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

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SOME ANGULAR CORRELATION FUNCTIONS FOR SUCCESSIVE NUCLEAR RADIATIONS BY FOREST GENE HESS THESIS SUBMITTED IN PARTIAL FULFILMENT THE REQUIREMENTS FOR THE DEGREE OF MASTER OF. - ARTS i n the Department of Physics We accept this thesis as conforming, to the standard required from candidates f o r the degree of MASTER OF ARTS Members of the Department of Physics THE UNIVERSITY OF BRITISH . COLUMBIA July, 1951 Abstract Let J*, J, J" represent the t o t a l angular momenta of the i n i t i a l , intermediate, and f i n a l states of a nucleus respectively and J , J 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 thi s notation, the following results can be found i n this thesis. <*-y and tf-fr c o r r e l a t i o n functions have been calculated 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 following conditions: f i ) J ' = J+Jlt J= J t t+J 2 f o r a r b i t r a r y J x , S2 - 1, 2r* ( i i ) J'= J - J x , J= J t t - J 2 f o r a r b i t r a r y J ^ , J g= 1, 2:; ('ill) J»- J ^ - J , J = J M + J 2 f o r a r b i t r a r y J ^ , J g-r 1, 2. (iv) J ' - J - J x , J = Jg-J° f o r J - ^ 1, 2, a r b i t r a r y Jg. These are called the "spec i a l t r a n s i t i o n s " i n the text. tX-mixedJf co r r e l a t i o n functions have been tabulated 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 with t o t a l angular momentum 1 or 2 and a photon corresponding to a mixture of e l e c t r i c quadrupole and magnetic dipole radiation. For an (X p a r t i c l e with t o t a l angular momentum 3 the tx-mixedy co r r e l a t i o n functions can be obtained from a table which l i s t s the sums of products of angular momentum Coef-f i c i e n t s appealing In these c o r r e l a t i o n functions. These cor-r e l a t i o n functions are too clumsy to be expressed e x p l i c i t l y In terms of cos^9 In general, however they can be f a i r l y e a s i l y evaluated once numerical values of the angular momenta of the nuclear states are prescribed. Table of Contents. c - Page Introduction 1 I . General expression 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 functions 15 A. Symmetry of the-summations 15 B. Correlation functions f o r special 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 functions S'7 A. *(1) -mixedlT(2) c o r r e l a t i o n functions 27 B. °<(2)-mixedY(2) c o r r e l a t i o n functions 29 C. <x(3) -mixedy(2) co r r e l a t i o n functions 29 Tables: : -~ 1. Some angular d i s t r i b u t i o n functions 32 Jm 2. E x p l i c i t expressions-for some Cjtt^uj M 3 3 3. A tabulation of the c o e f f i c i e n t s , a ^ 34 4. Abbreviations used i n Tables 5 and 6 35 5. «(1) -mixedV(2) c o r r e l a t i o n functions 36 6. c<(2)-mixedY(2) c o r r e l a t i o n functions 58 7. The s u c t i o n , , Z ( o J & V ( c f t £ ^ ) * 41 8. The ^ t l o n s . Z i i ^ ' ) ^ ^ . ) 2 49 9. The s u M m t i o n a . ^ C c J ^ j ^ J y ^ o J ? ^ 5g Appendices: A. The normalized angular momentum co e f f i c i e n t s B. To show that ( r f ? ^ )2 i s a polynomial i n C. The summation formulae D. -.The proof of formula (51) Bibliography Acknowledgement I wish to thank Professor W. Opechowaki f o r suggest-? ing the research problem and f o r h i s advice and encourage-ment throughout the performance of the research. I am grateful to the National Research Council of Canada f o r the donation of a Bursary (1949-50) and a Studentship (1950-51) i n support of the research. INTRODUCTION:. --If a nucleus emits two p a r t i c l e s or photons i n quick succession, there w i l l be a certain angle 0 between t h e i r d i r e c t i o n s of emission. The function, , representing the r e l a t i v e p r o b a b i l i t y f o r an angle© between the directions of emission of the p a r t i c l e s or photons i s calle d the d i r e c t i o n -a l or angular c o r r e l a t i o n function. The general expression f o r W(0) was f i r s t derived by Hamilton 1 f o r the successive emission of two photons (lT-T c o r r e l a t i o n ) . He calculated W(0) e x p l i c i t l y f o r the cases i n which the multipole orders .of the emitted photons are eit h e r quadrupole or dipole. Goertzel 2 , extended the theory of T -Y correlations by considering the e f f e c t on W(0) due to the presence of an i n t e r n a l atomic f i e l d or an externally applied magnetic f i e l d . He showed that the e f f e c t of the extra-nuclear electrons on the angular c o r r e l a t i o n between the successive nuclear emissions can be neglected provided the r a d i a t i o n width of the intermediate state of the nucleus i s much greater than the hyperfine s p l i t t i n g of that state. This was also shown by Hamilton. Goertzel also showed that an externally applied magnetic f i e l d may be used to reduce the e f f e c t of the extra-nuclear electrons on t he angular c o r r e l a t i o n . F a l k o f f and Uhlenbeck^ calculated c o r r e l a t i o n functions i n parametric 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 with angular momentum 1 or 2. Ling and F a l k o f f 4 then extended the theory to Include transitions i n which mixtures of multlpoles are emitted* They tabulated tf-mixedtf' c o r r e l a t i o n functions, where Tf' refers to dipole or quadrupole r a d i a t i o n emitted i n the f i r s t t r a n s i t i o n and mixed y t o mixed e l e c t r i c quadrupole and magnetic dipole r a d i a t i o n emitted i n the second t r a n s i t i o n . F i n a l l y , Spiers" 6 has Shown how the general angular c o r r e l a t i o n function f o r any successive emissions may be derived using the quantum mechanical addition of angular momenta. The same r e s u l t was shown by Lloyd^ using group theoretical methods. The above i s a resume of some of the theor 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 preparation of this thesis. For a more complete survey of such papers the reader i s referred to reference 3. This thesis consists of three main parts. In the f i r s t part, the general expression f o r the c o r r e l a t i o n function i s derived by following Spier's method. The general expression i s then written i n a form useful f o r c a l c u l a t i o n s I n the second part, a method of evaluating the summations which appear i n the formula f o r 1(8) i s presented. The method permits one to calculate angular c o r r e l a t i o n functions f o r any angular momentum f o r the f i r s t emitted p a r t i c l e or photon provided the transitions involved s a t i s f y certain special conditions* Some <X-rand Y c o r r e l a t i o n functions are given e x p l i c i t l y f o r these special cases. In the thi r d part, tables are given from which <<-mixedir c o r r e l a t i o n functions can be obtained f o r o an o*. p a r t i c l e with angular momentum 1, 2, or 3, and a photon corresponding to mixed e l e c t r i c quadrupole and magnetic dipole r a d i a t i o n . 1 I. GENERAL EXPRESSION FOR - T H E DIRECTIONAL CORRELATION P u H C T l O N . . The c o r r e l a t i o n function f o r the case i n which two p a r t i c l e s (not photons) are emitted i n quick succession by a nucleus i s derived below. E s s e n t i a l l y , the derivation due to S p i e r s 5 i s followed. The following notation Is used throughout this thesisJ J,ml,Jm,Jumu,J-Lm^,Jgmg - represent the t o t a l angular momentum and Its z component f o r the I n i t i a l , intermediate, and f i n a l states of the nucleus and the t o t a l angular momentum ( i n t r i n s i c plus 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. "vVjm- i s the normalized wave function f o r the nucleus i n the state represented by t he quantum numbers Jm. 4>jm- i s the normalized wave function f o r an emitted p a r t i c l e with quantized t o t a l angular momentum Jm. Henceforth the types of cor r e l a t i o n between p a r t i c l e s and photons with given angular momenta w i l l be denoted as follows: °t(J^)-5"(Jg) means that an «partide with angular momentum Jj. i s emitted i n the f i r s t t r a n s i t i o n and a photon with angular momen-tum J g corresponding to a 2 a m u l t i p o l e i s emitted i n the second. «(J-^)-mixedT(Jg) means the same as above f o r the f i r s t t r a n s i t i o n , but indicates that a photon corresponding to a mixture of 2 & • J,-1 e l e c t r i c and 2 6 magnetic attltipole r a d i a t i o n i s emitted In the second t r a n s i t i o n . In the derivation, i t i s assumed that the e f f e c t of the extra-nuclear electrons on the co r r e l a t i o n can be neglected. If this were not the Case, the t o t a l angular momentum of the nucleus would precess about the f i e l d due- to the electrons and hence could change i t s value. However, this value i s assumed constant throughout the derivation. This assumption i s V a l i d i f the rad i a t i o n width of the nuclear state i s much larger than the hyperfine s p l i t t i n g of that state. . _ If l '^i ,represents the state of the system consisting of the Intermediate nucle us and the f i r s t emitted p a r t i c l e then, using the quantum mechanical addition of angular momenta, Cra; ' 1 ^ a U , i ^ i i / W ( . ( D Here the Q.^ are the p r o b a b i l i t y amplitudes f o r the various possible values of J, ( |J f-JJ i= Jj £J '+J) which the f i r s t emitted p a r t i c l e can have; the bracketed expressions are the normalized angular momentum c o e f f i c i e n t s which may be considered defined by (1). Using (1), one may represent the two transitions of the nucleus by; HV^ < Or* '^^'-'^j-^^W, Equation (2) may be written i n the form ( 2 ) ( 3 ) where C l l w ^ ^ , ^ ' " ^ ^ ^ ^ ( 4 ) Prom (3) i t i s seen that the f i n a l state, Ij^,, of the sys-tem consisting of the f i n a l nucleus and the two emitted p a r t i c l e s i s given by • 2L it <C <£L* -^w . ( 5 ) Now, at any time,|l|>^l|2'dVuaV | dV z i s the p r o b a b i l i t y that, with the i n i t i a l nucleus i n the state J'm', p a r t i c l e 1 i s In the volume dV, about the point r ^ w i t h I n t r i n s i c angular momentum and i t s z component or , p a r t i c l e 2 i s i n the volume dV^ about the point r 6^with i n t r i n s i c angular momentum and i t s z component , and the f i n a l nucleus i s i n the volume dVtt about the point r t t&V. dV" and rtt0u4)tt represent symboli-cally-the volume elements and coordinates of a l l the nucleons. In this thesis, only the directions of the two emitted p a r t i c l e s w i l l be of in 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 dVt at. ijO,^ and that p a r t i c l e 2 i s i n dV^ at r ^ ^ i s desired. - To get this i t i s necessary to sum the p r o b a b i l i t i e s f o r the various spin orientations o f , o f the two emitted par-t i c l e s and to Integrate the nuclear coordinates over a l l space. Normally, the i n i t i a l nuclei w i l l be randomly oriented i . e . the 2J'+1 degenerate states i]/^ , are equally populated. The average co r r e l a t i o n f o r a l l nuclei 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 states. In the normal case, this i s done by summing over m1 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 function W(r(8(4> »  Ti%$i) between the dir e c t i o n s 0 , ^ and 9^ may be defined by where r t and r t are. taken to be large but constant. Only the re l a t i v e v a r i a t i o n of W with 6 ; a n d 6^ i s of i n t e r e s t experi-mentally and so, f o r this reason, factors independent of these angles w i l l be omitted from W. Prom equations(5) and (6) one obtains (omitting j^pj ) using the orthogonality of the H^-^'s. The summation over m can be brought outside the square modulus when0,=4| = O. For, the wave function of an emitted p a r t i c l e , which has t o t a l angular momentum J and z component m and i n t r i n s i c angular momentum S with z component o"", can be written, using the addition of angular momenta, as where .X i s the wave function representing the i n t r i n s i c ang-u l a r momentum and ^(r)Y"*u°(8<$) that representing the o r b i t a l angular momentum of the p a r t i c l e . The quantities b L are prob-a b i l i t y amplitudes f o r the various possible o r b i t a l angular momenta the p a r t i c l e can have. Since Y/^"^00) = 0 unless m-fT, i t i s seen that ^ ^ ( r , 00<Jp)-0 unless m,=cr|, and hence that C0* (r, 00<r; ) = 0 unless m'-m-a^  from (4). Thus, i f 0, = <f. = 0 i n (7) * there i s only one value f o r m which gives a non-vanishing term once m1 and cr- are given and the summation over m can be discarded, 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', m, and ft" knowing that the terms of the summation w i l l vanish unless m'-aut^ , and so I t i s possible to write W ( S 0 0 . ^ ) . £ w S | « ^ ( , 0 0 , ^ * 2 1 0 ^ . ( ^ ^ ) 1 * . (9) W(r, OO.r^Q^) does not depend on because of the aver-aging processes used i n obtaining i t . I t depends only on the angle 6^ -0 between the directions of emission of the two par-t i c l e s . Prom equations (3) i t i s seen that P^fO) and P ^ p ) defined by represent respectively the p r o b a b i l i t y that the intermediate nucleus i s i n the state Jm with the emission of the f i r s t par-t i c l e i n the z d i r e c t i o n (and at r, ) and the p r o b a b i l i t y that the f i n a l nucleus i s i n the state Jl,my with the emission of the second p a r t i c l e at an angle 0 to the f i r s t (and at r 2 ) . - u~sing -(9) and (10), one may write 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 ) , (11) where i t i s understood that r ( and rz have large but constant values. I t should be noted that i n equation (7) there exists ant. Interference between the various ways i n which* a transition-can occur from a given sublevel J'm' to a f i n a l sublevel. jVmu v i a d i f f e r e n t intermediate sublevels Jm because the p r o b a b i l i t y amplitudes are summed over the intermediate sublevels before squaring rather than a f t e r and cross terms appear. As seen i n equation (9), I t i s possible to remove this interference by taking the d i r e c t i o n of emission of the f i r s t p a r t i c l e along the axis of quantization. This r e s u l t was also obtained by Lloyd and discussed by Lippmann . If the d i r e c t i o n of emission of the second p a r t i c l e instead of the f i r s t had been taken along the z axis of quantization, then W(9) would have been written i n the form W(0)=2L P..feiPjo). i (12) The two expressions f o r W(0) i n (11) and (12) must obviously be equal. - • -By substituting (4) into P ,.(©) of (10) one obtains The above formulae have been derived f o r the case-that p a r t i c l e s and not photons are emitted. Ling and P a l k o f f ^ have treated the case f o r the emission of a photon corresponding to mixed 2— e l e c t r i c and 2^' magnetic multipole r a d i a t i o n . Their r e s u l t s are given by Zvd* > ck 2 A - A W r A k * Z l JA^  I * • (14 ) where A_a A x- L A Y, A„= A e , Atl=Axtt A Y and A ^ ( j » ^ m « m J jtij^ J m ) A ^ ( j i ^ H/3(J uJ rlm M m J J M£-lJm) AjJ(^-l,nJ . Here A£'(J, ,m,), A M('J-1, m-;) are the components of the normalized r T-l vector potentials f o r a 2* e l e c t r i c and a 2 * magnetic mul- . tipole respectively; ei and represent the p r o b a b i l i t y ampli-tudes f o r each multipole. Lloyd* has shown that and /5 can be made real by a prop er choice of the nuclea# phases. How-ever, i n some calculations, i t may be useful to have complex values, hence the formulae are l e f t i n the above form. The r e s u l t (14) can be incorporated i n t o formula (13) i f i s taken to be 1 (tr=l,0,-l) and ^ a r e replaced by /nPrkA^(J,,m,) and firrkA*(J-l,m,) respectively. Of course, o c * ^ and /&* A. • Henceforth (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, the proper ^'s being substituted In each case. At this point one can see that the dependence of P ^ i ^ and P^^and hence W(0) on r, and r t can be factored out and thus -omitted. For, i n (13) the <j>'s (and A's) depend on r through the functions fL (r) , which f o r large r are proportional to (-L) e_ (spherical wave), where k i s the magnitude of the propagation vector. I f only one value of Jz i s possible f o r the p a r t i c l e or photon emitted i n the t r a n s i t i o n Jm -*-Jum.a then the equation (13) takes the form the angular momentum co e f f i c i e n t s being r e a l (see equations (18) and (19)). Omitting the dependence on r , which can be II factored out, this has the form This formula can be .found i n Falko f f and u*hlehbeck*s paper. In this thesis only transitions i n which an * p a r t i c l e or a photon i s emitted w i l l be considered. Since an <* p a r t i c l e has no i n t r i n s i c angular momentum one can write F ^ ( 0 ) = |Y^"(0 <Px)\l f o r i t , using (8) and (13). General expressions f o r the F r s f o r a photon a r i s i n g from pure and mixed multipole tr a n s i t i o n s are given by Ling and Fa l k o f f ^ who substituted expressions i n terms of spherical harmonics f o r the A fs appearing i n (13) and (14). A 2 J * - e l e c t r i c and a 2^''siagne t i c multipole have the same F > ( 9 ) . For a n y p a r t i cle or photon, F ^ l ( 0 ) * PT'te) (see *i •>»> reference 3) • For a mixed tf(J2) t r a n s i t i o n , Ling and Falkoff** have given the following formula: W"' M + *^<*%r''TW^|J , ,TvT~X^^ (17) Here 2R(K6*) - <*/}* + ^ ( J , and the ^ ) a r e t h e angular d i s t r i b -u t i o n functions f o r the interference contributions to P,^ .. (0) a r i s i n g from the m i J E i n g of the 2"^  e l e c t r i c and 2^ -"' magnetic multipole f i e l d s . The form of the r e s u l t (17) follows from (13) and (14). Some P-J*** (0) are l i s t e d i n Table 1. The angular momentum c o e f f i c i e n t s are given 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 ) where J ^ J U * J x - At; 0*= \tmlnlmum of 2 J y , 2 J a ; l ^ + M ^ ; and V"/-" = * J . Ur'4-")! (7"-~")[ fa fc*v~0'J * ! j r a1'*-*").1-., Cr"-~")! ( n W . [ A (19) [M- h*- ' The summation over »C i s carried, out with the under-standing that each f r a c t i o n (A^0i>£ 0) appearing i n the summand i s to be-written 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 containing the f r a c t i o n —AL w i l l (A-flV. vanish. Using J=JU+JX-X,. and mrrattfm;l, i f i s e a s i l y shown that jo^Mj^must l i e within the smallest range given by the following conditions: -J*m±J, -J^fcrn^ J", -J^ fe mxfr , ^ (J t t-»-^ L-^)i,m M+m 1tJ , ,+-J i-^ x, - ( J - J A + - ^ ) 4 r m - m ^ J - J x + 7 . t , -(J-J ui-Ajfcm-m u 4:J-J u + A l. (20) The C's, defined by (16), have the following symmetry propertied* C y - M r ^ = ^ r . ^ " r t-^. (21) c J ^ - x > „ / „ a v c J ^ ^ (22) « J « A IJ^ - K / + + - r ^ ^ ^ i . ' (23) The C's used i n this thesis are l i s t e d i n Table 2. Using (1-1), (16), and (18) one can obtain the d i r e c t i o n a l c o r r e l a t i o n function W(Q) f o r a p a r t i c l e (J f ) or photon (Jj) 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 ) or photon (J A ) emitted i n the second t r a n s i t i o n . The r e s u l t i s wiaw21 [ n (cf~r*Tfcr"r^ , TJF? (^ fT(0), (24> Here, the normalization factors f o r the angular,momentum' co e f f i c i e n t s have been factored out and omitted; the summation over m1 (-m+m,), m, and mu (= m-ma) - has been replaced by one over m, , m, and m^ . The F's, as stated before, vary with the type of p a r t i c l e emitted* The angles 0 and 0 can be inter-changed ( C f . - ( l l ) and. (12)). The p a r t i c l e (J, )-mixed ^(J^) or 2f( J, ) -mixed KJ^} cor-r e l a t i o n function i s given by A n , < M t J — J ) * * , / ^ J " ^ . ^ Jt ^ J / " * - w c t /~ t t j t where common factors have been omitted. The formulae (24) and (25) are the ones which w i l l be used i n this thesis to calculate c o r r e l a t i o n functions. - The formula (25) i s v a l i d except f o r AA-0 and Ax" 2J^ . For, from considerations of the vector addition of angular momenta, i t can be seen that only a pure 2 ^ e l e c t r i c instead of a mixed 2^* e l e c t r i c and 2^' magnetic multipole t r a n s i t i o n w i l l occur f o r these two cases. For 0,2^ the formula (24) w i l l be used. A method of evaluating the summations appearing i n the square brackets of (24) and (25) i s given i n Section I I . of thi s t h esis, once these summations are known, i t i s a f a i r l y simple matter to obtain W(0) i f Jj or J*z i s small. W(0), as given by (24), i s also the co r r e l a t i o n function f o r the reverse t r a n s i t i o n scheme J t t — — ^ J * with the• emission of the p a r t i c l e s or photons occurring i n the reverse order. For, by making use of the symmetry property (23) and the re-l a t i o n F^ C d ) - F"**(6) , and changing the summations over m, and mt to summations over -m( and -mi, one can show that (24) i s equal 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 function f o r the reversed process. I f also J,= J Z , then W(0) i n (24) further represents the d i r -e c t i o n a l c o r r e l a t i o n function f o r the reverse t r a n s i t i o n scheme J t t—>J — * J f but with the emission of the p a r t i c l e s or photons occurring i n the given order i . e . as i n (24). This l a s t r e s u l t i s obtained from (26) by interchanging m( and mz, which i n t e r -change i s possible since m, and m^run over the same range of values (J, » J z ) . These results have been proved already, a l -though i n a d i f f e r e n t way, by Falkoff and Uhlenbeclr*. -These authors have also shown that the F^'s are polynomials of degree at most J i n cos*©• Since the expressions (11) and (12) are equal, I t i s seen that W(0) i n (24) i s a polynomial i n cos a© of degree at most the minimum of J, and Jz (see also Yang*). •"' II I I . CALCULATION OF-pIRECTlOKAL^^ -In order to c express 1(0) in-(245 e x p l i c i t l y i n terms of cos a6 and the angular momenta involved,, i t i s necessary ,to -evaluate the summations over m appearing i n the square brackets, to substitute expressions f o r the F's i n terms of COB^Q , and then to carry out the summation over m, and m^ . I t i s the object of this section to present formulae which one can use to simplify and perform-such calc u l a t i o n s . A. Symmetry of the Summations. I t i s possible to reduce the amount of c a l c u l a t i o n re-quired to obtain W(Q) e x p l i c i t l y by making use of the symmetry properties (21) and (22). Applying (21) to the summation ZL(^'*m V ( C ^ * » - ) 2 , (27) appealing 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 given, the formula (24) can be s i m p l i f i e d since the F m'(0) w i l l vanish • J , but f o r certain values of nij • Because <x.-y and r-Y correlations are the subject of this thesis, the formulas f o r t h e i r cor-r e l a t i o n functions w i l l now be obtained. If an <x p a r t i c l e l s : emitted i n the f i r s t t r a n s i t i o n along the axis of quantization then F^' (O)=0 unless m = 0. Using this r e s u l t and applying the relations (28) and F>(0) = P" T f f l i (&] to (24) one obtains ~ - (29) f o r the <K.(J, ) - p a r t i c l e ( J a ) or ^ ( J , ) - t f " ( J x ) c o r r e l a t i o n function. The interference summa t i o n appearing i n the mixture term of (25) f o r ^ ( J , )-mixed -^(J^) c o r r e l a t i o n functions can also be written i n a si m i l a r form. The common f a c t o r F^(o) i s omitted. The emission of a photon i n the f i r s t t r a n s i t i o n along the adis of quantization requires that m,*tl i n (24) since F1?' (0) = 0 unless m,=ll f o r a photon. From (24) and (28) then, one can write f o r y.(Jj ) - p a r t i c l e ( J ; ; L ) or &TJ, )-t(J1L) c o r r e l a t i o n functions. Also, f o r if ( J ( )-mixed *(3Z) c o r r e l a t i o n functions, the summation i n the interference term of (25) can be expressed i n a s i m i l a r form. The, common fa c t o r 2"p\(o) i s omitted.-B. Correlation-Functions f o r Special Transitions. The'summations (27) can be evaluated quite e a s i l y f o r some special transitions determined by the following considerations. The expressions f o r C j u m u j m are; the simplest when 0 , 2?z, 2 J M as one may check using Table 2 and the symmetry properties ( 2 2 ) , ( 2 3 ) . In each of the (Cj^'s i n (27) the f a c t o r (J+m)! (J-m)! w i l l appe ar, eithe r i n the numerator or denomi-nator, i f the values of \ and A aare chosen from "the set 0, 2 , 2J, and the set 0, 2Jg, 2J M respectively* The simplest 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 the f a c t o r (J>m)! (J-m)i w i l l be cancelled out. Of the combinations pos-s i b l e only four permit such a cancellation. These are! (i) . A, = 0> ^=0. i i i ) . V 2 J X , V 2 J 2 * ( i i i ) .*,-2J, \=0. t (iv) . 2J X, ^ 2 J » . Henceforth, the transi tions s a t i s f y i n g these conditions w i l l be c a l l e d the "special transitions' 1. The summations (27) f o r these four cases are evaluated below. The results may be used to calculate any cor r e l a t i o n functions f o r the special t r a n s i -tions. In p a r t i c u l a r , some and y-y co r r e l a t i o n functions are given. ( i ) . > = 0, A *0 or J'-J+J , J=J utJ . -For this case the t o t a l angular momenta of the p a r t i c l e s and nuclei 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 condition (i) that J and J are the JL Cf smallest angular momenta that can be emitted compatible with the angular; momenta of the nuclear l e v e l s J ' , J, J B . The summation (27) Is evaluated as follows: Z /PJfm+mi0\2/rJm0 \2_ K JmJ.m. ' v ^m-mjJ.m ' ^ T J , + m,)i (J,-m,)! ( J « - mt+m)«. (Jtt+ m -^m)! (J t+mj l (J x -m (from Table 2) o-(rv-~v) / J , f Ja.+m. + mA / J,t JA-m( - m A V " / JU-m.rm V J'-m.-m \ (from (20)) -""'tt-,) _ /J, +- Ji-t-m, + m.A/ J,+ J^-m, -mA^P/<!,+• JA+m,-f-m -^fv) / 2(J-Ja.)+" J. -fJ^ J,vm, A J i - m i ^ J ^ + ^ / l J,-t-J t-m,--m-av (using m=v-JfJ^trn^) _ I J, f JI+m, -H m A / J, + J*, -m, -mA f 2 J ( 3 1 ) Using the - summa t i on formula (45) i n Subsection C below. One can substitute this r e s u l t into'- (24) to obtain the corr e l a t i o n function between a p a r t i c l e (J^) or photon (J^) and a p a r t i c l e (J^) or photon ( J ^ > subject to the condition (i) on the angular momenta. In 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 long the axis of quantization i n the f i r s t t r a n s i t i o n , the co r r e l a t i o n functions are ebtained by substituting (31) Into (29) and (30). The resu l t s are: <*(J, ) - p a r t i c l e ( J A ) or ti(J, )-lr(J z|:: W(0)= ^ ^ ^ O f S ) ^ ! ^ ' * ^ (32) y(J, ) - p a r t i c l e ( J A ) or V( J, )-4T( J J : W(&-) --2. (J' +J^+1^\ | J i + Ji-l-a.\pm*(e) , (33) where the-factor (^gjit*) has been omitted. I n the case that a photon i s emitted i n the second t r a n s i t i o n with angular momentum J ^ - l (dipole) or Jx=2 (quadrupole) one can obtain the corr e l a t i o n functions <X(J*, ) - r f l ) : W(0)_- l - ^ — ^ c o s ^ »(J, )-*(U : WW) * l - j | p ^ c o s * 0 ( M > y(J t)-if(2)-. W(0) " 1 + ( J ) + i ) ( 5 J ^ 2 3 J l + 3 0 ) C O S ^ (j | +ir (5J^23J ,^0) C O S using the d i s t r i b u t i o n functions l i s t e d i n Table 1. Non-zero factors ( i . e . those common faetors which are polynomials i n J, having no i n t e g r a l roots) have been omitted from these f o r -mulae . ( i i ) . >=2J, , AJL=2Ji.or J'=J-J r , J ^ J " - ^ . • - By using the thi r d symmetry property (23) together with Table 2 one can sum (27) as i n part (I) to get y / r J»m+m, 2J, \2trJva2Jz \2 /J,+ Jr+m,+mA/J, + JA-m, -mA/2JTu+l\ / ^ ^ J m J . m , ' / ^ " m - f f l ^ m /  z \ J,4-m, A J , J { 2J ' / / Since (^ gj*"1) c a n b e factored out of W(£) just as ^ gjt^) was i n part ( i ) , then, comparing the results (35) and (31), one can see that the co r r e l a t i o n functions f o r this t r a n s i t i o n scheme are the same as those given In part (i) i . e . the same as equations (32), (33), and (34). ( i i i ) . v 2 J » V ° o r J , = J. - J* J«J M +J 4» : Prom this condition i t i s evident -that J,?s J - i n fact, J, i s here the largest angular momentum which can be emitted compatible with the angular momenta J • , J . The Calculation of corre l a t i o n functions f o r this t r a n s i t i o n requires the Use of the summation formula (46). Thus, (27) becomes y / C J 'm+m, 2J\ 2 / cJmO \ 2 , ^ JmJ, m, JBm-maJAm,/ -TV" (J,* m,)1. (J t -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! (from Table/ 2 ais£=£) _ / J, + m, \/ J, -m A y\ J • * Jn+ m, + m*\ / J»+J U -m, -mt) »/J,-*-m,\/ J, -m,\ y/j'+JVm^mA/ J'+J"-m,-mA _/J,tm,\/j, -m,\/2JH2J u\ , "* "(jx-aJl^OL : ? ( 3 6 ) using the summation formula~(46) i n Subsection C below. This re s u l t with (24) gives the cor r e l a t i o n function between par-t i c l e s or photons subject to condition ( i i i ) . I f an * p a r t i c l e i s emitted i n the f i r s t t r a n s i t i o n * then the corr e l a t i o n fun-ction i s obtained from (36) and (29) i . e . w<e)= tiffim^t (jJOU^y^'6'• ( 37> I f a photon i s emitted i n the f i r s t t r a n s i t i o n , then from (3^)) and (30), I f a photon i s also emitted i n the second t r a n s i t i o n with- angu-l a r momentum Jg= 1, or 2 then, as In (i) , one obtains the follow-ing c o r r e l a t i o n functions: *(J,)-.r(S): W(6) . ( J | - l ) [ ( J l - ^ i f J ^ c o s ^ J i f y t M^eos^ ( 3 g ) r(Ji ) - i r(2): from (37), (38), and Table 1. Non-zero common factors have been omitted from the formulae (39). For those values of which make the factors appearing outside the square brackets vanish the t r a n s i t i o n can not take place. For, using the Con-d i t i o n - ( i l l ) , i t i s seen that J^J'+J^+Jg I.e. the equation J ^ J g must be s a t i s f i e d i f a t r a n s i t i o n occurs. Thus, f o r OtJ^Jgno t r a n s i t i o n producing an <X(J^) p a r t i c l e can occur and f o r l- J-j^- J2 n o t r a n s i t i o n producing a photon (J^) can occur, ( i v ) . A ^ J , X 2 a 2 J " or J»*J-J , J = J 2 - J U . Here JgSj^+J'+J^ and so i t i s la r g e r than any of the other angular momenta. One cannot calculate c o r r e l a t i o n func-tions i n terms of cos 2© f o r this case with a r b i t r a r y since by the l a s t equation Jg must also be a r b i t r a r y . However, c o r r e l a t i o n functions can be calculated with J 6 a r b i t r a r y arid having some small value. Such cor r e l a t i o n functions are given below. -To evaluate (27) f o r ~ t h i s case the summation formula (46) i s used with Table 2 and the symmetry properties (22) and (23) Following the same method employed i n ( i l l ) one can show that 1 / JiT'^ZT'K fcond t r a n s i t i o n , then, omitting the fa c t o r 2Fj(0)^ J , I f a photon i s emitted along the axis of quantization i n the set the c o r r e l a t i o n function w i l l be w W - S f ' 1 y^-MF^'Ce) , C4i) from (21). Comparing (41) with (38), one observes that the tf(l)-&(Jg) and lf(2)-8(J 2) c o r r e l a t i o n functions f o r this case are the same as the Y(J^)-y(l) and Y(J^)-Y(2) c o r r e l a t i o n functions respectively i n (39) with J replaced by J 0 . The corresponding correlations are calculated from (41) using Table 1. The results are: * ( l ) - * ( J g ) : W(Q) s i t cos 2$ * ( 2 ) - t f ( J j : V(0)* C^-l) [(Ji-2) t 2 I ^ j ^ f 5 6 ) cos 29 + S ( j ^ 2 y ( g t J i b : i 6 ) e o 8 4 J tf(l)-*(J,); W(©) r l - iSli^ficos2© ( 4 2 ) 3 J*-u A -1 * ( 2 ) - r ( J z ) : w(9)=MJ*-i)k+ - 3 ( £ ^ ^ ^ - ' ° ) c o s * Q l omitting non-zero f a c t o r s . By the same reasoning as that i n ( i i i ) , one can show that no t r a n s i t i o n occurs f o r J g - l i h the <x(2)-*(J 2) and *(2)-}f(J 2) c o r r e l a t i o n functions of (42). C. General Method. In the preceding Section c e r t a i n summation formulae ((45), (46)) have been used, which w i l l be obtained now. To this pur-pose a general method of evaluating the summation (27) i s presented here. The method can also be used to evaluate summa-tions of the type appearing jtn the interference term of (25) . A very d i r e c t method of evaluating (27) i s as follows. In o Appendix B i t i s shown that the (C ) appearing i n (27) are polynomials i n m and hence the summand of (27) i s a polynomial i n m. Thus one can perform the indicated summation over m y d i r e c t l y by using the known res u l t s f o r the sums 2m*', where k i s a p o s i t i v e integer. I f the numerical values of m^  and mg are also prescribed, then the f i n a l r e s u l t i s a polynomial i n J . By f i n d i n g the r a t i o n a l roots of t h i s polynomial i n J , one can express i t i n a p a r t i a l l y factored form and some of the factors can be factored out of W(©) . However, f o r J and J *2, the summand i n (27) i s a polynomial i n m of degree *8 and so the d i r e c t procedure of evaluating (27) and then c a l c u l a t i n g W(9) becomes clumsy. In the following method one does not have to expand the complete summand as a polynomial i n m but only part of i t . Furthermore, the f i n a l r e s u l t i s automatically factored i n A. terms of some of the r a t i o n a l roots, i n some cases a l l of them (e.g. the f a c t o r (^gJ"1) l h (31) Is factored out of W(e)) . These advantages have permitted the c a l c u l a t i o n of c o r r e l a t i o n functions as given i n part B which W o u l d be d i f f i c u l t to get using the d i r e c t method. The method i s based on the f a c t that (27) can be, reduced to the problem of summ Ing either or A=Z/PVQ )(*"vrtA . v * - 1 * . . . ^ } (44) TnQ ^Av/Ln-v/ r „ r-1 0 , a f t e r substituting expression s f o r the C's and t r a n s l a t i n g the summation over m to one over v. In (44), v takes on a l l those values f o r which (v)»(n-v) " &° n o * vanish, i . e . i t i h a s the range defined by n-Qfevfcn and Ofev^P. The summation (27) can always be placed i n the form (43), since at the worst, p=0 and q=0, the binomial c o e f f i c i e n t s become 1, and (43) i s then the d i r e c t summation of (27) whose summand Is expanded into a polynomial i n v=m+J. The summations (43) and (44) have been used to evaluate the summations (27) f o r the special trans-i t i o n s In B. In Appendix C i t i s shown that imrrhM^rr1;.«« (45) and that Z ( P v ) ( n ? v ) v k M P ^ ) « *0 (46) where the a„ 's are obtained from Table 4. Substituting these results into (43) and (44), i t i s e a s i l y shown that S £ n q S | [ i A ^ (47) Ef A a A 17/V^+H+An M. (48) FnQ * /3*J(P-«).' l n - ^ / „ 0 \ n / ' only the term containi ng AQ appearing i n each formula f o r the case rsO. If p and q are set 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 before, t hese sums are used In the d i r e c t method. The c o e f f i c i e n t s a k e t i l i s t e d i n T a b l e d , are obtained from the induction f o r mula a k e J = 1 f o r k>l,<< =1 £2" , \ (49) - 2 1 ^ ^ - ^ " - I ) F O R to*'**2' proved i n Appendix G. For a t r a n s i t i o n scheme next to a special one, i . e . a t r a n s i t i o n scheme, characterized by ^ , % , f o r which the quantity d i f f e r s from that f o r a special t r a n s i t i o n by 1, the evaluation 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 special case. As an example, take the t r a n s i t i o n scheme A,= 0, 1 or J U J t J , , JeJ^+J^-l (next to the spec i a l t r a n s i t i o n ( i ) ) and evaluate (27) as follows. ^^cJrm4-m, 0)2( CJml _ ^ 2 A W JmJ | m ( J u m-m^  J a ma V ( J N-m,+ mf? (J *-»m, -m)V 4lmJA-mx(j"-i-Ja,i3 ( l i m i t s from (20)) (using m^v-J^m^) '+1 2& = KJjf (J,f J^m,+mjd-l)(Jj+' J z +m | + mJ^JJ+ ( t f - 2 J A J u ( m j ) (J ( + J a fm,+mj (gjj^)+J t t x ( J x * ^ ^ j u 1 ) using (47)). Here K - , ' " ' " t /^^^m^/j^J -mA ,(50) ( Prom (20), -(J,+ J A -1)^m^+m^feJ, -J-J^-1, hence the denominator of K does not vanish. The re s u l t (50) i s awkward to use unless numerical values of the angular momenta are given. As the values of A, , get farther away from those of the special t r a n s i t i o n the summation results w i l l become more awkward. I I I . #-MIXEBY CORRELATION.FUNCTIONS. .. In this section, tables 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 functions can be calculated are given f o r an Oi. p a r t i c l e having angular momentum 1, 2, or 3. In parts A and B i t i s shown how the and ci(2) -mixed (2) co r r e l a t i o n fun-ctions are obtained from the Tables I I . and I I I . of Ling and Falkoff * f o r d'(l)- and (2) -mixed^(2) co r r e l a t i o n functions respectively. In part C, those summations appearing i n the <V (3) -mixed t(2) c o r r e l a t i o n functions are expressed i n the form from which the corr e l a t i o n functions can be f a i r l y e a s i l y evaluated once the values of the angular momentum are pre-scribed. Some <X- f c o r r e l a t i o n functions are l i s t e d i n references 3 and 11. Some curves of * of- \C c o r r e l a t i o n functions are given i n reference 13. Devons 1 2 has l i s t e d some «(.-mixedy c o r r e l a t i o n functions - however, some of his results do not agree with the results obtained i n this t h e s i s . Since Hamilton's notation i s used to tabulate the angular co r r e l a t i o n functions l i s t e d In several papers, I t i s used to tabulate the cor r e l a t i o n functions appearing here. The tables are l i s t e d i n terms of Aj and defined by J^J-a ' j and J"= j+aj, i . e . *\-3^ and ^ J ^ g - J g . A. V ( l ) -mixedV(2) co r r e l a t i o n functions. The ^ ( J , )-mlxedot2) co r r e l a t i o n functions can be calculated from where W . 2 ( e ) . - Z [ § ( (JmJ^ I J J X J 'm+i^) ^ ( 9 ) ] [ < £ ( J ^ m - m ^ |J«2Jm) (0)] W J 1 1 ( 0 ) ^ [ ^ ( ^ i 3 1 ! l J J i J , m f m i j 2 p ^ ( e ) ] [ ^ CJttm-m2lm2| J^Jm) ^ ^ ( O ) ] J X ( Jt,m-m2 m2| J"2Jm) (^m-mglmg| J ulJm)P|j;(0)] . In the formulae (5«U> the angles 0 and 9 may be Interchanged. Common, non-zero factors In W(9) w i l l be omitted. cCand fb represent 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 dipole r a d i a t i o n respectively. (This notation agrees with the text of Ling and Falkoff's paper, but i n the i r tables I I . and I I I . , <K and ^ have been interchanged f o r some unexplained reason.) In Appendix D i t i s shown that from the jf(l) -mixedtf(2) co r r e l a t i o n functions, W(9) * Q+ Rcos 3^, l i s t e d i n Table I I . of Ling and Palkoff's paper, the <fc(l) -mlxedr(2) c o r r e l a t i o n func-tions, W(9) = Q» f R'cos*9, can be calculated using Q»= Q+R 2 and R»=-R. ^. ^  (53) The o((l)-mixedV(2) c o r r e l a t i o n functions thus obtained have been tabulated i n Tabled". Common, non-zero factors have been omitted. B. 0((2) -mixed f(2) c o r r e l a t i o n functions. By the method used i n A i t can be shown that from the • ot 2)-mixed ^ (2) c o r r e l a t i o n functions, W(e) ^%+Rcos 29 VScos 4©, l i s t e d i n Table I I I . of Ling and Palkoff's paper the oC(2)-mixed tf(2) cor r e l a t i o n functions, W(9) » +-R*cos29-f S*cos 49 can be calculated using Q « = i[6Q-(2R+5S j? , Rf --3?[2R+3S} , and S' = -S* The oc(2)-mixedV(2) c o r r e l a t i o n functions thus obtained have been tabulated i n Table 5. Common, n o n - z e r o factors are omitted from w(e). A misprint was noticed i n Ling and P a l k o f f f s Tab&e I I I . In the y T 2)-y(l) c o r r e l a t i o n functions l i s t e d , ,Q and R are polynomials-of the same degree i n J f o r a l l t r a n s i t i o n s but A J - 1 a n d & j r - i . Calculating the Y(2)correlation function f o r this case from Palkoff and Uhlenbeck ' 8 ° paper one can ob-ta i n W(9) - !+-• MlMlSM^ll—cos2©. This shows that i n Ling and • ' l l O J S 269J4-174 Palkoff «s Tabift l(J+2) (H0J i+269J+174) i n Q should be replaced by |j(J-t-2) (110^+269^174) . This has been done to o b t a i n the #.(2) -mixedVT2) c o r r e l a t i o n function f o r this t r a n s i t i o n i n Table 6 G. fl/(3) -mlxedJT(2) c o r r e l a t i o n functions. ' •The (jfcafcxaxx^xtxfcxBBXxax) summations /?](C , J , * ' * , )  2(0 J3 „ ) ^ Jm30 Jum-m22mg ^ ° J m 3 0 ' ( CJ um-m 2lm 2 ) > ^ 0 J m 3 0 ' CJ Mm-12l CJ«m-lll a r e l l s t e d i n Tables 7,8,9 respectively. They have been placed i n the form (43) and tabulated f o r a l l values of A j , AJ. They can then be d i r e c t l y evaluated using the formula (47). The c o e f f i c i e n t s , , are too clumsy to tabulate f o r general Values of the angular momenta J ' , J , J " . However, since the A^'s are obtained from products of binomials and monomials as i n the equation V r + A r - 1 v r " 1 + ' * * + A0 * [ a 2 v 2 + a l ? + ac][ V * + : b l V - + boJ' * • [V + *o] the set of co e f f i c i e n t s a^, a^, a^? bg, b^, b^;... ;£^,H^.can be tabulated instead. Once numerical Values are assigned to «Xr, J, J u the A t 's are quite e a s i l y obtained and the summations can then be evaluated using (47). The o/(S)-mixed2T(2) co r r e l a t i o n function i s obtained i n terms of cos 29 by substituting the evaluated sums together with the expressions f o r the P's given 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)) . For *(3)-tf(2) or k(3)-X(l) c o r r e l a t i o n functions the formula-(29) i s used. t An example showing how to xsuumt read the Tables 7, 8, and 9 i s now given. For the t r a n s i t i o n i n which AJ = 1, Aj -2 or 7it~ 5, 3 i n Table 9 one obtains 2 s (J-2) ; ^ = 2 , b Q = -(J-4) . Using (43), i t i s seen that this summation i s equal to 16fsSV)(2J"444"T)[T8-2'J-2>'+<J-2'2j[2l'-<J-4»j' which can now be evaluated using (47). If m^  has not been spec i f i e d numerically In Tables 7 and 8, I t means that the results are true f o r a l l possible p o s i t i v e and 3/ zero values of nig. In Table 9, aig-rrl-since =^ 0 unless fflg ~ ±1 and only nig - 1 i s required from these two p o s s i b i l i t i e s to calculate the <X(3)-mlxedo'(2) c o r r e l a t i o n function. 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 Functions, F^Ce) OC P a r t i c l e : J * l . Ff * 2cos*0 F* 1=l-cos 1e J-2 F j = 1-6COSH4-9COS*0 F*1 = 6 0 0 3 * 0 -6 cos^Q F^=| (l-2cos i0 + cos46 ) F*(0) = 0 unless m=0. Photon: (These d i s t r i b u t i o n functions are properly weighted so that the correct r e l a t i v e e f f e c t of each multipole i s rep-resented 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 dipole. Ff = |(2-2cosH) F?1 = i.(l+cos^e) - i £=2. E l e c t r i c or magnetic quadrupole. Ks f (6cos ae-6cos*0) F*'r I(l-3cos i&+4cos 40) H&z* £(l-cos^O) F?(0) s0 unless m«±l. Mixed e l e c t r i c quadrup ole and magnetic dipole d i s t r i b u t i o n functions. F^W-O Pf1, (e)«±ig(3cos^-l) For any J, FJ^.,(&)= - P ^ , (0) . For any p a r t i c l e or photon F^ (&) = p y ( & ) . J3 Table 2. E x p l i c i t expressions f o r some C j u ^ j m (equation (19)) . pJmO ["' '(J+m).' (J-m)! rJml , of (J+m)! (J-m)! l^ Jm»'t -m Tu1 GJ«m"J Lm x - 2 | (J"+m H)l (J^-m")! < & * M J ! (J,-m,)!J ^ J * M * J J , °J"m»JJLJ-ia»" V X ; [(2 -vj! (2Jx-x> UJu-AJ123X- A, /J VI*. cJmO - f _ (J+m)! (J-m)! _ !*• j-lm-m^lm^ [_(J-ltn^-m)! (J-l-%.+m)< (ln-m^i tl-m a)IJ Jm-m^lmx [(J-mi+m)! (J+m^-m)1, (ltm z)! (l-mi).'J L x a  rJm2 _ / -.x1-na, [ (J»l-ma»m)! (J+l»ma-m)l "1^  Jflm-milm1'* v ' L( J+m) I (J-m).< (l-m a) J ( l r m j i j J 4s 2: cJmO _ [" (jVm).' (J-m)! 1 ^ J-2m-m;t2mx * [(J-2-m4+m)J (J^m^-m)} (2t m*)l (2-ma)jJ CJ-lm-m12m1 " *{( J-l-m^m) J (J-lw^-ni) • (8+*^) i (2-mJjj L2m-m*< J + 1 l l ^mfm-'-tl^^^^r^J cJm2 ^ f3 (J4m)!.(J>2-mJn^ Jm-222 * (Jm-2)/ (J-m)!J , rJm3 _ /_•,xl4maQr(Jvi-m,tm)! (J+l4m»-mi! " I 1 ^ ^ jf CJflm-m A2m/ { 1 } • . r 2\(Jtm)! Jj-m)! (2+mj! (2-m.V g m m ^ pJm4 * * _ / ^ mi, (J-H3-m,t+m)l (J*2+ma-m)' U J+2m-m12mi' 7 (_(J + m)f (J-m)< (2*-mJJ (2-mA)!J J, = 3, m,*0: cJ4-3raO,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-my»(*, Jm30 " j_(J+m)! (J-m)j J 0Jm30 2 I £ ( J ^ 1 + m ) ( J * 1 " " 1 ? 1 1 C5mz-J(J42)] cJjjSg -|m[5mM3^43 J-l)] C J ; 3 ? e 4£( j + m ) ( j - m f l * f 5 m ' ^ ^ f l CJ-2m5 _ [(J+m)? (J-m)!"1* Jm30 " (j J+m-2)l (J-m-2)!J , GJ-3m6 _1 f(j4-m)! (J-m)! 1^ Jm30 = 3! L(J+m-3)! (J-m-3)U Table 3. A tabulation of the c o e f f i c i e n t s a<cl defined by (4$) . 10 9 8 7 6 5 4 3 2 1 1 • * * 1 2 • • • • • 1 1 3 • • • • • • • 1 3 1 4 • • • • • 1 6 7 1 5 • • • • 1 10 25 15 1 6 • • • 1 15 65 90 31 1 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 following abbreviations are used i n Tables 5 and 6. d 1 = Qi5J(jr+^ )]i d 2= (aj-l)[l5J(J+-2)J^ d 3= [5(2J-l)(2j+3)]^ d 4 r (2J+3)[l5(j 2-l)J^ d 5= [15(J2-1)J^ d g = (J-5)[15J(J+2)J ^ d ?= (2J-3)(2J+5)[15J(J*2)J^ d g = (J+2) (2J-1) f Jf6)Ll5J( J+2)j ^  d g r (j+2H2J-l)[l5J(J+2)] ^ d 1 Q= 3[5(2J-1) (2J+3)]^ d l l = 3 ( J - 5 ) L 5 ( 2 J - 1 ) (2J+3)]^ d 1 2 = (2J-3)(2J+5)[5(2J-l)(2J+3)]^ d 1 3 =3(Jt6)X5(2J-l) (2J+3)]^ d1 4 " 3[5(2J-1) (2J+3)ji ; d 1 5 ^ ( J - l ) ( 2 J + 3 ) [ l 5 ( ^ . D ] i d16 S ( J " 1 ) ( 2 J + 3 ) ( J - 5 ) L l 5 ( J 2 - l ) j ^ d 1 ? = |(2J-3) (2J+5)[ 15( J 2 - l ) ] ± dl f t= (J+6)Cl5(J2-l)]£ Table'5 . #•(!) -mixed5(2) co r r e l a t i o n functions, W(©) - Q» + R»cos 2©.* 0 lecl2- 2 R 1 13 1 R 1 3 0 R' 16 J-7 -3(2J-7) -1 Q' R» 26J*V71J+42 3J(2J-1) 1 Q 1 _5(29J+6) 7J i d , 3 1 R» -i(j+6) 7 - J d l 0 Q' _£(26J 2+17J+6) 21 J(6J+7) I d 2 3 R» 1(J+6)(2J-1) 7.. -S(58J2+-151J+78) 21 J(2J-1) - d2 -1 Q1 l4j 2+33J+20 3 2 R» - | ( J + 6 H 2 J-1) - J(2J-1) d2 1 Q ' l(20J 2-8J-5) 7 (2J-1)(6J-1) d3 R» -I(2J+5)(2J-3) 17 (2J -1H2J+3) 0 Q' 5(8J2+8J+6) 8J 2+8J-1 - S R* l(2J-3)(2J+5) 7 -(2J-1)(2Jt3) 3 d 3 -1 Q» |(20J2+48J+23) (2J+3) (6Jt7) d 3 R» -1(2J-3)(2J+5) 7 (2J-1)(2J+3) -=>d3 •Correl a t i o n functions f o r those transitions i n which only a pure e l e c t r i c multipole (fS*0) i s emitted, have been included i n the Tables 5 and 6 f o r the sake of completeness. These were ob-tained from reference 3. J/31* 2R(*/3#) -1 1 Q ' ^|(58J 2-35J-15) 14J2-5J+1 3 d4 R » -|(2Jf3)(J-5) -(J*l)(2J^3) - d4 0 Q1 -§('26JS+35J+15) 21 (m) (6J-1) 3 4 R ' i(2Jt3)(J-5) 7. -|(29J+23) (J+l)(2J-K5) d4 -1 Q 17. (J+l) K R -f(J-5» -(j+D " d5 -2 1 Q R ' 26J 2^19J-3 ' 3(J+1) (2Jt3) 0 Q R ' 16J+-3 ' -3(2J+3) -1 Q R ' 13 ' 3 Example * • For-AJ = -1, Ai = 1, one obtains Qt- |_(58J 2-35J-15 ) M A ' * (14J2-5J+l)l/S|* + |d 42R(^*) R « 5 -|(2Jt3)(J-5)|«|'a : -(J+l)(2J+3)|/3l A' -d 42R (A / s*) , from which W(9) = Q'+R'cos29 i s e a s i l y obtained. 3? Table 6. U(2) -mixedJf(2) c o r r e l a t i o n functions * W(0) = Q» 4-R»cos2© 4-S*cos4©. 6.3 AJ' 2 2 15 0) R» 6. S' -1 1 Q.' -2(SJ+2) R* 2(J-3) Sf - 2 (2J-3) 3 0 -4(2J-KL)(Jt2) 4(2J-3) S' i (J - l ) (2J-3) ; -1 Q' - (12 J3*-54 J2-K78 J+30) R» 2(2J-l) (J2-r2J4-6) S» - £ ( J - 1 ) ( 2 J - 1 ) ( 2 J - 3 ) 3 -2 Q» -(20J4-rl48J3+391J2+437J+168) R' -2J(2J-1)(2J2+7J+9) S» •Lj(Jiil) (2J-1) (2J-3) 1 2 Q' 5(3J+1) R» -5(J+4) S' |(2J+5) 1 Q» 5(5J2+5J+6) R • 25'(J+3) (J-2) S' -|2(2J-3)(2J+5) 0 Q' 5(4J3+8J2+3J+6) R» -5(2J-3)(2J+5)(J-2) 1/3 I4" 15J -3 J 3J(9J+-1) 3J(J-5) i(52J3+136J+99) 3 J(2J*3)'(2J+5) -d. 3d, -3d, id. 3 7 31 Table 6 (continued). S' |£(2J-3)(2J+5)(J-1) ? \ -1 Q' 5(10J 4+47J 3+62J 2+16J+12) 3J(J+2) (18J2+43J+30) d8 R» 5(2J-1) (5J 3+6J 2+4JH8) 3J(J+2) (J+6) (2J-1) -3d 8 S' -|°(J-l) (2J-3) (2J+5) (2J-1) > -2 Q' 15(2J 3+9J 2+13j+5) 3(J+2) (10J2+23J+14) " d9 R 1 -5(2J-1)(J 2+2J+6) -3J(J+2)(2J-1) 3d 9 S» i(2J-l)(J-1)(2J-3) 3 2 Q» 15(2J+1)(J-1) 3(2J-1)(4J-1) d i o R» 15(2J+5) 3(2J-1)(2J+3) -3d 10 S» -5(J+2)(2J+5) • 1 Q» 15(4J 3+4J 2-J-7) 3(2J-1)(10J 2+7J-5) - d u R» -15(2J-3) (2J-J-5) (Jt3) -3(2J-1)(2J+3)(J-5) S' 20(2J-3) (2J+5) (J+2) 0 Q» M(16J 4+32J 3+40J 2+24J-63) , i(2J-1) (2J+3) (32J2+32J-15) -d, 0 R 1 5(2J-3) (2J+5) (4J 2+4J-9) -(2J-1)(2J+3)(2J-3)(2J+5) 3 d l g S' -20(2J-3)(2J+5)(J-1)(J+2) -1 Q 1 15(4J 3+8J 2+3J+6) 3(2J+3) (10J2+13J-2) R» -15(2J-3)(2J+5)(J-2) -3(2J+3)(2J-1)(J+6) M 1 3 S' 20(2J-3)(2J+5)(J-1) -2 Q r 15(2J+1)(J+2) 3(2J+3)(4J+5) d14 R» -15(2J-3) 3(2J+3)(2J-1) - M14 S.' -5CJ-1)(2J-3) >. 2 Q » 15(2J3-3J2+J-KL) 3(J-1)(10J 2-3J+1) d15 R» -5(2J+3) (J%5) -3(J-1)(2J+3)(J+1) " 3 d16 S» |(2J+3)(J+2)(2J+5) Table 6 (continued). to A J A J 1 Q R S 0 Q R S -1 Q R S -2 Q R S 2 Q R S 1 Q R < S 0 Q R S -1 Q R 3 -2 Q R S -2 5(10J 4-7J 3-19J 2+7J+21) 5(2J+3) (5J 3+9J 2+7J-45) --|°(2J+3)(2J-3)(2J45)(J+2) 5(4J 3+4J 2-J-7) -5(2J-3) (2J+5) (J+3) M(J+2) (2J-3) (2J + 5) 5(5J2+5J+6) 5(5J-10) (J+3) -|2(2J-3)(2J+5) 5(3J+2) -5(J-3) |(2J-3) -(20J 4-68J 3+67J 2-19J-6) -2(J+-1) (2J+3) ( 2 J 2 -3J+4) I(J+1) (2J+3) (J+2) (2J+5) -6(2J 3-3J 2+J+l) 2(2J+3) (J 2+5) -|(,2J+3) (J+2) (2J+5) -4(2J 2-J-1) -4(2J+5) i(J+2)(2J+5) 3 -2(3J+l) -(J+10) -|(2J+5) 15 6 -1 1,31* 2R(^#) 3(J+l)(J-1)(l8J 2-7J+5) -d. 3(J+l)(J-1)(2J+3)(J-5) 3d I(J+1) (52J2-32J+15) 3 (J+l) (2J-3) (2J+5) 3(J+l)(9J-2) 3(J+1) (J+6) 15(J+l) -3(J+1) 16 16 ^d 17 3d 17 -d 18 3d 18 -3d, Table 7. The summa t i ons £ (CJm30 ' ) 2 < ^ m-i^m^ 2 • J ' - J-A3, J=* J f t AJ Aj mg KSp n q v 2 v 1 v°sl 2^  (4 l^4im2)4stffl2,2J.4(4+m2 1 -2(J-2) (J-2) 2 "2 ^  1 " T^^-rng, 2Je.2,3+m2 : 5 -10(J-l) (J-l)(4 J-6) 5 -10(J-l) (J-l)(4J-6) 0 ."8 f^-ing.SJ.a+iiig 5 -!0 J (J-1)(2J-1) 5 -10J (J-1)(2J-1) 1 -2J J 2 - 1 m2 "H-ma.SJ.^ -g 5 -10J 2JC2J-1) 5 -10J 2J(2J-1) 1 -2J -2 J - l - 2*2 ^-mg.aj.af-g 1 " 2 J - 4 J * 4 1 -2J -2J-l i -a j j 2 -3 0 is! o r , 1 3 2 4 3,2J,3 1 -4J-3 (2J+-1) (2J+2) "|S3,2J,3 1 - 2 J " 2 " 2 J - 3 1 -4J-3 (2J4-1)(2J-K2) "|S3,2J,4 1 - 4 J " 5 (2J+2)(2J-K3) 1 -2J-1 4 -4(2J-m2J-6) (ZJ-n^J-Q)2 Table 7 (continued). A J m2 K S j n q y2 y l v°*l 1 3 2 8 0 S 3 , 2 J - 6 , 6 1 2 0,1 •HX^.W^.S*^ 1 -2(J-2) 2 160 S 2,2J-4,5* 4 1 1 m 2 S2-m2,2J-2,2+-m2 5 5 4 -4(2J-m2J-4) -2(J-2) 1 -10(J-1) -10(J-1) 4(2-2J+m2J) * 5 -10J 2 i ^ S 6 g 1,2J—1,3 - 1 ° " S!,2J,1 3 1,2J,2 5 1 4 5 5 1 5 5 1 1 5 5 4 -10J -2J -4J(2-mg) -10(J-1) -10(J-1) -2(J-1) -10J -10 J < -2 J -2J -10J -10J -4 J 1 (j-sr (2J-m 2J-4) : ( J - 2 ) 2 2 (J-1)(4J-6) (J-1) (4J-6) (2-2J+m 2J) 2 (2J-1)(J-1) ( 2 J - l ) ( J - l ) J 2(2-mg) 2 (2J-l)(J-6) (2J-1)(J-6) ( J - 1 ) 2 J(4J-2) j(4J-2) -2J-1 J 2 J(4J-2) J(4J-2) J 2 -2J-1 Table 7 (continued). * J A 1 * 2 KS;nq V 2 v l .. V0=1 1 . 1 2 -^ UJ-^ O 5 ( S J - l ) f ^ 5 ) 5 -10(J-1) C'BJ-1) (2J-S) 1 -2J -2 0 -16SS . 1 j 2 '2,2 J, 2 1 -2J J 2 1 -2J -2J-1 0 3 0 I w 3 -6<J-3) 3,2J-6,4 1 '2,2J-4,2 1 "f S2,2J,2 1 ~ 2 J J 4 -4J 1 - 4 J - 3 (2J+1H2J+2) 2 2 4 S3,2J-1,3 1 " 2^-D (J-1)* 1 " 4 J"1 2J(2J+1) • 3 ° - 4 S3,2J,3 1 ^ . -2J-1 1 -2J j2 3 S3,2J,3 1 - 4J"3 (2Jf2)(2J+l) 4 -4J j2 3 4,2J-1,3 1 " 4 J " 3 (2J+1) (2J+2) 1 -2J •. , r2 2J -1?J+27W 3 , 2 J " 6 , S 2J2-19j+27„ 3 -6(J-3) icStefcrJcJdk&c&x 1 6S 3 _ , 4 -4(2J-5) (2J-5) 2 2 30S1 2J-fi c 1 5 6 , , 2J 2-13J+12 J -2 0 4S° rtT . 0 3 -6(J-2) ifc*x3$M&&k 3 -6(J-2) 2J 2-13J*12 1 1 -2(J-2) ( J - 2 ) 2 Table 7 (continued). J J »2 K S p n q ' V l V ° = 1 2 ° , 2 1 1 8 S2,2J-4,3 1 " 2 ^ 2 > C J - S J 4 -4(2J-3) (2J-3) 2 1 2 2 7 2 S2,2J-4,4 1 -SfJ-S) ( J - 2 ) 2 1 0 A , 2 J-2,1 1 3 2 i s 8 . , . , 5 -10 (J-1) (J-l)(4J-6) 5 -10(J-1) (J-l)(4J-6) 3 -6(J-1) 2J 2-7J+3 3 -6(J-1) 2J 2-7J+3 1 |s^ 2J-2,2 5 -IG(J-I) (J-l)(4J-6) 5 -10 (J-1) (J-UC4J-6) 4 -4(2J-1) (2J-1) 2 : ' i - i 2 Isf 0 '„ 5 -10 (J-2) 4J*-20J+21 2 2,2J— 3,3 5 -10(J-2) 4J2-20J+21 1 2 0 0 i s 1 0 r t , 5 -10J (2J-l)(J-l)s 9 0,2J,0 5 -10J (2J-1)(J-1) 3 -6J J(2J-1) 3 -6J J(2J-1) 1 -2J J 2 1 | s 8 ' T , , 5 -IO(J-I) ( 2 J - l ) ( J - 6 ) 3 1,2J-1,1 5 -10(J-1) (2J-l)(J-6) 1 -2(J-1) (J-1) 2 4 -4(2J-1) (2J-1) 2 7 (continued) . m KS?L V 2 2 °pnq 2 Is6 5 -10(J-2) (2J-1)(J-21) 3 2,2J—2,2 5 -10 (J-2) (2J-1)(J-21) 1 -2(J-2) (J-2) 2 0 Is8 5 -10J J(3J-2) 4 1,2J,1 5 -1GJ J(3J-2) 3 -6J J(2J-1) 3 -6J J(2J-1) 1 -is7 5 -10(J-1) (2J-1)(2J-5) 4 2,2J-1,1 5 -10(J-1) (2J- l)(2J-5) 4 -4(2J-1) (2J-1) 2 1 -2J 2 - i s 5 5 -10 (J-2) 4j2-22J4-20 2 3 f2J—2,2 St 5 -10 (J-2) 4Js?-22J+20 > 1 -(2J-1) 0 4 S _ _ 3 -6J J(2J-1) 2,2J,2 3 -6J J(2J-1) 1 -2 J J 2 1 —18S_ 4 -4 (2J-1) (2J-1) 2 3,2J-1,2 1 -2 J 2 7 2 S4,2J-2,2 1 -2(J-2) i (J-2)2 1 -(4J-1) 2J(2J-1) 0 S 4 3 ^ 2 J ^ 3 3 -6J J(2J-1) 3 -6J J(2J*1) 1 " 6 S4,2J-1 , 3 4 -4(2J-1) (2J-1)2 1 -2J" Table 7 (continued). pnq v v l v°=l 2 ° " 3 2 3 0 S f , 2 j - 2 , 3 • 1 - f 4 J - l J 1 5 1 2 2S 5 2J(2J-1) 1 3 S3,2J. 6,3 4 ^ 3 J - 5 ) (3J-5) 2 6 2 - S 3 1 3 3 3,2J-6,4 2 3 2 0 - 1 6 S | , 2 j _ 4 > 2 1 -2(J-2) ( J - 2 ) 2 1. -2(J-2) ( J - 2 ) 2 1 -2(J-2) -4(J-1) 1 | S2,2J-4,2 1 " 2 ( J - 2 ) ( ' - 2 ) 2 1 3 2 4 -12(J-1) 9 ( J - 1 ) 2 2 2 4 S3,2J-5,3 1 - 2 ^ - ^ 1 5 6 • ° " S1,2J-2,1 5 -10(J-1) ' (J-l)(4J-6) 5 -10(J-3>) (J-1) (4J-6) 1 -2(J-1) ( J - 1 ) 2 1 -2(J-1) -(2J-1) 1 1S2,2J-3,1 5 - 1 0 ( J - 2 ) 4J 2- 20J+21 5 -10(J-2) 4J 2-20J+21 4 -12(J-1) 9 ( J - 1 ) 2 1 2 '3,2J-4,2 5 -10(J-3) 4J2-30J4-46 5 -10(J-3) 4J 2-30Jt46 1 3 Table % {continued) . A J Aj m2 Kfi£ v°=l -1 0 0,1 2 , s f L n r A ' 9(4-m^) l+m2,2J-2,l-mg 5 -10(J-l-m 2) 2 l ^ S 6 9 3,2J-3,1 5 1 4 5 5 1 -lOfJ-l-mg) -2(J-l-xng) -4(J-l)(2+mg) -10(J-3) -10(J-3) -2(J-3) 5(J-l-ia 2) 2-(3J 2+-3J-l ) i (J-l-ittg) ( J - l ) 2 ( 2+ 1 ^ ) 2 2J 2-35JH6 r 2 . (J-3) 2J -33J+46 2 -1 nu S 2 2-»-m2,2J-2,2-m2 5 5 4 -10(J-l-m 2) -lOfa-l - B l g ) 5(J-l-m 2) 2-J(J+2) 5(J-l-m 2) 2-J(Jf2) (J-l) 2 ( 2 « i g ) 2 -2 m2 4(9-m2) S34.m2,2( J-l) v3-m 1 ^ ( J - l - n i g ) (J-l-mg) 2 4 -4(J-l)(2+m 2) ( J - l ) 2 ( 2 4 m 2 ) 2 -3 m2 S 2 4 2J-2,4-m2 -4(J-l)(2+m g) ( J - l ) 2 ( 2 + m 2 ) 2 2 3 0 l s 4 4 3,2J-6,3 1 -2(J-3) -3(2J-3) 1 -2(j-3) -4 (J-2) 1 -is4 6 3,2J-6,3 1 -2(J-3) -3 (2 J-3) 1 3 2 2 l s 3 6 4,2J~7,3 1 5 1 6 4 2 0 «6 D2,2J-4,2 1 -2(J-2) ( J - 2 ) 2 1 -2(J-2) -4(J-l) 1 -2(J-2) -2J-K5 Table 7 (continued). A j A j m 2 KHj V 2 *1 V ^ - l -2 2 1 -2S 5 2 J - 5 g 1 -2(J-3) -3(2J-3) 1 -2(J-3) ( J - 3 ) 2 1 2 2 2S 4 _ rt 1 -2(J-4) ( J - 4 ) g 4,2J-6,2 1 7 12 X n "L ~ i s ^ ' 4 2fm ,2J-4,2-ittg 5 -10(J-2-mg) 5(J-2-mg) 2-(J 2-l) 5 -10(3-2--^) 5(J-2-m a) 2-.(J 2-l) 1 -2(J-2-m ) (J-2-m ) 2 - J 2 2 T S 5 rtT c , 5 -10(J-4) 4J 2-40J+81 4 4,2J-5,1 5 -10(J-4) 4J2-40J4-83i 1 4 0 m0 i s 6 2 9 2+m2,2J-4,2-m2 5 -lO(J-2-m 2) 5( J-2-m 2) 2-(3J 2 -t-3J-l) 5 -10(J-2-m 2) 5(<T-2-Hig) 2-(3J 2+3J-l) ; 1 -2(J-2-mg) (J-2-ttgf -1 m (9-mj)c4. 2 4 3M 2,2J-4,3-iH2 5 -lO(J-2-m ) 5(J-2-m ) 2 -J(J+2) 5 -10(J-2-m2) 5 (J-2-m 2 ) 2 -J(Jf2) ,2 * \ 2 */\ 2 7°4tm 2 ,2J-4,4-m 2 -2(J-2-m2) (J -2-m 2) 2 -2 m 4 ( 4 ^ 2 ) ( 4 - m 2 ) -3 m2 ( 53 M 2 ) ( 5 ; M 2 J ^ 2 J - 4 ^ ^ Table 8. The summations ZiCj^o > ^  cPm^lm^ ^ ^ , S J - A . j , J - J y - A j or A J =^-2, A j ~ A - 3 . AJ Aj mg 1 3 m-, 2 J ~ 1 " p n q KS r v 2 V 1 v°=l V3 ^ 3 <7^4-m2,2«T-6,4+m, 2 m 2 ( 9 - 2 ) ( 4 - m 2 ) S 2 _ m 2 j 2 J _ 4 > 3 ^ 1 -2(J-2) ( J - 2 ) 2 1 ra2 "4^ S2-m 2,2J-2, 2+m2 5 -10(J-1) 4J 2-10J+6 5 -10 (J-1) 4J 2-10J+6 0 m o , m 1 ~ 2 J j 2 2 9 1-m ,2J,l+m2 5 -10J 2J 2-3J+1 5 ' -10 J 2J2-3J+-1 _n vn -lo6 x 2 4"^l-m2,2J,l+m2 5 -10J 2J(2J-1) 5 -10J 2J(2J-1) 1 -2J -2J-1 -2 0 -4SA 1 -2J J 2 '2,2J,2 1 -2J -2J-1 1 a s l > 8 J s 2 1 - 8 J j 2 1 -4J-3 (2JM.) (2J+2) - 3 0 - S3,2J,3 1 1 |s2 „ 1 -4J-3 (2J-rlK2J+2) 2 3,2J,3 0 3 0 4S3,2|--6,3 1 -2 (J-3) ( J - 3 ) 2 1 8 S3,2J-6,4 1 3 Table 8 (continued). j m2 KS r pnq v2 V 1 v° 1 2 0 1 6 S 2 , 2 J - 4 , 2 1 - 2 ( J - 2 ) ( J - 2 ) 2 1 - 2 ( J - 2 ) ( J - 2 ) 2 1 2 4 S 2 , 2 J - 4 , 3 1 - 2 ( J - 2 ) 1 ( J - 2 ) 2 2 a o Ms8 ™ 0 , 2 J , 0 1 - 2 J J 2 1 - 2 J J2 5 - 1 0 J 2 J 2 - 3 J t l 5 - 1 0 J 2 J 2 - 3 J U i 8 5 6 9 ' 1 , 2 J - 1 , 1 1 - 2 ( J - 1 ) 5 - 1 0 ( J - l ) 2 J 2 - 1 3 J + 6 5 - 1 0 ( J - l ) 2 J 2 - 1 3 J 4 - 6 i a b l , 2 J - 2 . n l 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 ) 2 I S 5 1 , 2 J - 2 , 2 5 -IO(J-I) 2 ( J - 1 ) ( 2 J - 3 ) 5 - 1 0 ( J - l ) 1 * 2 ( J - l ) ( 2 J - 3 ) 1 - 1 ^2 S1-HR0,2J,1-- 2 0 1 6 S ' - 1 0 ( J - m 2 ) - lOU-nig) - 2 J ( l + m 2 ) - 2 J - 2 ( J - 1 ) - 4 J -3 m i/5+m2V3-m2U2 2 9 V 2 J\ 2 / 5^2,23,5-^2 1 - 2 J ( l t m ) 1 1 2 S 2 , 2 J , 2 4 3 , 2 J , 1 5 5 1 1 1 1 1 5 ( J - m 2 ) 2 - J ( J + 2 ) SfJ-mg) - J ( J f 2 ) J 2 ( l+-mg) 2 J 2 ( J - l ) A T 2 4 J J 2 ( 1 4-m ) 2 s Table 8 (continued). m2 KSr pnq v2 v 1 - 1 3 0 -s 2 -3,2J-5,3 1 -2(J-3) 1 1«2 "2* 3,2J-6,3 1 5 2 0 -4S 4 2,2J-4,2 1 1 -aCr-2) -2(J-2) 1 % 2 J - 4 , 2 1 -2(J-2) 1 -(4J-5) 1 m 2 -lgS 4 l4/mrt,-2J-2fl-" m2 5 - 1 0 ( J - i 0 m I s 6 - 1 m i i i s l i s 4 2 4 2+m2,2J-2,2-m2 1 -2(J-l-m 2) ( J - l - m 2 ) 2 -3 m2 V 3 \ 3 / 4+m2,2J^2,4-in2 v°^i -3(2J-3) ( J - 2 ) 2 -4 (J - l ) ( J - 2 ) 2 (2J-2)(2J-3) 2) 5( J - l - m g ) 2 - ( J 2 - l ) 5 -lO(J-l-mg) 5 ( J - l - m 2 ) 2 - ( J 2 - l ) 1 -2(J-l-m 2) ( J - l - m 2 ) 2 - J 2 2 9 ltm ,2J-2,l-m 5 -10(J-l-m 2) 5(J-l-m 2) -(3J24-3J-1) 5 -10(J-l-m 2) 5(J-l-m 2) 2-(3J 2+3J-l) 1 -2(J-l-m ) (J-l-m ) 2 -10 (J-l-nig) 5(J-l-m 2) 2-J(J4/2)' 5 -10 (J-l-nig) 5(J-l-m 2) 2-J(Jt2) Table 9. The summations Zf4^) ' ^ 2 1 ^ 1 1 1 ' , J * J M-AJ o r A J ^ - S , Ajs a, . 3 . A J A.J KSp n q v 2 V 1 v°=l 1 3 || S3,2J-6,5 ^ -<JT-S) 2 16:43S 3^ 2 J_ 4^ 4 1 -2(J-2) (J-2) 2 1 S»5 2 1,2J-2,3 2 - -(J-4) 5 -10(J-1) 2(J-l)(2J-3) 5 -10(J-1) 2(J-1)(2J-3) 2 -(J-2) 8„7 . _ 2 0 6,2J,2 5 -10J *2 2J -3J-HL 5 -10J 2J2-3J+-1 1 -2J J 2 2 - J - 1 ^ 1 , 2 J , S 5 " 1 0 J 2J(2J-1) 5 -10J 2J(2J-1) 2 -5J-2 J(2J-KL) ^2,23,2 1 " 2 J j 2 1 -4 J-3 (2J-KL) (2J+2) 2 - J " 3 ^3,2J,3 1 - 4 J " 3 (2J+l)(2J4r2) 0 3 4 f 3 s | > 2 J _ ^ 4 2 -(2J-11) -3(2J-5) 2 ISflfcj^g 3 1 -2(J-2) ( J - 2 ) 2 2 -(2J-7) -2(2J-3) 1 f S l , 2 J - 2 , 2 5 -10(J-D 2(J-l)(2J-3) 5 -10(J-1) 2(J-l)(2J-3) .(2J-3] 2 xkikaZf. -(2J-1) Table 9.(continued). A j 41 KS r v S V 1 v°»l pnq o o |psI,2J-i,i 5 -io(J-i) (2J~l)(J-s) 5 -10(J-1) (2J-l)(J-s) 1 -2(J-1) (J-1) 2 2 -(2J-1) -1 ^Pi,2J-l,l 5 -10(J-D (2J-l)(2J-5) 5 -10(J-1) (2J-l)(2J-5) 2 -(6J-1) 2J(2J-1) -2 -12|3Sg 2 j - i , 2 1 -2(J-1) (J-1) 2 2 -(6J-1) . 2J(2J-1) -3 -4^3S 2 > 2 J_ 1 ) 3 2 -(6J-1) 2J(2J-l) -1 3 is5 _ I' 1 5 6 3 3,2J-S,3 : i ; 2 -(3J-5) *H.2J-*,2 1 " 2 ( ^ 2 ) ( J " 2 ) 2 1 3 2 2 / -3(J-1) 1 ^ P S f 2 J - 3 , l 5 " 1 0 ( ^ 2 ) (2J-3)(2J-7) 5 -10(J-2) (2J-3)(2J-7) 2 -(3J-7) -6(J-1) °9#2,2J-2,0 1 - 2 ( J " 2 ) ( J - 2 ) " 5 -10(J-2) (J-1)(2J-21) 5 -10(J-2) (J - l)(2J-2l) 2 -3(J-1) \ - l f s 5 ) 2 J _ 2 A 5 -10(J-2) 2(2J2-11J4*10) 5 -10 (J-2) 2(2J 2-lu4iO) 2 -3(J-1) Table 9 (continued). A j Aj KS r v 2 V 1 v°=l "° pnq - i -2 i s T 3 s f j 2 J - 2 > 2 1 wg(j-S) ( J - 2 ) 2 : 2 -3(J-D -» #5,^-2,3 2 - 3 ( J - 1 5 Appendix A. The Normalized Angular Momentum C o e f f i c i e n t s . In equations (A3) and (A4) below, two expressions are given 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 easier to evaluate than that i n (A3), an expression f o r the fact o r f (JJ t t J^V-is desired so that (A4) can be used i n the calculations. I t i s the object of this appendix to show how the normalization f a c t o r f(JJ t tJg) can be obtained and then to show that the two formulae (A3) and (A4) a£e the same by proving the summations i n each formula are equivalent. / Prom el-ther of these two results the equations (18) and (19) can be obtained. . . . The normalized angular momentum c o e f f i c i e n t s (J t ,J 2m um 2|j"J 2Jm) are defined by ^•^^V"^I^ J 8^W*J,-,• ( A 1 ) where i s the normalized wave function describing the state of the system c o n s i s t i ng of a nucleus (angular momentum J") and an emitted p a r t i c l e (angular momentum Jg) which are i n the states represented by the normalized wave functions l//„ M , J"m m respectively. Jm refers to the t o t a l angular momentum and i t s z component, r espectively, of the system. I f 2 dV=l i s formed i t can be seen that the normalized c o e f f i c i e n t s s a t i s f y ^ ( j u j ^ m g l J ^ J m ) 2 * ! . (A2) This follows from the orthogonality of the wave functions and the f a c t that the co e f f i c i e n t s are .real numbers (Cf A3). The expression given by Wigner"1'4 f o r the normalized c o e f f i c i e n t s i s (using d i f f e r e n t notation) I J-J 9m«Bu | J«J 9Jm) - R 2 J W ^ - ^ 2 ^ 2 L (2J+VDJ J r (T+««M (r-~i)\ 1x r/-i)ot4-T^ <^2Ji-rJ''t^»-^-^!(j^^vat)! where \ z i s defined by J=J M4Jg-A^twhich means i t s values must l i e i n the range b^Afcminimum of 2<Xrt, 2 J g since )J H-J 2]4: J t J t t f J g ; m=mtt+mg; stakes a l l those values'for which negative arguments do not appear i n the f a c t o r i a l s (01 = 1) ( A l l summation indices which do not have t h e i r range- given w i l l be summed i n this manner.). This formula i s not easy to use because i t i s not 15 very symmetrical i n the J*s and m's. Van der Waerden,, gives a symmetrical formula f o r the unnormalized c o e f f i c i e n t s . Using this formula the normalized angular momentum co e f f i c i e n t s may be written i n the form (^'Jgrn^nigl J»J 2Jm) = f (J J U J 2 ) * ^ f r f t f W ' M (T"-^»)( U i i - H a M tr^i)! r cr+»»v. rf (JJ«J )cJf*\ , (A5) 2 J umMJ 1m A using (19). Here f (JtT'Jg) i s the normalization f a c t o r neces-sary f o r (A4) to s a t i s f y (A2). The normalization f a c t o r f is,now calculated following If? the method used by K e l l e r . By substituting (A5) into (A2) with m*J one can obtain j " •fB2KcJ?,k j . ^ ) 2 .1. (A6) the range f o r m" being determined from the relations i n (20) The summation i n (A6)- i s evaluated as followsi f i r s t , express the summand e x p l i c i t l y i n terms of m" using Table 2; then change the summation over m" to one over vr&u-jr\;i4 > f i n a l l y , use the summation formula (45) with k-0 to get the sum. Combining the r e s u l t with (A6) one obtains f ( J J " J i ) r (2J+1) 12 J a * Ax) ! (2X>)'!"Aa •' (A7) (2J*-AA+l)! The equivalence of (A3) and (A4) i s now shown by proving that « (2J L -a i -«)! (J^Jifm^m^-Ai-^) 1. ( J t t - J * « \ (~i)*(Wm^)! (j»-ma j) ( J ^ ) ! ( J ^ - m a ) l ' - ( A 8 b ) ( J ^ m " - ^ * ) ! (JM-mM-*)! (J^m^-ot)! (J^-m^-A Af*)f « ! ( ^ ) ! Obviously, (b) cannot be obtained from (a) by a l i n e a r s u b stitution forc<. I t i s necessary to prove (a) = (b) by showing that both can be obtained from ( c) . To get (a) from (c) : and perform the Indicated d i f f e r e n t i a t i o n using A l \ s = K! V«--S -Ux) X (—)! * • (A9> To get (b) from (c) : (using Leibnitx* theorem f o r the d i f f e r e n t i a t i o n of a product of two functions) (using ( « ) ) . then form ( f Y ) J L " ~ 1 L f ) * " " * " ^ " J ft (using Leibnit2 ' theorem) Y U ^ . m ^ ) ! ^ ! (J«+m»-/j)i. (/5-Xx+t)! (using (A9)). Setting Y - l i n the l a t t e r summation leaves only the term f o r which /3 = A £ - ^ « Substituting these formulae i n (c) one obtains (a). Thus (a)=• (b). This equality has also been shown_by 17 Racah by a d i f f e r e n t method. The expression f o r f ( J J u J g ) , given i n (A7) could have been obtained by using the r e s u l t (A8) with (A3) and (A4). Appendix B. To show t hat ( r f * * + m > M > 2 and ( c i f M A « - T w ) 2 are polynomials i n m. If the l a t t e r quantity i s a polynomial i n m then so i s the former. Hence a l l that i s necessary to show i s the proof that •Tin ?v i 2 ^ CJ am-mJ m ^  i s a polynomial In m and this i s given below. Prom the symmetry properties (21) , (23) one can see that i f (CJII^JJ^J m J i s a polynomial i n m f or O t ^ - J g and Oim^Jg then i t i s a polynomial i n m for-any A.g and m^  i n the ranges 0t^gt2Jg, -Jg^g-Jg* I n the d e f i n i t i o n (19), i t i s seen that the summand i s a polynomial i n m. I t must be shown that the factors (j^4mtt.)l (J u-m t t)! appearing i n the denominator of 2 (C ) are cancelled out f o r 0-*g-Jg» ° f e m2 f c Jg t o complete the proof. Now, Jw+mw= J+m-(J2+m2-^) and i s £J+m f o r O^AgfeJg and m^O. Hence (Ju+mM)! divides (J+m)( f o r Oi:Ag^Jg and (Kmgtjg. Jtt-mH = J-m-(J2-mg-A2) and i s £J-m i f Jg-mg-Ag^O. For the case that J o-nu-a o^0, the f a c t o r (j-m)'"""_ (j-m)1-2 ^ 2 (J»-m")( - (J-m-(J a >-m, rA < L))l, i s cancelled by the common fa c t o r (J f t-m a)! , (J f l-m a)f (J-m)l (J^-m^J^-m^-"^) which appears i n a l l the terms, ( ja-m") 1 , of the summand (JM-ml,-oc)\ In (19) since J^-m^-AA+«l^O f o r a non-zero term. Appendix C. The Summation Formulae. To derive the formulae (45) and (49)'. I t w i l l now be shown that j T ^ £ v ) ^ + q - v j v k j - ^n+p+q+lj i f k = 0 (Bl) r ^ a . .l£lllL/n+P+q+i\ i f k i l (B2) k * pi [ n-* / where a k c ( ( = 1 i f k * l , * = l *<r*+' / k - l \ v ( B 3 ) - g a ^ ^ . ^ ^ i f k.2,.,2. These formulae are the same as (45) and (49) of the text. The r e s u l t (Bl) w i l l f i r s t be proven, (B2) and (B3) then follow from i t by induction. -To prove ( B l ) , one forms (l-X) -P-^l-X) " q - 1 = (1-X) "P'S" 2 and expands eash binomial to get Collect i n g the c o e f f i c i e n t of X n on both sides of this equation one arrives at 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\ /n+q-v\ v JyT (ptv)!/n+q-vU (r> +i) ? /P+l+w\ /n-lfq-w\ v ^ l p / l q ) ~£r (v-1)! p! (, q / P fe(p*lA q / (p+l)(n+P+5+1) using ( B l ) . Now, assume (B2) i s true f o r a l l values of k up to k=r-l (r=>2) . Then5^/P*vUn*1"vWr* T J&ziLl /n+q-v\ v r - l fe{ p A q / ^ ( v - l ) l p ! ( q / - (p+1>UJppaw)(n"1q<3",1 (using the induction assumption) - ( p + l ) Z A*/ . * A a " * C ' < | (p+-U! ( n - ( ^ l V + l P + 1 ) I n - 1 / --Zf2ay_ M|] l £ l ^ / n ^ ^ l W ( + 1 )/a+p +q +l\ " K F , r ^ - - 1 Ir-flJI p i ( n - W + ( P + 1 ) l -'f'a (Pt«)! /n+pfqtl\ +p+q+l\ n£L J <3*l = 1 i f - r i l . where a r f t This completes the proof of (B2) and (B3). To derive formula (46) : -The formula (46) may be derived by using the same method of proof as that already given f o r the formula (45) s t a r t i n g with the equation (1-X) P(L-X)Q- (1-X) P +Q instead of ( l - X j ' P - ^ - d - X ) - ^ 1 = (1-X) -P-q- 2 and expanding the binomials using the formula (1-X) (-X) v instead of ( 1 - X ) - P , [ X ( - v ) ( - X ) ^ | l ( - l ) v ( P 4 + V ) ( - X ) V . However, the r e l a t i o n ( - P) - (-1) v i s a 1 1 that is-needed to get (46) from (45). I f one sets -p-l-P and -q-l=Q In (45) then (46) can be obtained using this r e l a t i o n . The l i m i t s f o r the summa-tions are determined from the range f o r which the binomial co e f f i c i e n t s i n the summand do not vanish. Appendix D. The proof of formula (53). The oc(l)- and Y(l)-mixed y(2) c o r r e l a t i o n functions can be calculated from (*>f) and (52) . The second bracketed, exp-ression i n each W(©) i n (5X) i s independent of ©. 1^(9) can thus be written symbolically as (see reference 3) W i 2(9) r gGF°_(©) -fdGPj(9) , (Bl) 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-tained from (SO.) * The «,(l)-^(2) c o r r e l a t i o n w4 function W 1 2(9) w i l l f i r s t be obtained from the *(l)-X(2) c o r r e l a t i o n function W. (9) which-are tabulated i n Ling and Palkoff '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) ray one obtains • 0C(1) -1T(2) : Wlg (9) - dG 4- (2gG-dG) cos 29 (D2) |f(l)-V(2): W 1 2(9)-^ 2gG4-dG+-(dG-2gG)cos29. - -V (D3) Ct>3) The expression. Is given i n Table II. of Ling and Palkoff's paper i n the form tf(l)-lf(2). W 1 2(9) x Q-rRcos 2©, (D4) with e r 0. , M Hence, 2gG+-dG + (dG-2gG). -K(Q+-Rcos © ) , (D5) where K i s a possible common f a c t o r that has been omitted. Equating the co e f f i c i e n t s of cos 1©, one obtains two equations from which one can solve f o r gG and dG i n terms of K, Q, and R. Substituting the r e s u l t In (D2) gives .2, <X(l)-tf(2): W l g(Q)r K(Q4-R - Rcos^Q) from which K can be omitted. This r e s u l t i s exactly the same f o r W 2 2(9) andWj.(9) , a d i f f e r e n t G- being used i n each case* The common fa c t o r K i s the same i n each case* Hence the oCClJ-mixedtf^) c o r r e l a t i o n functions 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 Ling and Palkoff'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. Goertzel, Phys. Rev. 70, 897(1946). / 3. D. L. Palkoff and G. S. TJhlenbeck, Phys. Rev. 79, 323(1950). 4. D. S. Ling J r . and D. L. Palkoff, Phys. Rev. 76, 1639(1949). 5. J . A. Spiers, Phys. Rev.- 80 , 491(1950) . 6. S. P. Lloyd, Phys. Rev. 80, 118(1950). 7. B. A. Lippmann, Phys. Rev. 81, 162(1951). 8. S. P. Lloyd, Phys. Rev. 81, 161(1951). 9. C. N. Yang,-.Phys. Rev. 74 , 764(1949). 10. D. L. Palkoff, Phys. Rev. 82, 98(1951). 11. W. R. Arnold, Phys. Rev. 80, 34(1950). < 12. S. Devons, Proc. 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, 1931. 16. J. M. K e l l e r , Phys, Rev. 55, 509(1939). 17. G. Racah, Phys. Rev. 62, 438(1942). 

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