EEE‘E ENE’ELSTEIJAEEUA EEF TE EfiE " , '1} E EEEH'" TEON GEE N ==Z EkEuLLl THE ‘23 Ed SHELL Thesis Ear the Begree of Pi}. i3. MECHZQME STATE ‘Né‘iERSETY EEA‘E’EEGND L KOZUB 1%? ““13 ‘3“; ' .I ' e......s . “.7: ' L [B RA R Y Michigan State University This is to certify that the thesis entitled An Investigation of the (p,d) Reaction on N = Z nuclei in the 2s—ld Shell presented by Raymond L. Kozub has been accepted towards fulfillment ‘ of the requirements for Ph.D. degree in Physics 0 -,/ . (V . r [‘4 '0“ Major profgssor [hm~ August 9, 1967 ABSTRACT AN INVESTIGATION OF THE (p,d) REACTION ON N=Z NUCLEI IN THE es—Id SHELL by Raymond L. Kozub An investigation of the (p,d) reaction on N=Z nuclei in the 2s—ld shell has been made to obtain SpectrOSCOpic information and to study the JL=2 J-dependence for the (p,d) reaction. The experiments 24M 28 40 were performed with g, Si, 328, 36Ar and Ca as target nuclei. An enriched (>992» 24Mg target was used to study the 24Mg(p,d)23Mg reaction, and a natural SiO foil was used as a 288i target. Hydrogen sulfide and enriched (>992) 36Ar gas targets were used for the 32S(p,d)3lS and 36Ar(p,d)35Ar experiments, TOCa(p,d)390a experiments were per- respectively. The formed with evaporated foils of natural calcium. The 33.6 MeV protons were accelerated by the Michigan State University sector-focused cyclotron, and the deuterons were observed with (dE/dx)-E counter tele- scOpes. The overall deuteron energy resolution ranged from 95 keV for the 28Si(p,d)27Si reaction to 130 keV for the 36Ar(p,d)35Ar reaction. Virtually all of the 28-ld shell hole strength was observed for each target studied. Deuteron angular distributions for strongly excited 23 2781, 31S levels in , 35Ar and 390a were measured Me, for laboratory angles from 163 to 1550. Excitation energies were also measured. The J—dependence for the pick-up of an.Qn=2 neutron appears mostly in the forward angles of the angular distributions and seems to follow a systematic trend through the 2s-ld shell. Unique Spin assignments are suggested for levels in 318, 35Ar and 390a on this basis. There is some evidence for a correlation between the effects of J- dependence and the nature of the nuclear deformation. An attempt is made to reproduce J-dependence effects with calculations in the distorted—wave Born approxi- mation. Configuration mixing is found to exist in the ground state wave functions of all nuclei, and DWBA SpectrOSCOpic factors are extracted for the strongly excited levels. Of particular interest are the ,enzl levels excited in the 24Mg(p,d)23Mg and 28Si(p,d)27Si reactions, which could arise from either lp or 2p pick— 36 40 ups. The ground states of Ar and Ca are observed to contain appreciable mixing with the f7/2 shell, and evidence exists for a small [2p]2 admixture in the 40Ca ground state. The level orders of the residual nuclei and the DWBA SpectrOSCOpic factors are compared to the strong coupling rotational model and to Nilsson model wave functions. Evidence for rotational band mixing is apparent in many cases. AN INVESTIGATION OF THE (p,d) REACTION ON N=Z NUCLEI IN THE 2s-ld SHELL by Raymond LT Kozub A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1967 ACKNOWLEDGMENTS I wish to express my sincere appreciation to my thesis advisor, Dr. Edwin Kashy, for his guidance and helpful discussions concerning this work. The support and assistance of the entire staff of the Michigan State University Cyclotron Laboratory is gratefully acknowledged. Special thanks go to Mr. Lorenz Kull and Mr. Phillip Plauger for their help in taking the data, and to Dr. William P. Johnson for his assistance in operating the cyclotron. I acknowledge the financial support of the National Science Foundation for the experimental program. For three years of my graduate work, I was supported by a National Defense Education Act Fellowship. Very Special thanks go to my wife, Sandra, for her encouragement and understanding during the past two years, and for cheerfully typing both the rough and final COpies of this thesis. ii TABLE OF CONTENTS Page Acknowledgements ............. . ......... . ........ ii List of Tables ...... . ........... .... ............ Vii List of Figures ..... .......... .................. viii Chapter 1. Introduction......... ........... .... 1 Chapter 2. Nuclear Models...................... 4 2.1. The Shell or Independent Particle Model............................. 4 2.2. Collective Models and Deformed Nuclei ............. . ............. . 9 2.2.1. Collective Motion............... 9 2.2.2. General DevelOpment of the Unified Model......... ...... .... 15 2.2.3. Single Particle States in De- formed Nuclei................... 21 2.2.4. Information fnam the (p,d) Reaction........................ 28 Chapter 3. The Distorted Wave Theory for Pick- up and Stripping_Reactions.......... 31 3.1. Analogies in the Distorted and Plane Wave Formalisms............. 31 3.1.1. The Plane Wave Born Approxi- mation............... ..... ...... 31 3.1.2. The Distorted Wave Born Approximation... ........ ........ 33 iii Chapter 3.2. Extraction of (p,d) SpectrOSCOpic Factors ........................... 3.3 Discussion of Bound State Form Factor ............... . ............ 4. Experimental Methods.... ..... . ...... 4.1. Experimental Arrangement .......... 4.1.1. Cyclotron and Beam System ....... 4.1.2. Scattering Chamber Set-up ....... 4.1.3. E1ectronics........ ..... . ....... (a) (p,d) Reaction ............. (b) Proton Monitor............. 4.2. Lithium—Drifted Silicon Detectors. 4.3. Targets....... ....... ............. 4.3.1. 24Mg ........ . ............ . ..... . 4.3.2. 2831..... ......... . ............. 4.3.3. 39s... ...... ..... 4.3.4. 36Ar ............................ 4.3.5. 4OCa........ ...... . ........ ..... 4.4. Data Analysis ..................... 4.4.1. Determination of Excitation Energies ........................ 4.4.2. Measurement of Differential Cross-Section................... 4.5. Measurement of Beam Energy........ 4.6. Estimate of Errors................ 4.6.1. Measurement of Energy Levels.... iv Page 35 39 42 42 42 45 47 47 51 51 54 54 54 55 55 57 57 58 6O 63 65 65 Page 4.6.2. Energy Resolution ............... 68 4.6.3. Cross—Section Normalization..... 69 Chapter 5. Experimental Results.... ............ 71 5.1. 24Mg(p,d)23Mg........ ............ . 71 5.1.1. Results and Interpretation ...... 71 5.1.2. J-dependence .................... 80 5.1.3. Summary .......... ....... ........ 80 5.2. 28Si(p,d)27Si ........ . ........... . 82 5.2.1. Results and Interpretation ...... 82 (a) Positive Parity Levels ..... 82 (b) Negative Parity Levels ..... 88 5.2.2. J—dependence........ ........... . 89 5.2.3. Summary............. ..... ....... 91 5.3. l60(p,d)15O.... ..... .. ........... . 91 5.4. 32S(p,d)3lS .......... . ............ 93 5.4.1. Results and Interpretation ...... 93 5.4.2. J-dependence .................... 99 5.4.3. Summary......................... 99 5.5. 36Ar(p,d)35Ar..................... 101 5.5.1. Results and Interpretation ...... 101 5.5.2. J-dependence ..... . .............. 107 5.5.3. Summary ............ . ............ 107 5.6. 40Oa(p,d)390a......... ....... ..... 109 5.6.1. Results and Interpretation ...... 109 5.6.2. J-dependence .................. .. 115 5.6.3. Summary ........... . ............. 115 Page 5.7. Summary of Experimental Results... 117 Chapter 6. Analysis with the Distorted Wave Born Approximation and Comparison to Theory ...... ............ ........ . 120 6.1. Optical Model Parameters.... ..... . 120 6.2. DWBA Analysis of J—dependence..... 125 6.2.1. 40Ca(p,d)390a ................... 126 6.2.2. 36Ar(p,d)35Ar........ ...... ..... 127 6.2.3. 328(p,d)318 ........ ............. 127 6.2.4. 2881(p,d)27Si ................... 130 6.2.5. 24Mg(p,d)23Mg....... ........ .... 133 6.2.6. Summary of J-dependence Analysis. 133 6.3. DWBA SpectrOSCOpic Factors ....... . 138 6.3.1. 24Mg(p,d)23Mg ..... .. ...... . ..... 140 6.3.2. 28Si(p,d)27Si ................... 144 6.3.3. 323(p,d)3lS..................... 147 6.3.4. 36Ar(p,d)35Ar................... 149 6.3.5. TOCa(p,d)39Ca...... ............. 151 6.3.6. Summary............. ............ 154 Chapter 7. Summary and Conclusions..,,,,. ..... . 157 Appendix A. Calculation of (p,d) Spectroscopic Factors from the Nilsson Model.... 161 Appendix B. Transition Amplitude for the (d,pl and (p,d) Reactions............... 165 References......... ............... . ............. 174 vi Table LIST OF TABLES Page Optical Model Parameters for DWBA Analysis................................. 124 Summary of Neutron Parameters for DWBA Analysis of J—dependence............ 136 SpectrOSCOpic Factors for the 24Mg(p,d)23Mg Reaction................................. 142 SpectrOSCOpic Factors for the 2881(p,d)2781 Reaction ...... . ..... . ...... .............. 145 328(p,d)318 ......... 148 SpectrOSOOpic Factors for the Reaction...OOOOOOOOCOOOOOOCOOOOO 36 35 SpectrOSOOpic Factors for the Ar(p,d) Ar Reaction.. ........... . ............ . ...... 150 40C SpectrOSOOpic Factors for the a(p,d)390a Reaction................................. 153 vii . A’s Ira Figure 2.1 4.2 4.3 4.4 4.5 4.6 4.7 LIST OF FIGURES Diagram showing the coupling of particle and rotational angular momenta in the unified model........................................ Nilsson diagram of single particle levels in a deformed well........................... Schematic diagram of cyclotron and beam system.................................. ..... Experimental arrangement in the 36 in. scattering chamber........................... Block diagram of electronics used for (p,d) experiments. A diagram for the proton monitor is also Shown........................ Geometry and package system for lithium- drifted silicon detectors.................... 36 Schematic diagram of cell for Ar gas target....................................... Deuteron calibration spectrum taken at 30 from a 0.00025 in. Mylar target.............. Calibration curve and experimental arrange- ment for measuring beam energy by the range-energy methOdooooooooooooooo ooooo .0000. viii Page 18 26 43 44 48 53 56 59 64 5.2 5.3 5.4 5-5 5.6 5.7 5.8 5.9 Nilsson diagram of single particle levels in a deformed well... .......... .... ......O... ... 72 24 23 Deuteron Spectrum from the Mg(p,d) Mg reaction at GLAB=3O . . . . . . . . ).+. C . . C . . . O . O . . . . O 73 2 Deuteron Spectrum from the Mg(p,d)23Mg reaction at eLAB=9d""""°'° ..... . ......... 73 Deuteron angular distributions for the 0.00, 0.45, 5.32 and 9.63 MeV levels of 23Mg from the 24Mg(p,d)23Mg reaction.... ..... . ...... ... 75 Deuteron angular distribution for the 2.06 MeV level of 23Mg from the 24Mg(p,d)23Mg reac- tion........ ......... . .................... ... 75 Deuteron angular distributions for the 2.35 and 4.37 MeV levels of 23Mg from the 24Mg(p,d)23Mg reaction........... ..... ...... ..... .......... 77 Deuteron angular distributions for the 2.71, 3.79 and 6.02 MeV levels of 23Mg from the 24Mg(p,d)23Mg reaction....................... 79 23 Mg level diagram. (a) Results from previous work on A=23 nuclei. (b) Results from the present experiment. 0 . . C C . C . . . . C . . O O C C . . . O O . . O 81 28 27 Deuteron Spectrum from the Si(p,d) Si reaction at eLAB=25 ................O...‘.... 83 Deuteron Spectrum from the 2881(p,d)2781 0 reaction at eLAB=80 .....C..... ..... ......... 83 ix 5.11 5.12 5-13 5.14 5-15 5.16 5-17 5.19 5.20 Deuteron angular dflstributions for the 0.00, 0.952, 2.647 and 2.90 MeV levels of 27Si 28Si(p,d)2781 reaction .............. from the Deuteron angular distributions for the 4.275 and 6.343 MeV levels of 2781 from the 28Si(p,d)27Si reaction ................... .... Deuteron angular distributions for the 0.774, 4.127 and 5.233 MeV levels of 2731 from the 2881(p,d)2781 reaction........... ........... . Plot of (p,d) reaction Q-values t0 fln=l levels in N=Z nuclei versus target mass number...... 28 27Si levels observed in the Si(p,d)27Si reaction .................... . ................ Deuteron angular distributions for the 0.00 and 6.16 MeV levels of 150 from the l60(p,d)150 reaction.. ................................... 32 31 . Deuteron Spectrum from the S(p,d) S reaction 0 at eLAB=3O .......... . ....... ... ............. 32 31 . Deuteron Spectrum from the S(p,d) S react1on at eLAB=9O ........................ . ......... Deuteron angular distributions for the 1.24, 2.23 and 4.09 MeV levels of 313 from the 32S(p,d)3lS reaction ......................... Deuteron angular distributions for the 0.00, 3.29, 4.72 and 7.05 MeV levels of 315 from the 328(p,d)31S reaction.. ....... . ..... . ......... 84 85 87 90 92 94 95 95 97 98 5.21 318 levels observed in the 32S(p,d)3lS reaction ............... . ............. . ....... 100 5.22 Deuteron Spectrum from the 36Ar(p,d)35Ar 0 reaction at eLAB=30 ........ .. ............... 102 5.23 Deuteron spectrum from the 36Ar(p,d)35Ar 0 reaction at 0 85 .. ....... ................ 102 LAB— 5.24 Deuteron angular distributions for the 0.00, 2.60, 2.95 and 6.82 MeV levels of 35Ar from the 36Ar(p,d)35Ar reaction ................... 103 5.25 Deuteron angular distributions for the 5.57 and 6.01 MeV levels of 35Ar from the 36Ar(p,d)35Ar reaction........ ....... ........ 105 5.26 Deuteron angular distributions for the 1.18, 3.19, 4.70 and 6.62 MeV levels Of 35Ar from 36 the Ar(p,d)35Ar reaction... ................ 105 5.27 35Ar levels observed in the 36Ar(p,d)35Ar reaction...... ....................... . ....... 108 5.28 Deuteron Spectrum from the 40Ca(p,d)390a reaction taken at SLAB=3d’................... 110 5.29 Deuteron Spectrum from the TOCa(p,d)39Ca reaction taken at CLAB=9U’................... 110 5.30 Deuteron angular distributions for the 0.00, 5.13, 5.48 and 6.15 MeV levels of 39Ca from 40 the 0a(p,d)39ca reaction.... ..... . ......... 111 xi . r s p a p . r 5.31 5.32 5.33 6.3 6.4 6.5 6.6 6.7 Deuteron angular distributions for the 2.47, 2.80 and 3.03 MeV levels of 390a from the 40Ca(p,d)39Ca reaction......... .............. 39Ca levels observed in the 40Ca(p,d)390a reaction......................... ..... . ...... Summary of experimental results for forward angle J-dependence ........... ................ 36 36 Optical model fit to Ar(p,p) Ar elastic scattering data ............ ... ............ ... Optical model fits to the 26Mg(p,p)26Mg elastic 36 scattering data with Ar and 26Mg parameters.. DWBA fits to the,fln=2 J-dependence for the 5.13 (5/2+) and 0.00 (3/2+) MeV levels of 39Ca excited in the 40Ca(p,d)390a reaction... DWBA fits to the Eg=2 J—dependence for the 2.95 (5/2+) and 0.00 MeV (3/2+) levels of 35Ar excited in the 36Ar(p,d)35Ar reaction... DWBA fits to the fg=2 J—dependence for the 2.23 (5/2+) and 1.24 MeV (3/2+) levels of 31$ excited in the 328(p,d)313 reaction...... DWBA fits to the 4:22 J-dependence for the 0.00 (5/2+) and 0.952 (3/2+) MeV levels of 28Si(p,d)27Si reaction... 2781 excited in the DWBA fits to the 12:2 J-dependence for the 0.45 (5/2+) and 0.00 (3/2+) MeV levels of 23Mg excited in the 21+Mg(p,d)23Mg reaction... xii 113 116 119 122 123 128 129 131 132 134 , F 6.8 6.9 DWBA fits to 325/2+ angular distributions for levels excited in the (p,d) reaction on 2881, 328, 36Ar, and 40Ca..... .......... ..... 137 DWBA fits to deuteron angular distributions for different values of fig ................... 139 xiii Chapter 1 Introduction Since its original Observation by Standing in 1954 (St54)*, the (p,d) reaction has proven to be a valuable tool for the experimental investigation of nuclear properties. This has proven to be particularly true at higher bombarding energies, where the direct reaction theory is most successful (T061). This model of the reaction is characterized by the assumption that the incident proton interacts only with a single neutron in the target nucleus, thereby forming a deuteron and leaving the residual nucleus in some excited state. As was shown by the plane wave stripping theory of Butler (Bu51) for the inverse (d,p) stripping reaction, the shape of the angular distribution of the emitted part- icles is determined by the orbital angular momentum (1;) 0f the transferred neutron. Thus the parity (fl? and possible values for the total angular momentum Of the final nuclear state (J) can be deduced directly from the experimental data. The widely used theory of direct reaction pro- cesses is the distorted-wave Born approximation (DWBA) * References are denoted by the first two letters of the first author's name and the year of the publica- tion. (T061, Sa64) and, to the extent that one can trust the DWBA calculations the (p,d) reaction provides a direct measure of the overlap of the target wave function with the wave functions of the excited states of the residual nucleus. These overlaps represent the coefficients of fractional parentage (c.f.p.) for the ground state wave function, and therefore contain information about the nuclear structure. Configuration admixtures in the tar- get nucleus are thus easily detectable by this reaction. Also, the excitation of levels in the residual nucleus is quite selective, since the coefficients of fractional parentage will be small if the final state is not a hole state. In 1964, it was observed by Lee and Schiffer (Le64) that the angular distributions for ,Qn=l levels excited in the (d,p) reaction showed a dependence on the total angular momentum of the final nuclear state (J), even though theyn values were the same. Similar effects have also been observed in the (p,d) reaction (Sh64, 6165 (a), Wh66, K067). At present, there still is not a satisfactory theoretical explanation for J-dependence in the (p,d) reaction. Most of the previous investigations with the (p,d) reaction have been with nuclei in the 1p and lf7/2 shells. However, relatively few (p,d) experiments have been per- formed on the N=Z targets in the 2s—ld shell. This probably reflects the very negative reaction Q-values (-13 to -15 MeV) and the close level spacing involved, which require both a high bombarding energy and good resolution in order to observe the level structure over a reasonable region of excitation in the resi- dual nucleus. The subject of this thesis is the investigation 2881, 328, 36 of the (p,d) reaction on 24Mg, Ar and 400a. The primary objectives were to study the con- figuration mixing in the ground state wave functions of the target nuclei and the,Tn=2 J-dependence in the deuteron angular distributions. In addition, inform- ation concerning the level structure of the residual nuclei (23Mg, 275i, 318, 35Ar and 39Ca) was obtained. These experiments were performed with 33.6 MeV protons accelerated by the Michigan State University cyclotron. This bombarding energy was low enough to be compatible with the use of commercially available high resolution semiconductor detectors; at the same time it was high enough to expose 10 - 12 MeV of excitation in the residual nuclei, which was suffi- cient to achieve the goal of this work. P: In Cy u .CE Chapter 2 Nuclear Models A number of models have been constructed to explain the results of experiments in nuclear physics. Many of these have been introduced because of their success in other branches of physics. For example, the formal- isms of the shell, Hartree-Fock and rotational models are used in atomic and molecular physics. Also, the superfluidity model results from techniques developed in superconductivity theory. Each model accounts for some of the observed nuclear properties but no one descrip- tion successfully explains everything. This chapter concerns some of the models that are commonly used to explain internal nuclear structure. A brief discussion of the shell models for spherical nuclei is given, and the single particle and collective models for non-spherical nuclei are described in more detail. The prediction of these models for the (p,d) reaction are discussed, and are later compared to the results of the present experiments (Chapters 5 and 6). 2.1 The Shell or Independent Particle Model The shell model assumes that each nucleon is bound in a spherically symmetric potential and moves indepen- dently of other nucleons. A commonly used form for this potential is the isotrOpic harmonic oscillator .14 F) (_.1 ' ‘- 5 V= ii {0 (ad-I) - ,Q{.€+I)-S(s+l)j Eq. 2.12 Since jabl/Z, we have j .. “Q; “j =j+ l/2_ '5 1: 4 > 43*”). g":- 1" 72 Eq. 2.13 Since V80 (r) in Eq. 2.10 is less than zero, the spin- orbit force is attractive for j=le/2 and repulsive for 129-1/2. The J=fl+l/2 level thus lies lower in energy than the 331-1/2 level, which is generally consistent with the observed level order of spherical nuclei. The spin-orbit theory also correctly predicts the magic numbers: N or 2:2, 8, 20, 28, 50, 82 and N=126. Also, the level spacings are no longer equal. This simple model still assumes that a nucleon moves in a potential VC?) without being affected by the presence of other nucleons. Also, when there are two or more nucleons outside a closed shell, the j-j coupling of these nucleons leads to states of different total J, all of which are degenerate in the above model. Such is not the case in reality however, so various perturbations have been introduced to split these de- generacies. This perturbation is often some form of a two particle residual interaction. Also, the inter- mediate coupling model, which involves mixtures of the j-J and L-S coupling schemes, is used to predict the structure of light nuclei (Ku56, 0065). Both of these approaches result in wave functions which are super- *‘e 9 positions of (j-j coupled) shell model states. This is referred to as ”configuration mixing”, and the weighting factors in the expansion are called coeffic- ients of fractional parentage (c.f.p.). The spherical shell model fails to account for the large quadrupole moments and extensive configuration mixing observed in.deformed nuclei. The treatment of such cases is described in the next section. 2.2 Collective Models and Deformed Nuclei The formal description of collective nuclear- models and single particle motions in deformed nuclei has been presented by several authors (Ni55, 3053, Br64, Pr62, Ga64, He64). The treatment of Preston (Pr62), in conjunction with notes from the lectures of Gordon (6065) is given in Sections 2.2.1 — 2.2.3. The first section concerns collective motions in general, and the contributions of particle motion are discussed in Sections 2.2.2 - 2.2.4. 2.2.1 Collective Notion A.number of nuclei in various mass regions are found to have unusually large quadrupole moments. These large moments cannot be accounted for by assuming the nucleons are bound in.a spherical potential, and there- fore must arise from the coordinated motion of many nucleons. For example, a spherical shell model predicts ti Ila to It for the 10 that the quadrupole moment for odd A.nuclei is due only to the effect of the last odd nucleon, which implies that the moment for odd Z nuclei is much great- er than for those with odd N. These two moments are found to be comparable in the above mass regions how- ever, and much larger than the spherical prediction. It is therefore assumed that these nuclei are non- spherical. The surface of a deformed nucleus is described by an equation of the form «'9 ; 4:“) = 2?. 1 1+ S Gaff) 7 "Is; 419] a”... 1 Eq. 2.14 where ET'and Cy are variables of a space-fixed coor- dinate system and Bo is constant. If 3 =0, the nuclear volume is allowed to change; we assume the nucleus is incompressible. Center of mass motion is implied by the X =1 terms, and this does not correspond to inter- nal degrees of freedom. We are therefore interested only in terms involving R_=2, 3, ...., which correspond to quadrupole, octupole, ...., distortions respectively. It will be assumed here that 71:2 only. Then, trans- forming to the body fixed, principal axes (1,2,3) of the nucleus, Eq. 2.14 becomes +2 ('(5) ¢)= P°II+ S 6119‘) X”(e)¢)] J's-3, EQe 2el5 ‘0... Q '0... ya pr. eEar 11 where the two coordinate systems are related by a manor-fl’c/WQVQ'W) Eq. 2.16 In this system, the inertia tensor is diagonal so that a21=a2’_1=0. This leaves just two coefficients, which are defined in terms of the parameters ‘3 and 3‘: afiemr a -.-.- a '55 4a and 3,, 2-2. [if Eq. 2.17 The five (21+l) coordinates needed to specify the system are chosen as 6 , 3) and the three Euler angles (9,§,'f’) which define the transformation to the body fixed system. The parameter I) specifies the degree of symmetry of the system; if xkis a multiple of 7173 a symmetry axis exists. For example, if dk=0 or”fl'the axis of symmetry is the 3-axis and a22=0. (3.is a measure of the ”amplitude” of the deformation and, assuming 8‘=o, is positive for a prolate (football) shape and negative for a oblate (pump- kin) shape. If (3 or ris not constant in time, vibrat- ional modes of motion exist. These are referred to as 'F-vibrations" and " r-vibrations" respectively. I If the collective motion is assumed to be similar to an irrotational flow of nuclear ”liquid”, the total energy is written as (Pr62) 12 E: ng/O'éfllzat 92 C é log/cl: = ’45 32+ 72553 ’/z idem}; “I Eq. 2.18 .1 z, for X\=O. The (3 and @ tenms correspond to the kinetic and potential energies of a vibrational mode of frequency ‘IC7S: where the potential term arises from the Coulomb and surface energies of the nucleus. The JLK in the last term of Eq. 2.18 are not moments of inertia in the usual sense, but arise from surface waves that make the nucleus 132; like it's rotating with angular velocity 53 in the space-fixed system. For example, if p is small, a small amount of nuclear matter is involved in the ”rotation” andoq; will be small. Since 64:0 here, there is no contribution to the rotational motion from rotat- ions about the 3-axis (axis of symmetry), so 313:0. Also, since there is a symmetry axis, £1: 2, and it can be shown that (Pr62) J‘tol'zoflzt-J’Bfiz EQe 2019 The irrotational flow argument is quantized by writing the Hamiltonian for the total energy (Eq. 2.18) in the form H ‘3’- Hy;b(@) + #rot' 3 P4: =1 9°“. Eq. 2.20 rn r—‘I 13 where the Rk are the angular momentum components in the body fixed system. If @is nearly constant (i.e., a static deformation) Hrot will dominate in Eq. 2.20 and R2, R2 and R3 will be good quantum numbers: (W): Jame-Oi" < P2> ; Mflf < 35> 3K2“? Since axial symmetry was assumed,.23=0, and thus R3:KR=O. ’“x Eq. 2.21 This implies that JR must be integral. Settinngl=42=&, the rotational part of Eq. 2. 20 is 1.55% +fi):941(R2 93:5 first Eq. 2.22 which results in the energy eigenvalues 2' -— E (ma) = %— Ig(~)a"’) :2 2 Eq. 2.23 Because of the symmetries of the nuclear shape, only states of even parity and even JR can exist, i.e., J:;O+, 2+, 4+, etc., and the model applies only to even-even nuclei. If the nucleus is asymmetric (3)790), Jgaofl 2+, 3+, 4+, 5+, etc. and KR is restricted to even values for quadrupole distortions, except that KR=0 is not allowed for odd values of JR. Calculations involving asymmetric nuclei have been made by Davidov and Filippov (Da58). The parity is (-)7‘ in genera1,,so octupole (71:3) dis- tortions must be introduced to obtain odd parity levels 14 in a rotational model. In the above description it was assumed that 6 and 3‘ were constants so that the rotational part of the Hamiltonian could be treated separately. Such a division is also justified when @— and 5“ vibrations exist, if the rotational level spacing (Eq. 2.23) is much less than that of the vibrational levels. An equivalent assumption is that the fre- quency of the rotational motion is much less than the vibrational frequency. The addition of vibra- tional modes gives rise to rotational bands based on vibrational levels. For example, in the case of a @ -vibration where X20, a KR==O rotational band is based on each vibrational state. The results of the collective model are found to be in reasonable agreement with the low-lying level structure of heavy, deformed nuclei. The model describes light nuclei somewhat less success- fully. Collective modes by themselves can be applied only to even-even nuclei however, since they predict only integral spins; even the vibrational quanta are bosons. The descriptions of deformed odd-A.nuc1ei can also be included if the single particle motions are considered in addition to the collective proper- ties. This coupling is discussed in the next section. 15 2.2.2 General Development of the Unified Model In the unified model the motions of individual particles are coupled to the collective modes of motion: 3 ' at “noon. Eq. 2.24 These ideas were first deve10ped mathematically by Bohr and Mottelson (B052, B053), who assumed a particle Hamiltonian of the form .3“; qu1= 2E;({7; +'V/UPKX§’C?1§ §3))’ l Ego 2025 where the summation is over the nucleons outside of some specified core. The potential V is taken to be r A ."—* V= Wheaten) ’1") a) where V°(r;flb§) is just a spherical shell model potent- Eq. 2.26 ial and f is a form factor which becomes unity for spherical nuclei (6:1;0). For axial symmetry and quad- rupole distortions F If}. 0; e¢)= /+ @ >179») Eq. 2.27 i.e., it is just the radial form factor used in the preceding section (Eq. 2.15). The coordinates (r,9,¢) refer to the body-fixed system. Since Hp contains the De 16 parameters @and x: the Hamiltonian (Eq. 2.24) clearly contains coupling of the collective and single particle motions. There are two extreme cases for which the problem can be solved. One of these is the so-called "weak coupling“ case, where p is a small perturbation and the collective modes are small vibrations about a spherical shape. This solution is given in Ref. Pr62. The other extreme is the ”strong coupling” case, which assumes a large permanent deformation and results in rotational bands based on single particle levels. This solution is outlined below for axially symmetric cases. Assuming @ and Xto be nearly constant, the collect- ive part of the Hamiltonian (Eq. 2.24) is purely rotational: -.-.- . 3 1? 2' Hull. AIM.“ '- é 3f; EQO 2028 This is identical to the Hrot in Eq. 2.20. Using Eqs. 2.25 and 2.28 in Eq. 2.24, the Schroedinger equation for the nucleus can be written as [He 1‘ 6 (7: + v (m, a 1%))??? gay» I EQe 2029 Defining '3', B'- and '3 as the particle, rotational and total angular momenta respectfully, we have an ‘* «+ I = E *3 ...IA '1" 7 - j? v: EQe 2030 17 The solutions to Eq. 2. 29 are the CFUMKJI) where fM= and OK corresponds to other quantum numbers for the particles. KR=0 in the axially symmetric case, so K=n+xR=n. A diagram showing the coupling of the angular momenta is shown in Fig. 2.1. Since the 2 potential is not spherically symmetric, j is not a con- stant of motion. It then follows that R2 is not a good quantum number either. Assuming axial symmetry Gfll=12=3), Eq. 2.28 can be written Hat: 53.2,,“ J— [{3} 3)1‘(33 03)] :J-i [(3:0)] :SLEIE mull-{1] EQe 2031 in analogy with Equation 2.22. The term involving j2 involves only particle motion and can be included in the particle Hamiltonian: '2. = 7" J- f£2—— lflp ‘QJL£J EQe 2032 The matrix element of the remainder of Eq. 2.31 can be written {143‘- fi'fi> 33% [NM—2 K 733%; 1411'.) EQ0 2 033 The second term in Eq. 2.33 indicates that the rotational and particle coordinates are not completely separable. ............ (LL/chic: quDmUzq \ 18 .HmUoE Umfiufic: exp CH mpcmEoE smflsmcw HmCOfiQMpos Ucm maoapsmm mo mcflflmsoo map wcfizonm Emswmfim .H.m mssmfim 82: >89 3%: mouse N JMOOE sznzz: mo... 45.29205. 14.5024 l9 Denoting this term by RPC (rotation-particle coupling) we can write 2P0: 3’3 [333'— *1“ Eq. 2. 34 where J i=J1-4J2 and jtzjl"j2 are raising and lowering Operators. The J: and 3:0 operators change the value of the K quantum number to K-l when RPC operates on the total wave function @(JMK). Thus RPC mixes states of different K, so that K is not really a good quantum number. In the cases of axial symmetry, however, RPC can be treated as a perturbation. Its matrix element then vanishes by the orthogonality of the‘yEZJMK), except when Kal/2. The Schroedinger equation (Eq. 2.29) can be re- written as [1.(3- Al ,3‘3)+H.* RPCJQF= E‘E’“ Eq. 2.35 where Ho and RPC are given by Eqs. 2.32 and 2.34 respect- ively. The solutions to Eq. 2.35 involve both rotational and particle wave functions, where the normalized particle (intrinsic) wave functions are defined by H° Zoe-=6 “*7 K“- EQO 2036 Note that Zka‘contains the coordinates of all the particles 20 outside of the assumed core (See Eqs. 2.25 and 2.32). If RPC=0, the solution to Eq. 2.35 is explicity ”333W WW *9 ‘Eié‘ijé‘n’téfl Eq. 2.37 J . where the,DMK are functions of the Euler angles. Yij is just some wave function having constant 1, and is defined by the expansion '1 -= 66- 70*) Ki 5 JK 3" Eq. 2.38 The total energy (Eq. 2.35) can now be written 1 I. qux: exp-fin") +£‘3’(I./) + EFF? SK V1 p—v—J Sick—J E? 5""- Eq. 2.39 where Sm Eng") = £45035,” ($*”Z)§W Eq. 2.40 and 31'"; ,_ a: ’46“) (“VJ/QM Eq. 2.41 The total angular momentum (J) has the values K, K + l, K + 2, etc. It is seen from Eq. 2.39 that states of +K 21 and -K are degenerate. Equation 2.39 also shows that states of different K are actually different particle (or hole) states, each of which is the basis of a rotational band. If Ki?él/2, the excitation energy wathin a band increases like J(J+l) sincel Eq. 2.54 amfl.is equal to,fl if EE=O. In the case of axial sym- metry"23 is a constant of motion, which gives rise to another set of eigenfunctions that are also solutions of Eq. 2.47. They are the [Nn3A> states, where <23): atA Eq. 2.55 With the introduction of spine, bar-=15 Eq. 2056 and 1335-? K EQe 2057 Again. A2 and :2 in the spherical limit. are not constants of the notion, except This model gives a poor description of spherical 25 nuclei, since it reduces to the spherical harmonic oscillator (Sec. 2.1) as S~approaches zero. However it does provide a suitable set of wave functions with which to improve the calculation. This was done first by Nilsson (N155). who solved the equation {flied} He {-Cj'g + DR‘J/szfy Eq. 2.58 for the case of axial symmetry (320). The spin orbit term is included in analogy with the spherical descrip- tion, and the DA2 term splits the l-degeneracy that arises in the spherical harmonic oscillator. The Hh.o. term is just the spherical oscillator Hamiltonian, while He is the Hp of Eq. 2.50. The solutions depend on the values of C and D, which are chosen to fit the data on spherical nuclei (Seo). The wave functions and energies (which now depend on.K) are then calculated for various values of‘S. The results are illustrated by the well known Nilsson diagram. The portion of this diagram relevant to the present work in the Zs-ld shell is shown in Fig. 2.2. The wave functions on the right are the [Nn3AJ states, since for large distortions n3 is nearly a constant of motion. The values of K.and the parity are also shown. For large distortions, the nuclear volume tends to change so that «)0 is no longer indepenn ent of 5 3 it is then convenient to introduce the para- meter l‘ ‘l L .L-LIQAVQ m1~02~m ZOWWJ\Z 26 .czo;m no: mfl Hm>mH Am\HmHV m\HuM pmaflm mzp ”machpsmc 03p 6cm mCOpoaa 03p Ufiog cwo Hm>mfi Loam .Ammflzv flaw: Umeaommv m CH mHm>mH wHOHpamQ mdwcflm mo Emsmmflw commHHz .m.m maswflm ? e e w e .N- w- e- m ed who . o . «.0... to- ”Essa: «Ra -mNN \d . To;-~\_\\‘ .. . noumTN: / mam .oom Eves; m T .mexm «BU m .mmfi w H. EH3: :m 0m.m HCMMHIN\_ N\n m H~o~H+~B u . €873. ‘ m: mu m WomHmem / n 22%. mdfij 30.qu “.3026 283.2 «Eu Roam? : _ 27 42 38 waSZ C Eq. 2.5, which is also shown. The salutions to Eq. 2.58 (£1?) may be expanded in a number of representations, e.g., 35K = Zara/NjAé» where IN1A£>are the spherical oscillator solutions and EQe 2e60 A+£=L The quantum numberd, defines the particular Nilsson orbit for a given value of N and K. Also, as mentioned above, for large deformations ’& x anaAK) Eq. 2.61 which are just the solutions for the Hamiltonian of Eq. 2.50. Chi (Ch66) and others (3966, Ba67) have expanded the?“ in terms of the shell model states {Max} (See Sec. 2.1): 3 max. If = €C.(d)/N1jk> K 5: J" EQe 2062 jmax is determined by N , and jajtl/Z is restricted 1 by 1114-) -<-)". Equations 2.60 - 2.62 involve only single particle where .k. ‘ *l 28 wave functions, and in general more than one particle outside a closed core is included. In this case, the 3232; particle wave function is (in the notation of Eq. 2.36) 52,. =KZKZ K """ q ; } 3"}‘1/263“ Oink Hlfiare EQe 2.63 I'M L for q nucleons. If q is even, K=O in the ground state, and K=Kq if q is odd. The lg"? thus act as particle “creation operators" operating on leoré}. The equilibruim shape (i.e., the deformation para- meters) may be found for a particular nucleus by includ- ing all the particles in Eqs. 2.44 or 2.58. The total nuclear energy is then deduced from the particle energy «9(3 method is discussed in detail in Ref. Pr62. and is minimized by requiring theta—56:320. This The Nilsson model has been quite successful in explaining the lowhlying levels of some nuclei in the 2s-ld shell and in the rare earth region. This cal- culation was done for a symmetric nucleus, and has been extended by Newton (Ne60) to include asymmetric cases (3&0) . 2.2.4 Information from thefip,d) Reaction* The (p,d) reaction excites mainly hole states in 7—-The author is indebted to L. Zamick for his helpful explanation of this subject. 29 the residual nucleus by a direct reaction mechanism. Since the transfer of a single nucleon is involved, the information obtained is related only to the particle wave function, independent of the collective motion. All of the target nuclei in this study have J'“=o+ in the ground state so that KR=K=JR=j=0. Therefore, when a neutron is removed, the spin and parity of the final nuclear state (lg) is equal to that of the picked up neutron (jg). We have seen, however, that 3n is not a good quantum number in the deformed single particle wave function (Secs. 2.2.2 - 2.2.3). The removal of a neutron from a nonpspherical nucleus may be represented by a destruction operator (”hole' wave function) operating on the target ground state (Eq. 2.63): 1’: I; V V ”V ~16 W’Core “K” it} 5,}, k3”, n kn fut. Eq. 2.64 which is just the particle wave function of the final 7. nucleus. The 5‘ cancels the /V of the picked up neutron, leaving the final nucleus wave function with KfaKn and energy depending onKn and.o(n. It is seen then, that the spin of the final state can be any of the jn's in the expansion qyy‘ := 2%?Maahc Odu)l[N/J?n1..(<::> dnKn ' ‘K Chkn Jn' \ Eq. 2.65 30 yr which is similar to Eq. 2.62. For example, if Kn=l/2+ and N22 (e.g., Nilsson orbits 6, 9 or 11 in Fig. 2.2), Eq. 2.65 becomes V =“4.0120/2‘1,,>+C/,€j‘~)/223/,"9+[(in))M/) ant/2.67122. W 2 Sj/t I d 3’1— , Ail!- Eq. 2.66 Direct neutron pick-ups from one of the iw;l/2+, N=2 Nilsson orbits thus excite three levels having spins and parities l/2+, 3/2+ and 5/2+, which also correspond to members of a rotational band based on the Nilsson orbit. The experimentally obtained (p,d) spectroscopic 2 factors (Chapter 6) are therefore measures of the 101:? in Eq. 2.65. The calculation of theoretical spectroscopic factors from Eq. 2.65 is outlined in Appendix A, and the values obtained from the wave functions of Chi (Ch66) are compared to experiment in Chapter 6. Chapter 3 The Distorted Wave Theory for Pick-up and Stripping Reactions The theoretical formalism for direct reactions in the distorted wave Born approximation has been treated in detail by Tobocman (T061) and Satchler (Sa64). Some of the relevant results are presented here. Particular attention is given to a discussion of form factors for the nuclear bound states and the extraction of spectro- scopic factors. The direct interaction model for nuclear reactions is based on the assumption that the reaction is a one step process, without the formation of an intermediate nuclear state. Direct reactions were first described by the plane wave Born approximation, which predicted a cross-section that was strongly peaked for small angles of the emitted particles (Bu5l). This theory treated the interaction as a perturbation of plane waves, while the distorted wave Born approximation (DWBA) involves the perturbation of elastically scattered waves. The predicted DWBA cross-sections are similar to those of the plane wave theory and are more consistent with experimental results. 3.1 Analogies in the Distorted and Plans Wave Formalisms 3.1.1 The Plane Wave Born Approximation Consider the simple case of scattering from a 31 32 potential well V(?). The exact asymptotic solution for the total wave function in the center of momentum system is -L A ‘ ’o-H ..L, ..., Jet? YCIZIE-eéwt— gig—2.58,; lrVn‘") ”1”(I,Y‘),(r]% Eq. 3.1 where,/‘ is the reduced mass of the scattered particle. The Born approximation consists of setting e 4’ 2.x! A wi‘r 7( If N 8 Eq. 3.2 in the integral of Eq. 3.1. Defining the quantity in brackets as -f(6), we then have .A A ,Ltr _, slim ‘3:— 79(1) 0") e + {(9 ( Eq. 3.3 where f(9) is the plane wave scattering amplitude. The angular dependence is determined by the momentum transfer :'-i It can then be shown that the differential cross- section is given by ‘L 21199): /F(9)/ om Eq. 3.4 In this case, the potential acted as a perturbation to an incoming plane wave to produce outgoing spherical waves, i.e., it was an interaction between plane wave States. To predict the cross—section, one must evaluate 33 f(60 for some specified V(?). 3.1.2 The Distorted Wave Born Approximation The total wave function for the reaction A(d,p)B is given asymptotically by* H3 Lek (ream-+45 ”Va ‘injfle 17$ rP X) Eq. 3.5 where, in the exact case, the transition amplitude po is ,(gujaJZPYIPRJIVP; WV Mia-g)» 1502,34) Eq. 3.6 The second term relates only to elastic scattering and ..h- thus vanishes for the (d,p) reaction. The vectors ki’ 51 refer to the momentum and position of particle i in the center of momentum system. The internal wave functions of the particles 7f ( 5 (1:41.) have been included; frefers to the internal coordinates of the sub-units of nucleus i and the?jk are the relative coordinates of these sub- . - '* a- + . units. The Iiw 24mm 024. ZOmEbJU>o 44 oHQm>oE one .pcoEcmHHm Emop homosm mnzmcfi 0p poms mm; kHQEommm pHHm .pmpsmso wcfipoppmom .CH mm map CH psosowsmshm ampsosfihomxm .m.: ohswfim .o, . Pa «.55 N m z/ . wt W. 33... 22.50 I-- -o- -..- o-:---|-- II. - -- --mF---a--be--I IIIII I II _n_ 50.0 \ ESP FF \ / . \ a ..o. \ £36.84 \ ‘ Em 03262 \ scam...“ 5.2 .2500 “H w I 00 \ .|\_ c u x T .23» 2 2205.382. Zwkm>m KMHZDOU 024 mum—2410 oz_mw._.._.23 32.3 .3222. o... u n .0500 MOB—LO m ...-"3)" - fl _ .3 _u..< .. _ n .200 «SE ~.<.o.m “ >ooo_+ “ cow E< " 033a a -- u :0... :85 u .32.... m I _ >o_oo . --- .o._:o“ " .0! coo aE< . _ .2 oc::F _nEom " _r .22.. 22:8 t. w a-- u A >o_oo .o.=:oo . _ u< . aE< 3.6380 .8on 23 2:33 aE< v n _ .35.. .o ozm 69.: 4 9:35 xNN , v .0500 4 03.35 >ow+ sci: mm . i: - h Aconov_r<.o.w _ .oacaoo m aE< .ocso 385 200 aE< aE< -b 9502 xNN DA .85.. .28 I .85.. d .326 >+ 354. m 22.... rIIII mo_zom._.om-_m 49 energy AF in the AE-counter and has total energy E+AE, it can be shown that (0064): T/a = (E+AE)1°73 - (AE)1-73 Eq. 4.2 where T is the thickness of the AE counter. The quan- tity T/a, the value of which defines a particular part- icle, is determined by the identifier as explained below. After the E and AE pulses were properly shaped in the amplifiers they were fed into the mixer unit of the particle identification system (Fig. 4.3). A coincidence between the detectors was also required within the ampli- fier systems to reduce the pulse rates entering the identifier. The height of the output pulse from the mixer is proportional to E for about the first 1.5‘/usec and proportional to the total particle energy (E+AE) for an additional 1.5 /usec, i.e., the pulse has the shape of a step function. This signal from the mixer is fed into the function generator, where the entire wave form is raised to the 1.73 power. The sample amplifier picks out only the height of the step in this wave form, which is proportional to T/a in Eq. 4.2. The timing generator gets its start signal from the coincidence circuit in the amplifiers and stops with the baseline crossover of a doubly differentiated AE pulse. This provides the 1.5 ,flsec delay to produce the step in the wave form, while synchronizing signals are sent to the function generator and sample amplifier. 50 The "mass" output (T/a) of the identifier was fed into the 4096 channel analyzer and was displayed on an oscilloscope as the horizontal axis of a 64x64 array. The energy signal was displayed on the verical axis. This resulted in a contour plot of the counts per channel for each type of particle, with easily distinguishable groups of protons, deuterons, tritons, etc. An intense group appeared as a tail on the low mass side of the proton group. This corresponded to the elastically scattered 33.6 MeV protons, which were not stopped by the-detectors. The deuteron selection was made by set- ting a gate on the T/a signal with a single channel analyzer (SCA2 in Fig. 4.3) such that only pulses corres- ponding to the deuteron group were available to satisfy the coincidence requirement with the energy signal. In order to be certain that all of the deuterons were incl- uded in the gate, a very low background of protons and tritons was allowed in some cases. After the gate was set, a one dimensional deuteron energy spectrum was recorded and stored in each of four 1024 channel groups. The memory was then punched out on paper tape and print- ed on photographs by an Optical readout system. An overall electronic noise width of about 50 keV was attained with both detectors under bias, while the overall deuteron energy resolution was typically be- tween 95 and 130 keV. 51 (b) Proton Monitor A block diagram of the circuit for the proton moni- tor is also shown in Fig. 4.3. The pulses from the photo multiplier tube were preamplified and fed into a linear amplifier. A discriminator level was set such that only pulses corresponding to the elastically scattered pro- tons reached the scaler to be recorded. This setting was made by displaying the amplifier output on an oscillos- cope (CRT) which was triggered by the output of the discriminator. The ratio of the number of counts record— ed (times the sine of the target angle) to the charge indicated by the current integrator remained constant to within a few percent for all data points. 4.2 Lithium-Drifted Silicon Detectors Most of the lithium-drifted detectors used for these experiments were commercially purchased, although some of the preliminary data was taken with counters made in the laboratory. The method by which these detectors were constructed is discussed below. A.P-type silicon wafer of about 2 cm. diameter and 5 mm. thickness was lapped until smooth with 1000 grit lapping compound. Lithium was evaporated to one surface, and diffused into the surface by heating the wafer at a temperature of about 400°C. for approximately ten min- utes. The wafer was then etched for about ten minutes in a solution of 3:1 HIO :HF, and rinsed thoroughly with 3 52 de-ionized water. The diffused lithium formed an N— type surface to which a positive potential ofIV500 V. was applied. This caused lithium ions to drift through the crystal, forming an intrinsic region. With a power dissipation in the crystal of.~1 watt, about three weeks were required to drift to a depth of 3 mm. The depth was checked periodically with a staining solution that deposited copper only on the intrinsic region. After the drifting process was completed, the wafer was cut into a shape similar to that shown in Fig. 4.4 (a). All surfaces were lapped and etched, and gold was evaporated to the surfaces indicated in the diagram. This provided a good electrical contact between the P—type and intrinsic regions. The crystal was then mounted in a capsule consisting of a brass case with a lucite filler as shown in Fig. 4.4(b). A spring-loaded electrical contact was inserted at the rear of the capsule, and a 0.00025 in. Mylar window was pla— ced over the front aperture to keep the crystal free from dust. A copper block was designed to serve both as a mount for the capsule and a cooling device. Although several of these counters failed to work satisfactorily, an overall energy resolution of 100 - 130 keV with elastically scattered 25 MeV protons was obtained with some of these devices. This resolution is comparable to that obtained under similar conditions 53 Detector and Packaging System (Approximately to Scale) x I Gold Plated Surface 0 I P Type Reqlon (o) Detector Geometry (b) Detector Capsule and Mount Coollng Tube Brace Cece . 4 0-975 Gold Plate 2&4 . 1.9! I/4 mll mylar ' HV 7 ' Contact Coollnq Block Emm m (Brace Case) Figure 4.4. Geometry and package system for lithium-drifted Silicon detectors. 54 with the commercially purchased counters. The major difficulty encountered was that the quality of per- formance decreased rapidly with time, even without large exposures to radiation. 4.3 Targets The nuclear bombardment targets used for each ex- periment in this study are described below. Most of the nuclei have a high natural isotopic abundance, and were therefore fabricated in the laboratmry from ord- inary materials. 4.3.1 24Mg The target used for the 24Mg(p,d)23Mg experiment was a commercially purchased rolled foil of 1.07 mg/cm2 which was enriched to >99% 24Mg. An oxygen contaminant corresponding tovv0.02 mg/cm2 was observed in the (p,d) spectra (Chapter 5). 28 4.3.2 Si Chemically pure $10 of natural isotopic abundance (92.2% 288i) was evaporated from a tantalum boat to a 0.0001 in. nickel backing. The nickel was cooled dur- ing the evaporation by passing cold water through a copper block, which decreased the expansion of the nickel caused by heat from the boat. This therefore decreased the contraction of the nickel when the boat 55 was cooled, and prevented the extremely brittle SiO layer from cracking. A 2 cm2 area of the nickel was then dissolved with HN03, exposing both sides of the SiO layer. A thickness of 0.88:0.04 mg/cm2 was obtain- ed from the weights and areas of several pieces of the foil. 4.3.3 33% A gas target containing H23 was used to study the 32S(p,d)3lS reaction. The gas cell consisted of a 5 in. diameter by l in. high 00pper cylinder having 0.001 in. Kapton walls. The cell was filled to a pressure of 45 cm Hg with natural H28 (95.0% 32S). This pres- sure corresponds to an equivalent target thickness of .vl mg/cm2 at a counter angle of'30o with the colli- mators described previously (Sec. 4.1.2). The pressure was monitored continuously during the experimental runs with a mercury manometer. 4.3.4 3bAr A leak-proof cell was constructed for the purpose of making a permanent, isotopically enriched.(>99%) 36Ar gas target. A sketch of the cell is shown in Fig. 4.5. The frame was milled from a single piece of brass, and a Havar window of 10 mg/cm2 thickness (250.0005 in.) was attached with epoxy cement. A compound pressure gauge, which was readable to within an error 1 1 cm of Hg, was 56 GAS CELL FOR 3"5.4-r Protective Cover isthmus swam we" l-lovor Foll (~l/2 rnll) Broee Frame ( l piece l Valve Preeeure Gouge Hovor Foil Supp", VOW. Poet ‘- 7. c‘" Mount (with actual \\ " //’r Gouge arrangement ‘— ’ ot componente) 36 Figure 4.5. Schematic diagram of cell for Ar gas target. 57 mounted on the bottom of the cell. A helium leak-tested Circle Seal valve was also included. The cell was filled with 36Ar to a pressure of 45.1 i 1.0 cm. of Hg at 25° C., which corresponded to a 1 mg/cm2 thick— ness at a laboratory angle of 30°. The strength of the 0.5 mil Havar window proved to be sufficient to withstand the above pressure and the flexing encountered when the cell was moved from atmo- spheric pressure to the evacuated scattering chamber and back again. The target has existed for a period of eight months with no detectable change in pressure. 4.3.5 400a 40Ca) were used for Natural calcium foils (97.0% the study of the 4OCa(p,d)39Ca reaction. These targets were made by evaporating calcium metal on a 0.001 in. tantalum foil. The heat from the evaporation process caused the tantalum to expand and, when cooled, the two foils separated due to their different coefficients of expansion. Calcium foils as thin as-l mg/cm2 could be obtained with this method. Experimental data was obtained with foils of 1.10, 1.67 and 2.27 mg/cm2 thick— ness. 4.4 Data Analysis The deuteron energy spectra were stored in 1024- channel groups of the 4096 channel analyzer as mentioned earlier. After every four data points, the memory was 58 punched on paper tape and printed on photographs by the .optical readout system. The information from the tape was then transferred to computer cards, and plots of the spectra and listings of the counts per channel were obtained through the use of a computer program. The photographic prints permitted some preliminary data analysis while the run was in progress, which helped to determine if the experiment was proceeding normally. 4.4.1 Determination of Excitation Energies The ground state (P-d) reaction Q-values for the 2881(p.d) 2781 reaction to -12.86 MeV for the 32S(p,d)3lS reaction. d)ll nuclei studied ranged from ~14.95 MeV for the The Q-values for the l60(p,d)150 and 12C(p, C reactions are in the same region (-l3.44 MeV and -16.50 Meu.respec- tively) and the 150 and 11 C energy levels are well known (La6l). A (p,d) spectrum from a Mylar target was there- fore an ideal calibration device. Figure 4.6 shows a Mylar (p,d) spectrum obtained from a 0.00025 in. foil at a laboratory angle of 30°. 15 The deuteron energies for the levels excited in 0 and 11 C in the (p,d) reaction were determined from rel- ativistic kinematics by a computer program and plotted versus channel number. This effectively served to cali- brate the electronics and counter system. All the points on this plot fell within 1 10 keV of a straight line, except that on some occasions the 150 ground state appear— ed.at a channel number which was too low. This was OON 59 .UomeHUcH ohm COHpomoa onHAU.QV 6cm o AU.QVQ 03p cfi Uopfioxo mHo>oq oma HH ma .pompmp swam: :mmooo.o m Eosm hm.pm coxmp Esppoon soapmspfiamo cosopsom .m.: Tasmam mmmsSz nuzz3. 0072.13... m m / r l ( too. 3 H V N 9... N 00M u m 3 r ... Om_ _I- i"; (0) o... 3.30.. Pumas. Eq.-E); _ 0.. 8.30.. CON 60 possibly due to a lack of charge collection efficiency in the detector for the higher energy deuterons. In such cases, the deuteron energy from the ground state of the nucleus under study was plotted for several consecutive laboratory angles to determine the shape of the curve in this region. The deuteron energies corresponding to the excited levels in the residual nucleus were determined from the curve by their posi- tions in the deuteron spectrum, and the excitation energies were obtained from a kinematics calculation. 4.4.2 Measurement of Differential Cross-Section Deuteron energy spectra were obtained for labor- atory angles from 10° to 155° for each residual nucleus. The total number of events (deuterons) in each peak was determined for each angle, and a reasonable back- ground was subtracted. The differential cross-section was then calculated from the formula J6'__ M _ “Dumb. dJL rah/:49— '3" Eq. 4.3 where N1 is the net number of counts in the peak, n is the number of target nuclei per cmz, N2 is the number of incident protons and.451is the solid angle. The quantity N2 is calculated from the total charge obtained with the current integrator. For foil targets, 61 the solid angle is defined by the collimator directly in front of the counter telescope (C2 in Fig. 4.2), and n is determined by the equivalent target thickness exposed to the beam (which depends on the target angle 9T)’ In this case, Eq. 4.3 can be written in the form G£552= hso cowpmspfifiwo 32.5 ....m V m N _ - q _ 1 890... m: 2392?. 14... $80on v.02: . .8534. .2028 Exam .2ccEtmaxm By 9230 5:05:00 A3 F2m2mm3m0mwzm 2.4mm O .1 09.0» Non mdm V0» 0 .0m mdm Non .w.: oaswfim 65 on the counter arm in the scattering chamber and bombard- ing a polystyrene (CH) target with protons from the cyclotron (Fig. 4.7 (b)). The energy (Ep") of protons elastically scattered from 12 C was measured after they had passed through the absorber, and Ep' was determined from the calibration curve (Fig. 4.7 (a)). The energy of the incident beam (Ep) was then determined from the appropriate kinematics calculation for a given scatter- ing angle. The results obtained from this measurement varied between 33.6 and 33.8 MeV for different experi- ments, with an estimated error of i 200 keV for each measurement. The error was due mostly to uncertain- ties in the scattering angle (i0.5°) and the position of the broad peak in the Ep" spectrum. Due to the lower value of 33.4 MeV obtained by the cross-over method, the proton energy for all the experiments in this study is estimated to be 33.6 i 0.2 MeV. 4.6 Estimate of Errors 4.6.1 Measurement of Energy Levels The energies of the excited levels for the residual nuclei in the (p,d) reaction were measured by assuming 11C and 15O as standards as described the levels of earlier (Sec. 4.4.1). The only exception to this was in the measurement of the levels of 313, where the gas cell was filled with air to approximately the same pres- sure as the H23 gas used in the experiment (45 cm of Hg), 66 and the calibration levels were those of 15O and 13N. The position of the peak in the deuteron spectrum for the ground state of the nucleus being studied was used as a reference point for determining the excitation energies. This point was chosen at the same laboratory angle that the calibration spectrum was taken, with an estimated error of :5 keV. The error in the calibra- tion curve due to the :O.5° uncertainty in the counter angle was therefore negligible, since the spacing of levels in the deuteron spectrum is a very slowly vary— ing function of angle. The uncertainty in the slope of the calibration curve is estimated to be 0.05 keV/chan- nel, which corresponds to an error of about :10 keV at 4 MeV of excitation. The excitation energy for each level was determined from the spectra at a number of laboratory angles, and the average of these values cor- responds to the result quoted in Chapter 5. The standard deviation for each set of measurements was determined from the formula .— 2.. 6:; ==4azg7.1/é; (Ea? £5“) Eq. 4.8 where the Ex. are the measured values, F; is the average value and N is the number of measurements. The quantity (E reflects the errors due to choosing the postions of the peaks in the spectra and the uncertainty in the lab- oratory angle for angles other than that corresponding to 57 the calibration spectrum. The sources of error quoted in Chapter 5 for levels of 23 31 Mg, S and 39Ca can be summarized approximately as follows (typical case): SlOpe of curve: 10 keV (Ex 4 MeV) 6.8. Peak position: 5 keV (57%: 10 keV (5 measurements) e :t (Ia)‘-.L(‘3)’+(/ci)2 = i 15 kev~i 20 keV m, Eq. 4.9 28 (1)27 In the Si(p, Si experiment, the constant pre- sence of the well-known 150 levels in the spectrum, in addition to the usual calibration spectrum from the Mylar target, enabled the measurement of 27Si energy levels with an error of i 10 keV in many cases. The positions of the ground and 6.16 MeV levels of 150 were plotted versus deuteron energy for several laboratory angles. Since these two levels occur at opposite ends of the 27Si deuteron spectrum, the slope in the region of interest was well determined in this way and the corresponding error in energy measurement was reduced. Also, the over- all resolution of 95 - 100 keV permitted an accurate positioning of the levels at a relatively large number of angles, thereby decreasing the value of6; to ~5 keV. For all of the above experiments the target used in the investigation was similar enough to the respective calibration target so that no corrections were necessary 68 to account for differences in energy loss. Such was not 36 the case for the Ar(p,d)35Ar experiment, however, where the calibration was made with a Mylar target and the 36Ar target was the gas cell described previously (Sec. 4.3.4). Corrections were made for the proton and deuteron energy losses in the 10 mg/cm2 havar windows of the cell and in the gas itself. These corrections are slowly varying functions of particle energy and are summarized below: 33.6 MeV Proton loss in Havar 114 keV 33.5 MeV Proton loss in 3°Ar 36Ar 44 keV Deuteron loss in 110 - 148 keV Deuteron loss in Havar 287 — 380 keV Here, the corrections corresponding to the 35Ar ground state and 6.82 MeV level are shown for the deuterons. Each correction is estimated to be accurate within :5 keV, with some of the error due to the 1200 keV uncer- tainty in bombarding energy. This results in another :10 keV error to be added in quadrature to etot. in Eq. 4.9. 4.6.2 Energyfiesolution The overall deuteron energy resolution varied be- tween 95 and 130 keV during this investigation. The major contributors to the energy width are listed below 28Si(p,d)27Si reaction: for the Electronic and counter noise 50 keV Target thickness 25 keV (0.88 mg/cm2 SiO) 69 kinematic broadening 21 keV (AB =0.92°) Other 74 keV —\ 1/250)2 + (25)2 + (21)2 + (74)2 =95 keV Eq. 4.10 The electronic noise and kinematic broadening contri- buted similarly to the energy spread in the other foil targets, while an additonal width due to straggling in the gas and windows of the gas cells should be added in quadrature for the 32S(p,d)3lS and 36Ar(p,d)35Ar experi- ments. These values are approximately 25 keV and 50 keV 328 and 36Ar, respectively. The quantity labelled for "Other" in Eq. 4.10 includes the energy spread involved in the charge collection in the detectors and the reso- lution of the incident beam, as well as other unknown sources. 4.6.3 Cross-Section Normalization The errors involved in determining the absolute differential cross-sections for the foil targets are estimated as shown-below: Target thickness measurement: 2 - 5% Measurement of solid angle: 2% Current integration: 1% Target angle error (i2°): 2% Counter angle error (10.5°): 3% This results in errors of approximately t5%, t7% and 70 i5% for the normalization of the data from the 24 Mg, 810 and Ca targets, respectively, in addition to the statis- tical errors shown on the curves in Chapter 5. The estimated errors in target thickness are assumed to be mostly due to nonuniformities in the targets. The error due to uncertainty in the counter angle arises from the slope of the differential cross-section versus angle, and the above number corresponds to a typical case for an,£L=2 angular distribution. For the experiments involving gas targets, the errors in target thickness depend on the uncertainties in measurement of the pressure and temperature of the gas in the cell and the counter angle. These errors are summarized below for the 36Ar and H23 gas targets at GLAB=30 2‘- 0.5°: 36 112.8. Ar Temperature: rvl% ’Ul% Pressure: 1% 2% Counter Angle 1 . (target thickness): 2% 2% Measurement of solid angle: 3% 3% Current integration: 1% 1% Counter angle . . (slope of cross-section): 3% 3% The result is an error of approximately : 5% for both the 32S(p,d)313 and 36Ar(p,d)35Ar reactions, again ex- cluding the statistical error. Chapter 5 Experimental Results In this chapter the deuteron spectra and angular distributions obtained from the (p,d) reactions on 24Mg, 2881, 16 0, 328, 36Ar and 40Ca are discussed. Since all of these targets have Jfla0+ in the ground state, the spin and parity of the final state is always equal to that of the picked up neutron. The curves on the angular distributions represent only the general trend of the data, while the error bars refer only to statistics. Properties of the.[n=2 J-dependence are also described. Since evidence exists that most of these nuclei are deformed (St65), a Nilsson diagram (N155) show- ing single particle energies in a deformed potential well is presented in Fig. 5.1. This model was dis— cussed in more detail in Chapter 2. 5.1 24Mg(p.d)23Ma 5.1.1 Results and Interpretation Typical deuteron spectra from the (p,d) reaction on an enriched (>99%) 24Mg target are shown in Figs. 5.2 and 5.3 for laboratory angles 300 and 900, respectively. Several strongly excited levels of 23Mg and levels of 150 at 0.00 and 6.16 MeV of 71 72 Ho>oa Am\HmHV .m\anm pmsflm one mmcoscsm: or» men mco.oaa gap 6 .Ammazv Hams awesomov m CH mHo>oH oaoaohma maacflm Lo c. w w w o .N- ...- m- m. to- ~..o . o . «.0... 1...? 83-2. «he .mm.~ la . Toaumxm gm room H_o.T~\_\ .mwd 9373. /v m .OO.m c .meB «an m .mm.m H; _MDLTmW\_ \JW 1 . 6mm”— ..Nx. «\niul m Om m WON—...an 0 105m 837$. ‘ ._ WON—+3» / , n «b. anon—-35 : __ 2.3.... .. .CZOSm pOC mH Hoe cmo Hm>ofi comm smegmau Commfiaz .H.m masmflm 51.3; m4m>mq mnofikdi mqoza zowmiz 73 .o. O a. o q . - W % . _ - m I .h I a. t a n A . 8521. m .NB 9.0/ - .NR 86 4H,... .... . - . . .01, a 1 e O R r . .0 mum.) .... em... -..... ..tnhn . . uAN\n.N\:s_.~u.N|-o~udhwwwi|lnt W.“ W - n¢.0/ I 000.0 - Ir . )e - - on.» #33 .~\: 22»! U d s 8.0.. ~ Sch N 3w mom .7 \_ Nuns? - .o /\ [surly ..~h.~.mn_.owww./ L mg 5.8? l. ...RHHHI... :..nl\. ...Nm Wm . 2......) .1 -. $.13: moolx. . e s 0:14.114 N h Nnvlbpl.!ut m A t am. 1.. o g 5 H m g «flu/.906 W M V C mm M V momlnmx. e on. a 3 e .. .N....:.o.. a M mm a: M ......s ) a . I. . 1. d . "o. la. 6 o AON\n §N\nv ”mm o m WW. 0.! 60 00 .IoWOoo’o p . O t t 3 .. ”o. 3 3 c n I.\ 9 ... a. . ( 3 $\. 3 8 e 0 g 3 .\ m. .. .. ...... M .. ... ‘ p u n 4 Ep el— 2 E e n e 2 . . . O m 00 s L . 0 no 0 .u no nun" eww Au 0. .v o. 0%“ O O 4 as m m .mmw w w .mh m JMZZoz m.m cam h.m pm mason opfimoQEoo map hoe mcoflpsnfispmflv one one Eosm mzmm mo mfio>ofi >oz mo.© ocm mb.m .Hn.m esp pom mcofipsnfippmflo smfizmcm comopsom .Edm on. ow. on. Om. 0.0. pm pm 0..» pm 0 .COHpomom wzmon.va2¢m .w.m omsmflm “0.0 - a >22 mmfi ...mh.» o >22 NEN INN x 79 -.~\m.~\:>oz New I... -8233 >22 in .. IAN\m.N\: >m2 _~u.N I m_m>m1_ _ u e». >22 mom .. .m oznuAndvoEen 1 1111111 1 1 1111111 I _.0 .39: 5.0 d Ammo 0.. O. 80 5.1.2 J-dependence J-dependence for the 24Mg(p,d)23Mg reaction is observed in the angular distributions for the ground (3/2+) and 0.45 MeV levels of 23Mg (Fig. 5.4). The 5/2+ distribution has a steeper overall slope versus angle than the 3/2+ distribution, while the forward maximum for J=5/2 seems to occur at a slightly smaller angle than for J=3/2. Although the J-dependence here seems relatively weak, the effects are generally Oppo- site to those observed for most of the other nuclei investigated in this study. 5.1.3 awn—arr The results obtained from previous work on the A=23 mirror nuclei are summarized in the level diagram (a) of Fig. 5.8 (Br62, 1365), while the results fran this experiment are shown in Fig. 5.8 (b). Errors in the energy measurements are indicated wherever the agreement with Fig. 5.8 (a) is not exact. In addition to the levels shown here, several othershave been observed recently in 24Mg(3He;X)23Mg experiments (J066, Du66, Ha67).. The energies for four of the levels are in agreement with those found by Ref. Ha67 at 4.362, 5.286, 5.7 (doublet) and 5.986 MeV, respectively. In general, there appears to be extensive config- uration mixing of shell model states, and even some .1, 1 81 23 Mg Level Diagram ML (axis/2“) “gm-*(I/ZJ/ZI- W Mme/21* AL ' ’2 4.37* 0.03 4 I [2" - _ .191/5/2 .19]. 4579 - - - —&—fi§;§. Wis—i. (v2.3/2) 2,9 312, 5/2)‘ 11:23—94 3/2, 5/2 - _2.77 - A 1/2“ 7/2‘ .2.Q§_*.Q.OZ__.7,2+ 0.450l 25/2" {I EAR 3/2+ (a) Figure 5.8. 23Mg level diagram. 9.1L 45/2“ Géi r 4.3/2* (b) (a) Results from previous work on A=23 (b) Results from the present experiment; the heavy dots indicate the levels for which angular distributions were measured. nuClei. Errors in energv measurement are given wherever the agreement with (a) is not 82 mixing of Nilsson orbits, in the ground state wave function of 24Mg. This is not surprising, since the deformation of 24Mg is believed to be large (St65). The level order of 23Mg, at least for low excitations, seems to be consistent with a rotational model and a prolate deformation. 5.2 2881(p.d)27Si 5.2.1 Results and Interpretation Typical deuteron spectra from the (p,d) reaction on the 810 target are shown in Figs. 5.9 and 5.10 for laboratory angles 250 and 80°, respectively. Several excited levels of 2781 and 150 were observed with an overall energy resolution of 95 - 100 keV. For emis- sion angles eLAfi'75' the deuterons from the 150 ground state had energy less than or equal to those from the 2781 ground state. The differential cross-sections for the 27Si levels were obtained at the kinematic cross-over angles by interpolating both the oxygen and silicon angular distributions. (a) Positive Parity Levels 28Si(p.d)27Si Six deuteron groups from the reaction are observed to arise mainly from an.£n=2 neutron pick-up (Figs. 5.11 and 5.12). Spin and parity assignments of 5/2+, 3/2+ and 5/2+ for excitation energies 0.00, 0.952 and 2.647 MeV, respectively, are 400 20 . 27 . I SI (p.d) SI S 6,833.6 MeV 2 + . 52 N «I [d 300-'19LA81325P g; B E: — z: '/ 8 E! 2: 15 o g L IA 3‘1" + /% o zoos g ggg g ’23 +1 .. \ s a as 9+ 3- 0) ,g \ - \~\ «12! ~.e F 3 -1 2 : 2: an ”R I z 6 9° :2” 238't~~d 8100s 3 9'3 seam-eh‘ifal .. 0 0 0000 CDIOIO ¢¢8T2~NOA . I III) -- I-l 0 . L: 1 fig 1 II 250 500 750 IOOO CHANNEL NUMBER Figure 5.9. Deuteron spectrum from the 28Si(p,d)27Si reaction 27 . Si at GLAB=25°. A Si0 target was used, and levels of 15O and are indicated. 500 28 27 l I SinJi) SH :3 a. .2 0 a 9. " 3400- E, 33.6 MeV ‘9 90 8. _ g o - o “2" Gus 80 V J I: . ‘1 300'- , s I: L) ‘7 m m m {2 200- :3» “l - => 3'33 :33“ vs at O 559 1095.2 8.0.- 02:.» g) KHJ~ ;?g¥3 ¢%9 _ ‘ H» II\ III I ' ;. ‘07:“. MM} ‘ 0L..df*lu’ '? 1 ° I 250 . 500 750 IOOO CHANNEL NUMBER Figure 5.10. Deuteron Spectrum from the 2831(p,d)2731 reaction at eLAB=80 (SiO target). 81+ I0.0 283i (p.d) ”Si 1" '\ E,=33.6 MeV '\ if 2 Levels I.O I- .\ __ IH*\I\ \\ filiugdlglm I Q \ \ e f .’ (do '°° I “HF“. \" ‘ do cm. _ “A \,.§_§/k§\§ .. "lb/5' \. E, = 0.952 MeV \.~e..,.‘ EX: 2.6 47 MeV "x 4": 5/2+ O.| — S\ _. O \§—§~§\ 1 \H 5.: 2.90 MeV .1'= (was/2)" (2 Levels) l l l l l l l o 20 40 so so I00 120 I40 160 ISO 6 OJ“. Figure 5.11. Deuteron angular distributionstor the 0.00, 0.952, 2.647 and 2.90 MeV levels of 27Si from the 2 Si(p,d)27Si reaction. 85 .COHpomos Hmsonsavfimmm map soam Hmwm mo mHo>oH >oz m:m.m Ucm mnm.: one mom mCOHpsnfiapmHU amasmcm coaopsom .mH.m oasmfim on. on. oe. ON. 00. on on oe on o . . . . 4 q . s . q . . d n a . _ a .00 .AN\m.~\BuL. >.2mem.mu.m T /m.*lm 1 .33 .mxnv «u... r >22 33.5% / I 8 *\II* m/ : *25/m w/w *\ Lm\nF_ * 1* .. .56 a. r /m mM/ .1 ANIOIWV Im m* I VS . a £961. N u m 41,: . a 1 >223? m g 1 .. 0.. _ d _ .mss 5.. 86 obtained from corresponding levels.of known spin and parity in the 27Al mirror nucleus (En62). The 2.90 MeV excitation is known to consist of two levels which are separated by about 40 keV, one of which has been assigned 56:3/2+ (En62). Since the FWHM of this group in the spectrum (Fig. 5.9) is about 40 keV larger than that for the other levels, it appears that both levels may be strongly excited. Therefore, since the gum of their angular distributions retains the.2n=2 shape with no ambiguity, the assignment if¥(3/2, 5/2)+ may be reasonable for both levels, unless one of them is not excited by a direct process.* The angular distri- bution for the 4.275 MeV level (Fig. 5.12) indicates some admixtures from other unresolved levels having ka2, but the main contribution appears to be from the ld shell, resulting in an (3/2, 5/2)... assignment for this level and the level at 6.343 MeV excitation. This assignment has also been obtained for a level in 2881(d, 3He)27Al reaction 27A1 at 4.403 MeV from the (W167). The excitation of the 0.774 MeV level is evidence for a 291/2 admixture in the 2881 ground state. The .1n=0 angular distribution for this level is shown in Fig. 5.13, and the 1/2+ assignment is consistent with iA recent 28$i(d, jHe)27nl experiment where the mirror of this doublet is resolved has indicated that the ang- ular distribution for one of the levels is relatively isotropic (6067). IO.C I I I 87 I.o— */\§ To ...... \ mb/sr 0.| - Ep = 33.6 MeV in = 0,| Levels Ex=0.774 MeV = I/2‘ RV“ § § \gzy/ E= 4.|27 MeV 0\ J =(l/2, 3/2) '\.../ \‘b \ ,H L; E =5 233 MeV [\J'=(|/2, 3/2) NA I l J 2"3i (p,d)"Si l l 0.0 I Figure 5.13. and 5.233 MeV levels of 27 l l l l O 20 4O 60 80 4 I00 l20 I40 Gfll Si from the 28 |60 |80 Deuteron angular distributions for the 0.774, 4.127 Si(p,d)27Si reaction. 88 27A1 (En62). that for the first excited state of The fact that the 2731 ground state has its/2+ would suggest a spherical or prolate shape (520) on the basis of the Nilsson model (Fig. 5.1), but the existance of the 1/2+ level at low excitation energy (0.774 MeV) suggests a pick-up from Nilsson orbit 6 with an oblate deformation G§<0). Also, the 0.952 MeV 3/2+ level could be excited by removing a neutron from orbit 7 with either a positive or negative 8. It is thus difficult to interpret the ground state of 2881 in terms of the simple Nilsson model. This is not inconsistent with the result of a recent Hartree-Fock calculation, which suggests that the 283i nucleus undergoes shape oscillations (Mu67). (b) Negative Parity Levels The angular distributions for two.2n=l levels of 2781 at 4.127 and 5.233 MeV excitation are also shown in Fig. 5.13. The corresponding levels in the mirror nucleus have been observed by Wildenthal 2881(d, 3He)27Al proton pick-up and Newman in the reaction (W167). The predictions of Hartree-Fock calculations (Da66) and the conclusions of proton knock—out experiments (Ri65, Ja66) indicate a separa- tion energy difference between the 1p and 28-1d shells of 10 to 20 MeV in this mass region. This led to the conclusion that, because of their low 89 excitation energy, these lewels are excited mainly by a pick-up from the 2p shell (Wi67). It seems worthwhile to note, however, that in the analysis of (p, 2p) data, it is difficult to distinguish be- tween nuclear shells for which flio (R165). For example, the data corresponding to a proton knock- out from the lpl/Z shell ccnld look very similar to that for a d shell knock-out. In Fig. 5.14 5/2 the (p,d) reaction dealues to the.1L=l levels of lowest excitation are plotted for the N=Z, even- even nuclei for A=l6 - 28 (La6l and this thesis). Since the transition to the 150 ground state is due to a pick-up from the lpl/2 shell, it seems reasonable, from the observed trend in the neutron separation energy, that the.£n=l levels in the other nuclei could also be excited by a 1p pick- up. The strengths of these excitations are of little help in resolving this ambiguity, since a DWBA calculation predicts a.much lower spectro- scopic factor for a 2p pick-up than for a lp pick- up (See Chapter 6). One can conclude from the plot in Fig. 5.14, however;that the evidence is at least as strong for a lp hole state as for a 2p admixture. 5.2.2 J-dependence The angular distributions for the ground (5/2+) and 0.952 MeV levels (3/2+) (Fig. 5.11) seem to be 9O (p,d) Q-Values for N=Z Nuclei x Ground State -22 _ ‘ First .2": I LEVGI - ' Second 3,, = I Level '20 L II. J ’l” A (I/2,3/2)’ Q(p,d) -l8 - e _ MeV , _ (I/2,3/2) -|6 l- _ .. (l/2 t x * -|4 - I/z" - A -IZ -- -I “'0 l 1 l l A IS 20 24 28 TARGET 0 Ne Mg Si Figure 5.14. Plot of (p,d) reaction Q-values to £L=1 levels in N=Z nuclei versus target mass number. Straight lines are drawn for comparison. 91 very similar in shape, although there seems to be a flattening of the cross-section for the 3/2+ level for ecig. 60°. The angle at which the forward max- imum occurs in the 3/2... distribution is about the same, or slightly smaller than for J=5/2. The rela- tively poor statistics for the 3/2+ level make this judgment difficult. 5.2.3 Summary The excitation energies, spins and parities of 28$i(p,d)27Si reaction are the levels excited in the shown in Fig. 5.15. The energy measurements for the first six excited levels are in agreement with Ref. En62. Due to the constant presence of well-known 150 levels in the spectrum, energy measurements with :10 keV error were possible in many cases. The level order of 2751 is difficult to inter— pret on the basis of a simple rotational model, especially if the ground state transition is included. The large number of levels excited by the direct pick-up process indicates that the configuration 28 mixing in the Si ground state is complex. 5.3 l60(p.dll50 The ground and 6.16 MeV levels of 150 are strongly excited in the (p,d) reaction on the SiO target (Figs. 5.9 and 5.10). These levels have 92 27 . . SI Level Diagram 634330.020 a (3,2 5,2). 5.523200g0 5.233to.0I5 4 W2 3,2). t + 4.I27t0.0l04°275 0'03: 8:33:33" 2L9§2 + 2.5.7.99“, _. (333'5’2’ QGHODIO a 0.9239439 + 9174:00I0 4 3:. G—'$' 4 512* l 2"Si - I495T2°S i+ p-d Figure 5.15. 27Si levels observed in the 28Si(p,d)27Si reaction. The heavy dots indicate levels for which angular distributions were measured 93 Spins and parities 1/2_ and 3/2- reSpectively, corre- Sponding to 1pl/2 and lp3/2 neutron pick-ups. The J- dependence in their angular distributions (Fig. 5.16) l6Q(p,d)150 reaction is similar to that observed in the with 35 and 40 MeV protons (Gr66). There is generally more structure in the J=(L-l/2 distribution than for sza+l/2, which is also the case for ,(n=2 pick-ups 328, 36Ar and 40Ca (See Secs. 5.4 - 5.6). The from well known 150 doublets (La6l) at excitation energies 5.2, 9.5 and 9.6 MeV are indicated in the Spectra (Figs. 5.9 and 5.10). A level at EX=10.55f0.05 MeV in 150 is also observed. 5.4 32S(p,d)3lS 5.4.1 Results and Interpretation Figures 5.17 and 5.18 show deuteron Spectra from the (p,d) reaction on the H S gas target at laboratory 2 angles 300 and 9d). The overall resolution is about 120 keV. Many of the thirty-nine levels observed in a 328(3He,0()318 experiment (A366) are accounted for, in addition to a very weak level at EX=3.05t0.02 MeV which had been observed earlier (We65, Ne63). No strongly excited 318 levels are observed for EX>7.05 MeV. The deuteron angular distributions from levels of 318 at 1.24, 2.23 and 4.09 MeV excitation energies .mwso .mom mp Uo>sompo pmgp 0p smHHEHm mH mocowcomowuh one .COHpomos omafloqmvoma one Eosm onH mo mHo>oH >02 ©H.0 was 00.0 030 now mCOHpSQHspmHU smaswcm concusom .mfi.m osswflm .Ed 0 Om. cm. 03 ON. 00_ 0m 00 ov ON 0 . . . . q q . q _ q a a a _ q q _ . .0 .\ ... I ”IN ‘sss‘llo‘vcla‘o/o/O I It . z o a, . fagv/ 5x 9: 1| IIIIA IN\m H h..- ‘ai‘o/ IL O._.E.UAnQIU.v >os_m_.mu.m .... .\ so in.a} II A IN: use xe. . Owchm UCSOLO "(a . >22 can ... ..m ,, . e h — P — b lb L b _ p — _ p p _ 0.0— 95 400 t t 32 S (p,d)!”S t. . N E ' 33.6 MeV B 2 .J ' 8 300l- o "’ ' - '2' Gun. 30 2 o‘ 2: 1 ’/ a“: ... , I 82°“ g ‘3 E ‘3 ‘ In 0 '0 m (\f v *2 a” as. :5 AV n .— 8 IOOP 3". +“‘l c) T‘°“'-E.- 7 $3.9. 0¢¢ V . . ... . 1 ¢)_.ipwfivfiwunge¢~uridwv* . 250 500 750 IOOO CHANNEL NUMBEg Figure 5.17. Deuteron spectrum from the S(p,d)3lS reaction at GLAB=30°. No strongly excited levels are observed for Ex>7'05 MeV. |5W I T 3 2 3| _’ 55(Pud) s 3 m E, - 33.6 MeV a? 8 2 ol- e - 90° ‘5 3 '0 L» g l I: c) 1 33 § \ 8 n _. U) f l l 55 50» g l - :> ' . A O "f '- .° " ‘:” (fiflkdjax ’I (I l but l ' . . 250 500 750 IOOO CHANNEL NUMBER Figure 5.18. Deuteron spectrum from the 32S(p,d)31S reaction at O GLAB‘90 ° 96 are shown in Fig. 5.19. All of these distributions correspond to an.£;=2 neutron pick-up, and the respec- tive assignments 3/2+, 5/2+ and 5/2+ are consistent with those for the corresponding levels in the mirror nucleus 31F. Angular distributions were also mea- sured for the ground, 3.29, 4.72 and 7.05 MeV levels and are shown in Fig. 5.20. The ground state corres- ponds to an.£L=0 pick-up and has the expected spin and parity of l/2+. The distribution for the 4.72 MeV level could also be due to an 11:0 pick-up, although the statistical errors are quite large. The 3.29 MeV level is excited mainly by an~XL=2 pick-up, although the angular distribution indicates the presence of other admixtures. Ajzenberg-Selove and Wiza (A366) have reported a level at 3.35910.015 MeV, which would be unresolved here. The (5/2)+ assignment is consistent with the assignment for a possible 31F at the same energy (En62). The mirror level in 7.05 MeV distribution also corresponds to an.1;=2 pick-up, indicating that dr=(3/2, 5/2)+ for this level. The first three levels of 318 are the 0.00 (l/2+), 1.24 (3/2+) and 2.23 MeV (5/2+) levels, whose spins: and parities are consistent with a rotational band based on a neutron hole in Nilsson orbit 9 with either a prolate or an oblate deformation (Fig. 5.1). This possibility will be investigated further in Chapter 6 by comparing the experimental spectroscopic factors 97 '0 t ' V V I I I l I E 328 (p,d)3ls - i E, = 33.6 MeV h 3,. = 2 Levels LCF- . T j TITII o—-o—-c - E,=2.23 MeV (g_0') .Of J'=5/2+ a m . i: . mb/sr r]! Ey- 4.09 MeV J's 5/2" r I (llit q E 5,: l.24 MeV : N153”? . \H O-Or l t 1 5 ' 5 ‘ i ‘ g 1 : I I 1 n 1 0 20 40 60 80 I00 l20 I410 I60 I80 eOJ“. Figure 5.19. Deuteron angular distributions for the 1.24, 2.23, and 4.09 MeV levels of 318 from the 32S(p,d)318 reaction. The J—dependence is similar to that observed by Ref. 0165 (a). 98 I0.0"" t I I I I I I I _\ 32s (p,d)3'S 4 /'\. Ep 3 33.6 MBV \. £n= 0.2 Levels I, _ |.0 - A |.0 - P- - 1 \t‘it \\ Ground State do- - - {4 .I's l/2"' . (-__-) \b 1%? \\ \\:\:.(|le +) \§+§ LO? =.329 MeV J' =(5/2)" - If“. \: - ‘ \ I \.‘\§ \ \ .\. r‘ all \o’“'\.. E,=7.05 MeV _ \.., .I'= (3/2, 5/2)" . §\\\§$ a t N\L§ (l‘” 1 1 i 1 1 I n L 0 20 40 60 so l00 I20 I40 |60 |80 Eian. Figure 5.20. Deuteron angular distributions for the 0.00, 3.29, 4.72 and 7.05 MeV levels of 318 from the 32S(p,d)3lS reaction. 99 with theoretical predictions of Nilsson model wave functions. 5.4.2 J-dependence The 323/2+ and 5/2+ angular distributions shown in Fig. 5.19 exhibit a very striking example of J- dependence in the 1d shell, which is similar to that observed in the (p,d) reaction with 28 MeV protons (0165 (a)). The forward maximum of the distribution for the 1.24 MeV level (3/2) in 318 occurs at a smaller angle than the distributions for either the 2.23 MeV or 4.09 MeV levels (both 5/2), and dr0ps off much more rapidly to the first minimum. The oscillatory structure is much more pronounced for J=3/2; in fact, the second maximum (0:45.) is barely noticeable in the angular distributions for the 5/2+ levels. It is evident that the 4.09 MeV level, whose distribution is also shown in Fig. 5.19, could have been assigned drs5/2+ on the basis of J-dependence alone. This is an example of the usefulness of J-dependence as a spectroscopic tool. 5.4.3 Summary Figure 5.21 shows the 318 levels indicated in the deuteron spectra (Figs. 5.17 and 5.18). The energy measurements are in agreement with the results of Ref. A366, to within about :20 keV for each level. Due to the spins and parities of the strongly lOO 3' 5 Level Diagram l '-’-—°— - (3/2,5/2)“ 604C) 599 $.17 }4 Levels 5.7.0 }5 Levels 5.42 T225 4 Mat) 4.09 ‘ 5/2'.’ .2 3+0: 0.02 ‘ ‘5’” 2.23 4 5,2+ L24 d. 3/2* is- - 4 I/2+ SIS - I 2.8 6I32$+p-d Figure 5.21. 313 levels observed in the 328(p,d)318 reaction. The heavy dots indicate levels for which angular distributions were measured. 101 excited levels, it appears that the configuration mixing in the 328 ground state is less complicated than in 24Mg and 28 Si (Secs. 5.1 and 5.2). In fact, the only confirmed admixture of sizeable strength is a [dB/212 configuration which results in the strong excitation of the 1.24 MeV 3/2+ level. The spins and parities of the ground and first two excited states are consistent with the strong coupling rota- tional model. 505 36Ar(P2d)35Ar 5.5.1 Results and Interpretation Thirteen deuteron groups, corresponding to levels in 35Ar, were observed in the 36Ar(p,d)BSAr reaction. Typical spectra from the gas target described in Chap- ter 4 are shown in Figs. 5.22 and 5.23, where the overall resolution is about 130 keV. Angular distributions were measured for ten levels of 35Ar and are shown in Figs. 5.24, 5.25 and 5.26. The.JL=2 distributions for levels at 0.00, 2.60, 2.95 and 6.82 MeV excitation (Fig. 5.24) show that they are excited by neutron pick-ups from the d3/2 and d5/2 shells. The spin assignments for the 2.60 (3/2+) and 2.95 MeV (5/2+) levels are made on the basis of the observed J-dependence discussed in Sec. 5.5.2. The ground state assignment of 3/2+ corresponds to the spin and parity of the 35Cl mirror nucleus (En62). 102 300 41 1 3“AI (no) ”AI .N Ep=33.6 MeV 5 + Q d eue'30° +3 13 .3 E 200- 3 g / , _ <1 of 03 Q I: * * 3 cu m 8 e “s: l 7 8. I [-2 $8 ‘4' 8 S S l 0 I00- 0 -J "‘ " ‘9 9 - - O 0161- ONO at O O 090.0. 10.". -. ‘9 "' 0 owmAImnv Io 'cxi -' Kl ‘l‘ H l l . . 1 OI -2. .0 t .6 - ’In 1} A ' It . I‘ 250 500 750 IOOO CHANNEL NUMBER Figure 5.22. Deuteron spectrum from the 36Ar(p,d)35Ar reaction at GLAB=30.. It is apparent that all of the 2s-ld shell hole strength has been observed. 40 T 3“Ar(p,d)3 Ar 3 g E,=33.6 MeV J 5’ d 30 heLAB: 85° 8 "‘ 2 I: ~‘ * 4 .6 (l l 5 than ‘ ‘\.zo - ”‘9‘? OI ‘ O - O l9 1} "£53 2! Fl . :3 . . -L . .Q J C) L -. .W. I . 2'. .::_.-."°. 7; 1I "l . o .. . .._ gig”.-. 250 .0 500 750 IOOO CHANNEL NUMBER Figure 5.23. Deuteron Spectrum from the 36Ar(p,d)35Ar reaction at Q BLAH—'85 . 103 '000 I ' I I I I I l ”A! (p,d) ”A? E,= 33.6 MeV ' I ‘\ if 2 Levels lllllll IIIII |.O IIIIIIII 2'“ x/ "I .../ llllllll ,I’I\ \ [A‘R - l} K! \ _ I\ Ixf/i Ground State (31) 0. '2— K1,}? \1\ J = 3/2 —I a °""' I ("K \IJ'H I - "lb/:3r ‘1’ I \I\} \I—I\ '-° 5' \ it £32.60 MeV E : \ \{_I\l J'= (3/2)’ : : i kk 1\ 3 _ .3 l . \ .. m 1%}? . \! E,=2.95 MeV O.I :- \ J'=(5/2)* -. E t {Jr} [/1 E - 6. 82 MeV - - J'= (3/2 5/2)‘ \ I/ - (ICN I I I I I I I l 0 20 40 60 80 IOO I20 I40 ISO IBO (alum Figure 5.24. Deuteron angular distributions for the 0.00, 2.60, 2.95 and 6.82 MeV levels of 35Ar from the 36Ar(p,d)35Ar reaction. Spin assignments to the 2.60 and 2.95 MeV levels are made on the basis of - 104 The spin of the 6.82 MeV level is uncertain but must be either 3/2 or 5/2. The distributions for the 5.57 and 6.01 MeV levels also correspond to an.gn=2 pick-up (Fig. 5.25) and are therefore assigned J“; (3/2. 5/2)". The angular distributions for the 1.18, 4.70, 6.62 and 3.19 MeV levels of 35Ar are shown in Fig. 5.26. The first three of these correspond to an 1L=0 pick-up and therefore have dw=l/2+. The distri- bution for the 3.19 MeV level peaks at ecfiso’ , which indicates that this level is excited by the pick-up of an£n=3 neutron. This indicates config- uration mixing with the 1f shell in the 36Ar ground state. The level is assigned Jr=7/2- since a spin of 7/2 is most probable from a shell model stand- point, and the mirror level in 35Cl is believed to have the same assignment (En62). The spins and parities are 3/2+, l/2+, (5/2+), (3/2)" and (5/2)+ for levels of 35Ar at 0.00, 1.18, 1.70, 2.60 and 2.95 MeV respectively, where the assignment for the 1.70 MeV level is assumed from the mirror level in 3501 (En62). By inspection of Fig. 5.1, this level order is consistent with that of rotational bands based on neutron pick-ups from Nilsson orbits 8 (first and third levels) and 9 (second, fourth and fifth levels) if -0.2<€<0 (oblate). The 4.70 MeV level (l/2+) could then be excited by a pick-up from orbit 6, with at least 105 _ 36 as g Ar (p,d) Ar l.0 ? yrk! Ep= 33.6 MeV 4: : f“ \, I; 2 Levels 3 I— \! -l I— \! -l \ P 1*“?! E,=5.57 MeV - K ’ RN .1": wad/2)" l.0 — — dc; : : (da)c.m. : /!_§ \I/I—‘I\I : "WN/gr, _ i;} \X\ ‘\1//1 q r / l OJ :— qu-I. E,=6.0I MeV ‘5 - I\ .I'= (3/2,5/2I’ - : N\ : _ N\ / - ' {’l\ l l " ,_ 1\/ g 0.0| l l l l l l J l 0 20 40 6o 80 mo l20 I40 I60 |80 9cm. Figure 5.25. Deuteron angular distributions for the 5.57 and 6.01 MeV levels of 35Ar from the 36Ar(p,d)35Ar reaction. 106 IQG l I I I l l l l zmAr (p,d) z”Ar E,= 33.6 MeV fl; 0, 3 Levels- IIITT 1111111 I M/-d \. Ito/ I I I llllll '70—! w, m/ I ’a/ ‘ U. ( \ H / / x I I LIIIIH \1 LI—l-kI In} \I 5‘: |.|8 MeV 1g 0.l :— \ l\ .I'= I/2* —__- (““4 5 Ni 5.324%; E mb/s r : \ 4% J'= I/ 2" Ivtfit _ ' {\I/ }\1\ l \l _ i W. W’ ' {\1 E,=4.70 MeV : ” H‘I\ \1‘ .I'= I/2+ _ / a _ i I {‘1 o" :— \{~1/ \M’Ix E : l\ E,=3.l9 MeV - : I\le’{\ ‘1': 7/2- : I- I\I\I -- \ _ {\I _ \l~ 0.0I 1 l l l I I I 4 O 20 4O 60 8O IOO l20 I40 ISO lth .m. Figure 5.26. Deuteron angular distributions for the 1.18, 3.19, 4.70 and 6.62 MeV levels of 35Ar from the 36Ar(p,d)35Ar reaction. 107 three remaining candidates (5.57, 6.01 and 6.82 MeV levels) for the 3/2+ and 5/2+ levels of this band. The excitation of a third 1/2+ level (6.62 MeV) and a 7/2- level (3.19 MeV) indicates some configuration mixing of Nilsson orbits, possibly to orbits 11 (K: 1/2) and 10 (K-7/2), respectively. 5.5.2 J—dependence The J—dependence observed in the 36Ar(p,d)35Ar reaction for 3/2+ and 5/2+ levels (Fig. 5.24) is very similar to that for the 32S(p,d)3'-LS reaction. The angular distributions for the 2.23 and 4.09 MeV levels of 318 and the 2.95 MeV level of 35Ar are practically identical in shape for ecf;.9d’ (Figs. 318 levels have 5.19 and 5.24). Since both of the J=5/2, the 2.95 MeV level of 35Ar is assigned J"; (5/2)+. The distributions for the ground and 2.60 MeV levels in 35Ar are similar to the distribution for the 318 1.24 MeV level (3/2+), although the oscillatory structure is not quite so pronounced. The existance of the 450 maximum and the relatively small angle for the forward maximum in the distribu- tion for the 2.60 MeV level results in its assignment of i“;(3/2)+. 5 .5 .3 Summary The diagram in Fig. 5.27 summarizes the infor- mation obtained about the level structure of 35Ar 108 35 . Ar Level Diagram 6.62 : 0.03M; eggs/2f M3 4 (3125/2? 5.40: 0.05 5‘5720‘034 (was/2)” 5.07: 0.04 4.70! 0.94 fl ”2+ sletoog - 2354M; 1.5%? W a (312? I 1030.0 (5’2). |.|81’Q.Qg #9 ”2+ (3.8. ' 4 3,20- 35 J, Ar ’l3.04 T36Ar+p—d Figure 5.27. 35Ar levels observed in the 36Ar(p,d)35Ar reaction. The heavy dots indicate levels for which angular distributions were measured. 109 from the (p,d) reaction. The large deuteron energy losses G~3OO keV) in the 10 mg/cm2 Havar windows of the gas cell partially contributed to the errors in measuring the level energies. The energies of the first five excited levels correspond closely to levels in the 35Cl mirror nucleus. The ordering of the 35Ar levels appears to be qualitatively consistent with an oblate deformation in the strong coupling rotational model. 5.6 4OCa(p,d)39Ca 5.6.1 Results and Interpretation The deuteron spectra shown in Figs. 5.28 and 5.29 are similar to those obtained from the 400a (p,d)39Ca reaction by Glashausser, et. al., with 27.3 MeV protons (6165 (b)). A natural calcium foil of l .10 Ing/cm2 was used where a resolution of 100 keV was obtained, and the normalization was checked with 1.67 and 2.27 mg/cm2 foils. An oxygen contam- inant of about 3% is indicated by the 150 ground and 6.16 MeV levels, but tlis interfered with the data analysis only at the forward angles (efifiB 15°). Although about 12 MeV of excitation in 39Ca is observed, there appears to be no excitation of appreciable strength beyond the 6.15 MeV level. The angular distributions for the 0.00, 5.13, 5.48 and 6.15 MeV levels shown in Fig. 5.30 llO '50 I 4- ' 4°C 39 + ‘3 an). «r * m E,‘ 33.6 MeV ‘3 ‘3 ‘3 “i g é I00 - em- 30° 2:22 ‘ o 4 4 0: la la , g I -. .-. \ 0 “on I) '10.. 1 \ ‘3 QQ‘Q‘ f2 50 :3 J ‘33" z ‘6 888 - D 2 livid 9 O l 0 o 0 2 o . . A 1 Ln—nn-‘fi—M 250 500 750 CHANN EL NUMBER Figure 5.28. Deuteron Spectrum from the 40Ca(p,d)390a reaction taken at GLAB=3d° (1.10 mg/cm2 natural calcium target). It is apparent that all of the 28—1d shell hole strength is observed. IOO I T ”0o (p,d) 39Co § .1 80 I- Ep ' 33.06 MeV ‘ E eua' 90 Z z; 1% 60 r N 0 t ‘\ '0') 4O '- 0 0'0 0 Z '1 V... G! D '0 no N‘ o 1 l ‘ 0 20 - , t 3 . . .0. o n CHANNEL NUMBER Figure 5.29. Deuteron s ectrum from the 400a(p,d)390a reaction taken at eLAB =90. (2.27 mg/cm natural Ca target). . I III I“ I." Ill|‘l‘1 ||I"l|.’ lll I0.0 I I T 4‘0 T l I I _ y.-.\ Ca (p.d) 259Ca _ \ E, = 33.6 MeV . in = 2 Levels l O - {\VI’*\,\,/{'\' A - 4 i \ of \.\ r \§ 'v '\ GROUND STATE ‘ \k& y/*\\ J'=3/2+ ‘LL N \ \ i }\ \\‘//* I.O —- i I _ k \ E, = 5. I 3 MeV (d0) " H‘} \f\{\ J" (”2” iiarcnl *\ \\P// mb/sI ‘ % \{-§\ I . 0 L *\ /H\ — k} 5,: 5.43 MeV ‘}\} .I'= (3/2, 5/2)“ a \ \H—I \{ O.| *- f\} E‘8 6.|5 MeV — .I'= (5/2)+ * \{/}“{\}V} 0.0! l L 1 l l L l I O 20 4O 60 80 IOO IZO I40 I60 IBO (2on1 Figure 5.30. Deuteron angular distributions for the 0.00, 5.13, 5.48 and 6.15 MeV levels of 390a from the 40Ca(p,d)39Ca reaction. 1- The 5.13 MeV level is assigned J =(5/2)+ on the basis of J-dependence 112 all correspond to anqfln=2 pick-up. The ground state has dfl=3/2+, which corresponds to the spin and parity of the mirror nucleus 39K. The peak differential cross-section obtained at 33.6 MeV bombarding energy was about 6.0 mb/sr, and all three normalization checks agree within the statistical error. Glashausser et. al. (6165 (b)) obtained a value of¢v3.0 mb/sr at 27.3 MeV bombarding energy, while Cavanagh et. al. (Ca64) measuredvv4.5 mb/sr for the peak cross-section with 30 MeV protons. This indicates that the absolute differential cross-section may be quite sensitive to the bombarding energy. The 5.13 MeV level is assigned Jfl;(5/2)+ on the basis of the J-dependence observed in the angular distributions (See Sec. 5.6.2). Although the angular distribution for the 5.48 MeV level has the basic IQ=2 shape, which results in the (3/2, 5/2)... assign- ment, it also appears to contain contributions from unresolved levels corresponding to different.gg values (e.g.,.flL=O). The angular distribution for the 6.15 MeV level is very similar to that for the 5.13 MeV (5/2+) level and, from shell model considerations, a strongly excited 3/2+ hole state of this excitation energy is not expected. The 6.15 MeV level is there- fore given the tentative assignment lfl;(5/2)+. Figure 5.31 shows the angular distributions from 390a levels at 2.47, 2.80 and 3.03 MeV. The K10 _' I r I ' 4°00 (o.d)3§(30 I * EID = 33.6 MeV ‘ v1.)- zn= 0,I,3 Levels \ .I'\ I0— / '\ — I V . E,=2.47Mev - é \ /"\ J'= I/2‘ (fl " o \} ‘ do cm ”3"." \* mb/sr '\ '\ E,=2.80Mev "*‘Hk J'= 7/2' ‘ \§ § \l’ \i\ II * 0.I- {\ E,=3.03 MeV _ {\ /H~, § .I'=(I/2,3/2I' #4 \4*‘I\ H 0-0'0 2'0 4b 60 8b I60 I20 I40 I60 l80 EBCJTI. Figure 5.31. Deuteron angular distributions for the 2.47, 2.80 and 3.03 MeV levels of 390a from the 40Ca(p,d)390a reaction. 114 2.47 MeV level corresponds to a neutron pick-up from the 2s shell and thus has iwzl/2+. The 1/2 shape of the distribution for the 2.80 MeV level is very similar to that for the transition to the 3.19 MeV level in 35Ar (Sec. 5.5.1), which corres- ponds to an £L=3 pick-up. Therefore, configuration mixing of the form [dB/2].2 [f7/2]2 is apparent in the 40 Ca ground state, in qualitative agreement with the results of Ref. G165 (b). The 3.03 MeV level has been variously quoted as corresponding to.eL=l andnég=2 neutron pick-ups in 4OCa(3he,oC)39Ca reactions (0165, B065), while the mirror level appears to be excited by an.1;=l pick-up in the 4OCa(d, 3He)39.K reaction (Hi67). The present results indicate thatthis level corres- ponds to an.9n=l pick-up and has Jfl;(l/2, 3/2)”, in agreement with Refs. 0165 and H167. Even though the statistics are poor, the angular distribution seems to be peaked at 95?;Jl50, while the forward maxima of the.2n=2 distributions are peaked at 20? to 25° at this bombarding energy. These results indicate that there is some configuration mixing with the 2p shell as well as the f7/2 shell in the 400a ground state. The strengths of these admixtures are given in Chapter 6. Since there is no apparent evidence for rot- ational bands at low excitation energies in the 39Ca 115 level structure, an interpretation of these levels in terms of the Nilsson model seems meaningless. The strong excitation of the 3/2+(g.s.) and l/2+ (2.47 MeV) levels is in accordance with spherical shell model predictions. 5.6.2 J—dependence The J-dependence in the angular distributions for the 0.00 (3/2+), 5.13 (5/2*) and 6.15 MeV (5/2+) levels in 390a (Fig. 5.30) is very similar to that observed in the 36Ar(p,d)35Ar and 32S(p,d)3lS reactions (Secs. 5.4.2, 5.5.2). The spin assignment J“=(5/2)+ to the 5.13 and 6.15 MeV levels was partially based on this similarity. 5.6.3 Summary A summary of the excitation energies, spins and parities of the 390a levels observed in the 4OCa(p,d) 390a reaction is shown in Fig. 5.32. The energy measurements are in agreement with those of Refs. G165 (b) and En62, with an experimental error of about 120 keV. The level order and strengths of the levels excited in the (p,d) reaction indicate that the 400a ground state is probably less deformed than the other nuclei studied. However, the presence of lf and 2p shell admixtures is also evident. 116 ”Ca Level Diagram 6"5 4(5/2? 5°48 . I3/2,5/2)* ii? 4 Ion)“ 3‘81? 4 II/2,3/2)' ' 1 7/2' 2.47 4 ”2+ (22.3- 39 4 3/2" 1 C0 -I3.4I 1:‘°Co+p-d 40 Figure 5.32. 390a levels observed in the Ca(p,d)390a reaction. Angular distributions were measured for all levels shown. 117 5.7 Summary of Experimental Results The excitation of hole states in Ze—ld shell nuclei via the (p,d) reaction has shown that the configuration mixing of shell model states is appreciable in the ground state wave functions of all the target nuclei investigated. The strong coupling rotational model seems to qualitatively explain some of the level structure of 23Mg, 315 and 35Ar. An interpretation of the level order of 27Si in terms of rotational bands is not obvious. The doubly magic 40 Ca nucleus, although apparently more spherical than the others, contains admixtures with both the 1f and 2p shells. The J-dependence observed in the forward angles of the.£;=2 angular distributions seems to follow a systematic trend through the Zs-ld shell. The forward maxima of the 3/2+ angular distributions occur at smaller angles than the maxima of the 5/2+ distributions for (p,d) reactions on 4OCa, 36Ar and 328, while the opposite effect is observed in the 24Mg(p,d)23Mg reaction. The J-dependence is slight in the 28 Si(p,d)27Si reaction. Since the quadrupole deformation changes sign in the mass region A228 (Mu67, Cr66), it appears there may be some correla- tion between J-dependence and the nature of the nuclear deformation. The effects for A>28 are similar to those observed in other pick-up and stripping 118 reactions in the 1f and 2p shells (Wh66, Sh65, Le64, 6165 (a)). The J-dependence at large angles appears to follow no definite pattern. For 9c?;.90°, all of the 5/2+ distributions are very similar in shape, while the 3/2+ distributions sometimes undergo very distinct changes from nucleus to nucleus. The use- fulness of J-dependence as a spectroscopic tool has been demonstrated by the assignment of spine to levels 13, 35Ar and 390a. in 3 The experimental results for J-dependence have been summarized by plotting the difference in positions of the forward maxima of 3/2 and 5/2 distributions versus mass number (Fig. 5.33). The vertical axis is in terms of the momentum transferred to the residual nucleus, which reduces whatever phase differences that may arise from the two levels of a given pair having different reaction Q—values. (The differential cross- section for each of these levels is plotted versus momentum transfer in Chapter 6,) The 3/2+ and 5/2+ distributions represented correspond to the level of each respective spin having the lowest excitation energy. A.DWBA analysis of J-dependence and the extraction of spectroscopic factors to provide quanti- tative information on configuration mixing is carried out in Chapter 6. .comwsmmé sow csmso ma mCHH pgmfimspm m was qczogm ma posse mo owmsfipmo c< .chm use mwpm .mmom Sosa Uocflmpoo on .W coaomesomoo as Home on» pom mozfiw> .COHospflspmfio A+m\mv +m\m m mo Eseflxma mezsou one sow commons. sopcmsoe who ma A~9mwv +hm~ .oocmoconUIh cameo osmzsom pom mpHSmos HmQCmEHQoQXo mo massesm .mm.m ossmflm m_.OI mNdI mmdl mdev 5.0+ n .xoaa4 0v mm mm mm VN < u q q u q I l m- .\ mNOd- . I 1 .VI 1 l l. l. t. o 119 , .. .. 2.24.3 02.0 ON Chapter 6 Analysis with the Distorted Wave Born Approximation and Comparison to Theory The experimental (p,d) angular distributions discussed in Chapter 5 were analyzed in the distorted wave Born approximation (DWBA) with respect to J-dependence and the calculation of spectroscopic factors. The calculations were performed with the Macefield computer code in the zero range approximation. The optical model parameters used in this analysis are discussed in Sec. 6.1; the results are presented in Secs. 6.2 and 6.3. 6.1 thical Model Parameters The optical model parameters used to generate the incident channel wave functions for the DWBA calculations were obtained from 26 )26 Ms(p.p Me and 36Ar(pvp)36Ar elastic scattering experiments at 33.6 MeV bombarding energy. A best fit to the angular distributions was obtained by varying the parameters in an optical poten- tial with surface absorption, V(r) = - V,> For)"; 4411'“; lgrfcx‘fl- Vang}? Eq. 6.1 where f(x) is the Woods-Saxon form factor: 120 121 For) = //(/ + e“) x: (MI/aye, mz mmnz e:mn< aommm wm.o No.0 om.H om.H mm.H mo.mm 0.00 mcoaopsom no.0 no.0 wH.H mH.H mH.H sm.m fi.ss mQOposm sm.o sm.o mH.H mH.H mH.H om.m m.ms i.evee i.ev . i.ec . i.even i.evot Aetsvs lemsvoe meanness amen sod nsoooeesod Hoeoz Heoaemm. ”H ngmH 125 DWBA fits to the data. These parameters are also listed in Table 1 and were used for all the targets in this work. An imaginary surface well depth (W) of 25 MeV resulted in slightly better fits than the depth of 20 MeV obtained from the elastic scattering. It was neces- sary to further increase this depth to 35 MeV in order to obtain reasonable fits to the data from the 24Mg (p,d)23Mg reaction. 6.2 DWBA Analysis of J-dependence It was observed experimentally (Chapter 5) that the J-dependence in the ’eh=2 angular distributions is quite pronounced in many cases. An attempt is made here to obtain DWBA fits to the forward maxima of the 3/2+ and 5/2+ angular distributions by using different radii for the neutron well. This procedure has been suggested by Pinkston and Satchler (Pi65). The Woods-Saxon potential V(r)= -' V/[l 1" eKF( Lfl'fnhl7] Eq. 6.3 was used, where the depth V0 is determined by the neutron binding energy. As was mentioned earlier, we were unable to include spin-orbit effects. Also, it was seen in Chap- ter 3 that a spin-orbit potential in the neutron well predicts an effect that is generally opposite to the observed J-dependence at the forward angles. The radius corresponding to the pick-up of a d5/2 126 neutron was kept constant at 3.79 F. =1.25(28)1/3 F. for 2881 all targets having A228. This effectively assumes a core that does not change in physical size. The radius parameter ron was then varied in an attempt to obtain a1r DWBA fit to the positions of the forward maximum in a J = 3/2+ angular distribution for each nucleus, and to simul- taneously fit the slope following the forward maximum. The diffuseness (an) was kept constant at 0.65 F. for all cases shown here, except for the fit to the 3/2+ distri- bution in 31$ (See Sec. 6.2.3). It was observed during this analysis that changes in diffuseness produce effects similar to changes in radius in the DWBA calculation. A pair of levels from each nucleus, one having J“; 3/2... and one 5/2+, was selected for this analysis. The experimentally observed effects are discussed in more detail in Chapter 5. The strongly excited level correspon- ding to the lowest excitation exergy was chosen for each spin, and the differential cross-sections of these levels are plotted versus the momentum transfer Lg] in Figs. 6.3 - 6.7. The 5/2+ distributions always appear at the top of the figure, and are renormalized as indicated. A momentum transfer of 300 MeV/czl.5 F-l corresponds to a center of mass angle of approximately 70?, while the forward maxima occur at sea; 20°- 25°. DWBA fits to the data, which are also shown, are discussed below. 6.2.1 400a(p,d)390a 127 Figure 6.3 shows DWBA fits to the distributions from the ground (3/2+) and 5.13 MeV (5/2+) levels of 390a. The neutron well radius parameter of 1.11 F. for the J: 5/2 calculation arises from the assumption of a constant d5/2 radius (RS/2=3°79 F.), and seems to give a reasonable fit to the data. As is shown by the dashed curve, this same radius does not result in a good fit to the forward maximum of the ground state (3/2+) distribution. A larger radius (ron=l.35 F.) predicts the oscillatory structure of the J=3/2 distribution more accurately for 1%15 1.2 F71 but it appears that the general flattening 0f the 3/2..- distribution for larger values of H.’ cannot be accounted for by a simple change in radius. 6.2.2 36Ar(p,d)35Ar The J-dependence observed in the distributions for the ground (3/2*) and 2.95 MeV (5/2+) levels of 35Ar (Fig. 6.4) is very similar to that from the 4OCa(p,d)39Ca reac- tion, and similar neutron parameters were used in the DWBA calculation with about the same degree of success. The assumed d radius of 3.79 F. corresponds to an r 5/2 on of 1.15 F., while the ro for J=3/2 (1.35 F.) is the n same as that used to fit the 39Ca ground state distribu- tion. 6.2.3 32$(p,d)318 As was described earlier (Chapter 5), the J-depend- 128 I I00 - 1 fi 4:00 (p,d)3r9Co .. \W ‘4 ./ ;/0 DUI E CD > “n :5 OD o =-|8. 54 MeV }, J= 5/2+ = I II F do (d—' Mi . ‘1 cm. i t mb/sr . Q=-l3.4l MeV } (gs) (XIO) J'= 3/2+ ‘ A 0-' 7.": I35 F — ‘ i = I.I I F ------- (F) 0.5 l.0 L5 0 I00 200 300 400 500 Figure 6. 3. DWBA fits to the 41:2 J- -dependence for the 5.13 ( 5/2+ ) and O. OO ( 3/2+ ) MeVn levels of 39Ca excited in the OCa(p, d)39Ca reaction. '0 O “Ar(p,d) 35 ' x l0) Ep = 3.36 MeV ‘ \‘LK [DVVEMA FWlS Q =-l5.99 MeV ' .I'=(5/2)* = |.|5 F |.0 \ q (51.9.) \I , do ‘~. . m {113. 9 mb/sr t/ OJ 0 =-l3. 04 MeV (gs) - J'= 3/2+ ...... = |.35F — , We": I.I5F -—-- “‘ (FT' 0.5 |.0 \u/gfi 0 ICC 200 300 400 500 MeV I‘ll , Figure 6.4.1)th fits to the 41:2 J- dependence for the 2.95 (5/2+) and 0. 00 MeV ( 129 in the 36Ar(p,d)35Arreaction. 3/2+ )n levels of 35Ar excited 130 once for the 2.23 MeV (5/2+) and 1.24 MeV (3/2+) levels of 31$ is very pronounced. The DWBA fits to the data for these levels are shown in Fig. 6.5, where it is seen that the fit to the Jfle5/2+ distribution is quite good. The 3/2+ distribution drops much more rapidly from the forward maximum than the 5/2+ data, and this is not reproduced by using the assumed d5/2 neutron well radius (ron=l.20 F.) in the DWBA calculation. The radius para- meter was increased to 1.65 F. and the diffuseness to 0.75 F. in order to fit the data for 1%]:6 0.8 F71; how- ever, this resulted in a wrong prediction for the position of the second maximum at /¥J~’0.9 F71. A calculation where r°n=l.75 F. and an=0.65 F. (not shown) fits the forward angles equally well but results in an even worse prediction for the second maximum. 28 6.2.4 Sigp,d)27Si The distributions for the ground (5/2+) and.0.95 MeV (3/2+) levels of 27Si are shown in Fig. 6.6 where it is observed that the J-dependence is relatively weak. The position of the forward maximum of the 3/2+ distribu- tion appears to occur at a slightly smaller 1%] than the 5/2+ maximum. However, the slope following the maximum is, if anything,llgg§ steep for J=3/2 than for J=5/2. These two effects compete when the neutron radius or dif- fuseness is varied in the distorted wave calculation, so no variations were made in this case. The DWBA prediction 131 I0.0 I I T I I 3’23 (ad) 3'3 ' E, = 33.6 MeV ‘ o DWBA Fits Ur\‘\ l.0 .‘ ’ Q =-|5.09 MeV ‘ \ ~#A\\ \.\ J'= 5/2+ ‘\” ’ =|.20F . (d—° ’ \.‘~ \ '°“ do)“, ‘ \ ' \ \\‘ mb/sr ) . 0.| '- . Q =-I4.l0MeV .I'= 3/2+ '0": LBS F —— = I.20F ---- 00' (FT' 0.5ll LO. I I 4L 0 ICC 200 300 4 0 500 IQI —M§V Figure 6. 5. DWBA fits to the “fin-:2 J- dependence for the 2. 23 (5/2+ ) and 1.24 MeV (3/2+ )n levels of 313 excited 32 31 in the S(p,d) 8 reaction. 132 IC)I) I I I I I 288i (p,d)ZTSi \././"\ E, = 33.6 MeV ‘ \\ [)VVEBIX Frtts '\ . o =-I4.95MeV(g.s.I I 0 _ - .I'= 5/2+ _ ' . run= l.25 F \«n. do ' .. ' d (at)... \ly‘\ \. mb/sr ‘\ A A, ‘ . 0.| - ‘ ““ _ 0 =-l5.90MeV \/\A ‘ . .I'= 3/2+ ‘5‘ ‘ . to“: I25 F AA 00' (F)"0.5n I.O L5 2.0“ 2.5” 0 I00 200 300 400 500 MeV |§1| C Figure 6.6. DWBA fits to the.1;=2 J-dependence for the 0.00 (5/2+) and 0.952 (3/2+) MeV levels of 27Si excited in the 28Si(p,d)27Si reaction. 133. is shown normalized to the data for both levels, and is in excellent agreement with the ground state (5/2+) dis- tribution. 6.2.5 24Mg(p,d)?3Mg As was seen in Chapter 5, the experimentally observed J-dependence effects in the distributions for the ground (3/2+) and 0.45 MeV (5/2+) levels of 23Mg is generally opposite to that observed in the (p,d) reaction on most of the other nuclei studied. The DWBA fits to these dis- tributions are shown in Fig. 6.7, and the calculations were made with an imaginary deuteron well depth of 35 MeV. A depth of 25 MeV, which was used for all of the other nuclei, resulted in a curve having much less structure and no relative minimum for small I%/ . Also, the d5/2 radius used here is 1.15 (24)l/3=3.32 F. instead of the 3.79 F. used for all the other nuclei. It is seen that the value for ron of 1.15 F. is too large to yield an acceptable DWBA fit to the 3/2+ distribution. This was decreased to 0.75 F. to obtain a reasonable approximation to the forward maximum and overall slope of the data. 6.2.6 Summary of J-dependence Analysis The use of different radii in the neutron form factor for d3/2 and d5/2 pick-ups was at least partially success- ful in predicting the,fL=2 J-dependence at the forward angles. It must be emphasized however, that this analysis serves only to illustrate the extent to which the radius 134 Io.o . . . . . 24Mg (p,d) 23Mg Ep = 33.6 MeV DWBA Fits Q = - I4.76 MeV '0 " .I'= 5/2+ -I .\ f0": l.l5 F xx. . q (do)cm ' mb/sr 1“ “a... 0' " \ ‘\ .. Q“l4.3lM€V(9..S)\ z"“\«‘ “ ‘ 'J'= 3/2 ’ A - r0: l.|5F — ‘ ‘ r0": 0.75 F (Fi' 0.5 I0 I. 5 2.0 2.5 0-0' 4 4r Al 4 -: 0 I00 200 300 400 500 Figure 6. 7. DWBA fits to the 1:=2 J- -dependence for the 0.45 (5/2+ ) and 0. 300 (3/2+ ) MeVn levels of 23Mg excited in the 24M g(p,d)2 3Mg reaction. 135 must be changed. The results are summarized in Table 2, where R3/2 and R5/2 correspond to ronAl/3 for d3/2 and d5/2 pick-ups respectively. The radial change is quite large for most of the targets studied and, as can be seen from the ratio 6"(R3/2)/o"(R5/2) in Table 2, a large change in radius results in a large change in the magni- tude of the calculated DWBA cross-section. This gives rise to problems in extracting absolute spectroscopic factors, as will be seen in Sec. 6.3. It is interesting to note however, that the ratio of the d3/2 radius to the d5/2 radius (R3/2/R5/2) is in general closer to unity than the ratio of the semi-axes of the nuclear ellipsoid (NV/2,8 )/R(0,8 )) if one assumes reasonable values for the deformation parameter 8 (Table 2)? The only exception to this is the case of 40Ca. The effect of using a constant radius for a d5/2 neutron is summarized in Figure 6.8, which shows a com- parison of the DWBA fits to the 5/2+ angular distributions ‘for R =3.79 F. and 1.25A.l/3 F. The fits to the dis- 5/2 tributions for all the levels are reasonable at forward angles with the smaller radius (3.79 F.), while the pre- 1/3 dictions for a radius of 1.25A F. is in less agreement with the data as A increases. The effect is not large however, and the quality of the DWBA fits for R5/2=3.79 F. ‘X’ The radii R(9,$) in Table 2 were calculated from the nuclear surface formula given in Chapter 2 (Eqs. 2.15, 2.48 and 2.51). odam> 0085mm¢ao soon .oomxn .0090: mmflsamnpo moods: mopm .mmm song was m.mo modam>flo 136 n.a >H.H onH.0I mm.a n>.m no.4 no.0 soos n.H 0m.a nnm.0u na.a ms.m oe.e no.0 eaom A>M.Hv Am>.mv mm.m m>.o 0.n oe.a nm.0I os.H mn.m nn.n no.0 mmm 0.H an.a om.0I 00.H nn.m an.m no.0 amom n.0 nn.0 Ho.0+ no.0 Nn.m oa.m no.0 mssm Q east ads No. themes flames Sasha. ... a. Hassle moaoosomoolw mo mamhaws< cmuflamauosmmfim 142 mn.o mo.a A+N\m hm\mv mo.m Ammv ¢¢.o om.o Amav so.m unm\m .N\Hv No.0 pa.o sm.o +A~\m .m\mv mm.m sooo.o +N\H oa.o +m\a sm.v Ammv mm.o ms.o s N\H um\m mo.o Andy mo.a uAm\m .N\Hv m>.m Agmv om.o mm.a s «\H nm\a sm.a Andy ov.m uAN\m .m\av H>.m am.o +N\H om.o +m\a mm.m mm.m +m\m ms.a s m\m +m\m m>.H oo.m +m\m mv.o mm.o a m\m +m\m mm.o m>.o +m\m oo.o Ilsm .uh ||.m|..ozopflnso IIMII as Imamqnn. uldmwll .ph >ms em Iwfimwfll 3.0 + "3329 Ta 34 u ofllmm sowpomom mammac.mvw2¢m on» mom whosowm oafloomohvoomm um magma 143 to that of the single level predicted by the Nilsson model. The ratio of the spectroscopic factors for the 2.35 and 4.37 MeV levels is in poor agreement with the ratio obtained from Wildenthal's calculation for levels at 1.80 and 4.81 MeV. It is also difficult to deduce the configurations of the levels at 5.32, 6.02 and 9.63 MeV on the basis of a simple rotational model, although the 5.32 MeV (3/2, 5/2)+ level could belong to a K=l/2 rotational band based on the 4.37 MeV level. Spectroscopic factors corresponding to a pick-up from the 2p shell were also calculated for the.£;=l levels and, as can be seen from Table 3, these values are smaller than those for a lp pick-up by more than a factor of four. Therefore, it is possible to obtain reasonable values for SE by assuming a lp shell hole state, a 2p admixture in the target ground state, or a combination of the two. The ambiguity discussed in Chapter 5 is therefore difficult to resolve on the basis of spectroscopic strengths. The spectroscopic factors obtained from the wave functions of Chi are generally in good agreement with the relative experimental values for low excitation energies. This assumes of course that the jL=l levels at 2.71 and 3.79 MeV are due to lp pick-ups. The con- figuration mixing giving rise to levels of higher excit- ation is not well explained by a simple Nilsson picture. The excitation of two l/2+ levels of comparable strength 144 is evidence for configuration mixing of Nilsson orbits, which could be due to some of the strength of orbit 7 n- being shared with orbit 9 (K =l/2+). 28 6.3.2 Si(p,d)273i As was seen from the experimental data (Chapter 5) the level structure of 27Si is not easily explainable in F1” terms of a conventional rotational model. The s-d shell strength should all be contained in Nilsson orbits 5, 6 and 7 if 5g +0.15 (Fig. 5.1). The spin of 5/2 for the 27Si ground state indicates a prolate or spherical shape, while the excitation of the low-lying 1/2+ level (0.952 MeV) suggests an oblate shape. As can be seen from the spectros00pic factors list- ed in Table 4, approximately half of the 2s-1d shell strength is contained in ground state (dB/2) transition. The SE of 3.45 agrees to within the estimated error of f 20% with values of 3.9 and 3.97 obtained from 28Si(d, 3He)27Al proton pick-up reactions (W167, G067). This value is too large to correSpond to a pick-up from a single Nilsson orbit, since each orbit can hold only two neutrons. It has been suggested (En62) that the 27Al mirror nucleus has a prolate deformation where the first few excited levels correspond to a rotational band based on the K=l/2 [211] Nilsson orbit (Fig. 5.1). The excitation of the correSponding levels in 2781 by a direct pick-up process (such as the (p,d) reaction) 145 28 Table 4: Spectroscopic Factors for the Si(p,d)27Si Reaction §x_$M£!l ij ‘SE (rcn=l.25 F.} 0.00 5/2+ 3.45 0.774 1/2+ 0.64 0.952 3/2+ 0.34 2.647 5/2+ 0.47 2.90 (3/2. 5/2)+ (0.81) 4.127 (1/2, 3/2)‘ 1.20 (1p) 0.2l (2p) 4.275 (3/2. 5/2)+ 0.34 5.233 (1/2. 3/2)‘ 1.67 (1p) 0.28 (2p) 6.343 (3/2. 5/2)+ 0.45 146 would then indicate configuration mixing with Nilsson orbit 9 in the 28 Si ground state. However, there seems to be no prolate value for 8‘ (>0) for which the K=l/2, [211] wave functions of Chi (Ch66) are in reasonable relative agreement with the experimental values for the 0.774 (l/2*), 0.952 (3/2”) and 2.647 (5/2”) levels (Table 4).. Since the 5/2+ assignment for the 2781 ground state is inconsistent energy-wise with any other deformation (Fig. 5.1), it appears that no simple form of the strong coupling model can.explain the results and no comparisons are made here. The large spectro— scopic factor for the ground state transition is some indication that the average deformation is probably small, and that the mixing of rotational bands is ex- 28Si nucleus tensive. It has been suggested that the undergoes shape oscillations since the energy minima for the prolate and oblate solutions are nearly equal in a Hartree-Fock calculation (Mu67). The spectroscopic factors for 3(n=2 levels at 2.647, 2.90 and 6.343 MeV are also shown in Table 4. Since the experimental data showed some indication that both levels are excited in the 2.90 MeV doublet, the meaning of the spectroscopic factor obtained in this case is not clear. DWBA calculations were made for the.2n=1 levels at 4.127 and 5.233 MeV by assuming both 1p and 2p pick-ups. As was mentioned earlier (Sec. 6.3.1), the values of the spectroscopic factors 147 are of little help in determining the origin of the trans- ferred 9n=l neutron. 6.3.3 32$(p,d)3lS The 328 nucleus should have the last two neutrons in the K=l/2 [211] orbit if the deformation is not large frof 1/2... for the 313 ground state (Fig. 5.1), and the J is consistent with this assumption. DWBA spectroscopic factors were measured for the transitions to seven levels of 31$ and are shown in Table 5. Values of 0.94 and 0.18 were obtained for the 1.24 MeV (3/2+) level with the different neutron well parameters used in the analysis of J-dependence (Sec. 6.2.3). This is an example of the ambiguity involved in extracting,fln=2 spectroscopic factors when the J-dependence is strong. It is doubtful that either of these values can be trusted; in any case the uncertainty in the strength of the [d3/2]2 admixture is large. The error for the other,£n=2 transitions is estimated to be t 20%, and is probably larger for the.£L=0 levels. The interpretation of the level order for the. ground and first two excited levels of 31$ in terms of a rotational band based on Nilsson orbit 9 would seem reasonable. However, the values of the spectro- scopic factors for these levels are too large for all of them to arise from the same Nilsson orbit. The wave functions of Chi are in agreement with experi- 148 OO.N wH.H No.0 om.o m>.o Hm.a mo.o >.N ena.o em.o ¢O.H ~36 .. umv pawoosa g ~.a om.anoaqmm .mozo .mmm Bohm vopmazoamoan Q .m mNoO N 3w ..m m©.H H Chad +Am\m .m\mv mo.e A+N\.C N>.¢ +m\m mo.e aw\mv mm.m +~\m mm.m +~\m em.a +m\H 00.0 as >62. m soaeooom mamfloeawmmm one how mkopomm camoomohpommw um manna 149 ment for the ground state strength if values for S of approximately -0.11 or +0.14 are chosen. The spectro- scopic factors predicted for 8: --0.11 are listed in Table 5, and the agreement is fair for some of the observed levels if it is assumed that the 3.29 MeV level is the 5/2+ member of the ground state rotational band. No conclusion can be drawn about the agreement for the 1.24 MeV 3/2+ level due to the pronounced J- dependence in the angular distribution. The large experimental Spectroscopic factor for the 2.23 MeV level and the theoretical prediction of a second strong- ly excited 1/2+ level (which is not observed) are indi- cations that the ground state of 32$ is not easily explainable in terms of a simple Nilsson picture. Recent Hartree-Fock calculations (Mu67, Ba66) predict that the 328 nucleus is asymmetric. 6.3.4 35Ar(p,d)35Ar The shell model predicts that the 36Ar ground state should have a [d3/2]2 configuration for the last two neutrons. The DWBA spectroscopic factors for levels excited in the 36Ar(p,d)35Ar reaction were calculated for both of the neutron well radii used in the anal- ysis of.1L=2 J-dependence (Sec. 6.2.2), and are listed in Table 6. The value corresponding to the best DWBA fit for each level is denoted by an asterisk (*), except that no decision could be made for the.2n=0 fits. 150 ¢N.H 00.N H0.0V 05.0 mm.a m>.0 00.0 NN.0 00.0 00.H mm.H AHHV AHHV o m\a m m\m o m\a o m\a Aoav s m\m m «\H s m\m m m\a m m\m m m\a w m\m .oz passe m 32.." B snow: +N\m +nxm .mxm +N\H +m\m .N\m +m\m .mxm .m\m +N\H .N\m .uh .0030 .wmm Scum msoHPmHSonoAm .eam smra enoa wsa>am weapon woe osao> x Hm.o N>.o mH.O VN.O m®.0 *QH.H mN.H *>>.H mm.o ov.o *NM.O v0.0 mm.H *HM.N *mm.0 N¢.0 H.0V mo.H mN.H *0>.H m0.m : a .a 3.7 6.... in. m m soaposem am.m 0>.¢ md.m mm.N 00.N 0>.H 0H.H 00.0 >ms m 151 Although approximately 12 MeV of excitation in 35Ar was observed (Chapter 5), no strongly excited levels were found to exist above the 6.82 MeV level. This indicates that most of the s—d shell hole strength has been observed. It is interesting to note that the "best fit" values for.2n=2 in Table 6 seem to be in best agreement with the total expectedaen=2 strength (V8). The 3/2+ assignment for the 35Ar ground state is consistent with an oblate deformation GS<0) in the Nilsson model (Fig. 5.1), and the 1.70 MeV level is probably the 5/2+ member of the ground state rotational band. As was the case for the ground and first two excited levels in 318, an assumption that the 1.18 (l/2+), 2.60 (3/2+) and 2.95 (5/2+) levels are all members of the K=1/2, [211] rotational band is incon- sistent with the large spectroscopic factors observed for these levels (Table 6). The "best fit" spectro- scopic factors for most of‘!n=2 levels are in fair agreement with the Nilsson model for S =—0.10, if one makes the orbit assignments listed in Table 6. The agreement with experiment for the 1.18 MeV 1/2+ level is also quite reasonable. The missing 5/2+ level from orbit 7 suggests that the d5/2 strength from orbits 7 and 9 may be combined in the 2.95 MeV level. 6.3.5 4°Ca(p,d)390a The DWBA spectroscopic factors for levels excited 152 in the (p,d) reaction on the doubly magic 4OCa nucleus were calculated for both of the neutron well radii used in the analysis of J-dependence (Sec. 6.2.1) and are shown in Table 7. As was the case for the 36Ar(p,d)35Ar reaction, the radius corresponding to the best DWBA fit seems to result in the most reasonable spectroscopic factor (on the basis of total strength) for,2n=2 and £n=3. Again, the .ano fit is inConclusive. The excitation of the 2.80 MeV level (1L=3) is an indication of configuration mixing with the f7/2 shell. The spectroscopic factor of 0.58 is in good agreement with the value of 0.53 obtained by Bock, et. al. (B065) from the 4OCa(3He,0C)390a reaction and with the result of 0.5 obtained for the excitation of the mirror level in the 4OCa(d, 3He)39K reaction (Hi67). Glashausser, et. al. (Gl65(b)) extracted values of 0.14 and 0.28 by assuming different neutron separation energies in the 4OCa(p,d)390a reaction at 27.5 MeV bombarding energy. The excitation of the.£n=l level at 3.03 MeV is assumed to be due to a 2p shell admixture with an estimated spectroscOpic factor ofvv0.02 as compared to a value of 0.04 - 0.05 obtained by Hiebert, et. al. (H167) for the mirror level in the (d, 3He) reaction mentioned above. Both of these results are considerably smaller than the value 0.11 obtained by Cline, at. al. (0165) from the 4OCa(3He,¢()390a reaction. The spectroscopic factors for the.£n=2 levels at ll ll: Ill I'll Il- 153 Table 7: Spectroscopic Factors for the 40 _x_ J“ 3021-11 0.00 3/2+ 7.11 2.47 l/2+ 2.31 2.80 7/2‘ 1.04 3.03 (1/2. 3/2)‘ 5.13 (5/2)+ 2.08* 5.48 (3/2, 5/2)+ 0.97 6.15 (5/2)+ 2.15* a)Shell model predictions 5 __ST8 3.70* 4 00 1.82 2 00 0.58* 0.02 1.43 6.00 0.67 1.48 * Values for radius giving best DWBA fit. Ca(p,d)390a Reaction 154 5.13, 5.48 and 6.15 MeV indicate that these levels prob- ably represent most of the d5/2 strength, especially if the smaller neutron well radius is assumed. However, the strength obtained with a radius parameter of ran: 1.35 F. is in agreement with the d5/2 proton strength obtained by Ref. Hi6? in the (d, 3He) reaction for the 5 - 7 MeV region of excitation in 39K. It was there- fore suggested (Hi67) that other d5/2 hole states exist at higher excitation energies (E£>8 MeV). Although about 12 MeV of excitation is observed in the present work, no other strongly excited levels are found to exist above the 6.15 MeV level in 390a (Fig. 5.281in Chapter 5). In fact, no level structure at all appeared in the deu- teron spectra for Exa39 MeV. As mentioned earlier (Chapter 5), the data from this experiment indicates that most of the configuration mix- ing occurs with higher shells, and does not lend itself to analysis in terms of the Nilsson model. 6.3.6 Summary The DWBA spectroSCOpic factors obtained for the (p,d) 2851, 323, 36Ar and 40Ca appear to be reason? reaction on able from a shell model point of view. There is some evidence that the more trustworthy value for the,gL=2 spectroscopic factors may be obtained by using the neu— tron well radius that was assumed in the J-dapendence analysis for each respective spin. The uncertainty in 155 the DWBA results are still quite large, however, espe- cially if the J-dependence is strong. Configuration mixing of shell model states was found to exist in the ground states of all the target 24Mg and 28Si ground states con— nuclei studied. The tain mixing of the 81/2 and d3/2 shells, with the possi- bility of 2p shell admixtures as well. The major admix- ture in the ground state of 323 appears to be a [d3/2]2 configuration, while the.&n=3 transitions in the (p,d) reaction on 36Ar and 40Ca indicate appreciable mixing with the f7/2 shell. There is also evidence for a small [2p]2 admixture in the ground state of 40Ca. The comparison of the experimentally measured spec- troscopic factors with the predictions of the Nilsson model has shown that the description of 2s-ld shell nuclei in general is not so simple as one might expect from considering only the observed level order (Chapter 5). The strong excitation of low-lying 5/2+ levels in the (p,d) reaction is evidence for rotational band mix- ing in the residual nucleus, corresponding to a small deformation in the target. This effect is particularly 288i(p,d)27Si reaction. The spectroscopic strong in the factors obtained from the Nilsson model are in fair agreement with experiment for some of the levels of 23Mg and 35Ar, although it is not obvious that all the Nilsson orbit assignments are meaningful. The situation is even more ambiguous for the levels of 31$. The 156 nature of the configuration mixing in these nuclei therefore seems to be very complex, although it may still be explainable in terms of a strong coupling rotational model if band mixing is included. Chapter 7 Summary and Conclusions The investigation of the (p,d) reaction on N=Z nuclei in the 2s-ld shell has provided new information about the level structures of the 23Mg, 278i, 313 and 35Ar residual nuclei, while previous results for the A=39 mirror nuclei (Hi67, 0165 (b), 0165) have been 4OCa(p,d)390a reaction. The 33.6 MeV confirmed by the bombarding energy and particle detection techniques have permitted the observation of 10 — 12 MeV of excitation in the residual nuclei with the interesting result ‘7: that virtually all of the observed 2s—1d shell hole strength exists at excitation energies 48 MeV. (.A possible exception to this is the 9.63 MeV level in 23Mg.) It is therefore apparent that most of the 2s-1d shell hole states have been excited, the DWBA spectro— sc0pic factors obtained here qualitatively provide an additional confirmation of this fact. The forward angle J-dependence observed in the .9n=2 angular distributions appears to vary in a syste- matic way with mass number (for N=Z targets) (Fig. 5.33) and may possibly be correlated with the nature of the nuclear deformation. It was noticed (Chapter 5) that the shapes of all the 5/2+ distributions were very similar, while the distributions for J=3/2 had a ten- 157 158 dency to vary from nucleus to nucleus. Spin assign- 31 ments for levels in S, 35Ar and 390a were suggested on this basis. The attempts to reprocuce J-dependence effects by varying the neutron well radius in distorted-wave calculations (Chapter 6) were partially successful, although very large changes were necessary in most cases. The results obtained by assuming a constant total well radius (3.79 F. = 1.25 (28)”3 F.) for a d5/2 neutron pick-up are in slightly better agreement with experiment than those obtained with a constant radius parameter (rO 21.25 F.) (Fig. 6.8). Large variations in the nefitron radius parameter produce correspondingly large changes in the magnitude of the calculated DWBA cross-section and lead to uncertainties in extracting spectroscopic factors. However, there is some evidence that the radius parameter correspon- ding to the best DWBA fit to the data also results in the most trustworthy value for the spectroscopic fac- tor. It is apparent from the results of this and other investigations (Le64, Sh64, Sh65, 0165 (a), Wh66) that additional experimental information and more theoret- ical work are necessary to obtain an understanding of J-dependence. Configuration mixing of shell model states in the ground state wave functions was found to be appreciable 159 for all the target nuclei in this study. The strong excitation of the.p;=3 levels in 35Ar and 39Ca indi- cate the presence of a large [f7/2]2 admixture in each of these nuclei. There is also evidence for a small amount of mixing with the 2p shell in ground state of 40Ca. The mixing to higher shells in the 32S ground state seems to be concentrated mainly in the excitation of the 3/2+ level at 1.24 MeV excit- ation in 31S. The ground state wave functions of 24Mg and 2831 contain admixtures with both the 2sl/2 and ld3/2 shells. Of particular interest is the excitation of low- lying (Ex=2.7 - 6.0 MeV),Qnél levels in 23Mg and 27Si. Since proton knock-out experiments (hi65, Jabo) and Hartree—Fock calculations (Da66) predict a 10 - 20 MeV energy separation between the 1p and 2s-ld shells, the possibility of 2p shell admixtures exists (W167). However, a plot of (p,d) reaction Q-values for the excitation of the first £;=l level versus mass num- ber (Fig. 5.14) seems to be strong evidence that these levels are 1p shell hole states. As was observed in Chapter 6, this ambiguity is not resolved by extract- ing spectroscopic factors, since the calculated DWBA cross-section is much larger for a 2p pick-up than a lp pick-up. The ordering of the first few levels in 23Mg, 160 31 S and 35Ar seems to be qualitatively consistent with rotational bands based on neutron holes in Nilsson orbits (N155 and Fig. 5.1). However, the extraction of DWBA spectroscopic factors (Chapter 6) has shown that the explanation is not nearly so simple. The large spectroscopic factors measured for the excitation 31 of low-lying 5/2+ levels in S and 35Ar is an indica- tion of considerable rotational band mixing. This effect is even more pronounced in the excitation of the 2181 ground state (also 5/2+), where a spectro— scopic factor of 3.45 was measured for that transition. The nuclear deformation is expected to be small for nuclei near closed shells, and the results from the (p,d) reaction on the doubly magic 4OCa nucleus indi- cate that this is the case for the 2s-1d shell. Appendix A Calculation of (p,d) SpectroscOpic Factors from the Nilsson Model"r In Chapter 2 it was seen that the Nilsson wave function for a single neutron could be expanded in terms of shell model states: lwx. “’15 = 5.56m 44 we Eq. A01 Here 4" determines the Nilsson orbit when Kn and N are given. Restricting ourselves to one value of N, we define £1. /N17' K) :- 4K0“) Eq. A.2 in the body-fixed (intrinsic) system. If the symmetry requirements of the wave function are neglected, the total wave function of the target nucleus is 3: . 3" ‘31!“ aka/”1’3 7 MK: “T “4‘": *7 where the total particle wave function can be written as 3 1m: E 35K! ) aar‘>h‘nhsfg c 7:! f Eq. A03 Eq. A04 * The author is indebted to L. Zamick for his helpful explanation of this subject. 161 162 The wave function for the residual nucleus for a given final J is then :r___ " ,:r T (Erik - 8%: Jam: [Al-(“"7131 K E.A.5 1.. Q where ’2? u hole and acts as a destruction operator. 1‘ is the Nilsson wave function of the neutron 6 For the pick-up of a neutron having quantum numbers (zn, jn’ In) in the laboratory system, the observed spectroscopic factor is defined as simaht-Kt WM 4‘43» awn“ J (2:, +1)! :77 xffimw: 321,9 Ignaz/2 EQO A08 163 The integral is (de Sh63) in _ I a. . . . , . . SBZZvafimK 42-% (3'16 MAM: IIM)(0'\:¢ VK‘ (3K) Eq. A.9 and the 431nt ) are given by the inverse of the expan- sion A.l: —/ 3“)“ — (d ') a), (at) ‘ {Z City)" Kg)» Eq. A.10 By orthogonality of the 3:” then, we see that only the term with VaK (anddgadn) is retained in the summation in Eq. A.8. Therefore, the substitution of Eqs. A.9 and A.10 in Eq. A.8 gives the result 1. z —a 3:6.) ‘-= :2 (5.3": MM; 11M) (73.1, K, k, )IK) Eff”); Eq. A.ll This gives the spectroscopic factors in terms the (nor- Ialised) inverse expansion coefficients 033(99), which are obtained from the given coefficients (Ch66) of Eq. 1.1 by a simple relation for real, unitary matrices: 0‘1 =- c r: ’6":- 8” Eq. A.12 i.e., it is just the transpose of the original matrix. If Jga-O, i.e., if the target is an even-even nucleus, Eq. A.ll reduces to 3:61,.) = 2. /C;-;’ld.)/L qu A013 164 This equation, together with the wave functions of Chi (Ch66), was used to calculate the theoretical spectro- scopic factors given in Chapter 6. Appendix B Transition Amplitude for the (d,p) and (p,d) Reactions The transition amplitude used in Chapter 3 is derived here according to the procedure of Satchler (Sa64), with notation similar to that of Glashausser (Gl65(a)). The zero range approximation is used for the direct reaction mechanism, and effects due to isospin, Coulomb phases and the Exclusion Principle are ignored for sim- plicity. Since the (d,p) and (p,d) reactions are time reversals of each other, the final results are simply related (T061), so the A(d,p)B stripping reaction is assumed throughout the calculation. The expression for the transition amplitude given in Chapter 3 (Eq. 3.10) can be rewritten in the form -k w a r 1 +A1 11.418 1347.‘1}:t)[54f)§¢m>%n449454440) Eq. B.l where J is the Jacobian of the transformation to the relative coordinates";p and TE. The quantity in brackets, which will be referred to as Q(ffin), is of particular interest. It represents the matrix element of the direct interaction between the initial and final internal states, and contains all the nuclear structure information and angular momentum selection rules. Efrepresents all coordinates independent of the relative displacements r? 165 166 and rd. Thus Q(ffin) is a function of ?% and?d and represents an effective interaction between the elastic scattering states. This factorization pr0perty separates out the dynamics of the reaction and permits one to con- sider only the rotational properties of Q(? ). pn Introducing spins for all nuclei we have T‘ Mfi .s * 3p P A / A ‘4’” Q(t.>=9?’tr'?€t.)§ ” 409524454 3.4V “9 EQe B02 To eValuate this matrix element, Q(f§n) is expanded into terms corresponding to the transfer of a definite angular momentum 3 to the residual nucleus, where .s. ...). J- A .3 .5 a .5. J = JB - JA = ,(+ s, s = 5d - 3 Eq. B.3 Then, using Clebsch-Gordan coefficients corresponding to this coupling, we can write , .S~l\ . ro' '(ta~.,~.—mt4) , ,4 5.10 X (54 SP "‘47 “Pk". W'W)a‘n "‘7 mi"?! 1.. Ms’Mh) qu B04 where m=MBéMA+mp-md. G . (F F) depends on the various nuclear quantum 13‘1", 4) P numbers, and is defined by the inverted form of this expansion. It transforms under rotation like the spher- 167 t ical harmonic T? and contains the parity change of the nuclear transition which, if the deuteron D state is neglected, is (-) The value of G is determined by the reaction being considered and the interaction V(fhp). An explicit form will be derived later, but for now it is convenient to write amt) 4,, WW?) gal/81m 1%.: 130.1)“ I“ Eq. B.5 4 .L where A ,is a spectroscopic coefficient and {3(GJWE) fie ”J 18.1310 is an interaction form factor. The zero-range approxima- tion assumes that the neutron and proton coordinates coincide at the point of interaction. In this case‘Fc‘fq) 5.1:?“ can be written fiflim): g( {-4 ti)ffi€5*%m)4$ a1,“ * £3011” ‘- 2 F'(r m . "‘ ‘- F542”: :0); (e,,¢,)g(r9 g9) Eq. B.6 whereF (rd) is a scalar radial form factor, and A and If.“ B refer to the reapective nuclear masses. This makes G (and thus Q(fhp)) proportional to a delta function, so that the six-dimensional integral for po (Eq. B.l) is reduced to three dimensions, and is much easier to eval- uate. Using Equations B.6, B.5 and B.4 in Eq. 3.1 and ignoring spin-orbit coupling in the elastic scattering wave functions 2; , we obtain lilllll. r' Ill 1’31 168 all"; .Zumm Eng”: §,Wflfs,a' (Edmuna Mall/:1- Ma)fi1.m Eq. B.7 where fllnfllnp 5.3 s ”L ‘ flm ___(__) ' 91*’(15‘n nvj' 'PI11M'~P+MJ)(:S "'4,- PIS m‘-M')é‘b EQe B08 and 3(5)); (edi‘EfiWJ V4) EQe B09 Equations B. 8 and B. 9 have the property )83'7’1/7?” S “‘«1 Mvnd EQO B010 The differential cross-section is then given by 4.615;...” ,3 I 517'2132)’: d All (mi ‘44)" (2: ”Mn-w!) u, m, P M, m, Eq. B.ll In terms of the spectroscopic coefficient A 1‘.‘ A 2 :éaf759:= gulifl £é;_lL_:fl@;ZL.fir:}i(eod “('9' 235+! .qu' 254” " f 169 where, making use of Eq. B.10, the "reduced" cross- section is r79)= ”‘4” J“ 55217" I") ' (”77" «I445 5 Eq. B.13 Before these cross-sections can be calculated, an explicit expression for the reduced amplitude $3 is needed. This is done by expanding the distorted waves 2r into partial waves, and is discussed in Ref. Sa64. An explicit (d,p) form for the 01:3?) in Eq. B.4 will now be derived. Consider first an expansion of the wave function of the residual nucleus in terms of the eigenstates of the target: 30M” 33M . c r): (Sakaédlot/ffiflfl (7; ,3?)( ram/“(38%) 1,314; Eq. B.14 Here, C%r.is a fractional parentage coefficient and is the wave function for a neutron in an ((Cj) orbit. This wave function can be expanded in terms of the sphere ical harmonics of the ghA system: Sam» 'M“ 4 .1 ... , 91:17, mfg"): §» )1 (9,“,0'9711077;)Z(})(Ism.\0,49 Eq. B.15 where mn=fl—m is the neutron spin projection, and 5' involves only the neutron spin coordinates. Since the internal coordinates of nucleus B and the neutron have 170 been separated from rnA in Equations B.l4 and B.15, the integral for Q(rkp) (Eq. B.2) may be evaluated. Re- writing Eq. 3.2 explicitly and assuming Vnp is a central potential we have In“: Q(r) Fffl$J€J?/V(fi ’9'; §)z;"'V,,,(£P,§°§’) “WM 34mg x471] (3)5”? w?)/%(fir Eq. B.16 The integration is performed by first substituting Eq. B.15 into Eq. B.l4, and then Eq. B.l4 into Eq. B.16. Due to the orthogonality of the internal wave functions of nucleus A, only the terms with JA=JA and MA=MA in Eq. B.l4 will contribute to the integral. Again, f p and Y involve only the proton and neutron spins, respectively, so the "integration" over these variables is just a sum over spins. The result is 25 43"qu Y" can» 43,021 ‘9'(’;A’)D(rp ) xa,($ 5.. m MMMWXI’ MM.\4fl)(IjMfifl/35 Ma) Eq. B.l7 where//Z=MB-MA=mn+m and Eq. B.18 From the symmetry relations for the Clebsch-Gordan co- efficients we have l 71 3 ’m (SpSanmnlsd my): (-) P '3 £31.11 (€45, nappy] 5;»,5) 23,1?! 9 B.l Then B. 17 becomes Q Cir): 12:0. 4-11294.” 25’ ‘fi'ly Wani¢na)w1;” M) Dag] x cm Ar(I:a Mp, MIMA IJBM.) * (a Wrap/5.. Mu)(1am.m.~w!4‘,M.—M.) Eq. 3.20 Now, a direct comparison of this with Eq. B.4 shows that . . . . . '1 (F t the quantity in brackets in Eq. B.2O 18 Just I 6 4%), or 1:“)... Gud'rr)— I“ W1) 2:4:- Y (9 Hydra)?! (59.0%,) 13;Jna ””f Eq. B.21 In the zero range approximation, D(§;p) is proport- ional to a delta function: 007‘ )—.= 2,802,):- D,T’3 (7;—- £3) Eq. B.22 Using Eq. B.6 in Eq. B.5 and Eq. B.22 in Eq. B.21 indenti- fies the spectroscopic coefficient as Ajay: {32:4— 09. D. 2g+l 0 Eq. B.23 172 so that Eq. B.12 becomes (sn=l/2 only) time) _ 2(23‘11-1) g n [d T 2 —---.‘ — . In D 6‘. (0) am 4,, @I,r0(2s.+/) 41- *1 0 ° 1‘, 4,. Eq. B.24 Do is related to the asymptotic (rpfi-><>O ) normalization of the deuteron wave function and can be obtained from the effective range analysis of n — p scattering. For 2 oz 1.5xio4 Mev2 fm3 (Sa64)- the (d,p) reaction, then, D The number of nucleons in an orbit, nflj’ has been included in Eq. B.24. This factor arises when the nucleon wave functions are anti-symmetrized. The spectroscopic factor is then given by z. =I7.[0(1'] S37, “‘3 9 Eq. B.25 So that Eq. B.24 becomes 4; = 192.3% 2‘. 2”&-%"9)4, d—(L J? (23A+D(2S"+ I” 9 Eq. B.26 Recalling Eq. B.5, 60925) = A1, - ¥(rllrr) 15“ it“ .7 [yléim Eq. B.27 Equations B.23 and B.21 imply that the portion of Eq. B.21 associated with the form factor is 173 f a m = * A A fi‘ujam x (Q'A’gfi)%j (CM) S(rp- % (4‘) Eq. B.28 Here,‘l[14.(rnA) is the radial wave function for a neutron bound to nucleus A in an (nifi) orbit, and is referred to as the neutron "form factor". This is determined in an actual calculation by specifying the radius and shape of the neutron well and the neutron binding energy. The essentials of the DWBA calculation for stripping and pick-up reactions in the zero range approximation have now been outlined. The method by which this theory is compared to experiment (Chapter 6) is outlined in Chapter 3. 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