written: Wte) - i — o.u%4co$zfr - (5.010 a 2 , cos** We therefore obtain the proper correction to be applied to W(9) 35. to obtain W(9) as follows: A possible procedure i s to plot W(0) - 1 and obtain an approximate value f o r a which may be used with s u f f i c i e n t accuracy i n the correction formula above to obtain W(0). In actual practice the observed coincidence rate, NQgg, i s p a r t l y due to accidental coincidences. This accidental coincidence rate, N^, may be calculated from the i n d i v i d u a l counter rates N-j_, Ng, as follows: N A = 2 T A/, A/t - Z T j V being the mixer resolving time (0.13 yu-sec). Consequently Nggg i s given by: " NOBS = Nt&) t t i A = N o e ^ W ® ) +2T/Vol6,6z . Now NQgg may vary s l i g h t l y due to fl u c t u a t i o n s i n H.T. supply and amplifier gain, but t h i s may be corrected f o r by d i v i d i n g by (Nj )(N 2) since t h i s gives us a quantity which i s independent of the counter e f f i c i e n c i e s as follows: Woes _ ' Wfo) 0 ^ — — t 2.T • A/, • Nt A/, I f C i s the t o t a l number of coincidence counts and T i s the t o t a l time, and l e t t i n g A = N, T, and B * N2T, be the t o t a l single channel counts i n time T, then from the l a s t two equations i t follows that: Vfl(35_ CT AB For convenience of computation we define a quantity C^(9) as follows: c (9) =• - - — — . I f the source decays appreciably during the experiment, i t w i l l be necessary to apply a correction f o r t h i s as well. Since, C T(9) = i t w i l l increase as the source decays, so 1 No the correction consists of multiplying C T(9) by the proper decay factor, f. APPENDIX I I I . COMPTON SCATTERING OF GAMMA RAYS BETWEEN COINCIDENT COUNTERS. The operation of a s c i n t i l l a t i o n counter as a gamma detector depends upon the gamma ray in t e r a c t i n g with the c r y s t a l i n such a manner as to tran s f e r part, or a l l of i t s energy to one or more electrons. Pair production, photo e l e c t r i c effect and Compton scattering are possible methods f o r such an i n t e r a c t i o n to take place. For gamma rays with energy of the order of 1 Mev, Compton scattering i s the most probable i n t e r a c t i o n process. The electron of course produces a s c i n t i l l a t i o n i n the c r y s t a l which i s detected by means of a photo-multiplier. The scattered photon from the Compton process has a lower energy than the incident photon, but may e a s i l y escape from the c r y s t a l , and may i n fact be scattered i n the d i r e c t i o n of the o other gamma counter. When scattered through 180 the gamma photon has an energy of about 200 Kev ( i n i t i a l energy 1 Mev), so these scattered quanta may e a s i l y give r i s e to a spurious count. In fact the e f f i c i e n c y of the counter may be more than twice as great fo r the scattered photons than f o r the incident photons due to the higher t o t a l cross section of the c r y s t a l f o r lower energy photons. The magnitude of t h i s "scattered coincidence rate", C_(9) may be estimated as follows: i 1 Cs(fr) - Z A/sC-fr) Uf t where N g(0) i s the number of photons scattered through an angle 0 per unit s o l i d angle per second, £± i s the counter e f f i c i e n c y for the scattered photon, and U) i s the s o l i d angle subtended between the counters. The f a c t o r 2 occurs i f we assume that the gamma counters are i d e n t i c a l , and that the gamma rays i n cascade have nearly the same energy.. N g(0). i s given by: Nt& i t /y $ where i s the d i f f e r e n t i a l Compton cross-section, i s the t o t a l Compton cross-section and i s the single channel counting rate. The true coincidence rate, C^ ,, i s given by: Cp - Z A/0 61 tf_ - hJ( 62. , so that we obtain f o r the ra t i o of scattered to true coincidences: where ^ i s the e f f i c i e n c y of the -gamma counter f o r the incident photons and CO i s the s o l i d angle subtended by the counter at the source. Assume that the counters and source are arranged' syraetrically i n a straight l i n e , so that ct? r 4 u;. For CI 1 Mev incident gamma ray the scattered gamma (180°) has an energy of 200 Kev. The Compton cross section of the c r y s t a l s f o r 200 Kev gammas i s about twice that for 1 Mev gamma rays. Therefore i f the discriminator i s set low enough to count the smaller 200 Kev pulses the e f f i c i e n c i e s are i n the r a t i o — 2 . j i A a ^ may be determined from the formula and tables i n Heitler, Quantum Theory of Radiation, the value f o r 1 Mev gamma scattered through 180° being °^^/^ r- 0.04- Using these values we obtain Cscatt = ^ 0.04 0.5. This r e s u l t gives °true an order of magnitude only, since the photo e l e c t r i c cross section has been negleeted, however e f f e c t s of t h i s magnitude have been observed. 39. APPENDIX IV. THE STANDARD DEVIATION IN W(Q) Experimentally we calculate W(0) from the formula: c r (?) W(O) = " 0« (see Appendix II) The r e l a t i v e standard deviation i s accordingly given by: i »<*)1 [ C T t f / CTf90') The r e l a t i v e error i n Q T(o) may be evaluated from the t o t a l number of counts obtained, by the means of the well known Poisson Formula. In t h i s case the t o t a l count obtained, C, i s pa r t l y due to accidental coincidences, C, .so that: C_ C-C. , and A T A accordingly: ^ r ) V C * CA 40. APPENDIX V. CALIBRATION OF THE KICKSORTER CHANNELS  IN TERMS OF BETA ENERGY. According to Hopkins (25) who used a beta counter of s i m i l a r construction to the one used here, the pulses produced are proportional to the beta energy, at least i n the range 0.1 - 3-0 Mev. In order to c a l i b r a t e the kicksorter • i t i s only necessary to establish one point i n the energy spectrum, the endpoint of the 2.37 Mev beta group being the most p r a c t i c a l to use. Accordingly, before each run, the beta pulse spectrum was obtained on the kicksorter. It i s then necessary to estimate the endpoint of the spectrum and construct a Fermi pl o t . I f the endpoint was selected c o r r e c t l y , a straight l i n e Fermi plot r e s u l -ted as i n Fig. 17- The Fermi plot was concave upwards or down-wards depending on whether the o r i g i n a l estimate of the endpoint was too high or two low. In t h i s manner the endpoint could be determined to within 5 channel width or. £ 50 Kev. Referring again to Fig. 17, the Fermi plot i s seen to be p r a c t i c a l l y a straight l i n e i n the range 1.7 to 2.37 Mev, with a sharp break at about I .65 Mev. This i s the endpoint of the second beta group (see F i g . 11) i n Sb 1 2^ and i t s presence i n the Fermi plot lends considerable confidence to the c a l i b r a t i o n . The c a l c u l a t i o n of the fermi plot (Table VII) includes the correction f o r the forbidden shape of the spectrum, although t h i s i s hardly necessary considering the accuracy of the endpoint determination. TABLE VII  Fermi Plot of S b 1 2 ^ Spectrum 625 2.467 28 .0 17996 6 4 . 2 7 8 . 0 2 3 . 9 7 2 0 . 2 675 2.664 3 1 . 0 14973 4 8 . 3 0 6.95 . 4.04 17.2 725 2.861 3 4 . 2 14048 4 1 . 0 8 6 . 4 7 4-13 15.7 775 3 . 0 5 9 37.5 10982 29.29 5.41 4 . 2 4 1 2 . 8 V 825 3.256 4 2 . 0 7694 18.32 4 . 2 8 4 . 3 6 9 . 8 875 3.454 45.5 6555 14.41 3.80 4 . 5 0 8 . 4 925 3.651 49.0 5067 1 0 . 3 4 3.22 4.65 6 . 9 975 3.842 5 2 . 0 3139 6.13 2 . 4 8 4.80 5.2 1025 4 . 0 4 6 55.5 1751 3.15 1.78 4 - 9 8 3 . 6 1075 4.243 5 9 . 8 849 1.42 1.19 5.17 2 .3 1125 4 . 4 4 1 65.O 297 O.46 0 . 6 8 5 .34 1.3 1175 4 . 6 3 8 67.5 0 0 . 0 0 0 . 0 0 0 : 0 V = Kicksorter Channel Bias. £ = Beta Kinetic Energy i n units of mc . f(52 H) , The Fermi function f o r z = 52. Beta Spectrum. / 2 C = Total beta energy i n units mc . J\ = Beta Momentum i n units mc. C = ^l(60'i)r + ri~ i s the " OL -type" forbidden shape factor. z £' = - 2 APPENDIX VI. CALCULATION OF BETA-GAMMA ANGULAR CORRELATION COEFFICIENT 41. I f we now consider C to be the t o t a l number of coincidence counts i n time T we have: C = C f 3 ^ C }fif +" ^ A' w n e r e t n e subscripts r e f e r respectively to beta-gamma, gamma-gamma, and accidental coincidences respectively. Retaining our d e f i n i t i o n of Appendix I I : M 9 ) ~ l l — 2 T and introducing c T < r = ° 1B '~ we have: cf i c r ( n ) - c T r ( n f l - f c T ( f ) - G , / r / ] LcT<*) - c T f ( V l The gamma-gamma coincidence background rate i s about 15 - 20% of the (3-Jcoincidence rate. ™ c x r ( ? ) = D . C T ( f ) , than " where Ol i s the measured c o r r e l a t i o n c o e f f i c i e n t and "p^ i s the gamma-gamma cor r e l a t i o n c o e f f i c i e n t . C^("^ i s m e a s u r e d separately by absorbing the beta p a r t i c l e s with aluminium, which gives us . The co r r e l a t i o n f o r angular resolution may now be applied i n the way described i n Appendix I I . This procedure i s also applied to the data f o r each kicksorter channel when measuring the d i f f e r e n t i a l c o r r e l a t i o n 42. c o e f f i c i e n t , cX ( E) • As an example of the procedure followed, Table VIII gives the experimental data f o r one run i n the determination of i 0 ( E ) f o r Sb 1 2^. The notation used i n Table VIII i s consistent with the notation used i n the formulae i n t h i s Appendix. The "Bias" i s i n a r b i t r a r y units and the "Energy" r e f e r s to the energy i n Mev. corresponding to the kicksorter channel midpoint bias. The "Spectrum" i s the pulse spectrum obtained from the beta eounter, while the "gamma-spectrum" i s the pulse spectrum obtained from the beta counter when the beta p a r t i c l e s were absorbed i n aluminium. The kicksorter channel rates are i n each case expressed as a f r a c t i o n of the t o t a l counting rate obtained from the beta counter f o r that p a r t i c u l a r spectrum measurement. The standard deviation "S.D." refer s to the s t a t i s t i c a l error in£f(£). Following the procedure i n Appendix I I , 0^(0) i s calculated and m u l t i p l i e d by the appropriate decay f a c t o r , F. The recorded data consisted of the time T, the t o t a l coincidence counts C i n each channel, and the t o t a l single channel counts scaled by 640, A and B. C T(0) as given i n Table VIII i s i n these units m u l t i p l i e d by 1 0 ' . Twice the resolving time, 2 ft i n t h i s —2 —7 system of units i s 407 (hrs x 64O x 10 ') which i s .' 407 x 3600 x 640~ 2 x 1 0 " 7 =r • 0.356 x 10~ 6 sec. Sim i l a r l y , f o r T 2 -11 convenience the t a b l e records — i n units of (hr. x 640 x 10 ), AB The t o t a l number of true coincidence counts recorded i n time T i s therefore: C T ( 0 ) . I f x 10> -r-Total T AB T A B L E a(E) f o r S bu * B i a s r.o 5.5 6.0 6.5 7. a 7.5 8.0 '9.0 9.5 ,0.5 115 Enerav(Mev. 0.94 i.02 i.W no i.29 i.38 /.48 LSI 1.66 1.79 /.97 2.14 c ^ 1.0 0.IZ4 0.110 0.107 0.081 o.ogi "075 lOtf 0.053 0.039 0.059 me P.(?I7 i 4C7 so.s 44.*; 43. 35.3 32-3 30.7 262 Zl.b 16.1 24.: as 6-8 C T (goVf 12.30 1500 52.7 163.0 IS&4- 1480 I3&0 133.0 113.2 973 81.5 106.6 51.5 28.9 12-75" 1490 683 1675 !<?4.0 151.0 143.0 128.8 H4.5 92.0 65.7 98.5 45.7 22.7 Jl.VO )5"00 6)2 175.0 2032 101.0 158.0 1378 135.0 IM.6 92.7 66.0 90.7 445 ZQ9 C T(i8o)-f 13.20 !14! 48 Z 1644 160.0 I2Q5 fOf? 58.3 835 64.8 475 67.4 2 7.7 /4.3 li'.50 ;i48 565 I4Q8 174.6 160.6 116> Ilf[3 102.3 £0.4 646 437 (o\& 252 15.0 11.75 Jot? 134.0 1645 150.7 120.4 II 1.4 M 0.8 93.3 66;6 47.1 68.3 31.9 12.30 II47 ,45.2 1650 ISQ2 H8.0 107./ 946 80.0 7Q9 524 6 7.5 3QI 1 6.7 ,'2-00- ;i54 570 138.9 i'648 IW 118.0 I03J 103.2 819 66.5 458 654 33.1 16.3 C~T(90°)f (Av c r a ae) It 85 mi 193.1 I5Z,? 139.6 /32>3 113,2 94,0 7U 98,6 472 24.2. ( I3R7 1583 122.7 ioe,s I0l£ 81.8 66.7 47.3 66J 29.6 15.5 -aCE") 0,177 o.ito O.I80 0.114 au3 0.230 0.27T 0.290 0.334 0.330 0,372 0.362 - a(E) 0.193 0.170 0.190 0.215 0.245 0.255 0.316 0.337 0.37/ 0.393 D.424 0.420 S.D. I 0- 01-5 0.0/6 0.017 0-0/7 0-017 0.019 0-021 0 02^  0.026 0.023 0.02a 0.038 - a w 0.174 a zoo 0.224 0.254 0.276 0.330 0.357 0.3S7 0.4O9 0442 0.437 -^Spectrum 1.0 0.//O O.iOl 0.099 0-085 0.033 0.0$3 0.076 0.060 0.044 0.057 0.022 0.013 2-T 45.0 40.8 AOS 363 33. £ 34-0 31.0 24.4 17.9 235 9.1 £-.5 Cr^{90)-f 17R3 199.0 ZI59 169.6 1762 168.6 (54.9 1263 92.9 113.9 41.8 £3.8 154.1 177,8 1726 153/ 1512 1453 138.0 1114 .71,2 1047 38.1 Z19 D 0.181 0.171 0.191 0/190 0.216 0218 0.234 0.230 0.223 0.197 0.JSi 0.169 0.134 am 0.200 0.O98 0.142 0.138 0.109 0.1/8 0.22C a 081 o.m 0.080 • As an example take the f i r s t 12.3 hr. run at 9 0 ° , f o r which F — 0 . 9 9 . The number of true coincidence counts received i n channel 1 was: while the number of accidental coincidence counts was x 1 0 4 - 950. The standard deviations are calculated following the procedure i n Appendices IV and VII. APPENDIX VII. CALCULATION OF STANDARD DEVIATION. IN THE BETA-GAMMA CORRELATION COEFFICIENT From Appendix VI we obtain f o r the co r r e l a t i o n c o e f f i c i e n t : 7T = I P • a " a ' i-t> " V '/-]> Let t h i s be written a - W- b - P r Q • then the deviation £ q~ i s given by ; The most important term i s ( t A f l ) since i t involves the error i n the main experimental quantity Ol . Of the other terms only ( Q&fj) i s important since the remaining two both involve which i s quite small. Therefore we obtain i n s u f f i c i e n t l y close approximation f o r the standard deviation i n cf ' APPENDIX VIII  CALCULATION OF a FROM a(E) Let N(E) dE represent the beta spectrum, that i s the number of beta p a r t i c l e s whose energy l i e s between E and E-f dE. The r e l a t i o n between the d i f f e r e n t i a l beta gamma c o r r e l a t i o n function W(9E) and the integrated c o r r e l a t i o n function W(0) i s then as follows: W(0) /NIC) J£ For S b 1 2 4 W(9E) i s of the form: W(9E) - 1 -f a(E) cos 20 where a(E) i s the d i f f e r e n t i a l c o r r e l a t i o n c o e f f i c i e n t . We have assumed W(9E) to be a pro b a b i l i t y , so i t may be normalized as follows: , , 4 | J J ' - where cmz i s a s o l i d angle element. Consequently we have: W(9,E) - ) "t- C\LE) C O ^ V or a f t e r performing the / ( i + afe)tosx-e)<lSL int egration: w(9E) - _J_ / + a i t ) Co$\ + ir 3 -talE) Hence we may obtain W(9) as follows: W(9) = By analogy with the equation f o r W(9): 2. we obtain : 3 + a(E) to £ 3 + otfE) Je The range of integration extends from some minimum energy, E ^ n , (corresponding to the beta counter cutoff) to the maximum beta energy EQ. 47. APPENDIX IX. EVALUATION OF THEORETICAL BETA-GAMMA-ANGULAR  CORRELATION COEFFICIENTS. The theory of Fal k o f f and Uhlenbeck (23) was used to predict the value of the c o r r e l a t i o n c o e f f i c i e n t a(E Q) f o r each of the possible t r a n s i t i o n schemes l i s t e d i n Section VIII. Referring to (23) i t i s necessary f i r s t to specify the angular M d i s t r i b u t i o n function, F^ (Q(W), f o r the required i n t e r a c t i o n , and then to evaluate the parameters involved f o r the p a r t i c u l a r beta energy involved. For the a x i a l vector i n t e r a c t i o n , f i r s t forbidden t r a n s i t i o n with matrix element B^j, we have: F 2°(0, W) = J ( 2 p % + q Z ) + T-p*~C0SZ-& . JJX -r COS + ju^ COS & , giving p{ - ^p*>2^ ) p v A N D /^L - 0 , where \\/ i s the t o t a l beta energy (in units mc 2), WQ i s the maximum beta energy and Q^ = W0-W , p * W ~~ I It i s now possible to evaluate the parameter (f>Zi i ^ l ^ i ) J U t ' /\) from Equation (31) ( 1 5 ) . The d i f f e r e n t i a l c o r r e l a t i o n function i n t h i s case i s W(9,E) r 1 4- R/Q COS29 where a(E) = 2 . R Q -r i s given by Equation (29) ( 1 5 ) , and may ea s i l y be evaluated from the p a r t i a l sums l i s t e d i n Table IV (15) or the sums i n Table II ( 1 5 ) . As an example take the t r a n s i t i o n 1 — > 1 9 0 , • L x =. 2 L 2 ~ 1 48. In Falkoff and Uhlenbeck's notation t h i s i s written : J — A J ^ J > J-j- A J. so that A j - 0 A J = -1 ^ o/"cf J = 1. The parameter - f o r \V - \A/0 Then (>„//,/\) = y ('O/V+l) i A ' * /I - | Referring to Table II (15) and equation (29) (15) ,* (2 J -3 ) (2 J-rS-) - I p / _ ' v Q ' / 4 - J ( 2 J-\) (j>u +(% J * - (o T t f ) ~ z f a + l =• -0.447. S i m i l a r l y the other p o s s i b i l i t i e s are evaluated, the re s u l t s being l i s t e d i n Table VI. 4<7 E L E C T R O N I C C I R C U I T S Linear Ampl i f ier 100 X PL 5819 Preamp. 3X L i n e a r A m p . IOOX / 1 1500 v. L—Anthracene B L O C K D I A G R A M OF C O I N C I D E N T C O U N T E R C I R C U I T D i s c r i -minator l_ From duplicate channel A.I.C*IOI-A Scale of 6 4 s ha per Coinc. cct. r - Count I Mechanical switch registers I I o I I I I 1^ £>~ I Reset clock 0- | hour j 6AL5 6AC7 6AG5 6AL5 6AC7 DISCRIMINATOR*}* 5,~ i w R o 5 5 1 PWL 6AC7 I DISCRIMINATOR Input ^ i Coinc output To M.V. plate Hammond 165 + 320 v. 150 m.a. Resolving time - O.I7>i/s. Dead time - 3 jus. Input range- +K+IOO v. COINCIDENCE M I X E R AN D P O W E R S U P P L Y Scope 4 sync. Free-running blocking oscillator JL 02//5 delay line JL Inverter "U IT 0-15 met. +.+ 1.5 v. pu Ise Invertinq feedback amplifier —1.5 v. pu i s * Mul t i -vibrator 3~l50/js. Differcn- Triggered blocking o s c i l l a t o r IL Inverter f iafor loo ma. t j O O v. p u l s e P R E C I 5 I 0 N P U L S E G E N E R A T O R Raytheon UXT350 r Pulse transformer 6A67 lOOv. pulse Single 1.5 v . J»T ° Double 1.5 v. " 6AK5 6 A K 5 6AK5 | -1.5 v. - , A r f »J°>< -ISov. I 'iiilrJj Ic.i At 1 enuator for I.5 v. pulses To Power Supply + 300 V. reg. @> I 5P rtia - (50 v.reg <ffl 5 ma 6.3 vac. (2) 6.2 a. e.t connected to 0 PRECISION P U L S E G E N E RATOR 50. APPENDIX XI ON THE INFLUENCE OF ABSORPTION ON THE MEASURED VALUE OF a * The measurement of the integrated c o r r e l a t i o n function was important i n that i t was shown that no appreciable term i n cos^O e x i s t s . However the problem of comparing the measured integrated c o r r e l a t i o n coefficient- with theory i s quite complicated, due to the fa c t that the true beta spectrum i s some-what distorted by the aluminum absorber used to remove the lower energy beta p a r t i c l e s . -It i s sometimes assumed-(31') that beta p a r t i c l e s of energy E.are transmitted through aluminum according to a law of the form:. A = 1 * I (1) Rmax K X J where A i s the transmission, R i s the thickness of aluminum and R m o v corresponds to the maximum range of beta p a r t i c l e s of energy E. R max a n (^ ^  a r e S i y e n by Feather's r e l a t i o n which has been improved by Glendenin (31): • R m a x = 0.542E - 0.133 where R i s i n g/cm2 and E i s i n Mev. (E > 0.8 Mev). I f the above relations are assumed, i t may be possible to take the d i s t o r t i o n of the beta spectrum into account and so obtain a suitable comparison with theorv. We may, however, check the consistency of the integrated and d i f f e r e n t i a l c o r r e l a t i o n measurements by estimating the I would l i k e to thank Dr. L. Katz f o r bringing to my attention some aspects of the problems discussed i n t h i s appendix. 51. integrated c o r r e l a t i o n c o e f f i c i e n t a from' a(E) as shown i n Appendix VIII, but applying a correction to K(E) on the assumption that beta p a r t i c l e s i n a small energy range, E to E-hdE, are trans-mitted according to the above r e l a t i o n (1). In t h i s manner we obtain the value, a =• -0.27 - 0 .02. relationship as given in equation (1) for the transmission of an absorber i s a rather poor approximation. The transmission i n our-experimental set up i s undoubtedly considerably greater than i s indicated by equation (1). This, fact was checked by performing a measurement using an absorber of 440 mg/cm2.of Aluminum, which corresponds to the maximum range of beta p a r t i c l e s of energy E c 1.06 Mev. The t o t a l absorbing thickness* used i n the experi-ment was 310 mg/cm corresponding to E c = 0.&2 Mev. The r a t i o K of the counting rate with E c - 1.06 Mev to the counting rate with E C -=0.S2 Mev. was K - 0 . 5 5 - I f we assume that the absorber stops a l l beta p a r t i c l e s with energy below E c and transmits a l l beta p a r t i c l e s with energy greater than E c ( i . e . i d e a l cut off) we may integrate the spectrum N(E) from E c t o - E Q and thus obtain f o r the above r a t i o , the value K r O .65. On the other hand, i f we assume that the transmission i s given by equation (1), we may and estimate the r a t i o f(. In t h i s way we obtain the value K- 0.1+8. It i s apparent that the actual transmission i s greater than that given by equation (1), but i s of course le s s than the i d e a l There i s some evidence (29)(30) that the l i n e a r perform the integration dE x In order to be counted,a beta p a r t i c l e had to penetrate the absorber, the counter window, and the counter c r y s t a l , which totaled 310 mg/cm2. 52. transmission. Therefore our estimate of the measured integrated cor r e l a t i o n c o e f f i c i e n t a as obtained from a(E) gives the r e s u l t that a must l i e between the values: a = - 0 . 2 4 - 0 . 0 2 and a i -0.27 - 0 . 0 2 . The value a = - 0 . 2 3 1 0 . 0 1 i s ac t u a l l y measured and deviates from the above estimate by about - 0 . 0 2 to ± 0 . 0 3 . REFERENCES 1. J . V. Dunworth, R.S.I. 11, 167 (1940). 2. D. R. Hamilton, P.R. 58, 122 (1940). 3. R. Beringer, P.R. 63, 23 (1943). 4- W. M. Good, P.R. 70, 978 (1948). 5- E. L. Brady and M. Deutsch, P.R. 74, 1541 (1948). 6. R. L. Garwin, P.R. 76, I876 (1949). 7- J. R. Beyster and M. L. Wiedenbeck, P.R. 79, 411 (1950) 8. R. M. Steffen, P.R. 80, 115 (1950). 9. G. Goertzel, P.R. 70, 897 (1946). 10. D. S. Ling and D. L. Falkoff, P.R. 76, 1639 (1949). 11. S. L. Ridgway, P.R. 78 {821! (1950). 12. J . R. Beyster and M. L. Wiedenbeck, P.R. 79, 176 (1950) 13. • D. T. Stevensen; and M. Deutsch, P.R. 78 640. (1950). 14- T. B. Novey, P.R. 78, 66 (1950). 15- D. L. Falkoff and G.E. Uhlenbeck, P.R. 79, 323 (1950). 16. E. K. Darby and W. Opechowski, P.R. 676 (195D 17- E. K. Darby, Can. J . Phys. 29, 569 (1951). IB. D. T. Stevenson and M. Deutsch, P.R. 83, 1202 (1951). 19. S. P. Lloyd, P.R. 83, 716 (1951). 20. P. G. Hess, i n print , Can. J . Phys. 21. U. S. National Bureau of Standards, C i r c u l a r 499. 22. H. Frauenfelder, P.R. 82, 549 (1951). 23- D. L. F a l k o f f and G. E. Uhlenbeck, P.R. 79, 334 (1950). 24- M. Fuchs and E. S. Lennox, P.R. 79, 221 (1950). 25. J . I. Hopkins, R.S.I. 22, 29 (1951). 26. L. M. Langer et a l , P.R. 79, 808 (1950). 27. S. Nakanawa et a l , P.R. £3, 1273 (1951). 2B. J. R. Beyster and M. L. Wiedenbeck, P.R. 79, 723 (1950). 29. J. S. Marshall and A. C. Ward, Can. Jr. Research A (15,39(1937) ). 30. Rutherford, Chadwick and E l l i s , Radiations from Radioactive Substances, Cambridge University Press, p. 414. 31. L. E. Glendenin, Nucleonics 2, 12 (194^). . x The following abbreviations have been used: Physical Review: P.R. Review of S c i e n t i f i c Instruments: R.S.I. i 

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