I'll-IIIIIII"{ I I I I I I I f .b. Ir '4 If!!! Cl. ‘I .' ll. 1'! I. ‘i‘Iil‘ II‘AI .ly \ l92,l94,l96,l98 A STUDY OF Pt USING THE (p,t) AND (p,p') RWWTIONS AND THE INTERACTING BOSON APPROXIMATION MODEL by Paul Thomas Deason, Jr. A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1979 ABSTRACT A STUDY OF 192'194'196'198Pt USING THE (p,t) AND (PIP') REACTIONS AND THE INTERACTING BOSON APPROXIMATION MODEL by Paul Thomas Deason, Jr. The low—lying collective states of the 192'194’196'198Pt nuclei have been studied with the (p,t) and (p,p') reactions at Ep = 35 MeV. Approximately 45 levels were populated below 3 MeV for each nucleus in both reactions, with many levels seen for the first time. Angular distributions have been obtained in each reaction using a delay-line proportional counter in the focal plane of an Enge split-pole spectrograph and accurate energies were obtained from high-resolution, nuclear emulsion plate data. Distorted—wave Born approxima- tion (DWBA) calculations were performed for transitions in the l94'196'198Pt(p,t)192’194'196Pt reactions, and were used, in addition to empirical angular distribution shapes, to make J" assignments. No new levels were seen below 1.5 MeV excitation. .A new state with J" = 0+ at 1.628 MeV was found in 192Pt and new levels tentatively assigned J" = 4+ were seen in all three final nuclei near 1.9 MeV. In 196,198 Pt(P:t), these 4+ levels are populated with 15% of the ground state strength at 7°. Emhancement factors, 6, were calculated for simple two-neutron pickup configurations. A comparison is made between experimental (p,t) strengths and those calculated in the 0(6) limit of the interacting boson approximation (IBA) model for L = 0 and L = 2 transitions. The calculations included a small, quadrupole-quadrupole symmetry breaking term and allowed for both neutron and proton bosons. The calculations are generally in good agreement with the data, particularly the prediction that the second excited 0+ level should be more strongly populated than the first excited 0+ level. In the inelastic proton scattering studies, high 194'196'198Pt. 2n} 198Pt resolution data are presented for thirty-eight of the forty-four levels seen are reported for the first time. Assignments of J1T were made in each reaction by comparisons of angular distributions to those for states with well-known spin and parity. Coupled channels calcula— tions have been performed for each reaction using a deformed Optical model potential for the radial form factors, with relative matrix elements obtained from the IBA model near the 0(5) limit. Good fits to the data are obtained for the ground band (up to spin 4) and second 2+ level. Best fits are 194'196Pt with a negative value for the Obtained in interference term p3 (= M02M22,M02.), in agreement with recent (apa') studies of 194Pt and with the predictions of the 0(6) limit of the IBA model. To explain the strength of the second 4+ state in all three reactions a large value for M04, is needed, indicating a: strong, direct E4 transition competes with the various multi-step paths. The quadrupole (E2) and hexadecapole (E4) potential moments have been calculated from the parameters of the deformed optical model potential used in the inelastic proton scattering. The moments are in better agreement with the charge moments from Coulomb excitation in each case than those from the potential moments calculated from alpha scattering. This may indicate that inelastic proton scattering is a more reliable method for extracting potential moments than a-scattering due to the more complex intrinsic structure of the alpha particle. To my family: Kimberly and Debbie, Ed, Pam, Nancy, Mom, and Dad ii ACKNOWLEDGMENTS I am most thankful to Dr. Fred Bernthal, who served as my research advisor, for his friendship, advice, and genuine interest in this project, myself, and my future. His guidance and advice will always be welcome. I am also pleased to acknowledge the teaching and guidance of Drs. Chuck King and Reg Ronningen, who provided the day-to-day interactions and discussions which have contributed so much to my graduate school education. I consider myself very fortunate to have worked with such authorities in this area of nuclear science. Many thanks are due Drs. Teng Lek Khoo and Jerry Nolen, for their many hours of help in taking the data for this dissertation and sharing their expertise on the experimental electronics and the operation of the MSU cyclotron. I would like to thank Dr. Henry Blosser and the staff of the MSU Cyclotron Laboratory for providing and maintaining a truly unique experimental facility and research environment. I am most grateful to Richard Au and the computer staff for keeping the Sigma—7 alive and well and for those extra hours of computing time on maintenance days. Financial support in the form of a Quill Fellowship from the MSU Chemistry Department and the Eastman Kodak Company iii and numerous research assistantships funded by the Energy Research and Development Administration and the National Science Foundation is gratefully acknowledged. Thanks to the Nuclear Beer Group and "Athletic Club" for those memorable Friday afternoons and unforgettable night games - but mostly for your friendship and good times during the last five years. I would also like to thank Norval Mercer for his special kindness to myself and family and for teaching an ACC fan the merits of the Big 10. A special thanks is due Earvin and Greg for making the Winters of '78 and ‘79 more than just long and cold. Many thanks are in order for Mrs. Carol Cole for her e f forts in the typing of this thesis. Her skills are evident th roughout the entire text. Her willingness to complete this t ask before her "deadline" as well as mine is appreciated. I especially thank my parents, my lifelong teachers, for their encouragement, advice, and love. Most of all, I wish to thank my wife, Debbie, for her “‘3 hy sacrifices, for her understanding of my hours away from home, and especially for her love and unselfishness for Kim a “d myself . iv TABLE OF CONTENTS LIST OF TABLES JLIST OF FIGURES Chapter II. INTRODUCTION . . . . . . . . . . . . . . . . A. Orientation . . . . . . . . . . . . . . B. Motivation . .'. . . . . . . . . . . . . C. Organization . . . . . . . . . . . . . . .Ifjt - THEORETICAL BACKGROUND - NUCLEAR MODELS . . A. Introduction . . . . . . . . . . . . . . B. The Vibrational Model . . . . . . . . . C. The Symmetric Rotor Model . . . . . . . D. The Asymmetric Rotor Model . . . . . . . E. The Interacting Boson Approximation Model . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . 2. The SU(S) Limit - Vibrational Nuclei 3. The SU(3) Limit - Rotational Nuclei 4. The 0(6) Limit - Transitional Nuclei IJIII. THEORETICAL BACKGROUND - NUCLEAR REACTIONS . A. The Distorted—Wave Born (DWBA) For Transfer Reactions . . . . . 1. Introduction . . 2. The Transition Amplitude, Ta b I 3. The Nuclear Form Factor Approximation Page 10 12 12 l4 16 22 24 24 30 40 44 53 53 53 55 65 '- Chapter II‘I. B. The Coupled Channels (CC) Method For Inelastic Scattering . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . 2. The Coupled Equations . . . . . . . . 3. The Effective Interaction: The Optical Model Potential . . . . . . . 4. The CC Form Factor . . . . . . . . . 5. Solving the Coupled Equations . . . . C. Charge, Mass, and Potential Moments . . . EXPERIMENTAL PROCEDURES AND DATA ANALYSIS . . A. Introduction . . . . . . . . . . . . . . . B. Proton Beam and Transport System . . . . C. Dispersion Matching - . . . . . . . . . . D. Particle Detection - . . . . . . . . . . . E. Targets . . . . . . . . . . . . . . . . . F. Data Analysis . . . . . . . . . . . . . . RESULTS FOR THE 194:196.l98 Pt(p,t) REACTIONS A. General Analysis . . . . . . . . . . . . . B. DWBA Analysis . . . . . . . . . . . . . . C. L = 0 Transitions . . . . . . . . . . . . D. L = 2 Transitions . . . . . . . . . . . . E. L = 3 Transitions . . . . . . . . . . . . F. L = 4 Transitions . . . . . . . . . . . . G. L 5 Transitions . . . . . . . . . . . . Iv H. Relative Reaction Strengths . . . . . . . vi Page 70 70 74 77 80 82 85 88 88 89 95 96 107 113 119 119 133 137 141 145 146 150 151 Chapter \fI. RESULTS FOR THE A. Page General Discussion of (p,t) Results . . . 155 1. L 0 Transitions . . . . . . . . . . 155 N I.“ ll 4 Transitions . . . . . . . . . . 157 3. L = O, 2 Transitions in the Inter- acting Boson Approximation . . . . . . 160 194'196'198Pt(p,p') REACTIONS 167 General Analysis . . . . . . . . . . . . . 167 1. Elastic Scattering and L = O Transitions . . . . . . . . . . . . . 170 2. L = 2 Transitions . . . . . . . . . . 173 3. L = 3 Transitions . . . . . . . . . . 174 4. L = 4 Transitions . . . . . . . . . . 177 5. Transitions With L 3 5 . . . . . . . . 179 Coupled Channels Analysis of the Inelastic Scattering Data . . . . . . . . . . . . . 185 1. Introduction . . . . . . . . . . . . . 185 2. Optical Model Analysis . . . . . . . . 186 3. The Extraction of Deformation Parameters . . . . . . . . . . . . . . 195 4. Sensitivity of CC Calculations to Higher Order Couplings and Selected Matrix Elements . . . . . . . . . . . 214 a. The Sign of B4 . . . . . . . . . . 214 b. Investigation of a 86 Deformation 217 c. The Effect of y-Band Couplings . . 222 d. The effect of the 2+. State on the BA'S O O I O O O O O O O I O I O O 227 e. The Sign of P3 . . . . . . . . . . 229 vii Chapter Page f. The M E4 Matrix Element . . . . 233 04' 9. Comparison of Charge and Nuclear Potential Moments . . . . . . . . 239 VII. CONCLUDING REMARKS . . . . . . . . . . . . . . 247 LIST OF REFERENCES . . . . . . . . . . . . . . . . . . 253 viii Table II-l 1:1-2 I V—l V~1 V~2 VI~1 VI-2 LIST OF TABLES The Relationship Between the Coefficients of Equation (II—20 and (II—23) . . . . . . . . . . The Relationship between the Coefficients of Equation (II—26) and the CL Coefficients of Equation (II—25). The factors for Equation (II—26) were obtained from Ar 76 . . The Relationships Between the Coefficients of Equation (II-40, (0(6) Limit) and Those of Equations (II-20, 39) (Full IBA Hamiltonian) . . Target Composition . . . . . . . . . . . . . . States Populated in the 194Pt(p,t)192Pt Reaction S ates Populated in the 196Pt(p,t)194Pt and 4Pt(p,p') Reactions . . . . . . . . . . . . . igates Populated in the 198Pt(p,t)196Pt and 6Pt(p, p' Reactions . . . . . . . . . . . . . Optical Mbdel Parameters Used in DWBA (p,t) calculations 0 I O O O O O O O O O O O O O O I O Inte rated Cross Sections for Transitions in 194' 96'198Pt(p,t) and Cross Sfftion Ratios Relative to the Ground State of Calcu- lations in the 0(6) limit of the IBA model for the 1193 0 and L = 2 transfizzions, normalized to the Pt g.s. and the 21' transitions, respectively, are also shown . - - - - - - - - - Relationship Between the “d and Quantum Numbers for States in the SU(5) Limit (Harmonic Vibrator) and 0(6) Limit of the IBA . . . . . . . . . . . States Populated in the 198Pt(p,p') Reaction . . Optical Model Parameters Used in ELC.I.S. calcu1ations O O O O O O O O O I O O O O O O O I ix Page 29 34 46 112 120 123 128 134 152 163 168 188 Table Page VI-3 Relative Matrix Elements, , Used for Initial Optical Model Searches , , , , . . . 193 VI-4 IBA Parameters Used in the Perturbed 0(6) Cal- culations. A, B, and C are the coefficients of Equation (II—40) and K is the strength of the quadrupole-quadrupole interaction. N is the total number of bosons for each nucleus . . . . 197 VI-5 Relative E}. IBA Matrix Elements for 194Pt Cal- culated Using 0(6) Symmetry and K‘= 0.0375 keV 198 VI—6 Relative E). IBA Matrix Elements for 194Pt Ca1- culated Using 0(6) Symmetry and K’= 0.5375 keV 200 VII-7 Relative EA.IBA Matrix Elements for 196Pt Ca1- culated Using 0(6) Symmetry and K'= 0.025 keV 202 VI—8 Relative EA IBA Matrix Elements for 198Pt Ca1— culated Using 0(6) Symmetry and Kr: 0.016 keV 203 ‘7712—9 Deformation Parameters and Potential Moments Obtained from 0-2-4-6 Coupled Channel Calcula- tions for 194'195’198Pt. Values from calcula- tions with and without a spin-orbit (L*S) inter- action are included . . . . . . . . . . . . . . 211 VI—lo Comparison of Initial and Final Values for B). and x2 Between a o+—2+—4+—6+ and 0*;2+—4+-2+' Calculation, Both With L*S = 0 and K = 0.5375 keV . . . . . . . . . . . . . . . . . 228 ‘7'3E._11 Summary of Values for + ' Matrix Elements and B(E4) 153‘19 Pt . . . . . . . . . . . . . . 24o 192,196,198Pt ‘7.3E-—12 E2 and E4 Moments in . . . . . . . 242 Figure I-l 131-1 JE JE—3 I 32-4 311-6 LIST OF FIGURES Systematics of Experimentally Determined Energy ve r Positive Parit States in 134.136.158Pt y A 600 Sector of the B-Y Plane. The point P represents an asymmetric shape with magnitude 8 and an asymmetry parameter y. . . . . . . . . . Schematic Plots of V(B,y) for (a) Spherical, (b) Axially Symmetric Prolate, and (c) Asymmetric Shapes. Also shown are typical energy level spectra associated with these potentials. Taken from Reference [Pr 75] . . . Energy Levels in the IBA With No Boson Inter- actions. This spectrum is identical to that for a harmonic vibrator. The energies are cal- culated from Equation (II-25) . . . . . . . . . Sample Energy Level Spectrum Calculated With the SU(5) Limit of the IBA With Interboson Forces. The states are labeled by the quantum numbers J" (nd, V, n ). The energies are calculated from Equation II—29). Taken from Reference [Ar 76] The Effect of Finite Dimensionality on Relative B(E2) Values in the Ground Band of an SU(5) Nucleus. These values are plotted as a function of L' and nd for an N = 6 and N = 12 nucleus. Taken from Reference [Ar 76] . . . . . . . . . Sample Energy Level Spectrum Calculated With the SU(3) Limit of the IBA. The states are separated into bands denoted by the quantum nutmbers (1,11) and further identified by K and J . The energies are calculated from Equation (II-34). Taken from Reference [Ia 78] xi Page 18 20 32 37 39 42 Figure II-7 III-l III-2 IV-l IIfl—Z IVe3 71‘7—4 IV-S Sample Energy Level Spectrum Calculated With the 0(6) Limit of the IBA. The states are denoted by the quantum numbers (0, ‘l' , VA, J). The energies are given by Equation (II-42). Taken from Reference [Ar 78a] . . . . . . . . . . . . Coordinate System Used for Pickup Reactions , Denoted by A(a,b)B, Where b = a + x and A = x + B O O O O O O O O O O O O O I O O O O 0 Various Coupling Routes in the Coupled Channels Method of Inelastic Scattering. The "up" transition of a, b, and c can be calculated in the DWBA . . . . . . . . . . . . . . . . . . . Experimental Vault Area at the MSU Cyclotron Laboratory. The experiments in this study were conducted in Vault 3 with the Enge spectrograph and associated beam optics . . . . . . . . . . Schematic Drawing of Scattering Chamber and Enge Split—Pole Spectrograph. The detectors and plates are placed inside the camera box in the focal plane of the spectrograph . . . . . . . . Triton S ectra. Recorded at 70 for the 194'196'1 Pt(p,t)192'194'195pt Reactions. The data were obtained with nuclear emulsion plates in the focal plane of an Enge split-pole spectrograph. Peaks with an "*" above them indicate peak height has been cut off at the maximum value on the vertical axis . . . . . . InelastiE Proton Spectra for the 194'196' 98Pt(p,p') Reactions at E = 35 MeV; The data were obtained with nuclefi emulsion plates in the focal plane of an Enge split-pole spectrograph. The elastic scattering peaks are not shown because they were too intense to scan. Peaks with an "*" above them indicate peak height has been cut off at the maximum value on the vertical axis or were unscannable . . . . . Schematic Cross Section of Delay-Line Counter. The labeled parts are: (A) window frames, (B) anode support, (C) separator foil frame, (D) anode wires, (E)IAE anode wire, (F) pickup board, (G) frame for the delay line and board, and (H) delay- line. Taken from Reference [Ma 75] . . . . . . . . . . . . . . . . . . . . xii Page 48 61 73 91 93 98 100 103 Figure IV-6 IV-7 IV-8 IV-9 IIV‘lO V-—2 V-~3 VI—l VI-Z Schematic Top View of Delay-Line Counter. Taken from Reference [Ma 75] . . . . . . . . . . . . Schematic Diagram of Electronics for Wire Counter Experiments . . . . . . . . . . . . . . Triton Spectra for the 194’196’198Pt(p,t) Reactions at Ep = 35 MeV. The data were obtained with a delay-line proportional wire counter in the focal plane of an Enge split-pole spectrograph . . . . . . . . . . . . . . . . . Sample Spectra) for the 194’196’198Pt(p,p') Reactions at 85 . The data were obtained with a delay-line proportional wire counter in the focal plane of an Enge split-pole spectrograph Comparison of Wing Band and Data Band Spectra Taken With the Delay-Line Proportional Wire Counter. The ratio of counts in the wing band to counts in the data band is 1:10 . . . . . . L = 0 Angular Distributions for the 194'196'198Pt(p,t) Reactions. The curves are the results of DWBA calculations. Energies are given in keV . . . . . . . . . . . . . . . . . L = 2 and L = 3 Angular Distributions for the 194'196'198Pt(p,t) Reactions. The curves are the results of DWBA calculations. Energies are given in keV . . . . . . . . . . . . . . . . . L > 4 Angular Distributions for the 192h'J-95'198Pt(p,t) Reactions. The curves are the results of DWBA calculations. Energies are given in keV . . . . . . . . . . . . . . . . . Elastic Scattering and L 0, 2 .An ular Ibis- tributions Seen in the 194,195: 98Pt(p,p') Reactions. The curves are the results of DWBA calculations using a collective model form factor. Energies are given in keV . . . . . . $93,196?19QDQUIar Distr1butions Seen in the Pt(p,p') Reactions. Energies are given in keV. . . . . . . . . . . . . . . . . . xiii Page 104 106 109 111 118 139 144 148 172 176 VI-3 VI-4 VI-S VI-6 \II I \l \II-8 VI—9 VI~10 Figure ular Distributions Seen. in the 19% 196 1n9 Pt(p, p' ) Reactions. Also shown in the third and fourth column are angular dis- tributions with a unique but unidentifiable shape. Energies are given in keV . . . . . . . Angular Distributions Seen in the 19 195 198Pt(p, p' ) Reactions With Unknown L Transfer. Energies are given in keV . . . . Coupling Schemes Used in the Coupled Channels calcu1ations I O O I O O O O O O O I I I O O 0 Data and Coupled Channels Calculations for 94Pt(p,p') With and Without the Spin-Orbit Interaction. The calculations included the couplings shown in Figure VI-5b, the matrix elements of Table VI-5, and DOMP parameters from Tables VI—2 and 9 . . . . . . . . . . . . . . . Data and Coupled Channels Calculations for l95pt(p,p') With and Without the Spin-Orbit Interaction. The calculations included the couplings shown in Figure VI-Sb, the matrix elements of Table VI—7, and DOMP parameters from Tables VI-2 and 9 . . . . . . . . . . . . . . . Data and Coupled Channels Calculations for 198Pt(p,p') With and Without the Spin-Orbit Interaction. The calculations included the couplings shown in Figure VI-5b, the matrix elements of Table VI-8, and DOMP parameters from Tables VI—2 and 9 . . . . . . . . . . . . . . . Data and Coupled Channels Calculations in a 0-2-4-6 Space for 194Pt(p,p') With Positive, Negative, and Zero Values for 84. These calculations used the IBA matrix elements of Table VI-5 and the DOMP parameters of Table VI—2. No spin-orbit interaction was included. 82 = -0.l72 . . . . . . . . . . . . Comparison of lgita and Coupled Channels Ca1- culations for Pt(p, p' ) With the Couplings of Figure VI-5C, and Three Values for 85 The calculations included the matrix elements of Table VI—5, and the DOMP parameters of Table VI—2 with no spin-orbit interaction. 82 = —0.l72, B4 = —0.0567 . . . . . . . . . . . xiv Page 181 184 191 206 208 210 216 219 Figure Page VI-ll Data and Coupled Channels Calculations for 94 Pt(p, p') With Search on 82, B4, and With 86 = 0 (Dashed Curve) and for Search on 82, B4, and 86 (Solid Curve). No spin- -orbit interaction was included. Both calculations had the same initial values for 82 (= —0. 172) and 84 (—0.0567). The matrix elements were taken from Table VI—5 and DOMP parameters from Table VI— 2 221 VI-12 a and 0- 2— 4— 2'-3— 4' Calculations for 19 Pt(p, p' ) With the Spin-Orbit Interaction and Two Sets of IBA. Matrix Elements, Table VI-6 (Solid Curve), and Table VI-5 (Dashed Curve). The couplings included those of Figure VI-5e. The appropriate DOMP parameters were taken from Tables VI—2 and 9 . . . . . . . . . . . . . . . 224 VI-l3 Data and Coupled Channels Calculations for 194Pt(p, p') With Positive (Dashed Curve) and Negative (Solid Curve) Values for the Inter- ference Term, P3. Note that the data and theory for the 2+ state have been multiplied by 2. The couplings of Figure VI-Sd were used with 82 = —0.151 and B4 = -0.0453 . . . . . . . . . 232 VI-14 Data and Coupled Channels Calculations for 94pt(p,p') With K = 0.0375 and 0.5375 keV. The dashed curve represents calculations with the full set of IBA matrix elements, while the solid curve represents calculations with the best fit value for M04. included. The calculations used the couplings of Figure VI—5e and the DOMP parameters of Tables VI—2 and 9 . . . . . . . . 236 VI-15 Data and L*S Coupled Channels Calculations for 195'198 Pt(p, p' ). The dashed curve represents a calculation with the full set of IBA matrix elements, while the solid curve represents a calculation with the best fit value for M04, included. The calculations used the couplings of Figure VI— 5e (except 198Pt(p, ;>') did not include 3+ couplings) and the DOMP parameters of Table VI- -2 and 9 . . . . . . . . . . . . . . . 238 VI-16 Plot of Quadrupole and Hexadecapole Moments for 194'1 5' 8Pt Given in Table VI—12 . . . . . . 244 XV CHAPTER I INTRODUCTION A. Orientation The Pt isotopes lie in a shape transitional region, between well-deformed rare-earth nuclei and the spherical nuclei near doubly magic 208Pb. Since the Pt nuclei are not well described by simple model limits for collective nuclear motion (e.g. the symmetric rotor or the harmonic vibrator), they provide a valuable testing ground for more current models. Figure I—l shows the low-lying energy levels for three of the four Pt nuclides studied in this work. The first item one notes is the similarity of the levels from 192Pt to 196Pt. Most of the levels decrease or increase in energy very gradually in these nuclei, part of the evidence for the slowly changing shape characteristics in this region. Within a particular nucleus the most obvious features are the nearly equal spacing between the first three levels, and the low- lying second 2+ state. Understanding these relatively simply systematics, along with the many electromagnetic branching ratios, has provided the framework for comparing the predictions of several nuclear models. 1 u ad mmumum a mmwemmaémm . 3.3m mZDHmom no“ mHmqu >mumcm cmcfiELmumo maamucme no xm mo Owuméoumwm .HiH meow?“ 11mm_ o Ilye wmm I+N mow Ilwm mum Ills...— 92 Info. mm: II... row" [7... omrq low em:m_ O [+9 mNm I+N NNw |+N 2m Ill..r rmNm ll+m OMN— llu+r kmwa l|l+o N2: Ill+m Samar O [+9 mum II+N Nam I1|+N ka |+r J—Nm l+m 8N— IIL. mm: '3 mwm~ |l+m com 83 .HIH musmflm [A91] ASHBNB NOI1V113X3 It has been known for several years that a transition from prolate (negative quadrupole moment) to oblate (positive quadrupole moment) shapes occurs among the heavier Os and the lighter Pt nuclides [G1 68, G1 69, Pr 70]. For the heavier Pt nuclides (192.198 Pt) the quadrupole deformation parameter, 8, has a value [Ba 76, Ba 78] of approximately 0.15, or about one-half the value determined for the well-deformed rare earth nuclei. Consequently, these Pt isotopes exhibit few rotational features. Some aspects of the lowest energy levels of these nuclides, such as the nearly equal level spacings and a small 2: + 0: transition, can be interpreted in terms of a harmonic vibrator. However, a notable problem with this picture is the lack of a candidate for the 0+ member of the 2—phonon triplet. Moreover, the platinum isotopes are farther away from closed shells than those nuclei for which vibrational models have been applied most successfully. Because of the difficulty with vibrational models, various other collective models have been explored, such as the Y-unstable [W1 56] or the asymmetric [Da 58] rotor models. These models predict only one 0+ state (the ground state) and allow the second 2+ state (2:) to be degenerate with or below the energy of the first 4+. Figure I—l shows that the Pt nuclides display both these features. The asymmetric rotor model [Me 75, To 76, Do 77, Pa 77, Sa 77] has had considerable success in describing the odd—A nuclei in the Pt region by coupling the odd particle to a triaxial core. However, the predictions for this approach have been shown to be experimentally indistinguishable thus far from those obtained by a variety of other particle core couplings [Do 77, Pa 77], including that for a y—unstable core. Several attempts [Ku 68, Gn 71, Le 76] have been made to treat this region by solving the full Bohr-Hamiltonian, beginning with the work of Kumar and Baranger [Ku 68] in which the parameters of the Hamiltonian were determined by using the pairing-plus-quadrupole (PPQ) model. The PPQ model has demonstrated considerable success in predicting the prolate-to-oblate shape transithmu, but only addresses the properties of lower energy levels. It also predicts the potential energy surface to be y-soft. This is one of several predictions of y-soft potentials in the Os—Pt region. (Another model with similar predictions for the Pt nuclei is the boson expansion technique of Kishimoto and Tamura [Ki 72, Ki 76].) One problem with these more complete treatments of the collective properties of heavy nuclei is the rather complex numerical methods needed to solve the Hamiltonians. A simpler description of the nuclides in the Pt-Os region has recently emerged from the interacting boson approximation (IBA) model of Arima and Iachello [Ar 76, Ar 78, Ia 78, Sc 78]. In this model the emphasis is on the symmetries of the nuclear structure rather than the geometry, while also including the finite dimensionality of the subshells. The IBA.mode1 treats the nucleus in terms of a set of bosons, one for each pair of neutrons or protons outside a closed shell. The bosons can be in either of two states, denoted by their angular momentum, L = 0 or L = 2 (s or d bosons), and are allowed to interact. The most general Hamiltonian describing such a system possesses an SU(6) group symmetry. Particularly simple descriptions are possible when the Hamiltonian is symmetric with respect to subgroups of SU(6). Analytical solutions have been found for both the energy levels and electro- magnetic transitions for three subgroups, SU(5), SU(3), and 0(6). These symmetries are applicable at the beginning, middle, and end of shells respectively. The SU(5) subgroup corresponds approximately to the vibrational limit of the collective model, and SU(3) to the rotational limit. The third limiting symmetry, 0(6), is most like the y-unstable model of Wilets and Jean [W1 56]. It has been shown by Cizewski et a1. [Ci 78] that this limit is capable of accounting for most of the energy and decay properties of all positive parity levels below the pairing gap for 196Pt. Moreover, it predicts no 0+ level with 2—phonon components near the 4+ and 2; level, and it has no equivalent to the 3—phonon 2+ level. In fact, the structure of 196Pt and most of the lighter mass even-even Os and Pt nuclides can be understood [Ca 78] by adding a small but gradually increasing symmetry-breaking term to the Hamiltonian as one goes further away from the 0(6) limit. This A—dependent deviation from Fl. 1... IO. y—l A“. AA. 'V nu- [by 9“ PH. NHL i b. .. ~\U the 0(6) limit is predicted to occur within the more complete SU(6) representation of the IBA. The majority of the experimental information on the heavier Pt isotopes has come from'y-ray studies following the 6,8: decay of Au and Ir isotopes [Be 64, Ny 66, Ja 68, Be 70, F1 72, Cl 76]. There have also been several publications on y-decay following neutron capture [Gr 68, Sa 68, Su 68, Ci 79]. More recently the nature of the high-spin levels of the platinum nuclides up to spin 20 has been studied by (a,xny) in-beanx'y—ray spectroscopy [Ya 74, Fu 75, Hi 76, Sa 77]. There have been numerous inelastic scattering experi- ments [G1 68, Ba 76, Le 77, R0 77, St 77, Ba 78, Ba 78a] performed on the Pt nuclides, primarily by Coulomb excitation of the first 2+ states. However, three more detailed Coulomb excitation studies have recently been performed yielding conflicting conclusions when the results are interpreted in terms of various models. Lee et a1. [Le 77] see evidence for a stable triaxial Shape, Baktash et a1. [Ba 78a] favor the PPQ model, while the third study proved inconclusive [St 77]. The bulk of the transfer reaction data is from one-neutron transfer studies of the odd-A platinum nuclei [Mu 65, Ya 76, Sm 77, Be 78, Ya 78], but there have been searches for strong 190-196 L = 0 transitions :hi the Pt(p,t) reactions [Ma 72, Ve 76, Ve 78]. B. Motivation When this study began, most of the information on the Pt region was obtained from decay works and Coulomb excitation of the first 2+ states. The intention of this study was to utilize the high resolution capabilities of the Michigan State University cyclotron in collecting transfer reaction data on the even-even Pt nuclides to complement the existing data. The (p,t) and (p,p') reactions were chosen because both reactions populate primarily the collective levels of a nucleus and it is this type of state that most models of heavier nuclei attempt to explain. Specifically, the (p,t) reaction was chosen for the distinctive, diffractive angular distributions of tritons from the L = 0 transfers, which populate J" = 0+ levels in the residual nucleus when using even-even targets. Information on the spin and parity of these and other states seen in transfer reactions are obtained by interpreting the Shapes of the transitions in terms of the distorted-wave Born approximation (DWBA) reaction formalisms. The low-lying 0+ states play an important role in any attempt to distinguish the models men- tioned above, although additional information on branching ratios from the decay of these states is also necessary. In addition to locating the 0+ states, the strength of the transition populating such states in a (p,t) reaction can also provide information on the shape of a nucleus, as was seen in the Sm isotopes [Bj 66]. If the ground states of Os or Pt nuclei are relatively rigid in the y-direction, and y varies rapidly, strong L = 0 transitions. are expected no populate excited O+ levels, which have shapes similar to the target ground state. However, Sharma and Hintz [Sh 76] observe no strong L = 0 transitions in the Os nuclei, possibly because the y shape parameter appears to be changing slowly. The present study would extend this search into the Pt isotopes. The (p,p') reaction study was initiated as a probe of the macroscopic structure of the transitional Pt nuclides. Techniques have been established for determining deformation parameters and charge/potential moments from such inelastic scattering studies. These experimental parameters, obtained using matrix elements from some nuclear model, can then be used to determine the shape of the nucleus. The procedure has primarily been used for well deformed nuclei in (a,a') experiments. The work described here, along with (p,p') experiments on well deformed rare-earths and actinide nuclei at this laboratory [Ki 78], is an attempt to extend the technique from alpha [He 68, Ba 76] to proton scattering (which is now thought to be a better probe of the nuclear matter distribution than high energy (a,a') experiments [Ha 77]). The most widely used code for analyzing inelastic scattering data, ECIS [Ra 73], is capable of calculating several different sets of matrix elements, thus affording an excellent means of testing several models by using them to interpret scattering data. 10 The (p,p') study also complements the two-nucleon transfer study because each reaction populates collective states, particle-hole types in (P,P') and two—particle or two-hole configurations in (p,t). Due to these similarities, one would expect to see some correlation between the levels populated in each reaction and thus aid in the assignment of spin and parity for new levels. C. Organization The next two chapters of this dissertation are an introductory discussion of the theoretical background used in the study. Chapter II deals with the primary nuclear models used in the remainder of the text, while in Chapter III the theoretical methods of describing nuclear reactions and calculating transition strengths and shapes are presented, including the standard Distorted-Wave Born Approximation and the method of coupled channels for inelastic scattering. Chapter IV’ discusses the experimental. procedures and the methods used in the data analysis. In Chapters V and VI the experimental results are 194,196,198 192,194,196Pt presented for the and 194,196,198 Pt(p,t) Pt(p,p') reactions, respectively. Also included in each chapter is an interpretation of the results, primarily with the 0(6) limit of the interacting boson approximation model. The last chapter is a summary of the 11 results for both sets of reactions and a discussion of the effectiveness of the IBA model in explaining the data. CHAPTER II THEORETICAL BACKGROUND - NUCLEAR MODELS A. Introduction When one is studying a complex problem or situation where all the forces or variables involved are not known, some type of model is usually used to gain insight, answers, and ultimately predictions. The study of the nucleus is an excellent example of this situation. First, the nucleus must be handled as a many-body problem (A > 2, of course), which has no analytical solution. But the situation is even more complex than the atomic many-body problem because of the complicated nature of the nuclear force. The force between nucleons is very strong, with many complexities. In the atomic case the force involved is the simple electrostatic force between charged particles. Second, the dynamics involved in the nucleus are extremely complex due to the high particle density and the short range of the nuclear force. These complexities thus compel one to use a model to describe the nucleus and to predict experimental quantities. In the past forty years there have been numerous models developed in an attempt to understand the nucleus. For the 12 13 most part, these models work well only for limited special cases because of assumptions that are necessarily made to make the problem tractable. There are basically two kinds of models: one describing the particle-like or intrinsic features of nuclei; the other deals with the collective features often exhibited by heavier nuclei. This section will concern itself with the latter kind of model because of the collective nature of the Pt nuclides and also because the reactions employed in this study, (p,t) and (p,p'), are probes of collective structure. Extensive detail is not necessary as there exist many classic works and review articles on the subject [Bo 53, Da 58, Bo 69, Pr 75, Ra 75]. The most widely used model for collective nuclei is the phenomenological, hydrodynamic model of Bohr and Mottelson [Bo 53, Bo 69]. The nucleons are treated as a deformable liquid drop, whose macroscopic properties such as surface tension, volume, and Coulomb energy are included in the total Hamiltonian. Two idealized limits arise, one due to vibrational modes and the other from rotations. The vibrational model applies in first. order to spherically Shaped nuclei, where the nuclear excitations (of a collective nature rather than intrinsic) are assumed to be small— amplitude, harmonic vibrations about. the equilibrium spherical shape. In the rotational model, which best describes well deformed nuclei in the middle of shells, two collective modes of excitation are considered, viz. rotations and small vibrations of a permanently deformed system. - Inn- 1 I Uvu‘ Q "R IV. . . I a“. o”_ b.‘ 5- ‘\ 14 B. The Vibrational Model The usual parameterization of the surface of a distorted body involves an expansion in terms of spherical harmonics, R = RO[1 +xzpaku(t)Ylu(e'¢)] , (II-1) where the Ylu are the spherical harmonics expressed in terms of the laboratory angles 8 and ch, and the “Au are time- dependent parameters defining the distortion of a sphere of volume 4/3 nRg. The kinetic energy, T, can be expressed as B Z 2 -2 A” 1|“xul (II ) and the potential energy V as, 1 2 V=§ | 2 where the BA are "mass parameters" and the C1 are the vibra- tional "force constants". ‘Values are determined either empirically or via some microscopic model. The total Hamil- tonian for this system is given by _ 1 z ' 2 2 - H - 2 Ml[Bxlawl + Cilaxul J , (II 4) which corresponds to a set of uncoupled harmonic oscillators with frequencies, “A w)‘ = (CA/BA)l/2, 1 3 2 , (II-5) Where 1. is associated with the order of the excitation and also the magnitude of the angular momentum. The modes 1 l I . .n-I- l' ‘ ODDUO" . y'- fiA c V on. Q I‘D v JUVOU d I- 'CA. . M‘. :41.- fr: "vs . "AnA‘ t”""’ ‘vn ._ Q ‘PV._. h e ”V.‘ . :‘I. t. u ..,‘ y, .15 4“; 1- . ‘b (II c I “I 15 arising from A = 0 and A = l have been neglected Since A = 0, the compression mode, corresponds to a volume change of the nucleus and )(= 1 refers to a translation of the center of mass of the nucleus. The energy levels of this simple harmonic oscillator are given by E). = hwk f1 (“Au + 1/2) , (II-6) where nMl is the number of vibrational quanta, called phonons, which have angular momentum A and parity (_1)1. Thus, the energy spectrum of a simple vibrator has a ground state with zero phonons and spin and parity Jfl = 0+. The first excited state has one A = 2 phonon with an energy of hwz and J1‘ = 2+. Since one A = 3 phonon has nearly the same energy as two A = 2 phonons, the next excited state, which appears at Zhwz, could be either a 3- state or the three degenerate states formed via the coupling of two quadrupole phonons. In practice these states are not degenerate due to anharmonicities, so the third excited state could be either the 3— state mentioned above or a 0+, 2+, or 4+ state from the 2—phonon triplet. Some of the electromagnetic properties of this model include: (i) strong E2 transitions from the first excited 2+ state to the ground state as well as from the members of the three phonon triplet to the first 2+ state, (ii) no E2 transition from the second 2+ state to the ground state, and (iii) no quadrupole moment for the first 2+ level. The 16 nuclei which should exhibit these energy and decay properties lie near the closed shells, far from the strongly deformed regions and should be nearly spherical. C. The Symmetric Rotor Model Measurements of the quadrupole moments of nuclei have shown that the larger values occur in the region between major shell closures. This implies that these nuclei are not spherical, but ellipsoidal, and one would expect to see energy and decay properties quite different from those of a spherical vibrator. These strongly deformed nuclei exhibit a rotational structure much like that seen in molecules. To describe such nuclei in the Bohr—Mottelson model the nucleus is assumed to be strongly deformed with axial symmetry. The expansion of the nuclear surface is handled in a fashion similar to the spherical case except that the aku parameters are replaced by aMl values related to the body-fixed system and the Euler angles (61, 62, 83) of the principal axes of the nucleus with respect to the space-fixed axes. Thus, for quadrupole (k = 2) shapes the nuclear surface can be expressed as .. I _ R - Ro[1 + 5 aqu2u(9 ,¢')] . (II 7) The axes of the body-fixed frame are chosen such that a21 = a2'_1 = 0 and a22 = a2'_2. So for the rotational system, a20' a22 along with the three Euler angles are used 17 to describe the system rather than the five <1"u variables. For later convenience, a20 and a22 are defined in terms of two new parameters 8 and y: - __fi- - a20 - Bcosy, a22 - s1ny . (II 8) 2 The significance of B and y can best be shown by the value of R(8',¢') — Ro along each axis _ _ V5 an _ R(6',¢') - R0 - ORR - m BROCOS(Y - T), k - 1,2,3 , (II-9) where k represents the body—fixed axes, x', y', and 2'. From (II-9) one sees that B is a measure of the size of the deformation,the departure of the nucleus from sphericity. The value of Y determines the type of ellipsoid. For values of Y = 0 or multiples of fl/3, there is one axis of symmetry. With 8 > 0 and y = 0, 2n/3, 4n/3 a prolate (football) shape is obtained, while for E3> 0 and Y = fl, fl/3, 5fl/3 an oblate (doorknob) shaped ellipsoid is formed. If B < 0 the two Shapes would be reversed in the examples above. For other values of Y there is no symmetry axis in the nucleus. This case (asymmetric rotor) will be discussed in the next section. Now one can discuss the shape of a nucleus in a two dimensional B-Y plane, which can be reduced to a 600 sector of polar coordinates due to the symmetries about fl/3. Figure II—l shows a schematic view of this 600 sector. To derive an expression for the energy levels of a symmetric rotor the terms of the Bohr-Mottelson Hamiltonian 18 Sphere Prolate Figure II-l. A 60° Sector of the 8-7 Plane. The point P represents an asymmetric shape with magnitude 8 and an asymmetry parameter y. 19 must be recast in terms of the body-fixed frame. After this transformation the expression for the kinetic energy is given by ll MW -1'2 2-2 1 T - 2 B(B + B Y ) + 2 Ikwi , (II-10) k l where w is the angular velocity of the principal axes with respect to the space-fixed axes and I, the effective moment of inertia, is given by IR = 4BBZsin (Y - 2%5) . (II-11) The first term.of Equation (II—10) represents the vibrational energy and the second term is the rotational contribution. The potential energy can be expressed in terms of the B-Y plane mentioned earlier so that the total Hamiltonian has the form H = TB + TY + TR + V(B,Y) . (II-12) From this general Hamiltonian, solutions for both the rota- tional and vibrational limits can be obtained. For a Spherical nucleus, V(B,Y) + V(B) = % C82 (II-13) is obtained straightforwardly from Equation (II-3). Sub- stituting this expression into (II—12) leads to the same energy spectrum obtained in the previous section for a vibra- tional nucleus (see Figure (II—2)). If one assumes that B and 7 change very little, which implies a very steep V(B,Y) 20 '60‘ (b) y I Axially whom: you". — — —_‘. r 4‘ —3: — . __°O 7. -—2‘ mar-:03! W” —— 3° 6. so 20 f—-? OI. m 4. 46 . -——r ‘——-r Wfimno £3335 '——-W Figure II-2. Schematic Plots of V(B,Y) for (a) Spherical, (b) Axially Symmetric Prolate, and (c) Asymmetric Shapes. Also shown are typical energy level spectra associated with these potentials. Taken from Reference [Pr 75]. .pa .bv‘ I!“ \ H g... n" uv- LIA Iii ‘u 1". EMU s 21 about the equilibrium value of B and y, the Hamiltonian simplifies to 1 3 3 H = ‘2' E Ikwk = E T'- . (II-l4) For an axially symmetric nucleus, the moment of inertia about the symmetry axis is zero as no rotations about this axis are observable. From (II—ll) one sees that the two remaining Ik values are equal in this case. This leads to the simple energy expression 2 I 3' E = J J(J + l) , (II-15) N with the total angular momentum J = 0, 2, 4... for the ground band. An example of this band structure is shown in Figure II—2. If small vibrations are allowed about V(BO,Yo 2 0) two additional low-lying bands may be seen in a strongly deformed nucleus. These are called {3 and y vibrations. The 8 vibration causes a distortion from the equilibrium value of (30 while preserving axial. symmetry (Y = 0, for a prolate example). The y vibrational mode involves oscillations about y = 0 with 8 fixed. This type of oscillation disrupts the axial symmetry. These vibrational states will usually have a rotational band built on them which have the same J(J + l) spacing (see Figure II-2). The quantum number K, the projection of J onto the symmetry axis, is often used to describe these bands. The B vibrations preserve axial symmetry so K = 0, while the y vibrations have .__-- 22 K = 2 since they represent motion out of the symmetry plane. It is possible to have a two y—phonon vibration, much like the simple 2-phonon states described in Section II-A. This would result in a K = 0 and K = 4 bandhead, but these would be expected at approximately twice the single Y—phonon energy. This type of state will be discussed further in Chapter V. D. The Asymmetric Rotor Model The asymmetric rotor model of Davydov and Fillippov [Da 58] is actually not too different from the basic framework of the simple rotor except that the zero-point value of Y is allowed to be non-zero. This allows for rotation about all three axes and thus three different Kk values. Cast in its simplest form, this model has a potential V(B,y) with a steep minimum about a fixed 8 and some y f 0. If this condition is relaxed, similar 8 and Y vibrations will be predicted at lower excitation energies, much like the symmetric rotor case of Section II-B. The Hamiltonian is identical to Equation (II—14) except that each IR is allowed to be non-zero. The solution of the Hamiltonian is somewhat more involved in this case, however, since the quantum number K, the projection of J onto the symmetry axis, is no longer a good quantum number because the non-zero y—value destroys the axial symmetry. The eigenfunctions are thus mixed in terms of K and can be expressed as 23 = 2 J where the mixing coefficients gg. will be derived from Equation (II—17) and the |JMK> arelthe eigenfunctions of an axially symmetric rotor [Pr 75]. Solving the Hamiltonian in the |JMK> basis shows that only one 0+ state, two 2+ states, one 3+ state, three 4+ states, etc. are formed. This is due to the allowed values of J and K as a result of symmetry considerations K = 0, 2, 4... ,K, K + 1, K + 2...if K # o (II-17) K=0 J = .0, 2, 4... if Solutions for the energies of these states are obtained by solving the following equation for each value of K - 0 k=l 2T; - JM - - (II—l8) Figure II—2 shows a typical energy level pattern and V(B,Y) surface for an asymmetric rotor. The level spacing is quite Similar to the simple J(J + 1) rule of a symmetric rotor. In fact, if one "softens" the asymmetric rotor to permit B-vibrations following Davydov and Chaban [Da 60] there are actually very few differences between the electromagnetic properties and the energy levels predicted from this model or from an axially symmetric model with the rotation-vibration coupling terms included [Fa 65]. Yamazaki has shown [Ya 63] there are really no discernable differences in the two models 24 unless one studies the higher-lying bands, in particular the K = 4, two-y-phonon band. Thus, distinguishing between the two models on the basis of data related to the ground band and lowest K = 2 band is quite difficult. E. The Interacting Boson Approximation (IBA) Model 1. Introduction All the models previously mentioned in this chapter and in the introduction have a common framework in the hydro- dynamical model of Bohr and Mottelson. The only differences between the models lie in the assumptions made in deriving certain parameters or in those assumptions which simplify the Hamiltonian. In this section an entirely different approach to the modeling of the nucleus will be described. Arima and Iachello [Ar 76] have proposed a group-theoretical approach to explain the collective properties of nuclei with A.: 100 except those near closed shells. The cutoff of the model at A3 100 is due to the proximity of the Shell closures in lighter nuclides and the small number of particles outside the core available for this type of collective interpreta- tion. This discussion will be confined to even-even nuclei, although the model is capable of describing odd-A nuclei as well [Ar 76a]. In this model pairs of particles (protons or neutrons) are treated as bosons, which can occupy two levels, a ground state, associated with s bosons, and one excited state .-~vV--' -'—"" ‘ An”; 25 occupied by d bosons. All possible interactions are allowed between the two types of bosons. The energy and angular momentum of s bosons is es and L = 0, and for the d bosons, Ed and L = 2. The total number of bosons for a particular nucleus, N = nS + nd, is a fixed parameter. N is determined from the number of neutron and proton pairs (or holes) outside their respective closed shells. For the nuclides lgz-lggpt114_lzo the nearest shell closures are 126 :flor neutrons and 82 for protons. Thus, the number of neutron pairs ranges from 6 to 3 while there are 2 proton pairs, (82—78)/2 = 2. This gives values of N for 192’194'196'198Pt of 8, 7, 6, and 5 respectively. The following derivation of the many IBA expressions follows very closely the formalism of Reference [Ar 76]. The simplest Hamiltonian for the IBA which contains only one- and two-body terms is + + N H=ess+e2dd+zv.., (II-19) 3 dm mm i cL = v2 = V5/2 v = 53 (II-21) o 2 U2 = V5 Uo = The complex Hamiltonian of Equation (II-20) actually only has 4 types of terms. ‘The first two terms are just s and d boson counting terms. Terms with the coefficients CL and 27 UL do not change the number of s or d bosons, ns and ha. The third type of term has the coefficient V2 and changes nd by one unit, while the fourth type, with Vo as its coefficient, changes nd by two units. The dominance of one or more of these terms allows for the simplification of the IBA Hamiltonian and in three special situations will even allow analytical solutions to be obtained. These three limits are denoted by the group designations SU(5), SU(3), and 0(6), and are attained by requiring the Hamiltonian to contain terms that are generators of each subgroup. In the case of the SU(5) limit, the Hamiltonian would contain only terms that conserve nd. For the SU(3) limit, both one and two d—boson number changing terms are included, while only the two d boson changing terms are allowed in the 0(6) system. The basis states for the IBA Hamiltonian can be written as C dikgTN-k 0 "NZ |¢> = |Nndn n JM> = % |0> , (II-22) 8 A k k where |0> represents the closed shell, and A is a normaliza- tion constant. The d1 operators are coupled to some definite angular momentum J,M. Here nd, n8, and nA are the number of d bosons, number of pairs of d bosons coupled to spin zero, and the number of zero-coupled triplets of d bosons respectively. The full SU(6) Hamiltonian of Equation (II-20) can also be written in terms of the specific boson-boson interactions 28 H=€Zd$dm-klz.61-6.-k'2L1+k"ZPl, J 3 i - 2 nd(nd 1) = L(L + 1) - 6nd (II-27)

= (nd - v)(nd + v + 3) , where v is the seniority number for the bosons, v = nd - 2n . B This gives an eigenvalue expression for the complete SU(5) limit: E(N, nd, v, n , L, M) = en +<1§ n (n - l) d d d A + B(nd - V)(nd + v + 3) (II-28) + y[L(L + 1) - 6nd] . 34 Table II-2. The Relationship Between the Coefficients of Equation (II—26) and the CI. Coefficients of Equation (II-25). The factors for Equation (II-26) were obtained from [Ar 76]. Equation (II-26) Equation (II—25) C0 C2 C4 a --- 4/7 C2 3/7 C4 B 1/10 C0 -l/7 C2 3/70 C4 Y --- -l/14 C2 1/14 C4 35 These extra terms in the energy expression now lift the degeneracies shown in Figure II-3 and produce a spectrum similar to the one shown in Figure II—2 for the hydrodynamic vibrator. Figure II—4 Shows how the energy levels can now be arranged into bands with the maximum spin determined by N. All three limits of the IBA display this "cutoff" aspect, evident in both the energy and decay properties. 'This results from the finite dimensionality built into the IBA, in contrast to the geometric models which have N + 0° for the liquid drop. The effect of finite dimensionality is most evident in the branching ratios within the "ground band" as shown for the SU(5) limit in Figure II-5. In all cases of a simple vibrator, the IBA predicts a ratio less than the geometrical predictions. This is in agreement with current experimental measurements. As mentioned earlier, one of many properties which can be calculated within the IBA is the electromagnetic decay of a nucleus. The most general form of the IBA transition operator is given by (R) _ + + (2) + (1) T -a (352,2(ds+sd)m +B£(d d)m m 1 1 (0) _ + Ynoazo‘smo‘s S’0 ' (II 29’ where 1 is the multipolarity of the transition with projection m, and a, B, Y are the coefficients of the various terms. The exact form of the coefficients will depend on the 36 .mmh udu wocmuwwom EOuu cmxme .AmNIHHv cowumsvm EOum pmumHDOHmO mum mwfimumcm one .A .ocv eh mumoEDC Eoucmsv on» an ooawoma mum wmumum one .wwOLoh comonuwucH eefiz amH use Lo Leena Amcom one cues omuuasoano aneuumom HmSma Smooch maeeum .vIHH mosmfim 37 th .>x and .o 1.648% 8.0.3 + .vIHH musmflm 8849:. 1 81.1.1: .N . .ll 0 AO.NJMwI+N o I 09 + 88.3.1. 8.34% J Ao._u...mlvu+m V Aofifiv +¢ 8.3. .o . . v I 1.! o e v . o 8 N3 .N 1 V 8.3%.“. 8.3x +¢ 8.1V E +8 8.3% (A3W)3 38 mo pcsouu onu aw mooam> .moe ed; oocwuwmwm Boom cmxma .mooaosc NH u z can m n 2 cm uOu a can .q no cofiuocom m mm owuuon mum mosam> omoce .momaosz Amvam cm mo comm Ammvm o>fiumawm co mafiamcofimcoefla wuwcfim mo uommmm one .mIHH 8.33m 39 .n. NON :3. mm N. .282 u 2 0. 3:23.53 85:52.00 .mlHH musmwm V. 0. w h n n Snufmuhducm ANIUCNHI— ft:Nu.l_.N 1 _ mvm 40 specifh: limit of the model, SU(3), 0(6), and SU(5). The form of this operator 'must also be a generator of the particular subgroup, which simplifies Equation (II-29) and allows for an analytical solution. For E2 transitions in the SU(5) limit Equation (II-29) reduces to TéEZ) = a(d+s + 51a)é2) , (II-30) where v,_ a = —% , 5 being the quadrupole operator. This operator leads to a selection rule of And = i l, the same as in its geometrical counterpart. The trends of several branching ratios are given in References [Ar 76, Ar 78, Sc 78]. 3. The SU(3) Limit - Rotational Nuclei In situations where the Q-Q and L terms of Equation (II-23) dominate, the boson energy, e., and the pairing force, Pij' analytical solutions can be obtained from the simplified Hamiltonian by using the symmetry properties of the SU(3) group. Many of the results are similar to a special type of symmetric rotor. A new set of basis states is used to solve the SU(3) eigenvalue equations H|N,(A,u)KLM> = ElN,(x,u)KLM> (II-31) and -_£.+-!£ - H - k ij 61 Qj k ij Lij , (II 32) 41 where (Mu) are quantum numbers that label the represen- tations of SU(3), L, and M are the angular momentum and its Z—axis projection while K distinguishes states with identical (l,u), L. The solution of (II-31) is given by E(N,()\,IJ)KLM) aL(L + 1) - BC(X,)J) (II-33) Here a = ‘i’ K — K', B = K, and C(l,u) is the quadratic Casimir operator [Ar 78] of SU(3) C(k,u) = 12 + “2 + An + 3(1 + u) . (II-34) The (A,u) can assume values of (2N,0), (2N — 4,2), (251— 8,4) ...etc. Figure II—6 shows a typical energy spectrum obtained within the SU(3) limit with K' = 0. Note that the level spacing is proportional to L(L + l) and that a K = 0 "B-band" and K = 2 "y—band" arise quite naturally from this limit along with higher lying bands of a 2—8 or 2-Y nature. One notable exception in these bands compared to those of the symmetric rotor is that states of the same spin, L, belonging to the same (Lu) representation, are degenerate in the strict SU(3) limit. The E2 operator in the SU(3) limit is given by + v— + TéEZ) e a2[(d+s + s d)é2) - -% (d d)é2)] , (II-35) where a2 is the effective E2 charge and 82 in (II-29) is now — -—;0.2. This expression predicts very strong transitions within a band but in the strict limit transitions are not 42 E (MeV) 3‘ r ((2.0) (8.2) (4.4) (0.6) ($0) (2.2) (0.!0) ' “ ; / \ o'— l / \ .- / I \ I ‘._ a: — . ‘.- ..- 2... 2‘ |0.-- 7— o .. 0“: r- | O-‘.- ‘ _ — ..— ‘ 5°—5=— 2.-— 2.- .. o‘—n‘— ‘ —‘°:4‘— 0" 0"" r2 ’_ 2,_ ._ c'-—c‘— . 5'.— I D— ‘.-‘:_ )- 32:3.- 0‘— 5°— 4‘— r— _ O] o - SU(3) Figure II-6. Sample Energy Level Spectrum Calculated With the SU(3) Limit of the IBA. The states are separated into bands denoted by the quantum numbers (A,u) and further identified by K and .3". The energies are calculated from Equation (II-34). Taken from Reference [Ia 78]. 43 allowed between states with a different (x,u). Thus intraband transitions are allowed for the B- and Y—bands, but no ‘transitions are predicted from either of these bands (1,u) = (2N — 4,2) to the ground band O(,p) = (2N,0). This is contrary to the predictions of the geometrical symmetric rotor model. Examples of other decay systematics are given in Reference [Ar 78]. Closed form expressions can also be obtained for certain types of transitions. Of particular interest is the expression for transitions within the ground band. For the SU(3) limit [Ar 78] (L + 2)(L + 1) _ 1 B (E2. L + 2 * L) - a 4 (2L + 3)<2L + 5) IBA 2 2 x (2N - L)(2N + L + 3) (II-36) for the E2 decays, and _ _ 1§1 1/2 L _ for the static moments. If these expressions are compared with those of the geometrical model [Bo 69] one obtains B(E2; L + 2 + L) = (2N - L)(2N BBM(E2; L+ 2 + L) (2N + L + 3) )2 + 3 ° (II-38) '2' This ratio illustrates the finite dimensionality present in the IBA model. As N + m, BIBA + BBM' However, the "correction factor" differs from unity for finite N values and approaches zero as L + 2N. This implies that the IBA model predicts decreasing values for the transition proba- bilities between states with higher L values compared to a 44 single value predicted by the geometrical model. This cutoff effect has been seen in 20Ne decay [Ar 78]. 4. The 0(6) Limit - Transitional Nuclei For this third limit of the IBA, the dominant interboson force is the pairing interaction. The analytical solution to the IBA Hamiltonian is achieved by requiring the terms of the Hamiltonian to be generators of the subgroup 0(6). This produces a Hamiltonian of the form - I l 1/2 I (L) (L) (0) H - e g dmdm + € 2 (2L + 1) CL[(d d) (dd) ] + % VO[(d+d)(0)(ss)(0) + (s+s+)(0)(dd)(o)](o) (II-39) + % UO[(s+s+)(0) (ss)(o)](0) . Recall that the V0 (pairing) term results in a change in the number of d bosons by r 2. This causes considerable mixing of the wave functions if the basis set shown in (II-22) is used. If the basis denoted as INJU'tv LM> is used, the A Hamiltonian can be expressed in terms of operators which are diagonal in this new basis. New quantum numbers are then obtained: 1) o characterizes the totally symmetric irreducible representations of 0(6), with o = N, N — 2...0 or 1 for N even or odd; 2)T’ characterizes the totally symmetric irreducible representations of 0(5), where T = o, o — l...0; 3) vA is the number of zero-coupled boson triplets, with the relationship I = BvA + 1, VA = 0,1..., where L is allowed to 45 be L = 21, 21 - 2,...1 + 1,1. The Hamiltonian is conveniently expressed as H = AP6 + BC5 + CC3 , (II-40) where P6 is the pairing operator and C5 and C3 are the Casimir operators of 0(5) and 0(3) respectively. The presence of operators for several subgroups in this expression results from the group reduction 80(6):: 0(6):: 0(5):: 0(3) Table II—3 Shows the correspondence between the new coeffi- cients in (II—40), A, B,and C with the CL, VL' and UL coeffi- cients of (II-20,31). The eigenvalues resulting from (II—40) are [Ar 78a] E(N, o, T, VALM) = %(N - o)(N + o + 4) + % 1(1 + 3) + C[L(L + 1)] . (II-41) Figure II—7 displays a typical energy level spectrum for the 0(6) limit with N = 6 and A, B, C > 0. The signs of the three terms can considerably alter the level pattern. In particular, a positive value for C drops the 2; state below the 4:, a feature of the Pt nuclides. The grouping of energy levels in Figure II—7 is similar to those in Figure II-4 for the SU(5) limit with a few exceptions. The most notable of these are the "missing" 0+ and 2+ levels in the T = 2 and T = 3 groupings in 0(6)(o =°max) which correspond to the 46 Table II-3. The Relationships Between the Coefficients of Equation (II—40), (0(6) Limit) and Those of Equations (II-20, 39) (Full IBA Hamiltonian). Equation (II—20) Equation (II—40) 0(6) Limita Full IBA A B C e 0 4B 6C C0 5/2 A -BB -12C C2 0 2B -6C C4 0 28 8C v0 —<5/16)1/2A 0 0 V2 0 0 0 U0 1/2 A 0 0 U2 0 0 0 aValues from [Ca 78a]. 47 .mmmh Lfim mum mowmuwco one .An .<> .e .ov mewo23c Enucmsv on» an cmuocmp mum mmumum one .wq wmumcm OHQEmm . nu: 9.33m 48 llo are) . I momma n E .m :2: .30 . IION .uh 1|+N +v N.» JO m +¢ IIwO 0"» +0 m.» II+N .C IJN all + 1 JN +m ill...» N.» u ...llo c..— +0 O F JO +N A. .m .e JN .«b JV .9“ lll+m Io mub. IN llo o.» + S + +N ID V Nah .III+N '0 + + I? + + Illn .n ..Ilv .luo. n; +w + + +m +® Ill+m cu» Ilws Iluo 1 low INO— +N. m.» (A3“) 3 49 nd = 2 and nd = 3 states in SU(5). These states in the SU(5) limit effectively have been pushed higher in energy than their nd neighbors due to their nB = 1 character, while the other states in each group have nB = 0. This should be expected since n is the number of boson pairs coupled to 8 zero, and the difference between the SU(5) and 0(6) limits is the introduction of a pairing interaction. An additional feature of this limit is the set of 0 — 2+ - 2+ groupings with strong E2 cascades, and level spacings proportional to T(T + 1). Several closed expressions for EZ transition rates have been obtained, again by using a transition operator which is a generator of the limiting symmetry, in this case 0(6). The operator satisfying this condition is TéEz) = a2(d+s + s+d)(2) , (II-42) where 012 is a strength factor. This operator requires selection rules of A0 = 0 and AI = i l. Theta selection rule is a direct result of the operator being a generator of the 0(6) group and thus cannot connect states with different representations. This is analogous to the A(A,u) = 0 selection rule for the SU(3) limit. The AT restrictions arise from the fact that the operator can create or destroy one d boson. Several examples of these closed expressions are given in Reference [Ar 78a]. Two informative branching ratios are 50 + + + + B(EZ; 41 + 21) = B(E2; 22 + 21) + + + + B(EZ; 21 * 01) B(E2; 21 + 01 =lflfl‘1-1HN4'5) -l_0_ 7 N(N+4) N+fiao 7 which are two of several in this limit identical to those of an asymmetric rotor with Y = 30°. The major deviations from experiment occur for nuclei with small values of N — another finite dimensionality effect. The closest geometrical analogue to the 0(6) limit is the y-unstable model of Wilets and Jean [Wi 56], which also has a repeating sequence of levels and the 1(T + 1) energy dependence. But in this model 3 level can never fall lower in energy than the 4: state, the 2 a property which is a straightforward result of the 0(6) limit when C > 0. As is the case in most nuclear models, very few real nuclei can be described in great detail by the first order or special limiting cases. However, there are several good candidates for the three IBA limits which provide a starting point for more detailed calculations. Some examples are 112’116w and Xe for the SU(5) limit [Ar 76], 156cm and several Sm nuclei for SU(3) [Ar 78], and a recent candidate [Ca 78] for an 0(6) example, 196Pt. For more realistic comparisons one usually begins with the appropriate limit of the IBA and introduces one of the boson forces or varies the boson energy, 6, as a perturbation to the group symmetry. For example, in studying the samarium isotopes, Arima and Iachello [Ar 78] vary the boson energy as a function of N, 51 such that as 6 increases from the heavy to light samariums, the boson energy term overshadows the Q-Q terms, thus providing a transition from the SU(3) to the SU(5) limit. This‘perturbation causes the wave functions to mix so that ‘most. of the transition rate selection rules are broken. Casten and Cizewski [Ca 78] have carried out a similar procedure to describe the 0(6) + rotor transition for the Pt-OS region. This is accomplished by introducing a logarithmically increasing (with N) quadrupole-quadrupole boson interaction, while varying the 0(6) parameters, A, B, and C, to account for the changes in mass. Because the Q'Q force is comprised of a one d boson operator, all wave functions become mixed and both the 0 and T selection rules are broken. This study of the Pt nuclides with the (p,t) and (p,p') reactions has for the most part used wave functions similar to those of Casten and Cizewski [Ca 78], derived from the slightly perturbed 0(6) limit. The intention of this study as well as [Ca 78] was not to find a set of "best fit" parameters for the 0(6) limit, but rather to use a consistent set of parameters to adequately describe the Pt region. Another reason for not fine tuning the parameters is the absence of any explicit proton-neutron interaction between bosons [Ar 77]. Recall that when determining N, the total number of bosons in a system, the particular type of boson, p or n, was not a factor, only the sum of each type. The role 52 of this proton-neutron boson interaction appears to be most critical near the end of a shell — the region where the 0(6) limit should be most effective [Ar 78a]. Preliminary results indicate that the energy predictions will improve, while the branching ratios remain unchanged [Ci 78]. CHAPTER III THEORETICAL BACKGROUND - NUCLEAR REACTIONS A. The Distorted-Wave Born Approximation (DWBA) for Transfer Reactions 1. Introduction Current methods used to calculate cross sections for direct nuclear transfer reactions fall into three major categories, the DWBA, coupled channels Born approximation (CCBA), and the multi—step sequential transfer approximation. Although each method has a somewhat different domain of applicability, the DWBA is by far the least elaborate of the three methods and actually provides the basic framework for the latter two. The CCBA is usually used to describe reactions which involve very collective type nuclei, such as the rare earths. In these cases the levels in the rotational bands are so coupled that a one-step (one channel) analysis. like DWBA, cannot reproduce the data. The sequential transfer approximation is primarily used in multi-nucleon transfers, where several intermediate steps in the transfer process may be necessary to interpret the results. The degree of computational difficulty increases very dra- matically from DWBA to the sequential transfer approximation 53 54 for even the Simplest one—nucleon transfer reactions. For a two-nuclear transfer reaction, such as (p,t), these computa- tional problems are magnified. For this reason, and the lack of collectivity in the Pt region, the theoretical inter- pretation of the (p,t) reactions was limited to the DWBA method. The theory of direct two-nucleon transfer reactions has been developed by several authors [To 61, G1 62, Sa 64 Au 70] utilizing plane-wave (PWBA) and distorted-wave Born approximations (DWBA). The basic idea underlying these studies is a one step mechanism involving an interaction between the projectile and the transferred nucleons. The transfer process itself is assumed to be weak so that it can be treated in a first order Born approximation. The following derivation is presented as an outline of the general DWBA theory primarily to show the basic assumptions and approximations used to calculate direct reaction cross sections. The actual reaction process in DWBA can be broken down into three stages: 1) The projectile and target nucleus approach each other within their Coulomb and nuclear potentials. 2) The transfer is accomplished by a one-step mech— anism - for a (p,t) reaction the two neutrons are transferred at the same time, not sequentially. 55 3) The outgoing particle and the residual nucleus move apart under the influence of their Coulomb and nuclear interactions. Direct nuclear reactions are often divided into two types: stripping and pickup. In a stripping reaction, part of the projectile is transferred to the target nucleus (e.g. 24Mg(3He,d)25Al), while in a pickup reaction the projectile "picks up" particles from the target nucleus (e.g. 24Mg(d,3He)23Na). Although the theory is identical for these two categories of reactions, the conventions and notation differ slightly; For convenience the remaining discussion will use the conventions for jpickup reactions since the results will be applied to (p,t) reactions. 2. The Transition Amplitude, Ta b I For any nuclear reaction A + a-* B + b, commonly denoted by A(a,b)B (A = B + x and b = x + a, where x is the transferred particle(s)), the Schrodinger equation can be written (H-E)W = 0 , (III-1) where T’ is the complete wave function which describes all possible results of the collision, direct reactions as well as elastic scattering, multistep and compound nuclear reactions. The Hamiltonian can be expressed as , (III-2) 56 where Ka (Kb) is the Hamiltonian outside the range of the interaction potential in the entrance channel, Va (or exit channel, \QQ. Ka includes the kinematics of the system as well as the internal motion of the various particles and nuclei involved. The eigenstates of Ka can be defined as Ka ¢a = E ha . (III-3) The transition amplitude or T—matrix for the reaction can now be represented by [To 61] z (-) _ Ta'b _ . (III 4) This is the "prior" form of the T—matrix which is used for pickup reactions. The (-) superscript is used to indicate there are plane waves and incoming spherical waves in the b channel of the reaction. With this expression for the transition amplitude one can determine the reaction cross section u u k do _ l a b b 2 (2Ji + 1)(28i + l)(2nh ) a ’ Here “a and “b are the reduced masses in the entrance and exit channels and k3 and kb are the wave numbers. In general, for a reaction, with non-polarized beam and target, one must average over the spins of the initial channel and sum over the spins in the final channel; thus, the additional factors in (III-5), where J1 and Si are the spins of target and projectile, respectively. 57 The basic problem now is to solve for‘y(-), the complete wave function, which is very complex in its own right, by using potentials which are not fully understood. In practice, (13 obtain solutions to Equation (III—4) one must truncate 9“.) by replacing the exact interaction potential with an effective, phenomenological interaction. The potential that is universally used is the optical model potential (Ua), whose exact form is determined empirically. More details of the optical model will be discussed in the coupled channels section of this Chapter. The two basic restrictions of Ua are that Ua + 0 as ra+ m and that Ua be diagonal in the "a" channel. The eigenfunctions of this potential, usually called the "distorted waves," are determined from the expression (Ka + Ua - EH;a = 0 . ' (III-6) Introducing this new potential using a Gell—Mann, Goldberger transformation for scattering from two potentials [To 61], Equation (III—4) becomes _ (-) _ (+) - Ta,b - (Yb |Va Ualga > , (III 7) where the (+) superscript indicates plane waves plus outgoing spherical waves in the "a" channel. Since the dominant process in most reactions is elastic scattering, it follows that the major component of the interaction Va should reflect this. If Ua is taken to be the observed optical potential, 58 then Va- Ua =an becomes the perturbing residual inter— action responsible for the reaction. Thus, in general form, Equation (III—7) becomes (+) T xa‘ga > -)|V - (III-8) ( a,b = <“lb Now with a small interaction potential, an, perturbation techniques can be used, in this case the first-order Born (-) approximation. In the Born approximation l’b can be expressed as .+ + ikr (—) Ik-r _ ‘i’b - e + Ta,b r , (III 9) 1k * where e r is the incident plane wave. A more physical expression contains the distorted waves rather than the plane waves, because of the presence of an interaction potential as the incident particle approaches the nucleus. The first- order Born approximation with distorted waves consists of truncating Pg-) to (‘) (-) vb e 5b , (III-10) so that the expression for Ta b becomes DWBA ~ (-) (+) _ Ta'b - . (III 11) From Equation (III-2) Ka can be defined as Ka = Ha + HA + Ta , (III-12) with 59 Ha== internal Hamiltonian for the incident particle a , HA== internal Hamiltonian for the target nucleus A, and Ta== kinetic energy of relative motion. The associated eigenfunctions for these Hamiltonians are derived from the following expressions A Ji + (HA - EJiMA (A) - 0 (III-13a) a 81 + (Ha - Esi)a (a) = 0 (III—13b) a 2 (Ta + Va - Ei)xai 0 , (III-13c) + + where A and a are the internal coordinates and A a a E = EJi + ESi + Ei . (III—l4) Since each Hamiltonian spans a different space, the distorted waves can be expressed as a product of target and projectile internal wave functions along with an expression for the relative motion involved. Thus J S géf) = ¢AI(X)931(3)X;T) (III-15a) l 1 and J S (-) - f * f * (-) Ebf — ¢B (3)9b (b)xbf . (III-15b) In a typical pickup reaction the relative motion wave functions are a function of the relative momentum and a relative coordinate between the particles in each channel. Figure III-l displays the coordinate system used in this 60 Figure III-l. Coordinate System Used for Pickup Reactions, Denoted by A(a,b)B, Where b = a + x and A = x + B. 61 COORDINATE SYSTEM FOR PICKUP REACTIONS A(a,b)B Figure III-l. 62 derivation for pickup reactions. The expressions for the relative coordinates of Xa b are I + m +m E = ara xrx ' m + m a x and (III-16) + ' = + mx + ra - ra _ m + m rx ' x B where mx, ma, and mB are the masses of x, a, and B, + + _ respectively. Thus, Xé+) is a function of ka and ré, and)(é ) depends on Eb and R. Now, after substituting Equation (III-15) into (III—11) DWBA_ + + (-)*+ + Ta,b - J'dradrxxb (kb,R) Jf Sf Ji Si x <¢B Tb |Vax|4>a Ga > (III-l7) (+) * +. x Xa (ka, ra) Next, a parentage expansion, relating A to B and b to a, can be defined for 4A and O b ¢:1(A) = z F£0J(rx)[¢g'(§)[Y£(fx)o:(§)lJ]J loJJ' i (III—18a) and Sf + S' + A o * ¢b (b) = 1025' EAOS(rax)[¢a (aflyxuaxmxuflslsf . (III-18b) where the brackets represent angular momentum couplings: m 21¢§(§) . (III-19) + a + M _ [Y£(rx)¢x(x)]J = m2 (lmzomo|JM) y 1mg 63 (rx ) and E ) are called the nuclear and projectile F9.0J lo s(rax form factors. These will be discussed in more detail later. The quantity in parentheses is a Clebsch-Gordan coupling coefficient, the Y?“ are the spherical harmonics and 41(3) is the internal wave function for the transferred group, x, with angular momentum o- The quantum numbers 2 and 1, defined in Equation (III—l8), denote the angular momentum of "x" within A and "b" respectively. By inserting Equation (III-18) into + + + the expression for TDWBA and integrating over B, a, and x, one obtains DWBA Ta. b = 2 [(2L + 1)(2s + 1)]1/2(J MJMIJ.M.) i f Mf 1 1 LSJ x (SiMiSM|SfM f)(LM LSMIJM)FLSJ , (III-20) where M _ * * (- ) (+) *. PLSJ - fdradrx Xb bf (k b,R)fLSJ(ra ,r H)x ai (k a,ra). (III-21) The form factor, f contains all the nuclear structure LSJ' information involved in the reaction, l—m _ A 2 ( 1) (AmxnmzlLM) 201 fLSJ(ra 'rx) x W(loLJ:S£)F£ ) (III-22) oJ(rx x)V(ra x)Y X MAOS( where W(loLJ:S2.) represents a Racah coefficient. In the 64 expressions above, J, L, and S are the transferred total, orbital, and spin angular momenta, respectively. The angular momenta and spins along with their couplings are defined by the following expressions: + + + J. + J = J 1 f + + + S + S = S 1 f + + + L + S = J + + + (III-23) A + 2 = L + + + X + o = S + + + 2 + 0 = J . These relations are used in the determination of the selection rules for a particular reaction. Substituting (III-20) into Equation (III-5) one obtains the following expression for the DWBA cross section: QQDWBA = uiuf kg 1 an (2nh2)2 ka (23i + 1)(2si + 1) x z |TDWBA|2 (III-24) mimf MiMf Now, by utilizing the orthogonality relations of the Clebsch- Gordan coefficients and Equations (III-20, 21, 22), DWBA _ uiuf kb (25f + 1) M 2 5° " "7'": r arr—I'm— ’3 ' ’3 I'm' (”H-5’ an (2nh ) a 1 JM LS With this final expression of the DWBA cross section, the 65 remaining task is to evaluate the form factor, fLSJ , which contains all the nuclear information. 3. The Nuclear Form Factor From Equation (III—21) one sees that the expression containing the form factor contains a six-dimensional, nonseparable integral. This integral can be simplified by making a further assumption about the mechanism of the actual transfer of the particle group, x. It has been stated earlier that the process is assumed to take place in one step rather than by a sequential method. This is often referred to as the cluster transfer method. At this point, it is practical to assume that particle b is emitted at the same point at which particle a is absorbed. This is known as the zero-range approximation. Physically this implies that -;a = Ex' which effectively reduces the integral in (III-21) to a tractable, three dimensional integral. This approxima- tion has been shown [Ba 71] to be especially suitable for Os-shell projectiles (p, d, t, 3He,a.), which implies "a" and "x" are in a relative s-state or 1 = 0 in (III-23). The primary effect of the use of this approximation is an overall renormalization of the strength of the calculated cross sections. Equation (III—22) can now be reduced with the zero-range approximation using the relation + - + + 25 Vax(rax)EXOs(rax) - Do(r)6(rax) , (III- ) 66 + where Do(r) is the zero-range function, and 5(Eax) is the Kronecker delta function. Also, ? = ?2 - fl, which is the distance between the two neutrons in the case of a (p,t) reaction. The zero-range function can be written as D + - D + 27 O(r) - o¢b(rx) , (III- ) where, for (p,t), ¢b is the triton wave function. So, substituting (III—26, 27) into the expression containing the form factor, ZR _ D + (..) + + + -> rLSJM ‘ ° f draxb (kb'ra)FL3J(ra)¢b(ra) vxzs + l)(2L + 1) M m L A (+) + B + x yL (rama (ka, fig‘:‘fi; ra) . (III-28) DO can be determined for a particular reaction by sub— stituting the appropriate expressions for Vax and Elos' In the case of (p,t) reactions the form of Vax and E205 lS taken to be a gaussian shape [Ba 73]. Note that the variable of the distorted waves have also been simplified because of :a = Ex which implies + r a + R: I and I. —_ a m +1“ a + 3 ti! H+ The distorted-waves are usually described in terms of a partial wave expansion, 67 ma ma*R (k ,r >Y (f )Y ( ) a a L M a a a a La a La a (III-29) and similarly for)(é—). Here La(Lb) represents the angular momenta of the partial waves, the 0L are the Coulomb phase- a L , fL are the radial functions as determined a b from the solution of the optical model eigenvalue equation. shifts, and f The number of partial waves in a particular reaction at these energies (35 MeV) is primarily a function of the Z of the projectile and nucleus (or the range of their Coulomb interaction). Making these partial wave expansions for the distorted-wave gives .L L -L -L 1 rfisJ = 2 (i) a b exP(ioL ) L L a a b aMb 4n(2L + l) . b 1/2 x exp(1c )[ Lb (2La + l (III-30) x (LbOLOILa0)(LbeLMILaMa) M M x YLb(kb)yLa(ka)Ifisg b a b a and ILSJ I”? < )f (k )f (k m3 )d+ = r pr ’-__r I: o LbLa 0 LSJ a Lb b a La a mx + m8 a a (III—31) Note that the Clebsch-Gordan coefficients require Lb + L = La A_________;:l-IIIIIIIIIIIIIIIIIII-IIllIllllll-ll---_f. 68 and that (LBOLOILaO) vanishes unless L + La + Lb is even. This result, along with the conservation of parity, L Lb nanA(—l) a = nan(-l) , (III-32) requires that for the parity change of the reaction L+L An = n n n = (-l) b = (-l)L . (III-33) a A an This condition is often termed the natural parity selection rule. This is a direct result of the zero-range approxima- tion and is not true for a full finite-range calculation. Returning now to the explicit evaluation of the form factor for (p,t) reactions, F (rx) can be expanded in terms LSJ of single-particle wave functions, Uq , where the qi are the i neutron shell-model quantum numbers, nifiiji' and 1/2 + + F (r ) = 2 s U (r )U (r ), (III-34) S J l 2 L J x qlqz q1 q2 1/2 . . . . where SJ IS the spectroscopic factor def1ned by deShalit and Talmi [De 63] for (p,t) reactions: J. J l/2 _ l 2 J (2Ji + l) + + . Here, an and a are creation operators for the two neutrons 1 n2 Ji Jf and w are the wave functions of being transferred, and w the target and residual nuclei. The single particle wave functions are determined by solving the Schrodinger equation for a bound particle in a Woods-Saxon shaped potential, where 69 the well depth is adjusted to give each neutron one—half the two-neutron binding energy. In general, determining the form factor for two-particle transfer reactions can be very difficult. Unlike the procedure for single nucleon transfer reactions, the form factor cannot be broken up into a radial part and a strength factor because of the coherent sum taken over many possible particle couplings (see (III-34)). In single nucleon transfer the summation in (III—34) disappears since the transfer usually involves only one particular orbital of a shell. A similar situation can be obtained in a two-particle transfer reaction by assuming a pure two-particle pickup configuration giving Si/Z = l and thus allowing the radial dependence of F to be determined separately. This LSJ procedure is often used in (p,t) reactions when accurate shell model wave functions are not known [Ba 73, Br 73]. This simplification was used in this study and is discussed further in Chapter V. Two interesting features of two-nucleon transfer reactions which have made them a widely used method of examining nuclei are the distinctive diffraction-like shape of the L = 0 transitions and the concentration of the total cross section in the ground state to ground state transition for even-even nuclei (except near closed shells). Most of the angular distributions are expected to have some diffrac- tive-type shape, since they result from a superposition of 70 waves scattering from all parts of the nuclear surface. However, the L = 0 transitions display this feature much more so than higher L—transfers. This can be traced in part to the limited number of couplings that are allowed between the incoming and outgoing partial waves. Thus there are fewer terms in the superposition of the waves that can interfere and destroy the diffraction pattern. The explanation of the strong ground state to ground state transitions can be understood in terms of the pairing model of the nucleus [Br 73], where the ground states of even-even nuclei are considered to be "superconducting" states composed of many coherent pairs of nucleons. Thus, a two-nucleon transfer reaction is quite sensitive to these pairing correlations. The large cross sections result from the coherent sum taken over the many configurations that may be involved in the transfer (see Equation (III—34)). B. The Coupled Channels (CC) Method For Inelastic Scattering 1. Introduction In most inelastic scattering experiments the common starting point for analysis is usually with the DWBA [Ba 62], using a collective model form factor (see discussion in Section B.4). One of the major advantages of this method is its relative simplicity, as it deals with only one channel of the reaction and thus the numerical methods involve the 71 solving of uncoupled equations. The reason for the success of the DWBA method is a direct result of the comparatively weak interaction which causes the reaction and allows for the perturbation treatment. However, in some scattering reactions there are strong couplings between various levels of the target nucleus. When this occurs multi-step effects become significant in accounting for the varying strengths and shapes of the angular distributions. This feature is observed in scattering studies with the collective, well- deformed nuclei where the majority of the inelastic strength appears in the strongly coupled states of the ground band. Improvements in the DWBA method can be obtained by using second or higher order Born approximations. However, the computational difficulty is greatly increased. More improvement can be achieved by using the coupled channels method, where the interaction is treated to all orders and the number of levels that can be coupled is basically cietermined by computer size. Figure III—2 depicts the 'various excitation routes available in the CC and DWBA inelastic scattering methods. The DWBA method can only account for the "up" transitions of a, b, and c, the direct transitions. The CC method can calculate all transitions in Figure III—2 to all orders of the allowed angular momentum coupling (i.e. d has L = 2, 4, 6 components). For example, in the population of the 4+ level, the CC method includes pathways such a + d, c + e, a + g + d (where 9 represents the 72 . .NIHHH wusmfim 74 reorientation effect), and b, plus higher order multi-step paths, all of which can be included in one calculation. Although the computational problems encountered when solving many coupled equations are quite involved, their solutions result in a more complete treatment of the scattering problem. 2. The Coupled Equations The basic scattering problem is identical to that shown in the previous section on the DWBA. Solutions of the full Schrodinger equation, + + (H - E)W(r: A) = O, (III-36) will be sought for the scattering system with appropriate boundary conditions and an expansion of Ta (A). Defining H J A as the Hamiltonian for the nucleus of mass A and with + internal coordinates A, the nuclear eigenfunctions are defined by ‘P + -0 (HA - EGJ) GJ(A) ' r (III-37) where EaJ is the energy of state aJ and 0: represents all remaining quantum numbers needed to describe the state. For the system of the nucleus and scattered particle (in this derivation, a proton) the full Hamiltonian can be expressed as + + H = HA + T + V(r, A) (III-38) 75 where T is the kinetic energy of the proton, and V is the interaction of the proton with the nucleus. The spin-orbit functions of the proton are given by m _ A m stj _ [Y£(r)x(0)]j , (III-39) or explicitly showing the couplings m - .2 . A - stj — 2 1 (zmlsms|jmj)Y2(r)x(o) , (III 40) mamS where X is the spin function for the proton. For clarity, all the quantum numbers which describe a particular state of the nucleus and proton and their relative angular momenta before the collision will be denoted by C = aJlsj and all other states formed as a result of the collision by c'. By combining the wave functions for the nucleus and proton, the functions of total angular momentum, I, and parity, n, are obtained M M _ ¢ _ ¢c“I(r. o, A) [Ylsj a3]: , (III 41) with + 1 + l I = 3 + J, n = (-1) n2 . (III—42) + + Now an expansion ofHMr, A) in (III-36) can be performed in terms of the total angular momentum and parity, A + CHI g'n1(r' O, A) (III-43) M + + l V r c' CHI(LA) = (r)¢ 2 u c! Inserting this expansion into (III—36), multiplying from the left by the complex conjugate of (III—41), and making use of 76 the orthonormality of the ct functions, a set of coupled equations for the radial functions u(r) of the scattered proton is obtained for each I and n of the system. For each channel c', the coupled equations have the form I I (T.+V..(r)-E.)U.(r)=- Z V...(r)u"(r), c c c c c c"#c' c c c (III-44) where h2 d2 2(2 + 1) 2 2m Tc' = 5m ( 2 + 2 )' kc' ='_§ Ec' ' dr r h EC. = E - Edj . (III-45) with E denoting the incident proton energy. Note that the potential, V, has been broken up into the diagonal terms, on the left side of (III-44), and the off-diagonal, coupling terms on the right. Also I _ 5+ ++ A+ _ VC.C.(r) - <¢C.fl1(r.A)IV(r,A)|¢C.flI(r.A) (III 46) where the integration is over all internal coordinates and the polar angles of I. As it stands (III-44) is a system of equations, infinite in number, which must be simplified to achieve any reasonable solutions. The approximation that is made is to limit the number of inelastic channels to those with a large cross section and/or other special states of interest. The problem is now finite and can be solved numerically. As in the DWBA, the interaction will need to be adjusted to account for 77 channels that were explicitly eliminated. In fact (III-44) can be essentially converted to DWBA by simplifying to one scattering channel. This eliminates the coupled terms on the right side of (III-44). 3. The Effective Interaction: The Optical Model Potential The introduction of a nuclear model at this point allows for a parameterization of the potential and provides the necessary wave functions, that] , for computing the matrix elements in the form factor. Also at this point it is assumed that the orientation of the nucleus does not change significantly during the time the proton is within the range of the interaction. This is referred to as the adiabatic approximation [Cl 673. The standard form of the interaction potential that is used in most reaction problems is the complex-valued optical model, developed in 1953 by Feshbach, Porter, and Weisskopf [Fe 53]. The shape of the potential is usually that of a diffuse Woods-Saxon, which has the form f(x) = (1 + ex)'1, x = r ' R , (III-47) where R(= rRAl/3) and aR are the radius and diffuseness parameters. The actual form of the optical potential used in proton scattering is given by V(r) = -(V + iW)f(x) — 4iwDaIf'(xI) ++ 21 - VSO(0°£)X f f'(XSO) (III-48) 78 Here, V, W, WD, and VSO are the real volume, imaginary volume, imaginary surface, and spin-orbit well depths, xI and xSO are similar in form to (III-47) except with different values for the radius and diffuseness parameters. Also, X is _ df(x) - dr ' part of the potential represents an average interaction the n—meson compton wavelength, and f'(x) The real between the projectile and the nucleons of the target, while the imaginary part represents absorption of particles from the incident beam. Thus the imaginary part also fulfills the need to account for the channels not explicitly included in W after truncation. The scattering from a deformed nucleus is assumed to result from the interaction of the projectile with the part of the nuclear field arising from the deformation. 'To account explicitly for this, the nuclear shape is parameterized by the usual expression R = ROE1 + f; Bxuyxu(e"¢')] = R + SR , (III-49) with the sums over even values of A and11. The parameters 6' and ¢' refer to the body-fixed (symmetry) axis of the nucleus, and the BM1 are the deformation parameters. Sub- stituting this expression into (III—48), which along with the Coulomb potential, V Coul' interaction in (III—46), and expanding V about the spherical has been taken as the effective shape, one obtains for the nuclear part of V Q x 136R)“ a“v n. arn v (r - R) = v (r - R0) + nuc nuc (III-50) n=l 79 This new form of the optical potential is often called the deformed optical model potential (DOMP). Since 13 .. z . "' B Y ' (III-5].) Ro Au Au Au an expression in terms of the generalized deformation parameters, 6, can be obtained by using the addition theorem for spherical harmonics [R0 57], 6R n n (—-) = z 6 Y' RO LK LK LR 62K = 22 (£%%)1/2(A}JK|L1 - K)(AOA0|L0) (III-52) AkuA n-l x 5Au Bxk (for n > 1), where ' - = + 1. 6Au Bxk and 1 2L Now vnuc(r — R) = Vnuc(r - R0) + E; NLp(r)YLu(e ,¢ ) , (III-53) where °° ("'R )n anvg) - o n r NLp(r) - nil n: 6Lu 3:“ - (III-54) Likewise, a similar expression can be obtained in terms of the generalized Coulomb deformation parameters, so, for the Coulomb part of the potential (see Reference [61 67]) VCoul(r') where a variety of charge distributions can be used for p(r"). Using the D-functions [R0 57] to express the 80 potential in the laboratory frame and combining the nuclear and Coulomb parts of V, one obtains V[r - R(e,¢)] = V(r - R ) + 2: Y (e.¢) Z V (r) 0 LM LM K10 LK L L DMK + DM,-K x ' (III-55) 1 + 5K0 where VLK = NLK + CLK ' the nuclear and Coulomb parts. The first term of (III-55) is just the spherical optical model potential of (III-48) plus the Coulomb potential. This term has only diagonal matrix elements, the V' of (III-44), while the second part of c'c' (III-SS) is responsible for the excitations of the nucleus from one rotational state to another. 4. The CC Form Factor The coupling potential can also be written in a general form as IN Vc'c u(r) = i VLK(r)(QX-Yx) (III-56) where QA is an operator which operates only on the coordinates of the target nucleus. Comparing the second part of (III-55) with (III-56) yields a J a J l 1 2 2 V ' n = 2 C F I c LSJ SLJ LSJ (III-57) 81 where CLSJ is a geometric factor, defined in Reference [G1 67], and FLSJ is the form factor a J a J l l 2 2 _ FLSJ ‘ VLK ' (111‘58) where QL is the multipole operator of the form L L DMK + DM,-K I l + 5K0 for the symmetric rotor model. The problem has now been reduced to the evaluation of a series of form factors, which involves determining a set of optical model parameters as well as computing a group of matrix elements. As an illustration of the coupling nature of this problem, in particular the role of the deformations, BA' the form factor for an L = 2 transition in the ground band (K = 0) of a symmetric rotor will be derived. For clarity only the nuclear part of VLK will be used and the 6 values of (III-52) will. be taken to second order. Starting with (III-58) and substituting the simple expression for the matrix element [Pr 75] -L 1/2 (JIOLOIJZO) . (111—59) 1 J FL0L(r) = VL0(31) ('1) Taking the expansion of 6 to second order 6:0 = 2 [55—]1/2(A0A'0|L0)B B . . (III-60) 11' 4nfl A A where A and A' are cut off at 4 and again L = 2L + 1. Thus explicitly 82 52 = [2—331/2(2020|L0)Zs§ + 2E2—gll/2(2040|L0)23284 4flL 4nL + [5—311/2(4040|L0)282 . (III-61) 4nL Now consider a 0+ + 2+ transition with Jl = 0, J2 = 2, and L = 2. Inserting (III—61) into (III-S4) (with no Coulomb part, VLo = NLO) and combining with (III—59) one obtains J =0,J =2 _ 1 1/2 _ 8V F E 2 ‘ Ro‘ififi) [ B2 5? L-2 R 2 + o 3 V 2 2 -§ 5;: [5(2020|20) 82 (III-62) + 2(45)1/2(2040|20)28284 + 9(4040|20)233 The L = 2 collective model form factor mentioned earlier in the DWBA section is actually the first term of (III—62) EX .. qusz 8r . (III 63) 5. Solving the Coupled Equations It should be noted that there are considerably more coupled equations than there are nuclear states being considered. This is because "c" labels the nuclear state aJ and the angular momentum of the scattered particle. So for each value of I,n, the total angular momentum and parity of the system, and for each nuclear state aJ, there will be 83 scattered particles with angular momenta 2, j, that satisfy the conditions of Equation (III-42). For spin 1/2 particles there are 2J + 1 such couplings and thus for each I," value there are N = 2 (2J + l) (III-64) coupled channels, where the summation is over all states under consideration. For a spin zero projectile, N Z (J + 1). In a calculation of proton scattering on a nucleus where a 0-2—4—6 level sequence is used, there would be a maximum of 28 coupled equations for each I,“ and for alpha scattering, N = 16. With the angular momentum of the incident partial waves ranging in value from .2 = l to lmax = 2kR, where R is the radius beyond which the interactions are effectively zero, and 2j states for each A (for protons), the total number of differential equations that need to be solved is about 4kRN2. The usual method of solving coupled channel problems is to impose boundary conditions on the radial wave functions uc. of Equation (III-44); at the origin, where the functions must go to zero, and in the region where the nuclear potential is negligible, the equations become uncoupled and the uc's asymptotically become k _E_ 1/2 I (k) U , + GC'CIC(kCr) SCC' C OC'(kC'r) . (III-65) Cl (Here the I and 0 functions are incoming and outgoing waves respectively, which are represented as a linear 84 combination of the regular and irregular Coulomb functions, F and G. The Sic' are the elements of the S—matrix. Two sets of linear algebraic equations can be set up and solved for I cc" The S—matrix is related to the the matrix elements of S transition matrix, ch" by S = 6 - 2111TC CC. CC. .6(Ec - EC.), (III-66) C from which the cross section can be obtained (see Equation (III—5)). Recently, a new method has been derived for solving equations using a sequential iteration process [Ra 72]. The program which incorporates this method is called ECIS (equations gouplees en iteration sequentielle) [Ra 73]. The use of this method is increasing rapidly because of the drastic reduction in computational time in most cases, however, the storage requirement for this method is larger than for the usual methods for coupled equations. The ECIS method actually gives better results than DWBA codes after the first iteration, while the second iteration often gives the same results as the standard coupled channel method. The last iteration, defined by a convergence test on the phase shifts, is required to give the same result as the standard method. The details of both the integral and differential methods used in ECIS are given in Reference [Ra 72]. 85 C. Charge, Mass, and Potential Moments One of the most fundamental. quantities that. can be learned from inelastic scattering experiments is the shape of the nucleus. There are basically two methods used to study this property of nuclei with each sensitive to a different aspect of the nucleus. The first method involves low energy hadronic particles for Coulomb excitation, and electron scattering, where both serve as a probe of the charge (proton) distribution in the nucleus. The other group of measurements are made with higher energy hadronic particles that sample the mass distribution of both the neutrons and protons. With these two methods available, one may be able to detect differences in the charge and mass distributions within a nucleus, if they exist. This might be obtained by a comparison of the results from these two methods, however, this in itself is not a straightforward task. Because the Coulomb interaction is well understood, the measured charge moments can be related to the charge dis- tribution with only a few model dependent assumptions. The L-th moment of a charge distribution can be defined as _ . = A + + .- q). = M(EA,u) Ir Yx“(6,¢)pc(r)dr , (III 67) Where p is the nuclear charge density. This is also the same form for the electric multipole operator and can be related t<> the reduced matrix elements for y-decay in a collective nllcleus 86 = (2J + 1)’1/2(J M AuIJ M ) A f 1 i f f x , (III—68) By specifically introducing the reduced transition probability, B(EA), for quadrupole transitions on obtains -1 2 B(Ez,Ji + J = (2Ji + 1) || . f) (III-69) Thus, the charge moment, qx, is equal to the matrix element in special cases q: = B(EA.0 + Jf> = M(EA,0 + Jf) 2 . (III-70) In summary the measured experimental quantities are the qx's which can be related to the charge distribution by (III-67), where pc is the only model dependent term. This rather direct approach of relating the measured quantity to the underlying density is not available for the high-energy scattering measurements because the nuclear force is not well understood. This mandates the use of an effective interaction for the analysis of the data. One method is to use a deformed optical model potential (DOMP), which introduces additional uncertainties in any attempt to relate the deformation parameters of the DOMP to the more Physically relevant mass distribution. This study will follow the suggestion of Mackintosh [Ma 76] and discuss the reasults in terms of normalized potential moments. These are GQuivalent to the mass moments of the nuclear if the 87 underlying nucleon-nucleon interaction is assumed to be independent of the local density [Sa 72, Ma 76]. There is also evidence [Ma 74a] that this method is less model dependent and thus more fundamental than the usual deforma- tion parameter, BA’ or the deformation lengths, BXR (R = roA1/3). The L-th multipole of the DOMP is defined as KfV(r - R(e))r)‘Yx (6)dI qx E 3 . (III-71) fV(r - R(e))dr where V(r - R(e)) is the real part of the optical potential and K is chosen to be Z, the atomic number. This gives a charge component of the potential for comparison with the charge moments measured in electron scattering, or Coulomb excitation from low energy hadron scattering. With the results of the two types of measurements cast in a form to facilitate their comparison, one now must judge whether any difference between the charge and potential moments actually reflects a difference between the charge and mass distributions of the nucleus, or whether it is the result of ignoring the density-dependence of the nucleon- nucleon interaction, or one of the other assumptions used in this method. Calculations [Ha 77] for 154Sm indicate a 20—30% difference may exist between the potential and mass moments. This implies an even more detailed approach may be needed to extend these methods to measure differences in charge and mass distributions of deformed nuclei. CHAPTER IV EXPERIMENTAL PROCEDURES AND DATA ANALYSIS A. Introduction An important consideration in undertaking this reaction study was the quality of proton beams available from the MSU cyclotron. The MSU cyclotron is a variable energy, sector— focused machine with single-turn extraction capabilities. The single-turn extraction results in a very small spread in energy of extracted beam, typically 2 0.1% of the beam energy. This property, when combined with proper spatial correlations of the beam, allows one to operate the cyclotron-spectrograph system in the dispersion matching [Co 59, Bl 71] or energy-loss mode, which will typically result in 5—10 keV full width half maximum (FWHM) resolution for inelastic scattering of 35 MeV’protons, and 1.5 keV under ideal conditions [No 75]. Although the level density in the even-even Pt nuclides studied is not very great below 1.5 MeV, nevertheless, resolution and low background are very important in a reaction where the population of weak states is of prime concern, as was the case in the (p,t) reactions. 88 89 B. Proton Beam and Transport System The 35 MeV’ proton beam was used in both reactions studied, primarily for two reasons. In the (p,t) reactions, the maximum energy tritons that the Enge-split pole spectrometer can bend is approximately 30 MeV, so with the Q-values for these reactions near -5.0 MeV, a 35 MeV beam will produce the maximum energy tritons for the spectrometer. The highest energy tritons are desirable because a more oscillatory diffraction. pattern. will result. The .35 MeV proton beam is also one of the most reliable beams available on the MSU cyclotron, with highly reproducible set-up conditions for the cyclotron and beam lines. Beam currents are typically 1 to 2 DA on target. Figure IV-l shows the relationship of the cyclotron, the beam transport system, and vault 3 where the spectrograph is located. The transport system includes three bending magnets (M3, M4, and M5), two intermediate focus points (box 3 and box 5), and several sets of U am: on» um mmu< uasm> Hmucoefiuomxm .H..>H 853m 53% n o. \ fl . - V . 91 l u _fi_-,___.r-____ ‘ x a x n..5:<> .. I V V \ 1 ‘ J N V i I 1 I 1 H \. . \\\_ . . N \ I I k N h L U \ \..x x :36 no: 3...... IO . \ 200 0.90 32.23 Soc! .0 o )I. .1 f \\ I . x \ / \ / .5 {on .o _ ciao In .c can . Bio ’33.... s 0 x. o \\M\ «O b 0’ ‘0. x x4\, \\x 'a .. V ‘Y. 93... '1: )hv/ .\.\ \ , U . / \ us \5 t .Iom '50 m :‘fi'b D V\.LI . own.“ To U. -..mslm. .. mo 165 : K 86 x a :32: I : J! t Sui-lat: , grad.- .IQ. is about 8:23 2 52> 205386 I 9523 1 w . I R x . v .4 ( «000 :08 92 .nmmLOOLDUQO may mo mcmam Hm00w may cfi xoo mumEmo may wpfimcfi pmomam mum mmumam pom wHOuomumU one .cmmbmobowmm odomluzmm omcm can. LonEmnu mcfiuwuumom mo mewzmuo owumecom .NI>H 0.33m 93 .NL: 933R. .255 05.23% om ma. 5 63m no.6» 5201 5.8280 9.8525. to. :28: 22a -1 - 8.... 33:30 .. Soc 32.5 ,, . 82a 28 86> anm/ RR. R 94 which is used for visually adjusting the focusing and shape of the beam spot in the target plane. The amount of beam on the target was monitored with a 00 Faraday cup with a current integrator. The beam and target conditions were monitored with a well shielded NaI (T9,) detector placed at 90° to record elastically scattered protons. Two aperture sizes were used in the reactions: for the (p,t) study a 20 x 20 aperture corresponding to a solid angle of 1.15 milli- steradians, and for (p,p') the 2° x 20 as well as a 1° wide by 20 high aperture or 0.60 msr. The reaction products were analyzed with the Enge split pole spectrograph with particle groups of the same momentum focused at the detector plane. The use of a magnetic device in particle reactions allows one to select preferentially particles of given types and energies to be incident on the counter. Since different particles with varying energies have different magnetic rigidities (particle momentum divided by its charge), the magnetic field can be adjusted to study only particular groups of one particle type. This technique worked very well for the (p,t) reactions because of the large negative Q-value. However, in the (p,p') reactions, there is a small background due to a virtual continuum of high energy deuterons and y—rays entering the counter and even a discrete state from the 12C(p,d)11C (g.s.) reaction which appears near the ground state. But the majority of deuterons and tritons (from Pt(p,d) and (p,t) reactions) appear at another position in the focal plane. 95 C. Dispersion Matching The high-resolution data are obtained by using dispersion matching and kinematic compensation at the focal plane. The dispersion matching is achieved by adjusting the dispersion of energies in the beam to match the dispersion properties of the spectrograph. The energy spread must be coherent so that particles of energy E + AE go to one side on the target, and particles with energy E — AE go to the other side of the center, with intermediate energy particles in between these extremes. The dispersion of the spectrograph then focuses particle groups with the energy spread :tAE at the detector. The kinematic compensation corrects for the differing energies of particles across the aperture due to recoil of the target nucleus. This is done by moving the detector within the reaction plane. Experimentally this condition is achieved via a "tuning" procedure involving the beam transport elements, a small tuning counter in the focal plane, and a computer feedback sytem controlling the spectrograph magnetic field. The tuning is accomplished by adjusting the transport elements and focal plane positions to minimize the line width of elastically scattered protons from a thin target (usually 50 ug/cm2 Au foil). A 1 cm delay-line counter [RCMe], backed by a solid state counter, was employed. The computer feedback system is necessary to keep the elastic line on the detector. With the particles incident on the counter at 900 96 (versus the normal running mode of 450), line widths of 70 um have been obtained. D. Particle Detection Two types of particle detection were used in these experiments. The highest resolution data were taken with Kodak NTB 25)1 nuclear emulsion plates. For the (p,p') data, a 15 mil stainless steel absorber was placed in front of the plates to lower the energy and thus increase the ionization of the protons. The resolution achieved for the plate data ranged from 5 to 10 keV FWHM. Figures IV—3 and IV—4 show sample spectra for both types of reactions. Although plates do provide the maximum resolution and do not require an elaborate electronics setup, there are still drawbacks in measuring a complete angular distribution for a series of reactions. The data from plates are obtained by visually scanning the plates for individual tracks with an off-line microscope. This is a very time-consuming process, which, for high track density plates, may require several months to complete a scan for one angular distribution. For the six reactions in this study, the plates were used primarily for calibration purposes by taking a spectrum at two or three angles to insure that no peaks in the spectrum were obscured by impurity lines. However, in the (p,t) reactions, plate data were taken at the first and second maxima and the first minimum of an L = 0 transition as an aid in searching for weaker or unknown L = 0 transitions. Figure IV-3. 97 Triton S ectra Recorded at 70 for the l94r196'l 8Pt(p't)192’l94'196pt Reactions. 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The energy, TOF, and AE are used to provide a coincidence requirement and provide a strobe to the position TAC and AB linear gate. Thus, unnecessary signals from unwanted particle groups have been electronically gated and are not processed by the computer. Figures IV—8 and IV—9 show sample spectra taken with the wire counter setup for each reaction studied. The resolution for the triton spectra was typically :215—20 keV FWHM and for the proton spectra was 18-25 keV FWHM. E. Targets Two types of targets (for properties see Table IV—l) were used for this study. In the wire counter experiments rolled-foil Pt targets with a thickness of approximately 625 pg/cm2 were used [FKar]. For the data taken with photo- graphic plates, thinner targets were needed to minimize resolution degradation due to straggling and energy loss in the targets. For the 600 ug/cm2 targets the energy loss and 108 Figure IV-8. Triton Spectra for the 194"196'198Pt(p,t) Reactions at E = 35 MeV. The data were obtained with a delay-line proportional wire counter in the focal plane of an Enge split- pole spectrograph. 109 II L 7+ A I .. 9 J l 1’ 4H0 .H__ . r! .9 N- b 8 Im II ... e I .... r .N am? 9 I..J 1 no» ..Ho 4| p8 P I.... 2 uI P : 9N mum... 9 b .....J % Io. ITIO 1 9 s 0.0 73 m8. _ H. 3 ... mmmr P... % w 1 GI. .o 9m? ... mm: .9 SN: -m mum? F .8? 981 .o mme A a 2.8 mum? m8. . an“? mamas I .: mmmm- ...; .8me Km? wmdunvi mnwaomwmwomnmao 1 1 3 2 1 4m22 U!» 3p 2» 1+ {5P u!» 3» 2? b 5» ”lb 3+ 2» lb Oil 0 0 0 0 0 0 0 0 010 0 0 0 0 0 0 4|. 1|. 4|. 1 1|. 1|. 1 1 1 1 1 1 1 1 CHANNEL NUMBER Figure IV-9. 112 .mcflxomn m mm mom: was HHOM conumo ~20\m: om do .muOuMuonmq Hmcofiumz mmpfim xmo an cosmMCusm mommamc¢n .maw mag mocwuwummm mud mhw mm.mm ma.m mH.H m>.o Ho.o Ho.o mm.vl umwma omH omm mm.o Hm.>m hm.H mm.o Ho.o Ho.o hm.mI ummma OHH omm mo.o mm.o m¢.H av.nm mHo.o Ho.o mm.mI umwmfl w m m wmum on “mun: m mafiom ummma ummma ummma unema ummma umoma A> z. u a A~EU\m:V mmmcxofine uwmume mwsam>Io va omocmpcsn< OMQOuOmH .cofiuwmomeoo uowume .HI>H wanna 113 straggling particles contribute about 4-5 keV to the resolution, while» the thinner, sputtered targets cause .a 1-2 keV contribution. So for the wire counter the thick targets sufficed, but for plates, thinner targets were necessary. Targets [No 78] were made for this purpose by sputtering Pt metal onto a backing resulting in thicknesses of z 200 ug/cmz. 'The platinum used in each target was isotopically enriched to near 97%. This provided very clean targets for the foils but the sputtering process introduced several impurities (Sn, Ni, Fe, Cl, Na, Si) which limited the usefulness of the plate data for the (p,p') reactions. For the (p,t) plate data, however, the large negative Q-values allowed for very clean spectra. F. Data Analysis Peak areas and centroids were determined with two computer codes, AUTOFIT [JRCo] and SCOPEFIT [HDav]. Each program uses an empirical reference peak shape (usually derived from the ground state), with a user-supplied background to unfold the peaks of interest. The energy calibration was performed for each reaction by two methods, both using six or seven known excitation energies for platinum levels as given in Nuclear Data Sheets. The first method involved fitting a second—order polynomial to the levels and the second method used a kinematic routine to map out momentum versus distance along the focal plane for the 114 MSU spectrograph. Both methods generally agreed within 0.2% in the energies up to 3.0 MeV in excitation. Since only one excited level is known with any accuracy in 198Pt, a separate calibration experiment was performed to obtain energy information in this case. Plate data were taken at 75° and 196,198 20 43° for the Pt(p,p') and 6Pb(p,p') reactions. Spectra from all three reactions were recorded on one plate at each angle. Only the height of the plates was adjusted for each reaction, not their positions, thus assuring an accurate relative calibration. These two angles were chosen for minimum interference from the silicon, carbon, and oxygen impurity peaks. 196Pt was used because of the well-known low-lying levels, while 206Pb was chosen because of the strongly excited 3- level at 2.648 MeV. The known levels in 196Pt and Pb then provided the necessary calibration lines to bracket the 198Pt levels of interest. Due to energy losses from the different target thicknesses, the absolute energies of 196Pt and 206Pb cannot be used for calibration, so a value for the gain (keV per channel) was obtained for each l98PtH musmflm mmmzzz nmzz<1o 8mm 88 8.5 89 omml 82 0mm CD H 118 ____ 9:. +2 94d- ++1 1186'- ! (.II] 45%?! 498I .2 40h- $6- Dzt Previous Resultsa Ex J" o(7°) Exc J" (MeV) (ub/sr) (MeV) 2.132 7 2.1301 2.140 13 2.153 8 2.1494 2.166 19 2.188 7 2.204 17 2.271 18 2.308 25 2.330 (3) 1 2.3356 2.352 4 2.358 7 2.375 2 2.3755 2.389 2 2.411 6 2.4085 2.428 15 2.444 11 2.450 9 2.4533 2.467 13 2.4722 2.486 6 2.492 8 2.506 1 2.526 6 2.549 21 2.556 11 2.575 4 122 Table V-l (cont'd.). Present Experiment 194Pt(p,t)lgth Previous Resultsa EX J1T 0 (7°) eb Exc J" (MeV) (ub/sr) (MeV) 2.588 6 2.5853 2.605 7 2.624 6 '2.646 18 2.662 5 2.671 3 2.695 4 2.704 6 2.720 7 2.729 10 2.743 6 2.754 6 2.778 5 2.786 4 aReferences [Ny 66, Be 70, Ya 74, Hj 76, R0 77, Sa 77, Ba 78]. The enhancement factors were calculated with pickup con- figurations (0p ) for L = 0, (2p €>lf ) for L = 2, and (1135/2 ® 2p§//22) for L = 4. 3/2 ”2 cThe states above 2 MeV’ seen in this work and previous results are associated only because of similar energies. b dUsed as calibration point with energy taken from Nuclear Data Sheets B2, 195 (1973). 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'I".I| Ill"... .8.8.88088 8I> 88888 132 .88888 2888 m88>88 >8: 88888.8 8:8 .88888.8 .88888.8 .88888.8 888 8883 88088 8:808 80888888888 88 88888 .888808cc8 88 888088 .>mz 8.8 8>onm 88.o 82WI>8S 8.8 30889 >8x 8 >8wume8x08888 888 xmumcw co8umu80xw c8 m88uc8muumoco .8888: 888 .8m mumwcm 8888 8888082 20.88 cmxmu 888885 583 8:808 20888828880 mm 888: .mm8wumcm 8888688 mo 6 8858083 88cc 88888008m8 888 8888888 mso8>8u8 8cm x803 88:» c8 cmmm >82 8 m>onm mmumum 8290 n .mmh 8U .mh mm .mm mm .mm mu .mm 80 .mm :2“ wmucmumuwmm 808888.88 888.8 8.88 888888.8 88 888.8 88 888.8 88 888.8 888 888.8 88.8 88 888.8 +8 +8 888.8 88 888.8 8>828 888\n:8 8>828 888\n:8 8>828 x x x =8 o 8 808888 :8 m 88 808.8 88 m 8.8 .8 8. 888888 8888888 888888 88885888 mso8>888 '.l|lln'~'|' "'."|'I"""I' | ' I I "' "I'l‘l' I" '1' mucme8um8xm ucmmwu8 'I' ' Ill'll'ui l --a' ! .A.w.ucoov mt> manna 133 «distribution has a characteristic shape. of a jparticular L-transfer. In this study experimental shapes were compared to DWBA calculations and to shapes of angular distributions with well-known L-transfers. This study at 35 MeV is the first one in this mass region where L—transfers higher than L = 0 have been assigned (with the exceptitnx of Pb(p,t) [La 73, La 77]). The results of these shape comparisons are discussed further in the following sections for various L-transfers. B. DWBA Analysis The experimental angular distributions have been compared with standard, zero-range distorted waves calcula- tions using the code DWUCK [PDKu]. Table V—4 is a list of the optical model parameters used in analyzing the (p,t) reactions. Becchetti-Greenlees [Be 69] proton parameters were used in the entrance channel, and the triton parameters were taken from Flynn et al. [Fl 69]. The wave functions were calculated for a Woods-Saxon potential with the usual prescription for the binding energy of each neutron, 0.5 (S2n + Ex)’ Here S2n is the two-neutron separation energy and Ex is the excitation energy of the residual nucleus. In order to test the effect of small changes in the optical model parameters, calculations were carried out using the Becchetti-Greenlees proton parameters for 208Pb along .8888808 88888888 8:8 80 888888 8088888ox8 8:8 8:8 808888 8088888988 808888: 038 8:8 mo :58 828 88887880 888.80 £888 838 08 88888888 8.883 828888 8883 808888: 82.8. 134 m n o 088 H8 88 .888888 8608500 888 m8 8 .mWIII u .x 888 .m u x .m n x 888 8x8 + 80 n 8x88 mmlu mlu I mlu 8| 088 08 8 a 88883 8.88.8Iwu8888.88 > u lm8.8 8 388 I 8888838 + >8- u Auv> ”8888828888 80 80888888888 +4. 8 88888 In: 88.0 mm.8 mmu8 :1: sun In: III 88.0 m~.8 n > 82:08 88.8 nu: nu: vnu 888.0 888.8 III 8.88 ~88.0 88.8 0.888 88888 +8 88.8 88.0 80.8 m.8 888.0 mm.8 8.8 0.8 88.0 88.8 ~.mm 88888 +0 m 88.8 1:: us: 1:1 888.0 088.8 nun mm.~8 888.0 88.8 0.888 88888 +8 88.8 88.0 80.8 8.8 mm8.0 mm.8 8.8 0.8 88.0 88.8 8.mm 88888 +8 N 88.8 :uu 1'. sun 888.0 888.8 uzu 88.m8 N88.0 88.8 0.888 8m~m8 +8 88.8 88.0 80.8 8.8 888.0 mm.8 8.8 0.8 88.0 88.8 m.~m 88888 +m 8 8 cm 08 08 8 8 o m m 888 8 8 8 > 8 8 3 3 8 8 > 8888880 888858888 .mco8888so880 88.Q8 8888B 135 with the 206Pb triton parameters of Flynn et al. The results showed no major changes in the strength or shape for any of 196(p,t) reaction. the transitions calculated for the Since the platinum nuclides display low-lying collective excitations, one might expect that second-order or multistep effects would affect the strength and shape of the angular distributions in (p,t) reactions. This has been found to be true for reactions on well deformed nuclei [As 72, Ki 72]. Such effects are not accounted for in simple DWBA calcula— tions. But, there is an absence of such strong effects in the platinum nuclides, possibly because of relatively smaller values of the quadrupole deformation parameter (32 -.~. 0.15 for the Pt nuclides, whereas 82 2 0.3 for well-deformed rare earths). Since the strength of multistep couplings depends on terms involving various powers of 82, the smaller value of .82 may be responsible for the reduction of many of the second-order reaction steps. Further evidence for the predominance of the one-step mechanism is the absence, in all three reactions studied, of any strength (> lub) populating the unnatural parity states, in particular the 3+ level known to exist at = 950 keV in 194'194'196 Pt. Such transitions are forbidden to first—order in a one-step process. Previous work [As 70, Ud 74] comparing DWBA with two-step coupled channels calculations for 62 Ni(p,t) and Cd(p,t), nuclei with collectivity similar to Pt, has shown that there are very small differences between the two reaction models in 136 predicting shapes of angular distributions. The main effect of the two-step mechanisms was seen in the transition strengths. One method for obtaining spectroscopic information from two-nucleon transfer cross sections with DWBA calculations is to use an empirical normalization (Di) to define an enhance- ment factor, a , for the configuration which produces the strongest calculation for a given L transfer [Ba 73, Br 73]. The relationship between the experimental and calculated cross sections can be expressed as: -l LSJ 0 DW (9) . (v-l) Q = 2 2 (an exp 9.72 no a c (2J + 1) The factor D: is the normalization constant which results from making the zero-range approximation. A value of 2.2 x 105 MeVZF3 was used in these calculations [Ba 73]. The constant 9.72 is derived from the choice of the size of the outgoing triton used in DWUCK and the range parameters of the two-body interaction. The isospin coupling coefficient, C2, is unity for all transitions. The quantity J is the total LSJ is the differential cross section calculated in DWUCK. The angular momentum of the transferred neutron pair andcy factor 6 is a measure of the adequacy of the wave functions 1 would used in calculating the form factor. A value of s indicate an ideal wave function description if all other assumptions were valid. In the present case 2 represents the relative strength for a particular L-transfer expressed in 137 arbitrary units (oggJ calculated from the dominant 2-neutron configuration). This allows for the unfolding of kinematic factors which may favor a particular L-transfer. The configurations used for each L-transfer are in the footnotes for Tables V—l, V—2, and V-3. As mentioned above, these configurations produced the greatest calculated strengths for their respective L—transfer in each of the three reactions, 194’196'198Pt(p,t). No attempt was made to study the interference effects of using more than one term in the configuration. This could be an important factor as far as the strengths of the calculations are concerned since a coherent sum over these terms is involved. However, the enormous number of configurations precluded any neaningful approach to the problem. C. L = 0 Transitions As expected, the l.= 0 transitions were observed with the very characteristic diffraction pattern seen in most two- nucleon transfer reactions, allowing for reliable assignments of 0+ levels in the final nucleus. Eleven L = 0 transitions were observed in the three reactions, including the three ground state transitions and one transfer to a newly 192Pt. The L = 0 identified excited 0+ level at 1.628 MeV in transitions are shown in Figure V—l, along with the DWBA calculations. There are few differences from nucleus to nucleus, either in the phase of the distributions or in the 138 Figure V-l. 0 Angular Distributions for the 194' 196’ ’198Pt(p, t) Reactions. The curves are the results of DWBA calculations. Energies are given in keV. 139 r I I I I I |94p' (p. ”IQZP' L=0 h h b h _ I I III" \\ TN“ I [HUI O I TIIIIII' 0' I I I IIIIII da'ldfl (pb/sr) TV [IIII] I I IIIITII r I I IIIIIr 1 11111111 1 11111111 1 1111111] 1 11111111 1 1 1 1111' 1 11l1111l 1 1 Hum 1 111111 \ n r IIII II] I I I Inn] 9. 1 11l1111l 1 1 111111 111111] l 1 1 111111l 1 1111 1 1 1111111 1 1 1111111 1 1 111111] T I T T I '95P1(p. t)'9°P1 L=O 1 11111111 1 111111 11111 LJ 1 1 1111111 20 40 9mm. (deg) Figure V-l. 140 peak-to-valley ratios. The same is true for the calcula- tions, which show only slight deviations at forward angles. In general, the calculated shapes are independent of Q—value or choice of the simple 2-neutron configuration used in computing the form factor. The ground state transitions are by far the most intense transitions observed in each reaction. The strongest excited 0+ state in Pt is populated with only 8% of the strength of the ground state at 7°. The 0+ state at 1.195 MeV in 192Pt, previously seen [Be 70, Fi 72] in the decay of 192Au, was unresolved from the 4+ level at 1.201 MeV in the proportional counter data used for angular distributions. The spin of this level was confirmed using the three point angular distributions taken with nuclear emulsions. The new 0+ level seen in 192Pt at 1.628 MeV was populated with 5% of the strength of the ground state at 70. Three excited 0+ states were populated in the 196 4 Pt(p,t)19 Pt reaction. All three states were previously seen [Be 64, Be 70, Cl 76] in the decay of 194Au. The level at 1.479 MeV is very weakly excited (< 0.5% of the ground state at 7°) and was resolved only in the plate data. The L = 0 nature of the transition populating this state was also confirmed by the three point. angular distribution. The levels at 1.267 and 1.547 MeV were excited with considerably more strength, 3% and 6% respectively of the ground state 141 strength at 7°, and the 1.547 MeV level was the only excited 0+ state seen in the earlier (p,t) study of Maher et al. [Ma 72]. There are two higher energy 0+ levels known [Be 70, Cl 76] in 194Pt at 1.8936 MeV and 2.086 MeV. We observe a level weakly populated at 70 in the plate data with an energy of 1.892 MeV, but an angular distribution was not obtained. We populate no state within 20 keV of the 2.086 MeV level. Three excited 0+ states at energies of 1.135, 1.402, and 198 6 1.824 MeV were observed in the Pt(p,t)19 Pt reaction. All levels have been previously reported, although the spin of the state at 1.402 MeV was assigned as (0, 1) in the decay of 196Ir [Ja 68] and as (0+,2+) in the neutron capture experi- ment by Samour et al. [Sa 68]. A recent (n,y) study of 196Pt by Cizewski et a1. [Ci 78] also assigns a spin and parity of 0+ for the 1.402 MeV level. It is significant that there is E 0+ experimental evidence for levels below 1 MeV in any of the three Pt isotopes studied. D. L = 2 Transitions In contrast to the situation for (p,t) reactions measured at lower energies in this mass region [Ma 72, Sh 76], the L = 2 transitions observed in the present study appear to be sufficiently characteristic to allow spin assignments to be made. Transitions to the known first and + . . . . second 2 levels have quite Similar experimental angular 142 distributions. The major difference appears near 180 where the angular distribution for the second 2+ has a more pronounced oscillation than for the first 2+, as seen in Figure V—2. The angular distributions for the remainder of the excited 2+ levels have approximately the same shape as the second 2+ distribution. This small deviation in shapes, seen in all three reactions, may be indicative of some weak multistep effects. The sensitivity of the calculated angular distributions to changes in the two-neutron configurations is shown in Figure vez. This could also account for the variation in 2+ shapes. 194 In the Pt(p,t)192 Pt reaction, only two 2+ levels were populated with enough intensity to extract a complete angular distribution from the data. These were the well-known first and second 2+ states at 0.316 and 0.613 MeV. Two levels at 1.439 and 1.576 MeV’ which have been. previously assigned [Fi 62, Ya 74] as (1+,2+) and 2+ respectively, were weakly excited at forward angles. 196Pt(p,t)194Pt Four 2+ levels were populated in the reaction, with energies of 0.328, 0.622, 2.155, and 2.532 mev. The two lowest energy levels have been seen in earlier studies, while the state at 2.155 MeV may be the same state seen at 2.158 MeV and assigned (1,2)+ by Cleveland and Zganjar [Cl 77], and is tentatively assigned at 2+ in the present study. The new level at 2.532 MeV is also ten- tatively assigned 2+. This level may be part of a broad peak 143 3 Angular Distributions for the The curves are Energies are Figure V-2. = 2 and L = 194’ 195’ ’193Pt(p, t) Reactions. the results of DWBA calculations. given in keV. da/da (pb/sr) I \\ TV I [urrf I o _ ..° '1. O 10‘ E- Elanas' -13&56’ 1 1 1 l I {E b n u. 3’ C: N n * 11 11111 1 1 1111111 1 1111 1 11111111 O 20 144 1 1 1 T L=2 1 1 111111 (293/2 "*7/21 —— 1 1 1111111 1 1111111] \l Irlnq 1 1 1 111111 1 11111111 1 11111 1 1 1 111111 1 1 1111111 5 \ L‘3 — 5 : ~\ L=4 ""-: I o 9 \ : _ oo \ O N O Hemmeg) Figure V-2. I v I I Inn] V [1111] A \ T r l Illnq r t 1 [1m] 1 I T I '98Pt (p,1)'96P1 L = 2 1203,2011 7/2’ 0.355 2 \10'13/2) -- ' ' 1 1 1111111 1 1 1111111 1 1111 1 11 11111 1 1111111 1 1 1111111 1 1 111111 1111 6O 145 seen at 2.55 MeV in the 195Pt(d,t) data and 2.56 MeV in the 194Pt(d,d') study [Mu 65]. As in the 194Pt(p,t) reaction, several known 2+ levels were only weakly populated and angular distributions were not obtained for these. In addition to the first (0.355 MeV) and second (0.690 MeV) 2+ levels, two higher lying levels were populated 198 6Pt reaction. These levels are at 1.606 in the Pt(p,t)19 and 1.848 MeV and have been assigned as 2+. They have been confirmed in a recent (n,y) experiment [Ci 78]. E. L = 3 Transitions The 3— octupole vibrational state was populated in each of the three (p,t) reactions, as shown in Figure V—2. In 192Pt the 3‘ state at 1.378 MeV and the 6+ state at 1.366 MeV were not completely resolved, although the contribution to the cross section from the I.= 6 transfer is thought to be small. As shown in Figure V-2, the L = 3 DWBA fits are quite poor, missing the first maxima by as much as 10°. This may be the result of multistep effects, because the 3- state is strongly populated in scattering studies [Mu 65, R0 77]. In fact, this rather strong population of the 3- levels is somewhat unexpected. Such states are very weak in the Pb(p,t) reactions [La 73], which is understood because the 3— state is basically particle-hole in. nature, while (p,t) excites 2-particle, 2—hole states. 146 F. L = 4 Transitions The spin assignments from L = 4 transitions required special attention in this study, due to the seemingly uncharacteristic shape of the angular distribution populating the well-known first 4+ level in all three reactions. This shape differs from the shape seen in both the simple two- neutron DWBA calculations of L = 4 transfers, and the Pt(p,t) data of Lanford [La 73, La 77]. As shown in Figure V-3, the angular distributions for the first 4+ 1eVels have no distinct maximum at 15°, but continue to rise toward forward angles and also show a pronounced minimum at 30°. The angular distribution for the other known 4+ levels (1.229 MeV in 194m and 1.201 MeV in 192m) is characterized by a distinct maximum near 15°, more closely resembling that calculated in DWBA and in the Pb(p,t) reactions. It was the latter shape that was used to make spin assignments for possible high-lying 4+ levels. In addition to the first 4+ state, at least two more excited 4+ states were seen in each reaction, and in the 196pt(p,t) reaction six more 4+ states have been tentatively 192Pt, a known 4+ state, identified. The 1.201 MeV level in was not resolved from the weakly populated 1.195 MeV 0+ level, although the 4" angular distribution should be only slightly affected by the 0+ level. The plate data, in which the 4+ and 0+ are nearly resolved, support this. A possible third 4+ level in 192Pt was seen at 1.937 MeV, although its 147 mo>usu 0:9 .>ox cfi cm>flm mum mwfimuwcm .mcofiuommm Au.mvum mma.mma.wma .mcoflumasoamo ¢mzo mo muasmmu on» mom can no“ mcoflusofiuumwo smasmcd w M q .m1> musmwm 148 .1 O ... v N 2 3: g 8 «1 N O [1111vK r 'anIY I 1 ' I’IIVT I INVIIV Y 1 I L- “ 1 ’ 00-. -1 1 _ an 1 v v 1 m -1 105 (x 810$ .. o * 1 '0 N33 ON .4 1*- ._ . , r l . - N . 051.0 Q [*0 1v, ‘ cw . 1 ~ . -:—' ! Q. . {'0 .101 d 2 Q o- . 0' O! O 1111 1 111111 1111141111 111111 1 1 11111111 1 ‘N 9 " ' O Q 9 - 'YYYTT fT T IYVYY T T T 'IYIUYY TY .4 1111111 1\\1 1111 1 1 \\ 1111111 1 1 1111111. 1 1 [1111\1 1 1 1111111 1 1 N \\ \\~ \\ “ 9 <2 9 9 9 Q Q Q 9 "1111 1 I 111111 1 1 Y IIIIY r I 1" try I v ["1111 I O 1' .. v " 48 CL . 9 FN - O -1 m ' ° 2‘ 1 N O 1 ._' O h b . .4 .v'rn . Vt e» .. h- p 44 .. .1 Q 0 1—v .48 2 1- -1 . \ 1 1 1111111“ 11111111 1 Juu111 1 11111111 1 O \\ \\ o 9 9 <2 (IS/01*) zip/op Q 40 9C1“. (deg) Flgure V-3. 149 interpretation as a 3- state cannot be ruled out, as the DWBA calculations for L = 4 and L = 3 are quite similar. The assignment is tentatively made as 4+ because of the appearance of possible 4+ levels near this energy in 194Pt and 196Pt. Also, the empirical shape of the L = 3 angular distributions for the three known 3‘ levels is considerably flatter at forward angles (see Figure V—2). In addition to the known 4+ levels at 0.811 and 194Pt with L = 4 1.229 MeV, five new levels were populated in shapes, at energies of 1.911, 2.125, 2.246, 2.353, and 2.638 MeV, and thus have been tentatively assigned as 4+ levels. In 196Pt two levels were populated by transitions whose angular distribution shape is that of an L = 4 transfer. The level at 1.293 MeV may have been observed in inelastic alpha scattering (1.290 MeV) [Ba 76], but was not assigned a spin or parity. An L = 4 fit is not very good, but this deficiency is partially due to unfolding the contribution of the nearby 5_ level at 1.271 MeV. The trend in the other two Pt nuclei studied would suggest this is the second 4+ level. The level at 1.884 MeV in 195 Pt was populated very strongly with the shape of an 1.: 4 transfer, and completes a series of new, strongly populated 4+ levels seen at = 1.9 MeV in all three (p,t) reactions. This level may be the same one seen in the (d,d') study [Mu 65] at 1.88 MeV. 150 G. L ;,5 Transitions Only limited success was achieved with assigning J1T values to states populated by L transfers greater than 4. Although the DWBA calculations showed the first. maximum shifting approximately 50-100 towards backward angles as the transferred angular momentum increased by one unit, there was only one known higher spin state populated with a complete angular distribution for comparison. This was the 6+ level in 194Pt at 1.414 MeV. The other known levels with spins greater than 4 were either unresolved from other levels 194 196 (1.486 MeV 7— in Pt and 1.271 MeV 5_ in Pt), or too weakly populated for a complete angular distribution (1.517 MeV 7’ and 2.019 MeV 8+ in 192pt). Nevertheless, several spin assignments have been proposed for levels in 194Pt and 196Pt as shown in Figure V-3. In 194Pt a level at 1.374 MeV was populated, which has been assigned as a 5" level in (a,xn) reactions [Ya 74, Hj 76] and as (6+) or (4,5_) in 194'Au decay [Be 70] and triple neutron capture [Su 68]. From the present (p,t) results a clear distinction cannot be made between L = 4 and L = 5 transfer. ‘As a result, the state has been assigned (4+,5-) from the natural parity selection rule. The level at 1.414 MeV, a known 6+ state, is reproduced by the L = 6 calculation, particularly in the angular region about the maximum. This leads us to propose two additional levels to be assigned at 6+, at 2.566 and 2.700 MeV, as shown 151 in Figure v—3. Levels at 1.990 and 2.296 MeV have been assigned (1“ values (6+,7_1 and 7_,8+), respectively. A unique assignment was not possible because of the similarity of the shapes for the calculated L—transfers involved in each case. 19 In the 8Pt(p,t) reaction, two high-spin levels have been identified, at energies of 1.374 and 2.296 MeV. The first level was assigned as (6,7)- in the decay [Ja 68] of 1961r, and the (p,t) angular distribution data show it to be either a 6+ or 7— state. Thus, from the (p,t) natural-parity selection rule, this is therefore most likely a 7— state and may be related to the 7— state observed at 1.518 MeV in 192Ft and1.485 MeV in 194Pt. The second level, at 2.296 MeV, is assigned as (7_,8+). H. Relative Reaction Strengths The triton spectra shown in Figure Iv—3 for the three rections show many of the same overall features. The most notable are: strong population of the ground state and first 2+ level in the residual nucleus; several excited L = 0 transitions; and an increasing population of the 4; and 4; levels as the mass of the target increases. Table V—5 displays for each reaction the integrated differential cross section from 70 to 600 for the more strongly populated levels below 2 MeV. The values for the enhancement factors, 6, are listed in Tables V—l, V—2, and Ve3. Since these calculations 152 mo.o oa.~ nova mo.o om.~ mmoa «o.o mo.m ooma 1m ovo.o moo.o m~.o ONoma amo.o aoo.o a.o owama oao.o aoo.o a.o ommva Mm aoo.o ao.o hm.o moo mao.o «o.o ~>.a NNo omo.o mo.o oo.a mam MN va.o >~.o a.o~ mmm va.o oa.o o.ma mum ma.o ma.o o.aa mam MN 0 O O Ill-II "' 'Il' m 111 111 111 111 oao o moo o mm o onoa +o mmo.o oo.o om.m vuoa omo.o mo.o oo.v ooma ooo.o 111 111 111 No hoo.o mo.o om.~ mooa 111 moo.o m~.o ohva ooo.o mo.o >>.m omoa Mo ~ooo.o «o.o Nh.a mmaa mooo.o ~o.o «v.a howa vooo.o «o.o om.a mmaa Mo oo.o on.o o.om o.o oo.o mo.o o.ao o.o o.a o.a o.mn o.o Ho Go o 3x8 35 Sub: a: o Exec 3.5 98: at o 3:: 3.5 $9: mum mxo mxo Nmao\o «mao\o o m Noao\o ~mab\o o m «maO\o ~oab\o o m o = u o.o u o.o u .o mooaA o mooa umooaa o moma umumaau vunwoa .c3oam omam mum .aaw>auoommou chauamCMuu am ummma osu pom .m.m ummma on» o» omuaamauoc .w:0auam:Muu N n a can o u a on» 3m aoooe 49 on» no 35a .30 of Ca mcoauMasoamu .um mo 33m 0:595 on» 0» gay—mama mouauwm coauoom mmOuo new 3.9”; Ca macauamcmmm. new mmCOauomm mucuu cwumummuca .m1> manna maa.mma.ema 153 .4mH on» no uaeaa Aooo onu uOu macauma50amo mnu spa: comaummeoo uOu MN no wumum panama on» no umcu spas :Oauoom mmOuO amuOu Haws» wamow 0» com: coon m>mn chauoom mmOuO oh uaosu .uo>ozom .mam>oa +~ no +o mm ucweauwmxo was» :a coauaucopa no: wuoz moumum wwonao .mnomou mucouomwmn .mmana mum wwauCamuuooco .Ooo can 05 cwwzuon omEuOHuom scaumumwuch m~.o o.oa aooa o~.o o.oa aaoa ~o.o mn.a nmoa Me oo.o ms.o mama «o.o mo.m oNNa mo.o mm.~ aoma ma ao.o mo.a saw ao.o mo.o aam «o.o Ne.a was we loco onoo Aoeo A>oxo Aooo Amado Aoso A>oxo Aooo Amxoo Aoso a>oxo ~aab\o Noao\o oxoo m ~mab\o Noaoxo made a «aaoxo Noaoxo name a o o. u o.o u .m .m moaaA o mama uoeoalu ouoooa oomoaau .uoqoa .A.o.ucooo m1> manna 154 used only a simple 2—neutron wave function, values of a differing from unity suggest the absence of correlations in the wave function. As expected, the ground state transitions 19 are the most enhanced with an e;of 5.1 in 4Pt(p,t) and 3.7 in 198Pt(p,t). While the ground state population is decreasing with increasing A, the enhancement of the first 2+ level and third 4+ is increasing with A from 0.84 to 2.2, and 0.32 to 2.1 respectively. Although s was not calculated for the 4: level in 1'9th, Table V—5 shows the total cross section of this state also increases with A. In addition, the enhancement of the 4:, 2;, 3-, and the excited 0+ levels remains relatively constant in all three reactions. These same general trends, decreasing ground state population and a general increase in population of excited states with increasing A, were seen in the (p,t) reactions on the Pb nuclides [La 73]. This was interpreted as an indica- tion of an increase in the two-particle coherence of the wave functions as one moves away from the closed shell. The decreasing ground state population from 192Pt to 196Pt is not as dramatic as that seen in the Pb data, but this is under- standable from a simple pairing-vibration model [Br 73]. If the creation and annihilation operators for the two-neutron pickup are treated as boson operators, then the strength of the transitions is related to the number of pairs of neutrons (phonons) or holes, in the final state, relative to the 202,204,206 nearest closed shell. For Pb, the strengths of 155 the ground state transitions should be in the ratio 3:2:1, 194'196'198Pt the ratio would be 6:5:4. This is while for consistent with the experimental Pt ratio of 6.1:5.0:4.7, with 15-20% uncertainties on these numbers. Arima and Iachello [Ar 77a] have noted that both finite dimensionality effects and an increase in collectivity as one proceeds into a shell are important and give quantitative predictions for these effects with the IBA; however, the uncertainties on our measured ground-state strengths are too large for us to observe such an effect. I. General Discussion of (p,t) Results l. L = 0 Transitions One of the primary reasons for the current reaction study was to search for any low-lying 0+ states that could be interpreted as the "missing" 0+ state of the 2—phonon triplet in a vibrational model interpretation. Although (p,t) transitions to 2—phonon states are forbidden in first-order, these states have been seen [Co 72, Kr 77] in (p,t) reactions on Cd and Pd, probably due to two-step transfers and/or anharmonic terms in the vibrational potential. No evidence for low energy, L = 0 transitions populating a 0+ level is seen in any of the three (p,t) reactions. In fact, no new levels were populated below 221.5 MeV with a cross section 2 l ub/sr at forward angles, about 0.1% of the ground state population. (In Cd, generally considered a good example of a 156 vibrational nucleus, the relative (p,t) strength ratio for 0’5/0+ 9.5. is =0.25%.) A second result cf the (p,t) experiments is the absence of any strong L = 0 transitions populating excited 0+ levels. As mentioned in the Introduction, a strong L = 0 transition (==50—100% of the ground state strength) might have indicated a shape—isomeric level in the residual nucleus related to the Y degree of freedom. The strong transition would occur if there was a large overlap of the target, ground state wave function with the wave function of an excited 0+ state in the residual nucleus. For this region of nuclei, this would seem to imply a stable triaxial minimum in the potential of each nucleus. Since no transition was observed that was stronger than 10% of the total ground state cross section, the data seem to be consistent with an interpretation of these nuclei as being soft, with shallow minima in the potential surface. Tables V—l, V-2, and V-3 show three or four excited 0+ levels weakly populated in each of the (p,t) reactions studied. Most of these 0+ states are not easily interpreted within current models for this region. The energy is too high in the Pt region (2 1.2 MeV) for the first excited 0+ state to be a member of the 2-phonon triplet in a strict vibrational sense, although the pairing-plus-quadrupole model predictions of Kumar and Baranger [Ku 68] are quite reasonable: 1.207, 1.101, and 1.018 MeV for the first 192,194, 196 excited 0+ states in Pt respectively. The cross 157 section for populating the first excited 0+ state in (p,t) is 2—3% of the ground state in each reaction, rather weak for it to be considered the so-called "B-vibrational" state of a symmetric rotor; the typical cross section for the first excited 0+ levels in deformed nuclei is approximately 5-10% of the ground state. Some of the higher energy 0+ states may carry more of the B—vibrational strength, as they are populated by stronger L = 0 transitions. One interpretation of the 0+ levels may be that they are the K = 0, two y—phonon bandhead of a symmetric rotor, as 188,190,192 possibly seen [Ya 63, Sh 76] in Os. The energy of these 02 levels is quite close to the Bohr-Mottelson predic— tion of twice the single Y-phonon bandhead (==625 keV in Pt). Existing branching data for the decay of the first excited 0+ 190—196 state in Pt also supports this phonon interpretation with the ratio (B(E2)0; + 2:)/(B(E2)0: + 2:) > > 1. 2. L = 4 Transitions The new 4+ states near 1.9 MeV in excitation warrant particular attention because of their possible similarities to the third 4+ states seen in the neighboring Os isotopes. There have been two explanations offered for these states in Os. One possible explanation for these 4; levels may be that they are the bandheads for the K = 4 component of the 2-phonon y-vibrations of the symmetric rotor. These states 158 are seen in the Os(p,t) studies [Sh 76] at an energy near 1.2 MeV. There are problems with this interpretation for the Pt isotopes from an energy standpoint, however, as their energy (2 1.9 MeV) is much too high in the vibrational model, which predicts the energy to be about twice the energy of the l—Y phonon bandhead (2;), or about 1.2 MeV for Os and Pt. The energies are too low for the triaxial rotor model [Ba 58]. Here, the energies can be determined from the sum rule 3 2 E(4T) = 5 E(3+) , (v-2) i=1 1 giving E(43) z 2.5 MeV in the Pt nuclides. On the other hand, the decay properties of these 4; levels in the Os region [Ca 78] tend to support a 2—phonon interpretation as each level primarily decays to the second 2+ state rather than the :first 2+. However, additional problems arise from this interpretation due to the strength of the 4+ transitions in 196’198Pt(p,t). Because (p,t) transitions to 2—phonon states are forbidden to first order in a pure vibrational model, such states should be only weakly populated as a result of multistep effects and anhar- monicities in the vibrational potential. In the Cd region, the population of 2-phonon states is typically 1—5% of the ground state population. Similarly, in Os(p,t) the strength of the transition populating the 4; level is =:1-2% of the 196,198 ground state. However, in Pt(p,t) the strength of this transition is 2 15% of the ground state while only in 194Pt(p,t) is it as low as 1% of the ground state strength. 159 . . . . + Thus, a uniformly Simple interpretation of the K1! = 43 192,194,196 bandheads as vibrational states in the Pt isotopes is doubtful. Recently, Bagnell et a1. [Ba 77] have argued from the + 3 Os reactions that these states could be calculations explaining the strength of the 4 states in the 191'193Ir(t,a)190'192 described as single phonon, hexadecapole vibrations. This interpretation was also presented in a recent (a,a') study [Bu 78]. These 4; states may then be a mixture of both the Zy—phonon and the hexadecapole-type vibrations. If an attempt is made to extend this interpretation to the Pt isotopes, one must take into account the varying strength that is seen in populating these states in both the (p,t) and 194Pt(p,t),(p,p') reactions, the (p,p') studies. In the population of the 4; state is comparable to that seen in the analogous Os experiments. However, in the 196'198Pt(p,t),(p,p') reactions, the additional strength of these states indicates a change in their structure. Since to first order (p,t) reactions should populate only one-phonon states, the fact. that. the 4; states are jpopulated very strongly indicates only a small 2-phonon component. In addition, a large E4 component is needed to account for the 4+ strengths seen in the (p,p') reactions (see next Chapter). These observations seem to indicate that as the mass 4. 3 larger hexadecapole component in their wave functions. increases in the Pt isotopes, the 4 seem to be exhibiting a 160 In the 204Hg(p,t)202Hg reaction at E = 17 MeV, a strong P L = 4 transition was also seen with 40% of the ground state transition strength [Ma 74]. It was noted by Breity et al. [Br 75] that DWBA calculations for this seem to indicate large in-phase (2p3/2 ® lf 5/2) and (lf7/2 ® 2p1/2) neutron 206Pb components in the transfer form factor. By using the 4: wave function of Vary and Ginocchio [Va 71] with the (1f7/2 ® 2p1/2) amplitude enhanced by a factor of 2 to 3, the cross section for the 4; state surpasses that of the first 4+. In the present study, these same two configurations provided the greatest calculated strengths for a_l_l L = 4 transitions and the (2p3/2<3 1f5/2) configuration was used in calculating the enhancement factors. The suggestion that the lack of large 4+ cross sections in the lighter Hg isotopes may be due to a depletion of the Zpl/2 orbital [Br 75] may apply to the lighter Pt nuclides as well. Since these nuclides are farther away from the N = 126 shell closure, decreasing occupancy of the 1f5/2 orbital now becomes a factor rather than the 2p3/2 orbital. Thus, this same effect may explain the generally decreasing strength of the 4+ levels as A decreases (see Table V-S). 3. L = 0, 2 Transitions in the Interacting Boson Approximation It has been shown by Arima and Iachello [Ar 77a] that the IBA model provides a natural framework for a unified description of 2—nucleon transfer reactions across a complete 161 shell. The ease of associating the IBA with 2—nucleon transfer reactions is due to the inherent coupling in this model of pairs of fermions to bosons with angular momentum 0 and 2, or s and d bosons. It is also possible to treat higher L—transfers by coupling the bosons to form higher order operators, or alternatively by adding 9 bosons (L = 4). This discussion is restricted to the L = 0, 2 transitions. Reference [Ar 77a] investigates 2-nucleon transfer reactions in the SU(5) (vibrational) and SU(3) (rotational) limits, while this study presents features of (p,t) reactions near the 0(6) limit. The operators for the (p,t) reaction can be expressed in terms of creation and annihilation operators for the s and d bosons, s+(d+) or s(d), depending on whether one is near the end or beginning of a shell. This change of operators is due to a change from particles to holes in describing the system. For the L = 0 transitions in (p,t) reactions the operator, to first order, has the form [FIac] (0) Th) = 1‘ _ - 1/2 - aVS (9v Nv Ndv) . (V 3) In this notation a distinction is made between boson operators for neutrons (3:) and protons, as the calculations discussed below have been performed [OSch] using a code which allows for both neutron and proton bosons. Other quantities in the operator are a strength factor, av ; the effective neutron degeneracies for the sub-shell in question, 0v; the 162 neutron pair number, Nv’ and the neutron d boson number, ndv‘ The factor _ _ 1/2 [9v Nv ndvJ is a result of the finite dimensionality of the shells. The eigenfunctions in the 0(6) limit are denoted by three primary quantum numbers, 0, T , and VA’ which were discussed in Chapter II.E.4. The quantum number T is related to the expectation value of the number of d boson, ndv' _ N(N - 1) 1(1 + 3) <“d> - 2(N + 1) + ETE'1—I7 , where N is the number of bosons in the nucleus. Table V—6 shows the relationship between t and nd for the low-lying 0+ and 2+ levels in 194Pt. This analogy between.'rand the number of d bosons is made to show the correspondence between the change in'r (or nd) in the 0(6) limit for a (p,t) reaction versus the "change in phonons" terminology of the more common pairing vibration model [Br 73]. There is no one-to-one correspondence of the two descriptions due to the inclusion of finite dimensionality of the IBA. For L = 0 transitions there is a At = 0 selection rule which requires that the average number of d bosons does not change. Thus, the relative strengths of these transitions can be predicted by a check of the 0(6) wave functions for the 196 4 0+ states. For example, in the Pt(p,t)19 Pt reaction, the ground states for both nuclei have I = 0 or z 2, while 163 Table V-6. Relationship Between the “d and 1 Quantum Numbers for States in the SU(5) Limit (Harmonic Vibrator) and 0(6) Limit of the IBA. SU(5) 0(6) State a _ 9.5. a nd Ex/E + r Ex/E + 2 2 1 1 o: (9.3.) o o 0 15/7 0 o 2: 1 1 1 17/7 2/7 1 2:, 4: 2 2 2 20/7 5/7 2.5 3:, 4: 3 3 3 24/7 9/7 4.5 aFor SU(5) the energies are given by an , and for 0(6) they are (-). Values taken from [Ca 78]. 164 the first and second excited 0+ states in 194Pt have T = 3 I! ( =3) and T = 0 ( 0(6) limit the strongest L 0 transitions would be the ones populating the ground state and second excited 0+. This is indeed what is observed experimentally as shown in Table Ve5. For two of the three reactions in this study, the second excited 0+ state is more strongly populated. The exception is the 1479 keV level in 194Pt. However, it is believed that the 1479 Rev level in 194 Pt may not be a collective state, but single-particle in nature [FIac], since it lies near the pairing gap for the Pt nuclides. The cross sections reported in Table V-S have not been corrected for Q—value differences since this has been shown to be a small effect for (p,t) reactions with outgoing triton energies greater than 20 MeV [Sa 79]. The stronger population of the second excited 0+ state relative to the first in Pt(p,t) reactions has not been satisfactorily explained by any other model. The results of (0) +v Table V-S. The strengths are calculated in the 0(6) limit calculations using the T operator given above are shown in with a small quadrupole-quadrupole boson interaction which breaks the pure 0(6) symmetry and accounts for the changing properties of these nuclei as the 0(6) to rotor transition progresses. These calculations also reproduce the increasing strengths for the ground state to ground state transitions as A decreases, a trend which extends to the 190Pt(p,t)lBBPt reaction as well [Ve 76]. 165 For the L = 2 transitions the operator becomes somewhat more complex as a change in seniority of 0, t 2 is allowed. The L = 2 operator can be expressed as [FIac] (2) _ + T T L=2 T - (avdv + BVEdvd 5v] + L=2 +v + YvEdvdvdv] )A ' V where the change in seniority for each term is +2, 0, -2 B respectively. Here a and‘y.v are relative strength v' v' factors for each coupling of s and d bosons and A is a finite (0) dimensionality factor similar to that for T+\). The brackets represent angular momentum couplings. Calculations have been carried out [OSch] using only the first and second terms of this operator with 8v chosen to be 0.08. The results are shown in Table V‘S. The governing selection rule is AT = +1, which would allow only the population of the lowest of the first three 2+ levels in the strict 0(6) limit, since the ground state wave function has I = 0 and the first 2+ state has I = 1, while the 2: state has 1 = 2 and the 2*, r = 4. The addition of the small symmetry breaking term will allow some population of the other 2+ levels as well, due to the mixing of the wave functions. For relative populations within a nucleus, the general agreement of the IBA model calculations for the L = 2 transi- tions is very reasonable, but the calculations do not predict the proper trends from one nucleus to the next. For example, the model predicts a virtually constant 2: population, while experimentally the population of the 2: state increases 166 almost twofold from 192Pt to 196Pt. The calculations also predict a decrease in strength for populating the second 2+ as A increases, while experimentally one observes a constant strength. It is possible that the calculations could be improved with a morejudicious choice of the relative phases and magnitudes for the coefficients in the L = 2 operator, as well as investigating the effects of the third term. of Equation (V—S). It should be noted that the smaller values calculated for both the L = 0 and L = 2 transitions (i.e. 192Pt(0'2[):0.0004) may have large uncertainties, as higher (0,2) +v may provide a significant order terms not included in T contribution to these small values. CHAPTER VI 194,196,198 RESULTS FOR THE Pt(p,p') REACTIONS A. General Analysis Approximately 45—50 levels were populated in each of the three reactions up to z 3.0 MeV in excitation energy. In the case of 198Pt(p,p'), 38 of the 44 levels observed are reported for the first time. Only the energy of the first 2+ state was accurately known before this study. Tables V—2, V—3, and VI-l summarize the excitation energies, cross sections (8 = 30°) and assignments of J" in the 194'196’198}?t(p,p') reactions. The results from previous studies of these nuclei are included in addition to the results of the (p,t) reaction studies of 194'196Pt for comparison. The energy measurements for levels in 198Pt and 196Pt were made from plate data taken at eLab = 43°. Accurately known levels from the 206Pt(p,p') reaction, recorded on the same plate, were used as additional calibra- tion lines. The energy measurements for levels in 194Pt were taken from plate data (eLab = 25°) using internal, well-known level energies. Some levels do not have values for the cross section because these states were generally too weak to obtain a complete angular distribution. 167 168 198 Table VI-l. States Populated in the Pt(p,p') Reaction. Present Experiment a Previous Results l”Pt(p,p') Exb J" o(30°) Exb J" (MeV) (ub/sr) (MeV) 0.0 0+ 4.92x105 0.0 0+ 0.407C 2+ 3.24x103 0.4072 2+ 0.775 2* 55.1(40°) 0.775 2+ 0.984 4+ 1.05x103 0.991 4+ 1.246(3) (3*) 21.9 1.287 4+ . 252 1.305 1.367 (5') 142 1.445(3) 56.6 1.502(3) (7’) 82.8 1.657 119 1.682 3' 845 _ 1.722(3) 25.5(40°) } 1'722 (3 ) 1.785 (4*) 150 1.827(4) 1.900 113 1.949 1.971(4) 2.000 2.070 46.6(40°) 2.100 74.9 2.120 57.9 2.155 137 2.178 52.7 2.319 2.339 169 Table VI-l (cont'd.) Present Experiment a Previous Results 198Pt(PIP') Exb J" 0(30°) Exb JTI (MeV) (ub/sr) (MeV) 2.356 2.387 2.441 369 2.469 49.9 2.514 108 2.573 36.3 } 2.53 2.611 762 2.633 2.666 96.5 2.726 62.3 2.782 2.796 325 2.826 385 2.884 38.4 2.910 38.0 3.005 3.018 3.170(5) 3.197(5) aReferences [Mu 65, Br 70, Ba 76]. Uncertainties in the excitation energies are approximately 2 keV below 2.5 MeV and 0.1% above 2.5 MeV, except where indicated. cUsed as calibration point along with the 0.80310, 6.68408, 2.20023, and 2.64790 MeV levels from 206Pb. 170 An attempt was made to assign spin and parity to many of the states seen in each reaction using the DWBA code DWUCK [PDKu] and standard, collective model form factors (see Chapter II.B.4). Except for the ground state, and to some extent the first 2+ level, the DWBA calculations provided very poor fits to the data. The few spin assignments made relied on empirical shapes of angular distributions for states with well-known J". This method, in addition to energy and spin systematics in the Pt nuclei, has allowed several new spins to be proposed and a few assignments to be confirmed. These techniques were particularly successful for 198Pt since the spin and parity of only three levels had been' previously determined. The following sections will discuss many of the levels populated in all three reactions according to the particular L—transfer. 1. Elastic Scattering and L = 0 Transitions The Shape of the elastic angular distributions (Figure VI-l) is virtually the same in each reaction. The most notable features at this energy are the "plateau" around 40°, a decrease in cross section of three orders of magnitude within the angular range studied, and three distinct minima between 500 and 100°. The ratio of the elastic scattering to the classical, Rutherford scattering is shown in Figure VI—l 194 for Pt(p,p'). The Rutherford differential cross section is given by (92)... . (£132.)?- 1 d0 R 4Ec.m. sin4(0/2) I Figure VI-l . 171 Elastic Scattering and l.=»0, 2 Angular Dis- tributions Seen in the 194'196'198Pt(p,p') Reactions. The curves are the results of DWBA calculations using a collective model form factor. Energies are given in keV. 13111711111 C Q ”“191 b 103 2 10 lsqp* elastic 103 '2 2 < 10 ..Q .3 196p, £103 . ‘ ' elastic 0} i b 102 'o 101 0.1 9 f ‘sePt h if 1828(0’) 0.01 1f9‘1 .9 0,09 111111111, 720 ‘10 80 80 100 ec.m.[d99] 172 IIIIIIIII u“=2* 10 199p, 328 1.0 10 1.0 10 1.0 L111111111 20906080100 ec.m.[deg] Figure VI-l. I'I'I'ITI + o Jfl:2 . o 0.1 ’ o '. o ‘qut o o 622 0 O o o 0 1 0.... d , o O 196131 689 O ... ' o .0 0. 0.1 ‘ 198p, f ° 775 . o O .. O o 0.01 ’1» mp1 0 1670 o O . .0 O 0.001 . . 9 , 1 196.9, mm .0 9 1603 ° .0 9 O C 0.001 f ’ 71111111111 20 '10 6080100 ec.m.[d99] 173 where Z1 and 22 are the atomic masses of the target and projectile, E is the energy of the projectile, and 6 is the center of mass scattering angle. The large deviation from unity is evidence for the influence of nuclear forces in the scattering process, which produce an optical-like diffraction pattern. In 194Pt the two states at 1.547 and 1.892 MeV may be the well-known 0+ states seen in several decay studies and in the (p,t) reaction study described in Chapter V. Unfor- tunately these states are only weakly populated, and are not clearly resolved from nearby states in the wire counter data, so no angular distributions could be extracted. A level is seen at 1.826(3) MeV in 196 Pt which may be the known [Ci 79, De 79] 0+ state at 1.823 MeV. An angular distribution was extracted for this level (see Figure VI—l), however no definite spin assignment could be made due to a lack of any characteristic L = 0 shape. 2. L = 2 Transitions A total of eight known 2+ levels were populated in the three reactions, and in each case an angular distribution was obtained. Two different shapes were seen for the L = 2 angular distributions in the (p,p') reactions: one for transitions populating the first 2+ level (K = 0), and another, having more structure, for the second 2+ (K = 2) and higher-lying 2+ levels. The angular distributions (Figure VI-l) of the 0.328 MeV level in 194Pt, the 0.355 MeV 174 196 level in Pt, and the 0.407 MeV level in 198 Pt all display only a mildly oscillatory shape, while the 0.622 MeV level in 194pt, the 0.689 MeV level in 196 in 198Pt exhibit more pronounced oscillations. However, the Pt, and the 0.775 MeV level shape of the transition populating the 0.775 MeV level in 198Pt shows substantially fewer oscillations than those for 194Pt or 196Pt. This may be the result of the decreasing deformation as A increases. This is observable in the angular distributions for the first 2+ states populated in (p,p') scattering at 35 MeV on rare-earth nuclei [Ki 78]. There, 82: 0.23, and the shapes display even more diffractive shapes than the 2: states seen here. The assignment of a spin and parity of 2+ for the 196 1.603 MeV level in Pt supports the assignment made in the (p,t) study discussed in Chapter V. 3. L = 3 Transitions States with J1'= 3- were very strongly excited at 1.433 MeV in 194m, 1.447 MeV in 195m, and 1.682 MeV in 198Pt. The shape of the angular distributions for these states is very characteristic (see Figure VI—2) and thus enabled the assignment of the 1.682 MeV level in 198Pt as the 3- state. A state has been reported [Ba 76] at 1.722 MeV in 198Pt(a,a') and tentatively assigned as 3-, however the energies determined in [Ba 76] are systematically too high for most of the levels populated in 198Pt as well as in 194'1961Pt. This is particularly true for the 3- states in 175 Figure VI—2. L = 3, 4 Angular Distributions Seen in the 194'196’198Pt(p,p') Reactions. Energies are given in keV. IT 1 T I T T I I 13 5 453' a C 4 0 1'0 E- 0 o "E g ‘9"P1 E a .196 1932 I /h— o I—1 0 1.0 =- ° °°°0 -' E e E 2' ° 2‘ O 42 13% ‘96P) 10 0 1‘1”}? 0 1 E ~o 0°00 5 : ° ° 5 .. 00.00 0.00.1 L01 — ° - = O = \3: : .0 o. : .2 _ 198P'0 ....0.01 S —. 03 E " 5 "o : '5 b.001g « a r 922 m gr _ 0.01%- '1’ : ’95P: ’ 10H ”1’ , .001 § 0.01 1 1 1 1 1 1 1 1 1 1 20 L10 60 80 100 ec.m.[deg] 176 1'171'7'1 J'=H‘ 0 0 ‘3. ““Pt 00 0.1 - 9°. 811 0 1‘ °-o. 0 f 0‘. 196?? 0.1 00. 877 0 0 . °0.. 0 L0 _ 0. 198p, E 1' 989 _ 1.0 '. a: 00 Z .0“ -‘ . and 0.- .0 0 0.1 ‘I (9..“ 0 1229 00 0 1.0 .... 0 0 ° 1915p, 0.1 t 1293 .Q . ... .0 1.0 000 ‘5 ‘99P1 0.01 o. 1287 o 0 '0 0.. L 1 1 1 1 1 EL 20 L10 60 80 100 ec.m.[deg] Figure VI-2. l'I'TTT‘T o.o J'=H* ‘1’ ’“Pt 0,) .90. 1911 o0 00. 0 L0 0 ... ISSP‘ O. 1887 OJ ' foo. 0 00. 0 fl ‘, 0.1 01 139p' 0. 1:85 00 0 0 OJ 0”. ... 0 '5 19"P1 ’0 2126 0 1.0 ’0 00 0. .° ‘6 136m OJ 1’ 2008 0.. ' 0 '0‘. OJ 0 0 196p, 00 2280 0 0 .0 0.01 O. .0 7111L11111 20 ‘10 60 80100 ec.m.[deQ] 177 each reaction. Figure VI-2 includes the angular distribution for a 1.722 MeV level only weakly populated in the (p,p') reaction. The difference in strength of this level and the known 3— states in 194'196 Pt is a factor of 20. Although a definite assignment cannot be made for the 1.722 MeV level, the shape of the angular distribution for the 1.682 MeV level supports its assignment as the lowest 3— state. Two unnatural parity 3+ states are also seen in two of 194 the reactions studied. Both states, 0.922 MeV in Pt and 1.014 MeV in 196P t, are populated very weakly and thus complete angular distributions were not possible due to background from large impurity peaks (oxygen, carbon, and silicon). The partial angular distributions have been included in Figure VI—2 to illustrate their low cross section. A possible 3+ level was also seen in 198Pt at 1.246 MeV, however, this assignment is based on systematics and tentative agreement with recent (n,n‘) data [Ya 79] since a full angular distribution was not obtained. Unnatural parity states have no direct excitation mechanism via inelastic scattering unless a tensor component is included in the nuclear potential, but. may be populated by several multiple excitation routes, which explains the weak cross section. 4. L = 4 Transitions At least three states with J" = 4+ were populated in each reaction. Each displays a similar although rather 178 structureless angular distribution. The various transitions are shown in Figure VI—2. The major characteristics of the L = 4 transitions are a gradual decrease in strength towards backward angles, a slight plateau at 60°, and a forward peak near 35°. In 194Pt four 4+ states are seen and identified. Two of these, at 0.811 MeV and 1.229 MeV, were known from previous studies, while the level at 1.911 MeV was first seen in the (p,t) study (Chapter V). The state at 2.126 MeV is tentatively assigned 4+ and agrees with the assignment made in the (p,t) study. In 196Pt, five states with J" = 4+ were observed, including two previously known at 0.876 and 1.293 MeV. A third, at 1.887 MeV, may be the same state seen in the 198Pt(p,t)196Pt reaction (see Table V-3) although the energy is 3 keV higher for the state seen in the proton scattering (the energy uncertainties are about 2 keV for each level). Two additional 4+ states have been tentatively assigned at 2.008 and 2.280 MeV. These states may also have been observed in the (p,t) study. 198 - . + Three states in Pt have been ass1gned J“ 4 7 including the one known 4+ seen at 0.984 MeV in this study and at 0.991 MeV in (0,0') [Ba 76]. A state seen at 1.287 MeV is also assigned as 4+ and is probably the same state seen at 1.305 MéV in the (aqa') study [Ba 76]. The third 4+ in 198Pt is at 1.785 MeV excitation. This state completes a series of 4+ states seen at 2 1.8 to 1.9 MeV in l92-198Pt in both the (p,t) and (p,p') reactions. The nature 179 of these states is unknown, but their strength in the inelastic proton scattering may indicate a considerable hexadecapole component (see Chapter V). 5. Transitions with L ; 5 Five L = 5 transitions have been assigned on the basis of the empirical shape (Figure VI—3) for the angular distribution populating the probable 5_ state at 1.374 MeV in 194Pt. The 5— state seen in 196Pt at 1.270 MeV’ was previously assigned [Ja 68] as (4,5)— but the (p,t) study (Chapter V) prefers 5-. A new level at 1.367 MeV in 198Pt, seen for the first time, is assigned as (5-) on the basis of its shape and level energy systematics. The remaining two assignments (both tentative) were made in 194Pt for levels at 1.932 and 2.165 MeV. Only one angular distribution (Figure VI—3) was obtained for a level known to have J1r = 6+, the 1.412 MeV level in 194Pt. The error bars are quite large at most angles because of the weak population of the 6+ state and the proximity of the very strongly’ populated 3-' level at 1.432 MeV. 1k) attempt was made to extract an angular distribution for the 196 6+ level in Pt because the strongly populated 3- level is only 17 keV away. Angular distributions are obtained for levels seen at 194pt and 1.374 MeV in 195 194 1.485 MeV in Pt (Figure VI—3). The level at 1.485 MeV in Pt was previously known from in—beam y-decay studies [Ya 74] to have J" = 7_, thus affording an 180 _>mx :H cm>flm mum mmfimumcm .mmmnm odomfimflucwcfic: poo magma: m cues mcoflusofiuumwp amasmcm mum casaoo nuo50u pcm pugs» on» Ca czonm Oma< .mcofiuommm A.Q.mvummma.wma.vma on» CH comm mcofiusbwuumfio unasmc< m M a .m1H> wusmfim .mIH> musmflm 325.860 on: om cm or ow 363.660 2: 8 cm or am 181 4 fi q _ . _ u _ q . . _ _ _ 2 q q _ OECO CEOU Bog w 303 o 2: am am or em, 2: em 8 or am, ... . as o. ou.242._._ q.e...q_.d o 00 monm 000 0000 0 0 0 0 O O O O 00 000 0 «.o 0 0000.. 86 mm—w 0’ 00 «d g . 0 O O. WNQN O O. as. 00 0 No9 0 0 two». J 6 . . .M‘mmm . . . . — .... ..... C d H O ... o C 0 0 0 0 000 *0. ** 0 00.0 mowN . O . p 0 000‘0 —. c wmnw 0 c «d +* o 86 Nmm— ’000 ..o D 0 00000 O 00.. 0’ J {5.5- / . .. TKMd p 000 W 00 £09 0 o . 00 . NU BIN «.o o4 * q o 00 0 8 o] . ix: (0 1 2mm . o . w 000 0 1 ‘0 c 00 0 0 fl 00 0 m 0000 0 00000 0 00 S . 0 . ' 0 . 0 “mm" ‘ 0 00m07wm 00 o — 000 o a moi 0o 8 0 ion *‘ — owl O 0 T :mrmq .0 — 0 0‘ g 0 1 I 0 0 ... mm 1 I o 000 00 t, W -An...) 0000 0.00 mrwm 6“. 3 o. .0 3 we . as . I m 0 0 0000 I * + 000 4 .. .. .. n... r e 2 98. .. . :15 .o . m. * . .68. . o o e. S e * 8 e T 3 < m 9 .. . v 2... * . o O... ’ 1 «n. * .0. $3 0.1 im 2 m. .6. o. 0 1o _ b p _ b m C re V L _ . _» P p p p b m... 0.000 8.9 .. .763. 0. ru— .000 Kym.— b)... .b»..-F—»b 182 empirical shape for comparison in the 196.198 Pt(PrP') reactions. The level at 1.374 MeV in 196Pt had been previously assigned as (6+,7—), but its angular distribution has the shape for J“ = 7_. A tentative assignment of (6+,7_) has also been made for the level at 1.502 MeV in 198Pt. The level at 1.722 MeV in 198 Pt oculd not be given a definite assignment because of its weak population and uncharac— teristic shape. Figure VI—3 includes twelve seemingly’ unique (angular distributions, three from 194Pt, four from 196Pt, and five from 198Pt, all with essentially the same distinctive features. The most prominent of these is a peak at 500 on a gradually sloping curve. This shape is seen for transitions populating various new levels from 2.1 to 2.8 MeV of excitation, and in each case the transition is one of the strongest in the reaction. Unfortunately there are no known levels in any of the reactions with a Similar shape, so no spin information can be obtained. However, the strength of these transitions and their high excitation energy (approxi— mately 1 MeV above the pairing gap, = 1.4 MeV) may indicate these states are composed of highly correlated, particle-hole configurations. Further investigation is needed, though, before a definite characterization of these states can be made. Several additional angular distributions are shown in Figure VI-4. No spin assignments were made for these transitions due to the lack of any similar empirical shape or 183 csocxco Law: mcofluommm 121.35 .>mx ch cw>fim ohm mwfimuwcm .uommcmLe q o cw coo mcofiu: fluumfia um swam mma.oma.vma cu m o H . v1H> musmwm 184 3036.60 2: em 8 or am, _ . . _ _ — q _ d _ I I O O O . 00* wwwm o ["11]! 1 T 11111]! 111er I 1' [mun l O "[11 I O 0.00 mmw~ 0. e0%: PFLHL_»_._ 8.0 8.0 10 8.0 8.0 BmEédo 00“ 0m 0w 0+. 0N _ _ _ 2 _ _ 4 l . _ 00000 0 0 8.0 mmmw 0 20 3 0 0 00 0 00 O . 88 0 so 0 0.. 3 00 *0 0.0 mrwm 00* 8.0 f 0 ..o 0 00000 .0. wmh r 8.9 O O 0 00 . 000 S .0000 mm: 00 0 0 as 0 0 +0mml 3036.00 as 8 8 9. ON _ _ d d _ _ u d u 4 WI .006 .l O I O .l C O. M £8 II: F 1 0:00 ... w 2 ml 0kmN 0' O M000 05000 n. m. 000 O n $8. .I. w 0 m. 000 r m .0 m .. WI :0 - 88 0 n filfl0o0 W000 m. 000 T 000 n mmwm 0 m a... W 00 .0 w ... 0 NNNN 0 s T 0 I 0 * earml .+ L PL _ s b F b b 8.0 10 8.0 _.0 “.0 8.0 ".0 8.0 3036.60 02 8 cm or em, 2 _ 4 _ _ _ _ .4 _ W0000 0 1 ram 0 0 H (0* .II. 0 H000 W 1 88 00 0000 n 0... W1. 7 00 MI .. .... W 82 0 1 0’0. n. H. O 1W 00000 C .l * OKQH ...a n l 1 0 0 0 W0 0 00 T 0 0 H m~m~ m .. W 8 T.* *80* 00 o W 0 n. 0 00 m. wmb 000 1 0’0 r + 0 00 m arms C FL h b F _ _ . _ .01H> musoflm 8.0 8.0 8.0 8.0 185 because of large uncertainties in the angular distribution data. B. Coupled Channel Analysis of the Inelastic Scattering Data 1. Introduction Attempts to describe the inelastic scattering data with the one-step DWBA were not very successful. Only the shape of the first 2+ angular distribution for the three reactions could be reproduced by the DWBA. calculations (see Figure VI-l). Since there is only one form factor for a given L—transfer in the collective model approach, there is no means to account for the dramatically different 2; angular distribution. Hence, these limitations of the DWBA formalism suggest that a: more complete and complex coupled channels approach is necessary if one is interested in studying the higher energy states of the Pt nuclides. The procedures used in this study for the coupled channel analysis are very similar to those employed for the analysis of proton scattering from well-deformed nuclei [Ki 79]. Basically this method involves an iterative searching procedure to determine the best set of optical model parameters for reproducing the elastic scattering data. Next the deformation parameters, 81' are determined for the ground band by a similar searching procedure, where a "best fit" is obtained for the inelastic data as well as the 186 elastic. Both of these searches are performed with the coupled channel code ECIS [Ra 73], which is capable of carrying out a grid or gradient-type search. Additionally, the effects of several other features of the data and theory were investigated, such as spin—orbit effects, the coupling of the "y-band", the choice of matrix elements, and the effect of B6 on the extraction of 32 and 34. The details of these effects will be discussed later in this chapter. Two additional aspects of this analysis will be addressed: the use of matrix elements derived from the IBA model to describe the coupling between the nuclear states, and a comparison of the charge component of the nuclear moments extracted in the analysis with those previously determined by other methods. 2. The Optical Model Analysis The usual starting point of a DWBA or full coupled channel calculation is the determination of the best set of parameters for the optical potential which will reproduce the experimental elastic scattering. In this study the shape of the potential was assumed to be the standard Woods-Saxon form (as described in Chapter III.B.3) with the deformation parameterized via the expansion 2 o — R R0 1 +§ 89318.6) . (VI 1) and the Coulomb part of the potential is derived from a deformed uniform charge distribution with a sharp cutoff 187 _ 2 VC - 2122 e /r for r > Rc 2 (VI-2) v = ElEE-E— [3 - (£—)ZJ for r R c ZRC RC < c ' where Rc’ the Coulomb radius, is expressed in terms of the charge deformation parameters (8i) and spherical harmonics shown above. Thus, in practice, for proton scattering there are eleven optical model parameters that could be included in a search: V, W, W V r , aR, a1, and a . 0' so' rR' rI' r50' c so plus several deformation parameters, 8? and 8:. The procedure used in this study for determining the optical parameters limited the number of parameters to be varied to V, W and a (defined in Table V—4 and listed in D' VSO’ R' I Table VI—2), to simplify the searching process and avoid a well-known ambiguities [Au 702L However, the parameters held constant throughout the searches were not chosen arbitrarily. The radius parameters were not included since in this deformed optical model potential (DOMP) they enter the potential as a product with their respective deformation parameters. The Coulomb parameters, re, 8;. 8:. were taken from Reference [Ba 76]. The imaginary well depth was held constant primarily for simplification. The optimum parameters were determined by use of the automatic searching features in ECIS by minimizing 2 N - - 2 908. -21: (oth(ei) OexpwiH/Aaexpwi) (VI-3) where oexp(ei) 18 the measured, and oth(ei) the calculated differential cross section at angle 01, and Aoexp is the 188 Table VI-2. Optical Model Parameters Used in E.C.I.S. Calculations. Nuclide Va a W a V xz/N R D I SO (MeV) (fm) (MeV) (fm) (MeV) (ground state) 194ES L*S: K=0.0375 keV 53.40 0.712 4.70 0.663 6.497 7.0 K=0.5375 keV 53.12 0.723 4.82 0.658 6.466 8.6 L*S = 0: K=0.0375 keV 51.84 0.659 8.13 0.594 0.0 14.3 K=0.5375 keV 51.64 0.656 7.99 0.604 0.0 15.1 196Pt L*S 51.78 .786 5.46 0.666 6.551 17.1 L*S = 0 50.79 .734 7.86 0.644 0.0 29.9 lgBPE L*S 53.35 0.709 4.72 0.667 6.475 5.5 L*S = 0 52.53 0.624 7.60 0.611 0.0 10.2 aThe Bf and re values are taken from [Ba 76]. All radii were held constant, rR = 1.17 fm, r = 1.32 fm, r50 = 1.01 fm, and rC = 1.2 fm. Also, the spin-orbit and diffuseness, ago, was held constant (1.01 fm), and W retained the Becchetti— Greenlees value, 5.1 MeV for 194Pt and 5.0 MeV for 196'198pt. 189 error associated withcjexp. An error of i:3% was added in quadrature with the statistical errors from the peak fitting routine to account for uncertainties in relative normaliza- tion. The overall normalization of the data, N, was also varied to minimize x2 where N x =2; [(No 2 x x 2 9,5, 1 P] '1' [(N N )AN] , th — Oexp)/Aoex (VI—4) where Nx and ANx are the estimated value for the normaliza— tion and the approximate error in Nx to reduce the range of variation of 1L. Since the absolute normalization was not known (an approximate value was obtained from target thickness values determined from the (p,t) study), this feature of ECIS was used during the optical model parameter searches and the final value was used in computing the cross sections for the (p,p') reactions in Tables V-2, V-3, and VI-l. ECIS was used for these searches because of the need to include inelastic channels for non-spherical nuclei when calculating the elastic scattering. This allows one to account for flux usually absorbed by the imaginary part of the potential. In these searches a 0+-2+-4+ level space was used which includes the couplings shown in Figure VI-Sa. The parameter search was conducted by simultaneously varying either the three potential depths or both the diffuseness parameters. This avoided well—known ambiguities such as WD — al. The sequence of searches used for each Figure VI-S. 190 Coupling Schemes Used in the Coupled Channels Calculations. 191 [14+ 1 Qg't J 0+ (0) 1 1 A“ 1 A“ If__ul__&2+ 0+ (0) -—y~—a I; :21 a (8) Figure VI-5. 192 nucleus typically involved two iterations of a potential search followed by a search on the diffuseness parameters, one search on the deformation parameter, 82, while minimizing 2 x2+, then two more iterations on (V, W V and (aR, aI). D' $0) The search methods employed in ECIS are a "grid search" for a one variable search and a "gradient search" for two or more variables [Be 69a, Ra 72]. For 196Pt the starting 8? values have been er-scaled from values in [Ba 76], while for 194’198Pt the initial values for the 8:1 are those which produce the same quadrupole and hexadecapole moment as the N A the matrix elements used for the optical model searches are 8 's of Reference [Ba 76]. The initial relative values for shown in Table VI—3. The Coulomb and nuclear matrix elements are assumed to have the smae relative values, although each set is normalized to a different 0+ + 2+ and 0+-* 4+ matrix element“ These values are determined in ECIS from the deformation parameters by evaluating the integral L + + _ .L 32R 0 L+3 o M(El, o + L ) - 1 4n, Used for Initial Optical Model Searches. 194Pt 196Pt 198Pt Ii> |£> E2 E4a E2 E4a E2 E4a 0+ 2+ —1.0 -1.0 -1.0 0+ 4+ 1.0 1.0 1.0 + + b c c 2 2 -0.637 1.195 -0.657 1.195 -1.49 1.195 2+ 4+ —l.518 -1.140 -1.570d -1.140 -1.604 -1.140 4+ 4+ 0.815 1.207 1.529 1.207 1.529 1.207 aE4 values are from symmetric rotational model. bReference [Ba 76]. CReference [Cl 69]. dReference [Mi 71]: remaining E2 values are from symmetric rotational model. 194 step size of 2 0.33 fm, and a matching radius of 20.0 fm. The multipole expansion (Equation (III-53)) included couplings of 1 = 2 and 1 = 4 terms to L = 8. For simplicity, the nuclear deformations were equal for each portion of the potential. The results of the optical model parameter searches are given in Table VI—2 for a potential with a full Thomas form [Au 70] for the spin-orbit term (L*S), and one without (L*S = 0). Also reported are the values of chi-squared-per- point for the ground state, XZ/N, which is a measure of the "goodness of the fit", and is used for comparison purposes. The changes in the parameters from the initial Becchetti— Greenlees parameters are relatively small except for the real which diffuseness, a and the imaginary surface term, W R' 0' decreases 27% and 212%, respectively. There do appear to 196 be some problems with the parameters for Pt with spin- orbit as the changes in V, aR, and WD do not follow the trends seen in 194'198Pt. This is also reflected in the x2/N value, 17.1, which is almost three times the value for 194Pt and 198 Pt. The cause for this apparent discrepancy is unknown, although a more detailed grid search on V and a may be R necessary to bring the values more in line with the other two nuclides. The overall effect of these difficulties has a very minor influence on the moments extracted as noted in a later section. Changes in the optical model parameters should be expected, because the Becchetti-Greenlees parameters are an average set of parameters for spherical 195 nuclei. However, since the Pt nuclei are only slightly deformed (82 2 -.15), the couplings introduced in the searching procedure should result in only minor adjustments of the values, as was seen for most of the paraemters. 3. The Extraction of Deformation Parameters Once a set of "best fit" optical model parameters is obtained, attention can be focused on the extraction of deformation parameters of the nuclear potential. At this juncture one may input all previously measured matrix elements, with the remaining values taken from a collective model, then vary the 81's and several of the more crucial, previously unmeasured matrix elements to fit the inelastic data, or, one may use a complete set of matrix elements taken from some nuclear model and fit the data by varying only the BA'S. This study follows the second approach for several reasons. Since relatively little is known about the matrix elements which connect the low-lying states of the Pt 194Pt [Ba 78a]), one is forced to work with nuclides (except some model to make predictions for these matrix elements. This provides a special opportunity to stringently test the predictive qualities of a model rather than comparing energy levels and a few E2 matrix elements. With the recent success of the 0(6) limit of the IBA in the Pt-Os region for energy levels, E2 branching ratios [Ci 78, Ca 78], and (p,t) strengths (Chapter V) [De 79], the CC-analysis would provide a natural framework for further testing the E2 and E4 matrix 196 elements. This approach may not result in the best overall fit 13) the data, but one can judge the effectiveness of a complete set of matrix elements in describing the data rather than a partial set with uncertain significance. The matrix elements used in the following analysis were obtained from the IBA code, PHINT [Sc 77], which was modified to output matrix elements rather than B(El) values. The 0(6) parameters used as input to PHINT (see Equation (II-42)) are taken from Reference [Ca 78] and listed in Table VI-4. The values used in the 198Pt calculation have been extrapolated by extending the prescribed [Ca 78] relationships: A is held constant, B and C are varied linearly with mass, and K, the strength of the quadrupole-quadrupole interaction between the bosons, is varied logarithmically with mass. Casten and Cizewski [Ca 78] point out that the transition rates are really only sensitive to the ratio K/B, which specifies the location of the nucleus relative to the 0(6) or rotor limits, so the actual value of the individual parameters is not critical. The calculated matrix elements are shown in Tables VI—S, 6, 7, and 8. The values listed in Table VI-6 were calculated with a larger value of 1c (2 0.5375 keV). Calculations using these matrix elements will be discussed in the next section. The results of the CC-analysis of the states in the 194,196,196 ground band (up to the 6+ state) for Pt are shown in Figures VI-6, VI—7, and VI—8 as the solid line fits. 197 Table VI-4. IBA Parameters Used in the Perturbed 0(6) Ca1- culations. A, B, and C are the coefficients of Equation (II-40) and K is the strength of the quadrupole-quadrupole interaction. 11 is. the total number of bosons for each nucleus. Nucleus N A B C K K/B (keV) (keV) (keV) (keV) 194Pt Set 1 7 186 42.0 17.5 0.0375 0.0009 Set 2 7 186 42.0 17.5 0.5375 0.0128 196Pt 6 186 43.0 19.0 0.025 0.0006 198Pt 5 186 43.5 20.5 0.016 0.0004 198 194 Table VI—S. Relative E1 IBA Matrix Elements for Pt Ca1- culated Using 0(6) Symmetry and K = 0.0375 keV. Ii> |f> <£||M1E21Ili>a <£|lM(E4)||i>a 0+ 2+ -1.0 0+ 2; 0.0046 0+ 4+ 1.0 0+ 4: —o.00267 2+ 2* —0.0142 1.380 2+ 2; -1.156 -0.00370 2* 4+ 1.551 -0.0152 2+ 3+ -0.0061 0.559 2* 4; 0.0011 0.818 2+ 6+ —1.426 2: 2; 0.0142 0.656 2: 4* -0.0029 -1.721 2; 3+ —1.186 -0.00359 2; 4; -1.152 -0.0152 4+ 4+ -0.0127 0.0878 4+ 3+ 0.750 -0.00399 4+ 4; 1.098 -0.00572 4+ 6+ -1.913 0.0130 4+ 8+ -1.622 3* 3+ -l.767 3+ 4+ 0.00742 -0.748 199 Table VI-5.(cont'd.). Ii> |f> a a 3+ 6+ -1.452 4; 4: 0.00694 1.525 6+ 6+ -0.0101 0.965 6+ 8+ 2.120 -0.011 8+ 8+ —0.00752 1.133 aMatrix elements calculated with program PHINT [Sc 77]. 200 Table VI-6. Relative E). IBA Matrix Elements for 194Pt Cal— culated Using 0(6) Symmetry and K = 0.5375 keV. |i> |f> <£||M(Ez)||i>a a 0+ 2+ -1.0 0+ 2; 0.0627 0+ 4+ 1.0 0+ 4: 0.0369 2+ 2+ -0.l96 1.376 2* 2; -1.142 -0.0508 2+ 4+ 1.552 -0.209 2* 3+ —0.0838 0.557 2+ 4; -0.0155 -0.802 2+ 6+ -1.425 2: 2; 0.196 0.659 2: 4+ -0.0397 1.269 2; 3+ —l.186 -0.0494 2; 4; 1.155 0.210 4+ 4+ -0.175 0.883 4+ 3* 0.0748 0.0551 4+ 4; -1.090 0.0787 4* 6+ -1.914 0.180 3+ 3+ -l.765 3+ 4+ 0.103 0.744 201 Table VI-6 (cont'd.). |1> |f> a <£||M(E4)||i>a 3+ 6+ -1.446 4: 4; 0.0949 1.512 6+ 6+ -0.141 0.969 aMatrix elements calculated with program PHINT [Sc 77]. Table VI-7. Relative E culated Using 0(6) Symmetry and K 202 IBA Matrix 196 Elements for Pt Cal- 0.025 keV. I|i> NNN +hJ+104-N-+h)+ b J) +- -+hJ+ -+k:+ +- -Fh)+ ubO‘ubWO‘bWub O‘ -+h>+ -1.0 0.00242 -0.0073 -1.144 -1.535 0.00314 0.0006 0.0073 0.0016 1.155 -1.121 -0.0063 0.730 -1.069 1.862 -0.0064 0.0036 -0.0048 1.0 -0.00131 1.436 -0.0020 0.0081 -0.544 0.797 -1.388 0.694 1.818 0.0017 -0.00795 0.927 -0.0023 0.0030 0.0069 1.901 0.804 1.564 1.641 1.038 203 198 Table VI-8. Relative E). IBA Matrix Elements for Pt Cal- culated Using 0(6) Symmetry and K’= 0.016 keV. |i> |f> a a 0+ 2+ 1.0 0+ 2; -0.00123 0+ 4+ 1.0 0+ 4; -0.00060 2+ 2+ -0.0035 1.521 2+ 2; -1.127 -0.0011 2+ 4+ -1.512 0.00415 2+ 4; —0.00036 -0.763 2* 6+ -1.329 2; 2: 0.00353 0.751 2; 4+ 0.00085 1.970 2: 4; 1.073 0.00398 4+ 4+ -0.00295 1.004 4+ 4; 1.024 -0.00156 4* 6+ 1.783 —0.00346 4: 4; 0.00179 1.820 6+ 6+ -0.00204 1.152 aMatrix elements calculated with program PHINT [Sc 77]. 204 These calculations used the optical. model. parameters. of Table VI—2 and the IBA matrix elements of Tables VI—S, 7, and 8. Figure VI-Sb schematically shows all the couplings included in the calculations. These fits to the data were obtained by varying 82 and 34 simultaneously, while minimizing the total chi-squared xi, defined as the sum of the chi-squared for the ground state, 2+, and 4+. The x2 for the 6+ angular distribution was not included in the calculations since the effect of the 8+ couplings was not accounted for due to computer limitations. The absence of the 86 term in the expansion of the nuclear surface has consequences for the determination of 82 and 84 that will be discussed in a later section. Table VI-9 summarizes the deformation parameters and 194'196'198Pt. These corresponding moments determined for moments have been calculated using the methods described in Section C of Chapter III. The negative values for both 82 and B4 are consistent with previous measurements [Ba 76] for the Pt nuclides and are in qualitative agreement with theoretical predictions of Reference [Go 72]. A negative value for 32 also implies an oblate shape for the nucleus, which is consistent with the quadrupole measurements from Coulomb excitation studies. The overall quality of the fits is quite good considering no matrix elements were varied in the search. 196 The major problem appears in Pt between 700 and 900 in the Figure VI-6. 205 Data and Coupled Channels Calculations for 4Pt(p,p') With and Without the Spin-Orbit Interaction. The calculations included the couplings shown in Figure VI—Sb, the matrix elements of Table VI-S, and DOMP parameters from Tables VI-2 and 9. 206 F 1 1111”] 0.01 r I r1111] 1 1 111111 L 1 1111111 1 1 1 111111 1 1 1111111 1 1 LLLIII l f\) O 80 80 100 120 ec.m.[deg] Figure VI-6. Figure VI-7 . 207 Data and Coupled Channels Calculations for 196Pt(p,p') With and Without the Spin-Orbit Interaction. The calculations included the couplings shown in Figure VI—Sb, the matrix elements of Table VI—7, and DOMP parameters from Tables VI-2 and 9. 208 I I IIUII] I IIWITI \ I 0.01 I I IIIVII I 1 1 l LlllL L LLllllll 1 1 1L11111 1 1111111 1 1 111111 L N o 100 ec.m.[deg] Figure VI-7. 60 8'0 Figure VI-8. 209 Data and Coupled Channels Calculations for Pt(p,p') With and Without the Spin-Orbit Interaction. The calculations included the couplings shown in Figure VI—Sb, the matrix elements of Table VI—8, and DOMP parameters from Tables VI—2 and 9. 210 dddddqd 4 1d4ddd d J dedd-d d d .ududde 4 80 80 100 120 HO bps-Fb » LI b Vb-spphp p P 20 nU 11 TmVQEHGchU ec.m.[deg] Figure VI-8. Table VI-9. Deformation Parameters tions for 194'195'198Pt. tions with 211 and Potential Moments Obtained from 0-2-4-6 Coupled Channel Calcula- Values from calcula- and without a spin-orbit (L*S) interaction are included. Nucleus 82 B4 q2 q4 (b) (b2) 332: K = o 0375a L*S -o.154(2) —o.0455(1o -1.32(2) -0.156(7) L*S = o -0.l68(3) ~0.0566(17) -1.40(2) -0.184(12) K = 0.5375 L*S -o.151(2) -o.04s3(10) -1.30(2) -0.160(6) L*S = o -0.l64(3) -o.0550(20) -1.37(3) -0.181(12) 19.322 K = 0.025 L*S —o.142(3) -0.0485(13) -1.25(3) —o.202(11) L*S = o -o.152(5) -o.os73(21) -1.31(5) -0.226(16) 19.322 K = 0.016 L*S -o.119(2) -o.0422(20) -1.05(2) -o.177(7) L*S = 0 -0.128(4) -o.0479(30) -1.09(4) -0.181(18) aThe units for K are keV. 212 ground state angular distribution and near 500 in the 2+ angular distribution. The ratio-to-Rutherford calculation seems to be out of phase a few degrees between '700 and 90°, while the 2+ shape exhibits a pronounced minimum at 50° unlike the shape calculated in 194’198Pt(p,p'). These discrepancies may be the result of the problems with the optical model parameters for 196Pt mentioned above. Additional calculations were performed for each nucleus to investigate the effects of the spin-orbit interaction. Searches were performed on the optical model parameters with a spin of zero for the proton and with zero spin-orbit well depths. The major change in the parameters was in W which 0’ increased significantly to decrease the depths of the minima (increase absorption) from the deep ones which result if the average parameters are used with spin-orbit effects turned off. Table VI—2 lists the results of these parameter searches for each nucleus. One can see the comparison of the spin-orbit versus non-spin-orbit (L*S) = 0) calculation in Figures VI—6, 7, and 8. The most obvious difference in each figure is the more pronounced oscillations in the L*S = 0 calculations. Each. calculated shape idisplays this characteristic, with the 4+ shapes the most obvious. The poorer quality of these fits is borne out by chi—squared values which are nearly twice as large for each level as values from the calculations with spin-orbit. However, even though the 1&8 = 0 Bx's values are significantly different 213 from the spin-orbit ones, the moments calculated in each case are comparable. Although these calculations have shown the importance of spin-orbit effects in reproducing the data, there are still some practical advantages to investigating calculations without spin—orbit as long as one realizes their limitations. The major advantage of L*S = 0 calculations is the immense savings in computer time. As pointed out in Chapter III, the inclusion of a non-zero spin projectile in the CC calcula- tions can more than double the number of coupled equations that must be solved. For proton scattering this translates into calculations that will require ten times as much computer time as the same calculation with a spin-zero projectile. With fewer coupled equations in an L*S = 0 calculation, the effective size of the computer is also increased. For example, on the Xerox 2—7 computer at this laboratory, a 0-2—4—6—8—10 calculation is possible, while with spin-orbit included only a 2—2-4—6 calculation space can be used. 'The following section has capitalized on the advantages of calculations without spin-orbit to investigate the effect of a variety of couplings (some not possible with the inclusion of spin-orbit terms) and matrix elements on the quality of the fits and ultimately the extraction of the deformation parameters. 214 4. Sensitivity of CC Calculations to Higher Order Couplings and Selected Matrix Elements a. The Sign of 84 Earlier theoretical. calculations [Ge 72] and experi- ments [Ba 76]have indicated the need for a negative value of B4 to account for certain features of the Pt nuclides. A series of calculations were performed to investigate the sensitivity of the (p,p') data to the sign of 84. Figure VI-9 compares the result of a 0-2—4-6 calculation for 194Pt (without spin-orbit) with a positive, negative, and zero value for 84. The X2 values for the 2+ and 4+ angular distributions are also given. Although the fit to the 4+ level with a negative 84 value is quite poor (mostly due to exclusion of spin-orbit effects), the overall slope and the fit to the first maximum are in agreement with the data. However, in the case of a positive 84 (dashed line fit in Figure VI-9) the oscillations are almost completely out of phase with the example for 84 < 0, and the cross section is overestimated at backward angles. Also, the value of x§+ increases by a factor of 2. The calculation with 84 = 0 clearly fails to reproduce the data. The necessity of including a 84 component is discussed further in a later section concerning the second 4+ level. The present results seem to support earlier findings of a negative value of B4 in the shape of the Pt isotopes. Figure VI—9 . 215 Data and Coupled Channels Calculations in a 0—2—4—6 Space for 194Pt(p,p') With Positive, Negative, and Zero Values for 84. These cal- culations used the IBA matrix elements of Table VI-5 and the DOMP parameters of Table VI-2. No spin-orbit interaction was included. 82 = —0.172. 216 didddd 4 fl dddddd¢ d J ddddd dd 4 pth» p p + b bbhbprb F d djddlqqu + dqdqdda 1 d 80 100 120 90.m.[deg] 80 L10 1 l _ . 0 0 20 Figure VI-9. 217 b. Investigation of a 86 Deformation Several calculations have been performed to examine the need for a 86 component in the nuclear shape of the Pt nuclides. Figure VI-lO shows three situations for 194Pt(p,p') with 86 = -o.01, 36 = 0.0, and 86 = +0.01. These calculations were carried out in a 0+-—2+—4+-—6+-—8+ level space using the couplings shown in Figure VI—Sc, with no spin—orbit potential, and all angular momentum couplings up to L = 10. Although there does seem to be evidence in the rare-earth nuclei for a small 86 shape component [He 68], the data for the 6+ angular distribution for 194Pt do not indicate any preference for a nonzero value. Even with improved data for the 6+ state, it is doubtful whether any more information could be gained due to the similarities of each calculated 6+ shape. As Figure VI—lO shows, the inclusion of a 86 term also has very little effect on the shapes of the 2+ and 4+ angular distributions. A calculation was also performed allowing 82, B4, and 86 to vary simultaneously to study the effects of 86 on determining 82 and 84 values. Figure VI—ll compares this calculation with the final calculation of a 82, 84 search for 194Pt with the same initial Bx's and no spin- orbit potential. .Although the deformation parameters have changed significantly, and the fit to the 4+ is improved (at the expense of the 2+ fit), this is the result of a final value for 86 of +0.067, which seems physically unlikely (see e.g. [He 68]). Figure VI-lO. 218 Comparison of Data and Coupled Channels Cal- culations for 194Pt(p,p') With the Couplings of Figure VI—SC, and Three Values for 86' The calculations included the matrix elements of Table VI—S, and the DOMP parameters of Table VI-2 with no spin-orbit interaction. 82 = -0.172, 84 = —0.0567. 219 1 1111111 mb/sr] H dU/dSZ 10° r r I r rrtr] 1 1 1111111 1 1 1111111 1 1 1111111 1 J 1111111 1 J 1 111111 Figure VI-10. Figure VI-ll. 220 Data and Coupled Channels Calculations for 4Pt(p,p') With Search on 32, B4, and With 86 = 0 (Dashed Curve) and for Search on 82, B4, and 85 (Solid Curve). No spin—orbit interaction was included. Both calculations had the same initial values for 82(= —0.l72) and 84 (-0.0567). The matrix elements were taken from Table VI-S and DOMP parameters from Table VI—2. 221 Z _19L*P+[p,p°] L'S=0 i : BB:+0'087 .4 ’ ""'" 88:0'0 ‘ 1. " , 1 ‘1’ \, LP‘ ‘ 811 ‘ 10'23 a " 8+ I 1H12 10’3: 3 F . .. 8+ .. 2100 1 l 1 l 1 l 1 1 l 1 80 160 120 Gamideg] Figure VI-ll. l\) O .1: c3 07 C) 222 c. The Effect of y-band Couplings In order to test further the 0(6) IBA matrix elements, calculations were initiated to include the lowest states of the y-band, whose bandhead is the second 2+ state. Three previous inelastic scattering studies of the Pt nuclei have included results for the y—band. Only the 2+ state of the y—band in 194Pt was studied in References [Ba 76, Ba 79]. The former investigation used a combination of empirical and rotational matrix elements, while the later analysis was performed within a triaxial rotor framework. Reference [Ba 78] included both the 2+ and 4+ members of the y—band for 192Pt and also used empirical and rotational matrix elements. The later two studies also seemed to indicate that the inclusion of B4 deformations was important in obtaining a more thorough understanding of these nuclei. In the present study, data were obtained for the 2+, 3+, I and 4+ states of the Y—band in 194'196Pt and for the 2+ and I 4+ states in 198Pt. The calculations included both 82 and B4 deformations, and initially, a full set of IBA matrix elements. As discussed later in this Chapter, the IBA 0 + 4+. matrix element was extremely small and a larger value was necessary to reproduce the data. The results of these calculations for 194Pt are shown in Figure VI-lZ for two different sets of matrix elements. Both calculations included the couplings of Figure VI—Se with the L*S interaction and the appropriate deformation parameters Figure VI—12. 223 Data and 0-2—4—2'-3—4' Calculations for 194Pt(p,p') With the Spin-Orbit Interaction and Two Sets of IBA Matrix Elements, Table VI-6 (Solid Curve), and Table VI-S (Dashed Curve). The couplings included those of Figure VI—Se. The appropriate DOMP parameters were taken from Tables VI-2 and 9. 224 L35 (Dex—.1 DUI 00¢ 05 o 1' U [11"] T U 'llllr' fir T VITml 1...: O I N I IVIIIII] ._..1 CD I H W—r I [Irrl] fi—T I [11"] 0 e 0 375keV 375keV 1 11111111 1 11111111 1 11111111 1 11111111 1 1 1111111 1 11111111 1 11111111 1 1 1111111 1 1 11111 1 1 1111111 20 L+0 so 80 “160 90_m.[deg] Figure VI-lZ. 225 from Table VI-9. One set (K = 0.0375 keV) is the set mentioned earlier, which was derived from the parameters of [Ca 78]. The second set was obtained from PHINT by increasing the strength of the quadrupole force between the bosons (K = 0.5375 keV) while the remaining 0(6) parameters retained their previous values (see Table VI—4). The two calculations provide nearly identical fits to the ground band states (comparable fits to those shown in Figure VI-6 with no y—band couplings). The y—band fits, however, show distinct differences. The calculation with x:= 0.5375 (solid line) yields a definite improvement in the fit of the 2+. state, both in strength and the depth of the minima. However, even though both calculations underestimate the strengths of the 3+. and 4+. angular distributions, the solid line fit for K = 0.5375 introduces far more structure into the shapes of both angular distributions. This may be due to the order of magnitude differences in the matrix elements between the levels of the y-band, as well as between the 3+', 4+', and the first 2+ state. This changes the interference between the many coupling paths, and thus the shape of the angular distributions. As stated above, the difference between the two sets of matrix elements is a larger value of K, the strength of the quadrupole-quadrupole boson force. Small values for this parameter primarily affect the magnitude of transitions which were not allowed in the strict 0(6) limit due to the AT = t l 226 selection rule. As Figure II-7 shows, the affected transi- tions are M , M , M34. , and M2.4 , where 02' 23' ”24' M. jk = - Also affected are the reorientation matrix elements because of the increased mixing of the wave functions. Thus, both sets of matrix elements result in nearly identical fits for the ground band, since the intraband transitions are virtually unchanged. The value of K for the matrix elements of Table VI-6, which gives the solid line fit in Figure VI-12, was deter- mined by varying K'in the IBA code PHINT until the ratio of I I B(EZ; 2+ -+ 0+/B(E2; 2+ + 2+) approached the experimental value of 0.0031 [Ca 78]. This increase in M02. provides the necessary strength to fit the 2+. state, however, at the expense of the already poorly fit 3+ and 4+. states. The new value of K is not an unrealistic value even though it is about 20 times as large as the value taken from [Ca 78]. The larger value is still an order of magnitude smaller than the values used in the Os nuclei calculations of Reference [Ca 78], which show considerably more rotational character, the feature which K actually represents. A search on M02, was performed using the couplings of Figure VI-d and the K = 0.0375 keV matrix elements. The "best fit" value for M02, was determined to be 0.046, compared to 0.063 from the PHINT calculations when K‘= 0.5375 keV. 227 The implication for such small values of M02, is that the population of the 2+. state is achieved primarily through a two-step process,' 0 + 2 + 2+', where the two matrix elements involved are each two orders of magnitude larger than M02.. Another IBA matrix element that is very small compared to the experimental value is M22 , the reorientation matrix element. The present calculations are not very sensitive to this matrix element, because the reorientation effect is proportional to 22 of the projectile and produces the largest effect at backward angles. Thus, the proton scattering data are a relatively weak probe of this effect. For calculations involving the scattering of low energy alpha particles, where this effect is important, significantly larger values would be needed for K‘ than the values from Casten and Cizewski [Ca 78]. I d. The Effect of the 2+ State on the Bx's To test the influence the 2+ state may have on extracting the deformation parameters, a simultaneous search was made on 82 and 84 using a 0+—2+-4+-2+' coupling scheme and the IBA matrix elements with K = 0.5375 keV while minimizing the chi-squared values for each level. Table VI-lO compares the results of the above L*S = 0 calculation with those for a similar E§_ search also +_ + minimizing all four X2 values in a 0+-2+-4 6 level space. Both calculations had the same initial values for 82 and B4 228 mmma omma heme mmoa N mmm mew mew hum +Mx mam map mom was +Mx Hmm mum mam ham +mx ammo.o- memo.ou ommo.o- memo.on «m mama.o- maea.o- mmoa.o- mnhfi.ou mm mmsam> Hague mmsflm> HaauHcH mwsam> Hague mmsam> HauuacH cowumasoamo cowumasoamu 1+~I+ql+mi+o +ml+¢l+ml+o .>mx memm.o u y cam u m«u spas cuom .cofiumasoamo .+~I+¢I+~I+o PE +on+¢s+mr+o m 5253 x cam m home 331.5 35m EB 3315 no 820358 .31; flame N 229 . 2 2 . . . and in each calculationJ x; (= x0 +'x2 +-xi) minimized on the same final values within statistical uncertainties. However, the better fit was obtained in the 0+-2+-4+—6+ calculation as indicated by the total chi-squared, 1267, compared to 1355 for the 0+—2+-4+-2+ calculation. The relatively weak coupling of the 2+I state, and thus the entire 'y-band, indicates that calculations involving only ground band couplings are sufficient for determining the deformation parameters. e. The Sign of P3 One of the principal motivations for the earlier inelastic scattering studies on the Pt nuclides [Ba 76, Ba 78, Ba 79] was to determine the sign of the E2 inter- ference term P3 = M02,M02M22. , where M = - . jk The CC calculations are sensitive to the relative signs of these matrix elements, because of the interference between the one-step and two-step paths that can be used to populate the second 2+ state, 0 + 2' and 0 + 2+ 2', respectively. The studies of Baker et al. indicate that a negative value of P3 is needed to fit the data. This was an unexpected finding since both the asymmetric rotor [Ba 58] and pairing-plus- quadrupole [Ku 69] models predict P3‘> 0. These same experimental studies also conclude that large values of M04 and M04, are needed to explain the shape and strength of the 230 4+ angular distributions. However, in [Ba 79] it was shown that by including only the symmetric, E4 components to the usual E2 asymmetric rotor shape (Davydov model [Da 58]) the fit to the first 4+ level was improved and that a negative value of P3 was now consistent with the predictions of this "extended" asymmetric rotor model. In the present study calculations have been performed, with the L*S interaction, to investigate the sensitivity of the (p,p') data to the sign of this interference term. Since the proton scattering also indicates the need for large, direct E4 matrix elements, the first 4+ state and a B4 defor- mation were included in the calculations. The couplings used are shown. in Figure VI—Sd and the Bxfs were those from Table VI-9. The major differences in these calculations and those from Reference [Ba 79] are the use of 0(6) IBA matrix elements and a symmetric parameterization of the shape of the nucleus having a diffuse cutoff. Figure VI-l3 shows the 194 results of a 0-2—4—2' calculation for Pt performed with the matrix elements of Table VI—6. The sign of P3 was changed by changing the sign of M02.. The data are fit much better when P3 < 0, in agreement with earlier studies [Ba 76, Ba 79]. This is also true in similar calculations for 196Pt. Interestingly, the 0(6) IBA calculations correctly predict the sign of P3 in both sets of matrix elements calculated for 194Pt and for 196Pt. So, the present analysis and that of Baker et al. [Ba 79] both agree that P < 0 and yield 3 Figure VI-l3 . 231 Data and Coupled Channels Calculations for 194Pt(p,p') With Positive (Dashed Curve) and Negative (Solid Curve) Values for the Interference Term, P . Note that the data and theory for the 2+ state have been multiplied by 2. The couplings of Figure VI-Sd were used with 82 = -0.151 and 84 = ~0.04S3. 102 101 232 I I I I fi' I I I r 19L*P’r[p,p"] L*S K=O.S375kev P3<0 P3>0 ‘§ — J 1 1 1 1 b—’1 1 1 1111 1 J 1 1 11111 1 1 11111 J l 1 1 1111111 1 1 1111111 ‘ L10 80 80 100‘ ec.m.[deg] Figure VI-l3. 233 essentially the same g2 and q4 charge moments (see Section 3.4.9 of this Chapter), but with each method assuming a different shape for the nucleus. Thus, the scattering data analysis, much like the case of odd-A nuclei coupled to a variety of even-even cores, does not give unambiguous results as to the shape of these transitional nuclei. The scattering data predict no apparent distinction between a rigid, asymmetrically deformed nucleus, and a ir-unstable (0(6)) nucleus, "frozen" in a symmetric oblate shape. For 198Pt(p,p') the calculations with P3 > 0 provide slightly better fits to the data, even though the~ 0(6) calculations predict P 3 < 0. However, in these calculations, M02, is a small negative number, —0.0012. This change of sign compared to the 194'196Pt O(6) calculations is under- standable because in the strict O(6) limit M02, = 0 due to the AT selection rule. With the addition of a small perturbation (K), the selection rule is broken and small values of about zero are obtained. The reason P3 remains negative is because the sign of M02 unexpectedly changes as well. The cause of this sign change is not understood. To draw any conclusions about the role of P3 in the analysis of the data for 198Pt it seems best to secure first a better value for K because of its influence on the small matrix elements. f. The M E4 Matrix Element 04' The CC calculations obviously fail to reproduce the data 19 I for the 4+ angular distribution in 4Pt(p,p') with 0(6) 234 matrix elements (see Figure VI—12). This was true in similar calculations for 196 ' 198 Pt(p,p') (see Figure VI—lS, dashed curves). The major cause of this failure is the small, predicted E4 matrix elements between the ground state and second 4+ level, resulting in a calculated cross section which is more than an order of magnitude weaker than the data. A similar effect was seen in the (a,a') study of Reference [Ba 78]. To improve such calculations a search was performed on the 0—4+' matrix element to fit the 4+. angular distribution. The remaining matrix elements were the IBA values of Tables VI-S, 6, 7, and 8, scaled with the appropriate L*S = 0 Bx's from Table VI—9. The search was performed without the spin—orbit term so that a 0—2-4-2'-3-4' level space could be used with the couplings of Figure VI-Se (except for 198Pt where no 3+ couplings were included). Computer limitations precluded a search in this level space with a spin-orbit interaction. However, as shown in Figure VI—l4a, the shape of the 4+. angular distribution with and without spin-orbit are similar enough that the values of M04. from these L*S = 0 searches were also used in the cal— culations with spin-orbit included. The results of these calculations are shown as the solid curves in Figures VI—l4 194,196,198Pt and 15 for . For comparison the calculations with the full IBA matrix elements are also shown. The best fit is obtained in the 198Pt case, with the rather structure— less shape except for a maximum at 35°. The other three 5 3 2 .m can NIH> moanme mo wuwumEMumm mzoa on» cam wmlH> musmwm mo mmcflamsoo wnu com: mcowumasono one .pwbsaocfl .voz uOu osam> ufim ummn ecu cuflz mcofiumasono mucmmwumwu m>uso wwHOm on» waflcs .mucmewaw xwuume uso ponmmp one .>wx mhmm.o 0cm mnmo.o n y cups A.m.mvumvma qu mcoflumasoamo mawccmsu pmamsoo cam mama .vanfiw musmfim 236 32351.00 Hmwgécbo lawffiofbmi mmimriom lowi.o.owa.o.m.o.mw W 88 x. I m. .2. (005%. , ... \u. n I ._/ W --/, mmm 1 ./ w ..m e . r 00 1 a . co m 898 ”m «mm - +N I a . Pei __ 6V. o H U «(1.. m M as W a . 2m TOH 2m \ so u 0.?— OT 1: 2 311/ m. In ., , w 72 , \. e mmm mmm \ .w 1 .2 -.., ., , 3-0 zmm Fabian ,. w (mH :31 man. .1 "OH (mH :3... m.:. IIII W .115 imam m3 II U .115 from m3 Ill m >3mkmmsux r .3512 3 >$mkmosux Zaire N C) [Jg/qwksp/Dp H 1 h L h b b b b b F P . b F L L b we 237 .m can NIH> wanna mo muouwemumm mzoo mcu 0cm Ammcfiamsoo +m obsaocfl uoc can A.m.mvummma ammoxov wmIH> wusmwm mo nonwamsoo may pom: mcoflumasoamo one .cmpsaocw .eoz uom msam> new ammo ecu Lugs cofiumasoamo m mucmmmummu m>uso cwHOm mnu mawn3 .mucmewam xfiuume KmH mo uwm Hana on» sues cowumasoamo m mucwmmummu w>uso cmemmw was .A.m.mcummau.mmfi uou mcoaumasofimo mawccmzo cwflmsoo mwu new mama .mHIH> musmwm 238 IIIIII I I [IIIIII I I IIIIIII I I IIIIIIII I IIIIIII I I IIIIIII I I 'IIIIII I I IIIIIII I I 'IIIIII I I 'IIIIII I I I\ . m 1 N u—q O _ \ "N \ H b ’ 4 I I O .. . “O U) j' H __ I I .1 _I O \ b -1- I do "“ ‘0 CD I o.0 )- - CD 4 O. h: ..— ~Q. -c’ a, LO 0') 1- H o 4 H X _ , -c: 3' A ' .0 I- . d 11 1 11111111 1 I11 111 1 11111111 1 11111111 1 1111111 11111111 1 O N v—l C) 1 I I 1 O O O C) O O O O H H H H 0—1 H H H "III I I I [IIII'I I I [IIIIII I I lIIIIrI I I [IIIII I I T [IIIIII I I IIIIIrI I I '"I‘III I I IIIIIII I T [IIIIIII I _ m m _ m y + U) + l\ + a) N m 3' G) N (.0 F C) “(\J c—I >- -( .. ‘0 O U) 3- ._. L 0 f“ (‘0 (D Q a) . - m J O. H _' + do Q— (D (D 07 I- F4 4 1— .10 3" A 1- ° -4 1111 1 1111111 1 1 1111111 1 1111111 1 1 1111111 1 1 11111111 1 O c—a -—4 (\I N o-I O l I O C) O O H r—1 1—1 H .9. s [I /qw76p/Dp 9c.m.[deg] Figure VI-lS. 9amldeg] 239 calculations show a considerably more diffractive shape for this angular distribution. Table VI-ll summarizes the M04. and B(E4) values as well as these values from 192Pt(a,a') [Ba 78]. Baker et al. obtained a better fit with the posi— tive value for M04, , however, the (p,p') results give B(E4) values that are more comparable to the magnitude for the negative value for M04, . It should be noted that the sign of these matrix elements is relative to the two-step and three-step excita- tion routes for the 4+. level. The interferences between the routes may be causing the varying amplitudes of the oscilla- tions. This uncertainty in sign can also lead to an uncertainty in magnitude as is shown in Table VI—ll for 192Pt. The best fit values from this study are thus relevant only in regard to the signs and magnitudes of M02. , M22. r , M and M ”2'4' 4' ' 02' ° 9. Comparison of Charge and Nuclear Potential Moments As discussed in Chapter III.C, this study has followed the suggestion of Mackintosh [Ma 76] and reports the results of the CC calculations in terms of q, a potential multipole moment. Mackintosh, using a theorem due to Satchler [Sa 78], has shown that the multipole moments of the potential, q, are proportional to moments of the nuclear matter density, if the deformed optical model potentials (DOMP) for protons are derivable from a reformulated optical model potential, where one assumes the nucleon-nucleon interaction is independent of 240 Table VI-ll. Summary of 0 + 4' Matrix Elements and B(E4) Values for 192"198Pt. Nucleus M04. B(E4; 0-4') B(E4; 0-4) (ebz) (e2b4) (e2b4) lgtha +0.34 0.12 0.041 -0.20 0.040 0.041 194Pt K = 0.0375 -0.12 0.014 0.024 K = 0.5375 -0.11 0.012 0.026 196Pt +0.14 0.020 0.041 198Pt +0.21 0.043 0.031 aValues from [Ba 78]. 241 density, and the proton and neutron distributions have the same deformation. This multipole moment method is used here because in addition to being a more fundamental approach, it also facilitates the comparison of results from Coulomb and nuclear scattering experiments. The g2 and q4 calculated using Equation (III-71) and the DOMP parameters of Table VI—2 are listed in Table VI-12 and displayed schematically in Figure VI—l6, along with moments from Coulomb excitation [Ba 79, Ba 78a, G1 69], and those calculated from an (0,0') Coulomb-nuclear interference experiment [Ba 76], in which both charge and potential moments can be obtained. Also shown are the moments deduced from the theoretical predictions of 82 and 84 from Gotz gt 21; [Go 72] determined by a Strutinsky renormalization method. These moments were calculated using the parameters of the single particle potential for neutrons. The Coulomb deformation parameters obtained in [Ba 79] and [Ba 76] used a uniform charge distribution with a sharp cutoff for asymmetrically and symmetrically deformed shapes, respectively. The potential moments in the present study and in [Ba 76] used a deformed Fermi distribution. A comparison of the (p,p') results with those from the previous experiments indicates the g2 potential moments from proton scattering are in much better agreement with the charge moments from Coulomb excitation values than the potential moments of Baker et al. [Ba 76] determined by 242 Table VI-12. E2 and E4 Moments in 192'196'198Pt. Nuclide Method q; q: 2 2 (b) or (eb) (b ) or (eb ) 194Pt (p,p') -l.32(2) —0.156(7) at 35 MeVb Coulomb excitationC —l.273(6) (12c,12c:')d -1.269 -0.l486 (01,01')e P -1.52 -0.30 C -l.3l -0.12 Theoryf -l.468 -0.129 196Pt (PIP') at 35 Mevb -l.25(3) -0.202(ll) Coulomb excitationg -l.22(5) (a,a')e p -l.38 -0.24 C -l.l7 -0.097 Theoryf -l.380 -0.142 243 Table VI-12 (cont'd.). Nuclide Method q; g? (b) or (eb) (b2) or (eb2) 198PE (p,p') -l.05(2) -0.l77(7) at 35 MeVb Coulomb excitationg -1.00(3) (a:a')e P -1.12 -0.32 C -1.08 -0.14 Theoryf -0.579 -0.106 aThe units for the charge component moments are bx, AA: 2 or 4. The units for the electromagnetic moments are eb . bThese moments were obtained using the DOMP jparameters, including the spin-orbit interaction, contained in Tables VI-2, 9. CReference [Ba 78a]. dReference [Ba 79]. eReference [Ba 76]. The first value reported is for the potential moment (P) calculated from Baker et al. and the second value is for the charge moment (C). fReference [Go 72]. gReference [01 69]. 244 “Calculated and Measured -I.6- Quadrupole Moments 0 (p,p’), _| 5_ 1 Present Study a I (a.a’) 4.4- A (IZC,|2C’) _ i 0 Coul. Ex. (12 "'3 eon S x Theory 4.2- D -l l- '0 5 -4ID~ o -04- Calculated and Measured Hexadecapole Moments I -Cl3-‘ I q4 I -o.2- i i A! X——— \D .0'-4 a D 194' 196' 198' Pt Pt Pt Figure VI-l6. Plot of Quadrupole and Hexadecapole Moments 194,196,198 for Pt Given in Table VI-lZ. 245 (a,a'). This is also the trend of the g4 moments. In each nucleus the charge component of the proton potential moment is in better agreement with the charge moment from [Ba 76. Ba 79] and also the theoretical values of Gotz et al. These same theoretical values for the g2 moments, however, are nearer the values for the potential moments of [Ba 76], except in 198 Pt where the predicted q2 value is only one-half the smallest experimental value. This discrepancy between a scattering, potential moments, and Coulomb excitation values has been discussed by Mackintosh [Ma 76] and is thought to be an indication that the a scattering potentials cannot be derived from reformulated optical model potentials. If this is true it may mean that high energy a-scattering is not a reliable probe for measuring nuclear shapes. It has long been thought that a-particles should be an excellent surface probe because of their strong absorption by the nucleus and the simplifi» cations resulting from being a spin zero projectile. However, in most analyses the structure of the a—particle itself has been ignored. There is less of a problem of this type with a proton as the projectile. A fundamental question can now be asked about the data in Table VI—12. Do the differences between the potential (matter) moments from Coulomb scattering imply that neutron and proton distributions are not the same? Calculations by Hamilton and Mackintosh [Ha 77] indicate that differences of 246 20 to 30% could exist between neutron and proton moments in 154Sm. Even though each potential moment calculated in this study is larger than the charge moment for all three nuclei, the differences are statistically significant only for the g2 194Pt and the g4 value in 196Pt. This fact, in moment of addition to questions concerning the strict validity of this multipole moment procedure, prohibit one from seriously considering such small differences. Even though this approach to analyzing scattering data is believed to be the most fundamental phenomenological method available, there are several facets which must be investigated, such as: how should the imaginary part of the DOMP be included in the moments; are the DOMP's derivable from a reformulated optical model potential; are density dependent effects important in the interaction potentia; how does one scale Bxand r0? Once these are understood, it may be possible to interpret extracted moments with more confidence. CHAPTER VII CONCLUDING REMARKS The recent interest in the study of the shape transi~ tional Pt nuclei is because of the key role they possibly play in understanding collective nuclear behavior. These nuclei span a region between the relatively well understood limits of the standard hydrodynamic collective model, and thus provide an experimental test for new models in this region or for variations on the simple collective model. When the present study began most of the data on these nuclei were from (n,y) experiments or decay studies, except for two transfer studies [Mu 65, Ma 72]. Thus, the present high- resolution reaction study has contributed much new data on the Pt nuclides to aid in distinguishing between the variety of models applied in this region. Recently Casten and Cizewski [Ca 78, Ci 78] have shown that the 0(6) limit of the IBA model does a remarkable job in explaining the branching ratios and energies of all positive parity levels below 2 MeV. As shown in this thesis, some of the features of this model can also be used to interpret several aspects of the (p,t) and (p,p') data with reasonable success. In the (p,t) reaction studies, the strengths of the 247 248 L = 0 and L = 2 transitions have been compared to the predictions of the IBA model. This model provides a natural framework for discussing multi-particle transfer reactions because of its inherent feature of pairing fermions to form bosons. The general agreement of these calculations with the data is quite good, particularly in the trend of ground state strengths and in the preferential population of the second excited 0+ level over the first. For the L = 2 transitions the model correctly predicts the relative magnitudes for the cross section for the first and second 2+ levels within a particular nucleus, although the calculated trends versus changes in A are not seen in the data. These problems are possibly related to the choice of the coefficients in the L = 2 operator. The values used in these calculations (a = l, B = 0.08, and 1'= 0) were chosen without any previous knowledge of the (p,t) data. A more judicious choice of these parameters may give results in better agreement with the data. Also, by including some of the smaller terms of the L = 0 and L = 2 operators, some of the very small cross sections predicted for the third and fourth 0+ and 2+ states may be altered. The O(6) limit of the IBA has also been utilized in interpreting the inelastic proton scattering data“ Speci— fically, the matrix elements for both E2 and E4 transitions have been used in a series of coupled channels calculations to relate the strengths of the various coupling routes 249 between levels. These matrix elements were calculated using the 0(6) parameters of Reference [Ca 78]. The resulting fits to the data are quite good for the angular distributions populating states in the ground band up to J1r = 4+ and for the second 2+ level. In most of the calculations the full set of matrix elements was used without any searches on a particular matrix element. An additional set of matrix elements was used for 194 Pt(p,p') to investigate the sensi— tivity of the CC calculations to the value OfIC, the strength of the quadrupole-quadrupole force between. bosons. The primary effect of a larger value of K was to change a few small matrix elements, particularly the 0 + 2+, value, which improved the fit to the 2+. angular distribution. It is important to note that the fits obtained in this study do not rely on any special "best fit" group of matrix elements from the 0(6) limit, either from searching on energy levels and B(EA) values or from the CC searches. The sets used in these calculations were not determined individually for each nucleus, but were generated from one set, which is applicable in the entire Pt region with only a small mass dependence in the O(6) parameters. It would be worthwhile in the future to carry out these CC calculations with a better determined set of matrix elements from the IBA, preferably with the inclusion of neutron and proton bosons. Nevertheless, the (present. calculations. have jprovided sufficient information to confirm the results of earlier 250 inelastic scattering experiments, which found a negative I is needed to fit the 2+ data in 194Pt. The 19 value of P3 present calculations also indicate P3 < 0 for 19 6Pt(p,p') as well, but there are some uncertainties in 8Pt(p,p') that need further study. It is interesting to note that in each case the 0(6) limit also gives a negative P3 interference term. The present study also» is in agreement. with recent a-scattering studies [Ba 78] that indicate a strong E4 transition is needed to account for the strength of the 4+' angular distribution in 192Pt. Although the magnitude for this matrix element has been determined, there is still some uncertainty in its sign due to the competing multistep E2 routes that can populate the 4+. state. This sign dis- crepancy also results in a factor-of-2 uncertainty in the 192 magnitude of M for Pt [Ba 78]. It is unfortunate that 04' the third 4+ level seen in all three (p,p') reactions could not be included in the CC calculations, for these strongly excited states may be similar to those populated in 190'19205 [Ba 77, Bu 78], which were interpreted as hexadecapole vibrational states. Considering the cross section for these states compared to that for the first and second 4+ (including their rather large E4 component), this may indicate the possibility of a hexadecapole vibrational component in these states as well. One question not addressed in this study was the influence' on the CC Icalculations that. would result from 251 including the 3_ level in the couplings. This level is one of the more strongly excited levels in each reaction and thus might be involved in a number of coupling routes. Since the IBA code PHINT also can calculate El and E3 matrix elements, it should be possible to determine B(E3) values for 194’196'198Pt with a 0+-2+—3- coupling scheme and study the effects of the 3- level in more complex coupling schemes. An additional feature of the CC calculations was the determination. of moments for the deformed optical model potential. The method used to extract these values from the DOMP parameters proved to be most valuable in comparing the proton scattering values with moments determined by other methods. Many of the uncertainties and simplifications that were used in previous studies are overcome by this more fundamental method. Even though this method facilitates comparisons with other methods and gives moments that are in better agreement with the charge moments than an earlier (a,a‘) experiment, it is premature to draw any conclusions from the apparent discrepancies between the charge (proton) and potential (neutron) moments. More study is needed on the possible influences of factors mentioned in Chapter VI in determining the moments, such as Br scaling each potential and including the imaginary part of the DOMP. There are also many interesting alternative approaches to the analysis of the inelastic proton scattering data that might provide additional insight into the nature of these 252 nuclei. For example, CC calculations could be performed using form factors from the vibrational model with either IBA or vibrational matrix elements. This type of analysis would provide E2 and E4 moments that are uncoupled. It would also be of interest to analyze the (p,p') data within the extended triaxial rotor model framework available in ECIS, much like that carried out in the 12 C-scattering study [Ba 79]. It is possible that a comparison of moments calculated from the same data with differently shaped potentials may provide information on the equilibrium shapes of these nuclei. 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