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On alpha-decay in heavy nuclei Scherk, Leonard Raymond 1967

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ON ALPHA-DECAY IN HEAVY NUCLEI by LEONARD RAYMOND SCHERK B.Sc., University of British Columbia, 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the department of Physics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March, 1967 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that tfu: Library shall make it freely available for reference and study I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives.. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. n • • f PHYSICS Department of The University of British Columbia Vancouver 8, Canada Date OflSul 6y I f / 7 ABSTRACT The alpha-particle reduced widths f o r the ground state i n Po^ 1^ are calculated on the basis of the nuclear s h e l l model, employing the technique of Harada, but treating the nuclear surface i n a more d i r e c t manner. It i s contended that the calculations of previous authors, who have generally used a square-edge nucleus and a Coulomb ba r r i e r rounded-off by the nuclear potential of Igo, have, e s s e n t i a l l y , used the equivalent square-edge nucleus model of Vogt. Their J.W.K.B. estimate of the barr i e r p e n e t r a b i l i t i e s i s checked by an ana-l y t i c c a l c u l a t i o n in Chapter 3 and i s found to be reasonable. It i s shown i n Chapter 4 that, i n the scattering of an alpha-p a r t i c l e from the ground state of Pb^ o a, the diffuse nuclear edge considerably enhances the one-body reduced widths and, in a d i r e c t manner, that i t s i m i l a r l y enhances the one-body d i f f e r e n t i a l e l a s t i c scattering cross-section. In this man-ner, i t is demonstrated that the radius involved i n the equivalent square-edge nucleus model must be considerably larger than that of the diffuse-edge nucleus to which i t corresponds. This i s shown d i r e c t l y i n Chapter 5, where the v a l i d i t y of the equivalent square-edge nucleus model i n heavy nuclei i s examined. It i s contended that this explains the large r a d i i found i n previous c a l c u l a t i o n s . This i s demon-strated d i r e c t l y by repeating the c a l c u l a t i o n of Harada with the d i f f u s e nuclear edge being introduced in a d i r e c t manner. Although the effects of configuration mixing have not been d i r e c t l y examined, i t has been concluded that s h e l l model calculations can explain the major part of rates provided that the nuclear surface is manner. i i i the empirical decay-treated in a direct • iv TABLE OF CONTENTS Page CHAPTER 1 INTRODUCTION 1 CHAPTER 2 A DERIVATION OP THE DECAY CONSTANT 7 CHAPTER 3 EXPERIMENTAL REDUCED WIDTHS 21 3-1 THE REACTION P o ^ 1 2 » ^208 + 2 2 3- 2 COMPARISONS WITH J.W.K.B. AND SQUARE- 24 WELL ESTIMATES CHAPTER 4 INDEPENDENT-PARTICLE MODEL DECAY RATES 29 OF P o 2 1 2 4- 1 ONE-BODY REDUCED WIDTHS AND EXPERIMENTAL 32 SPECTROSCOPIC FACTORS 4-2 ONE-BODY DIFFERENTIAL ELASTIC SCATTERING 41 CROSS-SECTION OF Fb 208 FOR AN ALPHA-PARTICLE AT 90° CM 4-3 HARADA'S FORMULA FOR THE INDEPENDENT- 43 PARTICLE MODEL REDUCED WIDTHS 4- 4 NUMERICAL INDEPENDENT-PARTICLE MODEL 4 7 REDUCED WIDTHS CHAPTER 5 THE EQUIVALENT SQUARE-EDGE NUCLEUS MODEL 52 5- 1 THE EQUIVALENT SQUARE-EDGE NUCLEUS MODEL 53 5-2 APPLICATIONS TO HEAVY NUCLEI . 54 CHAPTER 6 : CONCLUSIONS 62 BIBLIOGRAPHY 65 APPENDIX A HARADA'S FORMULA FOR THE REDUCED WIDTH 66 FOR ALPHA-PARTICLE DECAY IN EVEN-EVEN NUCLEI APPENDIX B TALMI TRANSFORMATION COEFFICIENTS 74 APPENDIX C CALCULATION OF THE ONE-BODY DIFFERENTIAL 80 ELASTIC SCATTERING CROSS-SECTION LIST OF TABLES Page TABLE 1 EXPERIMENTAL REDUCED WIDTHS 25 TABLE 2 ONE-BODY POTENTIALS 37 TABLE 3 EXPERIMENTAL SPECTROSCOPIC FACTORS 38 TABLE 4 OVERLAP INTEGRALS 49 TABLE 5 COMPARISON OF DECAY RATES 50 TABLE 6 CALCULATED SPECTROSCOPIC FACTORS 50 VI FIGURE 1 FIGURE 2 FIGURE 3 FIGURE 4 FIGURE 5 FIGURE 6 FIGURE 7 FIGURE 8 FIGURE 9 LIST OF FIGURES SURFACE POTENTIAL l P b 2 0 8 J COMPARISON OF DECAY-RATE PARAMETERS WITH SQUARE-WELL AND J . W. K. B. ESTIMATES (Fb208) SQUARE AND DIFFUSE POTENTIALS ( P b 2 0 8 ) RESONANT WAVE FUNCTIONS FOR SQUARE AND DIFFUSE POTENTIALS ( P b 2 0 8 ) RESONANT WAVE FUNCTIONS: EFFECT OF WELL DEPTH RESONANT WAVE FUNCTIONS: EFFECT OF SURFACE THICKNESS Page 26 27 34 35 39 40 ONE-BODY SCATTERING CROSS-SECTION ( P b 2 0 8 ) 42 BARRIER ABSORPTION ( P b 2 0 8 ) 59 REFLECTION FACTOR ( P b 2 0 8 ) " • 60 v i i ACKNOWLEDGEMENTS I wish to thank Professor E. V/. Vogt for h i s continual encouragement and generous assistance i n the solution of this problem. This thesis was done while the author was supported by a bursary from the National Research Council of Canada. CHAPTER 1 INTRODUCTION The aim of this thesis i s to estimate the extent to which the independentrparticle model can explain alpha-particle decay rates in heavy n u c l e i . A considerable amount of e a r l i e r work has had this aim (c.f. Liang (1964)), hut past estimates of the alpha-particle decay rates have tended to be smaller than the observed decay rates. The calculated rates are a product of two factors: a) a spectroscopic factor, which accounts for the many-body aspects of the nuclear problem; b) a one-body decay constant, which accounts for the average int e r a c t i o n of the alpha-particle with the daughter nucleus. We f i n d that much of the remaining discrepancy between the calculated and the observed decay rates can be removed by a more d i r e c t and more accurate treatment of the nuclear surface in the calcu-l a t i o n of the one-body decay constant. We w i l l demonstrate that, apart from a difference in their r e f l e c t i o n properties, a heavy nucleus having a diffuse-edge behaves l i k e a consid-erably larger nucleus having a square-edge i n the analysis of alpha-particle decay rates or alpha-particle scattering cross-sections„ (The decay rates are related to the scat-ter i n g cross-sections in a simple manner; the decay constant i s proportional to the width of the e l a s t i c scattering cross-section.) Vogt (1962) has suggested that one may, i n general, replace a conventional" diffuse-edge nucleus with an "equivalent square-edge nucleus" which he defines in the following manner. He considers a simple one-body model in which the incident p a r t i c l e i s scattered by a potential w e l l . This i s the nuclear problem without the many-body aspects; i n fa c t , the spectro-scopic factor (which accounts for the many-body aspects of the problem) i s rather insensitive to the nature of the nuclear surface and to the size of the nucleus. He replaces the diffuse-edge nuclear p o t e n t i a l well with an "equivalent square-edge well" whose radius and depth are chosen so that i t ex-h i b i t s a resonance at the resonance energy of the d i f f u s e -edge well and so. that the resonant wave function of the square-edge well s a t i s f i e s the following conditions: a) i t has the same number of nodes as the resonant wave function of the diffuse-edge well; b) i t has the same amplitude as the reso-nant wave function of the diffuse-edge well at the radius of the square-edge well . Now the one-body scattering width (or decay constant) i s a product of two fa c t o r s : a) a one-body reduced width, which depends only upon the amplitude of the resonant wave function at the nuclear radius (and, hence, on the internal aspects of the nucleus); b) a pe n e t r a b i l i t y , which depends only upon the r e f l e c t i v e properties of the po t e n t i a l b a r r i e r outside of the nuclear radius. Apparently the only difference i n the one-body problem between the diffuse-edge well and the corresponding equivalent square-edge well i s the anomalous r e f l e c t i o n of the l a t t e r ; this difference in r e f l e c -tion i s assigned to a " r e f l e c t i o n f a c t o r " . Vogt (1962) has found for neutron scattering, and Vogt, Michaud, and Reeves (1965) for alpha-particle scattering from l i g h t n u c l e i , that the equivalent square-edge well has a con-siderably larger radius than the corresponding diffuse-edge well; i n general, i t has about the same depth as the d i f f u s e -edge well and the r e f l e c t i o n factor i s found to be small. In f a c t , the parameters of the equivalent square-edge well are found to be rather i n s e n s i t i v e to the nature of the reac-t i o n and to the many-body aspects of the problem so that the diffuse-edge nucleus does behave l i k e a larger square-edge nucleus. In thi s thesis, we w i l l show that these results apply to the alpha-particle decay rates of heavy n u c l e i ; i n f a c t , we w i l l show that the previous calculations have gener-a l l y used this equivalent square-edge well, thus hiding the true radius of the decay problem. It i s of considerable value to show that the discrepancies between the empirical and the calculated alpha-particle decay rates found i n the e a r l i e r calculations have l a i n in the method of c a l c u l a t i o n rather than i n the independent-particle model assumptions. The f a i l u r e of the independent-particle model would imply that the model does not introduce s u f f i c i e n t cor-r e l a t i o n s into the nuclear wave functions to account for the observed c l u s t e r i n g into a l p h a - p a r t i c l e s . The fac t that the Pau l i Exclusion P r i n c i p l e becomes less i n h i b i t i v e i n the surface region makes i t seem natural to a t t r i b u t e much of the clustering to the nuclear surface. Wilkinson (1961) has i n t e r -preted the discrepancy between the empirical and calculated alpha-particle decay rates as evidence that one cannot e a s i l y describe the nuclear wave function i n the surface region by shell-model wave functions; i n fa c t , he has suggested that - 4 -i t might he necessary to resort to a phenomenological clu s t e r model. A natural way i n which to introduce correlations into the independent-particle model wave functions i s through con-f i g u r a t i o n mixing; Harada (1961) and Zeh and Hang (1962) have found this enhances the calculated decay rates by an order of magnitude. We contend that the remaining discrepancy between the empirical and the calculated decay rates can be removed by a correct interpretation of the. radius involved i n the c a l c u l a -tions. The procedure used i n c a l c u l a t i n g the alpha-particle decay rates i s f a m i l i a r from the theory of nuclear reactions. In Chapters 2 and 3 of this thesis, we have derived a formula for the decay constant by this procedure. However, we have devel-oped the decay constant from a point of view which i s appro-priate to the decay problem rather than to the scattering problem. The formula for the decay constant has been devel-oped i n this manner by several other authors, p a r t i c u l a r l y Zeh (1963) and Mang (1964). Igo (1959) has used the e l a s t i c scattering data to empir-i c a l l y determine the modification of the Coulomb f i e l d s i n the nuclear surface by the r e a l , average interaction of an alpha-p a r t i c l e with a heavy nucleus. We use Igo's nuclear potential to round off the Coulomb barriers.~ The p e n e t r a b i l i t y through a rounded-off Coulomb ba r r i e r can be determined quite e a s i l y and, i n Chapter 3, the p e n e t r a b i l i t y i s evaluated for either 208 the scattering of an alpha-particle from Pb or of i t s decay from P o 2 1 2 . The results are compared with those obtained from the J.W.K.B. and square-well estimates; the l a t t e r estimates have been used i n most of the previous c a l c u l a t i o n s . We have previously noted that most of the effects of the nuclear surface upon the alpha-particle decay rates are as-signed to the one-body decay constants rather than to the spectroscopic f a c t o r s . In Chapter 4, we investigate the ef f e c t of the nuclear surface upon the one-body decay constants. By comparing the resonant wave function of a diffuse-edge nucleus with that of a square-edge nucleus of the same radius, we show that the diffuse-edge enhances the reduced widths of the one-body problem. In f a c t , i t i s seen from these resonant wave functions that the diffuse-edge nucleus corresponds to a con-siderably larger equivalent square-edge nucleus. This sug-gests that the one-body scattering cross-sections should be considerably enhanced by the diffuse-edge; i n f a c t , we show this i n a d i r e c t manner by ca l c u l a t i n g the d i f f e r e n t i a l elas-t i c s cattering cross-section for the scattering of an alpha-p a r t i c l e from a poten t i a l well appropriate to a P b 2 0 8 nucleus. We include the many-body aspects of the decay problem by repeating the independent-particle model c a l c u l a t i o n of p i p Harada (1961) for the alpha-particle decay rate of Po . Harada has used a square-edge well of radius ten fermis to to evaluate the amplitude of the one-body resonant wave func-tions; he has rounded-off the Coulomb barrier with the nuclear p o t e n t i a l of Igo (1959). He has chosen his model wave func-tions to be i n f i n i t e harmonic o s c i l l a t o r wave functions and has determined the harmonic o s c i l l a t o r size parameter by taking the amplitude of the mode of the r e l a t i v e motion to he equal to the amplitude of the one-body square-well wave func-t i o n s . While i t i s not obvious that this procedure r e a l l y represents a square-well c a l c u l a t i o n , i t might be suspected that i t does since the effects of the internal modes are rather i n s e n s i t i v e to the.size parameter. The'radius of Harada's c a l c u l a t i o n i s considerably larger than those believed t y p i -c a l of heavy n u c l e i ; i n f a c t , we contend that he has essen-t i a l l y used the equivalent square-edge well of a smaller diffuse-edge w e l l . In Chapter 4, we repeat his c a l c u l a t i o n but replace the one-body wave functions of the square-edge well with the one-body wave functions of a smaller d i f f u s e -edge w e l l . We show that the decay constant obtained i n this manner f o r Poc,-L* i s comparable to that which Harada has ob-tained with the larger square-edge well.. In fa c t , we c a l -culate the equivalent square-edge well of a t y p i c a l d i f f u s e -edge well i n Chapter 5 and f i n d that the difference in the r a d i i of the two wells i s similar to that found i n the pre-ceding decay rate c a l c u l a t i o n s . In addition to discussing alpha-particle decay, we use the b a r r i e r penetration calculations to study absorption pro-cesses. In Chapter 5, we give the extent to which the equiv-alent square-well model i s found to apply to the alpha-particle absorption of a Pb nucleus. The e f f e c t of the strength of the optical model potential in the electrostatic barrier is also studied. A summary of the preceding calculations i s given i n Chapter 6. CHAPTER 2 A DERIVATION OF THE DECAY CONSTANT The systems which w i l l he discussed in this thesis are nuclear systems which decay by means of simple alpha-particle d i s i n t e g r a t i o n . The formalism for describing such systems i s f a m i l i a r from the theory of nuclear reactions; i n f a c t , Thomas (1954) has derived the decay constant of such a system by the standard techniques of nuclear reaction theory. The purpose of the derivation provided i n this thesis i s to employ these techniques in a similar manner, but from a point of view which i s more natural to the study of decay problems. The most obvious method i n which to describe a decaying system would be to find the wave function of the system at some convenient time, say t Q . If at this time i t i s found that the system i s described by the quantum numbers,71 ( t 0 ) , then the temporal evolution of the system i s governed by the Schroedinger Equation; (2.1) £11^. I71.(t}> = Til7l(t)>. (Here H i s the Hamiltonian of the system and "h i s the crossed Planck's constant.) A natural choice for t would be at a time pr i o r to the formation of the system. If t 0 were chosen s u f f i c i e n t l y nega-t i v e , one could think of the system as consisting of the wave packets of the constituent p a r t i c l e s , with negl i g i b l e overlap and interacting through slowly varying potentials, and as con-verging on the point at which the system formed; the dynamics of the wave packets (at the speeds of interest) are then New-tonian so thatlTl ( t 0 )7 may be determined pr e c i s e l y . - a -The d i f f i c u l t y in this modus operandi i s that, af t e r having converged and formed, the system does not decay for a time many orders of magnitude longer than the t y p i c a l period of a constituent nucleon's motion within the formed system. The temporal fluctuations of the wave function of the formed system, which at f i r s t were strongly correlated to the forma-tion process, a f t e r a time "become s t a t i s t i c a l . It i s , therefore, more expedient to choose t 0 after the formation process and to construct a s t a t i s t i c a l l y determined wave function of the constituent nucleons, the nucleons being assumed to be within the nuclear volume. In practice, i t may be hoped to re a l i z e t h i s s t a t i s t i c a l d i s t r i b u t i o n by forming the nuclear wave functions from models such as the independent-p a r t i c l e model. For thi s reason, the wave function of the formed state i s commonly referred to as the "parent nucleus" wave function. Care must be exercised i n interpreting t h i s stationary state approximation. Such stationary wave functions are i r r e g -u l a r at i n f i n i t y , while the parent nucleus wave function i s not only regular at i n f i n i t y but e s s e n t i a l l y vanishes outside the nuclear radius. Therefore, the approximation is useful only within the volume of the formed system. Assuming a parent nucleus at t = 0, and assuming two fragment break-up, i t would seem reasonable to choose the wave function of the decaying system to be of the form, x 1 , s 1 , x 2 , s 2 , . . . x A , s A ) + ... deb c l C 2 ( 6 . t ) V c l ( q i ) ^ c 2 ( q ^ ) < i , C l c 2 ( R . t ) . 2£ai Sa » t ) where (xi»a.i,x2,Sg, ... ,2L A » ^ A ) = * * • ••• a ^ (xi..§.]_,Xg • x A , s A , 0 ) , and where ( x ^ s j are the s p a t i a l and spin co-ordinates of the i t h nucleon, respectively. Here, (j/ C l(q 1) a n d ( ^ C 2 ( q ^ ) are the wave functions of the f i r s t and second decay fragments; c-j_ and c 2 are the appropriate quantum numbers. ^ c (R, 6) i s a stationary state of the r e l a t i v e motion; £ i s the re l a t i v e energy and R the separation of the fragments. A theory of alpha-decay has been developed from this point of view by Mang (i960). In applications to the alpha-decay of heavy n u c l e i , he finds that, (2.3) a(t) # EXP( ( 6 - i * ) t ) . In f a c t , the formula which he obtains for 2f i s e s s e n t i a l l y the same as that which w i l l be obtained i n the c a l c u l a t i o n to be discussed. At large positive times, the parent nucleus w i l l have decayed and should be describable as an alpha-particle and a daughter nucleus at some large r e l a t i v e separation. Zeh (1963) has noted that, in a suitably small space-time region, the time dependence of the parent nucleus wave function may be taken to be that of Eq. (2.3) and the wave functions i n each channel of break-up, (2.4) G C l C 2 ( R . t ) « <clC2R|71 (t)> , may be taken to be those of purely outgoing fl u x . In this thesis we s h a l l follow this approach. The parent nucleus wave function w i l l be expanded in the nuclear volume in terms - 10 -of a complete set of stationary states which e x p l i c i t l y include the alpha-decay channels; the boundary conditions at the nu-clear radius w i l l be chosen so that the channel wave functions are asymptotically waves of outgoing f l u x . To determine the required expansion c o e f f i c i e n t s , i t w i l l be assumed that the parent nucleus wave function can be described by the independ-ent - p a r t i c l e model. Of course, this procedure i s well-known and has often been used i n the analysis of nuclear reactions. In the construction of the parent nucleus wave function, i t w i l l be assumed that non-local and many-body potentials can be neglected and that the Hamiltonian of the system, which w i l l be assumed to consist of A nucleons (N neutrons and Z protons), can be written i n the form, A (2.5) H » H i # where, (2.6) ' H i - ^ V ^ + * j 4 1 . V - ' l . ^ > . Here, "Vi s (ii»s-±), are the s p a t i a l and spin co-ordinates, respectively, of the i * 1 * 1 nucleon; the proton and neutron masses have been assumed to be equal. The construction of the independent-particle model wave functions i s well-known (Preston (1962)). One calculates the s p a t i a l average of the potential term i n Eq, (2.6) at the i ^ nucleon; from this average pote n t i a l one derives the one-nucleon o r b i t a l s . The parent nucleus wave function i s then formed of l i n e a r combinations of the Slater Determinants of - l i -the assumed configuration; the li n e a r combinations are chosen appropriate to the quantum numbers of the parent nucleus. The residual i n t e r a c t i o n does not, i n general, vanish; i f i t can be treated as a small perturbation, one can introduce the resultant mixing of the configurations by the techniques of perturbation theory. To find the nature of the stationary states of the Kam-i l t o n i a n commensurate with the system a f t e r decay, i t i s de-sir a b l e to introduce the approximations i n a manner convenient for constructing alpha-particle and daughter nucleus wave func-ti o n s . In the discussion of thi s chapter, i t w i l l be assumed that the alpha-particle and daughter nucleus wave functions have been antisymrnetrized with respect to the interchange of proton or of neutron co-ordinates; the parent nucleus wave function to be constructed from these wave functions w i l l , therefore, only be p a r t i a l l y antisymrnetrized. However, i t w i l l be shown that the parent nucleus wave function can be regarded as a solution of a Schroedinger Equation having a complex eigenvalue (Eq. (2.25)). Since the Hamiltonian is symmetric: i n the interchange of the co-ordinates of any two i d e n t i c a l p a r t i c l e s , the completely antisymrnetrized wave function w i l l be an eigenstate of the above Schroedinger Equation i f the p a r t i a l l y antisymrnetrized wave functions are solutions. Only the l a t t e r fact w i l l be used in the argu-ments to follow.' It w i l l be assumed that the system of nucleons lab e l l e d by (1234) constitutes the alpha-particle, where 1 and 2 are protons and 3 and 4 are neutrons; the remaining nucleons, - 1 2 -(56...A), w i l l be taken as constituting'the daughter nucleus. It i s convenient to make the d e f i n i t i o n s , 2L« = l2Ll.2£2-t2£3.2L4); a 12.5.2i6 2 L A ) ; s.^  = 8 2 , 8 ^ , 8 4 ) ; s s - ( 8 5 , 8 5 8 A ) ; and to introduce the r e l a t i v e co-ordinates, (2.7) - 4 (x]_+ 2L2 + 2S.3 ^ 2L4 ) I R , = (2L5 + 2L6 + * ' * + 2LAJ I R.CM = A" ( i i + 2L2 + *' * + 21A) J L i = 2Li - 2L2 I L2 = 2£3 ~ 2L4 ; C3 = i (2Li + 2£2) " -H2L3 + 2L4); L = ( L i . L 2 . L 3 ) ; R = R , - R < r • (The Jacobian of the transformation (x1.X2.2L3.2L4) — ^ ( R - . L i . 1-2» I-3) i s unity.) After decay, the nuclear system can be described by a "one-body" model; by this we mean that the interaction bet-ween the daughter nucleus and. the alpha-particle i s a func-tion only of t h e i r separation. If the calculations are per-formed in the center-of-mass frame, ^ ( M ) V RcM • . and i f A i s s u f f i c i e n t l y large that the r e c o i l in the daughter nucleus wave function can be neglected, 2lA^4)m R —f then one can write, (2.8) H = + H f +" H««- , where, u.9)a) ^ • 1 i 1 - | i v L ; + * A i 4 1 v l j ( v 1 . ^ j l - 13 -c) IW - V & + U ( R ) . Here, (2.10) M . 4 l A = 4 ± m ; P1 ZJ2 . I . , j a J . are the reduced masses with respect to the r e l a t i v e co-ordi--nates R, £ 2 , and ^ , respectively; U ( R ) represents the average i n t e r a c t i o n between the decay fragments. The ef f e c t of the residual interaction between the decay fragments can be accounted f o r by taking U(R) to be an o p t i c a l model p o t e n t i a l : (2.11) U(R) = V ( R ) + iW(R). (V and V/ are real.) It is well-known that the complex term, W(R), simulates many of the e f f e c t s of the residual interaction; i t e s s e n t i a l l y vanishes outside of the nuclear volume. In practice, both y(R) and W(R).are regarded as being phenomenological; i n genr e r a l , they w i l l be dependent upon the r e l a t i v e energy of the decay fragments and upon the channels being considered. We w i l l neglect this dependence. In t h i s thesis, we w i l l be concerned, only with the spher-208 i c a l nucleus, Pb . U(R) then depends only upon the distance between the decay fragments so that the states of r e l a t i v e motion are solutions of the Schroedinger Equation, (2.12) ( - ^ V 2 "T* U(R))(|/(R) » £ l | / ( R ) . 2p R The solutions of Eq. (2.12) have the form, (2.13) iy ( R ) • « Y ^ L ( R where Y ^ f L (~L_) i s a spherical harmonic and U T (R) is a solution L R u of the r a d i a l Schroedinger Equation, (2.14) - h 2. d 2' u T ( R ) -h (U(R)-f- h 2 L ( L f l ) ) u L(R) = ' " 2Jl dR2" u 2JJ R 2 • • • = £ L u L(R). - 14 -The eigenstates of the t o t a l Hamiltonian, H, are then of the form, T where \ij . m and (il .' are standard eigenstates of the angular momenta, and j.f , resps c t i v e l y , and are solutions of the Schroedinger Equations, (2.16) a) H . l } ^ , = ^ 1 ^ ; 7* and TJr are the remaining quantum numbers required to specify the states completely. The many-body aspects of the decay problem are contained within the nuclear volume. For the purposes of analyzing alpha-decay rates, i t i s convenient to define a set of func-tions i n the following manner. In each channel, one can define a boundary condition number, b c; (2.17) (R d u u(R)) * u c(R) dR Here, R 0 i s taken as the radius of the parent nucleus; the 8 *c R=RQ functions, u c(R), are taken to be normalized i n R^R Q. (We w i l l l a t e r show that the boundary conditions can be chosen in a manner which i s natural to the decay problem.) These boundary conditions, together with the d i f f e r e n t i a l equation, Eq. (2.14), y i e l d a set of s.olutions of Eq. (2.14) which are complete within R<R Q. These solutions may be taken to be, u£(R) , where n i s the number of nodes of u£ . Since the solutions of Eq. (2.16) (which are defined over a l l space) are complete, one can define a complete set of functions in the following manner. One defines the channel wave function - 15 -to be, (2.18) l | / c = | 0 i p £ m < l E M <i,L m,ML( J M ? V ^ Y L L . ' I where c denotes the channel quantum numbers, (j ^ ,mx,% , , L,J,M). The set of functions, U.w) l ( J n o « «• 1|/o. i s then complete within R R 0 so that one can make the expan-sion, where, (2.21) C n c = $VQ ^fo Nuclear systems exhibiting alpha-decay t y p i c a l l y have long l i f e t i m e s whence i t might be expected that they should, i n some sense, behave l i k e a stationary state. To see in what sense this i s true, i t w i l l be assumed that the c o e f f i c i e n t s , a(t) and ' b c 1 c 2 ^ » t ) o f E <1» ( 2» 2)» a r e o f t h e form, (2.22) a(t) = EXP(-~ E Qt)EXP(-|- t ) ; ^ W ' ^ = E X P ( - l E 0 t ) ( 1 - E X P ( - | t ) b C i C 2 ( * ) . . . (This temporal dependence has, i n fact, been j u s t i f i e d from theoreti c a l considerations by Mang (i960).) Accepting Eq.(2.22), the time-dependent wave function is of the form, (2.23) t£ (t) = EXP(- £ E t) ( 1 P Q E X P ( - I t ) " f t '•• (1 - EXP(- J t)V d). where tp ^  represents the system aft e r decay. The Schroedinger Equation of the' system then becomes', (2.24) i T i ^ ( t ) - ( E Q - i * ) ? ( t ) + i^EXP(4 E0t)f d = H l ? ( t ) . "bt Since ~& i s , i n general, small, a good approximation to the - 16 -parent nucleus wave function should be obtained by solving the complex eigenvalue problem, (2.25) H ? 0 = ( E 0 - i * ) f P 0 . Written in the form, i t i s seen that ~$ can be treated as a perturbation with the zeroth order approximation of Eq. (2.25) being a stationary state. From Eq. (2.25), (2.26) -2i#P0 = if0Hf Q - M t Defining the decay constant by, (2.27) " X - f £ . employing Eq. (2.9) and Green's Theorem, and integrating Eq. (2.26) over a l l space, Ro H _ jf r (2.28) J = i ^ o H ^ o " ^ o H V o JR^dRdldx^dadJl 7? 3 h V7 — ' < • • * < 3 l  U ) X ( j , ^ ' , T J. C n j : m;V j,7, L J ^ j ; ^ •) (2) n^T ./a ( R e a l " l J ^ 5 . °n^X j, ^  LJM . ..x rO^Lm^Mj JM>|1/^ fx( £ ? 5 T m ^ x . ) m^  1 <J(f m ' A ° _ i i»r.tT..T»MI • • «x IJ^ 'm; )dJ xds., - 17 -R = K 0 , Eq. (2.28) i s of the form, (2) (3) ?" L = R . I c ' d ^ : . /RJ=K0 where j_ denotes the pr o b a b i l i t y current i n the sp e c i f i e d channels. For heavy alpha-emitters, decay through channels other than simple alpha-decay i s generally negligible so that terms (1) and (2) i n Eq. (2.28) can be neglected. Eq. (2.28) can then be written i n the form, (2 ?Q) "X = - i l i . C* C (u c* d un« u c.du° ) {2.29} A - - j ^ ^ n . S i c S i ' c l U n ^  - > R = R O . A natural choice of the boundary conditions i s to match the r a d i a l wave functions i n each channel onto the wave of purely outgoing f l u x i n that channel. This i s accomplished by choosing, (2.30) b c .= S e(6 c,R 0) + i P c ( 6 C , R 0 ) ; (6 C= E Q - E ^ - E j ) S c and P c are. the nuclear s h i f t function and the nuclear p e n e t r a b i l i t y : (2.31) s C ( £ C , R 0 ) + i P c ( e C , R 0 ) = . . . . R . G C ( * C , R ) -f- iF»(6c , R ) I G C U C , R ) -h i F c ( 6 C , R ) I R = R 0 . Here F c and G c are those solutions of Eq. (2.14) which are asymptotic to the regular and irregular.Coulomb functions, J5'c and G c, i n the channel c. (The one-body potential i s essen-- 18 -t i a l l y the Coulomb potential outside of the nuclear surface.) The optical model potential, U(R), is essentially phenom-enological. Its behaviour in the nuclear surface has been determined from the elastic scattering data by Igo (1959), but the depth of the real and imaginary terms within the nuclear volume is largely arbitrary. (Values of 100-150 MEV for V(R) and 5-10 MEV for Y/(R) are currently fashionable.) To the extent that the residual interaction can be neglected, we need only retain the real term, V(R), in Eq. (2.14). It would then seem natural to choose the depth of the well in such a . manner that, for some n 0, (2.32) £ £ q = £ c . This one-body wave function should then represent many of the radial properties of the actual wave function of the decaying system. (The arbitrariness in the number of nodes reflects the ignorance of the well depth.) The one-body reduced width, n 2 , is defined by, By noting the boundary condition, Eq. (2.30), Eq. (2.29) can be written in the form, (2.34) 1 • Z » o t * £ „ - t t . > y ° a - , C „ „ C 2 ^ p n 0 n' n,n» nc^n'c |r£j2 Defining the one-body decay constants, (2.35) 2P c(e n o,R 0)l^nJ 2 and the spectroscopic factors, (2.36) S c J ^ C n c ^ | 2 Eq. (2.34) can be written in the form, - 19 -(2.37) J = ^ S c. Eq. 12.37) i s convenient for calculations of the decay constant from the nuclear models. It i s seen from Eq. (2.25) that we may approximate the parent nucleus wave f u n c t i o n , b y X 0, where X 0 i s the f a m i l i a r resonant state of reaction theory defined by, (2.38) HX0 = E 0 X 0 . It i s a r e s u l t of reaction theory that the width of the e l a s t i c scattering cross-section (as determined "by the Breit-V/igner • s i n g l e - l e v e l resonance formula) i s , (2.39) P o c - f ~K°0'cb' S, oc c oc "-"c. where S c i s the spectroscopic factor of Eq. (2.36) and "AQC^* i s the one-body width, (2.40) r ° c b * = 2 P c ( ^ o , R 0 ) | ^ ° J . Comparing Eq. (2.40) and Eq. (2.35), (2.41) T o c = "Xh, which i s an expression of the Uncertainty P r i n c i p l e . If the residual i n t e r a c t i o n i s s u f f i c i e n t l y small, then only the one-body wave functions, u n Q , are contained appreci-ably i n the parent nucleus wave function. Eq. (2.36) then reduces to, (2.42) S c = i i Q and Eq. (2.37).to, (2.43) -X = ? * 8 - H c n o 0 | 2 • To test the v a l i d i t y of the one-body model, we note that many of the effects of the residual interaction can be accounted for by the o p t i c a l model potential (Eq. (2.11)). The absorp-C n J ! - 20 -tion cross-section for such a potential i s of the'form, l 2 ' 4 4 ) 6 a b s = J " L ScT c; (k c = J W ) Tjj i s the o p t i c a l model transmission function i n the channel, c; L is the r e l a t i v e angular momentum in the channel, c; g c i s a numerical factor depending on the angular momenta of the parent nucleus, the daughter nucleus, and the alpha-particle. If W i s taken as the order of magnitude of the depth of the complex part of the o p t i c a l model p o t e n t i a l , then, in the one-body approximation about E 0 , the transmission function is of the form, (2.45) T C(E) r 4P L(£ L,R 0) c L jrxf (E - + W§ , c L being a numerical factor. Thus the r e l a t i v e residual i n t e r -action spreads the one-body statey-U r , through a width, WQ. Since i n t y p i c a l heavy nuclei, the s e p a r a t i o n s , ! ^ -£^«l f are greater than f i f t e e n MEV and reasonable values for W are less than ten MEV, i t would seem reasonable that the parent nucleus wave function should contain appreciably only the one-body Q wave f u n c t i o n , H n o , and i t s nearest neighbours. Hence, Eqs. (2.42) and (2.43) should be a moderately good approximation. The s t a t i s t i c a l factors ,jl'Cn J 2 , are reasonably i n t e r -preted as measuring the p r o b a b i l i t y that the resonant state can be represented by the appropriate decay products. The extent to which the independent-particle model correctly accounts for these correlations w i l l be shown to provide a a test of i t s v a l i d i t y i n the nuclear surface. - 21 -CHAPTER 3 EXPERIMENTAL REDUCED WIDTHS We have contended in the introduction that the calc u l a -tions upon alpha-particle decay rates performed by previous authors have, e s s e n t i a l l y , used the equivalent square-edge nucleus model of Vogt. They have customarily written the decay constant as a product of tv/o fac t o r s : a) a nuclear reduced width, which depends only upon the in t e r n a l aspects of the nucleus and which i s interpreted as measuring the frequency at which alpha-particles appear at the nuclear surface; b) a pe n e t r a b i l i t y , which depends only upon the potential b a r r i e r and which i s interpreted as measuring the ease with which an alpha-particle can penetrate the barr i e r and appear on the outside. The reduced widths have generally been calculated using square-edge n u c l e i , while - the .barrier has generally been rounded-off with the nuclear potential of Igo (1959). Hence, the anomalous r e f l e c t i o n of the square-edge nucleus has i m p l i c i t l y been removed; in f a c t , this cor-responds to a ca l c u l a t i o n with the equivalent square-edge model. In the present section, we w i l l check that previous J.W.K.B. estimates of the p e n e t r a b i l i t i e s are reasonable; i n the language of Vogt, this i s equivalent to checking that the r e f l e c t i o n factors have been calculated accurately. We have also included the pen e t r a b i l i t y estimates for the unmod-i f i e d Coulomb b a r r i e r . For completeness, we have checked the reduced derivative widths which have been used by some previous authors. is defined in terms of - 22 -the p r o b a b i l i t y amplitude, In the channels of alpha-decay, the channel wave functions are of the form, ^ • 2 ) ^ c = i o X l L ' L-] m ^ M ^ ^ i ^ j m j ^ . B j Y ^ ^ ) , where the single bound-state of the alpha-particle, which has zero angular momentum, has been denoted by .X (L» ft*) a n c^ where the bound states of the daughter functions, i|7 j ^ m < ( . (x^., s_tf) , have been denoted by l|/jmlx<r,s^) . It follows from Eq. [2.20) that, (3.3) Mi • • • Y L dfid£dx«.d.s, whence, 13.4) 2 n.n ' Sic n'c n n* It was noted i n Chapter 2 that a l l the c o e f f i c i e n t s , C be neglected except C n c . Then, 2 ..|Y,P' nc' may 13.5) 'n0c c l 2 and, from Eqs. (2.35) and (2.43), ( 3.6) \ = ^ 2 Pr-^n 0, »n? * The reduced derivative width has also been used by some authors, in p a r t i c u l a r , by Mang (1960). Thomas [1954) has defined the reduced derivative width as, (3.7) lcTc| 2 = S, 2 where S c i s the s h i f t function of Eq. (2.31). 3 - 1 : THE REACTION Po 212 In the subsequent discussion, only the p a r t i c u l a r l y simple 219 decay from the ground state of Po 6 to the ground state of 203 Pb plus an alpha-particle has been considered. Since the parent nucleus and both of the decay products have spin zero, i t i s seen that the o r b i t a l angular momentum of the alpha-208 p a r t i c l e r e l a t i v e to the Fb ° nucleus must be zero. From Eq. (3.6), the decay constant i s then simply, (3.8) \ : 2 P c 2 , where the subscript refers to the r e l a t i v e angular momentum of the channel. The r a d i a l Schroedinger Equation,. Eq. (2.14), then becomes, 2 2 l 3 ' 9 ) " I V d R * + l V R ) + V j , l R ) ) u = fr0 u f whex-e we have neglected the imaginary part of the opt i c a l model potent i a l and where we have set the r e l a t i v e o r b i t a l angular momentum equal to zero. Here V C(R) i s the e l e c t r o s t a t i c poten-t i a l as calculated from the P b 2 0 8 charge d i s t r i b u t i o n ; V^(R) i s the nuclear one-body p o t e n t i a l ; £ q i s the r e l a t i v e energy of the decay fragments. Rasmussen (1959) has stated a value of 8.81 MEV for the decay energy in the laboratory frame y i e l d -ing a r e l a t i v e decay energy of 8.98 MEV. In the present c a l c u l a t i o n , the e l e c t r o s t a t i c p o t e n t i a l has been taken as the Coulomb potential and the nuclear poten-t i a l as that derived by Igo (1959) from the e l a s t i c scattering data: (3.10) V N(R) = -1100 E X P i 1 ' 1 q > 5 ? 4 ) MEV. ( |vN(R)|£lO MEV) The nuclear potential i s somewhat uncertain. F i r s t l y , i t has been derived at higher energies (~40 MEV) and i t i s probably energy dependent. Secondly, i t w i l l be extended inward a l i t t l e beyond the range of v a l i d i t y determined from the 40 MEV scattering experiments. Thirdly, only the r e a l part of the o p t i c a l model po t e n t i a l has been used; the scat-tering data can, i n f a c t , only be f i t t e d with a f u l l o p t i c a l model p o t e n t i a l . It i s the imaginary term i n the opt i c a l model which brings i n the many-body aspects of the problem in a phenomenological way; i t s neglect in the pe n e t r a b i l i t y . i s j u s t i f i e d i f the imaginary potential i s ascribed to the nuclear i n t e r i o r . 5 - 2 : COMPARISONS WITH J.W.K.B. AMD SQUARE-WELL ESTIMATES The p e n e t r a b i l i t y and s h i f t function have been calculated from Eq. (2.31) using Eqs. (3.9) and (3.10). In heavy n u c l e i , G Q i s much greater than P Q at the nuclear surface so that Eq. (2.31) e s s e n t i a l l y reduces to, _ ( 3 . i i ) P 0 U 0 , R 0 ) = _ ° , ; s 0 ( e 0 , R 0 ) = R dR_ R=R0 G o ( R o ) G 0(R) We need therefore only evaluate the irre g u l a r function. The numerical calculation of the s h i f t function and the pen e t r a b i l i t y has been performed by evaluating the Coulomb functions at 24 fermis with an A i r y function expansion and then integrating the irre g u l a r solution of Eq. (3.9) back-wards to the nuclear surface. This c a l c u l a t i o n was performed by employing the Runga-Kutta method of order four with the use of the U.B.C. IBM 7040 computer. In Figure 1, the potentials defined in Eq. (3.9) have - 25 -"been plotted. The reduced width and the reduced derivative width have been calculated from Eq. (3.8) and (3.7), respectively; the res u l t s of the calculations are tabulated in Table 1. TABLE 1 EXPERIMENTAL REDUCED- WIDTHS ' Ro So P o r0 2 . J . ' (fermis) (e.v.) (k.e.v.) 10.5 -15.4 3.02 x 10" 1 1 24 5.9 10.0 -14.1 6.76 x 10" 1 2 111 , 22.1 9.5 -11.2 1.72 x l e " 1 2 436 55.2 9.0 - 4.0 6.71 x 10" 1 3 1119 17.8 212 -7 The li f e t i m e of Po has been taken as 3.04 x 10 sec. (Rasmussen (1959)). The square-we 11 approximation 'of the pe n e t r a b i l i t y and the s h i f t function are given by, v t ? dG D (R) I (3.12) P 0(* 0.R 0) = J S * - , I S 0 ( f 0 , R 0 ) = RdR ^o^o) where G 0 i s the irregular Coulomb function. The J.W.K.B. estimate i s given by, (3.13) P 0(^ 0,H 0) = qRQ EXP( -2 J q(R')dR« ; R, R=R, where R Q i s the.nuclear radius, r Q i s the outer c l a s s i c a l Is . ] turning point, and where q(R) « 2>;(Vu(R) +V N(R) - e0) In Figure 2, the pen e t r a b i l i t y , s h i f t function, reduced width, and reduced derivative width have, been plotted as - 26 9,0 9,5 J 0.0 10.5 I 1.0 11.5 F E R M I S FIGURE 2 p g Z 0 8 P A R A M E T E R S 4 PE I NET R A EM LITY T. P. • _ — , w ^ _ _ _ _ _ _ 1 1. A C T U A L S Q U A R E A C T U A L J . W . K , B 0 S H I F T F U N C T I O N 3 T.R R E D U C E D DERI VATI VE WIDTH \ - 28 -calculated "by each of the above methods. It would seem from Figure 2 that previous estimates of th empirical reduced widths using the 'J.V/.K.B. approximation, have not incurred serious error; in f a c t , the J.V/.K.B. approxima-t i o n i s seen to be quite good to within a tenth of a fermi of the c l a s s i c a l inner turning point. It i s also seen that the error incurred i n using the unmodified Coulomb bar r i e r is less than an order of magnitude outside of the c l a s s i c a l inner turn ing point. Bencze and Sandelescu (1966) have recently investigated the v a l i d i t y of the J.W.K.B. estimate of the p e n e t r a b i l i t y in the reaction P u 2 3 8 > U 2 3 4 f- o< . They f i n d that the J.V/.K.B. estimate i s low by a factor of from two to f i v e near the inner turning point, in agreement with the results pre-sented i n this chapter. In f a c t , they claim a much deeper one-body potential than i s customarily believed (-^  231 MEV) and, in this manner, obtain a further increase i n the pene-t r a b i l i t i e s . In Chapter 4, we show t h a t , . i f the nuclear sur-face i s treated i n a d i r e c t manner, i t i s not necessary to re-sort to such extreme well depths to obtain a reasonable agree-ment with the empirical decay rates. - 29 -CHATTER 4 INDEPENDENT-PARTICIE' MODEL DECAY RATES OF P o 2 1 2 In the present chapter, we w i l l examine the e f f e c t of the diffuse nuclear edge upon the calculated alpha-particle decay rates of heavy n u c l e i . In p a r t i c u l a r , we w i l l re-examine the square-well c a l c u l a t i o n of the alpha-particle decay rate of P o 2 1 2 performed "by Harada (1961). With the inclusion of configuration mixing, Harada has obtained reasonable agree-ment with the empirical decay rates, but has had to resort to larger r a d i i than those believed to be t y p i c a l of a heavy nucleus. It i s our contention that his c a l c u l a t i o n corre-sponds to a c a l c u l a t i o n with the equivalent square-edge nucleus model of Vogt; i n f a c t , we w i l l show that the radius involved i n this model i s much larger than that of the conventional diffuse-edge nucleus to which i t corresponds. V/e w i l l also repeat the c a l c u l a t i o n of Harada, introducing the d i f f u s e -edge into the c a l c u l a t i o n in a d i r e c t manner. The e f f e c t of the shape and surface of the nucleus upon the decay rates i s , primarily, a one-body e f f e c t . By this we mean that i n Eq. (2.37), (2.37) "X = f S c , the one-body decay constants, "X ' * , are quite sensitive to the size and to the nature of the surface of the nucleus whereas the spectroscopic factors, S c, are rather insensitive to these c h a r a c t e r i s t i c s . • It i s our contention that previous calculations have calculated the spectroscopic factors (and, hence, the many-body effects) correctly, but have misinter-preted the radius involved i n the calculation of the one-body - 30 -decay constant. It has "been customary in previous calculations to c a l -culate the alpha-particle decay rates from Eq. (3.6): ( 3 . 6 ) } = £ 2 r c f c S 0 t R 0 ) y i a e f t c' • The p e n e t r a b i l i t y has, 'generally, been calculated for a d i f -fuse-edge b a r r i e r , as has been discussed i n Chapter 3. On the other hand, the nuclear reduced width has, generally, been calculated by using the resonant one-body wave function of a square-edge well i n the following sense: one sets the ampli-tude of that mode of the nuclear wave function which describes the r e l a t i v e motion of the decay products equal to the ampli-tude of the resonant one-body wave function at the nuclear radius. It i s i n the sense of the l a t t e r procedure that the nucleus i s taken to be square. We have noted in Chapter 3 that these procedures for c a l c u l a t i n g the decay rates cor-respond to a c a l c u l a t i o n with the equivalent square-edge nucleus of Vogt (1962). It i s of h e u r i s t i c value to consider the freedom with which an alpha-particle can move within the nucleus. To see how far an alpha-particle can tr a v e l i n the nuclear volume before being absorbed, consider a square-well potential hav-ing t y p i c a l r e a l and imaginary parts, V Q and WQ, such that V o > > W o ' ^ o . Assuming the Coulomb effects to be incorporated in-V Q, the regular solution of Eq. (3.9) i s of the form, U ( R ) = sin K R , where, - 31 -J ~ T 2 J V 0 6 0 . (k = J^±L-~ 1.3 ) H For a t y p i c a l heavy nucleus., V Q may he taken to be about 100 MEV and WQ to be from 5 MEV to 10. MEV. The mean free path of an alpha-particle within the nuclear volume, (4.1) I n ~ / 3 S Z P _ N kW0 then ranges from about f i v e to about two fermis (respectively). There is some evidence that the imaginary potential i s large only in the surface region, so that the alpha-particle might have even greater freedom within the nuclear volume; i n gen-e r a l , V/ i s then larger i n the surface region so that here the alpha-particle has le s s freedom. Thus, to the extent that the-one-body model i s v a l i d , the alpha-particle moves rather f r e e l y within, the nucleus except i n the v i c i n i t y of the nuclear surface. I t can e a s i l y be shown from the J.V/.K.B. approximation that the mean free path of an alpha-particle in the bar r i e r region i s about, where V i s the height of the b a r r i e r . If we attribute the many-body aspects of the problem to the nuclear i n t e r i o r , we may take the imaginary potential (which accounts for the many-body aspects of the problem) as vanishing outside the nuclear radius, This i§ the assumption ma.d§ i n Chapter 3 , where we have attributed a l l the penetration effects of the barrier to the p e n e t r a b i l i t i e s , and i s probably extreme. With this assumption, i t i s seen from Figure 1 and Eq. (4 .2) that the mean free path i n the harrier and near the surface for a square-edge well i s considerably less than that for the d i f -fuse-edge well. It i s i n this sense that we expect a d i f f u s e - • edge well to behave l i k e a square-edge well of larger radius. In the discussion to follow, this e f f e c t i s developed from the following point of view. The e f f e c t of the d i f f u s e -edge on the one-body resonant wave functions i s examined and i t i s found that the diffuse-edge enhances the resonant wave function at and beyond the nuclear radius. Hence, the one-body reduced widths are enhanced in a similar manner. From Eq. (2.40), one would expect to observe this enhancement i n the e l a s t i c scattering cross-sections (the width of the elas-t i c - scattering cross-section i s proportional to the decay constant). We have demonstrated this i n a d i r e c t manner by calc u l a t i n g the d i f f e r e n t i a l e l a s t i c scattering cross-section of P b 2 0 8 at 9 0 ° C M . We have included the many-body ef f e c t s by repeating the independent-particle model ca l c u l a t i o n of the decay constant of P o 2 ^ 2 performed by Harada (1961) with the diffuse-edge of the nucleus.now being taken into account i n a d i r e c t manner. 4 - 1 : ' ONE-BODY REDUCED WIDTHS AND EXPERIMENTAL SPECTROSCOPIC  FACTORS In the x3^esent section, we w i l l examine the effects of diffuse-edge of the nucleus upon the one-body reduced widths. We a l s o p r o v i a a an est imate of the c o r r e c t i o n i n the e x p e r i -mental spectroscopic factors due to the diffuse-edge. The reduced mass in Eq. (3.9) has the value 5.9031 x 10~ 2 4 grams. - 33 -The e l e c t r o s t a t i c p o t e n t i a l , V C(R), has heen calculated from the P b 2 0 8 charge d i s t r i b u t i o n of H i l l and Ford (1955) : (4.3) Cl - £ EXP(R/6.7 - 10) i f R^-6.7 f . J>(R) • / \-k EXP(10 - R/6.7) i f R>6.7 f. . Then, . • V R (4.4) V C(R) = 82 x 2 x e 2 x £ Jj> (R')R« 2dR' + • •• RQ + J .P (R' )R'dR' R f °J:> (R* )R , 2dR* -1 The nuclear potentials, V N(R) , have been chosen to be of the Saxon-Woods form, (4.5) V N(R) = V Q x [ l + EXP( (R - r Q ) / a )] - 1 . In Figure 3, the potentials have been plotted corresond-ing to a nuclear Saxon-Woods shape of thickness (a) 0.5 fermis, radius (r ) 9.0 fermis, and depth (V Q) 105 MEV; we have also plotted a square-well of the same radius and of a similar depth. The depths of the wells were chosen so that the regu-l a r solutions of Eq. (3.9) were resonant and had the same number of nodes; these resonant v/ave functions have been p l o t -ted in Figure 4. The analysis of decay rates and scattering cross-sections assigns a l l of the effects of the i n t e r i o r of the nucleus to the reducd widths, these being evaluated in the nuclear sur-face. It i s exactly i n this sense that the diffuse-edge well can be replaced by a square-edge well of greater radius. If the diffuse-edge well i s replaced by that square-edge well having a radius such that t h e i r reduced widths are equal, the wells are indistinguishable v/ithin the nuclear volume; the difference i n the r e f l e c t i o n of the wells outside of the - 34 -FIGURE 3 F E R M I S - 36 -nuclear volume i s accounted for by a t y p i c a l l y small r e f l e c -tion f a c t o r . In f a c t , this i s e s s e n t i a l l y the equivalent square-edge nucleus model of Vogt (1962). It i s seen from Figure 4 that the radius of this "equiv-alent square-well" should be s i g n i f i c a n t l y larger than the radius of the diffuse-edge w e l l . This i s the basis of our contention that a diffuse-edge well should behave l i k e a larger square-edge well i n the analysis of decay rates. To exhibit t h i s c l e a r l y , we w i l l calculate the one-body reduced widths.with.a diffuse-edge well and compare them with the square-edge well estimate of Harada (1961). A single nucleon i n a heavy nucleus moves i n a p o t e n t i a l having a depth of about 50 MEV. It would, therefore, seem that reasonable depths for the potential well should l i e be-tween 50 MEV and 200 MEV, values of 100-150 MEV currently being fashionable. I f i s also seen from Eq. (3.10) that the Igo potential i s known inwards only to about 9.7 fermis so that the shape of the well edge may be chosen rather a r b i t r a r -i l y within this radius. I t , therefore, seemed reasonable to choose the potentials of the type of Eq. (4.5) in the f o l -lowing manner. The thickness of the surface, a, was chosen to have the values tabulated in Table 2.. The depth, V Q, and the radius, R 0, were then varied so that the potential was simultaneously equal to the Igo potential at 10 fermis and so that the regu-l a r solution of Eq. (3.9) was asymptotic to the i r r e g u l a r Cou-lomb function the condition for resonance. (There is - 37 -only one such potential for a given number of nodes.) By the discussion of Chapter 2, one of these potentials should be a good representation of the actual p o t e n t i a l . Exactly which one i s the actual potential cannot be decided u n t i l some further c r i t e r i o n i s established for deciding the well depths more p r e c i s e l y . Neither fundamental nuclear theory not the analysis of alpha-particle scattering data defines the well depth very c l e a r l y within the range 50 MEV to 150 MEV. Examples of the r e s u l t i n g potentials are summarized i n Table 2. The behaviour of the resonant wave functions as a function of the well depth has been i l l u s t r a t e d i n Figure 5 and their behaviour as a function of surface thickness i n Figure 6. TABLE 2 ONE-BODY POTENTIALS POTENTIAL V N NODES . n o THICKNESS a (fermis) DEPTH Vo (MEV) RADIUS (fermis) A 4 0.5 - 49.1 8.94 B 6 0.5 - 64.5 8.79 C 8 0.5 - 86.3 8.63 D 10 0.5 -114.5 8.48 E 12 0.5 -149.6 8.34 F 10 0.3 -103.4 9.12 G 10 0.7 -128.9 7.79 The one-body reduced widths have been calculated from the resonant wave functions (Eq. (2.33)) and have been used - 38 -to calculate the experimental spectroscopic factors (Eq. (3.5) and Table 1). TABLE 3 EXPKRIMENTAL S PEG TR OS COPIC FACTORS POTENTIAL EVALUATED AT 9.0 f. EVALUATED AT 10.0 f. ^ 2 "o (kev) exp (kev) C 2 n 0 7f 2 n 0 (kev) exp (kev) C 2 no A 105 1.12 1.07xl0" 2 11.5 0.111 0.98xl0" 2 B ••" 150 1.12 0.75 " 16.9 0.111 0.66 " C 188 1.12 0.59 21.8 0.111 0.51 D 223 1.12 0.50 26.3 0.111 0 .42 E 256 1.12 0.44 30.5 0.111 0.36 F 101 1.12 1.11 42.4 0 .111 0.26 G 170 1.12 0.65 15.5 0 .111 0.71 If the potentials chosen i n evaluating the experimental reduced widths and the one-body reduced widths had been the same, the experimental spectroscopic factors would be inde-pendent of the radius. The r a d i a l dependence has been i n -cluded to indicate the effect of the nature of the surface chosen and, for reasonable surface thicknesses of 0.5 - 0.7-fermis, this i s seen to be s l i g h t . Harada, using a square-edge well of radius 10 fermis, has obtained a value of 0.143 for the experimental reduced -2 width and a value of 0.08 x 10 f o r the experimental spectro-scopic factor. It would appear from Table 3 that the spectro-scopic factor might be about an order of magnitude larger than this value. - 39 -.0| F I G U R E 5 EFFECT OF WELL DEPTH 0 . 5 h E - 4 9 A -150 CMEV)-R r0 i.o o 6 FERMIS 8 10 12 l.Oi - 40 -F I G U R E . 6 EFFECT OF SURFACE "THICKNESS F 0 . 3 F, G 0 . 5 F. •o . 0 "0 4 6 FERMIS 8 10 - 41 -4 - 2 : ONE-BODY DIFFERENTIAL ELASTIC SCATTERING GROSS-SECTION OF P b 2 0 8 FOR AN ALPHA-PARTICLE AT 9 0 ° C M One of the most d i r e c t ways i n which to i l l u s t r a t e the effects, of the diffuse-edge on the decay rates i s to calculate i t s e ffect on the one-body e l a s t i c scattering cross-sections. (As has previously been noted, the one-body width i s propor-t i o n a l to the one-body decay constant.) In this section, the one-body d i f f e r e n t i a l e l a s t i c scattering cross-section for the scattering of an alpha-particle from the ground state of P b 2 ° 8 w i l l be calculated by d i r e c t and well-known procedures. What one does i s to study the behaviour of the logarithmic deriv-ative of. the wave function about the resonance; the one-body width can then be calculated in a straightforward manner. The procedure for c a l c u l a t i n g the cross-sections by these tech-niques i s discussed i n Appendix C. It i s worth noting that the c a l c u l a t i o n of the l o g a r i t h -mic derivative i s very tedious due to the very small widths involved; i n f a c t , the calculations performed in this thesis required seventeen place accuracy i n the potentials and re-quired many hours of computer time on the IBM 7040 computer used i n these calc u l a t i o n s . For.this reason, such d i r e c t calculations have previously been avoided by other authors. The P b 2 0 8 ( °C , ) P b 2 0 8 d i f f e r e n t i a l e l a s t i c scattering cross-section at 9 0 ° q j has been plotted i n Figure 7, both for a square-well and for a diffuse-edge well of the same radius. It i s seen that the diffuse-edge enhances the elas-t i c scattering cross-section (and, hence, the one-body decay - 42 -0.2 4 — FIGURE 7 O N E - B O D Y S C A T T E R I N G C R O S S - S E C T I O N PB 2 0 8 -0.04 6 8 I O " L 3 M E V o - 43 -constant) "by about an order of magnitude. The diffuse-edge well has here been taken to be of the Saxon-Woods type, Eq. (4.5), with the parameters, V 0 - - 48.9 MEV; r Q = 9.0 f . ; a = 0.5 f . ; the parameters of the square-well are taken to be, V s - - 47.8 MEV; r s » 9.0 f. . The depths of the wells have been chosen so that the wave functions are resonant at 8.9795 MEV and have the same number of nodes. The cross-sections have been plotted by studying the behaviour of the logarithmic derivative of the wave functions at 24.0 fermis. A width of 1.37 x 10" 1 3 MEV was found for the d i f f u s e -edge well and a width of 1.94 MEV for the square-edge well; hence, the enhancement i n the cross-section (or decay con-stant) due to the diffuse-edge i s a factor of 7.1. 4 - 5 : HARADA1S FORMULA FOR INDEPENDENT-PARTICLE MODEL REDUCED . WIDTHS A convenient technique for evaluating independent-parti-cle model reduced widths has been developed by Harada (1961). We provide here only an outline of h i s r e s u l t for the simple case of P o 2 1 2 ; a more complete discussion i s given i n Appendix A. It w i l l be assumed that the zeroth order approximation of Eq. (2,25) can be described by the ground state configu-r a t i o n of the independent-particle model. Employing the par-t i a l antisymmetrization scheme.of Chapter 2, and neglecting core excitations, the parent nucleus wave function can be - 44 -written in the form, (4.6) IP0(12...A) » i(l234) lpo 0(56...A) , v/here IJ^ QQ i s the wave function of the doubly magic P b 2 0 8 core. (The notation i s that of Chapter 2.) Prom Eq. (3.3), and noting that the parent nucleus, daughter nucleus, and alpha-particle have spin zero, (4.7) If = f f(!234) X(1234) Y ° ( ^ ) did*, . Harada has chosen the alpha-particle wave function to be of.the Gaussian type: (4.8, : X ( 1 2 3 4 ) . ( i | ^ ) 3 / a K P ( - | ( | ! + i f + ?!) ) ... (4TJ-3/2 - J ( 0 ( 1 2 ) X°(34); . where >CQ denotes the spin s i n g l e t function. Here, has . been written i n terms of i t s r a d i a l and angular components, fj_ and respectively.'^ Harada chooses the p a r a m e t e r s o that the r.m.s. radius of the charge density is equal to the, measured value (1.6 fermis); he finds a value of 0.44 fermis** 2 f o r Jb . In Appendix A, i t i s shown that Eq. (4.7) can be written in the form, / h 2R r ' " (4.9) XQ - .JWj [ T ( 0 0 ; J l O l ) ^(00; j 3 j 3 ) j • • • 7 « v ' s n ^ n o ^ l ^ l ^ 0 ^ ! 1 ^ ! 1 ^ ^ •^'^l'$2'^3'—* ' • n 3 ... <N 2On 20;0lV 3l 3V 3l 3;Q> </ 1 T 0 n 5 0 ; 0] ^ 0 3 ^ 0 ; 0 > ••• . ^ 8 ( l 2 ) ^ C ° ( 3 4 ) ] d £ l ^ ^ ^ | d ^ 2 d ^ 3 d s ^ . Here i t has been assumed that the one-nucleon wave•functions if The d e f i n i t i o n of \j_ d i f f e r s from Harada's. ( c . f . Appendix A) . - 45 -fo r the two protons {neutrons) i n the u n f i l l e d s h e l l are de-scribed by the quantum numbers ("LAIj-.m-. ) ( ( i C l j 7m ) ), o o o o where 1^ • ( V 3 ) f ± 1 ( l 3 ) , ( j 3 ) , and m1 (m3) denote the pr i n -c i p a l quantum number, the o r b i t a l angular momentum, the t o t a l angular momentum, and the magnetic quantum numbers of the one-proton (one-neutron) o r b i t a l s . In Eq. (4.9), the f i r s t brack-eted term contains the T c o e f f i c i e n t s of Rose' (1957) for trans-forming the j - j coupling scheme to an L - S coupling scheme; the second bracketed term expands the wave function in terms of the r e l a t i v e co-ordinates; the f i n a l bracketed term con-tains the spin functions. A s t a t i s t i c a l constant and a dou-ble parentage c o e f f i c i e n t , both of which are unity, have been omitted from the discussion. ^n^o ( 1 s 1»2,3 ) and are one-nucleon o r b i t a l s of zero angular momenta with pr i n -c i p a l quantum numbers n i ( i = 1,2,3 ) and N, respectively. The c o e f f i c i e n t s appearing in the summation of the second bracketed term are the Talmi c o e f f i c i e n t s (Talmi (1952.)); a formula for the Talmi c o e f f i c i e n t s incurred in the transfor-mation of harmonic oscillator.wave functions is derived i n Appendix B. The r a d i a l harmonic o s c i l l a t o r wave function i s of the form, where, . • L ^ M h r 2 ) = gQ\ n-h ) h'. j ' b being the harmonic o s c i l l a t o r size parameter. It w i l l be convenient to write Eq. (4.7) in the form, (4.11) l f c = j p o f ^ ( R o ) . - 4 6 -2T" 2 ^ , 2 ^ and to assume that the one-nucleon wave functions of the inde-pendent-particle model and the functions l ^ n i o ( * = 1 » 2 , 3 ) and IJ/^Q can he approximated by harmonic o s c i l l a t o r wave func-ti o n s . The overlap i n t e g r a l , ( 4 . 1 2 ) ' (RQ). = | ^ ( R 0 ) , is found by performing the integrations; the r e s u l t i s , ( 4 . 1 3 ) C ^ j ( R 0 ) A '^(OOJJ-LJ-L) T ( 0 0 ; j 3 j 3 ) n ^ ^N0n 30;0lK 10H 20;0> • ' • ^ 0 n l 0 ; 0 ) 1 ^  1 x ; 0> ^ N 20n 2 0 ; 0 1 ^ 1 ^ 1 ; 0> ( i ! ) - 3 / 2 ( n 1 + i ) I ( n 2 + i ) I(n 3^-) f n i'.n 2:n 3I * l2JTb \ 9/2-I ^  ^ b The r a d i a l functions, K0» define the center-of-mass dependence of the system ( 1 2 3 4 ) . If most of the contribution comes from a p r i n c i p a l node, N , one would expect the p r i n c i p a l function, (jy^Q , to be similar to the resonant solution of Eq. ( 3 . 9 ) having the same number of nodes. Since the only-free parameter of the harmonic o s c i l l a t o r wave function i s the size parameter, b, i t s choice must be somewhat a r b i t r a r y . Since only the amplitudes enter the determination of the reduced width, i t might seem reasonable to choose the amplitude of the p r i n c i p a l function at the nuclear radius to be equal to the corresponding amplitude of the one-body wave function as selec-ted from Table 2 . . Of course, such a solution w i l l , in gen-e r a l , neither y i e l d the correct energy nor s a t i s f y the boundary - 47 -c o n d i t i o n , Eq. (2.30). I t i s not c l e a r that connecting the independent-r^article model c a l c u l a t i o n with the one-body problem i n the above man-ner introduces the surface and the s i z e e f f e c t s of the one-body problem i n t o the c a l c u l a t i o n c o r r e c t l y . I t i s seen from Eq. (3.5) that the spectroscopic f a c t o r s should be r a t h e r i n -s e n s i t i v e to the surface and the s i z e of the nucleus so that these features should a f f e c t only the one-body aspects of the c a l c u l a t i o n ; i n f a c t , i n t h i s c a l c u l a t i o n , the spectroscopic f a c t o r s are connected to the one-body problem only through the choice of the s i z e parameter. Hence, i f the use of the har-monic o s c i l l a t o r one-nucleon f u n c t i o n s i s reasonable, the c r i -t e r i o n f o r the v a l i d i t y of the above procedure becomes that the spectroscopic f a c t o r s must not depend s e n s i t i v e l y upon the choice of the s i z e parameter. 4 - 4 : NUMERICAL INDEPENDENT-PARTICLE MODEL REDUCED WIDTHS In the present s e c t i o n , we w i l l c a l c u l a t e the independent-p 1 p p a r t i c l e model reduced width of Po by the technique of Harada, but employing the one-body resonant wave f u n c t i o n s of a diffuse-edge w e l l r a t h e r than those of a square-edge w e l l . In h i s c a l c u l a t i o n , Harada has assumed a square-edge w e l l of r a d i u s ten f e r m i s ; w i t h the i n t r o d u c t i o n of c o n f i g u r a t i o n mix-ing i n the parent nucleus wave f u n c t i o n , he obtains a reason-able value of about one-twentieth the e m p i r i c a l decay constant. A l a r g e r r a d i u s would produce a l a r g e r decay constant but, from Figure 1, i t would appear that over ten fermis i s too l a r g e a nuclear r a d i u s . In f a c t , as w i l l be shown i n the sub-- 43 -sequent discussion, this large square-edge well corresponds to a smaller diffuse-edge w e l l . Harada has chosen the size parameter so that the p r i n -c i p a l function should have an amplitude at the nuclear radius (ten fermis) equal to that of a square-v/ell resonant wave function; the square-v/ell has been chosen so that this wave function has the same number of nodes as the p r i n c i p a l func-tion and s a t i s f i e s the boundary condition, Eq. (2.30), exactly. The only aspect i n which the present calculation d i f f e r s from Harada 1s i s that the size parameter has been chosen so that the amplitude of the p r i n c i p a l function i s equal to the ampli-tude of the corresponding resonant function of a diffuse-edge well as selected from Table 2. This procedure i s intended to incorporate the properties of the diffuse-edge into the calcu-lated reduced widths i n a more d i r e c t way. It makes much clearer the role of the poten t i a l and the value of the radius of the resonant state. The contributions to the overlap i n t e g r a l from each node, OJJ(RQ). a^e given i n Table 4. Here, the p r i n c i p a l function has been chosen to have ten nodes and the results are tabu-lated for various nuclear sizes and surface thicknesses. The rates of contribution were found to be rather insensitive to the harmonic o s c i l l a t o r size parameter. The calculated values of the'reduced widths are tabulated in Table 5. A comparison v/ith the experimental values (Table l ) and, hence, of the decay rates, has also been included in the tabulation. - 49 -TABLE 4  OVERLAP INTEGRALS a) Surface Thickness: 0.5 fermis ( f 7 3 / 2 ) R 0 = 9.0 f . //b - 0.116 f 7 2 R 0 - 9.5 f . b = 0.095 f 7 2 R 0 » 10.0 f . b = 0.116 f r 2 5.68x10"8 2.73xl0**7 1.7 6x10 " 9 4.43xl0" 8 1.67xl0" 7 1.83xl0" 9 4.89xl0" 7 ' 1.43xl0*"6 2.75xl0" 8 7. 98x10" 6 1.75xl0" 5 6.35xl0" 7 6.06x10"° 9.62xl0~ 5 7.18xl0" 6 1.31xl0" 4 1.39x10"4 • 2.46x10"5 0~io 9 . 6 l x l 0 " 5 5 . 4 l x l 0 " 5 3.32x10'° O i l 1.59xl0" 5 1.08xl0~ 6 • 1.42xl0" 5 # This was the largest possible amplitude and i s slightly-l e s s than the resonant amplitude. b) Surface Thickness: 0.7 fermis ( f r V 2 ) R Q ='. 9.Of. b = 0.110 f 7 2 R 0 =.9.5 f . - . b = 0.092 f 7 2 RQ: =.10.0 f . b = 0.120 f r 2 O a 1.26xl0" 7 3.93xl0 - 7 9.27xl0" 1 0 8.84xl0" 8 2.2.8xl0-7. l . O l x l O " 9 8.75x10"7 1.84X10.**6 1.62xl0" 8 1.27xl0~ 5 2.10xl0" 5 3.99xl0" 7 8.36xl0" 5 1.06xl0" 4 4.82xl0~ 6 #9 1.51x1b"4 1.36xl0" 4 1.78xl0" 5 <?io 8.52x10"5 4.05xl0 - 5; 2.61xl0" 5 Oil 6.81xl0~ 6 4 .49xl0" 6 1.23x10"5 - 50 -TABLE 5 COMPARISON OF DECAY RATES Size of Nucleus Ro (f.) a (f.) ^£heor (ev) u exp (kev) ~X theor A exp ~8.5 f . 9.0 0.5 5.20 1.119 1/220 9.5 0.5 5.19 10.0 0.5 0.36 0.111 1/310 ~8 #0 9.0 0.7 6.16 1.119 1/180 9.5 0.7 5.18 10.0 0.7 0.22. 0.111 1/500 //Harada: ~10.0 10.0 0.0 1.3 0.143 1/110 # (Harada (1961)) We have previously noted that the. spectroscopic factors should be rather insensitive to the size and the surface of the nucleus. In Table 6, the spectroscopic factors found in the present c a l c u l a t i o n have been compared with those of Har-ada. TABLE 6 CALCULATED SPECTROSCOPIC FACTORS Size of Nucleus Surface Thickness Evaluated at 9.0 f . c 2 " O Evaluated at 10.0 1 . C 2  no ~8.5 f . 0.5 f . 2.4x10"5 1.3xl0" 5 -8.0 f . 0.7' f. 3.6x10"5 1.4xl0" 5 #Harada: ~10.0 f . 0.0 f . 0.9xl0~ 5 # (Harada (1961)) - 51 -I t Is' seen from Table 5 t h a t , by t r e a t i n g the one-body aspects of the problem wi t h a diffuse-edge w e l l , an agreement w i t h the e m p i r i c a l decay rates can be obtained that i s as good as that found by Harada employing an anomalously la r g e r a d i u s ; i n f a c t , the values suggested f o r the nuclear s i z e s i n Table 5 are rather modest s i z e s f o r heavy n u c l e i . Harada has found that the i n t r o d u c t i o n of c o n f i g u r a t i o n mixing increases the c a l c u l a t e d decay r a t e s by a f a c t o r of between f i v e and ten. I t should be p o s s i b l e to remove much of the remaining d i s c r e p -ancy -by choosing a l a r g e r , but s t i l l reasonable, s i z e f o r the nucleus. - 52 -CHAPTER 5 THE EQUIVALENT SQ.UARE-EDGE NUCLEUS MODEL In the present chapter, we w i l l d e f i n e , f o r a conven-t i o n a l ' diffuse-edge nucleus, a square-edge nucleus which, f o r the purposes of studying decay r a t e s , s c a t t e r i n g data, and absorption processes, e x h i b i t s many of the p r o p e r t i e s of the diffuse-edge nucleus to which i t corresponds. In f a c t , the "equivalent square-edge nucleus" which v/e w i l l define i s , es-s e n t i a l l y , that which has been used by previous authors, i n p a r t i c u l a r Harada (1961), i n the c a l c u l a t i o n of the alpha-p a r t i c l e decay r a t e s of' heavy n u c l e i . The usefulness of r e p l a c i n g a diffuse-edge nucleus with, an " e q u i v a l e n t " square-edge nucleus has been noted p r e v i o u s l y by Vogt (1962); he has defined and employed the "equivalent square-edge nucleus" i n the a n a l y s i s of the s c a t t e r i n g and a b s o r p t i o n of neutrons. In p a r t i c u l a r , he has found that the parameters' of t h i s model are i n s e n s i t i v e to the i n c i d e n t energy and channel and a l s o to the many-body aspects of the nuclear problem; hence, the equivalent square-edge nucleus i s defined i n a reasonably unique manner f o r a given diffuse-edge nucleus. Vogt, Michaud, and Reeves (1965) have found s i m i l a r r e s u l t s f o r a l p h a - p a r t i c l e s c a t t e r i n g from l i g h t n u c l e i . The u s e f u l -ness of the equivalent square-edge nucleus i s then two-fold: a) once having determined the parameters of the model, i t a l -lows one to e x p l o i t the simple a n a l y t i c p r o p e r t i e s of square-w e l l s i n subsequent c a l c u l a t i o n s ; b) i t provides a convenient set of parameters f o r studying the e f f e c t of the d i f f u s e nu-c l e a r edge on decay r a t e s , s c a t t e r i n g c r o s s - s e c t i o n s , and'ab-- 53 -s o r p t i o n c r o s s - s e c t i o n s . This i s p a r t i c u l a r l y h e l p f u l i n the study of a b s o r p t i o n c r o s s - s e c t i o n s at the low energies of i n -t e r e s t to a s t r o p h y s i c s where the c a l c u l a t i o n s are otherwise quite t e d i o u s . In the d i s c u s s i o n to f o l l o w , we w i l l provide the d e f i n i -t i o n of the equivalent square-edge nucleus model and w i l l check the extent to which i t a p p l i e s to a l p h a - p a r t i c l e s c a t -t e r i n g and a b s o r p t i o n i n heavy n u c l e i . 5 - 1 : THE EQUIVALENT SQ.UAIiE-EDGE NUCLEUS MODEL The equivalent square-edge nucleus of a diffuse-edge nucleus i s defined i n the f o l l o w i n g manner. The r a d i u s and depth of a r e a l square-well p o t e n t i a l are chosen so that the f o l l o w i n g c o n d i t i o n s are s a t i s f i e d : a) i t e x h i b i t s a r e s o -nant wave f u n c t i o n i n the channel, c, at the resonance energy, £c, appropriate to the decay, s c a t t e r i n g , or absorption prob-lem of i n t e r e s t ; b)the resonant wave f u n c t i o n of the square -w e l l has the same number of nodes as that of the correspond-ing resonant f u n c t i o n of the d i f f u s e - e d g e . w e l l which describes the average i n t e r a c t i o n between the i n c i d e n t p a r t i c l e and the t a r g e t nucleus i n the channel, c, and at the resonance energy, £c; c) the reduced widths of the two w e l l s , evaluated at the r a d i u s of the square-well, are equal i n t h i s channel and at t h i s energy. One defines the " r e f l e c t i o n f a c t o r " , which ac-counts f o r the anomalous r e f l e c t i o n of the square-well, as the r a t i o of the p e n e t r a b i l i t y of the diffuse-edge w e l l to that of the square-well at the square-well r a d i u s . I f the . many-body aspects of the nuclear problem have been accounted - 54 -f o r by choosing the diffuse-edge w e l l to be an o p t i c a l model p o t e n t i a l , the imaginary term i n the square-well p o t e n t i a l i s chosen to have the radius of the square-we 11 and the depth of the diffuse-edge w e l l . The e l e c t r o s t a t i c p o t e n t i a l i s chosen to be the same f o r b o t h ' w e l l s . The square-well defined i n t h i s manner i s c a l l e d the "equivalent square-well" (ESW). In g e n e r a l , the parameters•of the ESW are r a t h e r insen-s i t i v e to the channel and the energy. Since the ESW and the corresponding one-body diffuse-edge v/ell have the same pen-e t r a b i l i t i e s and reduced widths, they are interchangeable i n the one-body' problem. I t i s seen from Eq. (3.5) that the spectroscopic f a c t o r s should be rather i n s e n s i t i v e to the nu-c l e a r s i z e and surface since the nuclear reduced width and the one-body reduced width depend upon these aspects of the nucleus i n a s i m i l a r manner; i n f a c t , we have checked t h i s a s s e r t i o n in"Table 6. Thus the spectroscopic f a c t o r s , which account f o r the many-body aspects of the nuclear problem, should be about the same whether the nucleus i s chosen to have the s i z e and surface of the diffuse-edge v/el l or of the corresponding ESW. Thus we can i n a meaningful manner r e -place the conventional diffuse-edge nucleus w i t h the equiv-a l e n t square-edge nucleus f o r the purposes of a n a l y z i n g decay r a t e , s c a t t e r i n g , and absorption data. 5 - 2 : APPLICATIONS TO HEAVY NUCLEI In the present s e c t i o n , we w i l l examine the v a l i d i t y of the equivalent square-edge nucleus model i n heavy n u c l e i by c o n s i d e r i n g the a b s o r p t i o n of a l p h a - p a r t i c l e s . The technique - 00 -employed i s to evaluate the ESW f o r a fashionable o p t i c a l model p o t e n t i a l and then to study the dependence of the r e -f l e c t i o n f a c t o r upon the parameters of the problem by con-s i d e r i n g the transmission f u n c t i o n s of the two one-body poten-t i a l . The one-body r a d i a l Schroedinger Equation, w i t h an o p t i c a l model p o t e n t i a l , i s of the form, ( 5 . 1 ) : "W + ( V C ^ ) + % ( K ) + i % ( R ) + | ^ ^ J . ) = £ c u c D e f i n i n g the incoming wave i n the channel, c, by, ( 5 . 2 ) : I C ( R ) = e- i Wc ( G c t - i f c ) , one has, at s u f f i c i e n t l y large r a d i u s , that ( 5 . 3 ) u»(r) { r - I j T y = G C ( R ) + i D c ( R ) = Il.(R) - e^c* i*c) i t ( R ) I C ( R ) - e 2 i ( - c + i J c ) I C ( R ) where C c and D C are the r e a l and imaginary parts of the l o g -a r i t h m i c d e r i v a t i v e of u c and where <*c+- i$c i s the phase s h i f t , In g e n e r a l , J> i s very small v/hence i t i s e a s i l y shown from Eq. ( 5 . 3 ) t h a t , ( 5 . 4 ) •' # - 4 D C  T c ? 4 ^ c = ( c C G C - G C - D C F C ) 2 + (F« - C C E C - D C G C ) R = v/here T i s the o p t i c a l model transmission f u n c t i o n and where R m i s s u f f i c i e n t l y large that there i s no f u r t h e r absorption due to the o p t i c a l model p o t e n t i a l . I f there i s no absorption due to the o p t i c a l model poten-t i a l beyond the ESW r a d i u s , one can d e f i n e , - 56 -•to (5.5) a) P i e s - H k w D 0 . > , ( ^  = j ^ -b) s l c = -•^••lic0, * and write the optipal model transmission function i n the form (Preston (1962)), (5.6) 4P C MR0 f° 2 T n s Tq _ o. (•*>_ . iT>. \2 J W K ( R ) (S c - S i c ) 2 + (P C + i P i c ) 2 h 2 J u c(R) u c ( R 0 J dR. Assuming u c to he normalized in R 0, and noting that the sur-face, thickness i s small, one can write Eq. (5.6) i n the form, (5.7) 4Pc W0 c ' (s c - s i p) a+ (PC-+ i P i J 5 4 . f T i ^ y ••' where, (5.8) f c ( R Q ) = ]j£ •• i n " \ \ 2 u c ( R 0 ) 0 Choosing the nuclear radius, R Q to he the E S W radius, and not-ing that, at resonance, f ( R J J S W ^ i s t h e one-hody reduced width, one might expect i n general that, (5.9) fESw(%Sw) ^ 1. . f d i f f (RESW) 2 Numerical calculations show that the denominator, (S c - S i c ) + (P c +• P i c ) 2 , i s independent of the i n t e r n a l features of the well to within a few percent. Thus, (5.10) T c d i f f 't P c d i f f . T CESW * PcESW It w i l l he convenient to regard the transmission coef-f i c i e n t , T c, of Eq. (5.4), as a function of the radius of evaluation, R M . To study the ef f e c t s of the imaginary poten-t i a l within the b a r r i e r , one need then only consider the r a t i o of the transmission function of the diffuse-edge well to that - 57 -of the ESW. Within the b a r r i e r , G o > ^ F o . . D < < 1 s o that, (5.11) 'g(R) = T d i f f ( R ) Pdiff(R) # TESW^R' P E S W ^ ^ T One may then consider that the absorption due to the di f f u s e imaginary pote n t i a l ceases where g(R) becomes asymptotic to the r e f l e c t i o n f a c t o r , f . From Eq. (5.1), (5.12) a) dClRl I k 2 ( *c + % . l } _ c 2 ( r ) _ D 2 ( R ) f b) dD(Rl z k 2 % _ 2C(R).D(R) . ( L = 0 ) • d R . , . -The solution of Eq. (5.12) b) i n the b a r r i e r i s of the form, (5.13) D ( R E % _ _ conatant-e - ^ / Z r^-R , fr • -since, i n the b a r r i e r , P i s very small and '(J(R)'N-'k^p. -If V c/e i s s u f f i c i e n t l y large, the f i r s t term dominates the second so that the absorption from the o p t i c a l model poten-t i a l exceeds that from the b a r r i e r . The numerical example considered was that of s-wave scat-O A O terin-g. of an alpha-particle from a Pb u target. The poten-208 t i a l of the Pb - target was taken to be of the form, (5.14) V N(R) '+ iWH(R) = (V Q+ i W 0 ) - ( l 4;EXP( R-^a ) J " 1 , where the parameters where selected to be, V 0 = -99.58 MEV; r 0 = 8.556 f.; a = 0.5 f.; WQ = -10' MEV. The corresponding ESW was found to have the. parameters, V E S W = -99.99 MEV; R E 3 V, « 9.39 f . ; W E 3 W = -10 MEV; f = 1.9; ^ _ at an incident r e l a t i v e energy of 8.9795 MEV (Po decay-energy) . The technique employed was to solve the coupled d i f f e r e n t i a l equations, Eqs. (5.12) a) and b), for C and D. The e f f e c t s of the height of the Coulomb ba r r i e r are i l l u s t r a t e d i n Figure 8. It i s seen that, for heavy n u c l e i , the transmission of the b a r r i e r i s very sensitive to the t a i l of the imaginary potential within the b a r r i e r . This long-range absorption i s an i n t r i n s i c property of a l l o p t i c a l model ca l c u l a t i o n s , even though i t may not have a good phys-i c a l basis. Where i t becomes dominant, a modification of the absorptive pot e n t i a l (to remove i t s t a i l ) s h a l l be considered. In Figure, 9, the imaginary part of the diffuse-edge po-t e n t i a l has been chosen to be the same as that of the ESW. A r e f l e c t i o n factor of 1.9 i s obtained at the alpha-decay energy " of 8.9795 MEV, comparing with a value of 2.0 obtained by ana-l y t i c c a l c u l a t i o n of the p e n e t r a b i l i t i e s as discussed in Chap-ter 3. The value of the r e f l e c t i o n f a c t o r was found to be rather i n s e n s i t i v e to the height of the Coulomb b a r r i e r ; from t h i s , and noting that i t does not depend s e n s i t i v e l y upon the energy, i t i s seen that the ESY/ parameters should be rather i n s e n s i t i v e to the decay channel and energy. It was also found to be insensitive to the strength of the imaginary part of the o p t i c a l model p o t e n t i a l . 106, 104-FIGURE 8 10' I0; 102 10 " i t BARRIER ABSORPTION TDIFF/ TESW PB 208 10 M E V 13 14 DIFFUSE IMAGINARY SQUARE IMAGINARY 15 16 17 F E R M I S 18 19 2 0 ; - 6 1 -In the c a l c u l a t i o n of Vogt, Michaud, and Reeves (1965), the diffuse-edge o p t i c a l model potential was chosen to have a di f f u s e complex part. They obtain a value of 4.62 for the r e f l e c t i o n f a c t o r ; replacing the diffuse-edge complex part by the complex part of the appropriate ESW y i e l d s a value of 2.8. It might, therefore, seem that the r e f l e c t i o n factor has been overestimated by a factor of two. They also state a r e f l e c t i o n factor of 100 when the ESW i s replace with a square-well having the same radius as the diffuse-edge well. In the present c a l c u l a t i o n , the r a t i o of the p e n e t r a b i l i t y of the diffuse-edge well to a square-well was found to vary rather slowly with the radius selected. It i s seen.from the example studied i n this Chapter that the radius of the ESW i s about a ferrai larger than that of the corresponding diffuse-edge w e l l . It i s apparent from Harada's c a l c u l a t i o n that he has, e s s e n t i a l l y , used such an ESW for c a l c u l a t i n g the independent-particle model reduced widths. Thus the large radius i n h i s c a l c u l a t i o n i s consist-ent with conventional nuclear sizes, provided that i t i s i n -terpreted as the radius of. an equivalent square-edge nucleus. - 62 -CHAPTER 6 CONCLUSIONS In t h i s thesis we have shown that much of the discrepancy' between the empirical alpha-particle decay rates of heavy n u c l e i and those estimated from nuclear shell-model calcu-l a t i o n s can be removed by a more di r e c t treatment of the nuclear surface. We have contended that previous calculations on alpha-particle decay rates have e s s e n t i a l l y used the equiv-alent square-edge nucleus model of Vogt; we have shown by con-sidering the one-body problem that the equivalent square-edge nucleus has a considerably larger radius than the diffuse-edge nucleus to which i t corresponds. We have concluded that the large r a d i i which were found necessary i n previous c a l c u l a -tions on alpha-particle decay rates to obtain agreement with the empirical values were the r a d i i of the.equivalent square-edge nucleus model rather than the actual r a d i i of the decay-ing systems being considered. In f a c t , we have shown that the r a d i i of the corresponding diffuse-edge nuclei agree with the conventional r a d i i believed to be t y p i c a l of a heavy nu-cleus. - In our calculations,. we have checked that the J.W.K.B. and the square-well estimates of the nuclear p e n e t r a b i l i t y used by. previous authors are i n reasonable agreement with the a n a l y t i c values. In fa c t , we have found i n Chapter 3 that the -J.W.K.B. estimate i s quite good to within about one-tenth -of a fermi of the c l a s s i c a l inner-turning point and that the square-well estimate d i f f e r s by less than an order of magnitude at the c l a s s i c a l inner turning point. - 63 -We have demonstrated i n Chapter 4 that the: one-body 21P reduced widths appropriate to the decay of Po are enhanced considerably by a diffuse nuclear edge. In f a c t , we have checked the ef f e c t on the one-body decay constant d i r e c t l y by c a l c u l a t i n g the d i f f e r e n t i a l e l a s t i c scattering cross-section for the one-body scattering of an alpha-particle from Pb"^ 8 at 9 0 ° Q ^ . We have included the many-body aspects of the problem by repeating the c a l c u l a t i o n of Harada with the square-well one-body wave functions of his c a l c u l a t i o n being replaced with the resonant wave functions of a d i f f u s e -edge w e l l . We have found that his square-well c a l c u l a t i o n corresponds to a diffuse-edge well c a l c u l a t i o n of a smaller and more conventional nuclear radius. In f a c t , we have con-tended that his c a l c u l a t i o n corresponds to the equivalent square-edge nucleus c a l c u l a t i o n for such a diffuse-edge w e l l . We have examined the dependence of the equivalent square-edge nucleus model parameters i n Chapter 5 by c a l c u l a t i n g the one-body transmission functions f o r the absorption of an alpha-208 p a r t i c l e by a Pb nucleus. We have found that, provided the t a i l of the imaginary part of the o p t i c a l model p o t e n t i a l i s truncated at the nuclear radius, the equivalent square-edge nucleus.model parameters are rather i n s e n s i t i v e to the nature of the reaction and to the many-body aspects of the problem. We have from th i s concluded that the equivalent square-edge nucleus model should have some v a l i d i t y i n heavy n u c l e i . We have interpreted the res u l t s of our calculations as suggesting that the independent-particle model, with only - 64 -those correlations introduced by configuration mixing,' can s a t i s f a c t o r i l y account for the c l u s t e r i n g into complex par-t i c l e s i n the nuclear surface; i n p a r t i c u l a r , we believe that such a model can predict reasonable values for the alpha-p a r t i c l e decay rates of heavy n u c l e i . - 65 -BIBLIOGRAPHY Arima, A., and Terasawa, T., "Progress of Theoretical Physics 23, 115" (I960). Bencze, Gy., and Sandelescu, A., "Physics Letters 22, 473" (1966), Ford, K.W., and H i l l , D.L., "Annual Review of Nuclear Science 5_, 25" (1955). Harada, K, , "Progress of Theoretical Physics 26,. 667" (1961). Igo, G., "Physical Review 115, 1665" (1959), Mang, H.J., "Physical Review 119. 1069" (i960). Mang, H.J., "Annual Review of Nuclear Science 14_, 1" (1964) Messiah, A., "Quantum Mechanics, Volume I" (1962), (John Wiley & Sons, Inc., New York) Noya, H., Arima, A., and Horie, H., "Progress of Theoretical Physics, Supplement No. 8_, 33" (1959). Preston, M.A., "Physics of the Nucleus" (1962). (Addison-Wesley, Reading, Mass.) Rasmussen, J.O., "Physical Review 113. 1593" (1959). . Rose, M.E., "Elementary'Theory of Angular Momentum" (1957). (John Wiley & Sons, Inc., New York) Talmi, I., "Helvetica Physica Acta 25, 185" (1952).. Thomas, R.G., "Progress of Theoretical Physics 12, 253" (1954). Vogt, E.W., "Reviews of Modern Physics 34, 723" (1962). Vogt, E.W. 'Michaud, G., and Reeves, H., "Physics Letters 19, 570" (1965). Wilkinson, D.H., "Proceedings of the Rutherford Conference on Nuclear Structure, 339" (1961). Zeh, H.-D., and Mang, H.J., "Nuclear Physics 29, 529" (1962). Zeh, H.-D., " Z e i t s c h r i f t fur Physik 175, 490" (1963). - 66 -APPENDIX A HARADA'S FORMULA FOR THE REDUCED WIDTHS FOR ALPHA-PARTICLE DECAY IN EVEN-EVEN NUCLEI Harada (1961) has derived a convenient formula f o r eval-uating the independent-particle model reduced widths for the ground state alpha-particle decay t r a n s i t i o n i n even-even nuclei with the use of harmonic o s c i l l a t o r one-nucleon wave functions. In t h i s appendix, we present an outline of the d e r i v a t i o n of h i s r e s u l t . The reduced width amplitude f o r alpha-particle decay has been defined i n Eq, (3.3), • • • Y^ L dAdldx^ds. , ^ where: i s the parent nucleus wave function; X i s the al p h a - p a r t i c l e wave function;, IL/T i s the daughter nucleus wave function; Yj-^ i s the spherical harmonic describing the r e l a t i v e motion of the decay fragments. Here, "iX"and ^ j m j are assumed to be properly antisymrnetrized and I|J 0 i s assumed to be p a r t i a l l y antisymrnetrized in the sense defined in Chap-ter 2. We assume the parent nucleus and daughter nucleus to be even-even nuclei and to be represented by independent-particle model wave functions of s e n i o r i t y zero. We consider only the ground state t r a n s i t i o n s , so that the parent nucleus, daughter nucleus, and the alpha-particle have zero angular momentum; the r e l a t i v e o r b i t a l angular momentum of the decay fragments i s then also zero. w The notation i s that of Chapter 2 and Chapter 3. - 67 -It i s convenient to expand the parent nucleus wave func-ti o n i n the two-proton and the two-neutron wave functions which can be formed from the u n f i l l e d subshell of the parent nucleus configuration; i t i s these nucleons which we expect to contribute to alpha-decay. In order to make the above expansion, we require the double parentage c o e f f i c i e n t s , ( j m " 2 ( s j ) j 2 ( J 1 ) jfj j m S J ) . Here j m denotes the angular momenta of the m one-nucleon wave func tions of the u n f i l l e d subshell of the parent configuration; j denotes the angular momenta of the two one-nucleon wave func-tions of the u n f i l l e d subshell which are taken to constitute m-2 the two-nucleon wave function; j denotes the angular mo-menta of the remaining m-2 one-nucleon wave functions of the daughter configuration. S and J are the seniority and the t o t a l angular momentum of the parent nucleus,' respectively; s and j are the s e n i o r i t y and the t o t a l angular momentum of the daughter nucleus, respectively; J' i s the t o t a l angular momentum of the two-nucleon system formed from the u n f i l l e d subshell. Noting the p a r t i a l antisymmetrization of VJ0, using the subscripts 1 and 3 to denote proton and neutron quantum num-bers, respectively, and Invoking our previous assumptions, (A.l) - 68 -dx ds ( ^ ^ ( O O j j ^ O j o l B j ^ O O H J n ^ l O O j j ^ o J o l ^ j ^ O O ) AT-2, M / A 3 l 12 2 J p — »-p ( 2 j 1 + 3 - A x) ( 2 j 3 t 3 - A 3 )  L ( A 3 - l ) ( 2 j 3 t l J Here, (J)^ ( ( | ) N ) depends only upon the proton (neutron) c o - o r d i -n a t e s , (x]_, 2^2^ ^ (2^.3. 2L4) ) » A]_ ( A 3 ) denotes the nurnher of protons (neutrons) i n the u n f i l l e d proton (neutron) s u b s h e l l , j ( j n ) , of the parent c o n f i g u r a t i o n , the angular momenta of the protons (neutrons) i n t h i s s u b s h e l l being denoted by j" ( 0 ). In the f i r s t step of Eq. ( A . l ) , we have expanded the o narent nucleus wave f u n c t i o n ' i n the two-proton (two-neutron) m a l i z a t i o n f a c t o r a c c o u n t i n g f o r the f a c t that the two protons w v  f u c t i o n s of the u n f i l l e d s u b s h e l l ; i s a nor-(neutrons) m y be s e l e c t e d from the u n f i l l e d proton (neutron) s u b s h e l l i n ( (A^) ) ways. In the second step, the i n -t e g r a t i o n has been performed; i n the t h i r d s t ep, we have summe over the ways of s e l e c t i n g two protons (neutrons) from the u n f i l l e d p r o t o n (neutron) s u b s h e l l . In the f i n a l step, we have noted the r e s u l t of Noya, Arima, and H p r i e (1959), (A.2) ( j m - 2 ( 0 0 ) j 2 ( 0 ) 0 | 3 j m 0 0 ) = [ [ ^ ( g -- 69 -Noting, and Eq. ( A . l ) , and writing, Eq. (3.3) becomes, (2^-f 3 - A 1) U 1 - I ) ( 2 j 1 + 1 2 iFTsj3+ 3 - A 3) L_1_A3-I)(2j3+1) d-ad|_d_s 0<. We have defined the alpha-particle wave function, Chapter 4: i n (4 .8) X(1234) = • • ( 4 T ) - 3 / ^ 0 ( l 2 ) ^ ( 3 4 ) . To perform the integration in Eq. (A.4) i t i s , therefore, con-venient to transform the proton and the neutron wave functions to the L - S representation and to then transform the proton and neutron co-ordinates to the in t e r n a l and r e l a t i v e co-or-dinates of the alp h a - p a r t i c l e . To see the manner in v/hich to transform the two-nucleon wave functions to the L - S representation, we consider only the j - j coupled two-proton wave function, (R,(j2,0). (The neutron case i s analogous.) ,i - ,i representation: i { 2 ) = i l 2 ) + s < 2 ) : L - S representation ' S = s j 1 ^ s i 2 ) . J = L +- S = 0; the superscript ( l ) or (2) denotes the f i r s t or second proton, respectively. Both the ,1 - ,j coupled wave functions and the #Harada employs the co-ordinates =~2'i-i '• 12H3-~i-2» -£3H=i-3' - 70 -L - S coupled wave functions form complete sets so that we may make the expansion, •••|(l< 1»l« 2>)L.(.i 1>.« a's : . ' V/e adopt the notation of Hose (1957): <(ltl)l« 2))L . ( 4 l).« 2))S :™|(ltl).(l ) ) i U).(l<8 ) . ( 2 ) ) 4 2 ) : j l l > ; From Eq. (4.8), only that matrix element for which S - 0 (and, hence, L = 0) can contribute to the in t e g r a l in Eq. (A.4). We need, therefore, only consider c o e f f i c i e n t s of the form, T(00;jj"). We evaluate this c o e f f i c i e n t b y , f i r s t express-ing i t in terms of a 9-j symbol: / l 1 0s T(00; oj) = (2-J+-1) £ Oi the 9-j symbol can be reduced to a W c o e f f i c i e n t (or a 6-j symbol) which i s r e a d i l y evaluated. Conforming with Harada, we write the solution' i n the form, (A.5) T(00;oj) = ^ (2j+-l)W( jljl;£0) # - i / 2.1+1 ' "J2 J 21+1 . We write Eq. (A.4) in the L - S coupled scheme as, #The value of the matrix elements, T ( 0 0 ; j j ) , stated by Harada contains a misprint (being i n error by a factor of 1 / ^ 2 ) . This error does not appear to have been carried through in • h i s subsequent c a l c u l a t i o n s . - 71 -= y w / ^ S A l A 3 T(pO;J 13 1)T{00;J 3J 3)/' fp(L=0,S=0) ••• n^(L=0,S=0)5CdJid|ds.K . The one-nucleon wave functions are taken to he harmonic o s c i l l a t o r wave functions: (A.7) f ^ j l r . s ) = ( J 4 1 ( r ) | : < l i « . - * | J ^ ( | ) ^ . where, n ' h=0 \ n-n J h i ' ("b i s the harmonic o s c i l l a t o r size parameter.) Then, (A.9) i) (J)p(L = 0,S = 0) = ^ 1l 1(x 1)(/yi/ 1l 1(x 2 )g<l 1l 1 ^ 1 -^ 1 joq> • • • Y . i a Y ( X£):X°(l2); J - l X l 11 x 2 and 1^ (V*3 & nd 1 3) are the p r i n c i p a l and o r b i t a l quantum numbers, respectively, of the u n f i l l e d proton (neutron) sub-s h e l l . To evaluate the integrals in Eq. (A.6), i t i s convenient to write the co-ordinates, , in terms of the i r r a d i a l and angular components, 3^ and-ft-^, respectively. It i s also con-venient to define the co-ordinates, £12 = Kx-L-t-Xr,) ; r 3 4 = K x 3 t x 4 ) . Expressing the two-proton and the two-neutron wave func-tions in terms of these co-ordinates, - 72 --Pi (A.10) i)lpT5_l1(x 1)lpV 1l 1(x ) S 1<l 1l 1^ 1-^il00>Y^ 1(^)Y 1 1(^2) N -i i ) l ^ l 3 ( x 3 ) ^ l 3 ( x 4 ) l 3<l 3V 3-^|00>Yf 3(|i) Yj!|i) 3 4 n2'X2 6 <L • • • I < W 2 1 0 0 > Y r 2 Y T P (-^2) • 34 ^  Substituting Eqs. (A.9) and {A.10) into Eq. (A.6), and i n t e -grating over and (A.l l ) =^TFM^o c>lA 3 T t O Q j J i J j K O O ; ^ ) J1J3 ... f <N10n10;0|'V-1l1V1l1;0> ' l'^2'-3'-^ ' ' ' ^ 2 ° ( r 3 4 } ( ^ ( M } ^ 1° ( ? 1 } • " ••• l p n 2 b ^ 2 ) X d A ^ d 5 1 ' S | < i 5 a d i 3 d s K . Noting, (A.12) l[7 E l 0(r 1 2)^(^|)l|;H 2 0(r 3 4)Y°(Si) =... ••• = I<»0n30:0]Ml0N20;0>l||I0(Ro)Y00(A)i|;n3o(?3) 3 rfThe Talmi transformation c o e f f i c i e n t s , N 1L 1N 2L 2;L n ^ l ^ n g ^ ^ » are defined and evaluated in Appendix B. - 73 -substituting Eq. (4.8) and Eq. (A.12.) into Eq. (A.11), and integrating over 3 and (A.13) ^ = J i n 2 R ° S A X A 3 (J ^ JlJ3 where, (A.14) O(H0) = T( 0 0 ; J 1 J 1 ) T ( 0 0 ; J 3 J 3 ) (Jj^j / <N0n 30 ; 0\ U-jONgO \ O^N-jOn-jO ; 0 | " ^ l ] y i l 1 ; 0j> n l f n 2 , n 3 • ' -<N20n20 ; 0| ^ ^ ^ f j \ ) ( \)EXP( - f 5^ x •• :/H Jn 2o(^)EXP ( - | | | )5 |d5 2 n 3(^)EXP ( - | 5 | ) ^ d S 3 - l V K 0(R o) . Performing the integrations in Eq. (A.14), we can write the overlap integral in the form, (A.15) = I <y (s o) , where, (A.16) C H ( R 0 ) = T(00;J 1J 1)T{00;J 3J 3) >^ < N 0n 30 ; 0| 1^01^0 ; 0> n l ' n 2 . '•••<N On 0 ; 0 | V l V ~ l ;6>/N On 0 ; 0| ir 1 "V 1 ; 0> N 1 1 ' 1 1 1 1 1 ' / s 2 2 ' 1 3 3 3 3' / • - a 7* ( n ^ t ) t(n 2+t) ' . (n 3 H) n 1!n 2«.n 3i ^ / 2i/£b\ 9/2. /S +bJ .. . /JL=_k\ 11-,+ np+n, .3 + b NO ^ '0 ^ ' '1' "2' "3 The reduced width can now be calculated fromSq. (A.13) and Eq. (A.16). - 74 -APPENDIX B TALMI TRANSFORMATION COEFFICIENTS The Talrni transformation c o e f f i c i e n t s are the transfor-mation c o e f f i c i e n t s f o r expanding the s h e l l model wave func-tion of a two-particle system in terms of the wave function of their r e l a t i v e and center-of-mass co-ordinates. If the average f i e l d is taken to be an harmonic o s c i l l a t o r well, these c o e f f i c i e n t s can be calculated i n a simple manner. In this appendix, a recursive r e l a t i o n i s derived which i s con-venient for the computer evaluation of those Talmi c o e f f i -cients which have been used in Chapter 4 and Appendix A. A more general recursive r e l a t i o n for the Talmi c o e f f i c i e n t s appropriate to an harmonic o s c i l l a t o r v/ell has been derived by Arima and Terasawa (1959), and the technique employed in the present discussion i s based upon their c a l c u l a t i o n . In the discussion to follow, the wave functions discus-sed w i l l be assumed to be harmonic o s c i l l a t o r functions. The sp a t i a l harmonic o s c i l l a t o r functions are of the form, where: r and-^- are the r a d i a l and angular components of r, respectively; Y i s a spherical harmonic; b i s the harmonic o s c i l l a t o r function defined in Eq. (A.8). The s p a t i a l s i n g l e - p a r t i c l e states w i l l be taken to be 1 o s c i l l a t o r r a d i a l harmonic The s p a t i a l wave function of the two pa_rticles i s then, The r e l a t i v e co-ordinate and the center-of-mass c o - o r d i -nate w i l l he denoted by, £ s r x - r 2 ; R- = i ( r x -+• r2) ; r e s p e c t i v e l y . The o n e - p a r t i c l e wave f u n c t i o n i n the r e l a t i v e co-ordinate w i l l be denoted by 4^nl^» c)» a n d t 1 n e °^e-particle wave f u n c t i o n i n the center-of-mass by 4^Pi/!L> ^  • ^ i s e a s i l y shown th a t , c = £b; C s 2b. One can construct the t w o - p a r t i c l e wave fu n c t i o n s from the r e l a t i v e and center-of-mass wave f u n c t i o n s : Since both the set of t w o - p a r t i c l e s t a t e s defined by Eq. (B.2) and that defined by Eq. (B.3) are complete, we can make the expansion,. (B.4) = 2 <KLnl;L-M*|TTl - y i ';m); ; Y T ^ l ^ l g j j t L f I 1 ^ . 1 1-1 2-2 T ITLnl ' ii) ( f ) ™ = ^ l ^ l ^ L - M ' l NLnl;LM>4Vl,Vll : T I L n l 1 -1 1 2 2 ' 1 1 2 ^ L» ,M« where <NLnl ;L"M»1 V ^ l ^ l ^ I ^ i n d ^ j l ^ ^ I^'M" ( K L N L ""W^  a r e expansion c o e f f i c i e n t s . The p r o p e r t i e s of these c o e f f i c i e n t s were f i r s t s t u d i e d by Talmi, and the c o e f f i c i e n t s bear h i s name (Talmi (1952)). To d e r i v e a formula f o r the Talmi c o e f f i c i e n t s , we f o l l o w the procedure of Arima and Terasawa (1959). We f i r s t note t h a t , from conservation of energy, - 76 -(B.5) (27^ + l x + 2 ^ 4-12) = (2N + L f 2n +1). Moreover, from the orthogonality of the angular momentum eigenstates, L 1 = L ; M* = M. In f a c t , we assert that the Talmi c o e f f i c i e n t s are independent of the magnetic quantum number, M. To see the l a t t e r point, i t is convenient to choose _r and r_^ to be p a r a l l e l . Making this assumption, and noting tha t, mn , ^ , THr <lTlom nmo)UI/ r m 2-. ^ l o m ^ l I M ? ^ ! ^ ) Y 2 ( i l ) = l ' m 2 "* 1 2 <lll 200|L0> Y r ^ ) , / ( 2 ^ + 1)1212+- 1) J 4T (2U+-1) Eq. (B.2) can be written as, / (21, f 1) ( 2 1 ? f 1) ' • F .../ 1 _ ^ <i l 200/L0> Yr-^-) , ^ 4T (2L + 1) 1 ^ L and Eq. (B.3) can be written as, • • • -<L100| L0> Y^ (-O ) . Noting the l i n e a r independence of the spherical harmonics, Eq. (B.4) can be written as, (B.8) i) l j > 1 i 1 ( r 1 , b ) l ^ i r 2 l 2 ( r 2 , b ) x / (21-L + l ) ( 2 l 2 + - l ) <1 11 200| L0> = X ^ N L n l j I l I l ^ l ^ i g . l J I ^ ^ l R . C ) lp n l(r,c) N , L • - n ' 1 ••• v/ (2L+ 1)121+1) <L1001L0> ; - 77 -I I ) U J L ^ . C ) l|J n l(r,c)v/ (2L-t 1)(21+ 1) <L100|L0? = ^1', 1 a r2' 2 ••• ^ i r2 12( r2 » l 3 ) v / ( 2 11 + 1)(21 2 + - D OilgOOlLC^ It i s seen that the Talmi c o e f f i c i e n t s are, indeed, independ-ent of the magnetic quantum number,M; we w i l l henceforth deno them'by <NLnl ;L l ^ l ^ l g ;L> and O ^ l i 21 2;L!NLnl;L>. For the purposes of this thesis, only the Talmi c o e f f i -cients of the form, < N 0 n 0 ; 0 ] v i l i ^ i l i ; 0 ^ w i l l be required. Taking L to be zero i n Eq. (B.8) i ) , (B.9) l^'V- ii i(r 1,b ) l| /v il 1(r 2. ib) ( 2 1 ^ 1) O-^OOf00> -<NLnL;0['V1l1-V-1i1;0> 4 V R ' C ) ^ n o ^ . c) N,n., L • (2L + 1) <1L00(00> . Nov/, ( B . 1 0 ) . \ j / v l l l ( r 1 , b ) l ( i v 1 i 1 ( r 2 i h ) (21 x+ 1) ^111]_00 100> = ... 3 /2 - J g 1 t n 2 V 1 ^ - ^ ( 2 R 2 + r 2/2 ) * * ' 2 1 1 + 1 ( V ^ l ^ i ) ' . 6 2 j=0 k=0 p=0 q=0 W l - J / l^l-k / \ P J ... ( 2 k K L i \ M j L ^ k " q b j + k + 1 i P 2 ^ * 3 ^ 1 ^ - ( P + I ) . l q / 3 Ik1. . R P+q and, ( B . l l ) X <NLnL;0|Vl v- 1 ; 0^(j/ (.R t2b) tiAn-0(r ,£b) N,n, 1 L. ••• (2L t 1 ) <LL00\00> = - 78 -, w / 41\Tlnl(2b-.Vb)' • . . = 2_ <NLnL; 0) ^ i l i ^ l l ! ; 0>7 2L + 1' J { N t L + i ) . (f+L+i) « N,n, L - I 2 b l R 2 . i i b l r2 e ^ e ^ s = 0 t=0 ^ n-t s t (2b) . . , t + £L 2t+L _2s+L (ib) r K s I t i whence (eliminating common factors) Eq. (B.9) can be written i n the form, q=0 (B.12) V-2T^TT j^jyr Jo Jo ^o i n T x i r 2 # _ T >2(j^-k-tl 1) - (p* q) R p t q n 4N *.n I 2 2: ~ JZLTTJ t (n+Lt^)». s=o t=o /n^Tr-iA f - l 1 s+t+L s-t 2t-f-L 2s+-L A recursive formula for the required Talmi c o e f f i c i e n t s can be found by equating the c o e f f i c i e n t s of the monomials, r ° R 2 s , in Eq. (B.12). For these c o e f f i c i e n t s : L = t=0; p q=s; j + k+l-^s; p=2j + l 1 ; q=2k+l-L ; and we obtain the r e l a t i o n , 2V 1I ( B * 1 3 ) ^ V " 3 T ^ P W T ^ k ^ i = s V "1 ^ = < H O n o : o , T l W l , g r * iT*S 2 n j s 1 79 -where, Nm = 2 ^ i ^ . 2 V X t 1 2 From Eq. (B.13), i t i s straightforward to show that, (B.14) <sO(N m-s)0;0|^ 1l 11r 1i 1;0> 2~SV 21 T 1 V X I ' (V I 1 + t) i 2 _ (T^ - j) 1 (^i * (j + I ^ i ) : (k + i + i ) : j i k i - l 1L ^ <K0 (Nm-N) 0 ; 01 ^ I f ^ i l x ; 0> N=s 1 - l • The Talmi c o e f f i c i e n t s employed i n this thesis have been evaluated from the recursive formula, Eq. (B.14). - 80 -APPENDIX C CALCULATION OF THE ONE-BODY DIFFERENTIAL ' ELASTIC SCATTERING CROSS-SECTION In this appendix, we derive the s-wave d i f f e r e n t i a l elas-t i c scattering cross-section f o r the scattering of a p a r t i c l e from a spherical one-body p o t e n t i a l . The approach which i s adopted i s to express the cross-section i n terms .of the log-arithmic derivative of the resonant one-body wave function. This general method of approach i s fam i l i a r from the ©-matrix theory of nuclear reactions, the present case being, perhaps, the simplest example. We take the one-body Schroedinger equation to be, (c-1} [~%^2* ^£^n*)~\ * ( R ) = _ £ t o ) , where P is the reduced mass' of the one-body system; Ze and Z'e are the charges of the two interacting p a r t i c l e s . Here, (C.2) Y(R) = Y C(R) - Z ^ § i + V H ( R ) , where Vj-,(R) and V-^(R) are the one-body e l e c t r o s t a t i c and nu-clear potentials, respectively. Due to the short range of the nuclear force, V(R) i s t y p i c a l l y a short-range p o t e n t i a l . The scattering cross-section from such a po t e n t i a l i s w e l l -.known (Messiah (1962)): (C.3) 6(6) - |f (6)| 2 , where, (C.4) f(6) = - 7 1 . EXP(-i?Un(sin2ie) + 2io~) ••• • 2ksinHQ CO j , • . .. -JL_ ^ (21+1) e 2 i ^ l ( e 2 i 1 - D P^cosO) . 2ik 1=0 ; l v ' Here, (C.5) i ) k = ; - 81 -i i ) 7i = ZZ'e' <2t i i i ) S1 - arg P ( l + 1 •+ iTj.) ; and /-^  i s t 5 i e phase s h i f t of the 1 t h p a r t i a l wave. Defining, (C.6) i) P = 71 l n ( s i n H e ) ; i i ) f = —2^ 7 - . " sin^^e ' and neglecting contributions to the cross-section from a l l p a r t i a l waves other t h a n s-waves ( l = 0), Eq. (C.3) becomes, (C.7) tS(Q) = 4k 2 \ c o s U + / 0 ) s . i n c T + l i ^ It i s seen from Eq. (C.7) that, to evaluate the d i f f e r -e n t i a l e l a s t i c scattering cross-section, we need only evaluate the phase s h i f t , £ Q . The phase s h i f t is defined by the as-ymptotic property, (C.8) u 0(R) — A[(C- 0 - i E Q ) - e 2 i^°(G 0 -Y iF Q_)] , which holds since V(R) i s a short-range force. Here, E 0 and G 0 are the regular and i r r e g u l a r Coulomb functions of zero angular momentum; u Q(R) i s the r a d i a l part of the zeroth par-t i a l wave solution of Eq. ( C . l ) , being a' solution of the r a -d i a l Schroedinger equation, (C.9) - djuo + 2 2 ' e 2 + V(R) L R uo = £ o u 0 2P dR2 It i s convenient to define the CR function to be, (C.10) <R0 s U ° ( R ) R ^Uoill dR R=R, where R 0 i s a suitably large radius. It i s then e a s i l y shown that, R=Rf Pu?0(R,£«) - E 0(R, £')Af G 0(R, £')/<R - RGo(R, £•) Noting the Wronskian r e l a t i o n , (C.12) ]?«G0 = G*]?.0 -r k , Eq. ( C . l l ) can he written as, (C.13) cT(^t) = t a n " 1 kR G/ P - J o G. R=R0 6 a e» The cross-section in section 4 - 2 of t h i s thesis was plotted by numerically evaluating the logarithmic derivative of the one-body wave function of Eq. (C.9) about resonance. The phase s h i f t was then found from Eq. (C.13), and the cross-section from Eq. (C.7). The v/idth of the cross-section can be obtained from the Breit-Wigner one-level formula; a discussion of the one-level approximation can be found i n the book by Preston (1962). It can be shown from the formulas for'the scattering cross-sec-tion stated by Preston that, ^ 2 * c o s ( ^ D 0 ) s i n D 0 - j - s i n 2 p o (G.14) <S(Q) = 4 k' 2 k2 where, (C.15) i ) ' s i n 2 D 0 = ( 6 f/^ ^ ^ i i ) B -1 - d£ 6=6, Here, P i s the width of the scattering cross-section; 6Q is the resonance energy; (C.16) A : -S 0 7T 2 i s the l e v e l s h i f t (S being the s h i f t function and 7f the one-b o d y reduced width). By comparing Eqs. (C.7) and ( C . l l ) , i t - 83 -i s seen that we may i d e n t i f y D 0 w i t h the phase s h i f t , < 0^. The c o e f f i c i e n t , B, can then be found from Eq. (C.15) i ) by mak-ing a l e a s t squares f i t to the phase s h i f t , < 0^, To evaluate the width of the s c a t t e r i n g c r o s s - s e c t i o n from Eq. (C.15) i i ) , we r e q u i r e the energy dependence of the l e v e l s h i f t . For heavy n u c l e i , where there are lar g e Coulomb b a r r i e r s , t h i s energy dependence i s given to good approxima-t i o n by, In t h i s t h e s i s , the e f f e c t of the energy dependence of the l e v e l s h i f t has been estimated from Eq. (C.17).by c a l c u l a t i n g the energy dependence of Gfl, 

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