REACTiON MATRICES FOR FTNITE NUCLEJ Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY DAVID V. GRTLLGT 1967 THESIS This is to certify that the thesis entitled REACTION MATRICES FOR FINITE NUCLEI presented by David V. Grillot has been accepted towards fulfillment of the requirements for Ph.D. degree in Physics I Major professor Dam November 7, 1967 0-169 “ amomc av 3! "OAS & SUNS' BOOK BMBERY INC. LIBRARY BINDERS ABSTRACT REACTION MATRICES FOR FINITE NUCLEI by David V. Grillot A method for calculating the reaction matrix elements of a closed-shell nucleus is discussed. An improved treatment of the Pauli operator is proposed, and accurate solutions of the Bethe- Goldstone are obtained for the coupled and uncoupled integro- differential equations. The solutions are used to obtain the first order binding energy of 160 using the Hamada-Johnston and Brueckner-Gammel-Thaler potentials. The results indicate that the Hamada-Johnston potential gives too little binding and possible ways of improving the binding energy calculations are discussed. The reaction matrix is examined as a function of the single particle energies. It was found that the 351 states are very sensitive to the choice of single particle energies; whereas, the So states are relatively insensitive to this choice. Calculated singularities of the reaction matrix are also reported for both the 1s and 35 states. 0 1 REACTION MATRICES FOR FINITE NUCLEI by DAVID V. GRILLOT A DISSERTATION Presented to the Department of Physics and Astronomy and the Graduate School of Michigan State University in partial fulfillment of the requirements for the degree of Doctor of Phi1030phy November 19 67 I; 0‘ .» I on!) U\ ACKNOW LED GEMEN TS The author wishes to express his sincere appreciation to his advisor, Professor H. McManus, for suggesting the present problem and for his criticism and encouragement throughout the calculation. Appreciation is also extended to Professor P. Signell and Dr. Neil Yoder for suggesting numerical and computational techniques. The author is especially indebted to Professor M. Pal for many helpful and illuminating discussions and to Professor 1. McCarthy for pro— viding facilities and financial assistance during the terminal stages of the problem. Last but not least I am grateful to my wife, Anne, for her patience and encouragement. ii TABLE OF CONTENTS ACKNOWLEDGEMENTS . LIST OF TABLES . LIST OF FIGURES CHAPTER I. INTRODUCTION II. FORMULATION OF THE PROBLEM General Development Singlet and Triplet l=j states Triplet l=j-l states Approximate Solution of BG Equation prr-d III. NUMERICAL RESULTS . IV. CONCLUSION APPENDICES A. TREATMENT OF HARD-CORE . B. NUMERICAL METHODS C. PARAMETERS AND PROPERTIES OF THE NUCLEON-NUCLEON POTENTIAL D. DERIVATION OF BINDING ENERGY REFERENCES iii PAGE ii iv 11 25 63 67 70 81 85 9O LIST OF TABLES TABLE I. Comparison of diagonal t-matrices with and without Pauli Operator using HJ potential II. Comparison of t-matrices using the 160 Pauli operator and 40Ca Pauli Operator III. Matrix elements of 16O for several different values of Kw using the HI potential . IV. Matrix elements of 16O for several different values of {w using the BGT potential . V. Comparison of present calculations with the results of other authors VI. Parameters for Hamada-Johnston potential VII. Parameters for Brueckner-Gammel-Thaler potential . iv PAGE 34 44 52 55 60 83 84 LIST OF FIGURES FIGURE PAGE 1. ISO wave function without Pauli operator . . . . 28 2. 1So BG wave function . . . . . . . . . . . . 29 3. 381 wave function without Pauli Operator . . . . 31 4. 351 BC wave function . . . . . . . . . . . . 32 5. Comparison of 15 BG wave function with . 0 different center-of-mass quantum numbers . 36 6. Comparison of 1S BG wave function with . . 0 different energies . . . . . . . . . . . . . 37 7. Variation of 1So diagonal matrix elements with single particle energies . . . . . . . . . . 39 8. Comparison of 351 BG wave function with different energies................. 40 9. Variation of 3S diagonal matrix elements with single particle energies . . . . . . . . . . 41 10. ISO (n=l) BG wave function . . . . . . . . . . 47 ll. 381 (n=l) BG wave function . . . . . . . . . . 48 12. 1Pl BG wave function . . . . . . . . . . . . 49 13. 3P3 BG wave function . . . . . . . . . . . . 50 14. Binding energy per nucleon of 16O as a function of {w for Hamada-Johnston and Brueckner- Gammel-Thaler potentials with Pauli operator 59 15. Binding energy per nucleon of 16O as function of flu: for Hamada-Johnston potential without Paul Operator . . . . . . . . . . . . . . . 59 CHAPTER I INTRODUCTION The initial attempts to classify and explain nuclear structure relied upon certain models, viz. , the shell-model and the collective model. These models and generalizations of these models had some moderate successes; however, there were also some notable failures. The initial attempts to overcome the shortcomings of these models and to develop a unified nuclear perturbation theory were made by Brueckner, 1 Bethe, and Goldstone, 2‘ One of the major difficulties in the development of the nuclear perturbation theory was the strong nature of the nucleon-nucleon potential, v, which when used in the conventional perturbation theory gives rise to divergent matrix elements. The many body perturbation theory shows that the infinite matrix elements, (film-IQ) , should be replaced by the finite matrix elements, <<91t|49> , where t is a two body operator defined by t=nr-N,‘% <1) 1K. A. Brueckner and J. L. GammeL, Phys. Rev. 109 (1958) 1023, this work lists all earlier references. 2H. A. Bethe and J. Goldstone, Proc. Roy. Soc. (London), 238A (1957) 551. and 17149=A£V (2) In these expressions Q is the Pauli projection Operator which prevents two interacting nucleons from scattering into states occupied by other spectator nucleons. The energy denominator, e, is the difference between the energies of the pair of nucleons in their excited inter- mediate states and their initial states. Considerable confusion exists in the literature about the precise nature Of the energy denominator. The current belief is that this energy denominator should be comprised Of Hartree-Fock (HF) single particle energies. The first attempts to solve equations (1) and (2) were carried out for nuclear matter, i. e. , an artificial nucleus of infinite extent in which the Coulomb effects are neglected. One Of the major simplifica- tions for nuclear matter is that the intermediate state wave functions are plane waves. This is possible due to the translational invariance of nuclear matter. Another simplification for nuclear matter is that the intermediate state energies are the energies of free particles. Even with these simplifications, the calculations Of the binding energy 3 M. K. Pal and A. P. Stamp, Phys. Rev. 158 (1967) 924. of nuclear matter have oscillated throughout the past ten years and are still in a state of flux. Despite the uncertainties of the nuclear matter results, one of the more instructive conclusions to be shown is that the many body theory reduces in a certain approximation to the earlier shell model. This was shown to be the case by Gomes, Walecka, and WeisskOpf. 4 Their calculation using the t-matrix formalism of references 1, 2 showed in a convincing manner that, (a) the many body effects fundamentally alter the correlated motion of a pair of nucleons such that the perturbed wave function heals to the unperturbed wave function and (b) the hard core potential could be treated in a consistent non-divergent manner. The first conclusion is a result fundamental to both nuclear matter and finite nuclei, and the ”healing" of the correlated wave function to the unperturbed wave function can be used either as a calculational tool or can be viewed as a qualita- tive test for proposed two nucleon wave functions. The transition from early nuclear matter calculations using the t-matrix formalism to the present day calculations for finite 4L. C. Gomes, J. D. Walecka, and V. F. WeisskOpf, Ann. Phys. _3_(l958) 241. nuclei has proceeded quite slowly. The slowness of this transition is due basically to the fact that the simplifying assumptions of inter- mediate state energies and wave functions cannot be carried over readily to finite nuclei. In fact at the present time there is no firm basis for choosing the intermediate state energies and it will be shown in Chapter III that the t-matrix elements are dependent on the energy denominator. Moreover, in finite nuclei the Pauli Operator is difficult to handle because the bookkeeping devices, viz, Racah co- efficients and Moshinsky brackets5 are cumbersome to treat. ! As has been stressed by several authors the rigorously correct calculational procedure is a doubly self-consistent computa- tion. That is, using equation (I), a set of t-matrices are computed with some realistic nucleon-nucleon potential. Using these initial t-matrices, a self-consistent HF calculation is performed. The HF calculation generates a set of single particle energies and wave functions. These single particle energies and wave functions are then used to recalculate a new set of t-matrices. The double iterative 5T. A. Brody and M. Moshinsky, Tables 3f Transformation Brackets (Monografias del Institute de Fisica, Mexico, 1960). 6K. Kumar, Perturbation Theory and the Nuclear Many Body Problem (North Holland Publishing Co. , Amsterdam, 1962). procedure is then repeated until a self-consistent set of single particle energies and wave functions is Obtained. Even using high speed computers the problem of achieving double self-consistency is only now becoming feasible. In order to avoid the above-mentioned complexities several 7. 8. 9. simplifying assumptions are usually introduced which will be maintained in the present paper. The first assumption is that the intermediate HF wave functions are replaced by harmonic oscillator wave functions. This approximation has been discussed 7 , l by Kuo and Brown who argue that the results of HF calculations give wave functions which are similar to harmonic oscillator wave functions. The next major assumption is that the Bethe-Goldstone (BG) wave function,‘,V, can be separated into an unperturbed center- Of-mass wave function and a perturbed relative wave function. There is no justification for this assumption due to the nonlocality Of the Pauli Operator, but unless this assumption is made the BG equation 7 T. T. S. Kuo, and G. E. Brown, Nucl. phys., 8_5 (1966) 40. 8 C. W. Wong, Nucl. Phys. A91 (1967) 399. 9 A. D. Mackellar PhD Thesis, Texas A and M University, January 1966 (Unpublished). 0 1 L. Kelson and C. A. Levinson, Phys. Rev. 134 (1964) B269. is insoluble. The retention of this approximation will be necessary until a suitable method of solving directly the two-body BG equation is devised. The final assumption concerns the nature Of the intermediate state energies. As mentioned earlier there is no clear basis for choosing these energies; however, two choices are now popular. The first choice is based on nuclear matter results and uses plane waves; the second choice is to use harmonic oscillator energies. Both of these choices are questionable, and in this paper the harmonic oscillator is chosen. Before the complete self-consistent problem is attempted it is necessary that accurate solutions Of equation (1) be Obtained. Unless one has confidence in these fundamental solutions it seems unwarranted to proceed with the larger calculation. Recently there have been several convincing solutions to equation (I). The more notable of these have used the reference 12 spectrum method, 11 the separation method, and the Eden-Ems ry 13 method. Kuo and Brown7 have applied the first two methods to 11H. A. Bethe, B. H. Brandow and A. G. Petshek, Phys. Rev. 129 (1963) 225. lZS. A. Moszkowski and B. L. Scott, Ann. Phys. _l_1_ (1960) 65. 13 R. J. Eden, V. J. Emery and S. Sampanthar, Proc. Roy. Soc. (London) A253 (1959) 186. 18 the excited state Spectrum of 18F and 16 O and the binding energy of O. Mackellar9 has used the latter method to obtain the binding energy of 160. Both the separation method and the reference spectrum method use a perturbation expansion of the t-matrix which in the first approximation neglects the Pauli Operator. The Pauli operator occurs in the perturbation expansion and is ultimately calculated by using the angle average Pauli operator which was originally used for nuclear matter. Recently Wong8 has improved the Kuo-Brown treatment of the Pauli Operator. The separation method and the reference spectrum method differ basically in the choice of the first order t-matrix. In the separation method, which was initially used by Moszkowski and Scott12 for nuclear matter, one assumes that the potential can be separated into a short range and a long range part. The separation distance is chosen such that the short range attraction in some sense cancels the repulsive hard core part of the potential. Then onelis left with the problem of evaluating the long range part of the potential via the perturbation expansion for the reaction matrix. There are several minor assumptions such as averages of state dependent separation distances, but it is believed that these corrections are small. 14 The reference spectrum method was originally suggested by Bethe, Brandow, and Petchekll for nuclear matter. This method has particular utility for the odd state repulsive potentials because the concept of a separation distance is meaningless for these states. In this method the first order t-matrix neglects the Pauli operator and approximates the intermediate state energies with the free particle energies. With these approximations one gets an inhomogeneous differential equation for the first order t-matrix. The higher order corrections to the reaction matrix are then calculated in a manner analogous to the separation method. In both methods the question of convergence of the pertur- bation expansion is made plausible by explicitly showing that higher order terms are small. The third method was originally suggested by Eden and Emery in a series of papers, 13 and some calculations were performed using questionable numerical approximations. 9 Mackellar has recently redone the calculations in a precise manner for 160 using several different nucleon-nucleon 4 1 T. T. S. Kuo, to be published. potentials. There are two basic approximations in this approach to the problem which make the calculations possible. The first is to Observe that a tenable approximation exists for the Pauli operator based on the unique characteristics of the harmonic oscillator wave functions. The final effect of this approximation in 16O is to forbid scattering into one allowed state and permit scattering into a set of forbidden states. Despite these restrictions the method does include the essential features Of the Pauli Operator. The second approximation is the treatment of the HF self- consistency condition. By introducing a state dependent potential for the initial state energies, it is possible to obtain approximate self-consistency for the BG wave function. The overall method deviates considerably from the frame- work of the nuclear many body theory, and despite its ability to predict nuclear binding energies it is an artificial approach to the problem. In the present paper some of the above-mentioned approxi- mations are avoided. In particular it is shown that it is unnecessary to construct a perturbation expansion for the t-matrix since equation (1) can be solved exactly. At first glance this appears to be an involved calculation, but in actuality the calculation is simpler and avoids any question of convergence of the perturbation 10 series. Moreover, there is no necessity of calculating state dependent separation distances. In addition to this exact solution, it will be shown in Chapter II that it is possible to incorporate an improved treatment of the Pauli Operator. This improved treatment of the Pauli operator and the exact solution of equation (1) should increase the reliability of the t-matrix elements for finite nuclei, and it is anticipated that one could now proceed with the full self- consistent problem. CHAPTER II FORMULATION OF THE PROBLEM I. General DevelOpment Rather than solving equation (1) of Chapter I, the standard procedure is to calculate the two body correlated wave function, (P . By combining equations (1) and (2) of Chapter I, the following is obtained for the BG wave function: «y = r? + gr 3" (1) Or in terms of the wave Operator J'L: 1 +gfl—n- Once equation (1) is solved, the diagonal and off-diagonal reaction matrix elements are determined from (Wt-I43? = (4’1er (2) which is a finite quantity since W vanishes in that region where the potential, v, has an infinite hard core. 12 In Appendix A a detailed analysis of the hard core shows that the hard core contribution to the t-matrix is given approximately by Q 5;? I... . .tc . This contribution has been added to all matrix elements in equation (2). The task is to solve equation (1) which is the Bethe-Goldstone (BG) equation in as exact manner as possible. For complete generality the BG equation is written in the j-j representation even though this generality is not needed for closed-shell nuclei. In this representation the BG equation is xii/M1, (magma/m. = {am} (4013’; 74mm.) UNOCCUPl-‘D ’ I ' I I '1 i l , +5, (1 +5va m’l’j’) ((1.4 )j (44); 17.7w. 4 > ”xi, I’NJ’ UO” D ‘ t ’ me, (1:13,? ' x ({4’4.'):}'//.14.')1',’f'fi,oT'Nr'lflm—VM).7} (1.4.)1'. fem/m.) (3) 6 . + 6 O c -6 I ' «.41. Mi, 61’4’1,’ 14’1,’ where nlj designate the single particle quantum numbers, enlj are the initial state self-consistent energies, and én'l'j' are the energies Of the intermediate states. The summation in equation (3) is carried out over all initially unoccupied states and in principle these states should be the self-consistent HF single particle states. 13 Since the complete self-consistent problem, i. e. , the problem of doing a self-consistent HF calculation in conjunction with a self- consistent treatment of the BG equation, is an extremely involved computational exercise, the approximation of replacing the inter- mediate state single particle wave functions by harmonic oscillator wave functions is made. The initial state energies, which should also be determined in a self-consistent manner, are varied over a wide range Of physically interesting values. For the final calculation of the t-matrix elements, the initial single particle energies are Obtained from Hartree-Fock calculations. 3 The intermediate state energies 6 are chosen to be harmonic oscillator I Id + 6 t [.9 9 n11131 “21sz ' energies. Based on these considerations the problem Of double self- consistency as discussed by Pal and Stamp3 is circumvented. Due to the nonlocality of the Pauli Operator the BG equation is not separable into relative and center-of—mass coordinates. There- fore, the major approximation of this paper will be to assume that the BG wave function, hf , can be separated into a perturbed relative wave function and an unperturbed center-of-mass wave function. Similar assumptions are made for the initial state energies & +6. Thatis e , + e , _ rel. C.M. l 2 n11111 n21.232. " EnljS + ENL 14 where nljS are quantum numbers of the relative state and NL are center -Of-ma s s quantum number 3. Using these assumptions the BG equation can be reduced to an integro -differential equation by performing the following algebraic manipulations: (I) Operate with 49/. C." 4.6,. 6.". H, = E . + E 6011.1; *2/17; " M; m 0 (4) where H0 is the harmonic oscillator Hamiltonian. (2) Change the sum over the unoccupied states to one minus the sum over occupied states. (3) Use the standard transformation for the perturbed and unperturbed states, viz. , . . . '1 “4’91. (4'4") jar/77 Tnm’m‘> :: [1(1 4’ S‘V’I’, )mtIz 71.)] 1,1,, ' ' ' jn’r?’ X Z T 1’7‘ £/‘f'/) J x2: AM Z: C(jL/rglef/Z)luawlfisynfj-MNTMR (5) ILA , ”Vi 15 where a . Ij KL 1’ xiii/.1“ . , . IS,” - \/(25+/)(2A+/)(2Jl+0011+!) It 8. 4? (6) 5 J’ A J and . 1 5 51. - S+L+}*'/\ A”; =(-/) JM 3' L /\ (7) In these expressions the angular momentum jl and j2 couple to a total angular momentum, J. The spin of the two nucleons is S, and l, L, and /\ are the relative, center-of-mass, and total orbital angular momentum respectively. The T-coefficients are the trans- formation coefficients from j-j to L-S coupling. Aft-33 is the re- coupling coefficient of 1 and S to a total relative angular momentum, j. And the bracket < nlNL, /\ I n l n l > is the Brody-Moshinsky l l 2 2 transformation bracket. 5 (4) Eliminate the center-Of-mass energy . C.M. C-M- - by usmg (E NL .HO ) I LMLN> _ o. (5) Take the scalar product Of the transformed equation with (MI0431‘C4' I4Z’NL’ A°> trfid'frt. "M40. : 1., “’7' {ISJ’ ' . X 0-H)" ”JI’ 4") + ] Aux/1. CGLM‘M‘IM‘) (8) 1. ' . x :2: 42.2.7 C (J. “’9' "'1 I T» ”T. >/“size,«Nice/ne’bmiflrbem? This equation is valid for either l=j or l=j-1. 2. Singlet and Triplet l=j states. In this section only the case where l=j will be considered. Define a”. 4 1 (xi/Us);i””,-"'> : iii) 7?") (9) i 15 A. DeShalit and I. Talmi, Nuclear Structure Theory (Academic Press Inc. , New York, 1963). l7 and Mo .. v <,:/_{L|{IS)J'»~’./n>= AL 7!?” (10) .1 where ’g, ml is the standard angular momentum function which has jls been defined elsewhere. 16 Also introduce a dimensionless variable f =r/b. (half/ma) )- m. Then after taking the scalar product 3. le s with equation (8), summing both sides over m, and m , performing the sum over M J L Jo and Jo, and using the implicit energy relations of the Brody-Moshinsky bracket, 5 the following expression is obtained: 1‘ - {£1.22 - 2:3. w E 6.1,: +;’—,—. r r ,a, l ”3'; =[AMJJ'S_ ;,](Z£f) <&j [um/1'3) in: WPLIEQ’Ky / Mtg/h) %1.S(:')/;. ”VIM :J’ ’1‘ (12) 2. x { £(1L+I)(ZIH)]-IAZ (l/LH) ’(ntlA/LAIMM‘MI’A? ) E 16M. A. Preston, Physics o_f the Nucleus (Addison-Wesley Publishing Co., Reading, Massachusetts, 1962). 18 2 rel. wheren = (2n° +10 +2n° +1° - 2N-L-1)/2, /\ =——E _ , and o 1 1 2 2 nljS 47‘” nljS O A = 2(2n+l)+ 3. n1 Equation (12) is the final reduced form of the BG equation which has to be solved. It should be emphasized that the only major approxi- mation which has been made in deriving equation (12) is the separability Of the BG wave function. By explicitly writing out equation (12) it becomes Obvious that the form of this equation is identical to the equation derived by Eden and Emery. 13 However, there are several major differences. As- is 9 well-known the Eden-Emery approximation incorrectly includes and excludes scattering into certain intermediate states. This approxi- mation manifests itself in the number of terms in the sum in the reduced equation. Also, equal weights are assigned to all inter- mediate occupied states. In the present treatment Of the BG equation both of these difficulties are corrected. That is, each intermediate state is multiplied by the appropriate weighting coefficient and the summation is over the proper number Of terms. In the Eden-Emery method there is no clear prescription for extending the method to nuclei other than 160. It is now apparent how equation (12) can be extended to any closed-shell nucleus. In addition the Eden-Emery energy denominator differs from the present formulation. l9 3. Triplet l=j-1 states. In this section only the case where l=j-l is the dominant solution will be considered. Define ? (a) M") ’m/ y("& <5 l/ij'n'y m) = A jls (13) and . u (A) . y (slat/15);»3-w = «I 5 my“ 9’) + .111 {9 4°) 14 4/: .71’5 ( ) where unljS is the l=j-l component of the solution and wn 'l'jS is the l'=j+l component Of the solution. When unljS is the dominant solution, this implies that the unperturbed state is an l=j-l state. The procedure used in deriving the relevant reduced coupled equations is, in principle, the same as that used for the uncoupled equations. Starting with equation (8), the scalar product with first '5le and then ijs is taken. This leads to two equations which are summed over mj and mL. The summation over MJ and Jo can 0 then be performed and if the implicit energy relations Of the Brody- Moshinsky brackets are used, the following expressions are Obtained: 20 v!” I) u/) [A’Vi‘f will; it AI, 47"» (72$,M’3/12] Mfrs (”'17) >w’1!) =DM/5‘AL1]agd<fiv1/%,'s> ‘5: all: 19‘“"" (15a) actu Zf/so ') / ’) / +< 1”! 2°”! fiofi’fli’w fa! ’ [2%€JIMI;ZIS> 2%535 :17" :2: .015 $41,] og'lfm '1' I o I," xfiamxumj' ’22: (14“) I ] and J2 .. {(1.31}... ‘- 33' ' N” w’ (I [9"st +1? f‘ f fiw m”; (15b) aflofi, (f:——) 9. “f; (IV/”[2? [Ar/y >w’0') +MG')]J.’ [$1 M‘fm‘lo ”I!" ""1 j": (1’? all!!! I]: A”: Z x {Batu/)(u’Hfl-lg: (an. H)/ (,J’ff) :o 1!: 1150197: (21a) 23 and J‘ __ 1719‘") .— ‘ — 37-”- ,u— car?!) 0.437» -——-—-.. I may (2;. (21b) 33:. ° ) ‘0 'fiw bjuf;s 115 these equations which ignore the Pauli operator should be compared with equations 12, 15a and 15b which include the Pauli Operator. Although the form of the eigenvalue equation is considerably different from the correct form Of the BG equation, it was originally thought that the eigenfunctions would be reasonably good approxi- mations to the BG wave functions particularly for the higher angular momentum states. A priori it is difficult to assess the severity of neglecting the Pauli operator. Dawson, Talmi and Walecka, 17 based on some crude estimates of the effect Of the Pauli operator in 180, showed that the effect Of neglecting the Pauli Operator in the 1So states would result in about 10% error. It was hoped that the same would be true for the troublesome 351 states. The present paper . . l . . W111 show in the next chapter that for the So states their estimate 17J. F. Dawson, 1. Talmi, and J. D. Walecka, Ann. Phys. l§(1962h 339. 24 . 3 . of the error is reasonable. However, for the S1 states one 18 not quite so fortunate, and it is imperative that the Pauli Operator be taken into account. Moreover, in order to Obtain decent values for . . l6 . . . . the binding energy of 0 it IS necessary to include the Pauli Operator. In addition to the above quantitative considerations the qualita- tive comparisons of the solutions with and without the Pauli Operator demonstrates in a dramatic fashion the effect of the Pauli Operator on the nuclear wave function. CHAPTER III NUMERICAL RESULTS A computer program has been written to solve the coupled and uncoupled integro-differential equations exactly, i. e. , equations 12 and 15 of Chapter II. The numerical technique used to solve these equations is discussed in detail in Appendix B. The numerical accuracy required for the wave function, u, is gu/u < . 005 where Su is the computed change in the wave function at the end Of each iteration. Usually three iterations were sufficient for convergence. The method of solution was quite efficient, and ultimately it was faster to solve the full BG equation than the related eigenvalue problem. Disregarding compilation time, the computer time required to calculate one state and its off-diagonal matrix elements was 30 seconds and 60 seconds for the uncoupled and coupled states respectively using an IBM - 360 Model 50 computer. The nucleus of interest in this calculation was 160, but the method developed has equal applicability to other closed-shell nuclei. For O the summation in equations (12) and (15) is carried out over the os and op shell. For other closed-shell nuclei it is necessary to alter the range of this summation accordingly. 26 In solving equations (12) and (15), three pieces of information are needed: a phenomenological nucleon-nucleon potential, a set of HF single particle energies, and an oscillator parameter, ’Kw . The equations were solved using both the Hamada-Johnston18 (HJ) and the Brueckner-Garmnel-Thaler19 (BGT) potentials. Both Of these potentials contain a hard-core, tensor force and spin- orbit force. In addition the HJ potential contains a quadratic spin-orbit force. The properties and parameters of these potentials are given in Appendix C. The values of the HF single particle energies were -40 and -18 MeV for the os and Op states respectively. These were the values calculated by Pal and Stamp3 and should be correct to within 5MeV. It should be realized that these values will be altered somewhat when the doubly self— consistent calculation is performed. Since there is doubt concerning the specific dependence of the t-matrix on the initial state single particle energies a detailed examination of this dependence has been performed for the 1S0 and 381 states. 18T. Hamada and 1. D. Johnston, Nucl. Phys. 33 (1962) 382. 19K. A. Brueckner, J. L. Gammel and R. M. Thaler, Phys. Rev. 109 (1958) 1023. 27 The oscillator parameter used was based on electron scattering data20 which indicates that the value of Kw for 160 should be between 13 and 15 MeV. The t-matrix elements were calculated using three values of {w , viz., 13. 35, 14. 5, and 15. 5 MeV. Ultimately more values of if“) were needed to clearly define a minimum in the B. E. /A versus ‘50) curve. With these pieces of information the first equation solved was equation (20), i. e. , the eigenvalue equation which neglects the Pauli Operator. This equation was solved for all angular momentum states less than 3 and for both the singlet and triplet states. The resulting ISO relative wave function, ugo’is plotted in Figure l and is compared to the unperturbed harmonic oscillator, R00. The important qualitative feature to be noted is that there is no healing of the wave function 1180 onto the unperturbed solution. In this situation, the scattering of the two nucleons inside the harmonic oscillator potential can be viewed as a real scattering process in which the final wave function has undergone a phase shift. Figure I should be compare directly with Figure 2 in which the perturbed wave function, “00' is plotted. Here “00 is the 1So solution of equation (12) which includes the Pauli projection operator. 20 1961). L. R. B. Elton, Nuclear Sizes (Oxford University Press, 28 OO .Uoms mm? fimscouom coumcgoh. :35ng 05 was 0383030 65 3 A 0.0 m .cofioca o>m>> noumfidomo 3:05.25 05 515 Houmuomo Enema ofi muuofimoc £033 . 9 .COSSHOm mo 05 .«O :OmmummEOUIt A .wmh Qm O “E: a 0.0 0.? ON a. h . _ , . _ . O 00m m: .2 >22 ms. ... 3: aw: .. 0.. one am. OO .Ooms momma» Hmfizobom aaobmczowtmvgccm och. . m .soflocsu o>m3 Hobmzmomo 3:05am: mo of £32 . s 535:3 o>m>z OcomeoUtofiom mo 05 Go COmEonHEOUtt .N .mfm . 2...: c . afigrflithsaulnnlnfl .4. . . _ . .. . . it. _ . .O 29 00 00 >qu out saw + .w z. D \ 3...... 00.3 ”3: t. 0.. OHJ OHZ OH: Om. 30 The contrast of Figures 1 and 2 explicitly displays the effect of the Pauli Operator. The correct BG wave function, for all calculational purposes, has healed at 2. 5 fm, and the healing is complete at 4 fm. Similarly for the 381 coupled l = j-l dominate states, the solution without the Pauli operator is plotted in Figure 3 and the solution containing the Pauli operator is plotted in Figure 4. The same healing phenomenon for the dominate component of the solution occurs in the case of the coupled states at roughly the same healing distances as in the 1SO state. The minor component of the coupled solution is drastically reduced when the Pauli operator is included, and since the tensor force is strong, the off-diagonal tensor contribution to the matrix element (eff/VI)» is reduced accordingly. The healing of the BG wave function is Of fundamental significance in the nuclear many-body problem and sho‘vs precisely the effect of the Pauli operator. This phenomenon is well known and was originally investigated for the 1So state in nuclear matter by Gomes, Walecka and Weisskopf. 4 Similar results using different techniques and approximations have been reported by 9, 21 other authors for finite nuclei. Considerable importance has 21H. s. Kohler and R. J. McCarthy, Nucl. Phys. 86 (1966) 611. NO _ O 00 M .ucocomEoo HOEE 05 mm O? 0 one Omnnwcomacoo OcmEEOO on» ma 00: . m 59.35: o>m3 noumzmomo OficoEumfi m0 m5 515 .N03 was OD .COSSHOm “63:“ mo 65 .«O COmCmmEOOt- .m .mrw 2:: a co. Om. ow 0... on .. 5.0.2 >22 were: o... .m. OO 03 . 9 m 6 o cocomgou NO 00 00 not H0 0m 5w 5 .2058 65 2N0 3 OsmoumuocomgopucmcmEOU 65 mm a . m 505625.“ 6>63 68.632660 6508.35 65 515 . 3 can 5 50362.23 6>63 6220563001656m “653.5 60 65 mo GO6MH6QEOOtu¢ .wmh 2:3. 0.0. Ow 00 02» ON a. - . _ u . u d 11- o 32 No; 001 >22 om- "Nets 8: >22 Om.¢_u3ct 0.. o... 0.2 o... .m. . 33 been attached to the healing property since it implies that the wave function for nucleons inside the nucleus are independent and spheri- cally symmetric; this is precisely the assumption Of the classical shell model. So the validity of the shell model comes directly from the many-body effects of the Pauli Operator. Even though the aim of the many-body theory is to reproduce the experimental data and not to justify any models, it is encouraging to Ob serve that the many-body results, in some sense, do justify the classical shell model which has had some successes in classifying nuclear data. The healing of the BG wave function can be explained physi- cally by viewing the interaction of the two nucleons as a scattering process in which the energy and momentum distribution Of the two particles must be conserved. Since all the nearby energy states are Occupied by spectator nucleons, scattering can only occur into unoccupied states. But since energy is not conserved, the parti- cles must return to the original states with the same initial energy and momentum distribution. Hence there is no momentum transfer in the scattering process and hence the phase shift is zero. The entire process is considered to be a virtual process and the phase shifts are said to be virtual. 6 Since there are no real phase shifts, the wave function must heal, and only at very 34 short distances (r< 2fm) can be perturbed and unperturbed wave functions be different. In the past (references 7 and 12) healing Of the wave function has been viewed as a calculational tool. That is, healing has been imposed as an additional boundary condition and considerably reduced the computation involved. The point Of view of the present paper is that healing is a test of the validity of the solutions. More- over, it is believed that the numerical techniques of Appendix B are more efficient than those using the separation method, 1. e. , the method which imposes healing from the beginning, and it is unneces- sary to artificially force healing. The quantitative effect of the Pauli operator can be demon- strated by comparing the diagonal t-matrices calculated with and without the Pauli Operator. This has been done for several states and is shown in Table I. As can be seen from this table the effect TABLE I. --Comparison of diagonal t—matrices (in MeV) with and without the Pauli operator using the HJ potential. 1 3 l 3 3 3 S6 S1 p1 P6 P1 P2 Without Pauli Operator -8. 2 -15. 9 l. 7 _ -2. 0 l. 9 -l. 2 With Pauli operator —6. 4 - 7. 0 2. 0 -2. 4 2.1 -1. 0 35 . . . . 3 . of the Pauli Operator is quite large in the 5 state and less important 1 for all the other states. In the past there has been uncertainty concerning the dependence of the BG wave function on the center-of-mass quantum numbers and the initial state HF single particle energies. The results of this study show that for the singlet and triplet l = j states the center-of- mass and energy dependence are not crucial. In fact the final t-matrix results for the l = j states are extremely insensitive to variations in both of these quantities. Figure 5 compares two wave functions with comparable energies and different center-of—mass quantum numbers. The differences in the two wave functions are minimal. Since the center-of—mass quantum numbers determine the number of terms which are included in the summation in equation 12 Chapter II, it has been concluded that by far the most important term in the BG equation is the term ( A - .\° )4’<‘fl'}’> The additional terms which enter the complicated summation contribute at most . 5 MeV to the t-matrix. The energy dependence of the 1So wave function is shown in Figure 6 which compares two wave functions with the same center- Of-mass quantum numbers but very different energies. Again the differences in the wave functions are slight. In order to reinforce this point the t-matrix for the 1So, N = O, L = 0 state was 36 “moawuoco 633.260 “6356 ”22626326 322326 6:6 "£638.32 8.3226di 66.68 two 23:66 5:626:25 52>» 6:956de 6>63 622363001656m 56356 mo 025 00 2206360500.... .m .warm RE: 2 0.0. 0.m 0.0 042 ON e. |l . 1.). u - d — Id d - O >62 mhsuswtw / \ E 0.2 one 6...... 83 .. 6. >22 omtnsw .26 \\ o... 0.2 one am.) , .. o._ 37 .m6fiwu6c6 635.260 6356 5226.80.26 .66.» :33 6:036:30 6>6>> odoumOHOOt656m 56356 mo 03... mo dOmmuwmrcoU..- .o .wfim ES 2 . . . . 0 0m 00 0s . 0m 2 (13...... . A . _ . _ . , / . 1 / \ . // 8:\\ >22 owtnnwtw II / \ 1 >22 ON- "swim II 8: >22 91 63$ 1 o... 0.2 one 6.2 0._ N._ 38 calculated over a wide range of energy values and the diagonal matrix elements are plotted in Figure 7. This graph explicitly shows the insensitivity of the t-matrix to variations in the single particle energies. The final results vary less than 3% over the energy range -80 to -35 MeV which is the region of interest in 16O. For the 351 states the results are sensitive to the energy and insensitive to the center-of-mass quantum numbers. Figure 8 shows two 381 wave functions plotted for different values Of the single particle energies. The minor component of the solution changed appreciably for different energies and this is reflected in the t-matrix through the off-diagonal element. Therefore, it is imperative that the correct choice for the energies be made in the 351 states. Figure 9 shows that the final results vary by about 20% over the energy range -80 to -35 MeV. The mathematical reason for the insensitivity to e, #61 of the 1So state is directly related to the healing property of the wave function. In the BG equation the energy enters as a factor multiplying ( u - R ), and since the wave function, u, approaches the unperturbed wave function, R, the term A( u—R ) is a small quantity when compared to other terms in the integro -differential equation. The overall effect is that the BG wave function and the t-matrix are extremely independent of the energy. 39 . .m6flmH6C6 guinea 6Hm56 Mooht663§mm 65 no 63565 cofiuomou Qua 56356 65 mo 66:6OC6Q6Q11 .n .wwm .222. «6 + ..6 0..» ON 0 ON. 0?. cm. om- . q a u u u as. .0. V 9 O __ . as- w __ 1 __ = . 8... w 2 v : w. L . .. ow- m r/IW M m .81 .( O OO . $333606?” GOSOCE 6>63 0m 65 mo 65262200806 HOEE 6:6 60.38 65 6.2.6 N >2 0:6 : 66326226 40 966.2635 >26> 53> mcofiocdm 6.263 6coumO~oOt656m 56355 no OB... no 2206260500-.. .w .mfim 25. oo. . Om 0.6 o... _ om .. . _ s t J T _ .. . .. / . _ . 00 / / // N0;\ /.\ l N NO?) I 2». >22 Om- "swim II / >22 Omtunwtw II / 83 .. >62 8.: u 3: / \ .. m on... 0.2 one .6. / \ s: 41 .662m26c6 6635.23 6356 Mooht66uuumm 65 co 63.365 22056.66.“ Tmnfi 561:5 65 00 662269262260-.. .0 .mfim 2.220 "6 + ..6 06 ON 0m. 0w: 0m. 0w. 1 . u _ - .. QmT = C. __ .. 0.2. .s G __ w ,2 -0..- m V ql 1 ON... .... W w. I. GOT WM m 8 .. 8. m. 1 Om- 1 0.6.. 42 In the case of the 381 state the l = j—l dominant component of the solution behaves in a manner identical to the uncoupled solution. However, the l' = j + 1 minor component does not heal to any unperturbed solution. The resultant effect is that the minor component of the solution is sensitive to the sum of the single particle energies. Although it was not the purpose of this investigation to study the singularities of the t-matrix, it is worth noting that they constantly occurred in all states whenever the energy was varied. Fortunately these singularities did not occur in the energy regions which are important for the calculation Of the ground state energy. Theoretical investigations of these singularities have been performed . 22, 23 in several papers ’ but to the author's knowledge they have never been encountered in numerical solutions Of the BG equation. The fact that they do exist is evident from a cursory examination of the form of the defining equation for the t-matrix, viz. , equation (1) Chapter I. The singularities shown in Figures 7 and 9 22J. S. R. Chishold and E. J. Squires, Nucl. Phys._l_3 (1959) 156. 3 V. J. Emery, Nucl. Phys. _l_2_ (1959) 69;_1_9_ (1960) 154. 43 occur roughly at those values of E which satisfy the eigenvalue equation (E-ft'”>5‘°=° This is not the only value of E which gives rise to singularities. They occur quite frequently for energies at and above the Fermi surface, i. e. , when «fie-e; > O . Since these are Of considerable theoretical interest a possible future detailed examination Of them is being considered. In the past there has never been a clear demonstration Of the variation of the t-matrix from one nucleus to the next. In order to study this effect the t-matrix was calculated using a Pauli Operator which is apprOpriate to 40Ca while the single particle energies and oscillator parameter, #00 , were those appropriate to 16O. The 40Ca exclusion principle can easily be incorporated by extending the summation through the lsod major shell in the BG equation. In Table II the results of this calculation are compared to the 16O results. From this Table it is seen that there is some slight effect of the Pauli operator in going from one closed-shell nucleus to the next. All of the intermediate lsod shell nuclei should 0 possess Pauli operators in between the extremes of 160 and 4 Ca. However, since the effect due to different lsod shell Pauli 44 TABLE II. --The first column gives the matrix elements for 160. The last column gives the value Of the reactioh matrix using a Pauli Operator apprOpriate to 40Ca while the single particle energies are those of O. The numbers in the last column are not realistic matrix elements and are listed only to display the effect of the Pauli Operator in going from one major shell to another. 16 40 0 Ca 61+ 62 n'1' n1 jS NL to» = 14.50 14.50 -80 oo oo oo 00 -6.445 -6.342 10 -4.897 -4.811 00 oo 11 00 -7.051 -6.646 10 -5.206 -4.945 02 -6.373 -6.364 12 -s.939 -3.s73 -58 oo oo oo 01 -6.572 -6.467 10 -5.o11 -4.924 00 oo 11 01 -7.519 -7.228 10 -5.571 -5.389 02 -6.403 -6.367 12 -9.035 -8.928 -36 oo oo oo 02 -6.616 -6.575 (10) 10 -5.057 .5.023 00 oo 11 02 -7.774 -7.637 (10) 45 TABLE 11. --continued 16 40 0 Ca n'1' n1 jS NL 4;... = 14.50 14.50 10 -5.808 -5.724 02 -6.407 -6.396 12 -9.051 -9.b12 oo 10 oo 00 -5.278 -5.241 10 -4.451 -4.420 00 1o 11 00 -6.767 -6.489 10 -5.576 -5.396 02 -3.102 -3.124 12 -6. 117 -6. 125 46 operators is slight, it would be adequate to use some interpolated values for these nuclei since a precise calculation for these nuclei is extremely difficult, if not impossible. Solutions for a few of the higher angular momentum and higher nodal states are shown in Figures 10, ll, 12, and 13. In general these solutions exhibit the same properties as the previously discussed solutions. The higher angular momentum states heal more rapidly than the S-states and are almost completely independent of the values of the single particle energies. Bethe et a1. 11 have shown that the t-matrix should be Hermitian. The He rmiticity was tested for only the 1So and 381 states. By actual calculation with the same particle energies for the relative Is and 05 states, the following results were obtained: ([0 It; {06190) = - H7 (wagon/M) = 4.21 and :-:-f.$’/ :2 *fl/ <00 I€I{00)l/0> 47 O“ £05536 mo 62% HOG :65 was.” 6.205 m“ mcflmofi 628... m 5056250 6>63 “2026:2660 65.05.26: 62 65 53> . 2: £05625.“ 6>63 628263001656m 62 65 no 2206360500-.. A: .mfm ON .2 -r- 0.0 >22 mm... .... aw + .6 / >22 62 use . / em 1 6. I I I 0 o-.. 0-2 I. m. //. \65 48 22635669662 2.53306 65 00 6562200506 .2055 No paw 22%.er 65 6.2.6 N03 626 Om: .qu .cofiocdw 6>6>> noumzdomo 6508.26: 65 523 . 3 was ~23 5036.23 6.262.» OCOumeoUt656m 561:3 m~ 65 .«O 2206260500.... .- .wfim ES 2 )b b D ll.tql i - I - d - 1 >22 onlwwtw >22 6.2 use. o... 0.2 re .6. 49 .20 .moumumtm 65 mo 2cm 5 6.65 33mm.“ 6.85 6266: 22056223 6>63 622.. m 50560.3 6>63 noumflmomo 62:05.26: 65 59> .HOD 505693 6>6>> 6:0265H001656m mo 65 00 2206266050011 .Nn .wmh 28.3.. 0.0_ 0.6 0.6 0... 06 .. . _ I)... _ _ . . _ . a . OO \\ _O:\ ..l N. \ \ am I ¢ / .. 6 / >62 mm- "aw + ..6 / I m. >62 003.100.... / 0.2 0.2 0.... .a. . . I 0. I. N.— 50 66:66.. 65 05. 63.62266: .6056 a... mo 3 5262200506 .2055 62...... .306.“ 65.9.6 mm m 50565.5 6>6>> noumzmomo 6.205.262. 65 5.3 cofloce 636.5 6coum~6HOUt656m 56155 no 65 mo COmwummEOUtn .mfi .mmrm 0.0. 00 0.0 >62 66.. ..6 +6 >62 6.: u 3: 0...._ 0.2 0...... .6... -x .22. ON _ m0; .0 m\ n \ h 0.. N. 51 A complete list of the t-matrices relevant for the ground state properties of 16O are listed for the HJ and BGT potentials in Tables III and IV respectively. The single particle energies, -80, -58 and -36 MeV, correspond to the two particles in the 0808, osop, and opop states respectively. These values are the results which were ultimately used in calculating the binding energies. Once a good set of t-matrix elements has been Obtained there exists a wealth of experimental data which should provide good tests for both the t-matrix and the numerical results. Unfortunately rigorous tests usually involve either shell-model or Hartree-Fock calculations. Each of these calculations involves considerable effort. However, it is possible to immediately calculate the approximate binding energy of 16O and the TOE-splitting in 17O. In Appendix D it was shown that the approximate binding energy Of 160 is given by the following 52 TABLE III.--Matrix elements, < n'1'] th(NL)I nl>, for HJ potential. {w él-téé .61' n1 jS N1. 13.35 14.50 15.50 -80 00 00 00 00 -5.973 -6.445 -6.838 10 -4.733 -4.897 -4.988 00 00 11 00 -6.617 -7.051 -7.397 10 -5.127 -5.206 -5.222 02 -5.578 -6.373 -7.089 12 . -7.892 -8.939 -9.873 -58 00 00 00 01 -6.101 -6.572 -6.962 10 -4.852 -5.011 -5.106 00 00 11 01 -7.080 -7.519 -7.867 10 -5.508 -5.571 -5.568 02 -5.612 -6.403 -7.115 12 -7.987 -9.035 -9.968 01 01 10 00 1.757 2.029 2.284 11 1.875 2.197 2.504 01 01 01 00 -2.144 -2.389 -2.599 11 -1.779 -1.902 -1.993 01 01 11 00 1.849 2.133 2.393 11 1.985 2. 279 2. 549 TAB LE III.- - continued 53 47w él-t xfl' n1 jS N1. 13.35 14.50 15.50 01 01 21 00 -O.834 -0.990 -1.135 11 -l.118 -1.310 -1.485 03 0.682 0.794 0.896 13 +0.924 1.062 1.185 -36 00 00 00 02 -6.150 —6.616 -7.001 (10) 10 -4.904 -5.057 -5.147 00 00 11 02 -7.363 -7.774 -8.094 (10) 10 -5.776 -5.808 -5 776 02 -5.617 -6.407 -7 117 12 -8.007 -9.051 -9.981 00 1o 00 00 -5.062 -5.278 —5.430 10 -4.426 -4.451 -4.425 00 10 11 00 -6.566 -6.767 -6.876 10 -5.631 -5.576 -5.454 02 -2.809 -3.102 -3.345 12 -5.470 -6.117 -6.673 02 02 20 00 -0.497 -0.589 -0.675 12 -0.552 -0.650 -0.740 TABLE III. - -continued 54 #7..) 511-6; nu. nl jS N1. 13.35 14.50 15.50 02 02 11 00 1.676 1.361 1.537 12 1.099 1.252 1.389 00 -5.571 -6.363 -7.076 10 -2.853 -3.141 -3.386 02 02 21 00 -2.114 -2.459 -2.772 12 -2.141 -2.443 -2.710 02 02 31 00 0.038 +0.044 0.049 12 0.025 +0.026 0.027 04 -1.039 -1.239 -1.427 14 -1.403 -1.652 -l.882 55 TABLE IV. --Matrix elements, , for BGT potential. .520 e] +62 tf1 n1 jS NI. 13.35 14.50 15.50 -80 00 00 00 (x) -6.222 -6.683 -7.061 10 -4.697 -4.795 -4.838 00 00 11 00 -7.817 -8.426 -8.931 10 -6.896 -7.199 -7.416 02 -5.603 -6.371 -7.055 12 -7.879 -8 791 -9.608 58 00 00 00 01 -6.343 -6.800 -7.174 10 -4.806 -4.897 -4.933 00 00 11 01 -8.317 -8.939 -9.453 10 -7.302 -7.594 -7.797 02 -5.627 -6.389 -7 067 12 -7.968 -8.881 -9.698 01 01 10 00 2.553 2.916 3.242 11 2.574 2.881 3.156 01 01 01 00 -2.983 -3.302 -3.575 11 -2.342 -2.479 -2.584 01 01 11 00 1.992 2.288 2.558 11 2.087 2.373 2.637 01 01 21 00 -1.336 -1.553 -1.752 TABLE IV. - -continued 56 4faa 614.62 n'1' nl jS NL 13. 35 1.4. 50 15. 50 11 -l.584 -1.823 -2.041 03 0.930 1.071 1.199 13 1.254 1.404 1.542 -36 00 00 00 02 -6.388 -6.814 -7.210 (10) 10 -4.853 -4.940 -4.917 00 00 11 02 -8.554 -9.150 -9.683 (10) 10 -7.519 -7.782 -7.459 02 -5.621 -6.381 -7 058 12 -7.969 -8.879 ~9.694 00 10 00 00 -4.928 -5.064 -5.143 10 -4.094 -4.038 -3.946 00 10 11 00 -7 848 -8.187 -8.439 10 -7.117 -7.214 -7.246 02 -2.518 -2.698 -2.841 12 -4.921 -5.360 -5.749 02 02 20 00 -0.668 -0.809 -0.942 12 -O.862 -1.015 -1.159 02 02 11 00 1.761 2.030 2.313 TABLE IV. --continued 57 in: +02 n'1 n1 jS NL 13. 35 14. 50 15. 50 12 1.804 2.031 2.270 00 -5.701 -6.367 -7.050 10 -2.653 -Z.815 -2.965 02 02 21 00 -2.298 -2.718 -3.106 12 -2.659 -3.060 -3.429 02 02 31 00 0.095 0.079 0.058 12 -0.087 -0.155 -0.269 04 -1 088 -1.312 -1.524 14 -1.656 -1.915 -2.161 58 ' 411m. B E ; /gfiw +fz; L (2.7+!){ZJ'+I)(I-(~I)’+$+r)(alsjltflthm/Sj) ' ' r57 2. OCCUPIED (aIH-I) [fit A l x : (2’44) l mlltatil. A The coefficients in this expression have been tabulated by Mackellar9 and have been used in this paper to calculate the binding energy per nucleon as a function of the oscillator parameter, k w. The final results are plotted in Figure 14 for the HJ and BGT potentials. For the HJ potential a minimum of -3. 6 MeV/A occurs at about {00 =14 MeV, and for the BGT potential a minimum Of -7. 8 MeV/A occurs at about {‘0 =18 MeV. The experimental value is -7. 98 MeV /A. 24 The BGT potential is known25 to be unacceptable as a fit to nucleon-nucleon scattering for the T = 1 states, but is as good as the HJ for the T = O, 381 state. Since the central force is not affected as much as the tensor force by the Pauli Operator and since the BGT triplet force has a strong central part, its Sl matrix elements give much more binding than the HJ potential (see TABLES III and IV). 2 4L. A. Konig, J. H. Mattauch, and A. H. Wapstra, Nucl. Phys. _3_1_ (1962) 18. 25p. Signell and N. R. Yoder, Phys. Rev. 132 (1963) 1707. 59 ‘30 " \ / HJ % -5.0 - g . Q .3: ~70 - BGT l l l I l l l IZO - l4.0 ISO 180 1‘1w(MeV) Fig. 14. --The binding energy per nucleon for 16O as a function of the oscillator parameter for the Hamada-Johnston and Brueckner-Gammel- Thaler potentials. 440 .- 2; -.6. - if :3 ~18!) - H.) 1 . I ' . I . L IZO l4.0 l6.0 l8.0 1101(MeV) Fig. 15. --The binding energy per nucleon for 160 when the Pauli Operator is neglected. 60 Moreover, it appears that the calculations of binding energies in finite nuclei will follow the oscillating path of nuclear matter calculations. 26 Clearly higher order effects need to be examined for finite nuclei as they have been for nuclear matter. A series Of papers7' 8' 9’ 26 16 for O by different authors using different methods reveal con- On binding energy calculations siderable discrepancies. Table V is a sample listing of the t-matrix and B. E. /A results of different authors. TABLE V. --Comparison of diagonal t-matrix, (n1 ( t.S(NL)l nl> . and B. E. /A with results Of various authors using the HJ potential. Kuo g, Kohler 8: Present Brown Wong MacKellar McCarthy Calculation {co l4. 0 13. 5 20. 7 10. 5 l4. 5 (00) t01(oo)' oo) -5. 61 -5. 7 -9. 57 -4. 36 -6. 44 (ooltl3(oo)l oo) -9.73 -6.2 -1l.22 -5. 37 -7.05 B. E. /A -5.5 -- -8.0 -2. 70 -3.6 26 S. A. Moszkowski, Rev. Mod. Phys. _3_9 (1967) 657. 61 Although the results of Mackellar fit the experimental B. E. /A quite well the re is considerable doubt about the Eden-Emery treatment of the intermediate single particle energies. Moreover, the value Of {u at which self- consistency is obtained is about 20 MeV and this seems at variance with the experimentally determined value of 14 MeV. 20 As suggested by Kohler and McCarthy27 it would be of interest to see the Eden- Emery calculations repeated treating the energy denominators in a more precise manner. It is interesting to observe that the calculations of Wong, Kuo and Brown, 7 and Kohler and McCarthyZ7 are similar to the results of this paper. This is somewhat remarkable since their calculations are quite different in both point of view and in calculational details. In particular the choice of the intermediate state energies in those papers are the free particle energies as opposed to harmonic oscillator energies used in this paper. It is difficult to compare the present calculation with that of Kallio and Day28 since they choose a potential which is radically 27H. s. Kohler and R. J. McCarthy, Nucl. Phys. 29(1967) 65. 28A. Kallio and B. D. Day, Phys. Letters, E (1967) 72. 62 different from the HJ potential and ignores the difficulties of the tensor force. It is interesting to compare the B. E. /A calculated with and without the Pauli operator. Figure 15 is a plot of the binding energy per nucleon calculated without the Pauli operator. There is no minimum as there is when the Pauli Operator is included and the nucleons are greatly overbound by about 8 MeV/A. As a final calculation the T‘S splitting in 17O was computed 29 using an approximate expression due to Nigan: [S If 15 I: E. -E, =/Jé—[90(4, +37%; Hug, +714] where and V15: V13“) 1...? The experimentally observed splitting is 5. 9 MeV and the calculated results are 4. 01 and 4. 56 MeV using the HJ and BGT potentials respectively. 29B. P. Nigam, to be published CHAPTER IV CONC LUSION The results of this study show that it is possible tO solve the Bethe—Goldstone equation exactly for finite nuclei. The Pauli operator for finite nuclei has been treated in an exact manner and its qualitative and quantitative effects have been examined in detail. It has been found that when the Pauli Operator is treated correctly the BG wave function heals properly to the unperturbed wave function. In addition it has been shown that the effect of the Pauli operator on the t-matrix is appreciable in the 381 states and moderate in all other states. Since the numerical methods developed to solve the BG equation with the exact Pauli Operator are so efficient it seems unwarranted in future calculations to make various questionable approximations when solving the BG equation. With accurate two body t-matrices available it is now possible to examine the higher order effects with more confidence. Since the current trend in nuclear structure calculations for 3, 30, 31 both the ground state and excited state properties relies 305. Das Gupta 8. M. Harvey, Nucl. Phys. A94 (1967) 602 31Ripka, Lectures in Theoretical Physics Vol. VIII B (1965) 237 64 upon HF calculations an examination of the t-matrix dependence on the HF single particle energies was performed. It was concluded that the t-matrix for the 381 state is sensitive tO the single particle spectrum, and in doing a HF calculation it is important that these single particle energies be chosen correctly. For other states the single particle energies are not critical, and it should not be necessary to treat these in a self-consistent manner. The calculation of the binding energy per nucleon in this study and in those of references 7, 8, and 27 using the HJ potential give 2. 7 to 5. 5 MeV as compared to the observed value of 7. 98 MeV. Kuo and Lynch32 report similar results using the Yale and Reid potentials. These discrepancies clearly show that it will be necessary to perform additional calculations, and there are several improvements in these calculations which need to be incorporated. The first possible improvement is to pursue the HF calculations with more vigor. Although reference 3 gives an excellent treatment of the HF problem the authors of that paper suggest the following improvements which could bring the binding energy in line with the experimental value: First, the problem should be done in a doubly self- consistent way, and second, more nodes should be 32R. P. Lynch and T. T. s. Kuo, Nucl. Phys. A95 (1967) 561 65 included in the expansion of the HF wave functions. Efforts along these lines have been initiated. The second explanation of the difference in the experimental and calculated results is directly related to the nucleon-nucleon potentials which are currently in use. Since the spectrum calculations of Kuo and Brown7 rely basically on the long range part of the potential and give excellent agreement with experiment, it is believed that the poor binding energy results may be due to the short range part of the potential. In particular a potential with a smaller hard core radius or a soft core would improve the binding energy results. Along these same lines it has been suggested by several authors 33’ 34 that one should by pass the use of potentials and attempt to calculate the t-matrix for finite nuclei by working directly with the experimental phase shifts. This avenue Of approach would be very desirable although new calculational techniques would have to be devised, and the attempts to date--though admittedly initial attempts--have not shed any light on how to handle the Pauli operator in conjunction with the phase shifts. 3'B’J. P. Elliot, H. A. Avromatis, E. A. Sanderson, Phys. Letters 24B (1967) 358 34Koltun, unpublished 66 Finally, there is the difficult question Of the convergence of the Bethe-Goldstone expansion and the effect Of higher order terms in finite nuclei. Bethe and Rajaraman35 have shown that three body correlations are important in nuclear matter and these contributions will also have to be examined for finite nuclei. In summary it is believed that the Hartree-Fock, higher order cluster effects, and better potentials individually would contribute at best 1 MeV per nucleon. SO collectively one may expect an improvement Of the order Of 2 to 3 MeV which would still leave the binding energy too small by roughly 2 MeV. These crude estimates indicate that it may finally be necessary to include specifically three body forces in order to get good a- greement with the experimental binding energies. 35 R. Rajaraman and H. Bethe, Rev. Mod. Phys. 39 (1967) APPENDIX A TREATMENT OF HARD-CORE In this Appendix an approximate expression for the hard core contribution to the t-matrix will be discussed. Once the BG wave-function, ‘f’, has been obtained the t-matrix is calculated from I (alt/Z) = ffiIMZ/f .-.— [ +1}_ (1) where _. Lin 6 I, " 6+0 ff“ Carcfijf (2) I.) tiijfiaA’li/f (3) I: The integral, 12, is straightforward to evaluate using standard numerical integration. The first integral, 11’ presents some diffi- culties since v is infinite for ffi f: ; however, by using the BG 68 From the BG equation it is possible to approximate vcore W n’ equation for {5 f we have 0 Cer - 57” = (EA-)4)?” - (am-amQ/(g) +Zc”£ (mar/)3.) (4) Using equation (4) in (2) gives 6 . (:7. z = U" { (Egg-71) 45. -€)<£lt>£.£ ] 1f 7 é—fe {+6 (5) .§ c‘<{lnrlg,>)Z/:J{G)/f( It has been found in actual calculations that the last two terms in equation (5) are small compared to the first term. This is directly related to the fact that list/r ’~‘= ’0': ‘6’ Therefore, I 3‘ Li») f£’{m—f{,)z’/f (7) 69 Since 5 n is zero for f we this integral is zero except at ’2 fc ‘ where d has a g-function singularity. Then, Jr‘ (:f‘ z 4. Live A" ,éfi/ I, : s-n {m :g, f‘ f (8) Integrating by parts gives f ‘ {‘5‘ f 2’ Li" 3:- {fa/c. ’ I‘d/n'1‘! J! I, ' 5+0 1M Z'jf ‘6‘ -‘77 ‘7? Since 5; and it are both finite quantities, the second term goes to zero as 6+0. Therefore, we have 2 jfif”. I, 371%.. {“77 A”; (9) Equation (9) can be evaluated numerically and 11 has been added to all calculations of the t-matrix. 2: APPENDIX B ' NUMERICAL METHODS In this appendix the numerical methods used to solve the BG equation are discussed in detail. The method used is basically an extension of the Fox-Goodwin36 method. In the past, the Fox- Goodwin method has been applied to differential equations, and to the author's knowledge, ithas never been used to solve integro- differential equations. After dr0pping all subscripts in the BG equation, the integro- differential equation which was solved is of the form U; 5;. - woju/r) = (A - X ) WINK/“P " Z C‘R“/!) (1) By expanding u and u' in a Taylor series and performing a little algebra, we get the following three-point equation * The author is indebted to Dr. P. Signell for initially suggesting the method for solving differential equations which was ultimately extended to treat integro -differential equations. 36L. Fox and A. Goodwin, Proc. of Cambridge Phil. Soc. 1? (1949) 373. 71 A . ,—. ”)+(u viiuf’)“ (2) (M ..__ ' )— (103' +74 “J JL, ’1 J” where o is the mesh width and j denotes the jth mesh point. From equation (1) expressions for u',‘ , u" , and u',‘ j'+l 1 can be obtained, and when these are substituted into equation (2), we get . .-+L(~ C'. : -—°..4.‘ OR'FR R!“ “J4"; * $51 ,-, J (A A)». U}, *’ J x—zk > I a, a. “ “ " I“ (3) '7’;ch MI,» W} WM ’N‘ >1 where z 5 : I +fi'(/\—JH) BJ : ..2 *f6‘(’\‘g (4) «(A H s? IA F ’\ > | 1‘ { V 72 Let u* be a trial wave function and 5 u be the change in the trial wave function. Then M... = Ha. +501, (5) 4r * +8.5”. +0.5“, 60;“ 4—ng +$L$ +55“. 1,! JJ/ 0 n,— mum/s + Mam +3 <6» ..gfl’; 6"(5: ,wol? +/§,,)][(R" 'N'l“) +0? [xv-Isa) At this point the plausible assumption is made that q- << (RIM > and (RIM/5U) << «IN/W 73 Even though a given uj may be comparable to a given Suj, these assumptions imply that on the average Su is much less than u*. The validity of these assumptions is ultimately justified by the rapid convergence of the solutions. When convergence is obtained, Ju = O, and there is no question of approximation. The final results can be made as accurate as desired, limited only by the accuracy of the computer. After dropping the small terms in equation (6) we get (87' +- 5 503+! + @5037 + (3.] “ff-I a 0 (7) where k . 0" QJ. E J 69:, +§ u; f (3.09,, " (A‘A >;—i[@.H+/O@.+ ’5;,] z fly to Go ~ I. * +;°;[{:c 03,, + My +6-,)] (3) Next let (9) 74 Then equation (7) becomes 4} +55%, +93%. +9.0? +H. 565,.) -= o (10) The Aj' Bj' C. and Qj- are well defined numerical quantities, and the Fj and Hj can be determined provided F0 and Ho are known. F0 and Ho can be determined from the boundary conditions at the hard core, viz. u(Core) = u*(Core) = 0. Then S“ g 0 = I: + ILLS“, (13) Since F0 and Ho are independent quantities F0 and Ho must be zero in order to satisfy equation (13). . . . . 3 Since the ultimae aim 18 to calculate each of the Suj from equation (9), we now need to know SuN where N denotes the 75 last mesh point. 5 uN is found from the asymptotic form of the BG equation. From equation (1), we have {MN-0.721 J}:- (14) U °( 6 : e N and 4m}, (15) u °< C ”H From equations (14) and (15) we get 'd‘(/'1N)/Z (16) M” = Sum e Combining equation (16) and (9) gives “mafia/z)” (17) S” =-’:(”N-e N+I Since the Fj s and H, S and Suj+l S are known, all of the Suj J I can now be obtained from equation (9). Once the Suj 8 are known a new trial wave function u** can be computed using equation (5). After u** is obtained the whole procedure is repeated with u** replacing u*. The iterative process is terminated when ZSuj is less 76 than some arbitrarily small number. Usually the convergence is quite rapid if one is sufficiently careful in choosing an initial trial wave function. Even with ”bad” trial wave functions the convergence is satisfactory. In practice the initial trial wave function is chosen to be of the form m) = (I - t/r) W1“) (is) This form of the wave function fulfills the necessary requirements at the hard core and asymptotically. The method outlined above is applicable to both coupled and un- coupled equations. For the case of coupled equations the only differ- ence is that the wave function, u, is a vector quantity and the expression for the A's, B's, C's, H's, F's etc. are matrices. For completeness the method used to solve the ordinary differ- ential equation will be outlined. The form of this equation is ‘L [,\ +fi. «WNW/r) =0 (19) where A is now an eigenvalue to be obtained along with the wave function, u. Equation (19) is used to find u',’ , u',’ and u',‘ and J 3+1 J-l these expressions are substituted into equation (2) to give .U..+..=0 20 L9+I€lflg MC} () where Aj, Bj, and Cj are defined by equation (4). Now let u* be a trial wave function and )8“ be a trial eigenvalue. Then (,3, : L3” + Suj- (21) A : A” +$/) (22) after substituting equations (21) and (22) into equation (20) we get +59% +0.”. =0 (23) where terms of the order {USA have been dropped and - ,p u * f f * Q] " ujH 5’ + L; g ‘P (31/ C} (24) : Oz ' f ’9 1.)" " 72[”1'+z H“? " 3M] ‘25) 77 78 Next let {a}, = g + (9569., +5“ (26) Then substitute equation (26) into equation (23) and rearrange terms to get \\ t Ir *‘ M,- = — [5;- +9.51, +553.,+(9'+95.)“7 <2“ Q‘Q. where E +— C' I4 41 {3} J J" (28’ By comparing equation (27) and (26) term by term we get _—. - " F. . 5' (g; + 5 r/ > /‘$ * (29) H. = ' ‘3' ‘5' f " — . o + ' I 79 The Aj, Bj' Cj' Dj and Qj are well defined numerical quantities and the Fj’ Hj’ and E2j can be determined if F0, Ho and E0 are known. F0, H0, and E0 can be determined from the boundary condition at the coare, viz., u(core)=u*(core)=0. Then 5a,:5 +H,«W, +5.3)” ‘30) and since F , H and E are independent quantities F , H , and E o o o o o 0 must be zero in order to satisfy equation (30). Since the ultimate aim is to calculate each of the éuj '3 and S) we need to know 5 u where N denotes the last mesh point. As was N done earlier Su is found from the asymptotic boundary condition. N For this case we simply require that the wave function be zero at large distances. This makes SuN and JuNH zero. Therefore, from equation (26) we get g) = ..g/t} (31) I The solution is now complete since all the Fj's, Hj's, and Ej 8 are well defined in terms of the trial wave function and trial eigen- value, and 8') and the Suj's can be calculated using equation (31) 80 and (26). Once the SA and Suj's are determined a new eigenvalue A ** and a new wave function u** can be computed and the whole procedure is repeated until Z S uj and 51/) are arbitrarily small. APPENDIX C PARAMETERS AND PROPERTIES OF THE NUCLEON-NUCLEON POTENTIAL The form of the HI and BGT potential ”(4) = /\é’(4) + /%(4-)j'§ + Afifd) Lu +Nr’fa) 5;; where I-§ = fEJYjH) —!/J+I) -$’($H)] - 2. L 2LT. +5?'o'fjifi+/)—{J°§) (in H1) I:- [J (— - 0 (70v BGT) I?» = lJEI—wflaf 2:) -—é7"-o'i" 82 The radial dependence of the HJ potential is given by ”(+)=+-O flgfi n40») = o°7(+4)(fi-?.)74)[/ May/047%] N’SHv) =4 6;:5 VIMD 4' 5,, WM] nag.) = 75“. 4:2. z(+)[/ +- any/7t) +6“ W60] where -+ WM = £— 20) = (I ti +5))’/+) ,7 is the pion mass (139. 4 MeV), and x is measured in ‘fi/qc = 1.415fm. The hard core radius in all states is 0. 343 fi/A/c , and corresponds to rc = . 485 fm. The parameters for the HJ potential are given in Table VI. 83 TABLE VI. --Parameters for Hamada-Johnston potential. State ac bc aT bT GLS bLS GLL aLL bLL Singlet +8. 7 +10. 6 0 0 0 0 -0. 000891 +0. 2 -0. 2 even Triplet -9. 07 +3. 48 -l. 29 +0.55 +0.196l -7. 12 -0. 000891 -7. 26 +6. 92 odd Triplet +6.0 -l.0 -O.5 +0.2 +0.0743 -0.l +0.00267 +1.8 -0.4 even Singlet -8.0 +12.0 0 0 0 0 -0.00267 +2.0 +6.0 odd The radial dependence for all states of the BGT potential is given by Ara.) = +00 Ag/tc -44, 3 ‘KL— A >/"c. ’74 The parameters for the BGT potential are given in Table VII. 84 TABLE VII. --Parameters of the Brueckner-Gammel-Thaler potential. 3 l 1 3 3 3 3 P . arity V 7c . V 4c V 7t V18 “’18 + 877.4 2.091 434.0 1.45 159.4 1.045 5000 3.7 - 14.0 1.0 -l30.0 1.00 -22.0 0.80 7315 3.7 85 APPENDIX D DERIVATION OF BINDING ENERGY In this appendix the expression for the total energy of a closed- shell nucleus is derived in terms of the reduced matrix elements. The potential energy to first order is given by Fifi” (1+4; . 1' MI' 441’1’9 I 11’) $ 0],, M34117." (1) X «Md/4401;”. m, Mir/(mg (4». )1; If; Mm.) Where both the ket and bra are the prOper normalized, anti- symmetric wave functions given by equation (4) Chapter 11. After substituting the transformed two particle state vectors into equation (1) and using the orthogonality of the center-of-mass wave functions the following is obtained: 86 new [on 41,11 P‘VZ Z CA! I ; [I—f-I) J S'A’M’J’ {’fl’f “I I:- x Z A: 7’/C(j’LMM/ TH ,')<(11)J.«M)t[NL)/(z’3’)j»9.m’> A’ 1"”? Next we split the wave function into its radial and angular parts, use the fact that the interaction is diagonal in S, j, mj, and utilize the orthogonality of the Clebsch- Gordan coefficients to obtain 87 pl; 5,, 2; (ar+/) Q'Iofi‘" "A‘J:’57L lir+r x «maln’l'mmfl I-M’m'lf PM J 171' x (”lgj'ltmulmll If); (zJ’+I)/\’M, (3) 4/.1'1' “if. x ' . T' .2: 7;” $437 «7.1; For a closed-shell nucleus the sum over jl jz can be eliminated by using the properties of the LS-jj transformation coefficients. If we also use the properties of the 6-j symbols, viz. , [5' 1’5’ . . £01,“) A“: An: = (1J+I)(2A+I)§Jt/(zjfl) (4) .7 then we get 88 P= i 2:, ZL/‘TWD -(-> >Wr-7/2fl') rsj AINLAI’ M (_z__4 H) x (alsj/tr~t)lfl”5i>2:t (7”,) (5) «(at/g l) x