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" 1111 _£J—-—v-_ r: £12.: L J “RM!" 1381! State University This is to certify that the dissertation entitled A STUDY OF 206Pb BY lNELASTlC SCATTERING 0F 35 MeV PROTONS presentedby JOSEPH EUGENE FINCK has been accepted towards fulfillment of the requirements for PLD. degreein PkyblL) / Majo rofessor Date /0/?//? z MS U is an Affirmative Action/Equal Opportunity Institution 0- 12771 RETURNING MATERIALS: IV1£3I_J Place in book drop to B remove this checkout from .liaififgfiL .your record. FINES will ' be charged if book is returned after the date stamped below. A STUDY OF 206Pb BY INELASTIC SCATTERING OF 35 MeV PROTONS BY Joseph Eugene Finck A DISSERTATION Submitted to Michigan State University in parital fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1982 ABSTRACT A STUDY OF 206Pb BY INELASTIC SCATTERING or 35 MeV PROTONS BY Joseph Eugene Finck Using high resolution techniques the inelastic scat- 206 tering of 35 Mev protons by Pb have been measured. A resolution of 6 to 9 keV allowed identification of approx- 206Pb with excitation energies up to imately 180 levels of 6.8 MeV. Angular distributions of most of these states are measured. L-transfers and deformation parameters are determined by comparison of the measured angular distribu- tions to collective model calculations. Strongly excited collective states are compared to analogous states in 207Pb and 208Pb and the overall distribution of inelastic 206Pb is compared to 208Pb. Microscopic cal- strength in culations of natural parity states are presented and allow a test of theoretical RPA and TDA wave functions. Unnatu- ral parity states with well determined wave functions are also studied microscopically and permit an examination of the central and noncentral forces in the effective inter- action. ACKNOWLEDGEMENTS I would like to thank Professor G. M. Crawley for his constant guidance and assistance during the time that this work was performed. Professor J. S. Kovacs is also thanked for his support and understanding during my graduate career. Dr. J. A. Nolen, Jr. is thanked for his aid with this project and many helpful comments. I would also like to recognize the staffs of the Michigan State Cyclotron Laboratory, Princeton Cyclotron Laboratory, and Northern Michigan University Physics Department. I am especially grateful for the aid and hospitality of Dr. R. Kouzes at Princeton and Dr. D. Fowler at Northern Michigan. Without these people this work would not have been possible. Working with Bill Wagner, Rick Steele, and Paul Smith as a novice graduate student at the lab was an invaluable educational experience. In particular, my escapades with Paul will never be forgotten. Bruce Hasselquist and Jim Duffy provided much compan- ionship and support as officemates, especially during prelims together. All my friends are thanked. In particular the members -- old and new -- of the Nuclear Beer Group. I am indebted to Reg Ronningen and Wayne Bentley for their ii "n4 Vu‘ -“r (1.3. kindkness and comradery. Special thanks must be given to Jim Carr and Carol Dors for their close friendship. These individuals made my days as a graduate student most enjoy- able and special. I am grateful to the communities of Michigan State University and East Lansing for providing a nonpareil academic and pleasant social atmosphere for me to study and live in. I owe a large debt to my parents for their support and prayers. To my wife, Deborah, I am grateful for her encouragement, understanding, and patience. Mostly, I am thankful for her love. My daughter, Elizabeth -- born during the preparation of Chapter IV of this thesis -- is recognized for her inspiration. This thesis is dedicated to my entire family. iii TABLE OF CONTENTS List of Tables . . . . . . . . . . . . . . . . . List of Figures . . . . . . . . . . . . . . . . . CHAPTER I. INTRODUCTION . . . . . . . . . . . . . . . REFERENCES FOR CHAPTER I . . . . . . . . . II. EXPERIMENTAL PROCEDURE . . . . . . . . . . REFERENCES FOR CHAPTER II . . . . . . . . III. EXPERIMENTAL RESULTS . . . . . . . . . . . REFERENCES FOR CHAPTER III . . . . . . . . IV. COLLECTIVE MODEL ANALYSIS . . . . . . . . A. B. C. D. E. F. Description of the DWBA Method . . . Elastic Scattering and Optical Model L-transfers and Deformation Parameters . 1. L=2 Transitions . . . . . . . . 2. L=3 Transitions . . . . . . . 3. L=4 Transitions . . . . . . . . 4. L=5 Transitions . . . . . . . . S. L=6 Transitions . . . . . . . . 6. L>7 Transitions . . . . Systematics of Collective States in Lead Nuclei . . Comparison of 206Pb and 208Pb Inelastic Strengths . . . . . . . . Summary of the Collective Model Results REFERENCES FOR CHAPTER IV . . . . . . . . iv PAGE vii viii 17 35 36 36 40 41 49 49 50 51 51 52 52 66 69 71 V. MICROSCOPIC MODEL ANALYSIS . . . . . . . . . . A. Description of the Microscopic DWBA Method for Inelastic Scattering . . B. Forces Used in the Microscopic Calculations . . . . . . . . . . . . C. Wave Functions Used in the Microscopic Calculations . . . . . . . . . . . . D. Results of Microsocpic Calculations . . 1. Natural Parity States . . . . . . . 2. Unnatural Parity States . . . . . . E. Summary of the Microscopic Results . . . REFERENCES FOR CHPATER V . . . . . . . . . . . VI 0 SUWRY O O O O O O O O O O O O O O O O O O O APPENDICES APPENDIX I. Analysis of the Data . . . . . APPENDIX II. Samples of DWUCK and DWBA-70 Input . . . . . . . . . . . 206 APPENDIX III. Pb Angular Distributions . . APPENDIX IV. Abstracts of Publications . . . A. B. C. D. E. F. G. A Survey of the (3He, 7Be) Reaction at 70 MeV . . . . . . . . The 54Fe (p, d) 53Fe Reaction at 40 MeV and the DWBA Analysis . . . . . . . Extraction of the Deformation Parameters from Inelastic Proton Scattering . . Inelastic Proton Scattering from Lanthanide and Actinide Nuclei . . . A study of the 54Fe (p, d) 53Fe Reaction at 40 MeV . . . . . . Octupole States in 63Cu and the Weak- Coupling Picture . . . . . Multipole Moments of 1548m, 175Yb, 232Th, and 233U from Proton Inelastic Scattering . . . . Core Excitations in 63Cu by the 63Cu (9. p ') and 65Cu (p, t) 63Cu Reactions . . . Multipole Moments of 232Th and 234: 236: 233U from Proton Inelastic Scattering . . . . . . . . . . . . . Sysematics of Collective States in Lead Nuclei from Inelastic Proton Scattering . . . . . . . . . . . . . V 73 73 76 78 78 79 94 101 104 106 109 112 117 132 133 134 135 136 137 138 139 140 141 142 K. A study of 206 Pb by Inelastic Scattering of 35 MeV Protons . fl from 175Yb and 1545m . (p, p') Reaction . M. Deformation Parameters via the N . Proton Scattering at 35 MeV to Ground Band States in 152 186w 232Th and 2380 O. Proton Scattering at 35 MeV to Ground Band States in 232Th, 176Yb 236, 2380 , , P. Multipole Moments from Proton Inelastic Proton Scattering and 234, Scattering at 35 MeV to Ground State Band States 330 . . REFERENCES FOR APPENDICES vi 236, 154Sm' 143 144 145 146 147 148 149 TABLE III.1 IIIOZ IV.1 IV.2 A.II.1 A.II.2 A.III.1 LIST OF TABLES PAGE Energy Levels, L-transfers, and Deformation Parameters for 205Pb. A Comparison is Made With Previous Results . . . . . . . . . 18 Energy Levels, L-transfers, and Deformation Parameters for 06Pb O O I O O I O O O O O O O O O O O 24 Optical Model Parameters Used in DWBA Calculations . . . . . . . . . . . 42 Comparison of The Strongly Excited Collective Levels in 06Pb, 207Pb, and 208Pb . . . . . . . . 62 Sample Input to The Program DWUCK For The 3‘ State at 2.648 MeV of Excitation . . . . . . . . . . . . . 113 Sample Input to The Program DWBA-70 For The 6‘ State at 2.385 MeV of Excitation . . . . . . . . . . . . . 114 Cross Sections of the 206Pb (p, p') Reaction . . . . . . . . . . . . . . . 118 vii FIGURE II.1 II.2 III.1 III.2 III.3 IV.1 IV.2 IV.3 LIST OF FIGURES PAGE Typical spectrum of protons scattered by 206Pb obtained with a photo- graphic plate. States of well determined spin and parity are identified. The resolution is about 6 keV . . . . . . . . . . . . . . 10 Typical spectrum of portons scattered by 20 Pb obtained with the pro- portional counter . . . . . . . . . . . 14 Measured inelastic cross sections. The lines drawn through the data points are included to guide the eye and do not rep- resent theoretical fits to the data . . . . . . . . . . . . . . . . . 30 Same as Figure III.1 . . . . . . . . . . . 32 Same as Figure III.1 . . . . . . . . . . . 34 Comparison of the measured elastic angular distribution with the DWBA calculation explained in the text . . . . . . . . . . . . . . . 44 Collective model fits for identified states. Displayed with the fits are the excitation energy of the state and the deformation param- eter, BL, corresponding to orbital angular momentum transfer L . . . . . . 46 Same as Figure IV.2 . . . . . . . . . . . 48 viii FIGURE PAGE IV.4 Angular distributions for positive- parity excitations in proton scattering from 206Pb. The solid lines represent collec- tive DWBA calculations. The dashed lines represent interpo- lation of corresponding levels in 208Pb. The excitation energy, Ex (MeV), indicated for each state is the value determined from the present data with uncertainties given in the text . . . . 55 IV.5 Angular distributions for negative- parity excitations in proton scattering from 206Pb. The solid lines represent collective DWBA calculations. The dashed lines shown with the 2.648 and 3.772 MeV states are interpolations of corresponding levels in 208Pb (Ref. IV.7]. The excitation energy, Ex (MeV), indicated for each state is the value determined from the present data with uncer- tainties given in the text . . . . . . 57 IV.6 Levels for which angular distribu- tions were measured together with those measured in Ref. IV.15. The numbers give the transition strength . . . . . . . . . . . . . . . 60 IV.7 Single-particle and single-hole levels in the lead region. The indicated energies are those at which these levels are fixed experimentally . . . . 64 IV.8 Results of collective model fits of 06Pb compared to 208Pb. The deformation parameter, BL, is plotted against excitation energy for a number of L-transfers . . . . . . 68 ix II"- FIGURE V.1 PAGE Microscopic model fits for low- lying natural parity states using RPA wave functions. The solid lines correspond to cal- culations done with Force A; the dashed curves show results using Force B. The asterisks indicate only direct calculations. The curves without asterisks indicate calculations including exchange effects . . . . . . . . . . . Same as Figure v.2 with TDA wave functions used in the calcu- lations O O O O O O O O O O 0 Microscopic Model fits for higher- lying natural parity states using RPA wave functions. The meanings of the curves and asterisks are the same as in Figure v.1 . . . . . . . . . . Same as Figure v.3 with TDA wave functions used in the calcu- lations O O O O O O O O O O 0 Comparison of measured angular distributions with the cen- tral and noncentral parts of Force A and Force B . . . . Same as Figure v.5 . . . . . . . . Same as Figure v.5 . . . . . . . . 81 84 88 90 93 96 100 88' tCl clc CHAPTER I INTRODUCTION The lead region has always been an attractive area to test nuclear models. Nuclei in this mass region have been studied both experimentally and theoretically. The bulk of this work has involved the structure of the doubly- 208 magic nucleus Pb. Many of these studies have also ex- 207 tended to the single-hole structure of Pb and the 20931 which are now well- 206 single-particle structure of established. An examination of Pb is a further step toward the more complex structure that exist away from closed shells. 206Pb include Experiments previously performed on inelastic scattering [Refs. 1.1, 1.2, and I.3I which has given information about the strongly excited states. Information about the microscopic structure of many of the low-lying states has been provided by decay studies [Refs. 1.4, I.5, and 1.6], transfer reactions [Refs. 1.7, 1.8, 1.9, and 1.10], and isobaric analog resonance experiments [Refs. 1.11 and 1.12]. The spins and parities of many higher-lying levels have also been determined by these 206 experiments. Using the shell model which describes Pb 208 as two neutron holes in the Pb core, energies and wave 1 fu C‘J ca bu ti in st nul pm the 206 functions of the low-lying levels of Pb have been cal- culated [Refs. 1.13 and 1.14]. With this background a detailed study of inelastic proton scattering from 206Pb, including collective and microscopic calculations, has been undertaken. A direct reaction, such as inelastic proton scatter- ing, can be used as a means to obtain spectroscopic infor- mation. In direct reaction experiments a beam of par- ticles with a certain energy is focused on a target. The number of outgoing particles of a certain energy as a function of the angle between the incoming beam and the outgoing particles (the angular distribution) is measured. From these data the energy of the levels of the investi- gated nucleus can be directly determined; information about the spin and parity of the excited levels, and the spectroscopic strength for the excitation of these levels can be obtained by comparing the measured angular distri- butions with calculations assuming a specific configura- tion for the level considered. Of all the direct reactions available proton scatter- ing is the most appealing reaction to investigate nuclear structure and interactions. Almost all levels in a nucleus can be excited by means of inelastic scattering of protons with a beam energy for above the Coulomb barrier. However, the cross sections can be low. The structure of the states can be determined from comparison of the ca'. ava are PYOI dis OVEJ sta‘ experimental angular distribution for the level with that obtained from macroscopic or microscopic distorted wave calculations. When there are accurate wave functions available from model calculations microscopic calculations are preferred because these provide a better check on the proposed structure of the state. As the shape of angular distributions at forward angles and the magnitude of the overall angular distribution are very sensitive to the different configurations in the wave functions of the state, microscopic calculations are a suitable test for the wave functions. The possibility of finding states which have not been seen before, together with the possibility of comparing the experimental angular distributions of the states with model calculations provide the motivation for a high reso- 206 lution (p,-p') experiment on Pb. 206Pb has A proton inelastic scattering experiment on been reported [Refs. 1.1, 1.2, and 1.3] at 24.5 MeV bom- barding energy with an energy resolution of approximately 25 keV. This experiment identified 30 levels, and spin and parity assignments for the most strongly excited states were made. Using a collective model calculation, the reduced transition rates for some of these states were extracted. In this experiment, the angular distributions were compared only with the collective model predictions. The theoretical tools for a microscopic analysis were not tr} 0'! well developed at the time that this experiment was done. Furthermore, only states below 4.6 MeV of excitation energy were observed and in this region states weakly excited were not extracted. In addition, the resolution limited the number of states that could be analyzed unam- biguously. This represents the most extensive study of 206Pb (p, p') to date. With the availability of particle accelerators with increased intensity and improved resolution, along with advances in magnetic spectrographs and particle detection devices, weakly excited levels and close lying excited levels can be resolved and studied. The microscopic description of nucleon-nucleus scattering has also pro- gressed. Now with a better understanding of exchange effects and the nucleon-nucleon interaction, microscopic inelastic reaction theory can be used to study nuclear properties [Ref. 1.15]. This thesis reports a study of 206 Pb (PI P.) per- formed at 35 MeV with an energy resolution of 6 to 9 keV. Experimental procedures are described in Chapter II. 206Pb with excitation ener- Approximately 180 levels of gies up to 6.8 MeV are observed. Measured distributions for 144 of these levels are displayed. In Chapter 111 these results of the experiment are presented. In the remaining chapters the theoretical models are compared with experimental results. The collective the COT mil am model is used in fitting many of the measured angular dis- tributions, and L-assignments and deformation parameters are obtained for these states. Systematics of strongly excited collective states in 206Pb, 207Pb, and 208Pb are examined, and inelastic strengths of 206Pb and 208Pb are compared. Microscopic calculations are performed for a number of natural and unnatural parity states. The micro- scopic examination of natural parity states permits the testing of wave functions since such transactions depend little on the noncentral two-body interaction. Wave functions obtained from the random phase and Tamm-Dancoff approximations are examined. Unnatural parity transitions to levels with well determined wave functions allow the two-body central, tensor, and spin-orbit forces to be studied. In this study two different sets of forces are employed for comparison with experimental results. In Appendix 1, the methods used in the analysis of the data are outlined. This includes a description of the computer programs used to perform data reduction, deter- mine excitation energies, extract angular distributions, and plot the results. Appendix 11 give examples of input to the distorted wave programs used in this study. Measured angular distributions of 206Pb (p, p') are tab- ulated in Appendix 111. Appendix 1V lists abstracts of published papers to which 1 have contributed while a student at Michigan State University. r4 REFERENCES FOR CHAPTER I J. Saundinos, G. Vallois, and 0. Beer, Nucl. Sci. Appl. 3 (1967), 22. G. Vallois, J. Saudinos, and 0. Beer, Phys. Lett. 24B (1967), 512. G. Vallois, Centre d'Etudes Nucleaives de Saclay, Report CEA-R-3500 (1968). J. C. Manthuruthil, D. C. Camp, A. V. Ramayya, J. H. Hamilton, J. J. Pinajian, and J. W. Doornebos, Phys. Rev. C Q (1972), 1870. J. E. Draper, R. J. McDonald, and N. S. P. King, Phys. Rev. C 16 (1977), 1594. D. F. Coope, L. F. Cannell, and M. K. Brussel, Phys. Rev. C 15 (1977), 1977. W. A. Lanford, PHys. Rev. C 16 (1977), 988. W. A. Lnaford and G. M. Crawley, Phys. Rev. C 9 (1974), 646. R. Tickle and J. Bardwick, Phys. Rev. 166-(1968), 1167. E. R. Flynn, R. A. Broglia, R. Liotta, and B. S. Nilsson, Nucl. Phys. A221 (1974), 509. J. Solf, C. F. Moore, E. Grosse, and P. von Brentano, Nucl. Phys. A139 (1969), 523. P. Richard, N. Stein, C. D. Kavaloski, and J. S. Lilley, Phys. Rev. 171 (1968), 1308. W. W. True and C. W. Ma, Phys. Rev. C 3 (1971), 2421. J. Vary and J. N. Ginocchio, Nucl. Phys. A166 (1971), 479. G. R. Satchler, Comm. Nucl. and Part. Phys. 5 (1972), 39. l.— 1a ba le CHAPTER II EXPERIMENTAL PROCEDURE The experiment was performed using 35 MeV proton beams from the Michigan State University and Princeton University sector-focused cyclotrons. The Michigan State cyclotron delivered a proton beam of between 500- and 1500-nA average current on target. The average current from the Princeton cyclotron was between 50- and 150-nA. Throughout the experiment 206Pb targets of about 0.1 mg/cm2 thickness were used. The targets were pre- pared by vacuum evaporation of the isotope, enriched to 97.22%, on a 20 ug/cm2 carbon foil with a support of two layers of formvar. This choice of target thickness was based on a study by Wagner [Ref. 11.1] which showed that lead targets of this thickness affect the resolution very little. In addition, targets of this thickness yield tolerable count rates, and skewing of peak shapes due to straggling of the protons in the target was reduced. The beam on target was monitored by measuring the total charge collected in the Faraday cup and by measuring the number of beam particles elastically scattered into a NaI(T1) detector placed at an angle of 90° relative to the incident beam. This angle was chosen because 90° lies 7 ne 2!] a. the twc sic prc the II. near a relative maximum of the elastic cross section for 206Pb and also gives a good separation of protons elastic- ally scattered from lead and light mass contaminants in the target. The relative normalization obtained by these two measurements agree to within 5%. The spectra from the part of the experiment at Michigan State were obtained using nuclear emulsions in the focal plane of the Enge split-pole magnetic spectro- graph. This plate data was taken with a 0.6 milisteradian (1° x 2°) solid angle at forward angles and a 1.2 mili- steradian (2° x 2°) solid angle at backward angles. A stainless steel absorber of thickness 0.25 mm was placed immediately before 20 inch Kodak NTB 25 um nuclear emul- sions. The absorber stopped all particles other than protons, and decreased the proton energy. This enhanced the proton tracks in the emulsion and did not signifi- cantly broaden the line width. On-line determination of the focal plane line width was optimized by adjusting the dispersion of the beam across the target using a "specu- lator" technique [Ref. II.2]. Once the dispersion was Optimized, the resolution remained constant throughout the experiment. The resolution of the plate data ranges from 6-9 keV (FWHM). Each plate run covers a range of excitation energies from the ground state to about 7.0 MeV. A typical spectrum of plate data is shown in Figure 11.1. States of well-determined spin and parity have been Figure 11.1 206 Typical spectrum of protons scattered by Pb obtained with a photographic plate. and parity are identified. States of well-determined spin The resolution is about 6 keV. unann- DO " ' “.7 '—"'—T_- *‘TifY—‘d—W‘ “' ' T— 1 9".) ring 1‘ T“- "7‘ V ‘_" T— Olnb ‘“ , ”Tl Mu V "V‘f' ”Af” ’4 USU! - Olo 3?! 10 f fr fl T I 1_.-—a-- l l T I P .. J! in m _ )- u— P‘oo m L- L CD a) b ‘0 If? . (V H *- b“! .o m .2 CD a) _ ._.. r z * \\\\\\\\\\\V (f) , . m ._ ' a: II ‘ ”:3 O. m h r ‘2. - *r :3 men, n b .9 O _ J: n: 3 ’ 3 ________ , R m .. p; _ 4‘! L 1 1 1 4 1 1 J O O O O O O a s: a s s a stunog Excuotion Energy Figure 11.1 CC ti th EX St 11 identified. Some strongly excited states, such as the ground state, the first excited state, and the 3- state at 2.648 MeV of excitation, produce proton tracks too dense to scan. However, their positions can easily and accurately be determined by the plate scanner. This aids in the energy calibration of the spectra. Most states below 4 MeV of excitation appear to be completely resolved. Of special interest in this study are the unnatural parity states. These states, being weakly excited, present an experimental challenge. In particular the 1+ state at 1.708 MeV requires very high resolution to be extracted from the shoulder of the strongly excited 4+ state. In all plate spectra the 1+ state was clearly separated from the 4+ state and could be easily extracted. The density of states above 4 MeV of excitation becomes increasingly large. Many of the states appear to be completely resolved. Peaks whose widths indicate possible multiplet structure were extracted by an intera- tive procedure using the program SCOPEFIT [Ref. 11.3]. A description of this program, and other programs used in the data analysis, is given in Appendix I. At Princeton the quadrupole-dipole-dipole-dipole (QDDD) spectrograph was used. When excited states were examined the solid angle was opened to about 3.6 mili- steradian (2° x 6°). The solid angle was closed to 1.2 is re st re ta '1‘? ar 12 milisteradian (2° x 2°) when observing the elastic peak. This was done to decrease the number of particles striking the detector, thus reducing dead-time losses. The detec- tor used in the focal plane of the QDDD was a 20 cm long resistive-division position-sensitive gas proportional counter backed by a plastic scintillator in coincidence. The data acquisition and analysis was performed by the program TOY [Ref. 11.4] on a Sigma-2 computer. The pro- gram gates the position spectrum from the gas proportional counter by a window on a particle identifier consisting of the total proportional counter signal versus the scin- tillator signal. In addition, the position spectrum was gated by the particle time-of-flight (TOF) spectrum (measured relative to the cyclotron rf signal). The TOF is an aid in particle identification and was used to reduce background in the proton spectra. Data taken with the proportional counter must be done in three passes because of the large dispersion of the QDDD. Each pass covers an energy range of 2.5 MeV. The three passes overlap encompassing levels from the ground state to states up to 5.5 MeV of excitation. The energy resolution of this data is 15 to 20 keV. Data at 40°, taken in three segments, are displayed in Figure 11.2. The only counter data used in the analysis in this study are levels labeled in Figure 11.2. Both methods of acquiring data offer unique 13 Figure 11.2 Typical spectrum of protons scattered by with the proportional counter. 206 Pb obtained HMO". 14 (. _ 4 d J I I _ . . _ . (q n u . 8: End I” mrmfi 1%. 8.8 33. 8mg.” .ll , RBeIL . , Dam . e mad.l 9 Au .(IIILL Rem u - MET: - 0 8.32 zem mmk I. o u. 83 . ll“ In :0 5 Gem . r r a m V Nome. (Ill 8m 1. 7mm... AU x 3 . . In ray: 10 .. a so... :3 ( Vx may... m V I momd e I. 3 M l A ._ V . .V 5 r w u «2 r ( 1 Q3 SNN (In __ r . . TM 0. 8.3—Unm H E HQ O=Unm f o M391 L m mefim “J .. v _ m p h .. ’ a 0.66 .I . .0 I .14 we: (IIILIL m DI .. 2 Ab . m I. u w I 855a _ _ P . F p _ . _ .M p _ 0 m m m w w w 3 2 l. .l mecsou Number Channel Figure 11.2 15 advantages. Data taken with the nuclear emulsions have far better resolution allowing weakly excited and close lying states to be analyzed. The linearity of the plates yield accurate excitation energies. The proportional counter gives more accurate cross section data for strongly excited states because it is not limited by the number of counts in a peak. Because energy resolution is not as curcial in this part of the experiment, a compara- tively large solid angle could be used and data was accumulated at a rapid rate. The counter data also has the advantage of livetime data taking, while the data taken with the photographic plates is passive. Due to the time required to scan a plate, the results from this part of the experiment were typically not known for months. II.1 II.2 II.3 II.4 REFERENCES FOR CHAPTER II W. T. Wagner, Ph.D. thesis, Michigan State University, Unpublished (1974). H. G. Blosser, G. M. Crawley, R. deForest, E. Kashy, and B. H. Wildenthal, Nucl. Instr. and Meth.‘gl (1971), 61. H. David and R. Fox, Unpublished. R. Kouzes, Ph.D. thesis, Princeton University, Unpublished (1974). 16 Fri ‘1. IE on Th CHAPTER III EXPERIMENTAL RESULTS Listed in Tables 111.1 and 111.2 are the excitation 206Pb observed energies of the approximately 180 levels of in this experiment. In Table 111.1 the levels with exci- tation energy below 4.6 MeV are compared with the results of a recent compilation [Ref. 111.1] and an inelastic pro- ton scattering experiment [Ref. 111.2] with an incident proton energy of 24.5 MeV. Above 4.6 MeV of excitation, the correspondence of levels seen in different reactions is uncertain due to the high level density and uncertainty of excitation energy. As a result the data in this region, displayed in Table 111.2, is not compared with previous results. To determine the energies of identified 206Pb states only the data taken with the nuclear emulsions are used. This data not only has the advantage of high resolution, but also is very linear along the entire length of the plate. All levels listed in these tables were clearly observed in the photographic plate data at a minimum of three angles. The energy calibration of each plate expo- 206 sure was determined using both Pb states and well-known levels of nuclei which were present as impurities in the 17 18 vsmm.m no moom.~ Hom.~ onoueo moo.Hmom.~ oum.~ +o mom.~ In mmoo~.~ Hmo. e ooH.~ «mo. e ammoo~.~ 1+N. ooo.~ moo. Am. moo.HHmH.~ +4 oeoo.o moo. o ooo.o boo. o neooo.fl +~ ooo.H ooo.a moo.nooe.o +H moa.o o+Hnoo moo.hooo.fl +4 Homo.H moo. o «oo.o omo. o Hoo.Hooo.H +~ ooo.a omo.H moo. m moo.wooo.o +m ooom.H omm.o o+mueo «oo.ooom.fi +o HmoH.H moo.hoeH.H +~ Hmoo.o moo. N moo.o ooo. N nomoo.o +o oo.o noo.o x A x q x x go a m o a a m o A a no“ u m .mmm H .mm« egos unmmmum oofluaaomeoo ..a.a. >62 om .muflsmom msofl>oum saws one: we conflummEoo a .nm uom mumuoamumm :owumEuOMon new .mummmcmuulq .mao>oq amumcm mom H.HHH mam49 19 «ma.m HHo. Am. moo.umm~.m moo.«mmu.m A+mv mma.m vm~.m .o+mu=n. moo.«-a.m mmo.«mmo.m um mvwfio.m Hmo. m omo.m mac. m moo.«vao.m mnm.~ moo.«mmm.~ ~oo.uowm.~ no ommm.~ +v mmm.m amm.~ one. o nmmm.~ In mvwm.~ moo.n~mm.~ m.|vv emmm.~ moo.uamm.~ um mmmmn.~ hm>.~ mmo. m nmmmmb.m um mmmm.~ um mbvo.~ mHH. m mvm.~ moa. m embvm.~ +~ vme.m moo. AN. mmo.«-v.~ Fo use no a mxm no A mxmooxm N .wom H .mom xu03 ucomoum :oflumaaaeoo ..m.m. >mz am .poscwucou H.HHH wand? I wewvzc .n . :ncr 20 oeo.m ooo.«moo.m moo.«mmo.m moo.m ooo.omoo.m mom.m um omom.m «mo. m oom.m NNo. m ~oo.uomm.m A+o.+m. on.m «Ho. .mo moo.umHm.m «oo.o ooo.aooo.m mmo.m oHo. m mmo.m omo. o Hoo.uomo.m um omoo.m mmo. m moo.m owo. m Hoo.«oom.m Anne mom.m ~oo.oeem.m moo.oo~m.m um moe~.m moo. m Hoo.«~h~.m eo~.m oHo. o Hoo.oem~.m Aloe moo~.m .uo.-o. mm-.m moo.uo-.m eo mxm no a axm no a axmohxm m .umm a .mmm xuoz uommoum aoouaaoaeoo ..a.a. >62 om .omsoflucoo H.HHH memes 21 ooo.o ooo..ooo. In.o. moo.oooo.o 1+~HV ~omo.o 1+4. ooo.o Io. moo.uooo.o ooo.m moo.nooo.m moo.oooo.m 1+o. moo.m .oo moo.omoo.m +oH oomo.m omo.m ~oo.m moo.fiooo.m moo.«moo.m mmo.m moo.«e~o.m Hoo.m ooo.«moo.m Aim. mob.m boo. m m ~oo.n~eo.m 1+H. «oo.o Ao+Hupoo ooo.oomo.m «oo.o m ~oo.nofle.m no axm do A q mxmooxm N .mmm H .mom xuoz ucommum oooumaomeoo 1.m.m. >mz om .coscaucoo H . HHH "wage 22 A+vv mmm.¢ hmo. v mom.v mmo. v Noo.nmmm.v 1+o. moM\. A: wmm . v mmm.v «do. Am. moo.«~mm.v .Imv mmm.v mac. Am. moo.umvm.v mH~.v mmm.v use. Avv moo.HmHN.¢ moo.nmmfl.v mmH.v HmH.¢ moo. Am. voo.nmmfl.e oma.v moo.nmea.v «mo. m moo.wm~a.¢ A+mv wHH.v woo. N mmfi.e bvc.o N moo.«hoa.¢ emo.v noo.umno.¢ Aim. hwo.¢ mno.v vac. Am. moo.ummo.q mmo.v Fo axe go A mxm go a mxmoaxm m .mm& H .mm& xuo3 ucomoum coflumaflmsou A.Q.mv >mz «N .coscwucoo H.HHH wands 23 .wom Eoum poDQOUM auHumm new :Hmm .o .coHuounHHmo hmuoco CH pom: Ho>oH .n .>oz :H can monHoco HH¢ .m A+mo «oo.o moo. m ooo.v mmo. m moo.«omm.v mmm.v vmm.v mHo. m voo.«emm.v Nov.v NHo..Hoo. Ho.mv moo.«omv.v m>¢.¢ oHo. m moo.«vhv.¢ omo. Am. moo.wome.¢ omv.v moo.uo~e.v mHo. Am. moo.HHmm.¢ A+oo omm.v Hoo. m omm.v vmo. o moo.«hmm.v on mxm no a use no A mxmohxm m .mom H .mom xuoz ucomoum GOHDMHHQEOO A.m.m. >oz em .pmacwucou H.HHH wnmfle 24 «mo. .4. ooo.4Hom.m ooo.4o-.m 4oo.«Hoo.4 ooo.444m.m moo.«oo~.m mHo. .m. moo.nooo.4 oHo. .4. ooo.ommm.m moo.uooH.m 4oo.4moo.4 ooo.«oom.m 4oo.oooH.m oHo. .o. ooo.«ooo.4 oHo. .4. 4oo.4mo4.m noo.«omH.m m~o. .o. 4oo.uo~o.4 4Ho. .m. ooo.«mo4.m 4Ho. .m. ooo.uo~H.m 4Ho..MHo. .o.m. ooo.uooo.4 oHo. .4. ooo.a~m4.m mHo. .4. moo.hHHH.m «No. m moo.4moo.4 hoo.4mm4.m «Ho..mHo. .4.m. 4oo.4~oo.m 4oo.noeo.4 oHo. .4. ooo.4-4.m ooo.«ooo.m moo.h~4o.4 ooo.umo4.m ooo.am4o.m ooo.no~o.4 ooo.hmom.m moo.om~o.m oHo..mHo. .m.4. 4oo.«oHo.4 HHo. .m. ooo.«~mm.m 4mo. .4. 4oo.uooo.m ooo.ooo.4 4oo.ooom.m oHo. .m. moo.hooo.4 HHo. m 4oo.«4oo.4 ooo.ooom.m oHo. .m. ooo.«ooo.4 moo.“o4o.4 moo.4oo~.m omo. .o. 4oo.nomo.4 4oo. .m. ooo.h4Ho.4 «mo. .m. ooo.“m4~.m mHo. .m. 4oo.floHo.4 moo.hmom.4 no a xmoomxm go a xmonmxm no A smoomxm xuoz Hammond xuoz ucmmoum xuoz ucmmon .nm mom HOm muouoEnumm cowumfiuommo one .muommcmuunq .mHo>oH mmumcm N .HHH "Luanda. 25 ooo.umm4.o ooo.aoHH.o ooo.4m~o.m ooo.«oo4.o ooo.«moo.o mHo..4Ho. .m.4. ooo.«ooo.m ooo.«~om.o 4oo.uHoo.o 4Ho. .m. ooo.«o>o.m ooo.«o4m.o ooo.umoo.o ooo.amoo.m ooo.u~mm.o moo.4o4o.o ooo.uo4o.m ooo.u4Hm.o boo.«m~o.o ooo.n~mo.m ooo.n~om.o moo.«Hoo.o eoo.«mHo.m ooo.h4o~.o ooo.uooo.m mHo. .m. ooo.4moe.m mHo. .m. ooo.«4o~.o moo.«4oo.m 4oo.4ooo.m ooo.oo-.o moo.aomo.m ooo.4ooo.m HHo. .o. ooo.nooH.o ooo.«o4o.m ooo.«mmo.m ooo.oHoH.o mmo. .4. poo.hHHo.m mHo. .m. ooo.ho4o.m ooo.oooH.o 4oo.omoo.m ooo.ooHo.m moo.o4mH.o ooo.«ooo.m moo.4oom.m ooo.oo4H.o ooo.4m4o.m omo. .m. ooo.4oom.m cm a xmoflmxm no A xmoomxm go A xmoamxm xuoz ucmmoum ¥H03 UCQmOHm— xuoz ucomoum .poDGHHCOU N.HHH mnmdfi 26 moo.amom.m moo.HHmh.m hoo.Hmmm.m HHo.Hmmm.m moo.«mmw.m OHo.vam.m moo.«hHm.m hoo.«mmm.m moo.w¢hm.o oHo.Hmvm.m moo.wv~m.m ooo.«mmv.w hoo.uomv.m hoo.wmme.w hoo.uvvv.m A xuoz ucomonm .coscflucoo N.HHH mam¢8 27 target. The 206Pb states used had focal plane positions clearly determined in this experiment, and had excitation energies established in other high resolution experiments [Refs. 111.3, 111.4, 111.5, and 111.6]. These calibration states are indicated in Table 111.1. The levels of 12C and 160 strongly excited by inelastic proton scattering, as well as the ground states of 35Cl and 37Cl were used whenever possible. The presence of impurity states in the 206Pb spectra also allows the scattering angle to be accurately determined by kinematics. The excitation energies given in Tables 111.2 and 111.2 include statistical uncertainties plus an additional error of 1 keV per 500 keV of excitation energy for states beyond 3.5 MeV of excitation energy. This systematic error is an estimate of both the interpolation error and the uncertainties in the focal plan positions caused by the high level density. Below about 4.6 MeV of excitation energy most states are well-resolved and the agreement with previous work is very good. Several new levels have been identified in this region including two relatively strongly excited states at 3.257 and 3.980 MeV of excitation energy. A level previously reported at 2.658 MeV is not seen in this experiment. Even if this level is excited in the present experiment it could not be resolved from the state at 2.648 MeV of excitation energy in the counter data, and 28 because the 2.648 MeV state is the strongest excited state in this reaction the particle density is too intense to analyze in the plate data. Other levels previously reported in this region, yet not seen in this experiment, are probably very weakly excited, and an upper limit of 10 ub/sr can be put on their maximum cross section. Angular distributions for inelastic states seen at four or more angles are shown in Figures 111.1 through 111.3. The cross sections are displayed with their cor- responding excitation energies. Error bars indicate sta- tistical errors and are drawn only when greater than the symbol size. The curves drawn through the data are included as guides to the eye and do not represent theo— retical fits to the data. These angular distributions are also tabulated in Appendix III. 29 Figure III.1 Measured inelastic cross sections. The lines drawn through the data points are included to guide the eye and do not represent theoretical fits to the data. dcr/dQ [mb/sr] 30 0.1 —I X H P7 10“2 “’ a L. 4 U) p—o [\L C) ._I UL o " 2.960 3.827 0. 0.803 - 11.20 115.0 115.0 115.0 11.50 12.0 10-2 ‘1“ i \\ a 1 \fl‘“ 1.170 2.200 2.888 4.123 0. _ 3.478 33-8”? _ No 12.0 11.50 12.0 ‘ 112.0 102 (“3V\~ ‘V‘U\ww ’ l . 1.344 2.385 3.014 3.515 3 883 4.188 0. _ 11.50 12.0 11.50 W20 11 50 1.488 2.422 3.12 3.558 3.880 4.218 0. , _ . 11.20 11.02 112.0 Mk 11.20 10'2 . -, _ _ - 1 888 2.848 3.183 3.803 3.887 4.242 0. , _ ,_ 112.0 11.50 11.50 1.20 (X 11.50 1.50 10-2 ’ \\ 1.708 2.782 3.257 3.718 4 008 4.282 0.. ,_ _ - 112.0 115.0 11.50 112.0 112.0 1 10 1.787 10‘ . , 2.881 3.277 - 3.737 4.044 4.333 11.20 ' .2 15.0 H 0, o N X N O )J}. m D ogff? N O f O 1.888 2.828 3.388 3.772 4.058 4.357 111,111,111 141 111 11 0306090 0306090 0306090 0306090 0306090 0306090 eCJh.[deg] Figure III.1 31 Figure 111.2 ' Same as Figure III.1. dU/dQ [mb/sr] 0.1 Jet/12.0 ‘12.0 N120 1 F130 1 ‘ 11 0 10‘2 lxnhb . 4.381 4.814 4.808 5.007 5.188 5.385 OJ - - 150 ”m“ d 12.0 10'2 . 4.847 4.828 5.025 5.403 0 4.420 f “5.180 1.50 112.0 112 0 3“ 12.0 112.0 11.50 ' 10-2 M N - ' f ' K : 4.458 4.884 4.880 5.045 .422 0 5.208 , L13” 1 112.0 11.50 10_2 "H,“ 7‘ y \’x 4.474 4.710 4.888 5.088 5.245 5.435 0 - _ 112.0 112.0 k 10"" . VL . . 01 . ~ f 4.728 4.818 5.082 5,279 5.452 0. 4.488 f _ _. 12.0 m V 112.0 112.0 10‘2 m w ._ - _ (m _ , 4.534 4.742 4.838 5.111 5.288 5.483 0. . __ 11.20 4H“. M20 12.0 10‘2 A _. W ¥ 4.580 4.770 4.880 5.128 5.308 5.485 0. p - _ 11.50 1.50 1.50 12.0 10'2 (L. "M. , \n.‘ 4.585 4.783 4.888 5.13 5.332 5.507 L I l l l l l 7 l l L 7 l l L l l l I l 0 30 60 90 0 30 60 90 0 30 80 90 0 30 60 90 0 30 80 90 0 30 60 90 ecxn.(deg( Figure 111.2 33 Figure III.3 Same as Figure III.1. 34 2.0 r X 6.959 I\‘\}2.0 6.980 7' ((01 8.284 .0 31.7% 5.715 n50» fig; Q10 xmadh 5.885 \J“ 5.533 01 10‘? (1.41.2 6.117 5.722 5.581 0 10'2 2.0 W. 332 8.529 2.0 121 ‘ “\n 0 V2.0 12.0 5.848 8.187 8:831 \P ° ‘ 112.0 112.0 i 112.0 5.858 5.819 _ 8.545 8.382 8.181 5.779 5.890 8.57% 115.0 112.0 ’ 12.0 12.0 ’ )'2.0 8.593 8.908 8.198 I ‘ X2. .93” 8.229 8 (:3.\ 8 289 1 l (. 0308080 0308080 0308080 0308080 0308080 0308080 5.97’-+ 5.798 5.853 10‘2— NW! \N' . 5.823 5 8"(3 1 112,0 8.817 .0 V2.0 6.639 1 8H33 XKFQS 8444 l l "2. "2.0 L )‘2.0 1 5 878 5.703 1 l [deg] eCJT). Figure III.3 REFERENCES FOR CHAPTER III 111.1 M. P. Webb, Nucl. Data 26, No. l (1979), 145. 111.2 G. Vallois, J. Saundinos, and 0. Beer, Phys. Lett. 24B (1967), 512. 111.3 W. A. Lanford and G. M. Crawley, Phys. Rev. C 2 (1974), 646. 111.4 W. A. Lnaford, Phys. Rev. C 16 (1977), 988. 111.5 M. Kanbe, M. Fujioka, K. Hisatake, Nucl. Phys. A192 (1972), 151. 111.6 J. C. Manthurathil, D. C. Camp, A. V. Ramayya, J. H. Hamilton, J. J. Pinajian, and J. W. Doornebos, Phys. Rev. C 6 (1972), 1870. 35 CHAPTER IV COLLECTIVE MODEL ANALYSIS The usual method of extracting information on the spin and parity of a state and on the transition strength for the excitation of the level in a direct reaction is by making a comparison of the measured cross section with the results of a calculation using the Distorted Wave Born Approximation (DWBA). With this approach the experimental angular distributions are compared to DWBA angular distri- butions which have a characteristic shape determined by the strengths of each L-transfer involved. The problem is sim- 206Pb has a 0+ ground state. Thus, in plified here because a one step direct reaction all natural parity transitions can involve only one L-transfer. For a spherical nucleus such as 206Pb, the collective vibrational model can be used to obtain the characteristic L-transfer shape. The DWBA method is described here only briefly. A detailed description of the DWBA method is given in Refer- ences IV.1, IV.2, IV.3. A. Description of the DWBA Method In DWBA the differential cross section do/dQ for the direct reaction A(a,b)B is proportional to the square of 36 37 the transition amplitude. DW _ 1+ + (-) -)- + (+) + + T — derafderB (kB,rB)xa (ka,ra). (IV.1) Here Ed is the displacement of the projectile a relative to the target nucleus A, f is the displacement of the out- 8 going particle b relative to the residual nucleus B, and J is the Jacobian of the transformation to these coordinates. The distorted initial and final waves are represented by (-) 1-) X B a respectively. The remaining factor in the and x transition amplitude is the matrix element of the inter- action causing the transition, taken between the internal states of the colliding pairs: = fwabvawadg. (IV.2) Here 5 represents all coordinates independent of Ta and ES. The potential V is equal to V -UB, where V is the inter- B 8 action between B and b and U8 is the potential that gener- ates the distorted wave XB' Usually one takes for UB the potential that describes the elastic scattering. For inelastic scattering two different approaches are possible: the macroscopic and the microscopic DWBA. The macroscopic DWBA will be examined here; in Chapter V the microscopic DWBA will be considered. In the macrosc0pic DWBA the collective model is used to describe the wave functions 0A and TB, and the potential U8 is assumed to be the optical potential, deformed similar 38 to the nucleus. In this case the equipotential surfaces can be characterised by ' m R(6,¢) = 80(1+£ aLmYL10.¢)). (Iv.3) m where are the deformation parameters and and are Lm the polar coordinates in the lab coordinate system. The potential 0' felt by the projectile can be expanded into a Taylor series according to 0'13) = U1RO) = U1R/(1+£maLmY§(e.¢))) U(R)-{ aLmY$(dU/dn)+ higher order terms. (IV.4) Lm The first term of the right hand side of equation IV.4 is the potential which gives rise to the elastic scattering. The term linear in aLm induces inelastic scattering to collective states of multipolarity L. The higher order terms are neglected in DWBA calculations. With this form- ula the DWBA cross section can be calculated. For excita- tion of vibrational states one finds [Ref. IV.4]: 2 8 do L a§(9) W 2L:T ULle): (IV'S) - + dU + + 2 oL(e) = g|fdrxé )(k8,r)ra;Y$x; )(ka,r)|' (IV.6) and 2 2 h 8L = ZlaLml = (2L+1)_;p. (IV.7) L For this excitation wL is the frequency of the vibration 39 and CL the "spring constant". For rotational excitation a similar expression can be written containing the frequency and moment of inertia of the deformed nucleus. The collective description for inelastic scattering is rather simple, since only the optical model potential is needed to perform the distorted wave calculations. The only adjustable parameter is 8:. The value of B: for exci- tation of a level is found by normalization of the DWBA angular distribution to the experimental one: 2.. 911 do BL ‘ (E§)exp/‘dQ)DW' (Iv’a) The deformation parameter BL may be used to determine the reduced transition probability G in single particle L units (s.p.u.). This relation is given by: (3+L)2 22 GL = 4w(2L+l) 2 BL, (IV.9) where z is the atomic number of the target. Another quantity of interest is the fraction of the energy-weighted sum rule (EWSR) limit for a particular multipole contained in the observed transitions. The sum of the observed energy weighted transition strength is given by: s = Z G E (IV.10) I L f Lf f where the sum is taken over all final states f of energy E reached by a particular multipolarity L. f Although equation IV.6 is only valid for the 40 excitation of collective states, the calculated angular distribution is also used generally to assign L-values. This is often possible because the shape of the angular distribution is mostly determined by the angular part of the matrix element (Equation IV.2). However, the L-values obtained should be treated with caution especially when the fit is not very good. B. Elastic Scattering and the Optical Model For comparison with the measured angular distribu- tions, the DWBA collective model calculations were per- formed using the computer code DWUCK [Ref. 1V.5]. A sample of the input to this code may be found in Appendix 11. The optical model parameters used in the analysis are the general set of Becchetti-Greenless [Ref. IV.6]. Because the Becchetti-Greenless parameters are functions of the particle energy, the energy dependence of the incoming and outgoing distorted waves is accounted for. The set of optical model parameters used is listed on Table IV.1. A comparison of the measured elastic scattering angu- lar distribution with a calculation using these parameters is shown in Figure IV.1. Since the target thicknesses were known only approximately, the normalization of elastic scattering to this calculation is used to deter- mine the value for the thickness of the different targets. Using this procedure the absolute cross sections are 41 believed to be accurate within ten percent. C. L-transfers and Deformation Parameters Angular distributions for natural parity states are very characteristic of the angular momentum transfer. Comparisons of collective model fits to identified states are displayed in Figures IV.2 and 1V.3 and discussed below. The L-transfers are determined by comparing the data with theoretical angular distributions, and with experimental cross sections of states with unambiguous L-assignments. The experimental cross sections used in 206 this comparison include both Pb states observed in 208Pb states observed in the 35 MeV this experiment and proton study by Wagner gt_al [Ref. IV.7]. The deformation parameters and L-transfer assignments for states with excitation energies below 4.6 MeV are given in Table 111.1 for comparison with the measurements of References IV.8 and 1V.9. Where possible those states with angular distributions of unidentifiable shape have JTr adopted from Reference IV.8. The L-assignments and deformation param— eters of levels above 4.6 MeV are given in Table 111.2. Since above 4.6 MeV so many new states are observed in the present experiment and the correspondence of levels seen in different experiments is uncertain, the results of this experiment are not compared to previous studies in this region. OPTICAL MODEL PARAMETERS USED TABLE IV.1 42 IN DWBA CALCULATIONS VR rR AR Wv WSF II In: -53.247 1.170 0.750 -5.000 -5.497 1.320 Out: -54.099 1.170 0.750 -4.415 -6.126 1.320 AI vso wso rso Aso rc In: 0.653 -6.20 0.00 1.010 0.750 1.189 Out: 0.653 -6.20 0.00 1.010 0.750 1.189 43 Figure IV.1 Comparison of the measured elastic angular distribution with the DWBA calculation explained in the text. —H-—-— q H dqqdqj d ——udqq5 H —-qudd - —HHH-fid 4 35 MeV 44 206Pb[p,p] Ep 1 100 80 80 I+0 20 ec.m. [deg] Figure IV.1 L 56:... _ p 33:; . _ 7C... _ . _:p#_hp _ _:.e.. _ L .2... S u. 3 0 0 1 0 0 0 1 1.. 1.. HLmVQEH GU\.HOD 45 Figure IV.2 Collective model fits for identified states. Displayed with the fits are the excitation energy of the state and the deformation parameter, 8 , corresponding to orbital angular momentum transfer L. (Nib/6!“) d(1‘/ (132 dG/dQ [mb/sr] 46 01 10'2 0 10‘2 ma 0 m”- 10'3 EX=S.483 ' 83=0.012 10'2 ' o 10'3 ‘ EX:S.6Ll0 l l _L l l 0 301m 80 0 301m 80 0 30 HJSO Figure IV.2 47 Figure 1V.3 Same as Figure IV.2 48 om o... om o DH 4 H ommffi 9 .Jr fie mmoouoo momduxu . No. Soot... . 861...; o / ..JQ.Q. L .W //J V.-o_ 8 QC... (. m A. . o TI , ..-. x I .70 .H De ...1 w 0 q rooms. - - oo.o”... our (N10— :oouoo . . .O ._. 88.4.: w.) v DD. Neo— .4. o-.. . :o ..1/ 88.4.48 ~-o_ . oo.ouoo » p _ 1° acqlia om om om om cm on o m.>H ousmwm 383 .86.. om om cm H 4 hill. lull-uni huh; 1 LU AA 1.11111 L MAJ-Ann \nmdux... oo.o” “L1 x LU in M02 . .88. "no N . . ...o . A m. 84..."; . ._o . omooueo m . .. .8...qu . ~-o. e H ~.o.oumo . moo 8.3.qu . u A . .~-o. . ioonno . m to . Nummuxmw O 51. N.c.o1 m~o.ou - om ow cm 0 4 H 9 H H H {.21me gonna r p P Om ow om o . . . O.N.ruxm 2mg...“ mood“... r o. a. moron; . 23.”; Leona 1.2. 6 . 81m"... /. A owes"... D www.muxu 846.6 msoueo :3". use"... . _.o .38."; omooué . H H r (49/qu asp/op 49 C-1. L=2 Transitions Six probable quadrupole states were observed. These states have 17% of the total expected strength given by an EWSR. Most of the L=2 strength is concentrated in the first excited state at 0.803 MeV and a state at 4.107 MeV. These two states have transition strengths of 11.7 and 5.9 s.p.u. respectively. A well known 2+ state at 1.469 MeV was observed. The excitation energy of this state is not in agreement with decay studies [Ref. IV.10] but is con- sistent with the energy measured in transfer reaction experiments [Ref. IV.11]. The states at 2.151 and 2.422 MeV were previously assigned a J" value of 2+. These levels are observed weakly excited in the present experi- ment but their angular distributions do show the charac- teristic L=2 shape. A state at 4.242 MeV was previously identified as a 5- state [Ref. IV.12]. This experiment suggests an assignment of L=2. 208Pb [Ref. IV.7] Inelastic proton scattering from also identified six L=2 states with approximately the same total strength. However, all six states observed in 208Pb had energies above 4 MeV of excitation. C-2. L=3 Transitions Previous experiments have only definitely identified one 3‘ state at 2.648 MeV of excitation. Two other states at 5.444 and 6.045 MeV of excitation were tentatively assigned a J1T value of 3- [Ref. IV.12]. The 2.648 MeV 50 level is the strongest excited state observed in the pres- ent experiment with a transition strength of 32.1 s.p.u. exhausting about 18% of an EWSR. The 3— strength is frac- tioned and many other states with a characteristic L=3 shape were observed. In particular, the levels at 3.718 and 5.245 MeV are relatively strongly excited. The angular distribution of the 5.092 MeV state is fit equally well with L=3 or L=4 shapes so that the L-transfer is not uniquely determined. The observed 3' states have 26% of the total expected strength given by the EWSR. C-3. L=4 Transitions The dominant 4+ state observed was the 4.333 MeV level with a transition strength 8.8 s.p.u. The other well known levels at 1.686, 1.998, and 2.928 MeV were observed with transition strengths of 2.6, 0.8, and 2.6 s.p.u. respectively. New 4+ states were identified which 'were not previously reported, notably the relatively strongly excited (>1 s.p.u.) states at 3.450, 5.007, 5.422, 5.561, and 5.911 MeV of excitation. ‘There is some .ambiguity in assigning the JTr of the levels at 4.710 and 5.796 MeV. Both of these levels could probably be equally well fit by an L=4 or an L=5 shape. The observed 4+ states have 24% of the expected strength given by the lfiNSR. 51 C-4: L=5 Transitions All known 5- states were observed, the strongest being the state at 3.772 MeV of excitation with 5.8 s.p.u. of transition strength. Previously unreported levels at 4.456, 4.793 and 5.588 MeV are all relatively strongly excited. The L=5 assignment of the level at 3.515 MeV is in disagreement with the tentative 3+ or 4+ assignment of Reference IV.11. Before the first maximum at 37 degrees, this angular distribution is not fit well by a L=5 shape as can be seen in Figure IV.3. On the basis of this fit an assignment of L=5 seems rather weak. However, it has been previously noted [Ref. IV.13] that the predicted collective model cross section for large angular momentum transfer is usually smaller than the measured data at forward angles. This difference between data and theory is amplified as the spin of the state increases. An example of similar behavior can be observed by examining two well known 5- states in 206Pb at 2.782 and 3.277 MeV. Both states show this phenomena where forward angle data tend to rise rela- tive to the calculation. Indeed the shape of the measured 3.277 MeV state is nearly identical to the state in ques- tion. As a result of the similarity of the 3.515 MeV state with established levels it has been tentatively assigned to be an L=5 transition. C-5. L=6 Transitions The only previously observed level seen in this 52 experiment is the strongly excited state at 4.357 MeV with 9.7 s.p.u. of transition strength. New L=6 levels observed include relatively strongly excited states at 3.257, 4.123, and 4.939 MeV. C-6. L37 Transitions Only two states were unambiguously identified as involving L=7 transitions. The established level at 2.200 MeV was identified with 1.7 s.p.u. of transition strength and a previously unidentified state at 4.828 MeV is tenta- tively assigned an L=7 shape with transition strength of 1.9 s.p.u. Transfer reaction experiments [Refs. IV.11, and 206Pb, and this IV.14] have found more L=7 strength in experiment identifies states at similar excitation ener- gies. However, these states are weakly excited and their angular distributions do not contain enough information to make reasonableL-assignments. Two 8+ states were observed with the strongest being the level at 4.580 MeV. This state has a transition strength of 4.2 s.p.u. D. Systematics of Collective States in Lead Nuclei The strongest states excited by direct reactions in the doubly magic nucleus 208Pb are the collective 3-, 5‘, 2+, 4+, 6+, 8+ levels between 2.5 and 5.0 MeV of excita- tion. Many experiments have been performed to examine the 207 corresponding weak coupling states in Pb 53 [c.f. Ref. IV.15 and referenced contained therein]. Sev- eral experiments [Refs. IV.9, IV.16, IV.17] have also been performed on 206Pb to observe the analogous collective states in this nucleus. These experiments show a strong correlation in both energy and strength of the collective states in the three nuclei with the exception of the L=5 states. These states are now examined in the present experiment, and an explanation of the anomaly in the 5- strength is sought. Angular distributions of the even parity collective states are shown in Figure IV.4. The data are compared to the empirical angular distributions for the analogous 208 states in Pb [Ref. IV.7] and to collective model calcu- lations. The agreement between the calculated angular distributions and the experimental results is generally very good. The shapes of the angular distributions of 206 208 corresponding states in Pb and Pb are similar, but 206 the states of Pb are all weaker by approximately 30%. There is, however, still a one-to-one correspondence of 206 the strong collective positive parity states in Pb with those in 208Pb, ie. the fractionation is not significant. The angular distributions of the 3- and 5- collective 206Pb are shown in Figure IV.5. Two states at states in 3.193 and 3.515 MeV excitation energy, which were not pre- viously identified as 5. states, have angular distribu- tions which are fit best by an L=5 shape. The L=4 54 Figure IV.4 Angular distributions for positive parity excitations in proton scattering from 206Pb. The solid lines represent collective DWBA calculations. The dashed lines represent interpolation of corresponding levels in 208Pb. The exci- tation energy, Ex (MeV), indicated for each state is the value determined from the present data with uncertainties given in the text. do/dSZ [mb/sr] OJ OJ 55 [IITI' I IIII' 1 111111 J I r I V r l I I I \ 2+ 6+ 1‘ Ex='-+.107 Ex=‘+.357 . 32:.047 36:.054 lllLll " Ex=‘+.580 ‘ _ 98:.033 . \ I‘.\-“"~ -‘ . I i! ‘\ L" l ‘ 1 I- \\ 4 P- . - i- \ -4 .- ’\. b db \ - \ I p ‘1. t, -4 I l 1 l 1 1 l 1 l 1 0 40 80 0 40 80 Figure IV.4 56 Figure IV.5 Angular distributions for negative parity excitations in proton scattering from 206Pb. The solid lines represent collective DWBA calculations. The dashed lines shown with the 2.648 and 3.772 MeV states are interpolations of cor- responding levels in 208Pb [Ref. IV.7]. The excitation energy, Ex (MeV), indicated for each state is the value determined from the present data with uncertainties given in the text. do/dQ [mb/sr) ID I... 57 ”\A 1o" 00‘ UI I 1 r tvvvvr m X II E" N \l \l 4 1|. . n. l. 5: 15 “:E' in, 3:" 1: i: j; Ex=3.399 j: L I I I ITIvl ‘\\ 1 All IIIVVV' \‘V 1 A 141111 I 1 11 r v I TVII' I \ 1"" I'IIIT (L=5) Ex=3.515 85:.012 4 #411 A 11111:! A d d d ‘ d ‘ 1_1_ 1 9cm. [deg] Figure IV.5 58 calculation is also shown for comparison. The 3- state at 2.648 MeV of excitation is the strongest excited state in 206pb and, like the even parity states, has a similar angu- lar distribution with about 70% of the strength of the 2.615 MeV 3' state in 208Pb. However, in contrast with the states discussed above, 206Pb with significant strength as opposed to two such states in 208Pb. The 5— 206 there are eight 5- states in Pb at 3.772 MeV of excitation probably corre- 208 state in sponds to the 5' state in Pb at 3.709 MeV of excitation [Ref. IV.17], and is slightly stronger than that state. However, it has also been observed that there is no single 208Pb at 3.198 MeV of excitation 207 analog to the 5_ state in 206 in either Pb [Ref. IV.17] or Pb [Ref. IV.15]. Wagner, et a1 [Ref. IV.15] have observed six L=5 states in 207Pb with a total of 85% of the core 206 this region in 208 Pb. Pb data shows seven possible L=5 strength in states in this region of excitation summing to 80% of the 208Pb core strength. The summed transition strength is given by the relation . . __ 2 *5 Summed tran51tion strength - [X BL(Ei)] . (IV.11) . i In figure IV.6 the levels of interest here of all three nuclei are plotted with their relative strengths shown. .A comparison of these collective levels with transition strengths for all individual levels is presented in Table IV}2. These results suggest a spreading of the L=5 59 Figure IV.6 Levels for which angular distributions were measured together with those measured in Ref. IV.15. The numbers give the transition strength. :‘ o T Exch‘oflon Energy [MeV] (A) in 3.0 2.5 f 1 60 ls/zz____i] 177’2'-——-- '037“" 8* 040 8* 033””’,,,/~ .p 6... OSH/ 1V2- I I.063 * t+*-—-——.oss 7/2'.9/2-__ .067 __..__—————-—- ‘* 087 + [:ii/2:______i] 2 ..___.0‘+7 3/ 2 __ .058 2+ 053 5'____.0'+3\ \ \\ , 5' 03"} \\ ,z” \ — 11/2*___‘ , , ’ _ __ L 9/2+____ .036’ - (L=5)__- (L=5)— _ (L=SL______. S __ (L=5)— 5- .098 ----- L=5 .oso ------- - s- 058 5- 5- d _ 9/2+___ - " 7/2*___'7 3 108 _ 5/2. .120 fi 3- ._.120 ZOSPb 207Pb 208Pb Figure IV.6 61 207 206 Pb and trons removed from the 208Pb core. These phenomena may be strength in both Pb with only one or two neu- explained at least qualitatively by examining the wave functions of these states. 206 The wave functions of Pb have been calculated and examined for the region below 2.6 MeV of excitation, but very little is known about the wave functions of the higher energy levels. Fortunately, much work has been done in 208 this energy region on the nucleus Pb [c.f. Refs. IV.18 and IV.19 and references contained therein]. Figure IV.7 is a schematic representation of the location of the single-particle and single-hole neutron and proton shell model orbitals, above and below the N=126, Z=82 magic, shell closing energy gaps. Examining the 5- state at 3.198 208 MeV of excitation in Pb shows that the wave function has a large aplitude (s 0.8) neutron g9/2,p1/2-1 component. 206 Hence, if one takes the simplest picture of Pb as hav- ing an empty p1,2 neutron shell then one would expect there to be no 5- state with significant strength corre- sponding to the 3.198 MeV state in 208Pb. On the other hand, 206 Pb is not so simple and is known to have a p1/2 neutron in the ground state with a probability of about 40% [Ref. IV.20] so that some strength will remain. Thus in practice, it appears that the 5- strength is substan- 206 tially fractionated over at least seven states in Pb and, surprisingly, the total strength is only about 20% 62 TABLE IV.2 COMPARISON OF THE STRONGLY EXCITED COLLECTIVE 206 207 208 LEVELS IN Pb, Pb, and Pb 206Pb 207Pb 208Pb Ex J" BL Ex J" BL Ex J1T BL 2.648 3‘ .108 2.628 5/2+ .076 2.615 3' .126 2.663 7/2+ .087 2.782 5' .026 2.728 9/2+ .024 3.198 5‘ 058 3.014 5‘ .016 3.223 =5 .013 3.193 (L=5) .010 3.384 (L=5) .027 3.277 5' .015 3.429 (L=5) .016 3.399 5' .020 3.476 (L=5) .013 3.515 (L=5) .012 3.509 (L=5) .025 3.558 5' .022 3.772 5‘ .043 3.583 9/2+ .023 3.708 5' .034 3.620 11/2+ .028 4.107 2+ .047 4.103 3/2' .036 4.086 2+ .058 4.140 5/2‘ .045 4.333 4+ .055 4.313 7/2’ .067 4.323 4+ .067 9/2’ 4.357 6+ .054 4.364 11/2' .042 4.424 6+ .062 4.404 13/2' .047 4.580 8+ .033 4.630 17/2' .028 4.610 8+ .040 4.671 15/2' .025 63 Figure IV.7 Single-particle and single-hole levels in the lead region. The indicated energies are those at which these levels are fixed experimentally. ——-3.64-——— 3pU2 ———3.'2— 3p3/2 .____.2. ____ _.2 5|../ “an 83 2f5,z 247 ‘ 297/2 —2.0I 4SI/2 __|.5g 365/2 —|.6l—— “us/2 _— .4 —\ . ' '115/2 —O.79 Iiu/z 0'90 2f7/2 Ton 299/2 — °~° "‘9/2 2.79 MeV N=|26 Z=82 3.55 MeV 0.0 39:72" — 0.0 3302" _ —- o 35-\ -| —o.67 215,2 ' 263/2 --—0 89 _. —\ 393/2. -' —I.34—- Ihwz —I.64 n.3,," ——l.67-\ 345,2" -——2.34 me'" _3-43 "‘912" -—3.48—— lo7,2-' 64 Figure IV.7 65 less than that observed in 208Pb. This suggests that as well as the 99/2,p1/2-1 particle-hole component there might be a more complex collective component present in the 3.198 208 MeV state of Pb, which is difficult to detect in trans- fer reactions. This behavior is similar to that observed in the 40Ca - 48Ca region [Ref. IV.21], where both the 3- and 5- strengths were reduced and fractionated as one moved away from the closed d3,2 shell and populated the f7,2 shell with neutrons but again the decrease in the 5- strength was less than expected on the basis of the par- ticle-hole model. The contrasting character of the 5- state at 3.772 206 MeV in Pb can also be understood by examining the wave- function of the corresponding state in 208Pb. The 3.709 208 MeV level in Pb has been shown to be a mixture of many configurations [Refs. IV.18 and IV.19], none of which is dominant, and including only a small amount of p1/2 strength. Hence, one would expect this state to behave 206 like the other strongly excited states in Pb. However, 206 this state is significantly stronger in Pb than in 208Pb' suggesting that this state is possibly gaining collective strength from the fractionated L=5 states at lower excitation energy. The total L=5 strength in all 206Pb is 96% of the total L=5 strength of the two 5- states in 208Pb. eight 5- states observed in 66 206 208 E. Comparison of Pb and Pb Inelastic Strengths The results of the collective model fits are presented in Figure IV.8 and are compared with the resutls of a simi- 208 lar experiment on Pb [Ref. IV.7]. Here the strengths for each L-transfer ranging from 2 to 6, and L37 has been displayed according to excitation energy for each of the 206 208 two nuclei, Pb and Pb. The definite correlation in both energy and relative strength of the five strong col- lective 3-, 2+, 4+, 6+, and 8+ states in the two nuclei is again evident from this plot. In addition, this figure 208 also shows that for the first two strong 5- states in Pb there are no similar states in 206Pb. The distribution of L=3 inelastic strength is quite similar in the two nuclei. This suggests that the octupole strength in this lead region is rather insensitive to the p1,2 neutron population. However, the distribution of 206 208 total inelastic strength in Pb and Pb is quite dif- ferent for the other L-transfers. In 208Pb all the L=2, L=4, L=6, and L37 strength is above 4 MeV of excitation energy. In 206Pb there is a significant excitation of all these L-transfers observed below 4 MeV. The L=5 strength is quite fractionated in 206pb, especially below 4 MeV of excitation energy. Furthermore, there is relatively little L37 inelastic 206 strength observed in Pb. These results are in contrast to the study of 67 Figure IV.8 Results of collective model fits of 206Pb compared to 208Pb. The deformation parameter. 8 is Plotted against LI excitation energy for a number of L-transfers. 68 w, t, .9 m I-u- mow wow m3 .. fiwgggi .. mow 8N I ll _1 m . 0.3- x39. m.>H mucosa «>02. >o¢mzm 2034.593 his .g1g. ._ fl m _. m m t. . r mom . wow mun. "3.2m— _ s .w m m .r s. r mom qua—d _ 1 _ . mom FTTLf f T” m A ‘9. "N b.- _ :— wow U 8313WVUVd NOIlVWHOJBO 69 inelastic strength of the three nuclei, 207Pb, 208Pb, and 2093i, by Wagner et al [Ref. IV.15]. They showed clearly that the distribution of inelastic strength is quite simi- lar in these three nuclei. This suggests a sensitivity in this lead region to the p1/2 neutron population of all inelastic strength with the exception of the octupole strength. F. Summary of the Collective Model Results Almost one hundred-fifty angular distributions have 206Pb (p, p') experiment. For been measured in the presen half of these transitions L-values have been determined using a macrosc0pic DWBA analysis. At excitation energies below about 4 MeV, previous studies have identified most of the levels and the results of this experiment agree quite well with these earlier results. A few new levels in this region have been identified in this experiment, and most of the L-assignments for states above 4 MeV of excitation were previously unreported. The strongly excited even parity and 3- states in the stable lead nuclei appear to be insensitive to any single particle structure. The 5- states behave rather differ- 208Pb is principally depen- ently. When the core state in dent on the p1/2 neutron single particle level, and these two neutrons are removed, the L=5 strength is fraction- ated, implying that the 5‘ wave functions are probably more complicated than those suggested by the simple shell model. 70 IV.1 IV.2 1V.3 IV.4 IV.5 IV.6 IV.7 IV.8 IV.9 IV.10 IV.11 IV.12 IV.13 IV.14 IV.15 IV.16 IV.17 REFERENCES FOR CHAPTER IV G. R. Satchler, Nucl. Phys. g; (1964), 1. N. Austern, DirectNuclear Reaction Theories, Wiley, New York (1970). P. E. Hodgson, Nuclear Reactions and Nuclear Structure, Clarendon Press, Oxford (1971). R. H. Bassel, Phys. Rev. 149 (1966), 791. P. D. Kunz, University of Colorado, unpublished. F. D. Becchetti and G. W. Greenlees, Phy. Rev. 182 (1969), 1190. W. T. Wagner, G. M. Crawley, G. R. Hammerstein, and H. McManus, Phys. Rev. C 33 (1975), 757. M. P. Webb, Nucl. Data gg, No. 1 (1979), 145. G. Vallois, J. Saudinos, and 0. Beer, Phys. Lett.- 24B (1967), 512. J. C. Manthuruthil, D. C. Camp, A. V. Ramayya, J. H. Hamilton, J. J. Pinajian, and J. W. Doornebos, Phys. Rev. C Q (1972), 1870. W. A. Lanford and G. M. Crawley, Phys. Rev. C 2 (1974), 646. E. R. Flynn, R. A. Broglia, R. Liotta, and B. S. Nilsson, Nucl. Phys. A221 (1974), 509. M. Lewis, F. Bertrand, and C. B. FUlmer, Phys. Rev. C 1 (1973), 1966. W. A. Lanford, Phys. Rev. C lg (1977), 988. W. T. Wagner, G. M. Crawley, and G. Hammerstein, Phys. Rev. C 11 (1974), 486. J. Saudinos, G. Vallois, and 0. Beer Nucl. Sci. Appl. 3 (1967), 22. J. Alster, Phys. Lett. 25B (1967), 459. 71 IV.18 IV.19 IV.20 IV.21 72 W. W. True and C. W. Ma, Phys. Rev. C 3 (1971), 2421. H. Heusler and P. von Brentano, Ann. of Phys. lg (1973), 381. W. A. Lanford, Phys. Rev. C 11 (1975), 815. A. M. Bernstein and E. P. Lippincott, Phys. Rev. Lett. 31 (1966). 321. CHAPTER MICROSCOPIC MODEL ANALYSIS Shell model calculations have been performed on 206Pb [Refs. V.1 and v.2]. These calculations predict both the energy and wave function of low lying natural and unnatural parity states. Since natural parity states are excited primarily by the well understood central force, a compari- son of the measured angular distributions to those predic- ted by microscopic calculations allow a suitable test of the wave functions. .Unnatural parity states are excited by central and noncentral forces. PerEOrming microscopic calculations on unnatural parity states with well deter- mined wave functions permit an investigation of the reac- tion mechanisms for exciting these states. In this chapter a description of the microscopic DWBA is given first. Then the interactions and wave functions used in the calculations are discussed. Finally predicted cross sections of natural and unnatural parity states are presented and analyzed. A. Description of the Microscopic DWBA Method for Inelastic Scattering In the microscopic DWBA one tries to understand inelastic scattering starting from the nucleon-nucleon 73 74 interaction and the motion of the individual nucleons. The interaction potential V is assumed to be the sum over two body interactions between the projectile p and the target nucleons i, so V=X vip' i The interaction vip has a central part, a tensor part, and a two-nucleon spin-orbit (L . S) part [Refs. v.3 and v.4]. The central part of vio can be written as (c) _ + + + .+ vip - Vogo(r)+VOoiopgo(r)+VTT. TpgT(r)+ + + + T )(Ti°Tp)gOT(r), (v.1) 0+ P VOT( i p where spin and isospin operators are represented by c and I, or as = v + v + v + v , (v.2) where SE stands for singlet-even, etc. The interaction vip can be given a certain shape (for instance a Yukawa shape) and then the strengths Vb, V0, VT, and VOT can be adjusted in order to fit experimental data (phenomenological point of view). A more fundamental approach is to take an effective nucleon-nucleon inter- action, such as the long-range part of the Ramada-Johnston potential [Ref. v.5] or a Reid soft-core potential [Ref. V.6], and afterwards make a test with selected experimental data. For the tensor part of vip the following form is used 75 (t) = ' (t) (t) vip g=0'1VT g (rip)sip’ (v.3) with _+.+ ++ 2_+.+ Sip — (0i rip)(op rip)/rip (oi op)/3. (v.4) An analogous form is used for the spin—orbit part of Vip with Sip in Equation v.3 replaced by L - S: the two— nucleon spin-orbit operator [Refs. v.7 and v.8]. The effect of exchange has to be taken into account [Refs. V.9]. This leads to the formula [Ref. V.3] for the antisymmetrised form of the transition amplitude (Equation IV.1) which can be written as (-) T = A. (v.5) | 0 01 JiMi with A being the number of nucleons in the target. Par- ticle 0 is the incoming particle and can be exchanged with one of the target nucleons. The exchange is explicitly included here by the term with xé-). The ¢J and ¢J are i f now fully antisymmetrised wave functions. Since the interaction V is a two body interaction, 01 the contributions to the matrix element occur from those parts of the initial and final wave function @i and 4f which can only be connected by a single-particle transi- tion, therefore, Equation v.5 can be written as 76 (-) (-) T. = <0 la. +a. [0 >. (v.6) The first term on the right hand side of Equation v.6 is the spectroscopic amplitude S (af is a creation operator and a is an annihilation operator). In this way the tran- sition amplitude T can be written as a weighted sum of all inelastic scattering amplitudes in which a single bound nucleon in the j1 shell is promoted to the j2 shell. The value of the spectroscopic amplitude must be obtained from shell model calculations. B. Forces Used in the Microscopic Calculations In this study two different forces are employed for comparison to experimental results. The first set of interactions (Force A) uses the Serber exchange mixture for the central part of the interaction. This effective force has been found [Refs. v.10, v.11, and v.12] to be a good representation of the phenomenological force deter- mined by fitting definitive reaction data. The Serber mixture had strengths of Vo= -30: VO=10: VT=10: VOT=10 MeV, and the radial form was taken to be a 1 fermi range Yukawa. The tensor force was taken from the works by Crawley §E_3£ [Ref. v.13] and by Fox and Austin [Ref. v.14], and resulted from fitting the crucial (1+, T=0) to 1 (0+, T=1) transition in 14N(p,p') 4N(2.31 MeV) with a tensor force of the one pion exchange potential (OPEP) 77 with a rz-Yukawa shape. The range was obtained by matching the OPEP and the strength adjusted to fit the nitrogen data. This study assumed that the tensor isoscalar portion was zero. The L - S force was taken from studies by Fox and Austin [Ref. v.14], in which the spin-orbit potential was obtained by matching the cutoff Hamda-Johnston poten- tial. The radial shape was given by two Yukawas with respective proton and neutron strengths (ranges) of 29.1 and 20.1 MeV (0.577 fm) and -l496 and -752 MeV (0.301 fm). This set of interactions was used in a previous study of 208Pb (p,p') by Wagner g£_§; [Ref. v.15]. The second set of interactions (Force B) is from a study by Bertsch 2533g_[Ref. v.16]. The force is derived by fitting to the harmonic oscillator matrix elements of the Reid [Ref. v.6] or Hamada and Johnston [Ref. v.17] nucleon-nucleon potentials. Several choices for the indi- vidual terms in the interaction are given in Reference V.16. The present calculations have utilized the sum of the interactions labeled 1, 4, 11, 14, 16, and 18 in Table l of that paper. This set is obtained mostly from the Reid interaction, and is the set preferred by the authors of Reference V.16. Similar sets of interactions were pre- viously utilized in a study of 40 MeV protons inelasti- 24Mg [Ref. v.18], and in a study of cally scattered from unnatural parity states of 88Sr excited by 17.2 MeV pro- tons [Ref. v.19]. The latter study is of particular 78 88 interest here because Sr, being two protons removed from the doubly magic nucleus 90Zr, is similar in structure to 206Pb, which is two neutrons removed from the doubly-magic 208 nucleus Pb. C. Wave Functions Used in the Microscopic Calculations 206 The wave functions of the low-lying levels of Pb 208 are described by two neutron holes in the Pb core. Shell model calculations based on these two interacting 206Pb with both the neutron holes have been performed for Tamm-Dancoff approximation (TDA) and the random phase approximation-(RPA). In the present microscopic calcula- tions wave functions derived from both methods are uti- lized. The TDA wave functions have been obtained from the work of True and Ma [Ref. v.20], who employe a phenomeno- logical nucleon-nucleon interaction of a Gaussian central force plus a weak-coupling force, with a conventional shell-model calculation. RPA wave functions come from the work of Vary and Ginocchio [Ref. v.21] who use a central interaction. In general the energies predicted by the TDA are in slightly better agreement with experimental results than the RPA predictions. However, electromag- netic transition rates are given more accurately with the RPA. D. Results of Microscopic Calculations Microscopic calculations were performed for identi- fied unnatural parity states and a number of low-lying 79 natural parity states with the code DWBA-70 of Schaeffer and Raynal [Ref. v.22]. The code utilizes the helicity formalism [Ref. V.23] and allows the treatment of real interactions with central, tensor, and spin-orbit compo- nents, and an exact treatment of "knock-on" exchange. A sample of the input to DWBA-70 may be found in Appendix II. D-l. Natural Parity States Microscopic calculations of the angular distributions of natural parity states predicted by both sets of inter- actions and by both sets of wave functions are shown in Figures V.1 through V.4. Both direct and direct-plus- exchange calculations are presented. An asterisk indi- cates the direct calculation. For these microscopic cal- culations, the results with Force A are given by the solid curves while the dashed curves indicate results using Force B. Considered first are states of normal parity lying below the dominant 3- level at 2.648 MeV. Displayed in Figure V.1 are cross sections predicted by the RPA wave functions of the first excited 0+ state, the first five 2 states, the first two 4+ states and the first 7- state. The strongest state in this region of excitation is the first excited state of 206Pb, the 2+ state of 0.803 MeV. The shape of this angular distribution is reproduced well by the calculation. However, its magnitude is underesti- mated by about a factor of three. Levels of moderate 80 Figure V.l Microscopic model fits for low-lying natural parity states using RPA wave functions. The solid lines correspond to calculations done with Force A; the dashed curves show results using Force B. The asterisks indicate only direct calculations. The curves without asterisks indicate cal- culations including exchange effects. dc/dQ [mb/srl 81 MSUX—OZ-IZT Figure V.1 TWfiYI'I' 732.151 1 1111 1 1 1 111111 1 11111111 1 11111 . I1 11111111 1] \ I 1111 1 1111111] 1 1111114 1‘ 1 1 11111] x) 82 strength in this region of excitation energy include the 2+ state at 2.469 MeV, both 4+ states, and the 7- state. The shape and strength of these angular distributions are very well reproduced, especially by the calculations using Force B. Force B does better than Force A in matching the magnitude of the angular distributions and in reproducing the shape of these states of moderate strength. The success of Force B is especially clear at forward angles. The weakly excited 2+ state at 2.151 MeV of excitation is best fit by Force A. The remaining weak 0+ state and 2+ states are overestimated by these calculations, however, the shapes are well reproduced in general. Figure v.2 shows measured angular distributions of these same low-lying natural parity states compared with microscopic calculations using TDA wave functions. The 2+ state at 0.803 MeV is underestimated by about an order of magnitude. The data for the collective 1.686 MeV 4+ level is also stronger than predicted. In general the angular distributions of these states calculated using the TDA wave functions reproduce the weakly excited states as well as the RPA calculations, but give poorer agreement than the RPA calculations for the strongly and moderately excited states. Examining the wave functions in detail reveals some differences between the RPA and TDA predictions. For all the 2+ states, the 0+ state and the 7- state examined 83 Figure V.2 Same as Figure v.2 with TDA wave functions used in the calculations. 84 I .D r ) m fide q duquuuqq u 1:.qqa q :1. q q qqqq-qqd - 1:.- q :qqqq d qquq-qqq q -.~..‘-‘ 4 I x I I r m 0 2 v + I I I I v I Y I I m 6 O 1 I + u. i l 9 . « ll l . m c l o I I I. U 3 I U o M o 0 1| 0 o . 1 w. s I ..., T rl war I o I s o v o o I C '14 I Y _ - m... 1. J . I. In a. o a o o 1 t.\9t_ os\hs Figure v.2 85 here, the main particle-hole component is consistently larger in the RPA wave functions. As an example, consider the 2+ state at 0.803 MeV of excitation energy. The two predictions give: _ -1 -1 -1 prA ' '79IP1/2'f5/2>+°51IP1/2'p3/2>+'2°If5/2'f5/2>+ -1 -1 -1 '13193/2'93/2>+'17IP3/2'f7/2>+'12[f5/2'93/2>' wTDA - .71Ipll2,f5/2>+.54[pl/2,p3/2>+.27[f5/2,f5/2>+ -1 -1 -1 '21lP3/2'P3/2>+°2°'93/2'f7/2>+°17lf5/2'P3/2>° The result is an improved fit with the RPA wave functions for all these levels except the 0+ state. This suggests that these states have primarily a single particle-hole configuration, and that the 0+ state is probably a mixture of several particle-hole components. There is a major discrepancy between the predicted wave functions for the two 4+ states. Both the TDA and the RPA predict these 4+ states to have a configuration which is a combination of the |f5/2,f;}2> and IfS/2,p;}2> neutron particle-hole components. One of these components is always paramount while the second is of moderate strength. Other particle-hole components contribute only :modestly to these wave functions. The RPA predicts the Iconfiguration of the 4+ state at 1.686 MeV to be dominated by the |f5/2,f;}2> component, and the 4+ state at 1.998 MeV is dominated by the lfS/2,p'5'}2> configuration. The 86 TDA wave functions for these two states have the same con- figurations of these particle-hole components, only reversed. The angular distributions of these 4+ states are clearly predicted better using the RPA wave functions. Examined next are several highly collective natural parity states with excitation energies around 3 MeV. In Figure v.3 measured angular distributions of the 4+ state at 2.928 MeV, the 5- states at 2.782 and 3.014 MeV, and the 6+ state at 3.257 MeV are compared with calculations using RPA wave functions. The shape of these angular distribu- tions are all reasonably well reproduced. The first 5- state and the 4+ state are underestimated by the calcula- tions. The magnitude of the second 5- state and the 6+ state are accurately predicted by the calculations, espe- cially by Force A. Shown in Figure v.4 are calculated angular distributions of these states using TDA wave func- tions. These wave functions yield very different results. With the exception of the second 5- state, the predicted magnitudes fall far short of the data, by as much as a factor of 30 in the case of the 6+ state. As was the case with some of the low-lying states the systematic differ- ence between the two sets of wave functions is that the RPA wave functions have a larger concentration of strength in the principal particle-hole component. The RPA pre- dicts the wave functions of both 5- states to have an almost pure single particle-hole configuration, and the 4+ 87 Figure V.3 Microscopic Model fits for higher—lying natural parity states using RPA wave functions. The meanings of the curves and asterisks are the same as in Figure V.1. do/dQ [mb/sr] 0.1 10-2 p-o P H 10'2 8&3 MSUX-Bl-265 1 1111 1 1 11111111 11111111 1 1 111111 11111111 1 — C - - - - 1 1111111 1611 ec.m. [deg] Figure v.3 89 Figure v.4 Same as Figure V.3 with the TDA wave functions used in the calculations. 90 MSUX— BI- 266 1:: . a u 1d-J—d C — 1:: _ . q 12:4 4 . I 11 -/’\\ m ‘10 80 100 60 20 l+0 80 80 100 20 I. \\ I I \ 1 \ PCs—p» _ _::__»? —d~—~—qd d —d-~‘dj V‘ \ \ \ O o l I o o I r o o o x O \ .l O o o I . I II . ’ 9 , . , I I u _ /1.. .1 I I .21.. _ 7:: _ p p _::_r p _ 35b» _ b r::tb 1 J 2 1 J 0 .0 0 1 Hem\nr£ GU\.OU ec.m. [deg] Figure v.4 91 state to be about 93% pure. The TDA predicts these states to have a configuration of several particle-hole components with the largest component having about 75% of the strength. Here also is another major difference between the two sets of wave functions. The TDA predicts the largest neutron particle-hole component for the 6+ state to ‘1 The corresponding RPA wave function pre- 5/2' 7/2>' dicts this state to be an essentially pure Ig9/2,g;}2> be If f particle—hole state. In figure v.5 the measured angular distributions of the 2+ state at 1.469 MeV and the 4* state at 1.686 MeV are compared with angular distributions using Force A and Force B. Calculations using both the total forces are presented, and the force is broken down to show the contri- butions of the central and noncentral component parts independently. The calculations show clearly that the angular distributions of these natural parity states pre- dicted by the total forces are dominated by the central part. For both states the central force is predicted to be larger using Force B. This increased central contribu- tion results in a calculation with the complete Force B which is about fifty percent stronger than the calculation using Force A. Although the noncentral forces contribute weakly, it is instructive to note that the shapes of the tensor and spin-orbit calculations using Force A and Force B are very similar, and the predicted magnitudes are 92 Figure V.5 Comparison of measured angular distributions with the cen- tral and noncentral parts of Force A and Force B. do/dSZ [mb/er] 93 0.1. 10'? I I III!!! I I I IIIIII I I I lllll I I I I” I I I I I I + 53 1.HII9 Force A True NF --—L-S \ ‘J \ \ I ’\ X; \ ’1 \\ ’ \ ’ \ i \ i C+T+Lys _ ----Centr~ol 1 --- Tensor ‘ \ I mau424m3 I I I I ”1* I + ‘* 1.6!“; Force A RPA NF 0 O \\. O 0.1 10‘2 p p D b - p - I I IIIIII I I I III!" I 10"3 L l 1 l 1 l 1 l 1 T I I I I I I T I 1 + : Force 8 _ True NF ._ 1 1 IIIIIL 1 1 111111 I I [I III

, [pl/2,p;}2>, and [pl/2,iI;/2>. Calculated angular distri- butions using both Force A and Force B, and broken down into their central and noncentral constituent parts, are compared to the data in Figure v.6. The calculations using Force A reproduce the general shape of the data but the magnitude of the 3+ and 1+ levels is underestimated by about a factor of three. The data of the 6- state falls of more rapidly than the calcu- lations predict. For all three states the tensor part of Force A is the dominant interaction. The central force is the weakest interaction and its contribution to the total predicted angular distribution is observed to be 95 Figure v.6 Same as Figure V.5. 96 m.> whooflm 3mg .Edo _ on: 8 cm a: oN o [IIIIII r I lannnlg A l d-dd44fi14l4. s — «IJ d 8. am on or ow o 1; 1 1. IT .I LII m as p W m 8.8... H m auto“. H m auto“. 0. I wand 1.. . mom; 1 37m.— , -w H L “m +m “0 p p p h p — u p b b p nu p — p % h F F p P h . u r h r p F n r _ h p p n a _ 4 i as q. a a q \ u a a 1 a q 1’ fl . _ a — -~ ~ 1 q A d a v a — ] ’\ I I \ ~ In , . i J. . i w I . x \ - . O. .l / \ 4i \ ‘1 .I I \ A I \\ \ \ l / I (\ It ' ‘ /\ I. _I Am \ l S ' A \ l n u n. J I l v. ’I I\\\ '- MIO— [ - 73-59.. T I I 1I u H mm H m n n ”n u ..l V mum 1...! -3 I I - Jr 1 a 1 - maolul H ”U I...“ LO¢CO.-v II: n um mm _OLoCOUIIII n. 1% HI m.._+h+o _.e I 1' LT I < GOLD“. 1.. < 00.50“. L. < DULCE .l o 11 OK." [I o n mam Nuw u... o +— “n :5 _+m n_.—._._._.ue—spp—pbb.+nIPLL_Lbh-.bp On 97 less significant as the total angular momentum of the states examined increases. The only significant contribu- tion by the spin-oribt force is over a limited angular range for the 1+ state. When Force B is employed the predicted cross sections are all improved. The shape of the 3+ state is matched well and the theory underestimates the data by only about a factor of two. The enhancement of the magnitude is caused principally by the increased strength of the central part of the interaction. The central part is observed to be nearly equal in strength to the tensor part. Since these two components are out of phase, the angular distri- bution predicted by the complete Force B has less structure than is predicted by Force A. Thus, the prediction of Force B compares better with the data. Force B also gives an improved estimate of the magnitude of the l+ state. For this state the central force dominates the tensor force at forward angles, and, in fact, the total calculation is larger than the data in this region. However, past thirty degrees, where the central and tensor parts give contribu- tions of similar strength, the shape and magnitude are predicted quite well. The fit to the 6- state using Force B approaches the slope of the data more closely than the fit obtained using Force A. Once again it is the increased strength of the central part which is respon- sible for the improvement. The contribution of the 98 central part of Force B is also observed to decrease sig- nificantly as the angular momentum of the state increases. A similar phenomenon is observed with Force A but to a lesser degree. The predicted shape and magnitude of the spin-orbit interaction with both forces is similar for all states. In addition to the three established unnatural parity states there is evidence for two more unnatural parity states which should be seen in this experiment. A 3+ state with a pure [pl/2,f;}2> neutron particle-hole configuration is predicted at 3.193 MeV of excitation energy by the TDA, and at 3.156 MeV of excitation by the RPA. The level observed at 3.121 MeV has been suggested by a (p, d) experiment [Ref. v.25] to be this 3+ state. The RPA also predicts a 1+ state with a |f5/2,p;}2> configuration at 3.963 MeV of excitation. Two 1+ states are predicted by the TDA at 2.317 and 3.759 MeV of excitation energy with pure |f5/2,p3}2> and |f5/2,f;}2> configurations respec— tively. The state observed in this experiment at 3.737 MeV has been shown [Ref. V.24] to be a possible candidate for a 1+ state. Angular distributions for these two possible unnatu- ral parity states have been calculated with both forces. The results are displayed in Figure v.7 together with the experimental data for the 3.121 and 3.737 MeV states. The magnitude of the cross section for the 3.121 MeV state is 99 Figure v.7 Same as Figure v.5. do/dSZ [mb/sr] 100 1, MSUX-BZ-IZQ I I I I I I I T I I 7 I I I I I I T ] fi I r I 3+3.121 1“3.737 Force A Force A 1 144111 1 C+T+L-S 0'1 ----Centr‘ol I -- Tensor - - - L-S 1 J 1111111 10' 1 1 1111111 3 4L 2 m m ) , ’\ ' 3 -- /-_ \\ I V \ ,' -~\\ I \ f,‘ I /\ \\ I \ I \ d ’I I \ ‘7‘§\ , hut \ 1 1 .l 1 l 13 L L’J l\-\l 1 1 l 1 l 1 l J\Ll L I 1 ' ' I T I I r I ' I ' ' I r I I I ' I r T T : 3+ 1* : 3.121 3.737 - Force 8 Force 8 : 0.1 1 I S I d I! k I 1 - I 10 2 ' : ”\\ '3 1 I d } .4 -3' 10 \\ .1 \ . \\ —1 \\ J \sl’# 1 l L l 1 l L L 4— L L [1 l 1 \L J\l 1 L 1 l 1 0 20 '40 60 80 100 20 ‘IO 60 80 100 12 9mm. [deg] Figure v.7 101 underestimated especially at forward angles by Force A which excites this state principally by the tensor part of the force. The calculation done with Force B is somewhat better in reproducing the magnitude. However, the fit at forward angles is still poor. The improved fit is caused by both a slightly larger contribution by the tensor part and a contribution by the central part that is nearly an order of magnitude larger than the central force contribu- tion of Force A. The calculations performed for the 1+ state have used the configuration proposed by the RPA. This state lies on the shoulder of a relatively strongly excited L=3 state and is extracted at only a few angles. As a result it is difficult to compare predicted shapes of the angular dis- tributions to the data, but there is enough information to suggest that the magnitude of this state is best repro- duced with Force B. E. Summary of the Microscopic Model Results Microscopic calculations were performed on most low- lying natural parity states and all unnatural parity states identified in this experiment. These calculations were executed with the program DWBA-70 using realistic interactions and shell model wave functions. "Knock on" exchange contributions to the cross sections were included. The calculations allowed a test of wave functions pre- dicted by the Tamm-Dancoff approximation and the random 102 phase approximation. Two realistic interactions were employed and compared. Force A used the Serber exchange mixture for the central force, and empirically determined noncentral forces. Force B was derived by fitting to the harmonic oscillator matrix elements of the Reid potential. Natural parity states with peak cross sections z0.1mb/sr were best fit with Force B. These states were shown to be excited principally by the central part of the interaction, and the central contribution of Force B is as much as fifty percent larger than the central contribution of Force A. The tensor and spin-orbit forces gave little enhancement to the cross sections of natural parity states. The predicted shapes using either force or wave function were found to be very similar to the measured angular dis- tributions. These natural parity states were in general reproduced best with the RPA wave functions. In all cases where the predicted wave functions were significantly dif- ferent the RPA clearly gave a better fit to the data. Furthermore, it was observed that the RPA wave functions for most of these states had a larger concentration of strength in the primary particle-hole component. With the exception of the O+ state this tended to improve all pre- dicted angular distributions. This suggests that these states may be described by a rather simple single particle- hole configuration. Unnatural parity states are not excited collectively 103 and the wave functions of these states are predicted by both the RPA and TDA to have a simple single particle-hole configuration. Thus, calculations of these states permit a unique examination of the forces. With Force A calcula- tions of all observed unnatural parity states underestimate the magnitude of the data, while generally reproducing the shape. The calculations have shown that only the tensor part of Force A gave a substantital contribution to the predicted cross section. The results obtained with Force B were all much better. This improvement is caused to some degree to a small increase in the tensor strength for all states, but mainly because of a contribution by the central part which is similar in magnitude to the tensor force. This central force was seen to be most influential for the lower spin states. The resulting calculations using cen- tral and noncentral interactions in general match the mag- nitude of the data reasonably well. This central force also added to the tensor force in such a way as to smooth out the structure in the calculated angular distribution and thus give better agreement with the data. V.6 v.7 v.14 REFERENCES FOR CHAPTER V C. W. Ma and W. W. True, Phys. Rev. C 8 (1973), 2313. J. Vary, R. J. Ascuitto, and J. N. Ginocchio, Nucl. Phys. A185 (1972), 349. H. V. Geramb and K. A. Amos, Nucl. Phys. A163 (1971), 337. '-‘_' G. Bertsch, J. Borysowicz, and H. McManus, Nucl. Phys. A284 (1977), 399. W. G. Love, L. W. Owen, R. M. Drisko, G. R. Satchler, R. Stafford, R. J. Philpott, and W. T. Pinkston, Phys. Lett. 29B (1969), 478. R. Reid, Ann. of Phys. g; (1968), 411. R. de Swiniarski, Dinh-Lien Pham, and G. Bagieu, Can. J. Phys. §§ (1977), 43. H. Hori and K. Sasaki, Prog. Theor. Phys. EE (1961), 471. N. Austern, Direct Nuclear Reaction Structure, Clarendon Press, Oxford (1971). W. T. Wagner, G. R. Hammerstein, G. M. Crawley, J. R. Borysowicz, and F. Petrovich, Phys. Rev. C 8 (1973), 2504. G. Love, L. Parish, and A. Richter, Phys. Lett. 31B (1970), 167. S. M. Austin in The Two-Body Force in Nuclei, eds. S. M. Austin and G. M. Crawley, Plenum Press, New York (1972). G. M. Crawley, S. M. Austin, W. Benenson, V. A. Madsen, F. A. Schmittroth, and J. J. Stomp, Phys. Lett. 32B (1970), 92. S. H. Fox and S. M. Austin, Phys. Rev. C'gl (1980), 1133. 104 v.21 v.22 v.23 V.24 V.25 105 W. T. Wagner, G. M. Crawley, G. R. Hammerstein, and H. McManus, Phys. Rev. C 12_(1975), 757. G. Bertsch, J. Borysowicz, H. McManus, and W. G. Love, Nucl. Phys. A284 (1977), 399. T. Hamada and I. D. Johnston, Nucl. Phys. 34 (1962), 382. B. Zwieglinski, G. M. Crawley, W. Chung, H. Nann, and J. A. Nolen, Jr., Phys. Rev. C 18 (1978). F. E. Cecil, R. P. Chestnut, and R. L. McGrath, Phys. Rev. C 12.(1974), 2425. C. W. Ma and W. W. True, Phys. Rev. C 8 (1973), 2313. J. Vary and J. N. Ginocchio, Nucl. Phys. A166 (1971), 479. "“' R. Schaeffer and J. Raynal, unpublished. J. Raynal, Nucl. Phys. A97 (1967), 572. M. P. Webb, Nucl. Data 36, No. l (1979), 145. W. A. Lanford and G. M. Crawley, Phys. Rev. C 9 (1974), 646. CHAPTER VI SUMMARY Using 35 MeV proton beams from the Michigan State University and Princeton University cyclotrons the nucleus 206Pb was studied by measuring the scattered protons. High resolution techniques were utilized to identify approxi- mately 180 levels in 206Pb. Below 4.6 MeV of excitation energy the agreement with previous studies is very good although several new states were also observed in this region. Many new levels above 4.6 MeV were also measured. Angular distributions are presented for 144 of these states. Angular distributions predicted by the collective and microscopic models are compared to the data. The collec- tive model calculations allowed the extractions of L-values and deformation parameters. For states where results from other studies were available the agreement is quite good. The L-assignments for most states of high excitation energy were previously unreported. The collective model results for 206 208 Pb were also commpared to the core nucleus Pb. This comparison showed a similarity between the two nuclei for some strongly excited states and the L=3 strength. However, the overall distribution of inelastic strength was quite different for multipolarities other 106 107 than 3. The strongly excited collective states are com- pared with analogous collective states in 207Pb and 208Pb. A strong correlation in both energy and strength of these collective states in the three nuclei was observed with the exception of the L=5 states. A possible explanation of the anomaly in the 5- strengths is given in terms of the core wave functions. Microsc0pic calculations performed on natural parity states indicated that these states were excited primarily by the central two-body interaction. RPA and TDA wave functions were tested in calculations. The RPA wave func- tions, which gave the best fit to the data, suggested that many of the states examined have primarily a single par- ticle-hole configuration. Microscopic calculations for unnatural parity states with well determined wave functions permitted the examination of the two interactions. The magnitude and shape of the angular distributions of these states was best represented using an interaction derived by fitting shell model matrix elements of the Reid poten- tial (Force B). This interaction is the sum of three Yakawas with the ranges chosen to reflect various meson exchanges. Central, tensor, and spin-orbit components were included. This force predicted the central and ten- sor contributions to the angular distributions of unnatural parity states to be similar in magnitude. It would be of interest to study inelastic scattering 108 of protons by 206Pb at higher bombarding energies. Based on the apparent simple wave functions of many states in 206Pb, this nucleus provides an ideal target to examine the energy dependence of the central, tensor, and spin- orbit forces. Also of interest would be a study, similar to the present work, on 88Sr. The low-lying states of 88$r are described by two interacting proton holes in the 902r core. Included in these low-lying states are the 1+ and 3+ levels described by very simple wave functions. The proton hole wave functions of these states are analogous to the neutron hole wave functions used to describe the low-lying 1+ and 3+ states of 206Pb. APPENDICES APPENDIX I ANALYSIS OF THE DATA The plate data was scanned in vertical strips whose height was dictated by the optical systems of the scanning microscopes. Each band was scanned so that extraneous background was excluded. For each exposure, the separate passes were combined using the program JABBERWOCKY [Ref. 1] written by S. Ewald. This program allowed combination of the separate vertical passes in two ways: straight addition or addition after shifting of the passes so that the centroids of specified peaks were alligned as closely as possible. The latter option permits compensation for skewness in the focal plane images or zeroing errors in scanning. The counter data was taken with the data acqui- sition program TOOTSIE [Ref. 2]. With the data in counts-versus-channel number form, the program SCOPEFIT was used for the data reduction. In extracting the area of the peaks, the shapes were assumed to be identical for all peaks and the areas were extracted by an iterative procedure. The shape of the strongest isolated peak in each spectrum was assumed to be represen- tative of all peaks. The low energy tail was varied to assure a best fit for all peaks of interest. This method allowed extraction of weakly excited peaks on the shoulder 109 110 of strongly excited peaks and permitted the separation of barely resolvable peaks. Peak stripping methods would not be able to extract the areas of peaks in these situations. With the reduced data, the programs DOALL [Ref. 4] and SKRUNCH [Ref. 5] were used for further analysis. The correspondence between excitation energy and focal plane position was found with the code DOALL which can perform a search on beam energy, scattering angle, and focal plane parameters to determine the best fit to the positions of peaks of known energy. For this data, the searches were limited to the angle and to the focal plane variables because, since particles other than protons were excluded from the emulsions, the beam energy could not be uniquely determined. Instead, the bombarding energy was determined using the bending magnets' nuclear magnetic resonance readings and a correction empirically established using the momentum cross-over technique [Ref. 6]. Beam energies can be calculated better than 1 part in 1000 with the correc- tion. The focal plane parameters from DOALL were entered into the program SKRUNCH [Ref.S]. This program transforms a counts-versus-channel number spectrum, which is assumed to be quadratic in energy, into a counts-versus-channel number spectrum which is linear in energy. With spectra in this form there is a one-to-one correspondence between channel number and excitation energy at all angles. This 111 simplifies greatly the extraction of angular distributions for all levels. To determine cross sections from the reduced data the program SIGPLT [Ref. 7] was used. Using the output of this program plots were contructed of angular distributions, angular distributions compared to collective model calcula- tions, and angular distributions compared to microscopic model calculations, using the computer programs [Ref. 8] PLOTTER, COLL MOD PLOTTER, and DWBA-70 PLOTTER respec- tively. APPENDIX II SAMPLES OF DWUCK AND DWBA-70 INPUT The DWBA analysis in this paper was performed using the programs DWUCK [Ref. 9] for collective model calcula- tions. In this appendix sample inputs for these two programs are listed including control cards for the Sigma- 7 computer. The cases examined are the 3' state at 2.648 MeV for the collective model example (Table A.II.1), and the 6- state at 2.385 MeV for the microscopic model example (Table A.II.2). The punched output of DWBA-70 is very cumbersome so an auxilliary program (DWBA70 MASHER [Ref. 8]) is used to output predicted cross sections in a more convenient format. 112 113 hmmm. thm. humm. Fume. hmmm. hmmm. oomm.a comm.H comm.H comm.H comm.H oomm.~ www.muxm um ..m.mvmmmom vhvm.vm mvH¢.vI mmmH.H vwmm.HN oooo.ml mmmH.H vmmm.HNI oooo.m mmmH.H oooo.~m oooo.~m oooo.~m comb. ooao.H comb. ooma.a oooo.momoooo.H comb. coao.H comb. ooma.a oooo.oo~oooo.a oomh. oow~.~ oooo.momoooo.H coco. oooo.o~ ooom.m AHAHhImOIQva ooow.vml .vl .N mmmo.¢ml .H whoo.fl omvw.ml ooom.¢ml .vl .N Ohvm.mmi .H whoo.H oooo.mm .MI on¢m.mm .N whoo.H coca. m m H mm coco. oooo.mh OOCMOOOHOOOH 23m“ .oz.moH>mo. 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Mo+a«o«. oo+a05«. momma oo+a~51. oo+am«0. Oo+anno. oo+amo«. oo+a~v~. oo+a«o~. oo+amo«. oo+amo~. oo+an«n. oo+amm~. 00+a001. oo+a051. oooawav. oo+an~v. acoannv. oo+a«vn. esoammm. oo+a«~n. oo+a5~n. oo+a«1v. oo+aoon. oo+a«01. oo+aomo. oo+aoma. 00+a5««. «taum 0.00 0.00 0.00 0.15 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.01 0.01 0.01 0.01 0.00 0.00 «.00 «.00 «.00 «.10 «.00 «.00 «.0« «.0« (5015 :0: 500.0uxu «moanom. ~o+aho~. 00+aoou. 00+a000. ~o+a~o«. momma 00+000«. 00+0001. 0000010. 00+0000. 00+U05«. Ctonm 0.50 0.00 0.11 5.00 «.00 05015 :0: 000.0uxu .0---”.0----.'.l-‘-.0.0.4.l.1-.'-1-I-I-I --.~-I--‘-.n-c-u-o----¢---¢-'-I---¢.--n-a---- ~o+ano~. Nooanom. uo+aoom. uo+ahom. No+aho«. mooaooa. «moauuu. uo+a«m«. uo+a«o«. mo+ama«. No+a~o«. flo+avo«. uo+ano«. mo+ano«. NO+avo«. uo+avo«. No+a5o«. momma :0! 00«.0uxu mo+amo~. umoaoou. 00+a5um. mo+a5m~. 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N0I005”. 0.”5 N00000u. «0|0N'u. N.?? N0I0”5v. «0I0NuN. N.0' N0I0N0”. «0I050N. 5.N0 C10H0 C5015 :01 '00.0ux0 N0+0NON. 00+0000. 0.”5 . N0+000N. 00+0u00. 0.05 . N0+0n0N. 00+0000. 0.5” . N0+0N0«. 00+0u0N. N.?? . N000”0«. 00+000u. N.0' . N0+000q. 00+00”v. 5.N0 . 10000 C1000 C5015 301 ”'”.0u10 N00000N. 00+0”55. 0.”5 . N0+0n0N. 00+000'. 0.5” . N0+0N0u. 00+005n. N.?? . N0+0”0«. 00+00NN. N.0' . N00000n. 00+0”0'. 5.00 . 00010 C10n0 C5015 D01 '1'.0ux0 . . N090u0N. 00§0NON. 0.05 . N0+0«0N. 00+05N0. 0.5” . N0+0NON. 0000n50. N.0” . N0+0N0«. 00+00uN. N." . N0+0”0«. 00+000N. 5.00 . 00010 Clean C5015 D01 0V0.0ux0 APPENDIX IV ABSTRACTS OF PUBLICATIONS On the following pages are titles and abstracts of published papers which I have co-authored while a graduate and undergraduate student at Michigan State University. My contributions to these publications are indicated by a letter key following the title. The key is as follows: A. Data taking. Analysis of data. Performing theoretical calculations. Assisting in the preparation of the paper. Principal writer of the paper. Presented the paper at a meeting or conference. 132 A. "A SURVEY OF THE (3H6, 7Be) REACTIONS AT 70 MeV" A' B' C' D W. F. Steele, P. A. Smith, J. E. Finck, and G. M. Crawley, Nucl. Phys. A266 (1976) 424. ABSTRACT A study of the (3He, 7 using a 70 MeV 3He beam. By surveying a wide range of target nuclides, namely 12' 13c, 160, 24' 26mg, 40' 42. 44Ca, 58, 60, 62, 64Ni, QOZr' 120, 124Sn, 1448m and 206 Be) reaction has been undertaken Pb, systematics of the a-clustering phenomenon were investi- 60 gated. In addition, masses and energy levels of Fe and 120Cd were measured. The 7Be particles were detected in a single wire proportional counter backed by a plastic scin- tillator in the focal plane of an Enge spectrometer to ensure adequate particle identification. Total energy resolution as small as 140 keV full width at half maximum was obtained, although in most cases the target thickness limited the energy resolution to larger values. Differen- tial cross sections as low as 20 nb/sr were measured. The finite range programs LOLA and LOLITA were used to calcu- late differential cross sections for comparison to data, assuming the reaction to proceed by a direct a-transfer. The spectroscopic factors which were extracted show a marked decrease with increasing atomic mass number, imply- ing a decrease in surface a-clustering for heavier nuclei. 133 B. "THE 54Fe(p. d)53Fe REACTION AT 40 MeV AND THE DWBA ANALYSIS" A' B' D T. Suehiro, J. E. Finck, and J. A. Nolen, Jr., Proceedings of Int. Conf. on Nuclear Structure, Tokyo, 1977. J. Phys. Soc. Japan 23 (1978) Suppl. 534. ABSTRACT Angular distributions of deuterons from the 54Fe (p, d)53Fe reaction were measured with 40.16 MeV protons using a split-pole spectrograph and position sensitive pro- portional counter. The measurement was done with 15 keV resolution. Peaks previously unresolved in the (p, d) reaction were clearly observed. Calculations performed with zero-range local DWBA and the adiabatic model are shown to give poor results. Fits are improved with Finite- range and non-local corrections for 7/2- states. By use of the effective binding procedure fits were much improved for the 1/2-, 3/2-, and 5/2- states. 134 C. "EXTRACTION OF DEFORMATION PARAMETERS FROM INELASTIC PROTON SCATTERING" A' B' C' D C. H. King, G. M. Crawley, J. A. Nolen, Jr., and J. E. Finck, Proc. Int. Conf. Nucl. Structure, Tokyo, 1977. J. Phys. Soc. Japan 53 (1978) Suppl. 564. ABSTRACT This experiment reports the measurement of the inelas- tic scattering of 35 MeV protons from the nuclei 154Sm, 176 232Th and 2380. Angular distributions were extrac- Yb, ted for the ground state rotational band. The data were compared with coupled channel calculations using a deformed optical potential and values of the deformation parameters 82 and 84 were extracted. These values, together with the multipole potential moments are compared to the results of Coulomb excitation, electron scattering, and inelastic a- sscattering measurements. In general, the potential moments extracted from the present (p, p') measurements agree better with those from the Coulomb excitation and electron scattering measurements than with the moments from (a, a'). However, the deformation parameters from (p, p'), corrected for the projectile size, agree much better with values obtained from high energy a-scattering than with deformation parameters extracted from Coulomb excitation and electron scattering experiments. 135 D. "INELASTIC PROTON SCATTEIRNG FROM LANTHANIDE AND ACTINIDE NUCLEI" A' B' C' D G. M. Crawley, C. H. King, J. A. Nolen, Jr., and J. E. Finck, Int. Symp. on Nuclear Physics at Cyclotron Energies, Calcutta, India, September 14-16, (1977) 239. ABSTRACT The inelastic scattering of 35 MeV protons is reported from the nuclei 154Sm, 176Yb, 232Th and 2380. Angular dis- tributions were extracted for the ground state rotational band. The data were compared with coupled channel calcu- lations using a deformed optical potential and values of the deformation parameters 82 and 84 were extracted. These values, together with the multipole potential moments are compared to the results of Coulomb excitation, electron scattering, and inelastic a-scattering measure- ments. The deformation parameters generally do not show good agreement for the different methods although the values obtained from the proton measurements are reason- ably consistent with the values from high energy a— scattering. However, the potential moments from the present (p, p') measurements agree better with those from the Coulomb excitation and electron scattering measure- ments than with the moments from (a, a'). 136 54Fe(p,;d)53Fe REACTION A, B, D E. "A STUDY OF THE AT 40 MeV" T. Suehiro, J. E. Finck, and J. A. Nolen, Jr., Nucl. Phys. A313 (1979), 141. ABSTRACT 54Fe(p, d)53Fe reaction was studied using 40 MeV The protons with a split-pole magnetic spectrograph. A total of 53 states were observed up to an excitation energy of 7.364 MeV in 53Fe. At least 29 of these states have not been previously reported. Angular distributions were measured from 6° to 90° for transitions to 35 of these states, and were analyzed with distorted-wave Born approx- imation calculations. Excitation energies, transferred L-values, spectroscopic factors and the implied JTr values are given. Difficulties encountered in obtaining a reliable set of spectroscopic factors are discussed in relation to various prescriptions in the DWBA calculations, and to the one-nucleon transfer sum rule. 137 F. "OCTUPOLE STATES IN 63Cu AND THE WEAK-COUPLING PICTURE" A' B' D Y. Iawasaki, G. M. Crawley, R. G. Markham, J. E. Finck, and J. H. Kim, Phys. Rev. c‘gg (1979), 861. ABSTRACT A high-resolution experiment of proton inelastic scattering by 63Cu at Ep = 40 MeV has resolved three octu- pole states at Ex = 3.81, 3.84, and 3.89 MeV for the first time, thus showing the existence of seven strong octuopole states in 63Cu. This finding is direct evidence that the traditional simple weak-coupling model in terms of one quartet 2p3/2 G 31 is adequate for the octuopole core- excited states in 63Cu. This is not evidence, however, that the weak-coupling picture in general is incorrect for the octupole states in 63Cu. It is shown that to be con- sistent with the present experimental data, the weak- coupling picture for the octupole states requires a ground- state wave function substantially different from the ground-state wave function of the conventional particle— core-coupling model. 138 G. "MULTIPOLE MOMENTS OF 1548m, 176Yb, AND 238U FROM PROTON INELASTIC SCATTERING" 232Th, A, B, C, D C. H. King, J. E. Finck, G. M. Crawley, J. A. Nolen, Jr., and R. M. Ronningen, Phys. Rev. C 2Q (1979), 2048. ABSTRACT We have measured the inelastic scattering of 35 MeV protons from the nuclei 154Sm, 176Yb, 232Th, and 2380. Angular distributions were extracted for J1T = 0+-8+ members of ground state rotational bands. These data were analzyed using coupled channels calculations for scattering from a deformed optical potential. Searches were made on some of the parameters of this potential, including the deformation parameters 82 and B4. The multipole moments of the poten- tial distribution were calculated from the parameter values and are compared to the results of Coulomb excitation, electron scattering, and inelastic, alpha-particle scat- tering studies. In general, these moments deduced in our investigation agree better with those from Coulomb excita— tion and electron scattering than with moments deduced from a-particle scattering. But we also find the moments from our study to be systematically smaller than those from Coulomb excitation. 139 H. "CORE EXCITATIONS IN 63Cu BY THE 63Cu(p. p') AND 65Cu(p. t)63Cu REACTIONS" A' B' D Y. Iawasaki, G. M. Crawley, J. E. Finck, Phys. Rev. C 3; (1981). 1960. ABSTRACT 63 Core excitations up to Ex = 4 MeV in Cu(p, p')63Cu 65Cu(p, t)63Cu at 40 MeV proton energy. The tran- and ferred angular momentum L has been determined for each transition on the basis of the angular distribution shape. A quartet-plus-doublet pattern is consistently observed for the groups of states corresponding to the 2+, 3;, and 4: states of the core nucleus 62Ni. This implies the exis- tence of doublets arising from the coupling of collective states of the core with 2p“2 proton orbital, in addition to the quartets from the coupling with the 2p3/2 proton orbital considered in the conventional weak-coupling excited-core model. It is pointed out that the.existenoe of a weak-coupling situation cannot be proved only on the basis of transfer-reaction data, and in this regard the importance of a comparative study of the inelastic-scat- tering and transfer-reaction data is emphasized. 140 232 234, 236, 238 I. "MULTIPOLE MOMENTS OF Th and U FROM PROTON INELASTIC SCATTERING" A' B' c, D R. C. Melin, R. M. Ronningen, J. A. Nolen, Jr., G. M. Crawley, C. H. King, J. E. Finck, and C. E. Bemis, Jr., Proceedings of Int. Conf. on Band Structure and Nuclear Dynamics, Vol. 1 (1980), 69. ABSTRACT We have measured the inelastic scattering of 35 MeV protons from 232Th and 234' 235: 238 U. Angular distribu- tions were extracted for JTr = 0+-8+ members of the ground state rotational bands. These data are being analyzed using coupled channels calculations for scattering from a deformed optical potential. Our preliminary values for the quadrupole and hexadecapole moments of the potential dis- tribution are compared to moments from Coulomb excitation, electron scattering, and alpha particle scattering, as well 232 238 as theory. Preliminary values of 86 for Th and U are given. 141 J. "SYSTEMATICS OF COLLECTIVE STATES IN LEAD NUCLEI FROM INELASTIC PROTON SCATTERING" A' B' C' E J. E. Finck, G. M. Crawley, J. A. Nolen, Jr., and R. Kouzes, Phys. Lett. 1073 (1981), 182. ABSTRACT From the scattering of 35 MeV protons from 206Pb accurate excitation energies and angular distributions have been determined for the strongly excited collective states. These states are compared to corresponding states in 207, 208 Pb. A possible explanation of the anomaly in the S- strength is given in terms of the core wave functions. 142 206Pb BY INELASTIC SCATTERING OF A, B, C, E K. "A STUDY OF 35 MeV PROTONS" J. E. Finck, G. M. Crawley, J. A. Nolen, Jr., and R. T. Kouzes, submitted for publication. ABSTRACT Using high resolution techniques the inelastic scat- 206Pb is measured. Approxi- tering of 35 MeV protons by mately 180 levels with excitation energies up to 6.8 MeV are identified and angular distributions of most of these states are measured. L-transfers and deformation parame- ters are determined by comparison of the angular distribu- tions to collective model calculations. Microscopic cal- culations of natural parity states are presented and allow a test of RPA and TDA wave functions. Unnatural parity states are also studied microscopically and permit an examination of the central and noncentral forces in the effective interaction. 143 L. "INELASTIC PROTON SCATTERING 176 154 A, B, C, E FROM Yb AND Sm" J. E. Finck, G. M. Crawley, and J. A. Nolen, Jr., BAPS 31 (1976). 662. ABSTRACT Because of the complementary nature of (e, e') and (p, p') in proving proton and neutron transition densities 154 and because there are existing (e, e') data on Sm and 176Yb, measurements of the (p, p') reaction on these nuclei was carried out with 35 and 40 MeV proton beams from the MSU Cyclotron. The protons were detected both with a delay line counter and with nuclear emulsions in the focal plane of the Enge spectrometer. States up to 8+ in the ground state band of both nuclei were observed and many levels in other bands were also seen. Angular distributions have been measured from 20° to 80°. Calculations of the angular distributions will be presented. 144 M. "DEFORMATION PARAMETERS VIA THE (P, P') REACTION" A' B' C' E J. E. Finck, G. M. Crawley, C. H. King, and J. A. Nolen, Jr., BAPS 21 (1976), 985. ABSTRACT The (p, p') reaction is being studied on targets of 1548m, 176Yb, 232 238 Th, and U at a beam energy of 35 MeV. Data have been obtained via a magnetic spectrograph with a position-sensitive proportional counter (8-10 keV FWHM) and with nuclear emulsions (5 keV FWHM). Qualitatively the angular distributions of the 0+, 2+, 4+, and 6+ members of the ground state rotational bands are much more structured than either those from (p, p') reactions on spherical nuclei or on deformed nuclei at lower bombarding energies. Coupled channel calculations including interference between direct and multiple step excitations, using the nuclear deformation parameters, 82, B4, and 86' from (a, a') work at 50 MeV, and using Becchetti-Greenlees global optical 154 238 model parameters, produce good fits to the Sm and U data, but do not do well for the 176 Yb. The present results will also be compared to those from previous studies of Coulomb excitation, Coulomb-nuclear interference, and inelastic electron scattering experiments. 145 O. "PROTON SCATTERING AT 35 MeV TO GROUND BAND STATES 23 234, 236, 238 A, B, C, D IN 2Tb, and U" R. C. Melin, R. M. Ronningen, J. A. Nolen, Jr., G. M. Crawley, J. E. Finck, and C. E. Bemis, Jr., BAPS 21 (1979). 837. ABSTRACT Angular distributions of elastically and inelastically scattered protons have been measured in the angular range of 20° to 144.5° in steps of 2.5° and 5°. A 35.3 MeV dis- persion-matched proton beam from the M.S.U. cyclotron was used. The scattered protons were detected in the focal plane of an Enge split-pole spectrograph with the 25 cm inclined cathode, delay-line detector. The angular dis- tributions for states in the ground band with J"=0+ through 6+ are being analyzed within a coupled channels framework. Quadrupole, hexadecapole, and possibly higher order mass moments will be presented. The results will be compared to moments from (e, e'), (a, a'), and Coulomb excitation studies. 147 2. 3. 4. 5. 6. S. R. H. G. W. G. 82 W. J. J. REFERENCES FOR APPENDICES C. Ewald, Unpublished. Kouzes, Unpublished. David and R. Fox, Unpublished. Hamilton and L. Vance, Unpublished. T. Wagner, Unpublished F. Trentelman and E. Kashy, Nucl. Inst. Meth. (1970), 304. F. Steele, Unpublished. E. Finck, Unpublished. Kunz, University of Colorado, Unpublished. Schaeffer and J. Raynal, Unpublished. 149