SOME OBSERVATIONS CONCERNING THE USE OF REALISTIC FORCES IN A MICROSCOPIC DESCRIPTION OF THE INELASTIC SCATTERING OF NUCLEONS FROM NUCLEI AT MEDIUM ENERGIES Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY FRED L. PETROVICH 1970 THESIS LIBRARY Michigan Stave University This is to certifq that the thesis entitled SOME OBSERVATIONS CONCERNING THE USE OF REALISTIC FORCES IN A MICROSCOPIC DESCRIPTION OF THE INELASTIC SCATTERING OF NUCLEONS FROM NUCLEI AT MEDIUM ENERGIES presented by Fred L. Petrovich has been accepted towards fulfillment of the requirements for Ph.D. degree in Physics A ' //} 1%Leg “Ah/”rm I‘ Major professor Dme January 22, 1971 0-169 ,. ». ... .ln p» .. e . L, .p: .p; . ‘ INC. t . I T ,4 n. e. . M. .r.. . .I L r; .r.. «Q L . r” .... p; v.. I. u. I. :‘ .1 2‘ ,u . .w« n. ,3 . . C :I . rt: .3 . ‘ n. e. 3.1 ha. r? n. . I. .C +e r? T. f. L . . u... a» .3 n. f: I: _.. u; L. um . ¢ « ~.I h. S. L» L C vi. ‘ V». r“ CI t. . 3‘ ~ , » .\ r.. ,r.. IV . a I; . . s ._ u u h TC . . r . e . r». ‘ 3. vv. . ABSTRACT SOME OBSERVATIONS CONCERNING THE USE OF REALISTIC FORCES IN A MICROSCOPIC DESCRIPTION OF THE INELASTIC SCATTERING OF NUCLEONS FROM NUCLEI AT MEDIUM ENERGIES By Fred L. Petrovich The problem of describing, in a microscopic picture, the process of inelastic nucleon-nucleus scattering at inci— dent energies in the 15-70 MeV range is of current interest. Of primary interest are the properties of the projec— tile-target interaction. In this work several models for this interaction are investigated by direct calculation. All of the interaction models considered are consistent with some portion of the data concerning the free two—nucleon, force; hence, the term "realistic forces" which appears in the title of this paper. To be specific, it is assumed that the projectile-target interaction is given by (l) a pseudo- potential derived from the impulse approximation, (2) the long range part of the Kallio—Kolltveit potential (K—K force) which is known to be a good approximation to the central part of the shell model reaction matrix, and (3) a Yukawa force derived from effective range theory. Fred L. Petrovich This study is restricted in that the local distorted wave approximation (D.W.A.) is used throughout and no consider- ation is given to components of the interaction with compli— cated spin dependence such as the tensor and 1-5 parts. Approximations are made to treat the exchange component of the D.W.A. transition amplitude which is non—local. This component appears because of the required antisymmetrization of the projectile—target wave function and it has been neg— lected in most recent work on this problem. These approxi- mations are discussed and some comparisons with exact calcu- lations are presented. Application is made to (p,p') transitions in closed and pseudo-closed shell nuclei. Random phase approximation (R.P.A.) state vectors are used to describe the states of the target nuclei. Studies of the (e,e') reaction and the (p,p') reaction (at incident energies in excess of 100 IeV) have shown that these vectors give a good description of the transitions considered; therefore, these calculations provide a test for the proposed interaction models. The results obtained with all three interaction models are shown to be in reasonable agreement with experiment, although the Yukawa effective range force appears to be somewhat poorer than the other two at incident energies below 30 MeV. The inclusion of exchange plays an essential part in giving this agreement. In most instances deficiencies in the shapes of the theoretical angular distributions are noted. Fred L. Petrovich Further application is made to transition involving low lying states in nuclei which possess one or two nucleons outside of a closed shell. The purpose is to study core polarization effects which are known to be important in these transitions. The effects are estimated in calcula- tions which use either a micrOSCOpic model or the macroscopic vibrational model to describe the core. Emphasis is on the completely microscopic calculations which assume that the core can be described by a zero order shell model Hamiltonian and that only the effect of simple particle—hole excitations of this core with energies up to roughly 2gb need by consid— ered. The coupling between the valence nucleons and the core is treated by first order perturbation theory and the K-K force is taken to be the coupling interaction. This model is essentially the same as the one used recently by Kuo and Brown in work on the spectra of nuclei of this type. Contributions to (p,p’) cross sections due to core polariza— tion are large. The relation between the effect of core polarization on the spectrum and in inelastic proton-nucleus scattering is examined. The microscopic model doesn't do too badly on the (p,p’) cross sections, i.e. mass polarization effects. The experimental data is underestimated somewhat. However, effective charges for corresponding y—transitions, i.e. charge polarization effects, are badly underestimated. One case is found where this model does badly on the mass polarization. This is explained by explicitly taking into account the effect of a highly collective state in the core nucleus. Fred L. Petrovich From this study it is concluded that a reasonable description of this class of reactions is obtained using 'realistic forces'provided the treatment includes the effects of (l) antisymmetrization and (2) long range correlations in the target nuclei, in particular, core correlations (R.P.A.) and core polarization. SOME OBSERVATIONS CONCERNING THE USE OF REALISTIC FORCES IN A MICROSCOPIC DESCRIPTION OF THE INELASTIC SCATTERING OF NUCLEONS FROM NUCLEI AT MEDIUM ENERGIES By 3y Fred L3 Petrovich A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department oszhysics and Astronomy 1970 ACKNOWLEDGEMENT I would like to thank Professor Hugh McManus for suggesting this problem and for his constant guidance and support during the time that this work was performed. Thanks is also due to Dr. G. R. Satchler for assis— tance in the initial stages of this work, to Mr. J. Atkinson and Dr. V. A. Madsen for assistance with a particular phase of the work, and to the many people at the M.S.U. Cyclotron Laboratory whose interest in this work provided a constant source of encouragement. Mrs. Julie Perkins is also thanked for typing the manuscript. Lastly, I would like to thank my wife, Paula, and my sons, Michael and Joseph, without whose patience and understanding this work would not have been possible. ii TABLE OF CONTENTS ACKNOWLEDGMENT . . . . . . . . . . . LIST OF TABLES . . . . . . . . . . . LIST OF FIGURES. . . . . . . . . . . Chapter 1. INTRODUCTION . . . . . . . . . 2. DETAILS OF THE DISTORTED WAVE APPROXIMATION. 1. mm DU.) N 7. D.W.A. Transition Amplitude and Cross Section . . . . . . . . Form Factors . . . . . . Integration Over Internal Coordinates Final Reduction of Partial Matrix Element . . . . . . . . Zero-Range Interaction . . . . Approximate Treatment of Antisym— metrization . . . . . . . Transition Densities. . . . . 3. IMPULSE APPROXIMATION PSEUDO-POTENTIAL N. THE PROJECTILE-TARGET INTERACTION . . 5. THE APPROXIMATE TREATMENT OF ANTISYM- METRIZATION. . . . . . . . . . l. Yukawa Function . . . . . . 2. Transitions in Zr90+p . . . . . 12 16 A0 3. Transitions in C O , and Ca + p A. Summary . . . . . . . . . . 5 o K-K Force 0 o 0 o o o o o o 6. Effective Range Forces . . . iii Page ii viii 10 10 12 19 23 33 3A 39 A2 A8 Chapter 6. STUDY OF INTERACTION MODELS IN D.W.A. CALCULATION. . . . . . . . . . 1. Section A . . . . . . . . . . 2. Section B . . . . . . . . 7. CORE POLARIZATION IN INELASTIC PROTON— NUCLEUS SCATTERING . . . . . . 1. Introduction . . . . . . . . . . .50 9O 2. Calcul tions and Results — T1 , Zr , and Y 9. . . . . . . . . . . 3. Single Proton lh9/2-li13/2(Q=—l 61 MeV) Transition in B1209 . . . . 8. SUMMARY AND CONCLUSIONS . . . . . . REFERENCES . . . . . . . . . . . . . . Appendix A. APPROXIMATE SERIES OF EXCHANGE COMPONENT OF D.W.A. TRANSITION AMPLITUDE. . . . . . B. TRANSITION DENSITIES AND FORM FACTORS. . . l. Harmonic Oscillator Wave Functions . 2. Macroscopic Vibrational Model. . . . 3. Reduced Matrix Elements and Transition Densities for Various Transitions. A. Note on Phases. . . . . . . . 5. Multipole Coefficients . . . . . . C. INELASTIC ELECTRON—NUCLEUS SCATTERING. . . D. CORE POLARIZATION . . . . . . . . . I. Introduction . . . . . . . 2. Macroscopic Treatment of Core Polariza— tion . . . . . . 3. MicrOSCOpic Treatment of Core Polariza- tion . . . . U Microscopic Empirical Formula. . iv Page '7 Q8 125 139 139 14A 209 216 218 227 232 232 23A 239 ( C2 pad PA} I Ah/ 5.3 5.A 5.5 5.6 LIST OF TABLES E ST6L from bound state matrix elements of Kuo and Brown . . . . Composition of core transition densities for L=2—8 transitions in Zr9O . . . . . Composition of core transition densities for L=2—6 transitions in Ti5O . . . . Composition of core transition densities for L=O transition in Zr9O . . . . . . Composition of core transition densities89 for transition to Q=-.908 MeV level of Y Optical parameters used in calculating the ngo, T150, and Y89 angular distributions . Decomposition of integrated cross sections corresponding to results shown in Fig. 7.3, 7014, and 705 o 0 Theoretical and experimental values for a: corresponding to the results shown in Fig. 7.3, 7.A, and 7.5 . . . . . . . Effective charges for electric 2L—pole com— ponents of transition amplitudes for Zr90, T150, and Y89. . . . . . . . . . . vi 9A 117 118 1A6 152 156 168 170 177 180 19A Table 7.10 7.11 7.12 Experimental and theoretical values for the normalized proton and neutron transition densities for quadru ole transitions in Zr90, T150, N159, and Pb207 . . . . . . . . Normalized proton and neutron transition densities as given by the particle-hole model and particle-hole model with renormalized force for L=2-8 transitions in Zr90 and for L=2-6 transitions in T150. Theoretical and experi- mental enhancement factors are also shown. For Zr90 the experimental a values are from Ref. 67. The T150 8) values are estimates Particle and hole orbitals used in micro— scopic calculations for Bi209 . . . . . . vii Page 20A 207 211 14 Cu. ». . o. w. A .1. r. 1. M . v '. _.— Yu. Vs K fl.» n a. w. . ...u t _ .. I. Y“ .C .u :. v. 3 C. :. .. . .. . .. rm 2 .1 _,. a a. 3 ,1 . ... I. ... u; . . a. 14.... _ .. a. Tr. + .u . .1 . h. C. n. . .. .x. .q. I. :D I.» H- ,.4 . u o t | I t sL .. —. . .n .pa —. a .x: _..J ~s . :- Figure 3.1 LIST OF FIGURES Real part of the % (3AO+A1) component of the free two-nucleon scattering amplitude as a function of q . . . . . . . . . . . Comparison of approximate and exact results showing the variation with energy and inter— action range of sex/o ir for several multipoles in the Zr90(p,p')Zr90 reaction . . . . . Comparison of Odir’ Oex’ and CT as function of L for the m=o.5 and 3.01?"1 cases of Fig. 5.1. Both approximate and exact results are shown for o and o . . . . . . . . . ex T Exact value of CT for a 2F range Yukawa force is compared with Odir for a 1F and 2/3F range Yukawa force as a function of L . . . . Direct and approximate exchange angular dis- tributions for 2F range Yukawa force for L=O, 2,A,6, and 8 transitions in Zr90+p at 18.8 MeV . . . . . . . . . . . . . Direct and exact exchange angular distribu— tions for Yukawa force with range somewhat greater than 1F for L=2 transition in Zr9O + p at 18.8 MeV . . . . . . . . Direct and exchange form factors correspond- ing to results of Fig. 5.2. . . . . . . dir’ ex’ and CT for L=2 transition in Zr90 calculated with lF range force as in Fig. 5.1. . . . . . . Energy dependence of o 0 viii Page AA 62 66 67 69 7O 72 7A Figure Page 5.8 Differential cross sections obtained with the K-K "equivalent" interaction for the L=3 transition in cl2+p at 28.05 and A5.5 MeV. Direct and approximate and exact exchange and total differential cross sec- tions are shown . . . . . . . . . . 79 5.9 Exact and approximate total differential cross sections calculated with K-K "equiva- lent" interaction for L=2 transition in C12 +p at 28. 05 MeV and A5. .2 MeV and for the L=3 and 5 transitions in Ca 0+p at 25 and 55 MeV . . . . . . . . . . . 80 5.10 Same as Fig. 5.9 for L=O transition in 012+ p at £8.05 and A5.5 MeV and L=3 transition in 01 +p at 2A.5 MeV. . . . . . . . . 81 5.11 Comparison of Fourier transforms of singlet even and triplet even components of K-K force with those of Gaussian, exponential, and Yukawa effective range forces . . . . 92 5.1' Comparison of exact results obtained with long range part of H-J potential with approx- imate results given by K-K force. . . . . 95 6.1 R.P.A. vector and transition densities for l+T=l(Q=-15.l MeV) level of 012 . . . . . 99 6.2 R.P.A. vector, transition densities, with theoretical and experimental inelastic electron scattering form factors for the 2+T= O(Q= —A. A3MeV) level of 012 . . . . . 101 6.3 Same as Fig. 6.2 for 3’T=0(Q=-9.63 MeV) level of c 2 . . . . . . . . . . . 102 6.A Same as Figa 6.2 for 3'T=0(Q=-3.73 MeV) level of Ca 0 . . . . . . . . . . . 103 6.5 Same as Figqo6.2 for 5-T=O(Q=-A.A8 MeV) level of Ca . . . . .' . . . . . . lOA 6.6 Differential cross sections obtained with impulse approximation pseudo-potential for L=O transition in C12 . . . . . . . . 108 ix Figure 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.1A 6.15 6.16 6.17 Page Same as Fig. 6.6 for L=3 transition in Cl2 0 o o o o o o o o o o o a 1.09 Same as Fig. 6.6 for L=2 transition in C12 and L=3 and 5 transitions in Ca“0 . . . . llO Same as Fig. 6.6 for K-K force . . . . . lll Same as Fig. 6.7 for K—K force. Decomposi— tion of complete differential cross sections into direct and exchange components is also shown. . . . . . . . . . . . . . 112 Same as Fig. 6.8 for K—K force . . . . . 113 Same as Fig. 6.6 for Yukawa effective range force. . . . . . . . . . . . . . 11A Same as Fig. 6.10 for Yukawa effective range force. . . . . . . . . . . . . 115 Same as Fig. 6.8 for Yukawa effective range force. . . . . . . . . . 116 Form factors for L=2 transition in C12 obtained with the K-K force and Yukawa effec— tive range force . . . . . . . . . . 123 Differential cross sections for egcitation of first two excited states of Li by 2A.A MeV protons. . . . . . . . . . . 127 Result obtained gith K—K force for L=2 transition in Li based on empirical transi— tion density determined from inelastic electron scattering data . . . . . . 128 Differential cross section for excitation of 2+T=1(Q=—l6. 1 MeV) level of 012 by A5.5 MeV protons . . . . . . . . . . . . l3O Differential cross sections for excitation of 3 T=O(Q=6. 13 MeV) level of 016 by 2A.? MeV protons . . . . . . . . . . 132 Figure (5.20 65.21 6.22 7'.3 70"} 7. 5 7. 6 Page Differential cross sections for excitation of 3 T= O(Q= _AO28 MeV) and 2’T=O(Q=—6.02 MeV) levels of Ca by 2A.5 MeV protons . . . . 13A Differential cross sections for excitations of 3'§Q=-2.62 MeV) and 5‘(Q=—3.10 MeV) level of Pb 08 by A0 and 2A.5 MeV protons, respec- tively . . . . . . . . . . . . . 136 Differential cross sections for excitation of l-T=o levels in 012 and CaAO . . . . . 138 Spectra of Ti50 and Zr90 showing pairing effect due to core polarization . . . . . 1A8 Structure of transition density for L=O transition in Zr9O . . . . . . . . . 165 Differential cross sections for L=2—8 transitions in Zr9O + p at 18.8 MeV. . . . 17A Differential cross sections for L=2 and A transitions in T150 + p at 17.5 and Ac MeV . 175 Differential cross sections for excitation of 9/2+(Q=-.908 MeV) level in Y89 + p at 18.9, 2A.5, and 61.2 MeV . . . . . . . 176 Direct form factor and total form factors obtained in microscopic and macros§8pic cal— culations for L=2 transition in Zr + p at 18.8 MeV. 0 O 0 O O O O O O I 0 0 182 Same as Fig. 7. 6 for L= 2 transition in Ti50 + p at 17.5 MeV . . . . . . . . . . 183 Same as Fig. 7.7 for T150 + p at no MeV . . 18A Differential cross section for L=O transi— tion Zr90 + p at 12.7 MeV . . . . . . . 192 xi Fi gure 7.10 Page Experimental relationship between ep(sn) and e (e ) for quadrupole transitions in Zr90, T150, Ni58, and Pb207 . . . . . . . 201 The experimental data compared with the theoretical results obtained with both sets of wave functions. The total differential cross sections and the (303) component are shown for both cases . . . . . . . . . 213 xii CHAPTER 1 INTRODUCTION There are several factors responsible for the current interest in the microscopic description of inelastic nucleon— nucleus scattering at medium energies, i.e. incident energies ranging from 15-70 MeV. Most important are recent advances in the theory of nuclear structure which provide a des- cription of a variety of nuclear states in terms of the motions of the individual nucleons which comprise these systems.l’2 The medium energy region is of particular inter- est primarily because it is the best source of data on these reactions. This is credited to the new sector-focussed cyclotrons and the large tandem accelerators. Much has been said in the literature about this problem. Ref. 3-7 are a representative sample of papers and a rea— sonably good bibliography is contained therein. These papers consider some of the formal aspects of the problem and discuss those features of inelastic nucleon—nucleus scattering which make these reactions valuable for studying nuclear structure. Emphasis is on the distorted wave approximation (D.W.A.); however, a good discussion of the coupled channels method is given in Ref. 5. The treatment of the non-local D.w.A. transition amplitudeT is discussed in Ref. 3. This is encountered when the required antisymmetrization of the projectile-target wave functions is taken into account. The results of several calculations are also avail— able.8—lu In these works the local D.W.A. is used and the question of antisymmetrization is ignored. It is assumed that the projectile—target interaction can be expressed as a sum of two—body interactions between the projectile and individual target nucleons. The two-body interaction is taken to be local and scalar, separately in spin, i—spin, and coordinate space. Various radial forms are used and the strength and range parameters are fixed by direct calculation and comparison with experiment. Simple shell model wave functions are used to describe the target nuclei. Application is restricted to the (p,p') and (p,n) reactions (a limitation imposed by the experimental data) and the transitions considered serve to isolate different components of the interaction. As far as the weak components of the force are concerned, the information extracted in this manner shows some consistency; however, these analyses yield a large range of values for the strength of the strong, non- "Spin-flip" components of the force. In addition these strengths are considerably larger than that expected from a knowledge of the free two-nucleon force. # 1In this work the terms local and non—local D.W.A. are USed to specify the character of the operator appearing in the D.W.A. transition amplitude, i.e. local or non—local in the projectile coordinate. In related calculations the description of the target nuclei is improved so as to introduce explicitly the effects due to core polarization in those transitions which proceed 15’16 The macro— through the strong parts of the interaction. scopic vibrational model is used to describe the core and a closure assumption makes it possible to fix the core para- meters from experimental y—transition rates. The effects are large and much smaller interaction strengths result when they are included. It is likely that core polarization can account for many of the inconsistencies noted in the earlier works. The effects due to antisymmetrization are contained in the exchange component of the transition amplitude which is necessarily non—local. Its properties are presently being investigated. Initial results indicate that it cannot be neglected and that its importance is a function of inci- dent nucleon energy, multipolarity of transition, radial form and exchange nature of the two-body force, and initial and final target states.17—19 This dependence places restric- tions on the two—body interaction and implies a re—evaluation of some of the conclusions obtained in analyses in which antisymmetrization is ignored. Considerable success has attended the use of "realistic forces" in the bound state problem.20—25 The theoretical fOundations of this approach are reviewed in several 1’26—29 (Ref. 27 due to MacFarlane is an excellent places. article.) The major step is this treatment is the intro- duction of the shell model reaction matrix, or G—matrix, as a. v. . E.: . - r. _. . v. u r\ . ~ \ .i _\o _ . so . . . I . . ..m - AV w n u w; .. .. "E r“ . .T .C L. ._ :1 a. . . .. .t .w. ... r. e. .n. .E nu . a:E LE 1 . e. .1 4.. .. _. a. _ :. .t w. ... 3. . m. w. E.m-. a. a. ta E “~ A I ~ . . ~l . u E. . . :. a: w. u _ . .n . ”C ». x . .C s . .Tw QC hm. . . Q.» .E ,. h... A.» p v .. e. ,.E _... .. , .. v. A: a» 9. :. 4E .c .. Q,» ~.. ~.. D Q. .1. a v. .. . .. . . . e a. . h h. p . v w ., s .n Ev. :— 1. A: w E E r. A: ~E & h .E . o . o .E E Q» Q~ o . n? L. _ . ». ... L. . r” n. :. 4E .. in E... h. ,4 SE e. .1 h. .J- yu‘ Va A.” v. .p.E. «Ev . .. . e .v ...E . e L» a. . E v. “E. w.. Q. 2‘ FEE s...- «a A» QC r. U“ .. . .. .rm rm .. .. ,. . «C ..u C. LC 2.. .C s L. «C r.. r.. H. s s . H. e . . r . -. .5. . . a . :. 2E .. a E h. ... . :E h. ,7. . s . u a u.» y i. ...E r n .9 v »\ s. u. \ Pit. the interaction between bound nucleons.1L This is obtained directly from a two-nucleon potential in a manner which takes into account the presence of other nucleons in the nucleus and eliminates the need for using wave functions with short range two-particle correlations. The G-matrix used in Ref. 20—25 is derived from the Ramada—Johnston (H-J) potential which fits the nucleon—nucleon scattering data up to 300 MeV.30 Application has been made to nuclei not more than two nucleons away from a closed shell. The success of this treatment of the bound state problem is very encouraging. Because of its fundamental nature, it avoids many of the difficulties associated with commonly used empirical methods where the interaction is essentially left free.20 The biggest difficulty is the compensatory relation between the particular calculation which is per— formed (the proper calculation is, of course, not known a'priori) and the interaction which is so determined. These remarks need not be confined to the bound state problem. As an example, note that the initial empirical efforts8—ll4 on the inelastic nucleon-nucleus scattering problem conceal the importance of core polarization and antisymmetrization. The purpose of this paper is to explore a parallel treatment of the microscopic description of inelastic nucleon— ‘ " +The G—matrix referred to here is often called the bare" G-matrix. This provides a means of differentiating between matrix elements of this operator and corresponding Matrix elements which implicitly contain effects other than Interaction of nucleons through this operator alone, e.g. Core polarization effects. nucleus scattering. Here, asserted a'priori, are three models for the projectile-target interaction. All of these re late directly to the free two—nucleon force. The word models is used because no attempt at a precise derivation of the projectile—target interaction is made. This hopefully can be done within the framework of the many body theory of these reactions in a manner analagous to that followed in the treatment of the bound state problem. In this work the as serted interaction models are simply investigated by direct calculation. In related works they are used to calculate optical potentials for elastic nucleon—nucleus scattering in the medium energy region.31’32 To be specific, it is assumed that the projectile— target interaction is given by (l) a pseudo-potential derived from the impulse approximation, (2) the long range part of the Kallio—Kolltveit potential (K-K force) which is known to be a good approximation to the central part of the shell mOdel reaction matrix, and (3) a Yukawa force derived from Effe ctive range theory. These interactions have the same f‘OI‘rrls, i.e. local, scalar, etc. . . . , as those used in previous investigations and all calculations are carried out using the local D.W.A. Any effects due to long range corre- lations-—for example core polarization effects--are included eXplioitly in the target wave functions. Antisymmetrization is treated approximately in the impulse approximation and the effects are contained implicitly in the pseudo—potential. For the case of the reaction matrix and effective range ;1riteractions a local approximation to the exchange component cpf‘ the D.W.A. transition amplitude is included in the calcu— lations. The impulse approximation33 is a free scattering approxi— rnertion which can be derived from the formal multiple scatter- irag; solution to the nucleon-nucleus scattering problem which was developed by Watson and collaborators.3u-38 This approxi— nuat;ion has been applied with success to inelastic proton- nliczleus scattering primarily at incident energies greater than 1CDC) MeV.39_uu It is generally assumed to be invalid at eriexrgies lower than 100 MeV; however, there are indications tragic it might give good results at energies as low as 5C) MeV.36’I45 The pseudo—potential is simply a fit to the Felixéier transform of the free two—nucleon scattering amplitude vdliczh is calculated from the H—J potential,off the energy shea].l, i.e. using nucleon—nucleus kinematics in place of ruchLeon—nucleon kinematics. The Kallio—Kolltveit potential contains a hard core and. has an exponential radial form.“6 It fits the nucleon— nuclxeon S-wave phase shifts up to 300 MeV. The long range part. of this potential is defined by the Scott—Moskowski sepaasation method,u7 i.e. a separation distance is determined (it turns out to be of the order of 1F) below which the 90terltial is set to zero. The separation method gives the leadiJig term in a perturbation expansion for the components or true reaction matrix which act in states of even relative OPbitEil angular momentum. This force is a good approximation to the central part of the G—matrix used in Ref. 20—2A. The latter is derived from a more complete potential and con- tains additional detail. Application of the K-K force to the calculation of the low energy spectrum of 016 in Ref. A6 was one of the first attempts to use "realistic forces" in the bound state problem. In evaluating bound state matrix elements it is assumed that the K—K force acts only in relative s-states. The impulse approximation pseudo—potential and K-K force are selected because it is possible that they are valid representations of the projectile—target interaction asymp- totically, i.e. far outside and deep inside the nucleus, respectively. Reference to the high energy features of nucleon-nucleon scattering is contained in the potentials from which they are derived. It is of interest to see how these interaction models differ from the forces of regular functional form which are obtained in the shape independent analysis of low energy nucleon-nucleon scattering data.u8”u9 To this end calculations are performed with a Yukawa effective range force. Consideration by way of discussion is also given to Gaussian and exponential effective range interactions. There is an imaginary division of the remainder of this paper into two parts. Details relating to the interaction models, D.w.A. calculations, and exchange approximation are contained in Chapters 2—5 and a few Appendices. Applications and results are presented in Chapters 6 and 7 with Chapter 8 reserved for final remarks. To be a bit more specific, a cij.scussion of the antisymmetrized D.W.A. is given in Chapter 2. (true approximation used to treat exchange is also developed tigers. The impulse approximation pseudo-potential is presented j_r1 Chapter 3. Chapter A contains some rough arguments con— (3earning the possible character of the actual projectile- 1;girget interaction and its relation to the K—K force and irnpnflse approximation pseudo-potential. The effective range fYDIsces are introduced in Chapter 5 where some of the pro— pEBITtieS of the "approximate” exchange component of the D.W.A. txaallsition amplitude are discussed and a few results obtained wixtri exchange treated approximately are compared with results 18,19 01‘ eexact calculations. At this point the K-K force and eififeective range forces are compared on the basis of this appbxeoximation. Applications, mainly to (p,p') transitions in closed anti pseudo—closed shell nuclei, i.e. C12, 016, Cauo, and , are considered in Chapter 6. Random phase approxima- tioru (R.P.A.) state vectors are used to describe the target nuceilifiouli)4 Studies of the (p,p') reaction at incident €Nuer~gfies above 100 MeVuO_uu and studies of the (e,e') Peac3tion55’56 indicate that these vectors give a good des— CPiIDtion of the transitions considered. These transitions Serve; to test the proposed interaction models at least to withinq the quality of the approximation used to treat anti- synmmrtrization. Some inelastic electron scattering results are Irresented in order to provide a frame of reference for exaflflJuing the (P,p') differential cross sections which are presented. Chapter 7 is devoted to the treatment of transitions :1r1volving low lying states in nuclei which possess one or 13v70 nucleons outside of a closed shell. Core polarization g3].ays an important part in these transitionslS’l6 and has earl equally important effect on the relative spacing of these l_an lying levels.20—25 The effects of core polarization are easitinmted in calculations which use either a microscopic nlcudel or the macroscopic vibrational model to describe the czcxre. Emphasis is on the completely microscopic calculations vvrrich assume that the core can be described by a zero-order 5116311 model Hamiltonian and that only the effect of simple peazsticle—hole excitations up to roughly 26w in energy need bee considered. The coupling between the valence nucleons aIlCi the core is treated by first order perturbation theory aJlCi the K-K force is taken to be the coupling interaction. lle_s is essentially the model first used by Horie and Arima ifl czalculating quadrupole moment357and it is the same picture ttuat; Kuo and Brown have used in Ref. 20-25. Differential crcuss sections for (p,p') transitions and Y—transition rates are calculated. For the most part, the K-K force is used for the projectile-target interaction. The completely micro— scoglic (p,p') calculations are amusing as they constitute a fiPEit attempt to calculate the observed cross sections directly from a knowledge of the two-nucleon force. The releition between the effect of core polarization on the spect- mnn and in transitionsis examined. Conclusions are drawn as to the validity of the particle—hole model. CHAPTER 2 \ DETAILS OF THE DISTORTED WAVE APPROXIMATION 1 . D.W.A. Transition Amplitude and Cross Section The antisymmetrized distorted wave transition amplitude for the inelastic nucleon-nucleus scattering reaction Eaa,A+I—&bb,B (where E is the relative momentum of the target and projectile, the small letters represent the internal projectile quantum numbers, and the capital letters are used to specify the state of the target) is given by3’l6’l7 _ Z + iDW— (-) (+) (+) Xmém6 (1) ma Iflnezre a local interaction model is implied and provision ifs Inade for the presence of spin—orbit coupling in the Optxical potential. In this expression t(O,l) denotes flue two—body interaction, the X's are the distorted waves Whixzh describe the relative motion of the projectile and tarxget under the influence of the optical potential, and CC‘factor expansions of the target wave functions are emplxoyed. The latter account for the presence of the crea— tiorl (destruction) operators a+ (a) and the single particle bourui state wave functions ¢in the relation. The arguments 10 11 C) and 1 refer to all nucleon coordinates and fix the manner jJn which the integrals are to be taken. The first integral in Eq. (1) is the usual direct matrix ezlement while the second is the exchange component of the txransition amplitude. In the former the same particle is Inrlbound in both the initial and final states, but in the liatter the particle which is unbound initially is captured :irlto the nucleus and a target particle is expelled into the ffiiilal unbound state. The distorted waves are solutions to a one body ESCJarodinger equation which contains the optical potential. Sgbjnn projection is not a good quantum number when this pcat;ential contains a spin—orbit term. In Eq. (1) Xm’ m 38. (XfiTfigmb) is the ma(mb) spin prOJection component of the SCD1JJtiOD with initial spin projection ma(mb). It is clear tiaeat; spin orbit coupling gives rise to "spin—flip" in inelastic sceatztering over and above that which occurs through direct intsezraction via t. Note further the standard use of the Stqaeerscripts on the distorted waves to Specify the boundary coruiitions which they satisfy. From the form of the transition amplitude it is seen that; inelastic nucleon-nucleus scattering is represented by a orue—body operator in the distorted wave approximation, i.e. traruaitions are allowed between components of the target wave :functions which differ only in the state of a single nuclecnl. A state of the target nucleus is defined by its tcnzal angular momentum, projection, and additional quanttun numbers; the internal state of the projectile nucleon 12 1J3 fixed by giving its spin and i—spin projection; thus AEOLAJAMA, aEmaTa, BEOIBJBMB, and bi—ZmbTb. Further a’ and b” “Kill be used for méTa and m6Tb’ respectively. Scattering experiments are most frequently performed vvifith unpolarized beams and targets. Under these conditions 'tlle differential cross section for inelastic nucleon—nucleus :sczattering is obtained by introducing kinematical factors zaruj appropriately summing and averaging over projections. (Ernie gives as. (__u >259. 1 2 IT I2 (2) d9 2fih2 ka 2(2JA+1) MAME DW mamb whuezse u is the reduced mass of the projectile—nucleus system. 2. Iform Factors Without loss of generality the j-j coupling represen- tatixDrl can be selected as the single particle basis, ie. p=n’£"j’m’T’ and r=n2jmT, which gives ’ m l = z I; ’ x .’ ’ 21 " l I l- I ¢p( ) min?“ 2 mimle m >d>1;’(I’l)l2 ms 2 T > (3) ¢ Cl)=: Z ¢m2(5 )ll m l T> r mzms 2 isJ IL 1 2 s2 wh . 58 . ere <:ab08|cy> is a Clebsch—Gordan coeffiClent. It IS convenixent to rewrite Eq. (1) as 13 i - ( TDIM—m’m’ffxm a b —)*- (+) — 3 3 gmb(rO)xméma(rl)d rod rl (A) where = Z .<£’l m’m’lj’m’> rp p r mimS 2 A s 2 A s mRms — - ”It" — m9, — 3 x{6(rO—rl)f¢£, (r2)¢£ (r2)d r2 mi* _ mg _ —¢2’ (rl)¢£ (r0)} (5) unitih.the bra—ket notation applying to integration over the :iriteernal coordinates only. The quantity , called the partial matrix element, ccnqt:ains all of the nuclear structure information for a paletcicular transition. It also contains the details of the irn;eq?action model and the selection rules which govern the rearrtitwu It is an effective one-body operator in the pP0jenctile subspace. Examination of Eq. (5) shows that is ruarr—local, that is it depends on E0 and 51' Here this nou—lc>cality arises due to the presence of the exchange comporleru;CM‘the transition amplitude; however, had t been assunmecl non—local the direct component of the transition amplitwide would also contribute to the non—locality in . CFhe general rotational properties of the partial matrix element; can be exploited to reduce the distorted wave calcu— lations. for all transitions to a common form. It can be Show“ qiiite generally that can be written3 1A 1/2—m’ = 2. i—LF b LsJ LSJ,M(rO’rl;b B5a A)("l) l .1 I .0 I .4 ; ,o x<2 2 ma,—mb|S ma—mb> (6) vqrjere M=MB-MA+mb-ma. FLSJ,M transforms under rotation of x _ t;r1e coordinate system as Y the i L insures convenient LM’ t;j,me reversal properties, and L, S, and J satisfy the vector re lations J=JB-JA sta—Sb—+S=O’l L=J-S. (7) th is clear that L,S, and J are the angular momenta trans— fkerfired to the target nucleus through t in the inelastic ccalfilision. If i—spin is considered to be a good quantum Ininflaer for the target nucleus Eq. (6) can be rewritten as -ELSJT1 FLSJ,M(rO’rl;a A,b B)(—l) x 2 2 a’mb a b ’ a b ’ B A LX (8) A T ’ T T B T 2 b’ a b 2 a A B A B Where I"".-_:_ —- =— -- = ._ = _ I TB TA ta tb+T 0,1 and MTB MTA Ta Tb. Eq. (8) reduces to Eq. (6) by defining F =AT _ 1 _1 LSJ.N1 T FLSJ,M<2 TTb’Ta Tb|2 Ta>' (9) This Etxpansion of the partial matrix element can be used to Write tune transition amplitude as 15 Z A LMmbma TDW=LSJJ(JAJMA’MB—MAIJBMB>BSJ (10) where J=[2J+l]l/2 and LMm m .-L 1/2-m’ b a— l _ b 2 ’_ I _ BSJ —mZm.—w— ( 1) a b J M, l l ’ ’ I I X<2 2 ma,—mb|Sma—mb> (-)* - (+) - 3 3 XIIXm’m (rO)FLSJ,M’Xm’m (rl)d rod r1. (11) b b a a The cross section, Eq. (2), then becomes £2 =( U )2 :2 2JB+1 | BLMmbmal2 (12) d9 2Uh2 ka 2(2JA+ mbma LS SJ JM with the interference between different S and L for a given J occuring as a direct consequence of the spin—orbit coupling in the optical potential. In practice this interference is found to be weak. As partial wave expansions of the distorted waves are used in evaluating the integral in Eq. (11) the multipole components of FLSJ,M(EO’E1) are needed. They are defined as follows: _ - - SJ * F O (r ,r )= A F (r ,r )Y InnI,M O 1 LaLb LLaLb O 1 Lbe MaMb 9(- (r )Y (r ) o LaMa 1 x (l3) 16 SJ _ 2 FLLbL (rO’rl)—M M (LbLaMbMaILM> a b a XIIFLSJ,M(FO’rl)YLbe(rO)YLaMa(rl)dQOdQl (1“) The reduction has been achieved. All of the "physics" SJ for a particular transition is contained in the F (r ,r ) LLbLa O 1 which are independent of projection quantum numbers and are functions of the radial coordinates rO and r1. Given these quantities the distorted wave cross section is obtained by LMm m computing and summing the BSJ b a as prescribed by Eq. (11) and Eq. (12). Unfortunately, the calculation is still not easy. It will be seen that each of these multipole components is a fairly complicated quantity as far as computation is SJ concerned. Further, FLL ) is associated with (r ,r L O l b a angular momentum transfer L, S, and J to the target nucleus with the projectile undergoing a transition from the state of relative angular momentum La to Lb' Even though only a few values of L, S, and J are expected to contribute to a transition there may be as many as twenty partial waves used in the calculation of a cross section in the energy region (if interest here. The point is that the FSJ (r ,r ) are LLbLa O 1 rum: only complicated, but many of them are required. For the direct,or local component of the partial rmatrix element an additional separation can be made. ~ 6(r —r ) L L Fig.1.(rb,rd)=FLSJ(Po){ O 21 ;—-(’l) b} (15) ba r0 7T P5133: - ‘1 I 17 Using result (15) in Eq. (13), recoupling a spherical har— monic, and using the closure property of spherical harmonics gives for this case .. _ ~ 9" A ... _ FLSJ,M(rO’rl)=F (rO)YLM(rO)5(rO_rl)' (16) When Eq. (16) can be used the calculation of the cross section is considerably easier because the "physics" is then contained in the FLSJ(rO) which are few in number and depend on only one radial coordinate. In addition explicit LMm m use of Eq. (16) in Eq. (11) gives an expression for BSJ b a which is much simpler than the one obtained by using Eq. (13) in Eq. (11). Computational difficulties associated with the treatment of non—local partial matrix elements have been the major reason for neglecting the effects due to antisymmetri- zation in the past. Fortunately, this problem is well on 16,17 its way to solution. In this work an attempt is made to account approxi— inately for antisymmetrization in an expression of the form (16). The calculations are then essentially reduced to LSJ + (r These have two 0). LSJ . (r0) which comes from the direct component of tflle transition amplitude and ELSJ(rO) which approxi- ccnnitructing the form factors F components - D Imitely'represents terms coming from the exchange component. 1 (50,51) which is properly referred to as It is FLSJ,M ~LSJ 21 forum factor. When using the local D.W.A. F (r0) is the esseruxial part of FLSJ,M(;O’?1)‘ In this work the term form factcn? will refer to FLSJ(rO). a 18 Two approximations are used--one with the impulse approxi- mation and another for the case of the K—K and effective range forces. The approximations differ only in detail—— not in spirit. They are discussed in Section 6 of this chapter. Explicit identification of ELSJ(rO) is made when using the K—K and effective range forces, whereas it is 1 implicit in the impulse approximation pseudo—potential. Returning to the discussion of the complete partial . matrix element Eq. (13) is used to rewrite Eq. (6) as = 2 i’L(-1)l/2_m6 LSJ A A’ B A B B 2 2 a’ b a b L M a a Lbe x 17* A F8 (l X Lbe(rO)Y La Ma (r1 ) LLbLa (ro’r1)' 7) In the next two sections of this chapter it will be shown that Eq. (5) can be written in the above form, thus allowing . . . SJ . . identification of the FLL La(r0,rl). The discussion is b restricted to the static interactions being considered in this work. These have the form ‘ “ )a-aee t<0’1)ztoo(r01)+t01(r01)30'01+tlo(r01)To'Tl+tll(rOl O l 0 l A+y =1 (—1) t (r ) s s T T ST ST 01 O_A(O)OA(1)T_y(O)Ty(1) (l8) Ky wherwe o§(13) are the usual spherical tensor components of the Spifl (i—spin) operator and 08=18=1. p-u l9 3. Integration Over Internal Coordinates Using Eq. (18) the following result is obtained for the integrals over internal nucleon coordinates in Eq. (5) , » =2 _ A+y 1 2 _ 1 . 1 1 , (b’p|t(0’l)la’r> ST( 1) tST(rOl)<2 sma’ AI2 mb><2 SIn5M2 ms> Ay 1 1 1 1 , X<§TTa,-y|§Tb><§TTYI§T > 1 s 1 1 T 1 2 xi<§||o |I§><§IIT ||§>} <19> , . + 1 1 . 1 . 1 . =ST(”1)A ytsT(r01)<2 sms”Al2 mb><2 maxl2 ms> A3' 1 l .1 I1 I X<§TT,-yl'2—Tb><§TTayi§T > 1 S 1 1 T l 2 x{<§||o ||§><§IIT ||§>I <20) where = is the convention adopted.for the Wigner—Eckart Theorem.58 The following recoupding identity 1 1 , l . l . _ “.2 1 1 1 1. . <25"2;”A|2 mb><2 smaAIE mSI—SZA’S w(2 2 2 2’ SS ) S’—S+A’+A+l l ) 2 I l I 1 z I l / x(-l) <5 8 ma,—1 l5 mb><§ s mSA |§ ms> (21) arxi its i-spin counterpart is used in Eq. (20). Then the fact that 1 S 1 _ <5: I (Sir I 12>-<%> is IJSEHi and summation indices are interchanged to give f 20 E , , _Z A+y l , l , -ST(—l) tST(rOl)<§Sma,-A|§mb> Ay l c l , l l x<2 cmSAIZ m ><2 TTa, yl2 Tb> 1 1 , 1 S 1 1 T 1 2 X<§ TTY|§T >{<§||0 ||§><§IIT |l§>} (22) where E - _ $$fiT5T2 1111» 1111» tST(r01)'sZT* l) S T w<2 2 2 2’S S”“2 2 2 2’T T) XtS’T’(rOl) (23) 58 with w indicating a Racah coefficient. Eq. (22) can be summarized as =. The coefficients in the expansion of t§T(rOl) in terms - . {S’T’ of tST(rOl) which is given in Eq. (23) will be called/XST , . E _ 2 as’T’ . . that 18 tST(rOl)_S’T°&ST tS’T’(rOl% They are given in Table l. TABLE l.-—Coefficients for expansion of tET(rOl) in terms of tS’T’(rOl)’ s’T’ géT’ ST OO 10 01 ll 00 t i?- i 19: w 1- 1 i- i- m 1 fi- 1— i ll , i- -% 13- ii- I F" 1 W317 21 It is not obvious from the above table, but the relationship between tE(O,l) and t(O,l) can be stated very simply. To see this note the following alternative expansion for t(O,l). t(0’1)=%s VTS(rOl)PTS (25) Here PTS=PTPS with PT and PS representing the usual i—spin and spin projection operators-— PO=% (1—50-51) and “' -‘““P~rfim P1=% (3+60-51) for the case of ordinary spin. Unlike the a previous relations for t(O,l) and tE(O,l), where the sub— scripts S and T referred to the unit of spin and i—spin which could be transferred from the projectile to the target nucleon through the corresponding part of the interaction, the subscripts on VTS(rOl) indicate that it is the component of the interaction which acts when the projectile and target nucleons are coupled to total spin S and total i—spin T. Commonly used is the notation VOO=VSO,V10=VSE,V01=VTE,V11=VTO where SO, SB, TE,and TO refer to singlet odd, singlet even, triplet even, and triplet odd components of the interaction, respectively. Expanding Eq. (25) and regrouping terms as in Eq. (18) gives the following relations between tST(r01) and VTS(rOl). _1 too‘TE‘Voo+3V01+3V1o+9V11) t =1 (-v +v -3v +3v ) 10 TE 00 01 10 11 (26) _1 t01’T6(‘Voo‘3Vo1+V1o+3V11) -1 _ _ t111—30700 V01 V10+V11) 22 Similarly using Eq. (23) it follows that: E _1 too‘1€(vo0‘3V01'3V10+9V11) E _1 t1o‘T€("Voo‘vo1+3v1o+3V11) (27) E _1 to1‘IE(‘V00+3V01'V10+3V11) 4—1-1333; E _1 t11'IE(V00+V01+V10+V11) The right hand sides of Eq. (26) and Eq. (27) differ only by the signs of the even state terms. For the case of an even state force tE(O,l)=-t(0,l) and for an odd state force tE(O,l)=t(O,l). Remembering that the transition amplitude is proportional to the difference between the direct and exchange components, it is clear (insofar as the integration over internal coordinates is concerned) that the exchange amplitude contributes constructively to the direct amplitude for the even components of the interaction and destructively for the odd components. This result is a direct consequence of the fact that the internal wave function of the two nucleons is symmetric for odd states and antisymmetric for even states. It (would have been seen more easily by coupling the internal coorwiinates of the projectile and target nucleons to good spirm and i—spin before integrating in Eq. (19) and Eq. (20). ThifiS‘WaS not done, however, since Eq. (19) and Eq. (22) have the: form that is needed for the remainder of this disucssion. It is interesting to note that because of antisymmetri— zatixon, "spin-flip" and "i-spin—flip" through direct inter— 23 action is allowed even if it is strictly forbidden by the form of the interaction. To see this note that a Wigner force t(0,l)=t E( _1 r01) leads to t 0,1)-H[t00(r01)+t10(r01)x OO( 80-61+t01(r01)¥0-¥l+tll(r01)EO-Elto'tl]. Such consequences appear formally because of the introduction of the pseudo- interaction tE(O,l) into the exchange amplitude, but it should be remembered that this is simply a convenient way of cataloging the manner in which the incident projectile L can be captured by the target with expulsion of a target nucleon into the final projectile state. To conclude this section note that the partial matrix element Eq. (5) can now be written as I — z + ’— I ’ ’ I l \V>—mEm;<£ 2 mlmslj m ><£2 mgmsljm> mgms — - mt* — mt — 3 x{6(rO—rl)f¢£, (r2)¢2 (r2)d r2- m£* — E mt — ¢£, (Fl)¢£ (r0)}. (28) 4. Final Reduction of Partial Matrix Element Since the components of which correspond to the tgansfer of total angular momentum J are of interest it is convenient to couple the creation and annihilation operator in Eq. (28) to good J. A I ’.o :2 I ,_ — J—m+ J‘FI(J T .11) mm,<3 Jm, mIJMJ>( 1) aj.m,T. J (29) a jmT 24 The phase factor (—l)3-m insures that AJM has the correct J transformation properties under rotation. If i—spin is considered to be a good quantum number additional coupling to good T is necessary. TM 2 < 1/2-T T ’c_ ll’_ >_ 0’ I. AJMJ(J J)‘Tt’ 2 2T : TITMT ( 1) AJMJ(J T ,JT) (30) For these two cases it immediately follows that X + _ Z j- m ., , _ - Mm (_ 1) <3 jm , mIJMJ> j m T JM. J X (31) and Z + '— _ J-m 0’. I _ — j%T( 1) <3 Jm , mIJMJ> x<§ SmSAIE ms><£§ mgmsljm> ’ mm SS A’AA2 ’ ’l , ’ -2 /2j jL L-£+£ X<2 2 mims Ij m >_LM T (-1) 1 1 Xx(j jJ; 2 £L; ,5 2 S), (33) Rearranging some Clebsch—Gordan coefficients, summing over indices if necessary, and comparing with Eq. (17) allows the identification * 2 YL M (90)Y; Ma(r l)FLL La (r0,rl)= L M b b b a a Lbe ml , m’* m x(—1)L’2+2 {5(EO—El)f¢£% (r2)tST(r02)¢££(r2)d3r2_ mi* - E m2 — In these relations X (abcgdefgghi) is a 9-3 symbol58 and IBLJT)=/§$ A T, T MT B T 2 b’ a b 2 a A B A B x (35) B B J A A for“the case of good i—spin or “2 l B(;VP)=T§,T <§Tr,Ta-Tb|%T’><%Trb,Ta-T TbI2Ta > (36) 26 when i—spin is not considered to be a good quantum number for the target. The meaning of the reduced matrix elements appearing in Eq. (35) and Eq. (36) can be illustrated by writing them Eckart Theorem the reduced matrix element appearing in Eq. in a somewhat more familiar form. By inverting the Wigner— ! (36) can be written as follows. [ L z ’. __ Z J—m —MAMJ(—l) x + a laJM> (37) x _ Z la J M >- (38) A A A where the complete set of antisymmetric states composed of n-l nucleons has been introduced. The reduced matrix elements in Eq. (35) and Eq. (36) are simply related to the coefficients of fractional parentage (c.f.p.)59 and all results can be put in the form of the usual fractional parent— age expansion.58 The definitive relation is 27 la J M >=nl/2 (a J M A A A p p plajmr . (39) Using results (38) and (39) in Eq. (37) and summing over pro- Jections gives: AA ’—1 =s (A0) J —J +J—j’ A A, , . _ p A X{JAJ w(JJ JAJB,JJp)( 1) }. For the case of good i—spin it follows that: T ’ _ . . A AAp—lA "l/2 —S(JAJBJ,TATBT,jJ )JJ T(2) S(JAJBJ;TATBT;33 )=aJT n EDD P x (U1) A A Jp—JA+J-j’ x{JAJ W(JJ JAJBgJJp)(-l) } l l . _ p A x{TA/§w(2 2 ATB,TTp)( 1) }. 1TH? spectroscopic amplitudes S which have been introduced are syimply partial sums of the complete fractional parentage expmndsion of the partial matrix element. They contain the weiéyiting imposed by the nuclear structure for the contribution to time transition due to a single nucleon going from the initiial state J(jI) to the final state J’(J’I’). The factors ".Oz~_ . '0. 28 appearing in Eq. (HO) and (41) which have not been included in S guarantee a convenient interpretation of the remaining factors in Eq. (3“). When i—spin is not considered to be a good quantum number it is useful to redefine the interaction by per- forming the sum over T when Eq. (36) is used in Eq. (3“), i.e. define r"m»‘umv-—.. -ZA2 l l ’ l l (rOl)—TT <2TT’Ta‘Tbl2T ><2TTb’Ta_Tb|2Ta>tST(rOl) (42) t , STaTbTT with a corresponding relation for tE (r01). Table 2 gives the coefficients in the above expansion for the various combinations of i—spin projections. The first entry in each calumn is the coefficient for T=O while the second is for T=l. I—spin projection equal to % denotes a proton and —% denotes a neutron. Incompatible projection combinations are indicated by a dash. The table simply shows that for inelastic proton or neutron scattering the proton—proton and neutron—neutron interaction is tSO+tSl while the neutron-proton interaction is tSO—tSl‘ Further it illustrates that only the iso-vector part of‘the interaction contributes to the charge exchange reactixmms. Since it will always be clear what reaction is beiru; considered no ambiguity should result if the subscripts 1-'%) are dropped from t in Eq. (“2). For the ©,p') reaction it if; also convenient to use the subscripts pp and pn corres— ' ==.— :_=l ‘pondijug to Ta T 2 and Ta T 2, respectively. 29 TABLE 2.--Interaction components when i—spin is not used. REACTION TaTb IT % % _% _% % _% -% % (p,p’) % % 1,1 ,—1 - _ (n,n’) -%--% 1,—1 1,1 - — (p,n) H .. - - 0,2 (n,p) -% % - — 0’2 _ L Now Eq. (UO)-(u2) are incorporated into Eq. (3“). In addition t(rOl),tE(rOl), and 6(50-51) are expanded in spherical harmonics. This expansion is defined by _E f . * A A (A ) f M x(—l) 3 (A5) 3O Contraction of Clebsch-Gordan coefficients and recoupling in the exchange component, as was done previously in inte— grating over the internal coordinates (see Eq. (21)), gives for the case of good i—spin SJ F (r ,r )= LLbLa O l l X A “ 1 1 jj./Z T<2 TTb,Ta-Tbl2Ta> . T 3:; AAA AAAA L—QI’—2I A - A . A .1 1 . .. A xi /§ 31A LSJTx(j JJ,2 2L,§ ES)S(JAJBJ,TATBT,JJ ) A A A A 6(r -r ) x(un)’l{LaL L-le £ 1 b STL O r2 O * Z L’“,2 , , , , -un,£,(rl)un£(rO)LA(—l) L W(2Lb2 La,L L) E (A6) xtSTL ,(r03r1)}. For the case i-Spin is ignored, SJ F (r ,r )= LLbLa C) l {Av/2'11””2 -2/3 jAA’LSJZ(j’jJ;£’£L;§ ES)S(JAJBJ;jJ’IT’) JJ TT’ A A 5(I’ -I‘ —1 A A —2 A A , 0 l X(1rn) {LaLbL ISTT,L(rO)<££ OO|L0> r2 0 * Z L’A’2 A , A A -un,£,(rl)un£(rO)L,(—l) L w(2Lb£ La,L L) A A E . x<2 L 00|Lao>tSTT,L,(rO,Pl)}. (”7) 31 In these expressions £,(r2)t (r03r2)un£(r2)r22dr2 (U6 ) ISTszun ’ STL and _ , 2 ' ISTT L fun ’ 2’(r2)tsTT’L(ro’r2)un2(r2)r2 dr2° (“7 ) A?" M «MW 1 Using the symmetry properties of the Clebsch—Gordan coefficients it is easy to see that the first term in Eq. (U6) and Eq. (“7) has the form indicated in Eq. (15). Identifica- tion of DLSJ(rO) follows directly. ~LSJ ' 2 1 1 D (r0): 33 «E T<2TTb,Ta—Tb|2T2> T XS(JAJ BJgTAT BT, 33 ) 6(rl —r ) H , 2 LSJ T 1 , 2 XItSTL(rO,rl)rldrl (A6 ) 2 DLSJ(I‘O )=jJ w/E S(JAJ BJgjj’IT’) TT’ 6(1” ’1”) u ., 1 2 TLSJ Xf tsT T L(rogrl )r21drl (“7 ) 2 The suiin-angle tensor has been introduced, i.e. LSJ_Z > L s TMJ —MA= I" 2 311‘”-2 (un)"1/2/§32LSJTX= I'2 1L+2‘£ (HN)—l/2/§j£LSJX(J’3J;2’£L;% % s) "I (“7 ) * xun,£,(rl)un£(rl). H H The /5 and i—spin factors in Eq. (46 ) and Eq. (“7 ) appear because the partial matrix element contains the integrations over internal projectile coordinates. Examination of the above relations shows that the direct conmnonent of the partial matrix element for a given L depends on cudly one multipole coefficient of the interaction while the exckmunge component depends on several of these coefficients. In guidition one of the more interesting consequences of anti— synnmetrization is noted. the IQ—transfer and parity change (An) in a transition. This concerns the relation between In inelimstic nucleon-nucleus scattering no change in the intrinsic pnxrityr of the projectile is involved, thus any change in the paltity' of the target during a transition requires a correspond— iru; ch£u1ge in parity in the state of relative projectile-target 9" "" ““"“"‘N 33 motion. This condition is displayed in Eq. (“6) and (47) L +L 1+2, where it is obvious that An=(-l) a =(—l) These same relations illustrate that the direct component of the transition amplitude vanishes unless An=(-l) Such a relation does not exist for the exchange component and there will be contributions to the cross sections when An#(—l) . :1 In the local D.W.A. one has the selection rules indicated : ...-r. n:.‘.-- . in Eq. (7) along with those given in Eq. (8) for i-spin and the additional relation between L-transfer and AH. For a given value of J, (LSJ) can take the values (J,O,J) and (J,l,J) or (J—1,1,J) and(J+1,1,J). With the inclusion of the exchange component all four triads are allowed for a given J. The contributions to the cross section with (—1) #AN are referred to as non-normal transfers. 5. Zero-Range Interaction A special case of some interest is an interaction of zerwm—range, i.e. t(POl)=T (FD-Pl) which leads to t(rogrl)- T 6(rO-rl) r2 O cmxsfficient of’the exchange interaction can be factored out of which does not depend on L; therefore the multipole the snnn over L’ in Eq. (”6) and (U7). The sum then yields — 12’ 22 2 . 2 2 2 2 _ g,(..1) L w<£ L OOILaO>— A A A_2 ’ LaLbL <22 OO|LO> whixflu gfikves the exchange component of FLLbLa (rogrl ) the same forvn as ‘the direct component. The following expressions are 3H obuflnedfbr FLSJ(rO) for the case of good i-spin and the camawhmui—spin is ignored, respectively. jLSJ _ Z “ 1 1 P (rO)JJ,/§ T<§ TTb,Ta-Tbl§Ta> T , . , E xS(JAJBJ,TATBT;JJ )(TST—TST) 6(r -r ) .. 0 2 LSJ T X (£18) I" o 01” ZLSJ _ 2 , , , E P (rO)-jJ,/2 S(JAJBJ,JJ TT )(TSTT,L—TSTT,L) TT’ 6(r -r ) ., o 2 LSJ X. (”9) r0 [t is clear that the effect of antisymmetrization in the limit if zero range is simply to renormalize the strength of the nteraction. Further, non-normal transfers are not allowed 1 this limit. Apprwxximate Treatment of Antisymmetrization TTNB approximations used to treat antisymmetrization in is wcnflc are based on the fact the exchange scattering, as lpal%3d tn) the direct scattering, is sensitive to a partic- .r Runnernrum component of the two—body interaction. To see s ruote tflne form which the basic integrals of the D.W.A. nsitxiorl ammlitude, Eq. (1), have in a momentum represen- ion. r1 :- Ax-- : l J 35 Idir-(2fl)-9 x/xb“’*<12;> ;(E1+E2—E£)t(|El—E£| )¢r(E2)xé+)(El)d1 (50) 16x . (2n)-9 (51) ! beIVRtp¢;(El+E2—Ei)t3(|E£—E2|)¢r(E2)xé+)(El)dr In the direct scattering the projectile goes from the initial state 121 to the final state E]: by transferring momentum §=El-El’ to the bound particle. the projectile is captured and transfers momentum iii—E2 In the exchange scattering to the bound particle thereby expelling it from the target with momentum El. Introducing the initial and final relative momentum 12' and 12’ of the two nucleons it follows that: (52) the extent that the scattering is governed by the kinematics the nucleon—nucleus system, i.e. on the average the bound ticle is initially at rest in the lab and for scattering L particular angle in the nucleon-nucleus center of mass average value of a is the assymptotic value, it follows =1‘ 21¢ (53) 36 where N is the number of particles in the target and it has been assumed that no energy is lost in exciting the target. For the case of the K—K and effective range forces the exchange integral is approximated by evaluating tE( lEl—E2l) 2 LAB representation Eq. (51) becomes at k and removing it from the integral. In a coordinate -32 <->*— *— —- - (+)- 3 3 IeX-t (kLABbe (r0)¢p(rl)6(rO-rl)¢r(rl)xa (r0)d rod r1. (5“) , . vLSJ , I‘he following express1ons result for E (r0). «w -2 A _ i _ .1. E (rO)-JJ,.72'T<2TTb,Ta Tb|2ra> T xS(JAJBJ;TATBT;jj’)Aé%)(kg) 6(r —r ) x (55) I’0 LSJ (56) I’o —ii-E ) 2 _ O]. E 3 . (57) A2=k2 =2ME M12 where M is the nucleon mass. 0 LAB LAB This is the simplest approximation which can be made to the exchange component of the D.W.A. transition tude. Comparing the above relations for ELSJ(rO) with BLSJ for (r0) in Section H of this chapter leads to 37 the following qualitative conclusions about the prOperties of exchange scattering as treated in this approximation. (i) The angular distribution for exchange scattering will fall off slower in angle than that for direct scattering. (ii) The importance of exchange scattering will increase as the energy decreases. (iii) The importance of exchange scattering with respect to direct scattering should increase with increasing L- : mm*-i_:‘i .I transfer. (iv) The direct and exchange amplitudes will be roughly in phase. These conclusions require assumptions regarding the behavior of the multipole coefficients and Fourier transforms of the inter— actions being considered, i.e. A(l)(Ag) increases with decreas— ing A5 and tL(rO;rl) falls off with increasing L. The assumed behaviour is typical and the qualitative observations are in agreement with the results of exact calculations.17'—19 Quantitative comparisons are made in Chapter 5. One can object to this approximation for two reasons: .) it does not preserve the possibility of non—normal trans— rs and (ii) the validity of taking tE(IEi-E2I) out of egral in Eq. (51) is strictly valid only at high energies e the importance of exchange scattering is diminished. quantitative comparisons in Chapter 5 serve as an 2r to the latter objection. Because of objection .t is necessary that non-normal transfers be unimportant liS approximation is to be useful. One reason for “ing normal transfers over non—normal transfers is that -atter only contribute through the exchange amplitude. 38 FM°Wecweof a Serber interaction and isoscalar, normal paiWimmmhions (this being a hypothetical situation flmflartommm of the actual cases considered in this work) A normal parity transition a stronger argument can be given. is the lowest ischfhwdbythe condition Aw=(—l)J£ where Ji allowed J-transfer. For this case the dominant normal trmmfmrisspecified by the triad (JROJ£) and the corres- fl.“ pomfihgrmnqwrmal transfers are specified by (J£:1,1,J£). Anisomxflartransition proceeds through the T=O multipoles For a Serber interaction tOO is three of the interacticni. times stronger than t10 which introduces at least a factor cfl‘ninecfliference in magnitude between normal and non-normal In addition collective effects in the target For the case of nuclei will be displayed in (JOJ) triads. J +1 i.e. An=(-l) 2 , the factor transfers. an abnormal parity transition, Neglecting the non—normal trans— of nine goes the other way. ‘ers may be serious here. By expanding tE( Iii—K2” in Eq. (51) in a Taylor series VLSJ(rO). These out Ag, additional terms can be included in E ll csoruwect :for the finite spread of momentum components in distorted and bound state wave functions and will intro- In .» some dependence on the local momentum transfers. ciple this series conserves the possibility of non-normal fers . It is presently being studied only with the hope The series proving the results for normal transfers. ~veloped formally in Appendix A, but will not be discussed .is paper. 39 The t-matrix for free two-nucleon scattering is a functnn1of q2, p2, and aofi. The dependence on 5 is related tOtmm fact that it includes the effect of exchange scattering. {Nmepseudo-potential used in this work is determined from those cxmmonents of the free two-nucleon t—matrix which are off i.e. Eq. (53). A the two-nucleon energy shell as prescribed above, On the average, exchange scattering is thus being treated in fl A. A-.." _ essentially the same way. The pseudo—potential is strongly energy dependent. It might have been better to include only the effects of direct scattering in this pseudo—potential and treat antisymmetrization in a consistent way throughout. 7. Transition Densities It is convenient to introduce the transition densities. These are ,LSJ,T _ 2 A _ l _ l 5 (rl)-JJ, 2 T<2 TTb,Ta TbI2T3> xS(JAJBJ;TATBT;jj ) 6(r -r ) XTT<2>IIJ§> (58) I' 1 nd LSJ 2 , . 5(r1’r2) LSJ IT’(r1)=jj’/2-S(JAJBJ;JJ n )- (59) l Deleting reference to the fractional parentage expansion iese Twilations can be rewritten as U0 LSJ,T _ A _ g _ i F (rl)-/2 T <2Trb,1a Tb|2Ta> A B A B N 6(r -r ) 1 i LSJ T (58') I’ 1 and. ’ 6(r —r ) ,LSJ _ Z 1 i_ LSJ . , _ PTT, rl)—/2 . (59 ) ; r I l i a The sum on i in Eq. (58') runs over all target nucleons while in Eq. (59') the sum on i runs only over those target nucleons consistent with the subscript IT’ on F2§{(rl). For example, in the (p,p') reaction this sum would run over either target protons or target neutrons. The form factors are related to the transition densities by the following expressions. ~LSJ _§ , LSJ,T 2 n F (rO)—TIWETL(rO,rl)F (rl)rl dr1 (58 ) ~LsJ _ , LSJ 2 n F (rO)—T§’IWETT’L(rO’rl)FTT’(rl)rl dr1 (59 ) In Eq. (58") and Eq. (59")3/L(ro;rl) represents either the apprmnxriate multipole coefficient of the impulse approximation pseudo—potential or 6(r —r ) . (l) 2 ___g__1_ tL(rO,rl)+A (A0) 2 r O wherltflse K—K or effective range forces are used. Note that in introducing the transition densities an additixnial partition of the inelastic nucleon-nucleus Al scattering calculations has been achieved. The first separated the details of the interaction model and nuclear structure from the distorted wave calculation. Here the details of structure are separated from the radial form of the interaction and the effects of antisymmetrization. Detailed formulae for calculating BLSJ(rO) and FLSJ(r1) for the cases of interest in this work are given in Appendix B. The manner in which the transitions densities are related to the inelastic electron—nucleus scattering form factors and the reduced matrix elements for y-transitions is discussed in Appendix C. This is important as it provides the means for calibrating the nuclear wave functions used in testing the interaction models in this work. The relation of the transition densities to these reactions is independent of the approximations involved in treating inelastic nucleon— nucleus scattering in the local D.W.A. Pr “1.- 0“..— IL CHAPTER 3 IMPULSE APPROXIMATION PSEUDO—POTENTIAL The free two nucleon scattering amplitude has the 33 form A 7n_= A + BOO-nol-n+C(oO+ol)-n+EoO-qu-q+FoO-pcl-p (1) AAA where q=q/|q|, q=k'—k; h=h/|h|, n=kxk'; and p=qxn. Here k and k' are the initial and final relative momenta of the two nucleons and q is the momentum transfer. The unit vectors (q,n,p) form a right handed coordinate system and the coefficients A,B,C,E, and F are functions of q2, q2+p2, and a-fi as well as iso—spin, i.e. A = Ill—(3A1+AO)+Ill—(Al-AO)¥O.:E1 (2) where A0 is the coefficient for the singlet i—spin state and A is the coefficient for the triplet i—spin state. The 1 free tnno—nucleon t-matrix is related to7fl_by 2 t = - “T‘Mh m. ‘ (3) Note that ”Qcan be written as follows 1fi== A + §22 am. ..i 52 on ..-... :2 ms. ...l 2.304310". 1 ace-r210“! AS The range m is determined roughly by the half maximum and the quantity X? is proportional to the value of the scatter— m ing amplitude in the forward direction (i=0). The constant of proportionality is~A1.5 MeV~F2 with the scattering amp— litude given in F and E? given in MeV-F3. m The strength and range parameters for the spin, 1- "--r.-;—- . spin components and p-p (n—n) and n—p spin components of .- ....- the pseudo-potential are given in Table 1 for ELAB=2O-6O a MeV. Two Yukawa fits to the scattering amplitude have also been made although they have not been used in any calcula- tions. Unlike the one Yukawa fits which only fit the scattering amplitude closely in the forward direction, these fit it quite well over the entire range of q displayed in Fig. 1. The parameters for these two Yukawa fits are avail- able but will not be given here. Inspection of Table 1 shows that the pseudo—potential has a large imaginary part. The real part of A cm> qw> am> m>ozim Ho> OH> oo> m>mzam<4m noosomo men go upon Hoop osu Lou ma amuse one mm.>oz CH mE\> ma m .Hmfiocop0dloosmmd cofipmefixopoam omHSQEH no mucoCOQEoo pom mnouoEMLmd owCML can zuwcmpumll.H mgm9w ma E was A7 and with corresponding components of other interactions real 1F range Yukawa "equivalent" to this interaction has been determined over the energy region from 20 to 60 MeV. This is given in Table 2. The real lF range Yukawa form has been selected as it is the form which has been popular in recent analyses. This "equivalent" interaction is no I more than a rough representation of the actual pseudo— I potential, i.e. in a calculation it won't reproduce pre— : cisely the multipole and state dependence of the prototype. From the table it is seen that the pseudo-potential is similar to a Serber force and the strengths of the components decrease fairly rapidly with energy. The latter effect is a direct result of the decreasing importance of the exchange component of the scattering amplitude. TABLE 2.—-Strengths for real 1F range Yukawa "equivalent" to impulse approximation pseudo—potential. All values are in MeV. ELAB V00 V10 V01 V11 V36 Vép Vgp Vhp 20 —86.9 33.6 u5.9 38.3 -53.2 59.5 -123 —17.2 30 —69.3 23.1 35.u 29.5 -38.6 no.1 -103 —12.8 A0 -56.3 15.5 26.0 20.5 -3u.9 30.1 -81.8 -10.9 50 -u8.8 11.u 22.1 15.9 -29.8 22.2 -67.u - 9.6 60 —u3.8 u.3 18.3 13.6 -25.5 18.0 -59.1 - 9.2 CHAPTER A THE PROJECTILE—TARGET INTERACTION By analogy with the bound state problem the two—body interaction to be used in nucleon-nucleus scattering cal— culations is given by the integral equation _ Q t— v—v e—ie t (l) where v is the nucleon—nucleon potential, Q is the Pauli operator, and e is the energy denominator defining the many body Green's function — defined in accord with the conven— 1,20 tions of Kuo and Brown. The presence of the is in Eq. (1) makes t complex. It is possible to express t in terms of the operator — Q t — v—v e t (2) which is real. This expression is t = tB—intBQ6(e)t. (3) If the imaginary part of t is small, and from the deformed optical.potential description of inelastic nucleon—nucleus scattering (see Section 2 of Appendix B) it is expected to A8 A9 be small with respect to the real part in the medium energy region, Eq. (3) can be approximated as tItB—ithQd(e)tB. (3') This argument, which is based on the relative magnitude of the real and imaginary part of the inelastic scattering form factors given by that model, is valid only in the region of the target nucleus where the form factor is appreciable. Eq. (2) formally is equivalent to the definition for 1’20 but it must be remem— the bound state reaction matrix, bered that the energy denominator, e, appropriate for the scattering problem is not the same as that for the bound state problem. Kuo and Brown have solved Eq. (2) for the 1’20 taking the H—J potential for the bound state problem nucleon-nucleon interaction. Using the Scott—Moskowski separation method,“7 they have shown that the attractive, even components of tB are well represented by tZV-v 3v (u) where v1 is the long range part of the H-J potential and sz is the long range part of the tensor component of this potential. The second term in Eq. (A) only acts in triplet states auui is given approximately by 50 _v82 22 ‘VTi eVT1“‘ VT2(r) + VT£(r)812 (5) where ng(r) is the radial part of the long range part of tensor component of the H—J potential, 812 is the "tensor" operator, and is a mean energy denominator which is highly state dependent. The state dependence of will be discussed in a moment. The first term on the right in Eq. (5) gives a very important contribution to the central, triplet even component of tB while the second term gives a small (10%) contribution to the even tensor component of tB. In writing Eq. (A) several terms in the Scott—Moskowski expansion have been omitted. They consist principally, for of a contribution tS from the repulsive core and various second order terms including a cross-term the H—J potential, between tS and v1. These additional terms are state dependent, but their’net effect is small. They will be ignored. Note that Eq. (A) comprises a local interaction in configuration space. The odd components of the nucleon—nucleon potential 2n%3 repulsive; therefore, the corresponding components of tB can rust be obtained from the Scott-Moskowski expansion since it (hoes not exist. Kuo and Brown use the reference spect— ruunxnethodl’20 to treat the odd components. This does not yielxi a configuration space interaction. In any event, tB is rwnoulsive in singlet odd states and has some attractive 'tripljat odd matrix elements. In binding energy calculations, [Tm—4"" r ‘n-‘W 51 the net effect of these odd state interactions is negligible, therefore, it is concluded that the average effect of the odd state interactions is small. The contribution to the triplet even component of tB contained in Eq. (5) is state dependent due to its dependence on . Equivalently it is density dependent. The mean energy denominator is state dependent because of its connection to the Pauli operator which appears on the left in Eq. (5), i.e. as the effect of Q is reduced as the density decreases,the strong tensor interaction between relative s and d states is felt more strongly and this must be accounted for by a decrease in . This effect is very clear in nuclear matter calculations. At low density contributions to the binding energy from relative 381 states are consider— ably larger than those from the lSostate, showing the full strength of the tensor force. At observed densities the two contributions are about equal. For high densities the 1S contribution is the greater — an effect which is an 0 important aid to nuclear saturation. An estimate of this effect can be obtained by compar— ixu; calculations of the bound state matrix elements for two free riucleons in a nucleus, without the presence of other rnncleons, and with those where the presence of other nucleons 16 ijs‘taken into account. In the first case, taking 0 as an exanmfihe§6the 381 matrix elements are ~l6 MeV, while with the l Paulj.IPrinciple taken into account they are ~9 MeV. The S0 matrix elements are ~8 MeV and very quite slowly with 52 density. Thus the average s-state matrix element, which is by far the largesg varies from ~8 1/2 MeV in the nuclear interior to ~12 MeV far outside the nucleus. This somewhat lengthy discussion of the bound state reaction matrix has been given with a View towards assuming that it is equivalent to t for the scattering problem, B i.e. differences between the propagator of Eq. (2) for the bound state problem and the scattering problem (in the energy region of interest here) do not alter tB appreciably. The stability of the separation distances (they remain essentially constant up to 30 MeV in the two-nucleon center of mass) for the important even components of tB supports this hypothesis. With this assumption, near the target E_E E E t — tB—ithQd(e)tB (6) where the superscripts E and 0 stand for even and odd, respectively, and E_T8 2 . tB — Vl- vT£(r) (triplet states) (7) = vi (singlet states) wheiwe‘the superscripts T and S denote triplet and singlet, respectively. In writing Eq. (6) the odd state components of t are being;rueglected and in writing Eq. (7) the second order 53 contribution to the tensor force has been dropped. The state dependence of the triplet component of tg could be incor— porated in Eq. (7) by defining to be a function of the local density. Now consider the region.far outside the target nucleus where the density is low and the effect of other nucleons is negligible. Here the propagator in Eq. (1), Q/e, becomes the propagator for two free nucleons, l/eO, and t is given locally by timpulse’ i.e. the pseudo—potential given in Chapter 3 which was derived from the free two—nucleon scattering amplitude. The tensor force now makes itself felt with full strength, but not in the real part of the interaction. The approximation of Eq. (3') is no longer valid, and the optical theorem forces the strength into the imaginary part of the interaction. The large imaginary component of the pseudo-potential is evident in Table 1 shown in Chapter 3. Combining these local arguments leads to a picture of a ferce which is primarily real inside the nucleus where the effect of the tensor in generating an effective central force is somewhat damped, going over to the impulse approxi- matixni at large distances, i.e. a force which has a large imaginary component. This asymptotic region is,however, likelgr‘to be at a density where all form factors are quite negligible. That is to say the picture of the force in the regitni of the target, which is where the scattering takes place, is of’prime importance. In summary, near the target t 5A is expected to be complex and density dependent. The real part is expected to increase outside the nuclear surface by about 50% on the average; the imaginary part to be quite small in the interior, peaked outside the nuclear surface, as all the form factors involved in evaluating tg Q0(e)tg are peaked at the nuclear surface, but small for incident energies up to about A0 MeV. At much higher energies this is not true. As the incident energy increases then the difference between Q and %— becomes less important and the impulse approximation bgcomes valid. However it should be pointed out that this approach asymptotically at high energies is quite slow. The impulse approximation is still a poor approximation at 150 MeV, even though it predicts cross- sections correctly. Its order of magnitude is quite good, but it's phgsg, i.e. the relative strength of the real and imaginary part of the interaction, is quite wrong as is shown by the fact that the ground state expectation value of timpulse does not give the optical potential (real and imaginary part), and that variables likely to be sensi- tive to the phase, like polarization, are by no means pre- dicted.successfully.uo'u3 It works better at 1 GeV, though the testxs then are not as stringentlfLI Therefore, t is to approach timpulse only slowly for the energy region we are consider- ing; On the other hand, as far as its magnitude is con- cerwmai, disregarding its phase, the impulse approximation mighi;lqot be too far out. 55 The arguments which have been presented above are very rough and, in fact, they could be wrong in detail. They are intended more as a suggestion than actual truth. The resolution of the points which have been made is a problem related to, but separate from, the purpose of this work which is to determine whether or not one make some sense out of inelastic nucleon—nucleus scattering using the inter- actions which are already available and convenient to use in D.W.A. calculations. These are, of course, the impulse approximation pseudo-potential and the interaction defined in Eq. (6) and Eq. (7). It should also be mentioned that an essentially identical discussion of t has been given, independently, by Satchler?7 He has also made some estimates of the imaginary part of t. The Kallio-Kolltveit potential is an s—state potential A withtnfimflem;even and singlet even components defined by 6 Vi (r) = +w r30 kk e—di(r—c) = —A r>c 1 Where AT=“75-0 MeV. aT=2.521AF-l, AS=330.8 MeV, aS=2.u021F‘1, enui c=0.AF. The long range part of this potential is known to give a good representation the central components of t? as defined in Eq. (7). In the calculations of this work, the nonrcentral parts and the imaginary part of tg are neglected and the K~K force is taken to represent its central part of tg. Fixed separation distances, ds=l.025F and dT=0.925F, are used throughout. The K—K force acts only in relative 56 s—states, but since this is an inconvenient restriction for D.W.A. calculations it is allowed to act in all even states. This leads to a slight overestimation of tg. Density depen— dent versions of the K—K force have been proposed by Green. These account for the variation of tg with in Eq. (7).68 These forces are not examined in this paper. In lowest order calculations, all of the bound state forces discussed here are found to give a reasonable account of the real part of the optical potential in the medium energy region; therefore, at least the spin, isospin averages of the monopole components of these forces are adequate for the scattering problem.3l’32 In detail the K—K force gives larger well depths, smaller mean square radii, and somewhat poorer agreement with phenomenological potentials than do the other forces. A rea- sonable estimate of the imaginary part of the optical potential has also been obtained with these forces. The impulse approxi- mation pseudo-potential failed to describe the optical potential in that it gives too small a real component and a very large imaginary component, i.e. its phase is incorrect. A real 1F range Yukawa "equivalent" to the K-K force (A) has been determined. It is compared with other "equi- valent" interactions in Table 1. These are the impulse approximation pseudo-potential for ELab=60 MeV (B), the empirical interaction of Ball and Cerny69 determined from studiens of the (He3, He3') and(He3,t) reactions in lp—shell nuclei. (C), the interaction used by Glendenning and Veneroni in studies of the Ni isotopes in the (p,p') reaction (D), and the interaction used by True70 in Nlu shell model calculations (E)- almost complete. timpulse’ force should be diminished here. tude of the strengths for interaction C are given. 57 The agreement between the forces is The lab energy of 60 MeV was selected for because the implicit exchange contribution to this Note that only the magni— The analysis did not give any conclusive information as to the actual exchange mixture of the force. Further, a guess of the magnitude of enhancement effects in the target nuclei was used in arriving at the value of VOO for force C. These effects are considerably smaller in lp-shell nuclei than they are in heavier elements. these forces is very satisfactory. The overall agreement of TABLE l.——Comparison of strengths of various real 1F range Yukawa "equivalent" interactions All values are in MeV. A is the K-K force, B is timpulse at ELAB=6O MeV, C is the inter— action determined Ball and Cerny, D is the interaction of Glendenning and Veneroni, and E is the interaction of True. IForce V00 V10 V01 V11 vgp Vgp vgn Vin A -36.2 6.30 17.8 12.1 -18.u 18.u -5A -5.75 B -A3.8 u.30 18.3 13.6 —25.5 18.0 —59.1 -9.20 C69 |30-A0] |11-27| |21| |17| — _ - _ D“ —u0.5 6.80 20.2 13.5 —20.3 20.3 -60.7 —6.70 1570 —A1.1 7.A0 20.0 13.7 -21.1 21.1 —61.1 -6.30 CHAPTER 5 THE APPROXIMATE TREATMENT OF ANTISYMMETRIZATION In this chapter some results obtained with anti- symmetrization treated approximately (as discussed in Section 6 of Chapter 2) are compared with corresponding results obtained with the exchange component of the D.W.A. transition amplitude treated properly. The exact results are due to J. Atkinson and V. Madsen.l7-19 A modification of the D.R.C. (Direct Reaction Calculation) code available at Lawrence Radiation Laboratory, Livermore, California has been used in obtain- ing these results.71 This code is restricted to inter- actions with radial dependence which can be easily expressed as a combination of not more than three Yukawa functions. Because of this all comparisons are for interactions with Yukawa radial form. No direct information concerning this approximation, is available for the interaction of primary interest in this work--the K-K force. The recently develOped non-local D.W.A. code at Oak Ridge National Laboratory has been set up to handle interactions of this type, i.e. which have a "hole" in them, and new results should be forth— 16 coming. 58 59 l. Yukawa Function The essential ingredient of the approximation under consideration is the Fourier transform of the interaction. For the Yukawa function V(r) = Ve-mr/mr (1) this transform is given by v<12> = (unv/m><12+m2>’l . (2) Table 1 gives the value of this transform as a function of the lab energy for m=0.5, 1.0, 1.5, 2.0, 2.5, and 3.0F’l. V has been taken to be 1 MeV and it is to be remembered that the lab energy and A2 are related by A2=2MEflh2. The last row in this table gives the ratio of the Fourier transform at 20 MeV with respect to that at 80 MeV. These energies span the region of interest in this work and this ratio is indicative, within the framework of this approximation, of the relationship between the range of the interaction and the energy dependence of the exchange component of D.W.A. transition amplitude. It is seen that this ratio decreases with the range and is approach- ing one in the zero range limit. 2. Transitions in Zr90+p Dependence on Energy and Multipole The ratio of the exchange integrated cross section to the direct integrated cross section has been given for the 90* L=0,2,A,6, and 8 transitions in the Zr90(p,p’) Zr reaction 60 TABLE l.--Fourier transform of Yukawa interactions of various ranges as a function of the lab energy. V(E)[MeV.F3] m(F_l) E(MeV) .5 1.0 1.5 2.0 2.5 3.0 0 101 12.6 3.72 1.57 .80u .A65 10 3u.0 8.AA 3.06 1.u0 .7A6 .AAl 20 20.5 6.36 2.60 1.26 .696 .420 30 1A.6 5.10 2.26 1.15 .651 .u00 uo 11.u u.25 1.99 1.06 .613 .382 50 9.33 3.65 1.79 .975 .578 .366 60 7.90 3.20 1.62 .906 .5u7 .351 70 6.85 2.8A 1.A8 .8A7 .520 .337 80 6.0A 2.56 1.36 .79u .u95 .325 90 5.u1 2.33 1.26 .7A8 .u72 .313 100 A.89 2.1A 1.17 .707 .451 .302 21291. 3.A0 2.A8 1.91 1.59 1.A1 1.29 V(80 at 18.8 MeV as a function of the inverse range of an interaction <3f Yukawa radial form.18 For the L=2 transition the Oex/Odir :ratio has been calculated as a function of energy with the renuge of the force fixed at IF.19 A Serber exchange mixture luas been assumed, and j-j coupling wave functions for two 61 protons in the lg9/2 orbit were used to describe the target.Jr The 18.8 MeV optical parameters of Ref. 8 have been used throughout. Results obtained approximately are compared with the exact results in Fig. l. The exact results are shown as dashed lines and the approximate results are indicated by solid lines. In the lower graph the corresponding results are bracketed and labeled with the L—transfer. The importance of exchange increases with increasing multipole and with decreasing energy. Note that Oex/Gdir deviates more from 1, the zero range value, as the range of the force increases. For L=6 and L=8 Oex is greater than Odir' The approximate values of Cex/Odir for L=A are about one, but the exact values are less than one. Qualitatively, the agreement of the approximate results with the exact results is quite good. The approximate results overestimate the exact results except for the case L=8. The agreement 'between the approximate and exact values of Oex/Odir improves withxincreasing energy. Thereis no pronounced change in the .agreemmnt as the force range becomes shorter except for the L=E3 case. The approximation is improving with increasing +To be more precise the 0+ ground state and 0+, 2+, A+, 6+ sand 8 excited states are considered to be due to the allowed ccnuplings of two lg9/ protons. The allowed normal transfers are: specified by the griads (J,O,J) and (J,l,J) where the trwuisferred J must be the same as the total angular momentum (of tflie final state. The (J,l,J) contribution vanishes due to a sixructure selection rule. Two non—normal transfers, (JifiL,l,J), are also allowed. Only the contributions due to ncmnnal transfers are being considered in the following discuss— iorr, therefore it is unambiguous to specify each transition by the: L-transfer implying the contribution due to the triad (J,O,J). I I I I I I I .4— ‘ Zr9°(p.p') L=2 mml ,3— --EXACT ‘ — APPROXIMATE t? \ b5 .2— "‘ I -— 1 1 1 1 I l J l 20 30 40 50 60 7o 80 90 IOO E (MeV) IQOE I.O:- _ E/ .... 22222 -‘ 6% f" 1 \ #- / d b; - .///' I // .l E— / 2.9%,.” E=I8.8Mev ‘5 -_- --EXACT I : / ——APPROXIMATE I _ / —— mugging)“: - I] 1 1 l I l l 1 .4 .3 1.2 L6 20 24 28 32 ulflflffi Figure l.-—Comparison of approximate and exact resultsshowing the variation with energy and interaction range of the ratio of the exchange to the direct integrated cross section for several multi- poles in the Zr90(p,p') reaction. 63 multipole which is very good since the contribution from the exchange component of the transition amplitude is becoming more important at the same time. The last effect is consistent with the fact that transitions of high multipolarity are not sensitive to the details of the nuclear interior which are ignored in the approximation. The result indicated by the center line and labeled L=0 in the lower graph of Fig. l is interesting. This approxi— mate result was obtained by considering the ground state and first 0+ state of Zr90 to be described by more realistic configuration mixed wave functions involving both the lg9/2 and 2pl/2 proton levels.8 The ratio Oex/Cdir for this case is quite different than the result obtained using only the unrealistic (lg9/2)2 configuration. This indicates that the contribution to cross sections due to exchange can be quite sensitive to the wave functions involved. Total Cross Section (Direct + Exchange) If maximum interference between the direct and exchange amplitudes is assumed it follows that the total integrated cross section (direct plus exchange) is given by 2 0ex 2 + /Eex) = (l + ——— ) o (3) 0dir =0. OT ~Wadi dir Odir. 1" This assumption is quite good. It has been shown that the direct and exact exchange amplitudes are essentially in phase except for extreme forward and backward angles.18’19 This is true to a greater extent in the approximate calculations. Table 2 6A displays the values of aExact and “Approx/aExact of L for meB and 2.0F—l. Eq. (3) has been used to determine as a function and oex/o values have been taken from Fig. l. The numbers dir in the table indicate a maximum error of A0% in the approxi- mate total cross section. This occurs for L=0 and m=.8F—l. To illustrate the rate of improvement of the approximation /aExact to 1.02 as E goes from 30 to 50 MeV for the L=2 transition with increasing energy note that a goes from 1.15 Approx with m=lF—l given in Fig. 1. goes from 2.17 to 1.95 O‘Exact over the same energy region indicating that the enhancement of the direct cross section due to exchange is decreasing fairly slowly with increasing energy for this particular multipole. TABLE 2.-—Comparison of approximate and exact values of a, the enhancement of the direct cross section due to exchange, for two values of the interaction range appearing in Fig. l. m=.8F'l m=2.0F"l L 0‘A rox OLA rox OLExact O‘Exact O[Exact O‘Exact 0 l.AA l.AO 2.A0 1.35 2 2.00 1.32 2.77 1.26 A 3.0A 1.35 3.33 1.29 6 5.66 1.22 A.58 1.19 8 1A.8 .8OA 6.82 .978 65 Fig. 2 shows the direct,exchange,and total integrated cross sections as a function of multipole for the cases m=.5F‘1 and m=3.0F“l of Fig. 1. Both the approximate and exact results are shown for oex and CT and maximum interfer- ence has been assumed in obtaining CT. The absolute normal- ization of the results is arbitrary, but the relative magni— tude of each,for each force range,is as shown. This figure illustrates how OT falls off slower with L than does Odir due to the contribution from oeX——an effect which is not very pronounced for m=3F—l——and how the fairly large errors in the approximate values of Oex/Odir are not so strongly reflected in CT” Fig. 3 compares the behavior, as a function of multipole, of cT(Exact) for the 2F range force with Odir for forces with m=1.0 and 1.5F‘l. The behavior is similar. A previous empirical analysis of these transitions, in which antisymmet— rization was ignored, led to the conclusion that a 1F range Yukawa interaction reproduced the observed multipole depend- ence of the cross sections.8 A longer range would have been selected had antisymmetrization been taken into account. Angular Distributions The direct (D) and approximate exchange (E) angular distributions for the 2F range Yukawa force are shown in jFig. A. All curves have been normalized to one at peak. With tflie exception of the L=0 transition the exchange angular 66 omeonmdam on» zoom .80 new we o mom Czech one mpHSmon pomxm one .H onswfim mo momwo Himo.m ocm m.ou8 on» o “.360 no comfimeEOOI|.m onswfim B .xm mom A no coapocsm mm b one OIIIJ or m c N T. N n 533:6 931.6 lab. .328336 I 5b ..le » IIIIIN too-u. b 9.2.14 2.283 .r. «lid 0 1: ...... o.» us .III a w .2332 .339 83g 259...; M . , J zinc?“ $6.38; 79m N m. a... \ I / s\\ n w w I III \\V n o. ollua o or m o c N on. N 203336 Ollie 182336 I .Bb x x n 23.6. ...b «Illa / 3233 #5 old / «.0. x x N //, \\\ p s m \ .... m I. :0. ..o. m a a n" x / . .... ll/HH n. rv/ n ~ I / I .///0 I III _. N _ win. 0.0 «E .825: 89.9 38... 2,9...» m a: 3...“. cases a _ i L o. a- (Arbitrary Units) —* 67 1° I I I I Zr9°(p.p') E=18.8 Mev 5 Yu kawa Force (Serber Mixture) 2 E 5 -- 0——0 0'1- (Exoct){m-.5F"‘} X ----- X 0dir(m3|.OF-1) 2 A----A adi,(m-l.5F") 1o“2 5 2 10'3 O 2 4 6 8 10 |_ —> Figure 3.——Exact value of o for a 2F range Yukawa force is compared with 0 i for a 1F and 2/3F range Yukawa force as a function of L. 68 distributions fall off only slightly slower with angle than do the direct andwfor the lower multipoles)the latter exhibit quite a bit more structure. Both the direct and approximate exchange angular distributions for the .33F range Yukawa force are essentially the same as the approximate exchange angular distributions shown in Fig. A. In Fig. 5 the L=2 direct (D) and exact exchange (E) angular distributions given by a Yukawa force with a range slightly longer than 1F are compared.18 Comparing these with the L=2 results in Fig. A indicates that the approximate exchange angular distributions may fall off faster with angle than do the exact exchange angular distributions. This also might be multipole dependent, but no comparison is available for the higher multipoles. The differences, for large angles, between both the direct and exchange angular distributions shown in Fig. 5 and those which correspond in Fig. A is attributable to the fact that the spin—orbit term in the optical potential has been excluded in obtaining the results shown in Fig. 5. Inclusion of optical spin~orbit coupling is found to have no effect on the ratios of inte— grated cross sections discussed previously. .Form Factors The direct and exchange form factors corresponding to tflue results given in Fig. 2 are shown in Fig. 6. The overall ruarmalization for each force range is again arbitrary with tflie relative scaling given correctly. For the short range 02 (mb/sfer)—' Zr’°(p .p') E - l8.8 mv Yukawa Force (Sorbet Mixture) macawl -——0 ---£ '9 »’~\ 1. x ‘”‘ \ W‘\ IO —N 0| NU 1 IO ‘2” I L'S N J // / I I I \\g \ 10"- 20 40 so so IOO |20 I40 I60 l80 ec(deg)-+ Figure A.-—Direct and approximate exchange angular distributions for 2F range Yukawa fo§8e for L=0 2,A,6, and 8 transitions in Zr +p at 18.8 MeV. aMMitrary Uni rs) —9 7O IO 1 I l l l 5 Zr9°I p. p') E :2 l8.8 Mov Yukawa Force (Serber Mixture) m 5. LG F" 2 — D ---E lh-Ing1 \‘x L=2 l/ 5 x \ I \4~~“ ' ' I —\ \— 2 \ \V ’7 \I I0" \\ \l 5 \jly/\\\-i / W 2 . It!"2 5 2 o 20 40' so .30 I00 I20 I40 I30 I90 9c (deg)-' Figure 5.¢-Direct and exact exchange angular distribution for Yukawa force with range somewhat greater than 1F for L=2 transition in Zr90+p at 18.8 MeV. 71 force (m=3.0F'l) all of the form factors including the exchange (zero range) form factor are similar in shape with the peak magnitude of the exchange form factor assuming a value inter- mediate to those for the L=0,2 and L=A,6,8 direct form factors. I), 3 This ordering is preserved for the long range force (m=.5F- however, the differences in peak magnitude of the direct form factors is much more pronounced. Here the direct form factors are also much broader than the exchange form factor and peak at larger radii. The differences in peak radii between the L=0,2,A,6, and 8 direct form factors is not very large for either force range. Examination of these form factors emphasizes again that a multipole independent assumption has been made about the exchange scattering and that the variation of Uex/Odir is due mostly to the changes in the direct scatter— ing. The exact results call for additional multipole and energy dependence in the exchange scattering. Relation of Energy Dependence to Interaction Form It is found for both the long range and short range Ytdcawa interactions that the approximate Oex/Odir ratios are ggiven to within 5% by taking the square of the ratio of the aixaas under the appropriate form factor curves. The area is deifiined as the product of the peak magnitude and the half wixith of the curve. To what extent this will be true for otflie12 transition densities and forces with different radial formns is not known. For example, it has been noted that .m osswfim mo muadmtp on mGHoQOQmohsoo mAOpowm Show owcmcoxo ocm pomsHDII.m ossmam TE: ON. v... m6. Nd. md 0d v.0 mu. NN mm ow v.0 QV NV on on VN Q. N. o. O a _ _ a 1 . _ _ . q . a _ . coco. oeoN IIII I Tm on u E .0532 .0908 cocoa 039....» so: mm. «m ..a. a. .N — — — q q A — q _ q _ _ — u d . a d u \ / \\ m "4 \\ I / w ... \ I x x - / v-4 \ / \ \ // \ I I'\ I1 I N "J :4 3:2 83 llll on; I .l Th 0 u E A 83.22 8909 Sean 0393» >02 mm. u w Ta. 3 om.N ‘——(;-'/’A3W)ej x(J)107='/ 73 L=0,2,A,6, and 8 form factors obtained from the (lg9/2)2 transition density with a Gaussian interaction of 2F range exhibit a much larger spread in peak positions than is seen 9 for the 2F range Yukawa force. Nonetheless, this observation indicates that the energy dependence of Odir/Oex for a given transition density and multipole can be written 0 Tex (E) = [ACWEH2 K (A) dir with K, the ratio of the integrated cross section obtained with a 6—function force of unit strength to the direct integrated cross section, being roughly constant. Eq. (A) implies that o 0 5335021) : £A(l)(El)/A(l)(E2)l2 591082) . <5) dir dir Fig. 7 gives Odir’ Oex’ and GT as a function of energy for the L=2 transition and the 1F range Yukawa force. The conventions are the same as those for Fig. 2. This figure simply illustrates that 0 drops off faster with energy than T does Gdir due to the behavior of Uex’ that the approximation is improving with increasing energy, and that large errors are not observed in 0 even at the lower energies. The result T indicated by the center line and labeled oex (F.T.) has been obtained by fixing K from the value of Cex/Odir at 25 MeV and then using Eq. (5) to get this ratio at the higher energies. 74 5 .\ :L: 5 C D s, 3 t 5 2 s S. \ b -I IO Zr9CIp|p') L=2 5 Yukawa Force (Serber Mixture) m = IF-| A—A a'T (Approx) A---A 0'7 (Exact) 2 X—X O‘dir 0—0 O’axIApprox) o—--o 0.x(Exact) 0—'—O O'ex‘FHT) .crz l O 25 50 75 IOO EIMev)—v Figure 7.——Energv dependence of o i , Cex’ and o for L=2 transition in ZP§O calculated Witg {F range force as in Figure 1- Both approximate and exact results are given for g and GE. Energy dependence computed from the Fourier transférm of t e force 5 a so indicated. 75 The value of Oex is then easily extracted given 0 The dir' agreement between Oex (Approx) and Oex (F.T.) is quite good.+ This relation can be used with Eq. (3) to estimate, for a particular transition and force, the ratio of the enhance— ment of the direct total cross section due to exchange at two different energies given the value of oeX/o at one dir energy. Table 3 gives the values of d(20)/d(80) which have been obtained through the use of these relations for the 90 L=0,2,A,6, and 8 transitions in Zr for Yukawa forces with —l m=.5, 1.0, 1.5, 2.0, 2.5, and 3.0F . The values of Oex/Cdir at 20 MeV are those shown in the lower graph of Fig. 1. The differences between these ratios for the different multiples increase with the force range. For the 2F range force the energy dependence of 0 should be quite different for the T various multipoles provided the direct total cross sections vary slowly with energy, i.e. CT for the higher multipoles should fall off faster with energy than for the lower multipoles where oex is making a smaller contribution. This is an example of an effect due to antisymmetrization which might be used to study the properties of the projectile— target interaction. Non-Normal Transfers The cross sections for the non—normal transfer specified by the triad (J-l,l,J) have been calculated for these five 19 transitions for a 1F range force. In all cases they were found to be smaller than the corresponding normal exchange +Note that oeX(Exact) and ceX(F.T.) will only agree if the extrapolation is for energies above A0 MeV; however, reasonable approximate values of a (see Eq. (31)) might be obtained over the entire energy region. 76 TABLE 3.--Approximate energy dependence of enhancement of direct cross section due to exchange as a function of multi- pole and range of a Yukawa force. mEF—l] a(20)/a(80) L=0 L=2 L=A L=6 L=8 0.5 1.31 1.60 2.27 3.26 5.10 1.0 1.56 1.7M 2.07 2.A7 3.22 1.5 1.5A 1.62 1.75 1.92 2.10 2.0 1.A3 1.A7 1.52 1.60 1.67 2.5 1.33 1.35 1.38 1.A1 1.A5 3.0 1.30 1.26 1.27 1.29 1.32 cross sections. Only for the L=8 transition, where exchange scattering is dominant, was an appreciable contribution to This 0, obtained. Here the (718) component gave 25% of o T T' is encouraging, however, it must be noted that the S=0 and S=l components of the proton—proton force are equal in magni- tude. If the component of the force contributing to the non-normal transfer was larger than that contributing to the normal transfer (as in the hypothetical situation outlined in Section 6 of Chapter 2) the non—normal transfer would clearly be more important for the L=8 transition and might be impor- tant for the lower multipoles also. 3. Transitions in 012, 016, and Ca“0 + p. As a further check,comparison calculations have been performed for some of the transitions which are being used 77 in this work to "calibrate" the effective interaction. These are the excitation of the 1+T=l (Q=-15.1 MeV), 2+T=0 12 (Q=-A.A3 MeV) and 3‘T=0 (Q=—9.63 MeV) levels of c by 28.05 and A5.5 MeV protons, the excitation of the 3—T=0 (Q=-6.l3 MeV) level of O16 by 2A.5 MeV protons and the excitation of the 3‘T=0 (Q=—3.73 MeV) and 5‘T=0 (Q=-A.A8 MeV) levels of 0a“0 by 25 and 55 MeV protons. The experimental results to be 12 at A5.5 MeV and for Ca“0 at 55 MeV have been 12 shown for C published72’u5 while the results for c at 25 MeV is the unpublished work of P. Locard and S. Austin. The experimental results for 016 at 2A.5 MeV and for Ca”0 at 25 MeV are the unpublished work of w. Benenson and C. Gruhn, respectively. In these calculations the interaction was taken to be the 1F range Yukawa'equivalent"to the K—K force which was given in Chapter A. The wave functions used in these calcula— tions are specified in the following chapter. For the time being it is sufficient to say that both the exact and approxi- mate results have been obtained in a consistent manner from these wave functions. Optical spin-orbit coupling has beem omitted and the optical parameters used are given in the next chapter. Only normal transfers have been considered. The targets being considered all have 0+ ground states, therefore the J- transfer must equal the total angular momentum of the final state. All of the transitions except the one ending at the 12 1+T=l state of C are of normal parity. For these only the 78 cross section specified by the triad (J,O,J) has been cal— culated while that specified by (J—l,1,J) has been calculated for the abnormal parity transition. The exact and approximate results are compared with each other and with experiment in Fig. 8,9, and 10. The total differential cross sections are shown in all cases. The dashed curves are the approximate results and the solid curves are the exact results. The direct differential cross sections and the approximate and exact exchange differential cross sections are shown only in Fig. 8 which gives the results for the L=3 transition in 012. Here the direct angular distributions are shown as center lines and dashed and solid curves are used to designate the approximate and exact exchange angular distributions, respectively. No ambiguity results from not distinguishing the exchange and total differential cross sections in this figure as the latter are always larger. Not much need be said about the differ— ences between the exact and approximate results. It is quite clear that no serious discrepancies have been introduced in treating exchange by this approximation. The differences that are observed are generally consistent with those noted 90 results, i.e. the approximate total in discussing the Zr differential cross sections tend to overestimate the exact ones at the lower energies by less than A0% and the differ- ences all but vanish at the high energies. The shapes are generally consistent with some deviations being noted at 79 IO 5 — Exact 2 1 --- Approximate a.“ --- Direct I ~ \; \I. b\. , / \ ' \ 4/ ‘X N .l V/ $28.5 Mcv \ ' 3'7-0 . 08-933 Mov \l\ .05 "" \\ dO/da (mb/str) Cl2+ p E O 28.05 HOV 3'TUQ0I-953 NOV 0 20 40 60 80 I00 I20 I40 |60 l80 9cm. (deg) Figure 8.—-— Differential cross sections obtained with the K-K "equivalent" interaction for the L=3 transition in the C12(p,p') reaction at 28.05 and I45.5 MeV. A decomposition of the cross sections into direct and exchange components is shown and a comparison of approximate and exact results is given. The direct component which is not affected by the approximation is shown as a center line. The lower of the two sets of exact and approxi— mate results shown is the exchange component and the upper set is the total differential cross section. 80 CI": E - macaw 2’7-0.0-—4.43Ihv do/dn (mb/str) O 20406080|00|20|40|60l80 9cm'(deg) Figure 9.—-—Total differential cross sections obtained with the K-K "equivalent" interaction computed approximately and exactly, for the L=2 transition in 012 at 28.05 and A5.5 MeV and the L=3 and L=5 transitions in Ca 0 at 25 and 55 MeV. o" 3' no (a - -s.25 M») E - 24.5 am 0; (mb /sfer) c'2 I+ T-l (o - -|5.| M") E - 28.05 Mu c'2 I’ I'll (0--l5.l M") E «5.5 Mcv 9c (deg) Figure lO.--T0tal differential cross sections obtained the K—K "equivalent" interaction, computed approximately and exactly, for the excitation of the 3'T=0(Q=-6.l3 MeV) level of 016 at 2A.5 MeV figdstfigvexcitation of the 1+T=1(Q=-15.l) level of Cl‘2 at 28.05 and 82 forward angles for the L=3 transitions in C12 and 016. The only peculiarity that is observed occurs for these same cases--here it is found that the approximate results tend to underestimate the exact results. In the light of the other results which have been presented this is not expected and an explanation is not readily available. The value of 0 and the approximate values of 0 dir l2 and CT for the L=3 transition in C at 28.05 MeV are 6.59, 12.9, and 36.5 mb, respectively. At A5.5 MeV the values ex 7.A8, 7.A6, and 28.8 mb are obtained. Note that Odir has changed only slightly with energy and that oex/odir (28.05) = 1.96 and oex/odir (A5.5) = .997 which are in the ratio 1.97. The value one would predict using Eq.(5) and Table 1 is 1.86. The comparison of the results with experiment is of some interest. It is found that this 1F range Yukawa force yields results which are in reasonable agreement with experi— ment at the lower energies but appreciably overestimate the higher energy data.+ Thus it is concluded that the experi- mental data favor an interaction whose range is longer than IF. A. Summary Although the results which have been presented do not constitute a complete study of the approximation it is felt that they demonstrate that it is probably qualitatively +The optical parameters used in the calculation for the L=3 transition in 016 were not very good. Better para— meters are given in Ref. 73, The results shown in Fig. 10 should be reduced by a factor of 2-3. 83 correct over the entire medium energy region and may be quantitatively valid at energies exceeding A0 MeV. For the lighter nuclei considered it was found that the shapes of the total differential cross sections computed approximately were in reasonable agreement with those computed exactly with the possible exception of the L=3 transitions in C12 and 016. Here differences are noted between the exact and approximate exchange angular distributions. Differences were also noted for the L=2 transition in ngo. The energy dependence of Oex and CT has been related to that of Odir through the Fourier transform of the force being used. Further, it would appear that the damping of exchange scattering in the nuclear interior, i.e. the correction terms discussed in Appendix A, would improve the approximate results. As the exact calcula- tion of the exchange transition amplitude is quite involved it is felt that this approximation and the relations based on it can be put to good use in any analysis of the effects due to antisymmetrization. 5. K-K Force The singlet even and triplet even components of the K-K force have the form V(r) ll 0 "5 IA 0.. (6) which lead to the following expression for the Fourier transform. 8A md 2 +12)'2{§1959[(m2+12)(ma-1)+2m2] V(A2)=Anve_ A (m +cosld[d(m2+k2)+2m]} (7) As this force acts only in even states the A(l)(A2) are given directly by the Fourier transform of the appropriate component of the interaction. Table A gives the strengths of all components of A(l). The notation ASE and ATE, AST’ and Agp and Apg is used. The last row in this table gives the 20 to 80 MeV ratios as was done in Table 1. The values of these ratios,as compared to those given for the Yukawa functions,are illustrative of the long range character of the K—K interaction. The results of Section 3 of this chapter indicated that a long range force is needed. The behavior of the A(l)(E) is, for the most part, regular. The extreme long range behavior of A10 and the fairly short range character of Aim indicate that a great deal of cancel— lation has taken place in constructing them. This leads one to suspect that these components of the interaction are not well determined. Unlike interactions with regular functional forms, the long range behavior of the K-K interaction is not reflected in its range parameter. It is attributable, instead, to the presence of the "hole" in the interaction. To see this it is only necessary to note that V(Ae) = Anfoj0(1r)V(r)r2dr (8) . . l *3. 85 mm.m sm.m ms.m ms.m sm.o ma.m :.m: sm.o om.m ms.m Aom\omv m.oau s.mm- m.mH m.mHI m.HH m.mm wmm. o.mmu mma I m.HmI om m.mHI 0mm- s.mm s.mm: m.sH N.Hm ms.s w.mmu cma I e.omI os omen FHA- m.em m.=mu s.mm m.He am.m m.osu sow I 03H: om m.wau smHI m.om m.omu 0.3m s.mm s.ma sea- mmm I How- om m.mmu com- o.mo o.mou m.me m.me s.mm mma- em: I mew: o: m.om- sow- o.sm 0.:mu H.0o m.om m.mm omHI mmm I osm- om s.om- mam- mma mma- m.ss moa m.ss ammI mas I oomn om s.mm- smsu moH mcHI ooa oma 0.:o Hem: cam I owe. OH m.HsI mmm- saw sfimn mma flea m.sm wmmI oomHI mom- 0 ewe ewe owe ama Afla Hoe Ode oo< mea mma mma.>ozaxmcxav< m>ozam .ooESmmm one moocmpmao coapmsmdmm ooxam m we monom xIx pom coapompopcfi owcmnoxm mo masocOQEOo mo mnpwcmnmeI.z mamas .mwmocm an on» no coapocsm 86 and to remember that the main envelope of the Bessel function appearing in this integral is confined to values of r~%. Since A and A8 are roughly 1 and 2 F‘l, respec- 20 O tively, while the cutoff radii are approximately 1F, it is clear that this main envelope is falling within the "hole". Continuous exponential functions with the same range ’ r parameters as the singlet even and triplet even components ‘ of the K—K force give (20/80) ratios of 2.05 and 1.96, 3 “”4 respectively, as compared with the corresponding values of 9.75 and 5.36 given in Table u. It is interesting to estimate the effect of the energy dependence of the separation distances (which is being neglected in this work) on the values of A(l) (E) given in Table H. To do this it is assumed that D. II 1.025 + (.05/60)(E—20) (9) 0.. II 0.925 + (.03/60)(E-20) where E is in MeV. These linear relations represent reason- ably well the energy dependence of the cutoff radii as cal— culated by Kallio and Kolltveit.u6 Table 5 contains the results obtained for A(l) (E) under this assumption. The values of A(l) (20) given in Table 5 are identical to those given in Table M as the separation distances have been fixed at this energy. It is seen that ASE is quite sensitive to this change while the effect on A is smaller insofar as TE the (20/80) ratios are concerned. The energy dependence of 87 mm.m oa.s H.wH OH.0H H:.w mz.m m.HmI Hz.m sm.m H.ma Aom\omv m.oaI m.~:I ss.s ss.sI mm.m 0.0m om.HI m.smI FHHI fi.HmI om m.mHI H.maI m.sH m.sHI m.ma s.wm mm.m m.m:I NFHI m.HsI os m.mHI moaI m.om m.omI m.mm H.mm mm.a m.moI mamI mmHI om m.mHI omHI o.®: 0.0:I a.mm s.am m.mH m.mmI mmmI mmHI om z.mm| Howl w.mw m.mol 0.2: o.ww m.mm :MHI was! mom: 0: m.mmI :mmI :.mm :.mmI m.mm o.mm H.mm maHI ommI osmI om s.omI mzmI mma mmHI m.ss moa m.a: :mmI mzeI oomI om m.mmI ozzI sea smHI Hoa smfl m.mm :omI mmmI smoI OH NH.:I mmmI Hmm HmmI HMH msfi 0.0m mmmI HNHI :wmI o cwa ema aw< am< HH< Hoa oaa ooa me< mma mmm.>mzHAmVAHV< m>mzim mmpocm Spas mhm> .Amv .vm ou wcaopooow mmocmpmflc coapmnmamm pomoxm : magma mm mammII.m mqmqa 88 the separation distance is somewhat more pronounced in the former case. The differences between Tables u and 5 taken with the related effect on the direct component of the transition amplitude are large enough to produce noticeable differences in calculations; however‘it is doubt- ful that they will be more important than the effects of the density dependence, imaginary component, and odd state components of the interaction which are also being neglected in this work. Further, most of the available data lies in the energy region from 20-50 MeV and none of the strong transitions observed are likely to isolate the singlet even component of the interaction where the effect is the largest. 6. Effective Range Forces It has already been noted that the (20/80) ratios for typical Yukawa forces are much smaller than most of those appearing in Table H. In fact, since the long range limit of the Yukawa function is the Coulomb potential, the maximum value of (20/80) for the Yukawa is U. The Gaussian function, its Fourier transform, and the relation for the (20/80) ratio are given below. _ 2 2 V(r) = Ve m r (10) 3/2 2 2 V(A2) - v 1—§— e—A /um (11) m A2 —A2 2 V(2O)/V(80) = exp(-§QZ—%g—) = e'73l/m (12) m 89 For an interaction of exponential form the corresponding relations are: V(r) = Ve—mr (13) 2 _ 8an V(X ) - m (114) (m2+l20)2 I V(20)/V(80) = 2 2 2 (15) (m +l20) ! 1 E Fa. Eq. (1“) follows directly from Eq. (7) in the limit as d goes to zero. Note that V(20)/V(80) for the Gaussian form can assume any value from 1 to m whereas V(20)/V(80) for the exponential function can vary only from 1 to 16. The Fourier transforms of the components of the K-K force clearly cannot be matched with a Yukawa function over the 0-80 MeV energy range. It was found that this matching could not be achieved with an exponential function either. For example, an exponential function with V=—59.7 MeV and m=.636F-l gives the same value for (20/80) as the singlet even component of the K-K force; however, it gives V(O)= -58HO MeV.F3 which is about six times the value given in Table N for the K—K force (Aéé)(0)). In addition, V(E) for E intermediate to 20 and 80 MeV are smaller than correspond— ing values of Aéé)(E). A reasonably good match can be obtained with Gaussian functions. Gaussian interactions with V: -3u.9 MeV and m=.567F"l and v=-67.3 MeV and m=.660 give the- same (20/80) ratios as the singlet even and triplet 90 even components of the K-K force, respectively. They also give V(O)=-1070 and -l300 MeV.F3 which are reasonable agree- ment with the corresponding K-K force volume ingetrals. Table 6 contains the pertinent data for Gaussian, exponential, and Yukawa forces which fit the scattering lengths and effective ranges which are sufficient to 5 characterize low energy nucleon—nucleon scatteringu8’u9. Fig. 11 shows the Fourier transforms of these forces compared manna—T i with that for the K—K force. The transforms for the K-K force are bowed slightly upwards on the graphs, while those for the Gaussian are straight lines. Both the exponential and Yukawa transforms are bowed downwards. From the figure it is concluded that the Gaussian effective range force is quite similar to the K-K force and that the Yukawa effective range force shows the greatest deviation from it. This is consistent with the remarks made in the preceding paragraph. In fact the strengths and ranges for the Gaussian functions given in Table 6 are nearly the same as those obtained by matching to the K-K force. These conclusions are not surprising. Like the effective range forces, the K—K force is consistent with the low energy nucleon—nucleon scattering data. It is evident from Fig. ll that all of the forces are similar (on the average) for small values of E(<20 MeV). The Gaussian func- tion has properties similar to the K—K force and when the two are forced to correspond over a small region (0—20 MeV) they 91 automatically are similar over a much wider region (0-80 MeV). On the other hand two dissimilar functions forced to corres— jpond over a small region will not correspond over a wider .interval. (TABLE 6.——Forces which are consistent with low energy nucleon— nucleon scattering data. Singlet Even Triplet Even Claussian V(MeV) -39.5 —7l.O m(F—l) .637 .676 V(O)(MoV.F3) -850 —1279 V(20)/V(80) 6.06 “.95 IBJcponential V(MeV) -l38 —186 m Q) _ E U) 2 '00 ’— \\---...__ —1 a: \ C) \. UL _ l 00 2: 1 L 1 l 1 1, 2L mzuAmVAHV< m>ngm zmnoco bad on» go coapocsm mm moaom mzwxsw pom cofiuomnmpcfi owcmnoxm mo mpsmcoasoo «mama m>fiuommmm mo mnpmcmnsmII.a mqm®H A>®Z mm.mlfl@vOHBIm %O% N mhdwfih mm memm||.m mhdwfim All rite ...IIII A It. on oao. n... 0.. no 0 n o n e n m o _ _ — q o q — q - q i I an .806 I .I No' H i w- ... -306an III! 0. x/ \ l \ o T //I\\\ _ -806 0.90.}. e o.momll N .-h. 0.0.... :. on NB 2 m3 2 «rennet... mom. "Eons... .... 8.. Na a. Q» o. 0 .Iu «.6..- moon: Sodom“. > x e a $02 mod. «9 on... In so ‘—— (pd)21x(1) 1‘ PS" :1 FLSJ'TIIIXIZIFi) _. Figure lI.--Same as Figure 2 for 3-T=0(Q=-3-73 MeV) level Of Ca 103 c040 3’ T30 (0' -3.73 MeV) p h X Y |f7/2 Id 512 -.378 :20I |f7/2 28 l/2 -.538 1236 If7/2 Id 3/2 -736 1222 293/2 Id5/2 -.I26 1085 2pm Id 3/2 -.2|5 -.I3O If 5/2 Id 5/2 .I99 .I07 If 5/2 25 V2 .233 .I29 If 52 Id 3/2 1285 -.I63 Zp I/2 Id 5w: 1 .I46 .087 -I L 1 l l l l l 2 3 4 5 6 7 8 9 r(F)-—’ IFIqH xnoz—~ 2 2 r”"°II)-I-I.Ize.°I3+.9osa°I’)5' ' 8 5 1%: F3'3'°Ir)-I-.23|2asr3+264a I )e c- .496 F“ I I II 0.2 - O I u——-o-—-a 1 i 0.5 q(F')—' 10A .ozmo Mo Ho>oH A>oz w:.:tuooouelm pom m onswfim mm oemmll.m oaswfim ATIIIIAnHvL Indie .... I . I: IN ml In I... e. w - o... [0. ~ 10. ...i so... ... «$63.66 mm. .- «Sodas. N. «6.00506 mm”... u Any-08m “>02 mid- «9 on... In 96 mNr mar NB 2 Ne. : Q... oNr NB 2 Ne. : Q. .m. NB 2 NB 2 > x s a ‘_ (I51) 31x (1)1‘PS‘IA 105 again it is seen that the transition densities peak inside J1J,O the nuclear surface and that the F (r) transition densi- ties can be neglected. The harmonic oscillator constant for “O is taken from Ref. 32. Ca Also contained in these figures is a comparison of the theoretical and experimental inelastic electron scattering form factors for the excitation of these levels. The calcu- lation of the theoretical (e,e') form factors from the transi- tion densities is discussed in Appendix C. These results are essentially the same as those contained in Ref. 55. They' have been recalculated more as a check than for any other rea— son and are shown for completeness. The overall agreement between theory and experiment is quite good, although the data for Ca140 is admittedly sparse. The ground state corre- lations are responsible for factors 1.5-3 in the theoretical form factors which are from four to an order of magnitude larger than results obtained in single particle-hole excita- 55 The enhancement effects are largest for the L=3 transition in Cauo. The mixing in the R.P.A. vector tion models. for this transition is evident from Fig. A. Looking at these results a bit more closely, it is seen that the L=2 transitions for 012 gives a result for |F(Q)l2 which is about 15-20% too small. The theoretical form factor for the L=3 transition in C12 has about the right magnitude, but it peaks at slightly too large a value of q. Ref. 55 extends the comparison of theory and experi— ment for these two transitions up to about q=3-5F—l- The theoretical results overestimate the data in this region 106 which could be an indication that the theoretical transition densities are too large in the interior of the nucleus. The experimental data for the L=3 transition in Ca“0 is ambiguous and it is seen that the theoretical result in Fig. A falls in between the two sets of data points which are in disagree- ment. The corresponding result of Ref. 55 has been obtained I with the R.P.A. vector given in Ref. 51 and it is in agree- ment with the upper set of data points in Fig. A. The experimental data for the L=5 transition is also not very definitive. It appears that the theoretical result here could be a little too large and might peak at much too large a value of q. It is concluded from this discussion that (l) the transition densities presented in Fig. l-5 should not be responsible for any gross discrepancies in the (p,p') results which are to be shown and (2) the effects of long range correlations, which are included in the R.P.A. vectors, are playing an important part in building up the magnitude of these transition densities. Better transition densities have been constructed for the L=2 and L=3 transitions in C12 by fitting directly to the experimental (e,e') data.75 These have not been used as they do not differ a great deal from those presented here and the differences are well within the uncertainties associated with the local reduction of the D.W.A. transition amplitude. The (e,e') data for Cal40 is not sufficiently accurate and complete to even allow con— sideration of improving the transition densities. 107 D.W.A. calculations have been performed for Cl2(p,p') * “O at 25 and C12* at 28.05 and 45.5 MeV and Cau0(p,p')Ca 55 MeV. The differential cross sections obtained for the above transitions using the impulse approximation pseudo— potential are compared with experiment in Fig. 6-8. Cor- responding results for the K-K force are shown in Fig. 9-11 and those for the Yukawa effective range force are given in Fig. 12—1u. The total differential cross sections for the _ L=3 transition in C12 are decomposed into direct and exchange components in Fig. 10 (K-K force) and Fig. 13 (Yukawa effec- tive range force). Optical parameters used in the calcula— tions are given and referenced in Table 1. The form used for this potential is given in Eq. (B.l3). A tabulation of the theoretical total integrated cross sections, CT, is contained in Table 2. Values of odir and Oex for the K—K force and Yukawa effective range force are also displayed in this table. A quick glance at Fig. 6-1A shows that all of these forces are giving a fair reproduction of the data. For the normal parity transitions it is found that the results obtained with the impulse approximation pseudo—potential and the K—K force best reproduce the data. The impulse approximation gives results which are slightly smaller in magnitude than the K—K force. These differences are no larger than 20%. The results for the Yukawa effective range force are found to underestimate these cross sections at the lower energies, but at the higher energies they are very close to the results 100 +___ai CIZ I+ T=l IQ= _-I5.I Mev) Y E = 28.05 Mev I\ I I i I c'2 I” T=l 2 (Q‘=-I5.I Mev) i \ E=45.5 Mev I I62 I # 1,\ I II . \ ,_._ ______.._ ..d — ——-1r——' ————I 2 I———-—-»—-II———- -"—'—---1P-<---—*-—v4k , - - ., II _. ______T-_ _. .- _ __1.._. II 40 I60 I80 -- F—nr—O—IP—d ’3 '0 20 4o 60 80 I00 I20 00 (deg) Figure 6.—-Differential cross sections for L=0 transition in C12. The theoretical results have been obtained with the impulse approximation pseudo-potential. 109 5 t 1 '\ ‘0‘) 2 E I C'2 + p ' .DI. é J E=45.5 M v Xv“ 3’T=O ; Q=-9.63 Mev C .05 \ E \\ 6 .02 \ P I0 CI2+ p 5 ...“. E= 8.05Mev 9;-———— 3'T=O; Q=-9.63 Mev 2 JV \\ \0—1 I .0 . 5 . \ .2 O 20 40 60 80 I00 I20 I40 I60 I80 ecm (deg) Figure 7.—-Differential cross sections for L=3 transition in C12. The theoretical results have been obtained with the impulse approximation pseuod—potential. 110 I 00 50 Cowv p E-KS MW 3‘ T‘QO"3.73 W 2;; 20 \ cit, p .0 IO sass W E Z‘TIOIQ"QA3 HOV v 5 '33 50 b ‘0 (15.009 | £660 HIV 3' T'OI°"3.73 WV Co‘oep .- 6-560 MW 02 S‘T-OIO~~44€HW .OI o 20 4o 60 80 I00 I20 I40 I60 |80 ec_m(deg) Figure 8.--—Differential cross sections for L=2 transition in 012and the L=3 and L=5 transitions in Cauo, The theoretical results have been obtained with the impulse approximation pseudo—potential. ‘ ‘. ‘I I 111 IO" 5 .R—r ‘I § 2 4.:I———A IO'I=——-—e=\ 0 III A 5 I I 2 \ \\ Io° \\ X ch II T" ‘2 an”; \ III...“ II... £3) 5 \ Y 926.05 M" \ , Q 2 E I\ b" I0'I 5 6'2 l" TcI i 2 (Q=-I5.IMOV) 1 5.45.5 Mev I0‘2 \I 5 II II I 2 IOSL II 20 4O 60 80 IOO I20 I40 I60 I80 0e (deg) Figure 9.—-—Same as Figure 6 for K—K Force. F' "Raft-5". 112 I0 5 ----- Direct —— Exchange Tbkfl 2 J4> | /’/‘-."x. O X vi. ar"fi‘\. I —\ 5 v1.1 ’1‘, \o‘ \‘ .‘ ,h - f / ‘. \ \ a) V / \_ \ C A 2 y \' . \ 1 @- /// C'2 + p \.\'\.. \ . . q -| E «5.5 Mev x5 " 3‘Ts0- (as-9.63 my .\ b ’ \ I‘\ \ '05 \\ \ 6 "- \ ‘o .02 E! \3 I0 Cad-p 5 E . 28.05 Mev 3"1'80 ; 03-9.63 Mev 2L=5Iflt hII‘WuF__ ’ I\ I ’1/ \ 0% i ’P/ ...-0"" ‘._~ \ . '5’ ’.’ .‘.\ \1 .5 ha “ '\ ‘\~‘ \.\o~.¢¢—°"°" ..... ‘.\ '2 :W\, o 20 40 60 80 I00 I20 I40 I6 I80 ecm(deg) Figure lO.--Same as Figure 7 for K—K force. In addition complete differential cross sections are decomposed into direct and exchange components. J 113 06”» E-asuov 3‘I-o.o--3.nu¢v 63+. [-455 m 2.7.0.0”4” W do/dn (mb/str) Co‘ofli £0560 W 5‘T- 0.0"4AO W 0 20 4060 80IOO|20I40I60I80 0C.m_(deg) Figure ll.—-Same as Figure 8 for K—K force. llLI l0° :\.. / (I 2 IO' #0 \ ' LQI—k 5 1 . \ 2 \\X 0'2. I+ T-l . \ '2 '0 fl \ Io . -l5.l Mev) g 5 .. Y E s26.05 m, ‘0 i \ Q 2 9 \ é \ I)” I0" 5 c"- I+ T-l I\§ (Q - -l5.| Mev) E «5.5 Mev ‘ \ I0’” ' Wfi I III \ I I I03L ’ 20 4o 60 60 I00 I20 I40 0e (deg) Figure l2.-—Same as Figure 6 for Yukawa effective range force. I l60 I80 may dax /dn (mb/sir) Figure force. 115 l3.—-Same as Figure 10 for Yukawa effective range IO 5 —-- Direct --- Exchange 2 0%\ Total e.\ | ~ \ / >“\ PR 5 ’ ’ sv\‘. 4. ' I s )\ 0 /o I ‘ \\ .b\ / )\ \x .2 ; - 1’ ‘ ‘1» r. 1’ Cu + p \ \\ . 2!” ~ I ‘ E-45.5 Mev fl i —"'"" .05 3 7'0; GIN-9.63 Mev \‘ \\ \ \ \ ‘ \ ‘ a '02 I \ a IN IO (2'2 + p 5 , . E - 28.05 my . 0 I" . 3'T-o;o--9.63II«II 23.1/ \lol‘; I ?\ ,/”—-~‘~~‘\ W Och-"\‘i \\‘~__ we \ .\. \\ .2 MW. \‘ O 20 4O 60 80 I00 I20 I40 I60 I80 96m (deg) Kimmie-6y —___. 116 IOO 20 IO “I 3’T00500-mw dc/da (mb/sh) § .02 .0 I O 20 4060 80 IOO|20I40|60|80 ecm(deg) Figure 1U.——Same as Figure 8 for Yukawa effective range force. 117 .zmmm cam uma< Log mpmumEmme >mz o: Eoge Umpmaoamppxmm mww om.H ow. wH.H om.> on. o:.H om. wH.H o m.w H.H: mm onmo ww om.H mow. NH.H mm.m mmm. wmm.a mow. NH.H mw.: mw.a m.>: mm came up om.H No.0 mm.a m.b on. 0:.H we. mm.H o m.: m.:m m.m: NHO we om.H mam. mmH.H mm.m can. mem.fi mam. mH.H mm.m o mo.m: mo.mm mac mocmpmwmm 0% 0mm 0mg m> .m mm m oh 03 3 > Am umwpme .m cfl mam mnmmemme mmmcmwzmmaw.ucm Hanan cum >mz CH mum mspamv HHm3 .xpoz wasp mo mcofiumHSUHmo ozmo cam mfio cfi 6mm: mpmpmempwa HmoHuQOII.H mam¢e 118 TABLE 2.--Integrated cross sections corresponding to results shown in Fig. 6-1U. Decomposition of integrated cross section, 0T, into odir and oex is given for the K—K force and the Yukawa effective range force. All cross sections are in mb. Target E(MeV) E J1T gForce Odir(mb) oex(mb) oT(mb) E KK 1.22 .970 3.18 1+ ' ER 1.33 .303 2.86 IA - - 3.05 KK 22.u 33.5 1.02 28.05 2+ ER 17.9 21.3 73.6 I IA - - 9u.0 1 KK 5.06 12.9 30.1 3’ i ER u.06 8.22 22.1 . 012 IA - - 26.9 KK 1.06 .150 1.99 1+ ER 1.30 .122 2.18 IA — - 2.08 KK 17.0 9.12 97.0 A5.5 2+ ER 13.0 7.A2 37.6 IA - - 39-7 KK b.75 “.28 16.1 3' ER 3.63 3.98 13.4 IA - — 13.9 KK 16.1 1u.6 58.2 3’ ER 12.2 9.19 39.6 25 IA — — 52.9 ‘I.' " : «can—- I 119 TABLE 2.-—Continued. Target E(MeV) J1T Force odir(mb) oex(mb) oT(mb) KK 2.21 8.35 16.9 5' ER 1.79 5.26 12.6 IA - — 1u.5 Cauo KK 15.5 2.51 29.4 3‘ ER 11.7 2.9M 23.7 IA _ — 22.6 55 KK 2.28 1.32 6.30 5' ER 1.69 1.28 5.6u 1w“ 120 obtained with the other two forces.- Differences between the K-K force and the Yukawa effective range force were also noted in RGf- 32, i.e. the Yukawa effective range force over- estimated the real well depth and the mean square radius of the real part of the optical potential, giving much poorer agreement with phenomenological potentials than the K—K force. The differences between the K—K force and the Yukawa effective range force for these normal parity transitions can be understood from Table 2 and/or comparison of Fig. 10 and 13. From Table 2 it is clear that the values of Odir for these two forces do not show any pronounced energy dependence. The K-K force gives slightly larger values of odir' The values of gex do vary significantly with energy, with those for the K-K force exhibiting the sharpest energy dependence. Because of the slower drop-off with energy of oex for the Yukawa effective range force, the magnitude of the total differential cross sections it produces catch up with those for the K-K force as the energy increases. Differences of this type were suggested in the discussion of these forces in Section 6 of Chapter 5. It was also .pointed out in Section 3 of Chapter 5 that forces of longer .rangerthan a 1F range Yukawa were necessary to reproduce tflna energy dependence of the experimental cross sections - 2a condition satisfied by both of the above forces. As a :result of the note added to Chapter 5 no conclusion will txa drawn concerning the significance of the differences bennveen these two forces in relation to the data. This .I 121 note indicates that the approximate treatment of anti- symmetrization is better for Yukawa forces than for the K—K force which would leave any conclusion open to question. Recently, Agassi and Schaeffer79 have obtained a good fit to the 55 MeV data for the L=3 transition in Cauo. In their calculation antisymmetrization was treated exactly and they used a Serber force of Yukawa form with a range of 1.37F. This force is similar to the Yukawa effective range force used in this work. They used the R.P.A. vector of Ref. 53 to describe this transition. Their result is con- sistent with this work. They also found that the force CAL, used in the calculation of the state vectors, fails to repro- ouce the data for this case. For the abnormal parity transition, Fig. 6, 9, and 12, the magnitude of the theoretical cross sections obtained with all three forces are in reasonable agreement at both energies. Actually, at the lower energy CT for the Yukawa effective range force is slightly smaller than CT for the K-K force. This situation reverses at the higher energy; therefore, the trend is the same as in the other cases. As this is an L=0 transition exchange is not as important. Further the 'values of odir for the Yukawa effective range force are lixrger than those for the K-K force which is a reversal of the results for the normal parity transitions. This is sijnply a reflection of the differencesbetween the forces at litrge radii. For the K-K force 0 decreases a little with dir ermzrgy and Odir for the Yukawa effective range force remains 122 almost constant. Unlike the normal parity transitions, there are noticeable shape differences in the theoretical differ- ential cross sections for this transition with the experi- mental data favoring the results obtained with the impulse approximation pseudo—potential. It is concluded that the cross sections for this transition are sensitive to the pre— cise shape and phase of the two—body force. The theoretical cross sections have a tendency to fall off too slowly with increasing angle and they don't show enough structure. No attempt has been made to try and improve the shape agreement between the theoretical and experimental angular distributions. It is known that better shapes would result if the theoretical form factors could be pushed out radially. The density dependence and the imaginary part of the projectile—target interaction might produce this effect. It has been observed in many cases that the direct cross sections computed with the K—K force show good shape agreement with the experimental angular distribution. This shape agreement is then lost When the exchange component is included. This does not happen with the Yukawa effective range force. The reason for this is that the direct form factors for the K—K force are more surfaced peaked than those for the Yukawa effective range fcrce. This is evident in Fig. 15 'where the direct form factors for the L=2 transitions in 012 are compared. Also shown are the complete form factors (with exchange) for 28.05 MeV. The total form factors peak 123 czonm ohm mum 6:» QN. Am 0: .>oz mo.wm pom ARV mpopome Ehom HmpOp mm Hamz mm mQOpomm Show pompam .monom ownmm m>apommum mzmxdw on» cam monom 3 cocampno mm mao ca coapfimcmnp mug how weepomm Shomll.ma onswam QC. All-Ant. IIFII I \ om Ow ON 8 On of QM QN 0.. o ._ ._ _ A _ , _ _ _ _ 1 . w. 0 ON 0... dz 0 )2 0m “V x IJ om z M II Emu ikT 23mm 00. M ITExx . +833. ON_ 4.3 223.3%. 322 21?. «S outm no 02 H Ow. Om_ _ b _ _ 12u well inside the direct form factors. The difference in peak positions for the direct form factors in this figure is about .HF, whereas this difference for the total form factors is only about .lF. The latter accounts for the similarity of the final results for the two forces. The long tail on the form factors for the Yukawa effective range force does not aid in giving better shapes. The cross sections shown in Fig. 10 and 13 are not WM "tr-M extremely good examples of the above point. Here the total cross sections show fairly good shape agreement with the data out to at least lOO degrees. The direct cross sections show too much structure. It is noted, however, that the K-K direct cross sections show more structure than those for the Yukawa effective range force which is consistent with form factor differences like those diSplayed in Fig. 15. It would appear that some of the deficiencies in the angular distributions of Fig. 6-19 are attributable to deficiencies in the transition densities. In particular, the fact that the angular distributions for the L=3 transi— tlfifll in C12 and those for the L=5 transition in Ca“0 peak at too large an angle appears to be consistent with the (e,ea') results which have been shown. The impulse approxi— matitni pseudo-potential and the Yukawa effective range force yielxi cross sections for the L=2 transition in 012 which fall Inader the data. The (e,e') results suggested this. The KGJ< cross sections do not reproduce this discrepancy. 125 As a result of the uncertainties in the approximate treatment of antisymmetrization it is suspected that the magnitude of the differential cross sections for the K—K force might be overestimated appreciably, at least at the lower energies. This effect will be greatest for the L=0 transition and will become less important with increasing L. It has already been suggested that the L=2 result is being overestimated from the comparison of the (e,e') and (p,p') calculations. It has recently been indicated that the tensor force might be important for the abnormal parity L=0 transition.80 Including it is found to improve the shape agreement between theory and experiment at 95.5 MeV, parti- cularly at forward angles. It may be that the approximate treatment of exchange is masking the need for this contri— bution to this transition. 2. Section B Target L16 The J",T values for the first three states of Li6 are 1+ 0; 3+, 0; and 0+, 1.1“ The second state is observed at ’ 2.113 MeV above the first which is the ground state. The thiIW3 lies 3.56 MeV above the ground state. Differential * crwxss sections have been measured for the Li6(p,p')L16 (Q=_42.18 and -3.56 MeV) reactions at 2A.A MeV.81 Theoretical crmnss sections have been calculated using the K-K force. Skmill model, LS-coupled wave functions have been used to descndibe the target and the value a= .‘581F"l has been assumed 126 for the harmonic oscillator constant.1u The optical para- meters are also given in Ref. 1“. For the Q=-2.l8 MeV transi- tion only the contribution from the triad (202,0) is important and only the triad (011,1) is allowed for the Q=-3.56 MeV transition; therefore, the components of the force which are involved are t and tll’ respectively. The results 00 are shown in Fig. 16. The agreement between theory and experiment is poor. The L=2 cross section is badly underestimated and the L=0 cross section is overestimated. In addition, the latter result does not show any of the structure displayed in the data. Similar agreement with experiment is obtained when these wave functions are used in the analysis of the (e,e') reaction.82 Fig. 17 shows a rough fit83 to the experimental (e,e') form factor8u for the L=2 transition. Adjacent to it is the result which is obtained using the transition density, empirically determined from this fit, to calculate the corresponding (p,p') cross section with the assumption that the transition still goes through the tOO part of the K-K force. The correspondence between the (e,e') and (p,p') results is good and it is concluded that the LS- coupled wave functions do not give a good description of the target. Excellent fits to inelastic electron scattering data have been obtained, for both the transitions under discus— 82 sion, on the basis of the cluster model. A parallel analysis of the (p,p') data is planned. This possibly could WW1" 127 .uowsmp on» moanommo Op com: o>mn mQOfipocdm o>m3 omadsoonmq nHmooa Haonm new mcoapmHSono Hecapomoonp map CH com: soon was bosom Mix 039 .mCOpono >02 3.3m me mad mo mmpmpm pmpfioxo ozp pmMAh one mo coapmpfioxm on» now mCOHuoom mmopo Hmapcohmmmfialn.ma onswfim 20% o2 3.. cm. oo. oo oo oe 0.... 0 20% b b p P P p p p P on. 0! ON. 00. oo oo 0* ON 0 b p k n p r P b - ITO— ...oBBE + , ++ .8535 .36 + ... III ... ... d6 .7 . .8 . .. as .7 _:0| + fl. ONON I|.. ... acoczeoaxw + _u.r.oo >22mm.m..uo >o2¢.¢wum.. 3» 2087.33 + olfm >§m_.~-uo >62 Yew u m a 3. 128 ‘ ¥i93.‘fia‘r .ma .me CH CBOSm pHSmon wCHUQOQmmpAoo on» po>o pcosm>onaefl one opoz .mopom Mix on» no psoCOQEoo 00» on» swsopnp meow coaufimcmhp map page coameSmmm on» spa: dump A.o.ov on» on paw one Eomm mpfiwsmo coapamcmap on» wcfimz coapomoh A.Q.av one 90% pocfimpno pHSmmm esp ma pcwfim 05p :0 czonm .omzmaamfio ma mag QH soapamcmpp mug one now popomm Show wcfimmppmom coppooao oapmmaonfi on» On paw Hmoapfidam nwsop w pmma one coul.ua muzwfim 60v. oON_ .00. com com gov cow 0.. m._ _._ m. h. n. n. _ _ _ _ _ _ _ To. 2 _ _ _ a _ _ no. 1 I1 o._ ”IO— .m mm b nozmaiaxzmmm "axuo . - I w I‘) >22 svuum w m a 2.524 dxmo I o. 1~-o_ fadv L erv “>22 mm..NnuOv Cub +m {A ‘——z|(b):ll 129 be extended to transitions which have been observed in neighboring nuclei. It may be necessary to improve the treat— ment of antisymmetrization and to include the tensor force in this work, particularly for the case of the L=0 transition. These points were previously made with respect to the L=0 F transition in C12 which was discussed in Section A of this chapter. L Target Cl2 12 at Q=—16.1 MeV. The There is a 2+T=l state in C triads (202,1) and (212,1) can contribute to the excitation of this level in the (p,p') reaction. The components of the projectile-target interaction which are involved are t01 and tll’ respectively. Both triads make appreciable contribu- tions to the cross section as is seen in Fig. 18. This is to be contrasted with the situation for normal parity T=0 transition where only the non-"spin—flip” contributions were found to be important. Here the impulse approximation pseudo—potential has been used with the R.P.A. vector of Ref. 50. The data is from Ref. 72 and all parameters are fixed as in the previous 012 calculations. The total cross section shown has been obtained by summing the (202,1) and (212,1) components incoherently. No significant change occurs when a coherent sum is performed. The magnitude of the theoretical result is in reasonable agreement with experiment, but the shape is quite poor. A comparable fit 130 '20 ; 5:455 MeV 3+ Q=-l6.l Mev'2*,T=I + Experiment --—- 202: —-—2I2I Sum P/ \. 10"3 * \ .+\ 99. do "lb/sir. I043 I0'3-4 I I 1 l l l 1 T l o 20 4o 60 80 ICC 120 I40 I60 9m Figure 18.--Comparison of theoretical and experimental differ— ential cross sections for the excitation of the 2+T=l(Q=—l6.l MeV) level of C12 by A5.5 MeV protons. The impulse approxi- mation pseudo-potential is used for the projectile—target interaction. ‘4 AJ' {pufm l . 1 131 to the 156 MeV (p,p') data has been obtained using this vector,u2’143 so this result is an indication that the tOl’ component of the "realistic" interactions is not unreasonable. Target 016 Fig. 19 shows the theoretical result obtained with the K—K force for the excitation of the 3’T=0(o=-6.13 MeV) level of o16 by 2u.7 MeV incident protons. The data is the same as that shown in Fig. 5.10. This is an L=3 transition which goes through the tOO component of the force. The R.P.A. vector of Ref. 50 was used in the calculation and the harmonic l oscillator constant was set at d=.559F- The agreement between theory and eXperiment is good; however, since this calculation was performed better optical parameters have been obtained and it has been shown that the Gillet vector does not give a good fit to the inelastic electron scattering form factor.73 Correcting these deficiencies leads to a theoretical result which falls about a factor of 1.5 below the data. An explanation of this discrepancy is not presently available. Target Ca“O Theoretical cross sections have been calculated for the eaxcitation of the 3”T=0(Q=-6.28 MeV) and the 2’T=0(Q=—6.02 Me\f) states in Cau0+p at 2A.5 MeV. These are preliminary resnilts which have been obtained in a study of the Cau0(p,p') * CaLlo data collected by C. Gruhn and collaborators. The min“ CW 132 '60 ; E= 24.7 MeV Q=-6.I3 MeV 3',T=O ++ + Experiment '0‘ ‘* + 3030 T fir I T I I l I 0 20 40 60 80 IOO IZO I40 I60 QCM Figure l9.--Comparison of theoretical and experimental differ- ential cros sections for the excitation of 3"T=O(Q=-6.l3 MeV) level of 0:L by 211.7 MeV protons. The K-K force is used for the proJ ectile—t arget interaction . 133 impulse approximation pseudo-potential has been used in these calculations, R.P.A. vectors are from Ref. 5“, and all para- meters are fixed as before. Only the triads (303,0) and (l12,0) have been considered and these transitions go through t and t respectively. 00 10’ The L=3 cross section is shown on the left in Fig. 20 where it is compared with the result shown previously for the excitation of the first 3-T=0 state in Cauo. There is a noticeable difference in the shape of the two experimental angular distributions. This difference is not related to the difference in Q for the two transitions. The magnitude of the cross section for the second L=3 excitation is an order of magnitude lower than that for the first. The theoretical calculations reproduce the data quite well. In detail the change in shape comes about because of differences in the dominant configurations of the R.P.A. vectors, i.e. the lf7/2—ld3/2 particle—hole pair is the largest component of the first state vector (see Fig. A) while it is the 2p3/2- ld3/2 particle—hole pair which is most important in the second. Because of the node in the 2p3/2 radial wave func- tion, the transition density for the second excitation is large and negative in the interior and has a positive peak .just outside the nuclear surface. From Fig. A it is seen ‘that the transition density for the first excitation is annall and negative in the interior and has a dominant positive tweak Just inside the surface. The former simulates a some- vniat larger diffracting object and hence the cross section for .COHuowCopCH pommmploafipoonopm Com com: ma HmfipCopoaloodomQ COfimeonmem omHCQEH 0C9 .ComHCmQEoo COM CzoCm omHm mac 0 mo CH Hm>oH ouBIm pmmfim mo Cofiumpfioxm Com mpasmom .mConCo.HQ >0: m.:m an 0 mo mo mHo>mH m>oz mo.m|nav onenm UCm A>oz mm.mlnav ouelm on» Co COfipmpHoxo one Com mCoa omm mmopo HprCopommHU fimpCoEHConm oCm HmOfipohooCBII om ohswfim 20% oo. o... om. oo. oo oo cc 8 o 20m _ OO. 0*. ON. 00. 00 OO O? ON O C C C C C C C C NIO- _ F p r b p p b - one .-m 1623.6-.. ITO— + .853 E d6 Mm -_ 8:]: .costoaxm + r .3325 oinm 3.23.610 g»; + do >22 new. um I so: \ Iowa. .... t o. Omomull «cocszaxm + >220¢N um .. oOoc 135 this case falls off faster with increasing angle. This is an amusing comparison as it demonstrates some sensitivity to a particular detail of the target wave function. The result for the L=l transition is shown on the right in Fig. 20. The magnitude of the theoretical cross section is seen to be in reasonable agreement with experiment, but there is no apparent correlation in shape. The R.P.A. says "1 that this state is almost a single lf7/2—ld3/2 particle-hole pair. It would be interesting to examine the effect of the tensor force in this transition. Target Pb208 Theoretical differential cross sections have been calculated for the excitation of the 3_(Q=—2.62 MeV) and 5—(Q=—3.ll MeV) levels of Pb208 at No.0 MeV and 2A.5 MeV, respectively. Experimental data for the former transition is given in Ref. 77 and 85 and in Ref. 11 for the latter transition. Optical parameters used in the calculations are to be found in these same references. The K-K force is used, the R.P.A. vectors are from Ref. 52, and d was taken to be .MOSF-l. The results are compared with the data in Fig. 21. The agreement between theory and experiment for the L=3 transition is not bad, but the L=5 result falls a factor of 2-3 below the data. The proton and neutron L=3 transition densities are almost identical; therefore, this transition tests the tOO component of the force. Using this same vector to calculate 136 ' 208 Pb; 5:245 Mev Q = -3.l0 Mev 0 Exp — 505 IO'| I- 5 l 208 Pb; E=4O Mev Q = -2.62 Mev 3" 96¢ -—— 303» 0'5 (mb /ster)—’ 1 IO'ZP- 1 l l I J l I 0 20 4O 60 80 IOO IZO I40 96 (deg)—+ Figure 21.—-Theoretical and experimental differential cross segggons for 3‘ (Q=-2.62 MeV) and 5' (Q=—3.10 MeV) levels in Pb by M0 and 24.5 MeV protons, respectively. For the L=3 transition the dots are the data points from Ref. 77 and the circles are the data points from Ref. 85. The K-K force is used for the projecti e-target interaction. 137 the form factor for inelastic electron scattering, a fit to the data is obtained which is comparable to that shown in Fig. 21.83 Nothing can be said about the L=5 transition as there is no (e,e') data available although the poor result is probably a reflection of a deficiency in the R.P.A. vector for this transition. “2— l-T=0 Excitations Cl2 is known to have a l_T=0 level at Q=—10.8 MeV and the same Jfl,T is assigned to the Q=—5.90 MeV level in CHO. R.P.A. vectors are available for these states in Ref. 50 and 5A, respectively. These vectors contain a spurious component which represent translational motion of the center of mass of the target rather than internal excitation of the target. These vectors have been "cleaned” by constructing the corresponding spurious state386’87 and projecting them out. Theoretical results obtained with the K—K force, using both the original vectors (spurious) and the clean vectors, are conmared with each other and with the data in Fig. 22. TTmexnagnitude of the cross sections is not given satisfactorily in.tfliese calculations, but it is interesting that the clean vectcnrs reproduce the shape of the experimental angular dis- trfilnitions quite well as compared to the spurious vectors. .As true projection technique is not rigorous it is difficult ‘to say nmue about these results. .oopmppmsHHH mH mumpm mCOHCCQm me pCo MCHpoonopd Co pommmo 0C9 .ozmo oCm Ho CH mHm>mH ouBIH mo COHuproxo Com pCmEHCodxo UCm mCoon Coozuon ComHCmQEootI.mm oCsmHm 20% on. on. em. ow. c... at. o... ow o AI 250on 00. ow ow _ _ _ H _ _ a To. «.0. ... ~IO_ ... ++ ...... 93 \I/ . . ..-o. .w/ \ I .7 q \\ l/ / \ z + S \\ I + If so an I ++ m Am : mVO_O_III // ... u£m\nE TO— II 2.8.8 06. II ,, CcoczCoaxw + ,, \ www oi.-. 6286-5 / .1 8 332.3205. ..I- >02 mém um C 003 l/ \C r. Acoflov 0.0. III I /\ .axm.+ I ouC.-. >62 mo... . o x 32088 C 8. x I. . \ x xx ///.\\\ CHAPTER 7 CORE POLARIZATION IN INELASTIC PROTON-NUCLEUS SCATTERING 1. Introduction In this chapter the calculations are extended to (p,p’) transitions involving low lying states in nuclei which possess one or two nucleons outside of a closed shell. The importance of core polarization on the low lying spectra of these nuclei and in these transitions has been discussed by many authors. Several methods have been used for estimating these effects which can be expressed most simply as a renormalization of operators acting on the valence nucleons, e.g. the effective two—body force between valence nucleons and the effective charge of a valence nucleon. One method is a perturbative treatment of the particle- Iuole excitations of the core which are induced by the valence rnnzleons. This is carried out to lowest order and particle- Iuole excitatiomsup to about 25m in energy are included. 111 following this procedure the interaction of one c0113 nucleon with another core nucleon is neglected, i.e. a.2u3roth—order shell model Hamiltonian describes the core. 139 1ND As the interaction between core nucleons is responsible for the existence of low lying collective states in the core nucleus, it is clear that this method does not include the contributions from these states. 20—25,5A This approach is used by Kuo and Brown in their attempt to explain the spectra of nuclei with one or two valence nucleons. They have shown, looking in a systematic way at nuclei in the vicinity of 016, Cauo, Cal48 Sr88, and Pb208, that core polarization gives rise to a . N156, strong pairing effect which is the major feature of the observed spectra. Horie and Arima were among the first to use this method in their calculation of quadrupole moments.57 Recently, Federman and Zamick have used this model to examine some of the properties of the effective charges for quadrupole transitions for nucleons outside of Ca“0 and Ni56 cores.88 These studies have been extended to other nuclei89 and additional efforts have been directed at estimating the validity of neglecting low-lying collective states of the core nucleus.9o’91 An alternative method is to use the macroscopic vibra- ‘tional model to describe the core.92’93’15’l6 The inter- aaction between the valence nucleons and the core is treated in a manner completely analogous to that discussed in Section 2 of Appendix B where the interaction of a pro- jectile with a nucleus, so described,was considered. The caigenstates of the macroscopic vibrational Hamiltonian need ruyt correspond to physical states of the core as the model jjs'used.as a vehicle for parameterization. Under the lul assumption that the core strength is at an energy large compared to any of the energy differences between the valence nucleons involved, the role of a given core multipole in the core polarization process is fixed by a single para— meter, C the stiffness parameter for multipole L. The L’ renormalization of the two-body forces between nucleons outside the core (bound and/or unbound) as well as the effective charge are easily expressed in terms of these parameters. Using this method, and fixing the CL on the basis of empirical effective charges, Love and SatchlerlS’l6 have demonstrated that core polarization can give a very important, even dominant, contribution to (p,p’) cross sections. Another variant is to consider the coupling of the valence nucleons to low lying physical states of the core. The macroscopic vibrational model can be used to param- eterize the physical core states, although more consistent calculations would use microscopic wave functions for the core states——such as R.P.A. vectors. The energies of these corwz states are often comparable with the energy differences ‘between valence nucleons and this has to be taken into account. This method has been used extensively in the lea.d:region.93’914 Calculations of this type are useful iJieexamining the particle—hole model with respect to :neglectim@;low—lying collective states of the core. 1&2 Atkinson and Madsen have given yet another procedure for relating the effect of core polarization in electro- magnetic transitions to the effect in the (p,p’) reaction.19 All these models are attempts to enlarge the vector space used in shell model calculations in an easy to handle way and, at the moment, rest on a very empirical rather than theoretical foundation. These models are discussed in more detail in Appendix D. At any rate the main purpose of the present chapter of this paper is to extend, to the scatter- ing problem, the microscopic perturbative calculation of Kuo and Brown. Due to the selection rules, transitions generally give more detailed information about the nature of core polarization than bound state calculations. For example, consider a nucleus with two like valence nucleons which are restricted to the (j)2 configuration. Such a nucleus will have a 0+ ground state. It is shown in Appendix D that the pairing effect on the ground state binding energy is due to the coherent effect of a number of core multi- pole excitations, whereas transitions between the states of the (j)2 configuration which start or end at the ground state depend essentially on only one core multipole. The (p,p’) reaction is particularly useful for studying core txolarization since the available experimental data, unlike ‘that for electromagnetic transition rates, is not limited jprinmrily to quadrupole and octupole transitions. 1A3 209 are the nuclei considered T150, zr90, Y89, and Bi in this paper. The first two have two valence protons and the last two have a single valence proton. In all cases the 3p—lh (or 2p—lh) components of the target wave func— tions are included as prescribed in Appendix D. The K-K force is used as the interaction between core and valence nucleons. Angular distributions for the (p,p’) reaction and effective charges are calculated and compared with experiment. In the (p,p’) calculations the K-K force is also used for the projectile target interaction. These calculations constitute an attempt to reproduce the (p,p’) experimental data from a completely microscopic model with the assumption that the projectile and target nucleons all interact via the same force which in turn is closely related to the free two—nucleon potential. As an example of a particularly convenient way to relate the effect of core polarization on the spectrum and in transitions, calculations are carried out for T150 and Zr90 using the macroscopic vibrational des— cription of the core and fixing the core parameters from the bound state matrix elements of Kuo and Brown. This procedure is discussed in Appendix D. All results are reviewed with respect to coupling to physical core states and in light of the empirical formula of Madsen and Atkinson. A very interesting result is obtained in the case of 81209 where it is found that the transition considered is dominated by a single core phonon. 1AA O 90 89 2. Calculations and Results — Ti5 , Zr , Y Macroscopic Vibrational Model and Relation between Core Polarization in Spectra and Transitions (Zr90 and T150) + In Zr90 the transitions from the 0 ground state to the 0+, 2+ 9+ 6+, and 8+ states with Q=—l.75, -2.18, $ ’ —3.07, —3.U5, and -3.58 MeV, respectively, for 18.8 MeV l1 incident protons are considered. The transitions from the i 0+ ground state to the 2+ and U+ levels of T150 with i Q=-l.55 and —2.68 MeV for 17.5 and A0 MeV incident protons are also treated. The two 0+ levels in Zr90 result from the mixing of the (189/2)2 and (2pl/2)2 proton configura- tions where the ratio of g to p amplitudes in the ground state is about three quarters. This ratio has been fixed both theoretically and experimentally.8’9’5u’95’96 The 2+ u+, 6 3 + , and 8+ states in question in this nucleus are due to two protons in the lg9/2 orbit. The states in T150 are described as two valence protons in the lf7/2 shell. There is also a 6+ state due to this configuration, but it has not been resolved in inelastic proton scattering experiments. For these cases the multipole decomposition of the . 3p-lh contributions to the <(j)2OI)éff|(j)20> matrix elements liave been given.25’5u Comparison of the decomposition with qu, (D.25) and Eq. (D.26) allows the extraction of the pnxrameters 20L. A knowledge of is required to deteimflne the parameters 0L which are needed to calculate 145 the transition matrix elements. Following Bohr and Mottelson?3 is taken to be 50 MeV in these calculations. Estimates of this quantity, based on reasonable finite potential wells, for various orbitals in several nuclei 15,16 produce values from roughly 35—75 MeV. Uncertainties in the value of are probably the major source of error I in making this comparison between the spectrum and transi- We J. tions. Table 1 gives the values of 6L deduced in this manner. For ngofkv>8 =.ll9, which is the same as the value 2 given in Ref. 15 and 16. The latter value was extracted from the effective charge and can be obtained without knowing . It should be pointed out that the potential wells used in these references had ~70 MeV. In the last column of Table l the parameter CL is tabulated. This parameter represents the effective stiffness of the core to 2L—po1e surface vibrations and is simply the inverse of BL. From the table it is seen that the core of T150 is somewhat softer than the core of Zr90 and the L=2 vibrations are most important in both cases. This is expected as is the indicated increase in core stiffness to higher order vibrations. The indicated core coupling is by no means negligible, however, even for the highest core multipole. Note the large mono— pole coupling indicated for 2p1/2 protons outside the Sr88 core. On the basis of nuclear compressibility, L=0 vibrations are not expected to be so important. 1A6 TABLE l.-—Extraction of 8L from bound state matrix elements of Kuo and Brown. ngo 2 2 0 CL L j <(J) 0||G3p_lh||(3) O>(MeV) ML 6L (Mev) 0 lg9/2 -.020 .0796 .0050A 9920 2 lg9/2 —.578 .0970 .119. A20 u lg9/2 —.359 .0900 .079 633 6 1g9/2 -.218 .0770 .057 877 8 lg9/2 —.122 .5u2 .0u5 1110 0 2pl/2 -.2Al .0796 .061 820 T150 2 2 0 CL L J <(j) 0||03p_lh||(i) 0>(MeV) ML 0L (Mev) 0 1r7/2 —.033 .0796 .00892 5610 2 1r7/2 -.753 .0950 .159 31A 9 lf7/2 —.u60 .0839 .110 A55 6 11‘7/2 —.233 .0602 .0775 6A5 ' WI-n- __ H I' 1A7 The admixture of a core excited component in a shell model configuration is proportional to 2/CLhwL (see Eq. (D.l")). Assuming the hydrodynamical values for the mass parameter D gives 9.2 and 10.8 MeV for the energies 2 of the effective quadrupole phonon in Zr90 and TiSO. Using these energies and the C2 of Table l in Eq. (D.l") leads to values of 12% and 1A% for the L=2 core admixtures in the ground states of Zr90 and T150. Admixtures this large are not completely tolerable in view of the per— turbative treatment being used. Ref. 15 and 16 report 7% L=2 core admixture in the ground state of ngo. The discrepancy cannot be accounted for by differences in the values of and C2 which have been used here and in those works. As an example of the pairing effect which is due to the core polarization, the results of shell 50 90 model calculations of Kuo and Brown for Ti and Zr are shown in Fig. 1. Theoretical results obtained with and without the inclusion of core polarization are com- pared.with experiment. For both of the spectra shown tflie zero of energy is that of two non—interacting protons le the lowest available orbit outside of the filled core. 'Dhe experimental energies have been plotted with the aiid of the mass tables of Mattauch et al.97 The experi— nmnqtal energies for T150 have been shifted by .U MeV 1A8 .poCOCwH mH COHumNH ICmHoQ choc Cons mpHCmon momemeoo 0 Hoan 0C» oHHC3 noosHoCH COHpmuHCMHom onoo Cqu mpHCmoC ohm CHIQmo + c oonan mapoodm .uCoanono Csz ooCmQEoo ohm omHB » oCm omCN MOO Csonm oCm 05M mo mCOHumHsonO Hopes HHoCm mo mpHSmomll.H oCCme Coach IIIIleIII I aCoogh o 538.0 an. 3., , a . o ...- no .o an. n. I I +0 I)! ..o I ~ +0 I N. I I +0 _ .0 .N .... IN I _- u .6 .. o +N I M +0 I .e .I w e e w + +® + l O M +0 +N I _ m l. N + + 1 + N I .w . e + + I _ v I +w Ho In N l N I n l n I. ¢ on. 00 (new) KbJoua 1U9 because the Coulomb interaction was not included in the shell model matrix elements. The figure clearly shows that core polarization gives a large attractive contribution to the J=O matrix elements, a small attractive contribution to the J=2 matrix elements, and repulsive contributions to matrix elements of higher J. In both cases the theoretical 2+ energy is too high. For ngo both of the 0+ states and the 8+ state need to be pulled down. The theoretical results for T150 are in better agree- ment with experiment than are those for ngo. The T150 results are for a full lf—2p shell calculation while only the 2pl/2 and lg9/2 orbits were included in the ngO calcu- lation. Note that the ground state energy in T150 is 2.90 MeV below the unperturbed value. <(lf7/2)20|6V;fTJ(lf7/2)20> has the value -2.068 MeV with —.869 MeV coming from the bare force and -l.l99 MeV as a result of core polarization. The additional -.832 MeV ground state binding energy is due to very small admixtures (less than 5%) of (lf5/2)2, (2p3/2)2, and (2pl/2)2 components in the ground state wave function. For ngo <(lg9/2)20|7/eff|(lg9/2)20>=—.57 MeV—1.01 MeV 2 ((2p1/2)20|&Veff|(2p1/2) 0>=-.l2l MeV-.0105 MeV 150 where the first number in each case is the bare matrix element and the second is the 3p—lh correction. An additional -.3 MeV is added to the first matrix element to account for excitation of the two valence protons to the (lg7/2)2 configuration and -.2 MeV is added to the second matrix element to estimate the effect of configurations with two 2p3/2 holes. A pure (1f7/2)2 calculation for T150 would probably also underbind the 0+ ground state. In summary, the perturbative treatment of core polariza— tion gives a dramatic contribution to the theoretical results; however, the underbinding of the 0+ and 2+ states indicates that the effect is being underestimated. It is uncertain how these deficiencies are distributed between the different core multipoles. Further, the choice =50 MeV may result in contributions to transitions from core polarization which are somewhat larger than the matrix elements of Kuo and Brown actually imply-—at least for ngo. 90 50) Microscopic Transition Densities (Zr and Ti 90 In the completely microscopic calculations for Zr and T150 (detailed formulae are given in Section 3 of Appendix D) particle-hole pairs have been taken from the following shells: ngo Particles: 2d, lg7/2,3s,lh,2f,3p,lil3/2,2g9/2 Holes: ld,2s,lf,2p3/2 (and lg9/2,2p1/2 for neutrons only) T150 Particles: 2p,1f5/2,lg,2d,3s Holes: lp,ld,2s and lf7/2(for neutrons only) 151 These orbits include all the particle-hole excitations with energies up to roughly 2fiw, except those proton-proton hole excitations for which the particle level is the same as the valence orbitals, i.e. in ngo the proton particleohole pairs lg9/2-jh and 2pl/2—jh are neglected as are lf7/2-jh proton excitations in TiSO. The single particle energy levels have been taken from the Nilsson chart at zero deform— ation. The parameter/hm which fixes both the harmonic oscillator wave functions and the energy denominators has been taken to be 9.1 and 10.5 MeV for ngo and T150, respectively. The composition of the core transition densities, FLOL(C) and FLOL p n the L=2—6 transitions in T150 are displayed in Tables 2 and (C), for the L=2—8 transitions in ngo and 3. The important particle-hole pairs are listed with their energy denominators. The amplitude of the state |[(jj)L(jpjh)L]o> in the |(j2)o> ground state, AG, and the amplitude of the state |[(jj)o(jpjh)L]L> in the |(J)2L> excited state, A are listed along with the fractional E’ contributions, %, of a particular particle—hole excitation to its respective core transition density (either proton or neutron). Observe that in Zr90 it is only the states with j=1g9/2 that are involved in the L=2—8 transitions. For the definition of the amplitudes see Eq. (D.39). The ratio LOL LOL of F (c) to Fp (C) is also given in each case——denoted by n N/P. smo. 3H.QI mso.I m.m m\mCH m\HHeH 152 mas. smo.- mCo.I mma. Has.I mms.- m.sa m\mas m\sen a me we a we as H>ozufleovm s o mCOCpCoz mCOpOhm Has.oal Aao.mua\zvsnn o:m.ue ssm.ue mom. mmH.+ msm.+ m.m m\mmH m\mom* mHm. oso.I mmH.I o.mH m\amH m\mHHH* Hmo. poo.I mHo.I o.mH m\mam m\sCm mOH. Hmo.I sHH.I mmm. om.OI mso.I o.mH mxmofi m\CwH mmH. mmo.I omH.I cam. mmo.I Hmo.I m.sH m\mCH m\mnH mom. Cso.I msH.I Ham. mmo.I mmo.I m.mH m\sCH mxHHeH a me as a me as H>oznAeavm e o WCOCHPSQZ WCOpOCHm Haw.pai mm.muia\zv mud omnN .omCN CH wCOHpHmCme wImnq pow mmHuHmCmc COHpHmCme onoo Co COHprOQEooII.m mqmozwfisovm r o WCOCHPSGZ mCOQOaHm mam.mg AH:.mnm\szuq m:n.HB HH®.NB HHH. mmo.I mmo.I o.ma m\mmH m\mnaae mmH. Hmo. mmH. m.m m\me m\mom* OOH. smo.I HCH.I 0.: m\me m\smfl* smo. Hmo.I moo.I o.sH m\Hom m\msH* sso. o.mm m\moH N\MHHH Hmo. omo. one. sac. o.mH m\Hnm m\CwH mmo. o.CH m\moH m\st mmo. o.mH m\moH m\st smH. mmo.- moa.I 3mm. m.OH m\mdm mxaHeH sOH. smo.I moa.I mOH. m.mH m\sCH m\HHeH UmCCHpCooII.N mqmozLAsdvm s o mCOCHPSQZ WCOPOCHAM Ham.mHO COO.OId\zV and mHm.uB me.nB HOO. sOO.I OHO.I m.: m\sCH mmeH* ass. mHm. ass. O.m mstH m\mom* OmH. OHO.I OmO.I 0.0m m\HOH m\mCH :sH. HOO.I smH.I OOm. :mO.I smO.I s.sH m\mOH mxst OHN. mOO.I OOH.I Hms. OOO.I mOO.I m.mH N\mOH mxme a ma OO a ma Os Hsazaieavn e a mCospsmz mCOpOCm HsO.mmO Amm.mni\zv mud OmHe .omHB CH mCOHpHmCme mlmnq Com moHpHmCoo COHpHmCme opoo mo CoHpHmanooII.m mqm¢e C C h. C ..L C C.\ II .V H2. is... as: a L I I I- II. .1 III . . {\- ... . . Home: :H from»). .m. £4221 moo.HnB m:m.ue 157 ONO. OOO.I OHO.I m.s N\OCH N\OCH* OOO. OHO.I OOO.I m.mH N\OOH N\OOH NHH. OHO. OmO. OOO. OOOO. ONO. 0.0N N\OOH N\NOH OOH. ONO.I OOO.I OOm. OHO.I OOO.I O.NH N\OOH N\OOH O ma OO O ma OO H>OzOAsovm e o mCospsoz mCOpopm HON.HHO COO.OIN\2O Ono OOO.Ie HOO.HIe IImmmw OOO.I OON.I .IIIIII O.O N\CCH N\OCH* OOH. OHO. OsO. O.m N\COH N\HON* OOH. OCO. JON. O.m N\NCH N\mdN* ONO. ONO. NsO. OOH. NOOO. ONO. O.NN N\OOH N\mCH smO. NNO.I OOO.I OOH. NOOO.I ONO.I s.sH N\OOH N\NOH OOH. NsOO. NNO. O.ON N\mOH N\COH OOH. OHO.I N:O.I s.NH N\OOH N\OOH .UoCCHpCOOII.m mqm<8 158 The transition densities are, of course, functions of radial position within the target nucleus. The radial dependence of the valence transition density, FgOL(D), is given by un£(r)un£(r) while the radial dependence of the contribution of a particular particle—hole excitation to its core transition density is given by unp£p(r)unh£h(r). The particle-hole excitations which give important contri— butions almost invariably satisfy the condition unpzp(r)unh2h(r)~un2(r)un2= .6I(1g9/2)2O>-.8I(2pl/2) 0> + 2 |0 (Q=-l.75 MeV)>=.8|(lg9/2)2O>+.6l(2p1/2) 0> The transition density has two components——a (lg9/2)2 com- ponent and a (2pl/2)2 component corresponding to the matrix elements 2 2 .98<(1s9/2) 0|T|(ls9/2) 0> '0 ((2pl/2) OITI(2pl/2) O> Strictly speaking the theory also allows for contributions corresponding to the matrix elements; 2 2 .36<(2pl/2) olTl o> 16H 2 2 -.6}4<(1g9/2) O'Tl(2pl/2) O> There is no valence contribution to these matrix elements as the initial and final valence configurations differ in the state of more than one particle. Further the 3p—1h inter- mediate states which can contribute must have two protons in the 1g9/2(2pl/2) orbit and a third proton in the 2pl/2 (lg9/2) orbit-~a11 coupled to a proton hole. These are neglected. Similar contributions, corresponding to the matrix elements 2 2 -.8<(lg9/2) LIT|(2pl/2) 0>. have been neglected in treating the other transitions in ngo. The structure of the transition density for the L=0 transition is illustrated in Fig. 2. Shown at the top are 000 000 — ooo _ 000 000 I2) (D)[D1, Fp (C)[p—p1. IE) CT)-Fp (D)+Fp and FSOO(T)=FSOO(C)[n—n] for the (1g9/2)2 configuration. (C)[D+p-13] : Corresponding information for the (2pl/2)2 configuration is shown in the middle. The complete valence transition density [D], the complete proton transition density [P], and the complete neutron transition density [N] are shown at the bottom. Here [D] is the sum of the two curves labeled [D] in the top two drawings, [P] is the sum of the curves labeled [D+p—p], and [N] is the sum of the curves labeled [n-n]. 1 6 5 TRANSITION DENSITY 2|"0 9 p 0’(Q--L75Mcv) .I6 _ I I _I C l .08- H +- -< L— H r- —I 0 C —I -03— H r- —I -.I6_ 1 I L 1 I l I I I I I I .15 I I I T I I I I I I I I I2;.,,)': —— 0 I r— _'- D.” —d .08— n-a _. r- -1 l —1 OF 11 - -1 C .1 — 4 .1 A ~ Le '08~ *4 ,3 " ‘I I” " I u NSF I I I I I I 1 I I I I 15 I I I I I I I I I I I _ C TOTAL: — 0 I ._.—— P _4 .08: """ N A :— :\\\\ .4 o 1‘.— I- -1 -_08I— , —4 r- \-/ -* -.|6- I 1 I I I I I I I l I I 0 I0 2.0 30 4.0 5.0 6.0 7.0 8.0 9.0 I00 ".0 l2.0 r(F)-‘ Figure 2.——Structure of transition density for L=0 transition in Zr90, 166 As in the previous cases, neutron core excitation are found to be more important than proton core excitations. The complete proton transition density does not differ appreciably from the complete valence transition density. OOO The interior minimum of Fp (D) has been increased, the surface maximum has been decreased and shifted slightly 000 C . p ( ) The core transition densities are oscillatory and are not outward, and a longer tail appears as a result of F too similar to the valence transition densitities. Only particle-hole pairs with jp=jh contribute to the core transition densities. As the available particle and hole levels with the same total angular momentum do not have the same principle quantum number, the oscillatory shape results. The small core transition densities for the (1g9/2)2 configuration is understood in terms of the poor overlap of ulu(r)ulu(r) with u (r) when annh’ The overlap (r)u nptp nhth of u2l(r)u21(r) with unp£p(r)unh£h(r)(np#nh) is better, which explains the larger core transition densities obtained for the (2pl/2)2 configuration. In the latter case, the radial integrals (see Section 3 of Appendix D) still have the sign of the two-body force and the difference in sign between the core transition densities and the valence transition density for large r is just the difference between u2l(r)u21(r) and un 2 (r)un 2 (r)(np#nh) at large r. The net effect of core p p h h polarization will be an enhancement of this transition, although it occurs as a result of inhibition of the (2pl/2)2 contribution to the transition. 167 A11 particle-hole pairs which contribute to this trans— ition are listed in Table 9 with their energy denominators and the amplitudes AG. AE is not given since it is equal to A for this case. Percentage contributions are not given G either since differences in radial shape between the various components do not allow such a comparison. Ground state admixtures are given in brackets as before. These are quite small. Excitations involving excess core neutrons are indi— cated with an asterik. They do not contribute to this trans— ition via small energy denominators and thus do not play a special role in this case. Microscopic Transition Density for Transition to Q=-.908 MeV State of Y59 89 The excitation of the Q=-.908 MeV level of Y for incident protons of 18.9, 29.5, and 61.2 MeV is considered. In the ground state of this nucIeus the valence proton is in the 2pl/2 shell and for the excited state being considered it is in the 1g9/2 orbit. The triads (LSJ) which can contri- bute to this transition are (319), (519), (505), and (515). None of these are forbidden in the simple shell model inter— pretation of this transition so there is no breaking of the valence transition selection rules because of core polar— ization. It is found that the contributions from (519) and (515) are small enough to be neglected. The microscopic transition densities for the (319) and (505) triads have ‘been calculated by taking particle—hole pairs from the following levels: 168 TABLE 9.——Composition of core transition densities for L=0 transition in Zr90. ngo L=O(lgg/2)2 [(05%] Protons Neutrons p h E(ph)[MeV] AG AG 2f7/2 1f7/2 17.0 .0008 .0120 3p3/2 2p3/2 18.0 .0009 .0007 295/2 ld5/2 16.5 —.0017 —.0298 381/2 281/2 18.0 -.0056 —.0196 2d3/2 1(13/2 17.0 -.0080 —.0238 2f5/2 1f5/2 17.0 .0066 .0093 *301/2 2p1/2 17.5 .0005 *2g9/2 lg9/2 18.0 .0975 L=O(2pl/2)2 [2.9%] Protons Neutrons p h E(ph)[MeV] AG AG 2f7/2 1f7/2 17.0 .0101 .0163 3p3/2 2p3/2 18.0 .0930 .1099 2d5/2 1d5/2 16.5 .0291 .0986 381/2 281/2 18.0 .0218 .0610 2d3/2 1d3/2 17.0 .0060 .0909 2f5/2 1185/2 17.0 .0005 .0155 *3pl/2 2pl/2 17.5 .0786 *2g9/2 1g9/2 18.0 .0265 169 Particles: 2d, lg7/2I 35: 1h: 2fs3p: 1113/2,2g9/2 Holes: 1f, 2p3/2(and 2pl/2, 1g9/2 for neutrons only) Note that this transition involves a change in parity-—thus the particle-hole pairs contributing to the core polarization 90 here are not the same as those involved in the Zr .transitions which have been considered. The orbits listed include all the particle—hole pairs with energies up to roughly 1hw with the exception of the 1g9/2—jh and 2pl/2-Jh proton exc1ta- tions. By includ1ng the 2g9/2 and 1i13/2 particle levels a few 26w excitations are brought in. The constantIhw has been 90 fixed at 9.1 MeV for this case-—the same as for Zr p F205(C) is given in Table 5. The format of this table is The composition of Fglu(C), Fglu(C), 9505(0), and the same as that of Tables 2 and 3. AE is the amplitude of the state |2pl/2(jpjh)J;9/2> in the |1g9/2> excited state and A is the amplitude of the state |1g9/2(jp3h)J;l/2> in G the I2pl/2> ground state. For the expression for calculating these amplitudes see Eq. (D.33). The J=9 ground state admix— tures are almost zero while the J=5 ground state admixtures are just slightly smaller than the L=6 ground state admixtures 90 'which were obtained for Zr F3lu(C) is larger than Fglu(C) as is indicated by the 1’1 p IT/P ratio of -.383. The minus sign indicates that Fglu(C) its opposite in sign to Fglu(D) while F31u(C) has the same saign as Fglu(D). The sign difference is a result of the 170 NNO. ONO.I ONO.I HOH. NNO.I OHO. O.OH N\NOH N\NOH O OO OO O OO OO H>czOAsOvm c d mCOCpCoz mCoponm flON.NQ Arm.mum\zvmuw mNm.uB mom.u.H_ OOO. OOO.I HHO.I O.HH N\OOH N\OOHO NOH. OOO.I HHO.I O.O N\OOH N\HHeH* NON. OOO.- HHO.I OON. ONO. NOO. O.OH N\OOH N\OHHH ONO. HHO.I OHO.I O.OH N\NOH N\OHHH NOH. OOO.- NHO.I OON. HNO. OOO. O.N N\OON N\OON OOO. NOO.I OOO.I ONO. OOO. NNO. O.NH N\NCH N\HnO OOO. OOO. OHO. OOH. OOO.- OHO.I 0.0 N\NOH N\OON HNO. OOO.I HHO.I HNO. OHO. NOO. 0.0H NxNOH N\NOH O OO OO O OO OO H>OOOAOOOO s a mCoppsoz mCOpoCm IIIIIIII I OOO. O AOOO.IIO\szIO OOO m» Oo Ho>6H >mz mOm.IN® Op. COHIIHHQOcHP .HOrH mMHPH CQU COHUnghp ®.HOO ..HO COHPHWOQEOOII.m MQm/TH. 171 Nom.ue OHN. Omo. mmo. ONO. OOO. ONH. mmo. mOo. omo.I ONO.I NmO.I OHO.I OOO. mmo. omH. NOO. Omo. OHO. mOO. OOH. OOO. ONO.HHB mHH. mwO. mNm. ONO. mmH. Hmo. wNH. MOO. NOO. OHO. MOO. OOO. NHo. Omo.I mNo.I OOO. moo. OOO. NHO. ONO. m.O O.NH 0.0H 0.0 O.NH 0.0 m.m O.N N\OOH N\OON N\NOH N\OsN N\NOH N\OCH N\NCH m\mmH N\HHCH* N\OHOH N\OHOH N\NOH N\OON N\OON N\mON N\NOH OoCCHuCOOIl.m mqm<9 172 repulsive character of the "spin-flip" component of the p-p force. The "spin—flip" component of the p—n force is weak and attractive which explains the sign and size of Fglu(C). These conclusions are based on the discussion of Section 3 of Appendix D. The contribution to the transition from the triad (319) is reduced as a result of core polarization. This is a well known result first used to explain the slow M9 y—decay of the Q=—.908 MeV level to the ground state.98’99 The results for F205(C) and F205(C) are similar to those obtained for the core transition densities describing the lF2-8 and L=2—6 transitions in Zr90 and TiSO, respectively. FZOS(CD, F205(C), and their major components are similar in shape and have the same sign as FSOS(D). F205(C) is larger than FgOS(C), but N/P=3.89 is considerably smaller than the values obtained for ngo and Ti50 core transition densities. The reason for this is the decreased importance of excitations involving the excess core neutrons. About 66% of FgOS(C) and about 96% of F205(C) is due to particle- hole pairs which satisfy the coupling conditions given before. Some fractionization occurs because the overlap of u2l(r)ulu(r) with u (r) is somewhat more ambiguous than in the (r)u nptp nhth case of ngo and TiSO. Essentially the same results would be obtained for all the core transition densities if an average energy denominator of 11.1 MeV is used without exception. This is slightly greater than 16w, the value appropriate for negative parity transitions. “v. C. .h\ C. E Q t C T .O .. NL : i ..W S 3 C «C ’qu .r t. flu he. O: 173 0, TiSO, and Y89) Angglar Distributions (Zr9 Figure 3 shows the angular distributions which have been calculated for the L=2—8 transitions in Zr90 in the (p,p') reaction at 18.8 MeV. The data shown is from Ref. 8. The results for the L=2 and 9 transitions in T150 for the (p,p') reaction at 17.5 and 90.0 MeV are compared with experi— ment in Figure 9. The 17.5 MeV data was taken from the liter— aturelooand.the 90.0 MeV data is the unpublished work of B. Freedom. Theoretical differential cross sections obtained 89 for incident for the excitation of the Q=-.908 MeV level of Y protons of 18.9 MeV, 29.5 MeV, and 61.2 MeV are compared with experiment in Figure 5. The data comes from Ref. 10, Ref. 101, and Ref. 10 2, respectively. In Figure 3 and 9 the solid curves are the results of the completely microscopic calculations and the dashed curves are the results of the calculations which use the macroscopic vibrational model to describe the core with the core para— meters fixed from the bound state calculations. The solid curves in Figure 5 are the complete differential cross sections and the dashed curve represents the L=5 component of this cross section. The L=3 component is shown only for the 61.2 MeV case where it appears as a center line. The optical parameters used in these calculations are given and referenced in Table 6. The notation is the same as used in Eq. (B.l3) and the same geometry is used for the volume and surface imaginary terms. 179 J Z r90+ p E=|8.8 Mev __ 2*;0=-2.I8 Mev ,z’ \ L=2 9 I \ ’x \‘ / {Of \:::1 ‘ (\\ '3 N «\‘3 \ Zr9°+p . I I E 0'05 E: I8.8 Mev I if 4 ;0=-3.07 M \' L=4 8v \ Tibia? 0.02 r s c x ‘0 o I \ \ \ T T 0.02 “““OO- ‘_ , N \ 0.0I — 2.90., \\ I E: l8.8 Mev \\ 0_OQ5N— l.6-=;60=-3.45 Mev ‘\ j \ I 1 \ \ 0.05 \ “\\ 0.02 I j ) \H§ 0.0I I " J11 - Zr9°+ p 1 IT l C... sues Mev I 0005 8+ I 0 = —3.58 Mev i C._-u—- L=8 -“I~\\“ ”’4_,_.._— --I 0.002 CO “’ ” // 0.00I 4-” 0.0005 0 20 40 60 80 l00 I20 I40 90m (deg) Figure 3.—-Differentia1 cross sections for L=2-8 transitions in Zr90 + p at 18.8 MeV. completely microscopic calculations. Dashed curves re results obtained using macroscopic vibrational gescription of core and solid curves are results of ' 175 IO é . 5 \A 0 Ti5°+p ‘\\ . E=40.0 Mev \x.\ . 2*;0=-I.56 Mev 2 \\\\\d L=2 \ \ ' ' l a\d:::::‘\\\\¥ \ 0.5 \ . \ I \ . C \ 20— Ti5°+p \\ \""‘x E=l7.5Mev \Ik \ \ —\ IOO— 2+;Q=“|.56M8V \fi x = ‘ I. L 2. I \\ x. \ 5 If \\‘j t H \ \. 2 ,ijumrr\;\ I——-—"’ \x 0 \\ I 6 I \-:\ I I \ . 5 i\ \X. \ \\‘~-—d \\\\\\\\\ 2 1' T \ I ”'_ I ’J o . . ‘\< 0.5 /” ~ J x / \\\ 0'2 [—- 225210?) Mev \‘ . \ I. 4*; 0=-2.68 Mev \‘)—?\\\ O.) —" L =41 I ‘\\ \ 5 I \ ‘ \ Ti50+p \\ \\ 2__ E=I7.5Mev \ \ 4+;Q=’2.68 Mev < I ”4 \O 0.5 .\‘§ I § 9 —————— db“\\\ \ 5 § 0.2 “"“ ““““ KO \ O 20 4O 60 80 I00 9C (deg) Figure 9.--Differential cross sections for L=2 and 9 transitions in Ti5O + p at 17.5 and 90 MeV. Dashed and solid curves are used the same as in Figure 3. 176 I I I I .5 T O I Y89+p V 9/2"'; Q = '.908 Mev 0-2 0 Total —-— L=5 O I L T% —-— L=3 4/ fl-1F\ § i E: l8.9 Mev 0.05 [/7 \‘k\ / O 02 I, \\x‘ ’1"-‘\:\ /r . K N._ \\\’ I’ x y’ ‘\ (15 T .93 O 2 ‘ \\\ a U) ° TIL \ 4 3 0| I . I , 5:245 Mev N . E /§/’ ‘Ih\ i E J \ \. 0.05 / \\ i I / U / \ ’~ 6 002 \"i \IR I k . .L ‘ If,“ \ i i \i\ ”Vi/‘1‘" I I \. i/ 0.2 _I;§§I:7é\—— \\ t I [I / A \\ 0° 11> I\\ \\\ (305 5’ \\ §\x Es - . p \ I \ \I q \'I\ i \\ \ (>02 ! :% §x\ Q\\\ . I ,\ ' “ ' \-\ \\\§ x I ‘I\ a ° \. \ \ 0.005, \\ ‘. \\I E = 6|.2 Mev '\. K \ L \. \x I \ 0°02 I \1\ \ \0 0.0005 ‘-\ \f .\¢::\\ 0.0002 “g. \\\\ \ .\j \‘ O'000'0 20 40 60 80 ICC 120 I40 I60 l80 6C (deg) Figure 5.--—Differential ggoss section for excitation of 9/2+ (Q=-.908 MeV) level in Y + p at 18.9, 2u.5 and 61.2 MeV. Compolete cross sections and L=5 component are shown in all cases and the L=3 component is displayed for the 61.2 MeV case. 177 .COHpoom 0 go who mEmpm am m m p mo 0m pom wpoumemhmmm mmopo oflpmmfim on pHm ooow mofl>opa op omfigm> 3 new > Spa: who: pom: .mp0: pow: 2m 9:0H mm.a mmw. :mo.a :o.m 0mm. nm.a mu. mH.H 2H.H mm.» mm.:: 0.0: Amvomfie MOH mmm.a om. 0mm.a 0.0H mm. me.H ow. mmm.a ww.oa 0.0 m.mz m.NH AmvomHB NOH om.H mm. om.H oo.m mm. o:.H mm. om.H :m.m ma.m m.mm m.Hw mm» >mz m.wH m< m2 \m mu m o.H Q3 3 > A>mzvm0: CH .mcowpznfippmfio amazwcm mmw new . . LN mnp wCHpmHSono cfi pom: mew meadow Ham: mew weapon do Q II. omwe om p H an 0 w mqm O z -40 \— N k 3: ..o t 8 “L -80 I -|OO I. I _. I I Ti5°+ p .. I I E: I7.5 MeV “'20— \‘ l L: 2 'I _. I.) I] - \‘ ~’I 0 -I4oI- "\ I, ---- D+C(Mocro) - . \ /./ ----- D+C(Micr0) _ \Y’ ~ I a a 451:. g 2, It r(F)—' Figure 7.——Same as Figure 6 for L=2 transition in T150 + p at 17.5 MeV. 1814 40- - 20- .. 0‘ -——'~ \ 0-- IMAGINARY I - \ . ° .. / r o‘.‘ -20- / I _ “.- / ’ > " ‘\ " / n 0 :3 -40- \ a w" I ~ - I. \ i - x I. \ l’ ~ “6°” \ \ I “ k 0 «>- _ \ \«—-REAL I.’ 3 .3 '\ \ I! IL -80- . \ [I q \. \ Ii )— \. \ ’l- -- 400*- \ \ I .I 1150+ p .. \ \/ .’ E =40.0 MeV I— .\ I L = 2 .. -I20- '\ 0/ _ °\./ 0 ' ---- D +C (Macro) -I -°-'- 0 + C (Micro) ”'40)- -I l I l I l l l l l 0 I 2 3 4 5 6 7 8 9 IO r(F)—. Figure 8.--Same as Figure 7 for T150 + p at U0 MeV. 185 Enhancement Factors In order to examine more carefully the role of core polarization in these results, the square of the enhancement factors obtained in these calculations are given in Table 8. These are simply the ratio of the integrated cross section obtained with core polarization to the integrated cross section obtained without core polarization. They are denoted by a: where the subscript p appears because the valence nucleons are protons in all cases being considered. Except for the abnormal parity L=3 component of the Y89 cross section, the values of a: are of the order of 10. This illustrates that core polarization plays an extremely important role in the (p,p') reaction. Experimental values of a: are given for the transitions in ngo and T150. For the L=2 and L=u transitions in Zr90 these have been obtained by normalizing the theoretical angular distribution for the valence transition to the data at U00. For the L=6 and L=8 transitions in ngo, 600 and 700 were used to compute a: For T150 at 17.5 MeV, 53 was determined by comparing the theory and data at “00, but at NO MeV the hump at 510 in the experimental L=2 angular distribution and the flat spot at 35° in experimental L=U angular distribution were used for the point of normalization. In all cases good "eye" fits to the data have been achieved. Experimental enhancement 89 factors have not been obtained for the Y transition because the cross section contains two components. 186 TABLE 8.—-Theoretical and experimental values for the square of the enhancement factors corresponding to the results of Fig. 3, U, and 5. For prescription used to calculate e§(Exp) see text. Target ELAB(MeV)§ L e§(Micro) 55(Macro) 5:(Exp) , 2 18.9 16.5 33.2 Zr90 I 18.8 ' a 12.7 12.2 20.3 ‘ 6 9.56 11.0 18.u 8 7.62 11.0 55.2 T150 17.5 2 17.2 10.8 18.9 12.8 9.1 19.4 T150 no.0 2 19.8 13.7 19.8 14.9 12.7 18.2 Y89 18.9 3 .629 - ' — 5 9.1“ _ _ Y89 2u.5 3 .6u1 _ _ 5 9-55 _ _ 389 61.2 3 .617 _ _ 5 10.3 _ _ '1 »- 4.31.“; i. . . .. _ 187 Beyond any uncertainties associated with normalizing the theoretical results to the experimental data, the experi- mental values of a: given in Table 8 are subject to any errors contained in the approximate treatment of antisymmet- rization used in this work. For example consider the results of Love gt a1?“ which were discussed in the note added to Chapter 5. Using the central part of H-J force for the projectile-target interaction and treating antisymmetrization ; exactly, for the L=2 valence transition in ngo they obtain L’ o r=.Oul2mb, oex=.00u15mb, and o =.O689mb with Oex/Odir='l' T =.O52umb, o =.O2u6mb, and r ex di The results of this work are Gdi 0T=.l50mb with sex/odir=.u70. The first set of results gives €§(Exp)=72.5 for the L=2 transition in ngo. Taking for Odir the values obtained in this work for the K-K force, using the Oex/Odir ratios of Love 33 _l shown in Fig. (5.1'), and assuming maximum interference (see Eq. (5.3)) suggests that a proper treatment of exchange 90 might lead to the following modifications of the Zr results which have been shown. (1) Values of eg(Exp)=5u.7, 33.8, 23.2, and 35.1 might result for L=2, u, 6, and 8. (2) The microscopic angular distributions for the L=2, U, and 6 transitions of Fig. 3 may be reduced by factors of 1.65, 1.66, and 1.26, respectively, while the L=8 angular distribution may be increased by 1.57. Here it has been assumed that the complete 188 differential cross sections will be effected in the same way as the differential cross section for the valence transition. This is not strictly true since neutron excitations contribute to the former and the n-p and p—p forces do not have the same radial shape. (3) The macroscopic angular distributed of Fig. 3 may be multiplied by factors of 1/1.12, l/l.lU, l/l.07, and 1.15 in the order L=2-8. These cross sections are more stable than the micrOSCOpic ones since the core contributions are not effected by the uncertain- ties in question. (U) Under the assumption of (2) the values of e§(Micro) will not be changed. (5) From (3) it follows that 65(Macro) will be 28.2, 17.8, 13.9, and 8.09 for L=2-8. The main point here is that the results of this work might be biased so as to improve the agreement of theory and experiment for L=2—6 transfers. The indicated modifications improve the consistency of theory and experiment for L=2-8, but at the same time result in somewhat poorer absolute agreement. With the modi- fications the microscopic L=2 cross section is too low by a factor of 2.9 while the L=8 cross section is a factor of 4.6 under the data. Inclusion of the L=7 non-normal transfer in the 8+ calculation might then remove most Of the discrepancy between the two results. Finally observe that the agreement 189 between the microscopic and macroscopic results is not strongly effected by the uncertainties due to the exchange approximation, although the microscopic results for the L=2, U, and 6 transi— tions will be shifted downward 20% with respect to the macro— scopic results while the L=8 results might be brought into essentially complete agreement. The fact that the macroscopic cross sections may be larger than the microscopic cross sections could reflect that a larger value of should be used in these calculations. The value of e:(Exp) for the L=2 and L=u transitions in T150 are found to be about equal, roughly 19, and the data provides no indication that this number varies with energy. It would be useful to have results with exchange treated exactly to check these points. Except for the magni- tude of a: it is expected that the observations will be up- held. Guessing that the cross sections for these valence transitions are being overestimated by the same amount as 90 for Zr leads to a modified value for e:(Exp) of about 31 at 17.5 MeV. It is found that e§(Macro) for the L=2 transition is a little larger than for the L=u transition and both are too small by about a factor of two at 17.5 MeV. They also increase a little with energy. e:(Macro) for the L=2 transi- tion at 17.5 MeV might be modified to a value of 13 which is about 2.3 times smaller than the modified experimental value. The fact that 63(M10P0) are larger than 53(Macro) 190 has already been explained. The values for €:(Micro) increase slightly with energy as is expected since the shorter range n-p force is a factor in the complete cross section while only the p—p force is involved in the valence transition. 67 Calculations have been carried by Satchler §t_al, for the single particle transition in Y89 at 18.9 and 6l.u MeV. The H-J force has been used and exchange has been treated exactly. Comparing their results with the results of this work indicates that the approximate treatment of exchange is not introducing any serious discrepancies here. This is expected as the dominant multipole is L=5 in this case. The comparison also indicates that somewhat smaller (less than a factor of 2) cross sections would be obtained with the H-J force. This is also true for the L=2 transition in Zr90 where the K-K force gives the modified experimental value, eg=59.7, while eg=72 is obtained for the H—J force from Ref. 7n. In any event it appears as if the results obtained here for this transition in Y89 are somewhat better than those obtained for Ti50 and ngo. L=0 Transition in ngo An experimental differential cross section is available 90 in for the excitation of the 0+(Q=-l.75 MeV) level of Zr the (p,p') reaction at 12.7 MeV.105 Ref. 8 also gives an upper limit for this cross section for incident protons of 191 18.8 MeV. This is about 20 ub between U00 and 60°. A calcula- tion of the differential cross section for the valence transi— tion at 18.8 MeV gives a result which is in agreement with this upper limit. The decomposition of the integrated cross =.ou53 mb, 6 =.l69 mb, section for this case is o x='0392 mb, 0 dir T and oeX/odir=.865. This ratio is much larger than Oex/Gdir=’22 e which is obtained when it is assumed that only 1g9/2 protons are involved in the L=0 transition (see Fig. 5.1'). This same effect was observed for Yukawa forces in the discussion of Fig. 5.1. Core polarization gives eg=9.35 for this transi- tion which destroys the agreement with experiment. Assuming that Oex/Odir is 10 times too large which is inferred from Fig. 5.1' leads to a result which is only about u times greater than the upper limit. A calculation with core polarization was made for compari- son with the 12.7 MeV data. Optical parameters were taken from Ref. 105. The direct and total (direct plus exchange) differential cross sections are shown with the data in Fig. 9. The shape of the theoretical cross sections are not in good agreement with the data and it is seen that there is a large exchange contribution. Again assuming that the effect of exchange is being overestimated leads to a result which is not very different than the direct differential cross section shown. This is in accord with the data insofar as overall Inagnitude is concerned. Love 33 a1?” have indicated a value of 10 is needed for a: based on their calculation of the clifferential cross section for the valence transition using I 5 Zr9°+p _.__ E 8 l2.7 Mev 0+ gov-L75 Mev 2% — Total "—_"‘ '00 \\ --- Direct I 5 '-‘~ 3 \ "\ \\\ \ k \ a): 2 \ \\ \ \ Q lo-I I \ \\ ' ...—v"— E. i J. 5 50 1. 2 [ _-—-1 I62 .. 5 21 '00 20 40 60 80 IOO T20 :40 l60 |80 9c (deg)—. Fi ure 9.--Differentia1 cross sections for L=0 transition in Direct and total (direct + exchange) Zr 0 + p at 12.7 MeV. cross sections are shown. Core polarization is included. 193 the K-B reaction matrix. This is roughly the value obtained in this work. Note that the data has a deep minimum at 600 — the region where the upper limit of the 18.8 MeV cross section was fixed - and observe that because of the poor shape agree- ment this point is badly overestimated. It has been suggested that the shape of the theoretical result can be improved by 9,105 An damping the form factor in the nuclear interior. angular distribution with a better shape has been obtained in Ref.IUfl5from a macroscopic form factor representing a breathing mode.65 Effective Charges Table 9 contains the effective charges for the electric 2L—pole component of the transition rates for the transitions under consideration. Experimental values are given for the L=2 transitions. These have been extracted from transition rates given in the indicated references on the basis of the harmonic oscillator wave functions used in this work. Note that there are two experimental values given for the quadru- pole effectivecharge in ngo. The two numbers do not agree with each other and the larger number is the most recent result. The results for eeff(Micro) are simply the square roots of the ratios of the B(EL) computed with the complete proton transition density, FgOL(T), to the B(EL) computed O L L( D). For the p with the valence transition density, F 19“ TABLE 9.--Effective charges for electric 2L—pole component of transition amplitudes for the transitions under consider— ation in Zr90, T150, and Y 9. Nucleus? L eeff(Micro) eeff eeff(Macro) eeff(Exp) 3 2 1.23 1.79 2.08 {2.31.U15’106 Zr90 1 u 1.19 1.65 1.73 3.2i°2 i 6 1.13 1.51 1.52 - 8 1.08 1.3M 1.u1 - T150 2 1.19 1.67 1.92 1.8:.2107 M 1.15 1.54 1.64 — Y89 5 1.18 1.u6 - - definition of B(EL) see Eq. (C.17) or Eq. (D.5A). Eq. (D.23) has been used to calculate eeff(Macro). In these calculations it has been assumed that R%/=1. Actual values of this quantity based on reasonable finite well wave functions for various orbitals in different nuclei vary from .6—l.5.15’l6’108 The quantity eeff is obtained by taking FEOL by g-[FgOmegOLmn The quadrupole effective charges given by the micro- (C) to be given scopic model fall far short of the experimental values. The macroscopic model gives reasonable agreement with experiment 90 i if the smaller value for the L=2 effective charge in Zr 5 assumed.to be correct. The values of eeff are in better agreement with eeff(Macro) and eeff(Exp) than are eeff(Micro). ERR; substitution used in calculating geff is strictly valid I. 195 only in the limit of iso-scalar core excitation — a condition which might be closer to reality than the microscopic calcu— lations indicate because of the correlations between core nucleons which are neglected in that picture. Note that the values for eeff(Macro) are subject to a assumption similar to the one made in calculating eeff, i.e. only the overall effect of core polarization is contained in the values of 20L extracted from the Kuo—Brown matrix elements and an indepen- dent assumption as to how this effect is divided up into neutron and proton components is made in writing down Eq. (D.23). It is concluded that the proton—neutron imbalance pre— dicted by the microscopic calculations is not consistent with experiment. Experiment appears to favor something more like iso-scalar core excitation. This point will be examined in more detail in a short while. It should also be pointed out that the inclusion of those proton—proton hole excitations where the proton is in the valence orbital will not be sufficient to remedy this situation.+ Finally, there is no information indicating that these calculations are giving a fair description of the relative variation of eeff as a function of multipole. Additional experimental y-decay data would prove useful in examining this point. Coupling to Physical Core States Collective model analysis of the first 2+ excitation 111 Sr88 which has been observed at 1.89 MeV in the (p,p') +See note at end of this chapter. ' l'.‘ .7 196 0 reaction at 18.9 MeV yields the value 82=.l3.l Several other low lying 2+ states are also observed. A 4+ state is believed to exist at “.05 MeV but it has not been resolved experimentally. The first 2+ (Q=-3.82 MeV) and first 4+ (Q=-6.33 MeV) levels in Ca“8 have also been observed in the (p,p') reaction at 25, 30, 35, and no MeV.109 Values of B2~.l7 and Bu~.09 have been extracted from a collective model analysis of this data. From Eq. (B.l7) it follows that C2=272 MeV for the Sr88 levels and C2=330 MeV and Cu=3516 MeV for the Ca”8 levels. From the experimental data it is esti- mated that first u+ state in Sr88 Cu 10“. These values of C2 are comparable to those which has Bu~.04 which gives appear in Table l of this chapter. The values of Cu given here are roughly an order of magnitude larger than the corres- ponding values appearing in that table. The appearance of phonons in the core nuclei which have strengths comparable to the effective core phonon associated with the uncorrelated particle—hole model introduces serious reservations concerning the use of this model. Kuo has already made this point.90 A case where such a core phonon is dominant will be discussed in Section 3 of this chapter. The general consistency of 82 values extracted from analysis of the (p,p') reaction and (e,e') experiments indicates that such phonons have comparable proton and neutron transition densities; therefore, they will give a better account of the charge and mass polarization effects in these L=2 transitions ‘than.the particle-hole model does. 197 The large Cu values for the first 4+ states in Ca“8 and Sr88 could be indicative that the particle—hole model might be better for the states of higher multipolarity; however, the results obtained for L=u transitions do not compare more favorably with experiment than those for L=2 transitions. No strong 5' state has been observed in Sr88. The results obtained for the single particle transition in Y89 compare quite well with experiment-—better than those for T150 and ngo. This may suggest that there is something quite different about negative parity and positive parity transitions; however, the differences are not so large as to allow an unambiguous con- clusion. Calculations with exchange treated exactly are needed to see exactly how big these differences are. Also the MA y-transition rate must be calculated as a check on the L=3 component of the cross section, although the shape agreement between theory and experiment at 61.4 MeV suggests that it is given fairly well. Microscopic Empirical Formula For a normal parity transition the microscopic empirical formula of Atkinson and Madsen, Eq. (D.63), provides a rela— tionship between the enhancement due to core polarization, 5, of a valence transition in the (p,p') reaction and the :nature of the effective charge. For a transition involving 'valence protons Eq. (D.63) is conveniently rewritten as 198 where fp=e is the observed effective charge, which is given p’ by the ratio of the total proton transition density to the valence proton transition density; fn is the ratio of the neutron core transition density to the valence proton transi- tion density; and a is the ratio of the strength of the n—p force to the p-p force. For the K-K force a is about 2.5. The effective charge gives a measure of the enhancement of a y—transition rate due to core polarization. It is clear that the corresponding enhancement factor for the (p,p') reac- tion should be much larger than the effective charge if fn is comparable to ep. This is simply a result of the fact that the K-K force gives more weight to neutron excitations than proton excitations in the (p,p') reaction. When the valence particles are neutrons Eq. (D.63) can be written 6 = f + f /a. n n p Now fp=en’ is the effective charge, which is given by the ratio of the proton core transition density to the valence neutron transition density and fn is the ratio of the total neutron transition density to the valence neutron transition density. 'The fact that proton excitations are given 2.5 times less ideight than neutron excitations in the (p,p') reaction is aagain clearly displayed in the formula. For fixed fp and fn 'the enhancement factors for the case of valence will be much annaller than for the case of valence protons. This occurs txecause a large weight has been assigned to the valence txransition when the valence particles are neutrons. 199 The smaller enhancement factors for neutron valence particles, as compared to proton valence particles, do not imply smaller core polarization effects. The iso-scalar and iso-vector effective charges are related to f and f by: p n eO = f if for proton valence particles ! } p n . 1 e0 = f if for neutron valence particles } n p l { An iso—scalar transition corresponds to the condition fp=fn which is equivalent to e =0. Transitions with iso—scalar 1 core excitation are defined by fp = fn+l which is the same n} p} as el=l. For proton valence particles and fixed ep, the iso—scalar condition implies larger values of ep than does the condition of iso—scalar core excitation, i.e. a larger neutron core transition density is implied by the first condition. For neutron valence particles and fixed en, the condition of iso—scalar core excitation implies a larger neutron core transition density and a larger En than does the iso—scalar condition. Both of these conditions imply strong correlations between protons and neutrons when core polarization is large. Whenever there is a great deal of core polarization the differences between the conditions will not be very significant. 200 The experimental relationship between ep(en) and ep(en) for the lowest quadrupole transitions in Zr90(A), T150(B), Ni58(c), and Pb207(D) is shown in Fig. 10. Values of ep(en) and ep(en) which lie within the boxes drawn in the figure are consistent with the experimental data. The experimental data for ngo and Ti50 has been discussed previously. The lower limit on e for these two transitions are the results ! of this work, i.e. they have been obtained from the K—K force with exchange treated approximately. The upper limit F 90 is obtained from the results of Love, Satchler, 67.74 on e for Zr p for the H—J force with exchange 90 and collaborators treated exactly. The intermediate value of ep for Zr (indicated by the horizontal line through the middle of the box) are the results for the K-K force, modified to correct for the deficiencies in the approximate treatment of exchange. 'This was also discussed previously. The upper and inter- mediate values of ep for Ti50 are estimates based on the 90 results. 207 corresponding Zr N158 and Pb are two other nuclei which have been considered in the course of this investigation. They have 207 not been discussed in this paper. The Pb results have 117 N158 been reported elsewhere. has been discussed by Zamick and Federman.88 Both of these nuclei have valence neutrons. The transition in Ni58 is from the 0+ ground state to the 2+ state at 1.33 MeV and the transition in Pb207 is the 3pl/2—2f5/2 (Q=-.570 MeV) neutron-hole transition. The 58 207 effective charges for Ni and Pb come from Ref. 88 and 201 IO- €j0£j)-*' lfiigure 10.-—Experimenta1 relationship between ea (en) for quadrupole transitions in Zr90(A), EE§3?%) and . N158. afhi Pb207(D). Results of theoretical calculations are also shown. "9 iLT‘i -. .W bl. lint fl 202 Ref. 94, respectively. The experimental (p,p') cross sections for Pb207 comes from Ref. 107 while that for N158 comes from Ref. 5. The lower limits on En are again the results of this work and the upper and intermediate values on an for Pb207 are based on the results of Ref. 67. The upper and intermediate values of en for Ni58 are only estimates. Also shown in Fig. 10 are lines corresponding to the iso-scalar condition and the condition of iso—scalar core excitation. The solid lines are for valence protons and the w:- “1“ 25‘.“ .c_ l . dashed lines are for valence neutrons. Observe that above the iso—scalar line you have more neutron excitation than proton excitation in the transition. Below the iso-scalar line this situation is reversed. The experimental data is not terribly definitive, but the boxes definitely tend to stay somewhere in the vicinity of the iso—scalar and the iso—scalar core lines, i.e. the data implies that there are strong correlations between pro- ton and neutron excitations in these transitions. For ngo, T150, and Ni58 the data says that the total proton transition density is equal to or greater than the total neutron transi- tion density. This is consistent with the findings of Schaeffer118 who has studied the (p,p') data and the y-decay data for the first 2+ and 3_ excitations in Sr88, ngo, and the Ni isotopes. For Pb207 the data implies more neutron excitation than proton excitation. It should also be pointed out that the results shown here are not inconsistent with proton and neutron excitation in the ratio Z/N which has been 203 suggested from comparitive studies of the (a,a') data and y-transition rates.119 For Zr90, T150, and N158 the Z/N condition is not too different from the iso—scalar condition and for Pb207 it implies quite a bit more neutron excitation than proton excitation. 207 The data favors the iso-scalar condition for N158 and the condition of iso-scalar core excitation for Pb . For E Zr90 and T150 it is difficult to distinguish between the two conditions from the data. The lower limits on ep imply that (" iso-scalar core excitation is required. The upper limits on ep favor the iso-scalar condition. In reaching this 90 conclusion the higher value of ep for Zr has been con- sidered suspicious. This is admittedly arbitrary. Recent experimental data on quadrupole y-transition rates in Ca“2 and Ti50 indicates that iso-scalar core excitation is favored 120 in the 1f shell. The results presented here are 7/2 consistent with this finding, but they do not substantiate it. In conjunction with Fig. 10, experimental values of fp and fn for these transitions are presented in Table 10. Two sets of values are given for each transition--one for the upper limits on ep and 85 and one for the lower limits. They are labeled 6) and e< , respectively. The results of the particle-hole calculations for ngo Ti50 207 , Ni58, and Pb are also given in Fig. 10 and 3 Table 10. In the figure these results are indicated by the points A, B, C, and D, respectively. For ngo, T150, and 20“ TABLE 10.--Experimental and theoretical values for the normalized proton and neutron transition densit$es for quad— rupole transitions in Zr90, Ti50, Ni59, and Pb2 . Experiment Nucleus €> e< Theory fp fn fp fn fp fn ngO 2.30 2.30 2.30 1.30 1.u1 1.34a 2.55 2.11b _ 0 i 115 1.80 1.80 1.80 0.80 1.22 1.114a i 1.81 1.61b !_ N158 1.90 1.90 1.90 1.90 1.20 1.140a r Pb207 2.13 1.13 2.13 1.13 1.30 0.145a aResults obtained from particle—hole calculation. bResults obtained with renormalized force. N158 the results for the particle-hole model fall very near 207 the particle-hole model gives a result near the iso-scalar core line.1L In all cases the iso-scalar lines. For Pb +The reader is warned not to attach too much significance to this particular result. For Zr90 and Ti50 the particle- hole model predicts much larger negtron core excitation than proton core excitation and for Ni5 there is much more proton core excitation then neutron core excitation, i.e. valence protons couple more strongly to core neutrons and valence neutrons couple more strongly to core protons. The small ratio of proton core excitations to neutron core excitations for Pb207 is a result of the fact that the same harmonic oscillator well was used for neutron and proton single particle orbitals. This is tantamount to assuming there is neutron skin for which there is no experimental evidence. The proton and neutron wells probably should be adjusted so that the low lying proton particle and hole orbitals have radii comparable to the valence neutron orbitals. This will improve the overlap between the low lying proton orbitals and the valence neutron orbitals and a larger contribution from proton core excita— tions will result. 205 the particle-hole model underestimates both the enhancement factor and the effective charge. From Table 10 it is clear that the particle-hole model does not do too badly for the neutron core transition density when the valence particles are protons, but it tends to underestimate the proton core transition density by a fairly large factor. For the case of valence neutrons the model does fairly well for the proton W lam .’" Y core transition density and tends to underestimate the neu- tron core transition by a substantial factor. This simply )- bears out what was said earlier, i.e. the neutron—proton imbalance in the core transition densities, which is pre- dicted by the particle-hole model, is not consistent with experiment. It is not too bothersome that the particle-hole calcula- tions do note produce perfect agreement with the experimental transition rates. It definitely gives a good qualitative estimate of the overall effect of core polarization. It has already been pointed out that it doesn't do a perfect job for the spectrum, and that the question of fairly strong, low lying core phonons cannot be ignored. Further, the coupling between the valence particles and the core is a little too strong (e.g. see amplitudes in Tables 2, 3 and 5 of this chapter) to allow one to take first order perturbation theory too seriously. The results of Kirson and Barret121 do, in fact, demonstrate that the perturbation series for the spectrum converges only slowly, if at all. 206 It is interesting to follow up on a suggestion due to Harvey122 to see if the results of the particle-hole calcula— tion can be improved in a simple way. He points out that 57 did not use the "bare" force (the K-K force Horie and Arima in this work) in calculating quadrupole moments within the framework of the particle-hole model. Instead they used a two—body force which was fit to the experimental spectrum, i.e. a renormalized force in our language. He argues that 'u. this procedure might give a much better estimate of effec— fl”. Jimanu-n‘mnw .. E tive transition operators than does the first order pertur— bative calculation using the "bare" force. Just how good this new estimate is depends on just how well the actual renormalized force, which is a complicated operator, can be represented by a two—body force determined from the spectrum. A calculation using this approach was made for the L=2-8 transitions in ngo and the L=2-6 transitions in T150. The renormalized force was taken to be of the form V = V + G3p-lh where V denotes the K—K force and G3p-1h was taken to be separable, i.e. G - k < )k ( ')6 26 Y*(") Y (“'> 3p-lh ' " v r v r Tl L L L r L r ' The 0L are given in Table l of this chapter. The additional assumption is made that G3p-1h only acts in T=l states. 207 Spectra typically require large renormalizations of the bare force only in T=l states.20’27 The results obtained for the quadrupole transition rates in Zr90 and T150 are shown in Fig. 10. They are labeled A' and B', respectively. The corresponding values of fp and fn are compared with the experimental values in Table 10. Table 11 gives a breakdown of the results for all the multi- 90 poles in Zr and T150 and comparison is made with the results of the perturbative calculation. Theoretical enhancement factors are also compared with the experimental values. TABLE ll.--Normalized proton and neutron transition densities as given by the particle-hole model and particle-hol model with renormalized force for L=2-8 transitions in Zr and for L=2—6 transitions in T150 Theoretical and experimental enhancement factors are also shown. For Zr90 the experimental a) values are from Ref. 67. The T150 6) values are estimates. Nucleus L ,p-h Model Renorm. Force Experiment fp fn ep fp fn 8p 6) e< 2 1.41 1.34 4.35 2.55 2.11 7.83 8.51 5.80 Zr90 4 1.25 1.06 3.56 1.62 1.37 5.05 7.45 4.52 6 1.14 0.84 3.09 1.30 0.95 3.68 6.19 4.52 8 1.08 0.60 2.58 1.16 0.61 2.69 6.30 7.48 2 1.22 1.14 4.24 1.81 1.61 5.84 6.35 4.36 T150 4 1.16 0.92 3.74 1.40 1.11 4.18 6.26 4.32 6 1.07 0.63 2.65 1.18 0.70 2.93 -- -- 208 The agreement between theory and experimental for the L=2 transitions is quite good. Both the proton and neutron core transition densities have been increased as compared to the perturbative results. The proton transition densities have gone through the largest relative change. Differences between the perturbative results and the results of the calculations with the renormalized force decrease with increasing multipole. The renormalized force gives a fairly hefty boost to ep for the L=A transitions and it produces a sizeable increase in the polarization charge for all multipoles. It is difficult to discuss the multipole depen— dence of 8 because of the fairly large uncertainties in the experimental values, i.e. E) and e< bracket a fairly large range of values. It would be useful to have (e,e') data for these transitions as it would provide information on the multipole dependence of the effective charge. In any event this procedure would appear to have some merit. The calculation reported here is quite rough and a more careful investigation of this approach is planned. W - flZM.‘--fl*vl . .‘W 209 . i _ =- . 3 S ngle Protogoghg/2 1113/2 (Q l 61 MeV) Transition in Bi The nucleus Bi209 has one proton outside a Pb208 core. The valence proton is in the 1h orbit for the ground 9/2 state of this nucleus. The first excited state (Q=—l.609 MeV) has the valence proton in the 1113/2 level. Twenty triads (LSJ) contribute to the transition between these two levels. The two most important ones are expected to be (112) and L (303). This is similar to the situation for the single proton transition in Y89 which has just been treated. One might expect these two states to be connected by an M2 y—transition. In exciting the li level in the (p,p') 13/2 reaction one might expect to observe a differential cross section which is composed of (112) and (303) components in analogy with Y89. Contrary to these expectations, the 1.609 MeV is observed to decay to the ground state by an E3 y—transition with B(EB)=(1.3-2.0)x10‘2e2b3,110:111 208 The core nucleus, Pb , has a highly collective 3' state at 2.614 MeV. This phonon is quite stable as a closely spaced septet of states 209 are observed in Bi at roughly 2.6 MeV. The septet results from the coupling of the lh proton to the 3- phonon of b208. 9/2 P Another septet, formed by coupling a 1113/2 proton to this same state, is expected at about 4.2 MeV. This is to be contrasted with the situation in Y89 where no strong 5" state is observed in the spectrum of the core nucleus, Sr88. 210 The Pb208(He3,d)Bi209 experiment has been performed112 and some (He3,d) strength is observed in the l3/2+ member of the septet at 2.602 MeV. Using the particle-vibration coupl— ing model Mottleson113 has estimated the mixing of the first 209. The admixture of the 2.602 MeV 2 two 13/2+ states in Bi state into the 1.609 MeV state is €2=4.8x10— In this calculation the coupling matrix element was obtained from 208 - the y-decay of the 3_ state of Pb The mixing of the : states accounts for the observed (He3,d) strengths. The 1.609 MeV state of B1209 has been excited in the (p,p') reaction at 39.5 MeV and a differential cross section is available.llu Following Kuo's suggestion90 that the particle—hole treatment of core polarization may not be adequate when there is the possibility of contributions from highly collective phonons of the core (which appears to be the case for this transition) the cross section is calculated in two ways: (1) including only 2p-lh components in the wave functions, and (2) replacing the components with p—h coupled to angular momentum JC=3 by components which contain the 3— 208Pb. In the latter calculation the macro- core state of scopic vibrational model is used to describe the core. The wave functions corresponding to calculation (1) will be designated Set I while those corresponding to calculations (2) will be called Set II. Particle-hole pairs are formed from the shells shown in Table 12. Harmonic-oscillator wave functions have been used, and the energy denominators were taken in part from 211 TABLE l2.——Particle and hole orbitals used in microscopic Calculation. The absence of total angular—momentum subscript indicates that both j=£il/2 orbits are included. Particles Holes Protons Neutrons Protons Neutrons 1h9/2 1111/2 18 1f 2f 2g 2s 2p I 3p 3d 1f 18 f‘i PE“ ! 11 4s 2p 2d E I g 4 28 13 lg 38 i 1 3d 2h 2d 1h “8 31‘7/2 3s 2f lJ15/2 1h11/2 3p 2h11/2 li13/2 experiment115 and in part from the Nilsson scheme at zero deformation. The size parameter ho is 6.8 MeV. I Ref. 113 gives =60 MeV and C 208 208 3=649 MeV. Analysis of the reaction Pb(p,p') Pb gives 8 ~0.l3 for this 3 state77’85’116 which is the only state with a large value of B in 208Pb. The relation 33:71/29hw2/2C3)1/2 implies C =543 MeV which is smaller than the value from Ref. 113 and 3 corresponds to an admixture e:2=5.5x10-2 of the 2.602-MeV, lg: state in the 1.609-MeV, 2Lgistate. The smaller value of C is used in this work. 3 212 In these calculations, as a matter of convenience, a pseudo-potential has been used for the projectile—target interaction. This pseudo-potential is known to give results consistent with those obtained using the K-K force and treating antisymmetrization approximately. The 2p-lh compo- nents of the cross section have been included only in the 8:0 terms in the cross section because it is only in these components that they add coherently. In using wave function Set 11 the components of the wave functions containing the core phonon contribute only to the (LSJ)=(303) component of the cross section. The remaining 19 components are the same in Sets I and II. Figure 11 shows the total differential cross sections obtained with wave function Set I and Set II. The (303) components are also shown for both cases. The differential cross section (II) gives a good fit to the experimental data. The (303) (II) component is dominant as forward angles. The enhancement due to core polarization, of (303) (II) is about 200. Because of this large enhancement the valence contribution to (303) (II) is small. Considering only this component and neglecting the valence contribution, the data 2 1 places an upper limit on 5 =10- . Wave function Set II gives B(E3)-2.4x10-2e2b3 which is slightly larger than the experi- mental values. The particle-hole model fails to reproduce the effect 208Pb Of the 3- phonon of . The enhancement of (303) (I) is q» 1 ans)“ iLC-ufiiqfi i§;' 213 l I I 5 35209 + p E 8 39.5 Mev '3/2+ 3 Q '-l.6l Mev 2 Total (Set In ‘ / --- Total (Set I) IO" . , . /.-' 'z, --------- 303 (sun) r~ . ... o. _._ s 'I E 5 .5 _ '>, f 303 ( c l f. /..o ’4’ ~§~ 2...”... Q; ,3 .../I \\ ... ‘1' '2 2 '3’ \l i «2 I, atx‘... i ‘\" I0'2 .' ~ I .f. E 1! \.\ 9 5l ‘ 1‘. - \' I \ .. E‘: 0.. be I, x _ \ -..- \:'- \ 2. f \ fig. -:‘\\\ / .... . " .o.. - / ' ~ '0 3 ‘35; “a. ‘l 5 ‘. \O \.a-\ 2 x. \O \ -4 -\ '0 0 20 40 so so ICC 120 I40 90 (deg) —. Figure ll.-—The experimental data compared with the theoretical results obtained with both sets of wave functions. The total differential cross sections and the (303) component are shown for both cases. 214 about 13 which is an order of magnitude smaller than the value obtained for (303) (II). This model predicts that many components make important contributions to the total differential cross section. In particular, (303) (I) is comparable in magnitude with (112) which involves the low- est allowed L and J transfers. As the lowest J transfer is highly favored in v-transitions, the particle-hole model predicts that the 1.609 MeV, 1—31 state will decay to the ground state predominantly by an M2 transition which is in contradiction to experiment. It is concluded that highly collective core phonons can play an extremely important part in the core polariza- tion process. This is another indication that care must be exercised in applying the uncorrelated particle-hole model for core polarization. NOTE added ingproof: A calculation was performed to estimate the effect of exciting proton particles from the core into the valence orbitals in Zr90 and T150. These excitations were treated the same as proton excita- tions into orbitals outside the valence space, but ampli- tudes of configurations with three particles in the same orbit were multiplied by (n-2)/n (where n=2j+l) to account for violations of the Pauli principle. Experimentally observed single particle energy denominators were used in the calculation. With these excitations included e =l.4l, 1.26, 1.14, and 1.08 for L=2—8 in Zr90 and eff q“ 1.; .1333: mi? '3“ 215 eeff=l'22’ 1.16, and 1.07 for L=2—6 in T150. These are not much different than the results shown in Table 9 of this chapter. The result for the L=2 transition in ngO shows the biggest change. Here quite a large contribution was obtained from the 2pl/2-2p3/2 proton particle—hole pair. These changes will not effect the (p,p') cross sections very much as they are primarily sensitive to the neutron excita- tions. Note that the values for e (Macro) are somewhat eff larger than those for 5 even if the effect of the above eff excitations are included. The assumption of the collective model is that the charge transition density is Z/A times the mass transition density. Thus one expects that eeff should be slightly larger than eeff(Macro). Coupling this to the fact that Ref. 15 gives R§/>l for ngo again suggests that a larger value of than 50 MeV should be used in these calculations. CHAPTER 8 SUMMARY AND CONCLUSIONS It is felt that the results of this work, which are E admittedly rough, clearly demonstrate the feasibility of using "realistic interactions" in describing the inelastic scatter— ing of 15-70 MeV nucleons from nuclei in a microscopic picture. The use of such interactions requires a fairly detailed description of the target nuclei and it is necessary to treat antisymmetrization. These two requirements are not objectionable as the former is precisely the motivating factor for the microscopic approach while the latter should yield useful information about the interaction as well as the nuclear wave functions. Three interaction models have been considered in this work and the majority of the calculations which have been performed provide information only about the strong central components of these forces. The results obtained are sensi- tive to the gross features of the force, i.e. strength and range, and the impulse approximation pseudo—potential and the K-K force appear to be somewhat better than the Yukawa effective range force. The first two contain information about the high momentum components of the free two nucleon force while the third does not. In work on the optical potentia1,3l’32 it was shown that the impulse approximation 216 217 pseudo—potential does not have the correct phase——a property which is not examined in the inelastic scattering calculations-- and that the K—K force was better than the Yukawa effective range force. A convenient approximate treatment of antisymmetriza- tion has been developed and used in this work. This approxi— mation has been shown to be qualitatively correct in general and gives good quantitive results for Yukawa forces of 1F range at incident energies in excess of 40 MeV. For Yukawa forces of longer range and at lower energies the approxi— mation is still fair, but it appears to be considerably poorer for the K-K force. Although the K-K force is favored theoretically, uncertainties due to this approximation make it difficult to say that it is better than the Yukawa effective range force solely on the basis of the inelastic scattering data. 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APPENDICES 226 APPENDIX A APPROXIMATE SERIES FOR EXCHANGE COMPONENT OF D.W.A. TRANSITION AMPLITUDE Expanding th(|ki—E2D in Eq. (2.51) in a Taylor series about Ag keeping only the first two terms and then trans- forming back to a coordinate representation gives IeX=-fxé_)*(FO)¢;(Pl){} o¢rxé+)(Pl>d3rod3rl (A.1) {}=A(l)¢r(rl){}¢p(r1>xé+)(r1)d3rl _ (1) 2 2 2 2 {}-A (AO>—B¢;¢rx;+)<51>d3r1 -Bfxé‘)*v¢;~vcrxé+)d3rl (A.4) - %B(Ag)f3(p,r)-3(b,a)d3r 1 In Eq. (A.4) A< §;r1 >= -A(l)— [A3— §<§ H)JB(A > (1.5) 2_ 2 22 —k - U(r ) K h2 1 'where U(rl) is the optical potential and ’ i * " * ‘ )v (' ) J-¢r(rl.v¢p-¢p=x;+)(El)Vxé‘)*-xé‘)*vXa 229 The first integral in Eq. (A.4) contains a dependence on the magnitude of the local momentum of the projectile and the second integral expresses dependence on the magnitude of the local momentum of the bound particle. These two integrals can be arranged to display the dependence on the momenta of the projectile and bound particle in a symmetric way; however, the form which is given is more convenient as it does not explicitly refer to the binding energy and potential of the bound particle. Both of these integrals can be easily handled in the local D.W.A. The third integral in Eq. (A.4) cannot be incorporated conveniently in the local D.W.A. Contributions to non-normal transfer come from this term which essentially takes into account the fact that locally the projectile and bound particle are moving in different directions. The integral averages over these directions and the contributions for normal transfers are expected to be small. In the plane wave limit it can be shown that the integral vanishes for normal transfers when op and ¢r are the same. Neglecting the last term in Eq. (A.4) it follows that ~ LSJ _~LSJ ~LSJ 6 E (rO)—El (r0)+E2 (r0) (A. ) ~LSJ where El (r0) is given by Eq. (2.55) or Eq. (2.56) with the . ~LSJ . replacement A(l)(Ag)+A(Ag;r0). E2 (r0) contains the contribution from the second integral in Eq. (A.4). For the case of good i—spin ‘ '. ..‘ I V; Iv" 230 ~LSJ _ E2 (r0)— “ 1 1 ,/2T<§TTb,T -r |§ta> J} T B a b . + _ ’ AAA/\A— .’ ’ ’ XlL Q 2 /2jLSJTX(J ngi £L;% %8)8(JAJBJ;TATBT33J ) x(4n)-l/ZBST(AS)T (11,7) % and when i—spin is ignored ~LSJ _ ' E2 (r0)- ' 2,721L+£‘£ /2jLSJZ(j’jJ;£’£L;% l . a .0 jj §S)S(JAJBJ,jj TT ) x(4n)‘1/ZRSTT.(AS)37. (A.8) In these equations " 9(- Z L” . <> <> 8F = (+),(-)('l) () "+-L()L()un’zcr0)unt(r0) xW(L()tL()t’;1L)(L()L()L) (A.9) 0 0 0 abC 58 I _ a'- = '— where (087)18 a 3 j symbol, L (i) 2 +1, L(i) 8+1, and uéP = W203;- + £98.29) u§;)(r> = (2+1>1/2<§; - é>un, 231 There are four terms in the above sum and n+_ is a phase which is positive for the (+)(+) and (—)(—) terms and negative for the (—)(+) and (+)(—) terms. The net effect of including these additional terms in ELSJ(rO) is to damp out contributions to exchange scattering which come from the nuclear interior. This is reasonable as the momenta of the projectile and bound particle are much larger in this region than they are outside the nucleus. The exchange scattering here should sample momentum components of the interaction much larger than 13- a value determined by considering the assymptotic conditions. APPENDIX B TRANSITION DENSITIES AND FORM FACTORS l. Harmonic Oscillator Wave Functions Throughout this work, the single particle bound state wave functions used are those for a particle bound in a harmonic oscillator potential. This is a necessity because a complex description of the target nuclei is being attempted. The radial part of these wave functions are given by60 —l/4 n+£+l , 2 2 2 (n—l).]l/2a£+3/2rRe—a r /2 P [(2n+2£-l)!! r) (B.l) (r)=w unR n£( where the principle quantum number runs from 1 to w and _ _ l l , n+l(_l)k <2n+22 l)-- (a2r2)k. (8.2) n—l k P = n8 kZ02 (n—k—l)!k!(2g+2k+l)!l 1 The size parameter, a, is given in F_ by Q=EM%]1/2 = [M%%]1/2 = 1/2 .156Qfiw) (B.3) where hm is the energy separating the major shells of the potential expressed in MeV. Eq. (B.l) and Eq.(B.2) are some- what more convenient than the more commonly encountered rela- tions which give un£(r) in terms of the associated Laguerre polynomials. The first few Pn£(r) are 232 233 P12(r) = 1 22+ 2 2 P2£(r)=——§; - c r . (3.4) P3z(r) = %{(2£+33(2£+5) - (22+5)d2r2+duru} From Eq. (2.58), Eq. (2.59), Eq. (2.46"'), and Eq. (2.47"') it follows that the transition densities can be written LSJ,T = Z LSJ,T , LSJ _ X LSJ , where LSJ,T . _ “ _ g l M (33 ) ‘/7T<2 TTb’Ta‘Tb|2Ta> AAA/\AA xS(JAJBJ;TATBT;jj’)iL+£-£ (4n)‘l/2/2 leSJT I\)|l—‘ xX(j’jJ;£’£L; AAA/\A M%§{(35’) = /2S(JAJBJ;jj’TT’)iL+£—£ (4w)-l/2/2 jQLSJ l\)|l—’ NIH xX(j’jJ,t’tL; Inspection of the above relations leads to the conclusion that the transition density can always be written in the following form when harmonic oscillator wave functions are used. 1 1 2S) (B-5 ) S). (B.6') 234 N LSJ 2 LSJ N+3 N —d2r2 F (r) =N CN a r e a. Na =(Q+g )min (B-7) Nb =(£+£ +2n+2n —4)max In writing this equation reference to T or IT’has been dropped for convenience. Na or Nb is determined by the contri- buting unfiun’fl’ which yield the minimum or maximum values, res- pectively,of the bracketed quantities. Note also that the transition density is an even or odd function of r as the parity change in the transition is plus or minus, i.e. only even or odd values of N are included in summing from Na to N0. 0 2. Macros00pic Vibrational Model Considerable success has attended the use of the macro- scopic vibrational model in describing inelastic scattering. There are numerous references to this approach in the liter- ature - Ref. 61 and 62 are but two of these. As there must be a rough correspondence between the microscopic picture and this macroscopic picture it is useful to review this model. A modification of this model is used in the treatment of core polarization which is discussed in Chapter 7 and Appendix D. The following discussion is restricted to even target nuclei which have ground state spin equal to zero. wan-sum 2‘54” m . . I ' -r 235 In this model the nucleus is likened to a quantized drop of an incompressible, non-viscous fluid. The primary excitations of this system are small surface oscillations (phonons) about spherical equilibrium. The surface of the drop is given by R(0,¢) = R0{1+LMQQLMYLM(e’¢)—(un)—1LzlaLMI2} (3.8) which conserves volume to second order in aLM’ the deformation parameter. The Hamiltonian for the system is 1 1C H = LM{2DL|”LMl + 2 LIO‘LMI (B.9) where DL is the mass parameter for excitations of angular momentum L and parity (—1)L, CL is the corresponding stiff- ness parameter, and w is the momentum conjugate to a LM LM' In terms of the operators which create and annihilate phonons, c+ and c the Hamiltonian is written LM LM’ +1) LM c+LM 2 (B'lo)‘ H = ZthL (c+ )1/2 where w = (CL/D is the frequency of the phonon L designated by L. + The CLM LM If the hydrodynamic description of the system is adhered to and c obey boson commutation relations. strictly, relations for DL and CL are easily obtained. In practice it is necessary to treat them as free parameters. and c are related as follows 0+ LM The n , a LM’ LM LM’ 236 f1 _ “L 1/2 + L+M aLM ‘ [13(2CL {‘iM+(‘l) CL,—M} .IhC - -l L 1/2 L+M + LM ath O‘LM' CL TrLM where [i] is l for L even and i for L odd. Equations (B.ll) are subject to the conditions that the phonon states transform under rotations and time reversal in the same manner as the single particle wave functions ¢E£(E) which were defined in Chapter 2 and that R(6,¢) has appropriate matrix elements in such a representation.63 Note that these equations are consistent with the classical reality condition d+ = (—l)Mx LM L,-M. It is then assumed that the interaction between a projectile and this liquid drop is only a function of the distance between the projectile and the surface of the drop, i.e. (r-R). Since only small vibrations are being considered it is reasonable to make a Taylor series expansion of the interaction about R=R . To first order in a this expansion 0 LM is U(r—R) = U(r-R0) — k(r)Z (B.l2) * A LMaLMYLM(r) where k(r) = ROdU(r—RO)/dr. U(r—R0) is identified as the optical potential which is spherical and describes the elastic scattering. Assuming the usual Woods—Saxon form this potential is written I U=—V(ex+l)_l-iW(ex +1)’1+uiw +(f1/mflc:)2vsr"l 9— +1)" L-E (B.l3) where x = (r-rOAl/3)/a,x’= (r—rOAl/3)/a , etc. and to which is added the Coulomb potential of a uniformly charged sphere of radius rCAl/3. The potential contains a real volume term, volume and surface imaginary terms and a real volume spin- orbit term. The diffuseness parameters are a, a’, . . . . and the radii are identified as R = r Al/3, R’ =r’A1/3 O O O O Neglecting the Coulomb and spin orbit terms in the potential I O 0 leads to the following expression for k(r) k(r)=(VRO/a)—-e—-—2+i{WRé/a’) e 2+Lli(wDR6’/a”)§——-—(-%-E—e7~—) (3,114) (l+e) (1+e’) (1+e )4 where e=exp(r—RO/a),. . . . Before completing this discussion by defining the form factor for inelastic scattering ESLJ(r), it should be noted that the prescription (B.l2) for treating the deformation is not the only one which appears in the literature6l’6u, although it is the one used most frequently. Futher Eq. (B.l2) only provides for the treatment of (L,O,L) triads for normal parity transitions. In this model the form factor for the excitation of a single phonon is §LOL = -iL/§ k(r) . ' (B.lS) 238 Using Eq. (B.ll) gives ~LOL ’fi (L) F (r) = -1L[ii/§k(r)< L 1/2 ——— , (B.l6) 20L thus inelastic scattering experiments provide a measure of the stiffness parameter. It is common practice to tabulate the root mean square deformation in the ground state due to zero point oscillations ’fiw 33 = <0|§|dLM|2|o> = (2L+l)(§6i (B.l7) which gives ~LOL L 8L F (r) = —i [i]/§k(r) 7— . (B.lB) L In this discussion only the matter distribution in the drop has been considered. This fact and the restric— tion to lowest order is why the description applies only to normal parity transitions. In addition the liquid drop described here can only have excitations of quadrupole order or higher. By introducing other variables, i.e. compressi- bility, spin, and charge, the model can be generalized to 63,65 encompass a larger class of vibrations. In Appendix C electromagnetic transitions are considered and the model is extended with the assumption of a uniform distribution of charge throughout the volume of the drop. - :rrm _-‘ a» , rIl 239 3. Reduced Matrix Elements and Transition Densities for Various Transitions In order to calculate the transition densities it is necessary to evaluate the reduced matrix elements of the one body operators which appear in Eq. (2.58') and Eq. (2.59'). In the occupation number representation a one body operator is written -2 -2 + O _ 1 Ci — OLBaaaB (B-l9) where a+ and a are the fermion creation and annihilation operators which were introduced in Chapter 2. They satisfy anticommutation relations. When using i-spin the operator of interest is N N 6(r-r ) LSJ,T _ Z i LSJ T _ Z LSJ,T o _ 1:1 —————————r2 T (1)“: (i) — 1:10 (i) (3.20) and when not using i—spin it is ’ 6(r—r ) ’ LSJ _ Z i LSJ _ Z LSJ oTT, — 1 ————§—— T (i) — i o (i). (3.21) I‘ In the form of Eq. (B.lQ) theseVbecome LSJ,T _ Z , , , LSJ,T + , O - jmT aJ’m’T’ajmT (B.2O ) J’m’T’ l LSJ Z , , LSJ + , oTT, - Jm <3 m |o Ijm>aj, ,T,aJmT (B.21 ) "r‘“LE' P‘" F T 240 In the discussion which follows RLSJ will be used for LSJI and a single subscript will be used on + a and a to represent the quantum numbers jmT. Single Particle Transition This is a trivial case and there is no need to introduce i—spin. The initial and final states are |A>=a$|C> and |B>=ai|C>, respectively, where |C> denotes a filled shell state. The following result is easily obtained. LSJ _ LSJ RTT’ _ (Jlllo llj2>6TT’,T T (B'22) 2 l flhe 6 , is used with Table 2 of Chapter 2 to determine TT ,T2Tl the force component which is needed. For example, consider a single neutron transition in the (p,p’) reaction. Then Tl=T2=—% and the transition goes through the proton-neutron 1 force. For the (p,n) reaction T must equal —T’= —5 in order for the transition to be allowed; therefore, the single particle must initially be a neutron and a single proton will be left in the final state. Single Hole Transition For this case the initial and final states are 3 —m J -m 2 2 l lal|C>, respectively. |A>=(—l) a2|C> and |B>=(-l) The purpose of the phase was mentioned previously. It follows immediately that 241 J+31_j2+1. A Rfiii = <—1> 323116TT»,T1T2- (8.23) Using the conjugation relation <32|loLSJlIJ1>=(—1>S+J+J2-313132’1 (B.2u> gives for Eq. (B.23) Rgiq = ’(‘l)s<31|IOLSJllJ2>5TT’,TlI2° (B’25) This relation shows that a neutron single hole transition in the (p,p’) reaction is the same as a neutron single particle transition except for the phase factor (—l)S which may have some effect when interferences is important. In the (p,n) reaction the initial state must be a proton hole and the final state is a neutron hole. This indicates the significance of the interchange of T and T in Eq. (B.25) l 2 as compared to the ordering in Eq. (B.23). Transitions to Particle—Hole States The simplest excitations of closed shell nuclei are particle-hole pairs. In light nuclei with equal neutron- proton number i—spin is usually assumed to be a good quantum number and a particle—hole state is written J T B B . l l |B>=|J M T M >= C <3 J m —m [J M ><— —T -T IT M > B B B B mpmh Jth p h p h B B 2 2 p h B TB TpTh Jth r ...—n .I: L-‘_—# 2U2 J —m +l/2-T + x(—l) h h hapah|C>. (B.26) States of this form are obtained by diagonalizing a shell model Hamiltonian in the space of particle-hole pairs. This procedure is referred to as the Tamm-Dancoff Approxi— mation (T.D.A.) and it assumes that the ground state of 1’2 The reduced such a nucleus is a filled shell |A> =|c>. matrix element describing transitions from the ground state to the states (B.26) is J T A A A B B . -l . l c /2 J [T J" ] <3 -—||o B B 2 h Jth p p LSJ,T= Z Jp LSJ,T 3 Iljhéo. (3.27) J Since the ground state is a filled shell the only allowed values of JT are JBlB. For heavier nuclei with unequal neutron—proton number i—spin is usually ignored and particle—hole states of the following form are obtained J T jh—mh + 8 B>= J M >= < m —m J M >(-l) a a C> (B.2 ) I I Jpjh p hl B B - p hl c 3 3 3 ' J 3h Jth mm h ’UM e-I'U where T distinguishes between proton—proton holes and neutron-neutron holes. In this case the reduced matrix elements RLSJ 2 cJBT 3 A—l (3.29) II Jth Jth p B p h “1...”: “tum y . J" 2MB are used. The subscript on R has T=T’ since the initial and final states considered here are states of the same nucleus. Note that the form factor has explicit proton and neutron components when i—spin is not used. Random Phase Approximation Vectors The R.P.A. goes a step beyond the T.D.A. in treating closed—shell nuclei. It takes into account in an approxi— ‘ ‘ _‘ .L-‘IEJ mate way that the ground state may have 2p—2h, Ap-Ah, etc. and that the excited states may have 3p—3h, Sp—Sh, etc. components in addition to lp-lh components.l’2 The excitations in the ground state are referred to as gound state correla- tions (G.S.C.) The inclusion of these higher excitations has an important effect on transition rates as they allow the excited state to be reached by destroying a particle- hole pair as well as by creating one. Disregarding i—spin an R.P.A. state vector is given by |B>=|JBMB>=Q3 M |E> 3 3 (3.30) J T J T J —M . + Z 3 + 3 3 3 (J J 1)} Q = {x . A (J J T)“Y . (—1) A _ p h JBMB Jth Jth JBMB p h Jth JB MB T where |C> is the generalized ground state and J -m + _ . h h + AJBMB(ijhT)-mgmh(—l) apah. (3.31) 2AA The second term in Eq. (B.30) represents the G.S.C. The necessary reduced matrix elements are obtained by following the procedure = =: (B.32) where the fact Q|C>=O is used in introducing the commu— tator in the third step. It is easy to show that Eq. (B.29) applies with the condition J T J T J BT C = X Jth Jth . (B.33) Jth Correspondingly for good i—spin Eq. (B.2?) prevails with JBTB _ JBTB S+T JBTB C — X + (-1) Y . (B.3A) Jth Jth Jth X and Y are generally in phase and they add in non—spin flip amplitudes (for iso-scalar amplitudes if Eq. (B.3A) is being considered) and the enhancement due to G.S.C. is apparent if it is noted that the vectors (B.30) satisfy the normalization condition 2(x9—Y2)=l instead of 202:1. Like the macroscopic vibrational model, the T.D.A. and R.P.A. are schemes directed towards the explanation of low lying vibrational states in nuclei. The states (B.26), (B.28), and (B.30) may be called phonons. Transitions Between States of j2 Configurations Forgetting about i—Spin the wave function for two nucleons of the same type in the j2 configuration is 245 |A>=|JAMAT>= /%mzm a :a2|C> (3.35) 1m 2 where T again differentiates between protons and neutrons. This wave function is normalized and vanishes unless J A is even. For a transition between two states of this type RLSJ T“ . =(-1)J12JJ J J J 6 A J J 3 (3.3 ) B A I:8 where {} is a 6-J symbol“ and T=1’ because of the restric- tion to like nucleons. The single particle reduced matrix element vanishes unless L is even and when J=L and S=l; therefore transitions starting at the state JA=O do not proceed by spin-flip. Transitions from lp to 2p—lh States In treating core polarization as presented in Chapter 7 and Appendix D transitions from a one particle to a two particle-one hole state are encountered. A two particle— one hole state is written |B>=IJ1(Jp3h)JC;JBMB>==mZm m J _ x(-l) h mhaiagah|C> (B.3?) where i-spin is not being used. The above wave function is not normalized when p=l. This is not important at the present time and will be discussed in Appendix D. The 246 reduced matrix element for the transition from a single— particle state, lA>=a:\C), to the state (3.37) is LSJ R ,=5. 5 5 a A A-l LSJ TT 3111,3212 J,Jc TT ,Tth pJ — + +J+J {53 T J T 5TT’ T T {—1)Jl J2 OJ 3 ' J J p p’ 2 2 ’ h l c 1 J2 h c J1J3J . LSJ x.} (B.38) An allowed transition is subject to the condition that J2T2=JIT1 and/or Jpr as expected. When JlTl=J212¢Jpr only the first term in Eq. (B.38) contributes and the reduced matrix element is the same as that for exciting a particle- hole pair. This is seen by comparing with Eq. (B.29) and noting thatt =I’ in Eq. (B.38) when Tp=Th which is the condition for a transition between states in the same nucleus. The second term in Eq. (B.38) differs from the first by recoupling factors which appear simply because the role of the active and spectator particle have been interchanged. Transitions from 2p to 3p:lh States A general expression for a transition from a 2p state to a 3p-lh state is somewhat cumbersome to write down and tedious to derive. Further fractional percentage must be considered when the three particles are alike and in the same orbit. For the core polarization discussion in Appendix D only a particular result is needed and this is all that 2“? will be given here. This result is for the case when the 2p state is that of two like nucleons in a j2 configuration. Eq. (B.35) gives the wave function for such a state. The 3p-lh states which are connected to the states (B.35) by a one body operator can only have one particle in an orbit other than 3. These particular 3p-lh states can be written |B>=IECJJ)JV,(Jp3h )J JJBM 3 _ l. 2 . _ — /§ mlm m2< jpjhmp mhIJCMC> mpm h M M VC 3 —m _ p h + + +. x( l) ala2apah|C> (3.39) which is normalized as long as jptp¢jt which is to be assumed. This state vanishes unless JV is even. The necessary reduced matrix element is RLS€=6 6 d , j J—1 (B.UO) where again it is seen that the result is the same as for exciting a particle-hole pair with the 2p state playing the role of a spectator. Also T=T’ when T =Th which hols for p transition between states of the same nucleus. A. Note on Phases In all of the formulas presented in this paper the phases of the bound state wave functions have been fixed by 248 demanding that they be invariant under time reversal and rotation of 180° about the y-axis.63 When iso—spin is involved an extended time reversal operation is defined and a rotation of 180° about the y-axis in iso-space must be included. Fixing the phases in this way is one, but not the only way, of guaranteeing the reality of the bound state matrix elements of many operators. This phase convention explains the appearance for the i1 in the definition of the single particle bound state wave functions given in Chapter 2. Further it plays a role in the conjuga— tion property of a matrix, e.g. Eq. (B.2H). Many workers do not use the ifi in their single particle wave functions which is also a satisfactory phase convention. Since the wave functions of various people have been used in obtaining the results of this paper the phase convention of the formulas was not strictly adhered to in the calculations. Of course none of the physical results have been effected. It is generally quite easy to convert from one phase con— vention to the other. This note serves simply as a reminder that some of the tabulated results which appear will not be consistent with the formulas as far as phases are concerned. 5. Multipole Coefficients For Yukawa interactions and Gaussian interactions closed forms exist for the multipole coefficients. These coefficients are defined in Eq. (2.“3) and Eq. (2.““) and appear as tSTL(rO;rl) and tSTI’L(r03rl) in Eq. (2.U6), Eq. (2.“7), and later equations. For the Yukawa interaction 249 -IT1I' _ Ol f(rOl)-Ve /erl 9 (B.Ul) , + fL(rO;rl)=uniVJL(imr<)h£ )(imr>) and for the Gaussian interaction ”1712332 = 01 f”'01) V9 (B.u2) L 2 ’m2(r8+ri) ' fL(rO,rl)-unVi JL(—2im rOrl)e In Eq. (B.ul) h£+) denotes the spherical Hankel function and r< and r> denote the lesser and the greater of rO and r1. A general force requires that Eq. (2.“M) be handled numerically. A reasonably fast routine has been written for the calculation of form factors for the case of an interaction of general radial form. APPENDIX C INELASTIC ELECTRON—NUCLEUS SCATTERING The electromagnetic interaction between an electron and a nucleus can be decomposed into longitudinal Coulomb, transverse electric, and transverse magnetic multipoles. The excitation of collective states in normal parity transi— tions in the (e,e’) reaction proceeds predominately through the Coulomb multipoles. Restricting consideration to these cases the differential cross section, in Born Approximation, is writtenSS’56 0(a) = oM(6)|F(q(6))|2 (0.1) where oM(e) is the Mott cross section, i.e. oM = u2)cos2, (0.2) e is the scattering angle, q(6) is the magnitude of the momentum transfer q=ki—kf, and k1 and Er are the initial and final momentum of the electron. The Mott cross section describes the elastic scattering of a high energy electron by a point charge. Most of the kinematics is contained in this term. F(q(e)) is the inelastic electron scattering form factor which contains all of the nuclear structure information. It is defined by 250 251 |F|2 = Jl2mimAlzéieiQ'rIA>d3rI2. (C.3) Eq. (C.3) contains a nuclear matrix element of the charge density operator which is written as follows N p = g iZlT§(i>al — 2JA+1 L|(n/2) 2 g JL(qr)3- (r>r drI (c.u ) LOL 2 LOL T r = F ’ r 8~ ( ) T pp ( ) 2J +1 2 _ B X 1/2 —1 m. LOL 2 2 , |F(q)| - 2JA+1 L|(21T) Z IOJL(qr)Fp (r)r drl (0.5 ) In these equations FggL’T(r) and FgOL(r) are the transition densities defined in Eq. (2.58’) and Eq. (2.59’), respectively. The transition density defined in Eq. (2.58”) is reaction dependent because of the Clebsch-Gordan coefficient which contains the i—spin projection quantum numbers of the 252 LOL’T(r) in Eq. (C.M’) projectile. The subscript pp on F serves to specify the transition density for the (p,p’) reaction, i.e. Ta=Tb=%. In Eq. (0.5’) the subscript p on LOL(r) defines the proton transition density, T=T’=%. It should be pointed out that for the transitions F under consideration only the lowest allowed L—transfer will be important. For transitions where more than one L—transfer is likely to be important the treatment will usually have to include the transverse multipoles as well as the longitudinal ones. In such cases the relationship between the(e,e’) and (p,p’) reaction is not as direct as that seen by comparing Eq. (C. U’) and Eq. (C.5’) with Eq. (2.58") and Eq. (2.59"). For this case the inelastic electron scattering form factor is related to the Bessel transform of the proton transition density while the inelastic nucleon scattering form factor is obtained by transforming the proton and neutron transition densities with the appropriate multipole coefficient of the two—body inter— action. In practice it is necessary to include two corrections in Eq. (C.H’) and Eq. (C.5’). This is accomplished by multiplying these relations by f2(q) where55 f(q) = expE-q2(a§-l/a2A)/ul. (0.6) This serves to correct for the finite size of the proton (first term) and for center of mass motion (second term) 253 which is necessary because the shell model wave functions are referred to the center of the oscillator well. The parameter a: fixes the size of the proton distribution which has been taken to be Gaussian, a is the harmonic oscillator constant, and A is the target mass. In the calculations of this work ag=.43F2 is used. In principle center of mass corrections should be included in the (p,p’) calculations also. This is difficult because the D.W.A. is being used and this small correction is ignored as a matter of convenience. A closed expression for the inelastic electron scattering form factor can be obtained when harmonic oscillator wave functions are used by inserting Eq. (8.7) into Eq. (C.u’) or Eq. (C.5’) and using the following integration formula 2 2 2 2 {”6“ r Jv(qP)I’U—ldr‘=l"[%(u+v)](q/2a)v[2aul‘(1+\)]'le’q flm xF(§(v—u)+l|v+llq /ua ) where f() denotes the f—function, F(||) is the confluent hypergeometric function, and JV is the ordinary Bessel function. The confluent hypergeometric function is defined by CD 2 (X)nZn MIDI“ = n=O W 11 AO=lgkn=A(A+l)...(A+n—l) n21 . (c.8) oo=lson=o(o+l)...(o+n-l) n21 25M and is a finite polynomial when A is an integer less than or equal to zero. The spherical Bessel function and orginary Bessel function are related by JL(qr) = /§%; JL+l/2(qr). (C.9) For a definite value of L it can be shown that 2J +1 2 IFI2=n2JB+1 2w2<%>2L x [<2L+1>!zJ'2r2exp<—q2/2a2> A Z N (0.10) b x{% CLOL(L+N+l)!!2—(L+N+2)/2F(%(L—N)|L+3/2|q2/Ua2)}2 a N where n=l/u when i—spin is being used and n=l when it is not used. The correction factor f2(q) is defined in Eq. (C.6). The macroscopic vibrational model might also be applied to inelastic electron scattering. The treatment is the same as that for inelastic nucleon scattering which was outlined in Section 2 of Appendix B, but deforma- tion of the charge density is considered in place of the deformation of the potential for nucleon scattering. The charge density expanded to first order in the aLM is dp(r-RO) * p(r-R)=o(r-RO)—RO dr LMaLMYLM (0.11) where p(r-RO) is the spherical ground state charge distri- bution. Assuming a Woods-Saxon form 255 r-R o —1 a )] (c.12) 0(r—RO) = pO[l+exp( and p0 is fixed by the condition fpd3r=Ze. In this model the inelastic electron scattering form factor for the excitation of a single phonon of order L is UMB2 |F(q)|2 = ———Q— m3 (qr)h(r)r2dr 2 (ze)2|{ L l . (C.l3) h(r) = R0 dr A normal parity y-transition involving a collective state will proceed predominately through a single trans- verse electric multipole. The long wave-length approxi- mation is valid for y—transitions and in this particular instance the inverse electromagnetic lifetime is given by56 2 w =8nc9— (L+l) k2L+lB(EL) Y he L[(2L+1)!!]2 A (C.lu) B(EL)=M%B|%erYLM(r)d3r|2 where L gives the multipolarity of the radiation, k is its wavenumber, and B(EL) is the reduced transition probability. Note that the latter quantity is directional in that 2JB+l B(ELgJA+JB) = EIXII B(ELgJB+JA). (c.15) 256 From Eq. (0.“) and Eq. (C.5) it follows that 2J +1 m + B(EL) = EJEII §|£ rL ZFLOL(r)dr|2 (C.l6) 2J +1 _ B l w L+2 LOL 2 B(EL) - 53;:f-2lfor Fp (r)dr| (C.l7) for the case when i—spin is and is not used. For the excitation of a single phonon in the macroscopic vibra— tional picture it follows that B m + B(EL;O+L) = (_%92|jorL 2h(r)dr|2 (0.18) which reduces to 3Z_L 2 B(EL;O+L) = (HFfioBL) (0.19) for the uniform charge distribution. These relations show that electric y-transitions provide information about the proton transition density, however, this information is not as valuable as that obtained in inelastic electron scattering experiments since the integrals in Eq. (C.l6) Eq. (C.l7) are most sensitive to the tail of the density whereas the Bessel transform of the transition density samples different regions of the density as q is varied. rraluv' nl APPENDIX D CORE POLARIZATION 1. Introduction In the opening paragraphs of Chapter 7, several approaches werenentionedfbr estimating the effect of core polarization on the properties of the low lying states of nuclei with a few nucleons outside of a closed shell. There is one essential point in all of these methods-- the basic configurations needed to describe the low lying states of these nuclei are not those of the simple shell model lAn>’ which Consist of valence nucleons dis— tributed about a filled shell, but the configurations given by first order perturbation theory ~ ... E “‘1 I IAn>—|An>+c (EA —EC ) |Cn> (D.l) n n n which contain admixtures of core excited states, ICn>. In Eq. (D.l) EA and EC are the unperturbed energies of the n n states lAn> and ICn>, respectively, and V’ is the inter- action coupling the valence nucleons to the core. In general, when there are more than one valence nucleons, the complete wave function for a particular state 257 258 in the nucleus is given by a linear combination of the configurations (D.l), i.e. 1% _ A ~ |A>-n=f4n|An> (D.2) where thewlfi are obtained by diagonalizing an effective Hamiltonian for the nucleus in the basis {lAn>;n=l, N}. Matrix elements of the effective two—body interaction between valence nucleons are defined by . _ Z _"l 2 I —+CnECn (D.3) The first term on the right in Eq. (D.3) is the usual shell model matrix element)where V is the two-body force between valence nucleons, and the second term contains the effect of the coupling of the valence nucleons to the core. The latter term is similar, but not equivalent, to the energy correc- tion dictated by second order perturbation theory. EC is n an energy characteristic of the core excitation in the state ICn>. It can only be approximately fixed in a state independent manner. No attempt has been made at being complete in writing down these formulas as they are discussed in detail in the references cited in Chapter 7. In the language of Kuo and Brown,(Veff is the renormalized G—matrix and there is no distinction between V and V’ which is identified as the "bare" G-matrix. In applying Eq. (D.3) to systems with two valence nucleons Kuo and Brown use an average energy 259 denominator for EC =E where E=—2hw for positive parity n states and —hw for negative parity states. Eq. (D.l) and Eq. (D.3) can be written somewhat more compactly as ~ _ "'1 z ; |An>—[l+(EAn—HO) PV ]|An> (D.l ) _ ,3 2 , _ (D.3 ) where H0 is the unperturbed Hamiltonian and - I P — Cn|Cn>—. (D.5) The first term in Eq. (D.5) is simply the direct action of T on the valence nucleons while the last two terms account for the possibility of the transition being affected through the intermediary of the core. This is analagous to Eq. (D.3) where the shell model matrix element contains the effect of the valence nucleons interacting through their mutual force and the second term allows them to interact through the core. The necessary formulae for the specific models used in this work will now be deveIOped. The macroscopic treat— 260 ment of core polarization will be considered first as the basic results are displayed in a somewhat more revealing form than they are in the microscopic treatment. 2. Macroscopic Treatment of Core Polarization When the macros00pic vibrational model is used to describe the core V’ is given by (D.6) 5; »_Z Z *" V —_ikv(ri)LMaLMYLM and IBn> in Eq. (D.5) are written IAnO> and anO>’ respectively, and the projection operator is -2 . . _Z P‘LCnlCnL,JMJ> in the wave function (D.l’) is (EA "EC ~th)_l times n n Jon’JAn Jon ,th 1/2 =6JJ [1](—1) A (2C ) A J L n A n H x. (D.l ) A little algebra gives the following result for Eq. (D.5) ~ ~ _ 2 2_ 2 -1 —+LM{2th[Q (nwL) J tth/2CL) xf(r)[i—LY* LM<§>J} (D.lO) ..-} 262 where Q=EA —EB and f(r) is either k(r) (Eq. B.l2) or n n h(r)(Eq. C.ll). This is the same as the result of Love and Satchler.15 Further it is easy to show that the second term on the right in Eq. (D3) becomes =Z ME 1(MwL /2cL ) CM n E .—L * “ x (p.11) where it must be remembers that the states ICn> are simply shell model states. Using closure Eq. (D.ll) becomes = n — n (D.l2) E'kv(ri)kv(r Z ——1 E Qth/2CL) J where the self energy terms i=j have been excluded as their effect is assumed to be incorporated into the shell model potential. Eq. (D.lO) and Eq. (D.l2) are the essential relations for the macroscopic treatment of core polarization. Note that the collective model Hamiltonian has only one single phonon state for each value of LM. As was pointed out in Section I of Chapter 7, this does not have any physical significance with respect to the actual core nucleus. The model is used here as a vehicle for parameterizing the core polarization effects. In some calculations/th and CL will characterize a physical core state and in others 263 they define an "effective" core phonon. In discussing the macroscopic vibrational model in Appendix B it was pointed out that only vibrations of quadrupole order or higher fell within the framework of the model. This restriction is ignored here with the note that generalizations required to bring in other vibrations may not preserve the form of kv’ k, and h65 which have been given previously. The inelastic proton—nucleus scattering form factor corresponding to the transition matrix element (D.lO) is ~ (X) LSJ n~LSJ(r)'éso9Lk(r)1§’fokv(r )nF LOL (r )r F (r)= F 2 (D.13) eLz— :fiwL) 2 El : E— (mi)2>>Q2 Q —GfiwL) L L where the superscript n indicates that only a transition between basic shell model configurations is being consid- ered. The sum on IT’ is necessary in general since the form factor may have neutron and proton components even when the initial and final states are simple shell model configurations. For example, think of a transition between states formed from a proton and a neutron in the same orbital. The fact that the subscript TI’ does not appear on kv(r’) amounts to neglecting any differences between neutron and proton wells for the same orbital. From Eq. (B.6) it follows that Eq. (D.13) can be written ~ SJ SJ L (r)= 5M L F (n ){I£ (r)— 580 k(r)} (D.lA) SIT L 6L 26A where 11’“, (r)= “h; , (r'r’)u (r’)u (r’)r’2dr’ (D 15) SIT L g SIT L ’ n’SL’ nSL ' as in Eq. (2.“7’) and _ w I 2 I ’2 I —fokv(r )un,2,(r )un£(r )r dr . (D.l6) In Eq. (D.lU) the contribution due to core polarization appears as a modification of the radial form factor. This modification is scaled by the factor 9 R’A STT’L L which has sign Opposite to that of I 2(P) at large r. Since k(r) is posi- tive it leads to enhancement of the transition. Further in this model the modification only appears in S=O amplitudes. Deforming the spin—orbit term in the optical potential would bring in the possibility of core polarization contributions in'Bpin—flip"amplitudes. Note that the form factor is proportional to M%:{(n) which is the geometrical factor characteristic of the transition from the shell model state lAn> to the state IBn>° As the selection rules for the transition are contained in this factor, it is clear that they have not been effected by these considerations. For normal parity electromagnetic transitions, inelastic electron—nucleus scattering,or y—transitions, FgOL (r) in Eq. (C.5’) or Eq. (0.17) becomes LOL _ 2 LOL p (r)—TT,MTT,(n){6T . I F lun,£,(r)un£(r)—eeLh(r)}. 2 I .1. 2 (D.17) 265 For the (e,e’) reaction the correction factor (C.6) would only be applied to the first term on the right in Eq. (D.l7). For a uniform charge distribution the expression for B(EL), Eq. (C017), 2J +1 3ZC _ B l 2 MLOL B(EL)"2JA+1 2 [TTM (n)(5 n”; l+eLE——c BL )326 2 2 where the subscript C denotes core and L _ w L 2 —[or un,£,(r)un£(r)r dr. (D.l9) These relations are completely analagous to those for the (p,p’) reaction and it is seen that there is a core contribution even when the valence nucleons are neutrons. The transitions are enhanced as e has the same sign as or un,£,(r) Results D.(lu) and D.(l7) can be obtained by restrict- ing consideration entirely to the valence configurations and assuming the interaction between the proton projectile and the ith valence nucleon to have the form V(r—ri)=1/(r ri )— k(r)kv (r. )26 LMe LY YLM(r)YLM(;i) (D'2O) or that the density operator for the ith valence nucleon is (p.21) p=e35(9-;,>_hkvgme eLY YLMYLM<§i> with ei=0 or I as the ith valence nucleon is a neutron or a proton, respectively. Further Eq. (D.l8) can be written 266 2J +1 _ B l 2 LOL L 2 B(EL)-2JA+l 2[TT,1VITT,(n)eTT,] (D.22) where eTT. is the effective charge defined by _ 3 L v eTT;-85 ’ :1. 1+1??? ZCeRC L 6L. (D.23) TT ,‘2' '5 (I’ > The above relations clearly display the renormalization of transition operators due to core polarization. The renormalization is dependent on the valence configuration. This dependence appears in kv or , , and in the BL. From Eq. (D.3) and Eq. (D.l2) it also follows that the renormalized force between the ith and jth valence nucleons is 'V _ _ _ _ _ l 'X‘ A A eff>Q so that BL=l/CL. They consider Love and Satchler core phonon with th transitions in the (p,p’) reaction and BL is fixed from an analysis of corresponding v—transitions. Another method which is used in this work is to determine the 6L from the spectrum. For example, consider a nucleus with two like valence nucleons and assume that these nucleons are restricted to the (3)2 configuration. The low lying states of this +,...(2j-l)+ and their energies nucleus will have J=O+, 2 will be related to the matrix elements <(j)2J]ngfl(J)2J>. From Eq. (D.2U) it follows that <2Jligffl2J>=<2JIVI2J>-2%eLmi (D.25) Mg=<-1)J’2332W2 (D.26) where E—lonwL/CL)=—l/CL=—6 consistent with the assumption L discussed above in regard to the transition matrix elements. Examination of the behavior with L and J of the Racah coef- ficient in Eq. (D.26) shows that the second term in Eq. (D.25) will give a strong attractive contribution to the 268 J=O matrix element and will give a repulsive contribution to matrix elements for higher J provided the a fall off L sufficiently slowly with increasing L. This is the effect 20-25 required to reproduce the observed spectrum which in turn can be used to fix the eL's. Note that in computing the renormalization of the bound state matrix elements by this prescription that no contributions from abnormal parity states of the core are included. This is a direct result of the form assumed for the valence core interaction, Eq. (D.6). In the microscopic 20—25 calculations of Kuo and Brown these contributions are shown to be small and repulsive. Nevertheless, the values of 6L corresponding to normal parity core excitations deter- mined from the spectrum will be somewhat too small because repulsive terms are neglected. In this work this difficulty is circumvented by determining the 6 from the decomposition L of the G3p-lh contributions to the J=O matrix elements 20-25 calculated by Kuo and Brown Deficiencies in their matrix elements should show up as corresponding deficiencies in the results of this work. This procedure can be extended to more complicated cases. The essential criterion for its applicability is that there are no more eL's to determine than there are matrix elements defined by the spectrum. In a more general case the GL'S which are determined may show some configuration dependence which is, of course, expected. The inverse of this process has been used to renormalize bound state matrix elements in the Pb206 calculations by True and Ford.9u 269 3. Microscopic Treatment of Core Polarization In the completely microscopic calculations the quantities to be determined are the proton and neutron transition densities; therefore, interest is in the reduced matrix elements of the operator defined in Eq. (8.21) and Eq. (B.2l’). One Nucleon Outside of a Closed Shell For a nucleus with one nucleon outside of closed shell the unperturbed valence configurations are the single particle states defined in Section 3 of Appendix B. The necessary reduced matrix element corresponding to the first term in Eq. (D.5) is given by Eq. (8.22) and will be called R%§{(D) where D refers to direct. The reduced matrix element corres- ponding to second and third term in Eq. (D.5) will be called Ri§{(c) where C refers to core. In calculating R£§{(C), V’ is the "bare" G-matrix or an approximation to it such as the K—K force and P projects onto 2p—lh states. P=P lh=(l+6 2p- |(JJp)Jv,3 sJ’M’><(JJp)JV,jh;J’M’| 33 h V (D.27) |(JJp)Jva3h3J’M’>= Z (ijmmpIJVMV> p mth J "mh h + + x(—l) a apah|C> (D-28) -“'F 270 For convenience reference to the quantum numbers T,Tp, and T has been suppressed. In Eq. (D.27) only distinct pairs h (ij) are included in the sum, i.e. (1,2) is not different from (2,1) and only even values of Jv are allowed when J: jp. It is not hard to show that P1 +22 (v.29) P2p-1h= 2p—1h 2p—lh where ng—lh includes only the terms in Eq. (D.27) with =' and J Jp 2 _ - . I I - . I 2 P2p_1h j Jpjhlmplhwcu M >— . (D 31) -l —- — LSJ =— EjhE <31IVIJ2(Jth)J;J1> T ‘I p h where E=Ep—Eh+EJ2:Ep_Eh and Tp and Th have been intro- troduced explicitly. In writing Eq. (D.3l) use is made cf Eq. (B.38) which shows that 3 must equal jaor jp and JC must equal J. Since the sum of jjp in Eq. (D.30) includes distinct pairs and J¢jp one is free to choose j=j2 and sum over Jp¢J2. Further, the matrix element of V vanishes unless 31ml=J’M’. In the occupation number representation V is written 26a _l V.2 aBY + + Baaayaé (D.32) where is an unsymmetrized two—body matrix element. Using Eq. (D.32) it can be shown that bound state matrix element in Eq. (D.3l) is given by 272 _ j +3 +J’A AA_ <31IVIJ2Jsjl>=§.<-1> p 2 J’2J511{J JLJLMJLJL.32J J’31j2 p; (D.33) )11/2 where'U(31Jh.323sz’)=[(l+éjlJ )(l+6 h p 32J xV(Jth,J2Jp;J’T=l) (TL=Th;T2=Tp) (D.3U) x%IV(JLJh,Jgjsz’T=0)+v(Jth,szng’T=l)} (T1#Th;T2#Tp) with V(Jljh,jejp;J’T) designating a two—body matrix element between antisymmetrized two—particle states coupled to total angular momentum J’ and iso-spin T. In deriving this result using Eq. (D.32) contractions leading to one—body potential terms in Eq. (D.33) are neglected. For the (p,p’) reaction and electromagnetic transition T When the =1 and T =T P l 2 h' valence nucleon and excited core nucleon are of the same type only the T=l part of the particle-core interaction is effective, whereas both the T=O and T=l parts of this inter- action are effective when these nucleons are different. The product of the matrix element (D.33) and -E(ph)"l is the probability amplitude for the (ph) component in the final state wave function. Combining Eq. (D.33) and Eq. (B.38) gives the following expression for the reduced matrix element corresponding to Eq. (D.31) LSJ Z —1 J2+3 +J’“,2? A-l R ,(c )=- a . E(ph) <-1) p J J IT 2 Jpjh TT ,Tth p l T Th ’ LNG J’) wn-nr if 273 x J Jth l/(JLJL.JZJP;J’>- (D.35> A similar expression can be obtained for the reduced matrix element corresponding to the second term in Eq. (D.5). This L is called RTSJ(C1) and differs from Eq. (D.35) by a phase and the interchange of jl and j2. The sum of the two contri— butions from core polarization to the transition density is I! RLSJ LSJ LSJ _ 2' -l . . LSJ ,= R (c >+R (c2) —, pjh E(ph) A(JlJ2phJ) Tp’l‘h . _Z J’+Jp+'jl “2“ A —l A+<- 1>S*J jpth °U(JpJ2,Jth;J’)l (3231 hJ i JLJ2J’ ..J where the double prime on the sum over 3 pjh indicates that the first term in [J is not included when 3 pr=le l and the second term in [J is omitted when jpr=j2T 2. It was pointed out previously that there is no breaking of the valence transition selection rules when the macro— scopic treatment of core polarization is used. Consider the transition where the valence nucleon goes from an s orbit 1/2 to the d orbit. Without core polarization this transition 5/2 can only go with L—transfer equal to 2. From Eq. (D.36) it can be seen that L=M is also allowed, i.e. assume |C> con— tains a filled p—shell and consider a f -1 particle- 7/2‘p3/2 27M hole pair which gives a contribution for L=U. This point is probably academic as the L=U contribution to the transi- tion is not likely to be as important as that from L=2, but it does indicate that core polarization can effect the valence transition selection rules. Two Nucleons Outside of a Closed Shell For two nucleons outside of a closed shell the only transitions considered are between states where the valence configurations are the allowed couplings of two like nucleons in the same orbit. The wave functions for these configurations were defined in Eq. (B.35) and Eq. (8.36) LSJ gives RTT,(D). For this case _ 1 _ I . . 7 , , - , , P-P3p_lh-Jgjhfl(JJ)Jv(Jth)JC]J M > JVJC J’M’ where reference to T,Tp, and Th is again suppressed and the sum excludes terms with j=jp and odd values of Jv' The state vectors appearing in the projection operator were defined in Eq. (8.39). Using the notation of the last section I LSJ _ —1 2 . . — RTT,(C2)-—jgjhE T T p h LSJ TT’ X<[(JJ)JA(Jp3h)J]JBIIO ||(J)2JA> (B.38) where Eq. (B.40) has been used along with the properties of V to eliminate some of the summation. 275 The matrix element of V appearing in Eq. (D.38) is <(J)2JBIVI[(JJ)JA(Jp3h)J]JB>= E J+J—Jh-J A “U2 - a , 2J,(-l) J JJ . (D.uo) Combining this result with the corresponding result for RLSJ(C1) gives the following result. RLSJ Z LSJ . (C)=- . E(ph) lA R11 Jth p TpTh (D.ul) J+j-j -J’A A l A P H»? JAJhtl+<—1>S+JJ§J J p’JHJJ 3 i \thJ JJBJAJ X )/r(jhvj :quj 3J’) 276 As in the valence transition L must be even and when J=L and S=l A(jphJ) vanishes; therefore, spin-flip is still forbidden for a transition starting from the state J =0. A The inclusion of core polarization does not lead to any breaking of the valence transition selection rules in these transitions. When J =0, L=J=J and 8:0, A(JphJ) becomes A A B’ lljh 2 j +3+JZ : — p ’2 . 7 o . . I A(JphJ) Ff; JA 1) J' §JthJ {U(3p333hJ,J > (D.u2> 13 J J’j Phase of Microscopic Core Polarization Contributions The formula obtained for treating core polarization using the macroscopic vibrational model to describe the core clearly displayed the relative phase of the core and direct contributions to transitions. The phase of RE§{(C) defined in Eq. (D.36) and Eq. (D.Hl) with respect to the corresponding RE:{(D) is not apparent from inspection. It is useful to examine this phase relation. To do this it is necessary to express the two—body matrix element in terms of multipoles of the two body force. inJhJ’} l J231J h L’s’J’ p31 J2 , l , S,A A J+j +3 ‘;(le,hJ2;J>=L,§AJA2£0J>J * Jle(-l) h l L’S’sz Tth xM 277 lehJ2 IS’L’Q =jup(ro)u1(rl)VSALAQ(rO3r1)u2(rl)uh(rO )r2r2dr0dr O l l (D.NM) +fup (rO )ul (r “)LfsA L AQ(rO;rl )u2 (rO )uh (r l)roridrodrl 2 l h p 2 { }£"2I:"' (rO 3r)=— IV S “L Q L",L"' XW(£2£ R R L"L )WUL2 £ 2 L" H l p l h’ p l h3L L )VE (rO grl) (D.45) S ’LHQ This expression for the two-body matrix element is obtained by following the procedure used in Chapter 2 for decomposing the D.W.A. transition amplitude Q designates the p—p(n—n) or p—n force as T T is the same or opposite to T h p 2T1 VEA AL AQ(rO;rl ) is the exchange interaction defined by Eq. (2.23), L; and MTTS J(jj ) are the geometrical factors for the single particle transition density defined according to Eq. (8.6) and Eq. (8.6’). The second integral in Eq. (D.MU) is the exchange integral which is expected to be in phase with the direct integral for a short range even state force since N'EALAQ(rO;r)=VSALAQ(rO;rl) for a zero range even state force. Using Eq. (D. U3) in the definition 2 J +J.p’UE" . - . . . A F (D.U6) leads to 278 +1A LSJ 31M T2T1 J1 F(r;jlj2JpJfiLSJ)=(—l) (J2J1)up(r)uh(r) xLZSA<-1)S+SJ and . A _ J J'1“ LSJ . F(r,J2JlJthLSJ)-(-l) {-1) MT2T1(JZJl)up(r)uh(r) s . A A . xLZSA<-1> D(JlJ2Jth,J,L s ,LS,Q> (D.u8) -f 4 where A A hj ’ 2 . . . , A A , _ —3/2.2 -2 p31 2 L s J . LSJ xM (J J )M Tth h p L’s’J . LSJ (J2Jl)/MT T T211 2 l(J2J1) (D.ug) With these relations Eq. (D.36) can be written LSJ LSJ . Z —1 R I =-M . ; TT (C) T2Tl(J2Jl)Jth6TT ’TpThE(ph) up(r)uh(r) TpTh 8+8 ’ o o . I I . xLZSA£<—1> +JJD. (p.50) The sum on S’ can be removed as only the term S’=S gives a nonvanishing contribution. This gives LSJ LSJ . . Z -1 R ’(C)=-M (J J )2 6 2 E(ph) u (r)u (r) TT T211 2 l Jth TT ,TpTh p h TpTh xEAD. (v.51) 279 A similar result is obtained for the R%§{(C) obtained by using Eq. (D.U2) in Eq. (D.Ul). This is JOJ 2 JOJ .. Z -1 R A(C)=—AM (JJ)2 . 5 A E(ph) u (r) (r) TT J TT ijh TT,TpT p 9h Tp'th xD(jijjh,J;JO,JO;Q). (D.52) There is no sum on L’ in Eq. (D.52) as the transition being considered here is of normal parity and has only one allowed value of J. Similarly in Eq. (D.51) only L’=J contributes to the triad (LSJ)=(JOJ). Eq. (D.51) and Eq. (D.52) have the form needed to see the effect of core polarization, as treated in this micro— scopic picture, on transitions. In both equations the negative of the geometrical factor for the valence transition appears as an overall multiplicative factor. This does not mean that violations of the valence transition selection rules are not possible since this geometrical factor also appears in the denominator of D. Only triads allowed in the valence transitions will be considered here. To see the phase it is only necessary to consider particle—hole pairs whose radial wave functions are similar to those of the active valence nucleon. The largest values lehJ2 S’L’Q integral will have the sign of the S'Q component of the two of I will occur in these instances and this radial body force. Inspection of Eq. (D.M9) shows that 280 path2. SLQ ’ for the triads (JOJ) and (JlJ) the direct and core polariza— D(3132jpjh,J;LS,LS;Q) has the same sign as I therefore, tion contributions will be in phase if the corresponding component of the two—body force is attractive. Only the "spin-flip" componentof the p—p(n—n) force used in this work is repulsive. Because of this when the valence nucleons are protons (neutrons), proton-proton hole (neutron-neutron hole) excitationswill decrease the (JlJ) transition amplitude. The same arguments hold for the triads (J:l,l,J) although there is an additional complication because the phase depends on the sum of two terms. When D(31323pth;LS,LS;Q) is dominant the conclusions above will hold. This is likely to be true for the (J—l,l,J) triad as D(LS,LS) will by proportional to the L=J—l multipole coefficient of the two— body force while D(ES,LS) will be proportional to the L=J+l multipole coeffecient. For the triad (J+l,l,J) this situation is reversed. u. Microscopic Empirical Formula Here a formula for computing, from the effective charges, the enhancement of a cross section in the (p,p’) reaction due to core polarization,is derived on the basis of micro- scopic considerations alone. The argument is orginally due 19 to Atkinson and Madsen and is given here in the notation of this paper. For a normal parity transition with some degree of collectivity the triad (LOL) gives the dominant contribution 281 to the cross section. The form factor designated by this triad is MLOL( )_ I: ”(r r )FLOL(r ’)+’y LOLCr ’)}r’2dr On L(r; r )F (D.53) as specified in Eq. (2.59"). FiOL and FEOL are the proton and neutron transition densities, respectively, and’V‘OpL and VOn L are the multipole coefficients of the non-"spin- flip"components of the p-p and p—n forces with the exchange interaction included. Correspondingly for y-decay Eq.(C.l7) gives 2JB+1 l w L+2 LOL —|f 1" _ 2 2 B(EL)-§-j;fi 2 F (1")dI’I e . (D.5U) P The neutron and proton transition densities have two components, T LOL(r)_ _D LOL LOL is) Fa>‘””'o where D is the direct or valence component and C is the (r) (D.55) core component. Two assumptions make it possible to relate, algebra— ically, the effective charges of the valence nucleons to analagous enhancement factors for the (p,p’) reaction. One is to neglect radial differences between the proton and neutron transition densities and their direct and core components. The second is to assume that different com- ponents of the projectile—target interaction have the same 282 radial form or that "equivalent" components with the same radial form can be defined. The IF range "equivalent" impulse approximation pseudo—potential given in Chapter 3 can be used in this context. The local approximation to the exchange component of the D.W.A. transition amplitude is an implicit uncertainty in the second assumption. The total proton transition density can be written -1 l _ Fp(T)-§{Fp(T)+Fn(T)}+2{Fp(T) Fn(T)} (0.56) where Fn(T) has been introduced so that Fp(T) is expressed in terms of iso—scalar and iso—vector components. An iso- scalar transition is defined by the condition Fp(T)=Fn(T). In terms of the iso—scalar and iso—vector effective charges, F (T):Fn(T) e = + (D.57) {E FpCD)-Fn(D) Eq. (D. 56) becomes Fp(T)=%eO{Fp(D)+Fn(D)}+%el{Fp(D)—Fn(b)} =e Fp(D)+enFn(D) (D.58) p where the proton and neutron effective charges are ??=%(eoiel) (D.59> 1’1 Correspondingly, 283 F~15Fp(T)+'VnFn(T)=tbO{Fp(T)+Fn(T)}+16l{Fp(T)-Fn(T)} (D.60) where the proton-proton and neutron—proton forces have been decomposed into these iso-spin components as prescribed in Table 2 of Chapter 2. The proportionality sign in this equation refers to the integration implied in Eq. (D.53). In terms of the effective charges Eq. (D.60) becomes ~ F~lbOeO{Fp(D)+Fn(D)}+}blel{Fp(D)—Fn(D)} ~{Hboeo+rblel}Fp(D)+{}’OO O—}61el}Fn(D). (D.61) Finally it is concluded that E(T)=ep§p(o)+enin(o) (D.62) where 31 e :11 e e =,80 27,01 1 (D.63) {a ’00 01 For the (n,n’) reaction Eq. (D.62) is still valid but the signs must be reversed in Eq. (D.63). These relations will be used in discussing the results of Chapter 7. "'Tliti‘fllliuijflllfivMailfifljiflimjfijmffs