THEORETICAL STUDY OF THE REACTION D(ptf)He by DAVID HA WARD REND ELL B . S c c , Dalhousie U n i v e r s i t y , 1956 K S c , Dalhousie U n i v e r s i t y , 1957 0 k t h e s i s submitted 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 the degree of Doctor of Philosophy i n the Department of Physics ','/e accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BitlTloH COLUMBIA Ar'KlL, 1962 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available f o r reference and study. I further agree that permission f o r extensive copying of t h i s thesis f o r scholarly purposes may granted by the Head of my Department or by his be representatives. It i s understood that copying or publication of t h i s thesis f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of The University of B r i t i s h Columbia, Vancouver 3, Canada. The University of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of DAVID HAYWARD RENDELL B.Sc. Dalhousie, 1956 M.Sc. Dalhousie, 1957 WEDNESDAY, APRIL 25th, 1962, at 3:00 P.M. IN ROOM 301, PHYSICS BUILDING • COMMITTEE IN CHARGE Chairman F.H. Soward C.W. CLARK A.M. CROOKER F.W. DALBY External Examiner: G.M. GRIFFITHS F.A. KAEMPFFER C.A. SWANSON E. Vogt Atomic Energy of Canada, Ltd. Chalk River, Ontario a x / GRADUATE STUDIES A THEORETICAL STUDY OF THE REACTION D(p#)He F i e l d of Study: ABSTRACT Experimental studies of the reaction D(py)He (Fowler, et a l . , 1949; Wilkinson, 1952; G r i f f i t h s and Warren, 1955; G r i f f i t h s , Larson and Robertson, 1961) indicate that two different transitions contribute to the total y i e l d . While the t r a n s i t i o n giving the larger y i e l d has been shown, by the angular d i s t r i b u t i o n and polarization of the gamma radiation, to be an e l e c t r i c dipole t r a n s i t i o n there i s i n s u f f i c i e n t experimental evidence to determine d e f i n i t e l y the characteristics of the other transition. Theoretical Physics Electromagnetic Theory Theory of Measurements Nuclear Physics Elementary Quantum Mechanics Theory of R e l a t i v i t y Group Theoretical Methods J In this theoretical study the angular d i s t r i b u tions of the gamma radiation are calculated and numerical estimates of the cross-sections are made for a l l transitions which might conceivably c o n t r i bute to the reaction. I t i s concluded that the larger experimental y i e l d comes from the e l e c t r i c dipole t r a n s i t i o n of a P-wave proton and deuteron to the state of He^. The smaller part of the y i e l d i s most l i k e l y from the magnetic dipole trans i t i o n of a S-wave proton and 3g deuteron to the ^S state of He-* although part of the y i e l d might be contributed by an e l e c t r i c dipole t r a n s i t i o n of a P-wave proton and ^S deuteron to the ^D state of He . 3 Related G.M. Volkoff A.M. Crooker J.B. Warren F.A. Kaempffer P. R a s t a l l W. Opechowski Studies: Analytic Matrix Theory Non-Linear Mechanics Numerical Analysis M.D. Marcus E. Leimanis C. Froese iv ABSTRACT A theoretical study of the reaction D(ptf)He^ i s made i n an attempt to explain the experimental data f o r the reaction obtained by Fowler et a l . (19^9), Wilkinson (1952), G r i f f i t h s and Warren (1955) and G r i f f i t h s , Larson and Robertson (I96D. The angular d i s t r i b u t i o n of the emitted gamma radiation, measured with respect to the incident proton beam, i s predominantly proportional to s i n 9. Measurements of the polarization of the radiation by Wilkinson (1952) indicate that 2 the s i n 9 component i s e l e c t r i c dipole radiation. i s a small, possibly i s o t r o p i c , component. In addition there The proportion of the t o t a l y i e l d coming from the smaller 'isotropic' component i s 0.035 t a a proton energy of 1 Mev, and this proportion increases with decreasing proton energy. 2 The s i n 9 component has been interpreted by G r i f f i t h s and Warren as coming from an e l e c t r i c dipole transition from an i n i t i a l 3 2 state of a P-wave proton (L = 1, L =0) and S deuteron to the S z ground state of He"^. calculations. This interpretation i s supported by the present They also suggest that the smaller 'isotropic* compo- nent coulu be either a magnetic dipole t r a n s i t i o n of S-wave protons to the ^S state of He"^ or an e l e c t r i c dipole transition involving spin-orbit coupling. In this present work the cross-sections are examined for a l l . p o s s i b l e channels which might conceivably contribute to the reaction. The channels considered are - 1. V - e l e c t r i c dipole transitions for 2 a. • P-wave protons to the S state k b. P-wave protons to the D state k c. F-wav'e protons to the D state 2. 3. e l e c t r i c quadrupole transitions for a. S-wave protons to the D state b. D-wave protons to the ^S state the magnetic dipole t r a n s i t i o n for S-wave protons to the ^S state. Three-body wave functions are constructed, following Verde (1950) and Derrick and Blatt (1956), making use of the symmetry properties i n spin space, i s o t y p i c spin space and i n ordinary space. In addition to the states of t o t a l isotropic spin T = # considered by Derrick and Blatt the states of t o t a l isotopic s'pin T = 3/2 are included. The radiation matrix elements for the above channels are calculated and are expressed i n terms of integrals over the three internal coordinates. These r a d i a l integrals are estimated by using very simple r a d i a l functions which are v a l i d outsiae the range of the nuclear forces and which also disregard coulomb forces. The cross- sections uepend on the unknown amplitudes ana r e l a t i v e signs of the various possible symmetry states. Therefore the s i z e , although not the angular dependance or the general energy dependance, of the crosssections can be used only as an order-of-magnitude estimate. vi By comparison of the size, angular d i s t r i b u t i o n and energy dependance of the calculated cross-sections with the experimental 2 data i t i s shown conclusively that the s i n 9 component of the r a d i - ation comes from the e l e c t r i c dipole transition of P-wave protons to 2 the 3 S state of He . The smaller ation comes from either protons to the 'isotropic' component of the r a d i - (a) an e l e c t r i c dipole transition of P-wave or (b) a magnetic dipole transition of S-wave D state, giving an angular d i s t r i b u t i o n proportional protons to the 2 S state, giving an isotropic angular d i s t r i b u t i o n . The observed energy dependence of the r e l a t i v e y i e l d of the small component suggests the interpretation transition. i n terms of the magnetic dipole The cross-sections of the other transitions examined are too small to explain the experimental r e s u l t s . vii ACKNOWLEDGEMENTS The author to acknowledge the wishe6 continued assistance and guidance of Dr. F. A. Kaempffer. The regular discussions with Dr. Kaempffer helped to c l a r i f y the many problems that were encountered. This assistance was invaluable i n the preparation of this thesis. Many helpful discussions were held with Dr. G. M. G r i f f i t h s concerning the experimental data. The author i s grateful to D . G r i f f i t h s , E. Larson r and L. Robertson for being able to use their l a t e s t experimental data before publication. The comments and c r i t i c i s m of Dr. P. D. Kunz were very nelpful and are greatly appreciated. Work on this thesis was started while the author held a studentship from the National Research Council of Canada. TABLE OF CONTENTS ABSTRACT iv ACKNOWLEDGEMENTS vii 1 INTRODUCTION CHAPTER 1. CHAPTER 2 . The Three-Body Problem 0) Statement of the Problem 1) Spatial Description A) Definition of the Coordinates B) Transformation Formulae C) Moments of Inertia D) Angular Momentum Operators E) The Laplacian Operator 7 9 10 13 15 2) Symmetries of the Three-Nucleon Wave Functions A) The Permutation Group of Three Particles B) Spin and Isotopic Spin Functions C) Euler Angle Functions D) Internal Functions 19 23 26 32 3) Wave Functions for Continuum and Bound States A) Separation of the Schrodinger Equation B) Continuum States C) Bound States D) Estimates of Importance of the Bound States 33 35 37 38 Evaluation of the Matrix Elements 1) Introduction 2) Exact Evaluation i n Terms of Radial Integrals A) E l e c t r i c Dipole Transition, P State to S State B) E l e c t r i c Dipole Transition, P state to D State C) E l e c t r i c Dipole Transition, F State to D State D) E l e c t r i c Quadrupole Transition, 1> State to S State E) E l e c t r i c Quadrupole Transition, S State to D State F) Magnetic Dipole Transition, S State to S State 3) CHAPTER 3 . 6 Approximations for the Radial Integrals Discussion and Comparison with Experiment ^+1 k3 *+5 k7 49 50 52 55 6k iii APPENDIX l o The Coordinate Transformation 70 APPENDIX 2o Degeneracy of the P r i n c i p a l Axes 7^ A P P E N L 1 X 3. Matrix Elements 1) 2) Matrix Elements of the K i n e t i c Operator Energy 77 Matrxx Elements o f the Spin and I s o t o p i c S p i n Operators A) I n t r o d u c t i o n B) The Spin Operators C) Proton and Neutron Operators D) Magnetic Moment Operators 79 80 82 85 E) 88 S i n g l e t and T r i p l e t Operators 90 APPENDIX k. Radial D i f f e r e n t i a l APPENDIX 5. I n t e r a c t i o n s with the Electromagnetic APPENDIX 6. Equations Field 1) E x p l a n a t i o n of the I n t e r a c t i o n 2) D e r i v a t i o n of the Matrix Elements A) E l e c t r i c M u l t i p o l e Matrix Elements 106 B) 109 Magnetic 95 M u l t i p o l e Matrix Elements 111 Euler Angle F u n c t i o n s BIBLIOGRAPHY FIGURE 1„ FIGURE 2o FIGURE 3• ll*f Schematic Experimental Arrangement f o r Studying the Reaction D(p^)He^ f o l l o w i n g page Showing the R e l a t i o n s h i p of the Three C o o r d i nate Frames f o l l o w i n g page Showing the I n t e r n a l Coordinates and the O r i e n t a t i o n of the T r i a n g l e FIGURE 4. Illustrating the Operators f o l l o w i n g page 1 7 8 for Infinitesimal Rotations f o l l o w i n g page 13 TABLE 1. The Representations of the Permutations 19 TABLE 2. P r o p e r t i e s of the S p i n - I s o t o p i c Spin F u n c t i o n s 26 TABLE 3. Numerical Values of the I n t e g r a l s R ( i ) 61 TABLE k. Numerical Values of the Cross S e c t i o n s 63 TABLE 5» Summary of the Experimental Data 65 D e r i v a t i v e s o f the Functions W 77 TABLE A l , • (-) INTRODUCTION The reaction D(ptf)He several observers: has been studied experimentally by Curran and Strothers ( 1 9 3 9 ) , Fowler, Lauristen and Tollestrup ( 1 9 4 9 ) , Wilkinson ( 1 9 5 1 ) , G r i f f i t h s and Warren ( 1 9 5 5 ) , G r i f f i t h s , Larsen ana Robertson ( I 9 6 I ) . and f i r e d at a D^O ice or gas target. The protons are accelerated The emitted gamma rays are observed by standard s c i n t i l l a t i o n counter techniques and the intens i t y of the emitted radiation i s measured as a function of the angle with the incident proton beam ( F i g . l ) o In this way an attempt i s made to obtain information about the bouna state of He"^, the continuum states which contribute to the reaction and the detailed mechanism of the reaction. Fowler et a l . (l9<+9) and G r i f f i t h s and Warren ( 1 9 5 5 ) have measured the y i e l d and angular distribution of the gamma radiation for proton energies from 0.2 to 2 Mev. More recent measurements have been made by G r i f f i t h s , Larson and Robertson (I96D. With the direction of the proton beam as an axis, the angular d i s t r i b u t i o n i s mainly pro2 portional to s i n e (Fig. 1 ) . There i s a small, possibly i s o t r o p i c , component which contributes a small percentage of the total y i e l d . Wilkinson (I952) measured the polarization of the radiation at 90° to the direction of the proton beam and found that the polarization was mainly i n the plane of the reaction, i.e„ the plane containing the directions of the proton beam and the observed radiation. This p o l a r i - zation and the s i n ^ 0 d i s t r i b u t i o n of the main contribution to the to f o l l o w page 1 FIG- I Schematic experimental arrangement reaction D( pfl )He 3 for studying '"the 2 reaction indicate that the radiation^ i s mainly e l e c t r i c dipole radiation. The isotropic contribution to the t o t a l y i e l d decreases with increasing proton energy, The exact angular d i s t r i b u t i o n and polarization of this smaller component have not been measured. G r i f f i t h s and Warren suggest that the simplest assumption for the reaction i s an e l e c t r i c dipole t r a n s i t i o n from a P-wave proton (L =1, L z = 0) and "^S deuteron ( t r i p l e t spin state, L = 0) to the ^S ground 3 state of He . This would give a pure sin 2 9 distribution. They also suggest that the smaller component could be either a magnetic dipole transition of S-wave protons to the ^S state of He^ or an e l e c t r i c dipole t r a n s i t i o n involving spin-orbit coupling which would introduce 2 a small component proportional to 1 + cos 9 . The purpose of the present work was to examine i n d e t a i l a l l channels which might conceivably contribute to the reaction and to close several gaps i n the existing l i t e r a t u r e on the subject. Previous works on the nuclear reactions of three-body systems have dealt with: 1. the mirror reaction D(n^)H^ (Schiff, 1937; Hocker, 19^2; 1951, 2. Burhop and Massey, 19^8; Verde, 1950; Austern, 1952), The photodisintegration of (Verde, 1950; Delves, I960), 3. and the photodisintegration of He^ (Verde, 1950; Delves, i960). - 3 - Both Verde and Delves have calculated the cross section for the elect r i e dipole transition between t h e 3 of He from a 2 P continuum and 2 S bound states 3 and H . k Verde has shown that the magnetic dipole t r a n s i t i o n S continuum state to a 2. S bound state i s unimportant because i t does not involve the part of the wave function which i s completely symmetric i n the i n t e r n a l coordinates. The symmetric part of the wave function i s expected to be the dominant part. This supports the c a l - culation of Burhop and Massey who found that, for the capture of neutrons i n the 1 Mev range, the e l e c t r i c dipole t r a n s i t i o n i s more important than the magnetic dipole t r a n s i t i o n . The neglect of the magnetic dipole t r a n s i t i o n by Verde and Delves may be contrasted the estimates given without detailed explanation capture of thermal neutrons. with by Austern for the Austern states that at this energy the magnetic dipole t r a n s i t i o n i s the more important while the e l e c t r i c quadrupole t r a n s i t i o n i s approximately 10 time6 smaller. The extent to which these estimates for nD capture may be applied to pD capture i s uncertain. considerable At low energies the coulomb barrier should cause a difference between the two reactions. In the present investigation the calculations of the above authors have been extended by calculating and numerically the cross sections for the following t r a n s i t i o n s : 1. e l e c t r i c dipole t r a n s i t i o n for a. 2 P-wave protons to the S state b. I—wave protons to the D state c. F-wave protons to the D state h k estimating 2. 3. e l e c t r i c quadrupole transition for a. S-wave protons to the D state b. D-wave protons to the S sta>te magnetic dipole transition for S-wave protons to the ^S state. Chapter I contains d. c l a s s i f i c a t i o n of three-body wave functions, making use of the symmetry properties of the functions i n spin space, isotopic spin space ana ordinary space, as done by Verde (1950) and Derrick ana Blatt (195&). The c l a s s i f i c a t i o n of Derrick and Blatt i s more detailed than Verde's, but does not include states of total isotopic spin T = 3/2. present worko These states are included in the The d e i i n i t i o n of the internal coordinates and the Suler angles d i f f e r from those of Derrick and Blatt, being less symmetrical but more convenient for the description of the continuum states for a deuteron and an unbound proton. In the calculation of possible radiation matrix elements, contained i n Chapter 2, integration and summation over the Euler angles and the spin and the isotopic spin spaces can be done in closed form leaving an integration over the three i n t e r n a l coordinates. Rough estimates of the r a d i a l integrals are also made in Chapter 2. To make these estimates, very simple r a d i a l functions were used: for the deuteron, an exponential 2), and function (e.g. Blatt and Weisskopf, 1952, Oh. for the third p a r t i c l e either a plane wave for the continuum states or an exponential are valid outside function for the bound states. These functions the range of the nuclear forces, the effects of which enter the wave functions only through the binding energies whicn determine the constants i n the exponential terms. The effects of the coulomb repulsion are neglected. A discussion of the results and comparison with experimental data are contained i n Chapter 3« It i s concluded that most important 2 i s the e l e c t r i c dipole transition for P-wave protons to the S state, o the next largest contribution coming either from an e l e c t r i c dipole it transition for P-wave protons to the D state or from a magnetic transition for S-wave protons to the state. dipole The sizes of the cross sections for the transitions depend on the amplitudes and r e l a t i v e signs of the various possible symmetry s t a t e s 0 To obtain this i n f o r - mation, i t would be necessary to assume some definite s p a t i a l dependence for the nuclear forces and to examine the solutions of the twentyfour coupled Schrodinger equations for the r a d i a l functions. was beyond the scope of the present work. This - 6 - CHAPTER 1 THE The approximation THREE BODY PROBLEM nuclear three body problem i n the i s basically of the Schrodinger non-relativistic the problem of f i n d i n g e i g e n s o l u t i o n s equation^"^ (1.0.1) Once "$" E i s known, matrix elements of the r a d i a t i o n operators be c a l c u l a t e d c The equation (1.0.1) i s i n t r a c t a b l e i n the form i n which i t i s w r i t t e n . I t becomes more t r a c t a b l e by choosing a co- o r d i n a t e systcfl which allows a s e p a r a t i o n of the motion and a p a r t i a l s e p a r a t i o n of the r i g i d system. centre-of-mass 'body' r o t a t i o n s of the In s e c t i o n 1, these c o - o r d i n a t e s are defined and l a c i a n and angular momentum operators are d e r i v e u . t a i n s the method of c o n s t r u c t i o n of antisymmetric making use of the s p i n and i s o t o p i c s p i n formalism 153). the Lap- S e c t i o n 2 conwave f u n c t i o n s , ( B l a t t &• IVeiss- kopf, 1952, po Schrodinger equation i s given f o r unbounu and bound s t a t e s . s o l u t i o n s are used may In s e c t i o n 3, the formal s o l u t i o n of the i n Chapter These 2 i n the c a l c u l a t i o n of the matrix elements of the r a d i a t i o n o p e r a t o r s . N a t u r a l u n i t s , -ft = c = 1 8 are used throughout t h i s work. This convention s i m p l i f i e s somewhat the appearance of the formulae. - 1. 7 - Spatial Description A, Definition of Co-ordinates It w i l l be assumed that protons ana neutrons can be treated as equal point uiajjsek. The system of three nucleons may then be described s p a t i a l l y as the location of three equal point, masses. i s convenient i. It to use the three ngut-nanded co-ordinate frames ( F i g . 2 ) : the laboratory frame, an i n e r t i a l frame fixed i n the laboratory; ii. the centre-of-mass irame, a moving, non-rotating frame with the origin fixed at the centre of mass of the three p a r t i c l e s ; in. the 'body' frame, a rotating frame with the origin coinciding with the origin of the centre-of-mass frame and with the axes coincicing with the p r i n c i p a l axes of the triangle formed by the three p a r t i c l e s . The notation for the cartesian components of a point i s : x', y and 1 z ' for the laboratory frame; x, y and z for the centre-of-.nass frame; and X, Y and Z for the 'body' frame. The pos/tion vectors of the three p a r t i c l e s i n the laboratory frame are denoted by r^, and r ^ . The folxowirig nine co-ordinates are adopted : i. the three co-ordinates of the position vector s = 1/3 (r_^ + + .£}) °* the centre of mass i n the laboratory frame; ii. the three iuler angles, -<. , p> and giving the rota- tion of the 'body' frame with respect to the centreof-mass frame (Fig. 2 ) ; p FIG ct O 2 O o Showing the relationship of the three coordinate frames- °% iiio the three internal co-ordmatet,, r. = £1 - £2 ana 9 (cos 9 = £_-_q_/'rq) giving the size and shape of the triangle, ( i i r • 3) . The choice of Suler angles i s that i l l u s t r a t e d by ',/igner (1959, P« 9 0 ) and may be definea a s a positive rotation of the 'body' frame, viewed from the laboratory frame, through the angles: about the Z axis, 3 about the new position of the Y axis, ana <X. about the new position oi the Z axis. The rotations are to be performed i n the indicated order, "hen viewed from the 'body' frame, the equivalent rotation i s : - Od about the Z axis, a nd (3 about the Y axis, - & about the Z axis, The choice of internal co-oraiuates i s p a r t i c u l a r l y convenient for the aiscussion of continuum states consisting of a deuteron and an unbound proton. They are defineu so that t i e three par- t i c l e s are i n the XZ plane and tnac, for large separation of the Jeuteron and the ^roton, tiie p a r t i c l e s aio on the Z axis. Neglecting temporarily the effects of permuting the position of the p a r t i c l e s (see sec. 2E), r i s the distance between the neutron and the proton ( F i j . 3, p a r t i c l e s 1 and 2), q i s the distance between t i e other proton (particle 3) and the centre of mass of the deuteron, and 9 i s the angle formed by the vectors 1 3. and 3 - as indicated i n Figure To orient the triangle i n the 'body' frame, the three p r i n c i p a l to follow page 8 3 Z 2 FIG 3 Showing the internal coordinates and the orientation of the triangle in the X Z - p l a n e of the body frame- The Y - a x i s is d i r e c t e d o u t of the paper- axes of i n e r t i a are computed. 9 - The axis associated with the largest moment of i n e r t i a i s defined as the Y axis, that associated with the intermediate axis. moment as the X axis ana the remaining one as the Z The directions of these axes are not yet fixed by this requiremento The positive Y axis i s defined as the d i r e c t i o n of motion of a right-handed screw rotating from p a r t i c l e s 1, d ana 3 The positive Z axis i s defined so that the angle fi , in that order. the angle between q and the Z axis (Fig. 3), i s within the l i m i t s )4 -3L - T £ • ^ n e positive X axis i s fixed by the require- ment that the X, Y and Z axes form a right-handed frame. for Formulae f£ are given in section 1C. B 0 Transformation Formulae With the d e i i n i t i o n s adopted i n the preceding section the following transformation i. rules are established by inspection. Transforming from tne centre-of-mass frame to the laboratory frame introuuccs a simple t r a n s l a t i o n : X ' = S X + X y ' = s +y y z' = s +z z (1.1.1) ii. Transforming from the •body' frame to the centreof-mass frame after the rotation of the three rotations: jU^j,*^ consists 10 /0>S<COSp,COi,Na-S\a'<-S\tOi -Sin.<\coS(3CosS 5in«< -COS < Svn^> - C O S K S H V X sirv.^, Sin. (bcos* \ cc*^ y (1.1.2) ^ 2.1 ^31 ^21 ^ZJ ^32- ^35 / y In matrix notation, x - R_X. The matrix for the inverse transformation i s the transpose of the above -1 T matrix, R =R iii.. Transforming 8, from the internal co-ordinates, r, q and to the cartesian 'body' co-ordinates, X, Y and Z, of the three p a r t i c l e s are: X = - 5l ( (1.1.3) C. Sin. <f> fx sm-(^ + e)V=o X,= sin.^ - r sm.(^+e) X = Sin. ^ Z =~ c o s ^ + 1 co^(<L\e) Moments of Inertia To show i n d e t a i l how the triangle formed by the three nucleons i s located ana orientated i n the 'body' frame, the three p r i n c i p a l moments of i n e r t i a are now calculated. Since by d e f i n i t i o n the 'body' axes coincide with the princ i p a l axes of i n e r t i a , the three products of i n e r t i a (Goldstein, - 11 - 195°* P» 1^5) are zero. u.i.<o Z n.x^ -£"VA - ^ W c = 0 Since the three nucleons are defined as lying i n the XZ plane, = 0 and the f i r s t two of the products of i n e r t i a (1.1.4) are i d e n t i c a l l y zero. By expressing the third of these equations i n terms of the i n t e r n a l co-ordinates (1<,1»3) useful formulae for j> are obtained as follows: =Z 0 M XZ. = M l l ^ S i r L ^ c o s ^ + T*sia(tf + e)cos(rf+e)«j (1.1.5) from which (1.1.6) sin. 2 ^ = "i. -r 1 sin.2.6 cos 2.a^ - W A * Ir'co^2e> A where (1.1.7) A * = (1.1. fe) -R A 1 Z^ + |> 1 (1.1.9) - R * - * ^ 1 = 3 L •^ V ' 1rc s,, v e = area of triangle. For both R and A the positive roots are used i n order that <f> be - 5L z the direction of the Z axis. r e s t r i c t e d to the range <f> £ ?L z , removing an ambiguity i n The three p r i n c i p a l moments of i n e r t i a may now be calculated i n terms of the i n t e r n a l co-ordinates. - - 12 =M (l.i.io) \ +C O S t - l T = h |i 6 (i.i.ii) v = t ( l + Cos2.^C6s2S-S>a.2^s.n.2.&)j ; + a iMK 1 - M | ! W * W ) (ia.i2) =^ 0 ~ c o s f) * ^Tl0' V z cos c o sZ e +s i a ^' ' *®)} s ,,t It w i l l be found convenient to use the quantities X defined as: (1.1.13) = MY = i ( *- ) U M A x , Xy a n d "X z Do 13 - Angular Momentum Operators In the centre-of-mass frame, the t o t a l o r b i t a l angular momentum i s considered to be that of a r i g i d body rotating with the 'body' frame. This i s possible, although the three nucleons do not form a r i g i d body, because, from the d e f i n i t i o n of the 'body' frame (sec, IA), the total angular momentum of the three nucleons i n the •body' frame i s zero. The operators for the components of the total angular momentum i n the centre-of-mass terms of the Euler angles. frame w i l l now be obtained i n For later use, the operators for the components of the total angular momentum about the instantaneous positions of the body axes are also given. The operators are obtained i n the usual manner (Edmonds, 1957> P. 13) by considering two equivalent rotations: i. the rotation j^^.xj through the Euler angles with the corresponding operators for i n f i n i t e s i m a l ii . A- , JL a«t ap and A rotations , ' and-the rotation ,«^vj, <<^| the angles being measured about the fixed centre-of-mass axes with the corresr ponding operators for i n f i n i t e s i m a l rotations , _JL and b ( F i g . k). The l a t t e r operators are proportional to the operators for the components of the total angular m6mentum i n the centre-of-mass follows: (1.1.14) I •—x a 6 U = -i aou frame as to follow page Illustrating the operators for infinitesimal rotations - Ik - Since for proper rotations the operators for i n f i n i t e s i m a l rotations can be treated as vectors, _JL_ of _b , 3> (l.lol5) 6 and The equations ( 1 . 1 . 1 5 ) a and ^ _ can be expressed i n terms i n the following manner (Fig. 4 ) . -5\ruX_^L. — , + Co»_jL ">ay be inverted and then substituted i n ( 1 . 1 . 1 - + ) to give: L , - cos* Sin. ^ Sin, % + (1.I0I6) COS IS *s J The equations ( 1 . 1 . 1 6 ) may then be transformed using ( 1 . 1 . 2 ) to give a set of operators which appear i n the'Laplacian. COS oL cot ft + sin. d. JL - c o j ^ . J L 1 2>«l (1.1.17) U s= d(3 L y = - 1 J - sm.cC cot a A , + Cos=C t 1 ^ For both sets of operators ( 1 . 1 . 1 6 ) *>«-(i + gin. °6 ^ «,«.ftH and ( 1 . 1 . 1 7 ) , the square of the angular momentum i s : j the operator for = 15 (l.l.13) The commutation rules (1.1.19) [LX,L ] = «.L> (1.1.20) - 3 The L . expressed For of-raass an th. a„gl« L' = - L arises i n terms o f i n the i s given f.t') and v sets of and (1.1.20) observer frame two a n c minus s i g n i n L^, a r e f o r the L' the cyclic * the permutations. fact that L , L, A I Y same E u l e r a n g l e s 'body' f r a m e , jdC.', (i , tf j are ..... = the , Ly rotation operators w h i c h have the and v as (-tf^-ji^-xl a ..t of result are: permutations. cyclic from by = - L„ operators of the , Thus, i f L . - usual and centre- L,. commutation rules, E. The Laplacian Operator ii Since operator, used for will here the be this To c the Schrodinger operator must be do i t i s necessary this, co-ordinates used: laboratory equation the frame i n v o l v e s the evaluated i n terms of cartesian co-ordinates y^, z^, x' , 2 the t o o b t a i n the d e f i n e d i n s e c t i o n IA. (x^, Laplacian of y' , z' 2 co-ordinates metric tensor The following notation the three 2 > x'^, nucleons y , in the z • ) are w r i t t e n i n t h i s o r d e r as , i = 1 ... 9 d the c o - o r d i n a t e s (s , s , s , <f~ , (4 , 5 , r , q, Q) a r e w r i t t e n i n t h i s o r d e r as p. x y z i i a i = l 9« c.. combined into (1.1.21) ^ The transformation nine -^(ft) equations of equations the form n ( 1 . 1 . 1 ) to (1.1.3) a r e , then - 16 - Using (1.1.21) the Jacobian of the transformation i s obtained: (1.1.22) 3 = det lipjl The detailed expressions for (l 1.21) and (Id.22) are given i n a Appendix 1, The metric tensor consists of the elements (1.1,23) £,j . ^nv and, written as a matrix, can be conveniently expressed i n the following manner (Derrick, i960). I o o o S" o o o I l 0 o o o 1 o O I I o N. O O (S-')T O O o I (1=1.24) Where tosOLcot(i (1.1.25) s = -SiivCXcat jb Sin, OC cos a cos SuvCX. Sin. O and N i s the symmetric 9 x 9 matrix: a S\r\ p> (i - 17 - (1.1.26) N, o • - ^ 0T ° where br (1.1.27) 0 = AgL icy A 5r\ A Z s\rv. 2.9 1 The derivatives of 0 obtained from (1.1.6) and ?l x ^ Xy and are defined i n (1.1.13). The conjugate metric tensory, g *', i s also needed. It 1 i s defined by S^g*"^ anc * ^ n e *aatrix representing the tensor i s obtained from the reciprocals of the matrices i n ( 1 . 1 . 2 4 ) . i n a similar manner i t i s : (1.1.28) o where M i s the 9 * 9 I o o 0 s o o o I symmetric matrix: Written - 18 - (1.1.29) 3FC A 1 if I 6my ina5sir>.2.© o A* 3. H-A 4 5(5r -H-a/)6-cos26) 2A . z Z The can be w r i t t e n e x p l i c i t l y (see appendix Laplacian pL i n terras o i the with the result 1) J- v -f z T +1 E i n which V2 X a* + M )'+ z. (1.1.32) %36 r c^ ^ z 2 Sin. 2© cos A" The i s now u s e f u l n e s s of the operators apparent. The terms M'', 1 M° and T L^, L<_ and Q o i n t e r n a l co-ordinates only. L^ ( 1 . 1 . I V ) are f u n c t i o n s of the 19 - 2o The Symmetries of Three~Nucleon A« - »Vave Functions The Permutation Groups of Three Particles For the c l a s s i f i c a t i o n of the three-nucleon wave functions use i s made of the symmetry under permutation of co-ordinates i n space, spin space and isotopic spin space. This symmetry property of nucleons i s well known (Schiff, 1 9 5 5 , Ch 0 9 ) and this section contains- a summary of the relevant r e s u l t s . The six permutations of three objects are l i s t e d i n Table lo The permutations are considered as operators changing the posi- tions of three ordered numbers„ For example, the operator P^ has the property P^(abc) = (132)(abc) = (cab), i e . the number i n the f i r s t 0 position i s replaced by that i n the third, the one i n the third by TABLE I The Representations of the Permutations Permutation Symbol Representations \2 Mixed ^Symmetric (1) (2) (3) (123) (132) (12) (3) (23) (1) (3D (2) l P P 2 1 1 (^Antisymme trie l ( t ) 1 -ft B) -ft ) 1 0 (-ft \ ft {5 (2 r> -I (S .? ) .—\ P 3 k 9 P 5 P 6 1 1 1 1 (-ft [ft JJ [-ft ft ft V3 1 -1 ft) 1 -1 ft J -1 V - 20 - that i n the second and the one i n the second position by that i n the first. The permutations form a group which i s isomorphic with the group [^ (e.g. Lomont, 195?) and has three irreducible representations , \z and and \$ for which matrix representations D^^Tj)' , T ) (V^ W are given i n Table I. The three irreducible representa- tions w i l l be referred to as the symmetric, mixed ana antisymmetric representations respectively. When the three numbers are the arguments of a function <fi 1959, X^, X^), the function (X^ X^, X^ ) i s interpreted (Wigner, Ch. 11) so that ~^(x;x,x ) - P(^.X,,X ) (1.2.D 3 ; where (X>0X;) (1.2.2) = y (x^Xz.Xs) In this way the following six functions are obtamea. (1.2.3) ? ^ ( X " X 4 ' X 5 ) = /(X3.X-Xz) A 5 ^(*i,x x ) = /(x„x x ) / z> 5 (> 3 3 v ? ^(^.x^.x,) - ^(xx,x)=- ^ 5 1; 3 i 5 When the linear functions combinations dimension which For ^ of , there 1 are are 21 all independent, t h e s e so sets 1. representations i. of Table it that, for of linearly t r a n s f o r m under p e r m u t a t i o n s the - 1. the according I, these is possible to form j - t h representation independent to the of functions rule: are: Symmetric (1.2.5) ii. Mixed and (1.2.V) = ^ iii (1.2.6) Assuming 0 = _L the <ft i s for there + ^ + normalized, When <f> i s Assume, i rz - -^3 - ^ + i^r + {A) Antisymmetric $™ ordinates, (r, - invariant c a n be o n l y example, that ^ _ ^ _ ^ these s i x functions are orthonormal. upon p e r m u t a t i o n o f two of three linearly independent ^(X^X^X^ ^ ^(X X ,Xj) L> k the co- combinations, or i.e. Then the l i n e a r l y independent combinations become: i. Symmetric ( I . < M O ) ^ + i (l 2,12) u ii. Mixed i Antisymmetric i o \ = ** ) S o The two sets of mixed functions become proportional, and a l l functions have been normalized. When ft i s completely invariant, only the symmetric functions remain do2.i5) r >= f In the following sections, combinations of properly symmetrized functions from different spaces w i l l be needed such that the combinations transform upon permutation according to ( l 2 . * 0 . u With the functions i n the two spaces being f ^ \ t n e combined functions are: Symmetric Ui) (1.2„14) 10 K (,) = f g r w.rg'- f ^ ^ and - 23 Mixed VL, (0 : k"' (0 (I) k , i« k (1.2.lp) r uv k, (?) k , (3) > /pn) W, w k c>o • ^ g, x k>> ID o a) , cY> Antisymmetric k (1.2.16) CO k ci") = * g By s u c c e s s i v e a p p l i c a t i o n of the above, t o t a l l y antisymmetric wave f u n c t i o n s are c o n s t r u c t e d from the f u n c t i o n s f o r s p i n , i s o t o p i c s p i n , i n t e r n a l c o - o r d i n a t e s and B. Spin and I s o t o p i c Spin the Ruler angles. Functions Linear combinations of the s p i n f u n c t i o n s of the three nucleons can be formed which are e i g e n s t a t e s of the t o t a l s p i n and d e f i n i t e symmetry p r o p e r t i e s . The t o t a l s p i n may have the values have S = 3 / 2 , giving rise to a set of orthogonal quartet ), and S = H functions 4 giving r i s e to two sets of orthogonal ) and X. . z d 1 functions p a r a l l e l spins, i . e . for S (S ) z When two of the nucleons have 0 = +)£ or S = •=}£, there are three l i n e a r l y independent functions for each value of S . r From these are obtained by linear superposition, as i n (1.2.10 and 11), three ^ a symmetric function OC^^ (s^) functions tions X$ *(i> ) and X ^ 2 1 o ({) 0 (S d z (l 2 17a)X, 2 ) -yi doublet orthogonal and a pair of mixed func- which are for S = z / (1^,(6, CC z 5 + i ( i , * ^ -oC.oC^,) S i m i l a r i l y , wnen a l l three nucleons have p a r a l l e l spins, i e„ S 0 = -3/2, or S = +3/2 there i s one l i n e a r l y independent function for each value of S „ For S = +3/2 this i s z z (1.2.17b) X " ( | ) ; = oC.c^oCj The single nucleon spin functions are o( and CT 2 ^ - " ( ^ ( ^ ^ ~ "i) ° ^ o r (lo2.1?) have tne same form with " = |?> where c r t f - = +°c(S^ i) and L a n c z and ^ ^z (i t = :: 2 t "-^Z^ 9 t i i e f u n c ti° n 6 interchanged. The treatment of isotopic spin i s exactly the same as that of the spin, except for notation. tions and (T ) 3 (Tj) T 1T = - Tr 5 There are a set of symmetric func- with T = 3 / 2 and two sets of mixed functions with T = The convention and for neutrons "V^ (T ) 3 used i s that for protons -y - + v (Blatt & Weisskopf, 1952, Ch, 3)o To describe the isotopic spin ol two protons and one neutron, only those states with T, = •=•)! are needeo. To obtain the isotopic spin functions for H 1V and V i n the above. 3 (T, = interchange Unless otherwise stated, a l l isotopic spin functions w i l l be those for He (T^ = <=)z) 0 Tne combined spin and isotopic spin functions can now be constructed using ( 1 . 2 o l ^ , 15 and 16)„ An abbreviated notation i s used, the values of T and S not being included i n the l a b e l l i n g . symmetry properties, and the values of the spin and isotopic spin functions "f are l i s t e d i n Table I I . The functions are: The - 26 - TABLE II Properties of £ pin-Isotopic .Spin Functions C c Symme try Total Spin symmetric 3/2 3/2 mixed 1/2 3/2 symme trie 1/2 1/2 mixed 3/2 1/2 mixed 1/2 1/2 antisymme trie 1/2 1/2 Total Isotopic spin Euler Angle Functions In this section, functions of the Euler angles which have definite symmetry properties and parity are constructed. In order to do this the effects of the permutation and parity operators on the rotation ioc are now determined. When the wave functions of the deuteron and the incident proton do not overlap there can be no permutations involving the incident proton; thus the internal co-ordinates q and r are then unique ( F i g o The i d e n t i f i c a t i o n of q and r i s made invariant 3 ) , upon permutation of the nucleons by the s t i p u l a t i o n that q and r change continuously as the wave functions of tne proton and deuteron overlap s u f f i c i e n t l y to permit permutations involving the proton. The Z axis i s defined so that for large separation, the incident proton i s on the positive Z axis. mutation of the nucleons. numbering of the nucleons. The Z axis i s invariant upon per- The Y axis, however, i s dependent on the The permutations P^, P^ and P change the Y axis but the permutations P^, P do not and P^, interchanging the positions of only two of the nucleons, have the effect of reversing the positive Y axis and hence also the X a x i s 0 This may also be considered as a rotation, through an angle of 160° about the Z axis, of the triangle i n the 'body' frame. If the orientation of the t r i - angle i n the centre-of-mass frame i s described by the rotation , then after any of the permutations P^, £ P or P the same 6' physical orientation of the triangle i s described by the rotation ( The effect of the permutations on the Euler angles may be summarized as follows: - 28 - When two of the p r i n c i p a l axes become degenerate the assignment of the 'body' axes i s not unique. This occurs when the three nucleons form an equilateral triangle and when the three nucleons are oo-linear 0 A method of handling these degeneracies, which i s due to Derrick (I960), i s discussed i n Appendix 2. The parity operation, i . e . the inversion x and z Y =?• <=y i s equivalent to a rotation of 160° about the 'body' The operations of f i n i t e rotations about different axes do axiSo not commute and the effect of the parity operator,^ tion ^-x, y ju^p^l (1.2.21) , on the rota- must be expressed as TT|a,^j = ^ - o c , iv - p , TT + * j = ^-o^ ir + ^ * j These rotations are established by examining the rotation matrix R. l J (1.1.2) and requiring that R. (ac.frtf ^ J for columns 1 and 3 when ) = -R. ( oC iJ ^ j ^ , ^ , * ^ = j ^ ' , ( i \ X 'j remains unchanged by this transformation but as , |3>'1} j' ) ° Column 2 = 0, this does not effect the parity operation. Use i s now made of (J.2.20 and 2 1 ) to construct normalized linear combinations of the functions definite symmetry and parity,, • > which have In these functions L i s the t o t a l o r b i t a l angular momentum, p' i s the 'body^* Z component of L, and p. the centre-of-mass z component of L s The properties of these functions, which w i l l be needed, are given below. (i 2 22 0 0 ) O^yKfM) ' e ^ A ^ p l e ^ - (i 2,23) 0 29 - A.(a) -Ee yg^)!(t-u)i( ,^'o^oT x L C 0 3 ^- s«u&\ l x x + r> where the summation over x i s between the zeros i n the denominator. The orthogenality and normalization i B given b,y the i n t e g r a l (1.^.25) j<U j o s ^ d f t l ^ ^ ; ^ , ^ ) ^ ^ . ^ ) ' \ o O U z V,^ V,^|^T ' The properties under permutation are: (1.2.26) TJ^yCa.fi.is) = ^ y C . | ^ , ^ ) + a Thus these functions are either symmetric (u' even) or antisymmetric (u' odd). The functions never transform according to the mixed representation. Under the parity operation, the properties are: (1.2.27) T £ ^ ( a M ) =C-) L > , ^yC*.(4^) The following orthogonal functions are thus defined to have even (*) or odd (-) parity and symmetry properties according to (1.2.26 ) o - W ; - ©1 (<*(.* U 0 30 / u. iv 1 For some values of ji * and ^ i , these functions may be related to the spherical harmonics and Legendre polynomials: (l 2o30) o . , b The Euler angle functions may now be combined with the spin-isotopic spin functions ^ i n the manner given i n ( 1 . 1 . 1 ^ , and 16)o These are then eigenstates of L, L 15 = u, S and S . From these combined functions, eigenstates of J = L + S, J = M, L and S 2 can be formed i n the usual manner making use of Clebsch~Goraon coe f f i c i e n t s (L, u, S, S J, M)o For later use, a l l possible ortho- normal eigenstates formed i n this manner having J = h and even parity x are given below. To aid i n the c l a s s i f i c a t i o n and manipulation of these functions, an abbreviated notation i s used, i n analogy d e f i n i t i o n of the spin-isotopic spin states '6 states (L = 0) (1.2.51) K i " " Mr- « W w ~ ct) W ^, (Sl) % L . with the - 31 - P states ( L = 1 ) X •£('^-*. lli- KM«,,(Sl) s n D states (L = 2 ) ( l o 2 o 3 3 ) ^ •S(«-M-si|i.MKw^,(Sj) - (l.2.33)^ 5 i l 32 -S ^ l ^ i , * ) The states ^ / ^ to yCV - ^,(S^ y are doublet states (S = #) and JUT are quartet states (S = 3/2). The states 1 , y a n d to ft ^ \\ 4 a r e symmetric on permutation of angle, spin ana isotopic spin co-ordinates, whereas states JUf ^ jS y J/Z^y JU/' , ? jSy JjJ\y transform according to mixed representation, and states a n d y ^ ^ ^ are antisymmetric. Jyt/* \$ and US ^, Jjt/"^, Similar sets can be constructed for any value of J, however they are of no interest for the present problem as the ground state of He^ has J = # and even p a r i t y . Also i t i s more convenient to use eigenstates of L, y., S and S^ i n describing the continuum states. D. Internal Functions Functions of the i n t e r n a l co-ordinates may from an arbitrary function i n the way be constructed indicated i n (1.2.5 to 8) or (1,2.10 and 11) to form symmetric, mixed or antisymmetric functions. These must then be combined with spin angle functions, i n the manner of (1.2.16) to form functions which are completely antisymmetric permutation of a l l the co-ordinates of any two nucleons. on The con- struction of these functions i s given for use i n the following sections, assuming that the i n i t i a l function, ft , i s symmetric oh interchange of the co-ordinates of two nucleons. Denoting this symmetry by the n o t a t i o n , functions $ 33 - (1, 25) = $ ( l i 32), the symmetrical are (1.2.1J and 11): (102,3*0 Assuming ^5 describing asymtotic to be a product o f a deuteron the t h i r d nucleon,, the f u n c t i o n s f u n c t i o n and a f u n c t i o n (1.2 = 3^) have the forms, ^-—r± ==i "-(T)v(a) d.2.35) r ; _ ^ 1* when s e p a r a t i o n ^ - . - j i ^ H ^ ) o of p a r t i c l e 3 from p a r t i c l e s 1 and 2 i s l a r g e com- pared with the s i z e of the deuteron. Here u ( r ) i s the i n t e r n a l deuteron f u n c t i o n and v(q) i s the i n c i d e n t proton f u n c t i o n . asynytotic form i 6 used i n examining This the continuum s t a t e s i n the f o l l o w i n g s e c t i o n and a l s o i n the approximate calculations i n Chapter 2. 3o Wave Functions f o r Continuum and Bound S t a t e s Ao S e p a r a t i o n of the Schrodinger Equation The Schrodinger equation i s now r e - w r i t t e n of the preceding s e c t i o n . u s i n g the r e s u l t s Using (1.1.30), the equation i s : - The function 3f i s a function of the s p i n c the nine s p a t i a l co-ordinates. the potential operator, co-ordinate, by writing £ ?>h - c the isotopic spin and In the absence of external forces, V, does not depend on the centre-of-mass s_. The centre-of=mass motion i s then easily separated E = $ (s) ¥ where i s independent of s. The resulting equations are: (1.3.3) I zi- f-[ vVl$' = r where £" + E' = Ec Using the results of section 2, 4 r may be formally written i n the following manner, where a(L, •, j ) are numerical c o - e f f i c i e n t s , F(L, fi\ internal functions (1.2.1M, W (l 2.2o) and 0 u u (-) j ) are are Euler angle functions are spin-isotopic spin functions (1.2.19). When (lt^o *) i s substituted i n (1.3o3), both sides of the resulting 2 , ,L * equation multiplied on the l e f t by W • (±) ^ \ , and the i n t e - gration over the Euler angles and summation over the spin and i s o topic spin co-ordinates performed, a series of coupled d i f f e r e n t i a l equations for the r a d i a l function F(L, ji', j) result. The coupling - 35 - arises from the off-aiagonal matrix elements of the k i n e t i c energy operator, Vc — ' (~f + T ) , and of the potential energy operator, In Appendix 3 are tables of matrix elements for the kinetic energy operator and for spin and isotopic spin operators with central f o r c e s 0 associated In Appendix k, some of the coupled d i f f e r e n t i a l equations for the r a d i a l functions are given. The precise form of these equations i s not needed for the present work. Only the symmetry properties and the asymjtotic form of the r a d i a l functions are used here. B Continuum States 0 For the continuum states, i t i s required that the wave functions, for large separation of the incident proton and the deuteron, become a simple product wave function of a deuteron i n a "^S state and an incident wave for the proton The spin function must be either 0 .X. ^^) 10 or "X'°(S^) as only these two have the form of a t r i p l e t spin function, for nucleons 1 and 2, combined with a single p a r t i c l e spin function, for nucleon 3o The form of X - (Sy) i s that of a singlet spin function and a z single p a r t i c l e spin function. be The isotopic spin function can only as the deuteron i s i n an isotopic spin singlet state. the spin-isotopic spin function must be either or (1.3.5b) O t ^ C Thus - 36 - The r e l a t i v e amplitudes of the doublet and quartet states are and J | respectively so that, for large separation, the spin- isotopic spin function must be: The asymptotic s p a t i a l functions are required to be of the form: where u(r) i s the S-state deuteron function with the Jf r^ 0 Js*n, © de | T M T U(T-) (1.3.6) normalization 1 = I and the terms i n the summation represent a wave function to unit density at i n f i n i t y . normalised The constants are such that i f the functions v (q) are replaced by the spherical Bessel functions lj J > 6 U m m a tion becomes the well-known expansion of a plane wave : The terms i n this expansion are referred to as S, P, D, F ... waves for L = 0, 1, 2, 3 ... respectively. For properly symmetrized r a c i a l functions which have the above asymjtotic l i m i t s the notation i s G^(L) where i refers to the spin-isotopic spin function momentum. and L i s the o r b i t a l angular The continuum wave functions must be of the following form: - 37 - For the doublet state (Io3ol0) SIV^C^H) W (c-) )f-LQ L 0o L 5 5,1 and for the quartet state d 3ai) 0 EtV^au o + W^Cc-^fs^G.)^, the asym|totic expressions (1.2.35) were used. In Appendix 4, details are given for the derivation of the d i f f e r e n t i a l equations for the r a c i a l functions and i t i s shown that the above asymftotic solutions are possible for q greater than the size of the deuteron. C. Bound States 3 The bound state , assumed to be the ground state of He must have J = and even parity. This i s accomplished by using the functions ^Uf^ defined in (1.2.31» 32 and 33). functions there i s a properly symmetrized that, when combined withJU/^ formed. r a d i a l function, F , so « a t o t a l l y antisymmetric function i s The bound state i s then a linear combination of the as follows: (1.3ol2) For each of these ^ lJ[ L - 38 - The functions F^, Fg, F^ and are symmetric, the pairs F^ ^ Q F 3,i' F 6,i' 7,i' 9,i F F F,, , and F, 11 9 F 13,i' l4,x F are antisymmetric . Id orthogonal as the functions a n d F 15.i a r e m i X 6 The combinations uT^ d a n d F l ' F, Uf ' V are are orthonorraal, and each enters with an amplitude a^ i n the following sense: for F^ symmetric or antisymmetric, (1.3.13) and f(F u()V K) L l y d r = jKf d r =a. for F. mixed, l with (1.3.15) \lF ,rdT = f | U ^ r L - where 16 (1.3.16) E l ! Do , Estimates of Importance for Bound States It i s possible to make q u a l i t a t i v e estimates for the 3 r e l a t i v e importance of the sixteen states i n the bound state of He , These f a l l into four groups. i. Using a similar c l a s s i f i c a t i o n of state for H"^ and He"'', but including only the states of isotopic spin, T = 34, Derrick, Mustard and Blatt (1961) have made a v a r i a t i o n a l calculation for the binding energies of H"^ and He^. - 39 - Their results were negative in that the potential U6ed, that of Brueckner and Gaminel (1953) could not f i t the binding energy and the coulomb energy d i f f e r e n c e The results did show, however, that the t o t a l probability of the P states was of the order of 7 x 10 . It i s believed that this probability would not change by several orders of magnitude i f a more accurate force were used. For this reason i t was P states could be safely neglected culations. nuclear f e l t that the i n the present c a l - Derrick, Mustard and Blatt also obtained a 7 per cent probability for the D states, the remainder being S states. ii 3 The states may be divided into two groups, one of T = 34 and the other of T = 3/2 0 The ground states of the i s o - topic spin quartet L i3 , He 3 , a3 and n 3 are expected to have a similar energy l e v e l , neglecting the coulomb 3 forceo 3 Li and n do not form bound states and one expects the lowest energy, T = 3/2, states in H and He 3 to be i n the continuum and well above the observed ground state energies. of iii. The most important states in the ground state and He 3 would then be T = Vz states. Of the possible transitions examined i n Chapter III, only 2 one has an angular d i s t r i b u t i o n proportional to 6 i n 9 to agree with the experimental data. This i s an e l e c t r i c dipole t r a n s i t i o n from a P-wave proton to the S state of - He^ kO - This, combined with the estimates of Derrick, 0 Mustard and Blatt ( 1 ^ 6 1 ) , indicates that the S states are the most important. iv. For the S states, the expectation value of the magnetic moment operator: ll I u (1,3.17) V , +u ifi-T i(i+r W, ] may be calculated using tables of matrix elements for these operators l i s t e d i n Appendix 4 . (1.3.1.) <V> - ( f 3^)a. 2 N K (|N i N ) « + + (l + + <L*) + <) + The result i s : + (- 1 nuclear magnetons The experimental value i s uCHe'') =-2.13 n m o 0 It i s seen that the best f i t i s obtained for a^ = a^ = a^ = 0, a^ = 1. This implies tnat the states ^J^\ » with a symmetric r a d i a l function i s the most important state. Some admixture complete agreement with of D states i s needed to provide experiment. Thus i t i s concluded that the state with the largest probab i l i t y i s the r a d i a l l y symmetrical state admixtures F^JuT^ „ There w i l l be of the other S states and D states but with much smaller probability. The P states may be neglected. - ~ kl CHAPTER 2 EVALUATION OF THE lo MATRIX ELEMENTS Introduction In this chapter the multipole matrix elements given i n Appendix 5 are evaluated using the wave functions of Chapter 1 for the continuum and bound nucleon states. multipole moments, the factors involving In the integrals for r Y (e, expressed i n terms of the i n t e r n a l co-ordinates and angles (see App. 6). The d.) the are Euler integration over the Euler angles and summation over the spin and formed exactly, the isotopic spin co-ordinates may the be per- leaving an integration over the internal co-ordinates These r a d i a l integrals cannot be evaluated u n t i l the correct functions are known or u n t i l approximate forms for the functions are chosen. In section 3, internal internal approximate internal functions are chosen which enable rough estimates of the r a d i a l integrals of the cross-sections to be made. 0 and In the multipole moments, the r a d i a l integrals are independent of the value of ra (the WignerEckhart theorem, see e.g. Edmonds 1957» P« 73)* Thus the angular d i s - tribution of the emitted radiation for each multipole t r a n s i t i o n be evaluated without knowing the r a d i a l i n t e g r a l s . may This i s done by making use of the angular distributions l i s t e d in (A5.1c21). The well-known selection rules involving parity, o r b i t a l angular momentum and spin l i m i t the possible transitions. elements must be invariant under the parity operation. The matrix Examination of the integrals for the multipole moments shows that for e l e c t r i c 42 mu l t i p o l e r a d i a t i o n ( B l a t t and Weisskopf, 1952, p. 5^7): (2.1.1) and + l f o r magnetic m u l t i p o l e (2.1.2) ^ a W =" where 1"L tt ' 1 and b radiation are the p a r i t i e s o f the f i n a l and i n i t i a l nucleon s t a t e s and of the operator Y^ i respectively. The s e l e c t i o n m r u l e s f o r o r b i t a l angular momentum are shown by the f o l l o w i n g i n t e g r a l over the Euler angles, 6, d e r i v e d i n Appendix (2.1.3) tt c-) L " M 7^C^a^0(^^)(^b^'T where is x l h m l :a 2 "3/ the Wigner 3-J symbol (Edmonds, 19?'/, p. 46) and i s r e l a t e d to the Clebsch-Gordan c o - e f f i c i e n t s by (2.1.4) m n»> = (-) rr\ Convenient t a b l e s of the 3-.i (1959). symbols are given This i n t e g r a l i s zero unless tt^ 1T*L Tr by itotenberg et a l , b = | . The Euler angle f u n c t i o n s W come from the f i n a l nucleon s t a t e , the operator T L I ( i ^t) e a n d *'^ ^ l l l e n t a nucleon s t a t e r e s p e c t i v e l y . a c t i o n H* (A6.1.32) does not c o n t a i n any 6 p i n operators not mix s t a t e s of d i f f e r e n t a or i i ^ . The proton The i n t e r and so does operator contained Ik) and in i t mixes states of different T as shown i n (A3o2.13 and (A3.2.16 and The interaction H" 19)o containing (A5.1.33) the magnetic moment operator mixes states of different S and of different T as shown i n (A3o2.23 - 2 7 ) 2o 0 Exact Evaluation i n Terms of Radial Integrals A E l e c t r i c Dipole Transition, P State to S State 0 In evaluating the e l e c t r i c dipole moment Q, the i n i t i a l state i s the P-wave part of (lo3 ll) 0 for The quartet state (Io3ol0). w i l l not contribute to the reaction. (A3.2.13) Including the factor the r e l a t i v e amplitude of the doublet state, the initial state i s (2.2.1) ^ V - V' >^) K o ^ { ^ s z ^ \ , )€ The f i n a l state i s the L = 0 part of (2,2 2) 0 V W > > [ ^ ^e The factor L ; 5 * ^ " ordinates. using ^ 6 ) ^ J b (l ]5ol2), 0 M J + ^ . Z ^ wV) • are given i n ^4 (1.1.3) The result i s (App. 6 ) i n terms of the i n t e r n a l co- The integration over the Euler angles i s now .(2.1.3). + i s expressed i n terms of the i n t e r n a l co- ordinates and the Euler angle functions W. where X^ and + The only non-zero terms occur i n Q, 1 performed and come from 9 U the f i r s t term in ( 2 . 2 . 3 ) o Performing these operations and introducing the notation Q, (P-S, S ) for the (l,m) e l e c t r i c dipole moment between X, m z the P and S states for an i n i t i a l spin value of S gives z <2 .2 .'O Q^(?-S,S,) = i e p * £ J V / > { 1. « (, -T s )(Wl(-)Z L -W;^X ) W > [ ] ch fc from which where In 1^, the summation over i , the s p i n and i s o t o p i c s p i n c o - o r d i n a t e s may be performed using the matrix elements l i s t e c 19) i n (A3.2.lb and to give (2.2*7) 2 0) 5 -i<%e^ z o|s ">^^|i2,|G.^-jk<q--iZ J K«> W ^(.2,.Z )| 1 S 5 j ( o)-i(F |.Z |q o t ! ! / Thus the r e d u c t i o n o f the matrix element to a s e r i e s of r a d i a l i n t e g r a l s i s complete. Nothing f u r t h e r can be done to the e x p r e s s i o n 1^, u n t i l e i t h e r exact or approximate forms are known f o r the r a d i a l functions. The p a r t i a l c r o s s - s e c t i o n a r i s i n g from „(P-S, £ ) i s 1,0 given by (A5.I.J0) and (A5.2.19), z o- (2 2 6) C 45 - 0 For an unpolarized proton beam and target, this should be averaged over i n i t i a l spin values giving cr (?-s) = i Z<p (T>-S,s,) (2 2 ) 0 li0 o 9 it |p ^JT . elk! (I,) 1 The total cross-section i s obtained a (2 2„10) 0 (-P-S) = Z by summing over m„ <r (v-S) = tr, (P-S) 1>nv 0 The angular d i s t r i b u t i o n of the radiation from this t r a n s i t i o n i s 2 given by -1,0 (2.2.11) X i,o (A5.1.21) and i s This i s the only one of the transitions studied which has an angular d i s t r i b u t i o n proportional to sin^Q. Bo E l e c t r i c Dipole Transition;, P State to D State The method of evaluating the e l e c t r i c dipole moment i s exactly the same as i n the preceding section, except that only the quartet continuum state contributes. Including the factor jl for the r e l a t i v e amplitude of the quartet spin state, the i n i t i a l state i s , from (1,3=11), (2.2.12) t -_ ,Z/*F W; C-)[<^(•)£,,, - < V ^ , . « } b o b and the f i n a l s t a t e i s the L = ( 2 . 2 . 1 3 ) Ya = { ^ The e l e c t r i c *Wi 46 2 - part of ( 1 , 3 . 1 2 ) ^(^^^)^(^r^) d i p o l e moment becomes * x dT where the Clebsch-Gordon c o - e f f i c i e n t ( l o 2 . 3 3 ) and ( 2 . 2 . 1 3 ) and ( 2 o 2 0 l 6 ) The p a r t i a l c r o s s - s e c t i o n becomes comes from the i^f functions - (2.2.17) ' ^ . ^ • ^ • ^ • <t7 - ^ \ - ^ ) X ' In the averaging over S^, each cross-section ' - - 1 ^ ^ (" "^) p m contains, apart from the common factors, the factor 1 / VO -m my ^ z \ m s» -n / v ° - (2.2.18) 3oo 5oo In the summation over m to give the t o t a l cross-section this factor becomes 1/30, (2.2.19) The t o t a l cross-section becomes o- (T-D) = e*k . { I T 5 -I,"] Adding the angular d i s t r i b u t i o n Lm (A5O1J21) with the r e l a t i v e weights (3:4:jJ) for m = (-1:0:1) gives an angular d i s t r i b u t i o n proportional to (2.2.20) Co ^|-J_cos e) l E l e c t r i c Dipole Transition, F State to L State The calculations for this t r a n s i t i o n are exactly the same as those i n section B. Including the factor J| for the r e l a t i v e amplitude of the quartet state,'the i n i t i a l state i s , from (1.3.11), (2.2.21) \ » W^tt^fc)^, -<^W} ~ and the f i n a l state i s V 48 - i n (2 -?.13). 0 The e l e c t r i c dipole moment i s ( c f . (2.2.14, 15 and 16) ) '3 (2.2.22) Q (F-3?,S^ - - « ' F ( 2 . - . l , ^ ) ( o L|A I Z. > -in m , where (2 2.23) 2 (2.2.15) with £i'(0^> replaced by G I_ = I_ (2.2.16) with Q (0^ replaced by G (3)^ I 0 4 = I L L (^ and (2.2.24) 3 o L L The p a r t i a l cross-section i s In the averaging over S , each cross-section cr ( F - D ) contains, apart from the common factors, the factor 'I'^W (2.2.26) * V O -m my X s ^ i 350 - _3 OT £c>r 111 = m =. O 35-o In the summation over m to give the t o t a l cross-section this factor becomes 1/70. (2.2.27) The t o t a l cross-section i s cr^F-DJ = 1*. ^ j s ! '/3 1+ + Iyj*" Adding the angular d i s t r i b u t i o n (A5.1.21) with the r e l a t i v e weights (1:3:1) for m = (-1:0:1) gives an angular d i s t r i b u t i o n proportional to 49 (202o28) / j - -C^S^B) Do Electric Quadrupole Transition, D State to S State Only the doublet part of the i n i t i a l state w i l l contribute so the i n i t i a l state i s , including the factor J_ for the r e l a t i v e VT amplitude of the doublet state, (202o29) and x . - j w W; C )^I(G O + 5 ( Z WI; ; i - V ^ ) + G ( 1 e ^ } b i n (2 0 2 0 2). the f i n a l state i s ^ 3. The factor appearing i n ^l^m ^ r* Y„ (&^ > T 6 e x P r e S 6 e cl in terms of the i n t e r n a l co-orainates and the Euler angles (App 6). c (2.2.30) r^Y (e ^/^WWzz'-xM-^W Wx The evaluation of W Z^fJll S^) proceeds i n the same way ae the evaluation i n section 2A. The only non-zero term i s (2.2.31) Q ip (Vs. i i ) =, -eft r <V„> | W > ) | W > ) ) l z where r. . * *4J[lif'- »x";- ;)lL j (2o2032) T x dT t - and f ( 1 ) , f (2.2,33) ( 2 x £(T-J) \f ( (<-X = 5 ) 2 50 - are defined ae i n (A3»2 20) with 0 Z L ) The p a r t i a l cross-section i s (2.2 34) 0 r (3>-S,±^ = Z . f i l i L I* I p Averaging over spins gives (2.2,35) and V(p-s) -iZ ^(3)-S,S^^,(>^U) the t o t a l cross section i s (2,2.36) The angular d i s t r i b u t i o n i s proportional to (2.2.37: E. (co^ © -cos"©) 1 E l e c t r i c Quadrctpole Transition, S State to D State Only the quartet part of the S state w i l l contribute i n transitions to the (quartet) D state. Including the spin factor the i n i t i a l state i s (2.2.36) and \ - the f i n a l state i s Q, (S-D, S ) are 2m z 9 [ V^*., V -V B ) i n (2.2.13). V} b The integrals for - (2.2.39)^-1),%) = 51 - eZiv£J{(^ where (2.2.1.0) (2.2..D <2.2.42) with \ -^,\^C|^, CP^-<^^rf"|S».,C^> *^^|f l *«* IS„^ -^^JLJCIG-./"^ > > - A ^ i ^ - ^ ^ ^ ^ < M -^i-KV'*)- & K K » > ^<f*/'if'*K,<°> , \ (2.2.43) ^( T t) = and \ ae i n (A3.2.20) and z in. Ij IIV If, In. ~L°> n , n 52 - M The p a r t i a l cross-section i s (2.2.44) 0 - ^ ( s - T > ^ ) = ^ . e l k . (a.-m.l.Ssli.M^ilT-^^I^ r Averaging over S , (2o2 45) 0 _E ; A (z,-l^iM) 2 1 E 2 (.„ ivZ -M 10 In the summation over m to give the t o t a l cross-section, this factor becomes # The t o t a l cross-section i 6 0 (2.2,46) oIS v j J 1 Adding the angular d i s t r i b u t i o n s X 2m with the r e l a t i v e weights ( 1 : 1 : 1 : 1 : 1 ) forra= ( - 2 : - l : 0 : 1 : 2 ) gives an isotopic d i s t r i b u t i o n . E 0 Magnetic Dipole Transition; S State to S State The magnetic dipole moment (A5.2.25) (2 2 4?) ri' 0 0 = -JL-EWY { ? M v k ^.O-VXAT can be s i m p l i f i e d by a p a r t i a l integration to give The gradient (2.2.-9) term i n (2 2 48) can be s i m p l i f i e d , o -jS^ 0 where 'X are the vectors —m a, (2.2.50) OC —° = .. = e "1 Expressed i n terras of (2.2.50) the spin i s = -Jz X , <7v_ JI %_ (2.2.5D + + The integral (2.2.48) then becomes (2.2.52) K nn. - i f I, The s p a t i a l integral contains only the r a d i a l parts of HC, Y and , with no factors (e.g. X , Z. , co-ordinates. -By "^ and e The function ) depending on the r a d i a l (2.2.2) i s a doublet S state of energy contains both doublet and quartet S states of positive energy describing the same system. Because of the orthogonality of wave functions d i f f e r i n g only i n energy, the doublet part of not contribute to the i n t e g r a l . The i n i t i a l state i s therefore taken to be the quartet S state as i n (2.2.2). (2.2.53) These states are V - W > v ^ W; «(v ?,,i - v > w } q will o) The magnetic dipole moments become b 54 - (2.2.54) The summations over i , spin and isotopic spin are obtained using the matrix elements (A3 „ 2.23-27 ) . The results are (2.2.55) MLfc-V^V i±.e. (u -u NT K ..(S-S.tO = + ' . £.(V - NI u where (2.2.56) l,o = <Fz,, The p a r t i a l cross-sections, averaged over i n i t i a l spin states become v (2.2o57) 4- ^ l m 1 1 The t o t a l cross-section i s (2.2.58) ^ ( - S ) = lit . e l _ k f 3 S U. -u YI" A l l m values enter with equal weight so the emitted radiation i s completely i s o t r o p i c . 3„ 55 - Approximation for the Radial Integrals In order to obtain an estimate of the r e l a t i v e magnitudes of the six cross-sections of the preceding section, the r a d i a l integrals 1^ ... 1^ must be evaluated. The exact evaluation of these integrals would require the r a d i a l functions F^ ». To obtain 0 the r a d i a l functions, i t would be necessary to assume forms for the nuclear forces and solve the coupled d i f f e r e n t i a l equations. was beyond the scope of this present work. It was This therefore decided to use very simple approximations for the r a d i a l functions which would provide a rough estimate for the magnitude and energy dependence of the cross-sections, to show which reactions would be most important i n the low energy range. More elaborate calculations could then be made for the cross-sections of those reactions which appear to be most important. The main approximation i s to use the asymptotic form (1.2.35) of the r a d i a l functions at a l l distances, and to use functions valid outside the range of the nuclear forces and also neglecting the coulomb forces. (2'.3a) / Q In the notation of (1.2.55) the function u(V) v(cp For the continuum states (2.3.2) v(cy) = J l (feo^ and for the bound states (2.3.3) v(q) = V ^ ) -Jzf z $ is - The deuteron function (2,5o4) a C r ) = a ^ r ) 56 = u(r) i s , for the continuum S states, ^ 3 For the bound S states, (2.3 = 5) A () - e l - U^T) R f t r and for the bound D states, -B'f (2 3o6) 0 U( >) = ^(V) ^ R ° The function u^(r) i s a deuteron D state function (Blatt and Weisskopf, 1952, p° 103) for which the integration cannot be extended into r = 0 but has the normalization Integrals over r i n which u^(r) appears are evaluated over the range R 4 r 4 00 The f i n a l results are not sensitive to the value of R. 0 The normalizations of (2„3o8) J u ( r ) and v (q) are 0 e j a ( r ) T s » n 6 d r d e =l and l s The constants ^ , j3* , ^ and k J v $ ty)cfd^=\ are obtained from the binding 3 energies of deuteron and He state (App. 4 ) . and the kinetic energy of the continuum To determine the range parameter p, , an equal d i v i s i o n of the binding energy of He^ between the three nucleons i s assumed. This assumption gives a deuteron function which i s more t i g h t l y bound i n the bound state of He^ than i n the continuum state. - - 57 In using the approximation (1.2.35) for the bound state functions, there arises a question regarding the correct normalization. For example (2.3.10) f ^ ^ u„(r) v (a) has the normalization (2.3.1D d ^ i This factor of ^ i n (2.3.10) appears i n a l l the r a d i a l terms. It was decided that the approximate functions, whatever their form, must have unit normalization. Accordingly i n a l l the approximate approximations of the bound state r a d i a l functions the factor 1//3 i s omitted. This, however, does not a l f e c t the r e l a t i v e sizes of the estimated cross-sections. rr rr (2.3.13) - j ^ it p , p e e p * = a Ug(r)v (cv) , ^ (T Thus the functions become t (z G u CD , Q^(L) =o s F - 58 - In the integrations, there i s one further approximation 2 2 made ' The factors Z^, X^, (2Z^ - X^) ... appearing i n the integrals 0 <f ( 1 . 1 . 6 ) . contain terms depending on the angle (jl!> on r, q and 9 i s rather complex. The dependence of However i n the range of v a l i d i t y of the approximate r a d i a l functions, i . e . q ^ size), $ i s slowly varying and approximately zero. r (deuteron For example for q = 1.5 r, J^| -9«6° for 0 ^ 9 ^ T f . It was therefore decided to z place / = 0 i n these approximate calculations. The effect of this approximation on the calculation was f e l t to be considerably less than the effect of using the asymtotic r a d i a l functions (1.2.35). 2 2 The factors Z^, X^, (2Z^ - X^) ... appearing i n 1^ ... I are expressed i n terms of the internal co-ordinates using (1.1.3). The results and approximate forms used are given below. In I , I.,, I-., 1^ and I 1 (2.3,13) Z,= .Q.COSCZS^2Q. ^ ^ Z 3 (- Z , + Z , J ) X 5 = -T-CDS (^ + e) z& - r c o s e = | 0^, sin. (-X, •+ X ) = 2 as -T O sin. + e) ^ -T S\CL Q In 1^ and I (2*3,14) ' ^ 9 = 6 y c l ("2. cos a <ws + 6) - sin. 0 sirv 9 ' ve)) ^ £> ^ %Tgy cos 6 7 In I 59 8 f '= ia s»a <j> cos$r> >r r^sia (^ + e) cos(jz5 + e) =&JL sin. 6 cos© z 9 ' 5 "3 6 6 (2.3*15) 9 ' 6/5 In I, 3 9 ' 5 (2.3.16) 9 6 R ^ = _i_ TCJ. snx ^ sin. S (cf> t e ) s~ o The e v a l u a t i o n o f t h e a p p r o x i m a t e i n t e g r a l s straightforward. The i n t e g r a l s \e Morse a n d F e s h b a c h (1953, p. 1575). j^(ka^c^dcj, Taking only i s now a r e taken the non-zero terms, the r e s u l t s a r e : = if|(^+a^^gK - - ^ r C ^ T ' 5 5 (2.3.18) 2 2 5 = l / l a.^JU^CT-)^^)^) ^ W J , (fecprysirted-rdcyd© 3V3 3/3" ( .3.i9) 2 i s ^ < *..ferC-x.-xOK.«>-<F*\^c- »^..^ F from x "T3 C.3.21) * 1 1 3 1,1 - 60 - C V ^ ) * *° * -j=< .*,.|^(- « ^ O I ^ * > < ^ | ^ ( - ^ . x ^ , o > > F x c + -s<r*i-r|s,,«>K »l "l^ > F ^ (2.3.23) 3 L 5 ; ^?) c^P:)' Ij^^<F jr-r|^.^> 15l (2.3.24) Ig* (2.3.25) ^ /I< g r - q ^ = - L , a. is ^ \ CT) V (cy) CV S i a e ) U (T) j 2 s jj;* S ^*) & (kta. ) r ^ s t a 9 dvda^de - r u) - 61 - = J± (a + a^j a (r) v z where u (T) J (fy) T Ysin. edvd^de 5 s q p«o "RcO - \ o^C^u (2.3.2?) * to = H(?) = C^Tr-tlr f t o 3', \ U.^CT) uO-)rdT5 S M u^Cr) U CT) T*dr are f a c t o r s depending on t h e l o w e r l i m i t o f the i n t e g r a t i o n s r. Some n u m e r i c a l v a l u e s o f t h e s e a r e g i v e n i n T a b l e I I I . TABLE I I I Numerical x Values of the I n t e g r a l s R ( i ) R(l) R(2) R(3) 0.4 0,30 0 = 22 0.20 0.6 0.26 0.2b 0.53 o.b 0.26 0.32 0.44 1.0 0.23 0.3-+ 0.54 1.2 0.21 0.32 0.63 -ft' Pv F(a,b,c,z) i s the hypergeometrie (2.5.2b) F(a.,b,c,y) = ' 1 1 c ' function bCb + Oy-j| cCc + 0 a(a+0 + over These are now substituted i n the expressions for the t o t a l crosssections. A l l energy dependent functions have been written i n terms of the nucleon momentumfe-. (2.3.29) <K?-s) = . SL .ftft'A. K ^ / . f a , + a u T < 2 . 3 . 3 0 > 0 , ( M > ) - M . £ ^ (2.^31>(N(F-1>)>^ ( 2 . 3 . 3 2 ) ^ s)- (2.3.3, e M * ( M ^ ( k V p ; ) r t » ti )^)=f-& ^ (P-P" N ) ^ ' .y To obtain numerical values of these cross-sections, they are now calculated assuming a laboratory proton energy of 1 Mev. The values used are (App. 4) E c " 2 / 5 E L = 0.6b7 Mev = 0.356 x 1 0 Pi I = 2.32 x 10 cm = 3.53 x 10 cm = 2.79 x 1 Q ft - k.io * 1 2 n 10 •1 •1 •1 cm" cm fe. = 1 1 1.46 x 1 0 1 2 cm" 1 e* = 1/137 M = 4.76 x 1 0 1 5 fip= 2.79 n.m. U N = -1.91 n.m. cm" 1 - - 63 The results are tabulated belowo TABLE IV Numerical Values of the Cross-sections Multipolarity Cross-Section at , -30. i n 10 of Radiation o (v-s) = <s.<r cm 2 Distribution |a +a y E l e c t r i c dipole cn. E l e c t r i c dipole ff. (?-T>) = 2 £ [(25)a - (c7)(V E l e c t r i c dipole Angular, = 1 Mev. 3 £ i 1% a-*)}* 1 -ico^e 7 (F-D) = o.a ^^^(.otXa^-jza^y Electric quadrupole o»p>-s) = 0.0/2 (a + a y Electric quadrupole «^f>T>>o./o jCaJa^Ho^a^^+coOa,^* e 5 + i -icos^e cos e - cos^e Magnetic dipole isotropic isotropic For transitions going to the "D state of He , the value was a r b i t r a r i l y assumed. l fi\ R = 1 Examination of Table III shows that the cross-sections concerned are r e l a t i v e l y i n s e n s i t i v e to the choice of p| Ro The results of (2.3.29 - 34) and Table IV are discussed and compared with the experimental results i n Chapter 3. - 64 - CHAPTER 3 DISCUSSION AND COMPARISON WITH EXPERIMENT In the cross-sections for the reactions studied i n Chapter 2, the angular distributions obtained are independent approximations made for the r a d i a l functions. of any The magnitudes of the cross-sections given i n (2 . 2 .29'=34) are greatly dependent on the form of the r a d i a l functions and of the amplitudes, a., of the states and i t i s seen that considerable cancellation may occur depending on the r e l a t i v e signs of the amplitudes. From the argu- ments presented i n Chapter 1, section 3D, i t i s expected that the S state of He^ having a symmetrical r a d i a l function, , would be the most important state, i . e . a^ *"*•>• 1 and for a l l other states a^ <(<( 1. Thus i n the cross-section (^ ^) and CFjL (D"S) i t >_ might be expected that the amplitude factor • I <X I a l l other amplitude factors ) |a, + 5 (Table IV). - ^ I & n & These arguments, while only indicating expected orders of magnitude for these factors, are useful i n comparing the numerical results of Table IV with the experimental data. The most recent data with which the results of Chapter 2 may be compared are those of G r i f f i t h s , Larson and Robertson This data i s tabulated i n Table V. (1961). In obtaining the cross-sections 2 for the sin 9 component (CT ) and the 'isotropic ' component ( Cr? p from the measured t o t a l cross-section (<J~ = <J~ + cr ? ),'. i t was assumed that the smaller component was completely i s o t r o p i c . * ) - 65 •- TABLE V Summary of Experimental Data <JV Energy ( E ) Angular D i s t r i b u t i o n T fc <5V P > L 0.3 va Mev sin 9 2 + ( .079 +. .010) 0 6 Mev s i n 9 + (.032 +_ .004) 1.0 Mev s i n 9 + (.024 + .003) <7u / ? -30 , • . 0.803 10 7m ' Et cm. 0.095 -0.12 2 0 2 3.13 0.11 0.035 Comparing the angular d i s t r i b u t i o n s i t i s seen that the main component, proportional to s i n 9 can only arise from an e l e c t r i c 2 dipole t r a n s i t i o n from a P-wave proton to the S state of He"^. The calculated size of <S\^ (?-S) 1 at E^ = 1 Mev i s , assuming ja.^+3.^=1 remarkably close to the experimental value considering the roughness of the approximations made. value of at E. = 0o3 Mev i s ov (P-S) = ov (?-$) t For further comparisons the calculated 5.7 * »° °l 5 + a-Tcm. 3 a 2 It must be remembered that the effect of the coulomb repulsion has been ignored i n these approximations. The e f f e c t of this repulsion increases as the energy decreases and would, i f included, lower the calculated value of the cross-sections. coulomb penetraction x c (3.1.1) - For example, the simple factor (Blatt and Weisskopf, 1952, p. 87) Try c v -» 2 has the values C Mev. • 2 = 0.26 at E, = 0.5 Mev and C = 0.59 at E = 1.0 - 66 In attempting to identify the o r i g i n of the smaller component of the reaction, the contributions from the e l e c t r i c dipole transition from an F-wave proton to the D state of Ee^, and from both of the e l e c t r i c quadrupole transitions may be neglected at this energy. The cross-sections c- ( -D)= E( (3.1.2) f o.,z = o x ^[(z^a^CoGXa^-laa.,^ 0093 x i ' 0 1 •> ( S - D " ) = E (3 1.3) 0/0 x .D' x lO 50 3 0 z cm ja.,51 (.wXa-w. -/la-^cnr, 2 j(.i7)a, + (c*Xa^-|2a)+ (oOa-lcm 1 i6 3 * 1 = o 003 are both far too small to explain the observed value o f (T = 0 . 1 1 x ? 10 ^ cm . The cross-section 2 (3oi.4) o- ^(p-s) = 0 0 / 2 . E x 1 0 " J^a. + S L ^ 50 Z 3 i s small, but more important, the angular d i s t r i b u t i o n , proportional 2 4 + to cos 9 - cos 6 = s i n 2 2 9 cos 9 i s zero at 9 = 0 and would not appear as an 'isotropic* contribution. It i s possible that this reaction could contribute to the t o t a l cross-section but the contribution would be small and d i f f i c u l t to detect as = 1 Mev. ^aA / - Q Qie B at The accuracy quoted by G r i f f i t h s , et a l . i s , at this energy, Cr t o t a l = (3*24 + 0.33) x l o " - 5 0 cm , or + 11%. 2 67 Either of the remaining cross-sections (3.1.5) % ^ " 3. * " ^ W*-*-p*J} ' cm 2 or (S-s)= (3.1.6) s-o x IO" 3 0 ja^a.^2^1 could be the correct size to explain the smaller component, for a suitable choice of the amplitudes a^. It i s thus necessary to look at the energy dependence of these two cross-sections, or rather the energy dependence of the ratios ^ ^ and °h. , These r a t i o s are (3.1.7) CP'S) ( y ^ 5 1 and (3.1.8) The ratio ( 3 . 1 . 7 ) i s independent of energy, as would be expected, both cross-sections a r i s i n g from e l e c t r i c dipole transitions for P-wave protons. The ratio ( 3 . 1 . 8 ) decreases with increasing energy, the valueB for E, = 0 . 3 Mev and 1 Mev being 1 (3.1.9) or *<?-s) (s+M E i = 0 75 Kl^T -for E, = 0 3 Mev. "L EL = 1 0 Mev - 68 - Between these two energies this r a t i o decreases by a factor of 2.4. Experimentally, the ratio f<y » decreases by a factor of 3.4. If the smaller component of the experimental cross-section i s i d e n t i f i e d with |a- z + 3--5^ ~ ('°5) 0" (S-S) , i t would be necessary to assume that thus supporting the qualitative argu- (^3 ments concerning the amplitudes a^. It thus i s more probable that the small 'isotropic • component of the reaction comes mainly from the magnetic dipole trans i t i o n of S-wave protons to the S state of He^. The observed cross- section i s small because the reaction does not involve the r a d i a l l y symmetric state,yCt/^. F^. , even before any approximations are made. However, the e l e c t r i c dipole transition of P-wave protons to the D state of He^ could also contribute to observed cross-section. Only these two transitions appear l i k e l y to contribute -to the smaller component of the cross-section. The numerical values of the cross-sections as given i n Table IV depend on the particular choice of (3, and ^ in ( 2 . 3 . 5 ) . The o r i g i n a l numerical estimates were made using {3* = (5, = 2 ZZ * x io cm \o iZ cm ' (3.1.10) (3^ = t£o ,Z The cross-sections obtained differea from those i n Table IV, the ' differences coming mainly from the variation of (J 2 . The differences were not s u f f i c i e n t to a l t e r the conclusions of this chapter. s - 69 - To extend these calculations i n order to obtain more valuable estimates for the three cross-sections of interest O-S) , ^(?-!>) use more accurate and 0^ (.S-S) i t would be possible to r a d i a l functions which include the effects of the coulomb repulsion and also, for the continuum states, the scattering phase s h i f t s . This would not remove the problem of the amplitudes of the bound states and u n t i l numerical estimates for the a^ can be made the absolute magnitude of any calculated cross-section w i l l be uncertain. The programme for future work should thus include some method for deriving numerical estimates for the amplitudes of the bound state. - 70 - APPENDIX 1 THE CO-ORDINATE TRANSFORMATION The detailed expression for the transformations *Ru(Pj) defined i n (1.1.21) i s OC V i z 1 = s \ % 3. 3 + T! X JLl ^ Z , + 1 - 3', - i x t 'R X l l + I ^„Z, t 3z ( A l o l c l ) % - \ 1 = S !)+ \ l X The definitions of £, R respectively. v 3 ^ 3 5 ^ 3 and X i and are given i n (1,1.1, 2 and 3) From tnis, the elements of the Jacobian are obtained by inspection. determinant The resulting 9 x 9 determinant i s then evaluated by standard methods to give (Al.1.2) J= det = Sin op, '-•nn e The columns of the matrix are then multiplied together, term by term, to give the symmetric metric tensor, (Al.1.3) 6M ilk ¥L\ - 71 - 7^, (Al.1.3) cr CoslX N73 I where ( A l o l . 4 ) §6& 3 C o s 1 ' a c s ^ X s m ' h 1- a^cos^r} - 72 This form of metric i n t e r n a l co-ordinates tensor i s q u i t e g e n e r a l i n t h a t the ( r , q, 9 ) may three c o - o r d i n a t e s i n which the be expressed, metric ij tensor (1.1.28). be r e p l a c e d by any i s now On obtained g other s e t of 'body' c o - o r d i n a t e s X^ and s u b j e c t to the c o n d i t i o n that X^ = 0. three matrices i n which in - The may conjugate from the r e c i p r o c a l s of i s expressed. The the r e s u l t i s quoted m u l t i p l i c a t i o n of the three matrices in (1.1.2d) the L a p l a c i a n (Ai.1.5) vZ--± may be e v a l u a t e d . of the matrix in gv>_L\ The - g -^_ y J Lill A + Aj£ 1. e v a l u a t i o n i s abided by n o t i n g that the rows S (1.1.21?) are the operators + L^, the c o - e f f i c i e n t s of the d e r i v a t i v e s L^ ( 1 . 1 . 1 7 ) . and i n t r o d u c i n g these o p e r a t o r s . The T h i s i s the reason for terms are examined i n groups of three as f o l l o w s . i. i = 1, 2, 3 These are j = 1, the only c o - o r a i n a t e s^. (Al.1.6) ii. ±_ _ ^ 3 ^ i = 4, s 3 terms i n v o l v i n g the centre-of-mass The _i_ _ ^ 3 i * + 2, result i s 3 ^ 5,6 = ± v 3 Z j = •+, 5, § 6 These terms, i n c l u d i n g d e r i v a t i v e s of the E u l e r angles o n l y , are s i m i l a r asymmetric (Ai.1.7) top. rf (±±f 5 to the conjugate The result i s + M 6 6 ^ metric tensor f o r an - 7 3 i i i o i = 7, b , 9 J = 7, 0, 9 The metric tensor i s d i a g o n a l i n these terms and g i v e s airec tly iv. i = 4, b , 6 j = 7 , o, ^ In the expansion of V * i n ( A l . 1 . 5 ) the f i r s t —^ (AI.I.9) , gives O t f ^ , _A_ 56 term, airectly M "!i_\ s M + N e g l e c t i n g those terms f o r which the d e r i v a t i v e of $3^ i s zero, the l a s t two terms i n v o l v e (Al 1 . 1 0 ) ( | f l 4 . + ff*A_ ^ j i A f - ! — A ( - n i c t , ^ ) _ L _ ^ . ( c o f l s ^ a ) l o \ b>dcy J[s»ifi 2>a i I bp ^ < J S l v v. i = 7, o, 9 As i n 7 + n i_ 56 + S + 1 S = M !A_\ 5 j while the l a s t two terms become ' f term g i v e s 2>r ; A 1 . 1 . 1 2 ) /Ji£) 0 3 = 4 , 5 , 6 i v . the f i r s t Uiaai)(WW iL f {A / V a W n') 4- A _ (VVs.aO M * U A / V ^ W 7 !iS5)Uu\ thus d e f i n i n g the term M . C o l l e c t i n g together these v a r i o u s terms gives the Lapl a c i a n as expressed i n ( 1 . 1 . 3 1 ) . M - 7t - APPENDIX 2 DEGENERACY OF THE PRINCIPAL AXES The degenerate two s i t u a t i o n s are the c o - l i n e a r f o r which the p r i n c i p a l axes are p o s i t i o n and the e q u i l a t e r a l (i960). This has been d i s c u s s e d by D e r r i c k By r e q u i r i n g f u n c t i o n s to be continuous f o r these t*o p o s i t i o n s , the k i n e t i c energy occurs when 0 i n t e g r a l s a r e avoided. = 0°, l80° position. the r a d i a l divergences i n The c o - l i n e a r position or a l t e r n a t i v e l y when A = J^rqsinQ = 0 0 e q u i l a t e r a l p o s i t i o n occurs when both 9 = 90° and q = Y> JSr or The 2 a l t e r n a t i v e l y when both F = = (2q 2 - 3/2r ) = 0 and G = 2rqcos9 = 0. 0 I t i s convenient to use A, F and G as the i n t e r n a l c o - o r d i n a t e s instead of r , q and 9 and then r e q u i r e that have formal r e l a t i o n s as A — > 0 and as F metric tensor g *' may be w r i t t e n the r a d i a l f u n c t i o n s »-0, *-0„ G The conjugate as i n ( I d . 2 9 ) with the matrix 1 M * given by 1 1 (A2 lol) 1 3 0 3 3 31? L A & _A 3* n/3 A AG 1 -W5 A HK3 1 - 75 - where A = 3(FSQ*) 1 The H8n* = ( A 2 o l . 2 ) +3(F^G ) 1 Laplacian i s 3 § t b where The q u a n t i t i e s F and G form a p a i r o f f u n c t i o n s o f mixed symmetry, A i s symmetric on permutation o f the c o - o r a i n a t e s . the r u l e s of ( I 0 2 . l 4 - l 6 ) other symmetric, f u n c t i o n s may be b u i l t Using mixed and antisymmetric up from A, F and G« Thus using the p a i r (F, G) f o r both p a i r s of f u n c t i o n s i n ( 1 . 2 . 1 5 ( 5 ) ) , another mixed pair i s (A2.1o> (- F 2.FG,) \ G \ Combining ( A 2 . 1 . 5 ) symmetric (A2ol.6) with (F, G) a c c o r d i n g to (1.2.16(3)) an a n t i - function i s 5 F ^ G _ G , 76 In the same manner, two symmetric (A2.1.7) (F%G, ) By c o n t i n u i n g f u n c t i o n s are and 1 t h i s process, f u n c t i o n s may be c o n s t r u c t e d other - (F3-3F<3 antisymmetric, mixed and symmetric but i t can be shown that a l l may be expressed i n terms of the f u n c t i o n s given Thus when the c o - o r d i n a t e s generalized symmetrized Mixed 0 A, F and G are used the most f u n c t i o n s must be o f the form: Symmetric (A2.1 7b) i n (A2„1.5» 6 and 7). u.) Antisymmetric where the f u n c t i o n s g. are s y m m e t r i c - 77 APPENDIX 3 MATRIX ELEMENTS lo Matrix Elements of Kinetic Energy Operator The evaluation of the kinetic energy matrix elements (CL^^ with respect to requires the derivatives of the functions the operators L^, L^ and L^e These are (A3 The derivatives of the functions evaluated W, (t) can now be anci are l i s t e d i n Table A l . TABLE AI Derivatives of the Functions o OO for L = 0, 1, 2 0 o 0 Win -w,yt-) Ou. 1/3 W^t-) o wT c-) u Wj. (-) / u. / WJ. (-) -W,>> -3W/t) +vf W ^w x -fW^o -H>> ysW^w-W^ - 7S The matrix elements between the W states of the kinetic energy operator, (±) \ * \ | ^ l (+)^ are given below, taking the states i n the order given i n Table A l . (A3olo2) A, 0 0 A, -3, o B, o /IB, A, 0 o where A J = T S - " M S B, = xM15A \ = >/5 ^ 5 - K + 2 M 8 A vzn 9 4 "^. hQf + ^ M ~ • o o o o 0 (A3.1.3) o + K\* 0 o A o -B, 0 A 7 7 - 79 - These matrix elements do not contain the factor 2 a ( li_ | Matrix Elements of Spin and Isotopic Spin Operators At, Introduc tion In the calculation of the radiation matrix elements and in the separation of the Schrodinger equation, the matrix elements between the spin-isotopic spin states of the following operators are needed 0 cr Spin operators i3 , cr = ^ c r n x r ^ + Proton operators L^l-Tj^ Neutron operator -O ^) . and cr_ = ^(^ - c<^) 41 i/l-T W u Magnetic moment operator Singlet operator and ' Cl + T. Wu i. (\ - Q~ <r-\ HA T r i p l e t operator " L ~J/ 1 f3-v <r cr") •I \ -L -J/ These matrix elements may be evaluated easily from f i r s t and the results are tabulateu i n matrix form below. (S^)| (Operator )j^ (S^ the matrices are rows and l a b e l l i n g the columns. functions t , -f^ and ^ )% order ^ , £, ut £ » The elements of l a b e l l i n g the For the three pairs of mixed , the matrix elements are taken i n the . For each type of operator there are three sets of matrix elements, e.g. there are three proton operators i(i-T ^ and 2 1 ^ v (S^)|i (I-T 3S )| principles 1 (| - T ) 33 ^ (S^)^ • •jj( l - T (5 > ) ' with their corresponding matrix elements These three matrices are connected by unitary transformations, U.(PL) the permutation, J) form the diagonal blocks and a l l the o f f - ) diagonal blocks are zero. , i n which the representations of - 8o Thus, for example (A3.2.2) U 0 - T „ ) | where and liW(iO-T,^ = = (12) i s the permutation \\0 I - ^L^))|i('"'W)|'£j(^)^ 1 s ^ • interchanging p a r t i c l e s 1 and 2, matrix of the matrix elements n e Thus one of the matrices may be calculated from f i r s t principles and the other two sets obtained by use of the transformation (A3.2.1). CT- and operators B. (T For the operators i^i-T^ , ^(uT ) l5 , only the matrices for i = 3 are given and for the + 2"-§j L only the matrix for i = 1 and j = 2 are given. The Spin Operators The matrix elements of the operators G70 and or + are required for the magnetic moment operators. (A3 .2.3) J 4 = (rpi('- 3) Y - i O T N + T ^ The properties of these operators on the single nucleon spin functions, °C and (A3.2 4) 0 , are fi> ^ O+ cr a o cr- <x=p = <x = o The matrices for ^i(!)K„|<j(i)> » * (Mk)KK(-'i)) (A3.2.5) (-4(1)1% 0 o o 1 o o CM-) are: - 81 0+) (3) (6) j_ •5 3 o (A3o2.6) (5) 0 » o o o 3 o o O o o 3 o o 1 (Mil % iM \ V3 3 O o 1 3 2 O 3 3 2. 3 o (3) (2) (I) (A3.2.7) 1 o o o o o 0« 2 "3 (*>) (5; -ft O o 0) o o (2) (i) v3 O ^3 (3) ~/3 (5) 1 (A3.2 8) 0 K 3 3 o o -» o O O 3 o o o 3 o O O o o o o o o o o 3 3 2. 3 3 '3 O 3 Z. 3 *3 - 82 - The matrices with blank upper halves are symmetric other values of S and for 0 0 The matrices for are obtained from the above by using the i d e n t i t i e s : 4 (- , 3 C e Proton and Neutron Operators The proton operator ifl + T- ^ l^i -T^ a nd the neutron operator occur i n the magnetic moment operator (A3.2.3), i n the coulomb potential operator: iO-T^iO-T^ (A3.2.10) y and also i n the radiation operator i n the integrals for the e l e c t r i c multipole radiation: (A3.2.11) where f ( r ^ ) i s a function of the s p a t i a l co-ordinates of the i - t h nucleon. The properties of these operators i n the single p a r t i c l e isotopic spin functions, V (A3.2.12) % K 1 and , are Z K 1 - 8 3 The matrices for <4(!)|i(^ )K(!)> and 55 w CO 2 3 (A3.2.13) < * ( ± ) | i ( . * T „ ^ 0) 3 O I CU C3> e (5? CG> 3 (A3.2o14) O o Z 5 o ill o o o o C3) o o o o O 1 " 3 o 3 o o I 3 j_ "3 (A3.2.15) CO 3 W2 0> z. C4) 3 o o _1_ 3 O I O O g o o O o o f5) 3 3. 3 «0 bk - m (s) (3) 1 (0 3 O 3 i O O i o ~3 0 3 O O O z 3 O O O O o O I 3 O " 3 O o t 3 O O 1 3 O O o O o O 3 O O o o (Z) ( A 3 o 2 o l 6 ) i J? 3 O I (3J i i 00 the matric es are the same as and S = 3/2 respectively c Because the f u n c t i o n s d e s c r i b e two z neutrons and one proton, there i s the i d e n t i t y : (A3.2.17) ( ^ ( s ^ i o -nOK-^oKjCs^) ^ ^ ^ o ^ O K j C s , " ) ) where ( 1 m n) i s any permutation o f ( 1 2 3 ) . The o p e r a t o r s ( A 3 . 2 . 1 1 ) , leons have the matrix when summed over a l l three nuc- elements: (A3.2.lft) p <^(i)l?K'- ») M^)>' T f -/C (I) - 85 (s> O) ft) (0 r o o (A3.2.19) o £10 pU) 2 r O O o o o o o nu) o o M r-Ul 1 X r o -r t -c o 3) O 0 o o o o o P«.> 1 (5) where f ^ i s symmetric and (f P. f ( 2 ) ) form a mixed pair. Magnetic Moment Operators The matrix elements of the magnetic moment operator (A3.2.3) can now be evaluated by multiplying the matrices for the spin operators by the matrices for the proton and neutron operators: (A3.2.21), using the notation - 86 - U3=2.22) Summed over a l l three nucleons, the matrix elements for are : (A3.2.23) (4) cu >K0 o o jiO) (4) - 87 - ll> 0 0 f3) (2> 0 M« O -Mis) o 3>W ( 0 o 3^0 o o o 5>W j y.CO o o o o o o to (5; u(.) 0 0 (SO O o 0 z 2^(,) to i o 0 o o 7N where (A3o2.27) jj.Cz) = - Matrix elements not l i s t e d may be obtained from the i u e n t i f i c a t i o n ; ^(-|)|S H T O ^(-D)»-^,(|)E, K ( 0 (A3c2.2d) <^-0|S^(-«>--<^(4)|S^|^> <^(-0lSrJ^-l)>'^t«|?N^> E. b8 - The Singlet and T r i p l e t Operators In the phenomenological description of the interaction of two nucleons, part of the potential i 6 taken to depend on the r e l a tive spin state of the two nucleons. There are two possible spin states, S = 0 (singlet) and S = 1 ( t r i p l e t ) and the spin operators which select these states are ±fi-cr.<0 4 respectively. That i s , i f \ -L Co) a n ^ j_ (3 + <r . err") -J ' and -1 - J ' if. \ 'X.LJCO a r e the singlet and t r i p l e t spin functions for the two nucleons, (A3.2.29) CO The matrix elements of the operator -('"^I-Sj") -( a n d 5 + < ^i\)) are. (A3 .2 .30) ^ (t a) i (, - oj o;)ta (i I)) t (1) CO » O <3> 0 • 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 (?) O O O _1 L 0 0 0 0 0 0 O O 0 0 4 0 ~k z 0 (5) - 69 \1> (0 (6) 13) 0 (A3o2.33) 0 o o o o o o o o o o o o (3) o o O I o O O O o o o o o O 0 I o o z o 7. o z O APPENDIX k RADIAL DIFFERENTIAL EQUATIONS The derivation of the d i f f e r e n t i a l equations for the r a a i a l functions can he done once the matrix elements of the operators i n the Hamiltonian have been evaluated matrix elements are given i n Appendix 3o 0 Examples of these In this appendix, the equations for the S states of the three nucleon systems are obtained* The Hamiltonian used i s (A*.l.l) H-^T E + T ) S(^ S + where V ( r ) i s the potential between two nucleons i n a singlet spin c state (S = 0), V p(r) i s the potential between two nucleons i n a r 2 t r i p l e t spin state (S = 1 ) and V^Cr) = e / r i s the coulomb potential between the two protons. The equations can be written more concisely by introducing ( A l f . 1 . 2 ) V, = i(V(r } *- V ( T ) - * V ( T ; J ) U, (J h where (see F i g . 3) r and there i s no antisymmetric combination V (3) assuming that the potential V(r) depends only on the magnitude of r . - 91 - In writing down the equations use can be made of the symmetry properties of the equations Thus, for example, i n the e equation for the antisymmetric function, F^, any coupling to the mixed functions F, ,, F, combination must enter through the antisymmetric (V^^ F, _ = V"l^ F, ) „ Then i t i s only necessary to n obtain the numerical constant which can generally be done by examining a single matrix elemento then the coupling of the states If V i s the t r i p l e t potential, F,V° (0"^ o a n c l j= (j^ t l 3 by this potential i s obtained from the matrix element - JSl ^'^j-^VTt (A3»2.33). From this i t can be seen that the coupling term must be In this way the following equations are obtained. f r ^ - E } F „ -4 s ( A * . 1.6) fCi; *v,: {T • v*.1}F„.a{v;c"F, -v-F,J S F } • 4 { ( v £ -V;-)F,, T - - V - ) F ^ . „ i{(*:»- v;;') „ *(v£ -v'SX,}.„ F (T^V^EJF^-VI^jt/Ij'V'F^.o • V - Ej F „ t ^ [V F, s ( V ^ E}F T J jv" F»], o + ft F < w where (A-.1.7) v -.iv,™ , i K c ^ ^ [V'F„ + VF„}.0 - 92 - Outside the range of the forces, a l l these equations go into the form {VE}F-O (A4 1 8) o 0 For q » r, may be replaced by 2 / r ^ and the equation i s then separable by writing F = u(r) v(q) w(9) 0 The separated equations are (A^a.9) fe £(^V I=i + HE'] aw =C (A4 lol0) 0 A_ ( a A ) + It ME"\v(c^=o i 0 r A. fs.n. e JL^ ( A ^ o l . l l ) (j where E' + E" = E -v n - A w(e) = o 0 For the continuum states, the requirement that u(r) be the asyratotic deuteron S state function leads to rn = 0, w(9) = constant and (A4 1 12) 0 E' b where = solution of ( A 4 l . b ) i s (A4.1.13) 2c226 = - c _fl~ 1 P i r T with (A^.l.lM iz (3>, D Mev i s the binding energy of the deuteron. 0 uW -B -- 2 3Z S IO cm"* The The t o t a l energy i s E = -B^ + the centre-of-mass frame. (A4„1.15) v ( -^ = J o ( ^ 93 - where Thus E" = E i s the kinetic energy i n c and a solution of (A^.1.10) i s ^ b with fe = i ME, C 1 (A4 ioi6) 3 0 Examination of the continuum states for higher angular momentum 1 a 1, 2, ... with the same approximations leads to the equation for v(q), (A^ol.17) for which the solutions are the spherical Bessel functions j^(kg)„ For the ground state of He'', the t o t a l energy i s (A4.1.18) E = =B 3 He where B„ 3 - 7.73 Mev i s the binding energy of He"^. The d i v i s i o n of Ha this energy between E * and E" i n (A^.1.9 and 10) i s somewhat a r b i t rary. I t was decided to set (A4.1.19) E' = -2/3 B 3 He E" . -1/3 B 3 He representing an equal d i v i s i o n of the binding energy between the three p a r t i c l e s . With this choice, the solution of (A^.1.9) i s ( A ^ l 2 0 ) U.'(T) = C' e" ' 1 0 o 1 r - with (A*t.l.21) 3 A ri i = , 3.53 |Z x _ | IO cm and the solution of (A4 l„10) i s o (A<+ lo22) v(^) = C*."!^ 0 with (Ai(olo23) '3 = 2.19 X lO I z lZ cm" 1 3h - - 95 - APPENDIX 5 INTERACTION 1 WITH THE ELECTROMAGNETIC FIELD Explanation of the I n t e r a c t i o n 0 Tne manner i n which the electromagnetic field i s handled must be d e s c r i b e d before d i s c u s s i n g the i n t e r a c t i o n of the e l e c t r o magnetic f i e l d with the nucleons,, •ft = c = 1 w i l l be u s e d 0 In the f o l l o w i n g the convention Thus energy, E, momentum v e c t o r s , k. and k, the nuclear mass, M and time have the dimensions of r e c i p r o c a l lengths. f . s , The c o n v e r s i o n f a c t o r s are E(cm-') = E (Mev) Only the r a d i a t i o n f i e l d w i l l of be considered, consisting t r a n s v e r s e waves, f o r which the coulomb (or s o l e n o i a a l ) guage may be used, i . e . (A5ol 2) 0 =o V A E -- H = V ' Av and (A5ol 3) 0 -J L A ^ where A i s the vector p o t e n t i a l and _E, _H a r e the e l e c t r i c and magnetic fields. Any s t a t i c i n t e r a c t i o n s a r i s i n g from the s c a l a r p o t e n t i a l ft are assumed to be i n c l u d e d i n the Hamiltonian In both the c l a s s i c a l and quantum mechanical it i s thought f o r the nuclear system. treatment o f the f i e l d , of as being enclosed i n a l a r g e but f i n i t e cavity - 96 - having perfectly r e f l e c t i n g surfaces. The boundary conditions then require that the tangential component of E i s zero, and this permits an enumeration of the allowed modes of the cavity. The electromag- netic f i e l d can then be expressed as a linear superposition of the allowed modes. There are many ways i n which the allowed modes may be described, e.g. plane waves, spherical waves, c y l i n d r i c a l waves, etc. The method used for any particular problem should be that which allows greatest s i m p l i f i c a t i o n and ease of calculation. The nucleon states with which the f i e l d interacts are eigenstates of angular momentum and have zero linear momentum i n the frame. centre-of-mass The electromagnetic f i e l d w i l l therefore be expanded i n terms of spherical waves contained i n a spherical cavity centred at the o r i g i n of the centre-of-mass frame. This can be done for either the c l a s s i c a l (Blatt and Weisskopf, 1932, App. B) or quantum mechanical (Heitler, 195*+i App potential may 0 1) description of the f i e l d . The vector then be written as + complex conjugate where A C L + 0 LI V/L(1 + I) ^ _!_ s»a (kr - kA kr t + * 97 - + and a, (k-), a (k-) are amplitudes which, i n the quantum theory, im im n become annihilation and creation operators for a photon of energy K, angular momentum 1 and l harmonics expressed z = m 0 The Y^ are normalized spherical m i n polar co-ordinates (9, <f> ) The vectors 2£^ are normalized vector spherical harmonics, ra (A5.1.9) r* Each of the terms A, (k-) are solutions of the source-free wave —im equation, ( A 5 o l . l 0 ) (y + W ) j \ ( k i ) =o z l u and the factors /M-ft' and jH-Tr have been included i n order that the terms A, (k-) have the normalization, lm ( A S . l . l i n U ^ C k t ) ! dr = M. In order to perform this integration, the asymtotic for kr y) 1 was used for a l l values of r approximation increases as R — ^ o o 0 form of j^(kr) The accuracy of this and a greater proportion of the f i e l d i s contained i n the region for which kr )) l c The boundary conditions at the surface of the sphere require that E^. ^ = 0, this i s the same as requiring that ^ . g = 0« A an the approximation for kr j>) 1, this leads to an Using - 98 - (A5« 1.12a) Sin.^'R - i l ) =0 for A (k,+), or lm ( A 5 d , 1 2 b ) Cos (k"R - ^ ) = ° for A, ( k , - ) lm Thus 0 ( A 5 o l . l 3 ) k "R - Ift n, ^ = nft where n i s an integer. - \£_ = (n+iV 2. and ' respectively It i s of interest to know the number of allowed modes, A n, i n the range k^ to k fi + A k. In the l i m i t as ^ . o o , both k and n may be considered to be continuous variables, R Thus the density of states, , of given 1 and m may be defined to be (A5ol.l +) P, , dk k.l.m / 1 = da .dk dk = l.Jk ^ Alternatively, the density, p^, i . t ) the number of modes of given 1, c ia i n the i n t e r v a l dE, may be defined (A5.1.15) p , d £ = dLn.- d E = IL • <* E The terms Ai ^»~) ^ m o r 1 = 1» 2, 3 ... are referred to as e l e c t r i c dipole, quadrupole, octopole, . . radiation as these are c the same as the asymtotic forms, at large distances from the source, of the radiation from o s c i l l a t i n g e l e c t r i c dipoles, quadrupoles, octopoles, „.. For example, i n the radiation from an e l e c t r i c dipole (Panofsky and P h i l l i p s , 1955i p. 222) the magnetic f i e l d i s proportional to (A5.1.16) H = C e^I kr s,rv6 ^(coskr + ^ Icr ^aWr) Q , k.r / ~ sm & F where e_, e g r ordinates. and 99 ^ are the unit vectors i n spherical polar co- Examining the magnetic f i e l d a r i s i n g from A, ( k , - ) n gives, assuming kr ^) 1, sin. ( k r kr -S) 5>n.(kr kr Stn.(k> e kr where X ' — - i ' » 'X.o + 1 (A5.1 18)2C , = - - l ( f i + i . v 2. 6 + x r = -_e_ n e un ^- t vectors, ) £ and are related to e_, _e ^ a r e and _e ^ by Q ^Sia8 e r -v c o s e e + te Q ^) /I (A5»1.19) X D = (cos9e r s\«vee ) - = _&- ^(sii-LSe Q +• c o s 6 e L Comparing at r e - i>e^) (A5d.l7) with the part of (A5ol.l6) regular the o r i g i n , i t i s seen that, apart from a constant phase factor, the two f i e l d s are the same 0 Similarly the terms A i ^ » ^ + m a r e r e f e r r ed to as magnetic multipole radiation as the f i e l d s a r i s i n g from these terms have the form of the f i e l d s a r i s i n g from magnetic multipoles at large distances from the o r i g i n . Both e l e c t r i c and magnetic multipole ation contributes to the" energy density terms proportional to radi- - 100 - 2 . in which the angular dependence i s contained i n (A5ol.20) 2 — Un. 21(1. Y, lm - H) U>0 Y Zl(Ui) Y, l,m + i w i l l be called the angular d i s t r i b u t i o n of the corresponding —lm multipole radiation. For 1=1 and 1 = 2 these angular distributions are, 3 (i-cos e) %1t 2 (A5ol.21) A , +. = _5_ fi +cos e) z 2,tl l£1> The d i s t i n c t i o n between e l e c t r i c and magnetic multipoles of the same order cannot be made by measuring the angular d i s t r i b u t i o n . polarization must be measured to make this d i s t i n c t i o n . The An observer, looking along the radius vector towards the o r i g i n , would measure the relative magnitude of e l e c t r i c vector along the e^ and e_ ^ direction. The expected polarization of a given multipole radiation may be obtained by expressing the f i e l d s i n terms of the components along 2. * .£Q A N D r 2. <f> instead of 2=.+ , X Q and 0(1 _ . In the expansion of the f i e l d s i n spherical waves, the direction of the polarization vector i s not constant throughout expansion. space as i t i s -for the plane wave At any given point, however, the polarization vectors of e l e c t r i c and magnetic multipole radiations right angles„ of the same order are at - In 101 - the quantum mechanical description of the electromagnetic f i e l d i n terms of photons, i t i s well-known that, as photons are bosons, the number of photons i n a given state may be zero or any given positive integer. In the spherical wave representation, the observables which may be simultaneously diagonalized are k, 1, m = 1 and the parity ( + K l a b e l the states. These are the quantities which w i l l be used to Had the plane wave representation been used the observables which could be simultaneously diagonalized would have been k^, k^, k^ and the polarization vector <. The electromagnetic f i e l d i s described by the number of photons n^(k,-) i n any given state, the corresponding state vector being The basic state i s the vacuum state numbers are zero 0 From | 0^ t | 0^> i n which a l l occupation any state may be constructed by repeated application of the creation operator ^j^^-)« in the state (k,l,m,=). a, (k-)„ lm a photon The corresponding annihilation operator i s These operators have the usual commutation rules and the usual properties when applied to a state vector. (A5clc22) (A 5 o L io23) ^ J c u t ) |..., . L m C ^ ) , .-.> • • , n CW*)-»,..-> u The operators for occupation number, angular momentum, m, and energy can be expressed i n terms of the creation and annihilation operators. For example j, the number operator i s (A5.1.24) N L m (k±) - 102 - a* (ki) a (kt) m U n - with eigenvalues <A5.1.25)<N (l<±)> - n. Ckt) = Lm o,\,Z,. Lm and the energy operator i s (A5clc26) °P Sir) ^ _ °P with t o t a l energy ( A 5 o l , 2 V ) /-H \ . 2 k( , 0^ n To introduce the interaction of the electromagnetic f i e l d with the nucleon system, the method due to Dirac ( 1 9 3 1 ) w i l l be used, The wave function - Ye^''^* 1 ^ t i c l e s i s separated into a function phase, and a phase factor of a system of charged parY of a e f i n i t e amplitude and e for which function of the co-ordinates of the i - t h p a r t i c l e . not have definite value. The |3 L need The d e r i v a t i v e s ( M i ^ , Aili. , _4£i. , ^>fi>- have a definite value although the condition not be s a t i s f i e d . ji^ i s a ° ft => ^ 3 This i s equivalent to the requirement ) need that only the difference i n phase between two neighbouring points be d e f i n i t e . Integrating the phase change for a single p a r t i c l e around a closed four dimensional curve given by the use of Stoke 'e theorem, - 103 - where the l i n e element d_s = (dx , dy, dz ,-dt) i s a ^ - v e c t o r , the s u r face element dS_ = (dxdy, dxdz ,-dxdt, uydz , dydt, dzdt) i s a 6-vector and .l} = fer&d. ^° any , <±£L_ b ( \ by V - ^ft ^ „ L I f the g r a d i e n t of it ' s c a l a r i s aaded to (grad ^ ) the i n t e g r a l w i l l be i n v a r i a n t . This i s a s i m i l a r t r a n s f o r m a t i o n to the gauge t r a n s f o r m a t i o n f o r the electromagnetic field with the p o t e n t i a l (A ,.A X .A^, 0 ). I f i t i s r e q u i r e d that the i n t e r a c t i o n of the electromagnetic be gauge i n v a r i a n t , t h i s w i l l be accomplished (3 i f the d e r i v a t i v e s of a t each p o i n t are i d e n t i f i e d witn the electromagnetic L field potential at that p o i n t . (A5 lc29)(Mc , , Mk 0 If operator and = f e l \ , e i , eA, , e $ * ) x s a t i s f i e s a wave equation i n v o l v i n g the momentum p = - L \7 will satisfy , - and the energy operator ft ). Thus i n an electromagnetic ot the p o t e n t i a l enters ^ through 4-e A) Thus, e.g., the Schrodinger (A5.1.3D i then the same equation with p_ r e p l a c e d by ( p. •+- V E r e p l a c e u by ( E - (A3.1.30) E = >-A and (.E-e^) equation becomes ) field, - 104 where e^ i s the charge on the i - t h p a r t i c l e and the potentials are evaluated at the position of the i - t h p a r t i c l e . In f i r s t order perturbation theory, only terms to the f i r s t order i n e are used. This i s generally v a l i d because of the smallness of the coupling parameter e^ = 1/137o I f the matrix elements, taken between stationary states of the system, of the perturbation be zero for any reason, i t would then be necessary to examine the matrix elements i n e^ the Stark effecto 0 This, for example, i s the case when examining In the present problem, the matrix elements of H' are not zero and i t i s not necessary, i n the f i r s t approximation, to examine higher order terms„ The transitions between stationary states caused by H' are accompanied by the emission or absorption of a single photon Q If the^ p a r t i c l e s have magnetic moments ju^ = ^ i — i ' there w i l l be an additional perturbation where the magnetic f i e l d i s evaluated at the i - t h nucleon Standard time-independent 0 perturbation theory i s used so the transition probability i s given by The f i n a l state of the nucleon system has only a single energy so that the density of energy levels p i s given by the density of - 105 - energy levels for the electromagnetic f i e l d (A5<>1.15)° The t o t a l cross-section i s obtained by dividing the transition probability by the flux of incident p a r t i c l e s . unit density at i n f i n i t y , The incident wave i s normalized to thus the flux i s equal to the r e l a t i v e velocity of the incident proton and the deuteron. This i s , non- re l a t i v i s t i c a l l y , ( A 5 c l c 3 5 ) v = 3_k, where 2/3 M i s the reduced mass of the system and k. i s the momentum given by (A5 lc36) = c E 0 and E^, i s the kinetic energy i n the centre-of-mass frame,, This i s related to the laboratory energy £^ of the proton by (A5.1 37) 0 E c = |E L The energy of the emitted photon i s determined by E in binding energies of He (A5dc3<3) k = E c + (3 H e 3 Defining the quantity (A5.1.39) £ the - ^ M K e - ^ 3 and the difference and deuteron (Bjj -3 and B^), e 3V) by -B ) D photon energy may be expressed i n terms of It and fi z (A5ol.*+0) k = Ju 4-h ( k. + (?) z ' x , 106 2o Derivation A of the Matrix Elements E l e c t r i c Multipole Matrix Elements 0 The matrix elements a r i s i n g from the perturbation H* (A3.1.32) and giving r i s e to e l e c t r i c multipole radiation can be obtained. of now The use of the coulomb gauge allows a s i m p l i f i c a t i o n H\ H' = -.£_ (A + p & p (A5.2.1) op - - 1 M Using H{ and "H" 7 r as the f i n a l and i n i t i a l p a r t i c l e wave functions, b a. j'^Ck-^ and ^ and jO^lk-^ as the f i n a l and i n i t i a l photon state functions, the matrix element becomes (A5o2.2) = - JL An integration by parts enables this to be written 2Mk*JV (A o2.3) 5 J l ' ; U Ha b a J where i s the probability current density and i s related to the probability density (A5o2. ) p 5 a b , ^ % - 107 - by CA5.2.6) V J a b = ck P a b Integrating by parts again, (A5.2.5) becomes The evaluation of the integral i s (A5.2.7) now follows the method of Blatt and Weisskopf (1952, p. 606). It i s assumed that the wavelength of the radiation i s long compared with the size of the source so that the approximation (A5.2o6) j (kr) = M l (zUi) ! 1 may be used. Using the d e f i n i t i o n of (A5.1.20) and the hermitisn property of _L, (A5.2.7) becomes (A5o2.9) H ^ = te f+iE __k__ 1 ^T Y L . (y» > ,>dr 1 1 a uuou J~[([7o J L m d M Evaluation of the integral gives (A5.2.10) -MUOJ^Y^^VT The matrix element i s then, r (A5.2.1D U. . HZ!! = CiUO" where J i r* v / — k L * * r Y, L V V dr - 108 These are referred to.as the e l e c t r i c 23-pole moments, When the system contains several nucleons, the integrals (A5.2.12) are replaced by where use i s made of the isotopic spin formalism and selects the charged p a r t i c l e s 0 The i n t e g r a l includes 1 - T^ 5 ) integration over the s p a t i a l co-ordinates and summation over the spin and isotopic spin co-ordinates. (A.>.2 o 1*+) r J U S = ilr p | Here the density field The t r a n s i t i o n probability (A5ol.3*+) then becomes HgJ* of energy levels i s the density for the radiation (A5.1.15) as the f i n a l state of tne nucleons i s the ground state of He^ which has only a single energy l e v e l . The interaction H" (A3.1.33) due to the magnetic moment can also give rise to e l e c t r i c multipole radiation. The matrix element i s where p. i s expressed i n terms of the nuclear magneton. This i n t e g r a l may be evaluated i n a manner similar to that used for the evaluation of (A3.2.2). The result i s : (AS.2.16) H , = -t e /H^W . /l+T W L Q, 1 109 where, a d a p t i n g the n o t a t i o n of B l a t t and Weisskopf, For the expression a system of protons and neutrons t h i s becomes, u s i n g (1,3.17) f o r the magnetic moment of the operator, I n c l u d i n g both o f these e l e c t r i c m u l t i p o l e moments, the t r a n s i t i o n probability (A5o2 ,1k) (A5.2.19)MT= Bo becomes K ' ' U vQLr \Q L Magnetic M u l t i p o l e M a t r i x Elements Both o f the i n t e r a c t i o n s H 1 and H" (A'3.1.32 and 33) may give r i s e to magnetic m u l t i p o l e radiation„ The c a l c u l a t i o n of the matrix elements proceeds s i m i l a r l y to that f o r the e l e c t r i c m u l t i p o l e matrix eleraents (A5o2.20) 0 The r e s u l t f o r the i n t e r a c t i o n 41>W / (1+0 \^ (V| where (A5.2.21) K _e_\r Y* l v 7 [f*Lt.)dT H' i s -110 - The result for the interaction H i s M (A5.2.22) *b H -J^w^K^U^^ 0*1+OH v "R v L L m where (A5.2.23)M^= ^ j r ^ V . ( \ r \ > The integrals are referred to as the magnetic multipole moments 0 For systems of several nucleons they become and (A5.2.25) M' u = -Z]^ Y*> ^vj^^ L The transition probability has the same form as (A5.2.19) (A5.2.26) ur = +0 l((2i+0'.!) 4 / - Ill- APPENDIX 6 EULER ANGLE FUNCTIONS In the evaluation of the integrals for the multipole moments, i t i s necessary to express r^Y^CO, & ) i n terms of the Euler angles and the internal co-ordinates. This i s done by using the representation of "^^ (9» ^ ) i n cartesian co-ordinates and r m then using the transformation (1.1.2) to express these i n terms of Euler angles and the 'body* co-ordinates X, Y and Z (1.1.3). Thus for 1 = 1, m = 1 = - /iff W|, (-)X +^?r Wo,(-)Z Similar calculations for m = 0 and ra = -1 lead to the general result (A6.1.2) ^Y (m Wl (oZ-/2ft V j - l X m For r^Y„ the calculation may be done i n an exactly 2,m J J similar manner to give the result (A6.1.3) T Y z -/I W l w ( z z ^ x ) - y ^ w ; c 2 l m m + HXz)^ >£M(X*) Alternatively, this result may be obtained from the calculations for(A6 lo2) by using the formula given below for the product of two 0 - 112 functions. For example (A6.1.^f) J l In the evaluation of the integrals for the multipole moments, products of the Euler angle functions appear such as and i t i s helpful to express these as a sum of single W functions Use i s made of the formula for the product of 0 two functions (Edmonds, 1957, p. 61) rw-, m { r*. r*^ l,m',rn x \ rr,', m' t my i m ( m ^ m and the complex conjugate (A60I.6) 0 , (*f^) = to •" £) , («^> For the V/ functions, the complex conjugate i s r n m V v +tf J ^ (A6ol.7) ^ V " " ^ V »>v -rr./ - m -m C-) ' W, L m -mrr«/ l (ft) £ov I integral - 1 1 3 - where ft- = parity of the function (1,2.26). For m 1 - 0, TT = ( The product of two W functions i s This result i s valid for nor (A6.1o7 = 0. and * 0 but i s not v a l i d when neither m In the l a t t e r case different 8 ) and the orthonormality (A6.1.9)(Wj /^W^ m terms appear,, Using of these functions gives L, U - 114 BIBLIOGRAPHY Austern, I N . , 1951 1952 Phys. Kev. 63, 672 Phys. Rev. 55., 147 tslatt, J . M. ii Weis.skopf, V. W., 1952 Theoretical Nuclear Physics, (John Wiley & Sons, New York) Brueckner, K. A. & Gammel, J . L., 195b Phys. Rev. 109, 1023 Burhop, E„ & Massey, H. S. W., 1948 Proc. Roy. Soc. A192, I56 Curran, S. C. & Strothurs, J . , 1939 Proc. Roy. Soc. A172, 72 Delves, L. M., i960 Phys. Rev. l i e , 131S Derrick, G. H. & Blatt, J . Mc, Derrick, G. H., Derrick, 195« Nucl. Phys. 6, 310 i960 Nucl. Phys. 16, 405 I960 Nucl. Phys. 10, 303 G„ H., Mustard, b. &- Blatt, J . M., I96I Phys. Rev. Lett. 6, 69 Dirac, Edmonds, Fowler, P. A. M., A. R., Proc. Roy. Soc. A133. 60 1957 Angular Momentum i n Quantum Mechanics, (Princeton University Press, Princeton, Lauristen, C. C. & Tollestrup, A. V., 1949 Phys. Rev. 76, 1767 W. A., Goldstein, H., 1931 1950 C l a s s i c a l Mechanics, (Addison-Wesley Publishing Co., Cambridge, Mass.) G r i f f i t h s , G. H. & Warren, J . B., G r i f f i t h s , G. M., Heitler, W., 1955 Proc. Roy. Soc. A6_7, 7&1 Larson, E. & Robertson, L., 1954 N.J.) 19ol Quantum Theory of Radiation, to be published (3rd edition), (Oxford University Press) Rocker, K. H;, Lomont, J . S., 1942 1959 Phys. Z. 43, 239 Applications of F i n i t e Groups, (Academic Press, New York) Morse, F. M. &• Feshbach, H., 1955 Methods of Theoretical Physics (McGraw-Hill Book Co., New York) 115 Panofsky, W. & P h i l l i p s , M., 1955 C l a s s i c a l E l e c t r i c i t y and Magnetism, (Addison-Wesley Publishing Co., Cambridge, Mass.) Rotenberg, M., Bivens, R., Metropolis, N. & Wooten, J. K., The 3-j and 6-j Symbols, (The M.I.T. Press, Cambridge, Mass.) Schiff, L. I., 1937 1955 Verde, M., 1950 1951 Wigner, E. P., 1959 1959 Phys. Rev. 52, ZkZ Quantum Mechanics, (McGraw-Hill Book' Co., New York) Helv. Phys. Acta 23, ^53 Helv. Phys. Acta 2^, 29b Group Theory, (Academic Press, New York)
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A theoretical study of the reaction D(P,[gamma])He3 Rendell, David Hayward 1962
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Title | A theoretical study of the reaction D(P,[gamma])He3 |
Creator |
Rendell, David Hayward |
Publisher | University of British Columbia |
Date Issued | 1962 |
Description | A theoretical study of the reaction D(pγ)He³ is made in an attempt to explain the experimental data for the reaction obtained by Fowler et al. (1949), Wilkinson (1952), Griffiths and Warren (1955) and Griffiths, Larson and Robertson (1961). The angular distribution of the emitted gamma radiation, measured with respect to the incident proton beam, is predominantly proportional to sin²θ. Measurements of the polarization of the radiation by Wilkinson (1952) indicate that the sin²θ component is electric dipole radiation. In addition there is a small, possibly isotropic, component. The proportion of the total yield coming from the smaller 'isotropic' component is 0.035 at a proton energy of 1 Mev, and this proportion increases with decreasing proton energy. The sin²θ component has been interpreted by Griffiths and Warren as coming from an electric dipole transition from an initial state of a P-wave proton (L = 1, L₂ =0) and ³S deuteron to the ²S ground state of He³. This interpretation is supported by the present calculations. They also suggest that the smaller 'isotropic* component could be either a magnetic dipole transition of S-wave protons to the ²S state of He³ or an electric dipole transition involving spin-orbit coupling. In this present work the cross-sections are examined for all possible channels which might conceivably contribute to the reaction. The channels considered are 1. electric dipole transitions for a. P-wave protons to the ²S state b. P-wave protons to the ⁴D state c. F-wave protons to the ⁴D state 2. electric quadrupole transitions for a. S-wave protons to the ⁴D state b. D-wave protons to the ²S state 3. the magnetic dipole transition for S-wave protons to the ²S state. Three-body wave functions are constructed, following Verde (1950) and Derrick and Blatt (1956), making use of the symmetry properties in spin space, isotypic spin space and in ordinary space. In addition to the states of total isotropic spin T = ½ considered by Derrick and Blatt the states of total isotopic spin T = 3/2 are included. The radiation matrix elements for the above channels are calculated and are expressed in terms of integrals over the three internal coordinates. These radial integrals are estimated by using very simple radial functions which are valid outside the range of the nuclear forces and which also disregard coulomb forces. The cross-sections depend on the unknown amplitudes and relative signs of the various possible symmetry states. Therefore the size, although not the angular dependance or the general energy dependance, of the cross-sections can be used only as an order-of-magnitude estimate. By comparison of the size, angular distribution and energy dependance of the calculated cross-sections with the experimental data it is shown conclusively that the sin²θ component of the radiation comes from the electric dipole transition of P-wave protons to the ²S state of He³. The smaller 'isotropic' component of the radiation comes from either (a) an electric dipole transition of P-wave protons to the ⁴D state, giving an angular distribution proportional to 1 – (1/7)cos²θ, or (b) a magnetic dipole transition of S-wave protons to the ²S state, giving an isotropic angular distribution. The observed energy dependence of the relative yield of the small component suggests the interpretation in terms of the magnetic dipole transition. The cross-sections of the other transitions examined are too small to explain the experimental results. |
Subject |
Helium Nuclear physics |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-11-18 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085411 |
URI | http://hdl.handle.net/2429/39149 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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