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A variational wave function for the ground state of He³, and its application to the D(p,y)He³ capture… Banville, Marcel Roland 1965

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A VARIATIONAL WAVE FUNCTION FOR THE GROUND STATE OF He 3, AND ITS APPLICATION TO THE D( p,Y)He 3 CAPTURE REACTION, MARCEL'' BANVILLE B.Sc, Universite de Montreal, 1956 M„Sc„ University of B r i t i s h Columbia, 1959 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Physics We accept t h i s t hesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA FEBRUARY, 1965 In p r e s e n t i n g 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 L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study* I f u r t h e r agree that per-m i s s i o n 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 h i s r e p r e s e n t a t i v e s . I t i s understood t h a t , c o p y i n g 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 w r i t t e n permission?, Department of *jp' The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada Date 2. C A l a-v-TABLE OF CONTENTS ABSTRACT v i ACKNOWLEDGEMENTS v ' i i i INTRODUCTION 1 PART I. CHAPTER 1„ The Hyperspherical Internal Coordinate System 15 CHAPTER 2„. The S, P, and D wave Functions 1) Separation of the Variables 20 2) The T r i a l Wave Functions i ) The 3-state Functions 23 i i ) The P-state Functions 25 i i i ) The D-state Functions 26 3) The Singular Configurations i ) The In-Line P o s i t i o n A=o 28 i i ) The E q u i l a t e r a l P o s i t i o n E=0, G=0 29 CHAPTER 3. The K i n e t i c and P o t e n t i a l Energy Matrix Elements 1) K i n e t i c Energy Matrix Elements 31 2) P o t e n t i a l Energy 34 PART I I . The Electromagnetic Transitions CHAPTER 4, The Wave Functions for the Continuum States 40 CHAPTER 5. 1) The Electromagnetic Interaction 51 2) The Magnetic T r a n s i t i o n (M1(S-*S) 52 •- 3) The E l e c t r i c T ransitions 56 - i i i -4) The E1(P-*S) T r a n s i t i o n 60 5) The El(P-D) T r a n s i t i o n 62 6) The E2CD-S) T r a n s i t i o n 64 CHAPTER 6, Evaluation of the Integrals and Approximations 67 CHAPTER 7, Numerical Results and Discussion 1) Radial Orthogonal Functions 72 2) P o t e n t i a l s Examined i ) Central P o t e n t i a l s with no Core - 73 i i ) Central P o t e n t i a l with a Hard Core 74 i i i ) P o t e n t i a l s Containing a Tensor Interaction 74 3) E f f e c t i v e P o t e n t i a l s 75 4) Wave Functions 76 5) The E l e c t r i c Dipole T r a n s i t i o n EHS-P) 80 6) The Magnetic Dipole T r a n s i t i o n MKS-S) 81 CONCLUSION Wave Functions and P o t e n t i a l s 84 APPENDIX 1. 1) The S-state Solutions 2) Symmetries of the Wave Functions 3) The Radial Equation APPENDIX 2. The D-state Solutions APPENDIX 3. 1) The K i n e t i c Energy Matrix Elements 102 2) The P o t e n t i a l Energy Matrix Elements 103 APPENDIX 4, 88 92 93 99 1) The Electromagnetic F i e l d 105 - i v -2) The Interaction 3) E l e c t r i c Multipole Matrix Elements 4) Magnetic. Multipole Matrix Elements APPENDIX 5, Spin and Isotopic Spin Matrix Elements 1) The Spin Operator 2) Proton and Neutron Operators 3) Magnetic Moment Operator APPENDIX 6, Reduction of the J Integrals 1) Magnetic Dipole T r a n s i t i o n 2) E l e c t r i c Multipole Transitions 3) Detailed Form of the Relevant Integrals BIBLIOGRAPHY 107 109 111 113 115 117 120 124 124 126 FIGURE 1, FIGURE 2. FIGURE 3„ FIGURE 4„ FIGURE 5. FIGURE 6„ FIGURE 7, FIGURE 8. FIGURE 9„ The Internal Variables following page 41 Mixed Symmetry Bound State Function f o r a Fixed r following page I The Radial Polynomials, F N(R) f o r 4=0, N=l i n Terms of the Parameters \ and o following page The F i r s t Three Orthogonal Polynomials f o r the Symmetric S-state f o r X=,738, v=,3 following page F i r s t Three Orthogonal Polynomials for Mixed Symmetry S and D-state f o r X=J738, u=,3 following page E f f e c t i v e P o t e n t i a l f o r Central C - l i n the Symmetric S-state following page E f f e c t i v e P o t e n t i a l f o r Central P o t e n t i a l with Core i n the Symmetric S-state.. following page E f f e c t i v e P o t e n t i a l f o r Central P o t e n t i a l with Core Reduced A r b i t r a r i l y i n the Symmetric ,,S-state « following page E f f e c t i v e P o t e n t i a l f o r Feshbach and Pease Tensor P o t e n t i a l i n the Symmetric S-state following page 69 72 72 72 75 75 75 75 -V FIGURE 10. The Symmetric S-state Functions, following page 78 FIGURE 11. The Mixed Symmetry S-state following page 78 FIGURE 12. The Mixed Symmetry S-state with i?=0.2 and 0.3, \=0.738 following page 78 FIGURE 13. The D-state Functions following page 78 FIGURE 14. He 3(Y,p)D Experimental following page 80 FIGURE 15. He 3(v,p)D following page 80 section f o r Central P o t e n t i a l \=.738; MS=4; MM=3 3 FIGURE 16. He (\,p)D Theoretical E1(S-*P) Cross-Section f o r Shorter Range following page 80 3 FIGURE 17. He (y,p)D Theoretical E1(S-«P) for Hu and Massey Tensor P o t e n t i a l \=.738, u=.3, MS=4, MM=2, MD=3. Zero Range and Non Zero Range Deuteron Function following page 80 3 FIGURE 18. He (Y,p)D Theoretical E1(S-»P) for Feshbach and Pease Tensor P o t e n t i a l . MS=6, MM=2, MD=4, Zero Range and Non Zero Range Deuteron Function following page 80 FIGURE 19. The Continuum Scattering Function f o r the S-state following page 83 FIGURE 20. The Isotropic Magnetic Dipole Cross-Section TABLE 1. Three Body Cal c u l a t i o n s with a Tensor Force TABLE 2. C l a s s i f i c a t i o n of the T r i t o n Wave Function TABLE 3. Results of the V a r i a t i o n a l C a l c u l a t i o n following page 83 following page 2 following page 14 following page 76 TABLE 4„ Central P o t e n t i a l s Expansion C o e f f i c i e n t s f o r the Wave Functions. following page 76 TABLE 5. Tensor P o t e n t i a l s Expansion C o e f f i c i e n t s f o r the Wave Functions. following page 76 TABLE 6, S'-state P r o b a b i l i t i e s following page 83 TABLE 7. Representations of the Permutation Group following page 114 - v i -ABSTRACT, The present work proposes t r i a l wave functions for the three-body problem i n nuclear physics taking into account the group t h e o r e t i c a l c l a s s i f i -c a tion of the states given by Derrick and B l a t t and by Verde. We s t a r t from the Schroedinger equation i n the i n t e r n a l v a r i a b l e s (the i n t e r p a r t i c l e distances) obtained by Derrick from a summation over the matrix elements f o r k i n e t i c energy and p o t e n t i a l energy extended over a l l variables except the i n t e r n a l v a r i a b l e s . An '-'.equivalent" Schroedinger equation i s setr.up using a p o t e n t i a l due to Eckart. This equation has the same form as the o r i g i n a l Schroedinger equa-t i o n i n the region outside the range of the nuclear forces. The v a r i a b l e s i n t h i s equation can be separated i n a hyperspherical coordinate system and the r e s u l t i n g separate equations can be solved. Then using a superposition p r i n -c i p l e the solutions of the o r i g i n a l equation are expanded i n terms of solutions to the "equivalent" equation. The Rayleigh-Ritz v a r i a t i o n a l procedure i s used to determine the coef-f i c i e n t s of the expansions with a given p o t e n t i a l . Because of the computa-t i o n a l labor i n v o l v e d ^ ' . s i g n i f i c a n t approximation i s 'done i n allowing only the leading terms i n the angular v a r i a b l e s to appear i n the expansions while keeping a s u f f i c i e n t number of r a d i a l terms to insure convergence. The present functions with a r a d i a l v a r i a b l e R = V r ^ + r23 + give l e s s than 1/2 of the binding energy predicted by B l a t t * Derrick and Lyness (1962) who used a r a d i a l v a r i a b l e R = + r ^ + . This shows that our appro-ximation with the former r a d i a l v a r i a b l e i s indeed too crude to predict a - v i i -r e l i a b l e value f o r the binding energy and that more angular terms must be included i n the expansions, at least f o r the prepbnderent symmetric S-state. Wave functions derived by the Rayleigh-Ritz v a r i a t i o n a l p r i n c i p l e 3 are used t o ^ c a l c u l a t e cross sections f o r the reaction D(p, y)He . The e l e c t r i c dipole cross section depends very s e n s i t i v e l y on the p o t e n t i a l used to derive the wave function and ai-comparison with experimental data provides a test of the various model assumptions used to describe the nuclear i n t e r a c t i o n . A r e a l i s t i c potential must contain a tensor p o t e n t i a l plus a hard core i n the central p o t e n t i a l . The tensor i n t e r a c t i o n couples the S and D states and i s 3 necessary to explain the quadrupole moment of He while the hard core produced the required mixed-symmetry S-state. The experimentally observed i s o t r o p i c component of the gamma ray y i e l d i s a t t r i b u t e d to a magnetic dipole t r a n s i t i o n between a continuum quartet S-state and the mixed-symmetry component of the ground state wave function. For a range of the v a r i a b l e parameter used i n the c a l c u l a t i o n comparison with experiment.requires a 5% admixture of the mixed-symmetry S-state i n the ground state wave function. - v i i i -ACKNOWLEDGEMENTS, Dr. P.D. Kunz made t h i s work possible by suggesting the special choice of coordinates that made the problem soluble. He collaborated i n many phases of the-'work, supplied the phenomenological central p o t e n t i a l s to te s t the method exposed here and made part of the numerical c a l c u l a t i o n s , mainly concerning the magnetic dipole t r a n s i t i o n c a l c u l a t i o n s . Very useful discussions with Dr. McMillan at the Univ e r s i t y of B r i t i s h Columbia permitted to draw a pertinent conclusion concerning the discrepancy between the binding energies predicted by our method and that of B l a t t et a l . I wish to extend my gratitude to Dr. G r i f f i t h s who acted as super-v i s o r during my stay at the University of B r i t i s h Columbia i n the absence of Dr. Kunz. The National Research Council supported the present work by providing an assistanship followed by a research grant. They also bore the cost of the numerical c a l c u l a t i o n s . A V a r i a t i o n a l Wave Function f o r the Ground State of He^, and i t s Ap p l i c a t i o n to the D(p,y)He^ Capture Reaction. Introduction. E a r l i e r t h e o r e t i c a l studies (1-7) of the capture reaction D(p,y)He , and i t s inverse, have been made using approximations to the deuteron wave function which neglected the d i s t o r t i o n and p o l a r i z a t i o n of the deuteron i n the f i e l d of the i n t e r a c t i n g proton. The continuum nucleon-deuteron state was represented by the product of the deuteron function and a plane wave f o r (1) (7) the free nucleon., except in'.the works of Burhop and Massey ^ and Eichmann Eichmann took into account the e f f e c t of the nuclear force on the continuum function. In order tQ'-calculate the e l e c t r i c dipole t r a n s i t i o n to the bound (6) s t a t e s the bound state wave function has been represented, except i n Rendell and Eichmann's work* by the f u l l y symmetric S-state only ( t h i s state w i l l be denoted by S i n the f o l l o w i n g ) . Rehdell considers i n addition the mixed symmetry S-state (to be denoted by S') and the D-states. Eichmann also consi-•->. ders the S'-state. The wave functions used by Rendell were products of deuteron and proton functions with correct asymptotic behavior f o r large sepa-r a t i o n s . The e f f e c t of symmetry was not treated exactly i n h i s cjioice of asymptotic r a d i a l functions f o r the ground state and f o r the continuum. The e f f e c t of the Coulomb repulsion has been taken into account simply by multi-p l y i n g the f i n a l cross-sections by the simple Coulomb penetration factor. The other authors used wave functions without the correct asymptotic behavior f o r large separations of the nucleons* - 2 -In the early c a l c u l a t i o n s f o r the bound state energy of the three-(8 9) bodyjsystem s only c e n t r a l forces were used y and the binding energy was seen to be very s e n s i t i v e to the shape of the two-nucleon p o t e n t i a l . Central forces derived from the twp-nucleon data were seen to overbind the t r i t o n e i t h e r with exponential or Yukawa wells* Subsequent studies took into account the tensor forces necessary to explain the deuteron quadrupole moment^^ , and most of them agreed on an admixture of D-states of about 4% i n the three-body wave function, Inclusion of the tertsor forces decreases the Calculated binding energy appreciably since i t s e f f e c t i n binding the three-nucleon system i s (14) much l e s s e f f e c t i v e than i n the deuteron. However j, more rigorous treatments produced overbinding with p o t e n t i a l s adjusted to f i t the two-body data at higher energies„ A summary of the r e s u l t s using a tensor force i s given i n (23) Table l s the l a t e r r e s u l t s of Blatt<> et a l are also displayed f o r reference. The t r i a l functions used i n those studies did not take into account the angular dependence of the wave funct i o n . I t has been shown by Feshbach and Rubinow^*^ and by Verde^*^ that p o t e n t i a l s averaged over angles must be used i n the equivalent two-body problems They have indi c a t e d the,procedure f o r the symmetric S-state only. The averaging over angles has a tendency to r a i s e the wells and therefore makes the " e f f e c t i v e p o t e n t i a l s " less binding* In order to obtain a better f i t to the binding energy with ce n t r a l f o r c e s , p o t e n t i a l s with a hard core have been used with a s a t i s f a c t o r y reduction (17 18) i n the binding energy«- Ohmura et a l 3 have shown that the larger the core radius, the more important was the reduction i n binding energy f o r the triton., 3 Another desirable e f f e c t was to reduce the c a l c u l a t e d Coulomb energy of He to bring i t c l o s e r to the experimental value (CL77 Mev); most previous authors found a Coulomb energy near 1 Mev. Yukawa wells produce more binding than simple expo-(To follow page 2,) Table 1 - Three-Body C a l c u l a t i o n s with a Tensor Force,, P o t e n t i a l Shape % S«-State t % D -State % of Experimental Binding Energy,, Gerjouy and Schwinger (1942) Square well 4 32 Feshbach and R a r i t a (1949) Square well Yukawa 1.7 21 85 Clapp (1949) Square well 69.9 Hu and Hsu (1949) Square well Yukawa 4 7 * 63-143 Feshbach and Pease (FP) (1952) 2.2-3.6 89-118 B l a t t et a l . (1962). FP-1 Yukawa ^68 3.0 . 132 FP-2 Yukawa .52 3.3 119 FP-3 Yukawa 101 HM Yukawa 1.2 3.6 GB .15 8,0 67 HJ .14 7*5 30.5 Yale. .23 7.4 ••• 29.4 By varying the range of the cen t r a l and tensor potentials,, n e n t i a l wells but the d i f f e r e n c e i n binding decreases as the hard core radius i s increased. For large core r a d i i s the binding energy i s much less s e n s i t i v e to the shape of the p o t e n t i a l . (19) Derrick used both tensor and s p i n - o r b i t forces with the p o t e n t i a l of Gammel and T h a l e r ^ c o n t a i n i n g a hard core of radius 0.4 Fermi; the P-sta-tes are then found to be completely n e g l i g i b l e (= 0.001% p r o b a b i l i t y ) . With a t r i a l wave function containing over 40 v a r i a t i o n a l parameters„ and With the (21) (22 23) p o t e n t i a l of Bruekner and Gammel , Derrick et a l 5 found 7% D-states -4 and only 6.5 x 10 P-state p r o b a b i l i t i e s . Their examination of the more modern p o t e n t i a l s ^ ^ ' ^ " ^ gives even smaller binding energies. The D-states p r o b a b i l i t y i s found td be between 3 and 8%; the S'-state p r o b a b i l i t y between 0.14 and 1.2%, and the P-state p r o b a b i l i t y i s only 0.03%. The c a l c u l a t i o n of three-body capture or photodisintegration cross sections» using the various wave functions which give the correct binding energy f o r the t r i t o n s provides a good test of these functions since the capture cross section at very low energies depends c r i t i c a l l y on the admixture of S f - s t a t e into the symmetrical S-state and on the shape of the wave functions. The amount of S'-state mixed i n may be very s e n s i t i v e to the core s i z e . This i s an e f f e c t we want to Investigate. Also-, the extrapolation of the capture cross section to thermal energies i s of astrophysical i n t e r e s t i n the energy production i n s t a r s , and a good model f o r the capture crdss section including both S and S J -states i s necessary to give a r e l i a b l e extrapolation to very low energies. The information given by the e l a s t i c s c a ttering of high energy elec-3 3 trons from H and He provides another c r i t i c a l test f o r the wave functions -4-obtained by using various models for the nuclear interactions since i t has been (26) shown by S c h i f f to depend s e n s i t i v e l y on the amount of S'-state admixture. However8 comparison of t h e o r e t i c a l predictions on e i t h e r the r a d i a t i v e capture or photodisintegration cross sections with experimental data can give more d e t a i l e d information on the r e l a t i v e p r o b a b i l i t i e s of various components i n the ground state wave function. Experimental studies of the capture reaction 3 (27) D(p, y)He have been made by G r i f f i t h s at a l with low energy protons. For proton energies from 0.275 to 1,75 Mev.P the S-wave cross section a varies s from 0.10 to 0.18 jj,b while the P-wave cross section goes from 0.87 to 4,74 u.b« The S-wave cross section gives r i s e to the magnetic dipole r a d i a t i o n and depends c r i t i c a l l y on the S ! - s t a t e admixture. 3 Measurements on the inverse reaction He (Y ap)D with gamma energies (28) from 8.5 to 22 Mev, have been made by Berman et a l , The r e s u l t s were found ( 2 3 ) to disagree with the t h e o r e t i c a l predictions based on Gaussian 5 or pure (4) 3 exponential wave functions f o r He . However good agreement was obtained (3) with the predictions based on Irving's wave function, . This wave function i s of the form 4£> =r A j* /R' ; fati+ni + rv ( i ) where the r are the internucleon distances and the si z e parameter ^ was adjusted to bring the peak of the cross section to 11 Mev. (1/u. = 2.6 Fermi). (29) Warren et a l have made measurements with low energy gamma rays to provide a comparison with the capture data; the agreement was found to be well within experimental e r r o r s s being an accurate check on the p r i n c i p l e of d e t a i l e d balance . . . 5 -The Yale group^*"^ have extended those measurements up:to gamma energies of 42 Mev. f i n d i n g again very good agreement with Irving's prediction,, also using 1/H = 2.6 Fermi. Most t h e o r e t i c a l predictions u n t i l now gave too large an estimate f o r th^. maximum cross section f o r the.photodisintegration reaction with the (2) exception of Gunn and Irving using wave function (1) and Verde with a Gaussian wave function, Verde predicts a maximum d i f f e r e n t i a l cross section da = 60 |j/b. (The symbol da w i l l be used here to mean da/dfl) compared to the experimental value of Stewart et a l ^ ^ ? da = 100|_b. The f i r s t , t h e o r e t i c a l study was made by Burhop and M a s s e y ^ using only a t o t a l l y space-symmetric S~state of exponential behavior; they predicted a maximum e l e c t r i c dipole d i f f e r e n t i a l cross section (El) da of 270u.b. (31) Austern finds that at small energies the magnetic dipole t r a n s i t i o n (Ml) to the mixed-symmetry S'-state accounts f o r the neutron absorbtion by deuteron while the e l e c t r i c quadrupole t r a n s i t i o n s E2(S -. S) and E2(D -» S) are completely (2) negligible,. Verde shows that the M1(S -> S') contribution i s n e g l i g i b l e compared to the e l e c t r i c dipole E1(P -» S) near the maximum of the cross section, (4) Rossetti neglects the M l - t r a n s i t i o n altogether., and using a'.triton function with an exponential shape s he predicts a maximum d i f f e r e n t i a l cross section for photodisintegration of 215 |_b at a gamma ray energy of 16»5 Mev„ f o r the EK S - P) t r a n s i t i o n . R e n d e l l ^ ^ made a more complete study of the various possible channels fo r the capture reaction i n v o l v i n g E l 9 E2, and Ml t r a n s i t i o n s at low energies. Using a crude approximation f o r the functions of i n t e r n a l v a r i a b l e s but t r e a t i exactly the Euler angle and the spin«isospin functions, he only approximately takes into account the symmetry of the wave functions, and therefore* of the exchange terms i n the matrix elements. In the M l - t r a n s i t i o n , i n f a c t , one finds c a n c e l l a t i o n s of the matrix elements due to the exchange terms. Rendell's conclusions can be summarized as follows; at 1 Mev. laboratory proton energy, E1(P -» S) i s found to be 3,77 |ib ( the experimental value 2 i s 3.13 p,b f o r the s i n 9 component). The other t r a n s i t i o n s studied give Ml (iS 4 S) = 0.014 |ib E l (P -» D) = 0.06 u,b E l (JF - D) = 0.002 |j,b E2 (D - S)••-•= 0.003 |ib E2 (S - D) = 0.0001 |ib i f one assumes 4% D-states and 0.5% mixed S'-state. The i s o t r o p i c components are MKS -• S 1) and E2 (S -* D), they are too small to explain the experimental i s o t r o p i c component (0.J.1 |ib)» however, i f we allow 4% mixed S'-state, t h i s would bring the magnetic dipole cross section to the r i g h t value to explain the i s o t r o p i c component . A A more rigorous treatment which does not include the D-states admix-ture to the ground state has been made by E i c h m a n n ^ with a ground state •< (32) function having a Gaussian shape derived by Volkel using central forces only. This wave function gives 78% of the binding energy and the correct Coulomb radius. Volkel gives, a 4% p r o b a b i l i t y f o r the S'-state* but Eichmann does the c a l c u l a t i o n s using a 1% p r o b a b i l i t y only, i n view of her r e s u l t s f o r the thermal neutron capture cross section which Is proportional to the S'-state p r o b a b i l i t y . * P.D. Kunz, private communication. Near the maximum of the d i f f e r e n t i a l dross section evaluated at 45° (E = 15,5 Me v « ) s the contr i b u t i o n of the e l e c t r i c dipole t r a n s i t i o n from Y the symmetric part of the S-state i s da = 48 |_b» Including the contribution of the e l e c t r i c quadrupole t r a n s i t i o n ^ r o m the S-state 9 the value of the cross section i s r a i s e d tender = 66 u.b„ The interference from 1% S'r-state admixture raises the cross section s t i l l f urther to dcr = 69 u.b. For gamma ray energies higher than 17.5 Mev., the interference from the S'-state tends to decrease the cross section instead by a few u,b. On the other hand, i n the t o t a l cross section* the interference term due to the quadrupole t r a n s i t i o n disappears leaving only the e l e c t r i c dipole interference due to the S'-state superposed to:the c o n t r i b u t i o n from the symmetric S-state. The magnetic dipole and quadrupole contributions from the S'-state are n e g l i g i b l e (da = 0.05 u.b., and 0.25 i_b r e s p e c t i v e l y ) . The e l e c t r i c quadru-pole t r a n s i t i o n E2(S'-» D) i s seen to'have a maximum for the d i f f e r e n t i a l cross 2 4 section of 27 \j,b near Ey = 22 Mev. Its angular d i s t r i b u t i o n ( s i n 9 - s i n 9) 2 has the e f f e c t of d i s t o r t i n g the angular d i s t r i b u t i o n i n s i n Q forward f o r 3 3 3 He and backwards f o r H The e f f e c t on the D(p»y) He d i s t r i b u t i o n i s 5 3 times as large as f o r the D(n, y)H rea c t i o n . The extrapolation to vgry low energies gives f o r the thermal neutron -22 3 capture cross section v..o_,= 1.49 x 10 cm /sec (where v„, i s the neutron velo-• N = N (33) c i t y ) compared to the experimental value pf Kaplan et a l : v N G g = 1.25 x "22 3 10 cm /sec. for v^ = 2200 m/sec„ It i s seen that the disagreement with experiment would be worse i f one takes 4% S'-state instead of a 1% p r o b a b i l i t y . Eichmann makes two separate c a l c u l a t i o n s . In the f i r s t , the free nucleon i n the continuum wave function i s represented by a plane wave 3 and i n the second the e f f e c t of the nuclear i n t e r a c t i o n i s taken into account by using a Serber p o t e n t i a l with a Gaussian shape from Ernst and Flugge . An i n t e g r o - d i f f e -r e n t i a l equation i s set up .from the Schrodinger equation by i n t e g r a t i n g over the Euler and space v a r i a b l e s , except the deuteron-nucleon distance, and t h i s i n t e g r o - d i f f e r e n t i a l equation i s solved using a v a r i a t i o n a l function which i s required to s a t i s f y the i n t e g r o - d i f f e r e n t i a l equation f o r a set of energies i n the range from 0 to 30 Mev. The d i f f e r e n c e i n cross section obtained bet-ween the two c a l c u l a t i o n s becomes noticeable around 3 Mev., the e f f e c t of the nuclear i n t e r a c t i o n being an increase In the maximumr.of the t o t a l cross section by about 20% while the t a i l decays f a s t e r . Comparison with experiment shows that the maximum of the cross section has the correct s i z e but f a l l s near 15.5 Mev. instead of 11.5 Mev.^®\ and the t a i l does not decay f a s t enough even when the nuclear i n t e r a c t i o n i n the continuum state i s taken into account, A l l t r a n s i t i o n s considered by Eichmann have no i s o t r o p i c components to the (27) angular d i s t r i b u t i o n while a 3% contribution i s observed by G r i f f i t h s et a l at low energies. The e l e c t r i c dipole t r a n s i t i o n from a continuum P-state to the D-states suggested by R e n d e l l ^ , and not considered by Eichmann, would y i e l d an i s o t r o p i c component around the maximum of the cross section where the magnetic dipole component, which i s also i s o t r o p i c ^ would be too small to be observed. The programme of the present study i s to examine the t r a n s i t i o n s most l i k e l y to contribute around the maximum of the cross section f o r the capture reaction.In view of the r e s u l t s of R e n d e l l ^ \ and E i c h m a m / ^ „ they are: - 9 -The e l e c t r i c dipole t r a n s i t i o n s E K P - S) and E1(P -> D). The e l e c t r i c quadrupole t r a n s i t i o n E2(D -» S), The magnetic dipole t r a n s i t i o n w i l l be examined at small energies to provide a better check on the S ! - s t a t e admixture t The e f f e c t of the s c a t t e r i n g phase s h i f t s of the continuum wave functions w i l l be examined, the exchange terms i n the matrix elements w i l l be treated properly, and a better ground 3 state wave function f o r He w i l l becconstructed from a three-body c a l c u l a -t i o n . Previous authors, except Rendell, used a bound state wave function without the proper asymptotic form i n the e x t e r i o r region where the main contr i b u t i o n to;.the E l and E2 capture matrix elements w i l l come. The 2 matrix elements are weighted by an " r " f a c t o r f o r E l , and an " r " f o r E2. , Further, the computational labor w i l l be g r e a t l y reduced by having orthogonal functions with only two i m p l i c i t v a r i a t i o n a l parameters i n the wave functio n , (35) (36) and also by using one-meson exchange p o t e n t i a l s from Bryan and Ramsay with a tensor force component. I t i s possible to take into account the e f f e c t of the hard core i n modern p o t e n t i a l s with a steep function of Yukawa shape with a very short range, the inverse range being proportional to the mass of the three-pion resonnance. Meson theory' predicts a repulsion f o r t h i s e f f e c t „ These p o t e n t i a l s besides g i v i n g a good account of s c a t t e r i n g data allow us to handle the matrix elements f o r the Hamiltonian by a n a l y t i c a l . meanso The method proposed here f o r the construction of the wave function corresponding to a given p o t e n t i a l w i l l hopefully convey a l l the physical information contained i n the p o t e n t i a l used. This information w i l l i n turn be displayed by the t h e o r e t i c a l predictions f o r the physical properties of the three-body system. I t w i l l therefore become possible to/.establish ar.one-te-one -10-cprrespondence between a. set of models f o r the nuclear i n t e r a c t i o n and a set of th e o r e t i c a l p r e d i c t i o n s . Comparison with experimental data w i l l help i n se l e c t i n g good models, and r e j e c t i n g the bad ones, and hopefully help to reveal what are the e s s e n t i a l c h a r a c t e r i s t i c s of the nuclear i n t e r a c t i o n . The physical properties of the three-body system which are most l i k e l y to be s e n s i t i v e to the d e t a i l s of the wave function w i l l provide most of the information on the nuclear i n t e r a c t i o n . Such properties a r e t f o r example, the electromagnetic t r a n s i t i o n s and the form f a c t o r s . We w i l l examine here the electromagnetic t r a n s i t i o n s 3 involved i n the reaction D(p,y)He and compare together the predictions obtained from a small set of nuclear i n t e r a c t i o n s that w i l l serve to i l l u s t r a t e the usefulness of the method. In the f i r s t part of t h i s work, an approximate s o l u t i o n to the three-body problem i s given and a set of orthogonal wave functions with only two i m p l i c i t v a r i a t i o n a l parameters i s porposed. R i t z v a r i a t i o n a l procedure i s applied to these functions; a l l the amplitude c o e f f i c i e n t s i n the t f i a l func-t i o n are determined by a-diagonalization procedure while the two remaining i m p l i c i t parameters are v a r i e d to minimize the expectation value of :the ground state energy. The advantage of such a procedure over a tedious parameter search with some 40 parameters cannot be over-estimated. In the second part, the e l e c t r i c t r a n s i t i o n s E1(P - S), E1(P - D), and E2(D -» S) are derived using the ground state wave function of part one. The magnetic dipole t r a n s i t i o n M1(S -» S) has been studied by Kunz using our ground state function and a continuum wave function f o r the p~D system which, P„D. Kunzj University of Colorado (pr i v a t e communication). without the Coulomb potentials corresponds to those obtained by n-D sca t t e r i n g (38) (39) ca l c u l a t i o n s of C h r i s t i a n and Gammel t and Burke and Haas , However.-, •* according to Seth who repeated the p-D sc a t t e r i n g measurements including some p o l a r i z a t i o n r e s u l t s . C h r i s t i a n and Gammel's r e s u l t s are "open to considerable q u e s t i o n " S e t h ' s data was i n considerable disagreement with the o r i g i n a l data of Sherr on which C h r i s t i a n and Gammel1s analysis was based„ No one seems to have done a phase s h i f t a n alysis of the new sc a t t e r i n g data. The c l a s s i f i c a t i o n of the states contained i n the ground state has been done by Derrick and B l a t t ^ ^ P and by Verde^*^ using Euler angle wave (41) functions., and by Cohen and W i l l i s who used the t e n s o r i a l properties of the wave functions instead of the Euler angle wave functions* They a l l assumed that the t o t a l i s o b a r i c spin T = 1/2 i s an absolute quantum number,, This i s c e r t a i n l y true f o r t r i t i u m - and the small admixture of T = 3/2 i n 3 (42) He i s seen to be n e g l i g i b l e from a S h e l l Model c a l c u l a t i o n by MacDonald 3 on very l i g h t n u c l e i ; extrapolation to the He case indicates that the cont r i b u t i o n would have a p r o b a b i l i t y of the order of 10 ^ 0 Among the ten components of the ground state wave function i n Derrick and B l a t t ' s c l a s s i f i c a t i o n ^ we have chosen to neglect a l l the P-states (22 23) in view of the r e s u l t s of Derrick et a l ' „ and the space-antisymmetric component of the S-state because there i s a high k i n e t i c energy associated with i t hence i s not expected to contribute s u b s t a n t i a l l y to the ground state Also, being interested i n the states that give the predominant contribution to the capture reaction* we need only keep the symmetric and mixed S-states and the D-states. K.,K< Seth, North Western U n i v e r s i t y , I l l i n o i s ( private communication) and Seth et a l 5 B u l l . Am. Phys. Soc. 8, 38, (1963) -12;-The Schroedinger equation f o r the three-body problem i n the s i x variables remaining a f t e r subtracting out the center of mass coordinates becomes soluble provided one replaces the nuclear i n t e r a c t i o n by an " e f f e c t i v e p o t e n t i a l " i d e n t i c a l with an Eckart p o t e n t i a l , and also the c e n t r i f u g a l b a r r i e r term by an expression which reduces to the correct form at small inter-nucleon distances. These expressions are t eYR/U-(2) 2 2 2 2 where R = + r ^ + ^ . The solutions to the Schroedinger equation obtained have the corr e c t asymptotic behavior outside the range of the p o t e n t i a l and also f o r small R; they are used to f i n d the wave function and the energy expectation value with two-nucleon p o t e n t i a l s having c e n t r a l and tensor parts. The L'S i n t e r a c t i o n i s neglected since the even state s p i n - o r b i t term i s s t i l l poorly (44) determined . I n c l u s i o n of t h i s i n t e r a c t i o n would r e s u l t i n a repulsibn.: f o r the diagonal D to D matrix elements. The Schroedinger equation i s a n a l y t i c a l l y summed over the Euler angles and the s p i n - i s o s p i n v a r i a b l e s following the method described by (45) Derrick . Then, our s p e c i a l choice of i n t e r n a l coordinates permits us to uncouple the set of d i f f e r e n t i a l equations with the substitutions described above f o r the e f f e c t i v e p o t e n t i a l and the c e n t r i f u g a l b a r r i e r term . This procedure has been suggested by P.D. Kunz s U n i v e r s i t y of Colorado, -13;-The set of variables used i s s i m i l a r to the hyperspherical coordinate/system; special cases of t h i s coordinate system have been used by others i n the three-body problem^ Variables are separated, and the solutions to the d i f f e r e n - * t i a l equations i n one of the angular v a r i a b l e s are ordinary exponential functions, and Jacobi polynomials are obtained for the other angular v a r i a b l e and f o r the r a d i a l s o l u t i o n . For each v a r i a b l e we thus obtain a complete set of orthogonal functions and each o r b i t a l i n the wave function i s taken to be a l i n e a r combi-nation over the three quantum numbers. The Jacobi pdjrynomials are functions completely symmetric with respect to permutation of the p a r t i c l e s and the symmetry of .the o r b i t a l s i s contained i n the exponential functions which provide a,.;basis f o r the c l a s s i f i c a t i o n of the states. We note further that f o r the D-state functions coupled by the k i n e t i c energy operator, the decoupling procedure, which i s exact within the approximation described" above, f i x e s the r e l a t i v e amplitudes of the three parts of the D-state, thus reducing the number of a r b i t r a r y parameters. PART 1. (1-1) CHAPTER I, The Hyperspherical Internal Coordinate System, The problem of c l a s s i f y i n g the states involved i n the ground state configuration of the t r i t o n has been examined in d e t a i l by various a u t h o r s a n d w i l l not be repeated here; only a summary of t h e i r r e s u l t s w i l l be given where needed* Derrick and B l a t t ^ ^ found the wave function of the t r i t o n to be a sum over ten p a r t i a l waves numbered one to {45) 10. We use the notation of Derrick where f^ denote the functions of the i n t e r n a l coordinates only and the ^ are products of Euler-angle and s p i n - i s o s p i n functions. We have f o r the t o t a l wave function + (. t - U Y>~ + n %, * A, V6,J * a K , % * k, Vr.J * k (- <, Vj Table 2 giy/es the quantum numbers and the symmetries of the various components where s, n, and a stand f o r symmetric, mixed and antisymmetric res-p e c t i v e l y i n such a way that the t o t a l wave function Y be completely 7 a n t i -symmetric with respect to permutation of the p a r t i c l e s . Derrick expresses the k i n e t i c energy operator — J'jjj ( Yf •*• + Y 3 J i n terms of the Euler angles, c*» B, y a n c * t n e t r i a n g u l a r i n t e r n a l coordinates and obtains (To follow page 14«) Table 2 - C l a s s i f i c a t i o n of the T r i t o n Wave Function*. Spectroscopic Permutation Symmetry„ C l a s s i f i c a t i o n L, S Internal Euler angles Spin-isospin M> 0 1/2 s s a 0 S/2 b 1/2 a s s 0 0 1/2 m s m 0 1 1 1/2 1/2 s a a a s a 0 0 1 1/2 m a m 0 1 3/2 m a m 0 2 3/2 m s m 0 D l / 2 2 3/2 m s m 2 2 3/2 m a m 2 -15-where {X } i s the set of the three i n t e r n a l coordinates r,„ s r „ 0 > r„,. the 1 p J 12 23 31 —» three Euler angles, o/» B,3 y> a n c * t' i e t n r e e centrer of mass coordinates r^ T = . 2 Y f i l l +L^-+ + -*£ 9 X ) u-3) L ^ s L,. and are the components of the o r b i t a l angular momentum re f e r r e d to the body-axes. The system of body coordinates (X, Y, Z) i s chosen following Derrick such that the t r i a n g l e formed by p a r t i c l e s 1, 2,3 l i e s i n the X-Y plane with the Z-axis i n such a d i r e c t i o n that going from p a r t i c l e 1 to 2 i s a counterclockwise walk aroUhd the Z-axis. The X-axis i s along the d i r e c t i o n of the lar g e s t moment of i n e r t i a and the Y-axis along the d i r e c t i o n of the smallest moment of i n e r t i a . The Euler angles orient t h i s body-frame with respect to the center of mass frame having axes p a r a l l e l to the laboratory frame. The matrix M i n equations (1-2) and (1~4) a r i s e s from the transformation of the coordinates and A i s the area of the t r i a n g l e formed by the three p a r t i c l e s . 2 'Z. *Z, -16'-The summation over the Euler angles and the spi n - i s o s p i n v a r i a b l e s has been c a r r i e d out by Derrick y i e l d i n g the matrix elements (1-6) Spin —IT o The r e s u l t i s a matrix operator i n the i n t e r n a l coordinates ' 2 3 Hft>7 g <t ID 1 / 2 3 2 1M ss r, (1-7) T, DP / e 9 /o that w i l l be made to operate on the vector whose components are the c o e f f i c i e n t s of the functions i n equation (1-1) ' ^ 3 1 1 ' T -'p? 7~ -3 1 ) 5" 7 e0(rs t- -M?) 4 * (1*8) (1-9) / o /? €0 A O Q 3 (1-10) -17.-where In discussing the conditions required to avoid d i s c o n t i n u i t i e s of the functions of i n t e r n a l coordinates at the singular configurations (the i n - l i n e ^ and the e q u i l a t e r a l positions of the t r i a n g l e formed by the three p a r t i c l e s ) 9 Derrick uses the following set of i n t e r n a l coordinates: A = area of t r i a n g l e F = + ^ ~ ) ( 1 ' U ) subjected to the conditions 0 £ A £ oo - oo £ F S G £ oo They are r e l a t e d to R and A by (1-12) *4 _ With t h i s set of v a r i a b l e s (A, F , G ) , the operator T takes a form s with no c r o s s - d i f f e r e n t i a l s : 9 £ 7 <i-i3) I t w i l l be seen to be very convenient to use another set of coordi-nates R, 8 and cp analogous to the spherical system of coordinates which also does not give r i s e to c r o s s - d i f f e r e n t i a l s i n T » The system i s defined by s -18-r F rs (1-1-4) We w i l l frequently use the notation x = cos 8 » . z = s i n Q (1-15) Given the d i f f e r e n t i a l equation Ha ^ (1-16) The t r i a n g u l a r coordinates are expressed i n terms of the hyper* sphe r i c a l coordinates i n Appendix 1. We w i l l make use the following r e l a t i o n s A — Rz z In the hyperspherical system, T takes the form + 2 . + _ J 2 ! ) x ax / - x* (1-17) the i n t e g r a t i o n f a c t o r i s r(x) / a Q (x) where r(x)=exp { J[a 1(x)/a 0(x)]dx}. -19-Hence s the volume i n t e g r a l i n our system of coordinates i s seen to b e 1 •> 2TT ' 5 dR f xdx J dcp JdV = J R 5 d / C1-I8) where r e s t r i c t i o n of x from 0 to 1 insures that A _ 0. In terms of the new v a r i a b l e s , T._._. becomes DD 8 10 o I L L . * l r + — z \P o ® ° / — . _J. _ _ _ _ _ _ _ )p (1-19) 10 -20 CHAPTER 2. The S, P, and D Wave Functions.. The Schroedinger equation i n the i n t e r n a l v a r i a b l e s i s obtained by adding to the k i n e t i c energy matrix (1-7) the matrix obtained by the same method f o r the po t e n t i a l ( r e p l a c i n g T by V i n equation (1-6)), and operating on the vector 1. Separation of the v a r i a b l e s . The S~state functions are coupled together, to the P, and to the D-state functions by the p o t e n t i a l ; no attempt w i l l be made to disentangle the mixing due to the p o t e n t i a l . However, i n the region where the coupling due to the po t e n t i a l can be neglected the S-state functions f^» f^* a n c * 3^ w i l l be "solutions of the d i f f e r e n t i a l equation 2M 1 >2 2M or, d i f i n i n g V = — r — V and 7— = - — E, solutions of (T + V + £-)f. = 0 (2-3) s 4 i This equation i s separated i n Appendix one into a product of functions i n R, 6 and cp r e s p e c t i v e l y i n the approximation where V = V(R). The functions V(R) that w i l l y i e l d a n a l y t i c a l ; solutions i n terms of spe c i a l functions of physics can be chosen to be invaria n t s of RIemann and of Wittaker* -21-Our choice w i l l correspond to the Riemann invariant i n the equation (23-i) of B o s e ^ 4 ^ f o r the s p e c i a l value of g = 1. Our method can be generalized i n p r i n c i p l e to include i n v a t i a n t s with g / 0 leading to wave functions vanishing at some point R ^ 0. Such functions would be suitable to study discontinuous p o t e n t i a l s with an i n f i n i t e hard core i n s i d e which we wish to have a vanishing wave function•» In the present work, we r e s t r i c t ourselves to functions corresponding to p o t e n t i a l s with no hard core hence the choice of g = 1 i n BoseJs invariant ( 2 3 - i ) . In p r a c t i c e , when we use the two-body p o t e n t i a l s , V i s a complicated function of the three i n t e r n a l v a r i a b l e s R, 9, cp. The wave functions obtained under t h i s assymption w i l l nevertheless have the correct angular dependence i n the region external to the p o t e n t i a l range. Equation (2-3) i s separated with the s u b s t i t u t i o n f = F(R) X (x) $ (cp) We obtain the two sets of normalized orthogonal functions L ~ (SL-vn + 2)/x using equations (Al-12 to 14), and 1 COS m<J ) ^ - / for m-l,*i,7,-(Al-24) 7 — , vn = O 1/2W — for nl=Z,Sl&,~" (Al-5) where Pochammer's symbol (A)^ = A(A +1) •*» (A + N.-1) i s used, and SL and m are the two quantum numbers l a b e l l i n g the functions of the sets. -22-X (x) contains Jacobi polynomials of degree ( L - l ) i n powers of x t while the functions $ (cp) embody a l l the symmetries under permutation of the p a r t i c l e s f o r various values of m<> The r a d i a l equation i s not d i r e c t l y solved but i s used to construct another equation y i e l d i n g solutions forming a complete set of orthogonal functions a l l having the same asymptotic behavior as the solutions to the o r i g i n a l r a d i a l equation, both f o r R -» 0 and R — »» They are the functions (Al-43) where C = 44 + 5, and N i s a quantum number corresponding to di s c r e t e values of the strength of the e f f e c t i v e p o t e n t i a l . I t i s e a s i l y v e r i f i e d from equation (Al-28) that the functions (Al-43) have indeed the corre c t asymptotic behavior., that i s (R) cc R * f o r small R N and 1 FJJ (R) cc g * f o r large R . R Since the volume i n t e g r a l (1-18) contains a f a c t o r R^ and the r a d i a l 5/2 functions (Al-43) a l l have the same f a c t o r 1/R j. we w i l l adopt the convention that the volume i n t e g r a l fcW= / " j * f XclX <M (2-5) be used with the corresponding k i n e t i c energy operator T g obtained from the s u b s t i t u t i o n (Al-26) where k = 21 + 3/2, -23-The number k has the same meaning as the f a c t o r i n Feshbach s arid Rubinow^*"^ 1 s equation (8). The r a d i a l function to be used with the volume i n t e g r a l (2-5) and the operator (2-6) w i l l be the functions (Al-43) 5/2 without the f a c t o r 1/R The actual t r i a l wave functions w i l l be l i n e a r combinations of products among the three sets of functions with the quantum number m selected to give the desired symmetry of the wave function under permutation of the p a r t i c l e s , < 2 - 7 > where the c o e f f i c i e n t s a may be determined by the application of RayMgh-Ritz v a r i a t i o n a l p r i n c i p l e minimizing the energy expectation value of the ground state„ 2o The t r i a l wave functions. i ) The S-state functions. As shown i n Appendix one, the completely symmetric function f^ w i l l contain the terras = Cof 1jlA<4 9 /A=0,l ,2 (2-8) and the most general form f o r the symmetric function f^ w i l l be | = Z I °-L.* FZM <V"; <2"9) the summation over JL, a- being taken such that L = -| (i - 3p. + 2) = 1, 2V 3» (2-10) -24-The completely antisymmetric function £^ w i l l have and 1=3 /*=' N-l such that L = 2 ( j i " 3 | a i + 2 ) = 1 ' 2 j 3> ° *" The function of mixed symmetry £ w i l l contain the terms ; y ever GJA+V*? (2-11) (2-12) (j,- =0,1,2,. ..;(2-13) and j oof (3/1+*)^ (3/n-z)^/ eo eo eo £ X ^ ^ ; (2-14) subject to the condition L = £ U - 3|A - K + 2) » 1,2,3,... 5 -25-In our numerical c a l c u l a t i o n s s we w i l l l i m i t ourselves to the leading term (JL = 0) i n f ^ 3 neglecting completely the antisymmetric function f and keeping the (X = 1) -term i n f^» This l i m i t s (_ to the value zero i n both f^ and f ^ . However,, a s u f f i c i e n t number of terms w i l l be considered i n the sum over N„ In t h i s approximation p the S-state functions w i l l be flf-l K=i K-l (2-15) K-t i i ) The P-state functions. The P-state functions are obtained again by neglecting the coupling due to the nuclear p o t e n t i a l . The r e s u l t i n g d i f f e r e n t i a l equations are not coupled together but the d i f f e r e n t i a l equations f o r the S-states and the P-states are s l i g h t l y d i f f e r e n t . They are., f o r the P-states obtained from equa-tions (1-9) and (1-16) A/? (2-16) -26-A f t e r separating the v a r i a b l e s under the same assumptions used f o r the S-state we have the same equations f o r the va r i a b l e s R and cps hut the x-equation i s d i f f e r e n t : The s o l u t i o n to t h i s equation i s found i n Appendix 1, I t corres-ponds to the value n = 1 i n equation ( A l - 7 ) . From (Al-17, 18) (2-18) The function f ^ i s symmetric and f,. antisymmetric and w i l l be repre-sented by equation (2-9),and (2*12) r e s p e c t i v e l y with X™(x) replaced by equa-t i o n (2-18), and the sum over i, s t a r t i n g with I ~ l,and X = 4 r e s p e c t i v e l y . The mixed functions f, and fn w i l l have the form (2-14) with b / X™(x) replaced again by X m**(x) and the sura over X s t a r t i n g with I = K + 1, The P-state functions w i l l be neglected i n our numerical c a l c u l a t i o n s , i n (17) (22 23) view of the r e s u l t s of Derrick , and Derrick et a l * j, who found for the P-states, a very small p r o b a b i l i t y . i i i ) The D-state f u n c t i o n s 6 The set of d i f f e r e n t i a l equations obtained by applying the operator T defined by equation (1-19) on the vector of D-state functions (2-19) i n the Schroedinger equation 2 i s the set of s i x coupled d i f f e r e n t i a l equations (2-20) (2-21) 2*t X 9 / - fio, 2 fit \ fl0?l In Appendix 2, a complete set of solutions i s found that s a t i s f y the system (2-21) exactly. The va r i a b l e s are assumed separable and the (p(•$)- function i s assumed to have the same form as i n the S-state solutions so that the r e l a t i o n s (Al-25) derived i n Appendix one f o r the operators — dcp 2 and d_ can be used. Solutions f o r a l l values of JL > 0 are found and fo r each JL, a l l functions can have the same r a d i a l part a s defined i n Appendix one by (Al-43). The r e l a t i v e amplitudes f o r the three D-state components are f i x e d by the mixing produced by the Hin e t i c energy operator and each one has a d i f f e r e n t angular dependence, Since the r e l a t i v e ampli-tudes are knowns the t o t a l D-state i s normalized by the f a c t o r (A2-11). In our approximations we keep the leading term only (JL = 1) -28-h,2 f fol,t flO,\ l/sYr f ASM 9 ^ — 1/3 A^n ^ - t/1 uni9 uniP (2-22) {•10, x 3» The Singular Configurations. The s o l u t i o n l i s t e d i n equations (2-15) and (2-22) can be shown (43) to be i n agreement with the conditions derived by Derrick to insure the r e g u l a r i t y of the wave function i n the i n - l i n e and the e q u i l a t e r a l p o s i t i o n s of the p a r t i c l e s i n the t r i a n g l e formed by the p a r t i c l e s . i ) The i n - l i n e p o s i t i o n A = 0. In t h i s case, the condition i s a vanishing area f o r the t r i a n g l e A -• 0, or i n terms of our hyperspherical coordinates, x 0, f o r which z = s i n 9 -» 1 and R, and cp are arbitrary,. The formal expansions (Derrick's equation (15)) i n the neighborhood •29-H are the same as the expansions of our s o l u t i o n i n powers of x, i i ) The e q u i l a t e r a l p o s i t i o n F - 0, G = 0» The neighborhood of F = 0,, G = 0 f o r A a r b i t r a r y corresponds to the neighborhood of z = 0, x = 1, R and cp a r b i t r a r y . Derrick's expansions expressed i n terms of the hyperspherical coordinates are ~ j[ + °(zUX Z'M * ^  zx«*^fj] The f i r s t of Derrick's equation (16) i s ambiguous since i t should be an expansion i n terms of factors completely symmetric with respect to permutation of any two p a r t i c l e s and the combinations of F and G which are completely symmetric are 3 (2-25) (2-24) -30-plus the combinations obtainable from these and the products of two completely antisymmetric combinations l i k e With t h i s i n mind, one can rewrite equations (2-24) as (2-26) Iua.2 <p ) t , frf.o * z [..^^J + o(z \ UKsy, ^ ^) which i s compatible with the approximation (2-22); the higher order terms being the same as i n equation (A2-16) of Appendix 2 f o r higher values of i,. CHAPTER 3 The K i n e t i c and P o t e n t i a l Energy Matrix Elements. 1. K i n e t i c energy matrix elements. I f we denote by |N:^  » the r a d i a l functions described by equation (Al-42), the matrix elements f o r the k i n e t i c energy operator (equation (2-6)) w i l l be < ^ > - < M | ^ r J / v > = l£LlyH). il + kCk±L>//v> 3 where the i n t e g r a l i s taken over x = uR. The f a c t o r k = 2X + — . The f i r s t term on the r i g h t hand side of equation (3-1) can be summed a n a l y t i c a l l y by making use of the d i f f e r e n t i a l equation s a t i s f i e d by the r a d i a l functions„ and of some r e l a t i o n s involving sums over binomial c o e f f i c i e n t s . Writing the r a d i a l function where ± K*i*>- e ' 1 (i-e-V (3-2) a. d i r e c t d i f f e r e n t i a t i o n of K ^ ( x ) gives where A* = t>(}>-0 2)K = /> ^ (3-4) -32-Since the k i n e t i c energy matrix i s symmetric, we need only evaluate fo r M _ N X , ^ ' iK+C-IJ' (3-5) AK 4 _eK 0 We note that the functions J„(x) are l i n e a r combinations of f a c t o r s N e'ier3L(l-e"lJ f o r j 4 J~£§+N-J. The sum i n equation (3-5) contains fac-C SL tors with J < — s and J M ( X ) f o r M £ N w i l l always be orthogonal .to any l i n e a r combinatio'risowith the highest power J < N but not toccombinations of factors with J < ^„ The f a c t o r s with J < ^ a r i s e from " e ^ ^* a n d B k ^ " e ^ ^ ° Only such terms w i l l contribute to the matrix elements. On the other hand, depends on the summation index K and a l l factors w i l l contribute when M = N, SL It i s possible to rearrange the terms such that the f i r s t i s <JN(x) while the remaining powers of (1 - e ) are rearranged i n a l i n e a r combination of polyno-SL mials J w ( x ) , The r e s u l t i s M (3-6) -33-The i n t e g r a l s appearing i n equation (3-6) are evaluated i n Appendix 3* We f i n a l l y have* using (A3~5) (3-7) 2 The second .term i n equation (3-1) a the expectation value of 1/R , and the p o t e n t i a l V w i l l be obtained from the matrix elements; with the transformation (3-9) where Q»„ i s the matrix of the c o e f f i c i e n t s f o r the r a d i a l polynomials ML n - £ : / )L~H(M-i \ c - i J u - n ( 3 - i o ) 2 and H1 stands f o r 1/R or V, and the functions i n (L| are the factors of i n the functions f o r the states i n equations (2-15) and (2-22)«. 2 For 1/R a the r a d i a l i n t e g r a l s i n equation (3-8) w i l l be Jo r=o (3-11) where c depends on the value of I f o r the states < M| and < N|„ The matrix 2 elements < M| 1/R |N > vanish between states of d i f f e r e n t JL since the operator 2 1/R involves only the v a r i a b l e R. -34-2» P o t e n t i a l Energy,, The p o t e n t i a l s to be used i n our numerical c a l c u l a t i o n s Will-be one meson exchange po t e n t i a l s s i m i l a r to those of Bryan ,. and Ramsay \/ l V ) =r T V ' ^ (3-12) where some terms with shorter range w i l l permit to adjust the depth of the pot e n t i a l and also provide a,:steep hard core f o r a p o s i t i v e c o e f f i c i e n t Wol. The matrix elements < M| V ( r 1 2 ) | N > can be obtained with the transformation (3-9) but the i n t e g r a l s appearing i n the matrix H^ K f o r the po t e n t i a l w i l l depend upon the var i a b l e s R»Z and cp. For a t y p i c a l term of (3-12) the matrix elements f o r f 3 f g i , f g S and f^Q w i l l be evaluated f o r aj three-body p o t e n t i a l : 1/= VtrlxJ + l / ( r 2 3 ; + / ( i O , ; ( 3 ~ 1 3 ) where )/tr^~{\-^.^)Ys +±13**7-%)%+$* V T (3-14) V g and V are the s i n g l e t and the t r i p l e t c e n t r a l parts of the potentials and s ^ i s the tensor operator 3 ~+ (3-15) (43) Derrick gives the r e s u l t of the summations over the Euler angles and the sp i n - i s o s p i n v a r i a b l e s s i m i l a r to equation (1-6) f o r the po t e n t i a l V ( r ^ ) - W e w i l l be interested here i n the r e s u l t s f o r the -35-ce n t r a l s i n g l e t and t r i p l e t ^ a n d the tensor p o t e n t i a l s (Derrick's equations ( l l ) a n d (13) ): . (s i n g l e t ) ss 1 3 1/2 1/2 \ 1 A/2 1/2/ 3 (cent r a l t r i p l e t ) ss f 1/2 -J./2 \ 1 1-1/2 1/2/ 3 (3-16) ( s i n g l e t ) D D = 0 (ce n t r a l t r i p l e t ) DD 10 8 9 10 and the tensor i n t e r a c t i o n matrix 1 3 10 1 0 -1 0 -a b 0 1 -1 0 -a 0 0 -b 3 0 1 0 a -b 0 0 -1 0 1 0 -a 0 0 -b 8 -1 0 1 0 1 0 -a b 0 0 -a 0 -a 0 -1 0 0 0 9 -a 0 a 0 -a 0 -1 0 0 b 0 -b 0 b 0 0 -1 0 10 0 -b 0 -b 0 0 0 0 -1 (3-17) -36-where A r„ In terms of R 5 z, cps, a and b are <X - f3 LtfZ If - Z / - ^ U * ^ > (3-18) Using the p o t e n t i a l matrix elements (3-16) and (3-17) and equations (2-15) and (2-22) for the state-functions the expectation value of the poten-t i a l between states i and j w i l l be . Between state 1,we have f o r a t y p i c a l term of the sums (V,, ) 11 n where V and V are the s i n g l e t and t r i p l e t p o t e n t i a l s r e s p e c t i v e l y . For s t each p o t e n t i a l , the r e s u l t i s derived i n Appendix 3, For the s i n g l e t part (2-20) where y •37-Byntbersame m e t h o d t h e terms i n the seven matrix elements are found to be of the form r^o w-t " K 0 J K -\ (3-21) where the values of V n and a are, f o r the seven matrix elements 0 m Lr vl, ; i 0 - ^ p . \/03 clrid Q. w = ( I ) K > = 5 ^ y T ^ J - «.~,*(3.4rHJ (3"22) I U „ „ ; = ~ ~ >ij- « « . = (3,-«, /o; where J stands f o r s i n g l e t , t r i p l e t or tensor. Knowledge of the angular dependence of each part of the ground state wave functiun permits one to define " e f f e c t i v e p o t e n t i a l s " which are functions of R only by averaging a given p o t e n t i a l V(r ) over angles. We obtain expressions -38-(16 ) s i m i l a r to the one found by Verde f o r the S-state„ Using the approximations defined i n Chapter 2 for the wave functions, we have f or a^central p o t e n t i a l U c ( r l 2 } <i / Ucirlt))3> ^±^[\Jj**)f^ oL\\-z*l)<t*- (3_23) and f o r a tensor p o t e n t i a l S ^ 2 U x^ r12^ (3-24) For exponential, and Yukawa type p o t e n t i a l s ^ these i n t e g r a l s can be expressed i n terms of Bessel, and Struve f u n c t i o n s ^ ^ ^ ^ 5 and t h e i r d e r i v a t i v e s i where I (z) and L (z) are the modified Bessel and Struve functions r e s p e c t i v e l y , n n The d i f f e r e n c e between the Bessel, and Struve functions i n equation (3-25) gives r i s e to a very slowly convergent seri e s f o r z larger Since the asymptotic expansions f o r the modified Struve functions are not known at present & the i n t e g r a l s -39-(3-23) and (3-24) are more r e a d i l y evaluated numerically. In our cases, our choice of wave functions enables us to do the i n t e g r a l s i n the energy matrix elements a n a l y t i c a l l y provided the r a d i a l i n t e g r a l i s done f i r s t . PART 2 The Electromagnetic T r a n s i t i o n s P CHAPTER 4, The Wave Functions f o r the Continuum States. The frame of reference used by Derrick, and B l a t t (40) and i n the > f i r s t part of t h i s work,is not convenient to describe the continuum states due to ^ complications i n v o l v i n g Euler-angle functions., In order to describe the electromagnetic t r a n s i t i o n s , i t i s more convenient to redefine the "body frame" with respect to which the t r i a n g l e formed by the three p a r t i c l e s i s (6) located . We w i l l assume that the incoming proton progresses along the laboratory z - a x i s T h e body frame coordinates w i l l s a t i s f y the following p r e s c r i p t i o n : -The body Z-axis i s chosen along the axis of minimum moment of i n e r t i a of the t r i a n g l e l y i n g i n the X-Z-plane; -The X-axis l y i n g i n the d i r e c t i o n of the next largest moment of i n e r t i a has i t s o r i e n t a t i o n chosen i n such a way that going from p a r t i c l e 1 to p a r t i c l e 2 i s a counterclockwise walk around the Y-axis„ z-axes w i l l l i e i n the same d i r e c t i o n when the proton i s f a r away from the deu-teron. A convenient set of variables (r,, q,l?) f o r the continuum i s defined i n f i g u r e 1. They are c a l l e d the i n t e r n a l v a r i a b l e s and define the "normal" p o s i t i o n of the t r i a n g l e . This choice of body Z-axis insures that the laboratory, and the body A l l the Y-coordinates being zero causes the harmonic polynomials -41-•i- ~ tr /SL^yv\ (4+2?) ' i f (-.If /Osvr\ (lj> + 1>) x3 = i f i V; = O t = /, 2, 3. - H •4- ~ r ocz ^  tf + 7?) - U uz $ if) + 2?J H L&1$ The requirement that the moment of i n e r t i a about the body Z-axis be a minimum defines the angle (j) shown i n Figure 1. I t provides the r e l a t i o n sz^ztb irx ^ T 2 A r SL A where (4-2) (4-3) = f ^ xSx*vj 2^  = area of the t r i a n g l e . (To follow page 41) FIG. 1 The Interna/ Variables -42-The choice of the p o s i t i v e roots for R and f\ removes the ambiguity i n the d i r e c t i o n of the Z-axis. The angle ^becomes r e s t r i c t e d to and - f <4* f OO The'actual" o r i e n t a t i o n of the t r i a n g l e with respect to the center-of-mass frame (moving p a r a l l e l to the laboratory f i x e d frame) i s obtained by 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 by means of the r o t a t i o n D(V ->^ f2 r) which takes a vector r = (X, Y, Z) i n the body-system into a vector ir1 = (x,y,z) i n the center-of-mass system. (IJ = T ) ^ ^ ; ( K (4-4) We note i n passing that the s o l i d harmonics^^ generated by -* a vector r i n the body system w i l l transform i n t o the center-of-mass system by the law i 1 (4-5) the Euler-angle functions i fcflx) have o r b i t a l angular momentum ^ , where m i s i t s body Z-component while m' i s i t s center of-mass z-component. The s o l i d harmonics generate three-body Euler-angle functions having d e f i n i t e p a r i t y , and symmetry under permutation of the p a r t i c l e s . Since Y.. = 0 f o r a l l p a r t i c l e s -43-and the transformation (4-5) can be expressed i n the form where y s ^ ^ j j , We w i l l make use of t h i s f a c t to convert the i n t e r -a ction operators described i n Appendix 4 i n terms of the three-body Euler-angle functions. The p a r i t y operation IF (the inversion) i s equivalent to a r o t a t i o n of 180° about the body Y-axis ?. or T T K ^ y ; = J - Y , TT+/S, Yj (4-8) and the permutation P^ defined i n Table 3 of Appendix 5 leave the Y-'axis i n v a r i e n t f o r i = 1, 2„3 (the cyclic),permutations) while the others produce a r o t a t i o n of 180° about the Z-axis (4-9) Linear combinations of the Euler-angle functions having d e f i n i t e parity,,, and symmetry under permutations can be constructed from the d e f i n i t i o n s \ * , y t f tat**)?u-»,ntt+w,'Ra-» 'M ],/x 2 l + V r \ - w L if A6~+W,'-Wi 5(4-10) (55) -44-and the orthogonality r e l a t i o n s J J 1 ' J m. m, 4./ x<C. V>1-, 8W ;(4-11) From the d e f i n i t i o n s (4-10) The properties under the p a r i t y operation f o r the Euler-angle functions are i@5mit*? = tuo) (4-13) and the permutations give they are seen to be e i t h e r symmetric or antisymmetric. The following orthogonal functions have d e f i n i t e p a r i t y ( 7T_-±J) and symmetry; they are normalized with respect to the three Euler-angles„ -45-. f o r some values of rat, and m1 they are re l a t e d to the spherical harmonics and the Legendre polynomials I* (l-l* ) = \/ ,n 1 (4-16) 0 m (4-17) The Euler-angle wave functions (4T15) are combined with the spin-i s o s p i n functions defined by V e r d e , and by Derrick, and B l a t t ^ ^ ; they are * Z fi V ' ' y (4-18) $3 = ?l3> •* °fUJ pty -rpil> *W3p , Ms - '/z - °<LI; °<(z) ^s^%. i s symmetric with respect to permutations of the p a r t i c l e s while q^, and form a mixed symmetry part. The functions :£or Mg = - 1/2, and -3/2 are obtained by interchange of°((i)s a n d ^ d ) . The i s o t o p i c - s p i n functions v^, v^? a n d v3 a r e defined i n a s i m i l a r way by replacing °f(i) byT T ( i ) , and/#(i) by ^ ( i ) i n equations (4-18) . -46-The t h r e e - p a r t i c l e s p i n - i s o s p i n functions are denoted K (P t >T,S) s' t a f t e r Derrick, and B l a t t , with Mg, the spin magnetic quantum number (projection on the body Z-axis), P denoting the symmetry type E f c = s s a 5 or m for symmetric, antisymmetric or mixed symmetry respectively) . = 1 or 2 when Pfc = m, and T i s the t o t a l i s o t o p i c s p i n . We w i l l assume T to be a good quantum number (T = 1/2) The six s p i n - i s o s p i n functions are The bound-state wave functions involved i n the t r a n s i t i o n s we wish to examine (the S and D-states) are -47-2 f o r the S ^ 2 - S t a t e > ar>d (4-21) f o r the 4 D ^ 2 ~ s t a t e ' (16) The p r e s c r i p t i o n given by Verde to construct the continuum wave functions consists i n symmetrizing the functions which evolve from the product of a deuteron function by a proton fun c t i o n . We assume here that the deuteron i s i n a pure t r i p l e t S-state f o r nucleons 1 and 2; i t can be formed only from the.spin functions q^, or q^ 5 i t s i s o t o p i c spin i s zero; t h i s i s s a t i s f i e d only by The only two possible s p i n - i s o s p i n functions f o r the three par-t i c l e s are the doublet-, and the quartet functions. ;(4-22) ;(4-23) The proton at large separations from thet deuteron i s represented by a plane wave, so that the wave function i n that region i s -48-5(4-24) where the f a c t o r -—— has been included to provide normalization over the N/2TT three Euler angles (the plane wave i s u s u a l l y normalized over the q-space). In terms of the three-body Euler functions we have ;(4-25) Let (i,j„k) represent any permutation of (1, 2$ 3); we define g^ functions of i n t e r n a l v a r i a b l e s symmetric with respect to interchange of p a r t i c l e j and k. For large values of q& they have the following proper-t i e s : 8 l - 0 g 2 - 0 (4-26) g 3 - ~u DJ x(kq) and are used to construct the symmetric, and mixed symmetry functions of the i n t e r n a l v a r i a b l e s (4-27) (4-28) -49-i n v i r t u e of (4-26), they w i l l behave f o r large q as w — * o These functions may be combined with the s p i n - i s o s p i n functions (4-22 s 23) to form the doublet, and quartet functions t f ({, M ) = ± G, 10 VH ( a i {)* { fa, ID r' i) (4-30) (4-31) Using the r e l a t i o n s (4-29), i t i s r e a d i l y v e r i f i e d that the following l i n e a r combinations w i l l behave f o r large separations q l i k e w h e r e ^ 1 / 2 m p are the proton spin functions 0r/(3) o r ^ (3) , a n d ^ ' ^ , the t r i p l e t deuteron spin functions. -50-The f a c t o r -p— comes from the symmetrization of $ 3 therefore the f a c t o r u.v j (kq) i n equation (4-25) w i l l evolve into© s and the continuum U 2. Jb ^ lit state wave functions w i l l have the form; *0 »ii »rls HsfofoUt} >(4"33) where we have s p e c i f i e d the continuum state by the quantum numbers £ 9 s s. the proton 5 and the deuteron magnetic quantum numbers. CHAPTER 5. 1 * The Electromagnetic Interaction. The i n t e r a c t i o n of nucleons with the electromagnetic f i e l d i s described i n Appendix 4. When the r a d i a t i o n propagation vector K i s taken along the z-axis of some coordinate system, the index m i n A„ (.Kt) i s the p o l a r i z a t i o n quantum number p = t l f o r c i r c u l a r l y polarized l i g h t . The other axes are so chosen that ( 5 3 ) -* i s the p o l a r i z a t i o n vector, ^  a r e u n i t vectors f o r the spherical b a s i s . The process i s ref e r r e d to the center of mass frame (x, y v z ) by the r o t a t i o n R = (<P0ffl that takes a vector k i n the center of mass frame into a vector i n the system (x ! y 1 , z') attached to the photonif i e l d . The angle (Q) w i l l be the angle between the d i r e c t i o n s of the proton and the photon. The X-multipoJe part of the vector p o t e n t i a l A^ w i l l be transformed (53) i n t o the center-of-mass system by the r e l a t i o n Ai,t (**)+</> A4? i<-> = ? > e . « ; ;(5-2) so that the angular d i s t r i b u t i o n of the t r a n s i t i o n proportional to \<fijL/m ( k ^ ) } l Z w i l l be given by Z\^A$,&,<£)\% -52-2. The Magnetic T r a n s i t i o n MKS -> S) . We wish to examine here the magnetic dipole t r a n s i t i o n MKS -> S) whose t r a n s i t i o n p r o b a b i l i t y i s derived i n Appendix 4: equations {A4-31, 32). The term M, w i l l not contribute since the i n i t i a l state i s an S--wave with l sm zero o r b i t a l angular momentum.,.. The other term M' can be further s l m p l i -13m f i e d by p a r t i a l i n t e g r a t i o n . We note that where the unit vectors of the spherical basis are I - s and the spin of the i f c ^ p a r t i c l e Using (5-4) we obtain where ^  stands f o r A ' = id- + kU+itJP* and i s the f i n a l S-state function (5-6) (5-7) (5-8) (5-9) -53-(5-10) M We now define a "reduced" t r a n s i t i o n p r o b a b i l i t y B„ (i . -» ) which i s an average over i n i t i a l states and a sum over the f i n a l states by Su b s t i t u t i o n of (5-9), and (4-33) into (5-7) y i e l d s f o r the ampli-tudes HI 15m - ^  For m = 0 S / . f l * . s _P , and W — Ue = "+ > using the matrix « M I * 0 ' J j ^ elements derived i n Appendix 5 t (A5-23 to 27)» summation over the s p i n s i s o s p i n v a r i a b l e s s and the Euler angles gives *<%,l \1t,,> - J /*V<A,zlC%r>]} -3.4-Since the bound S-state and the continuum S-state ^({^ Ms<J d i f f e r only by t h e i r energies, they w i l l be orthogonal; that i s ;(5-i2) using t h i s orthogonality r e l a t i o n , M' n can be expressed i n terms of the mixed symmetry component f . only. We w i l l assume the continuum functions G . and 3 ? i z $ i G to be uncoupled so that 3, i The reduced t r a n s i t i o n p r o b a b i l i t y may now be obtained using the orthogonality r e l a t i o n f o r the Clebsch-Gordon c o e f f i c i e n t s - Ss r ,A ;(5-i3) summing over f i n a l states we obtain * T 2 5(5-14) where ;(5-15) S i m i l a r l y , f o r m = ± 1, we have -55-the m = 1 case gives and where we have used the notation M' (M f)„ Averaging over i n i t i a l states and X/ ,m t summing over gives m M , , Q.X , i r a ;(5-l6) The r a d i a t i o n i s seen to be i s o t r o p i c ; the summation over the pola-r i z a t i o n s gives a f a c t o r 2 The d i f f e r e n t i a l Cross-section i s obtained, using (A4-34) of Appen-dix 4„: where i s the volume element f o r the radiation» The reduction of i n t e g r a l (5-15) i s c a r r i e d out i n Appendix 6 using the symmetry properties of f and G under permutation of the p a r t i c l e s -56-where i s defined by equations (4-27, 28). We f i n a l l y have f o r the cross-section 5(5-19) 3. The E l e c t r i c T r a n s i t i o n s . The neglect of the spin-current part of the e l e c t r i c i n t e r a c t i o n derived i n Appendix 4 (equation (A4-28)), con s t i t u t e s an approximation more e a s i l y j u s t i f i a b l e than the long-wave approximation f o r gamma rays up to 30 Mev. The r e l a t i v e s i z e of the amplitudes Q' , and Q , apart from factors JL ,m Jo ,m of order one, i s seen to be J _ L _ ^ Jl M — 5(5-21) 6? £ / m - € r B 2M which i s of the order of 1% near the peak of the c r o s s - s e c t i o n . Besides, i t can be v e r i f i e d that no interference appears between Q' , and Q. f o r the dominant part of c r o s s * s e c t i o n , E1(P -» S). Therefore, Q' can be safely * I ,m neglected:for a l l t r a n s i t i o n s . The following operator -a-x,„=? i ( ; n kK, ~ V - , >ti) (5"22) appearing i n the e l e c t r i c t r a n s i t i o n amplitudes derived i n Appendix 4 (A4-25) can be expressed i n terms of the three-body Euler-angle functions (4-15) :. using the r e l a t i o n s (4-7) f o r the s o l i d harmonics -57-" >o > It i s convenient to define an operator T , operating on the A $m i s o t o p i c spin: where a = ^ o r " 1 = 0 1 f o r m > 0 i n terms of which/), becomes t * a. 5(5-25) The t r a n s i t i o n amplitudes Q given by (A4-25) can now be written JO jin wh ere the i n i t i a l states are characterized by the quantum numbers s^, i^, m^, m^ (see equation (4-33)). The "reduced" p r o b a b i l i t y i s here defined as -58-that i s an average over i n i t i a l s t a t e s , and a sum over f i n a l states. Writing the f i n a l states irl the general form the f i r s t bracket represents an i n t e g r a t i o n over Euler-angles, a and the second, summation over spin, ' isospin v a r i a b l e s , and p a r t i c l e index. Since T s operates only on the i s o t o p i c spin v a r i a b l e s s i t w i l l connect only i n i t i a l s and f i n a l states of same S a and M g. The i n t e g r a t i o n over the Euler angles can be v e r i f i e d to y i e l d = TT (J'™ 'bMCHjMHfab*!)] k(& It 6fP l' ^ ) 5(5-29) subjected to the c o n d i t i o n . (5-30) -59-(5:4) This r e s u l t may be derived by using the r e l a t i o n s ( M l . I M 1 M i ' / I # M . * « _ m l nturt 1 1 " \*ntwixm/\*i,™xmj *"v* and (4-11). The 3-j symbols are re l a t e d to the Clebsch-Gordon c o e f f i c i e n t s by the d e f i n i t i o n I t i s r e a d i l y seen from (5-29) that we must have m' ^  = m1 m^  = -m f o r non-zero matrix elements Q„ : i sm *(i iMpU, MfJW -wi; o\h Defining the i n t e g r a l s the reduced t r a n s i t i o n p r o b a b i l i t y (5-27) becomes (5-34) ;(5-35) ;(5-36) -60-f u r t h e r , from the u n i t a r i t y of the Clebsch-Gordon c o e f f i c i e n t s , n l 2 ;(5-38) 2 where J i s independent of M f as can r e a d i l y be seen from the matrix elements JO . t (A5-17,18) of Appendix 5. 4. The E K P -»S) t r a n s i t i o n . For the e l e c t r i c dipole t r a n s i t i o n E K P -» S ) , the reduced p r o b a b i l i t y reduces to where only the m = 0 component i s seen to contribute using equations (4-30), (5-10) and (4-24), J becomes -61-And with the matrix elements (A5-17, 18) -<4j/r/c^> +</Mi/ris.M>} x i<5" 41> where ^ / ( / . ^ r ^ J - f f * / < 5 . 4 2 ) from equation (4-1). The d i f f e r e n t i a l cross-section w i l l have the angular d i s t r i b u t i o n leading to the d i f f e r e n t i a l cross-section -62-_r 3 2 ;(5-43) The i n t e g r a l i s reduced i n Appendix 6* Using (A6-7)» and (5-42) e/g"/ d^fo) _ jjr e K' I? «**l*3>)%«^t& , (5 .44 , where the i n t e g r a l 0^(4 + / ^ a s been ignored since i t vanishes i d e n t i c a l l y as w i l l be shown i n the next chapter, 5 .• The E l (P -> D) t r a n s i t i o n . For the e l e c t r i c t r a n s i t i o n El(P-» D), (5-38) gives the reduced t r a n s i t i o n p r o b a b i l i t y that i s s f o r m = 0, _ 1 (5-46) (5-47) -63-The components are. seen to contribute i n the proportions 3i4:3 f o r m = -1, 0, 1 res p e c t i v e l y g i v i n g an angular d i s t r i b u t i o n proportional to 2 (7 - cos 0 ): too T x t » 2 ^ 1 5(5-48) The i n t e g r a l i s again given by (5-36) I = §<{ % I IT Kf> *» <! in.li > using (4-21) and (4-31) f o r the f i n a l and i n i t i a l states j , = i f fa, icisMI M > * <h.t i ft: i «•_. (»> + f [- <f<:> / A f w > * / r ; 7 ft-(2) (2) where f ^ are given by (5-42) , and the h^ are ... ;(5-49) ;(5-50) o -64-l ^ - ( *t + * i ~ 2. X i ) - " J f ^ (5-51) • * * * zH + =-hh r/&™ ft*'7*) The i n t e g r a l s < ^ > , and < £ f f / j> f, of equation (A6-8) also vanish i d e n t i c a l l y as w i l l be shown i n the next chapter leaving The d i f f e r e n t i a l c r oss-section finally/becomes -1 </!«,,)rA~»(4 + vJh"3 >}'(7- <J(B) '(5-53; where the term g" i s associated with the function G„ , 2 ~*» . 6. The E2(D -• S) t r a n s i t i o n . For the same reason as E K P -» S), the t r a n s i t i o n E2(D -• S) w i l l have a r c o n t r i b u t i o n f o r the m = 0 term only O 6 I . 1 2*'e2 -r 2 ( 5 _ 5 4 ) leading to the d i f f e r e n t i a l c r oss-section •65-*<rf,0U-+q> _ e^'\z . . g ^ , , ^ ;(5-55) where Using the matrix elements of Appendix 5, can be written as ;<5-57) The reduction i s c a r r i e d out using equations (A6-2), and XA6-7) *• zo Using equations (4-1 to 3) the factors f ^ and f£2^ may be expressed i n terms of ( r , q»2>.) since (5-59) -66-(5-60) = | ( R l + 3 A) - 1 / 3 i 0 / 5 2 ) ' / ? ^ (5-61) (5-62) liiTi A y The d i f f e r e n t i a l c r oss-section becomes where the bracket i s the same as the one appearring i n equation (5-58). There w i l l be some interference between the t r a n s i t i o n s leading to the bound S-state, the d i f f e r e n t i a l c ross-section from a l l t r a n s i t i o n s considered w i l l be given by 5(5-64) where 1^ 9 I 2 ? and I^ are the brackets i n equations (5-44), (5?58), and (5-53) r e s p e c t i v e l y . -67-CHAPTER 6 Evaluation of the Integrals and Approximateons. We w i l l adopt f or the bound state wave functions involved i n the t r a n s i t i o n s studied i n Chapter 5, the approximations described i n part one. (6-3) (6-4) In the continum functions, the part depending upon the i n t e r n a l v a r i a -bles i s the product of a deuteron S - state function by a proton Coulomb function that may be deformed by the nuclear i n t e r a c t i o n with the deuteron. When the nu-c l e a r i n t e r a c t i o n i s neglected, the proton function i s simply Where F. and G. are the regular and i r r e g u l a r Coulomb functions r e s p e c t i v e l y , -68-and 6^ are the phase s h i f t s . For the deuteron function., we w i l l use both the zero - range, and the non - zero range Hulthen wave functions. The zero - range function normalized with r dr _s uD  = — r e. r ( /- <L J (6-v) and the Hulthen function A/p „-Pr r with the normalization constant We have adopted for £ and v both i n ( 6 - 6 ) and ( 6 - 7 )„ the values p = 0,232, V = 1.0 Since the i n t e g r a t i o n has to be c a r r i e d ont i n the ( r , q, v a r i a b l e s , i t i s necessary to renormalize the bound" state functions. The volume elements are re l a t e d by the Jacobian transformation = 2Hf3%rrxd$clr/&vKZ}'Jtf (6-8) therefore, we only need multiply the bound state functions ( 6 - 1 to 4 ) by the f a c t o r ^24v/3. The e f f e c t produced by neglecting the Coulomb repulsion w i l l be exami-ned byusing ordinary Bessel functions i n place of the. Coulomb functions. For the magnetic dipole t r a n s i t i o n , the shape of the wave function i n the region near the nuclear core (the deuteron's i n t e r i o r ) w i l l be very important -69-since the t r a n s i t i o n proceeds from an S - wave. The bound state function f has a node at a short distance as a function of q ( equation (6-12) ), i t therefore appears necessary to assume some nuclear model to determine the shape of the continum wave function since the cross-section w i l l depend s e n s i t i v e l y on the overlap of the wave functions i n that region. Kunz (private communication) found that the f i t to the phase s h i f t s seems to require a p o t e n t i a l model which w i l l give a a c a t t e r i n g wave function resembling the function of Figure 2, The nuclear model that reproduces the phase s h i f t s given by C h r i s t i a n and Gammel f o r the - waves has the form V(q) = - 16,5 Mev. 0 <i q < 3.2 F - q (6-9) = 2177 e_2 q £ 3.2 F q In a s c a t t e r i n g function p o s i t i v e everywhere as a function of q, the over lap would cause a c a n c e l l a t i o n i n the matrix element (5-19), I t i s noted that the phase s h i f t s are large even at small energies (-40° at 1 Mev. laboratory > energy) f o r the S - waves. In the case of the e l e c t r i c dipole t r a n s i t i o n E l (P-*S) which i s the do-minant c o n t r i b u t i o n to the cross - section, the phase s h i f t s stay small over a large ener gy i n t e r v a l up to about 20 Mev., and since the matrix elements are weighted proportional to q, the shape of the continum wave function f o r small q w i l l not be very important, i t i s therefore not necessary to assume a nuclear model f o r the nuclear i n t e r a c t i o n between proton and deuteron i n f i r s t approxi-mation. Because of the smallness of the phase s h i f t s , the regular Coulomb functions are expected to d e s c r i b l e c o r r e c t l y the e l e c t r i c dipole c r o ss-section. *-P. Dp Kunzy University of Colorado, (To follow page 69) Mzed Symmetry bound stale function for a f/zectr FiQ.2 -70-For the E l (P"*D) t r a n s i t i o n , the phase s h i f t s are not so small, but since the percentage of D - states i s small, and since the matrix elements (5-52) are weighted with the functions ( 5 - 51 ) which go to zero f o r increasing q (see equation ( 4 - 2 ) ), we can safely assume that E l ( P*"»D) w i l l be but a small c o r r e c t i o n to the E l (P-»S) t r a n s i t i o n . The e l e c t r i c quadrupole t r a n s i t i o n E2 ( D-*S ) proceeds to the symmetric S - state i n part, and since t h i s w i l l be the main contribution, we w i l l neglect the t r a n s i t i o n s to the mixed symmetry states i n equation ( 5 - 58 ). In the bound state wave functions, the parameter \ i s set equal to while the value of the parameter v i s obtained from the v a r i a t i o n a l c a l c u l a t i o n of part one. Using the f i r s t Jacobi polynomial i n the expansion ( 6 - 2 ), Kunz was (27) able to obtain a f i t to the experimental data of G r i f f i t h s , et a l . f o r the magnetic dipole cross - section to within experimental error. The f i t was found to be f a i r l y i n s e n s i t i v e to the value of u. The relevant i n t e g r a l s for the t r a n s i t i o n s examined are l i s t e d i n Ap-pendix 6. They embody the approximations defined here. To write down these i n -t e g r a l s the angular functions i n the bound state functions have to be expressed i n terms of ( r , q, #) A * I (6-11) & (6-12) The integrations over *^ are independent of the energy or the values of the parameters ^ s U j and X.. If we make the s u b s t i t u t i o n the i n t e g r a l s over #.take the form F(<**t>) xu«#<l it = f FtyJ <Jy (6-i4) these i n t e g r a l s vanish i d e n t i a l l y .-whenever F(y) i s a. odd function of y. This i s the case f o r the l a s t i n t e g r a l i n equation ( 5 - 44 ), and for the two i n t e r a l < ^ < U lfZfal>* a n d ^>,* / A? 7 2/> i n equation (A6 - 8).. The parameters f and h defined i n ( 5 - 45>. ) s and ( 5 - 51 ) when ex-pressed i n terms of (r> qs1^) become 3 (6-15). Ztft 1 ^ ' J ^ (6-16) ~ ~[ ; + X( *******+ ir x)]^ A * ~ ~ J % d (6-17) (6-18) CHAPTER 7 Numerical Results and Discussion,, .......... i 1 * Radial orthogonal functions. I Figures 3 and 4 display the r a d i a l orthogonal functions (R) (equa-t i o n Al - 42) f o r various values of V> and N, We show the only values of & involved i n our approximation: A = 0 f o r the symmetric 5 - state and = 1 f o r the mixed symmetry S ^ - state and the D - state functions. Figure 3 shows the e f f e c t of a v a r i a t i o n of X and u on the f i r s t r a d i a l orthogonal polynomial J° (R). A decrease i n X f o r a f i x e d u tends to favour the t a i l of the function while a decrease i n u f o r X f i x e d - produces a s h i f t on the t a i l of the function outwards. In Figure 4 we have shown the f i r s t three polyno-mials J° (R) f o r A. = 0„738j u = 0.3 and i n Figure 5 8the f i r s t three polynomials jjj (R) entering into the serie s expansions f o r the S' and the D - states. The value of A. was set to .738 ( equation 6 - 10 ) and u =„3 appears to be the value of u which gives the maximum binding energy when the series expan: -sions are cut - o f f a f t e r one or two polynomials i n the v a r a i t i o n a l c a l c u l a t i o n s . I t i s also the value g i v i n g the experimental Coulomb energy when only the leading term of the expansion i s used. On the other hand, as w i l l be shown below, the energy and the p r o b a b i l i t i e s of the various states are i n s e n s i t i v e to the values of \ and u over a large i n t e r v a l when a s u f f i c i e n t number of polynomials are i n -cluded i n the c a l c u l a t i o n . (To follow page 72) Fus-.4 The first thee orthogonal polynomial for the symmetric 5state for /\".7J8, V*.} -73-2. P o t e n t i a l s examined. The p o t e n t i a l s used a l l have the general form V = V(ir^) + V(rZ3)+ i/(ir3J ( 7 - i ) where V(r) are two - body p o t e n t i a l s adjusted to f i t the two - body data„ In order to i l l u s t r a t e the method, we have examined three types of p o t e n t i a l s , a l l having a Yukawa shape; they are ( i ) Central p o t e n t i a l s with no core. The two following c e n t r a l p o t e n t i a l s adjusted by KunZ to reproduce the correct binding energy of the deuteron; they are denoted i n the following by C - 1 and C - 2 respectively: C - 1 : (7-2) C - 2 the t r i p l e t - even part the s i n g l e t - even part , , - 0.82T r> / i J r-, n-otQ*rr / „ ( 7 - 3 ) Different e f f e c t i v e ranges have been chosen f o r the s i n g l e t - even part of the p o t e n t i a l i n order to investigate the e f f e c t produced by varying the range of the p o t e n t i a l , We w i l l therefore use the name " shorter range " c e n t r a l p o t e n t i a l f o r C *• 2 S and " longer range ." central p o t e n t i a l for C - 1. * P. D„ Kunz (Private communication). -74-( i i ) Central p o t e n t i a l with a hard core. (35) One cen t r a l p o t e n t i a l of the type proposed by Bryan , and by Ram-(36) say with three Yukawa terms corresponding to the one-pion, two-pion (s c a l a r meson), and the three-pion (rho meson) exchange res p e c t i v e l y . The scalar meson, i and the rho meson are assumed to have masses 4m and 5.5m res p e c t i v e l y . The TT n strength parameter f o r the rho meson exchange term i s chosen to give roughly a 0.4 Fermi hard core radius. In f a c t , meson theory predicts a repulsion f o r the rho exchange term. -o,7r> ~7<B v> -XBS'ir 3|/ =• ~//.0~ 13, ZOO — * 2%OoO e ~o,yy> -2.8 r - T f l V i * i i i ) P o t e n t i a l s containipg a tensor i n t e r a c t i o n . Two pot e n t i a l s with a tensor i n t e r a c t i o n which have been used by others are examined to provide comparison with previous theories (a) (F.P.) Feshbach and Pease's p o t e n t i a l no. 2 (L A-) (23) the r e s u l t s obtained by Feshach and Pease , and by B l a t t et a l are displayed i n Table 1. (Introduction)!. -75-(b) (H.M.) Hu.and Massey's p o t e n t i a l : (7-6) Viewer = " 40 e / o , W y» 3, E f f e c t i v e p o t e n t i a l s , (16) E f f e c t i v e p o t e n t i a l s , i n the sense of Verde , acting between the symmetric S - state wave function are shown i n Figure 6 to 9. The c e n t r i f u g a l 3fj2 ]_5 b a r r i e r term i s , f o r the symmetric S - state, equal to x ^ 2 * ^ e term V\ _ (R) i s the f i r s t of the i n t e g r a l s (3-23),, Veff i s the sum of V, _(R), and i n t . i n t the c e n t r i f u g a l b a r r i e r term. Figures 6, 7 ..and 9 show the e f f e c t i v e p o t e n t i a l s f o r the shorter range ce n t r a l p o t e n t i a l C-2, the c e n t r a l p o t e n t i a l with a hard core, and the Feshbach and Pease's tensor p o t e n t i a l r e s p e c t i v e l y . The curves (15) obtained are very s i m i l a r to the e f f e c t i v e p o t e n t i a l i n Feshbach and Rubinow's Figure 2. The s i t u a t i o n for the ce n t r a l p o t e n t i a l with a hard core however i s quite d i f f e r e n t since even without the c e n t r i f u g a l b a r r i e r term, the integrated p o t e n t i a l v i n t ( R ) i s everywhere p o s i t i v e and no binding i s possible for the t r i -ton using t h i s p o t e n t i a l . The e f f e c t i s i n the same sense as the one noted by Feshbach. and Rubinow for the Levy ce n t r a l p o t e n t i a l (case 1) for which the e f f e c -t i v e p o t e n t i a l i s almost completely cancelled out by the k i n e t i c repulsion. The e f f e c t obserbed here i n our case i s much more pronounced, and occurs because we use a symmetric S - state wave function F° (R) which i s f i n i t e at the o r i g i n . (To follow page 75) FIG. 7 fffatitf? potential for centra/potent/a/ with core in the symmetric S-state. -76-The s i t u a t i o n would be improved by allowing the symmetric S - state to be repre-sented by a series expansion over orthogonal polynomials which vanish at the o r i -gin l i k e FACR) or F 2 ( R ) , etc... ° N N A ce n t r a l p o t e n t i a l with a hard core which proved to be binding was derived from the p o t e n t i a l (7-4) by a r b i t r a r i l y reducing the strength parameter of the rho meson term from 39,000 to 30,000 (such a po t e n t i a l would overbind the deuteron) i n order to have a q u a l i t a t i v e idea of the e f f e c t of the core size on the predictions for the percentage of 3', and D - states. The corresponding ef-f e c t i v e p o t e n t i a l i s displayed i n Figure 8. 4. Wave functions. The r e s u l t s of the v a r i a t i o n a l c a l c u l a t i o n using the p o t e n t i a l described above are tabulated i n Table 3, 4 and 5. The r e s u l t s i n Table 3, show the binding energies, the percent mixed symmetry S - state, the percent D - states, and the Coulomb energies for the var rious potentials using d i f f e r e n t sets of 0lf?) and various cue - o f f for the series expansions. Table 4 contains the expansion c o e f f i c i e n t s cC^ l^sI, MS)&nd A^y (kzztjMh) o r equation (2-15) for the ce n t r a l p o t e n t i a l s . Expansion c o e f f i c i e n t s for the tensor p o t e n t i a l s are shown i n Table 5. A glance at Table 3 shows how the binding energy, the percent mixed sym-metry S - state, and the Coulomb energy for the cdntral p o t e n t i a l C-2 are p r a c t i -c a l l y independent of A, I? over a wide range = Raising the number of polynomials i n the expansions from MS = 4, MM = 3 to MS = 6 , MM = 4 changes the binding and the (To follow page 76) Table 3 •-• Results of the V a r i a t i o n a l C a l c u l a t i o n , Type of K M potential„ Fermi" 1 Fermi 1 MS MM MD ^^Mev 7o S mix % D E c Mev. C - l .,8168 ,7301' 4 3 0 13.668 .427 l«0726 .638 .3 4 3 0 14^,5953 ,3261 1.14838 ,>738 .3 4 3 0 14.5954 ,3274 U14815 ,838 ,3 4 3 0 14,5959 ?3283 1*14791 C-2 ,738 .2 4 3 0 14.5826 ,3290 K14821 .738 .2 6 4 0 L4.5959 .3275 1.14813 .738 .3 6 4 0 14.5967 .3267 .738 .4 6 4 0 14,5951 .3270 1.14758 Central .738 .3 6 3 0 32,15 .0002 with core .738 . „5 6 3 0 36.29 3.8 a r b i t r a r i -l y reduced .738 6 3 0 37,35 4.3 .738 .9 6 3 0 36.57 3.3 ,.738 .3 4 2 3 2.9168 .47 2.03 0.6675 Tensor .738 .3 2 1 0 ,1.2275 .22 0.6964 HM Derrick et a l (1962) 8.1 1.2 3.6 Hu & Massey (1949) 6,4 ,738 ,2 6 2 4 4.0699 ,198 2,116 0,8603 Tensor ,738 .3 6 2 4 4.1462 , .569 -2.87 0.8670 FP .738 ,4 6 2 4 4,1579 ,326 2,36 0,8728 No, 2 .738 '3 6 2 0 1.9357 ,016 0-8024 Derric' <. et a l (1962) I O ' . I .52 3,3 (To follow page 76..) Table 4 - Central P o t e n t i a l s , Expansion C o e f f i c i e n t s f o r the Wave Functions,, Type Central Central -2 Central -2 of P o t e n t i a l r -1 MS = 4 e MM = 3, MS = 6, MM = 4 \ .8168 ,638 .738 .838 .738 .738 .738 .738 M ,7303 .300 .300 .300 ,200 .200 .300 .400 ,9964 , 8617 ,9053 .9384 ,8371 .8373 .9052 ,9475 Symmetric - .0520 .4984 .4171 .3380 J5187 .5180 .4176 .3140 S-state .0116 .0750 .0540 .,0411 ' .1569 .1603 O0531 - .0066 -.0021 " ,0163 .0125 .0089 ,0475 .0.408 .0071 .0017 .0142 - ,0020 -.0009 ,0140 .0059 .0019 Mixed -.0648 - ,0501 - ,0525 -.0544 ~ .0463 -.0457 -.0521 - .0555 Symmetry .0074 -.0268 - .0220 -,0174 - .0304 -.0322 -,0231 -.0136 S-state... -.0031 -.0064 -.0053 - ,0046 - ,0152 - .0110 - ,0058 -.0033 --.0022 -.0003 .0013 (To follow page 76.,) Table 5 - Tensor P o t e n t i a l s , Expansion C o e f f i c i e n t s f o r the Wave Functions, Type ,of Potential„ Hu & Massey Tensor Feshbach & Pease. \ .738 ,738 .738 .738 .738 .738 U ,3 .3 .3 ,2 .3 .4 .9952 .9816 .9939 .9536 .9779 .9864 -.8545 -.0845 -.0012 .2389 .1300 .0381 Symmetric ,0611 ,0986 ,1016 .0606 .0460 S-state -.0202 -=0416 ,0014 -,,0158 -.0215 „0228 ,0073 ,0085 -0106 ~ .0078 -.,0016 -.0029 - .0036 Mixed Symmetry •. -0468 -.0674 - .0138 -„0128 - .0024 -.0327 - .0303 -,0381 -,0206 -.0412 - .0125 S-state -.0015 - .0004 .1362 .1071 .1238 .1342 D-states. .0368 .0231 .0842 .0345 ,.0378 .0685 .0234 .0177 .0514 .0148 .0088 -77-Coulorab energies by only 1 part into! 10., 000 and raises the percentage of S'-state by 1 part into 500, This shows the highly convergent character of the ser i e s ex-pansion at least f o r the central p o t e n t i a l s with no core. The central p o t e n t i a l with i t s repulsive core a r b i t r a r i l y reduced shows a much larger s e n s i t i v i t y to the values of P both in;the binding energy (showing a maximum around 37.4 Mev.) s and i n the percent Sri-state. The predicted percentage of S'-state i s ten times larger than the p r e d i c t i o n s of both the c e n t r a l and tensor p o t e n t i a l s with no core. Since our ce n t r a l p o t e n t i a l s with core would overbing the deuteron 5 we are not j u s t i f i e d to take that r e s u l t as being q u a n t i t a t i v e l y c o r r e c t 5 but i t shows an i n d i c a t i o n that a ce n t r a l p o t e n t i a l with a hard core would probably favour a larger S'-state admixture than would a p o t e n t i a l with no core. Since there i s much i n d i c a t i o n from other sources that the S'-state p r o b a b i l i t y should be close to 4% instead of l/2%> p o t e n t i a l s with core should be investigated f u r t h e r . The r e s u l t s using the tensor p o t e n t i a l s are the only ones that can be (23 compared with previous calculations,, Referring to the r e s u l t s of Derrick et a l l i s t e d i n Table 3 S our binding energies are seen tobbe more than 50% lower than those c a l c u l a t e d by Derrick et a l . ,, who used functions of the v a r i a b l e 0 = f12 + r23 + r31 ( 7 " 7 ) The same r e s u l t was obtained by M. McMillan using an e n t i r e l y d i f -ferent approach. Applying to the Schroedinger equation the v a r i a t i o n a l procedures described by Feshbach and Rubinow^1"^ and by V e r d e ^ 1 ^ he numerically integrated the equivalent two-body Schroedinger equations obtained with a few central poten-t i a l s . He observed a systematic diff e r e n c e between the binding energies for the two types of functions(with v a r i a b l e J> and R respectively) the binding energies f o r functions of R being always less than 50% those f o r functions of P. -78-It i s also seen that the binding energy, and e s p e c i a l l y the S' - state as well as the D - state p r o b a b i l i t i e s are much more s e n s i t i v e to the values of ^, y than with ce n t r a l p o t e n t i a l s . The reason for such a v a r i a t i o n i s seen i n the fact that both the S' - state and the D - state functions are serie s expansions with respect to the same set of orthogonal polynomials ( f o r I = 1), corresponding to a single value of 9. Since the S1 and the D functions are expected to have d i f f e r e n t shapes, both expansions must contain a larger number of polynomials than i s contained i n the present c a l c u l a t i o n . Our c a l c u l a t i o n contains i n fac t only two polynomials i n the S' - state p r o b a b i l i t y ( t h i s l i m i t a t i o n was imposed by the low speed of the computor used, the IBM 1620). This i s more c l e a r l y shown by the one before l a s t entry i n Table 3.where we i n s i s t e d that no D - states be • present and more polynomials i n the S' - state function were included. The re-su l t s show that almost no S' - state i s present, and we also note that i n that case the serie s expansion i n Table 5 i s highly convergent but nop for the other cases where MM = 2 oniy. It i s f e l t that a" r e l i a b l e v a r i a t i o n a l c a l c u l a t i o n in*-, eluding D - . ates would require about s i x polynomials i n the expansions f o r the three functions. Another s o l u t i o n that may reduce the number of polynomials re-quired for rapid convergence would be to allow d i f f e r e n t walues of the parameter y i n the 3' state, and the D - states r e s p e c t i v e l y . The wave functions contained i n Tables 4 and 5 are depicted versus R i n Figures 10 to 13. The fig u r e s show further featureSmot r e a d i l y apparent i n Tables 3 to 5-, Figure 10 shows that the tensor potentials favour the t a i l of the symmetric S - state function, that a s l i g h t change i n the range of the s i n g l e t part of the cen t r a l potentials C - 1 and C - 2 produces only a very s l i g h t chan-ge i n the symmetric S - state function while a larger d i f f e r e n c e i s shown i n the > 2 3 ^ . 7 6 7 f l <9 10 /? farm/' Fm-10 The symmetric S-state functions (To follow page 782. 2 7 4 1 6 7 8 f 7? F<srmi mined symmetry J-s/a/e wtiJt 0-2 ano'0.3 , "X-0.73& - 7 9 -S'-state function (Figure 11). In the tensor p o t e n t i a l c a l c u l a t i o n s , the. wave functions f o r the Fashbach and Pease p o t e n t i a l are simply s h i f t e d inward compa red to Hu and Massey's. I t should be noted that these two po t e n t i a l s d i f f e r mainly by t h e i r central parts„ the two tensor parts being almost the same. Figure 12 shows the v a r i a b i l i t y of the S'-state function with res-pect to I?.j a v a r i a t i o n of A by as much as 20% would hardly show on the f i g u r e . We have not shown the v a r i a t i o n s on the symmetric S-state i n Figure 10 since It would hardly show on the f i g u r e both f o r a v a r i a t i o n on 2t and on V . -80-5. The e l e c t r i c dipole t r a n s i t i o n E K S -> P). The experimental determinationsof the photodisintegration cross sectionsare grouped together i n Figure 14. One t h e o r e t i c a l curve i s included (Eichman's) f o r comparison since i t seems to be the analysis i n v o l v i n g the lea s t approximations before the present work. From the experimental r e s u l t s , (27) we have retained the r e s u l t s of G r i f f i t h s et a l f o r the low energy region (before the maximum), and those of the Yale group^ 3*^, t h e i r r e s u l t s are repro-(3) duced i n Figure 15> together with the t h e o r e t i c a l curve of Gunn and Irving adjusted to bring the maximum i n coincidence with the experimental curve (eg. A = 2.6 Fermi). shown i n Figures 16 to 18. We have used i n a l l cases two d i f f e r e n t wave func-t i o n s f o r the deuteron, the zero-range, and the non zero-range deuteron func-tions described i n Chapter 6. For a l l p o t e n t i a l s , the non-zerd range deuteron function gives the larger cross section; t h i s i s compatible with our expecta-t i o n s since the dipole cross-section depends very much on the s i z e of the deu-3 3 te r o n s and the He nucleus., that i s on the overlap of the deuteron and the He wave functions at r e l a t i v e l y - ' large separations. The cross sections are also given f o r two values of (0.3 and 0.4); i t i s seen that very l i t t l e e f f e c t i s produced when we use wave functions correcponding the ce n t r a l p o t e n t i a l C-2 (Figure 16) while more v a r i a t i o n i s obtained f o r wave functions derived from tensor p o t e n t i a l s . Howeverf the v a r i a t i o n i s less pronounced than the one a r i s i n g from d i f f e r e n t choices of deuteron functions. The t h e o r e t i c a l predictions from the 4 pot e n t i a l s with no core are The most i n t e r e s t i n g r e s u l t concerns the r e l a t i v e sizes of cross sec-tions predicted by cen t r a l and tensor i n t e r a c t i o n s r e s p e c t i v e l y ; the cen t r a l Eichmann (Theoretical) Yale (Stewart et al) lllinoh (Berman et al) Heidelberg Moscow British Columbia (Griffith eta/) FIG- /4. He ? (n,-p) D. Experimental (To follow page 80) Gunn *' Irving V . 3 end non zero range V° .4 and non zero ratife U* land zero range da'ieron de>tte< }ff function function <rvrr fancifon Hon zero ra^ge deuteron function By (Mev) He >(Xp)D Theoretical El(5-P) Cross-Section for shorter range centra/ potential. A=T3&: MS* 4; fiM-3. TIG. 17 tfe'(y,p)V Theoreticol El(S-P) for Hu e Massey tensor potential \ » J}6f Y~.? tf$*4, MM'Zj MD--J- Zero range e non zero range deuteron function Zero-range and non zero range deuieron funciion -81-forces predict cross sections very much smaller than experimental ones while ten-sor p o t e n t i a l s predict values much too large compared to experimental ones. Both ce n t r a l p o t e n t i a l s C - l s and C-2 give almost exactly the same cross section curves, t h i s i s the reason why we have shown only the r e s u l t s from potential C-2 on Figure 16. 6- The magnetic dipole t r a n s i t i o n MKS -» S). In the d e r i v a t i o n of the magnetic dipole t r a n s i t i o n , we have assumed that the doublet and quartet continuum wave functions are uncoupled therefore that they have an equal a p r i o r i p r o b a b i l i t y and the same form as f a r as the i n t e r n a l v a r i a b l e dependence i s concerned. This leads to the t h e o r e t i c a l cross-section (5-19). In our work, we do not include a contribution from a possible exchange magnetic moment operator (see our Hamiltonian (A4-10)). This ampli-tude i n t e r f e r e s with the magnetic moment operator amplitude (From y,av H) and may cause important deviations from t h i s work. This i s an aspect of the pro-blem that should be looked i n t o . The smallness of the magnetic dipole t r a n s i t i o n i s accounted f o r by the s e l e c t i o n rule which allows the capture rate to proceed mainly from the quartet continuum S-. state (4-31) to the doublet mixed spa c i a l symmetry S-state ( S J) with .a small and n e g l i g i b l e c o n t r i b u t i o n from the doublet continuum S-state (4-30) to the state S ! since the doublet continuum S-state i s orthogonal to the doublet bound S-state (see equation (5-12)). Kunz used our equation (6-2) f o r the: mixed symmetry S-state function with = 1 (the leading term of our r a d i a l expansion ) to c a l c u l a t e the theoretic c a l magnetic dipole cross section (5-19) with a value of \ = 0.704 F P.D., Kunz, Uni v e r s i t y of Colorado has presented these r e s u l t s at the Denver APS meeting i n 1964. -82-For the continuum wave function he took a product of the Hulthen deuteron function (6-7) and a S-wave s c a t t e r i n g s o l u t i o n f o r the incoming proton. This i s a no p o l a r i z a t i o n and no d i s t o r t i o n of the deuteron approximation. For the quartet S'-state these are good approximations. The s c a t t e r i n g function f o r the proton was taken from the v a i r a t i o n a l (38) s o l u t i o n of n + D s c a t t e r i n g phase s h i f t s of C h r i s t i a n and Gammel , The po-t e n t i a l model (6-9) was selected since i t reproduced the experimental phase s h i f t s and gave a zero energy wave function the same shape as the zero energy wave function of C h r i s t i a n and Gammel, The model was then modified with the addition of a Coulomb p o t e n t i a l f o r the p + D sc a t t e r i n g system. This model f i t t e d the p + D phase s h i f t s of C h r i s t i a n and Gammel. The sc a t t e r i n g lengths of t h i s model are n + D a, = 6,2 F (7-8) p + D a 4 =14,0 F Furthermore,, he cal c u l a t e d the n + D magnetic dipole capture rate with the wave function of C h r i s t i a n and Gammel with a more recent v a r i a t i o n a l (39) n + D sc a t t e r i n g function of Burke and Haas (see Figure 19) In the calculations,, the parameter \> was var i e d . The value of P which f i t s the Coulomb energy f o r the dominant symmetric S-state i s 9 = 0.30 F The f i t to the experimental points i s shown i n Figure 20, The f i t of the t h e o r e t i c a l curve depends very l i t t l e on the value of ^ except f o r extremely small values of t h i s parameter., The shape of the cross section c a l c u l a t e d with no phase s h i f t f o r the continuum function gives nearly as good a f i t as with a phase s h i f t over t h i s energy range. -83-The r e s u l t s f o r the p r o b a b i l i t y of the mixed symmetry state are shown i n Table 6,. They were obtained by comparison Of the t h e o r e t i c a l curve with the (27)* experimental r e s u l t s obtained by G r i f f i t h s et a l . ' Extrapolation of the cross section to energies i n the astrophysical region by means of the expression a = S m P -+- mD - P ' 2 t 1 ^ (7-9) 2 where 7] = e /Kv gives S = („128 - .00045 E„ x, (Kev)) ev-barn ± 25%. This i s to be compared s CM with a graphical extrapolation value by G r i f f i t h s s of S = (.12 ± 25%) ev-barn. (7-10) s The close agreement between the n + D and p + D p r o b a b i l i t i e s gives us confidence that the er r o r quoted for the S g f a c t o r can be reduced conside-rably. (to follow page 83) Table 6 - S'-State p r o b a b i l i t i e s . „ - F " 1 .25 ,30 .35 .,50 Coulomb Energy Mev A 700 *7/60 i82Q ; .950 p + D PS-Gammel - C h r i s t i a n .054 .077 .100 ,186 n + D p s . Gammel - C h r i s t i a n .051 .072 .094 .175 n + D PS' Burke - Haas .032 «050 .064 .120 p S' Electron Scattering £ . 0 5 0 £ . 0 5 0 £ . 0 5 0 £ . 0 5 0 p + D p S' no phase s h i f t . .010 .015 .018 (To follow page 83) (To follow page 83) -84-CONCLUSION, Wave functions and p o t e n t i a l s . The p o t e n t i a l s examined show that i t would be desi r a b l e tgoinvestigate* more r e a l i s t i c c entral p o t e n t i a l s with shorter range t r i p l e t parts. Our central p o t e n t i a l s C - l and C-2 would probably give a>binding energy i n the neighborhood of 30 Mev ; with a more sophisticated wave funct i o n . This has been shown to be (23) the case f o r the tensor p o t e n t i a l s used by Derrick et a l , by comparing our (37) r e s u l t s with t h e i r s s and also by McMillan i n the case of central p o t e n t i a l s . Central p o t e n t i a l s with no core should be used to investigate the convergence of the expansion (2-9) when angular terms corresponding to a. and Z > 0 are in c l u d e d } i t i s expected that t h i s would make the wave function versa-t i l e enough to y i e l d a binding energy equal or even larger than the c a l c u l a t i o n s with functions of j> (equation (7-7)) only. The present work c l e a r l y shows the o rapid convergence with polynomials J^(R-) when cen t r a l p o t e n t i a l s are used, When such a c a l c u l a t i o n (using more angular terms) i s done, i t should be s u f f i c i e n t to cut-off the r a d i a l expansions to 4 polynomials. When central po-t e n t i a l s with a hard core are used, i t seems from the present analysis to require a much larger number of polynomials. It would probably be much more desir a b l e i n such a case to use solutions from Schroedinger equation containing (49) an Eckart po t e n t i a l with g ^ 0 (see Bose.'s equation 23-i) as was suggested i n Appendix 1. This choice of solutions would probably provide much f a s t e r convergence when cores are present. -85-The i n t e r e s t i n studying the ce n t r a l potentials with a hard core i s prompted by the q u a l i t a t i v e f i n d i n g that such a po t e n t i a l predicts a percent S'-state much c l o s e r to what i s indicated by other properties of the three-body system (i,e„: ~ 4%H Convergence i n v a r i a t i o n a l c a l c u l a t i o n s with c e n t r a l p o t e n t i a l s containing some S ! - s t a t e can also be improved by allowing a d i f f e r e n t para-meter \ i n that state. This i s even more desirable f o r c a l c u l a t i o n s with tensor forces as i s shown i n Table 3 f o r the Fashbach and Pease p o t e n t i a l : allowing no T>-states i n the wave function reduces the percent S'-state from 0.57% to 0.016%. Independent adjustment of a fa c t o r y> for each state (S, S' and D) would y i e l d a percentage of states which i s more c h a r a c t e r i s t i c of the po t e n t i a l used. Our predictions are that a p o t e n t i a l that would predict the correct percentage of S t S 1 and. D states w i l l have to contain a tensor i n t e r a c t i o n and a hard core. 2. Electromagnetic t r a n s i t i o n s . The small cross sections obtained with our cen t r a l p o t e n t i a l s are compatible with the f a c t that they are}.largely overbinding since t h e i r wave functions are too much drawn inwards reducing the overlap with the proton-deuteron function. The v a r i a t i o n of cross-section with a.o.different value of can also be predicted by the v a r i a t i o n on the S'-state wave function (Figure 12). An increase i n produced a r i s e i n the t a i l of the wave function leading one to expect an increase i n the c r o s s - s e c t i o n . This i s seen to be the case -86-i n f i g u r e 16. On the other hand, one can predict that a / v a r i a t i o n of 2i would produce very l i t t l e e f f e c t on the size of the cr o s s - s e c t i o n , since the v a r i a t i o n on the shape of the S'-state wave function i s very small„ The present a n a l y s i s shows conc l u s i v e l y that the cross-section i s much more s e n s i t i v e to':.the type of potential used than tol'.the v a r i a t i o n of X and p . I t i s therefore a good t e s t of the v a l i d i t y of the po t e n t i a l used. It can also be concluded that the choice of .deuteron wave function has much les s e f f e c t than the use of various p o t e n t i a l s . When a comparative study of various p o t e n t i a l s i s done, r e l i a b l e d i s c r i m i n a t i o n between good ones and bad ones (giving r e s u l t s f a r or close to the experimental cross-section) can be obtained by using the simpler deuteron function (the zero-range function) keeping i n mind that the predicted cross-section would be underestimated. Also no v a r i a t i o n on the parameters !X and ? would be necessary since the e f f e c t i s even smaller than the e f f e c t of changing the deuteron fun c t i o n . The large cross-sections f o r the e l e c t r i c dipole t r a n s i t i o n |E1(S - P) obtained from the two tensor potentials can probably be traced to the small number of r a d i a l polynomials used i n the S'-state (two i n our case). Before drawing any conclusions i t would be worthwhile to do some more c a l c u l a t i o n s with a larger number of S'-state cjomponents. This i s substantiated by the large v a r i a t i o n observed i n the percentage of S'-state when no D-state i s allowed i n the wave fund £ ion's, c a l c u l a t e d with the tensor p o t e n t i a l s . On the other hand, i t i s possible to draw some conclusion on the e f f e c t of changing the range of the cen t r a l part of the i n t e r a c t i o n since the -87-two potentials have almost exactly .the same tensor part (see equations 7-5 and 7-6)a Hu and Massey's pot e n t i a l has a ce n t r a l i n t e r a c t i o n range which i s too large to be r e a l i s t i c while the Feshbach and Pease pot e n t i a l has a range much cl o s e r to the accepted one-pion exchange value. We see i n f a c t that Feshbach and Pease's p o t e n t i a l p r e d i c t s a cross-section c l o s e r to experimental than Hu and Massey's. The use of more sophisticated wave functions l i k e the functions used (23) by Derrick et a l , or functions of the type used i n t h i s work but containing more angular terms would probably lower the predicted cross-sections c l o s e r to the experimental values since the increased binding would correspond to a more concentrated wave f u n c t i o n u There i s some uncertainty as to i n what way the tensor interaction'; should be modified i n order that the S'-state admixture be promoted to the expected 4% by the i n c l u s i o n of a hard core. Our r e s u l t s do not permit to answer that question since both tensor p o t e n t i a l s used have almost the same tensor term. -88-APPENDIX I 1, The S - state s o l u t i o n s . We wish to solve the d i f f e r e n t i a l equation I5 faf* + * e*J " T* ^ 4 T T ^ ^ J ' ^ / / ( A I-1) with a s u b s t i t u t i o n of the type / = rctzj X (*; d ? / ^ ( A I-2) The equation (Al - 1) w i l l be separable i f we replace the p o t e n t i a l by a function of Ronly: V = V(R). Let M> be the separation constant, we obtain the. r a d i a l and the angular equations ( A I -3) t, . -rrV ( A I-4) w r i t i n g we wish to s a t i s f y the boundary condition §(<p + 2TT) = $(cp) f o r m. r e a l , there" fore (AI -5) -89-The equation-fo X(x) i s a special.case of a more general equation O'xVK^j + x'U^(ju- J^) K(,^o (AI-7) occurring i n the solutions f o r the P and the D - states. We w i l l therefore study equation (Al - 7) under the requirement that the solutions s h a l l be absolutely converging i n the closed i n t e r n a l [ o , l ] . The two points x = 0, x = 1 corres-pond res p e c t i v e l y to the so -> c a l l e d i n - l i n e and the e q u i l a t e r a l "singular con-(45) f i g u r a t i o n s " of the t r i a n g l e formed by the three p a r t i c l e s . Equation (Al - 7) can be rewritten into the generalized hypergeometric „. (46) , equation form iL« -i=*V i^-'Jit o (AI -8) 2 by the s u b s t i t u t i o n j> = x _ X ^ (AI-9) />-/ 'Upif-O Riemann ,P - function The generalized hypergeometric equation (Al - 8) i s represented by the (46) (AI-10) Hence, for our equation (Al - 9) the representation i s O CO I I Pi % "»/* A (AI-11) - ^ x -Wi ) - 9 0 -(Al-12) (Al-13) where we have defined u. = Jl[& + 2). Two l i n e a r l y independent solutions can be obtained by expressing (Al - 11) a l t e r n a t e l y by ( 4 7 ) \-y\ {(-A-tyn + vi) -m or / O CO j - "A WZ f Ci-f) ? " ill+ +z) 0 p I t may be r e c a l l e d that the P - function / O CO J \ U.« P j o a o f f (AI-14) 1 l-C b C-Oi-b J i s a representation of the hypergeometrie equation pCt-fJ U" 4 [c ~(CK.+ l) + 0lA.]uf - (xbu = O (Al-15) admitting a s o l u t i o n of the form (Al-16) The condition f o r absolute convergence i n the i n t e r v a l [o, l3 being that QD, (a + b * c) < 0. The s o l u t i o n obtained from (Al - 13) must be rejected since i t i s not regular at / = 0 ( i e : X = 0) since n^ O,. In (Al - 12) s the condition f o r absolute convergence requires both that n^os m*0 and 62s. (a <- b -*• c) = m < 0 „ The two - 9 1 -conditions f o r m being incompatible, i t i s necessary to terminate the series by the quantization r e l a t i o n (Al-17) The function (Al - 12) y i e l d s the solution (Al-18) (46) where we have used Magnus and Oberhettinger's notation k f v i ) («+ n_)n A K (Ai-19) and Pochhammer's symbol (Al-20) The normalization f a c t o r I n i s obtained with the help of the orthogo-n a l i t y r e l a t i o n s dun (Al-21:) I (Al-22) 2, Symmetries of the functions The r a d i a l solutions and the functions X m , n (x) are symmetrtal with I* respect to permutations of the p a r t i c l e s and the symmetry of the solutions w i l l depend only on the functions §m(cp). Looking at the t r i a n g u l a r coordinates ex-pressed i n terms of the v a r i a b l e s R, x, and cp ( z = »/i - x 2) : i t i s r e a d i l y seen that the s i x operations of the group of permutations of three 2 objects correspond to rotations by —— and r e f l e c t i o n s of the angle cp cp —• q> •+- ~ i s the c y c l i c a l permutation l - * 2 f 2 - * 3 , 3-»l, 4TT _. 1 _• 1 . * - 4 - ~_ 1 Q o ^ o o . 1 cp, exchanges p a r t i c l e s 2 and 3 cp exchanges p a r t i c l e s 3. and 1 exchanges p a r t i c l e s 1 and 2.. I) <P —4 <P i i ) 9  9 i i i ) <P -iv) «P 217 3 v) <P 4n 3 vi) <P — » -The pairs (cos cp3 s i n cp) and (cos cp, - s i n cp) transform according to the mixed representation of the permutation group (see table 6 i n Appendix 5) Cox 2-f) cos 3cp i s symmetric and transform^ according to the i d e n t i c a l representation, and -93-s i n 3 cp i s completely antisymmetric. This sequence repeats f o r 4(p, 5cp5 6cp e We therefore have the means of constructing i n t e r n a l wave functions with the required permutational symmetry. d d The e f f e c t of the operators — , o n *-he mixed p a i r s /[=( m=L4.Z-= - I for ht •= 2, 3~j 8 w i l l be as follows (Al-24) -Ii YYl \ COX w i f — — YV) USX. yr\ ^  (Al-25)' The operator - j ^ i s antihermitean and occurs i n the mixing of the D - states. 3„ The Radial Equation,, The asymptotic behavior of the so l u t i o n to the r a d i a l equation (Al -94-i s obtained by neglecting the p o t e n t i a l V, and the c e n t r i f u g a l b a r r i e r term. Substituting into equation (Al - 3) y i e l d s / & - * m % : " * - / « ; - £ j c<K) - o which i s s a t i s f i e d f o r large R by Q id) - e (A1-28) and f o r small R by = R (Al-29) The same asymptotic behavior w i l l be obtained i n the following d i f f e -(43) r e n t i a l equation containing an Eckart p o t e n t i a l . In dimensionless, form, i t i s l & 4 - HMe'*"' + V*e-*- <*2]\aii^0 (A1.30) ; where k = 2£ + 3/2 giving k(k + 1) = ki U + 2) + 15/4, a = A/u > 0 and \ = UR. The parameter u i s introduced to describe the e f f e c t of an e f f e c t i v e p o t e n t i a l -95-Equation (Al - 30) reduces to the form (Al - 27) f o r small R therefore both equations w i l l have solutions with the same asymptotic behavior f o r small, and large R. Since equation (Al - 30) was obtained under the assumption that the p o t e n t i a l i s a function of R only, i t i s not expected to give the exact form of the wave function. However, from a complete set of orthogonal functions s a t i s -f y ing the boundary conditions, i t w i l l be possible to approach the true wave function with a l i n e a r combination of the functions of the set (e.g.: a serie s expansion). Our purpose therefore w i l l be to derive a set of orthogonal functions having the same asymptotic behavior as the solutions f o r the equation (Al - 30). This can be done since our " e f f e c t i v e p o t e n t i a l " , the terms i n the square brackets i n equation (Al - 30)^ i s recognized to be a Schroedinger inva-(49) r i a n t i , a special case of a larger class of Riemann invari a n t s dicussed by Bose. Our inv a r i a n t corresponds: to h i s equation (23 - i ) for g = 1. This choice of g w i l l give solutions which may be f i n i t e : anywhere between o and °°; i t i s con-venient, for the study of pot e n t i a l s with no hard core or pot e n t i a l s which are con-tinuous together with t h e i r f i r s t d e r i v a t i v e . I t i s i n p r i n c i p l e possible to ge-ne r a l i z e our method to include the cases where g ^1 that would cause the funchions to vanish at some "hard - core" radius R. (note that Redoes not represent the i n t e r - p a r t i c l e distance and that such a r e s t r i c t i o n i s not s u f f i c i e n t to prevent any two p a r t i c l e s to occupy the same p o s i t i o n i n space). -96-Equation (Al-30) can be written i n the generalized hypergeometric equation form (Al-8) by the s u b s t i t u t i o n 1 - £ ; i t becomes 1 This can be represented by the P-function (Al-32) (Al-33) J O <X> I or -h. _ I * V (Al-34) O _ y?-^4|- - Y Functions (Al-33 and 34) represent two l i n e a r l y independent so l u t i o n s . (Al-34) must be rejected since i t i s not regular at the o r i g i n (k>0). The s o l u t i o n corresponding to the function (Al-33) i */;^;^/ ( A 1 " 3 5 ) cannot be absolutely converging i n the i n t e r n a l [0,1 ] .since (£?g(a+b -c) = «v/ not <0. One must terminate the s e r i e s With the condition A " - (Al-36) y i e l d i n g the polynomial s o l u t i o n /o J ( f " $ ^ f , i , , ) (Al-37) -97-In the notation of Magnus and Oberhettinger the Jacobi polynomials i s a t i s f y the orthogonality r e l a t i o n (Al-38) (Al-40) This shows that the function A/ i s orthogonal i n the i n t e r n a l (0,<») with respect to in t e g r a t i o n over x , The function w i l l have the same asymptotic behavior provided we define p and q by T (Al-41) with C= 2 k-h Z = +s~ instead of the d e f i n i t i o n (A1-37K We obtain the required set of orthogonal solutions f o r every value of Z: (Al-42) -98-From (Al-26) the orthogonal r a d i a l functions w i l l be -99-APPENDIX 2 The D - state solutions., In order to uncouple and solve the system of d i f f e r e n t i a l equations f o r the D - state obtained from the system (2-21) a f t e r performing the operations defined i n (Al-17) f o r the d e r i v a t i v e s with respect to cp : 2A_ . ,/ . (A2-1) i t i s convenient to define the abbreviations where and In the regions where the r a d i a l s o l u t i o n (Al -32) "approximately" s a t i s f i e s the operator 'J)^ £ — O '• (A2-4) -100-the system of coupled equations (A2-1) reduces to a system i n one va r i a b l e only, the r a d i a l part ^ ( R ) and the $ ^ 9 ) f a c t o r being assumed common to the three functions f 0 , f Q and f . Using (A2-6), the system obtained can be decoupled using the solutions to the d i f f e r e n t i a l equation (Al-7). Making substitutions f o r X Q, X and X . o y IU X«l = {J ( / - x V fat*) (A2-6) and s u b s t i t u t i n g the r e s u l t of the following operations i n t o (A2-11) and s i m i l a r l y f o r x 1 0 > the system becomes id. — _ % r«-v-«--Vxji - f * - £ r r Hivf-m'}* <A2"7) -101-and i s seen to be s a t i s f i e d ( i f we s e t ^ = l ) , by J^=-/ and JL=m, i n the f i r s t equation. The t h i r d equation gives ^=-^X » These conditions s a t i s f y the second equation i d e n t i c a l l y . The system therefore i s s a t i s f i e d by the set of solutions &-'•>*&:)>'!<'> where X 1,2,4,5, This may not give a l l the possible solutions to the system but i t w i l l be s u f f i c i e n t f o r our purposes since we w i l l l i m i t ourselves to the functions f o r XssI i n the numerial c a l c u l a t i o n s . We note that the solution, f o r the D - states f i x e s the r e l a t i v e amplitudes of the three components.. We may c a l c u l a t e only one amplitude f o r each X provided we normalize the complete D - state. The normalization f a c t o r f o r a given X w i l l be (A2-9) -102-APPENDIX 3 1 o The k i n e t i c energy matrix elements. The brackets appearing i n equation ( 3 - 6 ) are of the form j,<MIe'(,-e-*/"> - f ^ f r r — i/C-l/ (K + C-7XK+C-J+IJ- - (k+C-l) (A3-1) whre we have used equation (3 - 2), and the i n t e g r a l h / (A3-2) The sums f o r J=l and 2 can be-evaluated using the r e l a t i o n s t^'iT-l) ° ^ ( A 3-3) and " 1*1/M-l) I (C-TJ!(H+J-Z)! " 1+1/ M-l I (L-I). a+c-iXL4c-j+o--U+c-ij u-OHH-kC-i)! (A3"4) a f t e r expanding the c o e f f i c i e n t s (or•+c-+K-j), i n powers of (K>C-J). Equation (A3 - 1) becomes ^ 04l(i~e-VT~leia,x>= ±I?(ZM+«+C-Z; (A3.5) ; a n d X <x-3j(Mt «-\J f o r J=l, and 2 respectively* -103-2„ The p o t e n t i a l energy matrix elements. The r a d i a l i n t e g r a l i n equation (3-19) can be done d i r e c t l y ; f o r e i t h e r the simplet, t r i p l e t or tensor p o t e n t i a l s . For example r o o J v (A3-6) The change of v a r i a b l e w i l l transform the i n t e g r a l I - [ [ ^/l^Zi^^J/z ) JzJV Jil / I - S u a i f into the trac t a b l e form * -ll / 1-71 where K=Jl tr + *J Making the s u b s t i t u t i o n ii u - f t - 7% and i n t e g r a t i n g by parts ( A 3 - 7 ) (A3-8) (A3-9) U + L (A3-10) 0 *V -104-The l a s t i n t e g r a l can be evaluated by elementary i n t e g r a l s *of the form f(fx /*) d*. f J } 4*J /K/K^^ where X = -+(1- brZ) CA3-11) The forms Jfft Jjt and ^//Vx-will y i e l d only p o s i t i v e powers of ^ and w i l l contribute nothing to the sum over the binomial c o e f f i c i e n t s because of the i d e n t i t y (A3 - 3) Only the part T — 8JJL lr 3 • gift. 3 L o 3 o where (A3"12) and w i l l contribute to the i n t e g r a l -105-APPENDIX 4 1. The Electromagnetic F i e l d . The c l a s s i c a l d e r i v a t i o n of the amplitudes of the ra d i a t i o n emitted form the i n t e r a c t i o n of charged p a r t i c l e s with the electromagnetic f i e l d can be found i n B l a t t and Weisskopf Appendix B . A quantum (52) mechanical treatment i s given by H e i t l e r t Appendix 1. Natural u n i t s -fJ=c=l w i l l be used here, thus the energy, momentum, and mass w i l l have the dimension of r e c i p r o c a l lengths. The e l e c t r i c , and magnetic components of the r a d i a t i o n f i e l d -* o -*• (A4-1) are expressible i n terms of the vector p o t e n t i a l where CX.QM(K±J' a n c ^ ^ « w llCiJ&re amplitudes which, i n the quantum theory become a n n i h i l a t i o n , and creation operators f o r a photon of wave number^, angular momentum J> , and i , a in. The terms A „ (K±)are solutions of the source-free wave equation (VX-*KV f?*,mLK±) ~0 (M"2) and eigenfunctions of the operators J and J with eigenvalues X[Jt+1)» and m. - 1 0 6 -They are discussed i n d e t a i l with t h e i r r e l a t i o n to the e l e c t r i c , and (53) magnetic components of the electromagnetic f i e l d by Rose , and H An,n\ a r e referred to as e l e c t r i c , and magnetic 2 -poles re s p e c t i v e l y since they have the asymptotic behavior of the r a d i a t i o n form o s c i l l a t i n g i 2 - poles at large distances from the sources, = - -'<£jl-ZJ Yimi*.«) (A4-5) llClrJ = ^  Jt*i (Kr) (A4*6:).: for KrXi The The functions J t (Kr) a r e Half-integer Bessel functions, vectors Xjj are normalized vector spherical harmonics PPnx npjr and the f a c t o r s f~jz[~ , and v~£j£ have been included i n (A4 - 3 , 4 ) i n order that have the normalization J I A,)Wl IK±)I J? = Z 7 T (A4*.8) the i n t e g r a t i o n being c a r r i e d out i n the volume enclosed i n a sphere qf radius R centered at the source of the r a d i a t i o n ( the center of mass of the -107-nucleus). The sphere i s thought of as having a radius large enough to enclose most of the radiation./ and have p e r f e c t l y r e f l e c t i n g walls. The approximation (A4 - 6) f o r fcy »Ji can be used i f the sphere i s chosen to be large enough. The condition that the tangential component of the f i e l d vanishes on the surface of the sphere enables us to enumerate the normal modes with momentum between X , and /C + </ H • The density of states j*^ f o r a given j£.jand i s found to be A 1 (/ k = ^ <J hi (A4-9) m 2. The Interaction. The p a r t i c l e , and f i e l d Hamiltonian i s given by tf~e *A ff *Z A (A4-10) where the f i r s t term i s the Loventz force, a central p o t e n t i a l which does not depend upon the spin, v e l o c i t y or time. In the i n t e r a c t i o n of the spin with the external f i e l d , jtA i s expressed i n u n i t s of the Bohr magneton y^g » thus U. = y t L p — 2,79 vn . f o r a proton ^ (A4-11) is= JH_m = — m, f o r a neutron. The l a s t term of the Hamiltonian i s the energy of the r a d i a t i o n f i e l d . The Hamiltonian i s separated i n two terms ^io = energy of the p a r t i c l e s , and the f i e l d (A4-12) ' — i n t e r a c t i o n between p a r t i c l e s , and f i e l d - 1 0 8 -The t r a n s i t i o n p r o b a b i l i t y to an occupied state i s ^T. o ... z=2irl/ki 9 v - •+ IIJ' I n\>r P. . ( A 4 - 1 3 ) where the states are l a b e l l e d by ,X,m= z^, and the p a r i t y ; they are the obser-vables that can be diagonalized simultaneously i n the spher i c a l wave represen-t a t i o n . In f i r s t order perturbation theory, only terms i n the f i r s t power 2 of e are used, the coupling parameter e = 1 / 1 3 7 being small, The t o t a l cross-section i s obtained by d i v i d i n g the t r a n s i t i o n p r o b a b i l i t y by the f l u x of incident p a r t i c l e s . The incident wave being norma-l i z e d to unit density at i n f i n i t y , the f l u x i s equal to the r e l a t i v e v e l o c i t y of the incident proton, and deuteron, Non r e l a t i v i s t i c a l l y V - g - ( A 4 - 1 4 ) 2 w h e r e — i s the reduced mass of the rystem, and k, the momentum of the proton i n the center-of-mass frame i s given by 3k 2 •7M-= £ „ ( A 4 - 1 5 ) 4M c £ c» the energy of the proton i n the center of mass i s rela t e d to the laboratory proton energy by E = | E ( A 4 - 1 6 ) c 3 p The energy of the emitted photon i s determined by E , and the d i f f e -3 rence i n binding energy of He , and the deuteron -109-where we have neglected the r e c o i l energy of the deuteron which i s small i n the energy range up to 30 Mev, Using the solenoidal gauge„ the i n t e r a c t i o n Hamiltonian i n the f i r s t order of e can be written ra U ^ r> (A4-18) where A i s the vector p o t e n t i a l defined by (A4-3 to 6). 3 E l e c t r i c Multipole Matrix Elements. The e l e c t r i c multipole matrix elements are given by equation (A4-13) where the i n t e r a c t i o n Hamiltonian contains the terms A. (k-) f o r the vector *, m p o t e n t i a l t c » ~- A I U u( \ % (A4-19) where. H a b , and H'a1j are the Lorentz force i n t e r a c t i o n , and the spin i n t e r a c t i o n r e s p e c t i v e l y . * ^ (A4-20) This expression can be transformed into B l a t t and Weisskopf's c l a s s i c a l expression by averaging (A4-20) with the i n t e g r a l obtained by p a r t i a l integra-t i o n , and using the expression for the current density *\ / n . l ") (A4-21) - n o -te- obtain a further i n t e g r a t i o n by parts y i e l d s This i s an exact expression s i m i l a r to B l a t t and Weisskopf' s ^ " ^ . Now, the long-wave approximation Kr4tfL i s used to reduce the i n t e g r a l to i t s f i n a l form where (A4-23) (A4-24) When the system contains several nucleons, the i n t e g r a l s are to be replaced by where the operator ^ ( / - Z^)selects the charged p a r t i c l e s . The i n t e g r a l sign includes i n t e g r a t i o n over sp a c i a l coordinates, and summation over spin, and iso s p i n coordinates. The cross-sections f o r e l e c t r i c 2 -poles become f i n a l l y ( T ^ ^lElAliL. K2HI l n /* U4-26) where £?£, m i s obtained from H ' j ^ i n a manner s i m i l a r to m b y using the magne-t i c f i e l d operator f o r the e l e c t r i c r a d i a t i o n , and noting that -111-since m i l C - ) s a t i s f y Laplace's equation (A4-2), (A4-27) For a system of protons, and neutrons we replace t h i s by + %Kf;> IkjJc (A4"28) 4. Magnetic Multipole Matrix Elements. The c a l c u l a t i o n of the magnetic multipole matrix elements proceeds along the same l i n e s as the e l e c t r i c multipoles by using the vector p o t e n t i a l A^ ) ) M ( for the magnetic multipole f i e l d . (A4-29) -112-where ^-J^T'klr*<0Y'iCL V'r (M-30) (A4-31) S i m i l a r l y , f o r a system of several riucleons ' £ /V ^ l / * in / I « I / , \T n\ ) i (A4-32) and (A4-33) The cross-sections being -113-APPENDIX 5 Spin, and Isotopic spin Matrix Elements. In the evaluation of the r a d i a t i o n matrix elements, the following matrix elements between the s p i n - i s o s p i n states are needed: Spin operators — r to y Proton operator rt- - i d - ''<ZJ id- T, ?; Neutron operator { ( /'+ T • ) Magnetic moment operators and i CI + ZV3 ) cr£ /w (6) They were evaluated from f i r s t p r i n c i p l e s by Rendell , and w i l l be tabulated i n matrix form below for ready reference. Vector l^(Ht)^> w i l l be defined thus, Hi (*V> - fl4j v IV, t <*i V, V, , (* i UX,, (m {{) and For each type of operator , there are three sets of matrix elements, one f o r each p a r t i c l e . These matrices are r e l a t e d by u n i t a r y transformations U ( P i ) , i n which the i r r e d u c i b l e representat ions of the permutation group B i d e f i n e d by (4-9). form the diagonal b l o c k s , the o f f - d i a g o n a l b l o c k s , being a l l z e r o s . The i r r e d u c i b l e representat ions of the permutation group f o r three o b j e t S j , and are l i s t e d i n Table 7. Thus» f o r example where / o o -/ l o o -/ / o o -I exchanges p a r t i c l e s 1 and 2, and / f ('"" i s the matrix of ^ t'(WS /)/[(/-^J/j , j,(Mi^ Thus one of the matrices may be evaluated from f i r s t p r i n c i p l e s and the, other two sets obtained by means of the u n i t a r y t ransformat ions . T h e r e f o r e „ only the matrices f o r i « J need be given e x p l i c i t e l y . (To follow page 114)„ Table 7 - The Representations! ^ of the Permutation Group. Permutation. Representations. Symbol.„ symmetric T 2 mixed 1 antisym-3 metric. ( 1 ) ( 2 ) ( 3 ) (123) (132) (12) (3) (23) (1) (31) (2) 0 1 •1/2 -1/2 «/3 1/2N/~3 -1/2 / • i h 1/21/3 -l/2>7~3 -1/2 1 0 0 -1 •1/2 1/2-TV l/2<7~3 1/2 / •1/2 - l / 2 i / 3 •1/2-73. 1/2 - 1 -115-1, The Spin Operators. The matrix elements of: the operators 6"! .. and (57 are required f o r the magnetic moment operators (A5-2) The properties of the spin operators on the sing l e nucleon functions 0£ and/fare (A5-3) The matrices f o r ^ / 2 ; / ^ 0 / Jj (\)> (L) I ^ J ft & )>, <kll)lr3+l%LU>, and <jrcti)lir7+ / / o ) (A5--4) -116-/ 0 ( _ / o 1 ~ 3 o 2 o I Z 0 -1 o O -2 0 2 -/ o / O o 1 0 -/ - / 6 ~1 _ / -•x O -1 ~ 3 O -2 <? -/ I o -/ 0 0 -/ o l o I I 0 z o o I (A5-5) (A5-6) (A5-7) Where the square matrices are symmetric. The matrices f o r other values o f Ms, and -117-(A5-8). f ° r <5£_ a r ^ obtained from the above ones by means of the i d e n t i t i e s <K hi)I r,+ \h I- IP = < 1j ti)l / k I V> '<tt U)I r,. //,• ( \ ) > = <ijl\) lr„ ji, U)> <h I-01 e-3- l t j U ) > = <fr Lk)\rM IK i-ij> 2, Proton, and Neutron Operators, The proton and neutron operators o c c u r i - n the magnetic moment operator (A5-2), i n the Coulomb p o t e n t i a l operator, i ('" * W i(l--v^) £ l (A5-9) and i n the r a d i a t i o n operators * III' tCj) /!/n-J (A5-10) f o r the e l e c t r i c multipole radiations were -f(fc)is a function of the sp a c i a l coor-th diates of the i nucleon„ The properties of the operators on the single p a r t i c l e i s o t o p i c spin functions ^ and 7T are -118-(A5-11) The matrices f o r [Jfi(/- \)? , <^(i)U(h lk)>. \o 1/ (A5-12) o X 1 o 0 -/ o X o o o o o o o o (A5-13) <td\)\kU+ln)\^l\)> - ( V? o O o (A5-14) -119-I 0 1 - / 0 1 o I 0 / o 0 o o 0 o o o z o (A5-15) 0 For Mj = --| J and - ^  the matrices are the same as fb r hjs = £ , and 1 r e s p e c t i v e l y . Because the functions]?, describe two protons, and one neutron, we have the i d e n t i t y < k(Ms)I i (/ - T, 3 ) iU- Ti3)i$jt Ms)> where k)is any permutation of (123). 02. (A5-16) When the operators (A5-10) are summed over a l l three nucleons, making use of the un±ary transformations, we have -120-<X-U> I? id- ~*ki) f>*;/>; U)> where ' t r i 0 O 0 o 0 1 o o o 0 fin) + fir*) fO> ft) ^ ft** i S symmetric, and fV) forms a mixed symmetry pair . (A5-17) (A5-18) (A5-19) 3, Magnetic Moment Operator,, The matrix elements of the magnetic moment operator (A5-2) can now be -12|-evaluated by multip l y i n g the matrices f or the spin operators by the matrices f or the proton and neutron operators, and adding: (A5-20) Using the notation (A5-21) the matrix elements summed over a l l three mucleons for K^tid^l^ J^to jTj ( f )/> > are <*(V IfAo ih ll)> = 3/«.> (j ° ) (A5-22) •122-0 /** 0 0 0 o 0 0 o O o o : O 0 juuj ( A 5 - 2 3 ) o o -& 0 o o 0 \[2JAIO (A5-24) -'/ill) 0 o o 0 o o o o 0 o o o O ZjAlO (A5-25) -123-where (A5-26) jAtx) ^JkP Matrix elements not l i s t e d may be obtained from the , i d e n t i t i e s <hH^f>* lr,[-V> = - <h(i)IZ/M*o )hH)> (A5"27) -124-APPENDIX 6 Reduction of the J^ i n t e g r a l s , 1. Magnetic dipole t r a n s i t i o n . The i n t e g r a l (5-15) for the magnetic dipole t r a n s i t i o n can be written i n terms of the f u n c t i o n ^ . , and ^r. symmetric with respect to the exchange of p a r t i c l e s j , and k ; (i j k.) represents any permutation of (1,2,3,) (A6-1) by permuting the p a r t i c l e i n d i c e s , t h i s i n t e g r a l can be expressed i n terms of g^ only on the r i g h t hand side 1= ~<h + h~ *A > (A6-2) 2. E l e c t r i c Multipole t r a n s i t i o n s . The i n t e g r a l (5-43) for the e l e c t r i c dipole can be reduced i n a /it) (.o where h i i s a function of the p a r t i c l e coordinates only. Writing f^, f^ (A6-4) and G i n terms of the S, and J , the i n t e g r a l (5-43) i=<f,ir?i<;h,(»>-<fj A w /<a z < > w i l l become, after permuting the p a r t i c l e indices r = - ft * 4 / J - ' ^ <v// - * 4, / A > d'/A.* A, - a*, /&> + l , 3 r h r Zh,\% >} ( A 6 _ 5 ) where the primed functions r e f e r to the mixed symmetry states, and the unprimed to the symmetric states. Noting that, from (A6-3) (A6-6) the i n t e g r a l s become - G <S2>1 ) f ) 9g> (A6-7) The same method i s used to reduce the i n t e g r a l s i n (5-50) f or the t r a n s i t i o n (A6-8) -126-where, besides the r e l a t i o n s (A6-6) , we have used and (A6-9) and are f u n c t i o n s assoc ia ted with Cj^  (< 3. D e t a i l e d form pf the relevant i n t e g r a l s . The i n t e g r a l f o r the magnetic d i p o l e t r a n s i t i o n i s given by (5-19) terms of the wave f u n c t i o n s of Chapter 6. ( A 6 - I 0 ) the normal iza t ion f a c t o r s are •A (A6«ll) - 1 2 7 -where 0{ - W For E1(P-»S) we may write equation (5-64) as J - - s 1 ' - i I" with 0*1''= <f, If te/|^3> where and u r «(*+o • • • (<*+$-)!> ] ^ -1 i v J (A6-13) )A0 -128-For E1(P-*»D) we w i l l have (A6-15) (A6-16) where F i n a l l y , f o r the quadrupole t r a n s i t i o n E2(1»S) 1 ) F /Lr ) > ft~g ' ( S R 1 2 I •") (A6-17) v l 2 9 -w h e r e (A6-18) -130 REFERENCE S. 1 1 . E „ H . S . : . B u r h o p , and H . S . W . Massey , . P r o c . Roy,, S o c . A 1 9 2 , 156 (1947) 2 . M . V e r d e s H e l v . P h y s . A c t a 2 3 , 453 (1950) 3 . J . C . G u n n ; and J . I r v i n g , , P h i l . M a g , 4 2 , 1353 (1951) 4 . C , R o s s e t t i , Nuovo C i m e n t o , 14 , 1171 (1959) 5 . L . M . D e l v e s , P h y s . R e v . U 8 > 1318 (1960) 6 . D . H , R e n d e l l , t h e s i s . , U n i v e r s i t y o f B r i t i s h C o l u m b i a (1962) 7 . V . E . E i c h m a n n , Z e i t s . P h y s . 175 , 115 (1963) 8 . W. R a r i t a * and R . D . P r e s e n t s P h y s , R e v . 5_1, 788 , (1937) 9 . F . W . B r o w n , P h y . R e v . 5 6 , 1107 (1939) 10 . E . G e r j o u y a n d J . S c h w i n g e r , P h y s . R e v , 6 1 , 138 (1942) 11 . H . F e s h b a c h , . and W. R a r i t a , P h y s . R e v . 7 5 , 1384 (1949) 12 . R . E . C l a p p , P h y s . R e v . 7 6 5 873 (1949) 1 3 . T . M . H u i and K . N . H s u , P h y s , R e v . 7 8 , 633 (1949) 14. R . L . Pease , , and H . F e s h b a c h , P h y s . R e v . 88 , 945 (1952) 15 . H , F e s h b a c h . and S . I . Rubinow* P h y s . R e v . 9 8 , 188 (1955) 16 . M , V e r d e , Handbuch d e r P h y s i k s 3 9 , 144 (1957) 17 . T . Ohmura , M . M o r i t a , and M , Yamada a P r o g , T h e o r . Phys- , L 5 , 222 ( 1 9 5 6 ) ; 1 7 , 326 (1957) 18 . T . Ohmura , P r o g . T h e o r . P h y s . 22 , 34 (1959) 19 . G . H . D e r r i c k , N u c l , P h y s . 17 , 67 (1960) 2 0 4 - J . L „ Gammel, and R , M . T h a l e r , P h y s , R e v . 107, 291 (1957) 2 1 . K , A . B r u e k n e r ; , a n d J . L . Gammel, P h y s . R e v . 1 0 9 a 1023 (1958) 22 . G . H . D e r r i c k . , D . M u s t a r d , and J . M . B l a t t , P h y s . . R e v . L e t t . 6 S 69 (1961) 23 . J . M . B l a t t , G . H . D e r r i c k , and J . N . L y n e s s , P h y s , R e v . L e t t - , 8 , 323 (1962) 24 . T . Hamada,, and I . D . J o h n s t o n , N u c l . P h y s . 34 , 382 (1962) 25 . K . E . L a s s i l a , M . H . H u l l , H . M . R u p p e l , F . A . M c D o n a l d , and G . B r e i t , P h y s , R e v . 126,. 881 (1962) - 1 3 1 -2 6 . We w i s h t o t h a n k P r o f e s s o r S c h i f f f o r a p r e p r i n t 65 t h i s p a p e r , P h y s . R e v . 113, B 802 (1964) 2 7 . G . M . G r i f f i t h s , E . A . L a r s o n 0 and L . P . R o b e r t s o n s C a n . J . P h y s . 4 0 , 402 (1962) G . M . G r i f f i t h s , M . L a i , and C D . S c a r f e , C a n . J , P h y s . 4 1 . 724 (1963) 2 8 . B . L , B e r m a n , L . J . K o e s t e r , J r . , a n d J . H . S m i t h , P h y s . R e v , L e t t , 10, 5 2 7 , (.1963)., P h y s . R e v , 133 , B117 (1964) 2 9 . J . B . W a r r e n 9 K . L . . E r d m a n , L . P . R o b e r t s o n * D . A , A x e n , and J . R , M a c D o n a l d . P h y s . R e v . 132 , .1691 (1963) 3 0 . J . R , S t e w a r t , R . C , M o r r i s s o n p and J . S . O ' C o n n e l , c o m m u n i c a t i o n a t t h e APS m e e t i n g , New Y o r k , 1964. 3 1 . N , A u s t e r n , P h y s , R e v . 8 3 , 672 ( 1 9 5 1 ) ; 8 5 , 147 (1952) 3 2 . U , . V o l k e l , t h e s i s * Mabburg (1961) 3 3 . L„ K a p l a n , G . R . R i n g o , and K . E . . W i l z b a c h , P h y s . R e v . 8 7 , 785 (1952) 3 4 . G . E r n s t i and S . F l u g g e , Z e i t s . P h y s . .162, 448 (1961) 3 5 . R . A . B r y a n j C R . D i s m u k e s , . and W. Ramsay, N u c l . P h y s , 4 5 , 353 (1963) 3 6 . W, Ramsay, P h y s . R e v . 1 3 0 , 1552 (1963) 3 7 . M . M c M i l l a n , U n i v e r s i t y o f B r i t i s h C o l u m b i a p r e p r i n t . We t h a n k d o c t o r M c M i l l a n f o r s e n d i n g u s a p r e p r i n t p r i o r t o p u b l i c a t i o n . 3 8 . R . S . C h r i s t i a n , and T . L . Gammel, P h y s , R e v , 9 1 5 100 (1953) 3 9 . P . G . B u r k e , and F . A . H a a s , P r o c „ R o y , S o c , A 252, 17.7 (1959) 4 0 . G . H . D e r r i c k . - and J . M . B l a t t „ N u c l , P h y s , 8 , 310 (1958) ' • 4 1 . L . C o h e n , and J J B . j W i l l i s , N u c l . P h y s . 32. „ 114 (1962) 4 2 . W.,M. M a c D o n a l d , P h y s . R e v . 101 , 271 (1956) 4 3 . E . E c k a r t , P h y s . R e v , 3 5 , 1303 c (1930) 4 4 . J . M . B e r g e r , , N u c l P h y s . 16 , 405 (1960) 4 5 . G . H , D e r r i c k , N u c l . P h y s . 16 , 405 (1960 4 6 . W. Magnus..- and F . O b e r h e t t i n g e r , f u n c t i o n s o f M a t h e m a t i c a l P h y s i c s , C h e l s e a P u b , C o . (1949) 4 7 . E . G . C , P o o l e , I n t r o d u c t i o n t o t h e T h e o r y o f L i n e a r D i f f e r e n t i a l E q u a t i o n s * D o v e r P u b , (1960) -132^ 48. G, Morpurgo, II Nuovo Cimento,, 9, 461 (1952) 49. A,K. Bose, II Nuovo Cimento, 32, 679 (1964) 50. A. E r d e l y i and associates, Higher Transcendental Functions. V o l . 1, McGraw-Hill (1953) 51. J.M. B l a t t and V.W. Weisskopf. Theoretical Nuclear Physics, John Wiley (1952) 52. W. H e i t l e r , Quantum Theory of Radiation,, Oxford Univ. Press (1954) 53. M.E. Rose, Multipole F i e l d s , John Wiley (1955) Elementary Theory of Angular Momentum^ John Wiley^i! 1961) 54» A.R. Edmonds, Angular Momentum i n Quantum Mechanics, Princeton Univ. Press (1957) 55. DT J-ackson, Fourier Series and Orthogonal Polynomials, Open Coutt Pub. Co. (1941) 56. G.N„ Watson, T r e a t i s e on the Theory of Bessel Functions, Cambridge Univ. Press (1962) 

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