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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 UNIVERSITY FACULTY  OF BRITISH COLUMBIA  O F G R A D U A T E STUDIES  P R O G R A M M E OF T H E FINAL ORAL EXAMINATION FOR T H E DEGREE OF D O C T O R  OF PHILOSOPHY  of  EDSEL K E N N E T H  DARBY  B.Sc. (University of Saskatchewan) 1946 M.Sc. (University of Saskatchewan) 1948  T H U R S D A Y , A P R I L 24th, 1952, at 3:00 P.M. IN R O O M 303 PHYSICS B U I L D I N G  C O M M I T T E E I N CHARGE:  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 O F P U B L I C A T I O N S Negative Feedback. Dosage Rate Meter using a very small Ionization Chamber. H . E . Johns, E . K. Darby, J . J. S. Hamilton. American J . Roentgenology 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 K V P 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 E. K. Darby and W. Opechowski. Phys. Rev. 83, 676 (1951).  in Sb . 124  Some Studies in Angular Correlation. E . K. Darby. Can. J . of Physics 29, 569 (1951) .  THESIS S O M E 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 CORRELATION  Abstract: The  beta-gamma  angular correlation for Sb ' has been measured as a 12  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  a (E) =  W (180°, E) —W (90°, 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 correlation 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  00  and  Sc " performed with the same apparatus are in agreement with the previous 4  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 in PHYSICS  We accept t h i s t h e s i s as conforming t o the standard r e q u i r e d from  candidates  f o r t h e degree o f DOCTOR OF PHILOSOPHY.  Members o f t h e Department o f P h y s i c s .  THE  UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1952.  ABSTRACT The  beta-gamma angular c o r r e l a t i o n f o r Sb-^4 has been  measured as a f u n c t i o n o f the beta p a r t i c l e energy  i n the range  from O.B'2 Mev t o t h e end of t h e beta p a r t i c l e spectrum As a beta p a r t i c l e  spectrometer,  (2.4 Mev).  use was made o f a twelve  channel  k i c k s o r t e r and a t h i c k c r y s t a l b e t a p a r t i c l e s c i n t i l l a t i o n T h i s was connected counter.  counter.  i n c o i n c i d e n c e with a gamma r a y s c i n t i l l a t i o n  A c c o r d i n g l y , t h e beta gamma c o i n c i d e n c e counting r a t e  W(0,E), as a f u n c t i o n o f . t h e angle 0 between t h e counters, and the energy  E  o f t h e beta p a r t i c l e s , was observed.  angular c o r r e l a t i o n  The d i f f e r e n t i a l  coefficient:  a(E).Jw(l80 ,E) - W(90 ,En L W(90°,E) J O  o  was found t o vary smoothly from -0.17  at 1.0 Mev t o -0.44 at the end of t h e beta p a r t i c l e  spectrum.  When a(E) i s i n t e g r a t e d , n u m e r i c a l l y , over a l l beta p a r t i c l e g i e s g r e a t e r than 0.82- Mev., t h e value o f t h e i n t e g r a t e d correlation coefficient  a z -0.24 —  0.02 was found.  measurements o f the value of t h e i n t e g r a t e d angular c o e f f i c i e n t were a l s o performed,  ener-  angular Direct  correlation  and the r e l a t i o n t o t h e above  value & £ a c o n s i d e r e d . An attempt  has been made t o i n t e r p r e t .these  results  i n terms of the a n g u l a r momenta o f the p a r t i c l e s emitted, u s i n g the t h e o r y developed  by F a l k o f f and Uhlenbeck.  Experiments on  the gamma-gamma angular c o r r e l a t i o n o f Co^° and S c ^ performed w i t h t h e same apparatus of other workers.  a r e i n agreement w i t h the p r e v i o u s r e s u l t s  TABLE OF 'CONTENTS Page I II, III  Introduction.  1  Theory o f Gamma-Gamma-Angular C o r r e l a t i o n  7  Gamma-Gamma Angular C o r r e l a t i o n f o r Co^° and S c ^  IV V VI  11  b  Theory o f Beta-Gamma-Angular C o r r e l a t i o n Apparatus f o r Measuring t h e I n t e g r a t e d Beta-Gamma-Angular C o r r e l a t i o n I n t e g r a t e d Beta-Gamma-Angular C o r r e l a t i o n for S b ^ 1 2  VII  19 21  Beta Gamma Angular C o r r e l a t i o n as a F u n c t i o n of Energy f o r S b *  24  R e s u l t s and C o n c l u s i o n s Regarding t h e 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 f o r Sb **-  28  1 2  VIII  16  12  TABLE OF APPENDICES  Page  Coincidence Counting with D i r e c t i o n a l C o r r e l a t i o n Between t h e Emitted P a r t i c l e s .  21  Appendix I I  Procedure t o E v a l u a t e the True Coincidence Rate  34  Appendix I I I  Compton S c a t t e r i n g o f Gamma Rays Between Coincidence Counters  37  Appendix IV  Standard D e v i a t i o n i n W(0)  39  Appendix V  C a l i b r a t i o n o f the K i c k s o r t e r Channels i n Terms o f Beta Energy  40  Appendix VI  C a l c u l a t i o n o f t h e Beta-Gamma-Angular  Appendix I  Correlation Coefficient.  41  Standard D e v i a t i o n i n a(E)  44  C a l c u l a t i o n o f a From a(E) E v a l u a t i o n o f T h e o r e t i c a l Beta-GammaAngular C o r r e l a t i o n C o e f f i c i e n t s  45  Appendix X  Electronic Circuits  49  Appendix XI  On t h e I n f l u e n c e o f A b s o r p t i o n on t h e Measured Value o f a.  50  Appendix V I I Appendix V I I I Appendix IX  47  TABLES  Page  The Gamma-gamma-angular function f o r C o  correlation  Table I I  The gamma-gamma-angular f u n c t i o n f o r Se^°  correlation  Table I I I  The beta-gamma-angular c o r r e l a t i o n f u n c t i o n f o r S b * ^ i n t e g r a t e d over b e t a e n e r g i e s 0 . 9 - 2 . 4 Mev.  22  Averaging o f d i f f e r e n t i a l beta-gammaangular c o r r e l a t i o n c o e f f i c i e n t a(E) f o r S b 4 (Runs 1 t o 4)  26  C a l c u l a t i o n of 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 from t h e results of a(E).  27  Table VI  Values o f a ( E ) f o r p o s s i b l e transitions.  29  Tables VII  Fermi P l o t o f Beta Spectrum  40  Table V I I I  a(E) f o r S b  Table I  6 0  14  14  2  Table IV  1 2  Table V  0  1 2 4  (Run 4)  42  FIGURES Page Fig. 1  Disintegration  scheme f o r C o  1  Fig. 2  Coincidence c o u n t i n g  1  Fig. 3  General n o t a t i o n f o r s u c c e s s i v e nuclear t r a n s i t i o n s  7  D U  Fig. 4  Counter arrangement f o r gamma-gammaangular c o r r e l a t i o n  Fig. 5  Coincidence counting c i r c u i t  block  diagram  11  Fig. 6  Lead S h i e l d i n g  Fig. 7  E f f e c t o f a n n i h i l a t i o n r a d i a t i o n on W(0) Gamma-Gamma-Angular C o r r e l a t i o n f o r Co  Fig. 8  11  t o prevent s c a t t e r i n g  6 0  Fig. 9  Gamma^Gamma-Angular Sc4o  F i g . 10  Apparatus  12 13 14  Correlation f o r  f o r Beta-Gamma  Angular  Correlation  19  F i g . 11  Disintegration  Scheme f o r S b  F i g . 12  I n t e g r a t e d beta-gamma angular correlation f o r Sb 4 22 Apparatus f o r D i f f e r e n t i a l Beta-gammaangular c o r r e l a t i o n 24  1 2  ^  21  1 2  F i g . 13 F i g . 14  The Four Runs f o r a(E) f o r S b  F i g . 15  The Averaged R e s u l t s f o r a(E) f o r  1 2 / f  Sb <-  26  12/  F i g . 16  Counter  Fig.  Fermi P l o t o f the Beta Spectrum  17  26  Geometry  31 40  ACKNOWLEDGEMENTS  I am indebted t o P r o f e s s o r ¥ . Opechowski f o r v a l u a b l e d i s c u s s i o n s i n t h e course o f these experiments, and f o r help i n i n t e r p r e t i n g t h e experimental  results.  I am  a l s o indebted t o Dr. A. H. M o r r i s h f o r h i s suggestions i n the e a r l y stages o f the gamma-gamma c o r r e l a t i o n experiments and  to Mr. G. W. W i l l i a m s  f o r h i s a s s i s t a n c e i n design and  c o n s t r u c t i o n o f t h e apparatus.  I a l s o wish t o acknowledge  t h e k i n d a s s i s t a n c e o f other s t a f f members o f t h e P h y s i c s Department and t h e e x c e l l e n t l a b o r a t o r y f a c i l i t i e s by t h e U n i v e r s i t y o f B r i t i s h Columbia P h y s i c s T h i s r e s e a r c h was made p o s s i b l e by a grant  supplied  Department.  from t h e N a t i o n a l  Research C o u n c i l o f Canada, and t h e author was a i d e d by a F e l l o w s h i p from t h e N a t i o n a l Research C o u n c i l o f Canada.  INTRODUCTION. The  e m i s s i o n o f a beta p a r t i c l e from a r a d i o a c t i v e  nucleus o f t e n l e a v e s the product nucleus i n an e x c i t e d s t a t e . The e x c i t e d nucleus may then decay, by the emission o f one o r more gamma r a y s i n cascade,  t o t h e ground s t a t e .  fit) scheme o f Co ( F i g . 1A) i s a simple cascade.  The d i s i n t e g r a t i o n  example o f two gamma r a y s i n  The l i f e t i m e o f t h e i n t e r m e d i a t e e x c i t e d s t a t e s i s i n -12  g e n e r a l very small (10  sec.) so the two gamma r a y s may be  regarded  f o r most purposes.  as simultaneous  gamma r a y counters near the Co to  a circuit  source  I f we arrange two  ( F i g . 2) and connect  designed t o r e g i s t e r two simultaneous  pulses  cidence c i r c u i t ) , we w i l l be able to o b t a i n a double counting r a t e .  them  (coin-  coincidence  T h i s c o i n c i d e n c e r a t e i s much s m a l l e r than t h e  s i n g l e counting r a t e i n e i t h e r counter  since t h e p r o b a b i l i t y o f  counting both gamma r a y s s i m u l t a n e o u s l y i s s m a l l . of having two gamma rays i n cascade, f o l l o w i n g the emitted beta p a r t i c l e  I f , instead  we have o n l y one gamma r a y ( F i g . IB),- we can o b t a i n  c o i n c i d e n c e counts by u s i n g one beta p a r t i c l e counter and one gamma counter. Now the atoms i n t h e r a d i o a c t i v e s o u r c e * are i n t h i s case randomly o r i e n t e d so i n t h e double-gamma cascade t h e 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 o f t h e i n i t i a l  s t a t e must be l o n g enough f o r t h e  angular momentum v e c t o r s o f the n u c l e i t o become randomly o r i e n ted,  e i t h e r by p r e c e s s i o n i n a magnetic f i e l d due t o o r b i t a l  e l e c t r o n s or by thermal  agitation.  C o  6  0  . N i  Z  1.17  60  z  Mev.  Gannna  Rcny  ' • ••  1.33  <  Mev. FIG. 1  >  Disintegration B  A  G a m m a  •Z  Multiplier  Rays  C o  Photoand Co i n c  S c h e m e s  6  Source  Crysfa iolence  M i x e r  S e a  er  F I G - 2 C o i n c  i d e n c e  Courfti  ng  2. d i r e c t i o n , however t h e second gamma-ray i n some cases may have i t s d i r e c t i o n o f emission c o r r e l a t e d w i t h r e s p e c t t o t h e d i r e c t i o n o f e m i s s i o n o f the f i r s t  gamma-ray.  This e f f e c t i s c a l l e d  the a n g u l a r c o r r e l a t i o n between t h e d i r e c t i o n s o f t h e two gamma r a y s , o r more b r i e f l y , gamma-gamma angular c o r r e l a t i o n . measure t h i s gamma-gamma angular c o r r e l a t i o n - b y gamma-gamma c o i n c i d e n c e r a t e as a f u n c t i o n the  counters.  We can  observing the  o f 0, t h e angle between  The gamma-gamma c o i n c i d e n c e r a t e as a f u n c t i o n  o f 0 ( a f t e r a small c o r r e c t i o n  f o r f i n i t e counter s i z e ) i s propor-  t i o n a l t o t h e gamma-gamma angular c o r r e l a t i o n f u n c t i o n , W(9), which i s t h e p r o b a b i l i t y o f t h e emission o f the second gamma r a y i n d i r e c t i o n 0 with r e s p e c t t o t h e f i r s t gamma-ray's d i r e c t i o n of  emission. In a s i m i l a r manner i f we observe t h e beta-gamma  c o i n c i d e n c e r a t e we sometimes observe a beta-gamma-angular correlation. the  The r e s e a r c h e s d e s c r i b e d i n t h i s t h e s i s  concern  beta-gamma- and gamma-gamma- angular c o r r e l a t i o n observed i n  several  radioactive  nuclei.  As an example o f t h e i n f o r m a t i o n which i s a v a i l a b l e from these experiments, l e t us c o n s i d e r t h e case o f the gammagamma angular c o r r e l a t i o n i n more d e t a i l . are  Gamma r a y t r a n s i t i o n s  c l a s s i f i e d on the b a s i s o f the angular momentum c a r r i e d away  by t h e gamma r a y .  The m u l t i p o l e order o f the t r a n s i t i o n i s 2^  where L i s t h e z-component o f t h e a n g u l a r momentum c a r r i e d away by t h e gamma r a y .  Thus i f the gamma r a y has angular momentum  L = l ( i n u n i t s o f " f t ) , t h e t r a n s i t i o n i s c l a s s i f i e d as d i p o l e . The t r a n s i t i o n s a r e c l a s s i f i e d f u r t h e r not  t h e r e i s a change o f p a r i t y , t h a t  a c c o r d i n g t o whether o r i s whether o r not t h e r e i s  3. a change i n t h e symetry p r o p e r t i e s o f t h e wave-functions two  levels.  of the  I n t h e case o f d i p o l e 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 d i p o l e , i f i t does not change, it  i s c a l l e d magnetic d i p o l e .  I f we c o n s i d e r o n l y t h e case o f  "pure" m u l t i p o l e t r a n s i t i o n s (that i s o n l y one angular momentum a s s o c i a t e d w i t h a g i v e n t r a n s i t i o n ) then t h e t h e o r y i n d i c a t e s t h a t the form o f t h e a n g u l a r - c o r r e l a t i o n f u n c t i o n i s a s f o l l o w s :  X  \  I w h e r e Z . A i s t h e lowest m u l t i p o l e o r d e r o f t h e 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 o f the multipole orders o f  the t r a n s i t i o n s , and o f t h e angular momenta o f t h e n u c l e a r l e v e l s involved i n the t r a n s i t i o n s .  F o r many cases t a b l e s  ( 2  been prepared which g i v e n u m e r i c a l values f o r t h e d j .  ) have  Experi-  m e n t a l l y , we can determine H Ck' q u i t e a c c u r a t e l y , and.the v a l u e s L  of the i n d i v i d u a l  d{ approximately.  . I f we have a v a i l a b l e some  o t h e r i n f o r m a t i o n concerning e i t h e r t h e angular momentum o f t h e l e v e l s o r the m u l t i p o l e -order of t h e t r a n s i t i o n s , we can compare the experimental data w i t h t h e t a b l e s , and u s u a l l y o b t a i n a d e f i n i t e assignment o f angular momentum and m u l t i p o l e o r d e r s to the l e v e l s and t r a n s i t i o n s i n v o l v e d . As an example o f the" procedure the data l e t us c o n s i d e r t h e case o f C o ^ . Co  o Q  g i v e the c o r r e l a t i o n c o e f f i c i e n t  Moreover from the graph  i n interpretation of The measurements f o r  d; -  0.1 5 7 - 0.01  ( F i g . 8) o f the c o r r e l a t i o n f u n c t i o n we  see t h a t t h e f u n c t i o n i s p r o b a b l y o f t h e form .' 2 4" Although t h e accuracy o f t h e 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 ,  t h e f a c t t h a t t h e r e i s a term i n  Cos^Q means t h a t both t r a n s i t i o n s a r e o f a t l e a s t quadrupole order.  We have one o t h e r p i e c e o f i n f o r m a t i o n ,  namely t h a t 0  the angular momentum o f t h e ground s t a t e i s probably since t h e product n u c l e u s neutrons.  }  has an even number o f both p r o t o n s and  The t a b l e s may now be searched q u i t e q u i c k l y and i t  i s seen t h a t t h e c o r r e l a t i o n f u n c t i o n f o r t h e double t r a n s i t i o n * 2 2 4 —> 2 — * This gives  0  ^  is s  W(0) =  @t * &z  ~  1  0.125 c o s  -h +  eos^Q.  0 . 1 6 5 i n q u i t e c l o s e agree-  ment w i t h t h e e x p e r i m e n t a l value 0 . 1 5 7 * therefore  9 -+0.040  2  0.010.  t h a t t h e a n g u l a r momenta and m u l t i p o l e  We conclude orders f o r the  t r a n s i t i o n i n v o l v e d a r e v e r y l i k e l y t o be g i v e n by t h e f o l l o w i n g 2 -2. scheme 4 — ^ 2 — » 0 . The  r e s u l t s f o r beta-gamma a n g u l a r c o r r e l a t i o n may  be i n t e r p r e t e d i n a s i m i l a r f a s h i o n ,  leading t o information  *The a n g u l a r momentum o f a n u c l e a r s t a t e i s o f t e n r e f e r r e d to as t h e " s p i n " .  Where no c o n f u s i o n should a r i s e t h e t e r m  w i l l be used f o r t h e sake o f b r e v i t y . 2. a  spin  The t r a n s i t i o n scheme  4 — ^ 2 —-9 0 . i s t o be i n t e r p r e t e d a s f o l l o w s :  the i n i t i a l  s t a t e has a s p i n o f 4 ( i n u n i t s of -tl), and a t r a n s i t i o n t a k e s p l a c e t o an i n t e r m e d i a t e s t a t e having s p i n 2 .  This t r a n s i t i o n  i s accompanied by a gamma r a y having a n g u l a r momentum 2 so i t i s c l a s s i f i e d a s quadrupole.  ;A t r a n s i t i o n then t a k e s  place  from the i n t e r m e d i a t e s t a t e t o t h e ground s t a t e w i t h s p i n 0 , t h i s t r a n s i t i o n a l s o being c l a s s i f i e d as quadrupole.  r e g a r d i n g the  s p i n s o f the  l e v e l s involved,  the m u l t i p o l e o r d e r s  o f the gamma t r a n s i t i o n . The  o r i g i n a l suggestion t h a t  gamma-gamma-angular  c o r r e l a t i o n might e x i s t was due t o Dunworth (1);  Following  t h i s suggestion, Hamilton (2) i n 1940 presented t h e t h e o r y o f the  e f f e c t g i v i n g the  r e s u l t s t h a t have j u s t been  outlined.  Attempts (3) (4) t o d e t e c t 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) r e p o r t e d t h e successful  detection  four n u c l e i . of highly  o f a gamma-gamma a n g u l a r c o r r e l a t i o n i n  Their  success was due l a r g e l y t o t h e development  e f f i c i e n t gamma counters u s i n g c r y s t a l phosphors and  photo-multipliers.  During 1949 and 1950 numerous o b s e r v e r s  ( 6 ) ( 7 ) ( 8 ) r e p o r t e d c o n f i r m a t i o n o f t h e s e 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 .  F u r t h e r d e t a i l s o f the  t h e o r y had a l s o been  given by G o e r t z e l (9) ( e f f e c t o f an e x t e r n a l Ling  and F a l k o f f  (10)  magnetic f i e l d ) , and  ( e f f e c t o f "mixed" m u l t i p o l e t r a n s i t i o n s ) . *  Although many n u c l e i have e x h i b i t e d  a gamma-gamma-  angular c o r r e l a t i o n , numerous attempts to d e t e c t a beta-gammaangular c o r r e l a t i o n r e s u l t e d  i n failure,  1951  Rb^6) had y i e l d e d  only two n u c l e i  (Sb 4  gamma-angular c o r r e l a t i o n  1 2  j  By  a d e f i n i t e beta-  (11)(12)(13), moreover the  r e p o r t e d were not c o n s i s t e n t . reasonable i n t e r p r e t a t i o n  (11)(hi)(13)(14)•  results  S u f f i c i e n t t h e o r y to a l l o w a  o f r e s u l t s i n most cases had been  developed i n 1950 by F a l k o f f and Uhlenbeck (15).  Further  inves-  t i g a t i o n i n t o t h e 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 c o n t r i b u t i o n s  o f the  researches  d e s c r i b e d i n t h i s t h e s i s a r e t h e experiments on the  beta-gamma-  angular c o r r e l a t i o n i n Sb-^ ^.  the b e t a  2  I n p a r t i c u l a r since  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 , of the  angular c o r r e l a t i o n  on the  t h e dependence  energy o f the beta  was  the object of t h i s i n v e s t i g a t i o n .  was  predicted t h e o r e t i c a l l y  not  been observed p r e v i o u s t o t h i s i n a q u a n t i t a t i v e  by F a l k o f f  results of this investigation preliminary report  (16)  particles  Such an energy dependence and Uhlenbeck (15)  but had  way.  have been .published, f i r s t  and l a t e r i n a : f u l l paper (17).  The  in a Similar  r e s u l t s have l a t e r been p u b l i s h e d 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  w i t h the  experimental method, t h e agreement  present work being q u i t e good. The  gamma-gamma c o r r e l a t i o n  experiments were not new,  Brady and Deutsch (5) having observed t h i s p r e v i o u s l y . the  However  accuracy obtained i n t h e present work i s somewhat b e t t e r than  i n the  p r e v i o u s work.  l a t e r i n the of the  Moreover, t h e same apparatus was employed  beta-gamma c o r r e l a t i o n  gamma-gamma c o r r e l a t i o n  experiments, so the c o n s i s t e n c y  experiments l e n d s support t o the  subsequent beta-gamma c o r r e l a t i o n  results.  7. II. THEORY OF GAMMA-.GAMMA-ANGULAR  CORRELATION.  The theory o f d i r e c t i o n a l c o r r e l a t i o n between s u c c e s s i v e l y emitted n u c l e a r p a r t i c l e s h a s been developed i n c o n s i d e r a b l e d e t a i l i n s e v e r a l papers,  (2)(9)(10).  The problem  i n v o l v e s the d e t e r m i n a t i o n o f W(0), t h e r e l a t i v e p r o b a b i l i t y o f the emission o f p a r t i c l e 2 i n d i r e c t i o n k of  2  f o l l o w i n g t h e emission  p a r t i c l e 1 i n d i r e c t i o n k^_ t h e angle between k^ 5  9 and the t r a n s i t i o n t a k i n g p l a c e from A — ? B,  and "k^  B—?C  being  as shown  i n F i g . 3. Hamilton  (2) has a p p l i e d second order time dependent  p e r t u r b a t i o n theory t o t h e gamma-gamma problem and o b t a i n e d W(9) i n terms o f the s p i n s of the n u c l e a r s t a t e s i n v o l v e d and a n g u l a r momenta o f the emitted photons. assumptions, 1.  Hamiltons theory i n v o l v e s two  namely:  That t h e l i f e t i m e o f the i n t e r m e d i a t e s t a t e i s short compared w i t h t h e n u c l e a r p r e c e s s i o n a l p e r i o d .  2.  That the t r a n s i t i o n s a r e "pure" m u l t i p o l e t r a n s i t i o n s and not due to a mixture  G o e r t z e l (9) has extended  of m u l t i p o l e s .  Hamilton's  theory t o cover the case i n  which t h e r e i s a s t r o n g magnetic f i e l d assumption extended  above i s not f u l f i l l e d .  present  so t h a t t h e f i r s t  L i n g and F a l k o f f  the theory to take i n t o account  (10) have  mixtures o f m u l t i p o l e s .  F a l k o f f and Uhlenbeck (15) have g e n e r a l i z e d Hamilton's t h e o r y  (still  r e t a i n i n g 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  particles.  S p e c i f i c a t i o n o f t h e types o f emitted p a r t i c l e s determines the parameters and g i v e s W(Q) f o r the r e q u i r e d type o f c o r r e l a t i o n .  Choosing  k^ as the a x i s o f q u a n t i z a t i o n t h e g e n e r a l r e s u l t o f  the t h e o r y  (2) (15) il.s:,  XZ[5 l(BjH^)|C )rj 2  where  IfS. I H. L w l L p J  p  denotes .the m a t r i x element o f t h e  Hamiltonian d e s c r i b i n g t h e i n t e r a c t i o n o f t h e 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 s u b s t a t e of  the (nuclear) s t a t e  B  m  (a s t a t e as c h a r a c t e r i z e d by a d e f i n i t e  v a l u e of t h e t o t a l angular momentum quantum number) to t h e substate p o f the s t a t e i s of course  similar.  C.  The meaning o f ^/\^>  j |~J|(d)|  ^>yy\  S-^ r e p r e s e n t s t h e average o v e r a l l  d i r e c t i o n a l i n f o r m a t i o n ( s p i n s , p o l a r i z a t i o n ) f o r the f i r s t t r a n s i t i o n except  o f course t h e d i r e c t i o n k-^ w i t h a s i m i l a r  meaning f o r S£.  The q u a n t i t y .'  S , K A J H  l  ( W l B j r =  ,  i s the p r o b a b i l i t y o f the emission o f the f i r s t  particle  < 2 )  along  the a x i s o f q u a n t i z a t i o n f o r t h e t r a n s i t i o n between s u b l e v e l s Aj^ that  B  m  •  Pkv^ p &)  .  i  s  d e f i n e d s i m i l a r l y so  ¥ ( 0 ) i s g i v e n by '  In g e n e r a l c o n s i d e r i n g a t r a n s i t i o n between s t a t e s with t o t a l and  Z-  component angular momenta ~J yn }  and ~$ Yy\  r e s p e c t i v e l y and the e m i s s i o n o f a p a r t i c l e having a n g u l a r momentum quantum numbers L and M group t h e o r e t i c a l methods show that,  9.  (4)  can be w r i t t e n  , J L T  _  n (5)  a nol • W\ + M ^ J L J '  with Where t h e  j ' * T + L  ,  C-y  .  when evaluated y i e l d numerical  values  depending o n l y upon these angular momenta, and not upon the k i n d of p a r t i c l e emitted and t h e  [  are t h e a n g u l a r  d i s t r i b u t i o n s which depend on the type o f p a r t i c l e emitted. t u r n s out t h a t i f a l l t h e i n i t i a l  It  s u b l e v e l s a r e e q u a l l y populated,  and the d i r e c t i o n o f emission o f t h e p a r t i c l e i s s p e c i f i e d ,  then  the i n t e r m e d i a t e s u b l e v e l s w i l l i n g e n e r a l not be e q u a l l y populated. Since t h e i n t e r m e d i a t e s u b l e v e l s a r e not e q u a l l y populated, t h e 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 thus e x h i b i t i n g an angular The  correlation.  J- ^  may be expressed  i n t h i s case powers o f cos 0 .  parametric forms,  number o f independent  distribution,  parameters i s L.  i n suitable The maximum  F a l k o f f and Uhlenbeck (15)  o b t a i n s e v e r a l convenient parametric forms.  The parameters a r e  s p e c i f i e d a c c o r d i n g to t h e type o f p a r t i c l e emitted. ~ J L J ' only necessary to e v a l u a t e t h e  (j- ^  ,  I t i s then  which a r e independ-  ent o f the type o f p a r t i c l e emitted, and then t o o b t a i n the ¥ ( 0 ) as i n formula Hamilton L  —  1  (3).  The r e q u i r e d sums have been e v a l u a t e d by  (2) and a r e l i s t e d by F a l k o f f and Uhlenbeck (15) f o r o r 2.  For h i g h e r v a l u e s o f angular momentum quantum  numbers ( o f t h e n u c l e a r s t a t e s and t h e emitted p a r t i c l e s ) t h e computations  o f these sums become very d i f f i c u l t , and they have  10.  been evaluated o n l y f o r a c e r t a i n number o f s p e c i a l cases. example Lloyd  (19)  The  and  gamma-gamma a n g u l a r c o r r e l a t i o n f u n c t i o n s have  correlation function  gives tables  For  (20).  and Hess  been t a b u l a t e d i n a " c a n o n i c a l form" by Hamilton ( 2 ) . he w r i t e s the  '  i n the  That i s ,  form:  o f R, Q and S as f u n c t i o n s o f the a n g u l a r momenta  involved. The tables  same r e s u l t s may be o b t a i n e d r e a d i l y from t h e  o f F a l k o f f and Uhlenbeck (15)  by i n s e r t i n g the  proper  parameters f o r the gamma-gamma t r a n s i t i o n s . The  following  g e n e r a l r e s u l t s o f the t h e o r y  gamma-gamma c o r r e l a t i o n may be l i s t e d : 1.  The c o r r e l a t i o n f u n c t i o n  for j_  i s o f the form ; W ( £ ) = l + 2  where L i s the minimum o f L-^, l>22.  The c o e f f i c i e n t s the  01 depend on t h e angular momenta o f 1  t h r e e l e v e l s and on the  t r a n s i t i o n but n o t on p a r i t y  m u l t i p o l e order o f t h e change.  2  3.  The h i g h e s t power o f c o s 9 cannot exceed  2-J so t h e r e  i s no c o r r e l a t i o n i f t h e i n t e r m e d i a t e s t a t e has a s p i n of 0 , o r £.  . '  .  11. III. 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 o f Co^O which c o n s i s t s o f a 0.31 Mev beta group f o l l o w e d by cascade gamma r a y s o f e n e r g i e s 1.17  Mev and 1.33 Mev i s g i v e n i n F i g . 1 (21).  Sc^  6  has a  s i m i l a r decay scheme, (21), with a maximum beta p a r t i c l e  energy  of O.36 Mev, and cascade gamma r a y s w i t h e n e r g i e s o f 0.88 Mev and 1.12  Mev r e s p e c t i v e l y . The Co  source c o n s i s t e d o f a wire o f c o b a l t  metal  about '0.1 mm i n diameter and 4 mm i n l e n g t h which had been i r r a d i a ted  i n the Chalk R i v e r p i l e to an a c t i v i t y o f about 30,0 yu c u r i e s .  The  source:- was supported v e r t i c a l l y and e n c l o s e d i n a s u f f i c i e n t  t h i c k n e s s o f aluminium  to stop a l l Co^O beta p a r t i c l e s .  source c o n s i s t e d o f ScO i n an aluminium  The  Sc^  capsule.  The gamma counters employed 1 x 1 x i i n . anthracene c r y s t a l s cemented t o RCA-5819 p h o t o m u l t i p l i e r s w i t h Canada balsam. The c r y s t a l s were covered w i t h a 0.001 i n . t h i c k aluminium a light reflector.  f o i l as  The counters were mounted on a spectrometer  t a b l e , t h e g e n e r a l arrangement being as shown i n F i g . 4«  The  source was r i g i d l y supported and a c c u r a t e l y c e n t r e d on a t h i n aluminium  rod machined t o f i t t h e h o l e i n the c e n t r e o f t h e  spectrometer t a b l e . Fig.  The e l e c t r o n i c s arrangement i s shown i n  5 w i t h the d e r a i l e d c i r c u i t s being g i v e n i n Appendix IX.  The r e s o l v i n g time o f the c o i n c i d e n c e mixer i s 0.18 p - s e c . The angular r e s o l u t i o n o f t h e counters was measured u s i n g a n n i h i l a t i o n r a d i a t i o n from a C u ^ source. 4  subtended  An angle o f 20°  by the counter a t t h e source was s e l e c t e d a s g i v i n g a  sufficient,..' counting r a t e , without r e q u i r i n g too l a r g e a c o r r e c t i o n  A r r a n g e menf  A mi  D ISC  5819 ScalerMo) Coi rtC\olencj | Source •  IZOO.V  Mixer  S t o bil i zedl  p.n  Scqler(£>40) Amplifier D j sc r i m m a t o r  9ca lev (10)  FIG 5  C o i incidence C i rcuit  f o r angular  resolution. The t r u e angular c o r r e l a t i o n between t h e gamma r a y s  may  be masked by t h r e e s p u r i o u s e f f e c t s . 1.  These a r e as f o l l o w s :  Compton s c a t t e r i n g i n one counter produces a lower  gamma r a y which may be counted  i n t h e second counter.  Compton s c a t t e r i n g produces a count  i n the f i r s t  p r o c e s s r e s u l t s i n a c o i n c i d e n c e count.  energy  Since t h e  counter, t h e above  For t h e case o f two 1 Mev  gamma r a y s i n cascade,  a calculation  (See Appendix I I I ) shows t h a t  the c o i n c i d e n c e counts  so produced, which we c a l l t h e " s c a t t e r e d  c o i n c i d e n c e r a t e " , w i l l be about $0% o f the t r u e c o i n c i d e n c e r a t e , i f we assume equal counting e f f i c i e n c i e s f o r both t h e 1 Mev gamma ray and t h e gamma r a y s c a t t e r e d through  180°.  With t h e f o r e g o i n g  assumptions t h e r a t i o o f s c a t t e r e d t o t r u e c o i n c i d e n c e r a t e when the counters are p l a c e d a t 90  w i l l . i n c r e a s e , because t h e s o l i d  angle between the counters i s i n c r e a s e d , although t h e number o f s c a t t e r e d gammas p e r s o l i d angle i s s l i g h t l y  decreased.  T h i s s c a t t e r i n g from one counter t o t h e o t h e r may be prevented The  by means o f l e a d s h i e l d i n g such as i s shown i n F i g . 6.  source i s p l a c e d i n f r o n t o f the a p e r t u r e i n t h e l e a d s h i e l d  and t h e s c a t t e r i n g i s reduced  d i r e c t l y i n the proportion of the  area o f the a p e r t u r e t o t h e a r e a o f the counter c r y s t a l .  In t h i s  way  i t was p o s s i b l e to reduce t h e s c a t t e r e d c o i n c i d e n c e r a t e t o  Jqq  o f i t s value without  a shield.  T h i s method i s p a r t i c u l a r l y  s u i t e d t o low energy gamma r a y s , i . e . 500 Kev o r l e s s , i n which case.the  s c a t t e r e d gamma r a y i s not a p p r e c i a b l y lower i n energy.  For gamma rays o f the order o f 1 Mev, the backward s c a t t e r e d gamma ray i s about 200 Kev,  so i n t h i s case i t i s s i m p l e r t o merely  d i s c r i m i n a t e a g a i n s t t h e 200 Kev p u l s e s , the 1 Mev gamma r a y s  P M  F I G . 6  - L e a d  S h i e l d i n g  13.  being counted with o n l y s l i g h t l y reduced 2.  efficiency.  Compton and photo e l e c t r o n s may be e j e c t e d from t h e l e a d  s h i e l d i n g i f i t i s used.  Owing to t h e r e l a t i v e l y high  efficiency,  of the counters f o r c o u n t i n g beta p a r t i c l e s , t h e c o u n t i n g r a t e  due t o t h i s s c a t t e r i n g may be l a r g e compared w i t h t h e gamma c o u n t i n g ::• rate.  T h i s e f f e c t r e s u l t s mainly i n changing  angle o f t h e counters so i t s presence  the e f f e c t i v e  i s undesirable.  solid  I t may e a s i l y  be e l i m i n a t e d by p l a c i n g s u f f i c i e n t aluminium i n f r o n t o f t h e counters to stop photo e l e c t r o n s from the gamma r a y s i n q u e s t i o n . 3.  For gamma r a y s o f energy g r e a t e r t h a n 1.02 Mev, l e a d  s h i e l d i n g cannot  be used w i t h confidence s i n c e p a i r p r o d u c t i o n i s  then p o s s i b l e and t h e 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 stronglycorrelated.'  F o r example w i t h t h e s h i e l d i n g arrangement shown i n 60  F i g . 6 and a Co  source  (gamma e n e r g i e s :  1.17 and 1 . 3 3 Mev), t h e  c o r r e l a t i o n curve^shown i n F i g . 7 was o b t a i n e d .  With t h e b i a s o f  the d i s c r i m i n a t o r s set t o e l i m i n a t e p u l s e s corresponding t o gamma r a y s below 250 Kev and no l e a d s h i e l d i n g i n p l a c e , t h e curve^shown i n F i g . 7 was o b t a i n e d .  The curves c o i n c i d e up t o about 160° and  the sharp r i s e i n curve B from 160° t o 180° i s c h a r a c t e r i s t i c o f t h a t produced to subtend  by a n n i h i l a t i o n r a d i a t i o n w i t h t h e c o u n t e r s . s e t  an i n c l u d e d angle o f 20°. I t was concluded  from t h e p r e c e d i n g i n v e s t i g a t i o n t h a t  the most s a t i s f a c t o r y method o f a v o i d i n g s p u r i o u s e f f e c t s i n measu r i n g t h e a n g u l a r c o r r e l a t i o n i n Co^° and S c ^ was t o d i s c r i m i n a t e a g a i n s t Compton s c a t t e r e d gamma r a y p u l s e s .  Accordingly the  d i s c r i m i n a t o r s were c a l i b r a t e d u s i n g t h e C o  1.3 Mev gamma p u l s e s ,  D U  and  set t o e l i m i n a t e p u l s e s corresponding t o e n e r g i e s o f l e s s than  250  Kev.  The o n l y s h i e l d i n g employed was - " o f aluminium 16  immediately  i n f r o n t o f each counter t o stop b e t a The f o l l o w i n g procedure  particles.  has been adopted  i n the  e v a l u a t i o n o f the experimental data: 1.  The data r e c o r d e d c o n s i s t e d o f the time T ( h o u r s ) ,  the t o t a l c o i n c i d e n c e counts channel counts 2.  C(0), and the t o t a l  ( s c a l e d by 640)  A and  B.  Since the counter e f f i c i e n c i e s may  t o H.T. r a t e was  single  vary s l i g h t l y  (due  v a r i a t i o n s or moving one c o u n t e r ) , the c o i n c i d e n c e d i v i d e d by the i n d i v i d u a l channel counting r a t e s  to c o r r e c t f o r these s l i g h t v a r i a t i o n s .  Upon s u b t r a c t i n g  the a c c i d e n t a l c o i n c i d e n c e r a t e , c a l c u l a t e d i n the u s u a l manner, we  o b t a i n the q u a n t i t y :  ( 2 T J X I0 7 where 2T-4-07. AS The d e t a i l s are g i v e n i n Appendices I and I I . C (Q) T  =  3.  The u n c o r r e c t e d c o r r e l a t i o n f u n c t i o n ,  was  next  evaluated. was  c o r r e c t e d f o r angular r e s o l u t i o n by the  method o u t l i n e d i n Appendix I I t o o b t a i n the t r u e angular correlation function, '5.  The  W(9).  standard d e v i a t i o n was  calculated s t a t i s t i c a l l y  on  the b a s i s o f the number o f counts o b t a i n e d , (see Appendix IV). 6. W(9)  A comparison was  made w i t h the t h e o r e t i c a l v a l u e s f o r  c a l c u l a t e d from H a m i l t o n s T  tables (2).  The e v a l u a t i o n o f the experimental data i s shown i n Tables I and I I f o r Co°° and S c ^  respectively.  The  c o r r e c t e d p o i n t s (W(9)) are p l o t t e d i n F i g s . & and 9 for  Co ^ D  TABLE  I  Gamma-Gamma Angular C o r r e l a t i o n i n Co  9  C (9) T  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  150°  1028  0.109  0.114  0.010  1.117  160°  1040  0.130  0.136  0.010  1.142  170°  1053  0.153  0.159  0.012  1.158  1049  0.147  0.153  0.010  I.I65  ISO  0  TABLE I I  Gamma-Gamma Angular C o r r e l a t i o n i n S c ^  0  C (9) T  W(0)-1  W(9)-l  6  w(e)  S.D. T h e o r e t i cal.  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.I63  180°  718  0.152  0.15$  0.009  1.165  and Scr*"  The s o l i d curve i n F i g s . & and 9  respectively.  represents the f u n c t i o n :  1+  0.\Z5CQS&  +0.0+0  COS 4-  which i s t h e t h e o r e t i c a l f u n c t i o n W(9) c o r r e s p o n d i n g t o t h e 4~*2 ~*> 0  transition  as g i v e n by Hamilton  (2).  These r e s u l t s  are in. good agreement w i t h those o f Brady and Deutsch ( 5 ) . I t i s worth  rioting  that there i s a small systematic  d e v i a t i o n o f t h e experimental curves from t h e t h e o r e t i c a l v a l u e s i n both Go^° and S c ^ .  The c o r r e l a t i o n i s s l i g h t l y l e s s than  p r e d i c t e d i n both cases.  These d e v i a t i o n s are p o s s i b l y e x p l a i n -  , ed by t h e p r e c e s s i o n o f t h e n u c l e u s duriiig t h e l i f e t i m e o f t h e i n t e r m e d i a t e s t a t e , which would be expected t o reduce t h e angular correlation,  (9).  Such a p r e c e s s i o n might a r i s e due t o a mag-  n e t i c f i e l d a t t h e nucleus from an e x c i t e d e l e c t r o n i c s t a t e . product n u c l e u s may be i n an e x c i t e d e l e c t r o n i c time, f o l l o w i n g the beta decay process. has performed effect.  an experiment  The  state f o r a short  F r a u e n f e l d e r (22)  on In"*""^ i n an attempt  t o show t h i s  The r e s u l t s i n d i c a t e t h a t t h e e f f e c t i s l i k e l y s m a l l  i f t h e source i s m e t a l l i c , but may be a p p r e c i a b l e 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 . Co^O  The s m a l l d e v i a t i o n s o f t h e  and S c ^ c o r r e l a t i o n curves ( F i g s . 8 and 9) from t h e t h e o r e t i -  c a l v a l u e s . a r e i n q u a l i t a t i v e agreement w i t h t h e r e s u l t s o f Frauenfelder,  s i n c e t h e Co°° source was m e t a l l i c and t h e Sc^6 source was  i n t h e form o f SeO^. F u r t h e r i n v e s t i g a t i o n o f t h i s e f f e c t would r e q u i r e c o n s i d e r a b l e refinement o f experimental t e c h n i q u e .  16.  IV. THEORY OF BETA-GAMMA ANGULAR CORRELATION F a l k o f f and Uhlenbeck (23) have a l s o a p p l i e d t h e i r (15) to t h e p a r t i c u l a r  g e n e r a l t h e o r y on angular c o r r e l a t i o n of  t h e beta-gamma angular c o r r e l a t i o n .  r e s u l t s o f the f i r s t  case  In o r d e r to apply t h e  paper t o beta-gamma-angular c o r r e l a t i o n s i t  M ,x i s o n l y necessary to o b t a i n the a n g u l a r d i s t r i b u t i o n s F-^ (9) for  the v a r i o u s p o s s i b l e b e t a - n e u t r i n o i n t e r a c t i o n s .  i n t e r a c t i o n s i n v o l v e t h e beta energy, W,  However t h e  so the a n g u l a r  distribu-  M t i o n s a r e F^ tions.  (9 W), and are c a l l e d d i f f e r e n t i a l angular  distribu-  I f the d i s t r i b u t i o n s are i n t e g r a t e d over a l l b e t a e n e r g i e s ,  so c a l l e d i n t e g r a t e d a n g u l a r . d i s t r i b u t i o n s are o b t a i n e d :  Correspondingly, we have two k i n d s o f beta-gamma a n g u l a r tion functions:  t h e d i f f e r e n t i a l c o r r e l a t i o n f u n c t i o n s ; VV ("0^ £•)  (E i s t h e beta p a r t i c l e K.E.) correlation function. F a l k o f f and Uhlenbeck functions  correla-  and  W(•£-, E )  the i n t e g r a t e d  0  (E i s t h e maximum beta p a r t i c l e o (23) have d e r i v e d the necessary  distribution  F ^ (9,W) f o r t h e f i v e p o s s i b l e b e t a - n e u t r i n o  a c t i o n s , f o r the f i r s t  K.E.).  and second f o r b i d d e n t r a n s i t i o n s .  interIt i s  p o s s i b l e t h e r e f o r e t o compute the angular c o r r e l a t i o n f u n c t i o n s from the r e s u l t s o f t h e i r p r e v i o u s paper ( 1 5 ) . The f o l l o w i n g r e s u l t s o f the t h e o r y are u s e f u l i n the i n t e r p r e t a t i o n o f angular c o r r e l a t i o n data: (See F a l k o f f and Uhlenbeck (23) ) . 1.  There i s no angular c o r r e l a t i o n f o r allowed  transitions.  beta  17. 2.  In the case o f a f o r b i d d e n b e t a t r a n s i t i o n , t h e b e t a  s p e c t r i n must have a " f o r b i d d e n shape" f o r angular  correla-  tion to exist. 3. 1  F o r f i r s t f o r b i d d e n b e t a - t r a n s i t i o n s , W(9) has t h e form: p -T a  cos Q.  F o r second f o r b i d d e n b e t a - t r a n s i t i o n s a  term i n cos^Q may occur as w e l l . 4.  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 i s g r e a t e s t f o r  b e t a p a r t i c l e s o f maximum energy i n t h e beta-spectrum, and i s almost 5.  i s o t r o p i c f o r b e t a - p a r t i c l e s near zero K.E.  The above t h e o r y assumes t h a t a beta p a r t i c l e  leaving  the nucleus can be r e p r e s e n t e d by a plane wave, which amounts t o assuming t h a t t h e n u c l e a r charge  z  i s zero.  the t h e o r y g i v e s o n l y approximate r e s u l t s .  Consequently  Fuchs and Lennox  (24) have made some c a l c u l a t i o n s t a k i n g i n t o account t h e z-dependence, and have shown t h a t f o r low v a l u e s o f certain types o f i n t e r a c t i o n  z and  ( i n p a r t i c u l a r i n t h e case o f t h e  i n t e r a c t i o n which g i v e s r i s e t o t h e " B i j - m a t r i x element") t h e r e s u l t s o f the simple t h e o r y (z - 0) r e p r e s e n t a very good approximat i o n . S i n c e , i n g e n e r a l , t h e form o f the i n t e g r a t e d b e t a gamma angular c o r r e l a t i o n f u n c t i o n i s : W W it  =I + £  i s convenient t o i n t r o d u c e t h e i n t e g r a t e d angular  coefficient,  <* COS -9-, £  correlation  a, d e f i n e d by:  _  W(l*0*)  ,  S i m i l a r l y t h e 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 f u n c t i o n W(0 E) i s o f 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  a(E)-H  a (E)= k  c o e f f i c i e n t , a(E) i s d e f i n e d as:  »MZL _, W(n°E)  1 APPARATUS FOR MEASURING THE INTEGRATED BETA-GAMMA-ANGULAR CORRELATION The gamma counter as used i n t h e gamma-gamma c o r r e l a t i o n experiments i s q u i t e s u i t a b l e f o r t h i s case a l s o .  However,  s i n c e we wish t o measure t h e beta-gamma c o i n c i d e n c e r a t e i t i s d e s i r a b l e t o have a beta counter w i t h a low e f f i c i e n c y f o r gamma counting,  so t h e background gamma-gamma c o i n c i d e n c e r a t e w i l l be  as small as p o s s i b l e . anthracene  A c c o r d i n g l y a beta counter c o n s i s t i n g o f  f l a k e s o f a t h i c k n e s s o f 30 mg/cm  connected  t o a 5819  p h o t o - m u l t i p l i e r was used. The prevent  source was made as t h i n as p o s s i b l e i n o r d e r t o .  s c a t t e r i n g o f the beta p a r t i c l e s i n t h e source.  The  sources were d e p o s i t e d from s o l u t i o n on a zapon f i l m u s i n g to o b t a i n a uniform t h i c k n e s s . The  The weight was l e s s than 1 mg/cm^.  source was then enclosed i n a 5 i n . diameter  chamber to prevent  insulinc  p l a s t i c vacuum  s c a t t e r i n g o f the beta p a r t i c l e s i n t h e a i r .  The beta counter was p l a c e d immediately  o u t s i d e t h e vacuum chamber  I n f r o n t o f an aluminium window o f 7 mg/cm p a r t i c l e s to e n t e r the anthracene.  to a l l o w the beta  The g e n e r a l arrangement i s  shown i n F i g . 10. The absence o f severe s c a t t e r i n g e f f e c t s may be i n f e r r e d from an experiment performed on the beta-gamma-correlat i o n i n Sc^ .  The beta-gamma c o i n c i d e n c e r a t e i s i s o t r o p i c i n  t h i s case as Novey. (14) has r e p o r t e d .  With t h e s e t up as des-  c r i b e d i n t h i s s e c t i o n , t h e beta-gamma c o i n c i d e n c e r a t e was found to be the same i n the 90° and 180° p o s i t i o n s , w i t h i n t h e experimenx  I am indebted t o Dr. M. K i r s c h and Dr. K. Starke f o r advice on the p r e p a r a t i o n o f t h e sources.  20. t a l accuracy o f about 1%.  I t was  concluded t h e r e f o r e t h a t the  technique d i d not i n t r o d u c e any s e r i o u s s c a t t e r i n g  effects.  A  Line  Source  ToVacuum S y s t e m  M o u n t e d on Za^on  Al. W i n d o w  "s  Movable  Beta Counter A nthra cene es s  Gammoi C o u nte r F I G . 10  0 - / - A ngular Corre lot, o n  ,7  Energ  FIG. Disintegration  0^0  Sc/. erne f o r  VI INTEGRATED BETA-GAMMA-ANGULAR  CORRELATION IN S b  Many n u c l e i -have been i n v e s t i g a t e d angular c o r r e l a t i o n , to be i s o t r o p i c .  1 2 4  .  f o r beta-gamma  but i n most cases t h e d i s t r i b u t i o n was found  At the time t h i s r e s e a r c h was undertaken o n l y  t h r e e cases had been r e p o r t e d i n which the c o r r e l a t i o n was not isotropic,  namely Sb «-, (11) (12)  Tni ? ,  12/  1  0  (14)  d Rb^ (13). 6  a  n  There appeared t o be c o n s i d e r a b l e doubt a s t o t h e v a l i d i t y o f the  results f o r Tin ? 1  0  and R b  g 6  ,  K  w h i l e i n t h e case o f S b 4 t h e 1 2  r e p o r t e d r e s u l t s were c o n t r a d i c t o r y ,  but p o i n t e d t o t h e d e f i n i t e  e x i s t e n c e o f an a n g u l a r c o r r e l a t i o n r e p o r t e d the i n t e g r a t e d -0.17  Ridgway (11)  effect.  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"*"^" as a : 2  w h i l e Beyster and Wiedenbeck (12) gave a - - 0 . 2 6 .  hoped that a value  i f the integrated )a|>  0.20  I t was  angular c o r r e l a t i o n c o e f f i c i e n t had  i t would be p o s s i b l e  to make a measurement o f  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 t h e r e f o r e o b t a i n e d from Chalk R i v e r and an experiment performed to o b t a i n t h e i n t e g r a t e d The  disintegration  scheme o f S b ^ ^ (21) i s r a t h e r  complicated as shown i n F i g . 1 1 . t i o n involves and  angular c o r r e l a t i o n c o e f f i c i e n t . 2  The beta-gamma-angular  correla-  the h i g h e s t energy b e t a group w i t h 2 . 3 7 Mev endpoint  t h e f o l l o w i n g .600 Kev gamma ray.  I f we absorb the beta p a r t i c l e  w i t h energy l e s s than about 1 Mev, the beta-gamma c o i n c i d e n c e The  presence o f a beta-gamma-angular c o r r e l a t i o n i n R b  since  been confirmed by Stevenson and Deutsch  (18).  o u  has  rate  w i l l i n v o l v e mainly the 2 . 3 7 beta group and t h e 600 Kev gamma ray, w i t h a few c o i n c i d e n c e s due t o t h e 1 . 6 Mev beta group.  A thin  beta counter as d e s c r i b e d i n the p r e c e d i n g s e c t i o n was used. Of course the l a r g e number o f gamma r a y s g i v e s a r e l a t i v e l y h i g h counting r a t e i n the gamma counter, which c o n t r i b u t e s to t h e accidental coincidence rate.  ~ The r a t i o ,  beta-gamma-coincidences was 0 . 1 1 5 . r a t e shows l i t t l e  D, o f gamma-gamma t o  The gamma-gamma c o i n c i d e n c e  i f any angular c o r r e l a t i o n i n t h i s case,  i s o t r o p i c w i t h i n the experimental accuracy o f about 5%. source was prepared by d e p o s i t i n g sodium antimonate  being The  from.solution  on t o a zapon f i l m u s i n g i n s u l i n s to o b t a i n a uniform t h i c k n e s s . :.2-70 mg/cm  An aluminium beta absorber weighing o f the beta counter.  was used i n f r o n t  T h i s absorber removes a l l beta  o f l e s s than about 0 . 9 Mev energy,  particles  so we o b t a i n t h e a n g u l a r  c o r r e l a t i o n f u n c t i o n I n t e g r a t e d over a l l beta e n e r g i e s g r e a t e r than 0 . 9 Mev.  The v a l u e 0 . 9 Mev was estimated from R =-0.542E - 0.133  energy-range r e l a t i o n s h i p : range i n gms/cm  2  and E i s t h e energy  an e l e c t r o n which produced  i n Mev.  through  as developed  where R i s t h e The t o t a l range o f  a count was about 3 1 0 mg/cm .  The f u n c t i o n ¥(0) as determined Table I I I and F i g . 1 2 .  Feather's  f o r S b ^ ^ i s given i n 2  The e v a l u a t i o n o f the d a t a was c a r r i e d  i n Appendices  I I and VI and c o n s i s t e d o f t h e  following consecutive steps: 1.  The r e c o r d e d data c o n s i s t e d o f C ( 0 ) ,  c i d e n c e s i n time T, t h e s i n g l e channel  t h e number o f c o i n -  c o u n t i n g r a t e s (scaled by  640), A,&B, and D* the r a t i o o f gamma-gamma to beta-gamma c o i n c i d ence r a t e .  (D was determined  completely i n aluminium).  by a b s o r b i n g the beta  particles  TABLE  III  I n t e g r a t e d Angular C o r r e l a t i o n  0  Crn(Q)  W(0)-1  W(Q)  -1  W(6)  i n So  S.D.  90°  364  -0.000  -0.000  1.000  0.000  100°  364  -0.000  -0.000  1.000  0.012  110°  358  -0.017  -0.019  0.980  0.003  120°  350  -0.038  -0.043  0.955  0.003  130°  336  -0.077  -0.086  0.910  0.003  140°  320  -0.120  -0.134  0.860  0.003  150°  311  -0.146  -0.163  0.330  0.010  160°  .301  -0.173  -0.193  0.799  0.003  170°  296  -0.189  -0.212  0.773  0.011  180°  293  -0.197  -0.220  0.770  0.007  90°  120°  150° -e-  180°  23.  2.  The a c c i d e n t a l c o i n c i d e n c e s  were s u b t r a c t e d  the r e s u l t d i v i d e d by the product o f the  from C(0) and  s i n g l e channel  counting  rates giving:  C (*)-(  cc*>  r  T  3.  The u n c o r r e c t e d  c o r r e l a t i o n f u n c t i o n : Wfc) -  — T C (<?0°) T  was  computed. 4.  to  '  The c o r r e c t i o n f o r the gamma-gamma background was a p p l i e d  ¥(0),  t h u s o b t a i n i n g .' W(^)  -•  W(&) ' T^t>  *~  * T^D  *  (Appendix V I ) . 5.  The c o r r e c t i o n f o r a n g u l a r r e s o l u t i o n was then a p p l i e d  i n o r d e r t o o b t a i n W(Q), The  the true c o r r e l a t i o n function.  p l o t t e d r e s u l t s ( F i g . 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: where a = - 0 . 2 3 - 0 . 0 1 .  Unfortunately  W(Q) -  1. + a cos 0  i t i s d i f f i c u l t t o compare  t h i s r e s u l t d i r e c t l y w i t h the t h e o r y so the d i s c u s s i o n w i l l be l e f t u n t i l t h e next s e c t i o n 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  .  I t may be remarked however t h a t  this  r e s u l t i s i n f a i r l y good agreement with t h a t o f Beyster and Wiedenbeck ( 1 2 ) , p a r t i c u l a r l y s i n c e t h e exact range o f beta ray energies  counted i s somewhat u n c e r t a i n  i n both c a s e s .  24.  711 BETA-GAMMA ANGULAR CORRELATION AS A FUNCTION OF ENERGY FOR Sb +. 12l  The t h e o r y o f the beta-gamma-angular c o r r e l a t i o n p r e d i c t s t h a t the e f f e c t w i l l i n c r e a s e i n magnitude f o r h i g h e r energy beta p a r t i c l e s o f the spectrum.  In t h e case o f Sb''"^' 2  the i n t e g r a t e d a n g u l a r c o r r e l a t i o n c o e f f i c i e n t was found a value: 0.9  a c.-0.23,  to have .  o n l y beta p a r t i c l e s o f energy h i g h e r than  Mev ( i . e . up t o . 2 . 4 Mev) c o n t r i b u t i n g t o t h i s v a l u e .  effect  The  should t h e r e f o r e be o f 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 t h e r e f o r e d e v i s e d  to  beta-gamma-angular c o r r e l a t i o n  measure t h e d i f f e r e n t i a l  icient  coeff-  a ( B ) , d e f i n e d p r e v i o u s l y (p. 17 ) • In o r d e r t o measure  some type o f spectrometer  a(E) i t i s necessary  as the beta d e t e c t o r .  t o employ  I t has been  shown by Hopkins (25) t h a t the p u l s e s produced i n a t h i c k cene c r y s t a l  s c i n t i l l a t i o n counter  anthra-  by beta p a r t i c l e s a r e propor-  t i o n a l to t h e beta K.E.. energy, a t l e a s t i n the energy range 0.1  - 3 . 0 Mev.  spectrometer of  I t was t h e r e f o r e proposed  t o use a s c i n t i l l a t i o n  as t h e beta d e t e c t o r with a twelve  Chalk R i v e r d e s i g n as the p u l s e amplitude  channel  kicksorter  discriminator.  The  apparatus was arranged a s shown i n F i g . 13 and f u n c t i o n s as follows.  The k i c k s o r t e r and s c i n t i l l a t i o n  c a l i b r a t e d by determining the endpoint beta group p u l s e spectrum.  The twelve  counter  circuit i s  o f t h e Sb- - ^ 2 . 3 7 Mev 1  2  k i c k s o r t e r channels are  t h e n , a d j u s t e d t o 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 . 3 7 Kev. t h i s c a l i b r a t i o n are g i v e n i n Appendix V.  Details of  The c o i n c i d e n c e mixer  Vacuum Chamber  Gamma Counter  Beta Counter J >ou r c e  Discriminators . A n d C o i nci'dence Mixer  Gamma Channel S e a ler  Beta C h annei Scc( l e r  Beta Pulse Amplifier  Delay  Gat(  02/* -sec  y T o 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 d i s c r i m i n a t o r i n the beta channel set a t about 0.9 Mev, so t h a t o n l y those beta-gamma c o i n c i d e n c e s are d e t e c t e d f o r which the beta e n e r g i e s a r e g r e a t e r than t h i s v a l u e .  Each c o i n c i d e n c e  count operates the gate c i r c u i t which opens t o permit t h e beta p u l s e t o reach the k i c k s o r t e r .  I n t h i s manner t h e e n e r g i e s o f  each beta p a r t i c l e which has been counted gamma ray i s r e c o r d e d .  i n coincidence with a  By measuring the' c o i n c i d e n c e r a t e s a t  the 90° and 180°positions o f t h e two c o u n t e r s i t i s p o s s i b l e t o determine energy,  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 f u n c t i o n o f beta  that i s  a(E).  Since t h e beta counter now employs a t h i c k it  i s s e n s i t i v e t o gamma r a y s as w e l l .  crystal,  The r a t i o o f gamma-gamma  to beta-gamma c o i n c i d e n c e s D may be estimated making r e f e r e n c e t o F i g . 11. 1.7  The beta counter i s b i a s e d t o about 1 Mev so o n l y t h e  Mev gamma r a y which occurs i n 60% o f the d i s i n t e g r a t i o n s ,  c o n t r i b u t e to t h e gamma-gamma c o i n c i d e n c e r a t e .  The counting  e f f i c i e n c y i s estimated at about .02 f o r t h i s gamma ray. 2.4  Mev beta group comprises  will  The  20% o f the d i s i n t e g r a t i o n s and roughly  a h a l f o f the beta p a r t i c l e s a r e counted.  On the average  there-  f o r e we would expect t o f i n d a value o f :  0.OZ0.6 0.2  =  0  |  Z  0.5  The v a l u e s measured e x p e r i m e n t a l l y f o r D ranged from 0.15 t o 0.20 (see Appendix V I ) . Coincidence measurements were taken a l t e r n a t e l y i n the 90° and 180° counter p o s i t i o n s f o r p e r i o d s o f about 12 hours.  T h i s procedure  tends to average  out t h e e f f e c t s o f  small v a r i a t i o n s i n a m p l i f i e r g a i n 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 s i x days c o u n t i n g i n t h i s  26. f a s h i o n were averaged a(E) computed. to  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  coefficient  A group of r e a d i n g s such as t h i s w i l l be r e f e r r e d  as a run. The  page  same computation  procedure as was  d e s c r i b e d on  2.2, i n connection w i t h the i n t e g r a t e d c o r r e l a t i o n  coefficient  i s a p p l i e d i n t h i s case to each k i c k s o r t e r channel, w i t h two  minor  changes, as f o l l o w s : 1.  Since the source decays a p p r e c i a b l y i n s i x or seven days  (half l i f e  equals 60 days) a c o r r e c t i o n must be a p p l i e d t o  as i l l u s t r a t e d i n A p p e n d i x 2.  II.  In order to o b t a i n the a c c i d e n t a l c o i n c i d e n c e r a t e f o r  each k i c k s o r t e r channel, f i r s t spectrum  Cp(Q)  N(E)  o f a l l , t h e beta counter p u l s e  i s obtained on t h e k i c k s o r t e r by "Qpening" the  "gate" c i r c u i t .  Then the t o t a l a c c i d e n t a l c o i n c i d e n c e r a t e i s  computed i n the u s u a l manner and the a c c i d e n t a l r a t e s f o r each k i c k s o r t e r channel are computed as the product o f the t o t a l a c c i d e n t a l rate,and N(E). of  (  i s expressed as a f r a c t i o n  the t o t a l beta counter c o u n t i n g r a t e ) . The computational  for  N(E)  procedure  used i s i l l u s t r a t e d  one run, i n Appendix VI. Three o t h e r s i m i l a r runs were made, however the  computations  are not t a b u l a t e d here  the v a l u e s f o r  in detail.  a(E) f o r a l l f o u r runs.  F i g . 14 g i v e s  The v a l u e s were combined  as shown i n Table IV and F i g . 1$ t o g i v e a s i n g l e curve. p l o t t e d i n F i g . 15 are two p o s s i b l e t h e o r e t i c a l curves which w i l l be d i s c u s s e d i n the next It  Also  (dashed)  section.  i s p o s s i b l e to c a l c u l a t e the i n t e g r a t e d 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  correlation  TABLE IV The Beta-Camma-Angular  Correlation  a(E) f o r S b  Run  1  2  q  .  Average  1.06  1.16  1.26  a(E) S.D.  0.188 0.013  0.20$ 0.015  a(E) S.D.  0.211 0.014  a(E) S.D.  Coefficient  1 2 4  .1.36  I.46  I.56  1.66  0.219 0.016  0.2ft$ O.017  0.273 0.018  0.298 0.020  0.327 0.023  0.220 0.015  0.237 0.015  0.272 0.017  0.288 0.018  0.315 0.020  0.196 0.019  0.224 0.020  0.254 0.021  0.280 0.021  0.308 0.021  0.334 6.022  a(E) S.D.  0.188 0.016  0.218 0.017  0.250 0.017  0.280 0.017  0.314 • 0.346 0.019 0.021  a(E) S.D.  o.d)95 0.008  0.218 0.240 0.009 0.009  0.270 0.009  0.296 0.323 0.349 0.376 0.393 0.411 0.428 0.010 0.010 0.012 0.013 0.014 0.018 0.026  Energy  I.76  1.86  I.96  2.07  2.17  O.368 0.026  0.380 0.034  0.410 0.045  0.414 0.053  0.448 0.070  0.328 0.023  0.346 0.025  0.370 O.029  0.388 0.033  0.431 0.047  0.445 O.O65  0.360 0.025  0.382 0.025  0.400 0.030  0.412 0.030  0.380 0.024  0.408 0.026  0.420 0.023  0.436 0.028  0.440 O.O38  O.444  0.040  0.446 0.040  1  \  o  i  v  " -  C o r r e l a t i o n C o e f f i c i e n t - g(E) O  i  I  i <o  1 •  i <o  o  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 o f the method  are given i n Appendix V I I I where i t i s shown t h a t :  where N(E)  i s the b e t a s p e c t r u n P a s * m e a s u r e d on the k i c k s o r t e r . c  The i n t e g r a t i o n i s performed range from E a(E).  c  r  n u m e r i c a l l y i n T a b l e V, f o r the  0.8g£Mev. t o E  Q  ^ 2.37  The value so o b t a i n e d i s  Mev.  i n which we have measured  -0.24  a ^  %•  0.02.  The problem o f comparing t h i s value w i t h t h e measured i n t e g r a t e d c o r r e l a t i o n c o e f f i c i e n t the. f a c t t h a t the t r u e beta spectrum  N(E)  energy  a  directly  i s complicated by  i s d i s t o r t e d by t h e alum-  inum absorber used t o remove t h e lower energy b e t a p a r t i c l e s i n t h e case o f the d i r e c t measurement o f  a.  I t i s shown i n Appendix XI  t h a t the value o f a which we would expect t o measure d i r e c t l y , be estimated by making some assumption  may  f o r . t h e amount o f d i s t o r t i o n  o f N(E) and i n t e g r a t i n g a(E) i n a manner s i m i l a r t o Table V.  The  r e s u l t o f t h i s estimate i s t h a t a should l i e between the v a l u e s a  r  -0.24  £  0.02  a b s o r p t i o n o f N(E).. value,  a  -0.23  r  d e v i a t i o n may  -0.27  to  -  0.02,  depending on the amount o f  The d e v i a t i o n of t h i s estimate from the measured -  0.01  i s about-0.02  to t 0.03.  This small  be i n t e r p r e t e d as t h e . l i m i t o f t h e s y s t e m a t i c e r r o r .  T h i s systematic e r r o r i s due' t o s m a l l d r i f t s i n a m p l i f i e r g a i n and counter h i g h t e n s i o n . determination of  a(E)  The i n f l u e n c e of t h e s e d r i f t s on t h e i s i l l u s t r a t e d i n F i g . 14,  o f f o u r separate determinations o f  where the  results  a(E) are g i v e n .  To conclude, i t should be emphasized t h a t i t i s t h e v a l u e o f a as o b t a i n e d i n T a b l e V, i . e .  a  =  which has an unambiguous t h e o r e t i c a l meaning.  -0.24  1  0;02,  TABLE V I n t e g r a t i o n of  Beta Energy  Mev  a(E)  a(E) t o o b t a i n  a  "N/(E) 3 + a IE)  Spectrum N(E)  a(E) ' N(E] 3 + aTE]  -0.130  25000  3700  -1130  0.96  -0.160  23000  8100  -1296  1.06  -0.190  21000  . 7470  -1419  1.16  -0.213  19200  6900  -1504  1.26  -0.240  13000  6520  -1564  1.36  -0.270  14970  5430  -1430  1.46  -0.296  13100  4340  -1433  1.56  -0.323  10930  4100  -1324  1.66  -0.349  7690  2900  -1012  1.76  -0.376  6550  2500  -940  1.86  -0.393  5070  1940  -762  1.96  -0.411  3139  1230  -506  2.06  -0.423  1751  2.16  -0.446  0.86  349 Totals: a  =  631  -291  323  -143  61684  -14769  -14769  61634  = -0.24  28 YIII RESULTS "AND  CONCLUSIONS REGARDING THE  DIFFERENTIAL  BETA-GAMMA-ANGULAR CORRELATION FOR From F i g . 15 the value o f 2.37  Mev  beta-spectrum  bourhood o f  a(E ) Q  =  Sb  12/  *  a(E) near the end of the  i s seen to approach a value i n the n e i g h -0.44 -  0.05.  In order to compare t h i s  r e s u l t with t h e o r y the f o l l o w i n g a d d i t i o n a l i n f o r m a t i o n i s available. spectrum the f i r s t  According to Langer e t a l (26) the 2.37  Mev  beta  i s very l i k e l y o f the "alpha-type", c o r r e s p o n d i n g to f o r b i d d e n m a t r i x element^Bjj-'.  T h i s i m p l i e s a change  o f p a r i t y and the f o l l o w i n g p o s s i b l e changes of the momentum quantum number: product nucleus  A J - 0> -  1,-2.  angular  Moreover, the  Te ^" has a s p i n i n the ground s t a t e o f 0 12  since i t i s an even-even nucleus.  I f the m a t r i x 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 c o n s i d e r a t i o n s we get f o r e l e c t r i c d i p o l e  gamma r a d i a t i o n , t h a t i s L  2  - 1,  the f o l l o w i n g p o s s i b l e s p i n  changes: 1  3  and f o r e l e c t r i c  -  0  1-  ^  1  0  l -—>  0  —?  quadrupole  -— >  >  radiation  -— >  2 -  0  (L = ?  »• 0  T  0  <s  0  O  0  0  1  c  3 -  =3>  2 -—>•  •0  2)  29.  The t a b l e s and formulae  g i v e n by F a l k o f f and Uhlen-  beck (23) have been used t o c a l c u l a t e t h e value o f the 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 t h e maximum beta energy each o f the above cases.  F u r t h e r d e t a i l s o f t h e procedure  i n v o l v e d a r e g i v e n i n Appendix IX. obtained are l i s t e d ,  close  Table VI.  The v a l u e s f o r  a(E ) Q  so  The v a l u e s o f a ( E ) f o r t h e  1—>• 1 — * - 0 ,  t r a n s i t i o n schemes  Eo = 2 . 3 7 Mev f o r  0  3 — > 2 — a r e  relatively  t o t h e experimental v a l u e , so the complete f u n c t i o n s a(E)  were evaluated i n these two cases.  The curves so o b t a i n e d a r e  shown i n F i g . 1 5 . In i n t e r p r e t i n g t h a t i n t h e energy superimposed.  range  these r e s u l t s , i t must be remembered  1 . 0 M e v - 1 . 6 Mev two beta groups a r e  The 1 . 6 Mev group ..''.. presumably has no beta-gamma-  angular c o r r e l a t i o n ,  s i n c e the magnitude o f the c o r r e l a t i o n  c i e n t decreases r a p i d l y below 1 . 6 Mev. r e s u l t s f a v o u r the t r a n s i t i o n scheme  coeffi-  With t h i s i n mind t h e 1  > 1  > 0  s i n c e any  e r r o r would presumably tend to make the c o r r e l a t i o n more i s o t r o p i c . There  i s , however, a s e r i o u s d i f f i c u l t y i n a c c e p t i n g t h i s a s s i g n -  ment, because the beta t r a n s i t i o n d i r e c t l y t o t h e Te^ ^" ground 2  s t a t e would then be allowed, o r a t t h e most, f i r s t i s then d i f f i c u l t  t o account  Very r e c e n t l y p r o v i d e an a l t e r n a t i v e et a l ( 2 6 ) .  forbidden.  f o r the s c a r c i t y o f these t r a n s i t i o n s .  Nakamura e t a a l  interpretation  Nakamura has c o n s i d e r e d ,  s h e l l model i n h i s i n t e r p r e t a t i o n , to angular c o r r e l a t i o n data a t c a l l .  (27) have attempted t o  of t h e r e s u l t s o f Langer f t v a l u e s and t h e n u c l e a r  but has g i v e n no c o n s i d e r a t i o n His t h e o r e t i c a l conclusions  seem t o be i n c o m p a t i b l e w i t h our experimental r e s u l t s . further  It  However,  c o n s i d e r a t i o n i s b e i n g g i v e n to the problem i n an attempt  TABLE VI  Transition  1 —>  Q  0  -0.44  2  $ 1 —* 0  +0.64  3  * 1 —  0  -0.14  0  > 2  =50  +0.73  1  » 2  3>0  +0.86  2  > 2  —90  -0.15  2 —3> 0  -0.37  3 4  1  a(E )  *  > 2  ?  — 0  E x p e r i m e n t a l Valued  +0.23  -0.44 + 0.04  to o b t a i n a s a 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 experimental data which i s now a v a i l a b l e A few months a f t e r ' t h e ary r e p o r t  of a l l  c o n c e r n i n g Sb"*" ^. 2  publication  o f our  prelimin-  ( 1 6 ) , Stevenson and Deutsch (18) have p u b l i s h e d t h e  r e s u l t s o f t h e i r measurement angular c o r r e l a t i o n  o f the d i f f e r e n t i a l  beta-gamma  a(E) f o r S b ^ i n which they made use o f a 1 2  magnetic beta r a y l e n s spectrometer.  There i s e x c e l l e n t  agree-  ment between t h e i r r e s u l t s and ours, though the two methods are quite  different. I t i s worthwhile n o t i n g the present  gamma-angular  correlation  experimentation.  state of beta-  Angular  e f f e c t s have been reported'and the i n t e g r a t e d  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 measured f o r the f o l l o w i n g Sb  12/  * (12),  Only f o r S b  Rb 1 2 4  8 6  (13), I  2  6  (13), T n i ? 1  0  (11)  nuclei:  and K  4 2  (28).  ( 1 7 ) ( 1 3 ) and R b ( 1 8 ) has t h e d i f f e r e n t i a l 86  correlation-coefficient has  1  correlation  been r e p o r t e d .  No a n g u l a r  correlation  been found f o r most commonly o b t a i n a b l e r a d i o a c t i v e  including:  C o , N a , Na *-, C o ^ , 6 0  2 2  22  S c  46-  and I  1 2 Z |  -.  angular-  nuclei,  31. APPENDIX  I.  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 F i g . 1 B.  Let us assume t h a t counter 1 counts o n l y the f i r s t  p a r t i c l e emitted, and t h e second counter counts o n l y t h e second particle.  Let us denote t h e p r o b a b i l i t y o f the counter 1 f o r  d e t e c t i n g p a r t i c l e 1 whose d i r e c t i o n f a l l s w i t h i n 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 c h a r a c t e r i z e d by t h e  u n i t v e c t o r CO, as:  c  (10,;  ;  counting r a t e i n counter 1 i s  "^.-ff N  '  n  e  the single  n  £, , where o l source s t r e n g t h ( d i s i n t e g r a t i o n s p e r sec.) and n  =  T  N  1  N i s the o  the i n t e g r a t i o n being over t h e whole s e n s i t i v e area o f t h e counter. I n t r o d u c i n g a s i m i l a r n o t a t i o n f o r counter 2, we g e t , r e f e r r i n g to F i g . 16, f o r the d i f f e r e n t i a l c o i n c i d e n c e c o u n t i n g r a t e :  where  W(^i  ^2)  '  :  V  .  (^)  c o r r e l a t i o n f u n c t i o n , 0 being d e f i n e d by: M  i^Ll  1 1 6  cos 9 -z •  "—  we i n t e g r a t e between the l i m i t s i n d i c a t e d i n F i g . 16, making .  the assumption t h a t t h e counters 1 and 2 both  subtend  the same  angles at the source we o b t a i n the t o t a l c o i n c i d e n c e r a t e :  la  angular  00, - ^  w  °  The s u b s c r i p t s 1 and 2 r e f e r to counters 1 and 2, w h i l e  01,02  r e f e r t o the c e n t e r p o s i t i o n o f counters 1 and 2 r e s p e c t i v e l y . -s> —-7 — • > I f we assume £ , and € ^ a r e constant over d i r e c t i o n s u) t  and  i^i/l  , and i n t r o d u c e the v a r i a b l e s :flf,- (fl- (ft  <• '  and  l  C( -- V> -(S> z  Z  c  (47T)  Of  0Z  expression f o r A/  c  i  , we o b t a i n from the p r e v i o u s +4  +4  1  From t h e f a c t p- i s small tions that  cos  w i t h 0, t h a t i s :  1 0 -  Q  (10°)  we now i n t r o d u c e t h e approxima-  and 1?- can be n e g l e c t e d 0  i n comparison  +• C{ -fl z  I f we assume t h a t Wi(0) has the simplest form c o n s i s t e n t w i t h t h e theory:  elementary i n t e g r a t i o n g i v e s us: 1  +  a  , i - i  ^  4TT AsO^, ^  1  2  4ir "  become 'very small we o b t a i n i d e a l r e s o l u t i o n , f o r which  case the c o i n c i d e n c e r a t e N ^ (C0) i s : l  o  We have assumed t h a t counter 1 has an e f f i c i e n c y o f zero f o r the d e t e c t i o n o f the second p a r t i c l e , and s i m i l a r l y t h a t counter  2 has zero e f f i c i e n c y f o r the d e t e c t i o n o f the  f i r s t p a r t i c l e emitted. c a l l y , f o r beta-gamma  T h i s c o n d i t i o n can be obtained,  coincidences.  However i f these  practi-  efficiencies  are not zero, the c o i n c i d e n c e r a t e as expressed  i n the  previous  formulae must i n c l u d e a second term, s i m i l a r to the f i r s t but i n v o l v i n g these e f f i c i e n c i e s .  The  case i n which both gamma  rays are o f abeiit the same energy ( n e a r l y r e a l i z e d f o r Co particularly  simple,  one,  s i n c e the c o i n c i d e n c e r a t e s are t h e n  by the preceding formula w i t h an a d d i t i o n a l f a c t o r o f  two.  ) is given  34. APPENDIX  II.  PROCEDURE TO EVALUATE THE TRUE COINCIDENCE RATE. The So t h a t  angular r e s o l u t i o n o f t h e c ounters was chosen  2fl(=2^ = 2 0  =  o  that W(0) = l + o |  0.349 r a d i a n s  ( F i g . 16).  I f we assume  cos 0, which i s t h e simplest form c o n s i s t e n t  with t h e theory,.the  c o i n c i d e n c e counting r a t e f o r i d e a l r e s o l u -  tion i s : IV  &) *  C  A/„ 6, 6  [/ + 0LC0S &] Z  Z  .  (See Appendix I ) ,  Upon s u b s t i t u t i o n o f t h e above numerical  v a l u e s f o r t h e subtended  counter a n g l e s we o b t a i n f o r t h e observed  WUfl)  Vo  coincidence r a t e :  [• ^(0.02/ f 0.<?r* Co A)] +  t  We have assumed t h a t :  At Let us now d e f i n e the e x p e r i m e n t a l l y observed  angular  correlation  function as follows:  Nc (Wj  }  and  -  deter  mine the c o r r e c t i o n r e q u i r e d i n order t o o b t a i n W(0) from i t . Applying the formula f o r |\Z[-8"), we o b t a i n : c  i +• 0.01/ a  K 1  and  since  0|  Wte) - i —  i s g e n e r a l l y small (about 0.2) t h i s may be w r i t t e n :  o.u%4co$ fr z  We t h e r e f o r e o b t a i n the proper  - (5.010  a  2 ,  cos**  c o r r e c t i o n t o be a p p l i e d t o W(9)  35. to o b t a i n W(9) as f o l l o w s :  A p o s s i b l e procedure i s to p l o t W(0) - 1 and o b t a i n an approximate value f o r a  which may be used w i t h s u f f i c i e n t accuracy i n the  c o r r e c t i o n formula above to o b t a i n  W(0).  In a c t u a l p r a c t i c e t h e observed c o i n c i d e n c e NQgg, i s p a r t l y due t o a c c i d e n t a l c o i n c i d e n c e s . coincidence  rate,  This accidental  r a t e , N^, may be c a l c u l a t e d from t h e i n d i v i d u a l counter  r a t e s N-j_, Ng, as f o l l o w s : N  = 2T  A  A/, A/  t  - ZT  j  V  the mixer r e s o l v i n g time (0.13 y u - s e c ) .  being  Consequently Nggg i s  given by: N  OBS  " =  Now NQgg  N &) t  tti  =  A  N o e ^ W ® )  +2T Vo 6,6 l  /  z  .  may vary s l i g h t l y due t o f l u c t u a t i o n s i n H.T. supply  and a m p l i f i e r g a i n , but t h i s may be c o r r e c t e d (Nj ) ( N )  f o r by d i v i d i n g by  s i n c e t h i s g i v e s us a q u a n t i t y which i s independent o f  2  the counter e f f i c i e n c i e s as f o l l o w s :  Woes _ ' Wfo) A/, • N  t  0  ^  — — t 2.T • A/,  I f C i s t h e t o t a l number o f c o i n c i d e n c e time, and l e t t i n g  A = N, T, and B *  counts and T i s the t o t a l  N T, be the t o t a l s i n g l e 2  channel counts i n time T, t h e n from t h e l a s t two equations i t follows  that: Vfl(35_  CT AB  For convenience o f computation we d e f i n e a q u a n t i t y C^(9)  as  follows: c (9)  =•  -  -  — —  .  I f the source decays a p p r e c i a b l y d u r i n g the experiment, it will  be necessary t o apply a c o r r e c t i o n f o r t h i s as w e l l .  Since, C ( 9 ) T  1  =  No  i t will  i n c r e a s e as the source decays, so  the c o r r e c t i o n c o n s i s t s o f m u l t i p l y i n g C ( 9 ) by the proper decay T  factor, f.  APPENDIX I I I .  COMPTON SCATTERING OF GAMMA RAYS BETWEEN COINCIDENT COUNTERS. The o p e r a t i o n o f a s c i n t i l l a t i o n counter as a gamma d e t e c t o r depends upon t h e gamma r a y i n t e r a c t i n g w i t h t h e c r y s t a l i n such a manner as t o t r a n s f e r p a r t , o r a l l o f i t s energy t o one or more e l e c t r o n s .  P a i r p r o d u c t i o n , photo e l e c t r i c  e f f e c t and  Compton s c a t t e r i n g a r e p o s s i b l e methods f o r such an i n t e r a c t i o n to take p l a c e .  For gamma r a y s w i t h energy o f t h e o r d e r o f 1 Mev,  Compton s c a t t e r i n g i s t h e most probable i n t e r a c t i o n p r o c e s s . e l e c t r o n o f course produces a s c i n t i l l a t i o n i n t h e c r y s t a l  The  which  i s d e t e c t e d by means o f a p h o t o - m u l t i p l i e r . The s c a t t e r e d photon from the Compton p r o c e s s has a lower energy than t h e i n c i d e n t photon, but may e a s i l y escape from the c r y s t a l , and may i n f a c t be s c a t t e r e d i n t h e d i r e c t i o n o f the o o t h e r gamma counter.  When s c a t t e r e d through 180  the gamma photon  has an energy o f about 200 Kev ( i n i t i a l energy 1 Mev),  so t h e s e  s c a t t e r e d quanta may e a s i l y g i v e r i s e to a s p u r i o u s count.  In  f a c t the e f f i c i e n c y o f the c o u n t e r may be more than twice as great f o r t h e s c a t t e r e d photons than f o r t h e i n c i d e n t photons due t o t h e h i g h e r t o t a l c r o s s s e c t i o n o f the c r y s t a l f o r lower energy photons. The magnitude o f t h i s " s c a t t e r e d c o i n c i d e n c e r a t e " , C_(9) may be estimated as f o l l o w s :  i C (fr) - Z A/sC-fr) Uf s  1  t where N (0) i s the number o f g  photons s c a t t e r e d through an angle 0 p e r u n i t s o l i d angle p e r second, U)  £±  i s the counter e f f i c i e n c y f o r t h e s c a t t e r e d photon, and  i s t h e s o l i d angle subtended between the c o u n t e r s .  The f a c t o r  2 o c c u r s i f we assume t h a t the gamma c o u n t e r s a r e i d e n t i c a l , and t h a t t h e gamma r a y s i n cascade have n e a r l y t h e same energy..  N (0). i s g i v e n by: g  N&  i t  t  /y  $  where  d i f f e r e n t i a l Compton c r o s s - s e c t i o n , s e c t i o n and  i s the  i s the t o t a l Compton c r o s s -  i s t h e s i n g l e channel c o u n t i n g r a t e .  The t r u e  c o i n c i d e n c e r a t e , C^,, i s given by: Cp  - Z A/  0  61 tf_  -  r a t i o o f scattered to true  where ^  i  s  hJ 62. (  , so t h a t we o b t a i n f o r t h e  coincidences:  t h e e f f i c i e n c y o f 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 t h a t the counters and source a r e arranged' syraetrically i n a straight l i n e ,  so t h a t ct?  4 u;.  r  For  CI  1 Mev i n c i d e n t gamma r a y t h e s c a t t e r e d gamma (180°) has an energy o f 200 Kev.  The Compton c r o s s s e c t i o n o f t h e c r y s t a l s f o r 200 Kev  gammas i s about twice t h a t f o r 1 Mev gamma r a y s .  Therefore  i f the  d i s c r i m i n a t o r i s set low enough t o count t h e s m a l l e r 200 Kev p u l s e s the e f f i c i e n c i e s a r e i n the r a t i o j i a  in Heitler,  ^  2  .  may be determined from the formula  and t a b l e s  Quantum Theory o f R a d i a t i o n , t h e value f o r 1 Mev gamma  s c a t t e r e d through 180°  being  we o b t a i n  ^  Cscatt °true  =  an order o f magnitude o n l y , has been negleeted, observed.  — A  0  °^^/^ r- 0.04.04  0.5.  Using these  This r e s u l t  values  gives  s i n c e the photo e l e c t r i c c r o s s s e c t i o n  however e f f e c t s o f t h i s magnitude have been  39. APPENDIX  IV.  THE STANDARD DEVIATION IN W(Q) E x p e r i m e n t a l l y we c a l c u l a t e W(0) from t h e formula:  c W(O) The  relative  i »<*)1 The r e l a t i v e  =  due  (?) "  0«  (see Appendix I I )  standard d e v i a t i o n i s a c c o r d i n g l y g i v e n by:  [  C f90')  C tf/  T  T  e r r o r i n Q ( o ) may be evaluated from t h e t o t a l T  number o f counts obtained, Formula.  r  by the means o f the w e l l known  In t h i s case the t o t a l count obtained,  t o a c c i d e n t a l c o i n c i d e n c e s , C, A  accordingly:  ^r)  .so t h a t :  V C *  C  A  C_ T  Poisson  C, i s p a r t l y C-C. A  , and  40. APPENDIX  V.  CALIBRATION OF THE KICKSORTER CHANNELS IN TERMS OF BETA ENERGY.  A c c o r d i n g to Hopkins (25)  who used a beta counter  of s i m i l a r c o n s t r u c t i o n t o the one used here, the p u l s e s produced  are p r o p o r t i o n a l t o the beta energy, a t l e a s t i n t h e  range 0.1 •it  - 3-0 Mev.  In o r d e r t o c a l i b r a t e the k i c k s o r t e r  i s o n l y necessary t o e s t a b l i s h  the endpoint o f the 2.37 to use.  A c c o r d i n g l y , b e f o r e each r u n , the beta p u l s e  the endpoint o f t h e spectrum endpoint was s e l e c t e d ted as i n F i g . 17wards depending  and c o n s t r u c t a Fermi p l o t . l i n e Fermi p l o t  In t h i s manner the endpoint c o u l d be  the Fermi p l o t  s t r a i g h t l i n e i n t h e range 1.7  Referring  i s seen t o be p r a c t i c a l l y a  t o 2.37  Mev, w i t h a sharp break a t  T h i s i s t h e endpoint o f the second  i n Sb  1 2  beta group  ^ and i t s presence i n the Fermi p l o t  considerable confidence to the c a l i b r a t i o n . the f e r m i p l o t  resul-  The Fermi p l o t was concave upwards o r down-  t o w i t h i n 5 channel width or. £ 50 Kev.  F i g . 11)  I f the  on whether the o r i g i n a l estimate o f t h e endpoint  again to F i g . 17,  (see  spectrum  I t i s then necessary to estimate  correctly, a straight  was too h i g h o r two low.  about I . 6 5 Mev.  spectrum,  Mev beta group being the most p r a c t i c a l  was o b t a i n e d on the k i c k s o r t e r .  determined  one p o i n t i n t h e energy  The c a l c u l a t i o n o f  (Table VII) i n c l u d e s the c o r r e c t i o n  shape o f t h e spectrum,  lends  f o r the f o r b i d d e n  although t h i s i s h a r d l y n e c e s s a r y c o n s i d e r i n g  the accuracy o f the endpoint  determination.  TABLE V I I Fermi P l o t of S b ^  Spectrum  1 2  3.97  20.2  4.04  17.2  625  2.467  28.0  17996  64.27  8.02  675  2.664  31.0  14973  48.30  6.95  725  2.861  34.2  14048  41.08  6.47  4-13  15.7  775  3.059  37.5  10982  29.29  5.41  4.24  12.8  825  3.256  42.0  7694  18.32  4.28  4.36  9.8  875  3.454  45.5  6555  14.41  3.80  4.50  8.4  925  3.651  49.0  5067  10.34  3.22  4.65  6.9  975  3.842  52.0  3139  6.13  2.48  4.80  5.2  1025  4.046  55.5  1751  3.15  1.78  4-98  3.6  1075  4.243  59.8  849  1.42  1.19  5.17  2.3  1125  4.441  65.O  297  O.46  0.68  5.34  1.3  1175  4.638  67.5  0  0.00  0.00  V  =  K i c k s o r t e r Channel B i a s .  £  =  Beta K i n e t i c Energy i n u n i t s of mc  f(52 H) , The Fermi f u n c t i o n f o r Beta  T o t a l beta energy i n u n i t s  J\  =  Beta Momentum i n u n i t s  £'  0  z  =  -  2  r  0:0  .  Spectrum.  =  = ^l(6 'i)  V  z = 52.  / C  C  .  +  ri~  mc  2  .  mc.  i s the "  OL - t y p e " f o r b i d d e n shape f a c t o r .  41. APPENDIX VI. CALCULATION OF BETA-GAMMA ANGULAR CORRELATION COEFFICIENT  I f we now c o n s i d e r C t o be the t o t a l number o f c o i n c i d e n c e counts i n time T we have: C  =  C  respectively  f3^  9  )  +" ^A'  w  n  e  r  e  t  n  e  subscripts  refer  t o beta-gamma, gamma-gamma, and a c c i d e n t a l  respectively. M  }fif  C  coincidences  R e t a i n i n g our d e f i n i t i o n o f Appendix I I :  ~  l  l —  2 T and i n t r o d u c i n g  c T <  r  =  ° 1 B '~  we have: cf  i c  r  ( n ) - c  T  r  Lc <*)  -  T  The  -fc (f)-G,/r/]  ( n f l c  T  T  gamma-gamma c o i n c i d e n c e background  i s about 15 - 20% o f the (3-Jcoincidence  ™  c  where the  x  r  (?)=D.C (f)  Ol  ( V l  f  T  ,  rate  rate.  than "  i s t h e 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  gamma-gamma c o r 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  ured  s e p a r a t e l y by a b s o r b i n g the beta p a r t i c l e s w i t h aluminium, which g i v e s us be a p p l i e d  .  The c o r r e l a t i o n f o r angular r e s o l u t i o n may now  i n the way d e s c r i b e d i n Appendix I I . T h i s procedure i s a l s o  kicksorter  applied  t o t h e data f o r each  channel when measuring t h e d i f f e r e n t i a l  correlation  42. c o e f f i c i e n t , cX ( E) • As an example o f the procedure  f o l l o w e d , Table V I I I i  g i v e s the experimental d a t a f o r one run i n t h e d e t e r m i n a t i o n o f 0(E)  for Sb ^.  The n o t a t i o n used i n Table V I I I i s c o n s i s t e n t  1 2  with the n o t a t i o n used i n the formulae  i n t h i s Appendix.  The  " B i a s " i s i n a r b i t r a r y u n i t s and t h e "Energy" r e f e r s t o the energy i n Mev.  c o r r e s p o n d i n g t o the k i c k s o r t e r channel midpoint  The "Spectrum" i s the p u l s e spectrum  bias.  o b t a i n e d from the b e t a  while the "gamma-spectrum" i s the p u l s e spectrum  eounter,  o b t a i n e d from  beta counter when the beta p a r t i c l e s were absorbed  the  i n aluminium.  The k i c k s o r t e r channel r a t e s are i n each case expressed as a f r a c t i o n o f the t o t a l c o u n t i n g r a t e o b t a i n e d from t h e beta f o r t h a t p a r t i c u l a r spectrum measurement. "S.D." r e f e r s to the s t a t i s t i c a l  error  The  standard d e v i a t i o n  in£f(£).  i n Appendix I I , 0^(0)  F o l l o w i n g t h e procedure  counter  c a l c u l a t e d and m u l t i p l i e d by t h e a p p r o p r i a t e decay f a c t o r ,  is F.  The recorded d a t a c o n s i s t e d o f the time T, t h e t o t a l c o i n c i d e n c e counts C i n each channel, and the t o t a l s i n g l e channel s c a l e d by 640,  A and B.  C (0) T  u n i t s m u l t i p l i e d by 1 0 ' .  counts  as g i v e n i n Table V I I I i s i n these  Twice the r e s o l v i n g time, 2 f  —2  t  i n this  —7  system o f u n i t s i s 407 ( h r s x 64O x 10 ') which i s .' 2 7 407 x 3600 x 640~ x 1 0 " =r • 0.356 x 10~ 6 sec. Similarly, for T 2 -11 convenience the t a b l e r e c o r d s — in u n i t s o f (hr. x 640 x 10 ), AB The t o t a l number o f t r u e c o i n c i d e n c e counts r e c o r d e d i n time i s therefore: C  T  (0)  .  I f  x  10>  T  -r-  Total T AB  T A B L E  a(E) for S b * u  '9.0 9.5 ,0.5 115 r.o 5.5 6.0 6.5 7. a 7.5 8.0 0.94 i.02 i.W no i.29 i.38 /.48 LSI 1.66 1.79 /.97 2.14 0.039 0.059 0.IZ4 0.1100.107 0.081 o.ogi "075 lOtf 0.053 me P.(?I7 1.0 i so.s 44.*; 43. 35.3 32-3 30.7 262 Zl.b 16.1 24.: as 6-8 4C7 C ( g o V f 12.30 150052.7 163.0 IS&4- 1480 I3&0 133.0 113.2 973 81.5 106.6 51.5 28.9 151.0 143.0 128.8 H4.5 92.0 65.7 98.5 45.7 22.7 12-75" 1490 683 1675 !<?4.0 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 (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 Jot? 134.0 1645 150.7 120.4 II 1.4 M 0.893.3 66;6 47.1 68.3 31.9 11.75 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~ (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. 1583 122.7 ioe,s I0l£ 81.8 66.7 47.3 66J 29.6 15.5 ( I3R7 -aCE") 0,177 o.ito O.I80 0.114 au3 0.2300.27T0.290 0.334 0.3300,3720.362  B i as Enerav(Mev. c ^ T  T  T  - a(E) S.D. I -  a w  ^-Spectrum 2-T Cr^{90)-f D  1.0  0.193 0.170 0.190 0.215 0.245 0.255 0.316 0.337 0.37/ 0.393D.4240.420 0- 01-50.0/6 0.017 0-0/7 0-017 0.019 0-021 0 02^0.026 0.0230.02a 0.038 0.174 a zoo 0.2240.2540.2760.3300.357 0.3S70.4O9 04420.437 0.//O O.iOl 0.099 0-0850.033 0.0$3 0.076 0.0600.044 0.057 0.022 0.013 45.0 40.8 AOS 363 33. £ 34-0 31.0 24.4 17.9 235 9.1 £-.5 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 0.181 0.171 0.191 0/1900.216 0218 0.234 0.230 0.2230.197 0.JSi 0.169 0.134 am 0.200 0.O980.142 0.138 0.109 0.1/8 0.22C a 081o.m 0.080  •  As an example take the f i r s t F — 0.99. channel  1 2 . 3 h r . run at 9 0 ° , f o r which  The number o f t r u e c o i n c i d e n c e counts r e c e i v e d i n  1 was:  w h i l e the number o f a c c i d e n t a l c o i n c i d e n c e counts was x The  10  4  -  950.  standard d e v i a t i o n s a r e c a l c u l a t e d f o l l o w i n g t h e procedure  i n Appendices IV and V I I .  APPENDIX V I I . CALCULATION  OF STANDARD DEVIATION. IN THE BETA-GAMMA  CORRELATION  COEFFICIENT  From Appendix VI we o b t a i n f o r t h e c o r r e l a t i o n coefficient:  7T a  I  = "  a  P•  ' i-t>  "  Let t h i s be w r i t t e n  a  -  then t h e d e v i a t i o n  W-  b  V  '/-]>  -  P  r  Q  •  £ q~ i s g i v e n by ;  The most important term i s  ( t Afl)  e r r o r i n t h e main experimental q u a n t i t y only  ( Q&fj)  since i t i n v o l v e s t h e Ol .  Of t h e o t h e r terms  i s important since the remaining two both  which i s q u i t e s m a l l .  involve  Therefore we o b t a i n i n s u f f i c i e n t l y  c l o s e approximation f o r the standard d e v i a t i o n i n cf '  APPENDIX V I I I CALCULATION  OF a FROM a ( E )  Let N(E) dE r e p r e s e n t t h e beta spectrum,  that i s the  number o f 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 f u n c t i o n W(9E) and t h e i n t e g r a t e d c o r r e l a t i o n f u n c t i o n  W(0) i s  then as f o l l o w s : W(0)  /NIC) For S b  1 2 4  W(9E) i s o f the form: W(9E)  where  J£  -  1 -f a(E) c o s 0 2  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 .  have assumed  W(9E) t o be a p r o b a b i l i t y ,  as f o l l o w s :  , ,  element.  so i t may be normalized  |  4  ' -  J J  We  where cmz i s a s o l i d  angle  Consequently we have:  W(9,E)  -  ) "t- C\LE) C O ^ V /(i  +  o r a f t e r performing t h e  afe)tos -e)<lSL x  int egration: w(9E)  -  _J_ + ir  Hence we may o b t a i n  /+ ait) 3  Co$\  -talE)  W(9) as f o l l o w s :  W(9) =  By analogy with t h e equation f o r W(9): 2. we o b t a i n :  3 + a(E) to  Je £  3  +  otfE)  The range o f i n t e g r a t i o n extends from some minimum energy, E ^ n , ( c o r r e s p o n d i n g t o the beta counter c u t o f f ) t o t h e maximum beta energy  E. Q  47.  APPENDIX  IX.  EVALUATION OF THEORETICAL BETA-GAMMA-ANGULAR CORRELATION COEFFICIENTS.  and Uhlenbeck (23) was used  The theory o f F a l k o f f to p r e d i c t  the value o f t h e c o r r e l a t i o n  each o f t h e p o s s i b l e t r a n s i t i o n  coefficient  schemes l i s t e d  R e f e r r i n g t o (23) i t i s necessary f i r s t M distribution  a(E ) for Q  i n Section VIII.  to s p e c i f y  f u n c t i o n , F^ (Q W), f o r t h e r e q u i r e d  the angular interaction,  (  and then to e v a l u a t e the parameters i n v o l v e d f o r t h e 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 ,  first  f o r b i d d e n t r a n s i t i o n w i t h m a t r i x element B ^ j , we have: F ° ( 0 , W) 2  =  J  . giving \\/  JJ p  {  X  and Q^ = W -W  )  +  T-p*~C0S -& Z  + ju^ COS & ,  A  v  N  /^L - 0  D  ( i n units mc ), W p  * W  (31) ( 1 5 ) .  i n t h i s case i s W(9,E) R -r i s g i v e n by E q u a t i o n the p a r t i a l sums l i s t e d  Q  ~~ I  i s now p o s s i b l e to e v a l u a t e t h e parameter  from Equation  , where i s t h e maximum  2  ,  0  Z  COS  -r  - ^p*>2^ ) p  i s t h e t o t a l beta energy  beta energy It  (2p%+q  (f>  Zi  i^l^i JU )  The d i f f e r e n t i a l c o r r e l a t i o n r  1  4-  R  /Q COS 9 2  where  1  function  Q  (29) ( 1 5 ) , and may e a s i l y be e v a l u a t e d from i n Table IV (15) o r t h e sums i n Table I I  As an example take t h e t r a n s i t i o n —>  ' /\)  a(E) = 2 .  (15).  1  t  9  0 ,  • L =. x  2  L 2  ~  1  48.  In F a l k o f f and Uhlenbeck's n o t a t i o n t h i s i s w r i t t e n : J  —  so t h a t  A  J  A j  ^ -  The parameter Then  0  A  J  >  J =  J-j- A J.  -1 ^ o/"cf J = 1.  f o r \V - \A/  -  0  (>„//,/\)  =  y  ('O/V+l)  R e f e r r i n g to Table I I (15) and equation p / Q  i A'*  /I - |  (29) (15) ,*  _  (2 J - 3 )'( 2 J-rS-)  -I  '  / 4 - J ( 2 J-\) (j> +(% J * - (o T t f ) ~ z f a + l  v  u  =•  -0.447.  S i m i l a r l y the o t h e r p o s s i b i l i t i e s a r e evaluated, t h e r e s u l t s being l i s t e d  i n Table V I .  4<7  E L E C T R O N I C  C I R C U I T S  Linear Amplifier  100 X  Preamp. 3X  PL 5819 /  Linear A m p . IOOX  A.I.C*IOI-A Scale of 6 4  1500 v.  1  r -  L—Anthracene Discriminator  s ha per  Coinc. cct.  BLOCK OF  D I A G R A M  COINCIDENT C I R C U I T  I  From duplicate channel  I  I Mechanical registers  I I  I  I  COUNTER l_  Count switch  £>~  1^ I Reset clock 0- | hour  o  j 6AL5  6AC7  6AG5  6AL5  6 A C 7 DISCRIMINATOR*}*  5,~ i w  R o 5 5 1  WL  P  6AC7  I ^  DISCRIMINATOR i Coinc output  Input To M.V. plate  Resolving time - O.I7>i/s. Dead time 3 jus. Input range- +K+IOO v.  Hammond 165  + 320 v. 150 m.a. COINCIDENCE  MIXER  AN D P O W E R  SUPPLY  Scope  4 sync.  Free-running blocking JL oscillator  0-15 met.  02//5 delay JL line  +.+ 1.5 v. pu Ise Inverter "U IT  Invertinq feedback amplifier  Multivibrator 3~l50/js.  Differcnf iafor  Triggered blocking  IL  —1.5 v. p u i s *  Inverter  oscillator  loo ma. t j O O v. p u l s e  P R E C I P U L S E  5 I 0 N  G E N E R A T O R  lOOv. pulse  Raytheon U X T 3 5 0 Pulse transformer  6A67  Single 1.5 v . J » T ° Double 1.5 v. "  6AK5  6AK5  6AK5  r  |  -1.5 v.  - , A r f »J°><  I  -ISov.  'iiilrJj  Ic.i A t 1 e n u a t o r  for  I.5 v. p u l s e s  To P o w e r Supply  + 300 V. reg. @> I 5P rtia - (50 v.reg  <ffl 5 ma  6.3 v a c . (2) 6.2 a. e.t connected t o 0  PRECISION  PULSE  G E N E RATOR  50. APPENDIX XI ON THE INFLUENCE OF ABSORPTION ON THE MEASURED VALUE OF a *  The measurement of the i n t e g r a t e d  correlation  f u n c t i o n was important i n t h a t i t was shown t h a t no a p p r e c i a b l e term i n  cos^O e x i s t s .  However the problem o f comparing t h e  measured i n t e g r a t e d c o r r e l a t i o n c o e f f i c i e n t - with t h e o r y i s q u i t e complicated, due t o the f a c t t h a t the t r u e beta spectrum i s somewhat d i s t o r t e d by the aluminum absorber used t o remove t h e lower energy b e t a p a r t i c l e s .  -It i s sometimes assumed-(31') t h a t beta  p a r t i c l e s o f energy E.are t r a n s m i t t e d through aluminum a c c o r d i n g t o a law o f the form:. =  A  1  *  (1)  I R  max  K  X  J  where A i s the t r a n s m i s s i o n , R i s the t h i c k n e s s o f aluminum and R  m o v  E.  corresponds t o the maximum range o f b e t a p a r t i c l e s o f energy R ax m  a n (  ^^  a  r  e  Si  y  e  n  by Feather's r e l a t i o n which has been  improved by Glendenin (31): • where R i s i n g/cm  R 2  m a x  =  0.542E - 0.133  and E i s i n Mev.  (E > 0.8  Mev).  I f the above r e l a t i o n s are assumed, i t may be p o s s i b l e t o take the d i s t o r t i o n of the beta spectrum i n t o account and so o b t a i n a s u i t a b l e comparison with t h e o r v . We may, however, check the c o n s i s t e n c y of the i n t e g r a t e d 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 e s t i m a t i n g t h e I would l i k e to thank Dr. L. Katz f o r b r i n g i n g t o my a t t e n t i o n some a s p e c t s of the problems d i s c u s s e d i n t h i s appendix.  51. integrated correlation  coefficient  a  from' a(E) as shown i n  Appendix V I I I , but a p p l y i n g a c o r r e c t i o n t o K(E) t h a t beta p a r t i c l e s  i n a s m a l l energy  -  (1).  In t h i s manner we  0.02.  (29)(30) t h a t t h e l i n e a r  There i s some evidence relationship  as g i v e n i n e q u a t i o n (1)  f o r the t r a n s m i s s i o n of an  absorber i s a r a t h e r poor approximation. experimental  set up i s undoubtedly  i n d i c a t e d by equation  (1).  The t r a n s m i s s i o n i n our-  c o n s i d e r a b l y g r e a t e r than i s  This, f a c t was  a measurement u s i n g an absorber of 440  checked  c  1.06  ment was  Mev. 310  corresponding t o E  K of the counting r a t e with E with E -=0.S2 Mev.  was  c  K-0.55-  - 1.06  2  = 0.&2  c  Mev  I f we  stops a l l beta p a r t i c l e s with energy  i n t e g r a t e the  of  spectrum  N(E)  Mev.  c  c  absorber  ( i . e . i d e a l cut o f f ) we  to-E  c  ratio  and t r a n s m i t s a l l  Q  and thus o b t a i n f o r  On the other hand, i f we  assume t h a t the t r a n s m i s s i o n i s given by equation (1), integration  and estimate the r a t i o  The  t o the counting r a t e  below E  from E  the above r a t i o , the v a l u e K r O.65.  perform the  energy  assume t h a t the  beta p a r t i c l e s w i t h energy g r e a t e r than E may  performing  The t o t a l absorbing t h i c k n e s s * used i n the e x p e r i -  mg/cm  C  by  mg/cm .of Aluminum, which  corresponds to t h e maximum range of beta p a r t i c l e s E  assumption  range, E t o E-hdE, are t r a n s -  m i t t e d according t o the above r e l a t i o n o b t a i n the v a l u e , a =• -0.27  on the  we  may  dE f(.  In t h i s way  we  o b t a i n the value K-  0.1+8.  I t i s apparent t h a t the a c t u a l t r a n s m i s s i o n i s g r e a t e r than t h a t given by equation  (1),  but i s of course l e s s than the  x In order t o be counted,a beta p a r t i c l e had t o penetrate the absorber, the counter window, and the counter c r y s t a l , which t o t a l e d 310 mg/cm . 2  ideal  52. transmission. correlation that  Therefore  coefficient  a  our estimate  as obtained from  a must l i e between the v a l u e s :  a i -0.27  - 0.02.  of the measured i n t e g r a t e d  The value  a  =  g i v e s the r e s u l t  a = - 0 . 2 4 - 0.02  and  -0.23 1 0.01 i s actually  measured and d e v i a t e s from the above estimate ±0.03.  a(E)  by about  - 0 . 0 2 to  REFERENCES  1.  J . V. Dunworth, R.S.I. 11,  2.  D. R. Hamilton, P.R. 58, 122 (1940).  3.  R. B e r i n g e r , 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  6.  R. L. Garwin,  7-  J . R. B e y s t e r and M. L. Wiedenbeck, P.R. 79, 411 (1950)  8.  R. M. S t e f f e n , P.R. 80, 115  9.  G. G o e r t z e l , P.R. 70, 897 (1946).  167  P.R. 76, I876  (1940).  (1948).  (1949).  (1950).  10.  D. S. L i n g and D. L. F a l k o f f , P.R. 76, 1639  11.  S. L. Ridgway, P.R. 78 {821!  12.  J . R. B e y s t e r and M. L. Wiedenbeck, P.R. 79, 176 (1950)  (1949).  (1950).  13. • D. T. Stevensen; and M. Deutsch, P.R. 78  640.  (1950).  P.R. 79, 323  (1950).  14-  T. B. Novey, P.R. 78, 66  15-  D. L. F a l k o f f and G.E. Uhlenbeck,  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  19.  S. P. L l o y d , P.R. 83, 716  20.  P. G. Hess, i n p r i n t , Can. J . Phys.  21.  U. S. N a t i o n a l Bureau o f Standards, C i r c u l a r  22.  H. F r a u e n f e l d e r , P.R. 82, 549 (1951).  23-  D. L. F a l k o f f and G. E. Uhlenbeck, P.R. 79, 334  24-  M. Fuchs and E. S. Lennox,  25.  J . I . Hopkins, R.S.I. 22, 29 (1951).  26.  L. M. Langer e t a l , P.R. 79, 808 (1950).  (1950).  (1951).  (1951).  P.R. 79, 221  499.  (1950).  (1950).  27.  S. Nakanawa et a l , P.R. £3, 1273  (1951).  2B.  J . R. Beyster and M. L. Wiedenbeck, P.R. 79, 723  29.  J . S. M a r s h a l l and A. C. Ward, Can. J r . Research A (15,39(1937) ).  30.  R u t h e r f o r d , Chadwick and E l l i s , R a d i a t i o n s from R a d i o a c t i v e Substances, Cambridge U n i v e r s i t y P r e s s , p. 414.  31.  L. E. Glendenin,  N u c l e o n i c s 2, 12 (194^).  . x The f o l l o w i n g a b b r e v i a t i o n s have been used: P h y s i c a l Review:  P.R.  Review o f S c i e n t i f i c Instruments:  R.S.I.  i  (1950).  

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