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On the symmetric S- and D- state components of the triton wave function Best, Melvyn Edward 1966

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THE SYMMETRIC S- AND D-STATE COMPONENTS OF THE TRITON WAVE FUNCTION by MELVYN EDWARD BEST B.S c , U n i v e r s i t y of B r i t i s h Columbia, 19&5 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 August, 1966 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the Li b r a r y s h a l l make i t f r e e l y avai]able f o r reference and study. I furth e r agree that permission, f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by his representatives. I t i s understood that copying or p u b l i c a t i o n of" t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of___,__PjffS_ICS___ The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8 , Canada Date AUGUST 30, 1 9 6 6 . i i ABSTRACT Approximate forms f o r the symmetric S- and D- state components of the t r i t o n wave functions are found using the equivalent two - body approximation of Feshbach and Rubinow. Two coupled, ordinary d i f f e r e n t i a l equations f o r the components are obtained and, f o r comparison with previous work, are solved numerically with the Feshbach"- Pease two nucleon p o t e n t i a l s . A f u r t h e r approximation i n v o l v i n g one v a r i a t i o n a l parameter i s shown to y i e l d good r e s u l t s . D e t a i l e d expressions f o r the symmetric S- and Instate contributions to the charge form f a c t o r of the t r i t o n are found and the symmetric S-state contribution i s compared to the r e s u l t s of S c h i f f . i i i TABLE OF CONTENTS Page CHAPTER 1 INTRODUCTION 1 CHAPTER 2 THE DERRICK-BLATT CLASSIFICATION OF THE TRITON WAVE 4 FUNCTIONS CHAPTER 3 INTERNAL WAVE FUNCTIONS 9 3-1 THE VARIATIONAL PRINCIPLE 9 3-2 THE COUPLED FESHBACH-RUBINOW EQUATIONS 11 3.3 THE MODIFIED FESHBACH-RUBINOW EQUATIONS 15 CHAPTER If NUMERICAL SOLUTIONS TO THE INTERNAL WAVE EQUATIONS 17 4.1 THE MODIFIED FESHBACH-RUBINOW EQUATION 17 4.2 THE COUPLED FESHBACH-RUBINOW EQUATION 22 4.3 NUMERICAL RESULTS 24 CHAPTER 5 THE CHARGE FORM FACTOR OF THE TRITON 34 5.1 INTRODUCTION 34 5.2 DEFINITION OF THE CHARGE FORM FACTOR 35 5.3 THE S-STATE CONTRIBUTION TO THE FORM FACTOR 38 5.4 THE D-STATE CONTRIBUTION TO THE FORM FACTOR 44 5-5 NUMERICAL RESULTS 47 CHAPTER 6 CONCLUSIONS 52 BIBLIOGRAPHY 54 i v Page APPENDIX A THE SYMMETRIC GROUP OF ORDER THREE 5 5 A . l IRREDUCIBLE REPRESENTATIONS OF S ( 3 ) 5 5 A.2 PERMUTATION PROPERTIES OF FUNCTIONS UNDER S ( 3 ) 5 6 A. 3 PERMUTATION ADDITION COEFFICIENTS 5 7 APPENDIX B '. SPIN-ISOSPIN. WAVE FUNCTIONS 6 l B. l SPIN FUNCTIONS 6 l B.2 ISOSPIN FUNCTIONS 6 3 B. 3 SPTJJ-ISOSPIN FUNCTIONS 6 3 APPENDIX C • EULER ANGLE WAVE FUNCTIONS 6 5 C. l THE CO-ORDINATE SYSTEM 6 5 C.2 IRR, REPRESENTATIONS OF THE PURE ROTATION GROUP 6 6 C-3 PERMUTATION PROPERTIES OF D^^C^, 6 6 C.k EULER ANGLE WAVE FUNCTIONS 6 9 APPENDIX D TOTAL ANGULAR MOMENTUM - ISOSPIp WAVE FUNCTIONS 72 APPENDIX E EVALUATION OF ISOSPIN MATRIX ELEMENTS 7k APPENDIX F INTERNAL WAVE FUNCTION BOUNDARY CONDITIONS 7 5 F . l THE TRITON POTENTIALS 7 5 F.2 THE MODIFIED FESHBACH-RUBINOW EQUATION 7 6 F.3 THE COUPLED FESHBACH-RUBINOW EQUATION 7 6 APPENDIX G . COMPARISON OF EQUATION (5-2lTb)TO THE SCHIFF ( I 9 6 U ) , : 7 8 .'• .. FORM FACTOR LIST OF TABLES Page TABLE 1 ' PERMUTATION PROPERTIES OF THE WAVE FUNCTIONS 6 TABLE 2 PARAMETERS FOR THE FESHBACH - PEASE (1952) POTENTIALS 19 TABLE 3 NUMERICAL RESULTS FOR CHAPTER k 27 TABLE k OVERLAP INTEGRALS 28 TABLE 5 F*(q a) VERSES q 51 TABLE A - l IRREDUCIBLE MATRICES OF S ( 3) 55 TABLE A-2 DIRECT PRODUCT DECOMPOSITION OF S ( 3) 58 TjJBLE A-3 NON-ZERO PERMUTATION ADDITION COEFFICIENTS 59 TABLE k-k DIRECT PRODUCT FUNCTIONS 60 TABLE B - l SPIN FUNCTIONS 61 TABLE B-2 SPIN-ISOSPIN FUNCTIONS 6k TABLE E - l ISOSPIN MATRIX ELEMENTS 7h V I LIST OF FIGURES FIGURE 1 FIGURE 2 FIGURE 3 FIGURE 4 FIGURE 5 FIGURE 6 FIGURE 7 FIGURE 8 FIGURE 9 FIGURE 10 FIGURE 11 FIGURE 12 FIGURE 13 FIGURE 14 FIGURE 15 FIGURE 16 THE POTENTIALS Y,(A-), Y 7 ( / L ) , Y ( /L) FOR FP No.l . POTENTIAL THE ENERGY E VERSES 0( FOR FP No.l POTENTIAL U 0 and U VERSES H- FOR FP No.l POTENTIAL (WM) AND (WCC) VERSES £ FOR FP No. 1 POTENTIAL EIGENFUNCTIONS OF (4-3) FOR FP No.l POTENTIAL EIGENFUNCTIONS OF (4-3) FOR FP No.2 POTENTIAL EIGENFUNCTIONS OF (4-3) FOR FP g = 0 POTENTIAL EIGENFUNCTIONS OF (4-3) FOR FP No.3 POTENTIAL U 0 AND BEST TRIAL EXP. FUNCTION VERSES ft. FOR FP No.l Page 18 21 21 26 30 30 31 31 W AND W VERSES^ FOR FP No.l; POTENTIAL POTENTIAL 32 33 POSITION VECTORS OF THE 3 PARTICLES 36 THE VECTORS-G-, j ZJr^ ,J1^ AND Q kO " I a. F 1 ( q ) VS. q FOR U 0 AND EXP. FUNCTION FOR FP No.l _ _ POTENTIAL 49 F j ( q ) VS. q FOR U^ AND EXP. FUNCTION FOR FP No.2 POTENTIAL 49 F f ( q 2 ) VS. q FOR U 0 AND EXP* FUNCTION FOR FP g * 0 POTENTIAL 50 F * ( q 3 ) VS. q FOR U0AND EXP. FUNCTION FOR FP No. 3 ' POTENTIAL 50 i v i i ACKNOWLEDGEMENTS I would l i k e to thank Dr. J . M. McMillan f o r suggesting t h i s problem and f o r h i s generous assistance with i t . This t h e s i s was done while the author was supported by a bursary from the National Research Council of Canada. I would l i k e t o thank A l v i n Fowler of the U.B.C Computing Center and Raymond Viekson f o r help during the i n i t i a l stages of the numerical work. - 1 -CHAPTER 1 INTRODUCTION The t r i t o n i s the bound state of two neutrons and one protonj the experimental value of i t s energy i s -8.1+92 Mev. The ce n t r a l t h e o r e t i c a l problem concerning t h i s system, of course, i s to f i n d the t r i t o n wave fu n c t i o n and to compute i t s energy. We are immediately faced then with t r y i n g t o solve a three - body problem. Major steps i n t h i s d i r e c t i o n were taken by Derr i c k and B l a t t (1959, 1960a, 1960b) who have constructed a complete set of states i n terms of which the t r i t o n wave f u n c t i o n may be expanded, and derived a set of sixteen p a r t i a l d i f f e r e n t i a l Sck^bdinger equations i n three v a r i a b l e s f o r the expansion c o e f f i c i e n t s . The work of Derrick and B l a t t , while providing a great s i m p l i f i c a t i o n of the nuclear problem, s t i l l c l e a r l y leaves one with an extremely d i f f i c u l t mathematical problem. Indeed, r e l a t i v e l y l i t t l e has been done i n f i n d i n g i t s s o l u t i o n . Instead the common procedure here has been to compute t r i t o n wave functions and energies using a v a r i a t i o n a l parameter approach. That i s , a form of the wave f u n c t i o n containing various parameters has been assumed, and the expectation of the Hamiltonian of the system has been computed and minimized with respect to these parameters. This approach has obvious disadvantages: there i s no guarantee that the assumed fu n c t i o n approximates the a c t u a l wave f u n c t i o n , and there i s no d i r e c t correspondence between the wave fun c t i o n parameters and the parameters appearing i n the nucleon - nucleon p o t e n t i a l . Some work on a non-va r i a t i o n a l approach to f i n d i n g a t r i t o n wave func t i o n and energy has been done, i n p a r t i c u l a r , by Feshbach and Rubinow (1955) and by McMillan (1965). Indeed, t h i s t h e s i s i s an extension of the of the work done by McMillan (1965) on the symmetric S-state component of the t r i t o n wave fu n c t i o n . The Feshbach and Rubinow (1955) approach i s employed i n these c a l c u l a t i o n s . We note that, as has been stressed, f o r example, by McMillan (1965), t h i s has the advantage over the v a r i a t i o n a l parameter approach of allowing the c a l c u l a t i o n of the f u n c t i o n a l form of the approximate wave function. I f one were to continue the Feshbach - Rubinow procedure, one would end up with a set of coupled, second order, ordinary d i f f e r e n t i a l equations. However, the large number of components, and hence equations, makes the so l u t i o n d i f f i c u l t , so f u r t h e r approximations must be made concerning the r e l a t i v e importance of the t r i t o n components. For example, we could omit some of the components, or modify the Feshbach - Rubinow procedure to allow the introduction of some v a r i a t i o n a l parameters. B l a t t e t a l (1962) have shown that the symmetric S-state i s the dominant component of the t r i t o n wave f u n c t i o n followed by the D-state. Using the D e r r i c k and B l a t t (1958) k i n e t i c energy argument, one may expect that the symmetric D-state component be the next important, followed by the mixed symmetric components. (We adhere here to the notation of De r r i c k (1960b) f o r the D-states.) As pointed out by McMillan (1966b), the Feshbach -Pease (1952) r e s u l t s also i n d i c a t e t h i s ordering. We s h a l l r e t a i n here only the symmetric S- and D- state components of the t r i t o n wave funct i o n . Throughout t h i s t h e s i s , the De r r i c k ->.; B l a t t expansion of the t r i t o n wave functio n , rather than1' the Sachs (1953) expansion, i s used. The reasons - 3 -are twofold 1) the Sachs (1953) expansion i s incomplete, as has been pointed out by McMillan (1966a); and 2) the Sachs bas i s functions are not orthonormal, whereas the Derrick - B l a t t functions are. Chapter 2 i s a resume of some aspects of the Derr i c k and B l a t t (1958) and D e r r i c k (1960a, 1960b) c l a s s i f i c a t i o n of the components of the t r i t o n wave fu n c t i o n . Chapter 3 gives a d e r i v a t i o n of the coupled, ordinary d i f f e r e n t i a l equations which r e s u l t |rhen the Feshbach - Rubinow approximation i s used. Chapter 3 also gives an ordinary d i f f e r e n t i a l equation which r e s u l t s when an a d d i t i o n a l approximation, r e l a t i n g the symmetric S- and D- state components of the t r i t o n wave fun c t i o n through a single v a r i a t i o n a l parameter, i s introduced. Chapter if contains the numerical methods of sol v i n g the above two equations and also the numerical r e s u l t s . Chapter 5 gives d e t a i l e d expressions f o r the charge form f a c t o r of the t r i t o n when the symmetric S- and D- states and the mixed S- state components of the t r i t o n wave fun c t i o n are retained. The symmetric S-state cont r i b u t i o n to the form f a c t o r i s evaluated numerically to provide a check on the approximate wave functions developed i n chapter 3 and k. Chapter 6 gives the conclusions drawn from the c a l c u l a t i o n s , while appendices A to D give d e t a i l e d expansions of the D e r r i c k - B l a t t wave functions. - k -CHAPTER 2 THE DERRICK - BLATT CLASSIFICATION OF TRITON WAVE FUNCTIONS The t r i t o n ( H S ) consists of a "bound state of two neutrons and one proton. Experimentally, the t r i t o n has t o t a l angular momentum J = 2 and even p a r i t y . The t h i r d component of the i s o s p i n i s M r - and i t i s assumed to have the good i s o s p i n quantum number T = \-3 A complete expansion of the ground state wave fun c t i o n of H has been given by D e r r i c k and B l a t t (1958) and D e r r i c k (1960a). (We s h a l l not give a complete account of t h i s work here, but s h a l l r e f e r the reader to the o r i g i n a l papers f o r a l l d e t a i l s . ) Their construction proceeds i n a manner analogous to the B l a t t and Weisskopf (1952) construction of the Sj and D| spin - angle functions which are relevant to the deuteron wave funct i o n . Since the t r i t o n has t o t a l momentum J = \, and since the t o t a l spin angular momentum of three nucleons may be \ or J- , the expansion contains 2S., ZP, , V P ^ and ^D,- states. The wave f u n c t i o n of H consists of a l i n e a r combination of functions which depend on spin, i s o s p i n , and s p a t i a l cc-ordinates . These functions consist of one part, c a l l e d t o t a l angular momentum - i s o s p i n functions, which depends on spin, i s o s p i n , and on s p a t i a l co-ordinates through the Euler angles which s p e c i f y the 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 the three pa r t i c l e s ^ " and on one part, c a l l e d i n t e r n a l wave fu n c t i o n s , which depends on the three i n t e r p a r t i c l e distances. Further, See appendix C f o r the Euler angle wave functions. the symmetric group of order 3 has three i r r e d u c i b l e representations, and thus the expansion contains functions which transform according to the symmetric, antisymmetric, or mixed representation of So) under a permutation of the three p a r t i c l e s . These functions are given e x p l i c i t l y i n appendices B to D and D e r r i c k and B l a t t (1958). Table 1 gives the permutation properties of these functions. These functions are l a b e l l e d using the Derrick (1960a) notation. The notation of table 1 i s tha,t of Derr i c k and B l a t t (1958) and Derri c k (1960a, 1960b) with the addition that (m,k) r e f e r s to a function belonging to the K~~- row of the mixed representation, -(m 1) means that ) a n d ~3f/e,2 3 3 : 6 ^ h e relevant partner functions under j o i n t spin-i s o s p i n permutations. See appendix A f o r the properties of the symmetric group of order 3. - 6 -TABLE 1 PERMUTATION PROPERTIES OF THE WAVE FUNCTIONS Spectroscopic C l a s s i f i c a t i o n T o t a l Angular Momentum - Isospin Functions Permutation Symmetry Inter n a l Functions Euler Angles Spin-Isospin Euler Angles Spin-Isospin 2 s , \ s a a s 2. s s s a s m,l m,l m,2 s mi 2 mj2 Xtt.j 1 \ a s a s 2 Us- a a s a \> a m,2 m,l m, 2 a -(m.l) m,2 m i l J7. I a m(2_ m, 1 m, 2 I " '( l a -(m,l) m, 2 m, 1 \, . s m/1 m,l m, 2 s my 2 m.,2 m;l D , s m y l m/1 m/2 z s m,2 m,2 m/1 •1,1 a m,2 m, 1 m,2 a 1) nj, 2 m/1 The t o t a l wave f u n c t i o n of the t r i t o n must be antisymmetric. Thus the i n t e r n a l wave functions which D e r r i c k and B l a t t (1958) define must transform according to the l a s t column of table l.: under permutations of the three p a r t i c l e s . The t o t a l wave fun c t i o n ( y ) i s given i n Derrick and B l a t t (1958) and Derr i c k (1960a) and, f o r convenience, i s w r i t t e n here: D e r r i c k (1960a, 1960b) has f u r t h e r defined a new set of momentum-isospin 1 wave functions f o r the D-states as f o l l o w s : ( a - , c ) -y/3 , - G (i \ , - CSQ ^ ,-c.tlfa,) + f(i\7 - csci\x - con %*) where 2 /? = 2 „ * , * 0 -1 i ^ 3 - A>i 1 D e r r i c k (1960b) l a b e l s these J?/( to - 8 -and ( ; ^11,1 )> ( i ^ a r e P6^1"3 o f functions which transform according to the mixed representation of S/3) (see appendix A), transforms according to the antisymmetric representation of S/^j , and Tj/3 transforms according to the symmetric representation of S (2) • I t i s convenient f o r l a t e r work to record here the fol l o w i n g quantities defined by Derri c k (1960a, 1960b): A - A / V - i f S A a Using the new Euler angle-spih-isospin D-states and the d e f i n i t i o n f o r the t o t a l D-state wave function iin Derrick (1960b) one can rewrite the i n t e r n a l D-state wave functions ( f a ; j j .... f/i>n ) i n terms of the new i n t e r n a l D-state wave functions 1 / ••••• ) as (**«) ^ = -2K [fa^Jv^ + Ccsc 3-fu + F^A-fia - 1 - 9 -CHAPTER 3 INTERNAL WAVE EQUATIONS' 3-1 The v a r i a t i o n a l p r i n c i p l e D e r r i c k (1960a, 1960b) has given the set'of coupled p a r t i a l d i f f e r e n t i a l equations f o r the i n t e r n a l wave functions i n the Derr i c k -B l a t t (1958, 1960a., 1960b) expansion of the J = T = \ three nucleon wave fu n c t i o n . In the Feshbach - Rubinow approach, however, the d y n a m i c a l ~ ~ ~ statement i s a v a r i a t i o n a l p r i n c i p l e from which d i f f e r e n t i a l equations are obtained using the Eu l e r - Lagrange equations. In the problem, at hand where, as explained i n the introduction, only the symmetric S- and D- state components ( i . e . , components 1 and 13 i n the D e r r i c k (1960a, 1960b) notation) of the t r i t o n are retained, the relevant v a r i a t i o n a l p r i n c i p l e i s 1 This chapter i s e s s e n t i a l l y sections 2 and 3 of Best and McMillan (1966) and i s based on work done by McMillan. O where ^ That t h i s i s a su i t a b l e v a r i a t i o n a l p r i n c i p l e may be checked by noting that the corresponding Euler - Lagrange equations are the dynamical equations given by Derr i c k (1960a, 1960b). - 10 -and where throughout the D e r r i c k (l960a> 1960b) notation has "been adhered to and the above functions are defined i n equation ( 2 - 4 ) . Ln addition the two-nucleon p o t e n t i a l has been written as f o l l o w s 1 s • d where V (^12) i s the s i n g l e t spin p o t e n t i a l , V (^12) i s the c e n t r a l t r i p l e t p o t e n t i a l , and V (fl-u) i s the tensor p o t e n t i a l with S u the two p a r t i c l e tensor operator: • (3-»k) = .s(£r£n)t-£z-£>*) - &r&2 Jl.2 The f u n c t i o n s 2 ^ and f / 3 are both completely symmetric functions of the three i n t e r p a r t i c l e distances and are normalized according to (IS). Jit ft* = I 1 The Feshbach - Paase (1952) p o t e n t i a l s are of t h i s form. 2 Note here that i t follows from the work of McMillan (1966a) that the D e r r i c k functions f / and f ( 3 are p r o p o r t i o n a l to the Sachs (1953) functions f1 and f 7 r e s p e c t i v e l y . - 1 1 -3»2 The coupled Feshbach - Iftibinow equations We now come to the c e n t r a l approximation! we assume that f ( and f | 3 depend only on the sing l e symmetric variable- 1-(3-6). 1= i(Hn + n.l3 + i . e . , we assume that fs-7«.) -P, = -T t U-U) -T,^  = -f/3 CO Feshbach and jlubinow (1955) use approximation (3-?a)j i t may be regarded as a g e n e r a l i z a t i o n of the t r i a l exponential f u n c t i o n used by B l a t t and Weissfeppf (1952). Approximation (3-7b) may, i n the same way, be regarded as a g e n e r a l i z a t i o n of the dominant p a r t of the symmetric D-state component of the Feshbach - Pease (1952) t r i a l f u n c t i o n 2 When approximation (3*-7) i s made, some of the i n t e g r a l s i n (3-1) may be e a s i l y evaluated to y i e l d 1 Feshbach and Rubinow (1955) and McMillan (1965) denote t h i s v a r i a b l e by R. We r e f r a i n from doing so.here, i n order.to avoid confusion with the De r r i c k (1960a, 1960b) R. 2 One may consult McMillan (1966b) to f i n d the Feshbach - Pease functions w r i t t e n i n the D e r r i c k - B l a t t notation. - 12 where we have defined u and w b y 1 (»-,b) f„<,i) = -L so that the normalization i n t e g r a l (3-5) becomes Or: o where the p o t e n t i a l terms are ( 3 - U ) 1 / V ) = • 2 4 j ' c / Z z V l - 2 +jL}[-L[\/\n2)-i-VCt(ri2)]J 0 ^ i / V ) -J \ J 5 A/ 5 5- J N 6 t. The Euler - Lagrange equations which f o l l o w from the v a r i a t i o n a l p r i n c i p l e (3-8) are -^ McMillan (1965) has shown also that the Feshbach - Rubinow approximation (3-9a) i s more tenable than the Morpurgo (1952) approximation - 13 -Thus, assumption (3-7) has allowed us to reduce the problem of f i n d i n g the symmetric S- and B- state components of the t r i t o n wave func t i on to the so l v ing of two coupled, ord inary d i f f e r e n t i a l equat ions. Equations (3-15) are c a l l e d " the coupled Feshbach - Rubinow equat ions" . I t i s worth s t r e s s ing that the employment of v a r i a t i o n a l parameters has not been resor ted t o , ra ther the t r i t o n energy appears as the eigenvalue of the coupled Feshbach -Rubiaow equat ions. The s i m i l a r i t y of the coupled Feshbach - Rubinow equations to the well-known p a i r of equations f o r the deuteron 3s]_ and 3 % components should be noted. In p a r t i c u l a r , one sees that the t r i t o n Sj. and Dj. z z components are a l so coupled by the tensor fo r ce (d|,of course, i s we l l-known), and that the p o t e n t i a l term i n the D-state equation (3-15b) involves the d i f f e r ence of c e n t r a l - t r i p l e t and tensor p o t e n t i a l s . On the other hand, the p o t e n t i a l terms i n the S-state equation (3-15a) invo lves a s i ng l e t sp in p o t e n t i a l (which, of course, does not appear i n the corresponding deuteron problem), 'and a l so the " c e n t r i f u g a l b a r r i e r terms" involve [\ JR^) (Y^^^O f o r JL*o, -x r a ther than Jl(f+0-We c lose t h i s sec t ion by g i v i ng expressions f o r the D-state p r o b a b i l i t y ( P D ), and f o r the coulomb energy (E C O t //.) of He . I t fo l lows from normal iza t ion i n t e g r a l (3-10) that (3-/6) PD = j ^ ( n j d t i 0 - Ik -and from De r r i c k (1960a, 1960b) t h a t 1 J L. i/i ' tin J which when approximation (3-7) and d e f i n i t i o n s (3-9) are used becomes 00 1 1 N 5 and N3 are de f ined i n equation(2-4i ) . On comparing (3-17) and (3-5) one may th ink a term i nvo l v i ng N3 should appear i n both equat ions. We note however, that - 1 5 -3 « 3 The modified Feshbach - Rubinow equation The eigenvalue of the coupled Feshbach - Bubinow equations ( 3 - 1 5 ) w i l l , because the equations f o l l o w from v a r i a t i o n p r i n c i p l e ( 3 - 8 ) without approximation, be b e t t e r ( i . e . , lower) than a t r i t o n energy computed i n any other way with f, and f ^ of the form given by equations ( 3 - 7 ) . Nevertheless, there i s some value i n considering f u r t h e r approximations at t h i s point. Indeed, i f one were t o take i n t o account the remaining components of the t r i t o n wave fun c t i o n and f u r t h e r employ the Feshbach - Rubinow single symmetric^variable approximation, one would, by f o l l o w i n g the procedure of the preceeding section, a r r i v e at a set of coupled, ordinary d i f f e r e n t i a l equations f o r the components, the number of equations depending on the number of components retained. The s o l u t i o n of a large number of coupled eigenvalue equations would be d i f f i c u l t . We how give a modification of the procedure used i n the preceeding section which lends i t s e l f to a considerable s i m p l i f i c a t i o n , and which y i e l d s good r e s u l t s as we show i n the next section. We assume that the functions u and w defined by equations ( 3 - 9 ) are r e l a t e d as follows (3-18) u r ( l ) = * where e>\ i s a parameter. When approximation ( 3 - 1 8 ) i s used, v a r i a t i o n a l 1 In t h i s connection, we note that a l l of the Derrick - B l a t t i n t e r n a l functions can be expressed i n terms of symmetric functions, as has been pointed out by McMillan ( 1 9 6 6 a ) - 16 -p r i n c i p l e (3-8) contains a si n g l e function to be determined, and the Euler -Lagrange equation reads (3-/9) - i L * / W ) ^ - i l l i ^ ^ f y j / z ) + ; ^ Thus, assumption (3-18) has allowed us to reduce the problem of f i n d i n g the symmetric S- and D- state components to the s o l v i n g of a single ordinary d i f f e r e n t i a l equation. Equation (3-19) i s c a l l e d the "modified Feshbach - Rubinow equation"^ I t w i l l be noted that the eigenfunction u and the eigenvalue E depend upon the parameter*, and that the best t r i t o n energy provided with approximation (3- l8) i s the minimum E with respect to o( . The normalization i n t e g r a l , D-state p r o b a b i l i t y , and the coulomb energy of He i n t h i s case f o l l o w immediately from equations (3-10) (3-16) and (3-17) and approximation (3-18). One has 1 S e t t i n g *r<2 i n t h i s equation y i e l d s the Feshbach - Rubinow (1955) equation ( a l s o see McMillan ( l 9 6 5 ) ) . „ . 17 -CHAPTER 4 NUMERICAL SOLUTIONS TO THE INTERNAL WAVE EQUATIONS. 4.1 The modified Feshbach - Rubinow equation (3-19) For the purpose of the numerical i n t e g r a t i o n , equation (3-19) has been rewritten i n the fo l l o w i n g form: where use has been made of equations ( f - l ) U.= 0 For a l l the numerical c a l c u l a t i o n s j the p o t e n t i a l s appearing i n (3-4) are assumed to be of the Yukawa type: A11 4/1 A e 7 1 The parameters defined i n equations (4-2) w i l l be those used by Feshbach - Pease (1952) and are l i s t e d i n table 2. The corresponding functions Y?(n) , , Yc(fl) f o r the FP No. 1 p o t e n t i a l are, f o r i l l u s t r a t i v e purposes, shown i n f i g u r e 1. . - 1 9 -TABLE 2 PARAMETERS FOR THE FESHBACH - PEASE ( 1 9 5 2 ) POTENTIALS P o t e n t i a l (mev) (mev) Vote (mev) A ( ? " ' ) A ( F " ' . ) A e (*"" ') FP No. I -55-0323 -65.I839 -26.3907 .84459 .84459 .47125 FP Ho'. 2 -55.0323 -59.3456 -34.7892 .84459 .84459 •54735 FP g= o -55-0323 -55.0323 -43.2731 .84459 .84459 .59067 FP No. 3 -55.0323 -48.4469 -49.5169 .84459 .84459 .65232 Equation (4-l) i s an-ordinary d i f f e r e n t i a l equation with eigen-value A . In order to determine t h i s eigenvalue, one makes use of the condition that the eigenf unction u(^- ) and u (ft ), i t s d e r i v a t i v e with respect ton. , be continuous wherever the p o t e n t i a l s Y ( / l ) are f i n i t e . As pointed out i n appendix F, the p o t e n t i a l s defined i n ( f - l ) are a l l i n f i n i t e at the o r i g i n . Thus, the i n t e g r a t i o n outwards from the o r i g i n must be sta r t e d at a small f i n i t e value of 1 , which we have taken to be 0.00001F. The e x p l i c i t form of the boundary conditions f o r small /i i s given by equation ( f - 2 ) . S i m i l a r l y , the e x p l i c i t form of the boundary conditions f o r large fl i s given by equation (f - 4 ) ; these were applied at n = 30F. For a f i x e d value of the parameter * , the procedure used to f i n d the eigenvalue W and the eigenf unction u(1 ) i s : 1. Applying the boundary conditions ( f - 2 ) at/z = oioOOOOJF, equation (4-1) i s integrated outwards using the Runge - Kutta method t o n = 3F. (the matching point) and the logarithmic d e r i v a t i v e of u(A ) at the matching point i s found. - 20 -2. Applying the boundary conditions (f-4) at/7 = 30F f o r some a r b i t r a r y non-zero value of/) , equation (4-l) i s integrated inwards using the Runge -Kutta method t o tl =3F and the logarithmic d e r i v a t i v e of u(/L ) at the matching point i s found. 3 . One now computes the d i f f e r e n c e i n the two logarithmic d e r i v a t i v e s c a l c u l a t e d above; the two steps are then repeated f o r d i f f e r e n t values of 3 u n t i l the d i f f e r e n c e i n the logarithmic d e r i v a t i v e s i s a r b i t r a r i l y small (we chose t h i s to be l e s s than 10 j . The value of A thereby obtained i s the eigenvalue of (4-l) f o r that f i x e d value of °< chosen. 4. The eigenf unction u(fl ) i s made continuous by m u l t i p l y i n g a l l the values of u(/l ) from 3F. to 3 O F . by the r a t i o of u(3) c a l c u l a t e d from the i n t e g r a t i o n outwards t o u(3) c a l c u l a t e d from the i n t e g r a t i o n inwards. The c o n t i n u i t y of u(ft- ) then follows from the co n t i n u i t y of the logarithmic d e r i v a t i v e . F i n a l l y , one normalizes the eigenf unction u(/Z.) according to ( 3 - 2 0 ) . The above procedure i s repeated f o r a range of values of the parameters . One then p l o t s the energy eigenvalue E verses the parameter^-The minimum value of E verses <* i s the t r i t o n binding energy. Figure 2 shows a t y p i c a l graph of E verses <* f o r the p o t e n t i a l FP No. 1 . To show the dependance of the eigenf unction u(/l ) on the parameter* , figure. . 3 gives a graph of u(/l ) f o r = 0 and u ( l ) f o r the value of which gives a minimum 1 A l l numerical c a l c u l a t i o n s were performed on the IBM 7040 d i g i t a l computer at the U.B.C. Computing Center. i r~—i r~—i r — r ~ — T 1 1 T A(!) - 22 value of E v e r s e s n f o r the FP No.l p o t e n t i a l . Table 3 summarizes the r e s u l t s of these c a l c u l a t i o n s f o r the 4 Feshbach - Pease (1952) p o t e n t i a l s . This table contains the minimum value of energy, the D-state p r o b a b i l i t y (3-21), and the Coulomb energy of H e 3 (3-22). Also the eigenvalues f o r the case °< = 0 are included. 4.2 The coupled Feshbach - Rubinow equations (3-15) Again f o r the purposes of the numerical i n t e g r a t i o n , equations (3-15) have been rewritten i n the f o l l o w i n g form The p o t e n t i a l s used were again those defined i n equations (4-2) with the parameters l i s t e d i n table 2. Equations (4-3) are a p a i r of coupled, ordinary d i f f e r e n t i a l equations with eigenvalue /\ . To f i n d the eigenvalue A , one makes use of the condition t h a t the eigenfunctions u(/Z ) and w(/l ), and t h e i r d e r i v a t i v e s with respect to ft , u (/l ) and w ( A ), are continuous wherever the p o t e n t i a l s Y(ft) are f i n i t e . As with equation (4-1), equations (4-3) have t h e i r integrations outward s t a r t i n g at a small f i n i t e value of /I , which we have taken at/I = 0.01F. The e x p l i c i t form of the boundary conditions f o r small n are given by equations (f-5). S i m i l a r l y , the e x p l i c i t forms of the boundary conditions f o r l a r g e ^ are given by equations (f-7); these w e r e applied at 30F. One sees from the boundary conditions (f-5) and (f-7) t h a t t h e r e are two unknown parameters, AVAL and f> , besides the eigenvalue /I . O 0 - 23 -The procedure used to f i n d the eigenvalue A i s as f o l l o w s : 1. For f i x e d p and ft and applying boundary conditions (f-5) with an a r b i t r a r y value of AVAL at /I = 0.01F, equations (4-3) are integrated outwards using the Runge - Kutta method to a matching poi n t of tl = 3F. and the logarithmic d e r i v a t i v e s of u(Al ) and w(/l ) at the matching point are found. 2. For the same f i x e d values of p and % and applying boundary conditions (f-7) at fl = 30F, equations (4-3) are integrated inwards using the Runge -Kutta method to the matching point and the logarithmic d e r i v a t i v e s of u( tl ) and w(/l ) at the matching point are found. 3. Again with the same p and ^ , one now computes the d i f f e r e n c e i n the logarithmic d e r i v a t i v e s of u(H ) and w(/7 ) c a l c u l a t e d above. Applying the c o n t i n u i t y condition f o r u(/L ) and u (/J ) at the matching point, the above two steps are repeated, with the same P and A , f o r d i f f e r e n t values of AVAL u n t i l the d i f f e r e n c e i n the logarithmic d e r i v a t i v e s of u(/l ) ( UM) i s - 3 . a r b i t r a r i l y small (we chose t h i s to be l e s s than 10 ;. One now makes u(tl ) continuous, as i n step 4 of section 4.1, by m u l t i p l y i n g a l l the values of u(/l ) from tl = 3F. to tl = 30F. by the r a t i o of u( 3) calculated from the i n t e g r a t i o n outwards t o u(3) c a l c u l a t e d from the integrations inwards. As before, the c o n t i n u i t y of u {tl ) then follows. At the same time \r{tl) and w (tl ) from tl = 3F. to J~i = 30F. are also m u l t i p l i e d by the above r a t i o . Ln general, when t h i s i s done the d i f f e r e n c e i n the logarithmic d e r i v a t i v e s of w(/L ) (WM) and the d i f f e r e n c e i n the values of w(/Z ) from the inwards and outwards i n t e g r a t i o n ( ^ c c ) w i l l not be a r b i t r a r i l y small. 4. One now v a r i e s P and A , each time repeating the above three steps, u n t i l both w ^ and wcc are a r b i t r a r i l y small. In t h i s ease the smailness -2k-of wm and wee depends on the accuracy of the graphs Used when p l o t t i n g wm -3 and wee verses p f o r f i x e d ^ and |t>M| < 1 0 . For i l l u s t r a t i v e purposes, a t y p i c a l graph of wm and wee verses f} f o r . /UMj C 1 0 and f ixed/| f o r the FP No.l p o t e n t i a l i s given i n f i g u r e if. Two points should be noted from t h i s graph. F i r s t , the accuracy of the eigenvalue depends on the accuracy of the graphical p l o t s and second, there e x i s t s a unique set of parameters AVAL, P, and ^ which makes u(/l ), w(/z. ) and \X{A- ), w(^Z ) continuous. This value o f ' ^ i s the eigenvalue f o r equations ( 4 - 3 ) . 5 . The eigenfunctions u(/l ) and w(/Z ) are next normalized according to equation ( 3 - 1 0 ) . The matching point and step size? of the numerical i n t e g r a t i o n were v a r i e d by changing the matching point to A = 2.5F. to see the change ':\ i n the eigenvalue ^ and the D-state p r o b a b i l i t y ( 3 - 1 6 ) . No s i g n i f i c a n t change i n these quantities or i n the form of the eigenfunctions was found. The f i n a l value of AVAL changed, however, and we conclude that t h i s r e f l e c t s the i n v a l i d i t y of equations ( f - 5 ) * We attach no importance to t h i s f a c t ; indeed, we f e e l now that d e t a i l e d expansions of the functions near the o r i g i n i s unnecessary and that i t i s s u f f i c i e n t to use some parameter i n c o l v i n g the r a t i o of u and w. (The boundary conditions | f - 7 ) were changed by le a v i n g o f f the l a s t terms and again no s i g n i f i c a n t change was noted). F i n a l l y , the eigenvalue E, D-state p r o b a b i l i t y ( 3 - 1 6 ) , and Coulomb.: energy of He ( 3 - 1 7 ) are tabulated i n table 3 . 4 . 3 Numerical r e s u l t s From table 3 , the s i m i l a r i t y between the eigenvalue of the Feshbach - Rubinow equation and the energy c a l c u l a t e d with the best t r i a l - 25 -1 exponential f u n c t i o n used by B l a t t and Weisskopf (1952), and between the eigenvalues of the coupled and modified Feshbach - Rubinow equations has encouraged one to c a l c u l a t e various overlap i n t e g r a l s . More s p e c i f i c a l l y , we define 1,-- t± Uie*%(fi) = / X ^ / i a e «otn-)dn. Ml 0 ^ IS' Q The eigenfunctions u(A ) and w(/l) are the solutions t o (4-l) f o r the best value of <^  } the eigenfunctions u(A ) and w(A ) are the solutions to (4-3)> u 0 i s the s o l u t i o n to (4-l) with# =0. The solutions to (4-1) are normalized according to (3-20) and the solutions t o (4-3) are normalized according to (3-10). The exponential f u n c t i o n i s normalized according to * ? € -2K/1 (i+-D N \n e • • din = l where 2 N = J^r K . The value of K f o r the p o t e n t i a l s defined,by the parameters i n table 2 are found from equations (2.4) and (2.8) of B l a t t and Weisskopf (1952): \% m 1 This s i m i l a r i t y has also been noted by Feshbach and Rubinow (1952). The normalization (2.3) i n B l a t t and Weisskopf (1952) on page 196 seems inc o r r e c t . The r i g h t hand side should be J£. ^ - 3 0 X 1 0 TABLE 3 • The f i r s t column designates the Feshbach - Pease (1952) p o t e n t i a l s ; the columns headed "FP" contain the f i n a l Feshbach,- Pease (195 2) r e s u l t s ; those headed "coupled FR", the r e s u l t s of s o l v i n g ( 4 - 2 ) ; those headed "modified FR", the r e s u l t s of sol v i n g (4-1), f o r the best<* . The second.column i s f o r the best t r i a l , exponential function; the t h i r d i s the r e s u l t s o f , s o l v i n g (1+ - l ) f o r <* = 0. The experimental r e s u l t s f o r the t r i t o n energy and the coulomb energy of He are E = -8.492 mev and E c o u/. = 0.764 mev. P o t e n t i a l E (mev) P D ( « E c ^ l ^ e v ) Symmetric S-State only S + D FP coupled FR modified FR FR coupled FR modified FR Exp. Function FR FP coupled FR modified FR - FP No.l -5.45 -5.89 -10.05 -8.10 -8.03 2.2 1.9 1.6 1.088 1.061 1.059 FP No.2 -3.27 -3.77 - 9.O6 -6.29 -6.24 2.8 2.8 1.9 1.059 0.997 0.995 FP g = 0 -2.01 -2.47 -8.40 -5.37 -5.32 3-1 3.1 2.4 I.O38 O . 9 6 I O.96O FP No.3 - 0 . 2 -0.92 - 7.50 -3.26 -3.22 3,6 3.6 2.4 1.008 0.861 0.861 - 28 -where t>, the i n t r i n s i c range parameter defined cn page 56 of B l a t t and Weisskopf (1952) i s equal to 2.5096 f o r a l l four p o t e n t i a l s i n table 2, and where T(S) verses S can be found on page 198 of B l a t t and Weisskopf (1952). The parameter S i s r e l a t e d to the Yukawa p o t e n t i a l depth as fo l l o w s : s= ° - + y.t" where we have used equation (2.17) on page 56 of B l a t t and Weisskopf (1952), and the f a c t that the s i n g l e t and c e n t r a l t r i p l e t p o t e n t i a l s have the same range. Table If gives the overlap i n t e g r a l s f o r the fo u r Feshbach - Pease (1952) p o t e n t i a l s . TABLE V OVERLAP INTEGRALS P o t e n t i a l 12 •I3 FP No.l .81 • 97 .89 FP No.2 .83 .96 .91 FP g = 0 .85 • 95 • 93 FP No.3 .86 • 93 .95 Table if, along with table 3, provide a measure of the r e l a t i v e merits of the approximate i n t e r n a l wave functions. Thus, the nearness of 1/ to u n i t y provides a measure of the merit of the best t r i a l exponential wave f u n c t i o n and the s o l u t i o n to (4-1) with«< =0, and the nearness of I j and 13 to u n i t y provides a measure of the r e l a t i v e merit of the solutions of (4-1) and (4-3). 1 The eigenfunctions of (4-3) are p l o t t e d i n f i g u r e s 5 to 8. Figures 9 and 10 show the functions whose overlap 1/ and 13 appear i n the f i r s t l i n e of table 4. The eigenfunctions u(1 ) and u(/i ) are i d e n t i c a l to two f i g u r e s of accuracy and have not been graphed. ± No compensation has been attempted f o r the f a c t that the various functions correspond to s l i g h t l y d i f f e r e n t energies. .t. 3 0 -1 1——I 1— I FIGURE 5 FP I POTENTIAL CASE E « -8.10 MeV M<F> T _ j T _ j- j j , , 1 FIGURE 6 FP 2 POTENTIAL CASE E « -6.29 MeV A(F) - 3 1 ' - 33 -F I G U R E 10 FP I POTENTIAL CASE ACF> - 3k -CHAPTER 5 THE CHARGE FORM FACTOR OF H 3 5.1 Introduction E l e c t r o n s c a t t e r i a g from H ^ y i e l d s information about the charge and magnetic moment d i s t r i b u t i o n of the t r i t o n . The experimental r e s u l t s are preseated as functions which depend on the momentum transfered from the elec t r o n to the t r i t o n and are c a l l e d the charge and magnetic form f a c t o r s . Expressions f o r the charge form f a c t o r of the t r i t o n are developed i n sections 5-2 to 5-4. These expressions contain the charge form f a c t o r of the proton and neutron and i n t e g r a l s which involve the i n t e r n a l wave functions. Hence,, we have a means of t e s t i n g the reasonableness of the approximate i n t e r n a l wave functions developed i n chapters 3 and k. The contributions to the form f a c t o r from the antisymmetric S-state and from the P-states are completely neglected since these states occur i n the ground state with n e g l i g i b l e p r o b a b i l i t i e s (see B l a t t e t a l , 1962). The c a l c u l a t i o n of the charge form f a c t o r i s done using the Derrick and B l a t t (1958) expansion of the t r i t o n wave fu n c t i o n , whereas S c h i f f (1964) and Gibson and S c h i f f (1964) use the Sachs. (1953) expansion. I t i s shown i n appendix G that the S-state contribution to the charge form f a c t o r gives by S c h i f f (1964) i s i d e n t i c a l to ours given i a ' s e c t i o n 5.35-no comparison of B-state contributions has been made however. F i n a l l y i t i s worth^pointing out that the o r i g i n a l Derrick -B l a t t functions are a l l orthonormal while the D-state functions used by Gibson and S c h i f f (1964) are not orthogonal. Using the Derrick - B l a t t expansion, i t i s r e l a t i v e l y simple to compute the charge form, f a c t o r while i t appears that the computation involved using the Sachs (1953) expansion i s rather complicated../ - 35 -5-2 D e f i n i t i o n of the charge form f a c t o r e I f P ) , P . are the i n i t i a l momentum of the t r i t o n and e electron r e s p e c t i v e l y and P_p , P^ , are the f i n a l momentum of the t r i t o n and electron r e s p e c t i v e l y , then the change i n wave number of the ele c t r o n ) and the t r i t o n (K) are ' <*-!.). The charge form f a c t o r of the t r i t o n Fc (H^) , as defined i n S c h i f f (1964), i s ^ .<*•') Km* [e'*'-<a> dJ where 4.^ ") i s the expectation value of the charge density operator (5-3) Pc=f]|[(/+^^^ and (see section B-2; and operates on the /< p a r t i c l e i n i s o s p i n space, •f-£ and f-. are the charge density d i s t r i b u t i o n s of the proton and neutron r e s p e c t i v e l y . Figure 11 shows the vectors The t r i t o n wave fu n c t i o n "y • defined i n equation (2-l) does not include the center of mass motion. However, f o r the c a l c u l a t i o n of the charge form f a c t o r , t h i s center of mass motion i s necessary. We assume the center of mass wave function i s given by _ 3 6 -FIGURE 11 THE POSH ION VECTORS - 37 -where K ff i s the wave number of the center of mass motion and R g i s the center of mass vector. Hence, the t o t a l t r i t o n wave fun c t i o n including the center of mass motion i s Rewriting (5-2) i n terms of (5-5) y i e l d s where cx> oo CD -CD The operator £ does not change the t o t a l spin of the triton wave fun c t i o n hence the form f a c t o r can be written a s 1 (5 where 3 4- r r . ^ f Only S- and D- states are considered here since S- and D- state components alone were ca l c u l a t e d i n chapters 3 and k. 38 -and we have defined - 1 5-3 The S-state contribution to the form f a c t o r Consider F c (H ) and make the change of v a r i a b l e ^LK ~ X~—K With the a i d of „ which define the proton ( /"c ) 8 1 1 ( 1 neutron (Fc j form f a c t o r s r e s p e c t i v e l y , equation (5~9a) becomes Changing v a r i a b l e s to /z.^ — and using equation ( 5 - 1 2 ) becomes 1 The antisymmetric S-state has been neglected. 2 We usel/fi, ^ 9 and'b^ instead of ^ t l ) \ ^ f o r the reason given i n section 5 ' 1 - ' J -39'-with S3, n3 n i j t . a ' ( S- / 5 ) FJH )=A Z je %f/^)%^+# 2 i e where Figure 1 2 shows geometrically the new v a r i a b l e I t i s i n t e r e s t i n g to note that equation (5-lk) implies that the 9 momentum transfered from the el e c t r o n to H i s equal and opposite to the momentum transfered from the t r i t o n to the ele c t r o n ( i . e . , K = - %•)> Equation ( 5 - 1 5 ) i s e a s i l y s i m p l i f i e d with the help of the isospini matrix elements ^ / ^ ^ )> defined i n appendix E. Performing the in t e g r a t i o n over Euler angles (^y/3^ ^ e q u a t i o n ( 5 - 1 5 ) becomes (s-n) Fc(H') = -iVi fe^'/V-f +f/ P \ • O 4. 2- "I -\e K If +fkli^ - f i [^ )^ o^^^>j--Fl-G/ r^ /K//>-^ i^^ >| The Euler angles and the Euler angle wave functions are d e f i n e d i n appendix C. - ko -(./> / / £2 / / \4, / / 13) / FIGURE 12 - THE CHANGE OF VARIABLE J1H .. * .la -where J V t i s the i n t e g r a l i n (5-l6) when the i n t e g r a t i o n over the angles (""v /3/ ^  ) $ a s "been done. Performing the summation i n (5-l6) and using the table E - l f o r the isospim matrix elements, equation (5-1°) becomes Z 3 The transformation properties of the functions f / ,. f 3 / and t3 z may "be found from appendix A. As an example of the use of these permutation properties t o s i m p l i f y equation (5-l8), consider From f i g u r e 12, i t i s c l e a r that c h a n g i n g ^ to A-t i s equivalent to interchanging p a r t i c l e s 1 and 2. However, i s i n v a r i a n t under any permutation of the three p a r t i c l e s and the combination transforms according to the symmetric representation of S (3) (see appendix A). Hence the f o l l o w i n g i d e n t i t y holds: Using arguments l i k e t h i s about the permuatation properties of the functions f , , f 3 ( and f g ^ , one can rewrite equation (5-I8) i n the - 4 2 -simpler form F c V ) = {F;4'f<?>f!,(<t) + ±{FS-rS)F*(t-*) where J L V A/T J Although equation (5-21b) appears to d i f f e r from that of S c h i f f (1964), appendix G shows that they can be brought into i d e n t i c a l form by using the r e l a t i o n s between the S c h i f f (1964) and De r r i c k - B l a t t (1958) i n t e r n a l functions. Experimentally, only the average value o f ^ over a l l orientations *— 2. 2. i n space i s measured. I f Fj ( ) i s the average value of F j ( ) when averaged over a l l o r i e n t a t i o n s of the v e c t o r ^ with respect to the vector n.1- and s i m i l a r l y f o r F 2 ( ), then equations (5-21) become Since the dominant component of the t r i t o n wave fu n c t i o n i s the symmetric S-state component, i t i s u s e f u l to consider where F j (f- ) i s the contribution of the symmetric S-state component to the charge form f a c t o r . From f i g u r e 12,. i t i s c l e a r that In order to make use of the functions c a l c u l a t e d i n chapters 3 and kr, one makes the f o l l o w i n g change of v a r i a b l e s Applying t h i s change of variables'Ho equation (5-24) y i e l d s ' , 00 tL ji. ± Applying the changes of va r i a b l e s 1 - -± one a f t e r the other to equation (5-26) and using the f a c t that the integrand i s an even f u n c t i o n of y, equation (5-26) becomes ( 0 2 ? AJ ~ <J2 The Jacobbian of t h i s transformation i s 2-- hk -where as i n equation (3-9a). 5'4 The D-state con t r i b u t i o n to the form f a c t o r As i n the case of the S-state contribution, equation (5~9b) can be rewritten as with Using equation (5-10b), the matrix elements i n appendix E and i n t e g r a t i n g over E u l e r 1 a n g l e s , equation (5-31) becomes •+ See page c-5-- 45 -Performing the above summation and using table E - l y i e l d s 1 As f o r the S-state contribution in the preceeding section, some of the i n t e g r a l s may be s i m p l i f i e d using the permutation properties of the .internal wave functions to give A comparison with the work of Gibson and S c h i f f (1965) would require the technique used i n appendix G plus the r e s u l t s of McMillan (1966b). - 46 Averaging (5-34) over a l l orie n t a t i o n s of £ , as i n section (5.3) y i e l d s 1 2. 1 1 V ^ t t " + - f ^ i ) * * + ^ ti^V^+^'2 ) ^  One can also write (5-35) i n terms of the s i x D e r r i c k (1960a, 1960b) i n t e r n a l functions f U ) l , f / ) / 2 , f t z , f / 3 , f^, fi^lZ using equations (2-5) . Keeping only the symmetric D-state component, (f/a) equation (5-35) becomes''" where The expressions N3 , N ^ , N^- are defined i n chapter 1. - hi -Thus i n the symmetric S- and D- state approximation f o r the t r i t o n charge form f a c t o r developed i n chapters 3 and k, the t o t a l charge 3 form f a c t o r of H i s ( As a check on the symmetric S- and D- state c a l c u l a t i o n s we consider the case of zero momentum t r a n s f e r from the ele c t r o n to the t r i t o n ( i . e . , q = 0). I t i s easy t o see t h a t 1 +-F?f') ~ S f t V / f r - f g ] ^ where (5-39a) i s j u s t the r i g h t hand side of equation (3-5). Noting that fl(o)=lj F?(°) = o f o r q = 0, (see S c h i f f (1964)), and using (3-5), equation (5J38) becomes (5.-39) F c ( H 3 ) = 1 as required. 5-5 Numerical r e s u l t s The c a l c u l a t i o n of the contribution to the charge form f a c t o r from the S- and D- states of the t r i t o n wave fun c t i o n provides a step i n checking the approximate i n t e r n a l wave functions of chapters 3 and h. However, only the component F j (f ) has been computed numerically up to the present time. As pointed out i n section 5'3, t h i s i s the dominant We note that £AV"fo =• ^ A ^ - f ^ ^ t =- O - 48 -contribution to the charge form f a c t o r since the i n t e g r a l i n (5-23) depends on the i n t e r n a l S-state wave f u n c t i o n ( f j ) which i s w e l l known to be the dominant part of the t r i t o n wave funct i o n . The solutions t o equation (4-1) with o( = o and normalized according to equation (3-20) (LL0 ) and the best t r i a l exponential functions normalized according to equation (4-7) f o r the four Feshbach - Pease (1952) p o t e n t i a l s were used t o c a l c u l a t e equation (5-28). _ Figures 13 to 16 are graphs of F | (q ) verses q f o r the best r t r i a l exponential f u n c t i o n and the eigenf unction LL0 f o r a l l four Feshbach -Pease (1952) p o t e n t i a l s . We see that as the t r i t o n binding energy decreases S a, ( i . e . , as we go from the FP No.l to the FP No.3 p o t e n t i a l s ) , Fj (q ) decreases f o r each f i x e d value of q / 0. S i m i l a r l y , f o r approximately equal values of the t r i t o n energy but d i f f e r e n t wave functions ( i . e . , a comparison of the best t r i a l exponential f u n c t i o n and the f u n c t i o n U0 f o r any p a r t i c u l a r Feshbach - Pease (1952) p o t e n t i a l ) F/S (q) i s l a r g e r f o r the best t r i a l exponential f u n c t i o n f o r each f i x e d value of q ^  0. Indeed, the observations above are not unexpected* f o r as one sees from (5-28) Fif ( q 2 ) depends most on the wave fu n c t i o n hear i t s maximum as a f u n c t i o n of/l and not on the wave f u n c t i o n t a i l . From f i g u r e s 5 to 8, we observe that the maximum values of u(/£.) decrease as we go from the FP No.l to the FP No.3 p o t e n t i a l s and from f i g u r e 9> "the best t r i a l exponential f u n c t i o n has a higher maximum than the corresponding function u 0 f o r the FP No.l p o t e n t i a l . For completeness, table 5 gives the values of F j S ( q 2 ) verses q f o r the best t r i a l exponential and the f u n c t i o n Uj, f o r a l l four Feshbach -Pease (1952) p o t e n t i a l s . TABLE 5 The function F,(q 2) f o * the four Feshbach - Pease (1952) p o t e n t i a l s f o r the eigenf unction Uo and the best t r i a l exponential. q FP No.l FP No. 2 FP e = 0 FP No.3 Exoeriment Exp. U Exp. U Exp. U Exp. q K=i.o4o K=0.931 KO . 8 5 6 K=0.746 0.00 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.25 0.975 O . 9 8 I O.968 O.978 O.96O 0.974 0.934 O.968 0 . 5 0 0.910 0.934 0 .886 0.920 O.858 O.907 0.784 0.881 0-75 0.816 O.863 0.773 0.833 0.728 0.808 0 .6 l8 0.758 1.00 0«710 0 .774 0.651 0.730 0.595 0.691 0.473 0.622 1.25 0 . 6 0 3 0.677 0.536 0.620 0.476 0.572 0.357 0.490 1.22 0 . 5 1 8 1.50 0.503 O.58O 0.434 0.513 0,375 0.461 0.268 0.374 1.41 0.404 1.75 0.415 0.487 0.348 0.417 0.294 0.363 0.201 0.280 I . 5 8 0.328 2.00 0.340 0 .403 0.278 0-333 O.230 0.282 0.151 0.206 1.73 0.284 2.25 0.277 0.330 0.221 0.263 0.179 0.216 0.114 0.150 1.87 0.227 2.50 0.225 0.268 0.176 0.206 0.140 0.165 O.O87 0.109 2.12 0.158 2.75 0.183 0.216 0.140 0.161 0.110 0 . 1 2 6 0.066 0.079 2.24 0.134 3-00 0.149 0.173 0.112 0.126 0.087 0.096 0.051 O.O58 3.25 0.121 0.139 0.090 O.O98 0.069 0.073 0.040 0.043 3.50 0.099 0.111 0.073 O.O76 0.055 0.056 0.031 0.032 3-75 0 . 0 8 0 0.088 O.O58 O.O59 0.044 0.043 0.025 0.024 4 .00 O.O65 0.069 0.047 0.046 0.035 0.033 0.020 0.018 4.25 O.O53 0.054 O.O38 0.035 0.028 0.025 0.015 0.013 4.50 0.043 0.042 0.030 0.026 0.022 0.018 0.012 0.009 4.75 0.034 0.033 0.023 0.020 O.018 0.013 0.008 0.006 5 . 0 0 0.028 0.026 0.015 0.013 0.009 0.004 - 52 -CHAPTER 6 CONCLUSIONS We have i n t h i s t h e s i s extended the equivalent two - body method of Feshbach and Rubinow (1955) "to include also one t r i t o n D-state component. The t r i t o n energy and wave fun c t i o n f o l l o w from the so l u t i o n of eigenvalue equations, and not from the R i t z minimization procedure which i s more customary i n the t r i t o n problem. Indeed, the coupled Feshbach - Rubinow equations (3-15) which we have derived and solved are, as we have pointed out, very s i m i l a r to the well-known p a i r of equations f o r the deuteron wave f u n c t i o n components. A comparison of our cal c u l a t e d t r i t o n binding energies and those obtained by Feshbach and Pease (1952) with the R i t z procedure, shows, however, that ours are smaller f o r a l l f o u r F e s h b a c h - Pease p o t e n t i a l s . We a t t r i b u t e t h i s t o the f a c t that W e included only the symmetric D-state component of the t r i t o n wave f u n c t i o n along with the symmetric S-state component, whereas Feshbach and Pease include, as pointed out by McMillan (1966b), a l l the D e r r i c k - B l a t t D-state components.(We consider our assumptions (3-7) and (3-18) to be of l e s s e r importance i n t h i s comparison since, as we have pointed out, the dominant parts of the Feshbach - Pease approximate components are of t h i s form.) The D-state p r o b a b i l i t i e s which we have cal c u l a t e d are major f r a c t i o n s of the Feshbach - Pease q u a n t i t i e s , which f u r t h e r bears out our assumption that the symmetric D-state i s the dominant D-state, but f o r accurate r e s u l t s i t i s not s u f f i c i e n t to include only t h i s component. Even though the D-states occur i n the t r i t o n ground state with small p r o b a b i l i t i e s , t h e i r contribution to the t r i t o n binding energy i s appreciable. - 53 -The general s i m i l a r i t y between the r e s u l t s obtained from the coupled and modified Feshbach - Rubinow equations indicates that the approximation (3-l8) i s quite good.- As a matter of f a c t , t h i s approximation leads t o f a r l e s s e r r o r than the replacement of the s o l u t i o n of the Feshbach - Rubinow equation with the best t r i a l exponential function. In p a r t i c u l a r , one sees that the computed functions as w e l l as the cal c u l a t e d energies are i n good agreement. One i s thus encouraged to write the remaining D-states i n terms of symmetric functions as, f o r example, generalizations of the Feshbach - Pease (1952) functions given by McMillan (1966b), apply approximation (3-7) to these functions, and f i n a l l y apply approximations s i m i l a r t o (3-18) to each of these components. I t i s f e l t that i n view of the c a l c u l a t i o n s reported here, the r e s u l t s obtained with the l a s t approxi-mation should be quite close to those obtained from solving the coupled, ordinary d i f f e r e n t i a l equations which would be obtained when the Feshbach -Rubinow approximation i s applied. BIBLIOGRAPHY - 54 -Biedenhern, L. C , B l a t t , J . M., and Kalos, M. H. 1958 "Nuclear Physics 6, 359." B l a t t , J . M., and Weisskopf, V.F. 1952 "Theoretical Nuclear Physics" (John Wiley and Sons, Inc., New York). Condon, E. U. and Shortley, G. H. 1935 "The Theory of Atomic Spectra" (Cambridge U n i v e r s i t y Press). Derrick, G. H. and B l a t t , J . M. 1958 "Nuclear Physics 8, 310." Derrick, G. H. 1960a"Nuclear Physics 16, 405". 1960b"Nuclear Physics I B , 3 O 3 " . Feshbach, H. and Pease, R. L. 1952 " P h y s i c a l Review 88, 945". Feshbach, H. and Rubinow, S. J . 1955 "Physical Review 9J3, 188". Gibson, B. F. and Schif'f, L. I. 1965 "Physical Review 138, 826". Goldstein, H. 1959 " C l a s s i c a l Mechanics" (Addison Wesley Publishing Co., Inc., Cambridge, Mass.) McMillan, M. 1965 "Canadian Journal of Physics 4^ , 463". 1966a "On Expansions of the T r i t o n Wave Function" (U.B.C. preprint) 1966b "The Feshbach - Pease T r i t o n Wave Function i n the Derrick - B l a t t Notation" (U.B.C. prep r i n t ) Morpugo, G. 1952 "Nuovo Climento 9_, 4 6 l " . Sachs, R. G. 1953 "Nuclear Theory" (Addison Wesley Publishing Co., Inc., Cambridge, Mass.) S c h i f f , L. I. 1964 "Physical Review 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). APPENDIX A THE SYMMETRIC GROUP OP ORDER 3(S ( 3 )) A.l Irreducible representations of S(g) The symmetric group of order 3 is the group of al l permutations on three objects. This group has 3 classes of conjugate elements; hence it has 3 irreducible representations. S (3) is isomorphic to the crystall-ographic group D 3 , the group of al l proper notations which leaves an equilateral triangle invariant, and thus the irreducible representations can be found from D 3 . The 3 irreducible representations of S(j) are labelled 'T^  (symmetric representation), 'T>A (antisymmetric representation), and 2 T 3 (mixed representation) where the upper left index gives the dimension of the representation. The 6 elements of S( 3) are labelled G, = (l ) , G r= ( 1 3 2 ) , G 3= ( 1 2 3 ) , G^= ( 1 2 ) , G^= (3D, G,= ( 2 3 ) where the cyclic notation of Wigner ( 1 9 5 9 ) is used. We shall also use the notation TABLE A-l IRREDUCIBLE MATRICES OF S,3) Group Element G' - 1 1 G 2 •' 1 1 ±1-1 J*\ G=j 1 1 X -1 -JS) G if. = P|2. 1 - 1 Gr = P / 3 1 - 1 G 6 = P23 1 - 1 2 Ijl 1 J - 56 -A.2 Permutation properties of functions with Soisymmetry Let 0f//2./3)be an a r b i t r a r y f u n c t i o n of the three indices (1,2,3). Define P. operating on ^ ( l j 2 ,3 ) by x f^ttxhv) = <!>(<*(h*;*)} where G^ e S and Gfc0i^3) i s one of the 6 permutations of the indices 3) . Only those functions 0 (l, 2/ a) with the fo l l o w i n g transformation properties w i l l be considered: (1) <p(hZ,i) transforms according to V7/ (see equation a-2) under a permutation of the indices (1,2,3). In t h i s case ^  i s l a b e l l e d ^ ^ 2 y 3 J and i s c a l l e d a symmetric function v , ; (2) 0(hi/3) transforms according to '71 (see equation a-3) under a permutation of the indices (1,2,3). 3n t h i s case (j) i s l a b e l l e d ^ ^ ( ^ 2 , 3 ) and i s c a l l e d an antisymmetric fun c t i o n . ^ - 3 ) /? 0 ^ ) = ivy' ' > (3) The p a i r of functions j E ^ ^ j J a n d ^"(^2)3) transform according to 2 ' P3 (see equation a-3) under a permutation of the indices (1,2,3). The lower index r e f e r s to the row of the representation T 3 and these functions are c a l l e d mixed-one and mixed-two functions r e s p e c t i v e l y . - 57 k£4>? (h**) = -&fir(h*i3)+i<ff(h*,*) A.3 Permutation add i t i o n c o e f f i c i e n t s Suppose ^ (y /2 /3) , with P^ one of the 3 i r r e d u c i b l e \ representations of S(3) and Kg the row l a b e l of the representation P^ transforms according to the i r r e d u c i b l e representation of S(j) and p 3'J with Pj, and defined as above, transforms according to the i r r e d u c i b l e representation T t of ; then how do the product functions transform under j o i n t permutations of the indices ( 1 , 2 , 3 ) and ( 1 , 2 , 3 )? One approach to t h i s i s to consider the d i r e c t product decomposition of two i r r e d u c i b l e representations of S^. The c o e f f i c i e n t s are determined from the formula which can be found i n Wigner (1959)« i s the number of elements, i n the conjugate c l a s s , Xof (^7^) i s ^ e c i i a r a c t e r of the conjugate 58 c l a s s f o r the representation "^73 > and • & i s the character of t h e ^ conjugate cla s s f o r the d i r e c t product representation _ / ' . Table A-2 summarizes the r e s u l t s . From t h i s t a b l e , i t i s c l e a r that one can write (see D e r r i c k and B l a t t (1958)) where the c o e f f i c i e n t s a x e c a l l e d the permutation addition c o e f f i c i e n t s f o r the group S f3 ). They are unique up to a phase f a c t o r which one chooses t o make them r e a l . The permutation addition c o e f f i c i e n t s form a u n i t a r y matrix, hence an orthogonal matrix since the c o e f f i c i e n t s are r e a l , which transforms the d i r e c t product space i n t o a d i r e c t sum space. The orthogonality condition above i s written The c o e f f i c i e n t s may be evaluated using equation (a-7) and table A-2 and have been given by D e r r i c k and B l a t t (1958). For convenience, we have included them here i n table A-3. Table A-k l i s t s the 16 l i n e a r combinations of the d i r e c t product space. TABLE A-2 DIRECT PRODUCT .DECOMPOSITION OF S«) 'T, ft 'V »7\ ® '7Y = >T, * •TV (9 = 'T, 9 2v3 = 2 r 3 T,@ 'T2® 2T3 TABLE A-3 NON-ZERO PERMUTATION ADDITION COEFFICIENTS - 59 -(Vl s,)= ( III) (* M ) = ( a a s \ i i i J = / ' ( 1 K Pi j ~ f on. s >m \ l K 1 A ) ) - '"SKA / <X <rt\. /mA _ A 1 K I ) ~* ( K » * 7 r o K*A 1 k=ij Pi = 2 /O-w O n ovOi / ( a I = j = ( 2 2 ; J _ 1 - 60 -TABLE A-k DIRECT PRODUCT FUNCTIONS SYMMETRIC f ' s = i f f ANTISYMMETRIC MIXED - 6 1 -APPENDIX B SPIN - ISOSPIN FUNCTIONS B . l Spin functions The t o t a l spin operator f o r a system of three i n d i s t i n g u i s h a b l e p a r t i c l e s of spin \ i s £> = s, + _§ 4 + J 3 wherejs^ i s the spin operator f o r the/. p a r t i c l e . Label the eigenstates of js.ss = s and s ^ = + +• ,s 3^ by JS^P^ *<s) and the eigenstates of s^ and s/^ by jSx/ma'X- ' w i t h S * = ^' m / = ~ 2« F o r s i m p l i c i t y , l e t "*^ >/-9 1 1 ( 1 Iii?/ = ' There are eight eigenstates of s 1 and s ^ since s = \ or s = \ y and these are obtained from the double Clebsch - Obrdan series. Table B - l gives these 8 eigenstates when the Clebsch - Gordan c o e f f i c i e n t s are evaluated using the phase convention of Condon and Shortley ( 1 9 3 5 ) ' The eigenstates <j. and <fe are i d e n t i c a l t o those defined i n Der r i c k and B l a t t ( 1 9 5 8 ) . The spin functions are also functions of the TABLE B - l EIGENSTATES OF S AND S, s-i -1 - 62;. -indices ( 1 , 2 , 3 ) . (i.e., q • = q - ( l , 2 , 3 ) ) . Using the definition of Pff given in equation (a-l), the commutation relations (b-a«) £S* P*Jz 0 K--I, •>• , 6 a. follow. Thus simultaneous eigenstates of Pj , S , and can he constructed. From equations (b-2a) and (b-2b), i t is clear that 8 where ty/' (fx) are coefficients to be determined. These coefficients form an 8 dimensional matrix for each G M6So) . The matrices are easily calculated using equations (b-2) and table B-l, and the set of six matrices obtained from (b-3) form an 8 dimensional representation of Sf 3| which we label ~p . By inspection the decomposition of T 7 into a direct sum of irreducible representations of S ^ ) is The four spin functions (S = •§ ) each generate the representation 'T; while the two pairs of spin functions (S = \) generate the representation 2719 . Hence the reason for the label j S P^. Kj^i Pj. is the representation •under which the functions transform, while Kj is the corresponding row number. Thus q ^  , q^ , q 7 , q & , are symmetric functions under a permutation of ( 1 , 2 , 3 ) while the pairs (fa) {\\ j a r e m i x e < i functions under a permutation of ( l 2 3 ) * 1 • ^ (It) c o r r e s p ° n d t o ( ) of section A.2. - 63 -B.2 Isospin functions The t o t a l i s o s p i n operator f o r three p a r t i c l e s of i s o s p i n ^ i s 'M T = T j + T 2 + T^where T^' i s the i s o s p i n operator of the / p a r t i c l e . Label the eigenstates of T and T^ by | TM-p Ej- Kj> and those of Tx* and T ^ by |T/M/>/ where II J ^ - T r ^ ) and \i'0/=*(/). The eight eigenstates obtained using an equivalent equation to (a - l ) have the same permutation properties as the corresponding q's. The eigenstates are l a b e l l e d 1 ? ( to P f f a n d are obtained from table B - l by rep l a c i n g *U)loy TT(/), #'7 by ty), and q^ by PA\ B.3 Spin - Isospin functions The spin - i s o s p i n functions are l a b e l l e d JTSMJ. M 5 P t = ^V^iXtC/i/T^S) 8 1 1 , 1 correspond to d e f i n i t e values of S, T, S^. and T ^ The notation used f o r the V's i s s i m i l a r to Derri c k and B l a t t (1958) except here the value of My i s included. They transform according to the represent-at i o n P t of S( 3) and i s the corresponding row number of P t . One f i n d s these functions by using the permutation a d d i t i o n c o e f f i c i e n t s of table A-3« For T =-j , S = ^  the spin - i s o s p i n functions are where i , j , = 3,6,7,8 depending on the values of My and Ms . The spin -i s o s p i n functions are l i s t e d i n ta b l e B-2 f o r T = ^r, S = §r and T =~k , and S = 5. To f i n d T = -f-, S = •§•, replace p by q and q by p f o r the case T = a, S =-!• . We should note that the V's are orthonormal as are the q's and the p's. 1 In De r r i c k and B l a t t (1958), the functions c a l l e d v, and v 2 are here c a l l e d p ¥ and p^- r e s p e c t i v e l y . - 6k -TABLE B-2 SPIN - ISOSPIN FUNCTIONS T = | S = 3 a. Mr = | T = 1 s - l r M - 1 1 i, % ) = 1 ) = 4<p* ) = t -4 ) (**/ i % ) = ( ^ 4 ) - t 3 P* ) - i Vj-j . a i I i^/i \ ) = q 7 p , = q 7 p < t j - g. 7p* a 2.* ) = is = ^ P5 1 = i s = . 1 " 2 T ) = ) j -) - fit] j = a a * (s< ti ) = a 11 ( ) ) = ) ) = ) u 2 - 65 -APPENDIX C EULER ANGLE WAVE FUNCTIONS C . l The co-ordinate system The Euler angles (<*,£, X ^ d e f i n e d by De r r i c k and B l a t t (1958) depend upon the f o l l o w i n g d e f i n i t i o n of the body frame (X/fy/y )• The o r i g i n of t h i s frame i s at the center of mass of the three p a r t i c l e s . One chooses the X' - a x i s , up to i t s d i r e c t i o n , along the p r i n c i p a l axis of the l a r g e s t moment of I n e r t i a of the t r i a n g l e formed by the three p a r t i c l e s . The - axis i s chosen so that a r o t a t i o n from p a r t i c l e 1 to p a r t i c l e 2 i s a r i g h t handed r o t a t i o n about the ^ ' - axis. One chooses they.'- axis to make the body frame right-handed. Thus, once the d i r e c t i o n of the X/ - axis i s chosen the body frame i s f i x e d . The space frame (X 5 Y s z ) has i t s o r i g i n at the center of mass i of the three p a r t i c l e s . This i s a f i x e d , right-handed frame and the three angles which takes t h i s space frame in t o the body frame are the Euler angles (^/fyX ) defined by the convention of Goldstein ( l 9 f ? ) . I t i s c l e a r from the above d e f i n i t i o n that an even permutation i of the 3 p a r t i c l e s does not a f f e c t the body frame while an odd permutation changes the d i r e c t i o n of the ^'- axis and hence the ft* - ax i s . Thea:' - axis i s i n v a r i a n t under permit&tions of the p a r t i c l e s since the three p a r t i c l e s are assumed t o be of equal mass. 1 The angles (^/^/^ ) are i d e n t i f i e d with Goldstein's angles ( i ^ -Q-Y 0 ) as pointed out by D e r r i c k (1960a). - 66 -C.2 Irreduci b l e representations of the group of proper rotations The angular dependence of the t r i t o n wave fun c t i o n ( i . e . , the dependence of the t r i t o n wave f-unction or the E u l e r angles fyft ) has been gives by Derrick aTa& B l a t t (1958) and aire c a l l e d the Euler angle wave functions. Ctete defines the Euler angle wave functions i n terms of the representation c o e f f i c i e n t s of the irreducible representations of the group / of proper r o t a t i o n s . where the- sum i s e d i t i o n (<3-2) i s over the zeros of the denominator. L i s the o r b i t a l angular momentum, i s the ^  - component of o r b i t a l angular momentum i n the space frame mdyk i s the-^, - component of angular momentum i n the body frame. C3 Permutation properties of From section C . l i t i s cl e a r that only odd permutations of the three p a r t i c l e s w i l l a f f e c t the Euler aag3.es and- hence the representation c o e f f i c i e n t s foi ^ ) <• An odd permutation of the three p a r t i c l e s transforms the angles (<*iP\V ) i n t o the angles (-o^  TrtPj Y) • Define 0 1 - 67 -I t i s worth recording here that McMillan (private communication) has shown that where the r i g h t hand side i s the fu n c t i o n given i n (15-27) of Wigner (1959)' The r e l a t i o n s h i p between the Goldstein Euler angles ( "V^ -e-, ^  ) and the ( °<j f) % ) defined on Page 9° of Wigner i s ' 2 p = -a-- 68 -operating on the representation c o e f f i c i e n t s "by Hence 9 0 i s the operator a c t i n g on QMAIl (^ihlf) which corresponds to the operation of performing an odd permutation of the p a r t i c l e s . Lemma Is 0 d^/t»L Proof; By d e f i n i t i o n (C-4), O = ^^(Wi-fi) and hence using equation (C-2) we have Let %~L-h^\L- K . we note that % i s an integer since L i s ah o r b i t a l angular momentum. Tiros ^^i^tP) ^ccmes ^ (-I)LA ^d-^Jt) where we have used equation (C~3)° Theorem: 0 DL (±)Pit) = ^ ^ ^ P> *) Proof: oDJ^jAt) = ^ " W ) ^ ^ / ^ j > - 69 -C.k Euler angle wave functions The e f f e c t of the p a r i t y operator TT acting on the represent-ation c o e f f i c i e n t s D^^fajPifr) i s •Experimentally, the p a r i t y of the t r i t o n i s even and hence only those representation c o e f f i c i e n t s with e v e n a r e allowed f o r the t r i t o n wave function. Combining t h i s with the r e s u l t that the allowed values of L are 0, 1 and 2 (since the t o t a l spin i s \ o r ^ ( s e e appendix A) and the t o t a l aagular momentum i s experimentally), there are 5 Euler angle wave functions. They are given i n Derr i c k and B l a t t (195^) where we use t h e i r notation f o r the functions ( i . e . Y/^ (p^ |/</ ) ) where L i s the o r b i t a l angular momentum, JIA i s the body ^ - component of o r b i t a l angular momentum, M^is the space ^ - component of o r b i t a l angular momentum, and P e i s the permutation symmetry which i s e i t h e r symmetric or antisymmetric) We include a complete l i s t of the Euler angle wave functions f o r convenience. A glance at the theorem of section C«3 shows how the permutation properties of these functions a r i s e . F i n a l l y we note that these Euler angle wave functions are orthonormal. S - STATE X°(S, o) - ~ y P - STATE (%o) = Aj Zp' Urfti) f 7 7 2 2. M=0 YiC*,*)* *± [co£l-s,Sfil I \ -A }f A,-.-I V . / f V ) - _ i _ COS0 - -A-Q> SYMMSTEIS ^ ( V j = ^ 1>L = i f f ' 2 e " s^-s,^ COS&I i+rr z a. 2 'S J - S'T'^E i V p - s. S O M E O ) * j = t ) + fl» * ^) - zfe e [e Cos¥£ it *|j * = Xiao = ' ^ e ' V ^ y . e - ^ c ^ ^ Alt 2. M i I - 71 -STATE 2S ANTISYMMETRIC y i M = ~±-L- j Di/r> fat,*)-&iA, j M - - 2 - 72 -APPENDIX D TOTAL ANGULAR MOMENTUM - ISOSPIN WAVE.FUNCTIONS The spin - i s o s p i n wave functions VnrM5j< TJS) are now combined with the Euler angle wave functions ( t o form t o t a l angular momentum - i s o s p i n wave functions l a b e l l e d ^ ^ ^ ( J , L , S , P , P f ,P t \ju\ These 1^  's are i d e n t i c a l to those defined i n Derr i c k and B l a t t (1958). J i s the t o t a l angular momentum of the system, Mj- i s they- - component of t o t a l angular momentum, L i s the o r b i t a l angular momentum, S i s the spin angular momentum, and/*- i s the body -^ - component of o r b i t a l angular momentum. The functions 'Ij transform according to one of the i r r e d u c i b l e representations of S under permutations of the p a r t i c l e s and the l a b e l P indi c a t e t h i s repersentation. As usual K i s the corresponding row number of the represent-a t i o n p . The c o e f f i c i e n t s i ( £ are the permutation ad d i t i o n co* e f f i c i e n t s defined i n section A. 3 and the c o e f f i c i e n t s ^Z.SAU/V\ SJ 7M3) are Clebsch Gordan c o e f f i c i e n t s as defined by Condon and Shortley (1935)* In the case of the t r i t o n , J = T = -T = §. Without any l p s s of g e n e r a l i t y , one can assume M ^  = Counting a p a i r of mixed states as only 1 s t a t e , there are 10 states f o r the t r i t o n wave function. They are given e x p l i c i t l y below and t h e i r permutation properties are l i s t e d i n table 1. The functions ^ ( , } Tjl0 2 given below are identlcaK?td|vbhcse^ r e f e r r e d to by D e r r i c k (1960b). ) - 73 -% - , f e y v r ^ u V ^ y i » « V/^y^^w-^'^^ C i" i *"5 3I* ~?T - Th -AFPEND'IXyE EVALUATION OF THE ISOSPIN MATRIX ELEMENTS The i s o s p i n matrix elements needed f o r the form f a c t o r c a l c u l a t i o n s are those i n v o l v i n g I = |, M 7 = These are the R^ . and • (V functions given i n ta b l e B-2. To s i m p l i f y w r i t i n g the matrix elements of the operator \ y between these two i s o s p i n functions, the fol l o w i n g convention i s used. < p* I P«) ~ * 2 I K"> J CPs I Ps> * ' ^ I K 1 2 > '!' •1' Evaluation of these twelve matrix elements i s s t r a i g h t forward and the r e s u l t s are tabulated i n ta b l e E - l . TABLE E - l ISOSPIN MATRIX ELEMENTS 0 l|l> = _ 2 3 0 / 2 / 0 = -A 0 / 3 / / > = i U> - 3 OU/2> r - 3 <2|> P> = Oja / O = -.3 < 2 | ^2I*IZ> = 0 .1 - 75 -APPENDIX F INTERNAL WAVE FUNCTION • BOUNDARY CONDITIONS F . l The t r i t o n p o t e n t i a l s For convenience i n the numerical computations described i n chapter k, we define the fol l o w i n g q u a n t i t i e s : IS + (f-ib) yl(n) = +[V%)-VU)J -/ where 1/%), ^(ty^^i'V'fi) and E are defined i n equations ( 3 - H ) to ( 3 - 1 5 ) - The asymptitljc^ forms of the p o t e n t i a l s f o r small ft are y.C/Z ^ ' and f o r large -7 are - 7 6 -F .2 The modi f ied Feshbach - Rubinow equation (4-l) The e igen func t ion , i((n.) , of equation (4-1) wi th f i x e d parameter o( has, f o r smal l values of rt , the fo l l ow ing form ( - f - 2 ) urn-) = dnK where z For l a rge values of/i , U/i) has the form U(n)- hC That these are the asympti4|i^ forms of U.(U) f o r l a rge and smal l /Z. i s c l e a r from look ing at equations ( f -1 ' ) and (f-2 ). F .3 The coupled Feshbach - Rubinow equations (4-3) For smal l va lues of /I , .the e igenf unct ions u(IXj and w(n) t o equations (4-3) are assumed t o be Expanding the p o t e n t i a l s i n equations (4-3), subs t i t u t i ng equations (f-6) i n to (4-3), and equat ing c o e f f i c i e n t s to zero y i e l d s (i-sL) Uh(n)^ CL0BK ^ D / i i / i i-mL/i2^...j " 7 7 " where ISO • « Using a s i m i l a r technique f o r large/Z. y i e l d s «r(n)= ape L + where - 78 APPENDIX G EQUIVALENCE BETWEEN EQUATION (5.21b) AND THE SCHIFF S - STATE FORM FACTOR The i s o s p i n functions (iv/ /n^ ) f o r T = ^, Mj = -\ and the spin functions Of }'%1) f o r S = J, M s = | defined by S c h i f f (1964) d i f f e r from the functions (P^/^. ) and (^. ^ ) r e s p e c t i v e l y , defined i n appendix B, by the interchange of the indice s 1 and 3. A l l those functions of S c h i f f (19610 d i f f e r from the De r r i c k and B l a t t (1958) functions by the interchange of indices 1 and 3. Thus the functions u; , Vy , v 2 defined by S c h i f f (196*0 correspond t o An - - ^3 in (fr-IC) ATZ^ - P/3 -f,,a where f 3^ f , f 3 ^ 2 are defined i n chapter 2, /-^ i s defined i n appendix A, and the minus sign appears from the comparison of equation (2#'| ) to equation (6) of S c h i f f (196*0. Rewriting equation (5-21b) interchanging indices 1 and 3 gives One now applies equations ( g - l ) and uses the permutation properties defined i n equation (3) of S c h i f f (196*0 on the functions ( v . , v - ) to y i e l d which i s i d e n t i c a l to the expression defined i n equation (lO) of S c h i f f (1964), 

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