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Some angular correlation functions for successive nuclear radiations Hess, Forest Gene 1951

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SOME  ANGULAR  CORRELATION  SUCCESSIVE  NUCLEAR  FUNCTIONS  FOR  RADIATIONS  BY FOREST  THESIS  SUBMITTED  THE  REQUIREMENTS  GENE  IN  MASTER  HESS  PARTIAL  FOR  THE  FULFILMENT DEGREE  OF  OF. - ARTS  i n the Department of Physics  We accept t h i s t h e s i s as conforming, to the standard r e q u i r e d from candidates f o r the degree of MASTER OF ARTS  Members of the Department of P h y s i c s  THE  UNIVERSITY  OF  BRITISH . COLUMBIA  J u l y , 1951  Abstract L e t J * , J , J " r e p r e s e n t the t o t a l a n g u l a r momenta of the i n i t i a l ,  i n t e r m e d i a t e , and f i n a l  r e s p e c t i v e l y and J , J  s t a t e s of a nucleus  the t o t a l angular momenta of the  f i r s t and second emitted p a r t i c l e s .  Then, i n terms of t h i s  n o t a t i o n , the f o l l o w i n g r e s u l t s can be found i n t h i s  thesis.  <*-y and tf-fr c o r r e l a t i o n f u n c t i o n s have been c a l c u l a t e d o  e x p l i c i t l y i n terms of cos 9 f o r those t r a n s i t i o n schemes s a t i s f y i n g the f o l l o w i n g f i ) J ' = J+J  conditions:  J= J +J  f o r a r b i t r a r y J , S - 1, 2r*  t t  lt  2  ( i i ) J'= J - J , J = J - J  x  f o r a r b i t r a r y J ^ , J = 1, 2:;  t t  x  2  ('ill) J»- J ^ - J , J = J + J  2  g  M  2  f o r a r b i t r a r y J ^ , J -r 1, 2. g  ( i v ) J ' - J - J , J = Jg-J° f o r J - ^ 1, 2, a r b i t r a r y J g . x  These are c a l l e d the " s p e c i a l t r a n s i t i o n s " i n the t e x t . tX-mixedJf c o r r e l a t i o n f u n c t i o n s have been t a b u l a t e d e x p l i c i t l y i n terms of:• cos^Q f o r an & p a r t i c l e w i t h  total  angular momentum 1 o r 2 and a photon corresponding to a mixture of e l e c t r i c quadrupole  and magnetic  dipole  radiation.  F o r an (X p a r t i c l e w i t h t o t a l angular momentum 3 the txmixedy c o r r e l a t i o n f u n c t i o n s can be obtained from a t a b l e which l i s t s  the sums of products of angular momentum Coef-  f i c i e n t s a p p e a l i n g I n these c o r r e l a t i o n f u n c t i o n s . r e l a t i o n f u n c t i o n s are too clumsy to be expressed I n terms of cos^9 I n  These corexplicitly  g e n e r a l , however they can be f a i r l y  easily  e v a l u a t e d once numerical values of the angular momenta of the n u c l e a r s t a t e s are p r e s c r i b e d .  Table of Contents. c  -  Page  Introduction  1  I . General e x p r e s s i o n f o r the d i r e c t i o n a l c o r r e l a t i o n  4  function I I . Calculation  of d i r e c t i o n a l c o r r e l a t i o n f u n c t i o n s  15  A. Symmetry of the-summations  15  B. C o r r e l a t i o n f u n c t i o n s f o r s p e c i a l t r a n s i t i o n s  16  C. General method.  25  I I I . c<-mlxedy c o r r e l a t i o n f u n c t i o n s  S'7  A. *(1) -mixedlT(2) c o r r e l a t i o n f u n c t i o n s  27  B. °<(2)-mixedY(2) c o r r e l a t i o n f u n c t i o n s  29  C. <x(3) -mixedy(2) c o r r e l a t i o n f u n c t i o n s  29  Tables:  :  -~  1. Some a n g u l a r d i s t r i b u t i o n f u n c t i o n s  32  Jm 2. E x p l i c i t e x p r e s s i o n s - f o r some Cjtt^uj 3. A t a b u l a t i o n  3  3  M  of the c o e f f i c i e n t s , a ^  34  4. A b b r e v i a t i o n s used i n Tables 5 and 6  35  5. «(1) -mixedV(2) c o r r e l a t i o n f u n c t i o n s  36  6. c<(2)-mixedY(2) c o r r e l a t i o n f u n c t i o n s  58  7. The s u c t i o n , ,  Z(oJ&V(cft£^)*  8. The ^ t l o n s . Z i i ^ ' ) ^ ^ . )  41  2  9. The s u M m t i o n a . ^ C c J ^ j ^ J y ^ o J ? ^  49 5  g  Appendices: A. The normalized angular momentum ^ )  B. To show t h a t ( r f ? C. The  2  summation formulae  D. -.The p r o o f of formula Bibliography  (51)  coefficients  i s a polynomial i n  Acknowledgement  I wish to thank P r o f e s s o r W.  Opechowaki f o r suggest-?  i n g the r e s e a r c h problem and f o r h i s advice and encouragement throughout the performance of the r e s e a r c h .  I am g r a t e f u l to the N a t i o n a l Research C o u n c i l of Canada f o r the d o n a t i o n of a Bursary a Studentship  (1950-51) i n support  (1949-50) and  of the r e s e a r c h .  INTRODUCTION:.  --  I f a nucleus emits s u c c e s s i o n , there w i l l  two p a r t i c l e s or photons i n q u i c k be a c e r t a i n angle 0  d i r e c t i o n s of e m i s s i o n .  The f u n c t i o n ,  between t h e i r , r e p r e s e n t i n g the  r e l a t i v e p r o b a b i l i t y f o r an a n g l e © between the d i r e c t i o n s of  e m i s s i o n of the p a r t i c l e s or photons i s c a l l e d  a l or angular The Hamilton  was  f i r s t d e r i v e d by  f o r the s u c c e s s i v e e m i s s i o n of two photons (lT-T  correlation).  He  c a l c u l a t e d W(0)  i n which the m u l t i p o l e quadrupole or d i p o l e .  explicitly  f o r the  Goertzel  2,  He  c o r r e l a t i o n between the  s t a t e of the nucleus  the h y p e r f i n e s p l i t t i n g Hamilton.  to the  or an e x t e r n a l l y a p p l i e d  can be n e g l e c t e d p r o v i d e d  the intermediate  due  showed t h a t the e f f e c t of the e x t r a - n u c l e a r  e l e c t r o n s on the angular n u c l e a r emissions  either  extended the theory of T -Y  presence of an i n t e r n a l atomic f i e l d magnetic f i e l d .  cases  orders .of the emitted photons are  c o r r e l a t i o n s by c o n s i d e r i n g the e f f e c t on W(0)  of  direction-  correlation function.  g e n e r a l e x p r e s s i o n f o r W(0) 1  the  of t h a t s t a t e .  successive  the r a d i a t i o n width  i s much g r e a t e r This was  than  a l s o shown by  G o e r t z e l a l s o showed that an e x t e r n a l l y a p p l i e d  magnetic f i e l d may  be used to reduce the e f f e c t of the e x t r a -  n u c l e a r e l e c t r o n s on t he angular  correlation.  F a l k o f f and  Uhlenbeck^ c a l c u l a t e d c o r r e l a t i o n f u n c t i o n s i n p a r a m e t r i c form (the parameters depending on the types of p a r t i c l e s emitted) f o r p a r t i c l e s or photons w i t h angular momentum 1 or 2.  Ling  and  Falkoff  4  then extended the theory to Include  i n which mixtures of m u l t l p o l e s are emitted* tf-mixedtf' c o r r e l a t i o n f u n c t i o n s ,  yto  They t a b u l a t e d  where Tf' r e f e r s  quadrupole r a d i a t i o n emitted i n the f i r s t  transitions  to d i p o l e o r  t r a n s i t i o n and mixed  mixed e l e c t r i c quadrupole and magnetic d i p o l e  emitted i n the second t r a n s i t i o n .  radiation  F i n a l l y , Spiers"  how the g e n e r a l a n g u l a r c o r r e l a t i o n f u n c t i o n  6  has Shown  f o r any s u c c e s s i v e  emissions may be d e r i v e d u s i n g the quantum mechanical of angular momenta.  The same r e s u l t was shown by L l o y d ^ u s i n g  group t h e o r e t i c a l methods. the  The above i s a resume of some of  t h e o r e t i c a l papers on angular c o r r e l a t i o n which have been  used i n the p r e p a r a t i o n of t h i s t h e s i s .  F o r a more complete  survey of such papers the reader i s r e f e r r e d This t h e s i s part,  consists  then w r i t t e n  second p a r t , in  to r e f e r e n c e 3.  of three main p a r t s .  I n the f i r s t  the g e n e r a l e x p r e s s i o n f o r the c o r r e l a t i o n f u n c t i o n i s  d e r i v e d by f o l l o w i n g is  addition  S p i e r ' s method.  The g e n e r a l e x p r e s s i o n  i n a form u s e f u l f o r c a l c u l a t i o n s I n  a method of e v a l u a t i n g  the formula f o r 1(8)  the  the summations which appear  i s presented.  The method permits one  to c a l c u l a t e a n g u l a r c o r r e l a t i o n f u n c t i o n s f o r any angular momentum f o r the f i r s t emitted p a r t i c l e o r photon p r o v i d e d the  t r a n s i t i o n s involved  Some <X-rand for  satisfy certain special  conditions*  Y c o r r e l a t i o n f u n c t i o n s are g i v e n e x p l i c i t l y  these s p e c i a l cases.  I n the t h i r d p a r t ,  tables  are g i v e n  from which <<-mixedir c o r r e l a t i o n f u n c t i o n s can be obtained f o r  o  an o*. p a r t i c l e w i t h angular momentum 1, 2,  or 3, and a photon  corresponding to mixed e l e c t r i c quadrupole and magnetic d i p o l e radiation.  1  I.  GENERAL EXPRESSION FOR - T H E DIRECTIONAL CORRELATION P u H C T l O N . The  (not  .  c o r r e l a t i o n f u n c t i o n f o r the case i n which two p a r t i c l e s  photons) are emitted i n quick s u c c e s s i o n by a nucleus i s  d e r i v e d below.  Essentially,  the d e r i v a t i o n due to S p i e r s i s 5  followed. The f o l l o w i n g n o t a t i o n Is used J m ,Jm,J m ,J- m^,Jgmg ,  l  u  u  L  throughout  this  thesisJ  - r e p r e s e n t the t o t a l a n g u l a r momentum  and I t s z component f o r the I n i t i a l ,  i n t e r m e d i a t e , and f i n a l  s t a t e s of the nucleus and the t o t a l angular momentum ( i n t r i n s i c p l u s o r b i t a l ) and i t s z component f o r the f i r s t and second emitted p a r t i c l e s  respectively.  "vVj - i s the normalized wave f u n c t i o n f o r the nucleus i n the m  s t a t e r e p r e s e n t e d by t he quantum numbers Jm. 4>j - i s the normalized wave f u n c t i o n f o r an emitted p a r t i c l e m  w i t h quantized t o t a l angular momentum Jm. Henceforth  the types of c o r r e l a t i o n between p a r t i c l e s and  photons w i t h g i v e n angular momenta w i l l be denoted  as f o l l o w s :  °t(J^)-5"(Jg) means t h a t an « p a r t i d e w i t h angular momentum Jj. i s emitted i n the f i r s t tum J  t r a n s i t i o n and a photon w i t h angular momen-  corresponding to a 2 m u l t i p o l e i s emitted i n the second. a  g  «(J-^)-mixedT(Jg) means  the same as above f o r the f i r s t  but i n d i c a t e s that a photon corresponding to a mixture • J,-1 e l e c t r i c and 2  6  transition, of 2  &  magnetic a t t l t i p o l e r a d i a t i o n i s emitted I n the  second t r a n s i t i o n . In the d e r i v a t i o n , i t i s assumed t h a t the e f f e c t of the  e x t r a - n u c l e a r e l e c t r o n s on the c o r r e l a t i o n can be n e g l e c t e d . If  t h i s were n o t the Case, the t o t a l a n g u l a r momentum of the  nucleus would precess about the f i e l d hence could change i t s v a l u e . constant throughout if  due- to the e l e c t r o n s and  However, t h i s value i s assumed  the d e r i v a t i o n .  T h i s assumption  i s Valid  the r a d i a t i o n width of the n u c l e a r s t a t e i s much l a r g e r  the h y p e r f i n e s p l i t t i n g of t h a t s t a t e .  .  than  _  I f l^'i ,represents the s t a t e of the system c o n s i s t i n g o f the Intermediate nucle us and the f i r s t emitted p a r t i c l e u s i n g the quantum mechanical  Cra;  a d d i t i o n of angular momenta,  ' ^aU i^i ,  1  Here the Q.^ are the p r o b a b i l i t y amplitudes  i / W (  .  (D  f o r the v a r i o u s  p o s s i b l e values of J, ( |J -JJ i= Jj £J '+J) which the f i r s t f  particle  then,  emitted  can have; the bracketed e x p r e s s i o n s are the normalized  angular momentum c o e f f i c i e n t s which may be considered d e f i n e d by (1).  U s i n g (1), one may r e p r e s e n t the two t r a n s i t i o n s of  the nucleus by;  HV^<  Or*  '^^'-'^j-^^W,  (2)  E q u a t i o n (2) may be w r i t t e n i n the form (3)  where  Cllw^^, ^  '"^^^  ^  (4)  Prom (3) i t i s seen t h a t the f i n a l  s t a t e , Ij^,, of the sys-  tem c o n s i s t i n g of the f i n a l nucleus and the two emitted p a r t i c l e s is  g i v e n by  • 2L it <C <£L* ^-w .  Now, a t any time,|l|>^ | 'dV aV d V 2  u  |  l  z  (5)  i s the p r o b a b i l i t y  that,  w i t h the i n i t i a l nucleus i n the s t a t e J'm', p a r t i c l e 1 i s I n the volume dV, about the p o i n t r ^ w i t h momentum  and i t s z component  dV^ about the p o i n t and  i t s z component  dV  about the p o i n t  tt  r 6^with  I n t r i n s i c angular  or , p a r t i c l e 2 i s i n the volume  i n t r i n s i c angular momentum  , and the f i n a l nucleus i s i n the volume  r &V. tt  dV" and  r 0 4) tt  u  tt  represent  symboli-  c a l l y - t h e volume elements and c o o r d i n a t e s of a l l the n u c l e o n s . I n t h i s t h e s i s , only  the d i r e c t i o n s of the two emitted p a r t i c l e s  w i l l be of i n t e r e s t , i . e . the t o t a l p r o b a b i l i t y that p a r t i c l e 1 i s i n dV  at. ijO,^ and that p a r t i c l e 2 i s i n dV^ a t r ^ ^ i s  t  d e s i r e d . - To g e t t h i s i t i s necessary to sum the p r o b a b i l i t i e s for  the v a r i o u s  ticles  spin orientations o f , o f  and t o Integrate  Normally, the i n i t i a l  the two emitted p a r -  the n u c l e a r coordinates over a l l space.  n u c l e i w i l l be randomly o r i e n t e d  2J'+1 degenerate s t a t e s i]/^ , are e q u a l l y p o p u l a t e d .  i . e . the  The average  c o r r e l a t i o n f o r a l l n u c l e i i s obtained by f i n d i n g the weighted average over the 2J'4-1 I n i t i a l s t a t e s . t h i s i s done by summing over m  1  I n the normal case,  and d i v i d i n g by 2J'+1.  Prom  the above statements, i t Is seen that the average c o r r e l a t i o n f u n c t i o n W(r84> (  (  »  i%$i)  T  between the d i r e c t i o n s 0 , ^  and 9 ^ may  be d e f i n e d by  where r  t  and r  t  are. taken to be l a r g e b u t constant.  r e l a t i v e v a r i a t i o n of W w i t h 6 ; a n d 6 ^  Only the  i s of i n t e r e s t e x p e r i -  m e n t a l l y and so, f o r t h i s reason, f a c t o r s independent angles w i l l be omitted from W.  Prom equations(5)  of these  and (6) one  obtains ( o m i t t i n g j ^ p j )  u s i n g the o r t h o g o n a l i t y of the H^-^'s. The  summation over m can be brought  modulus when0,=4| = O.  outside the square  F o r , the wave f u n c t i o n  of an emitted  p a r t i c l e , which has t o t a l angular momentum J and z component m and i n t r i n s i c angular momentum S w i t h z component o"", can be w r i t t e n , u s i n g the a d d i t i o n of angular momenta, as  where .X  i s the wave f u n c t i o n r e p r e s e n t i n g the i n t r i n s i c ang-  u l a r momentum and ^(r)Y"* °(8<$) t h a t r e p r e s e n t i n g the o r b i t a l u  angular momentum of the p a r t i c l e . a b i l i t y amplitudes  The q u a n t i t i e s b  f o r the v a r i o u s p o s s i b l e o r b i t a l  momenta the p a r t i c l e  can have.  L  are probangular  Since Y/^"^00) = 0 u n l e s s m-fT,  i t i s seen t h a t ^ ^ ( r , 00<Jp)-0 u n l e s s m,=cr|, and hence t h a t C * (r, 00<r; ) = 0 u n l e s s m'-m-a^ from ( 4 ) . 0  Thus, i f 0, = <f. =  0  i  there i s only one value f o r m which g i v e s a non-vanishing once m  1  n  (7) * term  and cr- are g i v e n and the summation over m can be d i s c a r d e d ,  i.e.- . •  *  W(r.00,r,M,)=2I ^ I C ^ ,  (r, 0 0 ^ 1 ^  ( r ^ r j f  However-, f o r l a t e r convenience, one can s t i l l  .  SUM over  m, and ft" knowing that the terms of the summation w i l l  m', vanish  m'-aut^, and so I t i s p o s s i b l e to w r i t e  unless  W ( 0 0 . ^ ) . £ S | « ^ ( , 00,^*2 10^.(^^)1*.  (9)  w  S  W(r, OO.r^Q^) does not depend on  because of the aver-  aging p r o c e s s e s used i n o b t a i n i n g i t .  I t depends o n l y on the  angle 6^-0 between the d i r e c t i o n s of e m i s s i o n of the two p a r ticles. Prom equations (3) i t i s seen t h a t P ^ f O ) and P ^ p ) d e f i n e d by  r e p r e s e n t r e s p e c t i v e l y the p r o b a b i l i t y t h a t the i n t e r m e d i a t e nucleus i s i n the s t a t e Jm w i t h the e m i s s i o n of the f i r s t p a r ticle  i n the z d i r e c t i o n (and a t r, ) and the p r o b a b i l i t y  the f i n a l  nucleus i s i n the s t a t e J m l,  y  that  w i t h the e m i s s i o n of the  second p a r t i c l e a t an angle 0 t o the f i r s t (and a t r ) . - u~sing 2  (9) and (10), one may w r i t e the d i r e c t i o n a l  correlation, function  i n the form W(6)=Z1 P J O P J J ) , where i t i s understood  that r  (11) (  and r  z  have l a r g e b u t constant  values. It  should be noted that i n e q u a t i o n (7) there e x i s t s  ant.  I n t e r f e r e n c e between the v a r i o u s ways i n which* a t r a n s i t i o n t o a f i n a l s u b l e v e l . jVm  can occur from a g i v e n s u b l e v e l J'm' via  u  d i f f e r e n t intermediate s u b l e v e l s Jm because the p r o b a b i l i t y  amplitudes  are summed over the i n t e r m e d i a t e s u b l e v e l s before  squaring r a t h e r than a f t e r and cross terms appear.  As seen i n  e q u a t i o n (9), I t i s p o s s i b l e t o remove t h i s i n t e r f e r e n c e by t a k i n g the d i r e c t i o n of e m i s s i o n of the f i r s t p a r t i c l e the a x i s of q u a n t i z a t i o n . Lloyd  This r e s u l t was a l s o obtained by  and d i s c u s s e d by Lippmann . If  of  the d i r e c t i o n of e m i s s i o n of the second p a r t i c l e i n s t e a d  the f i r s t had been taken along the z a x i s of q u a n t i z a t i o n ,  then W(9)  would have been w r i t t e n i n the form  W(0)=2L P..feiPjo). The  along  i  (12)  two e x p r e s s i o n s f o r W(0) i n (11) and (12) must o b v i o u s l y  be e q u a l . - •  -  By s u b s t i t u t i n g (4) i n t o P ,.(©) of (10) one o b t a i n s  The above formulae have been d e r i v e d f o r the c a s e - t h a t p a r t i c l e s and not photons are e m i t t e d .  L i n g and P a l k o f f ^ have  t r e a t e d the case f o r the e m i s s i o n of a photon corresponding to mixed 2— e l e c t r i c and 2^' magnetic m u l t i p o l e r a d i a t i o n . T h e i r r e s u l t s are g i v e n by  ck A - A W r k * Z l JA^ I * •  Zvd* > where A_a A - A , L  x  Y  A„= A , A =A tt A e  and A ^ ( j » ^ « J jtij^ m  m  (14 )  A  2  x  tl  Y  ) ^ ( j ^ H / 3 ( J J l m m J J £-lJm) AjJ(^-l,nJ . M  u  J m  A  i  r  M  Here A '(J, ,m,),  A ('J-1, m-;) are the components of the normalized  £  M  r  T-l  v e c t o r p o t e n t i a l s f o r a 2* e l e c t r i c and a 2 * magnetic mul- . t i p o l e r e s p e c t i v e l y ; ei and  r e p r e s e n t the p r o b a b i l i t y  tudes f o r each m u l t i p o l e .  L l o y d * has shown t h a t  ampli-  and /5 can  be made r e a l by a prop e r choice of the nuclea# phases.  How-  ever, i n some c a l c u l a t i o n s , i t may be u s e f u l to have complex v a l u e s , hence the formulae The is  result  (14)  are l e f t i n the above form.  can be i n c o r p o r a t e d i n t o formula  taken t o be 1 ( t r = l , 0 , - l ) and  (13) i f  ^ a r e r e p l a c e d by  /nPrkA^(J,,m,) and firrkA*(J-l,m,) r e s p e c t i v e l y . Of course, o c * ^ and /&* A. • H e n c e f o r t h (13) w i l l be considered v a l i d f o r both •VI  p a r t i c l e s and photons, each  the proper ^ ' s b e i n g s u b s t i t u t e d I n  case. A t t h i s p o i n t one can see t h a t the dependence of P ^ i ^  and P ^ ^ a n d hence W(0) on r, thus -omitted.  F o r , i n (13)  the f u n c t i o n s f  L  (-L)  e_  and r  t  can be f a c t o r e d out and  the <j>'s (and A's)  depend on r through  (r) , which f o r l a r g e r are p r o p o r t i o n a l t o  ( s p h e r i c a l wave), where k i s the magnitude o f the  propagation vector. If  o n l y one value of J  z  i s p o s s i b l e f o r the p a r t i c l e o r  photon emitted i n the t r a n s i t i o n Jm -*-J m. u  (13)  a  takes the form  the a n g u l a r momentum c o e f f i c i e n t s b e i n g r e a l and  then the e q u a t i o n  (19)).  Omitting the dependence on r  (see equations (18)  , which can be  II  f a c t o r e d out, t h i s has the form  T h i s formula  can be .found i n F a l k o f f and u*hlehbeck*s  paper.  I n t h i s t h e s i s only t r a n s i t i o n s i n which an * p a r t i c l e photon i s emitted w i l l be c o n s i d e r e d .  or a  Since an <* p a r t i c l e has  no i n t r i n s i c a n g u l a r momentum one can w r i t e F ^ ( 0 ) = |Y^"(0 f o r i t , u s i n g (8) and ( 1 3 ) .  <P )\ x  G e n e r a l e x p r e s s i o n s f o r the F s  f o r a photon a r i s i n g from pure  r  and mixed m u l t i p o l e t r a n s i t i o n s  are g i v e n by L i n g and F a l k o f f ^ who s u b s t i t u t e d e x p r e s s i o n s i n terms of s p h e r i c a l harmonics f o r the A s appearing i n (13) and f  (14).  A 2 * - e l e c t r i c and a 2^''siagne t i c m u l t i p o l e have the same J  F>(9).  *i  For  a n y  parti  PT'te) (see  c l e or photon, F ^ ( 0 ) * l  •>»>  r e f e r e n c e 3) • F o r a mixed  tf(J ) 2  t r a n s i t i o n , L i n g and Falkoff** have  g i v e n the f o l l o w i n g formula:  W  "'  M  +*^<*%r''TW^|J TvT~X^^  (17)  , ,  Here 2R(K6*) - <*/}* + ^ ( J ,  and the  u t i o n f u n c t i o n s f o r the i n t e r f e r e n c e a r i s i n g from the multipole f i e l d s . (13) and ( 1 4 ) .  m i J E i n  ^)  a  r  e  t  h  e  angular  distrib-  c o n t r i b u t i o n s to P,^.. (0)  g of the 2"^ e l e c t r i c and 2^-"' magnetic  The form of the r e s u l t  (17) f o l l o w s from  Some P-J*** (0) are l i s t e d i n T a b l e 1.  The a n g u l a r momentum c o e f f i c i e n t s are g i v e n b y (see Appendix A, equations  (A4) and (A7))  (j"x. ^ ^ . | j ' ' j ; ^ , [ W > H t f ^ X x ) \ C ^ ^ i . i ?*\lfc  (  1  8  )  l  where J ^ J * J - A ; 0*= \tmlnlmum of 2 J , 2 J ; l ^ + M ^ ; and U  y  x  V"/-"  =  t  * J . Ur'4-")! (7"-~")[  jr [M-  h*-  '  a  fa  a '*-*"). -., Cr"-~")! 1  *  fc*v~0'J  !  [ A (19)  (nW.  1  The summation over »C i s c a r r i e d , out w i t h the under-  s t a n d i n g t h a t each f r a c t i o n  (A^0i>£ 0) appearing i n the  summand i s to b e - w r i t t e n i d e n t i c a l l y as A(A-l) ... (A-yj+1) . Then, i f k-&*-0, the term c o n t a i n i n g the f r a c t i o n —AL  will  (A-flV.  vanish.  U s i n g J=J +J -X,. and mrra fm , i f i s e a s i l y U  tt  X  ;l  shown t h a t  jo^Mj^must l i e within the  s m a l l e s t range g i v e n by the f o l l o w i n g  -J*m±J, -J^fcrn^ J " , -J^fe mfr x  -(J-J +-^)4rm-m^J-J A  x  + 7. , t  conditions:  , ^(J -»-^ -^)i,m +m tJ +-J -^ , tt  M  ,,  L  1  -(J-J i-Ajfcm-m 4:J-J u  u  u  i  x  +A .  (20)  l  The C's, d e f i n e d by (16), have the f o l l o w i n g  symmetry  propertied* Cy-Mr^ = c  J ^ - x >  ^r.^" „ / „ a v c  « J « A IJ^-K  /  +  The C's used i n t h i s t h e s i s are l i s t e d  J  r -^.  (21)  t  ^ ^  +  (22) (23)  -r ^ ^ ^ i . '  i n Table 2.  U s i n g (1-1), (16), and (18) one can o b t a i n the d i r e c t i o n a l correlation  f u n c t i o n W(Q)  for a particle  ( J ) o r photon ( J j ) f  emitted i n the f i r s t t r a n s i t i o n and a p a r t i c l e ( J ) emitted i n the second t r a n s i t i o n . A  wiaw21 [ n  ( J ) or photon  The r e s u l t i s  (cf~r*Tfcr"r^ , TJ ? ^ F  (  f  T(0),  (24>  Here, the n o r m a l i z a t i o n f a c t o r s f o r the angular,momentum' coefficients  have been f a c t o r e d out and omitted; the summation  over m  1  (-m+m,), m, and m  (= m-m ) - has been r e p l a c e d by one  u  over m, , m, and m^.  a  The F ' s , as s t a t e d b e f o r e , v a r y w i t h the  type of p a r t i c l e emitted*  The angles 0 and 0 can be i n t e r -  changed ( C f . - ( l l ) and. ( 1 2 ) ) . The p a r t i c l e  (J, )-mixed ^ ( J ^ ) o r 2f( J, ) -mixed KJ^}  cor-  r e l a t i o n f u n c t i o n i s g i v e n by  An,<Mt  J — J ) * * , /  ^ J " ^ . ^ J  t  ^  J /"* - w c  /~t j t  t  where common f a c t o r s have been omitted. The formulae  (24) and (25) are the ones which w i l l be  used i n t h i s t h e s i s to c a l c u l a t e  c o r r e l a t i o n f u n c t i o n s . - The  formula (25) i s v a l i d except f o r A -0 and A " 2J^ . A  x  F o r , from  c o n s i d e r a t i o n s of the v e c t o r a d d i t i o n of angular momenta, i t can be seen that only a pure 2 ^ 2^* e l e c t r i c and 2 ^ ' magnetic f o r these two cases.  For  e l e c t r i c i n s t e a d of a mixed  multipole t r a n s i t i o n w i l l  0,2^  occur  the formula (24) w i l l be  used. A method of e v a l u a t i n g the summations appearing i n the square b r a c k e t s of (24) and (25) i s g i v e n i n S e c t i o n I I . of t h i s once these summations are known, i t i s a f a i r l y simple  thesis,  matter  to o b t a i n W(0) i f Jj or J* i s s m a l l . z  W(0),  as g i v e n by (24), i s a l s o the c o r r e l a t i o n f u n c t i o n  t  f o r the reverse t r a n s i t i o n scheme J  t  t  — — ^ J *  w i t h the• e m i s s i o n  of the p a r t i c l e s or photons o c c u r r i n g i n the r e v e r s e o r d e r . F o r , by making use  of the symmetry p r o p e r t y (23) and  the r e -  l a t i o n F ^ C d ) - F"**(6) , and  changing  the summations over m,  m  and -m ,  one  to summations over -m  t  (  equal  i  and  can show t h a t (24) i s  to  •WSJ*k?Zr^W,V*?JV?v  K>a ^  which i s the c o r r e l a t i o n f u n c t i o n f o r the r e v e r s e d p r o c e s s . I f a l s o J,= J , then W(0) Z  i n (24) f u r t h e r r e p r e s e n t s the  dir-  e c t i o n a l c o r r e l a t i o n f u n c t i o n f o r the reverse t r a n s i t i o n scheme J —>J — * J tt  f  but w i t h the e m i s s i o n of the p a r t i c l e s or photons  o c c u r r i n g i n the g i v e n order i . e . as i n ( 2 4 ) . i s obtained from (26) by i n t e r c h a n g i n g m change i s p o s s i b l e s i n c e m, values (J, » J ) . z  (  This l a s t  and m ,  and m ^ r u n over the same range of already, a l -  by F a l k o f f and Uhlenbeclr*.  authors have a l s o shown t h a t the F^'s degree a t most J i n cos*©•  are polynomials  Since the e x p r e s s i o n s  are e q u a l , I t i s seen that W(0)  -These of  (11) and  i n (24) i s a p o l y n o m i a l i n  cos © of degree a t most the minimum of J, and J a  z  Yang*).  which i n t e r -  z  These r e s u l t s have been proved  though i n a d i f f e r e n t way,  result  (see a l s o  (12)  •"' II  I I . CALCULATION  OF-pIRECTlOKAL^^  I n order t o express 1(0)  -  in-(245 e x p l i c i t l y i n terms of  c  c o s 6 and the angular momenta involved,, i t i s necessary ,to a  evaluate the summations over m appearing i n the square b r a c k e t s , to s u b s t i t u t e e x p r e s s i o n s f o r the F's i n terms of COB^Q , and then to c a r r y out the summation over m, and m^.  I t i s the  o b j e c t of t h i s s e c t i o n to p r e s e n t formulae which one can use to s i m p l i f y and perform-such  calculations.  A. Symmetry of the Summations. I t i s p o s s i b l e to reduce q u i r e d to o b t a i n W(Q)  the amount of c a l c u l a t i o n r e -  e x p l i c i t l y by making use of the symmetry  p r o p e r t i e s (21) and ( 2 2 ) . A p p l y i n g (21) to the summation  ZL(^'*  m  V(C^*»-  (27)  ) , 2  a p p e a l i n g i n (.24) , and changing the summation over m to one over -m,  one can show that  f  As soon as the types of p a r t i c l e s emitted are g i v e n , the formula (24) can be s i m p l i f i e d s i n c e the F ' ( 0 ) w i l l v a n i s h m  •  but f o r c e r t a i n values of nij • are  J ,  Because <x.-y and r-Y c o r r e l a t i o n s  the s u b j e c t of t h i s t h e s i s , the formulas f o r t h e i r cor-  r e l a t i o n f u n c t i o n s w i l l now be o b t a i n e d . I f an <x p a r t i c l e l s emitted i n the f i r s t :  transition  along the a x i s of q u a n t i z a t i o n then F^' (O)=0 u n l e s s m = 0 .  the r e l a t i o n s ( 2 8 )  U s i n g t h i s r e s u l t and a p p l y i n g P"  ffl T  i(&]  t o (24)  one o b t a i n s  ~ f o r the <K.(J, ) - p a r t i c l e ( J )  or ^ ( J , )-tf"(J )  a  The  x  (29) correlation function.  i n t e r f e r e n c e summa t i o n appearing i n the mixture term of  (25)  f o r ^ ( J , )-mixed -^(J^)  correlation functions  w r i t t e n i n a s i m i l a r form. The  e m i s s i o n of a photon i n the f i r s t  F ?' (0) = 0 u n l e s s 1  t r a n s i t i o n along  that m,*tl i n (24)  m,=ll f o r a photon.  From (24)  since  and (28) then,  can w r i t e  f o r y.(Jj ) - p a r t i c l e ( J )  o r &TJ, )-t(J )  ; ; L  1L  A l s o , f o r i f ( J )-mixed *(3 ) (  Z  form.  The, common f a c t o r  B. C o r r e l a t i o n - F u n c t i o n s The'summations ( 2 7 )  correlation functions.  correlation functions,  term of (25)  i n the i n t e r f e r e n c e  2"p\(o)  i s omitted.-  f o r Special Transitions. can be e v a l u a t e d  The  expressions f o r j u u j  2J  as one may check u s i n g  C  m  (22),  (23).  the summation  can be expressed i n a s i m i l a r  q u i t e e a s i l y f o r some  s p e c i a l t r a n s i t i o n s determined by the f o l l o w i n g  M  can a l s o be  The common f a c t o r F^(o) i s o m i t t e d .  the a d i s of q u a n t i z a t i o n r e q u i r e s  one  and F > ( 0 ) =  m  are; the  considerations.  s i m p l e s t when  0,  2? , z  Table 2 and the symmetry p r o p e r t i e s  I n each of the ( C j ^ ' s i n ( 2 7 )  the f a c t o r  (J+m)! (J-m)! w i l l appe a r , e i t h e r i n the numerator or denominator,  i f the v a l u e s of \ and A are chosen from "the s e t a  0, 2 ,  2J, and the s e t 0, 2Jg, 2 J  M  respectively*  The s i m p l e s t  summand i n (27) i s obtained when a combination of X and A, i s chosen from the above values i n such a way that (J>m)! (J-m)i  w i l l be c a n c e l l e d  out.  Of the combinations pos-  s i b l e only f o u r permit such a c a n c e l l a t i o n . ( i ) . A, = 0>  iii).  the f a c t o r  These a r e !  ^=0.  V2J , X  ( i i i ) .*,-2J,  V  2*  2 J  \=0.  t  2J , ^ 2 J » .  (iv) .  X  H e n c e f o r t h , the t r a n s i t i o n s s a t i s f y i n g these c o n d i t i o n s w i l l be  c a l l e d the " s p e c i a l  transitions' .  The summations (27) f o r  1  these f o u r cases are e v a l u a t e d below. to c a l c u l a t e tions. are  The r e s u l t s may be used  any c o r r e l a t i o n f u n c t i o n s f o r the s p e c i a l  I n p a r t i c u l a r , some  and y-y c o r r e l a t i o n f u n c t i o n s  given.  ( i ) . > = 0, A *0 o r J'-J+J , J = J t J . u  For and  -  t h i s case the t o t a l angular momenta of the p a r t i c l e s  n u c l e i r e s u l t i n g from each t r a n s i t i o n are p a r a l l e l to one  another.  I t i s seen from c o n d i t i o n  (i) that J  and J  JL s m a l l e s t angular momenta t h a t the  angular; momenta of f  P  K  r  v  are the  Cf  can be emitted compatible w i t h  the n u c l e a r l e v e l s J ' , J , J  summation (27) I s e v a l u a t e d as f o l l o w s : / J m+mi0\2/ Jm0 \2_ JmJ.m. ' ^m-mjJ.m '  Z  transi-  B  . The  ^TJ,  m,)i (J,-m,)! ( J « - m +m)«. (J + m^-m)! ( J + m j l ( J - m tt  +  t  (from Table 2)  x  t  o-(r -~ ) v  v  / J , f Ja.+m. + mA / J,t J -m - m A V " / JU-m.rm A  V  (  (from (20))  J'-m.-m  \  -""'tt-,)  _ /J, +- Ji-t-m, + m.A/ J,+ J^-m, -mA^P/<!,+• J +m,-f-m^-fv) / 2(J-J .)+" J. -fJ^-m-av J,vm, A i- i ^ J ^+^ /l J,-t-J -m,A  J  a  m  t  (using m=v-JfJ^trn^) _ I J, f J m, -H m A / J, + J*, -m, -mA f 2  J  I+  (  3 1  U s i n g the - summa t i on formula (45) i n S u b s e c t i o n C below. t h i s r e s u l t into'- (24)  One can s u b s t i t u t e correlation function  to o b t a i n the  between a p a r t i c l e (J^) o r photon (J^)  and a p a r t i c l e (J^) o r photon ( J ^ > s u b j e c t to the c o n d i t i o n ( i ) on the angular momenta.  I n p a r t i c u l a r , i f an cL p a r t i c l e  or photon i s emitted a l o n g the a x i s of q u a n t i z a t i o n i n the first  t r a n s i t i o n , the c o r r e l a t i o n f u n c t i o n s a r e e b t a i n e d by  substituting  (31) Into (29) and (30).  <*(J, ) - p a r t i c l e W(0)=  ( J ) orti(J,) - l r ( J | : : A  z  ^ ^ ^ O f S ) ^ ! ^ ' * ^  y(J, ) - p a r t i c l e ( J ) A  (32)  o r V( J, )-4T( J J :  W(&-) --2. ( ' +J^+1^\ | J i Ji-l-a.\pm*(e) , J  The r e s u l t s a r e :  +  (33)  )  where t h e - f a c t o r (^gjit*) has been omitted. a photon i s emitted i n the second  I n the case t h a t  t r a n s i t i o n with angular  momentum J ^ - l ( d i p o l e ) o r J =2 (quadrupole) x  one can o b t a i n the  correlation functions <X(J*, ) - r f l ) :  W(0)_- l - ^ — ^ c o s ^  »(J, )-*(U : WW) * l - j | p ^ c o s * 0 W(0) " (  y(J )-if(2)-.  1 +  t  J ) +  i)(5J^23J 30)  (  C O S  l +  ^  >  (j ir(5J^23J,^0)  COS  |+  u s i n g the d i s t r i b u t i o n f u n c t i o n s l i s t e d i n Table 1. factors  M  Non-zero  ( i . e . those common f a e t o r s which are p o l y n o m i a l s i n J,  having no i n t e g r a l roots) have been omitted from these  for-  mulae . ( i i ) . >=2J, , A =2J .or J'=J-J JL  •-  i  , J^J"-^ .  r  By u s i n g the t h i r d symmetry p r o p e r t y (23)  Table 2 one can sum (27)  together w i t h  as i n p a r t (I) to get  y / J » m + m , 2J, \2t Jva2J \2 /J,+ J +m,+mA/J, + J -m, -mA/2JT +l\ / ^^JmJ.m, '/^"m-ffl^m/ \ J,4-m, A J , J { 2J' / / u  r  r  z  r  A  z  Since  (^gj*") 1  c  a  nb  e  f a c t o r e d out of W(£) j u s t as  i n p a r t ( i ) , then, comparing the r e s u l t s (35)  ^gjt^)  was  and (31), one  can see t h a t the c o r r e l a t i o n f u n c t i o n s f o r t h i s  transition  scheme are the same as those g i v e n I n p a r t ( i ) i . e . the same as equations  (32),  (33),  and (34).  (iii). v  2 J  » V °  o  r  J  ,=  J  . - * J«J J » J  :  M +  4  Prom t h i s c o n d i t i o n i t i s e v i d e n t -that J,?s J - i n fact, J, i s here the l a r g e s t  angular momentum which can be emitted The Calculation of  compatible w i t h the angular momenta J • , J .  c o r r e l a t i o n f u n c t i o n s f o r t h i s t r a n s i t i o n r e q u i r e s the Use of the summation formula  (46).  Thus, (27) becomes  y / J 'm+m, 2J\ 2 / JmO \2 ^ JmJ, m, J m-maJ m,/ C  ,  c  B  A  -TV" (J,* m,) . ( J -m,)! (j+m)'. (J-mj! ' " ' " " ' " " " ' ' ' " -, ^(J'+m, +m)«. (J'-m.-m)! (J+m)i (J-m)j (J^-m^m)! (J^m^-m)! (JitmJ! (J^mJ! 1  t  (from T a b l e / 2 ais£=£) _ / J, + m, \/ J, -m A y\ J • * J + m, + m*\ / J»+J -m, -m ) n  U  t  »/J,-*-m,\/ J, -m,\ y/j'+JVm^mA/ J'+J"-m,-mA _/J,tm,\/j, -m,\/2JH2J \  , "*  u  "(jx-aJl^OL  ( 3 6 ) :  u s i n g the summation formula~(46)  ?  i n S u b s e c t i o n C below.  r e s u l t w i t h (24) gives the c o r r e l a t i o n  f u n c t i o n between p a r -  t i c l e s o r photons s u b j e c t to c o n d i t i o n ( i i i ) . i s emitted i n the f i r s t  transition*  This  I f an * p a r t i c l e  then the c o r r e l a t i o n  fun-  c t i o n i s obtained from (36) and (29) i . e .  w<e)= If and  tiffim^t  (jJOU^y^' '•  a photon i s emitted i n the f i r s t (30),  6  transition,  (37>  then from (3^))  If  a photon i s a l s o emitted i n the second t r a n s i t i o n with- angu-  lar  momentum Jg= 1, or 2 then, as I n (i) , one o b t a i n s the f o l l o w -  ing  correlation functions:  *(J,)-.r(S): W( ) . 6  ( J |  -l)[  ( J l  - ^ i f J ^ c o s ^  JifytM^eos^  (  3  g  )  )-ir(2):  r(Ji  from (37), (38), and Table 1.  Non-zero common f a c t o r s have  been omitted from the formulae  (39).  F o r those v a l u e s of  which make the f a c t o r s appearing o u t s i d e the square b r a c k e t s v a n i s h the t r a n s i t i o n can not take p l a c e .  F o r , u s i n g the Con-  d i t i o n - ( i l l ) , i t i s seen t h a t J^J'+J^+Jg I.e. the e q u a t i o n J^Jg  must be s a t i s f i e d i f a t r a n s i t i o n o c c u r s .  OtJ^Jgno for  t r a n s i t i o n p r o d u c i n g an <X(J^) p a r t i c l e  l- -j^- 2  (iv).  J  J  n  o  t r a n s i t i o n p r o d u c i n g a photon  A ^ J , X 2J" a  2  Thus, f o r can occur and  (J^) can occur,  o r J»*J-J , J = J - J . U  2  Here JgSj^+J'+J^  and so i t i s l a r g e r than any of the  other angular momenta.  One cannot c a l c u l a t e c o r r e l a t i o n f u n c -  t i o n s i n terms of cos © f o r t h i s case w i t h a r b i t r a r y 2  by the l a s t e q u a t i o n Jg must a l s o be a r b i t r a r y .  since  However,  c o r r e l a t i o n f u n c t i o n s can be c a l c u l a t e d h a v i n g some small v a l u e . g i v e n below.  with J  6  a r b i t r a r y arid  Such c o r r e l a t i o n f u n c t i o n s are  -  To evaluate (27) f o r ~ t h i s case the summation formula (46) i s used w i t h Table 2 and the symmetry p r o p e r t i e s  (22)  and (23)  F o l l o w i n g the same method employed i n ( i l l ) one can show that  I f a photon i s emitted along the a x i s  of q u a n t i z a t i o n i n the 1 / JiT'^ZT'K  set fcond t r a n s i t i o n , then, o m i t t i n g the f a c t o r 2 F j ( 0 ) ^ the  ,  c o r r e l a t i o n f u n c t i o n w i l l be  wW-Sf' from (21).  y^-MF^'Ce)  1  ,  and  lf(2)-8(J ) 2  c o r r e l a t i o n f u n c t i o n s f o r t h i s case  the same as the Y ( J ^ ) - y ( l )  functions respectively corresponding Table 1.  C4i)  Comparing (41) w i t h (38), one observes that the  tf(l)-&(Jg) are  J  and Y(J^)-Y(2)  i n (39) w i t h J  correlations  correlation  r e p l a c e d by J .  are c a l c u l a t e d  The  0  from (41) u s i n g  The r e s u l t s a r e :  *(l)-*(J ): g  W(Q) s i t  cos $ 2  *(2)-tf(Jj: V(0)*  C^-l)  [(Ji-2) t I ^ j ^ f 2  W(©) r l -  tf(l)-*(J,);  5  6  )  cos 9 + S ( j ^ 2 y ( g t J 2  i b  :i6)  iSli^ficos © 2  e o 8  4J  ( 4 2 )  3 J * - u -1 A  *(2)-r(J ): z  w(9)=MJ*-i)k+  - 3(£^^^-'°)  c  o  s  * l Q  o m i t t i n g non-zero f a c t o r s .  By  (iii),  t r a n s i t i o n occurs f o r J g - l i h the  one  can show that no  <x(2)-*(J ) and 2  C.  the same reasoning  as t h a t i n  *(2)-}f(J ) c o r r e l a t i o n f u n c t i o n s of  (42).  2  General Method. I n the p r e c e d i n g  S e c t i o n c e r t a i n summation formulae  (46)) have been used, which w i l l be obtained  now.  To t h i s pur-  pose a general method of e v a l u a t i n g the summation (27) presented h e r e . t i o n s of the  The  ((45),  is  method can a l s o be used to evaluate  summa-  type appearing jtn the i n t e r f e r e n c e term of (25) .  A very d i r e c t method of e v a l u a t i n g  (27)  i s as f o l l o w s .  In  o  Appendix B i t i s shown that the  (C )  appearing i n (27)  p o l y n o m i a l s i n m and hence the summand of (27) i n m.  Thus one  are  i s a polynomial  can p e r f o r m the i n d i c a t e d summation over m  y d i r e c t l y by u s i n g  the known r e s u l t s f o r the sums 2m*',  is a positive integer. are a l s o p r e s c r i b e d , J.  I f the numerical v a l u e s  of m^  and  m  By f i n d i n g the r a t i o n a l r o o t s of t h i s p o l y n o m i a l i n J ,  f a c t o r s can be f a c t o r e d out of W(©) . the summand i n (27)  g  then the f i n a l r e s u l t i s a p o l y n o m i a l i n  can express i t i n a p a r t i a l l y f a c t o r e d form and  some of  However, f o r J  i s a p o l y n o m i a l i n m of degree *8  the d i r e c t procedure of e v a l u a t i n g W(9)  where k  (27)  and  one  the  and  J  and  so  then c a l c u l a t i n g  becomes clumsy. In the f o l l o w i n g method one  does not have to expand  the  complete summand as a p o l y n o m i a l i n m but o n l y p a r t of i t . Furthermore, the f i n a l r e s u l t i s a u t o m a t i c a l l y f a c t o r e d i n  *2,  A.  terms of some of the (e.g.  the f a c t o r  r a t i o n a l roots,  (^gJ" ) 1  (31)  l h  Is f a c t o r e d  These advantages have p e r m i t t e d the f u n c t i o n s as u s i n g the The  i n some cases a l l of  g i v e n i n p a r t B which  out  of W(e))  c a l c u l a t i o n of be  W o u l d  them .  correlation  difficult  to  get  d i r e c t method. method i s based on  to the problem of summ Ing  the f a c t that  (27)  can be, reduced  either  or  =Z/ V )(*"v tA„ TnQ ^Av/Ln-v/ P  Q  r  after substituting the  . v * -  r  A  r-1  1  * . . . ^ }  e x p r e s s i o n s f o r the  summation over m to one  over v.  those values f o r which (v)»(n-v) " &° the  range d e f i n e d by n-Qfevfcn and  can  always be p l a c e d i n the  p=0  and  q=0,  the  C's  In n o  and  (44),  *  Ofev^P.  form (43),  (27)  i n t o a p o l y n o m i a l i n v=m+J.  The  been used to evaluate the  translating  v takes on a l l  vanish, i . e . i t i h a s The  summation  s i n c e a t the  b i n o m i a l c o e f f i c i e n t s become 1,  then the d i r e c t summation of  i t i o n s In  (44)  0 ,  and  (27)  worst, (43)  is  whose summand Is expanded summations (43)  summations (27)  f o r the  and  (44)  special  have trans-  B.  I n Appendix C i t i s shown that  imrrhM^rr;.«« 1  (45)  and  that  Z( v)(n?v)v M ^) P  k  P  «  *0 (46)  where the a„  's are obtained from Table  4.  S u b s t i t u t i n g these r e s u l t s i n t o (43) and (44), i t i s e a s i l y shown that  S  £  n  q  S  | [ i A ^  (47)  A* a/3*J(P-«).' 1 /V^ H A M. ln-^/„ 0 \ n / '  EfFnQ  (48)  +  A  7  +  n  o n l y the term c o n t a i n i ng AQ appearing i n each formula f o r the case rsO. I f p and q are s e t equal to zero i n (45), the summation reduces to the sum of the k  power of the p o s i t i v e  integers.  As mentioned b e f o r e , t hese sums are used I n the d i r e c t method. The  coefficients a  k e t  i l i s t e d i n T a b l e d , are obtained  from the i n d u c t i o n f o r mula a  k e  J = 1 f o r k>l,<< =1 , \  £2"  - 2 1 ^ ^ - ^ " - I )  (49) F  O  R  to*'** ' 2  proved i n Appendix G. F o r a t r a n s i t i o n scheme next to a s p e c i a l one, i . e . a t r a n s i t i o n scheme, c h a r a c t e r i z e d by ^ , % quantity  , f o r which the  d i f f e r s from t h a t f o r a s p e c i a l t r a n s i t i o n by  1, the e v a l u a t i o n of the summation i s a l i t t l e more d i f f i c u l t than f o r the s p e c i a l case. scheme  A,= 0,  As an example, take the t r a n s i t i o n  1 o r J U J t J , , J e J ^ + J ^ - l (next to the s p e c i a l  t r a n s i t i o n ( i ) ) and evaluate (27) as f o l l o w s . ^^ J m4-m, 0 ) 2 ( J m l _ ^ 2 AW JmJ | m J m-m^ J m r  c  C  u  (  V  a  a  ( J N-m,+ mf? ( J *-»m, -m)V  ( l i m i t s from  4lmJ -m (j"-i-J ,i3 A  x  a  (20))  ( u s i n g m^v-J^m^)  = KJjf  (J,f J^m,+m -l)(J +' J m + jd  (tf-2J J ( m j ) u  A  z+  j  |  '+1 mJ^JJ+ 2&  (J + J fm,+mj ( g j j ^ ) + J ( J * t tx  (  a  x  (u s i n g (47)). Here K - ,  '  "  '  "  t  ^ ^ j u  1  )  ,(50)  /^^^m^/j^J  -mA  Prom (20), -(J,+ J -1)^m^+m^feJ, -J-J^-1, hence the denominator A  of K does not v a n i s h .  The r e s u l t (50) i s awkward to use u n l e s s  numerical values of the angular momenta are g i v e n . values of A, ,  As the  g e t f a r t h e r away from those of the s p e c i a l  t r a n s i t i o n the summation r e s u l t s w i l l become more awkward.  I I I . #-MIXEBY CORRELATION.FUNCTIONS. .. In this section,  t a b l e s from which &'-mixed'&'(2) d i r e c t i o n a l  c o r r e l a t i o n f u n c t i o n s can be c a l c u l a t e d  are g i v e n f o r an Oi.  p a r t i c l e h a v i n g angular momentum 1, 2, o r 3. i t i s shown how the ctions  I n p a r t s A and B  and ci(2) -mixed (2) c o r r e l a t i o n f u n -  are obtained from the Tables I I . and I I I . of L i n g and  F a l k o f f * f o r d'(l)- and respectively. <V (3) -mixed t(2)  In part  (2) -mixed^(2) c o r r e l a t i o n f u n c t i o n s C, those summations appearing i n the  c o r r e l a t i o n f u n c t i o n s are expressed i n the form  from which the c o r r e l a t i o n f u n c t i o n s can be f a i r l y e a s i l y e v a l u a t e d once the values of the angular momentum are p r e scribed. Some <X- f c o r r e l a t i o n f u n c t i o n s are l i s t e d i n r e f e r e n c e s 3 and 11.  Some curves of * of- \C c o r r e l a t i o n f u n c t i o n s are g i v e n  i n r e f e r e n c e 13. D e v o n s  12  has l i s t e d some «(.-mixedy c o r r e l a t i o n  f u n c t i o n s - however, some of h i s r e s u l t s do n o t agree w i t h the r e s u l t s obtained i n t h i s  thesis.  Since Hamilton's n o t a t i o n i s used to t a b u l a t e the angular c o r r e l a t i o n f u n c t i o n s l i s t e d I n s e v e r a l papers, I t i s used to t a b u l a t e the c o r r e l a t i o n f u n c t i o n s appearing h e r e . are l i s t e d i n terms of Aj and J"= j + a j ,  i.e.  The t a b l e s  d e f i n e d by J ^ J - a ' j and  *\-3^ and ^ J ^ g - J g .  A. V ( l ) -mixedV(2) c o r r e l a t i o n  functions.  The ^ ( J , )-mlxedot2) c o r r e l a t i o n f u n c t i o n s can be  calculated  from  where W . ( e ) . - Z [ § ( J m J ^ I J J J 'm+i^) ^ ( 9 ) ] [ < £ ( J ^ m - m ^ |J«2Jm) 2  W  (  J 1  ( 0 )  1  X  ^ [ ^ ( ^ i  3  1  JX( In  ! l  J  J  i  J  ,  m  f  m  i  j  2  p  ^  (  e  )  (0)]  ] [ ^ CJ m-m lm | J^Jm) ^ ^ ( O ) ] tt  2  2  J m-m 2m | J"2Jm) (^m-mglmg| J lJm)P|j;(0)] . t,  u  2  2  the formulae (5«U> the angles 0 and 9 may be Interchanged.  Common, non-zero f a c t o r s I n W(9) w i l l be o m i t t e d . cCand fb r e p r e s e n t the p r o b a b i l i t y amplitudes f o r the e l e c t r i c quadrupole and magnetic d i p o l e r a d i a t i o n r e s p e c t i v e l y .  ( T h i s n o t a t i o n agrees  w i t h the t e x t of L i n g and F a l k o f f ' s paper, but i n t h e i r II.  and I I I . ,  tables  <K and ^ have been i n t e r c h a n g e d f o r some u n e x p l a i n e d  reason.) In  Appendix D i t i s shown that from the jf(l) -mixedtf(2)  c o r r e l a t i o n f u n c t i o n s , W(9) * Q+ R c o s ^ , l i s t e d i n Table I I . of 3  L i n g and P a l k o f f ' s paper, the <fc(l) -mlxedr(2)  c o r r e l a t i o n func-  t i o n s , W(9) = Q» f R'cos*9, can be c a l c u l a t e d u s i n g Q»= Q+  R  2 and R » = - R .  ^. ^  (53)  The o((l)-mixedV(2) c o r r e l a t i o n f u n c t i o n s thus o b t a i n e d have been t a b u l a t e d i n Tabled". omitted.  Common, non-zero f a c t o r s have been  B. 0((2) -mixed f(2) By  the method used i n A i t can be shown t h a t from the •  ot 2)-mixed ^(2) listed  correlation functions.  c o r r e l a t i o n f u n c t i o n s , W(e) ^ % + R c o s 9 V S c o s © , 2  4  i n Table I I I . of L i n g and P a l k o f f ' s paper the oC(2)-mixed +-R*cos 9-f S*cos 9 can be  tf(2) c o r r e l a t i o n f u n c t i o n s , W(9) »  2  4  c a l c u l a t e d u s i n g Q « i[6Q-(2R+5S j? , R --3?[2R+3S} , and S' = -S* f  =  The oc(2)-mixedV(2) c o r r e l a t i o n f u n c t i o n s tabulated  i n Table 5.  thus obtained  have been  Common, n o n - z e r o f a c t o r s are omitted from  w(e). A m i s p r i n t was n o t i c e d i n L i n g and P a l k o f f s Tab&e I I I . f  I n the y T 2 ) - y ( l )  correlation functions l i s t e d ,  ,Q and R are  p o l y n o m i a l s - o f the same degree i n J f o r a l l t r a n s i t i o n s but C a l c u l a t i n g the Y(2)correlation  AJ - 1 a n d & j r - i . for  function  t h i s case from P a l k o f f and Uhlenbeck ' 8 ° paper one can ob-  !+-• MlMlSM^ll—cos ©.  This shows t h a t i n L i n g and • ' l l O J S 269J4-174 P a l k o f f «s Tabift l ( J + 2 ) (H0J +269J+174) i n Q should be r e p l a c e d t a i n W(9) -  2  i  by |j(J-t-2) (110^+269^174) . #.(2) -mixedVT2)  This has been done to o b t a i n the  c o r r e l a t i o n f u n c t i o n f o r t h i s t r a n s i t i o n i n Table 6  G. fl/(3) -mlxedJT(2)  correlation functions.  •The (jfcafcxaxx^xtxfcxBBXxax) summations  '  ?](C J *'* ) ,  /  ^  ^°Jm30  ' J m-m lm (C  u  2  ) 2  >^ Jm30 0  i n Tables 7,8,9 r e s p e c t i v e l y . (43)  using  ,  M  C  „  J  Jm30  ' J m-12l J«m-lll C  (0 3  2  )  J m-m 2m u  2  a  r  e  l  l  s  t  e  g  d  They have been p l a c e d i n the form  and t a b u l a t e d f o r a l l values  d i r e c t l y evaluated  ,  of A j , AJ.  the formula (47).  They can then be The c o e f f i c i e n t s ,  ,  are too clumsy to t a b u l a t e f o r g e n e r a l Values of the a n g u l a r momenta J ' , J , J " . However, s i n c e the A^'s are obtained  from  products of b i n o m i a l s and monomials as i n the e q u a t i o n V  r  +  A  r-1  v r  "  1  + ' * * + 0 *[ 2 A  a  v 2 + a  l ? c][ +  V*  a  +  l  b :  V  - oJ' + b  * • [V  + *o]  the s e t of c o e f f i c i e n t s a^, a^, a^? bg, b^, b^;... ;£^,H^.can be tabulated instead. J  Once numerical Values are a s s i g n e d to «X , J , r  the A 's are q u i t e e a s i l y obtained and the summations can  u  t  then be e v a l u a t e d u s i n g ( 4 7 ) . The  o/(S)-mixed2T(2) c o r r e l a t i o n f u n c t i o n i s obtained i n terms  of c o s 9 by s u b s t i t u t i n g the e v a l u a t e d sums together w i t h the 2  e x p r e s s i o n s f o r the P's g i v e n i n Table 1 i n the formula (25) ( s i m p l i f i e d by_ the r e s u l t (29)) .  F o r *(3)-tf(2) o r  c o r r e l a t i o n f u n c t i o n s the formula-(29)  i s used.  k(3)-X(l) t  An example showing how to xsuumt read the Tables 7, 8, and 9 i s now g i v e n . 7i ~ 5,  F o r the t r a n s i t i o n i n which A J = 1, Aj -2 o r  3 i n Table 9 one o b t a i n s  t  2 s (J-2) ; ^ = 2 ,  b = -(J-4) . Q  U s i n g (43), i t i s seen t h a t t h i s  summation i s equal t o  16fsSV)( " 4 " )[ - ' - >' < - ' j[ '-< - »j' 2J  4  4  T  T8  2 J  2  +  J  2 2  2l  J  4  which can now be e v a l u a t e d u s i n g ( 4 7 ) . I f m^ has not been s p e c i f i e d n u m e r i c a l l y I n Tables 7 and 8, I t means t h a t the r e s u l t s are true f o r a l l p o s s i b l e p o s i t i v e and  3/  zero v a l u e s of nig.  I n Table 9, aig-rrl-since  ^= 0 u n l e s s  fflg ~ ±1 and o n l y nig - 1 i s r e q u i r e d from these two to  possibilities  c a l c u l a t e the <X(3)-mlxedo'(2) c o r r e l a t i o n f u n c t i o n .  Table 1. E x h i b i t i o n of the Angular D i s t r i b u t i o n F u n c t i o n s , OC P a r t i c l e : J*l.  Ff * 2cos*0 F* =l-cos e 1  J-2  1  F j = 1-6COSH4-9COS*0  F* = 6 0 0 3 * 0 1  cos^Q  -6  F ^ = | ( l - 2 c o s 0 + cos 6 ) i  F*(0) = 0 u n l e s s  4  m=0.  Photon: (These d i s t r i b u t i o n f u n c t i o n s are p r o p e r l y weighted so that the c o r r e c t r e l a t i v e e f f e c t of each m u l t i p o l e i s represented when a mixed t r a n s i t i o n  occurs.)  #-1. E l e c t r i c or magnetic d i p o l e . Ff = | ( 2 - 2 c o s H ) F? = i.(l+cos^e) 1  -  i  £=2. E l e c t r i c o r magnetic quadrupole. K  f (6cos e-6cos*0)  s  a  F*'r I ( l - 3 c o s & + 4 c o s 0 ) i  4  H& * £(l-cos^O) z  F?(0) 0 u n l e s s m«±l. s  Mixed e l e c t r i c quadrup o l e and magnetic d i p o l e d i s t r i b u t i o n functions. F^W-O Pf , (e)«±ig(3cos^-l) 1  F o r any J , F ^.,(&)= - P ^ , (0) . J  F o r any p a r t i c l e or photon F^ (&) = p y ( & ) .  F^Ce)  J3  Table 2. E x p l i c i t e x p r e s s i o n s f o r some C j u ^ j pJmO  ["'  Jml  r  ' ( J + m ) . ' (J-m)!  , of  J«m"J m  G  L  (J+m)! (J-m)!  - | (J"+m )l (J^-m")! 2  x  H  °J"m»J J-ia»"  V  X  ( e q u a t i o n (19)) .  m  [(2 - v j !  ;  JL  l^Jm»'t -m T 1 u  <&*MJ!  (J,-m,)!J  ^  J  *  *  M  J  (2J -x> UJ -AJ123 - A, u  x  X  J  /J  ,  VI*. c  JmO - f _ (J+m)! (J-m)! _ !*• j-lm-m^lm^ [_(J-ltn^-m)! (J-l-%.+m)< (ln-m^i tl-m )IJ a  Jm-m^lm [(J-mi+m)! (J+m^-m) , ( l t m ) ! (l-mi).'J Jm2 _ / -.x1-na, [ (J»l-ma»m)! (J+l»m -m)l "1^ Jflm-m lm '* ' L( J+m) I (J-m).< (l-m ) J ( l r m j i j J s 2: 1  x  L  x  a  z  r  a  v  i  1  a  4  JmO _ [" (jVm).' (J-m)! J-2m-m 2m * [(J-2-m +m)J (J^m^-m)} ( 2 t * ) l  c  m  ;t  C  x  4  1^  (2-m)jJ a  J-lm-m 2m " *{( J-l-m^m) J (J-lw^-ni) • (8+*^) i (2-mJjj 1  1  L - *< 2m  m  ^mfm-'-tl^^^^r^J Jm2 ^ f3 (Jm-2)/ (J4m)!.(J>2-mJn^ Jm-222 * (J-m)!J , Jm3 _ /_•,xl4m Qr(Jvi-m,tm)! (J+l4m»-mi! " I Jflm-m 2m/ • . \(Jtm)! Jj-m)! (2+mj! (2-m.V Jm4 * * _ / ^ m i , (J-H3-m,+m)l (J*2+m -m)' U J+2m-m 2m ' (_(J + m)f (J-m)< (2*-mJJ (2-mA)!J J, = 3, m,*0: c  r C  {  1  }  2  A  g  r  p  t  a  7  1  i  J4-3raO,l_f (J»3»m)! (j43-m)i7* Jm30 '3! L (J>m}! (J-m)i J pj+-2ml .f(J>SUm)f (j^2- y»(*, Jm30 " j_(J+m)! (J-m)j J c  m  0  Jm30  2  I£  ( J  ^  1 + m )(  J* "" ? 1  1  1 1  C5m -J(J42)] z  cJjjSg -|m[5mM3^43 J - l ) ] C  J ; 3 ? 4£ - fl* f ' ^ ^ f l J-2m5 _ [(J+m)? ( J - m ) ! " 1 * Jm30 " (j J+m-2)l (J-m-2)!J , J-3m6 _1 f(j4-m)! (J-m)! 1^ Jm30 3! L(J+m-3)! (J-m-3)U e  C  G  1  a  =  (  j  +  m  ) ( j  m  5 m  m  m  ^ ^ jf ^  J + 1  ll  Table 3.  10  A tabulation  9  8  7  of the c o e f f i c i e n t s  6  1  5  4  •  2  •  •  3  •  • •  •  a  d e f i n e d by (4$) .  <cl  3  2  1  *  *  1  •  •  •  1  1  •  ••  •  1  3  1  •  •  1  6  7  1  •  1  10  25  15  1  1  15  65  90  31  1  4  •  •  5  •  •  6  •  •  •  7  •  •  •  1  21  140  350  301  63  1  8  •  •  1  28  266  1050  1701  966  127  1  9  •  1  36  462  2646  6951  7770  3025  255  1  10  1  45  750  5880  22827  34105  9330  511  1  •  3  Table 4.  The f o l l o w i n g a b b r e v i a t i o n s are used i n Tables 5 and 6.  d  Qi5J(jr+^)]i  1 =  d =  (aj-l)[l5J(J+-2)J^  d =  [5(2J-l)(2j+3)]^  2  3  d  r (2J+3)[l5(j -l)J^ 2  4  d =  [15(J -1)J^ 2  5  d  = (J-5)[15J(J+2)J ^  g  d =  (2J-3)(2J+5)[15J(J*2)J^  ?  d  g  d  g  = (J+2) (2J-1) f J f 6 ) L l 5 J ( J+2)j ^ r (j+2H2J-l)[l5J(J+2)] ^  d = 3[5(2J-1) (2J+3)]^ 1Q  l l  d  d  1 2  d  1 3  =  d d  J  5  5  2 J  - ) (2J+3)]^ 1  (2J-3)(2J+5)[5(2J-l)(2J+3)]^  " 3[5(2J-1) ( 2 J + 3 ) j i  4  ^  1 5  16  d  ( - )L (  =3(Jt6)X5(2J-l) (2J+3)]^  d 1  3  =  S  ;  (J-l)(2J+3)[l5(^.D]i ( J  "  1 ) (  2  J  +  3  )  ( -5)Ll5(J -l)j^ J  2  = |(2J-3) (2J+5)[ 15( J - l ) ] ± 2  1 ?  d  = (J+6)Cl5(J -l)]£ 2  l f t  Table'5.  #•(!) -mixed5(2) c o r r e l a t i o n f u n c t i o n s , W(©) - Q» + R»cos ©.* 2  lecl -  2 R  2  13  1 1  R  1  3 16 J-7  0  R' -3(2J-7) -1  Q' 26J*V71J+42 R» 3J(2J-1)  1  Q  1  7J  _5(29J+6)  id, 3  1  -J  R» - i ( j + 6 )  d  7  0  -1  0  1  Q' _£(26J +17J+6) 2  21 R» 1 ( J + 6 ) ( 2 J - 1 ) 7.. Q -S(58J +-151J+78) 21 R» - | ( J + 6 H 2 J - 1 ) 2  1  Q' l(20J -8J-5) 2  7 R» -I(2J+5)(2J-3) 17  0  J(2J-1) l4j +33J+20  Id  2  3  - 2 d  2  -J(2J-1) (2J-1)(6J-1)  3  2  d  2  d 3  (2J-1H2J+3)  Q' 5(8J +8J+6)  8J +8J-1  R* l ( 2 J - 3 ) ( 2 J + 5 )  -(2J-1)(2Jt3)  Q» |(20J +48J+23)  (2J+3) (6Jt7)  2  2  7  -1  J(6J+7)  l  2  -S 3 d  d  3 3  R» -1(2J-3)(2J+5) (2J-1)(2J+3) -=>3 7 • C o r r e l a t i o n f u n c t i o n s f o r those t r a n s i t i o n s i n which only a pure d  e l e c t r i c multipole  (fS*0) i s e m i t t e d , have been i n c l u d e d i n the  Tables 5 and 6 f o r the sake of completeness. t a i n e d from r e f e r e n c e  3.  These were ob-  J/31* -1  1  -2  14J -5J+1  R » -|(2Jf3)(J-5)  -(J*l)(2J^3)  Q  (m)  1  -1  1  /  Q ' ^|(58J -35J-15) 2  0  2R(* 3#)  2  -§('26J +35J+15) 21 R ' i(2Jt3)(J-5) 7. Q -|(29J+23)  17. (J+l)  R  -(j+D  S  34 d  -4 d  (6J-1)  3 4  (J+l)(2J-K5)  -f(J-5»  d  4  K  " 5 d  Q ' 26J ^19J-3 2  R ' 3(J+1) (2Jt3) Q ' 16J+-3  0  R ' -3(2J+3) Q ' 13  -1  R' 3 Example  * •  obtains  | _ ( 5 8 J - 3 5 J - 1 5 ) M ' * (14J -5J+l)l/S|* +  QtR«  For-AJ = -1, A i = 1, one 2  5  -|(2Jt3)(J-5)|«|'  A  a  2  -(J+l)(2J+3)|/3l ' A  :  from which W(9) = Q'+R'cos 9 i s e a s i l y 2  |d 2R(^*) 4  -d 2R(A s*),  obtained.  4  /  3?  Table 6.  U(2) -mixedJf(2) c o r r e l a t i o n f u n c t i o n s * W(0) = Q» 4-R»cos © 4-S*cos ©. 2  4  1/3 I " 4  6.3 AJ' 2  0)  2  15 R» 6. S' -1  1 Q.' -2(SJ+2) R* 2(J-3) Sf - ( 2 J - 3 ) 3 -4(2J-KL)(Jt2) 0 2  4(2J-3) S' i ( J - l ) ( 2 J - 3 )  ;  -1 Q' - (12 J *-54 J -K78 J+30) 3  2  R» 2(2J-l) (J -r2J4-6) 2  S» - £ ( J - 1 ) ( 2 J - 1 ) ( 2 J - 3 ) 3 -2 Q» -(20J -rl48J +391J +437J+168) 4  3  2  R' -2J(2J-1)(2J +7J+9) 2  S» •Lj(Jiil) (2J-1) (2J-3) 1  2 Q' 5(3J+1)  15J  -d.  R» -5(J+4)  -3 J  3d,  S' |(2J+5) 1 Q» 5(5J +5J+6) 2  R • 25'(J+3) (J-2)  3J(9J+-1) 3J(J-5)  -3d,  S' -|2(2J-3)(2J+5) 0 Q' 5(4J +8J +3J+6) 3  2  R» -5(2J-3)(2J+5)(J-2)  i(52J +136J+99) 3 J(2J*3)'(2J+5) 3  id. 3 7  31 Table 6 ( c o n t i n u e d ) .  S' |£(2J-3)(2J+5)(J-1) -1 Q' 5(10J +47J +62J +16J+12) 4  3  2  R» 5(2J-1) ( 5 J + 6 J + 4 J H 8 ) 3  S'  2  3  1  2  2  3J(J+2) (J+6) (2J-1)  2  S» i ( 2 J - l ) ( J - 1 ) ( 2 J - 3 ) 3 2 Q» 15(2J+1)(J-1)  R» 15(2J+5)  -3d  8  >  3(J+2) (10J +23J+14) 2  " 9 3d d  9  3(2J-1)(4J-1)  d  3(2J-1)(2J+3)  S» -5(J+2)(2J+5)  8  d  -3J(J+2)(2J-1)  -5(2J-1)(J +2J+6)  io  -3d  10  •  1 Q» 1 5 ( 4 J + 4 J - J - 7 ) 3  \  3J(J+2) (18J +43J+30)  -|°(J-l) (2J-3) (2J+5) (2J-1)  -2 Q' 15(2J +9J +13j+5) R  ?  3(2J-1)(10J +7J-5)  2  2  R» -15(2J-3) (2J-J-5) (Jt3)  - u d  -3(2J-1)(2J+3)(J-5)  S' 20(2J-3) (2J+5) (J+2) 0 Q» M(16J +32J +40J +24J-63) , 4  R  1  3  2  5(2J-3) (2J+5) (4J +4J-9) 2  i(2J-1)  (2J+3) (32J +32J-15)  -d,  2  -(2J-1)(2J+3)(2J-3)(2J+5)  3d  S' -20(2J-3)(2J+5)(J-1)(J+2) -1 Q  15(4J +8J +3J+6)  3(2J+3) (10J +13J-2)  R» -15(2J-3)(2J+5)(J-2)  -3(2J+3)(2J-1)(J+6)  1  3  2  2  M  13  S' 20(2J-3)(2J+5)(J-1) -2 Q  r  15(2J+1)(J+2)  R» -15(2J-3)  3(2J+3)(2J-1)  S.' -5CJ-1)(2J-3) 2  Q»  15(2J -3J +J-KL) 3  R» -5(2J+3)  3(2J+3)(4J+5)  2  (J%5)  S» |(2J+3)(J+2)(2J+5)  d  14  - 14 M  >.  3(J-1)(10J -3J+1) 2  -3(J-1)(2J+3)(J+1)  d  15  " 16 3d  0  l g  to  Table 6 ( c o n t i n u e d ) . A J  1,31*  AJ  1 Q  5(10J -7J -19J +7J+21)  3(J+l)(J-1)(l8J -7J+5)  R  5(2J+3) (5J +9J +7J-45)  3(J+l)(J-1)(2J+3)(J-5)  S  --|°(2J+3)(2J-3)(2J45)(J+2)  4  3  2  3  2  2  -d. 16 3d 16  ^d 17  I(J+1) (52J -32J+15) 3 (J+l) (2J-3) (2J+5)  3d 17  5(5J +5J+6)  3(J+l)(9J-2)  -d  R  5(5J-10) (J+3)  3(J+1) (J+6)  3d  S  -|2(2J-3)(2J+5)  0 Q  5(4J +4J -J-7) 3  2  2  R  -5(2J-3) (2J+5) (J+3)  S  M ( J + 2 ) (2J-3) (2J + 5)  -1 Q  -2  2R(^#)  2  -2 Q  5(3J+2)  15(J+l)  R  -5(J-3)  -3(J+1)  S  |(2J-3)  2 Q  -(20J -68J +67J -19J-6) 4  3  2  R  -2(J+-1) (2J+3) ( 2 J -3J+4)  S  I(J+1) (2J+3) (J+2) (2J+5)  1 Q R  2  -6(2J -3J +J+l) 3  2  2(2J+3) (J +5) 2  < S  -|(,2J+3) (J+2) (2J+5)  0 Q  -4(2J -J-1) 2  R  -4(2J+5)  S  i(J+2)(2J+5) 3  -1 Q R 3 -2 Q R S  -2(3J+l) -(J+10) -|(2J+5) 15 6 -1  18 18  -3d,  Table 7.  AJ Aj mg  The summa t i ons £ (CJm30 ' ) < ^ m - i ^ m ^ • 2  KSp  v  nq  v  2  v°sl  1  2^ (^li2)4st , . 4  J ' - J-A3, J=* J  2  4m  ffl2 2J 4(4+m2  1  "2 ^ " T ^ ^ - r n g , 2J .2,3+m  1  e  ."8  0  f^-ing.SJ.a+iiig  -  m  2  2  -10(J-l)  ( J - l ) ( 4 J-6)  5  -10(J-l)  (J-l)(4J-6)  -!0  5  -3 0  -10J  1  -2J  1  " -2J  i  -aj  2  1  - * -2J-l  J  4 J  j  -4J-3 2 J  "  2  -4J-3 4 J  "  5  1  4  4  2  2  -  1  2J(2J-1) -2 J - l  3  -  1  1 S  2  2JC2J-1)  -10J -2J  1  1  "| 3,2J,4  J  -  or ,  S  (J-1)(2J-1)  10J  is! 4 3,2J,3 "| 3,2J,3  (J-1)(2J-1)  J  5  "H-ma.SJ.^-g ^-mg.aj.af-g  2  5  5 1 - *2  (J-2)  :  2  5  1  -2(J-2)  -4(2J-m2J-6)  (2J+-1) (2J+2) "  2 J  -  3  (2J4-1)(2J-K2) (2J+2)(2J-K3) -2J-1  (ZJ-n^J-Q)  2  ft  Table 7 ( c o n t i n u e d ) . AJ  m  1  3  2  KSj  2  8 0 S  y2  yl  v°*l  n q  3,2J-6,  1 6  2 0,1 •HX^.W^.S*^  2  1  m  2  160 S 2,2J-4,5*  2  -4(2J-m J-4)  1  -2(J-2)  (J-2)  1  2  5  -10(J-1)  (J-1)(4J-6)  5  -10(J-1)  (J-1) (4J-6)  4  4(2-2J+m J)  2  (2J-m J-4)  :  2  2  2  6  °  4  2-m ,2J-2,2+-m  2 i ^ S g 1,2J—1,3  1  -2(J-2)  S  *  -  (j-sr  1  " !,2J,1 S  3 1,2J,2  2  (2-2J+m J)  2  2  -10J  (2J-1)(J-1)  5  -10J  (2J-l)(J-l)  1  -2J  4  -4J(2-mg)  J (2-mg)  5  -10(J-1)  (2J-l)(J-6)  5  -10(J-1)  (2J-1)(J-6)  1  -2(J-1)  (J-1)  5  -10J  J(4J-2)  5  -10 J  1  -2 J  -2J-1  1  -2J  J  5  -10J  J(4J-2)  5  -10J  J(4J-2)  4  -4 J  J  1  -2J-1  5  <  2  2  j(4J-2)  2  2  2  Table 7 ( c o n t i n u e d ) . *JA1*  2  KS;  V  nq  -^UJ-^O  1.12  vl  2  ..  5  (SJ-l)f^5)  -10(J-1)  5  1 -2 0  1  -16SS '2,2 J , . 2  S  j  1  -2J  1  -2J ~  1  4  2 4 S  3,2J-1,3  -  3  - 3,2J,3 4S  S  J  J  4  "  J  -  3  4 J  "1  ^ . -2J  1  - J"3  1  4  4 3 4,2J-1,3  -4J "  1  4  J  "  3  1  ,  2  J  "  6  (J-1)* 2J(2J+1) -2J-1 j2 (2Jf2)(2J+l)  3  (2J+1) (2J+2) - 2 J •. , 2J2 -1?J+27 r  I  3 0  (2J+1H2J+2)  j2  1  0  2  -2J-1  2  1 3 3,2J,3  J  " ^-D  1  1  • °  2  2  -4J  1  2  C'BJ-1) (2J-S) -2J  1  "f 2,2J,2  V0=1  ,  -6<J-3)  3  w S  W  2J -19j+27„ 2  6S _ , 3,2J-6,4 3  3  -6(J-3)  4  -4(2J-5)  icStefcrJcJdk&c&x  (2J-5)  2  1 2  2 0  30S  1  2J-fi c  4S° . '2,2J-4,2 rtT  0  1  5  , , -6(J-2)  3 3 1  -6(J-2) 1  -2(J-2)  6 2J -13J+12 ifc*x3$M&&k 2  2J -13J*12 2  (J-2)  2  J-  Table 7 (continued). J »2  J  K S  pnq  '  V  l  V  °  =  1  2 ° ,  2  1  1 8 S  2,2J-4,3  " ^ >  1  2  4  2  -4(2J-3)  72S  2,2J-4,4  -SfJ-S)  1  1  is .,., A , 2 J-2,1 8  1  0  1  |s^ J-2,2  0 0  1  3  is , 9 0,2J,0 1 0  r t  |s ' , , 3 1,2J-1,1 8  T  2  (J-l)(4J-6)  5  -10(J-1)  (J-l)(4J-6)  3  -6(J-1)  2J -7J+3  3  -6(J-1)  2J -7J+3  -IG(J-I)  5  2  2  (J-l)(4J-6)  5  -10 (J-1)  (J-UC4J-6)  4  -4(2J-1)  (2J-1)  i  i  ' 0  (J-2) 2  -10 (J-1)  2  Isf '„ 2 2,2J— 3,3  2  5  : 2  (2J-3)  2  1 2  CJ-SJ  -  2  5  -10 (J-2)  4J*-20J+21  5  -10(J-2)  4J -20J+21  1  2  5  -10J  (2J-l)(J-l)s  5  -10J  (2J-1)(J-1)  3  -6J  J(2J-1)  3  -6J  J(2J-1)  1  -2J  J  5  -IO(J-I)  2  2  (2J-l)(J-6)  5  -10(J-1)  (2J-l)(J-6)  1  -2(J-1)  (J-1)  4  -4(2J-1)  (2J-1)  2  2  7 (continued) .  V  m  KS?L 2 °pnq  2  0  1  2  Is  6  3 2,2J—2,2  Is  8  4 1,2J,1  -is  7  4 2,2J-1,1  -is 2  5  3  f2J—2,2  5  -10(J-2)  (2J-1)(J-21)  5  -10 (J-2)  (2J-1)(J-21)  1  -2(J-2)  (J-2)  5  -10J  J(3J-2)  5  -1GJ  J(3J-2)  3  -6J  J(2J-1)  3  -6J  J(2J-1)  5  -10(J-1)  (2J-1)(2J-5)  5  -10(J-1)  (2J-l)(2J-5)  4  -4(2J-1)  (2J-1)  1  -2J  5  -10 (J-2)  4j -22J4-20  5  -10 (J-2)  4J -22J+20  1  -(2J-1)  3  -6J  J(2J-1)  3  -6J  J(2J-1)  1  -2 J  J  4  -4 (2J-1)  (2J-1)  1  -2 J  >  0  1  2  0  1  4S  _ _ 2,2J,2  —18S_ 3,2J-1,2  72S  S  4,2J-2,2  4  3 ^ 2J ^ 3  " 4,2J-1,3 6S  2  2  2  2  St  s?  i  2  2  1  -2(J-2)  1  -(4J-1)  2J(2J-1)  3  -6J  J(2J-1)  3  -6J  J(2J*1)  4  -4(2J-1)  (2J-1)  1  -2J"  (J-2)  2  2  Table 7  (continued). pnq  ° "  3  2  3 0 S  v  2 f,2j-2,3  • 1  3 3,2J. ,3  1  v  S  -f4J-lJ  2J(2J-1)  ^3J-5)  4  6  1  (3J-5)  5  -S 3 3,2J-6,4  2  v°=l  l  1  3  6  3 2  1 2 0  1  -  1 6 S  |,2j_  | 2,2J-4,2 S  (J-2)  1.  -2(J-2)  (J-2)  1  -2(J-2) "  1  3,2J-5,3  1  2  " 1,2J-2,1 S  S  2S '3,2J-4,2  2 )  '-  (  2 ) 2  -12(J-1)  9(J-1)  2  - ^-^ 2  5  6 -10(J-1) '  5  1 2,2J-3,1  -  2  2  1  1 •°  2 ( J  2  -4(J-1)  3  4 24S  3  -2(J-2)  1 4 > 2  1  2  2  (J-l)(4J-6)  5  -10(J-3>)  (J-1) (4J-6)  1  -2(J-1)  (J-1)  1  -2(J-1)  -(2J-1)  5  -  1 0 ( J  -  2 )  2  4J - 0J+21 2  2  5  -10(J-2)  4J -20J+21  4  -12(J-1)  9(J-1)  1  2  2  2  5  -10(J-3)  5  5  4J -30J4-46 2  -10(J-3)  4J -30Jt46  1  3  2  Table % {continued) . A J Aj m  -1  v°=l  Kfi£  2  0 0,1  2  ,sf  L  n  r  '  A  9(4-m^) l+m ,2J-2,l-mg 5 -10(J-l-m ) 2  2  2  l^S 9 3,2J-3,1 6  -1 nu2  S 2-»-m ,2J-2,2-m 2  5  -lOfJ-l-mg)  5(J-l-ia ) -(3J +-3J-l)i  1  -2(J-l-xng)  (J-l-ittg)  4  -4(J-l)(2+mg)  (J-l) ( 2+1^)  5  -10(J-3)  2J -35JH6  5  -10(J-3)  2Jr .-33J+46  1  -2(J-3)  (J-3)  5  -10(J-l-m )  5(J-l-m ) -J(J+2)  5  -lOfa-l-Blg)  5(J-l-m ) -J(Jf2)  2  m  2  2  2  2  2  2  2 2  2  2  2  2  4 -2  2  2  (J-l) ( 2 « i g ) 2  2  4(9-m ) S34.m ,2( J - l ) v3-m 2  2  -3  2  m  3 0  1  S  2  l 4 4 3,2J-6,3 s  -is  4  6 3,2J-6,3 2  2 0  l 3 6 4,2J~7,3 s  «6 2,2J-4,2 D  1  ^(J-l-nig)  (J-l-mg)  4  -4(J-l)(2+m ) 2  2  (J-l) (24m )  2  2  2  2  2  2J-2,4-m  2  4  -4(J-l)(2+m )  (J-l) (2+m )  1  -2(J-3)  -3(2J-3)  1  -2(j-3)  -4 (J-2)  1  -2(J-3)  1  3  2  1  5  6  1  4  1  -2(J-2)  (J-2)  1  -2(J-2)  -4(J-l)  1  -2(J-2)  -2J-K5  g  2  2  -3 (2 J-3)  2  Table 7 ( c o n t i n u e d ) . A j A -2  j  m  V  KHj  2  2 1  -2S  g  5 2  J  -  5  V ^ - l  *1  2  1  -2(J-3)  1  -3(2J-3)  -2(J-3)  (J-3)  1 2  2S _ 4,2J-6,2  1  4  rt  X n "L '  (J-4)  7  g  12  ~ i s ^  4 2fm ,2J-4,2-ittg  4 S4,2J-5,1 T ,  2  2  -2(J-4)  1  2  5  rtT c  5  -10(J-2-mg)  5(J-2-mg) -(J -l)  5  -10(3-2--^)  5(J-2-m ) -.(J -l)  1  -2(J-2-m )  5  2  2  (J-2-m ) - J 2  2  4J -40J+81 4J -40J4-83i 2  2  1 0 m i s 9 2+m ,2J-4,2-m  2  a  -10(J-4) -10(J-4)  5  2  4  6  0  2  2  -1 m 2  2  5  -lO(J-2-m )  5( J-2-m ) -(3J -t-3J-l)  5  -10(J-2-m )  5(<T-2-Hig) -(3J +3J-l)  2  2  ; 1 -2(J-2-mg) (9-mj)c4. 4 3M ,2J-4,3-iH2  2  2  (J-2-ttgf  2  5  -lO(J-2-m )  5(J-2-m ) - J ( J + 2 )  5  -10(J-2-m )  5(J-2-m ) -J(Jf2)  2  -2 m  4 ( ^ ) ( - 2 ) ,2  -3 m  * \ 2 * / \ 2 7°4tm ,2J-4,4-m -2(J-2-m ) ( 3 2)( ; 2J ^ ^ ^  4  4  m  2  2  2  2  2  5  2  2  2  2  M  5  2  2  M  2  J  -  4  (J-2-m ) 2  2  Table 8.  A J Aj 1  m  ZiCj^o  or  ~A-3. ~ 1  KS  g  J =^-2, 2  A  AJ j  v  r  "pnq  3 m-, 2m  The summations  V3 ^ 3  <  2  ra  m  2  2 S  _  m 2 j 2 J  , m ,2J,l+m  9 1-m  vn  2  0  4J -10J+6  5  -10 (J-1)  4J -10J+6  1  2  ~  2  -4SA '2,2J,2  a  l  s  8  J  s  S  3,2J-6,4  2  -10 J  2J -3J+-1  5  -10J  2J(2J-1)  5  -10J  2J(2J-1)  1  -2J  -2J-1  1  -2J  J  1  -2J  -2J-1  '  -  1  2  8S  j  8  2  2  J  j  2  2  -4J-3  (2JM.) (2J+2)  -4J-3  (2J-rlK2J+2)  1  |s „ 2 3,2J,3 3,2|--6,3  2 J  2J -3J+1  2  - 3,2J,3  4S  2  -10J  1 >  2  2  2  0  3 0  ^  -10(J-1)  1  1  4 > 3  5  1  -lo6 4"^l-m ,2J,l+m  -2 0  3  _  2  5  -  v°=l  1  (J-2)  5  1  y  -2(J-2)  2  o  x  J-A.j, J - J -Aj  " 4 ^ 2 - m , 2 J - 2 , 2+m  2  _n  , S  S  2  0  ^  2  1 1  ^  7^4-m ,2«T-6,4+m,  (9- )(4-m )  2  V  2  > ^ cPm^lm^  1  -2 (J-3)  1  (J-3)  3  2  Table 8 ( c o n t i n u e d ) . j  m  2  0  r  1  1  a  4  S  S  V  v°  1  1  - 2 ( J - 2 )  ( J - 2 )  1  - 2 ( J - 2 )  ( J - 2 )  1  - 2 ( J - 2 )  ( J - 2 )  1  2  1  - 2 J  J  1  - 2 J  J  5  - 1 0 J  2 J  5  - 1 0 J  2 J  1  - 2 ( J - 1 )  5  - 1 0 ( J - l )  2 J  5  - 1 0 ( J - l )  2 J  5  - 1 0 ( J - l )  2 ( J - 1 ) ( 2 J - 3 )  5  - 1 0 ( J - l )  2 ( J - 1 ) ( 2 J - 3 )  1  - 2 ( J - 1 )  ( J - l )  5  -IO(J-I)  2 ( J - 1 ) ( 2 J - 3 )  5  - 1 0 ( J - l )  2 ( J - l ) ( 2 J - 3 )  1  1  2 , 2 J - 4 , 3  8  2  2  2  2  ™ 0 , 2 J , 0  8  5  6  9'  a  b  1 , 2 J - 1 , 1  l ,  S  2 J - 2 . n l  5  1 , 2 J - 2 , 2  ^2 1-HR ,2J,1S  0  1 6 S ' 2 , 2 J , 2  -3 m i/+2V-2U 5 m  2  9V 2  3 m  J\ 2  - 3 J t l  2  2  2  - 3 J U  - 1 3 J + 6 - 1 3 J 4 - 6  2  2  -10(J-m )  5 ( J - m  5  -lOU-nig)  SfJ-mg)  1  -2J(l+m )  J (l+-mg)  2  2  1  - 2 J  1  - 2 ( J - 1 )  1  - 4 J  2  )  2  - J ( J + 2 ) - J ( J f 2 )  2  J  1  4 1 2 S 3 , 2 J , 1  1  *  2  5  0  - 2  2  1  2 , 2 J - 4 , 2  o Ms  I  -1  6  2  i  i  v  KS pnq  2  2  2  ( J - l ) A  4 J  T  2  2  /  5^2,23,5-^2 1  -2J(ltm  J (14-m 2  )  )  2  s Table 8 ( c o n t i n u e d ) . v  KS pnq r  m  -1  2  -s  3 0  -  v  2  v°^i  1  -3(2J-3)  1  -2(J-3)  1«2 "2* 3,2J-6,3  1  5  -4S  1  -aCr-2)  (J-2)  1  -2(J-2)  -4 ( J - l )  1  -2(J-2)  (J-2)  1  -(4J-5)  (2J-2)(2J-3)  2  3,2J-5,3 1  2 0  2,2J-4,2  1  %2J-4,2  1 m  2  -lgS 4 l4/mrt,-2J-2 l" 2 5 f  2  2  m  -10 (J-i  2  )  5(J-l-mg) -(J -l) 2  2  5  -lO(J-l-mg)  5(J-l-m ) -(J -l)  1  -2(J-l-m )  (J-l-m ) -J  5  -10(J-l-m )  5(J-l-m ) -(3J 4-3J-1)  5  -10(J-l-m )  5(J-l-m ) -(3J +3J-l)  1  -2(J-l-m  2  2  2  2  2  2  2  m2 I 9s l t m ,2J-2,l-m 6  0  -1  4  m  2  -3 m  2  iiislis 4  V3 \  2  2  )  2  2  2  2  2  (J-l-m )  2  4  2+m ,2J-2,2-m 2  3  2  -10 (J-l-nig)  5(J-l-m ) -J(J4/2)'  5  -10 (J-l-nig)  5(J-l-m ) -J(Jt2)  1  -2(J-l-m )  (J-l-m )  / 4+m ,2J^2,4-in 2  2  2  2  2  2  2  2  2  Table 9.  The summations Zf4^)  ' ^ 2 1 ^ 1 1 1 '  , J * J - A J o r A J ^ - S , A j s a, . 3 . M  A J A.J K S p 1  v  nq  || 3,2J-6,5  3  V  v°=l  1  ^  S  2 16:43S ^ _ ^  1  3  2J  2  4  4  -<JT-S)  -2(J-2) 2  1 S»5 2 1,2J-2,3  8„7  - -(J-4)  -10(J-1)  2(J-l)(2J-3)  5  -10(J-1)  2(J-1)(2J-3)  2  -(J-2)  5  _2 2J -3J-HL  -10J  5  -10J  2J -3J+-1  1  -2J  J  2  2 - ^1,2J,S 1  *2  "  5  ^2,23,2  -J 1  0  2J(2J-1)  J  -10J  2J(2J-1)  2  -5J-2  J(2J-KL)  "  1  2  J  j  -4 J-3 2  ^3,2J,3  3  3 4f3s|  0  > 2 J  -  1  _^  2 ISflfcj^g  2  4  3  1 2  1  f l,2J-2,2 S  2  5  1  "  2  5  .  6,2J,2  0  (J-2)  4 J  5 2  (2J-KL) (2J+2) -J  "  3  -(2J-11) -2(J-2) -(2J-7) -10(J-D  5  2  -10(J-1) .(2J-3] xkikaZf.  (2J+l)(2J4r2) -3(2J-5) (J-2)  2  -2(2J-3) 2(J-l)(2J-3) 2(J-l)(2J-3) -(2J-1)  Table 9.(continued). Aj  41 K S  v  r  V  S  v°»l  1  pnq  o o |psI,2J-i,i  -  1  -io(J-i)  (2J~l)(J-s)  5  -10(J-1)  (2J-l)(J-s)  1  -2(J-1)  (J-1)  2  -(2J-1)  5  ^Pi,2J-l,l  -10(J-D -10(J-1) -(6J-1)  5  5 2 -2 -12|3Sg j - i , 2  -2(J-1)  1  2  -3 - 4 ^ 3 S  is  5  -1  3  2 > 2 J  _  1 ) 3  _ I' 3 3,2J-S,3  2  -(6J-1)  2J(2J-l)  1  5  6 2  ;  "  1  2 (  -(3J-5) ^  2 )  3  -lfs  5 )2J  _  2A  2 ) 2  -3(J-1)  5 2  -(3J-7)  1 0 (  -  1  "  ( J  2  " ^ -10(J-2)  5  f  °9#2,2J-2,0  2  2J(2J-1)  2 / ^PS 2J-3,l  (J-1)  -(6J-1) .  1  1  (2J-l)(2J-5) (2J-l)(2J-5) 2J(2J-1)  2  :i  *H.2J-*,2  2  2 ( J  "  2 )  2 )  (2J-3)(2J-7) (2J-3)(2J-7) -6(J-1) ( J  -  2 )  "  5  -10(J-2)  (J-1)(2J-21)  5  -10(J-2)  (J-l)(2J-2l)  2  -3(J-1) \  5  -10(J-2)  2(2J -11J4*10)  5  -10 (J-2)  2(2J -lu4iO)  2  2  2  -3(J-1)  Table 9  (continued).  A j Aj K S "°  -i  v  r  pnq  -2 i s T 3 s f  j 2 J - 2 > 2  V  2  1  :  -» # 5 , ^ - 2 , 3  2  1  v°=l  wg(j-S)  (J-2)  2  -3(J-D  -  3 ( J  -  2  1 5  Appendix A.  The Normalized  I n equations  Angular Momentum C o e f f i c i e n t s .  (A3) and (A4) below, two e x p r e s s i o n s are  g i v e n f o r the normalized angular momentum c o e f f i c i e n t s .  Since  the summation i n (A4) i s e a s i e r to evaluate than t h a t i n (A3), an e x p r e s s i o n f o r the f a c t o r f ( J J J ^ V - i s d e s i r e d so t h a t (A4) tt  can be used i n the c a l c u l a t i o n s .  I t i s the o b j e c t of t h i s  appendix to show how the n o r m a l i z a t i o n f a c t o r f ( J J J g ) can be t t  obtained and then to show t h a t the two formulae a£e  (A3) and (A4)  the same by p r o v i n g the summations i n each formula are  e q u i v a l e n t . / Prom el-ther of these two r e s u l t s the equations (18) and (19) can be o b t a i n e d .  .  .  .  The normalized a n g u l a r momentum c o e f f i c i e n t s ( J J m m | j " J J m ) are d e f i n e d by t,  u  2  2  2  ^•^^V"^I^ ^W*J,-,• J  (A1)  8  where  i s the normalized wave f u n c t i o n d e s c r i b i n g the s t a t e  of the system c o n s i s t i ng of a nucleus and an emitted p a r t i c l e  (angular momentum J")  (angular momentum Jg) which are i n the  s t a t e s represented by the normalized wave f u n c t i o n s l//„ , J"m respectively. Jm r e f e r s to the t o t a l angular momentum M  m  and i t s z component, r e s p e c t i v e l y , of the system. I f 2dV=l i s formed i t can be seen t h a t the normalized coefficients  satisfy  ^(juj^mglJ^Jm) *!. 2  (A2)  This f o l l o w s f r o m the o r t h o g o n a l i t y of the wave f u n c t i o n s and the f a c t t h a t the c o e f f i c i e n t s are . r e a l numbers (Cf A 3 ) .  The e x p r e s s i o n g i v e n by Wigner" ' 1  c o e f f i c i e n t s i s (using d i f f e r e n t  I J-J m«Bu| 9  ^  2  r  (T+««M  where \  J«J Jm) 9  L  2  (r-~i)\  1x  4  f o r the normalized  notation)  R2JW^-^ (2J+VDJ  J  r/-i) - ^^ 2Ji-rJ''t^»-^-^!(j^^v )! ot4 T  <  at  i s d e f i n e d by J=J 4Jg-A^twhich means i t s v a l u e s must M  z  i n the range b^Afcminimum of 2<X , 2 J  lie  rt  g  since  )J -J ]4: J t J f J ; H  t t  2  g  m=m +mg; s t a k e s a l l those v a l u e s ' f o r which negative arguments tt  do not appear i n the f a c t o r i a l s  (01 = 1)  ( A l l summation i n d i c e s  which do not have t h e i r range- g i v e n w i l l be summed i n t h i s manner.).  This formula i s not easy to use because i t i s n o t  15 v e r y symmetrical a symmetrical  i n the J*s and m's.  Van d e r Waerden,, g i v e s  formula f o r the unnormalized  U s i n g t h i s formula  the normalized  coefficients.  angular momentum c o e f f i c i e n t s  may be w r i t t e n i n the form (^'Jgrn^nigl J » J J m ) = f ( J J J ) * U  r  cr+»»v.  2  2  ^ f r f t f W ' M  r f ( J J « J )cJf*\ 2  u s i n g (19).  J m J m u  Uii-HaM  tr^i)!  (A5)  ,  M  1  (T"-^»)(  A  Here f (JtT'Jg) i s the n o r m a l i z a t i o n f a c t o r  neces-  sary f o r (A4) t o s a t i s f y (A2). The n o r m a l i z a t i o n  f a c t o r f is,now c a l c u l a t e d f o l l o w i n g If?  the method used by K e l l e r  .  By s u b s t i t u t i n g (A5) i n t o (A2)  w i t h m*J one can o b t a i n j"  •f2KcJ?,k j . ^ ) .1. B  2  (A6)  the range f o r m" being determined The  from the r e l a t i o n s i n (20)  summation i n (A6)- i s e v a l u a t e d as f o l l o w s i  first,  express  the summand e x p l i c i t l y i n terms of m" u s i n g Table 2; then  vr& -jr\;i u  change the summation over m" to one over use  the summation formula  4  > finally,  (45) w i t h k-0 to g e t the sum.  Combining the r e s u l t w i t h (A6) one o b t a i n s (2J+1) 12 J * Ax) ! (2X>)'!"Aa •' (2J*-A +l)! a  f(JJ"J )r i  (A7)  A  The equivalence of (A3) and (A4) i s now shown by p r o v i n g that «  ( 2 J - a - « ) ! (J^Jifm^m^-Ai-^) . 1  L  i  ( J  t  - J * « \  t  (~i)*(Wm^)! (j»-m j) ( J ^ ) ! ( J ^ - m ) l ' ( J ^ m " - ^ * ) ! (J -m -*)! (J^m^-ot)! (J^-m^-A f*)f « ! ( ^ ) ! a  -  a  M  (  A  8  b  M  A  Obviously, (b) cannot be obtained from (a) by a l i n e a r substitution forc<.  I t i s necessary to prove  (a) = (b) by  showing that both can be obtained from ( c) . To g e t (a) from (c) :  and perform the I n d i c a t e d d i f f e r e n t i a t i o n u s i n g Al\  =  s  Ux) To g e t (b) from  X  (  K!  V«-- -  —)! *  S  •  (A9  >  (c) :  ( u s i n g L e i b n i t x * theorem f o r the d i f f e r e n t i a t i o n of a p r o d u c t of  two f u n c t i o n s )  )  (using ( « ) ) .  then form  (f ) Y  J L  "~  1  L  f)*""* "^"J  ft  ( u s i n g Leibnit2 ' theorem)  Y U ^ . m ^ ) ! ^ ! (J«+m»-/j)i. (/5-Xx+t)! (using (A9)).  S e t t i n g Y - l i n the l a t t e r summation l e a v e s o n l y  the term f o r which /3 = A £ - ^ « one  obtains  (a).  Thus (a)=• ( b ) . Racah  17  S u b s t i t u t i n g these formulae i n (c)  This e q u a l i t y has  a l s o been shown_by  by a d i f f e r e n t method.  The e x p r e s s i o n f o r f ( J J J ) , g i v e n i n (A7) u  g  obtained by u s i n g the r e s u l t (A8) w i t h (A3)  and  could have been (A4).  To show t h a t ( r f * * > M >  Appendix B.  + m  and ( c i f  2  A M  «-  T  )  w  2  are polynomials i n m. I f the l a t t e r q u a n t i t y i s a p o l y n o m i a l i n m then so i s the former.  Hence a l l that i s necessary to show i s the proof that 2  •Tin ? vi  ^ J m-mJ m ^ C  i s a p o l y n o m i a l I n m and t h i s i s g i v e n below.  a  Prom the symmetry p r o p e r t i e s if  (CJII^JJ^J  J  m  (21) , (23) one can see  i s a p o l y n o m i a l i n m f or O t ^ - J g and O i m ^ J g  then i t i s a p o l y n o m i a l i n m f o r - a n y A.g and m^ i n the 0t^gt2Jg, - J g ^ g - J g *  I  (j^4m .)l (J -m )! tt  u  ranges  the d e f i n i t i o n (19), i t i s seen t h a t  n  the summand i s a polynomial i n m. factors  that  I t must be shown that the  appearing i n the denominator o f  tt  2 (C )  are c a n c e l l e d out f o r 0-*g-Jg» °  f e m  2  f c J  g  t  complete the  o  proof. Now, m^O.  J +m = J+m-(J +m -^) and i s £J+m f o r O^AgfeJg and w  w  2  Hence (J +m )! u  2  d i v i d e s (J+m)(  M  f o r Oi:Ag^Jg and  (Kmgtjg.  J -m = J-m-(J -m -A ) and i s £J-m i f Jg-mg-Ag^O. F o r the case t h a t J -nu-a ^0, the f a c t o r (j-m)'"""_ (j-m) is ^ (J»-m")( - (J-m-(J -m, A ))l, tt  H  2  g  2  1  o  2  o  2  a>  c a n c e l l e d by the common f a c t o r (J -m )! (J-m)l ft  which appears i n a l l the terms,  a  ,  <L  (J -m )f (J^-m^J^-m^-"^) fl  a  ( j -m") 1 , of the summand (J -m -oc)\ a  M  l,  In (19) since J^-m^-A +«l^O f o r a non-zero A  r  term.  Appendix  C.  The Summation Formulae.  To d e r i v e the formulae (45) and (49)'. I t w i l l now be shown t h a t j T ^ £ ) ^ q - v j k j - ^n+p+q+lj v  (Bl)  +  v  i  f  k  =  0  r ^ a . .l£lllL/n+P+q+i\ i f k i l k  where a  k c (  *  pi  [  (B2)  n-* /  ( = 1 i fk*l,* =l ' -*<r* g a ^ ^ . ^/ k^-il \f  v k.2,.,2.  +  (  B  3  )  These formulae are the same as (45) and (49) of the t e x t . The r e s u l t  ( B l ) w i l l f i r s t be proven, (B2) and (B3) then  f o l l o w from i t by i n d u c t i o n . -To prove  ( B l ) , one forms  (l-X) - P - ^ l - X ) "  q  -  1  = (1-X) "P'S"  2  and expands eash b i n o m i a l to g e t  Collecting  the c o e f f i c i e n t  of X  on both s i d e s of t h i s e q u a t i o n  n  one a r r i v e s a t the r e s u l t ( B l ) . The formula (B2) i s now proven f o r the case k-1.  j*7P+v\ v  /n+q-v\ JyT  (ptv)!/n+q-vU  v  ^ l p / l  q )  ~£r (v-1)! p! (, q  (p+l)( P+5+ ) n+  Now, (r=>2) .  1  /  (r> i) ? /P+l+w\ /n-lfq-w\ +  P  fe(p*lA  q  /  using ( B l ) .  assume (B2) i s true f o r a l l v a l u e s of k up t o k = r - l  Then5^/P* U * " W * v  fe{ p A  n  1  v  q /  r  T  J&ziLl  /n+q-v\v  ^(v-l)lp! ( q /  r-l  - (p+1>UJ pa )( " q " 1 p  w  n  1  <3  ,  (using the i n d u c t i o n assumption) -(p+l)Z  A*/ . * A " * C ' < |  (p+-U!  a  (n-(^lV  --Zf2a _ M|] l £ l ^ / n ^ ^ l W y  " K F , -'f'a  r  ^ - -  Ir-flJI  1  pi  (  (  n - W  + 1 )  +  (  +  l  P  I  + 1 )  n-1/  /a+p q l\ +  P  +  1  )  +  l  J  +p+q+l\ n£L  (Pt«)! /n+pfqtl\  where a  r f t  <3*l  = 1  if-ril.  T h i s completes the proof of (B2) and ( B 3 ) . To d e r i v e formula The formula  (46) :  -  (46) may be d e r i v e d by u s i n g the same method  of p r o o f as t h a t a l r e a d y g i v e n f o r the formula w i t h the e q u a t i o n (l-Xj'P-^-d-X)-^ u s i n g the formula  ( 1 - X ) ( L - X ) Q - ( 1 - X ) Q i n s t e a d of P  1  P +  = (1-X) -P-q-  and expanding the b i n o m i a l s  2  (1-X)  (-X)  (1-X)- ,[X (- )(-X)^|l(-l) ( 4 P  v  P  v  relation  ( ) - (-1)  from (45).  - P  (45) s t a r t i n g  v  i  s  a  1  1  v  + V  i n s t e a d of )(-X) . V  However, the  t h a t is-needed  to get (46)  I f one sets -p-l-P and -q-l=Q I n (45) then (46)  can be obtained u s i n g t h i s r e l a t i o n .  The l i m i t s f o r the summa-  t i o n s are determined from the range f o r which the binomial c o e f f i c i e n t s i n the summand do not v a n i s h .  Appendix D.  The proof  The oc(l)be  of formula  (53).  and Y ( l ) - m i x e d y(2)  c o r r e l a t i o n f u n c t i o n s can  c a l c u l a t e d from (*>f) and (52) .  The second bracketed, exp-  i n (5X)  r e s s i o n i n each W(©)  i s independent of © .  thus be w r i t t e n s y m b o l i c a l l y as (see reference  1 ^ ( 9 ) can  3)  W ( 9 ) r gGF°_(©) -fdGPj(9) ,  (Bl)  i2  where gG- and dG are the c o e f f i c i e n t s of P^(9)  and P^(9)  ob-  t a i n e d from (SO.) * The «,(l)-^(2) c o r r e l a t i o n w4 f u n c t i o n W ( 9 ) w i l l 12  be obtained  from the *(l)-X(2)  first  c o r r e l a t i o n f u n c t i o n W. (9)  which-are t a b u l a t e d  i n L i n g and P a l k o f f 's paper.  Substituting  i n (Dl) expressions  f o r the P's from f a b l e 1 f o r an oi(l)  p a r t i c l e and a tftl) r a y one obtains • 0C(1) -1T(2) : |f(l)-V(2):  W  (9) - dG 4- (2gG-dG) c o s 9  (D2)  2  lg  W (9)-^ 2gG4-dG+-(dG-2gG)cos 9. 2  12  -  -V (D3)  Ct>3) The expression. Is g i v e n i n Table I I . of L i n g and P a l k o f f ' s paper i n the form tf(l)-lf(2).  W ( 9 ) x Q-rRcos ©,  (D4)  2  12  w i t h e r 0.  ,  M  Hence, 2gG+-dG + (dG-2gG). -K(Q+-Rcos © ) , where K i s a p o s s i b l e  (D5)  common f a c t o r that has been omitted.  E q u a t i n g the c o e f f i c i e n t s of cos ©, one obtains 1  two equations  from which one can solve f o r gG and dG i n terms of K, Q, and R. S u b s t i t u t i n g the r e s u l t I n (D2) g i v e s <X(l)-tf(2): W ( Q ) r K(Q4-R - Rcos^Q) .2, l g  from which K can be omitted.  This r e s u l t i s e x a c t l y  the same  f o r W (9) andWj.(9) , a d i f f e r e n t G- b e i n g used i n each case* 22  The common f a c t o r K i s the same i n each case* oCClJ-mixedtf^)  Hence the  c o r r e l a t i o n f u n c t i o n s can be obtained from the  £(l)-mixed*(2) c o r r e l a t i o n f u n c t i o n s - l i s t e d i n Table 11. of L i n g and P a l k o f f ' s  paper by the r e l a t i o n (S3).  4  Bibliography.  .  ,<  1. D. R. Hamilton, fchys. Rey. 58, 122(1940). 2. G. G o e r t z e l , Phys. Rev. 70, 897(1946).  /  3. D. L. P a l k o f f and G. S. TJhlenbeck, Phys. Rev. 79, 323(1950). 4. D. S. L i n g J r . and D. L. P a l k o f f , Phys. Rev. 76, 1639(1949). 5. J . A. S p i e r s , Phys.  Rev.- 80 , 491(1950) .  6. S. P. L l o y d , Phys. Rev. 80, 118(1950). 7. B. A. Lippmann, Phys. Rev. 81, 162(1951). 8. S. P. L l o y d , Phys. Rev. 81, 161(1951). 9. C. N. Yang,-.Phys. Rev. 74 , 764(1949). 10. D. L. P a l k o f f , Phys. Rev. 82, 98(1951). 11. W.  R. A r n o l d , Phys.  Rev. 80, 34(1950).  <  12. S. Devons, P r o c . Phys. S o c , A, 62, 580(1949). 13. C. A. Barnes, A. P. Prench, S. Devons,-Nature, 166, 145(1950). 14. E . Wigner, Gruppentheorie, Braunschweig, Vieweg,  1931.  15. B. L. van der Waerden, Die Gruppentheoretische Methode i n der Quantenmechanik,  B e r l i n , Springer,  16. J . M. K e l l e r , Phys, Rev. 55, 509(1939). 17. G. Racah, Phys. Rev. 62, 438(1942).  1931.  

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