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The magnetic moment form factor of He³ Poffenberger, Paul R. 1975

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THE MAGNETIC MOMENT FORM FACTOR OF He 3 by PAUL R. POFFENBERGER B.S., Washington State U n i v e r s i t y , 1973 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of Physics We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1975 In present ing th is thes is in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f r ee ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for s c h o l a r l y purposes may be granted by the Head of my Department or by h is representa t ives . It is understood that copying or p u b l i c a t i o n of th is thes is for f inanc ia l gain sha l l not be allowed without my wri t ten permission. Department of ^ ^ V ^ ) ? f ^ S The Un ivers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date Oe.toW, Abstract The D e r r i c k - B l a t t expansion of the He J wave function i s used to derive an expression f o r the magnetic moment form f a c t o r f o r 3 He . The symmetric and mixed symmetric S states and a l l the D states of the expansion are retained i n the c a l c u l a t i o n . TABLE OF CONTENTS 1 Introduction 1 2 The Magnetic Moment Form Factor of He3 3 3 Conclusions 25 Bibliography 27 APPENDIX A The De r r i c k - B l a t t Wave Function 28 A . l Introduction 28 A.2 The Body-Fixed and Space-Fixed Coordinate Systems 29 A.3 The Symmetric Group S(3) 30 A.4 The Euler Angle Wave Functions 33 A.5 The Spin-Isospin Wave Functions 35 A.6 The Total Angular Momentum-Isospin Wave Functions ..39 A. 7 The Tot a l Wave Function 39 A.8 Relative Importance of the States 43 APPENDIX B The Magnitude and Angles of r ^ i n the Body-Fixed System 45 APPENDIX C The Representation C o e f f i c i e n t s CV, ^ ,^)^mL 57 LIST OF TABLES 3 * ' 1 The terms Y ^ e ^ y - * A * d S 1 8 A . l The non-zero permutation ad d i t i o n c o e f f i c i e n t s 32 A.2 The d i r e c t product functions 34 A.3 The spin eigenstates 37 A.4 The s p i n - i s o s p i n functions 38 A.5 The t o t a l angular momentum-isospin functions 40 A.6 Permutation properties of the wave functions 42 LIST OF FIGURES 1 The v e c t o r s Y , r , r , and r „ — — \ —i —3 2 The change of v a r i a b l e s r =R + r ' b — K —Q. — k B . l The v e c t o r s r_ and B.2 The coordinate system w i t h o r i g i n at the centre of mass, Z a x i s p e r p e n d i c u l a r to the plane of the t r i a n g l e , and p a r t i c l e 3 on the X a x i s B.3 The body f i x e d coordinate system v i Acknowledgements I would l i k e to thank Dr. J . M. McMillan f o r h i s suggestion of t h i s problem and f o r h i s assistance i n completing i t . 1 1 X n t r o d a c t i o n By s t u d y i n g l i g h t n u c l e i one hopes t o g a i n i n f o r m a t i o n about the n u c l e a r f o r c e . The n a t u r a l s t a r t i n g p o i n t i n t h i s s t u d y i s of c o u r s e the d e u t e r o n , b e i n g t h e s i m p l e s t n u c l e u s . Indeed much has been l e a r n e d from the s t u d y of the d e u t e r o n . For example, the nonzero quadrupole moment of t h e d e u t e r o n i m p l i e s the e x i s t e n c e o f the n u c l e a r t e n s o r f o r c e . The next s t e p i n c o m p l e x i t y then i s the s t u d y of the t h r e e -n u c l e o n n u c l e i , t h a t i s , the t r i t o n o r He . The t h r e e - n u c l e o n system i s i n many ways b e t t e r s u i t e d f o r the s t u d y o f t h e n u c l e a r f o r c e , b e i n g f o r example more t i g h t l y bound than t h e d e u t e r o n a l l o w i n g one t o probe the n u c l e a r f o r c e more c l o s e l y . D e r r i c k and B l a t t ( 1 9 5 8 ) have c o n t r i b u t e d s i g n i f i c a n t l y t o t h e u n d e r s t a n d i n g o f t h e t h r e e - n u c l e o n system, c o n s t r u c t i n g a comp l e t e s e t of s t a t e s i n terms o f which the t h r e e - n u c l e o n wave f u n c t i o n may be expanded. The c o n s t r u c t i o n i s analogous to the c o n s t r u c t i o n of the deut e r o n wave f u n c t i o n found i n B l a t t and Weis s k o p f ( 1 9 5 2 ) . F u r t h e r , D e r r i c k (1960a,b) has d e r i v e d a s e t of s i x t e e n c o u p l e d p a r t i a l d i f f e r e n t i a l e q u a t i o n s i n t h r e e v a r i a b l e s f o r the e x p a n s i o n c o e f f i c i e n t s (the s o - c a l l e d i n t e r n a l wave f u n c t i o n s ) . There has s i n c e been a g r e a t d e a l of work on t h e n u m e r i c a l c a l c u l a t i o n s of the s e i n t e r n a l wave f u n c t i o n s . I n v e s t i g a t i o n of the e l e c t r o m a g n e t i c form f a c t o r s p r o v i d e s one means of t e s t i n g a g i v e n s e t of i n t e r n a l wave f u n c t i o n s . Best (1966) has c a l c u l a t e d an e x p r e s s i o n f o r the charge form f a c t o r f o r the t r i t o n u s i n g the D e r r i c k - B l a t t e x p a n s i o n of the wave f u n c t i o n . Best r e t a i n e d i n h i s c a l c u l a t i o n t h e symmetric 2 and mixed-symmetric S s t a t e s and a l l the D s t a t e s . D e r r i c k and B l a t t have shown t h a t the r e m a i n i n g s t a t e s do not c o n t r i b u t e s i g n i f i c a n t l y t o the ground s t a t e wave f u n c t i o n . Best a l s o o b t a i n e d some n u m e r i c a l r e s u l t s u s i n g a m o d i f i e d Feshbach-Rubinow (1955) approach t o ap p r o x i m a t e t h e i n t e r n a l wave f u n c t i o n s . I n t h i s work we c a l c u l a t e an e x p r e s s i o n f o r t h e magnetic moment form f a c t o r f o r He3 , a g a i n u s i n g t h e D e r r i c k -B l a t t e x p a n s i o n o f the wave f u n c t i o n . L i k e Best we r e t a i n t h e symmetric and mixed-symmetric S s t a t e s and a l l the D s t a t e s . U n l i k e the charge form f a c t o r , however, the magnetic moment form f a c t o r c o n t a i n s c r o s s terms between S and D s t a t e s . These a r i s e because of the appearance o f the s p i n o p e r a t o r i n the magnetic moment d e n s i t y o p e r a t o r . Thus the magnetic moment form f a c t o r can be s e n s i t i v e t o t h e presence o f t h e D s t a t e s . S c h i f f (1964) and Gibson (1965) have a l s o c a r r i e d o u t a c a l c u l a t i o n of the He 5 magnetic moment form f a c t o r . They, however, have used the Sachs(1953) e x p a n s i o n of t h e wave f u n c t i o n , which i s l e s s c o n v e n i e n t t o work w i t h s i n c e the a n g u l a r momentum s t a t e s a r e not o r t h o g o n a l , wheras the a n g u l a r momentum s t a t e s i n the D e r r i c k - B l a t t e x p a n s i o n a r e o r t h o g o n a l . A l s o the Gi b s o n and S c h i f f c a l c u l a t i o n l a c k s t h e g e n e r a l i t y of our c a l c u l a t i o n as t h e i r e x p r e s s i o n i s based on a p a r t i c u l a r assumed form of the i n t e r n a l wave f u n c t i o n s . No p a r t i c u l a r form i s assumed i n our c a l c u l a t i o n . N u m e r i c a l r e s u l t s may be o b t a i n e d by e v a l u a t i n g the t h r e e -d i m e n s i o n a l i n t e g r a l s i n c l u d e d i n our f i n a l e x p r e s s i o n u s i n g any one of tht: a v a i l a b l e s e t s of i n t e r n a l wave f u n c t i o n s . 3 2 The Magnetic Moment Form Factor of He3 Consider the e l a s t i c s c a t t e r i n g of an e l e c t r o n with i n i t i a l momentum P-l from He having i n i t i a l momentum P-t . A t t e r the s c a t t e r i n g takes place the e l e c t r o n emerges with f i n a l momentum P^ . and the Hee r e c o i l s with f i n a l momentum P^ .. The momentum t r a n s f e r of the e l e c t r o n and the change K i n momentum of the 3 He are d e f i n e d as (2 .1) (2.2) These terms are shown p i c t o r i a l l y below: The magnetic moment Schiff(1964) to be the form f a c t o r f o r He i s taken by F o u r i e r transform of the e x p e c t a t i o n 4 v a l u e of the magnetic moment d e n s i t y o p e r a t o r where the magnetic moment d e n s i t y o p e r a t o r i s (2. 3) The 0"'s and T s a r e the u n i t a m p l i t u d e P a u l i m a t r i c e s o p e r a t i n g on the s p i n and i s o s p i n f u n c t i o n s r e s p e c t i v e l y . The V s are the s p a t i a l d i s t r i b u t i o n f u n c t i o n s f o r the moment d e n s i t i e s about t h e c e n t r e s of the n u c l e o n s , w h i l e the ^ u. s a r e the s t a t i c m a g netic moments of the n u c l e o n s . The v a r i a b l e s "X.* E\ » £\ / and are shown i n f i g u r e (1). The D e r r i c k - B l a t t wave f u n c t i o n d e s c r i b e d i n appendix A 3 i n c l u d e s o n l y t h e i n t e r n a l c o o r d i n a t e s o f He . For t h e c a l c u l a t i o n of the magnetic moment form f a c t o r , however, we need t o i n c l u d e the c e n t r e of mass c o o r d i n a t e s o f the n u c l e u s . D e f i n i n g the c e n t r e of mass wave number t o be K, and"the c e n t r e o f mass p o s i t i o n v e c t o r to be R t h e c e n t r e o f mass wave f u n c t i o n may be w r i t t e n as and the t o t a l wave f u n c t i o n f o r He3 i n c l u d i n g a l l c o o r d i n a t e s i s 5 6 then ( 2 . 6 ) The form f a c t o r may than be w r i t t e n + S 0 - T K > , C ^ ^ ^ JVJ\ , 2 . 7 , where ( 2 . 8 ) D e f i n i n g the r e s p e c t i v e neutron and proton magnetic moment form f a c t o r s as and ^ - ^ 6 1 ^ ^ ( 2 ' 9 ) TA*<^ V <J (2.10) 7 and making the change o f v a r i a b l e s u_ = X - r v e n a b l e s one t o w r i t e t h e form f a c t o r as 3 c /V With t h e a d d i t i o n a l change o f v a r i a b l e s r = R + r" , a s shown i n f i g u r e ( 2 ) , we may w r i t e 1 f 3 = /-Utte) F ^ ( \ W 1 ,2.12, where we have d e f i n e d ~ 7 M F. (2. 13) 1 and where TT TT TT co oo r ^ + C^ --TT -TT O ~ V J -IT TT o o ^ <2*14> The a n g l e s oi , § , and V a r e the t h r e e E u l e r a n g l e s r e q u i r e d t o s p e c i f y the s p a t i a l o r i e n t a t i o n of the t r i a n g l e . 1 The c a l c u l a t i o n s o f the m a t r i x elements ^ ~ ^ ~ K ^ ^ and the i n t e g r a t i o n s over the E u l e r a n g l e s may be done a n a l y t i c a l y . W r i t i n g the wave f u n c t i o n as a sum of a n g u l a r momentum s t a t e s e n a b l e s one t o perfor m the c a l c u l a t i o n s term by te r m , t h a t i s « * j . 1> ^ *V '3 ( 2 . 1 5 ) where t h e a r e d e f i n e d i n e q u a t i o n (A.7.2 ) . To i l l u s t r a t e how-each of t h e s e terms i s c a l c u l a t e d we w i l l d e t e r m i n e here t h e terms ^ ( O T ^ - ^ K ^ ' ^ K J ) ^  * A ^ o t h e r t e r m s a r e found i n a s i m i l a r f a s h i o n . To s i m p l i f y c a l c u l a t i o n s each of the terms ^ (TK ^ f e a n d C l|j ^ ^ a r e de t e r m i n e d s e p a r a t e l y , t h e n the suras and d i f f e r e n c e s a r e found. i See appendix A.2 f o r more d e t a i l s . 10 Now 'ft H V 1 1 B | ' O B | V • B . t o - * / • - • . ( 2 . 1 6 ) 3 We w i l l c a l c u l a t e f i r s t the term 1_ ^ v ^ N ^ A ^ "B\ o ^ i 7 - * N O W ^ . x ^ X S ) ^ ^ ; ^ ^ ^ ' ) i s z e r o u n l e s s = M* so we may w r i t e 3 5 L V (2. 17) where V 3 \ denotes (2. 18) Making use of the p e r m u t a t i o n p r o p e r t i e s of the f u n c t i o n s we now e l i m i n a t e the sum by c h a n g i n g the v a r i a b l e s of i n t e g r a t i o n t o 0) r Y<> \S>ty 1 a n ; l ^ £ a r e a l l symmetric under p e r m u t a t i o n s w h i l e T* g ^ b e l o ngs t o t h e f i r s t row of t h e mixed 11 r e p r e s e n t a t i o n . Permuting the v a r i a b l e s r' and r' r e s u l t s i n 1 (2. 19) and permuting the v a r i a b l e s r^ and r' r e s u l t s i n Hence e q u a t i o n (2.17) becomes (2.20) ^ v : W u - o ) ^^ -l ?^0*K-\v^ O-^^Vs e v 5 I ? ^ T £ M a - r < i x i (2.21) where V ^ ^ i . (2.2 2) and 1 See appendix A.3. and where we have w r i t t e n ( 2 . 2 3 ) ( 2 . 2 5 ) We use here the argument R t o denote the E u l e r a n g l e s (<*,(•!>,^ ) . To perform the i n t e g r a t i o n o ver the E u l e r a n g l e s we need t o c o n s i d e r the E u l e r a n g l e dependence of 6 . 1 expanding t h e e x p o n e n t i a l we have ( 2 . 2 6 ) where the a r e t h e s p h e r i c a l B e s s e l f u n c t i o n s and the "Yivn £ a r e the s p h e r i c a l harmonics. Furthermore and r ? denote the s p h e r i c a l p o l a r a n g l e s t h a t <^  and , r e s p e c t i v e l y , make w i t h 1 The E u l e r a n g l e dependence of t h e e x p o n e n t i a l seems t o have been o v e r l o o k e d by B e s t ( 1 9 6 6 ) , but h i s f i n a l r e s u l t s i n e q u a t i o n s (5.23) and (5.35) a r e c o r r e c t . 13 t i i e space f i x e d c o o r d i n a t e system. The E u i e r a n g l e s r e p r e s e n t a r o t a t i o n o f t h e body f i x e d system i n t o the space f i x e d system. From wigner (1959) we f i n d 1 so A,™ (2.28) where now 'VJJ^ . ^ - ' S^ 1 S a f u n c t i o n of t h e a n g l e s o f r_ 3 w i t h r e s p e c t t o the body f i x e d s y s t e m . 2 E q u a t i o n (2.21) now becomes T (2.29) where we have made use of the e q u a t i o n 3 1 We show i n appendix C -the r e l a t i o n s h i p between the r o t a t i o n a l r e p r e s e n t a t i o n c o e f f i c i e n t s of Wigner and of D e r r i c k and B l a t t . 2 See appendix B. 3 See Wigner(1959). ^ ' . . . M an 14 9iv t. (2. 30) The z a x i s of our space f i x e d c o o r d i n a t e system l i e s p e r p e n d i c u l a r to t h e s c a t t e r i n g p l a n e , hence cj. l i e s i n the x-y p l a n e . We can thus w r i t e ^ \J TV (2.31) so t h a t e q u a t i o n (2.29) becomes 1 XS® ] ^ ^  * -r. (2.32) By a s i m i l a r c a l c u l a t i o n we f i n d t h a t (2.33) so t h a t 3 I I 15 A l s o one can show t h a t (2 .34) - - T (2 .35) e (2. 36) and (2 .37) S i n c e c\~ a n d T, '^ V a r e I l e r m i t i a n a n d . s i n c e t h e m a t r i x e l e m e n t s 16 a r e r e a l we see t h a t Hence the c o n t r i b u t i o n t o the magnetic moment form f a c t o r from the c r o s s terms of ^ and i s (2.39) From appendix B we have t h a t h m -IT (2.40) so t h a t we may w r i t e 17 (2.41) . The D e r r i c k - B l a t t wave f u n c t i o n has been a p p r o x i m a t e d by u s i n g o n l y the symmetric S, mixed symmetric S, and mixed symmetric 0 s t a t e s of the t o t a l wave f u n c t i o n . T able (1) l i s t s a l l the terms c a l c u l a t e d . The n o t a t i o n used i n t a b l e (1) i s where £\VN d e n o t e s <rk/^  o r T n e f u n c t i o n s ^ and T a r e d e f i n e d i n appendix B. 18 CvfcV - - VH ^ " A - V v ( r ^ M*,,- J w , r $ c W v t n ^ o ? v v ^ Table 1 - The terms ^_ V f O ^ Y " * ^ ^ .^ c \ <^  19 Table 1 - continued 20 0.1U -  -(uv Table 1 - continued 21 Table 1 - continued 22 With the a i d of t a b l e ( 1 ) the magnetic moment form f a c t o r f o r He may be w r i t t e n as: w h e r e F, F5 v * s. F;, = P»- + \ F T O F H = V + V T o o x (2.KK, and 2a and where (2.46) The form f a c t o r could be c a l c u l a t e d n u m e r i c a l l y u s i n g , f o r example, the i n t e r n a l wave f u n c t i o n s of McMillan(1970) . 1 i The i n t e r n a l wave f u n c t i o n s of McMillan a c t u a l l y correspond to somewhat d i f f e r e n t D s t a t e s than ours. The r e l a t i o n s h i p between the D s t a t e s used by McMillan and the D s t a t e s we use i s given by D e r r i c k (1960b). 25 3 C o n c l u s i o n s U s i n g the wave f u n c t i o n e x p a n s i o n of D e r r i c k and B l a t t (1 958) we have c a l c u l a t e d an e x p r e s s i o n f o r the magnetic moment form f a c t o r of He . The symmetric and mixed-symmetric S s t a t e s and a i l t h e D s t a t e s were r e t a i n e d i n the c a l c u l a t i o n . The exchange moment c o n t r i b u t i o n t o the magnetic moment form f a c t o r has not been i n c l u d e d here. A s i m i l a r c a l c u l a t i o n of t h e charge form f a c t o r of the t r i t o n has been done by Best ( 1 9 6 6 ) some time ago. Our c a l c u l a t i o n i s more g e n e r a l than the c a l c u l a t i o n of the magnetic moment form f a c t o r o f He by S c h i f f ( 1 9 6 4 ) and Gi b s o n (1965). The Gi b s o n and S c h i f f c a l c u l a t i o n i s based on a p a r t i c u l a r form of t h e i n t e r n a l wave f u n c t i o n s , whereas no such c h o i c e i s made i n our c a l c u l a t i o n . U n l i k e t h e charge form f a c t o r , the magnetic moment form f a c t o r c o n t a i n s c r o s s terms between S and D s t a t e s . T h i s i s a consequence of t h e s p i n o p e r a t o r a p p e a r i n g i n the magnetic moment d e n s i t y o p e r a t o r . Through t h e SD c r o s s terms the D s t a t e t h u s can make a more i m p o r t a n t c o n t r i b u t i o n t o the magnetic moment form f a c t o r than to t h e charge from f a c t o r . As p o i n t e d out by M c M i l l a n and Landau (1974) , an analogous s i t u a t i o n h o l d s f o r the s c a t t e r i n g of low energy p i o n s by He3". That i s , terms s i m i l a r t o the charge form f a c t o r terms a r i s e from the n o n - s p i n f l i p p a r t of the p i o n - n u c l e o n i n t e r a c t i o n , and terms s i m i l a r t o the magnetic moment form f a c t o r terms a r i s e from t h e s p i n - f l i p p a r t of the p i o n - n u c l e o n i n t e r a c t i o n . Indeed, many of the r e s u l t s d e r i v e d hare and by Best can be c a r r i e d over 26 q u i t e d i r e c t l y t o g i v e the pion-He e l a s t i c s c a t t e r i n g c r o s s s e c t i o n i n the s i n g l e - s c a t t e r i n g form f a c t o r a p proximation.' Work i n t h i s d i r e c t i o n i s , however, beyond the scope of t h i s t h e s i s . BIBLIOGRAPHY Best, M.E., t h e s i s (1966) The U n i v e r s i t y of B r i t i s h Columbia. B l a t t , J.M. and Weisskopf, V.F. 1952. "Theoretical Nuclear Physics" (John Wiley and Sons, New York). Derrick, G.H. and B l a t t , J.M. 1958. Nucl. Phys., 8_, 310. Derrick, G.H., t h e s i s (1959) The U n i v e r s i t y of Sydney. Derrick, G.H. 1960a. Nucl. Phys., 1_6, 405. 1960b. Nucl. Phys., 18_, 303. Feshbach, H. and Rubinow, S.J. 1955. Phys. Rev., 98_, 188. Gibson, B.F. 1965. Phys. Rev., 139, B1153. Goldstein, H. 1959. " C l a s s i c a l Mechanics" (Addison Wesley Publishing Co., Inc., Cambridge, Mass.). McMillan, M. and Maroun, D. 1970. Nucl. Phys., A159, 661. McMillan, M. and Landau, R.H. 1974. TRIUMF Report, TRI-74-1. Sachs, R.G. 1953. "Nuclear Theory" (Addison Wesley Publishing Co., Inc., Cambridge, Mass.). S c h i f f , L.I. 1964. Phys. Rev., 133^ B802. Wigner, E.P. 1959. "Group Theory and i t s A p p l i c a t i o n to the Quantum Mechanics of Atomic Spectra" (Academic Press, New York and London). 28 APPENDIX A The D e r r i c k ^ B l a t t Wave Function A.1 I n t r o d u c t i o n D e r r i c k and B l a t t (1958) have c l a s s i f i e d a l l angular momentum-isobaric s p i n f u n c t i o n s which can be present i n the ground s t a t e wave f u n c t i o n of He . I f the nuclear i n t e r a c t i o n was c e n t r a l only, each of the o r b i t a l angular momentum L and spin angular momentum S would be good quantum numbers and the ground s t a t e wave f u n c t i o n of He would be L = 0 only. However with the i n c l u s i o n of non-central f o r c e s i n the nuclear i n t e r a c t i o n only the sum J = L + S remains a good guantum number, and the ground s t a t e wave f u n c t i o n must be w r i t t e n as a mixture of s e v e r a l angular momentum s t a t e s . D e r r i c k and B l a t t w r i t e the He ground s t a t e wave f u n c t i o n as a sum of products: (A.1.1) The f u n c t i o n s are t o t a l angular momentum-isobaric s p i n f u n c t i o n s , each with the experim e n t a l l y observed J = 7^, T = Vt_» even p a r i t y , and d e f i n i t e values of L and S, while the f u n c t i o n s are f u n c t i o n s of the three i n t e r p a r t i c l e d i s t a n c e s . Each of the f u n c t i o n s .^x. 1 S i n turn w r i t t e n as a sum of products, each product c o n t a i n i n g two f a c t o r s : 1) a f a c t o r y depending on the Euler angles which s p e c i f y the o r i e n t a t i o n of the t r i a n g l e i n space, 29 2) a f a c t o r V depending on the s p i n s and i s o b a r i c s p i n s of the t h r e e p a r t i c l e s . The t h r e e E u l e r a n g l e s and the t h r e e i n t e r p a r t i c l e d i s t a n c e s a r e the s i x c o o r d i n a t e s r e q u i r e d t o s p e c i f y the s p a t i a l p o s i t i o n s of t h e t h r e e p a r t i c l e s a f t e r s e p a r a t i n g out the c e n t r e of mass c o o r d i n a t e s . To be c o n s i s t e n t w i t h charge independence o f the n u c l e a r f o r c e each o f the f u n c t i o n s y , \J , ^ , and "9 a r e r e q u i r e d t o have a d e f i n i t e p e r m u t a t i o n symmetry, t h a t i s , each t r a n s f o r m s a c c o r d i n g t o one o f the t h r e e i r r e d u c i b l e r e p r e s e n t a t i o n s o f the symmetric group S ( 3 ) . The f u n c t i o n s a r e then combined so t h a t each of the p r o d u c t s -f - ^ O-^is o v e r a l l a n t i s y m m e t r i c w i t h r e s p e c t o t o p e r m u t a t i o n of any two p a r t i c l e s . A«2 The Body_-Fixed and S f i a c e ^ f i x e d C o o r d i n a t e Systems In o r d e r t o s p e c i f y t h e o r i e n t a t i o n i n space of the t r i a n g l e formed by t h e t h r e e p a r t i c l e s , D e r r i c k and B l a t t f i r s t s p e c i f y a body f i x e d c o o r d i n a t e system. The s p a t i a l o r i e n t a t i o n o f the t r i a n g l e i s then d e t e r m i n e d by the s e t of t h r e e E u l e r a n g l e s r e q u i r e d t o r o t a t e the body f i x e d frame i n t o the space f i x e d f r a m e . 1 the body f i x e d c o o r d i n a t e system i s chosen as f o l l o w s : 1) The t r i a n g l e w i l l l i e i n the x-y p l a n e w i t h the c e n t r e of g r a v i t y being the o r i g i n . 2) the x a x i s w i l l be chosen as the p r i n c i p a l a x i s a s s o c i a t e d i D e r r i c k and B l a t t r e f e r to t h i s as r o t a t i n g the t r i a n g l e from i t s "normal" p o s i t i o n to i t s a c t u a l p o s i t i o n . 30 w i t h the l a r g e s t moment of i n e r t i a . T h i s does not u n i q u e l y s p e c i f y a d i r e c t i o n f o r the x a x i s so t h a t some of the E u l e r a n g l e wave f u n c t i o n s c o u l d be double v a l u e d , however i t t u r n s out t h a t wave f u n c t i o n s of even p a r i t y a r e n e c i s s a r i l y s i n g l e v a l u e d so t h e r e i s no a m b i g u i t y . 3) the z a x i s i s chosen such t h a t w a l k i n g a path from p a r t i c l e 1 t o 2, then t o 3 and back to 1 would amount t o a c o u n t e r c l o c k w i s e walk around the z a x i s . 4) once t h e d i r e c t i o n of the x a x i s i s chosen the y a x i s i s chosen such t h a t t h e c o o r d i n a t e system i s r i g h t handed. We w i l l go i n t o more d e t a i l o f t h i s c o o r d i n a t e system i n appendix B. We choose the s p a c e - f i x e d c o o r d i n a t e system t o be a r i g h t -handed system w i t h o r i g i n a t the c e n t r e of g r a v i t y , w i t h x - a x i s i n t he d i r e c t i o n of t h e i n i t i a l e l e c t r o n momentum, and z - a x i s p e r p e n d i c u l a r t o t h e s c a t t e r i n g - p l a n e . A. 3 The Symmetric Grou£ SJ3]_ As the c o n s t i t u e n t wave f u n c t i o n s of t h e t o t a l He wave f u n c t i o n are r e q u i r e d t o have d e f i n i t e p e r m u t a t i o n symmetries i t i s u s e f u l here t o summarize some of the i m p o r t a n t p r o p e r t i e s of the group 5(3) of p e r m u t a t i o n s on t h r e e o b j e c t s . There are s i x elements i n t h e group S (3) . I n t h e c y c l i c n o t a t i o n of Wigner ( 1959) t h e s e a r e ( 1 ) , (1 32) , (123), ( 12), (3 1 ) , and (23 ) . The f i r s t t h r e e of t h e s e a r e even p e r m u t a t i o n s w h i l e the l a s t t h r e e are odd. There a r e t h r e e i r r e d u c i b l e r e p r e s e n t a t i o n s of t h i s group. One, the symmetric 31 r e p r e s e n t a t i o n , r e p r e s e n t s each group element by 1. Another r e r e s e n t a t i o n , the a n t i s y m m e t r i c r e p r e s e n t a t i o n , r e p r e s e n t s the even group elements.by 1 and t h e odd group elements by -1. The r e m a i n i n g i r r e d u c i b l e r e p r e s e n t a t i o n i s two d i m e n s i o n a l and of mixed symmetry. The mixed symmetric i r r e d u c i b l e r e p r e s e n t a t i o n r e p r e s e n t s the group elements i n the above o r d e r a s : Now suppose a f u n c t i o n §>\l, (\,1 ,~$) t r a n s f o r m s a c c o r d i n g t o the k^~ row of the i r r e d u c i b l e r e p r e s e n t a t i o n of S ( 3 ) , and a n o t h e r f u n c t i o n (\ ' ( , V ) t r a n s f o r m s a c c o r d i n g t o the k row of t h e i r r e d u c i b l e r e p r e s e n t a t i o n of S ( 3 ) . One may ask how t h e p r o d u c t f u n c t i o n s t r a n s f o r m under j o i n t p e r m u t a t i o n o f the i n d i c i e s (1,2,3) and ( l ' , 2 ' , 3 / ) . I n ana l o g y t o t h e C l e b s c h -Gordan c o e f f i c i e n t s of the r o t a t i o n group, t h e r e are a d d i t i o n c o e f f i c i e n t s f o r combining base f u n c t i o n s of S (3 ) . Sums of p r o d u c t s can be formed l i k e V V o X ^\ (A. 3. 1) which t r a n s f o r m a c c o r d i n g t o t h e k row o f the P i r r e d u c i b l e r e p r e s e n t a t i o n of S (3). The p e r m u t a t i o n a d d i t i o n c o e f f i c i e n t s K y.' y J are unique a p a r t from a r b i t r a r y phase f a c t o r s which one chooses t o make a l l the c o e f f i c i e n t s r e a l . A l l non-zero p e r m u t a t i o n a d d i t i o n c o e f f i c i e n t s a r e l i s t e d i n t a b l e (A.1). 32 Table (A.l) - The non-zero permutation add i t i o n c o e f f i c i e n t s . 33 t a b l e (A.2) l i s t s the s i x t e e n p o s s i b l e d i r e c t product f u n c t i o n s . A.4 The E u l e r Angle Wave Functions The E u l e r angle wave f u n c t i o n s which d e s c r i b e the : angular dependence of the He wave f u n c t i o n are simply the r e p r e s e n t a t i o n c o e f f i c i e n t s of the i r r e d u c i b l e r e p r e s e n t a t i o n s of the r o t a t i o n group. These a r e 1 where L i s the o r b i t a l angular momentum with z component M L and body z component yj,. The e x p e r i m e n t a l l y observed t o t a l angular momentum f o r the ground s t a t e of He i s J = fx. , so with a maximum p o s s i b l e s p i n of S = "V^ the o r b i t a l angular momentum L cannot exceed 2. Also i t can be shown t h a t the p a r i t y o p e r a t o r H a p p l i e d to the E u l e r angle f u n c t i o n ^ V)^ w g i v e s (A. 4. 2) so that only f u n c t i o n s with even fJL have the r e q u i r e d even p a r i t y . R e q u i r i n g the Eul e r angle wave f u n c t i o n s to be 1 See appendix C. 34 Symmetric Antisymmetric <?. • -(v\' - ) 3 = J N S _ v n ' tr \ s' w> s ' _ ^ < Mixed Symmetric <p; < ] <Y, Table (A.2) - The d i r e c t product functions. 35 o r t h o n o r m a l and r e a l under time r e v e r s a l y i e l d s the f o l l o w i n g f i v e f u n c t i o n s : We use here the n o t a t i o n (^pe i o r a i 1 E u l e r a n g l e wave f u n c t i o n w i t h p e r m u t a t i o n symmetry P e. As a consequence of the h i g h l y symmetric c h o i c e of the body f i x e d c o o r d i n a t e system each of the E u l e r a n g l e wave f u n c t i o n s i s e i t h e r symmetric o r a n t i s y m m e t r i c ; the mixed r e p r e s e n t a t i o n does not o c c u r . A. 5 The Sp_in-Isos£in Wave f u n c t i o n s The s p i n f u n c t i o n s f o r a t h r e e p a r t i c l e system may be c a l c u l a t e d u s i n g t h s double C l e b s c h - G o r d a n s e r i e s : w h e r e t h e I S i f ' V ^ - a r e e i q e n s t a t e s o f St- a n d o f t h e i * * -p a r t i c l e , and the \ S> W % ^Vw ; ) a r e the e i g e n s t a t e s of S-S = S and 36 - Sx„ • Si„ + S,. of the t h r e e p a r t i c l e system. k s and P £ denote the row number and i r r e d u c i b l e r e p r e s e n t a t i o n t o which the e i g e n s t a t e b e l o n g s . There a r e e i g h t p o s s i b l e e i g e n s t a t e s l ^ ^ K ^ f o r S = N / T _ or S- = "Vt. These a r e l i s t e d i n t a b l e (A. 3 ), where we have s e t ol(l') = i a " d ^>U^ = \ \ - \ ) - • The i s o s p i n wave f u n c t i o n s a re c o n s t r u c t e d i n a c o m p l e t e l y a n a l o g o u s way. With t h e ass u m p t i o n the t h e qround s t a t e of He i s T = o n l y and s e t t i n g T^ = # two i s o s p i n f u n c t i o n s a r e o b t a i n e d : P\ - VNTC W ^ ^ W ^ ^ (A- 5.2) (A.5.3) where T(i) = \h.~x) I r e p r e s e n t s a p r o t o n and = r e p r e s e n t s a n e u t r o n . From the e i g h t s p i n f u n c t i o n s and two i s o s p i n f u n c t i o n s we can form s i x t e e n l i n e a r l y i ndependent sums of p r o d u c t s o f s p i n and i s o s p i n f u n c t i o n s . The s i x t e e n ' s p i n - i s o s p i n f u n c t i o n s , denoted by Vn,. ^ ( V t T , S ) , a r e l i s t e d i n t a b l e (A. 4). P t and denote the i r r e d u c i b l e r e p r e s e n t a t i o n and c o r r e s p o n d i n g row number t o which the f u n c t i o n b e l o n g s . 37 .0 ^uHu ^ ^ S \> = ^ T a b l e (A.3) - The s p i n e i g e n s t a t e s . 38 ^ ^ * P * V . . ,(sAA > A . V ) >-%P*. Z. Table (A.4) - The spin-isospin funct ions. 39 A.6 The T o t a l A n g u l a r Momentum-IsosDin Wave F u n c t i o n s Haking use of t h e p e r m u t a t i o n group a d d i t i o n c o e f f i c i e n t s ^ \ ^ ) a n ( i Clebsch-Gordan c o e f f i c i e n t s ^S vS vXn ,M T A ) r the s p i n -i s o s p i n f u n c t i o n s V ^ ^ ^ i ^ t ^) a r G n o w combined w i t h the E u l e r a n g l e f u n c t i o n s \p^) t o o b t a i n t o t a l a n g u l a r moraentum-i s o s p i n f u n c t i o n s . These a r e (A. 6. 1) With J = V-v i =Vt , T = Vt , and T t = Vt. we o b t a i n t e n d i s t i n c t s t a t e s . These are l i s t e d i n t a b l e (A.5). The n o t a t i o n used i n t a b l e (A. 5) i s t h a t of -Derrick (1 960) . Each p a i r of mixed symmetric f u n c t i o n s i s counted as one d i s t i n c t s t a t e as both f u n c t i o n s must be combined l i n e a r l y t o o b t a i n one c o mplete f u n c t i o n . From t a b l e (A.5) one sees t h a t , ana ( u M | U . - i x ) r e p r e s e n t *-S s t a t e s ; VJ^M, , and r e p r e s e n t "Vp s t a t e s ; r e P r e s e n t s a l p s t a t e ' a a d ( ^ , L i e ^ ) » < ^ V / ^ % X ) , a n d r e P r e s e n t l ° s t a t e s . A « 7 The T o t a l Wave F u n c t i o n The t o t a l a n g u l a r momentum-isospin f u n c t i o n s ^ are now combined w i t h the i n t e r n a l wave f u n c t i o n s t o y i e l d an o v e r a l l wave f u n c t i o n f o r the ground s t a t e of l!eJ . We denote the 40 Table (A.5) - The t o t a l angular momentum-isospin f u n c t i o n s . 41 i n t e r n a l wave f u n c t i o n s by ^C^-i^,^, ^ 3 ) ' where k' and p' are the row number and i r r e d u c i b l e r e p r e s e n t a t i o n to which the f u n c t i o n f belongs, andX denotes any of the ten p o s s i b l e v a l u e s °f ; V , ?,> V t (Jz) • Using the permutation group a d d i t i o n c o e f f i c i e n t s and r e q u i r i n g t h a t the o v e r a l l wave f u n c t i o n be antisymmetric, the o v e r a l l wave f u n c t i o n i s w r i t t e n Using the n o t a t i o n of Derrick(1959) the t o t a l wave f u n c t i o n may be w r i t t e n , = ^ ^ * ^  • • ^ 6 • y<? • y; • ^  , ^ ( A . 7 . 2 ) Some important p r o p e r t i e s of each of the ten s t a t e s are summarized i n t a b l e (A.6). Permutation Symmetry 0 0 0 2 2 2 3h 'Internal 3 ^ Euler Angles s s a s m s s a a a m a m a m s m s m a Spin-Isospin a s m s a m m m m m 0 0 0 0 0 0 0 Table (A.6) - Permutation properties of the wave function 4 3 A . 8 Rftla t i v e lJ!!JJ2£liiiIl£i2 Q.L the s t a t e s For the purpose of f u r t h e r a p p r o x i m a t i o n i t i s of i n t e r e s t here to e s t i m a t e which of the t e n f u n c t i o n s a r e l i k e l y t o be dominant. In the absence of n o n - c e n t r a l f o r c e s the ground s t a t e would be S o n l y . F u r t h e r m o r e i f the i n t e r a c t i o n i s s p i n i n dependent the symmetry of the i n t e r n a l wave f u n c t i o n i s a good quantum number. Now a symmetric i n t e r n a l wave f u n c t i o n need not be z e r o f o r any shape of the t r i a n g l e . A p a i r of mixed symmetric f u n c t i o n s i s n e c e s s a r i l y z e r o whenever the t r i a n g l e i s e q u i l a t e r a l , t h a t i s , whenever r r i_ = r ^ = r ^ . An a n t i s y m m e t r i c f u n c t i o n must be z e r o whenever the t r i a n g l e i s i s o s c e l e s , t h a t i s , whenever r l r = r ^ , r > t = r v ^ , or r r i = r.^ . The more z e r o s a f u n c t i o n p o s s e s s e s , t h e more i t i s f o r c e d t o change, hence the h i g h e r i t s d e r i v a t i v e s , and hence the h i g h e r i t s k i n e t i c energy. Thus a p u r e l y symmetric f u n c t i o n has the l o w e s t k i n e t i c energy a s s o c i a t e d w i t h i t . In the absence of n o n - c e n t r a l and s p i n -dependent f o r c e s t h e ground s t a t e would then be symmetric S , t h a t i s , the wave f u n c t i o n would be of the t y p e f, . With the i n c l u s i o n of n o n - c e n t r a l and sp i n - d e p e n d e n t f o r c e s t h e symmetric S s t a t e i s s t i l l e x p e c t e d t o dominate. The next most i m p o r t a n t s t a t e s a re those which c o u p l e i n the f i r s t o r d e r t o ^ under the i n t e r a c t i o n . D e r r i c k (1959) has shown t h a t the s p i n exchange o p e r a t o r c o u p l e s the mixed symmetric S s t a t e d i r e c t l y t o , w h i l e the t e n s o r o p e r a t o r c o u p l e s the t h r e e mixed symmetric D s t a t e s ^ , ^ / and d i r e c t l y to y 7 , • F i n a l l y t h e L • S f o r c e s c o u p l e the P s t a t e s f^, H\ , and V-t 44 d i r e c t l y t o ^ , but t h e s e a r e c o n s i d e r e d u n i m p o r t a n t as the L • S f o r c e i s b e l i e v e d t o be o f very s h o r t range. With the i n c l u s i o n of t e n s o r and s p i n - d e p e n d e n t f o r c e s then the most i m p o r t a n t s t a t e s p r e s e n t i n the ground s t a t e w i l l be T"i , %, % / ^ , and t | 0 . The l e a s t dominant s t a t e s w i l l be t h o s e w i t h a n t i s y m m e t r i c i n t e r n a l wave f u n c t i o n s as t h o s e s t a t e s a r e a s s o c i a t e d w i t h v e r y h i g h k i n e t i c energy. U5 APPENDIX B The Ha.qnit.ude and An g l e s of i n the Bod^-Fixed System C o n s i d e r the t r i a n g l e formed by the t h r e e p a r t i c l e s w i t h s i d e s r ^ = r ( - r,^, r = r^ " I 3 / a n d £- c Z = - I-^- D e f i n e t h e v e c t o r s r = r ^ and P = r , ^ + r v 3 as shown i n f i g u r e (B.1). I t i s e a s i l y shown t h a t ^ = (^r^ * i r „ - ) (B.I) Now l e t t i n g the c e n t r e o f mass p o s i t i o n v e c t o r be R & '= i ( * V ^ + £\z + r u ) we have 1 3 (B.4) 46 Figure (B.l) - The vectors r and Thus the magnitude of r' 3 i s 3 = (B.5) Now c o n s i d e r a c o o r d i n a t e system w i t h o r i g i n a t t h e c e n t r e o f mass, Z a x i s p e r p e n d i c u l a r t o the p l a n e of the t r i a n g l e , and p a r t i c l e 3 on t h e X a x i s . The d i r e c t i o n of the Z a x i s i s chosen such t h a t w a l k i n g a path from p a r t i c l e 1 t o 2, then t o 3 and back t o 1 would amount to a c o u n t e r c l o c k w i s e walk around t h e Z a x i s . T h i s c o o r d i n a t e system i s shown i n f i g u r e (B.2). We have a l s o d e f i n e d Tj as the a n g l e (^,r) , which w r i t t e n i n terms of the t r i a n g l e s i d e s i s (B.6) I t can a l s o be shown t h a t Si/A?? c [ ( V ^ V ^ ( ^ (B.7) The c o o r d i n a t e s of r and £ i n t h i s c o o r d i n a t e system a r e 40 Figure (B.2) - The coordinate system with o r i g i n at the centre of mass, Z axis perpendicular to the plane of the t r i a n g l e , and p a r t i c l e 3 on the X axis. 49 (B.8) ^ - o ~ ° (B .9) Suppose now t h e c o o r d i n a t e s y s t e m i s r o t a t e d a b o u t t h e z a x i s by an a n g l e a s i n f i g u r e ( B . 3 ) . The c o o r d i n a t e s o f r and (3 i n t h i s new c o o r d i n a t e s y s t e m a r e (B. 1 0 ) r , = O ^ = 0 The c o o r d i n a t e s o f r' i n t h i s c o o r d i n a t e s y s t e m a r e hence (B. 1 1 ) - ' ^ 3 " (B. 1 2 ) F i g u r e (B.3) - The body f i x e d c o o r d i n a t e system 51 In p a r t i c u l a r suppose that the axes of the r o t a t e d system are the p r i n c i p a l axes, the x a x i s being a s s o c i a t e d with the l a r g e r p r i n c i p a l moment of i n e r t i a . This new system then c o i n c i d e s with the body-fixed system described i n appendix A.2. In t h i s coordinate system the products of i n e r t i a must vanish: ~ O (B.13) Now 1^ and are i d e n t i c a l y zero since z\ - 0, but 1 ^ - 0 i s an equation f o r ^ : So l v i n g the equation S w l ' C A 3 ^ ^ ' ^ ^ - O (B.15) 52 (B. 16) where we have d e f i n e d From e q u a t i o n s (B.15) and (B.16) one can show t h a t C o ^ - v /\ (B. 18) I f the s i g n o f s i n (2?) i s ( + ) • then th*? s i g n of cos (2"?) must be (-) , w h i l e i f the s i g n of s i n ( 2 t ) i s (-) then t h e s i g n o f cos (2£) must be ( + ) . To determine t h e s i g n s of s i n (2:') t h e moments of i n e r t i a I,,*, and 1 ^ I*x > 1 ^ . Now i cos(2£) we c a l c u l a t e impose the c o n d i t i o n I. 3 - U * ' ^ ) y * ( t > -M : ; ^ t t - * n * 4 ^ - 4 ^ \ 8 r 6 o ( 3 . 19) 53 By a s i m i l a r c a l c u l a t i o n we f i n d C o s ^ (B. 20) From e q u a t i o n s (B.19) and (B.20) we see t h a t I X K ^ I ^ i m p l i e s t h a t ^ c o s ^ V L c o 5 IL(7/ + ^ < o (B. 21) Now - \ •V 7 - \ V (B. 22) Hence i n o r d e r t h a t e q u a t i o n (B.21) be s a t i s f i e d the above s i g n i n e q u a t i o n s (B.16) and (B.18) must h o l d , t h a t i s 3 r (B. 23) 1. and 54 - O r * ' b ^ c . o s -1.7)) Cos. - / ; ^ ( B . 2 4 ) Consider now t h e f o l l o w i n g four v a r i a b l e s as d e f i n e d by-D e r r i c k (1 960 b): F 7 1- v i~ (B. 25) (B. 26) ( B . 2 7 ) (B. 2 8) 55 With t h e s e v a r i a b l e s we may w r i t e *d^rv\ — —:—~ f\^x ( B , 2 9 ) and v (B.30) Note t h a t D e r r i c k ' s d e f i n i t i o n of c> d i f f e r s somewhat from o u r s . In e q u a t i o n (2.27) we i n t reduced "YATl^ £%) • From the above i t f o l l o w s t h a t (B. 31) so t h a t {Id = - ^ ^ i /I i o (3-32) (B.33) 56 I t i s a l s o u s e f u l to d e f i n e (B.34) (B. 35) (B. 36) 57 APPENDIX C The Rep_resentation Coefficients_D^^§(_cp\^mj, D e r r i c k ' s and B l a t t ' s E u l e r a n g l e s (c\,^ > , f ) a c t u a l l y are the a n g l e s (f, 9 , <p ) of G o l d s t e i n ' s (1950) E u l e r a n g l e c o n v e n t i o n . We wish to show here t h a t the r e p r e s e n t a t i o n c o e f f i c i e n t s D ( ^ 8, ^ P^m L g i v e n by D e r r i c k and e q u a t i o n (A.4.1) o f t h i s work a r e e q u a l to those g i v e n by Wigner (1959) , who uses a d i f f e r e n t E u l e r a n g l e c o n v e n t i o n , t h a t i s . (C. 1) where $ , V^0/u ' A \_ ^ s t n e r e p r e s e n t a t i o n c o e f f i c i e n t g i v e n by Wigner. Wigner w r i t e s r' = Rr, where c o s * s U * o \ / C O S ^ O - 5 ^ n l A / c o S ^ S v ^ X 6 \ O - S v A ^ C O S ^ O ° t / \ s U $ O C o ^ / \ O O \J - ( - S l * i * CoS>§ Co** - COS« S W Y - S^NCX C o ^ Y * COS* Co% * S U * S U $ (C.2) a n d wiiere 5 8 O s < c ( £ 'Or O < ^ TT t O <C Y << cT\ ' ( C 3 ) G o l d s t e i n w r i t e s r ' = Ar, where cos iV S W A » V o\/ * ° O \ A = (-sUT C o S^V 0 ( o co<>£ COS (p O O G \/\o -SUB COS9/V O O \ co$^ cov^-sWf c o s e s c o s ' ^ v sU^+sU,H'cose» COS^ > S V / V ^ S W A S (C.U) and where 0 ^ ( r n f O <. 9 < -rr 0 ^ s< i TT (C .5) Now c l e a r l y R and A are e q u a l i f t h e i r r e s p e c t i v e matrix-e l e m e n t s a r e e q u a l . T h i s g i v e s us e q u a t i o n s to determine r e l a t i o n s between («, ^  rTS ) and (%&,<Q). In p a r t i c u l a r L o b w (C. 6) 59 A l s o Combining e q u a t i o n s (C.7) and (C.B) y i e l d s (C. 8) S^ols'^^V + CoScLCoS^' - COS V) = 0 (C.9) so ^ = V + t r y ^ o r a, = + 3 t y z . However = + OT/^does not s a t i s f y e q u a t i o n s (C.7) and (C. 8) so- we must have C\ •= + . Fu r t h e r m o r e (C. 1 0 ) b ^ ^ C o ^ , - 9 - > C O S Y = S^x cp (C.11) Combining e q u a t i o n s ( C . 1 0 ) and ( C . 1 1 ) y i e l d s 60 SOwX cosx c o s - C D S ( $ - Vs) = O (C.12) so <$> = V + u/-c or Q = ^ + l T/i.. However = X ^ / ^ d o e s not s a t i s f y e q u a t i o n s (C.10) and (C.11) so we must have <P = ^  +"^A,. F i n a l l y then we have the r e l a t i o n s (C. 13) (C. 14) ( C 15) That i s , Wigner's (d., ^  ) and G o l d s t e i n ' s ( f / v ^ , * ? ) r e p r e s e n t t h e same r o t a t i o n i f e q u a t i o n s (C.13) / (C.14), and (C.15) h o l d . Wigner w r i t e s X e v * c « u + w ^ - i v i$ - s ' ^ ^ - ^ i e . e 1 - ^ ( C . 16) 61 With the a i d of e q u a t i o n s (C.13), (C. 14) , and (C.15) we have e - e - e ^ ( c m and ^ C ~ ~ V ^ (C.18) Now (C. 19) Combining e q u a t i o n s (C.16), (C.17) , (C.13), and (C.19) we have ~ u v • ^ r - ^ u \Xw>-v.(v--^v.(L^WV.d-^y] -(C. 20) as s t a t e d . 

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