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Re-investigation of the excited states of Gd 154 Ng, Leung-Kai 1967

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The U n i v e r s i t y of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of LEUNG-KAI NG B.Sc  ( S p e c i a l ) , U n i v e r s i t y of" Hong Kong, 1961 M . S c , U n i v e r s i t y of Hong Kong, 1964  FRIDAY, SEPTEMBER 22, 1967 AT 3:30 IN ROOM 301, HENNINGS BUILDING  COMMITTEE IN CHARGE Chairman:  B„ N. Moyls  M.P. Beddoes G .Mo G r i f f i t h s K.C. Mann Research S u p e r v i s o r :  J.W. B i c h a r d D.L. L i v e s e y B.L. White K.C. Mann  E x t e r n a l Examiner: G.T. Ewan Chalk River Nuclear L a b o r a t o r i e s Atomic Energy o f Canada Ltd.. Chalk R i v e r , Ontario  RE-INVESTIGATION  OF THE EXCITED STATES OF Gd 154  ABSTRACT  The e x c i t e d s t a t e s o f Gd 154 o b t a i n e d f r o m the decay o f Eu 154 have been i n v e s t i g a t e d , , P r e c i s e measurements o f t h e e n e r g i e s and i n t e n s i t i e s o f t h e gamma t r a n s i t i o n s have been made, u s i n g L i t h i u m - d r i f t e d Germanium d e t e c t o r s ,  A r e v i s e d decay scheme i s p r e s e n t e d  i n w h i c h a l l t h e gamma t r a n s i t i o n a l e n e r g i e s agree w i t h the c o r r e s p o n d i n g energy d i f f e r e n c e s between l e v e l s t o w i t h i n 1 Kev.  A 1263.3 Kev l e v e l i s e s t a b l i s h e d by t h e .  presence o f a gamma t r a n s i t i o n o f 892,7 Kev,  Two o t h e r  gamma t r a n s i t i o n s o f e n e r g i e s , 903.6 Kev and 582.1 Kev from the n e g a t i v e - p a r i t y l e v e l s t o the 2 b e t a e x c i t e d +  l e v e l have a l s o been d i s c o v e r e d .  The proper l o c a t i o n s  of the 995.9 Kev and 1004,5 Kev i n t h e decay scheme a r e c o n f i r m e d by a gamma-gamma c o i n c i d e n c e method u s i n g a Germanium d e t e c t o r and a N a l ( T l ) s c i n t i l l a t o r .  The  e n e r g i e s and i n t e n s i t i e s o f t h e i n t e r n a l c o n v e r s i o n e l e c t r o n s and t h e b e t a t r a n s i t i o n s have been measured by an i n t e r m e d i a t e image s p e c t r o m e t e r .  Their values  are q u i t e - c o n s i s t e n t w i t h t h e e s t a b l i s h e d decay scheme.  T h e o r e t i c a l v a l u e s of t h e energy l e v e l s and the b r a n c h i n g r a t i o s f o r gamma t r a n s i t i o n s have been c a l c u l a t e d , u s i n g the Asymmetric R o t a t o r Model ( J . P , Davidson's t r e a t m e n t ) .  The ' s t i f f n e s s ' parameter o f  '  the n u c l e u s )A. and i t s asymmetry parameter TT o b t a i n e d are 0.402 and 11.52 degrees r e s p e c t i v e l y .  Comparison  of the e x p e r i m e n t a l and t h e o r e t i c a l e n e r g i e s of seven p o s i t i v e - p a r i t y l e v e l s g i v e s a root-mean-square d e v i a t i o n of 1.5 %.  Three out of t h e f o u r e x p e r i m e n t a l  b r a n c h i n g r a t i o s a r e i n good agreement w i t h the theor e t i c a l values.  The monopole t r a n s i t i o n  f o r the t r a n s i t i o n from the 0 the 0  +  +  probability  beta excited  state to  ground s t a t e measured a l s o agrees w i t h the  present t h e o r e t i c a l  calculation,  GRADUATE STUDIES F i e l d of study:  Nuclear  spectroscopy  Nuclear Physics  J.B. Warren  Theoretical  Mo M c M i l l a n  Special  Nuclear Physics  Relativity  E l e c t r o m a g n e t i c Theory E l e m e n t a r y Quantum Mechanics Advanced Quantum Mechanics Electronic  Instrumentation  H  0  Schmidt  P. R a s t a l l G.M.  Volkoff  F.A. Kaempffer M.P. Beddoes  AWARDS 1957-60  Hong Kong Government S c h o l a r s h i p  1964-67  Canadian Government S c h o l a r s h i p under the Canadian I n t e r n a t i o n a l Development A s s i s t a n c e Programs,  RE-INVESTIGATION OP THE EXCITED STATES OF Gd 1 5 4  . B.Sc.  (Special)  LEUNG-KAI NG  The U n i v e r s i t y  M.Sc. The U n i v e r s i t y  A T H E S I S SUBMITTED  • o f Hong K o n g , 1 9 6 1  o f Hong K o n g , 1 9 6 4  I N PARTIAL FULFILMENT OF  THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF  PHILOSOPHY  i n t h e Department of PHYSICS We a c c e p t t h i s required  t h e s i s as c o n f o r m i n g t o t h e  standard  THE UNIVERSITY'OF B R I T I S H August,  1967  COLUMBIA  In p r e s e n t i n g  for  that  this  an a d v a n c e d  the  Study.  thesis  thesis  degree  Library shall  I f u r t h e r agree  for  Department  at  in p a r t i a l  f u l f i l m e n t of  the U n i v e r s i t y of  make  it  the  requirements  British  Columbia,  I  freely available for  reference  and  that permission  for  extensive  copying  of  agree  this  s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my  o r by hils  or p u b l i c a t i o n of  w i t h o u t my w r i t t e n  representatives.  this  thesis  for  permission.  Department The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  Columbia  It  is  understood  f i n a n c i a l gain  shall  that  copying  n o t be a l l o w e d  ABSTRACT  The . e x c i t e d  states  o f Gd 154 o b t a i n e d  o f E u 154 h a v e b e e n i n v e s t i g a t e d . P r e c i s e  from t h e decay  measurements  of the  e n e r g i e s a n d i n t e n s i t i e s o f t h e gamma t r a n s i t i o n s h a v e "been made, u s i n g  L i t h i u m - d r i f t e d Germanium  detectors.  A revised  d e c a y scheme i s p r e s e n t e d i n w h i c h a l l t h e gamma t r a n s i t i o n a l e n e r g i e s a g r e e v/ith t h e c o r r e s p o n d i n g energy between l e v e l s t o w i t h i n established  1 K e v . A 1263-3 Kev l e v e l i s  by t h e p r e s e n c e  o f a gamma t r a n s i t i o n o f 892.7  K e v . Two o t h e r gamma t r a n s i t i o n s o f e n e r g i e s , 582.1 K e v f r o m t h e n e g a t i v e - p a r i t y excited level of  903-6 K e v a n d  l e v e l s to the 2 beta +  have a l s o been d i s c o v e r e d .  The p r o p e r  locations  t h e 995-9 K e v a n d 1 0 0 4 - 5 K e v i n t h e d e c a y scheme a r e  c o n f i r m e d b y a gamma-gamma c o i n c i d e n c e Germanium  detector  and a N a l ( T l )  method u s i n g  scintillator.  and  i n t e n s i t i e s of the i n t e r n a l conversion  the  b e t a t r a n s i t i o n s have been measured by an  image s p e c t r o m e t e r . T h e i r the  differences  established  decay  Theoretical  values are quite  The e n e r g i e s  e l e c t r o n s and intermediate  consistent  v a l u e s o f t h e energy l e v e l s and t h e  t h e Asymmetric R o t a t o r Model  treatment).  with  scheme.  b r a n c h i n g r a t i o s f o r gamma t r a n s i t i o n s h a v e b e e n using  a  The ' s t i f f n e s s '  calculated,  (J.P. Davidson's  parameter of the nucleus  ii and i t  - 1 1 1 -  asymrnetry  parameter  respectively. energies square  of  values. ion  Comparison  seven  deviation  branching  from  ratios  The  measured a l s o  0  +  are  o f the  o f 1.5 are  beta  agrees  i°• T h r e e  i n good  0.402 and  experimental  positive-parity  monopole  the  y obtained  levels out  of the  four  agreement w i t h  with  the  state  to  present  the  and  gives a  0  degrees theoretical root-meanexperimental  the.theoretical  transition, probability excited  11.52  f o r the +  ground  theoretical  transitstate  calculation.  -ivTABLE OF CONTENTS Page CHAPTER I  INTRODUCTION  CHAPTER I I THEORY 1.  1  . .. .  6  . ..  The C o l l e c t i v e M o d e l s  .  6  \  \  2.  The A s y m m e t r i c R o t a t o r Model  3.  The O c t u p o l e  11  Case i n t h e Asymmetric 17  R o t a t o r Model 4.  C a l c u l a t i o n s o f Reduced E 2 . T r a n s i t i o n . 19.  Probabilities 5.  O t h e r Forms o f E l e c t r o m a g n e t i c 23  Transitions  26  CHAPTER I I I THE DESIGN OF EXPERIMENTS CHAPTER I V THE GAMMA SPECTROSCOPY  CHAPTER V  '.' 2 9 .  1.  General  29  2.  I n t e r a c t i o n b e t w e e n Gamma R a y s and t h e  Considerations  the Detectors  32  3.  Gamma-ray D e t e c t i o n A s s e m b l i e s  34  4.  S o u r c e P r e p a r a t i o n and M o u n t i n g .......  42  5.  Experimental  6.  R e s u l t s and A n a l y s i s  Procedures  GAMMA-GAMMA COINCIDENCE SPECTROSCOPY 1.  General  Considerations  "45 50 62 '. 6 2  . Page 2.  The C o i n c i d e n c e S y s t e m  3.  Experimental Procedures  4.  Results  64 70  ... . ....  74  CHAPTER V I THE BETA SPECTROSCOPY 1.  General Considerations  2.  The I n t e r n a l C o n v e r s i o n and P a i r  74  77  Production • 3.  The B e t a  Spectrometer  • •••  82  4.  P r e p a r a t i o n o f Beta  5.  Experimental Procedures  88  6.  R e s u l t s and A n a l y s i s ...................  90  1.  The D e c a y Scheme  2.  Model F i t t i n g  CHAPTER V I I I CONCLUSIONS  86  Sources  CHAPTER V I I THE DECAY SCHEME AND MODEL F I T T I N G  REFERENCES  72  103 103 . 108 114  .. ..  117  -viLIST  OF FIGURES AND TABLES " Page  CHAPTER I V Figure  1.  T h e gamma s p e c t r o m e t e r  Figure  2.  Electronic  circuit  35  f o r t h e gamma 36  spectrometer Figure  3.  Cs 1 3 4 s p e c t r u m t a k e n f r o m t h e gammaspectrometer assembly  38  . . . . . . . . . . .  Figure  4.  Germanium d e t e c t o r a s s e m b l y  Figure  5.  Input  Figure  6.  Co 6 0 gamma s p e c t r u m  40  . . . . . . . .  stage o f t h e low-noise p r e a m p l i f i e r . from  1 . 5 c . c , Ge 43  detector Figure  7.  Co 6 0 gamma s p e c t r u m detector  f r o m 5 c . c . Ge 44  . .  Figure  8.  E u 1 5 4 gamma s p e c t r u m  Figure  9.  L o w - e n e r g y gamma p e a k s  Table  I.  Gamma p e a k s  Figure  1 0 . Gamma e n e r g y  Table Figure Table  II.  Gamma peaks  1 1 . Gamma  Figure  used  48 o f Eu 154  f o r energy  calibration used  intensity  (gamma  1 2 . Weak gamma p e a k s  calibration  curve  for intensity calibration  I I I . E n e r g i e s and I n t e n s i t i e s spectrum  4l  )  49 .  52  .  53  calibration  56  c u r v e . To f o l l o w 5 6  o f Eu 154 58  . . . . . .  i n t h e Eu 154 spectrum  .  60  Table  IV.  Gamma p e a k s b e l o n g i n g  t o E u 152 i m p u r i t y  .  61  o f gamma-gamma c o i n c i d e n c e .  .  63  . . . .  65  CHAPTER V Figure  13- Illustration  Figure  14- The gamma-gamma c o i n c i d e n c e  Figure  15- P h o t o m u l t i p l i e r  Figure  1 6 . The t u n n e l  Figure  1 7 a , b , c , d . The gamma-gamma c o i n c i d e n c e  output  diode  system  stages  discriminator  67 . . . . . . spectra  69 73  CHAPTER.VI Figure  18. B e t a  Figure  19- Magnet  Figure  20a. Beta  spectrum  o f E u 154, l o w - e n e r g y  Figure  20b. B e t a  spectrum  o f E u 154, h i g h - e n e r g y  Figure  20c. Beta  continuum  Table  V. '  of beta  Figure  20d- E x p a n d e d  Table  VI.  s p e c t r o m e t e r assembly  Data  current  83  control circuit  84  o f Eu 154  conversion  of K-internal  91  part  92  • • - -  peaks  . . . . . .  p o r t i o n o f E u 154 b e t a  Comparison  part.  spectrum.  Figure  VII.  21.  Energies  95 98  conversion  coefficients Table  93  100  and r e l a t i v e  intensities  of beta  transitions  10.1  The K u r i e  .102  plots  CHAPTER V I I Figure  22.  The e x c i t e d decay  Table  s t a t e s o f Gd 154 f r o m t h e  o f E u 154  VIII. Transitional conversion  intensities  . .'104 f o r gammas,  e l e c t r o n s and b e t a s .  . ... . . 105  Table  IX.  X.  -  Page  Comparison of the experimental theoretical  Table  v i i i  The  and  energies i n the quadrupole case.  b r a n c h i n g r a t i o s o f gamma  between p o s i t i v e - p a r i t y  levels  110  transitions I l l  -ix-  ACKNOWLEDGMENTS  I wish  t o express  my g r a t i t u d e t o D r . K.C. Mann  f o r h i s g u i d a n c e , encouragement and h e l p t h r o u g h o u t t h e work. I am a l s o i n d e b t e d Physice  D e p a r t m e n t , K a n s a s U n i v e r s i t y , D r . T. K a t o h o f  the Nuclear Dr.  Engineering  D. K i a n g  for valuable Van  t o Dr. J.P. Davidson of the  de G r a a f f  D e p a r t m e n t , Nagoya U n i v e r s i t y , and  o f t h e Physics Department, Dalhousie  University  i n f o r m a t i o n g i v e n t o me; t o t h e members o f t h e G r o u p , i n p a r t i c u l a r D r . G. J o n e s , D r . G.M.  Griffith  and M r . D.A. D a l b y f o r a l l o w i n g me t h e u s e o f some  of t h e i r  e q u i p m e n t ; and t o t h e C o m p u t i n g C e n t r e S t a f f f o r  t h e i r help  i n data  processing.  T e c h n i c a l a s s i s t a n c e b y M r . T. W a l t o n , M r . E . P r i c e , Mr.  A. F r a s e r The  Office, and  and M r . J . L e e s a r e h i g h l y  present  p r o j e c t i s supported  appreciated. by t h e E x t e r n a l A i d  Government o f Canada t h r o u g h a s t u d e n t  scholarship  the National Research Council through Grants-in-Aid o f  R e s e a r c h t o D r . K.C. Mann.  -1-  CHAPTER I  INTRODUCTION  The d e s c r i p t i o n  of a nucleus.is e s s e n t i a l l y  a  c o m p l i c a t e d many-body p r o b l e m w h i c h h a s n o t r e a c h e d  any  precise  phenomeno-  solution.  To s i m p l i f y  t h i s problem., s e v e r a l  l o g i c a l m o d e l s h a v e b e e n p r o p o s e d . The commonly known one i s 1) the  'single-particle  s h e l l model  1  , w h i c h views  each n u c l e o n  ( p r o t o n o r n e u t r o n ) i n a n u c l e u s t o be m o v i n g u n d e r t h e action  o f an a v e r a g e p o t e n t i a l  contributed  by t h e o t h e r  n u c l e o n s . As a q u a n t u m - m e c h a n i c a l c o n s e q u e n c e , t h e n u c l e o n s s h o u l d be m o v i n g i n d e f i n i t e  'shells' s i m i l a r  to the  electrons  i n a n a t o m . The number o f p r o t o n s Z o r n e u t r o n s N w h i c h c o m p l e t e s a g i v e n number o f s h e l l s T h i s model p r e d i c t s parities  o f most n u c l e i  i s called  successfully  a m a g i c number.  t h e s p i n s and  at t h e ground s t a t e .  For nuclei  Z and N v a l u e s a r e c l o s e t o t h e m a g i c n u m b e r s , good tative can or  agreement  i n low-lying  be o b t a i n e d . However,  energy l e v e l s w i t h  as Z  I t i s even worse  when one c o n s i d e r s h e a v y e v e n - e v e n n u c l e i A are w i t h i n  quanti-  experiments  t h i s agreement d e t e r i o r a t e s  N r u n s away f r o m t h e m a g i c n u m b e r s .  whose  whose mass  numbers  t h e r a n g e 1 5 0 - 1 9 0 o r g r e a t e r t h a n 228..'.In t h i s  -2case, the  i n d i v i d u a l n u c l e o n i c m o t i o n g i v e s way  predominent motions of the collective rotation  on  the•liquid-drop  and  n u c l e u s as  the  a whole, i . e .  m o d e l was  p r o p o s e d by of the  Bohr  the  interaction  d e f o r m a t i o n i n the  the  . Thus  of  single  equilibrium  drop.  p a r t i c l e and  shape of  the  nucleus i s accounted  hydrodynamic b e h a v i o u r of a l i q u i d  because of  more  v i b r a t i o n . A c o l l e c t i v e model based 2)  s u r f a c e o s c i l l a t i o n or v i b r a t i o n f o r by  to the  the  Also  closed  nucleus  shells,  from  3) spherical rise the  to  s y m m e t r y becomes p o s s i b l e  the  collective rotation  e n e r g y l e v e l scheme o f  l e v e l s which are On rigid The  rotator,  rotator  therefore, is  the  the  further  split  o t h e r hand, the  of the  nucleus.  nucleus consists into rotational n u c l e u s can  which gives r i s e to  be  gives  Consequently, of  vibrational  bands.  considered  pure r o t a t i o n a l  as  vibrational  l e v e l s . The  shape, t h a t  shape o f  Is the  c o l l e c t i v e model d e s c r i b e d  the  shape a t a b o v e has  a  levels.  i s then softened to y i e l d surface v i b r a t i o n s ,  equilibrium The  the  . This deformation  rigid zero  and  rotator  vibration.  been under  g r e a t t h e o r e t i c a l d e v e l o p m e n t i n t h e p a s t few y e a r s , n o t a b l y ; by A. F a e s s l e r , W. G r e i n e r , R.K. S h e l i n e , A.S. D a v y d o v and . 4)-16) J.P.  Davidson  d e t e r m i n a t i o n of vibration  . The the  current  interest  form and.magnitude of  lies the  i n t e r a c t i o n which accounts f o r the  '•between-energy l e v e l c a l c u l a t i o n s , and  in  the  rotation-  discrepancy  e x p e r i m e n t a l measure- -  -3-  ments. T h i s d i s c r e p a n c y i s e x p e c t e d for  nuclei  A ranges typical  h a v i n g n u c l e a r masses l y i n g  mentioned  the b e t a  states  Intense  beta  Similar B.V.  investigation  2 0  )  intensities  complexity impurity  have w i d e  o f t h e decay  those  favours source  2  1  ^.  c o i n c i d e n c e work.  The t r a n s i t i o n a l  authors  vary  a p p r e c i a b l y , and  o f u n c e r t a i n t y . The r e a s o n  scheme, a n d t h e p r e s e n c e scheme n o t l e s s be n o t e d  o f Eu 1 5 2 a r e s i m i l a r  o f Eu 1 5 4 , and t h a t  o f Eu 152'  complicated  that  i n energy  the neutron-capture  impurities.  so f a r b e e n u n a b l e  than  the e l e c t r o n and i n t e n s i t y process  t h e p r o d u c t i o n o f Eu 1 5 2 o v e r Eu 1 5 4 - O t h e r p r o d u c t i o n have  et a l " ^ ,  done by B.S. D z e l e p o v ^ " ^ ,  ranges  o f Eu 1 5 4 - I t s h o u l d a l s o peaks  a n d F.S- S t e p h e n s  a p p r e c i a t e d when c o n s i d e r i n g t h e  w h i c h h a s a decay  conversion  scheme was s t a r t e d  scheme was e s t a b l i s h e d  et a l  by t h e above  from  neutron-capture  a n d gamma-gamma  have b e e n  obtained  isotope being  on t h e d e c a y  et al"*"^  analysis  c a n be e a s i l y  using  decay  a n d 0. N a t h a n  determined  these  that  by J.M. C o r k  investigations  energies  for  detailed  spectral  Bobilin  their  of the  154 i s one o f t h e  1 5 4 , the l a t t e r  by S c h e i c h e n b e r g e r  1 9 5 7 - The f i r s t  using  t h e ends  o f Gd 1 5 4 a r e o f t e n  of'Europium  produced  independently  to  above, and G a d o l i n i u m  excited  decay  process. in  near  pronounced  examples. The  first  t o be more  methods o f  to e l i m i n a t e a l l  Beta-gamma and gamma-gamma c o r r e l a t i o n methods h a v e since  t h e n been employed  i nthe investigation  by  several  22)-30)  authors  , a i m i n g a t l ) t o c o n f i r m o r improve t h e  established  decay  scheme, 2 ) t o d e t e r m i n e more p r e c i s e l y t h e  r e d u c e d gamma t r a n s i t i o n p r o b a b i l i t i e s , and 3 ) t o f i n d t h e f o r b i d d e n n e s s o f each beta t r a n s i t i o n i n t h e decay  energy  continuum. These a r e e s s e n t i a l  f o r v e r i f y i n g any proposed  n u c l e a r m o d e l and f o r f i n d i n g  t h e m a t r i x elements o f t h e beta  t r a n s i t i o n s . O t h e r methods such as l i f e - t i m e measurements 3D f o r s p e c i f i c gamma t r a n s i t i o n s , 4 / | - c o u n t i n g f o r K- Y 32)  c o i n c i d e n c e work  , investigation  o f t h e decay o f s h o r t -  33)  lived  Tb 1 5 4 t o Gd 1 5 4  , and Coulomb e x c i t a t i o n f o r p r o d u c i n g 34),35)  higher excited  l e v e l s o f Gd 1 5 4  have been  However, a l l e x p e r i m e n t s performed resolution severely  o f t h e gamma d e t e c t o r s  hinders the analysis  exploited.  t o date s u f f e r  from  (Nal s c i n t i l l a t o r s ) ,  o f a c o m p l i c a t e d gamma  poor which  spectrum.  A l t h o u g h t h e u s e o f c o i n c i d e n c e methods may h e l p i n s u c h a n analysis,  i ti sonly practicable  when d e a l i n g  with r e l a t i v e l y 36)  s t r o n g gamma p e a k s . F i n a l l y , e x t e r n a l (the  b e t a c o n v e r s i o n method  p h o t o e l e c t r o n p r o c e s s ) has been used. I t improves t h e  resolution,  but again suffers  from l a c k  of intensity.... -  With t h e newly developed L i t h i u m - d r i f t e d detectors i nthe laboratory a.re-investigation  Germanium  o f t h e P h y s i c s D e p a r t m e n t ,U..B .C .,  o f t h e d e c a y o f Eu 1 5 4 becomes more  promising, since the r e s o l u t i o n of these detectors i s t e n times b e t t e r than that of the usual Nal s c i n t i l l a t o r s . I n t h e p r e s e n t w o r k , t h e a u t h o r a p p l i e d t h i s new t o o l w i t h an i n t e r m e d i a t e - i m a g e a c c u r a t e energy in  together  b e t a s p e c t r o m e t e r t o o b t a i n more  and i n t e n s i t y  d a t a f o r t h e gamma  t h e d e c a y o f Eu 1 5 4 . T h e s e d a t a w e r e t h e n u s e d  the v a l i d i t y o f t h e c o l l e c t i v e model proposed  and  transitions to  check  developed  10)-16)  by D a v y d o v , F i l i p p o v , will  and D a v i d s o n  be shown l a t e r , p r o v e d f r u i t f u l .  . The r e s u l t s ,  as  CHAPTER  II  THEORY §1.  The  C o l l e c t i v e Models. 2)  B o h r ' s h y d r o d y n a m i c a l model from the  liquid-drop  t h e o r y . By the  1)  nuclear  mass d e n s i t y .  can  be  the  nucleus  described  the  in  as  tension  surface, of  the  R(8,^,t) = R  and  i s the  Q  a{t)\^l It  less  be  distance  motion, the V =  nuclear  are can  (1 +5,  the  Coulomb  Y  A  radius  i s an  constant invariant;  oscillations  motions; to  oscillation  repulsion. written  as  an  expansion  and  (1)  from  a  spherical  (t)^^represent expressions  k i n e t i c energies  of  configuration symmetry. the  dimension-  simple  become, (2a)  •>t  i  T =  B. A K  A  which  and  the  when i n . s p h e r i c a l  d i s t o r t i o n s from  2  be  has  (8,$)  A  e a s i l y seen t h a t  —  liquid,  shown, t  p o t e n t i a l and C-  surface  can  present  charged  and  volume  harmonic  nucleus  i t s concept  apply:  contributing  the  Q( )  variables;  ±2  of  the  a drop of  nuclear  simple  and  s p h e r i c a l h a r m o n i c s as  where R  the  i s capable  small  of  i s incompressible  restoring forces  surface The  fluid  basis  be  immediately  Consequently,  3) the are  a nucleus to  assumptions  the  2)  model, forms the  considering  following  , which borrowed  (2b)  harmonic  respectively, constant  where Bx  I s t h e mass c o e f f i c i e n t  and C\ , t h e  for restoring forces.  3) It moving  was  shown by R a i n w a t e r  that  a single  particle  i n a p o t e n t i a l w e l l had a l o w e r e n e r g y i f t h e w e l l  was d e f o r m e d  than  i f i t was  spherical.  This 'single-particle(  11 shell  effect  makes i t p o s s i b l e f o r t h e ' l i q u i d  non-spherical  a t e q u i l i b r i u m . Hence we  p e r m a n e n t l y deformed S u b s e q u e n t l y , we the  three  can attach.a  principle  coordinate  (ellipsoidal)  z  1  c a n assume t h a t a  shape e x i s t s  s y s t e m S' c a n be r e l a t e d 6^,9  i n the nucleus.  set of coordinate  axes o f t h e deformed  s y s t e m S by E u l e r i a n a n g l e s , =  new  d r o p ' t o be  body. T h i s  axes new  to the laboratory  2,83,  such t h a t  along  coordinate  z'-axis  in S  1  ( 9 , 9 ) i n "S and z - a x i s i n S = z (0 , v-6 ) i n S' . The n  2  variables  i n s y s t e m S' c a n t h e n  be r e l a t e d  t o the  corresponding  37), variables  38)  V It not  i n s y s t e m S by a u n i t a r y r o t a t i o n m a t r i x  =  I  ^  1  should  be n o t e d  unique. R e s t r i c t i n g  ways o f c h o o s i n g  Considering time being,  !  with  This w i l l  o  }  =  ( e  i' 2' 3 8  e  )  ( 3 )  o f system S  systems, t h e r e  i t s axes a l o n g be d i s c u s s e d  only the quadrupole  Z R (l.  i  t h a t t h e assignment  and d r o p p i n g  becomes, R -  ( 8  t o right-handed  system S  axes, o f t h e n u c l e u s .  the  '  as shown  1  is  a r e 24  the p r i n c i p l e  later (§2). (i.e.X= 2),  case  for  t h e s u b s c r i p t A., e q u a t i o n ( l ) •. V  (Ma)  or  i nCartesian  coordinates,  R •= R (1+o-xx , ^ + y y 4 + z z ± + 2 a y ^+2°-yz a  Q  f c  with  ^ + 2 a z x =-^i) t  the relations, O-yy  + O-zz = 0  2]I ( x x - ayy 15  + 2ia-xy)  a  fa-zx + ia-yz)  8Tr  15  a  C 2 z z - a-xx - a )  = °  For  X  K  a  45  yy  t h e system t a  t yz  T h e r e f o r e , we c a n a  t  T  = a = 2 -2  t = zx  = 0  a  write  P sinT  Q  z/2  » _ ' Q  1  ' 1  w h e r e P ,y a r e n e w l y d e f i n e d  =  °'  T I o  variables  (6)  indicating the t o t a l  d e f o r m a t i o n o f t h e n u c l e u s and t h e d e v i a t i o n symmetry  from a x i a l  respectively. (2a),(2b) and ( 3 ) / and m a k i n g u s e  From t h e e q u a t i o n s  o f D^ (0_.) •.(  the  o r t h o n o r m a l and t i m e d e r i v a t i v e  properties  see  r e f . 37JP73) t h e p o t e n t i a l and k i n e t i c e n e r g i e s become:.  v>  A.  (7) T = lB ^_|^|  2  = | B Z ||,- (^D*„(8.) + ^b*„  IB '(P + P y ) + ± B „ L 4 P s i n ( y - k 2  (9,))  2  2  2  2  vib  2  2  rot  2g .8 2  2  2  (8).  -9-  w h e r e j , k = 1,2,3 Q  k  =2.^  a r e t h e s u b s c r i p t s f o r c o o r d i n a t e a x e s , and  6j i s t h e a n g u l a r v e l o c i t y  component a l o n g t h e p r i n c i p l e  a x i s k ( s e e ' r e f . 2,p.12). A l s o n o t e t h a t t h e f i r s t i s the v i b r a t i o n a l part  and t h e s e c o n d t e r m , t h e r o t a t i o n a l  The n e x t s t e p i s t o f i n d 39) w h i c h P a u l i ' s method  2  curvilinear I  3  s p a c e ; and  = LrB P sin (y-2Trk/3) 2  which from  the Hamiltonian operator, f o r  be t h e 5 - d i m e n s i o n a l  ,  2  2  R  (9).  (8) I s j u s t t h e k-component  Then t h e square o f t h e d i f f e r e n t i a l ds and  G..„ d l i „ d f l  =2.  2  part.  i s used as f o l l o w s :  ( i ^ p = ( P , y ,6^ ,0 , 0 )  Let  term of T  o f moment o f i n e r t i a  interval  of action i s  2Tdt'  = u  f r o m (8) = B dP  + B P dY  2  2  2  Therefore,  2  2  +  2 Ik^qgidB k 7 J J  o  B 2  B  2 ' P  ,  (10)  2  , 2  (11)  G =  ?  0  3 ^ 3  V  j  The d e t e r m i n a n t , | G | = 4 B P sin (y-2Tr/3) sin (T-4TT/3).sin (7-277)0(8^ 3  8  2  2  2  2  =  4 B ^ s i n 3 Y Q(B ) 8  2  ±  where Q(0i)  (12)  -10The  H a m i l t o n i a n c a n t h e n be n = .  The  V  +  potential  = -  |  energy  H  found.  > _ M L ^  +  y  ( 1 3  operator,  V - lC P  (14)  2  2  The  )  v i b r a t i o n a l energy  operator,  2 *  2B  The  2  (P*3f3  ^p  p  l S  i n 3 y 37  *y )  J  r o t a t i o n a l e n e r g y o p e r a t o r c a n be s i m p l y e x p r e s s e d  follows without T  using equation  =£_J= -^^._2  Ul  r 0 t  x  2  I  (13),  =  +  as  7  JS  (16)  8 B P £ J sin -(Y-2irk/3)  3 1  2  2  2  w h e r e t h e component a n g u l a r momentum  operator,  i . e . moment o f i n e r t i a X a n g u l a r v e l o c i t y , c . f . (8) and ( 9 ) • Hence t h e S c h r o d i n g e r  ( T . + T v  Its  b  solution So  + v )^(P,y,e ) i  (is)  i s deferred t o § 2 .  w h i l e A= 3 h a s been, t r e a t e d b y D a v i d s o n  be o u t l i n e d  corresponding  i n | 3 - No e v i d e n c e  exists  , which  f o r any l e v e l  toA?3.  F u r t h e r development  separated  = EcJr(P,y,e.)  f a r , we h a v e o n l y c o n s i d e r e d A = 2 c a s e . A = 1 g i v e s no 14).  contribution, will  r o t  e q u a t i o n becomes,  i n t h e c o l l e c t i v e m o d e l c a n be 4)-9).  i n t o two b r a n c h e s .  ed t h e d e f o r m e d n u c l e u s  F a e s s l e r and c o w o r k e r s  t o be an. a x i a l l y s y m m e t r i c  considerellipsoid,.  -11- •  :  io)-i6)  w h i l e D a v y d o v and c o w o r k e r s  t r e a t e d i t as a x i a l l y  a s y m m e t r i c . O n l y t h e l a t t e r w i l l be d i s c u s s e d , a s i t i s a g e n e r a l c a s e and r e p r e s e n t s t h e l a t e s t §2.  The A s y m m e t r i c  Rotator  development.  Model..  The  a s y m m e t r i c r o t a t o r m o d e l was p r o p o s e d b y D a v y d o v 10) and F i l i p p o v , and d e v e l o p e d b y D a v y d o v , D a v i d s o n and many 11)-16) others . I t i s a c o n t i n u a t i o n o f B o h r ' s m o d e l . The d i f f e r e n c e i n concept l i e s asymmetric Vis  on t h e r e c o g n i t i o n o f a f i x e d  n u c l e a r s h a p e . T h i s i m p l i e s t h a t t h e shape  parameter  no l o n g e r a v a r i a b l e . H e n c e , i n f i n d i n g t h e H a m i l t o n i a n  o p e r a t o r a s shown p r e v i o u s l y ( f o r A= 2 ) , dimensional  r: ( p , 8 ,0 ,0 ) . F r o m  s p a c e , i . e . (fl)  curvilinear  only a 4 -  we need  i  (13),  equation is  axially  L  e  i t f o l l o w s t h a t t h e v i b r a t i o n a l energy  i  operator  simply,  V  Note  I  2B  B  2  P  2(3  r  (19)  P(3  that B 2  G  =  T  *5  h  f  O  2  q  ij  i 2q 2  ?  (20) 2 j  I Z 3  q  3 j  O  The equation altered  r o t a t i o n a l energy operator  is still  t h e same a s i n  ( l 6 ) ; but t h e p o t e n t i a l energy o p e r a t o r t o take  equilibrium  i n t o account of t h e deformation  shape, i . e .  i s slightly from  spherical  V = 1 C (P  - P )  2  P i s the t o t a l  where t h e c o n s t a n t the nucleus  2  Q  i s not v i b r a t i n g  Consequently,  ( fl  ' ,• (21)  2  Q  3  J  \  p a r a m e t e r when  ( i . e . i n pure r o t a t i o n a l  the Schrodinger /  deformation  equation L£  ^  states).  becomes ^  ^  2B =  Since only the operators L angles 9 i , i =1,2,3,  Tj"(P,8.)  E  (22)  are f u n c t i o n s of the E u l e r i a n  k  t h e above e q u a t i o n i s s e p a r a b l e . L e t  -f (P,8i) = $ ( 8 ) eP (P) L  u  ±  (23)  u  then,  i 2,  .  L k ?  - £(L)1 -4(9.) = 0 J  fcSin (y-2Trk/3) 2  (24)  I  and r  2  <2 I f-fp 3 -^-) + |C„(P-P ) + X £ ( ) - E 1 c^(P) = 0 2B P ^PV J&J ' o t+B P ^ (25) £(L) g i v e s r o t a t i o n a l e n e r g i e s o f t h e a s y m m e t r i c r o t a t o r  L M_  L where in  2  3  2  2  ;  L  L  r j  2  2  units of K /(4B p ) . 2  2  2  (24),  To s o l v e e q u a t i o n  one n e e d s t o c h o o s e a s e t o f  basis v e c t o r s , the simplest of which i s that of n  L  MK  (8 } , t h e i  r o t a t i o n m a t r i c e s , because l  ?  K  t  3  D  MK^i.5 D  M K ^  0^(8.)  L  =  =  ^  M  L  +  D  L  )  : :-v.-  ;  :  D  (26a)  SKC8 )  M K ^  • K ir^CH.)  ±  ;  (26b) (26c)  ,where t h e s u b s c r i p t z r e f e r s t o S s y s t e m and 3 - z' r e f e r s t o S.' s y s t e m  (see r e f . 33  p.64).  -13Since t h e b a s i s vectors|LMK> = D J ^ C ^ ) .  are e x p l i c i t l y  known and so a r e t h e o p e r a t o r s L f j , t h e m a t r i x e l e m e n t s , • <XMK I L?|LMK')>  , f o r a g i v e n v a l u e o f L and o f M c a n be  c a l c u l a t e d . Consequently, parameter  y  f o r a g i v e n v a l u e o f t h e shape  ,  A <LMK |H |LMK'> rot  A < I L? 1LMK' > = !£, . 2 L M K  ..  ^  can be c a l c u l a t e d . However, s i n c e t h e n u c l e u s  i s not a x i a l l y  symmetric,  K I s n o t a good quantum number. H e n c e , t h e s t a t e v e c t o r s o f the nucleus  h a v e t o be m i x t u r e s  of thebasis vectors,i . e .  L  (28)  |LM > = 2 i A j L M K > What i s now l e f t and  i st o determine  t h e c o e f f i c i e n t s A^  t h e e i g e n v a l u e s i^(J-d i n (24). The e i g e n v a l u e s  c a n be f o u n d  A  by d i a g o n a l i z i n g t h e m a t r i x ^ matrix dimension  r o t  c a n be r e d u c e d  • But before  doing t h a t , t h e  a p p r e c i a b l y by c o n s i d e r i n g t h e  symmetry p r o p e r t y o f t h e c o e f f i c i e n t s A . A s was p o i n t e d o u t R  i n | l t h a t t h e r e a r e 24 ways o f a s s i g n i n g t h e s y s t e m S' t o t h e nucleus  s o t h a t t h e c o o r d i n a t e axes, o f S' c o i n c i d e w i t h t h e  p r i n c i p l e a x e s o f t h e n u c l e u s . T h e wave f u n c t i o n o f t h e n u c l e u s should  be i n d e p e n d e n t o f t h e a s s i g n m e n t . T h e r e f o r e , we  some symmetry r e l a t i o n s  among t h e A  1  K as  follows:  expect  s , w h i c h c a n be f o u n d  The  24 ways o f a s s i g n i n g  rotation operators, of-rr  3  about y ' - a x i s ; P , a - r o t a t i o n , o f  P , a rotation of from t h e three  MK V ((  axes  1 -> LMK  =  produces a r o t a t i o n  i t / 2 a b o u t z - a x i s ; . and 1  2TT/3 a b o u t t h e a x i s e q u i d i s t a n t  3  D  such t h a t  PL,P ,P , 2  S' c a n be i n t e r r e l a t e d b y t h r e e  1  ( r e f . 40; ,. p . 1 1 0 ) . A p p l y P ^ , P o n 2  >  w e  h a v e  P |LMK>  = exp i-rr(L+K) | LM-K>  P JLMK>  = exp(iTrK)  1  2  angularly  (29a)  | LMK>  As L and K a r e i n t e g e r s , a p p l y i n g  (29b)  on|LM> g i v e s  rise to four  cases: 1)  P |LM> = +|LM>  , Pg\LM>  = +|LM>  2)  P |LM>  , P |LM>  = + |LM >  3)  P I L M > = +|LM>  ,  P |LM>  =-lLM>  4)  P )LM>  = -]LM> ,  P?|LM>  = -|LM>  1  =-lLM>  1  2  1  1  2  These f o u r cases b e l o n g t o t h e T o u r of a D  irreducible representations  g r o u p , e a c h t o e a c h , and t h e r e f o r e  2  c a n n o t be m i x e d  together. For  case l ) ,  (29b),  from  K m u s t . b e e v e n and f r o m  (28),  (29a), A It  K  - (-1)  L  A_  K  - C  (31)  K  follows, f  o  r  L=0: K=0, A = +A =C ,. dimension o f H Q  Q  0  for  L=1: K=0, A = -A =0, N=0, 1LM>=0  for  L=2: K=0,2, A = +A =C  Q  r b t  ,N=1, | LM > = C JLM0> C  Q  Q  0  Qj  A = +A =C ,.'N=2', . 2  2  2  \LM> = C |LM0> + C (|LM2> + I LM-2 > ) /J2 o  etc .  2  (31a)  -15Similarly, the l e v e l 2)  .  i t f o l l o w s f o r cases 2),3)  and  s t r u c t u r e of even-even n u c l e i ,  Now  ( \ = 3 ). N  £ m N  back  i n t o the matrix equation, the  c o e f f i c i e n t s C „ c a n be d e t e r m i n e d . The Is thus  7040 To  s o l u t i o n of equation  completed.  (The c o m p u t a t i o n s t h e IBM  2)  A = 2 ), and  t h e e i g e n v a l u e s f ( L ) c a n be c a l c u l a t e d . T h e n by  substituting  (24)  However, f r o m  o n l y c a s e s l ) and  c a s e '(  apply, i . e . l ) f o r the quadrupole  f o r the o c t u p o l e case  4).  a b o v e and what f o l l o w s w e r e done by .  computer.)  s o l v e e q u a t i o n (25)  f o r the v i b r a t i o n energy  levels,  the sole technique i s t o t r a n s f o r m the equation t o a s i m p l e r standard d i f f e r e n t i a l v a r i a b l e s and 12 and  15.  Then (25)  e q u a t i o n by c h a n g i n g t h e  c o n s t a n t s i n t h e e q u a t i o n . T h i s was  I t proceeds  i s reduced T~o 2B ^ P 2  w i t h boundary  different  +  W  as  done i n r e f .  follows:  to  r CP) LM N  condition  - r u l Ur^ ) LN J LN  =  U  , and  E  1 3  (0) = 0  (33) •  0  the p o t e n t i a l  energy  1_/N  operator, W  LN  (P)  = | C (P-P ) 2 o  2 +  J l L i (L) + 4B P^ N 2  - ^L 8B P 3  2  ;  2  (34)  Let  be t h e v a l u e  PQ  ->w  of  when  P  W  i s a minimum, i . e .  (P)  L N  (P) =  C  2  ( P ; - P  O  (f (L)  -  )  N  P=P:  or  2  ' 'o  2  B  P  +  3/2)  =  0  O  2 P  o  =  P  L  o + 2 B  Then  P  2  + V2)  L  M  2  P J  (35)  3  i s a new t o t a l d e f o r m a t i o n p a r a m e t e r a t e q u i l i b r i u m .  Q  It differs obviously Further,  from  P  by  Q  ^ (£ (L)  which i s  +3/2)/(2B C PQ )  2  3  2  N  2  due t o t h e r o t a t i o n - v i b r a t i o n i n t e r a c t i o n . define . = t  z  = P; / ( P f O  -  The p a r a m e t e r ^  2  2  )  2  (36)  (37)  Q  (35). becomes  vibration,  /(B C PJ  ^  and  I  Z  _  3  (£  (L)  +  3/2)/2  =0  (38)  measures t h e s t i f f n e s s o f t h e n u c l e u s  and i s c h a r a c t e r i s t i c o f a n u c l e u s . f o r ^ , ( 3 8 ) c a n be s o l v e d  a s u i t a b l e value Also,  (f ( )  C  After  against choosing  numerically  f o r Z.  introducing z  l = ^  +  |  +  3  /  2  • C  )  = z ( P - P^)/Pi  y  (41)  L N  (33) i s reduced t o a standard  equation  d Du ( J 2 y)  ,  2  V  +  V 2  dy where v> =v> (Z,) n  n  K  I'  boundary c o n d i t i o n  (2V>  +  )  -,(i«o  L  (4~2 y) = u ( P ) the  3 9  form,  . \ 1  _ . 2 Y  )  D  J  >  (|2  y)  =  0  (42)  i s a quantum number d e t e r m i n e d by t h e . U  L  N  (0)  =  0  or.  (-\2  Zj-) .= 0 . . .  (43)  '-17Th e e i g e n v a l u e s f o r (42) a r e E  L  N  n  = JL  {(P  0  1) ( / Z ) 2  n +  Z l  f +  N-  ( L ) + 3 / 2  MZ  where  w  = J C  0  2  / B  £  , (10+3/2  ^  2  2Z  (MM)  2  2  v a l u e s \) f o r n = 1 and f o r n = 2 w e r e computed and n  The  tabulated refers  a s f u n c t i o n s o f Z^ I n r e f s . 14 and 4 l . H e r e n = 1  t o _ a l l t h e l e v e l s b e l o n g i n g t o t h e ground  b a n d , and n = 2, t o .'the l e v e l s It  ' [l+ N  i s also, c l e a r  The  levels  i nthe f i r s t  state  excited  beta  beta  band.  t h a t N i s t h e l e v e l o r d e r f o r a g i v e n L.  c o r r e s p o n d i n g t o N ? 2 and n = 1 a r e o f t e n c a l l e d 9 )  t h e gamma v i b r a t i o n a l E  LNn  calculated  levels  are i n arbitrary  the present  calculation,  each l e v e l  LNn  E  The O c t u p o l e  The  energy  the following  scale.  LNn  r a t i o was c h o s e n f o r  211  - E  Oil  " 011 E  o f t h e o c t u p o l e case  asymmetric r o t a t o r  model i s c o m p l e t e l y  quadrupole  \ = 2  (  give r i s e t o different  ( \ = 3 ) i n the  i n p a r a l l e l with the  ) . T h e r e a r e o n l y two p o i n t s  mathematical  given i n equation ± 1  =  Q ± 3  = 0,  which  expressions:  a) I n s t e a d o f t h e e x p r e s s i o n s f o r  a  Therefore, i n  Case i n t h e Asymmetric R o t a t o r M o d e l .  treatment  case  . The e i g e n v a l u e s  }  E  |3 •  i n another, model  =-2,-1,0,1,2)  ( 6 ) o f ^ 1 , we h a v e a Q  =  P cosY  a  ± 2  =(pyJ2.]  s i  .riY.  (MB)  T h e r e f o r e , t h e t h r e e moments o f i n e r t i a ti 1  1 TT  1  4 B  3  (  = MB P  '  3  = 4B P  Jl S 531117 C O s Y  TT p  TT  p  fl C  TT  i  +  C O S  Tt  (sin7 - i |  p 11  O 7  ^  TT  p TT  ^  s i n 7 cos7 + | c o s 7 )  (46)  TT  TTO  sln7  3  and t h e r o t a t i o n a l  energy o p e r a t o r ,  i n § 2 , t h e symmetry c o n d i t i o n o f t h e  • b) As m e n t i o n e d coefficients A 2)  TT  ~~2  Sin7  3  2  ti  1  2 "  Tt O =  become  1  K  s i n o c t u p o l e case i s d e r i v e d  o f e q u a t i o n (30).  from t h e case  H e n c e , f r o m e q u a t i o n s (28),  (29a),  (29b)  a g a i n , K must be e v e n and A  -K  =  (-V  L+1  A  K  ™  The  forms o f t h e p o t e n t i a l  are  t h e same a s i n e q u a t i o n s (19) ^  t?  ,  V = \  -  3  \ib The  „"  C (P  2B  3  and v i b r a t i o n a l  energy o p e r a t o r s  and ( 2 l ) , i . e .  p  (49)  P )^ Q  p  "3 ^  methods o f c a l c u l a t i o n  1  P  ( S 0 )  following  t h i s are exactly the  same as i n §2 . Finally, and  i t s h o u l d be n o t e d t h a t t h e shape p a r a m e t e r 7 "  t h e s t i f f n e s s p a r a m e t e r y."  general different as t h e s e t w o ' c a s e s  =  i n v a l u e s from  ^ T ^ P ^ B ^ Q ^ J  7 and  -1  are. i n  . i n t h e q u a d r u p o l e .case,  are quite unrelated. Also the p a r i t i e s of  t h e q u a d r u p o l e and o c t u p o l e negative  £4.  cases  are necessarily  respectively.  o f Reduced E 2 T r a n s i t i o n  Calculations  S i n c e we a r e c o n s i d e r i n g d e f o r m e d large to  Probabilities.  even-even  q u a d r u p o l e moments a r e t o be e x p e c t e d .  very favourable  electric  quadrupole  (E2)  This  calculations  when compared w i t h e x p e r i m e n t a l  a sensitive The  general  probability  test  of the v a l i d i t y  =  lead  of these'  data  of this  should model.  e x p r e s s i o n f o r t h e gamma t r a n s i t i o n  SvU+l) iJ{{2i+Y))\]H  where t h e e m i t t e d X j the angular  .  2J+1 /«\ l /  photon has energy  f  respectively;  e n e r g y i n d e p e n d e n t ".Is c a l l e d E2 transitions  BCf L.-*LJ-  and a n g u l a r and f i n a l  and B(j ,1, ->> L ) w h i c h I s f  t h e reduced  transition 2  probability from a  jL'M'N'n > , 1  B(E2;L,MNn-> L M N ' n ' ) ' = | ^ L M N n ' | Q ^ j L M N n > | T  momentum  l e v e l s are  due t o t h e q u a d r u p o l e moment Q ^  |LMNn> t o a s t a t e  (51)  f  c  momentum o f t h e i n i t i a l  g i v e n b y L.. and L  state  should  per unit-time i s  TU) i-if  For  nuclei,  transitions  b e t w e e n t h e v a r i o u s e n e r g y l e v e l s . The r e s u l t s  provide  p o s i t i v e and  T  t  f  ,  ,  2  2  Summing o v e r the  initial  a l l final states  states  w i t h r e s p e c t t o M' and a v e r a g i n g  w i t h r e s p e c t t o M, i n o r d e r t o o m i t  polarisation, B ( E 2 ; L N n —> L'N'n') = _ _ L ^ > L + J  »,A1  j/L M'N <n.' | Q T  /  | LMNn> |  2  2  (52)  -20-  The  electric  system fixed as  quadrupole  (S-system), system  moment  Q  s h o u l d be i n t h e l a b o r a t o r y  2[7  b u t i ttakes a s i m p l e r form  (S'-system),  i n the body-  the t r a n s f o r m a t i o n from  S  1  t o S being  usual •Q  2 p  D * , (a)  =  (53)  ( 3 ) ) « Nov/ a s s u m i n g t h a t t h e n u c l e a r  (c.f» e q u a t i o n  d e n s i t y i s a constant i n the nucleus, then  the quadrupole  and  from  Prom  (6)  )  Zy  0  (l)).  o f a£y(t) t h a n  L  5 ^ 1 Y ,(Q • , 0 •) s i n 0 ' d 9 ' d#'  where R ' ( 9 ' , ^ ' , t ) = R [_l+ ^ a ^ ( t ) Y equation  ze/( \rcR^ /')),  moment i n t h e S ' - s y s t e m i s j u s t  = constant J  (c.f.  i.e. ^ =  charge  2 V  (6',0»)  Therefore, n e g l e c t i n g higher order  the f i r s t ,  terms  v/e o b t a i n  r  _3_  (5^)  cos 0 . sin ^ 2  To fori>=  (55)  1j ±1 ±2  (53) therefore, ^  =  Subtituting  R  into  B(E2:  o ?[ ^ o  c o s  V +  +D*_ ). 2  s i n /z] X  ( 5 2 ) , v/e h a v e LNn — > L ' N n ) 1  ,2.2  mt^L,,/o  ^ / t « . ( f ) ! p | c P  M  ( p ) > <  I'M'"'!  (57) where t h e l a s t  two f a c t o r s r e p r e s e n t t h e v i b r a t i o n a l and t h e  y  -21rotational parts respectively. (31a)  Prom e q u a t i o n s | BW>.  C™  +  g  we have  3  C-1)  O f ( D ^  +  L D  £  K  >//?  Hence t h e r o t a t i o n a l p a r t c a n be c a l c u l a t e d . in  The e x p r e s s i o n  terms o f C l e b s c h - G o r d a n c o e f f i c i e n t s ^ ( j j j g j ^  v/hen p r o p e r l y n o r m a l i s e d  i sas f o l l o w s  ( V)| 1MN^> =  C (L2L •; M^M )  < L 'M ' N 1 1  |[ o' C  o  N , C  i [ ( l + (-1) ') C^' C^ 1  N I  N  > m-^rr^m^)  J^-±.1.  1  C(L2L';000) +  N  (58)  C ( L 2 L ' ,-KOK)] c o s V  fcc^'c™  +  C ( L 2 L ' : 2 ~ 2 , 0 ) + (l+(-l) )c£ c£ ' 6(L2L;02« L  N  W  S  +/2>~\ O ^ J C ^ C ( L 2 L ' ; K , 2 , K + 2 ) 4 - / 2 T^2 ^ '  ^ 5 C^ C(L2L«;K-2,K-2 j l s i n J Ife4 " (59) N  k  k  2  K  J  t h e v i b r a t i o n a l p a r t , we must e v a l u a t e ^ ) , , \p\$ S) = x. (IS) S^/ i n t h e n o t a t i o n o f D a v i d s o n R e c a l l i n g e q u a t i o n s (32), For  n  L  D  p  (37),  ( 4 0 ) and ( 4 1 ) , we c a n w r i t e ^ L N W  "3/2 J/  -P~ \(^y)  and  , (/>)  <J>  =  /T  3 /  %, ( ^ ' )  where y = Z-^/3/u.j^Z - 1 ) and y ' = Z «(/ ^|iZ» - l ) . 3  1  Thus, ^ ^ ^ / ^ ( M - Z - J and  ( p  L  ,  with x ^  Therefore,  N  ,  =p-3/2  Dj|/  [/2( x  ^ - ^ ^ ( ^ z ^  (61)  Z*)]  (62)  Z^/l^Z'  the matrix  (60)  element  <C(P  L  I I W  ((3)|pj ^  L N  (|S)^= (63)  where (h dfi i s t h e v o l u m e e l e m e n t , and t h e f u n c t i o n 2  - nd  =f\[w*  *  -  The n o r m a l i z i n g c o n s t a n t s 2  2  Thus, N ,  =  = J  N,., and N  are defined  v  from  2  (65) (66)  D2 (/2y)dy p  explicit  =  2  ( Z i ^ / Z ' Z ) ^ ^ ! ) ^ ^  (jupo) (Z!/z)(Z'/zi)3  = The  °  ^ Z / Z i ) / D (/2y)dy = 1  Hence ^' ' K^L'N'CPJIH^LN^))!  out  ( 6 i  = ( Z p ^ Z I J  2  with I„  d x  1  •Jo  N J D , (/2y)d^  ^  z  i  2  expressions f o r 1 ^  b y D a v i d s o n a s shown b e l o w and  and  2  I^  (67)  ^ were  worked  programmed  •to  = [z^ '}JK  /exp{-i(x +  +v  ~'  X  z  where R  R^x+Z^-Z-^  2  2/;  2  (x+Z-^XyCx). (68)  ( R ( x + Z ) - Z - [ ) dx z  1  = Z Z'/ZZ-[ ,  z  1  0 0  a n d  and  I  V  () X U;  X  ( Z  x  p  1  )  where and  I  V E  I  =  r  ^e  =  (  +  _i , i  X )  =  transition  (  1  _  L  ;  j(k-iQ(2x)  k  )  *yS-Ici+l) i J f [ ( 2 ^ ) - H C l + ^ 2 ) ] / ^ R - i > ) +  Zl^TC  y\ - Z c a s e .  ^  H  1  _ f , 2^t°±.^ expj-ZjC^Txr^)  returning  probabilities  i s only applicable  the  i  k  (69)  H(x) - d / d x ( l n R x ) )  Thus,  ground  r  (-D  t o E2  7  to equation  2  v , 2  X  (57)»  (Z^-XQ-^J-  t h e r a t i o o f two  c a n be c o m p u t e d . transitions,  Of c o u r s e ,  this  c o r r e s p o n d i n g to the  F o r t h e /\ = 3 c a s e , d i s c u s s i o n  i s omitted  on t h e  t h a t no t r a n s i t i o n b e t w e e n 7\-- 3 s t a t e s i s o b s e r v e d i n •  present experiment.  55-  O t h e r Forms o f  Two can  Electromagnetic  possible  forms of  Transitions.  electromagnetic  compete w i t h E 2 t r a n s i t i o n a r e  the  Ml  transitions  and  EO.  that  I t was  shown  10) by  Davydov  symmetric  et  al  that  Ml  t r a n s i t i o n was  ( i . e . when y = O ) J but  rotator  forbidden not  i n an  in a  asymmetric  42)-44) rotator. to  be  The  incorrect,  o n l y EO  EO  i s both  the  out  by  cases are  be  EO  L.  Since  no  Tamura  al  Therefore,  considered. b e t w e e n two  photon c a r r i e s  an  states  of  a n g u l a r momentum  t r a n s i t i o n o n l y works t h r o u g h the  c o n v e r s i o n p r o c e s s . The  et  forbidden.  t r a n s i t i o n i s only possible  same s p i n  zero,  that  then pointed  t r a n s i t i o n needs t o An  the  l a t t e r was  internal  electron  absolute t r a n s i t i o n probability  per  45) unit  t i m e was  g i v e n by C h u r c h e t T(E0) = J l ^  where fl i s t h e and  ^  the  in this  electronic  nuclear  case,  and  factor  factor. hence i l  al  (70) involved  Only the was  as  i n the  K-conversion  plotted  against  i s just  the  transition is  the  important transition  K  e n e r g y by  the  above a u t h o r s . ^  matrix .  corresponding (no  •_ *  the  r o t a t i o n ) . For  first (n=  to  excited  EO  state,  l ) , (see  ref. •  Only c o n s i d e r  f  vibrational  part  of  the  t r a n s i t i o n from a beta n=  2)  to  a ground  element •  *  E 2 transition band  rotational  (i.e.  the  state  16) •^LNn'C'P)|P lg'  transitions  2  LNo  (P)>  from beta  %  vibrational  Ql) band.  -2.4wliere z = a t o m i c Thus  f  number.  (3V4 r^(p ,, , ,(p)^^  =  r  L  o  •and  ( 3  2  T  n  A r c ) N  A  ^ [ | 2 ( ^  -  ^ =  (3zArc)^p Z/Z )^yi  or  j>  integral  ¥  )] 2 „[/2(M_ f  D  -  Z^dp  obtain X  2  D^(X-Z )]  dx  x  2  ?  2  )  D^Cx-Z-^Jdx  2  1  r- 2^+^')J  (  1^(0)  = /Dyj/2(x-Z )J x =  3  .C^Z/Z^ Y ^ M O )  2  3  The o v e r l a p  Z i  /D^^X-Z-L))  1  O  ( zArc)  / 3  w h i c h l e d t o e q u a t i o n (67)  b u t w i t h Z = Z', and Z-j_ = Z-£, we  =  (^)  oo  w i t h t h e same s u b s t i t u t i o n s  2  L N n  T C  jf(  F u n c t i o n s 1^(0)  x + Z  )2 2( ) x  x  (c.f.equation  d x  and I ( 0 ) a r e g i v e n  (68).  i n (69).  v  The m o n o p o l e t r a n s i t i o n p r o b a b i l i t y T(.EO) i s u s u a l l y compared w i t h t h e E2 t r a n s i t i o n p r o b a b i l i t y T ( E 2 ; 0 •(- -»2 + ).  T  (  E  2  )  From  f r o m t h e same  (51)  8rc. 3 VK<o~ ^ B ( E 2 j 0+ - » 2 + ) = 2(5oa)^ = —^TT—c— 75K c5  %l  6  r  B ( E 2 ; 012 \\\  Oil) \\\  LNn where E j , i s t h e e n e r g y o f t h e e m i t t e d and  w i t h e q u a t i o n s (57), (67), and T i l ) rr^2)  =  state,  M x T ^ W " ? 2.53xl0^V3  E r  L'Nh'  photon.  W i t h E ^ i n Mev  (72), we c a n w r i t e  (z/z )5 z /z'P_i >( ) i ^ ( Q ) 2  1  5  (  i  y  I  w  ( 0 ) l 2  v  (  2  )  P  rot (78)  v/here P - t i s t h e r o t a t i o n a l r 0  computed by (59).  part of the t r a n s i t i o n  probability  S i n c e EO t r a n s i t i o n s states  o f t h e same s p i n  be a l t e r e d  i  l  o f the f i r s t  the ground s t a t e  transition  b  '  J  *  equation  (78) may e a s i l y  F o r example, f o r t r a n s i t i o n s  b e t a band and s i m i l a r s p i n  b a n d , we c a n w r i t e  states,  12,(2)  A  P  r o f c  (16)). several  (78a) transitions  3) t o p o s i t i v e  o b s e r v e d , no t h e o r e t i c a l  states  f o r the r a t i o o f the  2.53xl09 V3Ef  b  Although  b e e n worked  o c c u r b e t w e e n a n y two  p r o b a b i l i t i e s b e W e e n t h e same  (see r e f e r e n c e  levels  and p a r i t y ,  to apply t o these.  between s t a t e s of  may a l s o  parity  from t h e n e g a t i v e  levels  (?\ = 2) have  parity been  t r a n s i t i o n p r o b a b i l i t i e s have s o f a r  out to account f o r these.  -26CHAPTER I I I  THE  In nuclear are  DESIGN OF  EXPERIMENTS  s p e c t r o s c o p y , the  commonly i n v o l v e d  are:  to  types of problems  construct  the  nuclear  scheme o f a r a d i o a c t i v e  isotope,  o f t r a n s i t i o n s f r o m one  nuclear state to another,  consequently t o understand the The be  energy l e v e l s i n the  n u c l e u s by  of the  The  physical  are  u s u a l l y measured, are  emitted  energy t r a n s f e r  only  model  the the  determined  k i n e t i c e n e r g i e s of t r a n s i t i o n s , the  particles either i n polarised  conditions,  the  between a p a i r of  time c o r r e l a t i o n or  emitted  p a r t i c l e s . The  positrons),  neutrinos  of the the  from I t s ground  d a u g h t e r n u c l e u s , Gd  d e c a y , an  e x c i t a t l o n of  electron  and  each e x c i t e d  the  I n t e n s i t i e s of or  unpolarised  (electrons  and  photons. .  I n t h e ' p r e s e n t e x p e r i m e n t , we 15^  states.  particles referred  alpha-particles, beta-particles  d e c a y o f Eu  by.making  directional correlation  to here are  and  cannot  i n a t r a n s i t i o n , which  e m i s s i o n of the and  and  fitting.  i n t r a n s i t i o n s between  observables involved  p a r t i c l e s during  be  decay  mechanisms  d e c a y scheme o f a n u c l e u s  measured d i r e c t l y . They can  use  to determine  that  15^.  are  s t a t e t o the  excited  the  states  D u r i n g each t r a n s i t i o n i n  a neutrino state  concerned w i t h  so  are  formed  e m i t t e d . The may  de-  result, i n  the  emission o f a photon, or an e l e c t r o n through conversion process pair  (see ^2, Chapter  i f t h e energy t r a n s f e r  6),  an i n t e r n a l  or apositron-electron  i s g r e a t e r t h a n . 1.022 M.ev. S i n c e  the n e u t r i n o c r o s s - s e c t i o n of nucleons i s next  the only measurable p a r t i c l e s using c o n v e n t i o n a l are t h e betas The  and t h e gammas  cm ) ,  techniques  (photons).  m e t h o d s f o r i n v e s t i g a t i n g t h e b e t a and gamma  t r a n s i t i o n s v a r y f r o m one e x p e r i m e n t ing  (~10  t o zero  factors f o r using the different  the nature  t o another.  The d e t e r m i n -  methods l i e p r i m a r i l y i n  o f t h e t r a n s i t i o n s o f t h e chosen i s o t o p e  energies of the emitted further examination a t i o n s such  as e x i s t  particles),  the specific  (e.g. t h e  need f o r  o f t h e i s o t o p e , and t h e t e c h n i c a l  limit-  o n t h e e f f i c i e n c i e s and r e s o l u t i o n s o f  t h e d e t e c t o r s a v a i l a b l e and o n t h e speed o f t h e e l e c t r o n i c system. Strong  e m p h a s i s h a s t o be p l a c e d a t t h e l a s t  because s p e c t r o s c o p i c work r e l i e s  point,  e s s e n t i a l l y on t h e p r e c i s i o n  o f measurement. As has  s t a t e d i n t h e I n t r o d u c t i o n , t h e d e c a y o f E u 154  been i n v e s t i g a t e d by s e v e r a l w o r k e r s ,  and c o n s e q u e n t l y  a  r o u g h s k e l e t o n o f t h e d e c a y scheme has. b e e n e s t a b l i s h e d f o r some t i m e . The p r o b l e m s l e f t l ) t o determine  over  now a r e :  t h e e n e r g i e s and i n t e n s i t i e s  of a l l  t h e t r a n s i t i o n s a c c u r a t e l y enough s o a s to,make. a- m e a n i n g f u l i d e n t i f i c a t i o n w i t h an a p p r o p r i a t e t h e o r e t i c a l  model.  -282) t o s e a r c h f o r new weak t r a n s i t i o n s t h a t h a v e n o t b e e n f o u n d b e f o r e , a n d , i f a n y , t o a s s i g n them t o t h e i r proper  p o s i t i o n s i n t h e d e c a y scheme; and 3)  t o check those  are disagreementsor  p a r t s o f t h e decay i n which t h e r e  uncertainties i nearlier  investigations.  W i t h t h e above c o n s i d e r a t i o n s i n m i n d , t h r e e o f i n v e s t i g a t i o n were employed f o r t h e p r e s e n t gamma s i n g l e s s p e c t r o s c o p y , and  beta  spectroscopy.  following three  work, t h e  gamma-gamma c o i n c i d e n c e  They a r e s e p a r a t e l y d i s c u s s e d  chapters.  methods  work, i n the  CHAPTER  THE §1.  General  It  IV  GAMMA SPECTROSCOPY  Considerations.  was  mentioned  i n the previous chapter t h a t the  e x c i t a t i o n o f an e x c i t e d s t a t e o f Gd be  carried  "out by t h e d i r e c t  The  this  t o a lower  internal pair  o t h e r two  The  processes  i n Chapter  will  be  considered  d e - e x c i t a t i o n corresponds  be t r e a t e d .  along with  beta  to the  i n the process  sudden a l t e r a t i o n o f  d i s t r i b u t i o n of the nucleus  i n order  acquire a lower p o t e n t i a l energy. In g e n e r a l , t h i s e q u i v a l e n t t o s u d d e n s w i t c h i n g on and  of electromagnetic  an  6.  change o f s t a t e s i n a n u c l e u s  charge-current  may  production.  c h a p t e r , o n l y t h e d i r e c t gamma e m i s s i o n w i l l  transitions  is  state  e m i s s i o n o f a p h o t o n , o r by  i n t e r n a l c o n v e r s i o n p r o c e s s , o r by In  154  de-  oscillators,  and  thus  o f f of a sending  of  the  to alteration combination  out  a  series  o f e l e c t r o m a g n e t i c waves w i t h v a r i o u s ; . m u l t i p o l e o r d e r s . T h e s e waves o r r a d i a t i o n s c a n be electric The  separated  m u l t i p o l e s (EL) and  i n t o two  groups,  the  t h e m a g n e t i c m u l t i p o l e s (ML) .'"  m u l t i p o l e o r d e r L i s j u s t t h e a n g u l a r momentum i n u n i t s K  carried  away by e a c h quantum o f r a d i a t i o n  p a r i t y o f an e l e c t r i c  (a p h o t o n ) .  m u l t i p o l e r a d i a t i o n , . i s •defined  The. as  E L V -('*) . F r o m M a x w e l l ' s V(r) E(r)=  - -^-H(r)  x  and and  equation,  (79)  V(-r)xE(-r)= - £ H(-r), since  Y(-r) = --V(r)  L, t h e e l e c t r i c  field  / i t implies that  and t h e m a g n e t i c f i e l d  f o r t h e same  are of  p a r i t y . Hence, t h e p a r i t y o f a magnetic m u l t i p o l e is  7T  =  M  Not  '(-1)  restricted  i n each t r a n s i t i o n . T h e i r  r a d i a t i o n s are  presence i s  by a few s e l e c t i o n r u l e s a s shown b e l o w :  1. I n o r d e r  t o c o n s e r v e a n g u l a r momentum, t h e f o l l o w i n g  t r i a n g u l a r r u l e must |ii -  hold,  I |^ L <  and f i n a l  (80)  I ]  f  w h e r e 1^, I f a r e t h e t o t a l initial  radiation  L  a l l t h e components o f t h e m u l t i p o l e  equally preferred  opposite  f  a n g u l a r momenta i n u n i t s H o f t h e  l e v e l s i n the t r a n s i t i o n r e s p e c t i v e l y .  2. C o n s e r v a t i o n  of p a r i t y during  the t r a n s i t i o n  must  be o b s e r v e d . 3.  From t h e e x p r e s s i o n  with multipole  order  for transition probabilities  L ( c . f . equations  (5l)  C h a p t e r 2 ) , t h e f o l l o w i n g f a c t o r c a n be Ui  I. c  2L  R" L  V  2  0  w h e r e / V i s t h e wave l e n g t h the  r a t i o R /A D  i s very  and  (56) i n  extracted,  -, 2 L  of the emitted  (81) r a d i a t i o n . Since  s m a l l , e v e n f o r s e v e r a l Mev  transfer  of energy, only t h e term w i t h lowest  order  of L i s significant  i n each t r a n s i t i o n . 4. A g a i n probability,  T  i-f  ( L ) becomes i n f i n i t e , w h i c h i s i m p o s s i b l e ,  H e n c e , no e l e c t r o m a g n e t i c multipole order.  (51), when L = 0, t h e t r a n s i t i o n  from e q u a t i o n  radiation  c a n be e x p e c t e d  with  zero  ^ '  5. 'For t h e same L , i t c a n be shown t h a t t h e m a g n e t i c radiation  i s weaker t h a n t h e e l e c t r i c  radiation  b y a' f a c t o r o f  (v/c) ( s e e r e f .47 p.592), w h e r e v i s t h e speed o f m o t i o n o f t h e charges i n t h e nucleus. In c o l l e c t i v e models, another rotational  selection  m o d e l , s i n c e K i s n o t a good  quantum number, t h i s K - f o r b i d d e n n e s s but does not f o r b i d it.The h i n d r a n c e K  According  i ~  (82)  f  However, i n asymmetric r o t a t o r  K  only hinders the t r a n s i t i o n  i s expressed a s , (83)  - L  f  t o t h e asymmetric r o t a t o r  model d e s c r i b e d i n  C h a p t e r 2, t h e p r e d o m i n a n t m u l t i p o l e r a d i a t i o n s o f Gd 154 a r e a g a i n  t h e above s e l e c t i o n types  due t o  quantum number K i s a d d e d , l , e . L > \K - K ±  transitions  rule  rules.  limited  Experimental  of multipole radiation  i n t h e gamma  t o E2, c o n s i s t e n t w i t h determination of the  r e q u i r e s t h e knowledge o f i n t e r n a l  c o n v e r s i o n d a t a , and h e n c e w i l l  be t r e a t e d i n C h a p t e r  6.  Meanwhile, Ing  §2.  we w i l l  proceed t o d e a l  t h e e n e r g i e s and i n t e n s i t i e s  w i t h t h e means o f measure.  o f t h e s e gamma r a y s .  I n t e r a c t i o n b e t w e e n Gamma R a y s and t h e D e t e c t o r s .  When a p h o t o n e n t e r s a d e t e c t o r ,  several things can  h a p p e n . The p h o t o n e n e r g y may be c o m p l e t e l y a b s o r b e d b y a n e l e c t r o n i n t h e d e t e c t o r . T h i s e l e c t r o n t h e n moves a b o u t detector producing i o n p a i r s o r luminescence u n t i l r e s t w i t h i t s energy  i t comes t o  s p e n t . The i o n p a i r s i f c o l l e c t e d  pair of electrodes w i l l  i n the  by a  g i v e a c h a r g e p u l s e whose m a g n i t u d e i s  l i n e a r l y p r o p o r t i o n a l t o t h e e l e c t r o n energy. S i m i l a r l y , the luminescence i f c o l l e c t e d  by a p h o t o m u l t i p l i e r w i l l g i v e a  pulse proportional t o the luminiscent  energy. T h i s process o f  complete energy t r a n s f e r from a photon t o a d e t e c t o r i s c a l l e d p h o t o - a b s o r p t i o n . The p e a k s  i n an energy  spectrum obtained  f r o m t h i s p r o c e s s a r e t h u s named p h o t o - a b s o r p t i o n p e a k s . I f t h e gamma r a y s a r e m o n o - e n e r g e t i c , a s i n g l e p e a k s h o u l d be observed which i s i d e a l l y G a u s s i a n i n shape. On t h e o t h e r h a n d , i f a p h o t o n i n t e r a c t s w i t h a n e l e c t r o n and  i sitself  scattered  out o f t h e d e t e c t o r , then t h e energy  of  t h e photon absorbed by t h e e l e c t r o n  of  this partial  energy  (i.e.  i s p a r t i a l . The amount  the electron r e c o i l  energy, E ) e  43)  v a r i e s w i t h t h e s c a t t e r i n g angle 9 as g i v e n i n t h e e q u a t i o n  ,  -33E (e)  = E [ i - 1/(1 + l l ( l - c o s e ) ) ]  e  (84)  mc2  where E ^ i s t h e i n i t i a l photon energy and 9 i s d e f i n e d as shown below.  -*  •Ey/ The  recoil  (scattered  photon)  e l e c t r o n t h e n m e e t s t h e same f a t e a s t h a t  i n the previous paragraph. This process of p a r t i a l energy i s c a l l e d  transfer of  Compton s c a t t e r i n g . S i n c e 0 c a n be a n y v a l u e  b e t w e e n 0 ° and l 8 0 ° , mono-energetic  described  t h e energy spectrum thus o b t a i n e d f o r  i n c i d e n t gamma r a y s i s known a s t h e Compton  continuum. T h i s continuum extends from  E  e  ( 0 ° ) - O  to E (l80°). e  H e n c e , t h e Compton edge ( u p p e r edge o f t h e c o n t i n u u m ) c o r r e s p o n d s to  t h e case i n which t h e photons  are back-scattered.  Another p r o c e s s which has been mentioned  before i s  p a i r p r o d u c t i o n . When a p h o t o n w i t h e n e r g y g r e a t e r t h a n two e l e c t r o n mass e n t e r s t h e E M - f i e l d o f a n u c l e u s i n t h e d e t e c t o r , the  e n e r g y may be c o n v e r t e d i n t o m a t t e r b y l i f t i n g  from a n e g a t i v e energy s t a t e t o a p o s i t i v e  an e l e c t r o n  e n e r g y s t a t e and  l e a v i n g a 'hole' o r p o s i t r o n b e h i n d . Thus an e l e c t r o n - p o s i t r o n pair  i s c r e a t e d . The p o s i t r o n i s o f t e n s t o p p e d and a n n i h i l a t e d  w i t h an e l e c t r o n i n t h e d e t e c t o r y i e l d i n g of  a p a i r o f photons  e n e r g y 0 . 5 1 1 Mev e a c h . T h e r e f o r e , a d o u b l e - e s c a p e  energy,  peak o f  -34E-  =  c a n be o b t a i n e d ,  ( E  y  -  i f both  1.022)  photons escape from t h e d e t e c t o r .  H o w e v e r , i f o n e o f them i s r e - a b s o r b e d single-escape  ( E- S 6  i s formed  (86)  Mev  \  The  use  0.511)  *  Gamma-ray D e t e c t i o n  intended  i n the detector, a  peak o f e n e r g y ,  E  §3.  (85)  Mev  first  gamma-ray d e t e c t i o n a s s e m b l y c o n s t r u c t e d and  f o r t h e present  of the principle  from t h e source  Assemblies.  1 . I t makes  w o r k i s shown i n f i g u r e  o f Compton b a c k - s c a t t e r i n g . Gamma r a y s  S were c o l l i m a t e d by t h e c y l i n d r i c a l  b l o c k s A and B, and t r a v e l l e d  through  lead  t h e aluminium tubing t o  a L i t h i u m - d r i f t e d S i l i c o n d e t e c t o r which has a d e p l e t i o n depth o f 3 mm. and a n a c t i v e v o l u m e o f 0.15,. c . c . were stopped of p l a s t i c The  b y t h e d e t e c t o r and b a c k - s c a t t e r e d  scintillator  plastic  to the block  m o l d e d i n t h e shape o f a c y l i n d e r .  S i l i c o n d e t e c t o r was c o o l e d  DeWar o f l i q u i d The  Some o f t h e p h o t o n s  by a copper brush  n i t r o g e n so as to. a c h i e v e  optimum  dipped  resolution.  s c i n t i l l a t o r was v i e w e d b y f o u r p h o t o m u l t i p l i e r s  w h i c h were a d j u s t e d  to yield  f a c t o r . The o u t p u t s  o f t h e p h o t o m u l t i p l i e r s were t i e d  cathode f o l l o w e r stage defined  i na  t h e same.charge  multiplication t o one  a s shown I n f i g u r e 2 . E a c h event, was  by t h e c o i n c i d e n c e  between a p u l s e . f r o m  the detector.  -36-  li-D  Si  Detector Lov;-noise Preamp.  - Plastic Scintillator Photomultiplier i  o  9  9  9  Cathode Follower  0  Cathode Follower  Driver 21  \2h  Driver B.H.T. -1100  v.  Fast  Variable Delay  Coin.  gate Kick  Fig.2  Sorter  Electronic circuit  innut  f o r t h e gamma  Low-noise Amplifier  spectrometer.  -37and  •  a p u l s e f r o m t h e p h o t o m u l t i p l i e r s . The o u t p u t f r o m t h e  detector corresponds t o t h e r e c o i l  energies of the electrons  i n t h e d e p l e t i o n r e g i o n . T h i s o u t p u t was s t o r e d  i n a .128-  c h a n n e l k i c k s o r t e r a f t e r b e i n g gated by t h e c o i n c i d e n c e p u l s e s . T h e r e f o r e , each count r e c o r d e d by t h e k i c k  sorter  corresponds t o a back-scattering event, the•back-scattering o  angle being l i m i t e d  t o a range  _  o  1 5 7 - 1 8 0 by t h e cone i n t h e l e a d  b l o c k C. A s p e c t r u m o f Cs 1 3 4 was t h e n t a k e n a s shown i n f i g u r e 3 . Prom § 2 , t h e peak p o s i t i o n s o f t h e s p e c t r u m a r e a l m o s t a t t h e p o s i t i o n s o f . t h e o r i g i n a l Compton e d g e s . I t c a n be s e e n from t h e spectrum that t h e r e s t highly  o f t h e Compton c o n t i n u u m i s  suppressed by t h e c o i n c i d e n c e system, which i s t h e  b e a u t y o f t h i s a s s e m b l y . The r e s o l u t i o n a s q u o t e d o n t h e f i g u r e i s f a i r l y good. I t i s achieved because i n t h e range o f backs c a t t e r i n g a n g l e s a c c e p t e d , t h e energy o f s c a t t e r e d  electron  is  drawback  almost independent".of s c a t t e r i n g  i n t h i s assembly  a n g l e . The o n l y  i s low e f f i c i e n c y .  By t h e t i m e t h e a b o v e a s s e m b l y was t e s t e d , drifted  Germanium d e t e c t o r s w e r e a v a i l a b l e  i nthis  Lithiumlaboratory.  S i n c e t h e l a t t e r f a r e x c e e d t h e f o r m e r i n r e s o l u t i o n and e f f i c i e n c y , t h e gamma e x p e r i m e n t s were t h e n p e r f o r m e d  by u s i n g  t h e Germanium d e t e c t o r s i n s t e a d . H o w e v e r , i t i s i n t e r e s t i n g t o n o t e t h a t t h e above a s s e m b l y  p r o v i d e s a method o f gamma  -3820  605 Kev  18  Cs 134 FWHM = 11.6 Kev (e  |6  recoil  energy)  If  Kev 803 Kev 797  a  10  CO  o H  O  o  563 Kev 569 Kev  / /  j I" 2%  38  62  28  101  Channel Number  Fig.  3  Cs 1 3 4 . s p e c t r u m assembly.  t a k e n from t h e gamma-spectrometer  -39-  d e t e c t i o n b a s e d on The Two  a quite different  Germanium d e t e c t o r  planar-type  detectors  h a v e b e e n u s e d one  d e p l e t i o n d e p t h o f 5 mm. the  s e c o n d one  7.5.mm. and  assembly  prepared  a f t e r the  case,  i s much b i g g e r , h a v i n g  a c t i v e volume, 5 c.c.  d e t e c t o r b i a s had  t o be  d e t e c t o r r e s t i n g on t h e from the  latter  the L i surface) a field a 500 the high  stage  input  with other stages(e.g. o u t p u t was t i m e and  as  by  (2N3823)  low-noise by  depthtof:..  P surface  electrically  and  This  p r e a m p l i f i e r u s e d . The the  field  the  insulated  through  presents super-  r a t i o when compared cathode f o l l o w e r  input  preamplifier  with  1 ^sec.  then to a low-noise  a 1024-channel k i c k s o r t e r . Using  of  effect transistor  c o n n e c t e d t o a T i m e C o n s t a n t Box and  of  the  N surface ( i . e .  figure  S i m t e c P - 1 0 ) . The  10 f - s e c . d e c a y t i m e ,  used,  a feedback lead  the peak-to-rnoise  and  was  w i t h the gate t e r m i n a l  preamplifiers using  101XL  while  a cold f i n g e r . In  Thus t h e  shown i n f i g u r e 5 .  increased  c.c.,  a  u s u a l , each d e t e c t o r  cooled  contact  Dalby  d e t e c t o r has  a depletion  s h e e t o f n y l o n . The  irnpedence p r o v i d e d  Ortec  first  c o l d f i n g e r was  i n spot  of the  low-noise  As  negative.  a thin  was  M A resistor  significantly  and  by  mounted by D.A.  s p e c i a l preamplifier c i r c u i t  effect transistor  first  and  4.  shown i n f i g u r e  a c t i v e volume, 1.5  and  because of the  is  o t h e r . The  mounted i n a vacuum chamber and this  principle.  rise  amplifier  a d e t e c t o r b i a s of  -600  v.  -40-  Dewar  r  R o u g h purap  Id Liquid  Nitrogen'  Supports  Li-d^ifted Detector  I o n pump  Cold  Fig.  4  finger  The G e r m a n i u m d e t e c t o r  assembly.  Ge Al Window  + 24  +12  v  v  O.lHf -24  Fig.5  Input  v  stage of the low-noise  connected  t o t h e Germanium  preamplifier detector.  for  the f i r s t  a b o u t 1.3 for  the  Mev  It  of energy  second  same e n e r g y  o b t a i n e d a r e s o l u t i o n o f 4 Kev  d e t e c t o r , we  -1000  or below; w h i l e w i t h a b i a s of  d e t e c t o r , t h e r e s o l u t i o n was  range. These are I l l u s t r a t e d  i s a l s o c l e a r that the  second  3.5  Kev  at v.  i n the  i n . f i g u r e s 6 and  7. .  detector having a larger  active  v o l u m e g a v e a much b e t t e r r a t i o o f p h o t o - a b s o r p t i o n p e a k t o Compton c o n t i n u u m .  %4.  S o u r c e P r e p a r a t i o n and  The  Eu 1 5 4  Mounting.  s o u r c e was  Europium i n oxide form, which The  o x i d e was  Ridge  about 5 m i l l i c u r i e s  o f Eu  154  by n e u t r o n - c a p t u r e  a m i n i a t u r e beaker  to  and  a few  drops  evaporated  remove t h e e x c e s s  to dryness  a c i d , and  dissolved  in distilled  water.  dissolved  in distilled  water  soluble  of pure  yield  process. A  introduced  solution.  under a t u n g s t e n  when, c o o l t h e c h l o r i d e I t was  into  concentrated  a c i d w e r e added t o o b t a i n a c h l o r i d e  s o l u t i o n was  The  s o u r c e was  re-evaporated  lamp was  and  then  again.  purpose of changing  the  153-  for 7  irradiated  National Laboratory reactor to  s m a l l q u a n t i t y of the i r r a d i a t e d  The  e n r i c h e d t o 9 8 . 7 6 $ o f Eu  was  s e a l e d i n a q u a r t z c a p s u l e and  d a y s i n t h e Oak  hydrochloric  from 2 m i l l i g r a m s of  prepared  i n s o l u b l e - oxide into  choride i s mainly f o r preparing the beta  sources.  the  40 h  C h a n n e l Number Fig.6  Co 60 gamma s p e c t r u m o b t a i n e d : f r o m t h e 1.5  c . c . Ge  detector.  Fig. 7  Co 60 gamma s p e c t r u m o b t a i n e d f r o m t h e S e c .  Ge  Detector.  -45-' But  i t i s a l s o c o n v e n i e n t t o p r e p a r e gamma s o u r c e s f r o m  form, because t h e r a d i o a c t i v e  s t r e n g t h c a n be e a s i l y  liquid  controlled  i n t h i s way.' A drop o f t h e l i q u i d a circular disc  was d r i e d  o n a n IBM c a r d c u t i n t o  o f 1 i n c h d i a m e t e r w i t h t h e drop a t t h e c e n t r e .  The s o u r c e was t h e n s e a l e d w i t h s c o t c h t a p e and mounted o n a l u c i t e r i n g a s shown. S e v e r a l s o u r c e s were p r e p a r e d w i t h v a r i o u s s t r e n g t h s . The o t h e r s o u r c e s u s e d energy ^5-  for  °  c a l i b r a t i o n were a l s o p r e p a r e d  i n t h e s i m i l a r , way.  Experimental Procedures.  The e x p e r i m e n t s w e r e c a r r i e d  out a t three separate times  u s i n g t h e Germanium d e t e c t o r s d e s c r i b e d i n | 3 , and t h e gamma sources prepared  a s d e s c r i b e d i n § 4. A t e a c h t i m e , a 4 0 - m i n u t e  r u n was t a k e n t o o b t a i n t h e gamma s p e c t r u m 1024-channel  o f E u 154 i n a  k i c k s o r t e r . This'was f o l l o w e d by 10-minute  for various calibration  runs  s o u r c e s i n c l u d i n g Co 57, Co 60, Cs 137;  Na 2 2 , Mn 5 4 , Ba 133, Cs 134, Y 88, RdTh, Hg 203, and Am 2 4 l . The p o s i t i o n was f i x e d  o f t h e source holder r e l a t i v e t o t h e d e t e c t o r  on an a l u m i n i u m  frame,  so t h a t e v e r y source c o u l d  be p l a c e d a t e x a c t l y t h e same p o s i t i o n . H e n c e , no c o r r e c t i o n f o r t h e d e t e c t o r g e o m e t r y was n e c e s s a r y .  (Actually, there i s  -46no way t o do t h i s ,  a s e v e r y Germanium d e t e c t o r h a s i t s own  p e c u l i a r e f f e c t i v e geometry.) Then, i n o r d e r t o o b t a i n  better  i n f o r m a t i o n on t h e weak p e a k s , a n o v e r - n i g h t r u n o f t h e E u 1 5 4 s p e c t r u m was r e p e a t e d . , and so a l s o a l o n g e r - p e r i o d  run f o r  each o f t h e weaker c a l i b r a t i o n s o u r c e s . From t h e b e g i n n i n g t o the end, t h e c o n d i t i o n s o f t h e e l e c t r o n i c amplification, threshold The  had  and b i a s v o l t a g e w e r e k e p t c o n s t a n t .  loss percentage indicated  t o exceed 5 i n a l l  system such as t h e  on t h e k i c k  s o r t e r was n o t a l l o w e d  t h e r u n s . I f i t d i d , a new s o u r c e  t o be l o c a t e d . S p e c i a l c a r e was t a k e n t o e n s u r e no t i m e  shift  i n the electronic  s y s t e m when t h e s h o r t - p e r i o d  w e r e t a k i n g p l a c e . T h i s was done b y u s i n g a s u i t a b l e source as a probe. of t h i s  runs calibration  (Na 2 2 was c h o s e n f o r t h i s p u r p o s e . ) S p e c t r a  ' p r o b e ' w e r e t a k e n a t t h e b e g i n n i n g and t h e end o f t h e  s h o r t - p e r i o d r u n s and a l s o b e f o r e t h e 4 0 - m i n u t e Any  position  shift  i n t h e t w o Na 2 2 p e a k s  1 2 7 4 . 6 Kev) would  r u n o f Eu 1 5 4 .  ( i . e . t h e 5 1 1 . 0 Kev and  indicate i n s t a b i l i t y  i n t h e s y s t e m and t h e  s p e c t r a t h e n had t o be r e - t a k e n . The o u t p u t d a t a f r o m t h e k i c k s o r t e r were p h o t o g r a p h e d  i n polaroid  f i l m s f o r immediate  i n s p e c t i o n , and p u n c h e d o u t i n p a p e r t a p e s f o r f u r t h e r . The f i r s t  processing  e x p e r i m e n t was done i n O c t o b e r , 1 9 6 6 , i n . w h i c h  t h e Eu 1 5 4 s p e c t r u m was t a k e n i n t w o p o r t i o n s b y means o f a b i a s a m p l i f i e r . Each c h a n n e l o f t h e k i c k s o r t e r t h e n c o r r e s p o n d s  -47to s l i g h t l y  •  l e s s t h a n 1 Kev.  o f t h e number and  These s p e c t r a gave a f i n e p i c t u r e  l o c a t i o n s of the  similar  s p e c t r a o f Eu  152  of t h i s  i s o t o p e I n Eu  154.  were t a k e n The  •  •  \  measurement was . . I n the  .  not  very  were  (1.5  smaller detector  c.c.  and  April,  •  case,  so t h a t t h e  third  logarithmic p l o t . In t h i s c o r r e s p o n d s t o a b o u t 1.7  19^7  r e s p e c t i v e l y , the.bigger e m p l o y e d , and  case, Kev  f a i r l y h i g h l e v e l . T h i s was high counting  In order  t o be  f r e e f r o m any -  i n t e n s i t y at the lower  k i c k s o r t e r as  were n o r m a l i s e d The  t o the  standard  from the  spectrum  of the  energy  on  seen from  able to  set at a eliminate  pulses.  p o s s i b l e d i s t o r t i o n i n energy to the threshold  shown i n f i g u r e 9, main spectrum. •  and  the  and  setting,  spectrum.was r e c a l i b r a t e d i n a  sources  results  kick sorter  k i c k s o r t e r was  done so as t o be  l o w - e n e r g y p e a k s due  p o r t i o n of the  154  o f e n e r g y . I t c a n be  r a t e of the low  detector  much b e t t e r  each channel  the f i g u r e that the t h r e s h o l d of the  the very  •  intensity  e x p e r i m e n t s w h i c h w e r e done i n  w e r e o b t a i n e d . F i g u r e 8 shows a t y p i c a l Eu  obtained  152  t o Eu  reliable.  c . c . e f f e c t i v e v o l u m e ) was  channel  impurity  •  used i n t h i s  s e c o n d and  D e c e m b e r , 1966  the  Also  \  e f f e c t i v e v o l u m e ) was  (5  the  expected.  t o check f o r the  peaks b e l o n g i n g  soon s o r t e d o u t . U n f o r t u n a t e l y , •  p e a k s t o be  400-  intensities  .  for intensity  c a l i b r a t i o n were  I n t e r n a t i o n a l Atomic Energy Agency,  Vienna.  -.000  LULriiNunotn ur LUUINISJ  2.000  4.000  • ^ — ++++++++ ^ J  a  6.000  8.000  10.000  12.000  _J  1  •  86.86 Kev 105 .32 K e v 122.93 K e v . ( E u l 5 4 +Eul'32)  o a a  247.63 K e v (Hul54 +3Sul52) 343.63 K e v ( E u l 5 2 )  ro  r°"  a a a  444.02 K e v  557.96 K e v 582.11 Kev 591.61 Kev  « I oo  6'92.02 K e v 722.90 Kev 756.71 Kev  2:  C—- *C i-  j  C?5 S3  •1  y  Ca CD  -815.02 K e v 872.62 Kev 892.74 K e v 903.60 K e v  rn ID  5  964.09 Kev ( E u l 5 2 ) 995.94 K e v 1004.50 Kev  a  CO  O cf pi  3  1085.47 Kev ( E u l 5 2 ) 1 1 1 1 . 9 8 K e v (Eu'152)  cn  O CD'  a a  1246.16 Kev ( ? ) 1274.43 Kev  a • •1408.16 K e v ( E u l 5 2 ) 1460.88 Kev ( ? ) 1493.68 Kev ( ? ) 1536.64 K e v ( ? )  CD •  b a a  1595.87 Kev  S-4 »  o a a  -817-  -49-  lt 122.9 3Kc.-v  16 •  IZ  O  # c  105.32Kev  O  86.86Kev  4  Z  i €0  •  70  i  i  30  10  i  i  100 I/O  <  UO  1  1—  130 IfO  1  ISO ISO  C h a n n e l Number L o w - e n e r g y gamma p e a k s o f Eu  1  1  170  1  l$0  —i—  1  If0  ., 154.  ZOO  . ' -50-  •  T h e y i n c l u d e Am 2 4 1 , Hg 2 0 3 , Co 5 7 , Na 2 2 , Cs 1 3 7 , Mn 5 4 , C o .60, and  Y  88,  c o v e r i n g an energy  They were a l l spot being performed,  range from 5 9 . 5 7  s o u r c e s , a n d , when t h e c a l i b r a t i o n was  these  s o u r c e s were a g a i n p l a c e d a t e x a c t l y  t h e same p o s i t i o n a s t h e E u 1 5 4 One i m p o r t a n t at  Kev t o l 8 4 l K e v .  source.  reason f o r performing  the  experiments  t h r e e s e p a r a t e t i m e s h a s n o t y e t b e e n m e n t i o n e d . By  examining  t h e Eu 1 5 4 s p e c t r a t a k e n a t t h e t h r e e  p e r i o d s , we f o u n d  t h a t some o f t h e p e a k s w e r e  different  shrinking  w i t h respect t o the o t h e r s , i n d i c a t i n g the presence short-lived  i m p u r i t i e s . These peaks,  o f some  o f c o u r s e , had t o be  excluded. The a n a l y s i s o f t h e s p e c t r a t a k e n w i l l in  §6.  the next  be d i s c u s s e d  section.  R e s u l t s and A n a l y s i s . A f t e r t h e v a s t amount o f s p e c t r a l d a t a w e r e  plotted  o u t by t h e IBM c o m p u t e r , t h e p e a k p o s i t i o n s and t h e p e a k c o u n t s w e r e e s t i m a t e d by g r a p h i c a l m e t h o d s . I t s h o u l d be n o t e d t h a t t h e p e a k s h a p e s were n o t G a u s s i a n , Gaussian  fitting  so t h a t t h e s t a n d a r d  c o u l d n o t be a p p l i e d i n t h e h o p e . o f o b t a i n i n g  more a c c u r a t e r e s u l t s .  I n s p i t e o f t h e . o n l y 3 . 5 Kev f u l l  at  h a l f maximum, a l o n g t a i l  existed  of  e a c h p e a k a s shown i n f i g u r e  at t h e lower energy  width . side  7 i n £ 5 . T h i s was p r o b a b l y due •  •-51- . to  d i s l o c a t i o n s o r i m p u r i t i e s i n t h e Germanium c r y s t a l ,  which  o c c a s i o n a l l y t r a p p e d some e l e c t r o n s c a u s i n g i n c o m p l e t e c o l l e c t i o n o f c h a r g e s i n each e v e n t . These difficulty in the  a tail  i n the intensity  t a i l s gave  some  e s t i m a t i o n . However, t h e c o u n t s  a r e o n l y about 5 p e r cent o f t h e c o u n t s under  r e l e v a n t peak. Assuming  the t a i l  counts are p r o p o r t i o n a l  to  t h e p e a k c o u n t s , t h e e r r o r s h o u l d n o t be t o o s e r i o u s  if  part of the t a i l  even  i s neglected.  A n o t h e r p o i n t t o be m e n t i o n e d  i s that,  since there are  so many p e a k s o v e r t h e e n t i r e Eu 154 s p e c t r u m , - i t i s q u i t e possible that  some o f t h e s e p e a k s m i g h t  s i t on some Compton  edges, o r s i n g l e o r double escape peaks. F o r t h i s , p o s i t i o n s o f t h e Compton e d g e s , s i n g l e and d o u b l e peaks r e l a t e d the  a l l the escape  t o t h o s e p r o m i n a n t p e a k s w e r e computed  by u s i n g  s i m p l e e q u a t i o n s ( 8 4 ) , ( 8 5 ) , and (86) g i v e n , i n f 2 . C a r e  was t a k e n i n a n a l y z i n g t h o s e f e w p e a k s i n t h e v i c i n i t y o f t h e mentioned For are  listed  positions. t h e e n e r g y c a l i b r a t i o n , t h e known-energy  peaks  used  i n t a b l e I . F r o m f i g u r e 1 0 , w h i c h shows one  c a l i b r a t i o n c u r v e f o r t h e Eu 154 s p e c t r u m , we s e e t h a t t h e • •. electronic  system i s f a i r l y  l i n e a r . H o w e v e r , we d i d n o t assume  any s i m p l e f u n c t i o n a l d e p e n d e n c e  o f t h e e n e r g y on t h e c h a n n e l  number. S i n c e t h e c a l i b r a t i o n c u r v e i s f a i r l y  linear, the  -52Radionuclide  Radionuclide  Energy(Kev)  59.57  C s 134  569.0  Hg 2 0 3  73.0  RdTh  583.0  Ba  81.0  Cs 134  604.65  Co 57  122.0  Cs 137  661.59  Co 57  136.4  RdTh  727.0  Hg 203  279.1  Cs 154  796.0  RdTh  238.6  ' r-Tn 54  835.0  Ba 133 '  276.0  RdTh  860.0  Ba 1 3 3  302.0  Y 88  897.5  Ba 133  355 .0  Co 60  1173.3  Ba 133  383.0  Na 22  1274.6  RdTh  511.0  Co 60  1333.0  Na 22  511.0  RdTh  1592.4  Cs 134  563.0  Y S8  1841.0  Am  Energy(Kev)  241  133  Table  1  Peaks  used f o r energy  callbration  - ( o b t a i n e d f r o m r e f . 49 and o t h e r  sources).  ENERGY ( Ke*) Fig.10  Gamma e n e r g y c a l i b r a t i o n  curve  f o l l o w i n g method o f l i n e a r  i n t e r p o l a t i o n was c o n s i d e r e d t o  be t h e b e s t way: L e t A,B,C,D be t h e known  energy  p o i n t s and X, t h e p o i n t t o be calibrated. Linear  interpolations  w e r e computed u s i n g t h e p a i r s o f known e n e r g y and X(3)  (B,D),  points,  ( B , C ) , (A,C)  and t h e s e g a v e t h e r e s u l t s f o r X a s X ( l ) , X ( 2 ) and  r e s p e c t i v e l y . I f these three values f o r X agreetto  w i t h i n 1 Kev, then X = X ( l ) . I f o t h e r w i s e , t h a t s e c t i o n o f t h e c u r v e was e x p a n d e d i n a g r a p h by c u r v e  above m e t h o d , o f c o u r s e , d e p e n d s c o n s i d e r a b l y o n  the accuracy  o f t h e i n d i v i d u a l c a l i b r a t i o n peaks.  b e c a u s e s o many known e n e r g y not d i f f i c u l t  However,  peaks were b e i n g used,  i t was  t o check whether any i n d i v i d u a l peak v a l u e  up t o t h e s t a n d a r d  present  determined  fitting.  The  was  p a p e r and X was  o r n o t . No p u l s e r was u s e d i n t h e  c a s e , a s b e t t e r a c c u r a c y was n o t e x p e c t e d  Provided that the energies l i s t e d  i n table  w i t h i n 0.5 K e v , t h e p e a k s c a l i b r a t e d  from i t . .  I are correct tp  i n t h i s way s h o u l d be  w i t h i n 1 Kev. The  e n e r g i e s o f those w e l l - d e f i n e d peaks i n each ,  40-minute spectrum  o f Eu 154 w e r e t h u s c a l i b r a t e d . As f o r t h e  -55weak p e a k s , c a l i b r a t i o n h a d t o be done o n t h e o v e r - n i g h t - r u n spectra. In this  c a s e , t h e s t r o n g peaks  o f t h e same  spectrum  acted as s u b s t a n d a r d s . Consequently, any time s h i f t not a f f e c t  the calibration,  have s h i f t e d The intensity  would  s i n c e t h e whole spectrum  would  altogether.  characteristics calibration  of the standard sources f o r  a r e g i v e n i n t a b l e I I , w h i c h were  a t z e r o h o u r GMT, J a n u a r y 1, 19^7• The r e l a t i v e  calibrated  s t r e n g t h o f e a c h s o u r c e a t t h e t i m e when i t s s p e c t r u m was half  t a k e n was c a l c u l a t e d ,  and s o w e r e t h e r e l a t i v e  c o r r e s p o n d i n g t o i t s gamma e n e r g i e s . The o b s e r v e d of The  intensities  t h e gamma p e a k s w e r e d i r e c t l y m e a s u r e d f r o m e a c h ratios  of relative  then plotted figure of  intensities  intensity  spectrum.  t o observed i n t e n s i t y  were  a s a f u n c t i o n o f gamma e n e r g y a s shown i n  11*. The c u r v e i s f a i r l y  the efficiency  smooth and i s j u s t  c u r v e . The window e f f e c t  the inverse  s e t s i n a t about  80 K e v . I t i s due t o t h e o b s t r u c t i o n o f t h e N l a y e r o f t h e detector the  ( a b o u t 0.5 mm. t h i c k ) and t h e a l u m i n i u m window o f  "vacuum chamber (0.25 mm. t h i c k ) . U s i n g t h e g r a p h  r a t i o a t 8 l . 0 K e v was o b t a i n e d b y t a k i n g a  *  The i n t e n s i t y  Ba  133 s p e c t r u m . The r e l a t i v e . i n t e n s i t y  was  plotted.,  o f 355 K e v o f Ba. 133  f o u n d f r o m t h e g r a p h . T h e n we m u l t i p l i e d  r e f , Ab ) t o o b t a i n t h e r e l a t i v e and h e n c e i t s i n t e n s i t y  ratio.-  intensity  i t b y 0.5 2, ( s e e  o f t h e 8 l Kev p e a k  Isotope  Strength  Half-life  (Kev)  .(uc) Am 2 4 1  10.66  Y-energy  485.110.5  Y-ray p e r disintegrati o n ($>)  59.5710.02  35.910.6  122.0  85 .311 • 5  279.110.05  81.5510.15  511.0  179.710.8  1274.610.3  99.94  661.59 10.076  84.610.6  835.010.3  100  1173.310.3  10010.012  1333.010.3  iooio.oo  897.510.5  92  1836.2  100 .  years Co 57  10.78  271.6+0.5 days  Hg 203  21.77  46.57+0.03 days  Na 22  11.46  2.60310.005 years  C s 137  10.68  29.8210.11 years  Mn  54  10.70  313  11  days Co 60  10.87  5.26310.003 years  Y  88  10.64  106.610.1 days  Table  II  Peaks used f o r i n t e n s i t y  0.3  calibration.  Energy  (Kev)  the  relative  intensities  f o r t h e c o r r e s p o n d i n g E u 15+  were found by m u l t i p l y i n g t h e observed corresponding The  ratios  calibrated  i n the  intensities  spectrum  to the  graph.  e n e r g i e s and t h e r e l a t i v e  intensities  w i t h e r r o r l i m i t s were t h e n p r e s e n t e d i n t a b l e I I I . The - e n e r g i e s g i v e n are a c c u r a t e t o w i t h i n 1 Kev. T h i s a c c u r a c y i s r e v e a l e d i n C h a p t e r 7 when c o n s t r u c t i n g t h e d e c a y The  relative  intensities  determined  scheme.  a r e f a r more c o m p l e t e  than  t h o s e o f p r e v i o u s w o r k e r s . The a c c u r a c i e s o f t h e e n e r g y m e a s u r e m e n t s compare v e r y f a v o u r a b l y w i t h t h o s e f r o m c o n v e r s i o n l i n e s measured w i t h h i g h r e s o l u t i o n s p e c t r o m e t e r s . Scanning through t a b l e I I I , I n a d d i t i o n t o the established familiar Kev,  well-  s t r o n g p e a k s , we f i n d t h a t t h e r e a r e a f e w p a r t i a l l y  peaks,  i . e . 86.86 K e v ,  692.02 K e v ,  815.02 K e v ,  IO5.32 K e v ,  and 892.74 K e v .  444.02 K e v ,  557-96  Among t h e m , t h e  557.96 K e v and 892.74 K e v p e a k s h a v e b e e n o b s e r v e d b y H a r m a t z 33) et a l i n t h e d e c a y o f Tb 154 t o Gd 154, b u t h a v e n e v e r b e e n o f E u 154. The gamma i n t e n s i t y o f the.' 36) 692.02 K e v was b a r e l y m e a s u r e d b y H a m i l t o n e t . a l u s i n g the-  observed  i n t h e decay  e x t e r n a l c o n v e r s i o n m e t h o d . The r e s t o f t h e t r a n s i t i o n s . w e r e 20),36) known f r o m t h e i n t e r n a l  c o n v e r s i o n data  •In a d d i t i o n t o t h e above, t h e r e a r e c o m p l e t e l y u n f a m i l i a r peaks,  i . e . 582.11 K e v , 903.60 K e v ,  1460 .88 Kev and 1493.68 K e v .  1246.16 K e v ,  I t i s quite.interesting.to  find  -58y-energy (Kev)  Relative  Remark  Intensity  *  15.42 ± 1 . 3 1  86.86  9.70  105.32 122.93  1 0 0 . 0 0 ±2.44  247..63  1 5 . 8 6 ±0.72  444.02  Eul52 I n t  f  E.ul52  ••1.56 ±0.14  557.96  0.867±0.163  582.11  1.73  ±0.16  591.61  1 0 . 5 1 ±0.54.  692.02  3.80 ±0.29  722.90  .47.30  ±2.05  756.71  10.36  ±0.60  1. 3 1 t0.22  815.02  29.38  872.62  ±1.36  892.74  1 . 2 3 ±0.20  903.60  2.04  964.09  1.82 ±0.25  995.94  2 4 . 6 9 ±1.12  1004.50  43.85  ±0.23 Eul52  ±1.92  1085.47  1.67 ±0.22  Eul52  1111.98  2.40  Eul52  1246.16  2.27 ±0.21  ±0.24  •**  92.00  ±4.13  1408.16  3.29  ±0.20  1460.88  0.386±0.052  #- *  1.71  #*  1274.43 .  . 1493.68  Eul52  ±0.11  1536.64  0.125±0.022  1595.87  4.67 ±0.22  Transitions 2 0  ^,  subtracted  Eul52 I n t . subtracted  ±0.29  3.76  343.63  *  i0.84  I  -known t o be b e l o n g i n g to. t h e d e c a y  but cannot  Weak p e a k s  **  be f i t t e d  w i t h unknown  into  t h e decay  second  scheme.  origin.  . Table I I I E n e r g i e s and I n t e n s i t i e s (The  of Eul54  figure  of Eul54  Spectrum.  after  each decimal i n  c o l u m n one i s p h y s i c a l l y  insignificant.)  '-•59-. ,. • that the f i r s t i n C h a p t e r 7.  two The  peaks f i t p e r f e c t l y others  still  into the  c a n n o t be  d e c a y scheme  fitted  to the  known  levels. Among t h e v e r y weak p e a k s , 4 4 4 . 0 2 Kev i n f i g u r e 8, s c a l e as  §5.  o t h e r s w e r e p l o t t e d a g a i n i n an  the  fluctuation.  I t was due  distinctly enlarged  shown i n \ f i g u r e . - j l 2 . T h e y .are a l l w e l l a b o v e  statistical  present  The  i s shown  noted  i n t a b l e I I I t h a t a few  to a small percentage  of t h i s  Eu  152  Impurity.  peaks were Comparison 4 ) 9  of t h e i r is  i n t e n s i t i e s w i t h those  given i n Nuclear  Data  Sheets  shown i n table'.'IV. T h e y a g r e e f a i r l y w e l l . We  also  expected  a small impurity mixing due  to the  121.8  Kev  c o n t r i b u t i o n to the has  p e a k s 1 2 2 . 9 3 Kev  and' 2 4 4 . 7 Kev  p e a k s o f Eu  i n t e n s i t y by t h i s  a l r e a d y b e e n s u b t r a c t e d . T h i s was  conversion data p e a k o f Eu beta  i n the  152  spectrum).  separable  from the  152.  247.63  Kev  The  i m p u r i t y i n each peak done by u s i n g  c a l c u l a t e d i n Chapter 6 was  and  ( s i n c e the  1 2 2 . 9 3 K:  internal 121.8  K-  1  peak i n t h e  -60557.96 Kev 582.11 Kev I7S  903.6 Kev  A \  170  \  X CO  %  I  892.74 Kev  i«4  ax  (60  40 3X0  W>  "33*  C h a n n e l Number  CN  IS5  CO  O  o 150  \ 145  /40  5*8  486  Channel Number P i g . 12  538  548  Channel Number  Weak gamma p e a k s i n t h e E u ^ s p e c t r u m  (see f i g  Gamma-energy (Kev)  3.76±0.29  3+3.63 964.09  '.  -I.82i0.25  Intensity ref.49  (29$)  24$  (14$) .  14$ 11$  1085.47  1.67±0.22  (13$)  IIII.98  2.40i0.24  (18$)  1408.16  3.29*0.20  (26$)  Table.IV.  Rel. from  Rel. Intensity obtained  Gamma p e a k s b e l o n g i n g  ;  15$  .  •  25$  t o Eu 152 i m p u r i t y .  -62-  CHAPTER V  GAMMA-GAMMA COINCIDENCE SPECTROSCOPY  §1.  General  Considerations.  The r e s u l t s  from t h e l a s t  c h a p t e r p r o v i d e many  ing aspects which require f u r t h e r be s t a t e d 1)  i n v e s t i g a t i o n . They c a n  as f o l l o w s , The 9 9 5 . 9 4  never been separated Therefore,  interest-  K e v and 1004.50 K e v p e a k s w h i c h h a v e by p r e v i o u s w o r k e r s a r e w e l l  i f gamma-gamma c o i n c i d e n c e  resolved.  s h o u l d be p e r f o r m e d  a g a i n , we e x p e c t t o g e t c o n c l u s i v e r e s u l t s r e g a r d i n g t h e proper l o c a t i o n s 2)  o f t h e s e two t r a n s i t i o n s  New p e a k s s u c h a s 9 0 3 . 6 0  f o u n d , w h i c h , as w i l l  I n t h e decay  scheme.  Kev and 5 8 2 . 1 1 K e v w e r e  be shown l a t e r ,  have s u i t a b l e  places  i n t h e d e c a y scheme. B u t t h e y need t o be c o n f i r m e d . 3)  New p e a k s s u c h a s 1 2 4 6 . 1 6 K e v , 1 4 6 0 . 8 8  1493-68 Kev w e r e f o u n d , b u t t h e i r 4)  origins  Weak p e a k s s u c h a s 8 7 2 . 6 2  444.02 Kev w h i c h have been r e p o r t e d m e a s u r e m e n t s ( e . g . r e f s . 20  a r e unknown.  K e v , 8 1 5 . 0 2 Kev and from beta  spectroscopic  and 3 6 ) , h a v e a l s o b e e n s e e n  gamma s p e c t r u m . T h e i r p o s i t i o n s be  K e v and  i n this  i n t h e d e c a y scheme need t o  confirmed. All  t h e a b o v e p o i n t s c a n be e x a m i n e d b y gamma-gamma  coincidence at least  i n p r i n c i p l e . However, i n p r a c t i c e .  .  . - 6 3 - •.  one r e q u i r e s much s o p h i s t i c a t i o n i n t h e d e t e c t i o n and e l e c t r o n i c systems such Germanium etc.  as a n i n c r e a s e i n t h e a c t i v e volume o f t h e  d e t e c t o r s , t h e use o f m u l t i p a r a m e t e r  I n view o f t h e time  consumption I n doing  kick sorters gamma-gamma  c o i n c i d e n c e w o r k and t h e present e q u i p m e n t a v a i l a b l e , we h a v e for  t h e time being  the f i r s t  limited  our coincidence experiments  to  o f t h e above f o u r p o i n t s . R e s u l t s o b t a i n e d a r e  q u i t e c o n c l u s i v e . The e x p e r i m e n t s w i l l , be d e s c r i b e d i n t h e n e x t sections. The  i d e a o f gamma-gamma c o i n c i d e n c e i s s i m p l e and  s t r a i g h t - f o r w a r d . S u p p o s e t h e r e a r e two gamma t r a n s i t i o n s i n c a s c a d e a s shown i n f i g u r e 13a, and t h e d i r e c t i o n s o f  ^  c  F i g u r e 13. e m i s s i o n o f t h e s e gammas a r e r e l a t e d b y t h e a n g l e 6 a s i n figure  13b. Now i f . w e p l a c e two c o u n t e r s  or detectors at the  p o s i t i o n s R and S r e s p e c t i v e l y , t h e t w o gammas w i l l be recorded  either  simultaneously or i na small fixed  o f t i m e . H o w e v e r , t h e most p r o b a b l e with different  pairs ^  o f 0 v a r i e s ..  and <( , s o t h a t t h e p o s i t i o n o f S  :  t  r e l a t i v e t o R h a s t o be d e t e r m i n e d maximum number o f p a i r s  angle  Interval,  i n o r d e r t o g a t h e r a"'  i n a g i v e n p e r i o d o f t i m e . I f t h e two  gammas were u n c o r r e l a t e d , t h e n a n y p o s i t i o n o f t h e c o u n t e r  S  -64relative pairs.  to R w i l l  g i v e e q u a l p r o b a b i l i t y i n c o u n t i n g the  In the c o r r e l a t e d  c a s e , the p r o b a b i l i t y  i s given  by  N (87) n=0 where A 2  n  a r e c o n s t a n t c o e f f i c i e n t s , and. P 2 ( c o s n  Legendre p o l y n o m i a l s , g i v e n e x p l i c i t l y P (cos  0) = 1  P (cos  0) = i-(3cps. e' - .1)  P^(cos  Q)  0  2  9) a r e  the  by  2  =  (35cos-^e - 30cos 9- + 3)/8  etc.  2  T h e o r e t i c a l e v a l u a t i o n o f the t r u e c o u n t i n g r a t e s i s not a problem  i n t h i s i n v e s t i g a t i o n , b e c a u s e no v a r i a t i o n  the c o r r e l a t i o n angle in  any  9 i s i n v o l v e d i n the e x p e r i m e n t ,  case, the r e s u l t s  that w i l l  unambiguous t h a t r e l i a n c e entirely  2.  The  Coincidence  l a t e r are  so  upon a t h e o r e t i c a l a n a l y s i s i s ^  System.  e x p e r i m e n t a l a r r a n g e m e n t s a r e shown i n f i g u r e  The  s o u r c e was  and  a l X l i - i n . diameter  d e t e c t o r was use  presented  and  unnecessary.  The  to  be  of  14.  p l a c e d between a Germanium-Lithium d e t e c t o r  used  Nal(Tl) scintillator.  in this  the 5 cc d e t e c t o r ,  case. except  The  1.5  cc. Ge  ( I t would have been p r e f e r a b l e t h a t i t . was  b e i n g used  i n other  -65-  Liquid  Nitrogen  Ge  Detector-  Nal •Ion Pump  Photo-  Fast  Multiplier  Preamp  ^rTrv-  Low-noise  Source-  Preamp.  Fast  Time Constant Box  Fast Discriminator  Discriminator A  Fast Linear Amplifier  B  Variable Delays &  v  Fast  Low-Noise  Coin.  Unit  S. C. A  Delay  a  Amplifier 1. Slow C o i n .  Kick  Fig.  Sorter  Unit  G a t e input? P u l s e (. Shaper  14'. The gamma-gamma c o i n c i d e n c e  Scaler  system,  • e x p e r i m e n t s . ) As u s u a l with a photomultiplier v o l t a g e of -1100 obtained  -66-  , t h e s c i n t i l l a t o r was (RCA 5 8 1 9 )  volts.  Outputs  i n photo  contact  which.was b i a s e d a t a  from the p h o t o m u l t i p l i e r  f r o m t h e anode and t h e e i g h t h • d y n o d e  were  a s shown i n f i g u r e  1 5 . The r e a s o n s f o r u s i n g t h e e i g h t h d y n o d e i n s t e a d  of the  last  dynode a r e t o have t h e o u t p u t p u l s e h e i g h t comparable  the  c o r r e s p o n d i n g one f r o m t h e anode o u t p u t and t o a c h i e v e  b e t t e r l i n e a r t y . The n e g a t i v e o u t p u t f r o m t h e . anode was f o r the purpose  of fast  cascade  emitter follower  drive  the fast  c o i n c i d e n c e . I t was  connected  i n the following  e m i t t e r f o l l o w e r d o e s n o t need good l i n e a r t y , a fast  used  to a  i n o r d e r t o a c q u i r e e n o u g h power t o  discriminator  demanded o f i t was  to  rise  time  (-^50  stage. This  a l lthat  nsec.)  was  and a l o w  impedance o u t p u t . The p o s i t i v e a white cathode  o u t p u t f r o m t h e e i g h t h d y n o d e was f e d t o  f o l l o w e r , w h i c h has a l i n e a r i t y  1%. W i t h t h e g i v e n i n p u t R C - c o u p l i n g , t h e r i s e t i m e were found This c i r c u i t well,  t o be 0 . 2 j K - s e c . and 30  accepts both p o s i t i v e  and h e n c e a n y o v e r s h o o t w i l l  psec  .  better t i m e and  linear  decay  respectively.  and n e g a t i v e p u l s e s e q u a l l y not cause  cut o f f or other  d i s t o r t i o n a t t h e o u t p u t . The o u t p u t p u l s e s w e r e t h e n by a f a s t  than'  amplifier with double-delay-line  amplified  clipping.  These p u l s e s were c l i p p e d  t o 1 j^sec. i n w i d t h and w e r e  fed  c h a n n e l a n a l y s e r .(S .C .A:.) / w h i c h  t o an anti-walk s i n g l e  then  -67-  selects the pulses corresponding t o the desired spectrum  fast  f o l l o w e r output  d i s c r i m i n a t o r connected i s shown i n f i g u r e  p u l s e was a m p l i f i e d  by t h e f i r s t  resultant  t o t h e cascade e m i t t e r  16. The n e g a t i v e i n p u t  two t r a n s i s t o r  discrimination, i s controlled by'the  The  i n the  t a k e n by t h e s c i n t i l l a t o r .  The  brought  energy  . The  2.5 K i l p o t e n t i o m e t e r . The  current t r i g g e r e d the tunnel diode  about  stages  (lN37l6).and  a s q u a r e p u l s e o f r i s e t i m e l e s s t h a n 10 n s e c .  l o w i m p e d a n c e o u t p u t was t h e n o b t a i n e d f r o m t h e t w o e m i t t e r  followers  i n s e r i e s . The Z e n o r  d i o d e 1N752A was u s e d  to limit  the p u l s e h e i g h t i n o r d e r t o match t h e i n p u t requirement o f the f a s t  coincidence unit.  The  l o w - n o i s e p r e a m p l i f i e r a t t a c h e d t o t h e Germanium  d e t e c t o r assembly The  i s t h e same a s t h a t d e s c r i b e d , i n C h a p t e r  pulses f o r fast  obtained shaped  4.  c o i n c i d e n c e on t h e d e t e c t o r s i d e were  d i r e c t l y from t h e p r e a m p l i f i e r output before being  by t h e t i m e c o n s t a n t box. These p u l s e s b e i n g  an i n v e r t e r  s t a g e was i n s t a l l e d  f a s t d i s c r i m i n a t o r . The shaped  positive,  at the input of the following p u l s e s from t h e time  constant  box w e r e f e d t o a 1 2 8 - c h a n n e l k i c k s o r t e r v i a a l o w - n o i s e amplifier. The  r e s t o f t h e u n i t s i n f i g u r e 14 c o n s t i t u t e a f a s t - ,  slow c o i n c i d e n c e system, which kick sorter.  chooses  .. ''.  the gate pulses, f o r t h e  +24v  +12v  Ov  Pig.  16  The  tunnel diode  discriminator.  §3.  Experimental  A source  o f E u 154 w i t h s u i t a b l e s t r e n g t h was s e l e c t e d  f r o m among t h o s e support.  procedures.  prepared  earlier,  and mounted o n a l u c i t e  The d i s t a n c e s o f t h e s o u r c e  d e t e c t o r and t h e s c i n t i l l a t o r  f r o m t h e Germanium  were chosen by t a k i n g t h e  single  c o u n t r a t e s f r o m t h e d e t e c t o r and t h e s c i n t i l l a t o r r e s p e c t i v e l y . With t h e source  c l o s e t o t h e a l u m i n i u m window o f t h e d e t e c t o r  a s s e m b l y and a t a d i s t a n c e o f 5 cm. f r o m t h e s c i n t i l l a t o r , t h e single  c o u n t r a t e s f r o m t h e d e t e c t o r and t h e s c i n t i l l a t o r  0.7 x 10^ cpm and 1.1  x 10^ cpm r e s p e c t i v e l y . T h e s e c o u n t  w e r e a b o u t t h e maximum t h e s y s t e m c a n t o l e r a t e . H i g h e r these  would r e s u l t  coincidence  r a t e was o b t a i n e d switched  pulse width  2 input  unit switch o f f , the delays  u n i t were v a r i e d u n t i l  a maximum  i n t h e s c a l e r . Then t h e c h a n n e l  on a g a i n , w h i l e t h e c h a n n e l  coincidence  of the fast  u n i t was s e t a t 50 n s e c . W i t h c h a n n e l  of t h e slow c o i n c i d e n c e  u n i t was s w i t c h e d  adjusted  f l u c t u a t i o n s with the help  o f a n o s c i l l o s c o p e . The c o i n c i d e n c e  fast  than  o f e a c h f a s t d i s c r i m i n a t o r was  e l i m i n a t e most o f t h e b a s e l i n e  coincidence  rates  i n p o o r r e s o l u t i o n on b o t h s i d e s .  The p o t e n t i o m e t e r to  were  i n the counting  2 i n p u t was  A input of the fast  o f f . Also t h e output o f the f a s t  l i n e a r a m p l i f i e r was c o n n e c t e d t o . t h e  k i c k sorter, input. i n  place of the low-noise required  energy  a m p l i f i e r . With t h i s  arrangement, t h e  p e a k f r o m t h e s c i n t i l l a t o r t o be u s e d a s t h e  k i c k s o r t e r gate  control  (hereafter referred  as t h e gate  c o u l d be l o c a t e d . T h i s was done b y v a r y i n g t h e b a s e  peak)  line(from  z e r o u p ) o f t h e S.C.A. and a t t h e same t i m e w a t c h i n g  the display  i n t h e k i c k s o r t e r , u n t i l t h e d e s i r e d peak p o s i t i o n i s r e a c h e d . T h e n t h e window  o f S.C.A. was n a r r o w e d down t o a b o u t 20 Kev i n  r  w i d t h and t h e b a s e l i n e was a g a i n a d j u s t e d t o a c q u i r e t h e p e a k p o s i t i o n and t o o b t a i n maximum s c a l e r c o u n t s . F i n a l l y , t h e c o n n e c t i o n s w e r e r e s t o r e d t o t h a t shown i n f i g u r e 14 and t h e experiment  was s e t g o i n g .  The s y s t e m all gate  was a l l o w e d t o r u n . f o r 24 h o u r s  t o g e t h e r f o u r r u n s were done c o r r e s p o n d i n g t o d i f f e r e n t s e t t i n g s . The t o t a l I t was c h e c k e d  counting the fast by p r i n t i n g  t i m e t a k e n was a p p r o x i m a t e l y 300  once i n a w h i l e f o r any t i m e  coincidence output  hours.  s h i f t by  and t h e S.C.A. o u t p u t , and  out t h e k i c k s o r t e r data. I f time  the v a r i a b l e delays i n the fast line  a d a y , and  shift  took place,  c o i n c i d e n c e u n i t o r t h e base  o f S.C.A. w o u l d h a v e t o be a d j u s t e d . H o w e v e r , t h e  a m p l i f i c a t i o n s , b i a s v o l t a g e s and t h e k i c k s o r t e r t h r e s h o l d , n e v e r were a l t e r e d  throughout  the four  runs,  $4.  Results.  R e s u l t s o f t h e f o u r r u n s a r e shown i n t h e f i g u r e s 17a,b,c,d.  F i g u r e 17a u s e d  722.90 Kev p e a k a s t h e g a t e  peak,  and t h e spectrum*, shows c l e a r l y t h a t 872.62 K e v and 995.94 Kev w e r e i n c o i n c i d e n c e w i t h 722.90 K e v . The t i m e f o r t h i s r u n was 77 h r s . 26 m i n s . When t h e g a t e s e t t i n g was b r o u g h t down t o t h e b a s e o f the  722.90 K e v p e a k o n t h e l o w e r e n e r g y  s p e c t r u m .shown i n f i g u r e  s i d e , we g o t t h e  17b i n 6 l h r s . 33 m i n s . T h i s  background  s p e c t r u m was composed of. t h e a c c i d e n t a l c o u n t s p l u s t h e c o u n t s i n c o i n c i d e n c e w i t h t h e Compton e v e n t s . Similarly,  t h e spectrum i n f i g u r e  17c c o r r e s p o n d s t o  t h e g a t e p e a k o f 591-61 Kev r u n f o r 85 h r s . 2 m i n s . , and t h a t i n f i g u r e 17d c o r r e s p o n d s t o i t s b a c k g r o u n d 67 h r s . 37 m i n s . I t l e a v e s no d o u b t  spectrum r u n f o r  t h a t t h e 1004.50 Kev p e a k  i n t h e s p e c t r u m i s i n c o i n c i d e n c e w i t h t h e 5 9 1 . 6 1 Kev p e a k .  872.62 Kev 995.94 Kev  ( ) gate a  <i  26  36  *6  56  66  76  g6  peak  722,90Kev  96  4  00 F i g . 17.  The gamma-  gamma c o i n c i d e n c e s p e c t r a (see t e x t ) . w  >6  *6  J6  +S  S6  i6  76  41  Hi  1004.50 Kev (c) g a t e  i£  »S  35  +6  S£  66  76  %6  l£  •(a)  ii  >-6  36  +6  S6  Channel  66  76  Number  86  f£~  peak  591-6lKev  CHAPTER V I  THE BETA SPECTROSCOPY  £l.  General Considerations  The  theory presented 1  transitions.  The r e a s o n s  i n Chapter  2 d i d not i n c l u d e beta  f o r t h i s are two-fold. F i r s t l y , the  m e a s u r e m e n t s made w i t h t h e b e t a s p e c t r o m e t e r g i v e i n f o r m a t i o n o n o b o t h t h e p r i m a r y b e t a r a d i a t i o n s b e t w e e n p a r e n t and nuclei  and t h e I n t e r n a l  concerned  daughter  c o n v e r s i o n e l e c t r o n s . We a r e m a i n l y  with the internal  conversion electron  peaks.  E v a l u a t i o n o f t h e e n e r g i e s and i n t e n s i t i e s f r o m t h e b e t a continuum  a r e o n l y f o r t h e purpose  scheme and a s a c h e c k attempt  i n t h e beta decay,  correlation this  upon t h e t r a n s i t i o n  h a s b e e n made t o d e t e r m i n e  elements  of establishing  w o r k as. w e l l .  t h e decay  i n t e n s i t i e s . No  the t r a n s i t i o n  matrix  since this requires directional  Secondly, the beta t r a n s i t i o n s i n  c a s e a r e f r o m a n odd-odd n u c l e u s  ( E u 154) t o a n  e v e n n u c l e u s , w h i l e t h e gamma t r a n s i t i o n s  even-  discussed i n Chapter  2 u s i n g t h e Asymmetric R o t a t o r Model a r e concerned  w i t h an  e v e n - e v e n n u c l e u s o n l y (Gd 154).. The two c a s e s a r e n o t t h e same. In t h i s of  section,  the beta t r a n s i t i o n s  we w i l l  outline  some o f t h e f e a t u r e s  o f a n odd-odd p a r e n t n u c l e u s and t h e n .  -75pass t o t h e d i s c u s s i o n of t h e i n t e r n a l c o n v e r s i o n  process.  S i n c e E u 154 i s a l s o a d e f o r m e d b o d y , t h e s e l e c t i o n r u l e s g i v e n i n Chapter take i n t o account  4,£l a l s o a p p l y . I n a d d i t i o n , we h a v e t o  a n odd p r o t o n and a n odd n e u t r o n .  of t h i s , t h e N i l s s o n Model f o r i n d i v i d u a l nucleons \ 51) deformed n u c l e i  i s required  H = H  where  H  o  n  + Cl.s + D l  Q  T / ^ r r /\  = -  2M  2  / M 2  i n a deformed n u c l e u s t o (88)  2  t  + ~(ao X + coy .Y 2 x V  i n strongly  . I n t h i s model, N i l s s o n c o n s i d e r s  the Hamiltonian f o r a s i n g l e nucleon be  Because  T  T  + GO Z z'  7  )  (89)  v  J  i s t h e o s c i l l a t o r H a m i l t o n i a n , and t h e o t h e r t w o t e r m s a r e the  spin-orbit  and o r b i t - o r b i t  approximation,  y  co  o  K  T  where £ i s a d e f o r m a t i o n  2 J  (X",Y",Z") = (X'  follows that  H  e  .  Hence H  x  such t h a t  Mco .  T  x"  H „ = o  Mco  = H „ + H „ + H o  r  (90)  p a r a m e t e r . The. v a r i a b l e s a r e t h e n  changed t o  6  symmetric,  = co ( 1 - 2£/3) o  z'  h  crude  c o n s i d e r t h e d e f o r m a t i o n , t o be a x i a l l y  x'  W  As a  u . - u ; = u (1 + kt)  i.e.  It  coupling terms.  y"  itf«v(-J^  ' z"  (91) K  •+ X " ) , e t c . 2  J  (91a)  i s d i a g o n a l i n t h e r e p r e s e n t a t i o n , !n.>|nA|n^\ I V \ 2'\ 3 ' H ,|n )= x ?  L  (n  1 +  i)Kco  x t  ,etc.  (92)  •'•  Consequently  H  With  Define  N = n  L  This solution  |n >|n >|n > = E |n >|n >|n >  q  ]  2  3  E  q  = (n >  + n  2  + n ,  E  -76-  Q  3  q  fa  2  .  3  !)tfco , + ( n + n  3  =  ]  z  n  L  + n  L  ((N+3/2)  + 1 ) ^ ,  2  (94a)  , n , =n and f r o m (90)  2  + £ ( n  (93)  z  x  T  3  - 2n „) /3)]  (94b)  z  i s known a s t h e a s y m p t o t i c s o l u t i o n i n t h e l i m i t  o f s t r o n g d e f o r m a t i o n , and i t i s o b v i o u s t h a t t h e e n e r g y r e l y o n t h e q u a n t u m numbers N,nj_, and n , z  f o r a given  t  levels  deform-  a t i o n parameter  £ . Now l e t A be t h e z"-component o f t h e o r b i t a l  quantum number,  then A  = n j _ , n - 2 , ... ,-n +2,-n_ x  L  Thus f o r a n odd-A n u c l e u s , we e x p e c t t h a t o n t h e q u a n t u m numbers N , A and Indeed  would  (9 5)  t  some s e l e c t i o n  rules  appear. T h i s i s  t h e case,' and t h e s e l e c t i o n r u l e s w e r e t a b u l a t e d b y  Alaga f o r t h e m a t r i x elements fS-transition p r o b a b i l i t i e s  o f t e n used  i n calculating the  a s shown i n r e f e r e n c e 52.  F o r a n odd-odd n u c l e u s , t h e s e l e c t i o n r u l e s  mentioned  a p p l y t o b o t h t h e odd p r o t o n and odd n e u t r o n . I n a d d i t i o n , t h e . c o u p l i n g r e l a t i o n s b e t w e e n t h e s e t w o odd n u c l e o n s initial  and. f i n a l  s t a t e s would  a l s o g i v e r i s e t o new ,53)  i n the selection  r u l e s . The d e t a i l s w e r e g i v e n b y G a l l a g h e r H o w e v e r , i f one c o m p a r e s t h e r e d u c e d probabilities  transition  o f t w o - t r a n s i t i o n s f r o m t h e same i n i t i a l  t o d i f f e r e n t members o f a r o t a t i o n a l f a m i l y , t h e n  state  the.Intrinsic  ,wave f u n c t i o n s o f t h e odd n u c l e a n s a r e c a n c e l l e d , o u t i n t h e ratio  shown b e l o w ,  and h e n c e t h e s e l e c t i o n r u l e s  involving  N, A, and n  x  do n o t e n t e r . T h u s ( r e f . 5 4 ) , CCliL  B(L,I ->I , j  C ( I L I ,;K.,K ,-K.,K ,)..  1  This  equation  mixtures |.A  BiLjIt^If.) •  K m  I .;%,%-%,%) f  f  f  i s foraxially  asymmetric case,  It  f  o f K terms should C(l LI 1  f m  1  V  ]  m  -K ,K t  between t h e emitted  )  n  J  of each  0  beta  e l e c t r o n and n e u t r i n o  i n a l l p o s s i b l e w a y s . As a r e s u l t , t h e b e t a  and  be i n c l u d e d , i . e .  ;K , K.,K )  i s .well-known t h a t t h e energy E  beta  spectrum f o r a  t r a n s i t i o n i s a continuum w i t h a d e f i n i t e  an energy range, 0 — E  of E  0  shape  . F o r more t h a n one t r a n s i t i o n , t h e  spectrum i s a s u p e r p o s i t i o n o f a l l t h e c o n t i n u a . determination  (96)  f  symmetric n u c l e i . F o r t h e  2, A K n C d i L I f n j K ! , ^  t r a n s i t i o n i s shared  single  2  BCL.I^If)  Experimental  i s u s u a l l y done b y t h e method o f K u r i e  plot  ( c . f . r e f . 5 5 ) • F r o m t h e K u r i e p l o t , t h e component c o n t i n u a  can  a l s o be r e s o l v e d . Hence t h e t r a n s i t i o n i n t e n s i t i e s c a n be f o u n d , ^ 2 . The I n t e r n a l  Conversion  I n C h a p t e r 3, excitation,  and P a i r  Production.  we h a v e d i s c u s s e d  the processes  which include i n t e r n a l conversion  p a i r p r o d u c t i o n . The i n t e r n a l c o n v e r s i o n transfer by d i r e c t  of energy from t h e nucleus interaction  electromagnetic  field  o f de-  and i n t e r n a l  process  involves the  t o an e x t r a n u c l e a r  between t h e n u c l e a r  electron  c h a r g e and t h e  o f t h e e l e c t r o n (Coulomb  interaction).  -78-  \; 56)  .  T h i s was shown b y T a y l o r and M o t t probability  :  .  by c a l c u l a t i o n , t h a t t h e  f o r internal conversion byaphotoelectric effect  ( i . e . b y e m i s s i o n and  a b s o r p t i o n o f a photon) i s very  compared t o t h e p r o b a b i l i t y energy t r a n s f e r .  small  f o r i n t e r n a l c o n v e r s i o n by d i r e c t  I t c a n a l s o be v i s u a l i z e d  from t h e presence  " • A of  e EO t r a n s i t i o n s  case  t h e r e i s no gamma c o u n t e r p a r t . C o n s e q u e n t l y ,  conclude  that direct  i n t h e conversion spectrum, as i n t h i s  gamma e m i s s i o n and i n t e r n a l  are two independent p r o c e s s e s The  energy c a r r i e d  conversion i s related E =  t o the  :  conversion  of de-excitation.  by t h e emitted  ¥ - B  e  we may  electron i n internal  d e - e x c i t a t i o n energy, W by (98)  x  where B^ i s t h e e l e c t r o n b i n d i n g energy f o r o r b i t X f r o m w h i c h t h e e l e c t r o n was e j e c t e d . S i n c e X,  i sdifferent  t h e r e i s more t h a n o n e v a l u e o f E  e  for  different  f o r a g i v e n ¥. T h e r e f o r e ,  the c o n v e r s i o n peaks i n an energy spectrum c o r r e s p o n d i n g t o one  ¥ are classified  i n t o X = K,L,M,N,-etc. A l s o s i n c e each o f  t h e L,M,N, e t c h a s more t h a n one e l e c t r o n o r b i t , X may be redefined  a s X = K,L1,L2,L3,M1, e t c . The r a t i o o f t h e e m i s s i o n  r a t e o f t h e c o n v e r s i o n e l e c t r o n s t o t h a t o f t h e gamma p h o t o n s i n a. g i v e n e n e r g y t r a n s i t i o n i s c a l l e d t h e i n t e r n a l  conversion  c o e f f i c i e n t d. , w h i c h may be e x p a n d e d i n t e r m s o f X ' s a s b e l o w o( . 4 + c < .*i + 4 + * +-'' (99) / K  L1 +  2  L 3  M 1  -79Also,  s i n c e t h e t r a n s i t i o n s depend o n t h e  angular  momentum L c a r r i e d away and t h e c h a n g e o f p a r i t y T T , o< c a n be expressed  as a l i n e a r  different  electric  as  combination of the terms a r i s i n g  multipoles  from  EL and m a g n e t i c m u l t i p o l e s  ML  shown,  . *  2sfa)24 (L)  s  x  where c< (L)and  +  2  $2(L)]L  ^ ( L ) are the conversion  x  X  t o EL and ML r e s p e c t i v e l y , and t h e contributions  corresponding  The c o e f f i c i e n t s # ( L ) v  £  (3 (L)  (100)  X  coefficients referred  's a r e t h e r e l a t i v e  t o t h e r e s p e c t i v e E L ' s and ML's. • and/S (L) v  , which are functions  57) o f e n e r g y , w e r e c a l c u l a t e d and t a b u l a t e d b y R o s e , and S l i v 58) • et a l f o r t h e f i r s t f e w v a l u e s o f X and L . R o s e ' s c a l c u l a t i o n A  I  X  was b a s e d on t h e a s s u m p t i o n o f a p o i n t account  the e f f e c t of screening  cloud, while  i n t h e case of S l i v  with modifications  nucleus taking  by t h e a t o m i c  electron  et a l , a f i n i t e - s i z e d  on t h e i n i t i a l  and f i n a l  into  nucleus  e l e c t r o n wave  f u n c t i o n s were a d o p t e d . Thus t h e second c a l c u l a t i o n i n c l u d e d the is  penetration  e f f e c t ( that  i s t h e i n t e r a c t i o n when t h e e l e c t r o n  i n s i d e the charge d i s t r i b u t i o n of the n u c l e u s ) ,  considered  t o be more a c c u r a t e .  experimental  data  Experimental  In general,  t o w i t h i n a few p e r  we used t h e s o - c a l l e d n o r m a l i z e d  i t agrees  with  cent.  evaluation of internal 59)  c i e n t s c a n be done i n s e v e r a l ways  and was \  conversion  . In.the  present  coeff-'-...  .work,  p e a k t o gamma'method . F r o m  •  t h e gamma s i n g l e  -  '  spectrum  8 o  -  and t h e j S s p e c t r u m ,  p e a k ( e . g . 122.93 K e v ) and a c o r r e s p o n d i n g  beta  ^  c o e f f i c i e n t . Using t h e tables' o f t h e o r e t i c a l oL  x  , (^x^st  experimental  c  ^  ^  n  e  internal  >  where I  and  eX  '  After unoccupied dropping  I  l  i  conversion coefficient  e  c  o  n  v  e  r  s  i°  n  values of Then t h e  f o r any o t h e r  ,  offers  o r bound s t a t e  intensity  .  another  the K x-ray.  together  with  . However, n o t a l l  by e m i s s i o n o f K x - r a y s . A s m a l l  f r a c t i o n o f t h e events goes through energy r e l e a s e d , i n s t e a d  down t o  a method o f a b s o l u t e d e t e r m i n a t i o n 32)  conversion coefficients  emitting a so-called  ^  thus r e l e a s e d i s c a l l e d  vacancies are f i l l e d  be used t o e j e c t  r  up e v e n t u a l l y . T h i s i s done b y  T h e r e f o r e , a knowledge o f t h e K x - r a y  K-shell  i  i s i n t e r n a l l y converted, the  from a free  The r a d i a t i o n  of t h e i n t e r n a l  e  :  electron  the gamma.intensity  h  are the r e l a t i v e i n t e n s i t i e s of the p a i r .  o r b i t must be f i l l e d  the K - o r b i t .  t  from  a K-orbit electron  another  e  ^ r ^ s t .,. '  eX'  y  e  obtained by i n t e r p o l a t i o n .  t r a n s i t i o n may be c a l c u l a t e d  b  e  t  i n t e n s i t i e s and (°fx^st  respective relative  c o n v e r s i o n peak  and ( l x ) ' s t  were c h o s e n a s s t a n d a r d s . L e t ( l . ) y  a s u i t a b l e gamma-  a different  c h a n n e l . The  o f i n t h e f o r m o f a K x - p h o t o n , may electron  from a h i g h e r o r b i t ,  Auger e l e c t r o n .  thus  The f r a c t i o n o f t h e e v e n t s  -81-  going through  the x-ray  channel  (known a s t h e f l u o r e s c e n t y i e l d )  Is a constant  f o r a g i v e n a t o m and c a n be e v a l u a t e d .  T h e r e a r e , o f c o u r s e , o t h e r x - r a y s , L,M,N, e t c . , b u t t h e y a r e much, l o w e r In Chapter  i n e n e r g y and i n t e n s i t y .  4, we h a v e d e s c r i b e d p a i r p r o d u c t i o n i n a  d e t e c t o r . The p r o c e s s same p r i n c i p l e  of internal pair production follows the  as t h e process  o f i n t e r n a l c o n v e r s i o n . The  b a s i c d i f f e r e n c e i s t h a t i n t h e second c a s e , t h e n u c l e a r i s t r a n s f e r r e d d i r e c t l y t o a bound a t o m i c  energy  electron, while i n  the f i r s t . p r o c e s s , t h e energy i s t r a n s f e r r e d t o an e l e c t r o n i n a ' n e g a t i v e e n e r g y ' s t a t e . However, s i m i l a r t o t h e i n t e r n a l . conversion c o e f f i c i e n t s , the emission rate of the i n t e r n a l p r o d u c t i o n i s a l s o expressed  i n r a t i o w i t h t h a t o f t h e gamma  p h o t o n s , and i s known a s t h e p a i r p r o d u c t i o n T(L)  . P o r t r a n s i t i o n energies exceeding  process energies  s h o u l d be t a k e n i n the present  pair  i n t o account.  coefficients,  s e v e r a l Mev, t h i s  However, t h e t r a n s i t i o n  i n v e s t i g a t i o n a r e a l l b e l o w 2 Mev, and 60)  the corresponding T ( L ) a r e l e s s than 5 x 1 0 ^ -  are w i t h i n t h e experimental an e n e r g y l e s s t h a n occur.  . Hence,  they  e r r o r and c a n be n e g l e c t e d . F o r '  1 . 0 2 2 Mev, s u c h p r o c e s s e s  cannot even •;:.- .  £3 .  The B e t a  Spectrometer.  An i n t e r m e d i a t e image b e t a as a d o u b l e - l e n s  spectrometer)  As shown i n f i g u r e  18/  spectrometer  (also  known  was u s e d i n t h e p r e s e n t  i t c o n s i s t s o f two  work.  water-cooled  m a g n e t i c l e n s e s , a c y l i n d r i c a l vacuum c h a m b e r , b a f f l e s and  A,B,C,  e x t e r n a l l y a d j u s t a b l e h o l d e r s f o r t h e s o u r c e and t h e  d e t e c t o r . The m a g n e t i c l e n s e s , a s t h e name i m p l i e s ,  focus  t h e b e t a p a r t i c l e s w i t h a n a p p r o p r i a t e momentum f r o m t h e s o u r c e o n t o t h e d e t e c t o r . The a p p r o p r i a t e momentum r e f e r r e d d i r e c t l y p r o p o r t i o n a l t o the. m a g n e t i c f i e l d vacuum c h a m b e r , w h i c h  strength i n the  i s i n turn p r o p o r t i o n a l t o the current  i n t h e l e n s e s . The c u r r e n t was g e n e r a t o r and c o n t r o l l e d  s u p p l i e d b y a 150  volt  b y a c u r r e n t r e g u l a t o r . The  c o n s i s t s o f two t r a y s o f 6AS7 t r i o d e s , a s t a n d a r d of  low temperature  system  ( f i g u r e 19)  to i s  coefficient  and a f e e d b a c k  d.c. latter  resistor  bias control .  with a high p r e c i s i o n potentiometer f o r  v a r y i n g t h e d . c . c u r r e n t . Thus t h e momenta o f t h e b e t a c a n be e x p r e s s e d  i n terms of t h e p o t e n t i o m e t e r  To a v o i d t h e e f f e c t .spectrometer  setting.  of the e a r t h magnetic f i e l d ,  the  was p l a c e d w i t h i t s a x i s a l o n g t h e m a g n e t i c  m e r i d i a n and b e t w e e n two c o m p e n s a t i n g c o i l s t o r e d u c e v e r t i c a l component o f t h e The s o u r c e be v a r i e d  particles  the  field.  p o s i t i o n t o g e t h e r w i t h t h a t o f b a f f l e A can  i n t h e p l a n e p e r p e n d i c u l a r t o t h e chamber a x i s  by  Detector :.Bias Power Supply  Vacuum Gauge  Detector Positioning Rod /  t  =jyn  Spot Source  J.  Source- . \C e n t e r i n g 'Control  i.P \  . Low Noise Preamp.  >r  T  PUTT Pump  Magnet Current  150 v o l t d.c Delay & Shaper  Time Constant :. Box  Bias Control Circuit  Anti-coin. Amplifier  Disc Output  Scaler  Unit  Fig.18  Beta  spectrometer  High Precision Potentiometer  assembly,  Standard Resistor  r  CO 00  Fig.19  Magnet c u r r e n t c o n t r o l c i r c u i t  .. means o f t h e two along the slit and  -85-  c e n t e r i n g d i a l s , w h i l e the d e t e c t o r i s movable  chamber a x i s by  t h e p o s i t i o n i n g r o d . The  In b a f f l e A defines the l i m i t s the beta  initial  angle  i n t e n s i t y t o the  those  b e t a s not  of t h i s  f o l l o w i n g the proper paths.  t r a c k s and  on t h e  d e p e n d s on t h e  d.c.  current  d e t e r m i n e d by m a g n e t i c f i e l d s e n s i t i v i t y of the The  current  s t a b i l i t y . The  s t r e n g t h , and  Firstly,  originating  beta  former i s latter,  by  the  e i t h e r i n the  of  scintillators  or  gas-filled  or i n the background. There-  chamber i s n e c e s s a r y .  i s unaffected  chamber, whereas i n t h e  w h i c h must accompany a s c i n t i l l a t o r , ..strong f i e l d  of t h i s type  i n s e n s i t i v e t o gamma r a d i a t i o n  source  lead s h i e l d i n g i n the  i n the  use  because of the t h i n d e p l e t i o n l a y e r of  surface b a r r i e r detector  field  The  conventional  the d e t e c t o r , i t i s p r a c t i c a l l y  the  of the  case i s p a r t i c u l a r l y advantageous, i f  compares i t w i t h t h e  f o r e , no  resolution  d e t e c t o r m e n t i o n e d a b o v e i s a t h i n s u r f a c e b a r r i e r :•  detector i n the present  counters.  the  from  regulator.  t y p e w i t h s u r f a c e d i a m e t e r o f 1cm.  one  curvature  beta  detector  The  tracks  central  of the passing  d e t e c t o r b a f f l e C p r o t e c t s the  spectrometer  beta  d e t e c t o r . The  b a f f l e B c o n t r o l s t h e momentum s p r e a d p a r t i c l e s . The  of the  annular  w o u l d be  by t h e  Secondly,  magnetic  case of a p h o t o m u l t i p l i e r the presence of t h i s  i n t o l e r a b l e . The  s i g n a l output of  :  the  -86d e t e c t o r is,.however> r e l a t i v e l y  small. Therefore,  p r e a m p l i f i e r w i t h a time constant the peak-to-noise  a  low-noise  b o x was r e q u i r e d t o i n c r e a s e  ratio.  The i n t r o d u c t i o n o f a n a n t i - c o i n c i d e n c e u n i t and a feedback device minimizing pulses  the effect  of noisy surroundings.  o c c a s i o n a l l y were o b s e r v e d  Each of these cidence  shown i n f i g u r e 18 was o n l y f o r t h e s a k e o f Sudden b u r s t s o f  i n the monitor o s c i l l o s c o p e .  bursts of pulses, a f t e r entering the a n t i - c o i n -  u n i t , were reduced t o j u s t  one l e a d i n g  pulse.  Consequently, the effect  became i n s i g n i f i c a n t when c o n s i d e r i n g  the higher  r a t e . The d e l a y  beta  counting  produced t h e r e q u i r e d b l o c k i n g pulse  and s h a p e r u n i t  f o r the anti-coincidence  unit.  ^4.  P r e p a r a t i o n of Beta  Sources.  The p r e p a r a t i o n o f a s p o t w i t h minimum  p o s s i b l e source  source  f o r beta  spectroscopy .  s c a t t e r i n g b u t w i t h maximum  possible radioactive strength i s a challenging task. methods h a v e b e e n t r i e d evaporation,  molecular  electroplating,  liquid  by p r e v i o u s  w o r k e r s , s u c h a s vacuum  deposition, electrostatic spraying, 61)-64) drop d e p o s i t i o n e t c  method h a s i t s own a d v a n t a g e s and d i s a d v a n t a g e s . the  liquid  Several  d r o p d e p o s i t i o n method i s t h e s i m p l e s t  . Each B u t by f a r , and most  f r e q u e n t l y u s e d among them a l l . limited  I t has t h e disadvantages o f  s t r e n g t h and c o m p a r a t i v e l y p o o r s u r f a c e u n i f o r m i t y .  But, i n a d d i t i o n t o being easy t o prepare, any  kind of t h i n backing. Therefore, having t r i e d  methods w i t h o u t •  almost  some o t h e r  s a t i s f a c t o r y r e s u l t s , t h i s method was f i n a l l y \  chosen.  • Before  had  i t accepts  s t a r t i n g t h e source p r e p a r a t i o n , a t h i n  t o be p r e p a r e d  purpose.  first.  V i n y l f i l m s were s e l e c t e d f o r t h i s  A b o u t 10 gms. o f B a k e l i t e v i n y l r e s i n powder  VYNS) w e r e m i x e d w i t h 100 m l . o f c y c l o h e x a n o n e b o t t l e and s t i r r e d was l e f t  (called  solution i na  u n t i l most o f t h e powder h a d d i s s o l v e d . I t  f o r a f e w d a y s w i t h o c c a s i o n a l s t i r r i n g . When a l l t h e  powder h a d d i s a p p e a r e d , diluted  backing  a s m a l l q u a n t i t y o f t h e s o l u t i o n was  b y a n equal'.amount o f c y c l o h e x a n o n e  on a watch  glass.  By u s i n g a g l a s s r o d , a d r o p o f t h e s o l u t i o n was i n t r o d u c e d to the surface o f a t r a y o f c l e a n water. spread  i n t o a tough  the r e f l e c t i n g and  The s o l u t i o n  quickly  t h i n f i l m . The u n i f o r m p a r t ( j u d g i n g f r o m  c o l o u r s ) was p i c k e d up b y a c o p p e r  placed on an aluminium  wire  r i n g . The f i l m t h u s p r e p a r e d  loop was  e s t i m a t e d t o be l e s s t h a n 10 pg/cm^- t h i c k . When t h e f i l m was dry, a t r a c e o f aluminium  was e v a p o r a t e d  f i l m t o s e r v e as a charge  conducting  i n vacuum o n t o t h e  l a y e r . Great  c a r e was '~  t a k e n n o t t o overheat? t h e f i l m d u r i n g e v a p o r a t i o n . S e v e r a l  -88-  . backings  were p r e p a r e d  i n this  To p r e p a r e t h e s o u r c e , and  plane-cut  way. a small glass j e t with a clean  n o z z l e was c o n n e c t e d t o a s y s t e m s u c h t h a t t h e  Internal pressure  o f t h e j e t c o u l d be v a r i e d  conveniently.  A t i n y q u a n t i t y o f t h e c h l o r i d e s o l u t i o n o f E u 154 s u i t a b l e c o n c e n t r a t i o n was backing  was t h e n p l a c e d  sucked i n t o t h e j e t . A  source  c o u l d be d e p o s i t e d  uniform  dried  source  was f i n a l l y  e t h e r was p r e p a r e d ,  dilute  trials,  obtained.  and a f a i r l y The s o u r c e  o f f . The s o u r c e . t o be mounted  t o p r o t e c t the source  was a g a i n r e p l a c e d i n . t h e vacuum  Experimental  was  intense then  hygroscopic  s o l u t i o n of c o l l o d i o n i n dry  and a d r o p o f i t was i n t r o d u c e d  surface of the d r i e d source  ^5.  a small well-defined  i n a d e s i c c a t o r ( s i n c e Europium C h l o r i d e i s a  compound). Then a v e r y  base.  onto the centre o f the v i n y l  f i l m . T h i s was done w i t h r e p e a t e d and  source  u n d e r t h e n o z z l e on a n a d j u s t a b l e  By c a r e f u l l y c o n t r o l l i n g t h e p r e s s u r e , spot  with  onto the  from  flaking  i n the d e s i c c a t o r , ready  chamber.  Procedures.  .Preliminary adjustments t o the spectrometer of changing b a f f l e s ,  and r e p e a t e d  consisted  v a r i a t i o n of the source  and  d e t e c t o r p o s i t i o n s as w e l l as t h e o r i e n t a t i o n o f t h e chamber, until  optimum i n t e n s i t y w i t h b e s t  obtained.  A beta  source  possible  resolution-was  o f Cs 137 was u s e d f o r . t h i s  purpose.  -89Details are  •• ••/•}^'- -:-:-'./\ :  o f t h e a d j u s t m e n t s and c a l i b r a t i o n o f t h e s p e c t r o m e t e r  g i v e n i n r e f . 65. The b e s t r e s o l u t i o n a c h i e v e d was 0.7 $  i n t e r m s o f p o t e n t i o m e t e r s e t t i n g o r momenta w i t h 0.96$ transmission. •The C s 137 s o u r c e was t h e n r e p l a c e d b y a b e t a s o u r c e o f E u 154 c h o s e n f r o m among t h o s e p r e p a r e d , and t h e chamber was c o n t i n u o u s l y pumped down t o k e e p away t h e w a t e r  vapour.  ("Unfortunately, i n t h e l a t e r r u n s , because o f repeated  removal  and r e p l a c e m e n t o f t h e s o u r c e , a s m a l l q u a n t i t y o f w a t e r was a b s o r b e d T h i s caused  b y t h e s o u r c e and t r a p p e d i n t h e c o l l o d i o n deterioration  of resolution  vapour film.  i n t h e low energy  p o r t i o n o f t h e spectrum, but d i d not a f f e c t t h e h i g h e r energy part.)  , I n each r u n o f t h e experiment, t h e p r e s s u r e o f t h e  chamber was k e p t a t a b o u t  10  mm. o f Hg. The p o t e n t i o m e t e r  r e f e r e n c e v o l t a g e was c a l i b r a t e d every f i v e the  cell  o r t e n m i n u t e s . The s h a p e o f t h e o u t p u t p u l s e s f r o m  t i m e c o n s t a n t b o x was o b s e r v e d c o n s t a n t l y w i t h a n  oscilloscope. varied the  against a standard  The c o u n t i n g t i m e f o r e a c h p o t e n t i o m e t e r s e t t i n g  from h a l f a minute t p e i g h t minutes depending on  c o u n t i n g r a t e . Ore s e c t i o n o f t h e s p e c t r u m was t a k e n . e a c h  time w i t h repeated runs, u n t i l such t h a t  the s t a t i s t i c s  enough c o u n t s were  accumulated  f o r ' e a c h d a t a p o i n t was w i t h i n t h e  • required for  limits  • -90-  ( i . e . 1 % f o r t h e energy  and l e s s  the peaks). Three experiments  of  continuum  t i m e . The f i r s t  have been performed  two e x p e r i m e n t s  the i n t e r n a l c o n v e r s i o n peaks.  at different  were m a i n l y concerned  A typical  spectrum  periods with  was p l o t t e d  o u t i n t w o p o r t i o n s a s shown i n f i g u r e s 20a,b. The t h i r d experiment  was d e s i g n e d  near t h e h i g h energy s o u r c e was u s e d  t o i n v e s t i g a t e t h e beta  end. I n t h i s  experiment,  continuum  a stronger beta  t o o b t a i n a g r e a t e r c o u n t i n g r a t e and b e t t e r  s t a t i s t i c s . The s p e c t r o m e t e r was s e t up t o u s e a l a r g e r  source-  d e t e c t o r d i s t a n c e and s m a l l e r i n i t i a l t r a j e c t o r y a n g l e s . T h i s was n e c e s s a r y i n o r d e r t o r e a c h e n e r g i e s n e a r t h e end p o i n t of  t h e E u 154 s p e c t r u m  c u r r e n t . The s p e c t r u m  w i t h t h e maximum a v a i l a b l e f o r t h e beta continuum  was shown i n  f i g u r e 20c.(Note t h a t t h e counts a r e d i v i d e d  ^6.  magnetic  b y t h e momentum.)  R e s u l t s and A n a l y s i s .  The b e t a s p e c t r u m  shown p l o t t e d  i n f i g u r e s 20a,b i n  t h e p r e v i o u s s e c t i o n was o b t a i n e d a f t e r a l l s h o r t - l i v e d had  decayed  to negligible  s i z e . They were p l o t t e d  mic  v a l u e s i n o r d e r t o show a s much as. p o s s i b l e o f t h e  features of a l l the conversion Since t h e beta  i n logarith-  peaks.  s p e c t r o m e t e r was f i r s t  a ,Cs 137 s o u r c e , t h e e n e r g y  peaks  correspondence  calibrated  o f each  using  potentiometer  CM r-IVC C\l C\l CO CO rH rH  Lf\ O iH  r-c--  (Al CM CM CM i—! r H  CM  •"3" vtCM CM  4-4- •  .1811 Fig.20a  .226 .268 POTENTIOMETER Beta spectrum  .310 SETTING  .352  o f E u 154, low-energy  .394 part.  .136  .1478  in  m in Lf  5"  Ui  ^  cn CM t-- cn VO vo  cn  •  CO  •  •  •  •  •  cn cn LT\ CM vo vo c-- f -  C\J CM C\J VO C\J  CM  r—  LP.  CD  Ui vo  cn O O C~- cn  O  CM  co  VO *  Ui Ui • • LP. "td-  cn o cno  CM  co  ID  CO I—  2:  ZD 2 Oco ^cV o f + + + + +.. «—>  * " " " " I I H I I I H + I  o  -"S cn  o  oo T  .500  ,542  .584  _ _ _  ,BZ6  .  R  .B58  POTENTIOMETER Fig.20b  Beta spectrum  .710  ,  —  1  .752  .794  SETTING  o f E u 154,  high-energy  part.  .836  6  CM  E 3  QJ E o  I  I  co +-> 3 C 3 3  O CJ  0  I Fig.20c  Continuum  Z Momentum P f o r beta  3 ( i n mc  transitions  )  o f Eu  154.  s e t t i n g was  roughly.known. Hence, the 154  peaks i n t h e Eu  c o n v e r s i o n p e a k was  spectrum then  conspicuous  conversion 722.90K  w e r e s o o n r e c o g n i z e d . The  s e l e c t e d as a s t a n d a r d  to  calibrate  the e l e c t r o n energies of a l l the other c o n v e r s i o n peaks. i n t e n s i t y measurement was each peak i n an e n l a r g e d s u b t r a c t e d , and potentiometer  done by  e s t i m a t i n g the area  linear plot  setting'at  t h a t p e a k . The  width at  f o r the c o n v e r s i o n peaks are presented  F i g u r e 20a  graphically  full  particulars  shows t h a t t h e  lower  i s d o m i n a t e d by t h e 122.9  Kev  i n this  energy range.  The  half  i n each case.  The  i n t a b l e V.  p o r t i o n of the  beta  conversion peaks.  t h e asymmetry o f t h e peak shape i n d i c a t e s  side of these  was  t h e n d i v i d i n g t h e a r e a by t h e momentum o r  a l s o determined  scattering  under  a f t e r the continuum  maximum was  spectrum  The  slight  tails  The  source  at the lower  s t r o n g peaks d i d i n t e r f e r e w i t h the  energy  intensity  measurement o f t h e w e a k e r p e a k s b e t w e e n t h e m as w e l l as t h e c o n t i n u u m b e n e a t h them. However, s o u r c e  s c a t t e r i n g begins  Column 1 o f t h e t a b l e g i v e s t h e e n e r g i e s of t h e  *  as d e t e r m i n e d atomic  s h e l l s . Column 5 g i v e s t h e t r a n s i t i o n e n e r g i e s Column 2 t o t h e a p p r o p r i a t e s h e l l  B  R  x  ( i . e .B  =  50.22 Kev  e t c f o r Eu  154,and B  binding K  -  to  transitions  by t h e gamma-ray m e a s u r e m e n t s t o g e t h e r w i t h  by a d d i n g  beta  the  calculated  energies  46.85 e t c f o r  E u . 1 5 2 ) . C o l u m n 6 c o m p a r e s t h e t r a n s i t i o n e n e r g i e s , from, gamma and  conversion electron data.  •""•'/"...'  -95Electron Peak Energy Name (Kev) ( f r o m Y)  Relative Intensity  FWHM  Y-energy Trans E n e r g y ': D e v i a t i o n (Kev)  86.9K  37.28  105.3K  54.54  10.5 ±0.4  1.21  104.8  0.48  122.9K  72.49  260.0±2.6  1.0  122.7.  0.16  121.8K  74.6  10.7 ±2.1  1.0  121.5  0.25  •86.9L  77.85  1.90 ±0.22  1.0  85.7  1.4  86.9M  83.05  0.37 ±0.08  1.0  84.6  2.6  105.3L  96.18  1.21 ±0.25  1.45  104.0  1.2  122.9L  115.20  170.313.5  1.0  123-0 .  0.081  122.9M  121.30  48.8 ±1.8  1.0  122.8  0.081  247.6K  196.56  5.71 ±0.12  0.7  246.8  0.32  247.6L  239.33  1.46 ±0.15  0.77  247.2  0.16  247.6M  245.78  0.30 ±o.03  0.74  247.3  0.12  343.6K  293.94  0.58 ±0.02  0.67  340.8  0.82  591.7K  540.57  0.26 ±0.03  0.78  590.8  0.15  679.4K  629-2  0.13510.014  0.73  679.4  692.OK  642.1  0.839±0.017  0.67  692.3  0.04  722.9K  672.68  0.47710.019  0.66  722 .9  0.  692.OL  684.5  0.127±0.025  0.80  •69-2.3  0.04  692.0M  690.9  0.09710.019  0.80  692 .4  0.058  756.7K  706.57  0.23810.020  .0.67  756.8  . 0.013  .722.9L  718.0  0.04810.010  0.67  725.8  0.40  872.6K  824.0  0.40810.018  0.70  874.2  " 0.18  872. 61  868.0  0.08410.017  0.67  875.8  0.57  995.9K  945 .0  0.23410.047  0.71  995 .2  0.07  1004.5K  953.2  0.39310.080  0.66  1003.4  0.11  1278,1.  0.29  12J4,4K  1227.9  Table V  '  1.4  0 .299 ± 0.070 1.64  .87.5  0.69-  —  Data o f P Conversion Peaks i n f i g u r e s  20a,b,  ( E x p l a n a t i o n g i v e n . i n t h e f o o t n o t e *, p . 9 4 ) .  -96-  ;  :  .  d i s a p p e a r when t h e e n e r g y i s a b o v e 121.30 K e v ( i . e . a b o v e t h e 122.9M p e a k ) ,  and a l l t h e c o n v e r s i o n p e a k s h a v i n g  greater than t h i s are f a i r l y  energies  s y m m e t r i c a l . T h i s c a n a l s o be s e e n  f r o m t h e v a l u e s o f t h e FWHM p r e s e n t e d  i n t a b l e V. The i n t e n s i t y  o f t h e 86.9K p e a k c o u l d n o t be e s t i m a t e d , b e c a u s e i t i s b a r e l y a b o v e t h e n o i s e l e v e l and t h e t a i l extends  below t h e d i s c r i m i n t o r  due t o s c a t t e r i n g  certainly  l e v e l o f t h e a m p l i f i e r . The  two humps b e t w e e n t h e 86.9K and IO5.3K p e a k s a r e s u s p e c t e d t o be A u g e r e l e c t r o n l i n e s b u t h a v e n o t b e e n p o s i t i v e l y It  i s i n t e r e s t i n g t o note  t h a t t h e 121.8K p e a k b e l o n g i n g t o  Eu-152 i m p u r i t y i s s e p a r a t e d and  t h a t t h e former  identified.  f r o m t h e 122.9K p e a k o f Eu. 1 5 4 ,  i s o n l y 4.12 % o f t h e l a t t e r .  Since  these  p e a k s a r e r e s p e c t i v e l y t h e s t r o n g e s t p e a k s i n E u 152 and E u 1 5 4 , the percentage  r a t i o was made u s e o f i n c a l c u l a t i n g a l l t h e  o t h e r Eul52 peak i n t e n s i t i e s p r e s e n t  i n t h e spectrum.'. F o r  e x a m p l e , t h e i n t e n s i t y o f t h e 244.7K o f E u l 5 2 e x p e c t e d present  i n t h e 247.6K p e a k o f E u 154 was c a l c u l a t e d .  t o be This  i n t e n s i t y was s u b t r a c t e d f r o m t h a t o f t h e l a t t e r p e a k i n o r d e r to find  the experimental  value f o rthe i n t e r n a l  c o e f f i c i e n t . The p r o c e d u r e  conversion  was a l s o a p p l i e d t o t h e gamma  p e a k s 122.93 Kev and 247.63 Kev a s m e n t i o n e d p r e v i o u s l y . From f i g u r e 20b, t h e main i n t e r e s t l i e s  i n t h e group  o f p e a k s f r o m 679.4K t o 722.9L shown i n a n e x p a n d e d s c a l e .  . in  :  '  '• • •  -97-  f i g u r e 20d. According  peaks namely 6 7 8 K other. I n this  t o Hamilton  and 6 8 2 K  spectrum,  36) et a l , two c o n v e r s i o n  were found  lying  we o n l y f o u n d  679.4K and FWHM o f 0.73  c l o s e t o each  a s i n g l e peak  labelled  %. I f t h e r e a r e i n f a c t t w o p e a k s a s  r e p o r t e d , t h e p e a k w i d t h w o u l d h a v e b e e n much l a r g e r t h a n of t h e neighbouring discussion w i l l  peaks. T h i s i s not the case.  be g i v e n i n t h e n e x t  t h a t n e e d s t o be m e n t i o n e d or beta  Further  c h a p t e r . The n e x t  i s t h a t we h a v e n o t f o u n d  c o n v e r s i o n peak c o r r e s p o n d i n g  g i v e n i n t h e same r e f e r e n c e . F i n a l l y ,  t o t h e 6 2 6 Kev  run of t h e beta the  experiments,  point  a n y gamma transition  i t s h o u l d be n o t e d  t h e 1274.4K p e a k g i v e n I n t a b l e V was n o t I n c l u d e d f i g u r e s 20a,b,d, b e c a u s e t h i s  p e a k was o b t a i n e d  those  that  i n the  i n the third  w h i l e t h e above f i g u r e s were  from  second r u n . From t h e l a s t  c o l u m n o f t a b l e V, t h e a g r e e m e n t f o r t h e  K p e a k s o f E u 154 i s g o o d . T h e r e l a t i v e  intensities  i n the.'  same t a b l e h a v e n o t b e e n c o r r e c t e d f o r t h e E u 152 i m p u r i t y . The to  c o r r e c t i o n was done l a t e r o n l y f o r t h e K p e a k s i n o r d e r " find  the K-conversion To  find  coefficients.  the K-conversion  •  coefficients,  the  1 2 2 . 9 3  Key  t r a n s i t i o n , . w h i c h was c o n s i d e r e d t o be p u r e l y E 2 , was c h o s e n as t h e s t a n d a r d . F r o m t h e i n t e n s i t y d a t a g i v e n i n t a b l e s I I I 58) and V, and t h e t a b l e o f S l i v e t a l . , we h a v e f r o m e q u a t i o n .(101),  7  •  '•  i  .  -99X = K  <Vst  =  (Wst  1 0 0  =  260  -  (from Table I I I )  0 0  '°  (from Table  (E2 c o n v e r s i o n - 1 2 2 . 9  ( % ) - = 0.652 Q 1  With  the.se v a l u e s and e q u a t i o n  conversion Table E2  V)  coefficients  (101),  t h e e x p e r i m e n t a l re-  were c a l c u l a t e d ,  VI together w i t h the t h e o r e t i c a l  and a r e l i s t e d i n  v a l u e s f o r E l , M l , and  transitions. The d a t a i n T a b l e  VI confirms  o f a l l b u t one o f t h e t r a n s i t i o n s  the expected  between p o s i t i v e  The one e x c e p t i o n i s t h e t r a n s i t i o n o f e n e r g y has a c o n v e r s i o n c o e f f i c i e n t This suggests component  ( calculated  the t o t a l  out.  692 Kev,  ) i s approximately 6  o f u n c e r t a i n t y do n o t q u i t e  assignments Figure  22.  attention  i n an  I t i s possible of course,  effort that  underestimated.  Kev and t h e 1 2 7 4 . 4  involve negative parity  Kev  E2 v a l u e , a l t h o u g h M l and E l c a n be  l i m i t s were somewhat  The 7 2 2 . 9  E2 v a l u e .  591.7  The  e x p e r i m e n t a l l y , and many c o u n t s were a c c u m u l a t e d  the e r r o r  which  percent  T h i s c o n v e r s i o n peak r e c e i v e d a l o t o f  t o a t t a i n good s t a t i s t i c s .  states  t h a t t h e S2 c o n v e r s i o n  EO + E2 t r a n s i t i o n p r o b a b i l i t y .  encompass t h e t h e o r e t i c a l  parity  the t h e o r e t i c a l  on t h e a s s u m p t i o n 0.0052  E2. n a t u r e  The m a g n i t u d e o f t h i s  h a s a n D C ^ whose l i m i t s  gamma-ray  ruled  ten times  a n EO c o m p o n e n t .  c o e f f i c i e n t i s indeed of  Kev)  Kev t r a n s i t i o n s  both  l e v e l s , and a r e most p r o b a b l y E l ,  t h a t a r e c o n s i s t e n t w i t h t h e d e c a y scheme shown i n The 1 0 5 * 3  Kev t r a n s i t i o n h a s n o t b e e n f i t t e d  t h e d e c a y scheme, b u t i t a p p e a r s  also  t o be E l o  into  -100-  K- I n t e r n a l  Peak Name  Experimental  C o n v e r si o n  Coefficients  (E2)  (Ml)  (El)  105.3K  0.272  ±0.033  1.002  1.469  0.216  122.9K  0.652 ± 0.022  0.652  0.941  0.142  24-7. 6K  0.085 i 0.006  0.0815  0.135  0.0218  591.7K  0.0065  0.0011  0.00803  0.0140  0.00281  692.OK  0.056 i 0.0055  0.00519  0.00954  0.00201  722.9K  0.00254+0.00021 0.00471  0.00857  0.00183  756.7K  0.0058 +0.0010  0.00426  0.00766  0.00167  872.6K  0.00347i0.00031 0.00314  0.00542  0.00127  995.9K  0.00238±0.00058  0.00238  0.00395  0.000988  1004.5K  0.0022510.00056 0.00234  0.00388  0.000971  1274.4K  0.0007li0.00023 0.00145  0.00219  0.000631  Table VI  ±  Comparison o f K - I n t e r n a l Conversion C o e f f i c i e n t s 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 E l , M l , and E2 t r a n s i t i o n s taken from reference ( 5 8 ) .  -101-  The a n a l y s i s K u r i e - p l o t method  on t h e b e t a  c o n t i n u u m was done by u s i n g  ( s e e f i g u r e 2 l ) , and t h e r e s u l t s a r e  presented  I n t a b l e V I I . ( F o r d e t a i l s o f t h e a n a l y s i s , see  reference  65.)  B e t a E n e r g y (Mev) .  Transition  Intensity  1.866+.012  10.8 ± 0 . 1 2  1.1 98 ±.060  0.67±0.49  0.976 + .030  4.6 ± 3 . 8  0 . 8 4 3 + .015  17-0 ± 3 . 9  0 .579 + .005  37-8 ± 3 . 5  0.274 ± .010  2 9 . 1 ± 2.5  Table V I I . beta t r a n s i t i o n s .  {%)  E n e r g i e s and r e l a t i v e i n t e n s i t i e s o f t h e  -102-  P i g . 102 The  Kurie  plots  N=counts/min. p=momentum(mc) f=reduced Fermi function 6=correction f a c t o r (see r e f . 55)  CfN/EKtjr  (MEV)  -103-  CHAPTER V I I ..  THE  •$1.  DECAY SCHEME AND MODEL F I T T I N G  The D e c a y Scheme.  Summing a l l t h e r e s u l t s and  gamma-spectroscopic work as w e l l  coincidence chapters, and  obtained  investigation  a s t h e gamma-gamma i nthe previous  three  t h e d e c a y scheme o f E u 154 t o Gd 154 was c o n s t r u c t e d ,  the excited  s t a t e s o f Gd 154 r e s u l t i n g  areshown i n f i g u r e  from t h i s  22. The r e l a t i v e i n p u t and o u t p u t  i n t e n s i t i e s with respect The  described  so f a r f r o m t h e b e t a -  decay transitional  t o e a c h l e v e l a r e p r o v i d e d . i n t a b l e VHT.  level structure  presented  whole I n agreement w i t h t h o s e  i nthe figure  constructed  i s as a  by t h e p r e v i o u s  36) workers, i n p a r t i c u l a r by Hamilton are  important  differences.  improvement i n t h e a c c u r a c y be  e a s i l y perceived  determined values  et a l  , although  I n general, there  i s a marked  of the l e v e l energies, which can .  from t h e f a c t that a l l t h e . e x p e r i m e n t a l l y  f o r t h e gamma t r a n s i t i o n a l e n e r g i e s  t h e d e c a y scheme w i t h e n e r g y d e v i a t i o n s l e s s t h a n The. t w o new weak t r a n s i t i o n s and  there  532.11 K e v , were a s s i g n e d  discovered,  to the transitions  f i t into  1 Kev.  i.e.  903.60  Kev  from t h e  • l e v e l s 1718.80 Kev and 1397-36 K e v t o t h e l e v e l '814.77 Kev .respectively.  ( T h e s e l e v e l s were, a l l a s s i g n e d , f r o m o t h e r , more ..  Gd  C22JL  1718.8 (2-)  591 61  [211] 722 L90  1397.4 (2-) 90!  I42JQ  126 3 .3 (4+)  60 582 11-  I3JJ1  1127.4 (3+) 99 5.8  154  1595 87 f - 1049 .0 (4+)  I22J1  (2+)  678 ^0 892. 74 Fig.22  The e x c i t e d  states  of Gdi54 obtained  814.8  756 71  from  (2+)  [012]  8 7 2 . h2  (The l e v e l and t r a n s i t i o n s  1274 43  9 9 5 94  680.1  (0+)  370.6  (4+)  1444 02  626 .0  692 .02  are o b t a i n e d from r e f .  815 02  • 36.)  557 96 679 4  A. 247 63 C211] [Oil]  i M O  -t" I  1004 50  the decay o f Eul54.  given i n dotted l i n e s  i  C212]  122.93(2+) 122. 93 0.0  (0+)  -105-  Transitional Intensities (Relative) Input  Level  y  Name  +  CE.  Output  Y  +  C.E.  Difference  B e t a from  ( in% )  Continuum  (Kev)  (%)  [212] " 1718.8  64.72+3.06 64.72+3.06 (26.33%)  29.1+2.5  [211]1397.4  93.80+4.30 93.80+4.30 (38.10%)  37.8+3.5  [421] 1263.3  +  1.23+0.20  [311] 1127.4  +  [221] 995.8  +  [411] 370.6  +  [211] 122.93  +  [212] 814.8  1.23+0.20 (0.50%)  10.58+0.55  54.3112.04 43.73+2.59 (17.80%)  17.0+3.9  47.43+2.06  54.25+2.50  6.82+4.56 (2.77%)  4.6+3.8  13.75+0.95  17.73±0.80  3.98+1.75 (1.65%)  .  192.85+8.93  220.014.44 27.15+13.37 (11.07%)  +  [012] 680.10 +  T a b l e VUI  3.77+0.39  6.9810.88  3. 21+1.27 (1.31%)  0.0  0.9010.17  0.90+0.17 (0.37%)  10.8+0.12 0.67±0.49  T r a n s i t i o n a l i n t e n s i t i e s f o r gammas, c o n v e r s i o n e l e c t r o n s K  and b e t a s .  L e v e l name n o t a t i o n , [LNnj  .  -106- . I n t e n s e gamma-rays.) The e n e r g y d e v i a t i o n s a r e l e s s t h a n f- K e v . T h i s energy  agreement i s e x c e l l e n t . I n a d d i t i o n , t h e s e  transit-  i o n s a r e p r e d i c t e d t o h a v e no c h a n g e o f a n g u l a r momentum b u t only  a  change o f p a r i t y  ( E l ? ) . F o r these, t h e beta conversion  p e a k s s h o u l d be much w e a k e r I n i n t e n s i t y t h a n t h e gamma  peaks.  T h i s i s t h e case i n t h e p r e s e n t i n v e s t i g a t i o n . A l t h o u g h t h e gamma t r a n s i t i o n s h a v e b e e n d e t e c t e d , none o f t h e c o n v e r s i o n peaks have been f o u n d . From t h e l o w o u t p u t i n t e n s i t y o f t h e l e v e l 8l4.77 K e v g i v e n i n t a b l e VI3I, i t i s o b v i o u s t h a t two  t r a n s i t i o n s a r e n e c e s s a r i l y weak, w h i c h  these  i s the reason that  t h e y have n o t been o b s e r v e d b e f o r e . C o n s e q u e n t l y , t h e s e two t r a n s i t i o n s c o n t r i b u t e t h e major  portion of the population of  t h e l e v e l 8l4.77 K e v .  36) Hamilton et a l  suggested  a l e v e l o f 1049 K e v o n  t h e b a s i s o n l y o f t h e e x i s t e n c e o f t h e 678K b e t a c o n v e r s i o n p e a k . I n t h e p r e s e n t w o r k , no s u c h p e a k was f o u n d the beta c o n v e r s i o n spectrum we c a n f i n d  either i n  o r i n t h e gamma s p e c t r u m .  no o t h e r t r a n s i t i o n f r o m t h i s  Also.  l e v e l t o any o t h e r  l e v e l o r v i c e v e r s a . A l t h o u g h t h e 1049 K e v l e v e l was w e l l -  33) established  i n t h e decay  r e a s o n s t o doubt  o f Tb 154 t o Gd 154 .  i t s presence  i n t h e decay  t h e o t h e r h a n d , a n o t h e r 4+ l e v e l o f e n e r g y found  due t o t h e p r e s e n c e  , we h a v e  o f E u 154. On 1263.3 K e v was  o f 892.74 Kev.gamma  transition.  -107-  As w i l l  be s e e n i n t h e n e x t  by t h e A s y m m e t r i c R o t a t o r  .. "  section, this  M o d e l . The l a s t p o i n t 36)  f r o m t h e scheme o f H a m i l t o n e t a l 626 Kev  l e v e l was  of difference  i s t h e absence of t h e  transition. I n t a b l e VTH, t h e sum o f t r a n s i t i o n a l  •  predicted  •  -  \  intensities  .  •, V  o f gammas p l u s were g i v e n .  conversion  e l e c t r o n s t o and f r o m e a c h  A f t e r allowing f o r the experimental  difference  should  be c o n t r i b u t e d  the ground  s t a t e o f E u 154 t o t h a t  are  also expressed  the  experimental  the  last  level  e r r o r , the  by a b e t a t r a n s i t i o n  from  l e v e l . These d i f f e r e n c e s  i n p e r c e n t a g e so a s t o compare them  values  f o r the beta t r a n s i t i o n s  c o l u m n . The r e s u l t  that a l l the data are q u i t e  of t h i s  given i n  c o m p a r i s o n shows  consistent.  with  clearly  |2.  Model F i t t i n g .  •  The: t h e o r y f o r c a l c u l a t i n g e n e r g i e s and t h e reduced  t h e gamma t r a n s i t i o n a l  t r a n s i t i o n p r o b a b i l i t i e s base on  t h e A s y m m e t r i c R o t a t o r M o d e l was g i v e n i n C h a p t e r proceed  t othe actual  c a l c u l a t i o n , we made u s e o f f i v e  by D a v i d s o n . .  first  program c a l c u l a t e s  t h e m a t r i x elements o f  t h e a n g u l a r momentum c o m p o n e n t s g i v e n e x p l i c i t l y (L(L+1)  •>2,  - K ) /2  t  -  i  l i t ( L ^ K - L ) ( L ^ K ) ( L + K + l ) (L±K+2)^j  . f o r K'=K+2  2  |LMK'> = ( ^  2  L  +  l  ^  ' ^  / 2  >  f  (L+K+2)J ^,  t t ( ( L + K - l ) (L+K) ( L + K + l )  <LMK/b3|LMK'>  and  thus  =  matrix i s diagonalized  in  r K  for  '  =  K  K'=K+2  t h ematrix elements o f the r o t a t i o n a l  (see equation  eigenvalues  °  ^^KK'  calculates  Hamiltonian  as follows,  f o r K' = K  2  <LMKf L H L M K ' > = <  <LMK|L  basic  41)  programs w r i t t e n The  2. To  £ (L) N  (27)) f o r a g i v e n y . T h e n , t h e l a t t e r  b y u s i n g t h e J a c o b i a n method. The  as w e l l  as the c o e f f i c i e n t s  CK a r e f o u n d  t h i s way. The  second program s o l v e s e q u a t i o n  p. by. i t e r a t i o n p r o c e s s  and c a l c u l a t e s  (38) f o r a g i v e n  Z, f r o m e q u a t i o n  (39) ••  . 41) The  quantum number )> i s a v a i l a b l e  linear  {  interpolation.  state  i n t h e ground  level  energy  (E n^ 2  Hence, t h e t h e o r e t i c a l  state a  i n tabular  r  e  rotational  calculated  form  by  r a t i o s o f each  band t o t h e f i r s t 2+-.  b y means o f e q u a t i o n (44)  •  •'•'v  and  -109-  .  (44a). The  t h i r d program i s the  same as t h e  t h e quantum number i s r e p l a c e d by  second except  ^ , which is. obtained  that  from  41) another beta  table  . T h i s g i v e s the energies of the f i r s t  excited  band. The  fourth  program j u s t  calculates  the  rotational  p a r t o f t h e E2  t r a n s i t i o n p r o b a b i l i t i e s . T h i s i s done  by  using equation  (59)  known  and  w i t h the c o e f f i c i e n t s  w i t h a Clebsch-Gordan C o e f f i c i e n t s The  C^  and  N  subroutine.  l a s t program t h e n c a l c u l a t e s  the v i b r a t i o n a l  of the p r o b a b i l i t y . T h i s program c o n t a i n s a l l the necessary  for solving i n (68)  integration i n t o 20  (68)  equations  y  (69).  and  i s done.by d i v i d i n g  the  part  subroutines  Note t h a t  integral  the  range  increments. All  ( A = 2  t h e above programs a p p l y t o t h e q u a d r u p o l e ) and  the o c t u p o l e case  In the present  calculation,  ( A = 3 the f i r s t  case...'  ) . three  programs  were combined t o g e t h e r t o y i e l d a l l t h e l e v e l e n e r g i e s f o r a given pair  o f 7 and  l e v e l energies are also deviation al  read  of the c a l c u l a t e d  ones i s found  l e v e l s having ••.the "two  . The  no  i n . The  root-mean-square  l e v e l energies from the  i n percentage experimental  p a r a m e t e r s y and  .which enclose  experimental values of a l l the  form.  In this calculation,  correspondence  \K. a r e v a r i e d  experiment-  within  those  a r e o m i t t e d .' T h e n suitable  a minimum r o o t - m e a n - s q u a r e d e v i a t i o n  of  ranges the  level  energies. F i n a l l y ,  iteration,  and  t h i s minimum i s d e t e r m i n e d  so a l s o t h e c o r r e s p o n d i n g  applying the l a s t  two  I X . The as  energies.  programs, the r a t i o s o f the  t r a n s i t i o n p r o b a b i l i t i e s are a l s o found. energy f i t t i n g s  level  i n the quadrupole  The  fy  ~  ,„Th  = l^ LNn  By  reduced  r e s u l t s of  case are p r e s e n t e d  root-mean-square d e v i a t i o n i n t h i s case  follows,  by  the  In table  i s defined  ^ E x p ^ U  E  m  -LNn > J  ( 1 Q 2 )  where t h e s u m m a t i o n s a r e c a r r i e d o u t o v e r t h e l a s t s i x e n e r g y l e v e l s i n t a b l e IX, - E ? ^ are the experimental l e v e l ( i n Kev)  determined,  the f i t t e d  and  t h e o r e t i c a l l e v e l e n e r g i e s . Note t h a t the  i n a n a r b i t r a r y e n e r g y s c a l e , and  a r e  t h a t t h e e q u a t i o n (103)  Th. obtained under the r e s t r i c t i o n of E Q - ^ = The  energies  Exo ^QJJ_  and  E  Th ^ =  is E  0  Exp ^ i i •  p u r p o s e o f t h i s r e s t r i c t i o n i s t o o b t a i n t h e same e n e r g y .  s c a l e - f o r b o t h E?i?  and E ^ . P  j_iNn  t h i s method o f f i t t i n g square f i t .  We  I t s h o u l d a l s o be m e n t i o n e d  i s d i f f e r e n t from the standard  RMS  least-,  b e l i e v e t h a t t h e a p p l i c a t i o n o f t h e l a t t e r method  w i l l meet w i t h g r e a t c o m p u t a t i o n a l d i f f i c u l t i e s * The  that  LNn  d e v i a t i o n D-r^jg r e s u l t i n g f r o m  "; :  the values i n .  t a b l e .IX i s 1.503 p e r c e n t , and t h e c o r r e s p o n d i n g v a l u e s f o r t h e asymmetry p a r a m e t e r T and t h e s t i f f n e s s JU. a r e r e s p e c t i v e l y 11.52  d e g r e e s and  0.402.  ' S i m i l a r (unpublished)  c a l c u l a t i o n s have. been.done by.  67) Davidson  using another  s e t o f e x p e r i m e n t a l d a t a . The  results  To f o l l o w page 110. '. . . ., o b t a i n e d f o r T and  a r e 11.62 d e g r e e s and 0.401  respectively;  i n g o o d a g r e e m e n t w i t h o u r s . A v e r y r e c e n t p u b l i c a t i o n by Aisenberg  et a l  6 8  ^using  T = 11.5 and  gives  t h e method o f D a v y d o v and C h a b a n ^ 1 2  Y =0.8.  The e x p e r i m e n t a l d a t a i n t h i s 34)  p u b l i c a t i o n have b e e n d e t e r m i n e d of  by Yoshizawa e t a l  b y means  Coulomb.excitation.  Experimental Ener  §  y  E  (Kev)  Sn  Theoretical Energy  E^  Deviation  L N n  (%)  n  (Kev)  0.00  0.00  0 1 1  122.93  122.93  2 1 1  995.75  1052.73  5.41  2 2 1  1127.35  1121.31  0.54  3 1 1  370.56  365.54  1.37.  4 1 1  1263.30  1211.96  4.24  4 2 1  680.10  676.49  0.53  0 1 2  814.77  831.85  2.05  2 1 2  Table IX  Comparison o f the e x p e r i m e n t a l and t h e o r e t i c a l e n e r g i e s i n the quadrupole parity).  _  " •'_ .• ..  case  (^=2,  positive  -111-  To f i n d t h e r a t i o s o f t h e r e d u c e d t r a n s i t i o n  probabilities  f r o m t h e e x p e r i m e n t a l d a t a , we made u s e o f e q u a t i o n ( 5 1 ) . Onlyf o u r r a t i o s were c a l c u l a t e d , transitions  f r o m t h e same i n i t i a l  of t h e ground are  each o f which c o n s i s t s  state  state  beta band. Assuming  o f two  to different a l lthese  levels  transitions  E 2 , t h e n f r o m e q u a t i o n ( 5 l ) , / = 2 ; and h e n c e B(E2,L-r -> L n )  (E72) 5  =  B(E2,L -^» L ) i  where  E  y  =  (E )  f 2  fa  n  gamma e n e r g i e s , t h e r a t i o s  Energy  2  (  B  and T E  2  > i L  B(E2,Li  i ntable  T (E2)  5  • , t h e gamma e n e r g y . S u b s t i t u t i n g  al relative intensities for T  below  T i (E2)  L  2  and t h e c o r r e s p o n d i n g  f i )  were found a s  Lf2)  X together with the theoretical  Ratio  Expt. Value  Name  t h e experiment-  values.  Theo. V a l u e  995.94 872.62  [221] [bit! [22LJ -> [211]  0.43410.040  0.443  444.02 692.02  (212] (411} (212] - » [2113  3.790+0.629  4.352  815.02 692.02  [212] — 1211]  0.152+0.014  0.373  1.026+0.104  1.004  [212]  [31.1] - f e l l ] [311] -* £411)  1004.5 756.71  :Table X  Con]  The b r a n c h i n g r a t i o s o f gamma t r a n s i t i o n s " b e t w e e n  positive-parity  levels.  -112-  Three o f the four, c a l c u l a t e d  ratios  i n Table  with- t h e measured v a l u e s w i t h i n t h e e x p e r i m e n t a l The t h i r d  ratio  then  i s less  than h a l f the value p r e d i c t e d .  from  t h e 6 9 2 Kev t r a n s i t i o n h a s a s t r o n g EO c o m p o n e n t .  i t was o n l y 6 p e r c e n t . i n Table  X on t h i s  To a c c o u n t  While  error limits, again t h i s  but predicted that  f o r the low 815/692  6 9 2 Kev t r a n s i t i o n  o f the t o t a l  The 6 9 2 Kev t r a n s i t i o n 444/692.  this,  The.  ratio  b a s i s a l o n e , t h e EO component w o u l d h a v e t o  e q u a l a b o u t 60 p e r c e n t  intensity.  i s i n v o l v e d i n one o t h e r  t h e o r y and e x p e r i m e n t  the t h e o r e t i c a l value  agree here  ratio,  w i t h i n the  i s on t h e h i g h s i d e .  If  by t h e 6 9 2 Kev EO c o m p o n e n t , i n t h i s  i s caused  case  o n l y r e q u i r e s . i t t o be a b o u t 12 p e r c e n t f o r t h e mean  experimental value i s not l i k e l y this  to match the t h e o r e t i c a l p r e d i c t i o n .  t h a t the low 815/692  ratio  So i t  c a n be e x p l a i n e d i n  way. Por  the o c t u p o l e  parity levels assigned  (1718.8  case,  and 1 3 9 7 . 4  {~212]~ and  them t o be  s i n c e we h a v e o n l y two n e g a t i v e Kev) both w i t h s p i n 2, ^21l]~ respectively.  T p-"  which  the s p i n assignment o f the l e v e l s i n v o l v e d ,  conversion c o e f f i c i e n t data indicated  it  uncertainties.  t h a t i f t h e 8 1 5 Kev t r a n s i t i o n i s p u r e E 2  This suggests seems l i k e l y  815/692  X agree  were f o u n d  t r a n s i t i o n was transitions  t o be 4 . 6 0 6 found  linking  it  and.the  d e g r e e s and 0 , 4 9 7 2 5 .  b e W e e n them, a l t h o u g h them t o p o s i t i v e  parity  (0 + — > Q ± j ) was n r  calculated  from  With  these  stiffness  No gamma  there are s i x levels.  The m o n o p o l e t r a n s i t i o n p r o b a b i l i t y T(E0) transition  we  f o r the-  equation  ( 7 8 ) . The  -1131^(0),  functions  I ,^(2),  Davidson's subroutines. parameters  I , ( 0 ) and I ( 2 ) were e v a l u a t e d  p  y  y /  The n u m e r i c a l v a l u e s o f t h e n e c e s s a r y  follow;-  E  = 0 . 5 5 7 9 6 Mev.  fl  = 0.335 x 1 0  1 1  Z' = 2.64444 sec"  1  (ref.  ( Z f 5  >)  Z-f = 2 . 7 5 5 5 5  p. = 0 . 4 0 2 1 5 8  1^(0)  ( r e f . (16)  f> =  0.303  z  =  2.53275  z-,  = 2.56668  0  From t h e s e  =. 1.726.0  1^(2) = 1.76-84  r o t = °«964775  p  v/ith  T(E0) T(E2)  data,  (0;>  Ognd)  v  1^(0)  = 6.. 3418  .1^(2)  = 1.5334  °«085  = =  from  (78)  0.039 ± 0.012 (experimental)  I t thus appears t h a t the p r e d i c t i o n s o f the Asymmetric  Rotator  m o d e l , w h i l e s u c c e s s f u l f o r t h e E2 r a t i o s , a r e n o t r e l i a b l e f o r monopole  transitions. We n e x t programmed  692 K e v .  fl  =  (78a) f o r the t r a n s i t i o n  The v a l u e s o f t h e p a r a m e t e r s u s e d  = 0.69202  E  equation  Mev  O.3.5 x: 1 0  1 1  were  Z = sec"  Z  1  x  = 2.7 5 5 5  |JL = 0 . 4 0 2 1 5 8  1^(0)  = 6.855  r o t = 0.2714  1^2)  =  p  B  1.236  = 0.303  0  °  Thence  TiEOj.  (  T(E2)  2  -->2j  +  n d  )  =  0.158  P*  f r o m w h i c h t h e p r e d i c t e d E0 component i s a p p r o x i m a t e l y c e n t o f t h e t o t a l 692 t r a n s i t i o n . too l a r g e (as above), results.  Assuming i t i s about  13 p e r twice  i t i s c o n s i s t e n t v/ith the c o n v e r s i o n  data  -114-  CHAPTER ¥111 CONCLUSIONS  • The i n f o r m a t i o n d e d u c e d f r o m t h e p r e s e n t  experimental  data as w e l l as from t h a t o f the other workers l e a d s t o a fairly Eu  complete l e v e l  s t r u c t u r e o f Gd 154 f r o m t h e d e c a y o f  1 5 4 . The gamma t r a n s i t i o n s  from o n e - l e v e l t o another  been e x h a u s t i v e l y s o r t e d o u t . S h o u l d transitions,  they  not  that there w i l l  expected  energies the  of the levels  presence  III,  must be e x t r e m e l y  t h e r e be a n y new gamma weak i n i n t e n s i t y .  It i s  be f u r t h e r m o d i f i c a t i o n t o t h e  t o any a p p r e c i a b l e e x t e n t . Because o f  o f some o f t h e w e a k e r t r a n s i t i o n s  w h i c h a p p a r e n t l y h a v e no p l a c e  h o p e d t h a t new l e v e l s w i l l these  have  be f o u n d  shown i n t a b l e  i n t h e decay scheme, i t i s i n the future to include  transitions. Comparison o f the energies  gamma t r a n s i t i o n  of the levels  and t h e .  p r o b a b i l i t i e s with the t h e o r e t i c a l values .  b a s e d on t h e A s y m m e t r i c R o t a t o r M o d e l d e m o n s t r a t e s a s a whole t h e v a l i d i t y even n u c l e u s remarkable.  o f t h i s model i n t h e p r e s e n t  (Gd 154).  The c l o s e f i t s  The e r r o r l i m i t  sition i s relatively  i n most c a s e s a r e  of the o —>o +  c h o s e n e v e n - •-.  +  monopole  large.. Experimentally,  tran-  re-investigation  -115-  o f the monopole t r a n s i t i o n s  i n t h i s n u c l e u s m i g h t be  best  carried  o u t by means o f Coulomb e x c i t a t i o n , b e c a u s e t h e  excited  s t a t e s i n the p r e s e n t i n v e s t i g a t i o n are weakly  ulated. great  Thus the t r a n s i t i o n  i n t e n s i t i e s were e s t i m a t e d  and  b e e n shov/n i n t h e  7 ),  ( Chapter  s e c t i o n on m o d e l  t h e s t i f f n e s s p a r a m e t e r JJL  i n v o l v e s the  energies.  I n t h i s m e t h o d , we  and  lowest energy  h a v e o n l y two  an a s s i g n m e n t . parameters  w o u l d h a v e t o be v a r i e d  o f the reduced JX .  One  ( i.e. deviation.  i n order to minimize  probabilities  than those  so t h a t t h e a c c u r a c y o f t h e method  values  i n order to evaluate  However, the u n c e r t a i n t y l i m i t s  a r e i n g e n e r a l much l a r g e r  the  certainly  c o u l d , o f c o u r s e , i n c l u d e t h e RKS  transition  such  additional adjustable  C o m p u t a t i o n by t h i s method w o u l d  be more c o m p l e x .  present  parameters  t o r e l a x the c o n d i t i o n o f  In consequence, Wo  deviation.  levels  determined  ) t o v a r y i n o r d e r t o a c h i e v e a minimum RMS  An a l t e r n a t i v e method -would be  and  fitting,  assignment  t o be e q u a l t o t h e c o r r e s p o n d i n g e x p e r i m e n t a l l y  y  with  the d e t e r m i n a t i o n o f the asymmetry parameter  o f t h e t h e o r e t i c a l e n e r g i e s o f t h e two  RMS  pop-  difficulty. As h a s  Y  beta  on  these  o f the l e v e l  ratios  energies  c o u l d be r e d u c e d  over  the  approach. Finally,  i t i s u r g e n t l y t o be  hoped  t h a t the  further  development o f t h i s model o f the n u c l e u s v / i l l i n c l u d e the p r e d i c t i o n o f the t r a n s i t i o n p r o b a b i l i t i e s between n e g a t i v e . p a r i t y and  positive  parity levels.  There are a  sufficient  -116-  number o f s u c h t r a n s i t i o n s i n e v e n - e v e n n u c l e i c a l c u l a t i o n s b o t h i n t e r e s t i n g and  useful.  t o make s u c h  -117-  REPERENCES  •  1.  K. S l e g b a h n , A l p h a - , B e t a - a n d  2.  A. B o h r , M a t . F y s . M e d d . D a n . V i d . S e l s k  3..-  J . R a i n w a t e r , P h y s . R e v . 79, 432 (1950)..  4.  A. F a e s s l e r & ¥. G r e i n e r , Z. P h y s i k 168, 425 (.1962) .  5.  A. F a e s s l e r & W. G r e i n e r , Z. P h y s i k 170, 105 (1962).  6.  A. 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