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" A. *1 ”“hém'j'M. ‘.¢H..‘.:~-.. ...; 3".“ _ 41;" ': -“ ‘ "3" - "'4 I'fiib’Jfi mummy Midligan State University This is to certify that the thesis entitled Eagwea m0 f/Q @300 flew/{7% presented by 924/ flé/am/ér Sm/f/c hfieen accepted towards fulfillment fl We requirements degree in Z 3/(1‘2 fix OVERDUE FINES ARE 25¢ PER DAY ' Y PER ITEM 9 Return to book drop to remove this checkout from your record. FEATURES or THE (p.a) REACTION BY Paul Alexander Smith A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1976 ABSTRACT FEATURES or THE (p.a) REACTION By Paul Alexander Smith Theoretical and experimental features of the (p,a) reaction are presented. A microscopic reaction theory of the 3-nucleon direct pick-up reaction using shell model wavefunctions is developed. This theory involves the calculation of microscopic form factors and spectroscopic amplitudes. Two form factor models are given—the first using harmonic oscillator single-particle wavefunctions , and the second using single- particle wavefunctions of a Woods-Saxon potential. The motion of the nuclear center of mass in a fixed center potential is considered and the microscopic form factors are corrected for this motion. Three nucleon spectroscopic amplitudes are related to two nucleon reduced matrix elements and single nucleon reduced matrix elements which can be calculated using existing shell model codes . Calculations for pure configurations are presented and the features of these calculations are discus sed . Paul Alexander Smith The Distorted Wave Born Approximation for the (p,a) reaction is investigated. Angular momentum mismatch, finite range effects , and j-dependence are discussed. The 52Cr(p,ct)49V and the 44Ca(p,a)41K areactions are pre— sented as the experimental side. These reactions have been studied with a beam energy of 35 MeV. The 52Cr(p,n)49V spectra show that the 7/2- ground state, the 3/2”, 0.748 MeV proton hole state, and the 1.646 MeV 1/2+ proton hole state are strongly excited. Many seniority three transfers are also observed. A comparison of the (p,a) results to (p,t) and (t,a) work is presented. This comparison shows that coherence in the (p,a) reac- tion is important, since some states found in the (p,t) or (La) results are not observed in the (p,a) spectra. Candidates for high spin states are found at 3.612 MeV, 3.745 MeV, and 4.797 MeV. T=5/2 proton hole states (analog states) are observed in the region from 6 to 9.5 MeV. Cluster model DW'BA calculations are shown to fit the data successfully. The normalization of the cluster model calculation to the T: 5/2 , 7/2- angular distribution relative to the corresponding T = 3/2 state is about ten times larger than is expected from simple isospin considerations. Microscopic model DWBA calculations are also given. The microscopic calculations are shown to fit the angular distributions reasonably well. The relative normalizations of the DWBA curves to the proton hole state angular distributions are seen to be in good agree- ment with simple shell model predictions . The general trend of Paul Alexander Smith experimental cross sections for the negative parity states is shown to follow the features of the microscopic model calculations assuming pure (0f7/2)3 transfer. The 44Ca(p,ar)41K spectra show that those states which contain portions of the sd-shell proton holes are strongly populated. The 0d5/2 proton hole is mostly concentrated in a new level at 3.520 MeV. The 7/2- state at 1. 294 MeV is also excited. T=5/2 proton hole states (analog states) are observed in the excitation region from 8 to 10 MeV. These states have 11r values of 7/2-, 3/2+, and 1/2+. Clear I. =2 j- dependence is observed for this target. Cluster model DWBA calcula- tions are shown to fit the dam well. The normalization of the cluster model curve to the 7/2-, T=5/2 angular distribution is ten times larger than would be expected from isospin arguments . Microscopic model DWBA calculations are shown. These do not fit the angular distribu- tions nearly as well as the cluster model calculations. The relative normalizations of the sd-shell hole fragments is seen to disagree with (d,3He) results. However, the sums of the fragments are in good agreement with simple shell model considerations . Microsc0pic cal- culations for the 7/2- , 1 .294 MeV state are found to be in good agree- ment with the (d,3He) result. The possibility of (0f7/2 0d3/2)0d3/2 pick-up is considered and found to be unnecessary for this transition. ACKNOWLEDGM ENTS Writing a dissertation is a special occasion for a graduate student. It represents the culmination of many years of hard work, sleepless nights at the cyclotron console, and self-imposed slavery to the project. The dissertation is the end and the beginning. This is my chance to explain what I have learned to the public. It is my chance to teach you and to thank you. Unfortunately, the lan- guage of nuclear physics is a difficult one, which is unfamiliar to the general public and many of my friends . I have, therefore, included an appendix where I hope that some basic nuclear physics is described in more common English. This is my way of saying thanks to my parents and the taxpayers of America and Michigan who have provided enor- mous resources for my education. This is also the time when I can say thanks to everyone who has helped me. Everyone means everyone—you have all helped. My teachers, from my parents to my advisor, have provided the informa- tion and stimulation necessary for me to learn what little I have. My friends , though many of them know little physics, have managed to delay this moment with good times that have preserved my insanity and made my graduate career a lot of fun. This thesis is for my friends ll whose love I have often been unable to share fully because of my com- mitment to this project. I am especially grateful to the Nuclear Beer Group, which I trust will continue to grow and prosper in the future. I would like to thank Gary Crawley for suggesting this project. I am especially thankful for the supervision of my advisor Jerry Nolen. Without his expertise and dedication, this project would never have become a reality. Roger Markham, Ranjan Bhowmik, Din Shahabuddin, and Ioe Finck deserve a great deal of credit for helping in many aspects of this work and plugging away at what has proven to be an immense project. Finally, no words can describe how the wisdom of the grand old man has influenced me. Few students will ever have the chance to know the golden age of physics the way I have. This thesis is dedi- cated to an oldtimer—my grandfather, Professor R. A. Wolfe—and his wife . ill TABLE OF CONTENTS PAGE ListofTables..................... vi List of Figures . . . . . . . . . . . . . . . . . . . . vii CHAPTER I. INTRODUCTION . . . . . . . . . . . . . . . . 1 REFERENCES FOR CHAPTERI . . . . . . . . . . . 12 II. THEORETICAL CONSIDERATIONS . . . . . . . . . 14 A. Introduction . . . . . . . . . . . . 14 B. Distorted Waves Formalism . . . . . . . . . 18 C . Decomposition of the Orbital and Spin Parts . . . . . . . . . . . . . 27 D The (p, a) Form Factor: Harmonic Oscillator Model . . . . . . . . . 32 E. The (p, a) Form Factor: Woods- Saxon Model................. 37 F. Correction for the Center of Mass Motion . . . 39 G. The Microscopic Basis for Cluster Model Spectroscopic Factors . . . . . . . . 43 H. The Use of Shell Model Wavefunctions . . . . 45 1. Sample Calculations . . . . . . . . . . 48 I. The DWBA and the (p,a) Reaction . . . . . . 68 K. Conclusions . . . . . . . . . . . . . . . 84 REFERENCES FOR CHAPTER II . . . . . . . . . . 89 III. EXPERIMENTAL CONSIDERATIONS . . . . . . . . . 92 REFERENCES FOR CHAPTER III . . . . . . . . . . 101 iv C HAPTER PAGE IV. FEATURES OF THE 52Cr(p,a)49V REACTION . . . . . 102 A. Introduction . . . . . . . . . . . 102 B. Experimental Method and Data . . . . . . . . 105 C. Comparison with Other Experiments . . . . . 116 D. j- Dependence . . . . . . 131 E. DWBA Calculations—Cluster Form Factors . . . 138 F. DWBA Calculations-M icroscopic Form Factors . . . . . . . . . . . . . . 150 G. Conclusions . . . . . . . . . . . . . . . 157 REFERENCES FOR CHAPTER IV . . . . . . . . . . . 159 V. THE 44Ca(p.a)41K REACTION . . . . . . . . . . 162 A. Introduction . . . . . . . . . . . . 162 B. Experimental Method and Data . . . . . . . 164 C. DWBA Calculations—Cluster Form Factors . . . 178 D. DWBA Calculations—Microscopic Form Factors................ 192 E. Conclusions . . . . . . . . . . . . . . . 199 REFERENCES FOR CHAPTERV . . . . . . . . . . . 201 VI.SUMMARY................... 203 APPENDICES APPENDD< 1. Elements of Nuclear Physics for Non-Technical People . . . . . . . . 207 APPENDD( II. An Introduction to Scattering Theory . . 234 TABLE 11.1 11.2 11.3 11.4 11.5 1V.2 1V.4 IV.5 IV.6 V.2 v.3 v.4 LIST OF TABLES PAGE AList of Symbols . . . . . . . . . . . . . . . . 21 Optical Parameters . . . . . . . . . . . . . . . 51 Typical Oscillator-Size Parameters . . . . . . . . 59 Structure Factors for (0f7/2)3 Configurations . . . . 61 Maximum Cross Sections for (0f7/2)3 Configurations . . . . . . . . . . . . . . . 64 T=5/2 Energies . . . . . . . . . . . . . . . . 116 Levels of 49V . . . . . . . . . . . . . . . . . 125 52 49 Levels Seen in Cr(p.a) V That Are Not in 51V(p,t)49v................ 129 Levels Seen in Either 51V(p,t)49V or 50Cr(t,a)49V Not Observed in 52Cr(p.a)49V . . . . . . . . 130 Optical Potentials . . . . . . . . . . . . . . . 140 Relative Spectroscopic Factors . . . . . . . . . . 156 Levels Observed in 44Ca(r>.ar)411( . . . . . . . . 181 Optical Potentials . . . . . . . . . . . . . . . 186 Relative Proton Hole Spectroscopic Factors . . . . . 197 Total Spectroscopic Factors . . . . . . . . . . . 198 vi FIGURE 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 LIST OF FIGURES Coordinates for cluster pick-up Nucleon coordinates in the target nucleus Nucleon coordinates in a fixed center potential. . . . . . . . . Form factors for 52Cr(p.a)49V going to the 7/2" ground state of 49V . . . . . . . . Zero range DWBA calculations for the 7/2- form factors shown in Figure II.4. Ep=35 MeV . Microscopic form factors for a seniority one 7/2" transfer in the 520r(p.a)49v reaction—Woods-Saxon potential . . DWBA calculations for the form factors shown in Figure 11.6. Ep=35 MeV . . . . . Center of mass correction for a 5/2+ seniority one transfer in the 24Mg(p,a)21Na reaction. The top two curves show the absolute magnitudes , while the bottom two curves have been shifted to emphasize the shape differences introduced by the correction Cluster form factors for a 7/2- transfer in the 52Cr(p.ar)49v reaction . . . . . . . . vii PAGE 20 31 42 50 53 55 57 67 70 FIGURE 11.10 11.11 11.12 11.13 11.14 111. 1 III. 2 1V.1 A comparison of an exact finite range DWBA calculation and a zero range DWBA calculation. The micrOSCOpic Woods- Saxon model form factor shown in Figure 11.4 was used in both calculations. The calculations were done without spin- orbit coupling in the proton channel. Ep=35MeV........... Reflection coefficients for proton and a-particle elastic scattering . . . . . . A plot of 'ALa' (as defined by Equation 11.15) vs. La. IALal is proportional to the contribution of the La, partial wave to the cross section. The calculations are for the SZCr(p,a)49V reaction at Ep=35 MeV DWBA calculations for the 52Cr(p,ar)49V reaction with the wrong sign for the spin-orbit potential in the proton channel. Ep=35 MeV . . . . . . . . . . . DWBA calculations for the 52Cr(p,a)49V reaction without the spin-orbit potential in the proton channel. The real well radius is varied to mock-up the effect of VSO’ Ep=35MeV............... Cross sectional view of the focal plane counter O O O O O O O O O O 0 O O O O O 0 Top view of the focal plane counter . . . . . . . The 52Cr(p.ar)49V spectrum at 16 degrees plotted on a linear scale. The spectrum was recorded on a photographic emulsion. Ep=35 MeV;FWHM~10 keV . . . . The same spectrum as in Figure IV.1 but plotted on a log scale . . viii PAGE 73 76 80 83 86 96 98 108 110 F IG URE PAGE IV. 3 The 52Cr(p,a)49V spectrum at 16 degrees recorded with the counter system. Ep=35 MeV; FWHM ~ 20 keV . . . . . . 113 IV.4 High excitation spectra showing the T=5/2 proton hole states in 49V . . . . . . . . . 115 IV.5 TheszCr(p,a)49V spectrum at 60 degrees showing candidates for high spin states at 3.612, 3.745 and 4.797 MeV . . . . . 118 52 49 . . IV. 6 Cr(p,a) V angular distr1but1ons . . . . . . . 120 52 49 . IV. 7 Cr(p,a) V angular distributions . . . . . . . 122 52 49 IV. 8 Cr(p,a) V angular distribution . . . . . . . 124 52 49 IV. 9 Cr(p,a) V L= 2 angular distributions . . . . 134 52 49 IV. 10 Cr(p,a) V L=3 angular distributions . . . . 136 IV. 11 Cluster model DWBA calculations . The real well geometrical parameters were ro=l.22 fm. and a=.72 fm. . . . . . . . 142 IV. 12 Cluster model DWBA calculations . The real well geometrical parameters were ro=1.22 fm. and a=.50 fm. . . . . . . . 144 IV. 13 Q-value dependence of cluster model DWBA calculations . . . . . . . . . . . 147 IV. 14 Wavefunctions for analogs of 49V hole states................. 149 IV. 15 Cluster model DWBA calculations for the T=5/Zstates............. 152 IV. 16 Microscopic model DWBA calculations . The form factors were calculated for (on/2) 3 , (or7/2) 20d3/2 , and (0f7/2)zlsl/2 configurations . . . . . . . 155 FIGURE V. l V.2 v.3 V.4 V.5 V.6 V.7 V.8 V.9 V.1 A.I. A.1. A.I. A.1. A.I. A.I. A.1. A.I. 0 1 2 3 4 5 6 7 8 The 44Ca(p,a)41K spectrum at 12 degrees. 44Ca(p,a)41K angular distributions . . . . 44Ca(p,a)41K angular distributions 44Ca(p,a)41K angular distributions 44Ca(p,a)41K L=2 j-dependence . T=5/2 proton hole states in 41K . DWBA calculations for the 44Ca(p,a)41K reaction using cluster form factors Wavefunctions for analog states in the mass 4lsystem......... DWBA calculations for the two 7/2- levels observed in the 44Ca(p.a)41K reaction using cluster form factors . . . . . . DWBA calculations for 44Ca(p,a)41K reaction using microscopic form factors A sketch of the famous Rutherford experiment............. The compound nuclear model of nuclear reactions 0 O O O O O O C O O O O O The direct nuclear reaction model . . . . Asimple cyclotron . . . . . . . . Ahot-filament ion source . . . . . . . . Aparallel plate accelerator . . . . . . . The experimental facilities at M.S.U. . . Bending magnet and slits for beam energy definition . . . . . . X PAGE 167 169 171 173 177 180 188 191 194 196 208 212 213 217 218 219 222 224 F IG URE PAGE A.I. 9 A sketch of a magnetic spectrometer . . . . . 226 A.I. 10 A charge-division focal plane counter . . . . 228 A.I. 11 Identification of the reaction products . . . . 230 xi CHAPTER 1 INTRODUCTION The field of physics may be divided into the study of bound state problems, such as the solar system, crystals, or atomic structure, and unbound problems , such as the scattering of light or neutron diffraction. The same division applies to nuclear studies. The bound state problem is the study of nuclear structure. The problem to be solved in this instance is the prediction of the energies of the nuclear states and their properties, such as angular momentum and electromagnetic decay rates. The scattering problem is the study of elastic scattering , inelastic scattering, or transfer reactions. Here the problem is to pre- dict the angular distributions and the strengths of the transitions. The division of the field to which I have alluded is easily made for macrosc0pic systems, such as the solar system, where the indi- vidual components of the system can be examined. The division of nuclear physics in this way cannot be total since we cannot observe the bound system microscopically. Nearly all the information about the details of nuclear structure must come from the observation of free radiation resulting from radioactive decay or artificially induced reac- tions . Thus bound state models and scattering models must be 2 connected in a way that allows the deduction of nuclear structure infor- mation from the results of scattering experiments. The shell model is commonly used for nuclear structure calcula- tions . This model is based on the idea that a nucleon orbit may be described as that of a particle moving in a central potential created by the average interactions with the other nucleons in the nucleus . In addition, the residual interactions between pairs of nucleons are included in the Hamiltonian. This has the effect of removing degen- eracies in the original orbits of the central potential. Nuclear states can be described as distributions of the nucleons in these perturbed orbits . During the last twenty years many direct reaction studies have concentrated on the deduction of nuclear structure information. As might have been expected, these studies began with the single nucleon transfer reactions and progressed sequentially to more complex reac- tions. The purpose of this dissertation is to present work aimed toward testing shell model wavefunctions with the direct transfer of three nucleons. Prerequisite goals are to develop a microscopic model for calculations of three nucleon transfer reactions , and to detail the qualitative features of the (p,a) reaction which is the specific three nucleon transfer reaction to be considered in this work. The simplest nuclear states are those which are described by a single particle or hole outside a closed shell. Such states give the 3 energies of the single particle orbits in the shell model central poten- tial. Consider 41Ca as an example. A “zero order" shell model would describe 41Ca as a 40Ca inert core plus a neutron in the 0f7/2, lp3/2, 0f5/2, or 1p1/2 orbit. Given this simple model, 41Ca should have a 7/2- ground state and 3/2-, 5/2-, and 1/2- excited states, all of which look like a neutron bound to a 40Ca core. Therefore, if a neu- tron is added to 40Ca in a nuclear reaction such as 40Ca(d,p)41Ca , the spectrum is expected to contain strong transitions to the 7/ 2‘, 3/2- , 5/2- , and 1/2- states of 41Ca on the basis of the zero order shell model assumption. In general, the zero order shell model is not sufficient to explain all the transitions observed experimentally. For the 40Ca(d,p)41 example, the zero order shell model not only predicts that 7/2- , 5/2_ , 3/2- , and l/2- transitions will be observed, but also that they will be the only transitions . The zero order shell model may be improved if the 400a core is allowed to contain components with two neutrons in the 0f7/2 shell. In other words, some of the time 40Ca may have vacan- cies in the 0d3/2 shell. Observation of 3/2+ states in the 40Ca(d,p)41Ca reaction gives a measure of the amplitude of this com- ponent of the 40Ca ground state. A complete description of a final state observed in the (d,p) reaction is =ZAU¢TT 4411 where 11:? is the target wavefunction, 41111 is the neutron wavefunction 4 (n1 represents the fact that there is no restriction on the major shell, although contributions from anything other than the lowest shell are usually negligible for low-lying states), and A1 is the amplitude for I this component. The Aij's can be deduced from experimental cross sections . In Appendix II and Chapter II the details of the reaction theory called the DWBA (Distorted Wave Born Approximation) are dis- cussed. For the moment, a short summary is all that is necessary to see the role of the Aij's in the cross section prediction. The idea behind the DWBA is that the operator which causes transitions is a small part of the total Hamiltonian that describes the scattering prob- lem. The transition operator is treated as a perturbation causing tran- sitions between elastic scattering states (distorted waves) which are the eigenstates of the rest of the Hamiltonian. The cross section is proportional to the square of the transition matrix element. The transi- tion matrix element can be "factored" (see Chapter 11) so that the over- lap of the target wavefunction and the final wavefunction is isolated. In other words , the cross section is proportional to I (leltPr) I 2 which is proportional to IA1 l2. A nice example of how the values of the Aij's l are determined is given by Barry Freedom in his notes on nuclear reac- tions (1. 1, page 72) . The simplicity of single nucleon transfer lies in the fact that only one term in the final wavefunction contributes to a given transition if the target wavefunction has IT 5 0. In other words , only one of the A1 j's is deduced for each transition. However, the signs of the amplitudes are not determined. 5 Two-nucleon structures are the next in order of complexity. There are not many simple two-nucleon structures observed experimen- tally. As an example, let us use 40Ca as a core again and consider the 40Ca(p,t)38Ca reaction. Final states with j": 0+ are relatively simple because the only way two neutrons can have zero angular momentum is if they both come from the same orbit. For our example, the two neutrons picked up can come from the 0f7/2, 0d3/2, Isl/2 , or 0d5/2 orbits. Because of the residual interaction, a given 0+ final state will look like 40Ca+A1(0f7/2)-2+A2(0d3/2)"2+A3(ls1/2)"2 + A 4 (0d5/ 2)-2. Other transitions may have more components since the neutrons can come from different orbits for final states that do not have + j"=0 . This discussion can be written more precisely as ~11? = jELA1k¢1T(*nj¢nk)L where the 41111 Mink) indicate that a neutron is removed from the jth (kth) orbit and ( )L indicates vector coupling of the neutron angular momenta to the value L. If we consider the overlap of the target and the final state wavefunction as was done before, it is seen that, in general, many shell model configurations contribute coherently to the cross sec- tion for a given transition in the (p,t) reaction. Therefore, it is not possible to extract the values of the A '3 directly from the data. jk Instead, we must work in the other direction by calculating the Ajk's as overlaps of shell model wavefunctions and using the results in a reaction model. Because many configurations contribute coherently to 6 the cross section, the phases of the A). 's are important. Whereas k single nucleon transfer cannot measure these phases, multi-nucleon transfer has the advantage of being sensitive to them. The ability to predict the data tests both the shell model and the reaction model simultaneously . The progression from single nucleon transfer to two-nucleon transfer results in an enormous increase in complexity, hence one might ask, "Why study three—nucleon transfer?" The answer lies in the hope that there might be simple features of reactions such as (p,a) which are less complex than two-nucleon transfer, as well as other specific reasons discussed below. Because of the problems previously encountered in the analysis of two-nucleon transfer reactions, detailed agreement between a microscopic (p,a) reaction model and the data is not expected. Hence, this thesis is primarily concerned with predic- tion of the qualitative features of this reaction and their identification in the data. One of the simple properties of the ,(p,a) reaction is the popula- tion of the same single proton hole states that are seen in single proton pick-up. The target 52Or can be used as a good example. Much of the time the neutron pairs in the closed shell will have an internal angular momentum of zero. If one of these pairs and a proton are removed, we will see the single proton hole states that would be observed in the 50Cr(d, 3He)49V reaction. This was one of the first observations made by the pioneers in this area (1.2, 1.3, 1.4, 1.5). Sherr (1.2) used the 7 58Fe(p,ar)55Mn reaction to demonstrate this feature. The 55Mn ground state has j”: 5/ 2". In a zero order shell model the neutrons would be coupled to zero angular momentum since there are an even number of them. Therefore, the 55Mn ground state must be primarily a seniority three—proton state. The term "seniority" means the number of particles that are not part of nucleon pairs that have zero angular momentum. The 55Mn ground state has a pair of protons that have nonzero angular momentum and an odd 0f7/2 proton, making it a seniority three-state. The 58Fe ground state looks like four lp3/2 neutrons and three pairs of zero coupled protons . Removal of a pair of neutrons and a proton will lead to a 7/2_ state which is not the ground state. The data showed that the ground state was weakly excited and that the 7/2- first excited state was very strong. The inverse reaction 5SMn(ar,p)58Fe, presented at the same conference , showed excellent agreement in that the ground state of 58Fe was not observed (1.6). Although the strongest states will be those that can be made via seniority one transfers (a seniority zero neutron pair and a seniority one proton), there will be seniority three pick-ups where the pair of neutrons does not have zero angular momentum. Some of the time a neutron from two different pairs will be picked up leading to a trans- ferred pair with nonzero angular momentum. If a pair of neutrons is removed with nonzero angular momentum along with a 0f7/2 proton, final states with spins different from 7/2- can be populated. Since the 0f7/2 neutron pair may have angular momentum as high as six, it is 8 possible to populate states in 49V with spins up to 19/2. In recent years high spin states have been studied intensively with (a,xn‘y) and (HI,xn‘y) reactions. The (p,a) reaction and other multi-particle trans- fer reactions may be useful for differentiating between high spin states due to simple shell model configurations and those which are collective rotations of a deformed nucleus. Four recent experiments have observed j": l3/2+ transfers which is the maximum (0d5/2)3 coupling. 13/2+ levels have been observed in 12C(a,p)15N, 16O(ar,p)19F, and 24,26 21,23 Mg(p,a) Na, while an 8+ state was observed in 23Na(p,a)20Ne (1.7, 1.8, 1.9, 1.10). In principle, the (p,a) reaction may also be used to locate states with isospin greater than the ground state (T) states) that are not allowed in the (d, 3He) reaction because of isospin selection rules. Thus single nucleon spectroscopic factors and Coulomb energies can be deduced for levels that cannot be made by single nucleon pick-up. This feature of the (p,a) reaction has not been confirmed in previously published experiments . The determination of both the total angular momentum and parity of a nuclear state can be a difficult problem. In many cases more than one experiment is necessary. Gamma-ray decay studies often deter- mine the total angular momentum but not the parity. When the state of interest can be made by single nucleon transfer, the orbital angular momentum but not the total angular momentum is usually determined. The parity in this case is given by (-1)£. One of the observations 9 made during the first (p,a) reaction studies was the marked dependence of the angular distributions on the value of j for Z = 1 transitions (1.11, 1. 12, 1.4, 1.5) . The angular distribution for the 1/2- transfer is characterized by deep minima, while the 3/2- distribution is feature- less. There is promise that the (p,a) reaction may be used to deter— mine both j and 11. Lee e331. (1.13) have shown that this effect is a consequence of spin orbit coupling in the proton channel. The j- dependence for I. = 2 and higher is currently not well documented and , in fact, different experiments have reached different conclusions (1.13, I. 14 , 1.15 , 1.16). This point is pursued in Chapters IV and V of this thesis. The (p,a) reaction has the potential for reaching states that cannot be reached by single proton pick-up in nuclei where proton pick-up can be done. It also can be used to reach single proton hole states in nuclei that cannot be reached by single proton pick—up because the targets do not exist or are hard to obtain, thereby per- mitting the measurement of single nucleon spectroscopic information in these otherwise inaccessible cases. If simple models explain the properties of the spectra observed in nuclei that can be studied with proton pick-up, then these models may be extended to new nuclei to further study the systematics of hole states, Coulomb energies , high spin states, etc. 10 Even though there are simple features that make three-nucleon transfer attractive for spectroscopic studies , detailed study of the reaction for all states is even more difficult than for two-nucleon trans- fer. The additional complication may be illustrated if the final wave— function for the (p,a) reaction is expanded as before If 1 IT £12 1 klJf Z ArstLp'Zl‘ (¢nr¢ns) 4lpt 1‘81. 312 2121 where 41.? is the target wavefunction, ‘l’nr (Lpns) is the wavefunction for a neutron hole in the rt].l (3th) orbit, £12 is the angular momentum cou- pling of the two neutrons, and \Ppt is the wavefunction for a proton hole in the tth orbit. There is a sum over shell model configurations as in two-nucleon transfer, but in addition there is a sum over all the allowed values of the neutron pair angular momentum. A discussion of the internal degrees of freedom associated with the (p,a) reaction and some of the consequences within the 0f7/2 shell has been given by Bayman (1.17). A microscopic model for the (p,a) and (a,p) reactions is pre- sented in Chapter 11, including the details of the connection between the shell model and the DWBA. The remaining chapters are devoted to testing this theory and documenting the qualitative features of the reaction. The experimental considerations are described inChapter III, with the fourth and fifth chapters being devoted to the 52Cr(p,ar)49V and 44Ca(p,a)41K reactions. The general features are discussed and 11 DWBA calculations using both phenomonological cluster form factors and microscopic form factors based on simple pure configuration wave— functions are compared to the data. Microscopic calculations using sd-shell wavefunctions are currently in progress and will be presented at a later date. These calculations are being done for the 26:24 23'21Na data (1.9). Mg(p.a) Two appendices are included to help less experienced readers to understand the body of the text. The first appendix is for the general public which has supported this project. The second is a dis- cussion of the basic reaction theory for the purpose of helping students who may wish to learn about nuclear reactions. REFERENCES FOR CHAPTER I 1. 1 Barry Freedom, Michigan State University Cyclotron Report, MSUCL-27 . 1.2 B. F. Bayman, F. P. Brady, and R. Sherr, Proceedings of the Rutherford Iubilee International Conference, 553 (1962). 1.3 R. Sherr, B. F. Bayman, H. O. Funsten, N. R. Roberson, and E. Rost, Proceedings of the Conference on Direct Reaction Mechanisms, 1025 (1962). 1.4 I. A. Nolen, Ir., Ph.D. thesis, Princeton University (1965) , unpublished . 1.5 I. A. Nolen, Ir., C. M. Glashausser, and M. E. Rickey, Phys. Lett. 21 (1966). 705. 1.6 T. Yamazaki, M. Kondo, S. Yamabe, Proceedings of the Con- ference on Direct Reaction Mechanisms , 1079 (1962) . 1.7 W. R. Falk, A. Djaloeis, and D. Ingham, Nucl. Phys. A252 (1975), 452. 1.8 K. van der Borg, R. I. deMeijer, and A. van der Woude, to be published in Nucl. Phys. 1.9 R. K. Bhomik, R. G. Markham, M. A. M. Shahabuddin, P. A. Smith, and I. A. Nolen, Ir. , Bull. Am. Phys. Soc. , private communication . 1.10 H. T. Fortune, P. A. Smith, I. A. Nolen, Ir. , and R. G. Markham , private communication. 1.11 L. L. Lee, Ir. , and I. P. Schiffer, Phys. Rev. Lett. _l_2 (1964), 108. 1.12 L. L. Lee, Ir. , and I. P. Schiffer, Phys. Rev. 1363 (1964), 405. 12 1.13 1.14 1.15 1.16 1.17 13 L. L. Lee, Ir. , A. Marinov, C. Mayer-Boricke, I. P. Schiffer, R. H. Bassel, R. M. Drisko, and G. R. Satchler, Phys. Rev. Lett. 14 (1965) , 261. L. S. August, P. Shapiro, and L. R. Cooper, Phys. Rev. Lett. _2_3 (1969) , 537. L. S. August, P. Shapiro, L. R. Cooper, and C. D. Bond, Phys. Rev. CA (1971), 2291. I. E. Glenn, C. D. Zafiratos, and C. S. Zaidins, Phys. Rev. Lett. _2_6_ (1971), 328. B. F. Bayman, Nuclear Spectroscopy with Direct Reactions II, Argonne National Lab. Report ANL—6878, 335 (1964) . CHAPTER 11 THEORETICAL CONSIDERATIONS A. Introduction The (p,a) and (a,p) reactions have been shown to proceed pri- marily by direct three-nucleon transfer for incident energies greater than about 17 MeV (11. 1 , 11.2, 11.3, 11.4) . These reactions have the attractive features of reaching nuclei and nuclear states that cannot be reached by simpler reactions. An important class of states that may be populated via these reactions are high spin states that are described by relatively simple shell model wavefunctions . Much of the Distorted Wave Born Approximation (DWBA) analysis that has been done in previous work has employed mass three-cluster form factors (11.5, 11.6, 11.7) . Calculations of this type can give good fits to the shapes of many angular distributions . Unfortunately, this simple model cannot produce different shapes for angular distributions for final states which have the same j” value. Differences of this type have been observed in the sd-shell (11.8) . Furthermore, the "spectro- scopic factors " which are derived from this model are not easily related to nuclear structure models . 14 15 The first microscopic form factor model was due to Bayman and Rost (11. 9). Although this model was never published, it may be found in Nolen's thesis (11.2) and Ginaven's thesis (11.10). This model used harmonic oscillator single-particle wavefunctions and a technique developed by Mang (11.11) to project out the "triton" internal and center of mass coordinates. Nolen (11.2) used this model to study the (p,a) reaction on Cu and Zn isotopes while Ginaven applied the model to the (d,p) reaction on the Ca isotopes (11. 10, 11. 12) . Nolen's analysis assumed that the two neutrons were coupled to zero angular momentum. The j-transfer was then the total angular momentum of the proton. This simplification was necessary because complete wavefunctions were not available for these targets. Since Nolen studied the same states that were strongly populated by the (d,3He) reaction, the assumption of simple seniority wavefunctions was probably reasonable. The study of proton hole states by the (p,a) reaction has resulted in models based on the seniority one assumption (II. 13, 11. 14) . It is assumed that the three nucleon wavefunction is the product of a di-neutron wavefunction and a proton wavefunction. The models differ in the way that the di-neutron wavefunction is calculated. Ginaven's calculations (11. 10, 11. 12) went a step beyond those of Nolen's to include the coherence in the di-neutron angular momentum 16 coupling. (0f7/2)3 M82 (11. 15) wavefunctions were used to describe the Sc isotopes with the exception of 51Sc. A semi-microscopic model which includes coherence over inter- mediate couplings and configurations has been devised by Smits gt__a_l. (11.16, 11. 17) recently. Their method is a weak coupling model. Good agreement has been obtained with Sn(p,a) data. This model has the drawback of using mass three-cluster form factors . Thus, shape dif- ferences in the angular distributions of states with the same j1r value cannot be reproduced. Such effects are apparently weak in heavier nuclei, but may be important in lighter nuclei. In addition, the model is limited to those cases where the weak coupling assumption is appro- priate. Falk _e_t__a_l. (11. 18, 11. 19) have developed a microscopic model which uses the principle of expanding single particle wavefunctions that are generated in a Woods-Saxon potential in terms of harmonic oscillator wavefunctions. The resulting oscillator wavefunctions are coupled together with a generalized Talmi transformation (11. 20) to make a "triton" wavefunction. The results for the 12C(a,p)15N reac- tion are quite good when the coherent sums over all allowed couplings are performed. In this work we present a theory for the (p,a) and (a,p) reac- tions which is derived from a generalization of the Bayman and Kallio method of calculating two nucleon form factors (11.21) . 1n the first 17 section the problem is formulated in terms of the single nucleon trans- fer theory as presented by Tamura (11. 22) . 1n the second section the spin part is separated from the spatial part of the wavefunction and the spin coupling factors that enter multi-nucleon transfer calculations are determined. In the third section harmonic oscillator single particle wavefunctions are used to rederive the theory of Bayman and Rost in a slightly different way. The fourth section shows how single particle wavefunctions generated in a Woods-Saxon well may be used. Sec- tion five is devoted to correcting for the center of mass motion in a fixed center harmonic oscillator potential. A method of deriving cluster model spectroscopic factors from microscopic considerations is given for the harmonic oscillator model in the sixth section. In section seven we show how to calculate spectroscopic amplitudes as sums of products of two nucleon spectroscopic amplitudes and single particle spectroscopic amplitudes. Sample calculations using form factors cal- culated with the models presented in sections three, four and five are presented in section eight. The last section is devoted to studying the features of DWBA calculations for the (p,a) reaction. Finite range cal- culations and zero range calculations are compared, angular momentum matching effects are investigated, and the j-dependence in the DWBA calculations is discussed. 18 B. Distorted Waves Formalism Since certain factors appear in the DWBA formalism for all trans- fer reactions, it will be useful to look at structureless cluster transfer to display the elements of the analysis that are present for all reac- tions. Then it will be possible to concentrate on those features which are unique to three-nucleon transfer. To develop the cluster transfer formalism, we borrow the elegant treatment of Tamura (11. 22) . The transition matrix element from a particular initial state to a particular final state is (-) T =9 Sdrdrx. (kg) MmeMAma .25 ~a "b 1m] “b b . . (4') (IBMBsbmbIVIIAb/ksama) Xm'ama(5a .36) 11.1 if spin-orbit coupling is present in both the entrance and exit channels. Table 11.1 contains a list of the notation used in this work. The coordinates for a pick-up reaction are shown in Figure 11. l . The internal matrix in Equation 11.1 is an integral over the coordinates 5 8' fix, and 5.6 which can be factored if the interaction is assumed to act between the center of masses of a and x. (IBMBsbmbIVIIAMAsama) = SUBMBIIAMA) (sbmb IV(r2) Isama )dgx. 11.2 The first matrix element under the integral in Equation 11. 2 is an inte- gral over £3 and the second is an integral over £6 , so that both are 19 Figure 11. l Coordinates for cluster pick-up. 20 .\> Figure 11. 1 21 TABLE 11. 1 A LIST OF SYMBOLS Symbol Explanation a, b, x, A, B Projectile, ejectile, transferred cluster, target, final nucleus 8 Iacobian for transforming to relative coordinates “Ea , “Eb Relative coordinates for the aA and Bb systems (See Figure 11.1) Isa , Eb Incoming and outgoing wave vectors :1: xfn ')m (1,5,.r) Optical wavefunction including spin-orbit coupling IB’ IA Spins of the residual and target nuclei MB' MA Magnetic substates for IB and IA sb, Sa Spins of the ejectile and projectile mb, m Magnetic substates for sb, sa é's Internal coordinates for the target nucleons relative to an origin located at the center of the core (See Figure 11.2) Crawl: s a j Expansion coefficient for the decom- 1 1 x x position of the target wavefunction in terms of a core and a cluster with quantum numbers (n ,1: , s , l l x a .i) x q) Wavefunctions—superscripts denote angular momentum—sub- scripts are for magnetic substates or principle quantum number 22 TABLE 11- 1"Mfl24 Symbol Explanation .t 8,, 1 £9 ”£1” (5.x) m Angular momentum vector coupling 1 1', 5x 1 [c9 ‘(Llhp (§X)] = "‘1 - Z, (£1mls Ijmj) m1 mX s .121 x - ‘pm,(£1)‘pmx(§x) (jlmijzmzljsms) £1 “£2 0.3 CM I I 112122 SXQ'X S djfls Vector coupling (Clebsch- Gordon) coefficients Orbital angular momentum transfer, spin transfer, total angular momentum transfer j_ =3 + E Principle quantum number (oscil— lator convention) Separation of the cluster from the residual nucleus and projectile (See Figure 11.1) Oscillator size parameters for target and a-particle Expansion coefficient for decompo- sition of the ejectile in terms of the projectile plus cluster Spin and other quantum numbers of the cluster when part of the ejec- tile (not necessarily the same as Sx'ax) ~ indicates time reversal conjuga- tion " 1‘. -m 12 to z = (-1) z 24) 2 m2 ’m2 ~ _ _ js'ma ‘(j1m1’2m2'13'm3) Spectroscopic amplitude for a ifs-transfer 23 TABLE II. l—Continued Symbol Explanation £1111 1‘.an f 2mg (£1.52) Finite range form factor W(£ 1‘. js;£s ) Racah coefficient related to 6-j l 2 x symbol by . _ _ £1+£2+j+s {£1 £2 2 } j s 5x {r11 21 ji} Set of quantum numbers which describe the single particle orbits £12, 512 . 112 Di-neutron angular momentum couplings r13 2 r q Transformation coefficients for l 2 12 . chang1ng j-s to E-s coupling— s1 sz 512 related to -j symbols by L_11 12 112 £1 £2 £12 _ S1 52 S12 = 11 12 112 = «231214) (2312+1)(211+1) (212+1) £1 £2 £12 51 52 S12 11 12 112 CE?!) I, j 13 Expansion coefficient for decom- i 1 1 12 posing the target wavefunction into 0x1 a core and three nucleons with quantum numbers {niiijiflnaxj x Spin functions 24 TABLE 11. l-Continued Symbol Explanation (11121112222212 1(n"V2)212V20:£12> 11 ((n'-V2)£12n3£3:£| V1.61V30 :fi >21 £312 Klsz J01 T's N's Transformation coefficient for changing from nuclear coordinates to internal and center of mass coordinates (See Figure 11.2) v's and 1's are the principle quan- tum numbers and angular momenta in the transformed coordinates Transformation coefficient to rela- tive and center of mass coordi- nates for the di-neutron Transformation coefficient from the di-neutron C.M. and proton to tri- ton C .M . and internal coordinates . Transformation coefficient for transforming two nucleon coordi- nates to relative and C.M. coordi- nates in a Woods-Saxon well Internal coordinates in a fixed center potential Isospin quantum numbers 2 projection of T's (N=+§ for a proton) Antisymmeterizer Reduced matrix element-reduced in spin only Reduced matrix element-reduced in spin and isospin 25 functions of §X. The core may be integrated away if the target is expanded in terms of a core plus the transferred nucleon cluster. This procedure yields: (IBMBIIAMA)= Mgcm n1}. 18x0 WOBM jmj I1A M[ (r )«p: :(é‘. X 11.3 sxaX where ax represents other quantum numbers that may be used to describe the cluster. It is in Equation 11. 3 that the structureless clus- ter assumption has been realized by separating the center of mass coordinate from the internal "triton" coordinates . We will return to this point later. The ejectile can also be expanded as a projectile core plus the transferred group . (smeIVEszamé) = 2 01131228 ax sb(s mi) srnS lsa m'a<1>1[::(r;2)cp:;(§xs)] nzfizs x $an 11.4 where 3: ~ 1,552) = VQgMPiZQLz) 11.5 and ~ denotes time reversal conjugation (see Table 11.1) . {$352) contains the separation of the projectile from the cluster center of mass. If this function is not taken to be a delta function (zero range approxi- mation) , then the six-dimensional finite range integral must be per- formed (Equation 11. l) . 1f the results of Equations 11. 3 and 11.4 are used in Equation 11. 2 and, at the same time, we change to 1-s coupling, the transition matrix element for pick-up is seen to be 26 (IBM 3 m' bIV(5 2)IIJD‘M‘A‘sama) = dJE s £1n1£2n2 B b f (111“?) £lnl£2n2 ng j£s{£1nl£znzd IB+sa+j+Zs+mS firth-MA '(- (23+l) (21 +1) (1 A-MAIB MBljmj) -a(s m' asb- m'lsms)(jmjsmS|£-m£). 11.6 The Quantities dfffi 11111sz and figg‘fiznth :52) are defined to be (1111;113:112: 5x2 anlastx 0 X1 Grizfizsxaxsbl)S+£Z-SXW(£1£2jS;E 5x) H'7 and ... fan1£znz(r£1,rz ) = [nil ‘(r )o 112%] r. 11.8 2mg Because expression 11.6 is exact, these are complicated looking 195 and {511111532112 equations. The meanin of d g £1n1£2n2 2mg (51,52) becomes clearer if we look briefly at their zero range cousins for, say, the (p,d) reaction. Under these conditions n=0;£=0:£=£;C(b) =1; , 1 nz£zsxaxs W(£1£zjs;£sx) = 1 jfls nd d a £1n1£znz film-[32112 3mg is seen to reduce to a single spectroscopic amplitude. (51 .32) reduces to the usual form factor 18m1 _ E ... E Since our particular interest is the (p,a) reaction, we will now write Equations 11.6 , 11.7, and 11.8 again in the specific form that applies to "triton" transfer. 27 For (p,a) we must have 2 l an£zs a s W(£1£zjs; 23X) = These conditions imply 1.25 fflnl (IBMBSbmbIVIIA MASa ma '=) 121::{E’dn1fm 21:0 ,~2)} 13+(3/2)+1+ms'MA 1 -1) 2(21B +1) (1 AMAIB- Bljmj) (jmjzmaIE-mz) 11.6a -2 2011125631 11.7a n1 _ 72' o fjm£(r~1,£2) — «pnlqaypogzx 11.8a The quantity on which we need to concentrate in the future is the quantity in brackets { } in Equation 11. 6. We will not specify whether the zero range approximation is being made or not. Instead, the func- tion 415(52) will be carried along through all steps . C . Decomposition of the Orbital and Spin Parts In the last section we saw that all the reaction information is contained in the quantities djfis and f £(r1,£2). The function Em f E mg (51 ,52) conta1ns only the spatial parts of the wavefunctions . In order to isolate f Em 2(51 ,52) for the (p,a) reaction we must, therefore, separate the spin and orbital angular momenta of the wavefunctions that are involved. If k nucleons are involved, there will be (k- l) 9-j 28 symbols which come from the transformation from j-j coupling to 1-3 coupling. We show this for the (p,d) reaction by making the expansion of the target wavefunction (Equation 11. 3) again _ (A) {011211121311 (IBMBIIAMA) _{n§.j,}c{n1£111}¢(§1§z§a) (IBMBjmleAMA). II.3a 1112611 1 ”an {n12 ij i} X The difference between Equations 11.3 and 11. 3a is that some of the quantum numbers that were previously part of ax have been written explicitly and (§X,r_1) have been replaced by (€1,§2,§3) . The change to 1-s coupling is accomplished by 112 1 212 23 £1 _31 £2 £127 cp{(jlj2) 13} = Z 512 31; é % g 512 (gigzgs) £125” {niflijih'x 15 L112 13 1_ L11 12 112‘ Fur/293122.14 {st-Was ’ Wags.) x 11.9 L{ni£iji} Choose particles 1 and 2 to be the two neutrons. Since they must go into the Os orbit in the a-particle, the Pauli principle requires =0 and 112:3 Using these conditions and Equation 11.9 in the S12 12' expression for (IBMBIIAMA) gives 29 Du 2. F 12. 2. 212-l _ (A) 1 1 1 1 (IBMBIIAMA1— 2 cm“) 0 2 2 2 2 o {"1281} 1 1 1 z 1 1 1 1 t £120,ij 2”an l. 12 3 __, L 1 z 12—J _ E E . o 1 j (I M n M ) {(212.1 ”2.1 {(11) :1} H 10 B Bjm1 A A (p(§1§z§3) X . . _- {1113111} Comparing Equations 11.10 and 11. 3, we see that they have the same form except for the recoupling terms. Therefore, the term in brackets in Equation 11. 6a can be written as d (r ,5 ). 11.11 £12 £12 The spectroscopic amplitude is given by _ a t. q ”1 (A) £12 £3 2 £1 £2 2,; d 2 = Z c o g $- % é o . 11.12 {nizlji} a {niziji} £12 x £120 1 [’12 13 1 L11 12 £12 n-E In order to write an expression for £13m} iji}(£1,~r2) , it is neces- sary to choose a wavefunction for the a-particle. We make the fol- lowing Gaussian choice «p = ne-B/Z (p122+p1§3+ar22) where g” and £123 are related to £12 and 5123, which are the "triton" internal coordinates that are shown in Figure 11.2. This choice allows some of the overlap integrals to be done analytically in the oscillator model. 30 Figure 11. 2 Nucleon coordinates in the target nucleus . 31 Figure 11. 2 32 Using this choice gives E 2 1 {11121111212 0 {(21132) 1 £3} _B/2( Z 2 = piz+P1231 fit/mg (£1552) ”@0152) S¢(§1§2§3) e d212d£123' {“15111} 11. 13 £1 is not explicitly shown on the right hand side of Equation II. 13 because (€1,§2,§3) must still be transformed to (312,13 This 12311)- is the subject of the next three sections. D. The (Day) Form Factor: Harmonic Oscillator Model In order to reduce the right hand side of Equation 11. 13 to a func- tion of (51,52) , the nuclear coordinates (€1,§z,§3) must be transformed to center of mass and internal coordinates (312’2123’51) (see Figure 11. 2) . {(2,22)“1223}£ The function <9 may be written in the anti-symmetric (g11g21§3)ax form 1 ' Z [ £111§ £212§ £111.: £212§ :‘212 £313g E {(2132) .223} 1%. < 11an < 2) “’n, ( 21¢“; ( 1) “’n. < 3)} (p = (519293) ‘12 (1+ bull-120312251112) {ni‘eiji}ax 11.14 where the convention that particles 1 and 2 are the neutrons is used. In this section the single particle wavefunctions are taken to be harmonic oscillator wavefunctions . We need to determine the expansion coefficients defined by Equation 11. 15 33 £111 £212 £111 £212 £12 £313 £ 11%, «:1»an (£21-co (2 Mn (11)] w (13)} n1 n3 V2” + 6n1n26£1£26 1112) x 1 x A £ V1 A1 n1 £1 [LP 1Q )P( 2 3) (P .P 1] _ 2 v2 A; 1‘12 22 V1 1 V2V3 ~12 "123 H 15 _ A _____ . Mvikzvz x 2122 2 (1+ 631n26£1£261112) k v A V3 3 n, 3 3 3 where 3 3 .Z(2ni+£i) = Z (2vi +11) = N 11.16 1=1 i=1 is required to conserve oscillator quanta. The transferred "triton" should have zero internal angular momen- tum if it is to fit into the a-particle. This condition implies that only the terms with 11 =0, 12:0, and A =0 should be saved. Terms with nonzero v2 and v3 need not be zero, however. If the size parameter of the a-particle (B) is different from the size parameter of the target wavefunctions (a) , there will be nonzero overlaps for the v2 , v3#0 terms . Equation 11. 15 may be written again including these conditions. £111 £21 flklj £12 £1 E p {[wn (t ”11,220.29 - “G. ‘20: )«o n:71: 1] @3231} V1 2 ml 21 _ V; 0 11; £2 (00)0 ‘ “z" o 2,, “’vM‘EiW vzv3 (”12'3123) 1 Z 3 V3 0 n3 £3 11.15a where P is a projection operator that picks out the portion of the wave- ‘B/z (P12413123) . function that has nonzero overlap with e 34 It is our goal to relate these expansion coefficients to Talmi- Moshinsky coefficients . Proceed by examining the two-nucleon part of the wavefunction £111 £212 £111 £212 £12 [(9111 (51110112 (€21'cpnl (€21an (81)] = 22 XRVR >\121’2 -< z I 2 z r ) XR 1‘2 ) 2‘2 :n 17 ”RXRV12112' 12 n1 1“2 2' 12 11 ¢vR(5£pv,(£12 11' ' where (2'2 -£ ;) N'=(N-n)+-——%—i— 11.18 vR = N' - v2 . The (n B n E :E lv 1 v E ) are the normal Talmi- 1 1 2 2 12 R R 12‘12‘ 12 11 Moshinsky transformation coefficients for equal mass particles . The subscripts 11 denote that the masses involved are both single nucleon masses . The vector R runs from the center of the core (B) to the center of mass of the di-neutron. The condition that there be no internal angular momentum implies that 1.12:0 and XR=£12° This result may be inserted into the left hand side of Equation 11. 15a to get 35 ZZ<[“ (R) < )2 Wm] V2”12"2011“22‘1211“’(N'—v,) 23,512‘pn V2 (0 )0 ) 11 19 o 2,, ‘9 (ri)"’v,v3 (912'3123' ' V1 V1V2V3 The function in brackets on the left hand side of Equation 11.19 can be expanded in a similar way 0 £12 ¢v2(p12)[W(N'-V2)(R)¢n:j3(§3)]£ - Z x1V1 X3V3 .( ' 0 >‘1 x3 2 V1"1"3*3 ”MN "’2”12‘“3“3 2‘21212,912)”12115211253122] ° 11. 20 The subscripts 21 denote that the mass on the left hand side of the bracket is two nucleon masses while the right hand side is for a single nucleon. Again the requirement of zero internal angular momentum can be applied so that 13:0 and 11 =2. Using Equation 11.20 in Equation 11. 19 yields the relation V1 2 “1 £1 V2 0 n2 2; 0 212 V3 0 “3 £3 = 2((N'-v2)212v20:12 2|:n1211'1222 212)11 °(v12v30:2|(N' -21v)2 233112 :2>2111.21 The ( l )21 coefficient is the Talmi-Moshinsky transformation for a mass two particle and a mass one particle (11. 20). 36 We may use Equation 11.21 in Equation II. 15a and then insert the result in Equation 11. 13 to yield {n-E 1'} f3 111 @1552) = \/2(1+6 2116 6 ) 2 m2 111112 2122 jljz v1vzv3 «(112-121212120 2 ”Inlfilnzfiz 21211 E O ' (v12v30:2 I (N -v2)212n323:2)21¢VIQ1)¢OQ‘2) -B/2(p‘+p‘ ) 6‘": (”12”: ("123)8 12 13113120123123). 1"” The integral in parentheses in Equation II. 22 can be factored into two integrals of the form 0 _ z I = Same B/Z " pzdp. 11.23 This integral can be done analytically if we use the oscillator function 2"! 3/2 2 O _ 2 2v+1) I la {-20} v 2K -a/2 p wv(p) _] 43;, v! éo{(21<+1):1(“>p }e . 11.24 The result is 1— 211421111 213/4 13m)" 1125 v12" (B+a)3/2 3+0! ‘ ° Equation II. 22 may now be written {1112111} 2 f 15 :5 ) = .fi '" Z 2mg 1 2 2 (1+ bnlnzbzlzzbjljz) VIV2V3 °<(N'-v2)212212|n1£1n2£2w12>11 ._ 2 o <"1’“3°°z'(N "2‘212“3£3 ‘21“ Av2v3 "v.91)"o(.52) II.22a 37 where .1. (2v +1)!!(2v +1)” 3/2 _ V2+V3 = We] 2 3 a B a) 11.26 V2 V3 V2"’ V3 (6+0) 3 3+0 vzlv3! 2 The last result is very similar to one obtained by Bayman and Rost (II. 9). It is purely a result of allowing the oscillator size parameters for the a-particle and the target to be different. If a=B, v2 and v3 must be zero and the problem is easily simplified. A recipe for the harmonic oscillator model calculation is as fol- lows . For allowed combinations of (v1 ,v ), calculate the A 2'V3 vzvs and the two Talmi-Moshinsky coefficients . The form factor can be calculated via equation II. 22a. The spectroscopic amplitudes are given by Equation 11. 12. The transition matrix element is then found by substituting these values into expression II. 11 . E. The (p.01 Form Factor: Woods-Saxon Model Once again we wish to reduce the right hand side of Equation II. 13 so that it is explicitly a function of £1 and 5 only. Again, the 2 triton wavefunction is taken to be as given in Equation 11.14 . We will work in the same general pattern as the last section. As before, begin by expanding the two-nucleon part of Equa- tion II. 14, but this time follow the Bayman and Kallio (II. 21) two- nucleon expansion 38 £111 £212 31h £252 £12 212 [mu] (alwnz (*‘szl-cpnl (@2an (Ell-J Z fX1:A(plz' R) ~12(1+5n1n25£1£25hj2) MzA 012R x .. - ”212 ~ [Y ”(pm/‘02)] . 11.27 Since the internal angular momentum is required to be zero, only terms with x12=0 and A=£ need to be saved. This result may be used in Equation II. 13 to yield = 0 2: "B/2 9123 ffimg @1052) “032) m (£12m1223m3'£m£)5d9123e 12m3 £92121:(p12 IR) 912R q{slag 3) (Sdplze'B/Z 0:2 Y:(5,Z)Yrfi::(§)> . II. 28 The inner integral in Equation II. 28 is the normal two-nucleon form factor that Bayman and Kallio calculate. Call this integral Pm (R)Y£111:(R) where g is the coordinate of the center of mass of the di-neutron as defined in the last section. Using this notation in Equa- tion II. 28 and recoupling the angular momenta yields f1{n l “j } 2 E - z E 2mg £15332)= 11° 092523312 393/2p‘23[P(2)(R)Y 12(RM 363)] . 11.29 The quantity in brackets can be expanded as E f (11:91.23) - E - ,_ B [1‘(2)(R)Y£"(R)cp£3(§3)] = 2 1.1231 [Yx‘z’mlzwlrufl] . 11.30 T191211 X123 39 As before, take only terms with x =0, and hence '1’ =2 , so that 123 {n 3.1 } _ 2 f2 (1319123) .. .. g i ii _ 0 B/me oil 0 ] fzmz (51°52) ' '1 @oQ2)Sd91233 rlpm [Yowizswjhfl ' II. 31 The integral in Equation 11. 31 has the same form as the two- nucleon form factor. However, one of the particles is a di-neutron. It is possible to redefine the coordinates in the Bayman and Kallio paper to handle the unequal mass problem. Roger Markham has written a code to perform this calculation for three-nucleon or four-nucleon transfer (II. 23) . If single particle wavefunctions generated in a Woods-Saxon potential are used, the prescription is to first calculate the two- nucleon wavefunction Pm (R)Y£”(R) then use this to calculate the three-nucleon form factor, which is given by Equation II. 31. The tran- sition matrix element can then be found by using the spectrOSCOpic amplitudes given by Equation 11. 12 . F. Correction forgthe Center of Mass Motion The steps leading to Equation II. 22a involved integrating over the coordinates of (IBMBI in Equation II. 3a. We implicitly assumed that all the coordinates were internal, which they are in the coordinate system whose origin is the center of the core (see Figure II. 2). How- ever, this is really incorrect since the core (B) is moving about the center of mass of nucleus (A) which is fixed in our reference frame. 40 Hence the coordinate system shown in Figure 11. 3 is the correct one. In this system the appropriate coordinates for (I M I are $3 and R ~B and we are interested in determining the effect of motion about BB on the form factor. We begin by expanding II MA) again II M > = 2 CW A A . {nifiiji} {1112111} lizaxl 1”an 1 1 1 1 1 1 1 I {13113.11ng )19 21121-11 ‘11211 2221] ‘2 311311)}1‘. 4211+6n11n26£ 3261.112) II.32 At this point we can change to LS coupling and apply the two Moshinsky transformations outlined in Equations II. 15 through II.21 to get 11 M ) = Z, 2 CW A A {1112111} V1v2v3 C” £111}~F(1+6n 11.52 2251112) Elzaxj jlzaxj 1 _ .1. I- 0 0 2 2 “N "2”12"0 2 2111112111222212111 11 12 £12 £12 13 l L. _J .— 3. 2. 2.? "2.. £3 27 l 5 Mr- —1 ._ o o (V12V30:£| (N V2)£12n3£3. £>21 ¢v2(£lz)¢v3(£123) . 110121 I We assume that there is no spurious center of mass motion which means that the center of mass motion does not have any oscillator quanta associated with it. In other words , 41 Figure II. 3 Nucleon coordinates in a fixed potential. Ks 42 C Figure II. 3 43 11131: 1 - 1131: 111002) 11 34 B’EB — B o ~B ' ° We may now perform another generalized Moshinsky transforma- tion on the term in brackets in Equation 11. 33 , where the masses involved are B and 3. 2 o o I. CPVIQ‘tMO (BB) (OOVIE .13 I OOVIE '“BB 190 (RAMVIB) A vl+£/2 E __ O (11.-3 (Po (RA) (111-3) «pvlggp. Similar analysis for cluster transfer has been presented by Ichimura e_t__a_1. (11.24) . Our result reduces to the cluster result if we require equal size parameters, which is the equivalent of the cluster transfer form factor. It is intriguing to note that the correction is different for Os, ls, 25, . . . internal motions for realistic size parameters, thereby changing the shape as well as the magnitude of the form factor. G. 1115 Microscopic Basis for (Luster Model Spectroscopic Factors In the harmonic oscillator model the importance of is , Zs , 35, . . . terms in the "triton" wavefunction is a function of the difference of oscillator parameters (a-B) . In principle, this difference is fixed by 44 the a-particle radius and the nuclear radius. However, it is instruc- tive to consider the case where a: B. If (oz-B) is zero, there is no contribution from the ls, 25, 3s . . . = v =0 and the triton is structureless . 2 3 Therefore, (or-B) = 0 implies mass three cluster transfer. Furthermore, terms. In this instance, v the center of mass correction reduces to the cluster center of mass cor- rection given in Reference 11. 24 . It was pointed out in the introduction that "spectroscopic fac- tors " deduced from a cluster model calculation are not easily related to nuclear shell structure. In the sense that a harmonic oscillator cluster model is given by (a-B) =0 , we can derive a spectroscopic amplitude by investigating the expression II. 11 and Equation 11. 22a. For single nucleon transfer, or cluster transfer, II. 11 becomes 12A 2 Liég C1 3 d fszQ‘I ,52) = d 4’0 (5sz (£1) . II. 36) L The spectroscopic amplitude, d122, is called the "spectroscopic fac- tor" and is usually denoted by s”. If, on the other hand, we require a=B in Equation II. 22a and use the result in expression 11. 11, we obtain F2.. 2. 2 F21 2. 2.? { 2 CW 0 i i i i 0 2.— {n I! j } {nigijiulzaxj 2 2 2 2 Jahoninzbflizzbhjz) 1 H £12 13 l 11 12 £12 Enax L _ L _J (OON'Z In E n 16 :2 (00NZ:£IN'£ n o E 12°£12 1 1 2 2 12)11 12 323.2>21 mogszgl) 11.37 45 where 2N'+1€1 = 2(n1+n2)+£1+£2. 2 Comparing the right hand side of Equation 11. 36 to II. 37 shows that the spectroscopic factor is given by e=1 1ee1 1 represents the term in brackets in 11.37 . A number of (p,d) and (d,p) studies have deduced cluster model "spectroscopic factors" (see II. S and II.25 for example). None of these experimental spectroscopic fac- 12 tors have been compared with theoretical S values calculated in the manner presented here . H . The Use of Shell Model Wavefunctions The nuclear structure information is contained in the expansion coefficients C(A) . In order to test shell model wavefunctions, a pre- scription for calculating these coefficients must be devised. We will do this by introducing an intermediate state expansion so that reduced matrix elements for two nucleon configurations and single nucleon con- figurations may be used. This method has the advantages of using the output of existing shell model codes (codes for calculating three nucleon overlaps have not been written as yet) and of allowing us to separate the two neutrons for easy anti-symmeterization as we have done. In addition, this method is intuitively pleasing since it divides the problem into two nucleon transfer and single nucleon transfer. 46 The expansion coefficients defined in Equation 11. 3 are given by I I I 012(1): 2.11 Him), 1 =<[¢ B{’4 «01119129121913? ] Alcp A>. 11.38 11 . A p-n formalism has been used and we have anti-symmeterized accord- ingly. However, the shell model wavefunctions which we wish to use (A) for the calculation of C are derived in the isospin formalism. It is necessary, therefore, to begin by converting the right hand side of Equation 11. 38 to the appropriate form for the isospin formalism. We do this by expanding the three nucleon wavefunction {)4<112¢ 125'2'>J121(p 13§}_ ___ -mVQ‘JIE‘p 122->1“) 1pj3 5}” + VT/—3{74ij‘%1ph%>lle 113132} llcpANA>] . 11.41 B A Here the reduced matrix elements are reduced in spin only. The Wigner Eckhart theorem can be applied to reduce them in isospin as well. 1 (A) 1212[ TB'NB TB 5 TA) IBTBH c = {—— -~/2 3 -1 {nigijihlzaxj ZIA+1 / ( ) "NB "% NA (q) .i.ijl 11 IT T—N T 3/2T .{4(¢112¢122) ’2 ‘Pj32}j21-[CP A A>m (_1) B BK B 1 A) -NB -2 NA 1 '1 .1. L i I T ((9 B B H{A(¢112¢122>1121¢132}j 3/2:[[¢ A 13>] , 11.42 These reduced matrix elements can be expanded using the rela- tion I I k k k (Junk! x #2151111) = 1417”” r—zm Z {1'1 z .} 7]" . I kl I" In I"; In (a 111 1117x1117 IIT 1111) 11.43 where '7 represents other quantum numbers which may be needed to describe the intermediate states. The resulting expression for cm) is I +j+IB 2T (A1 1—11A 21+1 2 [ A 1 1 C = Z (-1) (T N z-le N) {1112111}lean VZIA+1 2TA+1 a; IITI B B A A 7 .— 1 1 3 1 E '2' ZTA‘I']. 3 1 5 7 .{ '}+ (-1) (TBN 82";- § MNA){B T T'}] X 48 12 j j IBTB L .1. I I c I L I T Xjul 3 u}? H4K¢112¢122)1121H¢;T >QP'IYT leazflw A A). A IB 1' 11.44 This is the result we desire since it is really these reduced matrix elements that the shell model overlap codes provide. Notice that only even values are allowed for the two nucleon coupling if the target has a ground state spin of zero. This is so because the two nucleons are in an isospin one state and a relative 5 state. Thus the sum over I' is only over the values 0, 2, 4, . . . . 1. Sample CalciLations Figure 11.4 shows a comparison of three methods of calculating three nucleon form factors . The microscopic Woods-Saxon form factor was calculated assuming each was bound by 8 MeV. The oscillator form factor was calculated assuming a size parameter of .226 fmn2 (see below). A hankel tail was matched to the oscillator form factor to give the correct asymptotic behavior. The mass three cluster form factor was calculated using a Woods-Saxon well and the triton separation energy from 52Cr. The two form factors have similar shapes , although they differ a bit near the nuclear surface. The mass three cluster form factor has its nodes pushed out further than the other two form factors and does not display an accentuated last maximum. In addition, the cluster form factor does not have the same tail as the microscopic Woods-Saxon form factor. 49 Figure 11. 4 Form factors for 530r(p,a)49V going to the 7/2- ground state of 49V. Arbitrary Units 50 T IIIUI I I I VIIIII I j T l r I F arm Factor Comparison l L 114111 I 1 [111111 11111 Radius (fm.) Figure 11. 4 3 Mass 3 Cluster W.S.— - Microsc0pic H.O. --—- ' 5 Microscopic W.S. — '— i - 4 - -4 T I I I T f |.0 2.0 3.0 4.0 5.0 6.0 7.0 51 Zero range DWBA calculations for the form factors in Figure 11.4 are shown in Figure 11. 5 . Even though the cluster form factor is dif- ferent, the predicted angular distribution is similar to both microscopic calculations. The optical parameters for all the DWBA calculations which we present are given in Table 11. 2. TABLE II. 2 OPTICAL PARAMETERS Particle V r a V a) r a V r o W a) e r r so so so I 1 I ' s -43.22 1.22 .72 -25.0 1.01 .75 -5.0 1.32 .52 12.6 a 196.0 1.22 .72 -16.0 1.82 .38 aAs input for DWUCK72 (II.27)-—e.g. , includes factor of 4. 0. The effect of different neutron orbitals is shown in Figures 11.6 and 11.7. The (lp3/2)20f7/2 and (0f7/2)3 form factors differ in much the same way that the cluster form factor and the (0f7/2)3 form factor differ. The angular distributions are similar again, but the (lp3/2)20f7/2 transfer is predicted to have ten times more cross sec- tion. This feature can be tested by examining the 7/2- states in “’56 51'53Mn reactions. Fe (ma) The form factors which are shown in Figures 11.4 and II. 6 are for a particular choice of the a-particle size. We have used 1. 63 fm for the root mean square radius of the car-particle. This value was meas- ured by electron scattering (11.26). 52 Figure 11. 5 Zero range DWBA calculations for the 7/2- form factors shown in Figure 11.4. HP = 35 MeV. da/dn (Arbitrary Units) 53 TVII'\' \\ \ [TUIITI I I I DWBA Calculations Mass 3 Cluster I Microsc0pic W. S. — '— 111111 20 I 40 6'0 30 C.M. Angle (degrees) Figure 11. 5 1 I00 l20 54 Figure 11. 6 Microscopic form factors for a seniority one 7/2- transfer in the 5 Cr(p,a)49V reaction-Woods-Saxon potential. 55 : I 1 I j V I 1 C 2 q ; Form Factors: (om/21’. (lp3/2) Of7/2 ; C (ups/2)2 Of7/2 - 5 E (om/213 : >‘ I- Q . 5 . < " I 1 .. t .. Co 2'.o 310 470 25:0 6.0 7.0 Radius (fm.) Figure 11 . 6 56 Figure 11. 7 DWBA calculations for the form factors shown in Figure 11. 6 . Ep = 35 MeV. da/dI). (Arbitrary Units) 57 l I T 1' I l DWBA Calculations: (om/213, (lp3/2)2 arr/2 (lp3/2)2 Of7/2 —— (om/2)3 --- X l/IO 11111 2'0 4'0 go 810 30 I30 C.M. Angle (degrees) Figure 11. 7 58 To perform the Woods-Saxon form factor calculation, it is neces- sary to determine the n-n separation and the proton-di-neutron sepa- ration from the a-particle radius. The appropriate r.m.s. separations turn out to be A 1.537 fm n-n A = 1.331 fm. p-(nn) For the harmonic oscillator calculation, it is necessary to find the relation between B and the a-particle radius. The a-particle wavefunction is 4 ”ls/2 ’3 Ef'BCM)?’ 6 i=1 . (p = The sum may be transformed to internal coordinates only by 4 2__1_ 2 2 2 g 2 21(51BCM) ' 2 12+3 l1123+4r2' These variables are now Earlier we used the variables £12 and £123. seen to be _ 1 . _ iii 212 " (5512' 3123 ‘ 2”5123' £1234 " 2 52' The definition of the r.m. s . radius is °° 2 z z -B(p +0 +1) ) z z z 2 foe ‘2 123 123‘ (912+Piza+91234)Pizpizspizudpizdpizadpizu A = 1 a 4 °° -B(pz +pz +p7' ) z z 2 L e ‘2 123 "'34 P12912391234d912d9123d91234 If the integrals are carried out, the expression for B is 2 i = .42 fm’. 2 AC? aolco fl: 59 The nuclear wavefunction also requires a size parameter. For the 1 Woods-Saxon well the usual A3 prescription was used with r = l . 22 fm 0 and a diffuseness of .72 fm. The size parameter for the harmonic oscillator can be determined from the electron scattering r.m.s . radius measurements with the relation 21 (R Z Nn£(2n+2£+3/2) = _1 Zar n2 which follows from the virial theorem. The sum is over protons only and Nn is the proton occupation number for the nE-orbit. The center I. of mass motion is neglected in deriving this relation. Therefore, it cannot be expected to give accurate results for light nuclei. For example , the size parameter for the a-particle that was calculated exactly is 33 percent smaller than that which is arrived at if the above relation is used. Typical size parameters for the periodic table are given in Table II. 3. TABLE II. 3 TYPICAL OSCILLATOR SIZE PARAMETERS j — Nucleus Radius (fm) a) 0(fm-2 16o 2.718 .3046 40Ca 3.482 .2474 58m 3.764 .2420 9°an 4.265 .2062 208Pb 5.498 .1900 aRadius parameters are electron scattering results from Refer- ence 11. 26 . 60 The difference between the size parameters, a and B, is a meas- ure of the importance of the Is, 25, 35 , . . . internal triton states. A table of structure factors which show the effect for two nucleon trans- fer has been published by Glendenning (11.28) . Similarly, a three nucleon structure factor may be defined by F121, 13 Z F1. 1. 1.: 2 IE 1 1 1 1 _ Z s = 0 — — - — 0 v1 2 2 2 2 m1+6n1n26£1£261112) vzv3 L512 13 1 11 12 £12 - <(N'-v2)112v2012:1n111n212 112)“ . . ._ 3/2 E 9/4 (VIZVBOJI (N v2)£12n3£3: '23) 1(2 (2) szv3> where Av v is given by Equation 11.26. The form factor is given by 2 3 ._ ii 3 fzgl) _ Esv10v10/33511. Table 11.4 is a set of structure factors for all (0f7/2)3 configura- tions. In general the structure factors increase as the number of nodes in the radial wavefunction increases . They decrease as the intermedi- ate coupling value increases if j=£ + 5, but have the opposite behavior if i=1. - %. An alternation in the magnitude of the structure factors is also apparent. For example, the 19/2- transition has a larger structure factor than the l7/2- transition. Similarly the 15/2- structure factors are larger than the 13/2- values . The actual strength of a pure configuration is dependent on the reaction kinematics . Examination of the structure factors is not NONmmoHN. NNNNNNam. 1 NoomNmoa. moamommo. 1 _ o momNmmNH.H HNVNoNNm.H1 NNNNHNNN. oHNomooH. 1 _ v vaNmoom.H ommaNNmN.N1 ooNHHmNH.H oNNomNmN. 1 _ N omHamHNN.N Nmmmmvma.v1 HHNommmN.N HmoNNvNo. 1 _ o a m N H o 1 HH _ NHA 1N\N 1H NoNNHHNo.H NmmNoomN.H1 Nmommmmm. oHNommmH. 1 _ o NommoONo.H NNHommmm.H1 NonovNo. mvaHNaH. 1 _ a NNmHNONN. vommommm. 1 NoooomNa. HNNooHoH. 1 _ N v m N H o 1 HH _ NHA 1N}. 1H 1. movmovmo.H NHoNONNm.H1 vaoaNaN. NmoNHNNN. 1 NNomaoNo. _ a .a omHNNmNo.N oHNmoon.N1 mmNmHmHo.H vamNNNo. 1 onHNomo. _ N a N N H o 1 Ha _ NHA 1N\N 1H NmomoaHN.H omoONmmm.N1 ONoHNmoo.H NmmmNmNm. 1 HNOvaao. _ a a m N H o 1 HH _ NHH 1N\H 1H N\N u A: o .1 a2 1N\N .1. E o 122 1N\N u E o u 22 886. 1 Sam oooNN. u 932 25.3 x mmoHofiH mmoeoomam w. 3 mans. 62 HmmmNoNo.N NN¢NNN¢N.N1 _ o NNNHHmNm.o NNvoHNHN.o1 _ a a N N H o u _ NHN 1N\mH nH omNoNHmo.H NNNmNmoo.H1 _ o NoHoomoo.H NomaOoNN.H1 _ a a N N H o n _ NHA 1N\NH n H NvaomNH.H mvNNNNNN.H1 amNNomNm. _ o moNoNNNN.H NNNNNNNH.N1 NmHNoNNN. _ a vommNHHN.N mommmNNN.N1 mmNeooN6.H _ N a N N H o N _ NHH 1N\HH 1H NoomomNH.H maNNNNNN.H1 omNNvam. _ o NNNNoovo.H mNaHNNON.H1 NoaoHNov. _ a NNHmNNoN. NHNomNmm. 1 ONoaNHvN. _ N a N N H o u _ NHA 1N\m "H 1N\N .1. A: o .1. a2 1N\N .1. E o 1 22 1N\N 1 E o u 22 oooNa. 1 Sam oooNN. n 6&2 UmzcficoOlv .3 392... 63 ooNNNooo.HH _ N o z _ NH.H 1N\NH 1H oNNNNNHN.N _ o H. N N H o .1. z _ NH.H 1N\NH .1 ONE 1 n: o 1 a2 1N\N 1 E o u 22 1N\N .1 E o u 22 25$. 1 Sam RES. .1. 232 cmdcficoOIv .HH mum/HQ. 64 sufficient to investigate cross section ratios of transfers that have dif- ferent angular momentum characteristics. (0f7/2)3 configurations are tabulated in Table 11.5. TABLE II. 5 Peak cross sections for the MAXIMUM CROSS SECTIONS FOR (0f7/2)3 CONFIGURATIONS* 1T 1 112 = 0 2 4 6 1/2- (.283)1.132 3/2' (.398).796 (.107).214 5/2' (.057).076 (.114).152 (2.18).291 7/2' (1.0) 1.0 (.283).283 (.122).122 (.044).044 9/2" (.059).047 (.109).087 (.142).114 11/2' (.602).401 (.206).137 (.083).055 13/2' (.052).030 (.071).041 15/2' (.092).346 (.203).102 17/2' (.005) . 029 19/2' (.698).279 ( ’ = “DWUCK' N0 ( ) = “DWUCK/(ZI-fl)‘ *All values relative to the ]"=7/2-; £12 =0 maximum cross sec- tion. These values are the results of zero range calculations using (0f7/2) 3 Woods-Saxon form factors. The 7/2- seniority one transfer has been normalized to one and all other values are relative to it. The peak cross sections were taken to be the largest value irrespective of the angle at which it occurred. The exact values may not have much 65 meaning when comparing calculations with different angular momentum values because of the large angular momentum mismatch. However, the fact that the j> member always has more cross section than the j< member of a spin—orbit pair is general. In addition, the same inter- mediate coupling dependence that was noted for the structure factors is again evident. The effect of the orbital spins on the cross section is demon- strated in Figure 11. 7 where the (lp3/2)20f7/2 seniority transfer was seen to have ten times the predicted cross section of that of the (0f7/2)3 seniority transfer. This effect can be understood as being the result of a larger "S" component in the relative wavefunction for two nucleons in low spin orbitals than for two nucleons in high spin orbitals. The ZOBPb(D.a)205Tl and 20 8Pb(ar , p) 2 11Bi reactions should be ideal reactions with which to test this feature. Since the (p,d) reaction picks particles out of low spin orbitals , the cross sections to 205T1 should be large. The states in 21181 will be mostly three parti- cles in orbits with spins of 9/2 and should not be easily excited. The center of mass motion becomes important when comparing cross sections of states in different final nuclei. This is especially important for light nuclei. The center of mass correction is displayed for a (0d5/2)3 seniority one transfer from 24Mg in Figure 11. 8. Fri- marily the center of mass correction is a multiplicative factor, even though it could change the shape slightly. Slight shape changes are 66 Figure 11. 8 Center of mass correction for a 5/2"’ seniority one transfer in the 24M9(p,0:)21Na reaction. The top two curves show the absolute magnitudes, while the bottom two curves have been shifted to emphasize the shape differences introduced by the correction. Arbitrary Urits 67 I I I I I I —1 1': ‘- H - \ —- - ' r I/ \\ E .. \ \ .. _. / \ / \ . ’ I \ 1 I - I, :l 1 :4: E’ : _1 . '1 \ ‘ 1 1 ‘ \ .. \ Center of Mass Correction 1 1111111 / 1 (oats/.2)3 _ Corrected " "' - 1'0 2'0 3.0 41.0 53 6'0 2'0 Radius (fm) Figure 11. 8 68 evident on the first two maxima. For this case the cluster multiplica- tive factor (A/(A- 3))N+ 1/2 is sufficient correction. Since the shape change is small, it seems reasonable to apply this correction to all form factor models . If the size parameter for the a-particle is equal to the size parameter of the target potential well in the oscillator model, then the triton internal state must be a US state. In this instance, the structure factors are all zero except the one with the largest number of radial nodes . The radial shape becomes the same as that of a mass three particle in a harmonic oscillator well with a size parameter given by 30. A (0f7/2)3 calculation where both a and B were taken to be . 226 fm is compared with the Woods-Saxon mass three cluster form factor in Figure 11.9 . These two form factors are rather similar. Since the differences are small, it seems reasonable to calculate mass three cluster spectroscopic factors microscopically by the prescription given in Section G . I. The DWBJA and the (p,d) Rem Multi-particle transfer reactions are difficult to handle with the DWBA because of the large mass transfer and, hence, the large momen- tum transfer. Such circumstances give rise to finite range effects which often cannot be accounted for using the zero range approxima- tion. Most important of these effects is the recoil which is left out of the zero range approximation because the projectile, picked up cluster, 69 Figure 11 . 9 Cluster form factors for a 7/2- transfer in the 52Cr(p,a)49V reaction. Arbitrary Units 70 Radius (fm) Figure 11. 9 l- l l l I I T \ : / \ 3 : \ : Z \ I C ' Cluster Form Factors 2 ' W. S. _— 1 Ho.———- : 1- -i E E I I W W I I 1.0 2.0 3.0 40 5.0 60 7.0 71 and ejectile all move on the same line in the zero range theory. In addition, a large angular momentum mismatch can cause great diffi- culty since both the shapes and magnitudes of the angular distribu- tions are sensitive to the optical parameters , which are poorly deter- minted for composite particles , when the mismatch is severe. The finite range effects for the (p,a) reaction have been studied previously by Drisko and Satchler (11. 29) . They found very little shape dependence for the one case that they reported. We have per- formed exact finite range calculations using the code LOLA (II. 30) for a number of transitions . No spin-orbit potential is allowed in this code so the comparison to zero range calculations done with DWUCK72 (11.27) must be done without spin-orbit coupling in the proton optical parameters. The results of an 2 = 3 calculation using the microscopic Woods-Saxon (0f7/2)3 form factor from Figure 11.4 is shown in Figure 11.10. There do not appear to be strong shape changes that can be attributed to finite range effects . This is very fortunate since the spin-orbit coupling is necessary to reproduce the j-dependence that has been observed. The lack of finite range dependence may be attributed to the choice of optical parameters . To first order the largest finite range effect is to reduce the contribution of the nuclear interior to the transition matrix element. To some extent this may also be accomplished by a judicious choice of the optical parameters. It has been noted previously for the (d,a) reaction that finite range 72 Figure 11. 10 A comparison of an exact finite range DWBA calculation and a zero range DWBA calculation. The microscopic Woods-Saxon model form factor shown in Figure 11.4 was used in both calculations. The cal- culations were done without spin-orbit coupling in the proton channel. Ep = 35 MeV. 73 I j l l 1 DWBA : Finite Range vs. Zero nge IIIIIT 52Cr(p, a )49V L = 3 da' / d1). (Arbitrary Units) 11111 1111111 1 2'0 4'0 60 8'0 130 CM Angle (degees) Figure 11. 10 74 effects are minimized if the "well matched" optical potentials are chosen (II. 31) . The optical parameters which we have used have the same geometry and have real well depths of about 50 MeV for the proton channel and about 200 MeV for the a-particle potential. These are the well matching conditions. The angular momentum mismatch may be illustrated in a number of ways. A classical approximation which is frequently used is to assume that the reaction takes place at the nuclear surface and that the ejec- tile is emitted at 0° . Under this assumption the classical change in angular momentum is A2 = '1‘ (PE-Pa)l fi . For the 520r case which we have been considering, A2 is about 6. For the (p,d) reaction on Te, where the Q-value is more positive and the radius is larger, A2 may go as high as 9 or 10. Since most fl-transfers are between 0 and S , they do not meet the angular momen- tum requirements and are suppressed. Classically they would not occur at all. A quantum mechanical way of looking at this effect is to examine the elastic scattering reflection coefficients. Most of the reaction cross section comes from a few partial waves near the one which has ITLLI '5 . 5. A plot of the reflection coefficients vs . L is displayed in Figure 11. 11. The gap at I111! 2‘ .5 between the two curves is about 75 Figure 11. ll Reflection coefficients for proton and a-particle elastic scattering. 76 mamas: 083983. 896.813 mo... ww§m< Nb Figure 11. ll 77 80. In other words, this picture predicts an even larger angular momentum transfer is favored. Another point is apparent in Figure 11.11. The partial waves with L less than 11 for the a-particle are completely absorbed. Hence, they do not contribute to the elastic scattering. A study of 40 MeV a-scattering from Ni has shown that the lowest 14 partial waves can be ignored without changing the elastic scattering calculations by more than a few percent (11. 32) . If these partial waves turn out to dominate the (p,a) cross section, the DWBA may be sensitive to changes in the optical parameters and the nuclear interior. The magnitude of the contribution of the individual a-particle partial waves to the cross section can be determined by examining the radial integrals. This has been done by Stock gal. (11. 33) for the (3He,a) reaction. It is better to go one step further and look at the quantity L -L -2 A = | 10 p 0 l l i“ l 2 +1 La 121.2 (LP szllp2)(La(2 mHmlegH La ) p p L 0 L (L OEOIL 0)\/2(2 +1) (2L +1)(2L 1) £0 1 a IESj O + _ a p j a p I IprLaLa L9 5 I9 The radial integrals are 1131 S * ( 1231‘ 1 ( )d d = X r r x r r r . IprLaLa LaLa a) l Ipr p p a The quantity A includes all the radial matrix elements that go with the La partial wave. Previous workers have chosen to examine only one 78 term of this sum. For the sake of calculation, m can be taken to be 1/2. Figure 11.12 is a graph of A vs. La for a few angular momentum transfers . The two different curves in each box are for the two differ- ent 1 values that are associated with the E-transfer. The first thing to notice is that essentially all the important La values are the ones that are poorly determined by elastic scattering. The only exception is the Z = 9 transfer. Perhaps the most striking feature of these plots is the difference between the 1/2- and 3/2- curves. It is not surprising that the DWBA predicts these transfers to have very different angular distributions. The 1/2- transfer is dominated by the sixth and seventh partial waves , while the 3/2— transition is spread over a much larger group of partial waves. The radial matrix elements are the same for these two trans- fers since they do not know about the j-value unless the form factor depends on 1. However, some of these radial matrix elements do not contribute to the sum for the 1/2- transfer because they do not satisfy the selection rule implied by (L002 5| IP21 . The possible values for Lp are La+l and La-l. The possible values of Ip are Lp+§ and Lp-é. If 1p is Lp-é, then all the radial integrals with Lp=La-l will not be allowed since this implies that Ip=Lp- 3/2. Similarly for Ip=Lp+%, the set of radial matrix elements with Lp=La+l cannot contribute. For the 3/2- transfer all the radial integrals will contribute to the sum. 79 Figure 11. 12 A plot of IALal vs. La. IALa l is proportional to the contribution of the La partial wave to the cross section. The calculations are for the SZCr(p.a)49V reaction at Ep =35 MeV. 80 W (Arbutrcxy Units) w (Arbitrary Units) 3 3 )Al (Arbtlray UnItS) [A] (Arbitrary Units) 1 =3 11 11 1 =2 5‘ 11 \>.\ '0" \\'~.\ §\‘E\\ .- - . \ “x. 01/ \l’/ a: \. >. Riki --« < I ”,H ' ‘ l_ 4 .‘ ...... 35““ I D 5 ./ I, 75-) /.4—”” x-i /,-,-/-” S g ./ . , I 511/ : if I 3 3 Figure 11.12 ”*1 uo aouapuadag uouoag 55013 81 The j-dependence which is so apparent in the 2 =1 graph is much more subtle for the other three cases in Figure 11.12. This is because many partial waves contribute to each term. Even though one less pro- ton partial wave is involved in the j< sum, there are still enough par- tial waves involved to give a broad distribution of strength. This would seem to indicate that the j-dependence may be the result of simple angular momentum selection rules. To test this hypothesis the sign of the spin-orbit potential in the proton channel can be changed. Since the sharply peaked distribution for the 1/2— transfer is basically the result of angular momentum selection rules changing the sign of the potential should not change this distribution much. Therefore, the angular distribution should not change. Figure 11. 13 shows the angular distributions for the 1/2- and 3/2— transfers when the spin-orbit sign is reversed. The angular distribu- tions are nearly reversed. The oscillations are nearly gone in the 1/2- curve where the 3/2- curve now has peaks and valleys which were not previously apparent. Therefore, the j-dependence is not caused by the number of radial matrix elements that are allowed. The only other effect that the spin-orbit potential has is to change the shape of the proton optical potential. The effect is something like increasing the radius of the real volume term for the j> member and decreasing the radius for the j< member. The j-dependence can be investigated by turning off the spin-orbit force and changing the radius 82 Figure 11. 13 DWBA calculations for the 52Cr(p.ar)49V reaction with the wrong sign for the spin-orbit potential in the proton channel. Ep = 35 MeV. Arbitrary Units 83 T I r I C.M. Angle (degrees) Figure 11. l 3 - 4 _ I DWBA Wrong Sign for Vso :- l/2 - 3/ 2 _ I 1 2'0 40 ab 80 100 120 84 of the real well in the proton channel. The results of this procedure are shown for an 2 =1 transition in Figure 11.14. The 1/2- structure is apparent for small radii, while the washed out 3/2- shape appears for larger radius parameters. This appears to be the source of the j-dependence. j-dependence still remains a puzzle, even though we seem to have isolated the source. The above discussion seems to imply that there should be a larger j-dependence for the (p,d) reaction than is observed. It seems clear that the a-particle must play an important role in the j-dependence in the (p,a) reaction while the deuteron is less important to j-dependence in the (p,d) reaction. Qualitatively this may be understood, if the a-particles come from the nuclear sur- face , since changing the proton optical potentials in the way that we have causes the region of high proton flux on the back surface of the nucleus to change rapidly across the surface. K. Conclusions Two methods of calculating microscopic form factors for the (p,d) and (d,p) reactions have been found to produce consistent results . When hankel tails are matched to form factors calculated using har- monic oscillator wavefunctions, the form factors have been shown to be nearly equivalent. The correction for the center of mass motion has been carried out on the harmonic oscillator model. This correction is mainly a 85 Figure 11. 14 DWBA calculations for the 520r(p,a)49V reaction without the spin-orbit potential in the proton channel. The real well radius of the proton optical potential is varied to mock-up the effect of VSO' Ep= 35 MeV. da/ d11(Arbitrary Units) 86 leUYII No Spin - Orbit Coupling r0= LSOfm r0 = 105 fm. 1111 1111L I l l l 20 40 60 80 100 C.M. Angle (degreeS) Figure 11. 14 1 120 87 multiplicative factor. Hence, we conclude that a reasonable form fac- N+ [/2 where N is the tor can be corrected by multiplying by (A/(A- 3)) number of nodes in the form factor. It has also been shown that a mass three cluster spectroscopic factors can be calculated microscopically using the harmonic oscillator model. The form factor for a mass three particle in an oscillator poten- tial has been found to be similar to that of a mass three particle in a Woods-Saxon well if a hankel tail is matched to the oscillator form factor. We conclude that spectroscopic factors calculated by using either cluster form factor model can be calculated microscopically by the method given in section six. The cross section of pure configurations on the angular momentum coupling has been investigated. It has been found that the 1) member of a spin-orbit pair is always predicted to have more yield, if coher- ence is neglected. The use of zero range DWBA has been tested and found to be adequate. Finite range calculations indicated that angular distribution shapes are not affected by finite range effects. Angular momentum matching has been studied. The (p,d) and (a,p) reactions have been found to be severely mismatched so that detailed fits are not expected to be good and sensitivity to optical parameters is expected . 88 Finally the cause of j-dependence has been investigated. It appears that j-dependence is not a result of differences in the number of partial waves that contribute to the j< and j> transitions. The j-dependence seems to come from the change in shape of the real well in the proton optical potential which is caused by the spin-orbit force. II. II. II. II. II. II. II. II. II. II. II. II. 10 ll 12 REFERENCES FOR CHAPTER II R. Sherr, Proceedings of the Conference on Direct Interactions and Nuclear Reaction Mechanisms , University of Padua, edited by E. Clementel and C. Villi (1962) , 205. I. A. Nolen, Ir. , Ph.D. thesis, Princeton University (1965), unpublished. I. A. Nolen, Ir. , C. M. Glashausser, and M. E. Rickey, Phys. Lett. fl (1966), 705. L. L. Lee, Jr. , A. Marinov, C. Mayer-Boricke, I. P. Schiffer, R. H. Bassel, R. M. Drisko, and G. R. Satchler, Phys. Rev. Lett. EL (1965) , 261. R. I. Petersen and H. Rudolph, Nucl. Phys. A241 (1974), 253. Michel Vergnes, Georges Rotbard, Jacques Kalifa, and Genevieve Berrier-Ronsin, Phys. Rev. 010 (1974), 1156. Yong Sook Park, H. D. Jones, and D. E. Bainum, Phys. Rev. g_7_ (1973), 445. R. K. Bhowmik, R. G. Markham, M. A. M. Shahabuddin, and I. A. Nolen, 1r. , Bull. Am. Phys. Soc. 20 (1975), 1164, private communication. D. F. Bayman, Nuclear Spectroscopy with Direct Reactions 11, Argonne National Lab. Report ANL-6878 (1964) , 335. R. O. Ginaven, Ph.D. thesis, Massachusetts Institute of Technology (1966) . H. I. Mang, UCRL-893l Lawrence Radiation Lab. Report (1959). R. O. Ginaven and A. M. Bernstein, Nucl. Phys. A154 (1970), 417. 89 II. II. II. II. II. II. II. II. II. II. II. II. II. II. II. II. II. II. II. 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 90 S. H. Suck and W. R. Coker, Nucl. Phys. A176 (1971), 89. Tetsuo Kammuri and Hiroshi Yoshida, Nucl. Phys. A137 (1969) , 641. I. D. McCullen, B. F. Bayman, and Larry Zamick, Princeton University Technical Report NYC-9891 (1964) . I. W. Smits, F. Iachello, R. H. Siemsen, and A. van der Woude, Phys. Lett. 538 (1974), 337. I. W. Smits and R. H. Siemsen, Nucl. Phys. A261 (1976), 385. W. R. Falk, Phys. Rev. _C_§ (1973), 1757. W. R. Falk, A. Djaloeis, and D. Ingham, Nucl. Phys. A252 (1975), 452. Y. F. Smirnov, Nucl. Phys. 21 (1961), 177. B. F. Bayman and A. Kallio, Phys. Rev. _1_5_6 (1967), 1121. Taro Tamura, Physics Reports _1_4_C, 2. R. G. Markham, private communication. M. Ichimura, A. Arima, E. C. Hilbert, and T. Terawasa, Nucl. Phys. A204 (1973), 225. C. R. Dingham, K. van der Borg, R. I. deMeijer, and A. van der Woude, Bull. Am. Phys. Soc. (April 1976). C. W. de lager, H. deVries, and C. de Vries, Nuclear Data Tables fl (1974), 489. P. D. Kunz, unpublished. N. K. Glendenning, Nuclear Data Tables. R. M. Drisko and G. R. Satchler, Phys. Lett. _9_ (1964), 342. R. deVries, unpublished. R. M. Del Vechio and W. W. Daehnick, Phys. Rev. _C_6_ (1972), 2095. 91 11.32 R. M. Drisko, G. R. Satchler, and R. H. Bassel, Phys. Lett. 5 (1963), 347. 11.33 R. Stock, R. Bock, P. David, H. H. Duhm, and T. Tamura, Nucl. Phys. A104 (1967), 136. CHAPTER III EXPERIMENTAL CONSIDERATIONS The (p,a) reaction presents a number of experimental problems . The bombarding energy must be sufficient to insure that the reaction proceeds by direct pick-up. In addition, it is desirable to be looking at a-particles that are much more energetic than the boil-off and decay a-particles so as to not be hindered by background from these processes. Sherr £511. (111. l) have found that proton energies above 17 MeV are sufficient to observe direct reaction a-particles for nuclei in the nickel region. The density of final states is frequently very high. An energy resolution of 20 keV FWHM, or better, is desirable to resolve a reasonable number of states. Typical cross sections are of the order of lOub/sr. The strongest peaks may be as large as ZOOpb/sr. , while many weak transitions may be observed at the lpb/sr. level. In order to observe such small cross sections , a system to cleanly identify the a-particles in the midst of a sea of other reaction products is required. The 35 MeV proton beam was chosen because it satisfies the experimental requirements and because it is a reliable beam for the 92 93 M .S.U. cyclotron to produce. 35 MeV is sufficiently above the Coulomb barrier, even for lead, to insure that boil-off a-particles are not a problem. Many high resolution (p,p') experiments have been performed in the last couple of years using this beam. Energy resolu- tion as good as 1.5 keV FWHM has been obtained with this beam in test situations. The cyclotron and beam line settings for this high quality beam are highly reproducible. Furthermore, beam currents of 2 to 3 11A on target are obtained with relative ease at this energy. An Enge split pole magnetic spectrograph is an ideal instrument for studying low cross section reactions with good energy resolution. The spectrograph aperture may be as large as l . 2 msr. without degrading the resolution to worse than about 5 keV FWHM at 30 MeV particle energy. The small cross sections require the largest possible solid angles and the highest possible beam currents. The (p,d) reaction is a bit more difficult to study with a spectro- graph than most reactions because protons and a-particles have the same magnetic rigidity (m/qz) causing protons and a-particles of the same energy to be focused at the same place in the focal plane of the spectrograph. Most (p,d) reactions have Q-values near zero, some slightly negative and others a bit positive. Thus the region of the focal plane which contains the a-particle spectrum is riddled with strong proton groups. In addition, there is a continuum of lower energy deuterons and tritons in the same region. The focal plane 94 detector must be able to cleanly identify a-particles while rejecting protons at a high rate. Photographic emulsions may be used in the focal plane. These emulsions may be purchased with various sensitivities. Since a- particles are highly-ionizing, while protons leave a minimum ioniza— tion, insensitive emulsions are called for. Ilford K,1 emulsions (III. 2) were used for this purpose. These emulsions have a low enough sensi- tivity that protons and deuterons pass through without leaving a track. Unfortunately, it was found that tritons did leave tracks that could not be distinguished from the a-particle tracks. It may be that Ilford K,0 emulsions are better for this purpose. Photographic emulsions have the advantage of being the highest resolution detectors available. In addition, they do not require expen- sive electronics to operate. However, they must be scanned with off-line microscopes. This process usually takes a number of months for a complete angular distribution. In addition, the triton background can obscure very weak peaks . Therefore, a livetime counting system is desirable. Most of the spectra taken for this work were recorded with the focal plane counter developed by Markham and Robertson (III. 3) . Figure 111. 1 shows a cross sectional view of this counter, while Figure 111. 2 shows a top view of the front chamber. The ionization track left by the particle is multiplied at the anode wires. Charge is 95 Figure 111 . 1 Cross sectional view of the focal plane counter. /////////// \\\\\\\ \\\\ /\\\\\ /////\ 97 Figure 1H . 2 Top view of the focal plane counter. N . H: 0.59m awhznoo ..omdomImSS= “.0 A305 :2»: o_._.t_aE< A1018 55.61 11 I 1. .. 111w:10_.a1 gj _aoo... mo.._>> Ego \ \ N 205501. $29.09 1..\ 1.31.42... //Nao._. 2.3 >23 8. 30053 3333.... /e 20:80 99 induced on the cathode pick-up stripes below. The induced charge is collected by a delay line. The signal from one end of the delay line is used to start a time to amplitude converter, while the signal from the other end of the delay line is delayed for a period which is longer than the total delay of the delay line and then used to stop the TAC. The resulting pulseheight is the position of the track. The second chamber contains a conventional single wire proportional counter. The signal from this wire is proportional to the energy loss of the particles that pass through the chamber. The a-particles may be identified because they have a larger energy loss than protons, deuterons , tritons , or 3He ions . Further restrictions are necessary to eliminate all the pro- tons . Because of the tremendous number of protons , there are an intolerable number that have the same energy loss as an a-particle even though the probability of such events is very low. The added restriction is obtained by using a plastic scintillator as a third counter mounted behind the exit window of the proportional counter. The anode signal from the photomultiplier can be used to start a TAC which is sub- sequently stopped by the cyclotron r.f. The resulting signal is a measure of the flight time of the particle through the spectrograph. The time-of-flight requirement easily distinguishes protons from air-particles. Very clean spectra can be obtained if the energy loss and the time-of- flight requirements are used simultaneously. The system count rate is limited by the proton rejection rate if the elastics are on the counter. 100 The targets need to be S 100 pig/cm2 thick in order to keep the energy loss of the a-particles from degrading the energy resolution. The targets were made by reducing metal oxides and evaporating the liberated metal. The vapor was condensed on 20 pg/sz carbon foils. The details of each experiment are presented in the fourth and fifth chapters . REFERENCES FOR CHAPTER III III.1 R. Sherr, Proceedings of the Conference on Direct Interactions and Nuclear Reaction Mechanisms , University of Padua , edited by E. Clementel and C . Villi. III.2 Ilford Nuclear Research, Ilford Limited, Ilford Essex, England. III.3 R. G. Markham and R. G. H. Robertson, Nucl. Inst. and Meth. 129 (1975), 131. 101 CHAPTER IV FEATURES OF THE 52Cr(p,ar)49V REACTION A. Introduction The (p,d) and (d,p) reactions may prove to be very useful spectro- scopic tools. The qualitative features of these reactions are not well documented , with the exception of j-dependence for 2 =1 transfers (IV. 1 , IV.2, IV. 3, IV.4) . In this chapter the qualitative features of the (p,d) reaction as seen in the SZCr(p,a)49V reaction are investigated. Since three nucleons are transferred, it is possible to study final nuclei that are not accessible by other pick-up reactions either because the targets for these reactions are unstable or difficult to make. Final nuclei in this class are 47V (IV.5, IV.6), 51Mn(IV.6, IV.7, IV.8) , 5500 (IV.6, IV.9) , and 119Te (IV.10). To understand the spectra of these unknown nuclei, it is necessary to document the prOperties of the (p,a) reaction on nuclei that have been previously studied with simpler reactions. The 52Cr(p,ar)49V reaction is a good choice for such a study in the 0f7/2 shell because 49V has been studied by a number of others (IV.11, IV.12, IV.13, IV.14, IV.15, IV.16, IV.17, IV.18, IV.19, IV.5). Previous work in this mass region at beam energies above 17 MeV has shown that the simple proton hole states that are populated in 102 103 single proton pick-up reactions dominate the (p,a) spectra (see Refer- ence IV. 2, for example). These states seem to be described reasonably well with seniority-one wavefunctions (IV.2) . Therefore, we should expect the 49V spectrum to display strong peaks for the 7/2- ground state and the 3/2+ and 1/2+ sd-shell proton hole states. In addition to the T=3/2 proton hole states, the T=5/2 analogs of the neutron hole states in 49Ti should also be populated. These states are not isospin allowed in the 50Cr(d,3He)49V or 50Cr(t,cir)49V reactions. Experimental observation of analog states with the (p,a) reaction has not been previously demonstrated except for some tenta- tive assignments by Bardin and Rickey (IV.20) using Ti isotope targets. Multi-particle transfer reactions offer the chance to study high spin states. For some time now the (a,xn'y) reactions have been used to populate such states. Recently heavy ion induced reactions such as (19F,p2n‘Y) have been used to find high spin states such as the 12+ in 44Ti (ll/.21) . If two 0f7/2 neutrons and a 0f7/2 proton are picked up via the (9.0) reaction, it is possible to reach final states via I1r trans- fers of up to 19/2-. The (pm) reaction on 51V could, in principle, then, directly populate the 12+ in 48Ti. If the proton comes from the 0d3/2 orbit, 15/2+ is the maximum I1r transfer. A study of the 90'92'94'962r(p,ar) reactions has concentrated on this aspect 0f the reaction. Spins up to 15/2- were observed in that work (IV. 22) . The maximum coupling of (0d5/2)3, which is 123/2+, has been observed in 23Na(p,a)20Ne, 120(a.p)15N, and 160(a,p)19I-‘ «v.23, IV.24, 1v.25). 104 Lee _et__a_l_. (IV. 1) have shown that the j-dependence is a result of spin-orbit coupling in the proton optical potential. The j-dependence for the 1": 1/2-, 3/2- spin-orbit pair is reproduced by DWBA calcula- tions using mass three cluster form factors (for example, see Refer- ence IV. 22) . The reliability of j-dependence for higher E-values is still an open question. Studies of the 2 = 2 and 3 transfers are con- fusing (IV.4 , IV. 26) . Much of this confusion is apparently the result of important structure effects in the sd-shell. A study of the 24 ' 26 )21’ 23Na reactions shows that the angular distributions Mg(p,a for states with the same 17r values sometimes have very different shapes (N. 27) . It would seem that j-dependence will only be a useful tool if the shapes of the angular distributions are insensitive to the detailed structures of the states . This may be the case for targets that are heavier than those in the sd-shell. The most common method of using the DWBA to predict the shapes of angular distributions for (p,d) and (a, p) studies has been to do zero- range calculations which employ mass three cluster form factors. These calculations can fit the data reasonably well in many cases . In regions where nuclear structure does not affect the shapes of the angular dis- tributions, it may be possible to use these calculations to make I" assignments (IV. 22, IV. 28) . Recently a few microscopic reaction models have been developed (see Chapter II and IV.24, IV.29, IV.30, IV. 31) . Such models may 105 make it possible to predict both shapes and magnitudes of the angular distributions even when nuclear structure effects are important, pro- vided detailed wavefunctions are available. In the sections to follow we will look into the general features of the data , try to evaluate the reliability of the 2 =2 and 3 j-dependence for this case, check the use of the DWBA using cluster form factors , and test DWBA calculations based on the microscopic form factors described in Chapter II. B. Experimental Method and Data The 35 MeV proton beam from the Michigan State University isochronous cyclotron was used to bombard an isotopically enriched 52Cr target. The reaction products were momentum analyzed in an Enge split pole spectrograph and detected with the delay line counter developed by Markham and Robertson (IV. 32) . Position and energy loss information were taken from this counter, while a plastic scintillator placed behind the counter was used to obtain particle time-of-flight information relative to the cyclotron r.f. structure. The a-particles were cleanly identified by their energy loss in the counter and their time-of-flight. An over-all energy resolution of 20 KeV FWHM was obtained with this system. The target thicknesses were typically 20 to 40 ug/cmz. The thicknesses were measured by comparing proton elastic scattering on the second maximum of the elastic scattering angular distribution to 106 the results of optical model predictions . Targets S 100 pig/cm2 were necessary to keep the energy loss of the a-particles to a minimum. However, the thermal coefficient of expansion of Cr presented a further constraint on the target thickness. The targets were made by reducing SZCrZO:3 with tantalum and simultaneously evaporating the liberated Cr. The Cr was deposited on 20 ug/cm2 carbon foils. When the target reached a thickness of ~50 [1.9/sz , the backing would break thereby imposing an upper limit on the target thickness obtainable by this technique. Because of the thin targets , large solid angles and high beam currents were necessary. For the most part, a solid angle of 2.0 msr. and a beam current of 2.5 uA were used. A few spectra were also recorded on photographic emulsions in order to obtain more precise values for the excitation energies and better resolution. One of these spectra is shown in Figure IV. 1. The resolution is about 10 keV FWHM . The three strong peaks are the 7/2- ground state, the 3/2+ proton hole state at 0.748 MeV, and the l. 646 MeV, 1/2+ proton hole state. In addition, there is a tall peak due to the 1.95 MeV 5/2+ hole state in 29P which is the result of 32$ impurity in the target. The wide peak near channel 1650 is the 13N ground state, which is kinematically out of focus. A log plot of the same spectrum is shown in Figure IV. 2. It is immediately evident that there are a great number of weaker states in 107 Figure IV. 1 The 52Cr(p,ar)4‘9V spectrum at 16° plotted on a linear scale. The spec- trum was recorded on a photographic emulsion. Ep=35 MeV; FWHM ~ 10 keV. 108 H .2 239m .5952 3555 8.:w 8mm 8.8 82 89 8.: 8m; 82 8m 8m 8: com a p p a f «:2. A. . L , a... _ t . _ a .. 02‘? I”? 9m 88!? I £641. 6:: lien“ 0M0 W0 1 9B1 .30 mg >mL8dfo~m _ p _ p _ p _ b _ p _ _ _ . _ L P _ _ . b p b . com |auuoq3/9iun03 109 Figure IV. 2 The same spectrum as Figure IV.l but plotted on a log scale. 110 Inn I 1 I 111] I corw ooNN cooN comm b p L .- b L A a .80 m: N .2 was; .8952 .23on 82 8i 8- 82 8m 8m 8.. _ fl_ __ *8 _ , , E , _. m _, _,_ , m .... 22.33.58. am am W0 1 lllllll 111114 2 oo— leuuoq3/9wnog 11 1 addition to the proton hole states . The background counts are probably triton tracks on the plate that could not be distinguished from the Cir-particle tracks . Ilford K,1 emulsions (IV. 37) were used for this work. These emulsions are insensitive to the inelastic proton groups which struck the plate, but discrimination against tritons was not possible. A spectrum taken at the same angle with the counter system is shown in Figure IV. 3. This spectrum is much cleaner because the tri- tons cannot satisfy the time-of-flight requirement. In this case the resolution is about 20 keV FWHM . In order to look for the T= 5/2 proton hole states, the spectro- graph field was adjusted so that the high rho end of the counter was located at approximately 4 MeV excitation energy. This allowed for about one and a half MeV of overlap with the lower excitation spectra. Typical spectra for the high excitation region are displayed in Figure IV.4. At forward angles the break-up of 9B, made by the 12C(p,ar)QB reaction, causes a large background as can be seen in the top half of Figure IV.4. The bottom half of Figure IV.4 contains the 55-degree spectrum where the 9B has kinematically shifted out of the way. The three sharp peaks are the 7/2-, l/2+, and 3/2+ T=5/2 49 analogs of states in Ti. The excitation energies of these levels are compared with the corresponding levels in 49T1 in Table IV. 1 . 112 Figure IV. 3 The 52Cr(p.ar)49V spectrum at 16° recorded with the counter system. Ep = 35 MeV: FWHM ~ 20 keV. 113 ooom com— com— cord m .>H 930$ 3922 _accozo 8m. 82 a— can lauuoug/swnog 114 Figure IV. 4 High excitation spectra showing the T =5/2 proton hole states in 49V. 115 sow - .8 £32 mac—Eon? 30¢ 82 8...— _ r b . L - mm m n a - mg m a - 80 mm m. _ _ p _ b p . _ T b . b h p _ r . ... .. m m m m . m m m M - .moo rm _ . b p b F _ b L . _ . b . L r coaom coIotoxm £9: >mr§df0mm leuuoug/stunog Figure IV. 4 116 TABLE IV.1 T = 5/2 ENERGIES 49V Excitation EC Differential 4 9T1 (MeV) 7/2' 6.4461.020 7.8481.03 —-- 0.000 1/2+ 8.9471.025 7.8481.03 2.50 1.010 2.50 3/2+ 9.0881.025 7.8301.03 2.5421.010 2.66 NOTE: Ec/Z< = .357 for 7/2'. Candidates for high spin states can be identified by looking for large peaks in the back angle spectra. Figure IV.5 is the 60-degree spectrum. Peaks due to levels which have lower spins become weak as the angle increases , while the higher spin states have relatively flat angular distributions. The peak at 4. 797 MeV is a good candidate for a high spin state. In the spectrum at 12 degrees (Figure IV. 1) , it is comparable in yield to many other states, while at 60 degrees it is the strongest peak. The angular distributions that have been obtained are displayed in Figures IV. 6, IV. 7 , IV. 8. The high spin type of angular distribution is evident for the 4.797 MeV level. This angular distribution is even seen to rise in the low angles as the angle increases . C. Comparison with Other Experiments A summary of the energy levels observed in this experiment is presented in Table IV.2. The table also contains a summary of the data 117 Figure IV. 5 The 52Cr(p.ar)49V spectrum at 60° showing candidates for high spin states at 3.612, 3.745, and 4.797 MeV. 118 m.>H 939m LGQEDZ _oCCOr—U 8w 1 £2 . 8121 8.: 82 82 80 8o emu - i, . 4 .4 «I . . 2 _ m m. 1 n m a 0 e . 0 n .... . m a . a w 2 a .30 cm 31.3.2108. p p b b p b p b b p p p p p b F on: on“ leuuoqg/swnog 119 Figure IV. 6 52Cr(p ,a)49V angular distributions . 444 4 4 14444 4 4 144414 4 4 1444444 4 4 144444 4 4 1444444 4 4 11444 4 4 144414 4 4 144444 4 4 1.44444 4 4 14444 4 4 144444 4 4 14444 4 4 14444 4 4 V V V O V W V m 0 m m N 1‘. O M v u. o 3 .Y. 8 0 3 O 7 2 3 r m S 6 V. 7 o J 9 . v 9.. V 2 d n v m 0 ..u ..u 0 E m. ... . .. .. ... c v 0 V v V. V 3SMeV 120 52Cr[p,o<)49V Ep — o I. r V o 0 m V I. w 2 M 2 4.. H V / 0 M s l. 3 2 7 N In I O m I i m I 9 1 J O L f .1 ..u .m .. o a. .l C I i I M o o i o o E O h V. x m ”C 1! E 9" ... o ! . u «M. ... _r'lfblllo‘lrnfllndrp>>> > 1 I01» 1 r 1}???» > > 1 11>1 > . . o . . m . w w o w m w o o 0 V 2 v V H c I 2 m h V V O I. V C 0 v w .. .0. 5 .. .1 W a o J 0 «rs o '0 I o . I5 ' - O Q 0 V C . 0 0 c 0 c 0 J O V d m c v J J o .. v I. J i I I V I ! I .... C an V V c o o x E E .V. E on . V E ‘ “c ' ' c 'V C. V. 'V' 0 V I V m V ' " i... ! ' D1 1 1 DE?” D 1’ IFFDVD P 1 r ’1 V D {1” F k {1” 1 E FDFI art}, D “D1?" D 1 “I?” b 1 In 2 In an _ in . . I. In . c . . o .m. o .m ..n... o o m o w w ....\e£ iuu\bui Figure IV. 6 121 Figure IV. 7 52Cr(p.cr)49V angular distributions . 1 q q 1444444 4 4 144144 4 4 4 144444 4 4 1.444444 4 4 144441 1 14444444 4 1444444 4 11444444 4 1444444 4 4 14.444444 4 44444444 4 3SMeV 122 $2'Cr[p.o(]‘+9V Ep W v v v u ... ... m . m . w ... fl 6 o 0 8 8 7 V H: 8 2 9 s n 5 5 6 7 V 4 O r I C 9 9 B = V. 8 I V ' d 6 o o '- 0 V 8 n ..I.\ x x V L O m V E w E v E o E m m. . .. v . V O m .. ... e O V o v u art? 1'}! F 1 IUD}? ’ 1 Q r 7 D 1 1...! 7 1 1W1? b 7 1 . E7 1 D o o m m w v v w k ) V; 9 O 8 N V. H. o “u N V. H. 0 u. V. 9 ... . .. . ... 8 0 ll\ 1 .w m E E V m 0 o O V c 0 VV 0 C V O V JifflifflorhEb 1 10:11?» P r lrpu>> » u aripp b 0 . . m m o w v V v .0. m v m I. o v M .0. ... 2 v 1 n d n ...... o v. v e ... u I. a v a o J v x o r ... V V 3 V 3 O 9 .. V .. O V V 0 v . o 1 v a o .M a V x o X E ~ I V . O E v v x o v o E v E m V . V o .. . . .... .. ... V rift>b > > 1.0-11>» » p 4....»th 1%» 0 0 0 0 O .0 I. I. 0.. It ...}... a..\5.. Figure IV. 7 123 Figure IV. 8 52Cr(p,ar)49V angular distributions . 124 52Cr[p o<]L+9V10Ep= 35MeV 1 r T f f# E r f ff r Y Ex=S.OSH MeV Ex=S.HHS MeV 0.1 1 J'=7/2'.T=S/2 ’2 V V 0 000 10 I 3V" 0.1 %g 0 v 006 v ’g 1 1 \ Ex=5.l3H MeV Ex=8.SHS MeV D 0.1 0.1 “"1/2"T=5/2 E i V V "‘ 05° i ' C3 10"2 W 10’2 'V '7 U "o l v E 0 o o '0 1 W 1 Ex=S.282 MeV ‘ Ex=3.087 MeV J'=3/2*.T=S/2 v o “v i 0 Co 10"2 "’8 + 10"2 v 00° i 0 20 H0 60 0 0 0 H .0 :u 9c.m. [degrees] Figure IV. 8 125 TABLE rv.2 LEVELS or 49v #47 (p.415) (p,t)b’ (16)") <3He,d)°’ (pmd’ (awe) 1” 0.000 0.000 0.000 0.000 7/2_ 0.091 0 091 0.090 0.092 0 091 0 091 5/2_ 0 153 0 153 0.153 0.155 0.153 0 153 3/2+ 0.746 0.752 0.750 0.747 0.746 3/2_ 1.021 1.020 1.025 1.025 1 021 1.022 11/2+ 1.141 [1 148] 1.140 1.141 5/2_ 1.154 ' 1.155 1.155 9/2_ 1.513 1.516 1.5311» 1.514 1.515 5/2+ 1.602 1.603 7/2_ 1.644 1.643 (1/2+) 1.646 1.646 1.2_ 1.662 1.672 1.661 1.661 3/2 1.770(?) 1.7960) + 1.995 1.999 1.996 1.995 3/2+ 2.179 9/2 [2 161] 2.183 [2.169] [2.193] 2.183 7/2- 2.204 _ 2.235 2.235 2.241 2.235 2.235 5/2_ 2.263 15/2 [2.265] [2.263] [2.266] 2.265 3/2- 2.279 _ 2.306 2.306 2.314 2.317 2.309 2.310 9/2_ 2.354 2.350 2.356 2.353 9/2+ [2 394] 2.366 2.366 5/2_ 2.406 2.404 ' 2.406 7/2 2.673 2.666 2.661 2.671 _ 2.726 2.727 2.736 2.728 (15/2 ) 2.741 _ 2.766 2.766 (9/2,11/2) 2.611 2.612 2.606 2.611 2.661 2.661 3.020 3.017 3.1334 [3.133] [3.136] [3.132] [3.137] 3.1337 3.152 _ 3.241 3.241 3.246 3.246 3.237 7/2 3.259 3.305 3.330 3.332 126 TABLE IV. Z-Continued (0.416) (13.1)” (1,610) <3He,d>°’ (pmd) (a.pv)e) 1” 3.346 3.347 3.345 3.342 3.391 3.396 3.366 3.401 3.390 3.479 3.499 3.525 3.534 _ 3.612 3.609 :(11/2)a) 3.624 3.639 3.649 3.673 3.694 [3 665] 3.720 {_a) 3.745 3.746 3.744 :(9/2 ) 3.757 3.763 3.757 3.795 3.625 3.616 3.636 3.640 3.662 3.666 3.910 3.914 3.934 3.929 3.922 3.965 3.975 3.976 4.004 4.005 4.012 4.006 4.046 4.042 4.064 4.096 4.090 4.135 4.127 4.149 4.152 4.165 4.209 4.224 4.253 4.250 4.266* 4.277 4.260 4.305 4.326 4.375 4.375 4.373 4.400 4.402 4.436 4.448(?) 4.470 4.501 4.511 4.502 4.496 4.536 4.566 4.566 4.567 4.590 4.599 4.626* 4.639 4.646 4.645 127 TABLE IV. Z—Continued (p.a)a’ (p,t)” (no/)0) <3He.d)°’ (pmd) (6.11716) 1” 4.662 4.660 4.755 4.743 ) 4.797 (> 11/2)a 4.630 4.646 4.663 4.652 4.665 4.949 4.959 4.945 4.966 5.101 5.016 5.017 5.134 5.130 5.204 5.216 5.212 5.262 5.265 5.347 5.355 5.375 5.370 5.367 5.411 ) 6.446 7/2,T=5/Za 6.945 1/2+,T=5/2a) 9.067 3/2+,T=5/2a) NOTE: Errors are 1.003 MeV for states below 3 MeV, 1:.006 MeV for states above 3 MeV, and 1.025 MeV for the T=5/2 states 6This experiment. b Reference N. 11. c Reference N. 12. d Reference IV. 19. eReference N. 16. 'k The peak is an unresolved doublet. Its width is too large to be a single peak. [ ] Energy corresponds to a known multiplet that cannot be resolved. (?) Placement is unsure. 128 in the literature. The column of I” values is a consensus of the litera- ture. Although most I1r assignments are from the previous work, our results are consistent with those assignments. The assignments explicitly made in this work are the tentative "high spin" assignments and the IW,T assignments for the three analog states. To begin the discussion of Table IV. 2, consider the columns labeled (p,a) and (p,t) . Beginning at the top of the (p,t) column and working down, it is seen that the 0. 748 MeV state is the first one not seen in the (p,t) reaction. This is the 3/2+ state due to a proton hole in the 0d3/2 orbit. The next level not observed in the (p,t) reaction is the l. 141 MeV, 5/2+ state. Furthermore the 1.602 MeV, 7/2+ and 1. 646 MeV, l/2+ levels are not observed in the (p,t) data. All these levels are seen in the (p,a) experiment. A summary of the levels seen in the (p,a) data that are not in the (p,t) data is given in Table IV. 3. Table IV. 3 contains every known positive parity state in 49V except the 2.179 MeV, 9/2+ which cannot be resolved from the 2.183 MeV, 7/2- state. Furthermore there are no known negative parity levels in this list. This comparison indicates that parity assignments can be made with reasonable certainty by such a comparison. Reversing the comparison shows that there are levels seen in the (p,t) experiment that are not seen in the (p,a) spectra. The first of these is the 1.661 MeV, 3/2- state. In addition, all the 5/2- levels are so weak in the (p.41) reaction as to be virtually absent. If the (t,a) results are included in the comparison, it is found that there are levels 129 TABLE IV. 3 LEVELS SEEN IN 52Cr(P.a/)49V THAT ARE NOT IN 51V(p,t)49Va) Excitation Energy 117 1.746 3/2+ 1.141 5/2+ 1.602 7/2” 1.646 1/2+ 1. 995 3/2” 2. 366 5/2+ aReference IV. 11 . excited by the (t,a) reaction that are not in the (p,a) column. Some of these levels are in both the (p,t) and (t,a) data, but not in the (p,a) column. Behavior of this nature can only be explained by a microscopic model which contains the coherent sum over all the di-neutron cou- plings and all the three nucleon configurations. Table IV.4 is a sum- mary of the missing levels in the (p,a) data. (mm) research has resulted in a 15/2— assignment for the levels at 2.263 MeV and 2.726 MeV. Unfortunately, 17/2' and 19/2’ levels have not been found by gamma-ray spectroscopy. In the case of posi- tive parity, the highest definite I1r assignment is 9/2+. The two 15/2— levels are observed very weakly. The angular distributions are found in Figure IV.6. Although the high spin levels which are observed here have not been previously reported in a-induced gamma-ray coincidence 130 TABLE IV. 4 LEVELS SEEN IN EITHER 51v(p.t)49va) OR 50Cr(t,a)49Vb) NOT OBSERVED IN 52Or(p,a)49v Excitation Energy 17 I bobohbbwwwWQWQJNNNi—IH .154* .662* .766* .611** .661* .020* .305* .479* .624* .720* .625* .910* .046** .096** .165* .209* .305* 1/2: 3/2 _ (9/2, 11/21 (5/2. 7/2) 13/2’ _ (3/2. 7/2) 7/2’ aReference IV . bReference 1V . *Seen in (p,t) data. 11. 12. MSeen in (p,t) and (t,a) data. 131 experiments , we can be reasonably sure that the 3. 612 MeV level has (-) parity and the 3.745 MeV state is a positive parity state. Neither of these assignments is unambiguous since the 3. 612 MeV peak is too broad at forward angles to be a single state and the 3.745 MeV level has been reported in the (t,a) and (pxy) data, indicating a low spin state at this energy. The parity of the 4.797 MeV level cannot be dis- cussed since the (p,t) data does not extend to this excitation. Neither the 3.612 MeV state nor the 4.797 MeV state are particularly close to the M82 (IV. 33) predictions for high spin negative parity states. The predicted energies for 15/2' levels are 2.575 MeV, 3.544 MeV, 4.063 MeV, and 4.964 MeV. The predicted excitation for the first 15/2- is nearly 300 keV too high, while the second one is over-predicted by about 800 keV. It is not surprising that our high spin candidates are not near the M82 predictions given the poor results for the known 15/2- levels. The MBZ predictions for the two 19/2- levels with the largest "triton" components are 4. 331 MeV and 5.143 MeV. D. j-Dependence The striking j-dependence for 2 =1 transitions was not observed in this experiment because the only known 1/2- level is obscured by the 1/2+ proton hole state. The 3/2- level at 0. 153 MeV excitation does exhibit the usual featureless exponential fall-off. Two 3/2+ and two 5/2+ states have been observed. The 5/2+ state at 2. 386 MeV cannot be resolved sufficiently well to obtain its 132 angular distribution. The three 16 = 2 angular distributions are shown in Figure IV.9. The 3/2+ distributions have similar shapes. The 5/2+ is rather featureless. There is no strong j-dependence in this case, though the angular distributions are different. There is not enough data to determine if the difference is due to j-dependence or is the result of structure effects . Additional data in this mass region is necessary to document the stability of the 2 =2 j-dependence. The spin-orbit pair with 2 =3 is observed. Three 5/2- levels are populated weakly. The cross section is less than lub/sr at many points. The 0.091 MeV and 1.513 MeV levels were strong enough to obtain angular distributions. The angular distributions for these two levels appear to be different. The lowest 5/2_ level in many nuclei in this mass region is known to be primarily a seniority three proton state (IV.33, IV.34) and, hence, should not be populated in this experiment. Its weak population may indicate a more complicated reaction mecha- nism. The angular distribution for the 0. 091 MeV state is flat. The angular distribution of the 1. 513 MeV level is probably more repre- sentative of a 5/2- angular distribution. 7/2- states are observed at 0.000 MeV, 3.241 MeV and y.446 MeV excitation. Figure IV. 10 shows all the E = 3 angular distributions. The angular distribution for the 3.241 MeV level is very different from the other 7/2- levels. The 7/2- assignment for this state is unambiguous since it is based on the 2 =0 angular distribution observed in the 51V(p,t)49V reaction (IV.11) . Either 133 Figure IV. 9 520d? .a)49V L = 2 angular distributions . 0.1 do/dQ [mb/sr] 0.1 10"2 10‘3 134 SZCrtp.o<1“*9v 1:2 T W F r ' r r 1’ Ex=0.7 H8 MeV u‘=3/2" Ex=1.335 MeV 0":3/2‘ d“=S/2"' 20 ”+0 ‘ 60 ‘6'0 8c.m. [degrees] Figure IV. 9 135 Figure IV. 10 52Cr(p.ar)49V L=3 angular distributions. 136 76833 .E.o¢ D D DD D DDD D 'VT v hAAA‘A A N\wu._..uN\kuv_a >0! wrrduxm WVV 'vvv1111 v AAA A A AA v—fiv v 9 9 LLAAA as U as m . ...... H IN\K"V_J ... m >6: oywduxm A n H6 3 1 b b b b L L 1- h OH .3 0.50?” 33me .Eda Ev 1N\kuv_a >0! oooduxw N-e~ H6 a ”$833 .E.O¢ 194 Lnuln ‘7’ D H -~\mu .1 ..u N-ofi h u w L Heo W A . 6 up.» up .m- n e H W A I 1N\WNV_J 1 NIQH >0: “moduxw AAAIA A TINT-7'? v 137 this is an unresolved doublet or this is a case where structure effects can be as strong as j-dependence. If the abnormal angular distribution is neglected, the j-dependence appears to be manifested in the for- ward angles. The 5/2- tends to go down as the angle decreases, while the 7/2- rises. Two other 7/2- states are also populated. Unfortu- nately, they are members of close doublets. Only one 2 =4 transfer is resolved. This leads to the 7/2+ state at 1.602 MeV. The 9/2+ level at 2.179 MeV may be populated, but cannot be resolved from the 7/2- state at 2. 183 MeV. There are known 9/2- and 11/2- levels in the first two MeV of excitation. The 11/2- is observed, but there is no evidence for the 9/2-. Another 9/2- state located at 2. 354 MeV is barely visible in some spectra. Both j-values that go with I. =7 have been previously identified. The two known 15/2- states are weakly populated. The 13/2— level at 2. 861 MeV is not seen in this experiment. There have not been any previous spin assignments for states to be reached by :2 = 6, 8, or 9 transfers. An alternation of strength is clearly evident for negative parity states. For a given £-transfer, the j> member is the strongest. This is in qualitative agreement with the structure factors presented in Table 11.4 and the calculated peak cross sections found in Table II.5. In conclusion there appears to be some evidence for subtle j-dependence in the i=2, 3 transfers. This j-dependence may not 138 be a very reliable tool in this mass region, since structure differences may cause similar changes in the angular distributions . E. DWBjAfiCalculations-Cluster Form Factors It has been shown that zero-range DWBA calculations can be used to obtain reasonable fits to (13,0) angular distributions (see, for example, References IV.22, IV.28, IV. 36) . In addition, finite range effects have been found to produce only minor changes in the shapes of the DWBA calculations (see Figure 11.10). Mass-3 cluster form factors have been used frequently for these calculations because they are easy to generate. In addition, most researchers have found the radius and diffuseness parameters of the bound state well to be useful variables. These are usually varied to obtain the best overall fit to all the known levels. A wide variety of these parameters have been used (see Refer- ences IV.28, IV. 36) . Many different sets of car-particle optical poten- tials have been tried also. These vary from shallow real wells of about 50 MeV depth to deep wells of 200 MeV depth. For the most part, the choice of a-optical potential determines the values of the bound state well parameters that will best fit the data . A simple, consistent, method of generating reasonable calcula- tions is needed. We have, therefore, set out to find a general proce- dure that can be used to get first-order fits reliably. Since angular momentum matching is a problem for the (p,a) reac- tion, it seems reasonable to try the "well matching" procedure for 139 choosing the optical potentials and the bound state parameters . This procedure has been suggested by Dodd and Greider (IV. 37) and by Stock i611. (IV. 38) for reactions that are poorly matched. The method has been successfully applied to the (d,a) reaction (IV.39). The proton optical potentials were taken from Bechetti and Greenlees second best set (IV.40). This is the set with a 1.22 fm. radius parameter. The a-particle optical potential was taken to be a set with roughly 200 MeV real well depth. The well matching proce- dure requires that the radius and diffuseness of all the real potentials be the same. a-scattering data of Fernandez and Blair (IV.41) were refit to find an optical potential that met the well matching requirement. Since the a-elastic scattering could not be reproduced as well with a radius parameter of l. 17 fm. as with the 1.22 fm. choice, the second preferred proton set was taken. By this prescription the bound state wavefunction should be calculated in a well with r0= l. 22 £111. and a diffuseness of .72 fm. to agree with the other potentials. The well depth should ideally be about 150 MeV. However, the depth was allowed to vary to reproduce the triton separation energy. For the most part the appropriate depth was between 120 MeV and 140 MeV. The optical potentials are given in Table IV.5 . The fits that are obtained to the known energy levels using this procedure are shown in Figure IV. 11. In general they are satisfactory. The forward angle behavior of the 7/2- calculation does not increase 140 TABLE IV. 5 OPTICAL POTENTIALSa) V r a V r a W r a. W 0 so so so oi 1 sf Proton 43.22 1.22 .72 -25.0 1.01 .75 -5.0 1.32 .52 12.60 a 196.34 1.22 .72 -15.72 1.76 .42 aVSO and st include the factor of 4 necessary for using the code DWUCK45. as the angle decreases , but the data does. The dip in the 11/2- cal- culation comes at too small an angle. The "well matching" procedure has also been used in studies of Te(p,a) and 44Ca(p,a)41K with similar results (see Reference IV. 10 and Chapter V) . Most of the searching on the bound state that has been done by other researchers has resulted in smaller diffuseness parameters than were used above. If the diffuseness is decreased to . 65 fm. , the 7/2- calculation has the correct forward angle behavior. In other words , the forward angles are sensitive to the diffuseness. The fits obtained with this choice are shown in Figure IV. 12. Smaller diffuseness values were also investigated. In general smaller values were found to pro- duce more pronounced oscillations. Sensitivity of this kind is a characteristic of reactions that suf- fer from a severe angular momentum mismatch. The semi-classical matching value for the ground state Q-value is about six. The angular 141 Figure IV. 11 Cluster model DWBA calculations. The real well geometrical parame- ters were ro=l.22 fm. and a= .72 fm. da/dQ (mb/sr] 142 10”; 0.1 E 1 1 1 j’ v vvvvvv' 0.1 10'? 0.1 10'2 v v 'vvvv' v SZCrtp,o<]‘*9v Ep=35M ' V Ex.=0.000 Mov J“=7/2’ A AA‘AA .1 ‘ AAAAAAL Ex.=0.091 HOV “=5/2' A A ALLA“. Ex.=0.lS3 MeV J‘=3/2' Ex.=0.7H8 HOV .J'=3/2" Ex.=l.021 HOV J‘Ill/Z' j 0c.m. (degrees) Figure 01 10'21 eV;Clus+er Model 1 1 10'3 0.1 10'2 10'3 0.1 10'2 10'3 10°2. 0.1 0.1 - 1 1 1 1 rv "vvv" 10'2 IV.11 r V *V’ Y Ex.=l.lHl HOV A A-AAA“ .J'=5/2’ I.‘ 1‘ .. o i o 00 , 00 4 Q 9; Ex.=l.513 NOV '3 u'=s/’2‘ I ! 1 I 1 i 1 Ex.=l.802 HOV i -=7y/2‘ 1 -9 ; Q §i 1. Ex.=1.SH6 NOV 1 a. 0 O" J 1/2 Ex.=l.995 HOV 0'83/2’ 4 0c.m. [degrees] 143 Figure IV. 12 Cluster model DWBA calculations. The real well geometrical parame- ters were ro=l.22 fm. and a= .65 fm. 0.1 10" 0.1 0.1 da/dQ [mb/srl 0.1 0.1 144 52Cr[p,o<]L’9V Ep=35MeV;CIus+er Model I” fl 7 - 'v'vw 1 1 1 1 1 1 * vv'mv' fl 0 I N 1. C C Vfiv'v' ' Y— ’7‘ fi— ffi Ex.=0.000 HOV J'=7/2' Ex.=0.091 HOV J‘85/2‘ Ex.=0.153 HOV J'83/2' Ex.=0.7H8 HOV J'I3/2° £331.02] HOV J'Bll/Z' 6c.m. (degrees) Figure IV, 12 ‘— A AAAAAA.‘ A A AAAAAAA A A AAAAAAI A AAAAAAAA _A AAAAAAAI A A A‘AAA‘ 0.1 10' 0.1 10' U. V rv'fiwf v - 0.1 E 0.1 - 0.1 r v v v v v f v j’ Ex.=l.1Hl HOV J"=II‘.§1/2+ Ex.=1.513 HOV J'IIS/Z‘ Ex.=l.602 HOv J'I7/2" Ex.=l.6H8 HOV J'-1/2’ Ex.=l.39$ HOV J‘83/2’ 0c.m. (degrees) A A AAAAAA‘ A A AAAAAA‘ AAA-l A AAAAAA-L 4 4 4 4 4 145 momentum characteristics of the DWBA have been discussed previously in Chapter II. Angular momentum mismatch also implies that predicted strengths can vary strongly with excitation energy. The higher the excitation energy the lower the angular momentum matching value. In general, the better the match the larger the cross section. Even though cluster model calculations are not particularly useful when comparing the strengths of individual transitions, their energy dependence is mean- ingful. Figure IV. 13 shows a comparison of DWBA calculations assuming no excitation to a set calculated with 5 MeV excitation. When calculating the excited levels it is important to make both the Q-value and the binding energy more negative. No strong dependence on excitation energy is observed, though there are some small changes in predicted shapes. It is interesting to note that the change in shape for the 7/2— forward angle behavior is similar to what is actually observed for the 6.446 MeV state. In general the normalization of cluster model calculations to the data does not yield a number which is easily related to nuclear wave- functions. In certain cases, however, two states may be described by the same p-n formalism wavefunction. These states differ in their iso- spin quantum number. This is illustrated in Figure IV. 14 for the 7/2— ground state of 49V and the T=5/2, 7/2- state in 49V. Since the wavefunctions are the same, the spectroscopic factors are identical and the ratio of the T> normalization to the T< normalization becomes 146 Figure IV. 13 Q-value dependence of cluster model DWBA calculations. da/dQ [f "1.21 10'2 10'3 10'“ 10‘2 10'3 10' ‘ 10'“ 10'2_ 10" 10'? 10' ' 10‘“ 0 0 «a .: Oc.m. [degrees] 1117' Oc.m. (degrees) Figure IV. 13 0=-2.590 — o=-7.590 10.2‘ r I v I I l 10'3 1 5/2+ l 10'“' 10" 10'" 10'” 10'2 10‘" 10'” 10'“ 10'“ 10' O-e O-e O O I I J 0 20 H0 60 3" Oc.m. [degrees] 148 Figure IV. 14 Wavefunctions for analogs of 49V hole states. 149 49v 7/2' (T=3/2 p n xxx ooxxxxxx of 7/2 I 49Ti 7/2'u=5/2) T< a 49v 7/2'035/3) p n p n xx oxxxxxxx f772 xxx looxxxxxx f7,2 l 1 49v 3/2+ (T: 3/21 p n p n O OOXXXXXX f 7/2 + b XXX OXXXXXXX f 7/2 ° ' d 32 ° d 3/2 XXXX 49Ti 3/g“ (T= 5/21 n p n 9 xx Jxxxxxxxx f7,2 xxx oxxxxxxx f 7/2 a + '0 d 3,2 B 0 d 3/2 4“"v 3/2“ (T: 512) p " f p " f xx 7/2 7/2 + 3 ALL—421155533. 7 0 d 3/2 0 d 3/2 Figure IV. 14 150 1/(2T+1) which is the ratio of the isospin Clebsch-Gordon coefficients squared, where T is the isospin of the target. This ratio is independ- ent of the form factor model since the microscopic wavefunctions are the same. Since the Q-value dependence of the form factor is most easily handled with the cluster model, we will use this model for the comparison. For the 52Cr target, the expected ratio is . 2, while the ratio deduced from the fit to the 7/2- state at 6.446 MeV shown in Figure IV. 15 is 2.41. This kind of analysis cannot be applied to the 3/2+ and 1/2+ states, since the assumption of equal spectroscopic factors is not valid as is illustrated in Figure IV. 14. A similar enhancement of analog states in the (p,d) reactions has been observed (IV.43) . The effect has been explained via coupled channels calculations which may be approximated by using the same well depth for the analog calculation as for the T < calculation and allowing the bound state radius to vary to reproduce the correct binding energy (IV.44) . It remains to be seen if this approach will work for the (P . a) reaction . F. DWBA Calculations-M icroscopic Form Factors The cluster model is useful for studying the effects of optical potentials and the bound state well shape. However, the relative strengths of states are difficult to predict with such a model. A microscopic model is necessary to predict the relative strengths of states from shell model wavefunctions. Any microscopic form factor 151 Figure IV. 15 Cluster model DWBA calculations for the T= 5/2 states . do/dQ (mb/er) 152 520r[p,o<)”9V Analogs 10 . fl . . . Ex=6.L+L+8 MeV 1 J“=7/2' T=S/2 Ex=8.9‘+5 MeV J“=1/2* T=5/2 Ex=9.087 MeV wee/2" T=S/2 0 20 '+0 80 80 Oc.m. [degrees] Figure IV.15 153 will not have the shape flexibility that the cluster form factor has . Therefore, it is necessary to choose the "well matched" optical param- eters , since they are the least sensitive to form factor shapes . A microscopic model which uses single particle wavefunctions generated in a Woods-Saxon well has been developed previously (Chapter II) . The two neutrons are coupled together to make a di- neutron using the two nucleon form factor method of Bayman and Kallio (IV. 35) . The di-neutron is then treated as a mass two particle and coupled to the proton to make a triton in a Os internal state by a modi- fication of the two nucleon technique. DWBA calculations using these form factors are shown in Figure IV. 16. The fits are comparable to the cluster fits shown in Figure IV. 12. The calculation of strengths requires detailed spectroscopic amplitudes from shell model wavefunctions. However, the hole states may be described by simple seniority one transfers . We assume that the two neutrons are coupled to zero angular momentum. The j-transfer for the hole states is the proton total angular momentum. This assump- +, and 1/2+ proton tion makes the relative strengths for the 7/2- , 3/2 50 3 49 hole states the same as would be expected for the Cr(d, He) V reaction. The hole state relative spectroscopic factors are given in Table IV.6. The agreement with the expected values is very good, thus the simple seniority one assumption seems to be reasonable for these states . 154 Figure IV. 16 Microscopic model DWBA calculations. The form factors were calcu- lated for (0f7/2)3, (0f7/2)20d3/2, and (0£7/2)21s1/2 configurat-ions. da/dQ (mb/sr] 155 52Cr~(p,o<]'*9V Ep=35MeV:Microscopic Model 0.1 10'2 0.1 0.1 ’ ... °. N V v vyvvvv' ... 9 N v . -.--wr v tv'vvwv 0.1 v vrvvvv' r v v v ‘f v v Ex.=0.000 MeV J'87/2' EX.'0.091 MeV J'tS/Z‘ Ex.80.153 Nev J'I3/2’ 61.30.7‘08 MeV J'I'S/Z’ Emma: MeV mun/2' 0cm. (degrees) A A ALAAAA‘ A A AAAAAA‘ A AAA-AA. A AAAMAA. 0.1 10'2 0.1 0.1 10' EX.‘1.602 MeV J'I7/2’ " Ex.=1.l‘tl Nov 5 J'IS/Z‘ A A . AAAAAI A AA‘AA 5:31.513 MeV 1: J'IS/Z' £331.898 MeV J'Ii/Z‘ Eta-1.985 MeV J'Ia/Z’ Oc.m. (degrees) Figure IV. 16 156 TABLE IV. 6 RELATIVE SPECTROSCOPIC FACTORS Excitation ” Energy I Theory Measured 0.000 7/2' 1.0 1.0* 0.748 3/2+ 1.0 .84 1. 646 1/2' .5 .54 *Normalized to one. All other values relative to this. The same ratios should be expected for the T=5/2 proton hole states. Analysis similar to that given in the last paragraph shows that the model does not work for these levels. The seniority one assump- tion apparently is not a good one for these levels. Although the angular distributions of the high spin states may not be reliably predicted, the microscopic model can be used to predict the likelihood of observing a 19/2— level. Comparing the calculations shows that the ground state prediction is 48 times stronger than the 19/2- at 12". At 70° the 19/2- is expected to be 6 times stronger than the 7/2- seniority one transfer. Although these numbers may be off by a factor of 2 or so, qualitatively this is what is observed for the level at 4.797 MeV. The peak cross sections from microscopic model calculations for (Of7/2) 3 configurations have been given in Table 11.5 . A qualitative feature of this table is that the j> member of a given Z-transfer is 157 always larger than the j < member. This is exactly the trend that was notice for the negative parity states in the j-dependence discussion. G. Conclusions The proton hole states have been found to dominate the 52Cr(p,a)49V spectra at forward angles. This observation agrees with others that have studied the (p,a) reaction in the fp-shell. Many weak transitions are observed with average differential "' 10 ub/Sr. The (p,a) reaction has been shown to have some degree of selec- tivity. A number of levels that are observed in 51V(p,t)49V were not observed in the (p,a) spectra. Turning this comparison around has proven to be a useful tool for finding positive parity states, none of which appear in the (p,t) spectra. Large peaks in the back angle spectra have been observed. These states are candidates for high spin states. The likely spins are 19/2', 15/2', or 15/2+ for these levels. Little evidence for j-dependence for 1?. =2 and 2 =3 transfers has been found. If j-dependence exists , it is subtle and at a level that structure effects can be equally important. DWBA calculations using cluster form factors have been shown to reproduce the shapes of the angular distributions of the known levels reasonably well. 158 DWBA calculations for the T= 5/2 proton hole states are unable to reproduce the ratio of T =5/2 strength to T: 3/2 strength expected by analogy to single nucleon transfer. All the T: 5/2 strengths are too large . Microscopic form factors have been tested. DWBA calculations using these form factors have been shown to reproduce the shapes of the angular distributions with quality slightly inferior to the cluster model fits. Relative spectroscopic factors for the T: 3/2 proton hole states have been derived from the microscopic calculations and are found to be in agreement with assuming seniority one wavefunctions. However, similar calculations for the T= 5/2 proton hole states do not vvork. Finally a qualitative feature of j-dependent strength is observed in the data. The j> member of the 2 =1, 3, 5 transfers is always observed to be stronger than the j< member. The microscopic model based on (0f7/2)3 pure configurations also reproduces this qualitative effect . IV.1 IV.5 IV.6 IV.7 IV.10 IV.11 IV.12 IV.13 REFERENCES FOR CHAPTER N L. L. Lee, Ir., A. Mainov, C. Mayer-Boricke, I. P. Schiffer, R. H. Bassel, R. M. Drisko, and C. R. Satchler, Phys. Rev. Lett. 14 (1965), 261. I. A. Nolen, Ir. , thesis, Princeton University (1965) , unpub- lished. I. A. Nolen, Ir. , C. M. Glasshauser, and M. B. Rickey, Phys. Lett. g_1_ (1966), 705. L. S. August, P. Shapiro, and L. R. COOper, Phys. Rev. Lett. G. Brown, A. Macgregor, and R. Middleton, Nucl. Phys. A77 (1966), 385. I. E. Finck. I. A. Nolen, Ir. , R. Sherr, and P. A. Smith, private communication. G. Brown, S. E. Warren, and R. Middleton, Nucl. Phys. A77 (1966), 365. R. W. Tarara. I. D. Goss, P. L. Iolivette, G. F. Neal, and C. P. Browne, Phys. Rev. C13 (1976), 109. I. D. Goss, P. L. Iolivette, A. A. Rollefson, and C. P. Browne, Phys. Rev. C10 (1974), 2641. R. G. Markham, R. K. Bhowmik, M. A. M. Shahabuddin, P. A. Smith, and I. A. Nolen, Ir. , private communication. A. Saha, H. Nann, and K. K. Seth, private communication. D. Bachner, R. Santo, H. H. Duhm, R. Book, and S. Hinds, Nucl. Phys. A106 (1968), 577. N. H. Prochnow, H. W. Newson, B. G. Bilpuch, and G. R. Mitchell, Nucl. Phys. A194 (1972), 353. 159 IV.15 IV.16 IV.17 IV.18 IV.20 IV.21 IV.22 IV.23 IV.25 IV.27 IV.28 IV.29 IV.3O 160 Z. P. Sawa, I. Blomquist, and W. Gullholmer, Nucl. Phys. A205 (1973), 257. I. G. Malan, B. Barnard, I. A. M. deVilliers, I. W. Tepel, and P. Vander Merive, Nucl. Phys. A195 (1972), 596. B. Haas, I. Chevallier, I. Britz, and I. Styczen, Phys. Rev. C11(1975), 1179. S. R. Tabor and R. W. Zurmlihle, Phys. Rev. C10 (1974), 35. B. Haas, I. Chevallier, N. Schulz, I. Styczen, and M. Toulemoude, Phys. Rev. C11 (1975), 280. C. Rossi-Alvarez and C. B. Virgiani, Il Nuovo Cimento 17A (1973), 730. B. Bardin, thesis, University of Colorado (1964) , unpublished. I. I. Kolata, I. W. Olness, and B. K. Warburton, Phys. Rev. C10 (1974), 1663. R. I. Peterson and H. Rudolph, Nucl. Phys. A241 (1975), 253. H. T. Fortune, P. A. Smith, I. A. Nolen, Ir., and R. G. Markham, private communication. W. R. Falk, A. Djaloeis, and D. Ingham, Nucl. Phys. A252 (1975), 452. K. VanderBorg, R. I. deMeijer, and A. vanderWoude, to be published in Nucl. Phys. L. 8. August, P. Shapiro, L. R. Cooper, and C. D. Bond, Phys. Rev. _C_4 (1971), 2291. R. K. Bhowmik, R. G. Markham, M. A. M. Shahabuddin, P. A. Smith, and I. A. Nolen, Ir. , Bull. Am. Phys. Soc. _2_0 (1975) 1164. Michel Vergnes, Georges Rotbard, Jacques Kalifa, and Genevieve Berrier-Rossin, Phys. Rev. C10 (1974) , 1156. W. R. Falk, Phys. Rev. C_8 (1973), 1757. I. W. Smits and R. H. Siemsen, Nucl. Phys. A261 (1976), 385. IV. IV. IV. IV. IV. IV. IV. IV. 31 .32 33 34 .35 .36 37 .38 39 40 41 .42 .43 44 .45 161 P. A. Smith, R. G. Markham, and I. A. Nolen, Ir. , to be pub- lished. R. G. Markham and R. G. H. Robertson, Nucl. Instr. and Meth. 129 (1975), 131. I. D. McCullen, B. F. Bayman, and Larry Zamick, Princeton University Technical Report NYC-9891 (1964) . R. Sherr, Proceedings of the Conference on Direct Interactions and Nuclear Reaction Mechanisms , University of Padua , edited by B. Clementel and C. Villi (1962), 1025. B. F. Bayman and A. Kallio, Phys. Rev. 156 (1967), 1121. Yong Sook Park, H. D. Jones, and D. B. Bairum, Phys. Rev. _C_JZ_ (1973), 445. K. R. Greider and L. R. Dodd, Phys. Rev. 146 (1966), 671; L. R. Dodd and K. R. Greider, PhyS. Rev. 146 (1966), 675. R. Stock, R. Boch, P. David, H. H. Duhm, and T. Tamura, Nucl. Phys. A104 (1967), 136. R. M. DelVechio and W. W. Daehnick, Phys. Rev. (_36 (1972), 2095. F. D. Bechetti and G. W. Greenlees, Phys. Rev. 182 (1969), 1190. B. Fernandez and I. 8. Blair, Phys. Rev. g; (1970), 523. I. P. Schiffer, Isospin in Nucleg Physics, edited by D. H. Wilkinson. R. Sherr. B. F. Bayman, B. Rost, M. E. Rickey, and C. G. Hoot, Phys. Rev. 139 (1965), 81272. W. T. Pinkston and G. R. Satchler, Nucl. Phys. 1?: (1965) , 641. P. D. Kunz, unpublished. CHAPTER V THE 44Ca (P . (1)41K REACTION A. Introduction The (p,a) reaction has been used to locate proton hole states in nuclei that cannot be reached by proton pick-up (v.1, V. 2) . For many targets it is possible to extract meaningful proton hole spectroscopic factors by assuming that these states are populated primarily by the pick-up of a zero coupled neutron pair and a proton. Relative spectro- scopic factors calculated by this method have been shown to agree with the values obtained from the (d,3He) or (t,a) reactions for the 52Cr(p,oz)49V and the 922r(p,a)89Y reactions (Chapter IV and Refer- ence V. 3). Smits gt_al. (v.4. v.5) have found that this "spectator model, " which neglects coherent sums over three nucleon configura- tions and the di-neutron angular momentum, does not adequately describe the cross sections that are observed in Sn(p,a) reactions. Previous (p,a) hole state analysis has been done on nuclei where the hole strength is concentrated in one state. A logical extension of "spectator model" analysis is to study a nucleus where proton pick-up has shown that the hole strength is divided among a number of states. If the zero coupled pair assumption is reasonable, the relative cross 162 163 sections of the fractions and the relative total spectroscopic factors observed in the (p,a) reaction will agree with the results of the (d,3He) and (t,a) reactions. A recent survey of the (d, 3He) reaction on the Ca isotopes (V. 6) has shown that the Isl/2 proton hole strength is divided into three states in 41K. The 0d5/2 strength was found to be split among many levels. Thus the 44Ca(p,a)41K reaction is a good reaction to test the "spectator model. " Work on the 42Ca(p,a)39K reaction by Falk (v.7) produced good agreement for the ratio of the ls 1/2 spectroscopic fac- tor to the 0d3/2 spectroscopic factor, deduced assuming a (0f7/2)02 neutron configuration, with 4OCa (d, 3He)39K results (V.8) . Although this is an encouraging result, Falk also found that the 0d5/2 proton hole strength was four to five times too large. The beauty of the spectator model is its simplicity. However, because of this simplicity, it can be successful only for those few states which have large proton hole amplitudes in their wavefunction. In general, most states observed in the (lp,a) reaction are not of this type. Microscopic models have been developed recently to describe these more complicated transitions (Chapter 11, References v.5 and v.9). For sd-shell targets where complete shell model wavefunctions exist, it is possible to perform the completely coherent calculations (Chapter VI, Reference V. 9). For most of the rest of the chart of the nucleides, it is necessary to settle for qualitative agreement with pure 164 configuration calculations as has been done with the 52Cr(p,oz)49V reaction in Chapter IV. To date only neutrons from the same shell have been considered. The low lying 7/2- state in 41K provides a testing ground for mixed neutron configurations , since it can be popu- lated directly by (0d3/2 0f7/2)0d3/2 pick-up. 2p-2h excitations in the 44Ca ground state protons provide another process for making 7/2- final states. The strength of the 7/2- state in the 44Ca(d, 3He)43K reaction indicates that the 0f7/2 proton spectro- scopic factor is approximately . 85 (V. 6). If the (p,a) spectroscopic factor relative to the total (0d3/2+ Isl/2) spectroscopic factor, cal- culated assuming a spectator model, differs from (. 85/5 . 15) = . 165 , the discrepancy will be a measure of the mixed shell pick-up. Finally, it is worth noting that very few )1r assignments have been made for 41K. A number of 3/2+ and 5/2+ final states should be populated, thereby testing the 1:2 j-dependence. The j-dependence of the (p,a) reaction may turn out to be useful for making low-spin assignments . B. Experimental Method and Data A beam of 35 MeV protons was accelerated in the MSU cyclotron and directed to a target of isotopically enriched 44Ca. The reaction products were momentum analyzed in an Enge split-pole spectrograph. The particles were detected with the delay line counter developed by Markham and Robertson (V. 10) . Position and energy loss information 165 were taken from this counter. Particle time-of-flight relative to the cyclotron r.f. structure was obtained from a plastic scintillator placed behind the delay line counter. The a-particles were cleanly identified by their energy loss and time-of-flight. On the average, the energy resolution obtained from this system was 20 keV FWHM. The target was made by reducing 44CaCO3 with Zr and simul- taneously evaporating the liberated metal. The evaporant was caught on a 20 pg/cm2 carbon foil. The target thickness was found to be 85 pg/cm2 by comparison of 35 MeV proton elastic scattering to opti- cal model calculations using the Bechetti and Greenlees global proton parameters (v.11). Because of the thin target and small cross sections , a large solid angle and high beam current were necessary. The solid angle was 1.2 msr. and the typical beam current was 2. 5 uA. The spectrum taken at 12 degrees is shown in Figure V. 1 . The three large peaks are the 3/2+ ground state, the l/2+ state at 0.980 MeV, and a state at 3.520 MeV which is probably a 5/2+ state. The angular distributions are given in Figures V. 2, V. 3, and V.4. The angular distribution for the . 980 MeV level displays the deep minima which are typical of a l/2+ transfer. The angular distri- butions for the l . 590 MeV and 3. 758 MeV peaks , which have been assigned to be l/2+ in the (d,3He) work, also have this shape. In addition, a new candidate for 1/2+ assignment is found at 3.063 MeV, although the oscillatory structure is somewhat washed out. 166 Figure v.1. The 44Ca(p.a)41K spectrum at 12 degrees. 167 L0 ED OCCO 8..." 8N— n 82 _ .350. Swu Gama Q43; .1.? A. z . 1:62“! ASN m m... m ... a .80 S b L b p P r . — p _ p b F F P h _ >mz mmHQ VITHverHOUTT leuuoug/swnog Figure V. 1 168 Figure V. 2 44Ca(p,a)41K angular distributions . da/dQ [mb/sr) 1, - . - . D . . I Ex.=0.000 MeV ¢ . J'=3/2' < 0.1? 1 E W E : ' ' v : D ' 4 10'? r V 1 E w ' ' 1 E ' I . v v . 0",? m Ex.==0.980 mv 'g : o 8 o J'=1/2’ : r o . -2 1° 5' e» ‘2 0° 0 D o 1 P ,, 0 10'3? o 1 i 1 1r 1 0.1, Ex.=i.29-+ mv 1 E J'=7/2° g > V v V 1 V -2 v ' V 10 r ', i . ' 1 I v v 1 r ' V v i 001 r 1 i Ex.=1.559 MeV : J'=3/2* ‘ 10' on. 0 o I o . 0 I 10'3; 9 a 9 . 1 E f o 3 0,1 Ex.=1.590 MeV I q 9 14 ; J 31/2 , D ' 1 D w ’ ' 1 10-2.: ' 1 E "' ‘4 0cm. (degrees) 169 WCo[p,o<]'”K Ep=35 MeV 04, - . - . - . - . 0.1= r - . - - ; Ex.=2.l'-H MeV 3 f Ex.=3.063 MeV j 4":(3/2‘5/2’) 1 E '=(1/2*l « -2 -2 1° E on 3 1° E." v . 3 E 0 ° 0 o ‘ : ' : : 00 j : v' 'V ' 4 -3l 0 -3l 3 o O I I V 1 i f p ! 4 0.1:, Ex.=2.500 MeV ‘: 0'1? Ex.=3.163 mv 1: E : E J'=(9/2*) -2 -2 Z ' v 1 : 1!? i g 0 0 I v 10'3E V I) 10'-3|? 009 1 . 1i . : a o 0 3 ’ ‘ : Q . E Ex.=2.673 mv j E Ex.=3.198 mv 3 p J'gl/z. 1 > 1 - -z 10 2g a ' o 1. 1° 5’ 1. : O o ' I 1 -3( ° 0 -3 ! 1' 10 o 1 m I! 1. : o : ‘ c 9 : f! I . 0.1 E, 1‘ 0.1 1‘ i Ex.=2.706 mv 1 Ex.=3.250 mv 1 F 1 4 J'=[3/2’) -2 '2 10 E 1 10 d” o o I 5 fig ! i e 1 D ' 4 1 v o r i ! v 4 00° 4 “rmE i ! 1 10‘3’ 0 1 i 1 I I i O O 0 0 .i E I 1 : : . 0'1? Ex.=2.758 Hell 1 0,1 .- Ex.=3.‘iOS MeV 1 E 4'85/2' 3 5 I ‘ E 1 fl . 0 > ‘ i 00 j E 3 I I I V 1 r . ° 0 0 « I '1 ' 1 . . o 1 10'3 Oc.m. [degrees] Figure v.2 Oc.m. (degrees) 170 Figure V . 3 44Ca(ID.oz)41K angular distributions . do/dQ [mb/sr) 171 W(:otp.o()‘“+< 5p=3s MeV 0.1 , - - . 5 . 9.1— - . - . - - - . E 1 3 ; Ex.=3.‘-178 MeV j Ex.=‘+.053 MeV 1 E ‘ l 10'? E 1 10" 1 E i 3 J 0 O O O i P 1 1 E VI 11 1 0° . 10-3 1 I ! 1 10-3 O 1 ! 1 0g 5 1 o o 1 Ex.=3.520 MeV 3 Ex.=‘i.097 MeV 3 -2 °-‘ .0 e 1. ‘° 1 0 o : : ° 0, : 1 - 0 ‘ - II I v 1 ‘ I 0 Q I I 1 : ° ° ‘ : I i 1 ! : O I I I I 0.1F 5x.=3.sso MeV 1 10‘ Ex.=‘i.225 MeV 1 p 1 1 -3 - ° 10 r 1 no i § 9 Q 0' E e P I V ' 1 I o 1 , ' ' V 1 i 1 0.1 :F 1 001 a ’ 5:33.807 mv I 5x.=9.2so mv 1 10‘2 10' f 8 o i i I ' ' I 1: e ‘ V ‘ 9 ° '0 : '2! : 10' 9 10‘ l ! I ’g!. I 1"? : I : 1* > 1 W 0.1 E 1‘ 0.1 1! : 5x.=3.sss MeV : Ex.=‘-l.‘+'+6 Me)! i - -2 10 2'. ' 1 10 1 5'” 2 “1%. a - ’ '! V ' v ‘ r O O 0 0 ‘ 10 , v 1 to" e 1 1 1 0c.m. (degrees) 0c.m. (degrees) Figure V. 3 g 0.1 0.1 10' 0.1 F 0.1 10'2 '1' 'V v ' v ' 1 V v I 1. 8.89.825 mv 1 i , i Q!!! : i f 3 f 1 1. E’s-H.986 MeV i ""'v i ' v V I V ' 4 V V v V V v v m 3‘ ll ;‘ m w O 3 O < A A AAAAAAI 1" A A A AAAAA‘ A AAAA Ex.=‘i.778 MeV A A AAAAAAI A AAAAA‘J A AAAAAA‘ 9c.m. (degrees) 172 Figure v.4 440a (p , a)41K angular distributions . 173 $3.62: .918 on o. o o o b b D bub +N\HH:7 N\mfl._. 0 0090 898 e +N\mu:7 N\mu._. D L {P L b b D > ¢.> 839m ”$8me .845 on .. .+. o c 99 es . . ee ~-S 33.62: E 8 999 9.2 Hoo >02 Kfirgmfluxw HoO H H -N\mu .1 N\mu._. «-2 .. >3 TS D . D S :3» ~32 >0: rowfifliw . . . . . . . . ad a m: mmuam $18.38;. 1.3/qu esp/op 174 Most of the 0d3/2 proton hole strength is in the ground state. Other possible 3/2+ assignments based on their angular distributions are found at 1.559 MeV, 2.144 MeV, 2.500 MeV, 3.250 MeV, 4.053 MeV, and 4.446 MeV. Two of these levels have been reported in other experiments . The 2. 144 MeV state has been observed in 40K(n,'y)411( work (v.12). A 5/2+ assignment is favored on the basis of the 'y-decays of this level. A level populated with an 2 =1 transfer is observed at 4.444 MeV in the 4013.r(3He,d)41K reaction (V. 13). If the level at 4.446 MeV is the same state, then it must be a 3/2- level, since the angular distribution does not have the oscillatory nature of a 1/2- transfer. Since 3/2- and 3/2+ angular distributions are expected to have similar angular distributions at this excitation, it is not possible to distinguish between them. In addition, the 2. 706 MeV level has been reported to have an fl =2 angular distribu- tion in the 40Ar(3He,d)41K experiment. Its angular distribution does not look like either the 3/2” ground state or the 5/2", 3.520 MeV state. The 0d5/2 strength is largely contained in the 3.520 MeV state. Other 5/2+ shapes are seen at 2.758 MeV and 3.850 MeV. The 2 =2 j-dependence is well defined for this target. The 1- dependence, which is not as obvious as that of the 2 =1 1- dependence, is a small forward angle effect. This type of small- angle j-dependence has been reported previously by Glenn et a1. 175 (V. 14) . The 5/2+ angular distribution shows a distinct dip as the angle is decreased, a peak at about 14 degrees , and a flat shoulder, while the 3/2+ displays a smooth rounded fall-off as the angle increases. The j-dependence for 3 =2 is illustrated in Figure V.5 . The maximum spin that can be made from (0f7/2)20d3/2 is 15/2. A 15/2+ tentative assignment has been made from the results of a (H1,xn,yp,za, . . . ,7) experiment at 3.897 MeV (v.15). There is no evidence for a peak at this excitation in the (p,a) spectra . A tenta- tive 19/2- assignment has been made on the basis of results of the same experiment at 4.986 MeV. This level can be populated in the (p,a) reaction if there is a Zp-Zh component of the 44Ca ground state. A peak at this energy is observed. Its angular distribution is reason- ably flat as would be expected for a high spin state. There is another angular distribution that is associated with a peak at 4. 854 MeV that is nearly identical to the angular distribution of the 4.986 MeV level. The largest peak in the back angle spectra is a doublet located at 4. 34 MeV. It is likely that at least one of these states has a spin greater than 11/2. Unfortunately, the data for the high spin states is not definitive because the because the density of states in this region is so high and empirical shapes for large angular momentum transfers are not known. The only negative parity state that can be definitely identified in the low excitation spectra is the 1.294 MeV 7/2- state. It is pos- sible that the 4.446 MeV state is a 3/2' state. 176 Figure V . 5 44Ca (p , (1)4 1K L = 2 angular distributions . 177 L=2 Angular Distributions 1 Ex.=0.000 MeV 0 1 ..J"'=3/2+ vav .... O. N 4 "1'5 Ex.=3.520 MeV 11: + 0.1 6900 5/2 06o O da/dQ [mb/sr] ....- o. N @000 o 0 20 40 80 80 Oc.m. [degrees] Figure V. 5 178 In order to look for the T=5/2 proton hole states, the spectro- graph field was adjusted so that the excitation region from 6 MeV to 10 MeV fell on the counter. A spectrum of this region is shown in Figure V.6. The three strong peaks are the 7/2- analog of the 41Ar ground state and the 3/2+ and l/2+, T=5/2 proton hole states. This data suggests that the 1.03 MeV state in 41Ar should be assigned j1r = 3/2+. The angular distributions for these levels are found in Figure v.4. A summary of the levels observed in this experiment and others is given in Table V.1. The )1r values which are given in ( ) in the (Pay) column are assignments made on the basis of the (9.0:) data. C . DWBA Calculations-Cluster Form Factors Cluster model calculations have been used frequently because they are easily generated and because they can reproduce the shapes of many angular distributions . This type of calculation is especially useful for studying j-dependence and the effects of different optical potentials. Furthermore , continued study of cluster form factor parameters is desirable because of the wide range of choices that are found in the literature (see Chapter IV for a discussion of this point). Because of the consistent success of the well-matching proce- dure for (p,d) reaction calculations , the 44Ca ctr-elastic scattering data of Fernandez and Blair (V. 16) were refit to obtain an optical potential with real well parameters: V=200 MeV, ro= l. 22 fm. , 179 Figure V . 6 T: 5/2 proton hole states in 41K. 180 o .> 83$ .3952 .258 8.8 £2 . ewe t 8... a. oar 8N 00—...d - p P P p P P GEN 000‘ com— r . k p ‘ P a a .z/! A new leuuoug/siunog .Z/l .xw a9... .25 rm v_=.?.a.ooz. 181 «mm.m A+N\AAV A+N\A: NNmN NNNmN mNom.N $2; A+N\2 8m.N AN\N AN\N .N\: .N\: mNeeN NNAVN AN\N A+N\N A-N\N .N\$ .+N\3 moANN .-N\$ NANN BNN A-N\N A..N\N AN\N A-N\N . ..N\AV .N\mwv oeoAN .N\AV oeoAN .-N\: SAN A+ N}. A-N\N .+N\8 .NEWV AoeAN A+N\3 AooAN A+N\8 SAN NAN} A+N\N HaN}: AmmoA +N\$ ANmeA 3A: 8A: A+ N\N A-N\N.N\N A+ N\N .....NE A+N\t ANNoA NE in: +N\3 2:: 8A: ++N\A AN\mwv ommA +N\A 23A A..N\N AN\mwv NNmA A.. N}; oNNmA .-N\: NNmA SA A+ N\8 AN\mwv mommA A.. N\: ommmA A +N\8 ammA +..N\N -N N emNA N\N ommN. A +N\N ommN. A ..N\N moNA N\N NN. A +N\N omNA +N\A +N\A some. 0 +N\A 33. o +N\A some +N\A mm. o +N\A some +N\N +N\N o8. o +N\N coco +N\N 25. o +N\N 80.0 eh G2. En cx HE G? 5 A08 sway 33mm 3 Am? 3 AAA. .8 @523 >055 H.>,mHde 182 A-N\N A-N\N UN}; .-N\$ as; 8:” N86. AANN.N \ A A NE AN AN omN.N + 32$ + SAN . N AAoAnsoE oAN . N A+N\m A+N\m .+N\8 .+N\$ 2a NoA.N mooA.m meA.m N25 A+N\: A+N\AV 89m A..N\N A-N\N ...N\Av ANoo.N .-N\: 255 A+N\N: A+N\N: «RN A-N\A: A..N\A: NoNN NooNN rN\2 momNN .+N\8 SEN A+N\$ NmNN A+N\m A+N\m .+N\8 NNAN.N .+N\8 oAN.N eoNN mANo.N +N\A +N\A NNoN +N\A No.N +N\A EARN A..N\N A-N\N ...N\: .-N\: NomN 1 AAA» as ex AB AoAa 5 86 same 33$ 3 Avo.& nosemsoOIA .> 39% IlllllltAfllllliiruflflflfl 183 A+N\2 A+N\8 89A. rN\m A+N\m .+N\8 .+N\8 N84. A-N\N A-N\N .-N\AV .-N\Av e35 A+N\m: A+N\m: N23” A..N\N A-N\N .-N\$ ...N\3 N35 rNE A +N\3 8A: A-N\N A-N\N KN“: .-N\AV mANN, A \ V N 3 N m N86 AmeN + RN}; A..N\N ...N\e ntum A..N\m A..N\N .-N\: .-N\AV oNN.N owed A..N\N .-N\: A-N\N ...N\Av 22m A+N\3 A+N\$ on.N A..N\N A+N\m .-N\e 8: .+N\Nv 2; N2; 1 AAA». as ex :6 AAoAA. 5 Avo ommv AaAomN 3 Avo e oossAEoolA .> mAmfi 184 A-N\N: A-N\m: mNNé 8mg. AAvosoE NNNJ mNoé FN} A+N\m . +N\8 . +N\8 86 NoN.o emN.o 25o A..N\N A-N\N .-N\: .-N\: NNNJ A3383 Aeoé New; ommé A-N\N A-N\N .-N\AV ...N\Av N2; A-N\m A«N\2 .-N\: NE}. AAN\8 as; rN} A+N\m .+N\8 .+N\8 coo A-N\N A-N\N .-N\3 UN}: memo 8N6 A-N\m: A-N\m: mNN.o AAoAasoov NNNJ mNNé 03A. Nmoé eh a? 9» ex HE GA». A: A0? ommv AnAommcv 33.3 i try §IAS mama . 3 .> mucosomomm . H m . > mucouomomo . NH .> mocfiomomo . mH . > mucouomomo .m .> mucouomomn .NcoEHAoqxo 3:90 522 m.m can m coosfion >ox A: new coHuSono Ho >32 m 32mg >3. m mum monAoco coHAmuHoxo 3.8 co muotm £902 185 AN\mua. +N\: AN\mua. +N\AV AN\mua. +N\2 . AN\mn.A.. +N\$ AN\mua. ..N\t AN\mu.A. . -N\t NA: A-N\N...N\: A-N\N.-N\: 82m A-N\N.-N\: A-N\N.-N\AV 8N1... A..N\N...N\: A..N\N...N\Av EN.m A..N\N.-N\: A-N\N.-N\Av mNN.m ENJ. A-N\N.-N\3 A-N\N.-N\$ so; A-N\N.-N\: A-N\N.-N\: A85 A+N\m A+N\m.+N\Nv .+N\NV mod :9... A-N\N.-N\3 A-N\N.-N\2 was; 1 a? A: cx H8 8? 5 AUG ommv 333m 3 Am? 3 H k UGDCHHCOOI H .> mama 186 a = . 72 fm. The choice of proton parameters was determined by the success in fitting the a-elastic scattering data. Since the a- scattering was fit better by the choice rO =1. 22 fm. than r0 =1.17 fm., the r0= 1. 22 fm. Bechetti and Greenlees proton parameters were used (V. 10). Given these choices , the bound state well should be chosen so that ro= 1.22 fm. , a = .72 fm. , and V E 150 MeV. However, the well depth was allowed to vary to get the correct triton separation energy. For the most part, the depth was near 130 MeV. The fits to the data were not particularly good for the forward angles for this set of parameters . Since it was found that reducing the value of the dif- fuseness parameter slightly produced better fits to the 52Cr(p,ct)49V data (see Chapter IV) , this was tried here. The best fit was found to occur for a = .55 fm. , which is quite a bit smaller than was found to be necessary previously for the SZCr(p,ar)49V reaction. The calcula- tions are compared to the data in Figure V.7 . The optical potentials are given in Table V. 2. TABLE V. 2 OPTICAL POTENTIALS * 'k V IDo a vso rso 830 W rI a1 WD p -43.2 1.22 .72 -25.0 1.06 .68 -5.0 1.32 .57 12.2 -206.2 1.22 .72 -16.0 1.64 .57 *Values for Vso and WD include the factor of 4 required by DWUCK72 (V. 17) . 187 Figure V. 7 DWBA calculations for the 4403(p,a)41K reaction using cluster form factors . da/dQ (mb/cr] 0.1 10'?: 10'3 0.1 10'2. v 1 vvrvvv' V v va-vvv' 0.1 r V r E :.=0.000 HOV J'=3/2’ E:.=0.980 HOV J'81/2’ Bah-1.299 Nov 11 J'I7/2‘ 'V E:.=l.559 Nov .r-a/z‘ Era-1.5m HOV J'Il/Z“ 0c.m. [degrees] L*‘+C°[p. o<]‘“K 188 -2 10 E 10'3. 0.1_ 0.1 1 0.1 10'2 10'3 0.1 0.1 v vva ‘ V T EXE‘ZJ'“. HOV J'=3/2’ f E:.=2.500 ”0V J'83/2’ E:.82.673 HOV J'II/Z' 5:33.063 Nov J'Ii/Z’ 8cm. (degrees) Figure V. 7 Ep=35 MeV;Clus+e q l A AAA-AAA. A AAA- 0.1 r Mode #71 yvvvv' 0.1 v v v'vvvv' 10'2 0.1 to" 0.1 ' 1 ‘Y 8:53.163 HOV E:.=3.250 HOV 3":3/2‘ E:.83.520 NOV J'BS/Z’ E:.=3.807 mv ass/2‘ 8:33.850 Nov J'IS/Z’ ' l I I . I 0cm. [degrees] A AA‘A A AA‘AAAAL 4 Ah. Aagul - A A A A A A‘A‘AL A ALLA“ 189 The 2 = 2 j-dependence is predicted properly by the calcula- tions; however, the fits to the 1/2+ states are poor. It has not been possible to fit the 1/2+ and the 3/2+ and 5/2+ angular distributions simultaneously . In general the normalization of the cluster model is not easily related to nuclear wavefunctions . If, however, two states have the same wavefunction in a p-n formalism, then the ratio of normaliza- tions is independent of spectroscopic factors and reduces to 1/(2T+l) . This has been discussed previously for the ground state of 49V and 49V that are observed in the 52Cr(p,ar)49V the T=5/2, 7/2- state in reaction (see Chapter IV) . DWBA cluster model calculations where the well depth is allowed to vary to get the correct triton binding energy gave the analog state normalization to about 12.5 times too large for 49V. The two 7/2- states observed in 41K in the 44Ca(p,a)4lK are not related as those observed in 49V. They are, however, related by being an analog-anti-analog pair for the 41Ar ground state. This relation is illustrated in Figure V.8. If we assume that there is no mixed shell pick-up (see the section on microscopic calculations) , the second term will not be involved and the ratio of spectroscopic factors using any form factor model should be 190 Figure V. 8 Wavefunctions for analog states in the mass 41 system. 191 4'Ar 7/2’ (T=5/2) 1 XXX f7/2 T<3 xxoo I (13,2 4'K 7/2'(T=5/2) Xl XX f 1 XXX f7/2 a xxoo l d 2”: + B xxxo lo d 3,2 4'x 7/2" (T=3/2) x | xx f _ f I xxx 1‘ 7,2 xxoo I (lg/22 a xxxo Io d 3/2 a = .1/ 3/7 , B = 4/ 7 Figure V.8 192 The expected ratio of the normalizations, given these assump- tions , will be The ratio deduced from the fits shown in Figure V.9 is 1. 92 which is 12.8 times the expected value. Once again the analog state is anomolously large with nearly the same enhancement that was observed in the 52Cr(p,ar)49V reaction. D DWBA Calculations-M icroscopic Form Factors The microscopic form factors were generated by the prescription given in Chapter 11. (0f7/2)20d3/2, (on/2)21s1/2, (017/2)20o5/2 pure configurations were assumed for the fragments of the sd-shell hole states . The fits that were obtained are shown in Figure V. 10. In general they are not as satisfactory as the cluster model shapes . The calculations have been normalized to minimize x2. Relative spectroscopic factors assuming a "spectator model" for the sd-shell hole states are given in Table V. 3. The table has been normalized so that the total 0d3/2 spectroscopic factor is 1.0. For a zero order shell model assumption, the total ls 1/2 spectro- scopic factor should be .5 and the total 0d5/2 spectroscopic factor should be 1.5 . The fractions are markedly different from the results obtained by the (d,3He) reaction. The 3/2+ strength is found to be divided in the (p,a) data, though most of it is in the ground state. 193 Figure V. 9 DWBA calculations for the two 7/2- levels observed in the 4“lCa (p,a)41K reaction using cluster form factors . dc/dQ [mb/sr) 9 .... H O. N 194 Cluster Model '1fivirrr E Ex.=1.29'-l MeV bd“=7/2- T=3/2 g b 1. b va "_“w ‘V w W w W W v b d"=7/2’ T=5/2 T" l T rtvrn' T Tfirvwl 0 ‘2‘0 ‘le ‘6‘0 ‘8‘0' 9c.m. [degrees] Figure V. 9 195 Figure V. 10 DWBA calculations for 44Ca(p.cr)411( reaction using microscopic form factors . 0.1 10‘2 0.1 10’2 10'3 da/dQ [mb/srl 10" 0.1 10'2 lSlfi WCo[p.o<)L”K Ep=35 MeV:Microscopic Mode v vvvv vvvm' 1v 0.1 E:.=0.000 MeV J‘=3/2’ 8:30.980 MeV .. Jul/2’ . g o 5:31.298 MeV J'87/2' 5:31.559 MeV aha/2* 6:31.580 MeV J'-L/2’ 0c.m. (degrees) : 0.1 10‘2 10‘3 0.1 10'2 10'3 0.1 10'2 10'3 0.1 :- 10' V V v r Y7 r r 5:32.149 MeV J'83z’2’ 8:32.500 MeV J'83/2’ I!!!v E:32.873 mv J'UVZ’ 00° 0 0 o E:32.758 MeV J'lS/Z' 5:33.083 HOV J‘tl/Z’ Oan.(degrees]: Figure V. 10 A AAA A AAAAAA‘ A A A AAAAA‘ A AAAA A A A AAAA“ A AAAAAAA A AAA A A AAAAAA‘ A A AAAAAAA 0.1 r T V V ‘7 fir V fir 6:33.163 MeV lo-z 4"9V’2’ 10'3’ 0.1 r E E:.=3.250 MeV J'83/2’ io'2 10'3 O l Ex33.520 MeV ‘ J'IS/Z’ 0.1 v 10'2 V v v 0.1 " 6:33.807 MeV 10"2 \F'S/zQ 1. t o i 9° 10'3 § 9 ll".1 it 0.1 5:33.850 MW 0 10-, .r-S/z v I | '0 :1: :I Oc.m. (degrees) A A A‘AAA‘.‘ AA A ALAAAA‘ A A A AAAAA. A A AAAAAA‘ A A AAAAAl A A AAAAAAAA AAAAAAAI A AAAAAAAA A A A-AAAA‘ 197 TABLE V. 3 RELATIVE PROTON HOLE SPECTROSCOPIC FACTORS Excitation 7r 2 a) 3 b) Energy I Relative C S (d, He) 0.000 3/2+ .69 1.0 0.980 1/2+ .35 .22 1.559 3/2+ .11 .050) 1.590 1/2+ .12 .050) 2.144 3/2+ .08 2.500 3/2+ .03 2.673 1/2+ .05 .19 2.758 5/2+ .32 3.063 1/2+ .05 3.250 3/2+ .10 3.520 5/2+ 1.13 .24 3.807 5/2+ .10 3.850 5/2+ .06 aNormalized so that Z(3/2+) = 1.0. bFrom Reference V.6. c:Not resolved . Some of the weaker 3/2+ levels may be seniority three couplings. The Isl/2 strength is primarily located in the 0.980 MeV state in the (p,a) data, but is nearly equally divided with the 2.673 MeV state in the (d,3He) results. The grossest difference is in the 5/2+ levels where 70 percent of the 0d5/2 strength is found in the 3.520 MeV level in the (p,a) data. The (d,3He) data does not have a single strong 5/2+ transition. The authors choose to call all levels above 2 MeV 5/2+ levels with small spectroscopic factors in that study. 198 If all the levels of the same 17r in Table v.3 are totaled, the results shown in Table v.4 are obtained. The agreement with a zero order shell model is fortuitous given the quality of the fits. TABLE v.4 TOTAL SPECTROSCOPIC FACTORS Zero Order Hole Total Shell Model 0d3/2 1.0* 1.0 ls 1/2 .57 .5 0d5/2 1.61 1.5 1"Normalized to 1.0. Calculations for the 1. 294 MeV 7/2- state were performed for the configurations: (0f7/2 0d3/2)0d3/2 and (0f7/2)3. The proton occupation number for the 0f7/2 shell in 440a has been deduced from the 44Ca(d, 3He)431( reaction to be . 85 . Therefore the occupation of the (Isl/2 + 0d3/2) orbits is 5 .15 and the spectroscopic factor relative to the total (ls1/2+ 0d3/2) strength should be . 165 for the (0f7/2)3 configuration. A difference from this number would indicate a need for the mixed neutron configuration. The number which we get is . 157 which is consistent with the (d,3He) result. There does not appear to be a need for a mixed neutron configuration for this transfer. This also implies that the relative spectroscopic factor of the 0d5/2 199 hole states should be larger than 1.5 since the 0d3/2 occupation is less than 4. The result shown in Table v.4 is in fact larger than 1.5. 13. Conclusions The 44Ca(p,a)41K reaction has been found to strongly populate the sd-shell proton hole states that have been observed in previous 42Ca(d,3He)41K work. Fragments of lsl/Z, 0d3/2, and 0d5/2 holes have been identified. A number of new states have been identified as parts of the OdS/Z proton hole structure. The largest of these frag- ments is located at an excitation energy of 3.520 MeV. In addition to the positive parity states , the 7/2- state at l. 294 MeV is populated with about the same relative strength as seen in the (d,3He) spectrum. A comparison of the 3/2+ and 5/2+ angular distributions shows that there is a definable j-dependence for this target. This j- dependence has made it possible to make 3/2+ and 5/2+ assignments to a number of levels below 4 MeV of excitation. Three T=5/2 states with probable spins of 7/2-, 3/2+, and l/2+ have been identified. These are the analogs of the neutron hole states in 41Ar. DWBA calculations have been performed using mass three cluster form factors . The best fit has been found with the bound state parameters ro=1.22 fm. and a= .55 fm. The i=2 j-dependence which is observed is predicted by these calculations. 200 DWBA calculations using microscopic form factors and pure configurations have also been performed. The fits to the data are not as good as the cluster model calculations. The normalization of the theory to the data indicates that relative spectroscopic factors for the l/s 1/2 and OdS/Z fragments are quite different from those deduced from 42Ca(d,3He)4lK. This indicates a need to carefully include the coherence in the di-neutron coupling. The relative total spectro- scopic factors are, however, consistent with the (d,3He) result. The spectroscopic factor for the 7/2- state, deduced by assuming a (0f7/2)3 pick-up, relative to the total (ls 1/2+0d3/2) spectroscopic factor is consistent with the results of the 44Ca(d, 3He)43K reaction. There does not appear to be a need to include a (0d3/2 0f7/2)0d3/Z pick-up for this transition. .10 .11 .12 .13 .19 REFERENCES FOR CHAPTER V R. G. Markham, M. A. M. Shahabuddin,1& K. Bhowmik, J. A. Nolen, Jr., P. A. Smith, Bull. Am. Phys. Soc. 31 (1976), 634. J. E. Finck, J. A. Nolen, Jr., R. Sherr, P. A. Smith, private communication. S. H. Suck, W. R. Coker, Nucl. Phys. A176 (1971), 89. J. W. Smits, F. Iachello, R. H. Siemssen, A. van der Woude, Phys. Lett. 53B (1974), 337. J. W. Smits, R. H. Siemssen, Nucl. Phys. A261 (1976), 385. P. Doll, G. J. Wagner, K. T. Knopfele, G. Mairle, Nucl. Phys. A263 (1976), 210. W. R. Falk, Phys. Rev. Cg (1973), 1957. J. C. Hiebert, E. Newman, R. H. Bassel, Phys. Rev. 137 (1965), 8102. W. R. Falk, A. Djaloeis, D. Ingham, Nucl. Phys. A252 (1975),452. ““ R. G. Markham, R. G. H. Robertson, Nucl. Inst. and Meth. 129 (1975), 131. P. D. Becchetti, Jr., G. W. Greenlees, Phys. Rev. 82 (1968) 1190. D. F. Berkstrand, E. B. Shera, Phys. Rev. 93 (1971), 208. L. R. Medsker, H. T. Fortune, S. C. Headly, J. N. Bishop, Phys. Rev. Cll (1975), 1937. J. E. Glenn, C. D. Zafiratos, C. B. Zaidins, Phys. Rev. Lett. 26 (1971), 328. 201 202 v.15 P. Gorodetzky, J. J. Kolata, J. W. Olness, A. R. Poletti, E. K. Warburton, Phys. Rev. Lett. 31 (1973), 1067. v.16 B. Fernandez, J. 8. Blair, Phys. Rev. 21 (1970), 523. V.l7 P. D. Kunz, unpublished. CHAPTER VI S UM MARY The features of the (p,d) reaction have been investigated from both a theoretical view and an experimental approach. Neither is complete and many aspects and implications need to be pursued in future work. The theoretical side of the work has been directed toward developing a microscopic model for calculating angular distributions from detailed shell model wavefunctions. A microscopic form factor model which uses harmonic oscillator wavefunctions has been pre- sented. This model, which is similar to one used previously by other researchers , has the advantages of being analytic and easily general- izable to multi-nucleon transfer with heavy ion beams. The oscilla- tor has the disadvantage of not being realistic in the nuclear surface region. A microscopic form factor model which uses single particle wavefunctions generated in a Woods-Saxon well has been developed from a generalization of the popular Bayman and Kallio technique for calculating two nucleon form factors. This method has the advantage of being realistic in the nuclear surface region, but is restricted to 203 204 those reactions where the transferred particles are in a OS state of internal motion. A method for deriving spectroscopic amplitudes for the (p,d) reaction in terms of two nucleon reduced matrix elements and single nucleon reduced matrix elements has been presented. This method has the advantage of using the output of already existing shell model codes . It has the disadvantages of not being exact and not having simple sum rules. The motion of the nuclear center of mass in a fixed center potential has been investigated for the harmonic oscillator model. It has been found that this motion can be approximated by a multiplica- tive factor introduced into the form factor. Even though the analysis cannot be carried out in a Woods-Saxon potential, it is reasonable to assume that the correction is similar to the harmonic oscillator result. Part of the theoretical side of the investigation has concerned itself with the aspects of the DWBA for the (p,a) reaction. The angu- lar momentum mismatch has been found to be severe. The effects of the mismatch are minimized by choosing "well matched" optical parameters. In the process of investigating the angular momentum dependence of the DWBA a brief excursion into the source of j- dependence in nuclear deactions has been taken. No conclusion can be drawn about j-dependence. Further investigation into this ques- tion is needed . 205 The experimental part of this thesis has been directed toward discovering the features of the (p,a) data . These features may be summarized as follows: Proton hole states are strongly populated. Many seniority three transfers are observed. Many states that appear in the (p,a) spectra are not observed in (p,t) spectra or (t,a) spectra and vice versa indicating that coher- ence is important in the (p,a) reaction. There may be j-dependence for L=2 transfers, but this depend- ence is not consistent from target to target. Candidates for high spin states are observed in back angle spectra. Analog states are observed in the (p,a) spectra. These are the analogs of the neutron hole states in the neighboring nucleus. DWBA calculations using cluster form factors have been found to be capable of reproducing the data if the well matched optical parameters are used and the diffuseness of the bound state well are reduced slightly. DWBA calculations using microscopic form factors based on pure configuration wavefunctions are moderately successful in fitting the shapes of the angular distributions . The spectroscopic factors derived from these calculations have the correct behavior for the proton hole states observed in the 52Cr(p,a)49V reaction. The 206 . 44 41 agreement for the proton hole states found in the Ca(p,a) reac- tion is more limited. APPENDICES APPENDD( I ELEMENTS OF NUCLEAR PHYSICS FOR NON-TECHNICAL PEOPLE The desk beneath my elbow is a hard object, yet it is mostly empty space. With a powerful enough microscope it is possible to see that this surface is really a large number of small fuzzy balls that bind together to form what we know as wood. These fuzzy balls are the basic building blocks of matter. They are the individual atoms of hydrogen, carbon, sulfur, iron, and numerous other elements that make up wood fiber. Unfortunately, there are not any microscopes which allow us to look at the individual atoms in enough detail to see what they are made of. To answer this question, it is necessary to rely on evi- dence which is not nearly as direct as a picture of atoms . An early picture of atomic structure comes from the famous experiment shown in Figure A.I. 1 . In this experiment a-particles , which are the result of the radioactive decay of radium, hit the thin gold foil. Most of them pass through and strike the forward scintil- lator. A scintillator is a material that glows when it is struck by a-particles.‘ Small bursts of light indicate that a-particles are 207 208 SCI/(19% fort! PigureA.I. l A sketch of the famous Rutherford experiment. 209 striking the scintillator. When the experiment was done, it was found that bursts of light were seen in the backward direction also. This could only occur if the projectile had struck something solid and massive. From such evidence it was concluded that atoms have a small, hard core, which is called the nucleus. More refined experiments have shown that the nucleus is about one onehundredmillionth of the atomic size. It contains almost all the mass of the atom. The resulting picture is a small, positively charged hard core surrounded by a diffuse cloud of negative charge. Atomic physics and chemistry are the branches of science that are concerned with the properties of atoms and the cloud of electrons that make up the ball of negative charge. It is the goal of nuclear physics to learn about the properties of the nucleus itself. This dis- sertation concerns itself with some details which should add to our understanding of nuclear phenomena. Here are some of the questions that need to be answered in our quest toward understanding the nucleus: 1. What are nuclei made of? 2. Why do some of them decay, emitting radiation, when they do ? 3. What holds nuclei together? 4 . What happens when a projectile strikes a nucleus ? 210 5. What shape is a nucleus? 6. How can we probe the nucleus to answer these questions ? The last question was the first to be answered. There are many known ways to probe the nucleus and there may be more which have not been invented yet. The experiment that has already been dis- cussed contains an answer. To probe a nucleus you must hit it with a small fast projectile, such as an a-particle from a radioactive source. Unfortunately, radioactive sources do not emit fast enough particles to cause anything to happen other than scattering like bil- liard balls. To determine the constituent parts of the nucleus it is necessary to break it apart in a controlled way. So accelerators were invented to accelerate light particles such as hydrogen nuclei (pro- tons) and helium nuclei (or-particles) . In these accelerators the electrons are ripped from hydrogen or helium gas by an electric dis- charge. Then the free nuclei are accelerated toward a negatively charged piece of metal. There are a number of designs that do this. Later I shall describe how a cyclotron accomplishes the acceleration. Early experiments showed that the nucleus was made of two types of matter. One type carries a positive charge. The other is neutral. Both particles have almost the same mass. We believe the answer to the first question is probably answered. Nuclei are made of protons and neutrons . There are only partial answers to the other questions. 211 It is the fourth question that I have been working on during this project. It is this question to which the remainder of this part is addressed. Today there are three basic pictures of nuclear reactions . The first type of scattering is shape elastic scattering. This is the case where nothing happens . The projectile bounces off the target nucleus without causing any change in the state of the nucleus. This is the same thing that happens when two billiard balls collide. The second picture is the compound nucleus view. In this instance , imagine that the projectile is absorbed by the target nucleus to form a new nucleus . A very short time later this nucleus comes apart at the seams and emits other particles which the experi- menter detects. This model is illustrated in Figure A.I. 2. The third reaction model is called the direct reaction. In this picture the projectile makes a close pass , or grazing collision, with the target nucleus. As it passes by, a few protons or neutrons may be transferred, or the target nucleus may be made to vibrate or spin, without an intermediate nucleus being formed. Figure A.I.3 is a schematic of the direct reaction where x represents a cluster of nucleons which the projectile picks up as it passes by. How do we know which type of reaction occurred ? The answer is that we do not. In fact, there is often a great deal of controversy over this point in nuclear physics literature. In general, one usually argues in the following way. If a compound 212 \SIMSLn . 2.53002 .8205: no #0008 .8303 0:309:00 05. N .3“ 953m éuaozm use “9.2 decentrcou «nuke. file .1.... 0300. FLL 213 .8008 “53000.“ .8203: “00:0 05. m . H.< 0.59m fizen + 0:0 no we .6 3.463 obsteox 0:0 no ....em 0E0» Fax 0 & hMaSLW In 0.0 Nbfiv W30 03: flau\§\\ 7.51 .Mm Lei—u 214 nucleus is formed, it may fly apart in any direction. In this case, there will be lots of particles emitted in the backward directions. On the other hand, in the direct reaction the target nucleus is just barely grazed, so the reaction products should go primarily forward. The research presented in this study is on a direct reaction. Various target nuclei have been bombarded with protons. These pro- tons have picked up two neutrons and a proton from the target to make an a-particle as the out-going particle to be detected. There are two questions which you should ask at this point. Why do this ? What can be learned from nuclear reactions ? The answers lie in trying to find out whether question three can be answered using reactions as a tool. The ideal way to answer the third question would be to bounce protons off of protons and neutrons individually to determine the forces between them. One could use these forces to predict the prop- erties of nuclei. This has been tried with little success. Yet we are still interested in describing and predicting the properties of nuclei. So models have been invented that describe nuclei. One of these models is called a shell model. You can think of it in the following way. Imagine that you are at the bottom of a well which has shelves on the walls above you. You have twelve balls with you which you may throw up. If you throw one up and it stays on a shelf, the system 215 of twelve balls has gained some potential energy. This is called an excited state of the system. It will subsequently lose this energy to heat radiation when the ball falls back to the bottom of the well. Experimentally, it has been observed that nuclei behave in this way. The twelve balls represent a carbon nucleus . The normal state of carbon is when all the balls are at the bottom of the well. However, it is observed that a reaction that results in the production of carbon produces excited states. This is analogous to having one or more of the balls on the shelves . Sometime later the nuclear excited state decays, emitting 'y-radiation in the process. It is the job of the shell model to predict how many excited states a nucleus has, the energies of these states, the probability that they will decay in a particular way, and a number of other prop- erties including the chances of making any given excited state with a direct reaction. In other words, this model should predict the results of the work presented in this thesis. The reason for studying direct reactions is to test the validity of the shell model. _gmple Nuclear Experiment The basic parts of a nuclear experiment are an accelerator, a transport system, a target, and a detector. The accelerator produces the beam of projectiles that are to be used to bombard the target. The transport system serves the purpose of carrying the beam from the 216 accelerator to the target and also determines the velocity of the beam. The detector analyzes the fragments that result from the reaction. In this section each of these components will be briefly described. A Cyclotron—the Accelera;o_r A schematic top view of a cyclotron is shown in Figure A.I.4 . Positive ions, usually hydrogen or helium nuclei, are produced in the ion source in the center of the cyclotron. Figure A.I.S is a sketch of an ion source. A high current (about 400 amps) runs through the U- shaped tungsten filament. The filament becomes hot, just as the tungsten filaments in incandescent light bulbs do, and emits elec- trons. At the bottom of the cavity, which is filled with the gas that is to be ionized, is a block of tungsten which is at a high voltage compared to the filament. The high voltage causes the electrons to speed up. As they pass through the gas the accelerated electrons strike the gas atoms and knock off electrons. The result is that some atoms will not have any electrons. If the edge of the DEE which is just outside the ion source is negatively charged, the positive ions will be pulled out of the ion source through the hole in the side. In the last paragraph we have already talked about the basic principle of all accelerators. Ions are accelerated when they are in a region of space where there is a high voltage. In other words, posi- tive ions are accelerated toward a negative charge. The simplest possible accelerator is illustrated in Figure A.I. 6. 217 _ ////f//- m. 5. f Dene. Gene Qéarj ‘ ' Mk We 170A So 12ch DeHe’fM DEE 3 Figure A.I.4 A simple cyclotron 218 /0’"°o°$ k $400,410,103. a Fflun ent: Figure A.I.S A hot-filament ion source. 219 gliééCf) Figure A.I.6 A parallel plate accelerator. 220 It is hard to make a high enough voltage across a single gap to accelerate the ions to high enough energies. In the case of the research that is presented in this book, the voltage would have to be 35 million volts. In practice, it is practical to achieve 100,000 volts if there is good vacuum. The best way to attain high energies is to have many accelerations which add up. For example, the acceleration caused by 35 million volts can be attained by 350 100, 000 volt accelerations . This brings us to the second principle of cyclotrons. A charged particle that is moving between the pole tips of a magnet will follow a circular path. So now you can visualize how the machine works. First the ions are pulled through the hole in the side of the ion source by a negatively charged DEE. The DEE is hollow and between the pole tips of a magnet so the ions move in a circular path inside the DEE (see Figure A.I.4) . After half a circle they return to the gap between the DEEs . When they get there the alternating voltage source which is attached to the DEEs has reversed and given the opposite DEE the negative charge. Thus the ions are accelerated across the gap again. Again they drift in a circle inside the DEE, and once again they are accelerated across the gap between the DEEs . In our cyclotron at MSU the ions make about 210 turns or 420 gap crossings and accelerations. 221 Each revolution the ions move a little faster in an orbit with a little larger radius. When they have spiraled to the outside edge of the machine, it is necessary to get them out of the machine. This means they must be deflected from their circular orbit. In many cyclotrons this is accomplished by placing a negatively charged plate near the outside orbit and pulling the particle out of the orbit. In the MSU cyclotron even fancier things are done to obtain a better beam outside the cyclotron. The Transport System A layout of the current status of the MSU cyclotron lab is shown in Figure 11.1.7 . There are a number of experimental setups available. This section deals with what must be done to get the beam from the cyclotron to one of these experimental areas . This is really the simplest part of the system. We need only to apply the principle that charged particles follow curved orbits in a magnetic field. So every time the direction of the beam has to be changed, we pass it between the poles of a magnet. The magnets marked M3, M4, and M5 on Figure A.I.? may be adjusted to send the beam whichever direction is desired. While doing the accelerating and transporting of the beam, it is important to keep the beam in a high vacuum. Even a little air will result in a loss of beam because of collisions with the gas atoms. 222 CD 0 . O m 2 #0 $3233 #9008200me 05. 5c... 2:6; Smarties Mt/o. 189:»: as: do... 903%". 223 Besides transporting the beam to the experiment, the beam line may be used to define the beam energy. Consider the diagram in Figure A.I.8. The beam enters the magnet with some particles moving faster than others. The higher energy particles bend less than the low energy ones causing the beam to be spread out when it leaves the magnet. A set of metal jaws which only allow the central portion of the beam to pass through can then be used to define the spread in the energy . We; There are two important qualities that a target must have. It must be pure and thin. The purity requirement is different than just the _idea of chemi- cal purity. In this instance, isotopic purity is required. For exam- ple, the chromium found in nature is a mixture of four isotopes all of which have 24 protons but 26, 28, 29, or 30 neutrons. Although they are chemically the same, their nuclear properties are different. The thinness is important since even these fast particles are greatly slowed down as they pass through any matter. They will go through only about 1/16 inch of brass. Good targets are only a couple hundred atomic layers thick. You can often see through them. To make these thin targets the material must be made hot enough to melt and evaporate. A glass slide that has been coated with a water soluble salt may be placed above the hot metal so that 224 lu‘ A angry e ”8 (‘5) beam Figure A.I.8 Bending magnet and slits for beam energy definition. 225 the evaporating metal condenses on the slide. If the slide is slowly immersed in water at a later time , the soluble salt will dissolve leaving the target material floating on the surface. A metal slide with a hole in it may be used to pick up the floating foil in such a way that it is stretched across the hole. The experimenter should put the beam through the hole in the target frame, thereby hitting the target foil only. The Detector The detector must be capable of giving three pieces of informa- tion. It must tell you what kind of particle has been seen, how much energy the particle had, and in what direction the particle was emitted from the target. The detector used in this experiment is a rather complicated beast called a spectrograph. It works on the principle that fast parti- cles bend less than slow ones in a magnetic field. A simple spectro- graph sketch is given on Figure A.I.9. Only particles that leave the target with the correct angle "6" can enter the opening to the magnet. This requirement gives the directional information. The energy information may be determined by finding the posi- tion that the particle hits the counter. High energy particles strike one end and low energy reaction products strike the other end. 226 .oouoeohooom 03950.: 0 mo soaoxm .4. m.~.<0.39.m A0300 75F.» ¢vm§1r Q0583. 303 J :8” 1 Sirens 83.156 wfih it... 434 wmfiam “V cm... I «6.9 Efluuum 227 One type of counter is diagrammed in Figure A.I. 10. When a fast charged particle passes through a gas, it leaves a track of ionized gas atoms along its path. The resulting charge is collected on the wire that runs the length of the gas region. More charge travels to the closer end of the wire than to the far end of the wire. Therefore, the position is given by the ratio: endl end1+end2 ' The total signal will tell what kind of particle has been observed. You must keep in mind that all kinds of junk is being made in the target, but in this experiment we want to count a-particles only. The total signal (signal 1 + signal 2) is bigger for a-particles than protons, deuterons , tritons , and 3He-particles. So by counting only the large total signals and rejecting the rest, it is possible to pick out the a-particles . Doing thejExperiment The cyclotron is turned on and adjusted so that the beam goes down the beam pipe and strikes the target. This simple sounding operation may take half an hour or half a day depending on your luck. The reaction products are allowed to strike the counter at the end of the spectrograph. At this point the work of finding out what is happening begins. The signals from the counter are very small so they must be amplified. After they are amplified they have to be added and then divided. Performing these operations requires the 228 £3550 983 #008 :ofiflficuomomno 4 OH.H.< 0.59m domek mum WE 0.4.0 0.0.5 “23% Osakzm / $.30 .1-—fl . . 0 lIiI / /.U.:3 5% wzfifimuk \Nixafi 229 experimenter to process the signals with a number of electronic black boxes . One of these boxes adds the signals. The division is done by a computer. At the same time the computer will make a television display of the plot shown in Figure A.I. l 1 . The bands in the display represent the different particles coming from the target. The bottom band may be protons and the top band might be a-particles. The com- puter program which we are using allows us to choose whichever band we want to study. The desired result is a spectrum such as the one shown in Figure v.1. Each peak in the spectrum represents a different state of the nucleus that is left behind after the reaction has occurred. All the information about nuclei must come from spectra like this. To complete the experiment it is necessary to move the spectro- graph to many different angles and to record the spectrum at each of these angles. The relative sizes of the peaks at these different angles is important information. It is likely that about two days (48 hours) of beam time will be necessary to get all the necessary spec- tra for a given target. Aha lysis of thegQaLa The data of an experiment similar to the one that was just described are a set of spectra (plots of number of a-particles vs. the energy of the a-particles) . One of these spectra is shown in Figure V. l . [do ) [6: /n. (3.466» 64.? 230 .0 .: :2: ': : : ,: ....O; .... ‘0‘. ... . . . .' .. " .0... 5.3.3041... . deyé'eflfi ‘0. .: ‘: " 00. . .0 or n ‘. ‘0: 8:‘ ’:...a/ b‘nd g; i’ ‘13:: 1...... ‘..O~...o ..o: . band .. ...'°". ....0.;0 :;. Figure A.I. 11 Identification of the reaction products . 231 The analysis of this data begins by determining how much potential energy was left in the nucleus after the reaction for each peak that is observed. This list of energies can be compared with results from other experiments (see Table IV. 2) . It may also be com- pared with a list of energies predicted by the shell model. There is also a lot of information contained in the sizes of the peaks . The relative sizes of the peaks changes as the detector angle changes. Some peaks become larger as the angle is increased, while others become smaller. This suggests that we should make graphs which show how big each peak is at each angle that a spectrum was taken. Such graphs are given in Figure V. 2. Many years of experimentation has shown that these shapes are characteristic of a property of the nucleus called spin. You can think of spin classically if you wish by considering the collision of two billiard balls. In general, both balls will be spinning after the colli- sion. The spin is particularly evident if one of the balls is striped. Each time a nuclear reaction takes place, the resulting nucleus is caused to spin. The states of the final nucleus which we observe as peaks in the spectrum have various amounts of spin. The reasons the angular distributions (graphs of peak size vs. detector angle) are dif- ferent is because the nucleus has different amounts of spin for each peak. With some reaction data, the spin of a particular nuclear excited state may be determined. When this is done, the list of 232 energies and spins may be compared to the predictions of the shell model. One of the principal reasons for the research presented in this thesis was to investigate the possibility of using the (p,d) reac- tion (incident proton, outgoing a-particle) to determine the spin of nuclear excited states. There are two approaches we can take toward determining the utility of the (p,d) reaction for determining spins. First we may make a table of shapes for states whose spins have been determined previ- ously. The table of known shapes can then be compared to data for new states and used to determine the spins of the new states. This method works only because it turns out that only certain amounts of spin may be realized , just as we saw that there were only certain energy shelves in the well. The second method is theoretical. If the shell model works , it predicts both an energy and a spin value for all nuclear states . If a theory is developed to predict the angular distributions, given the shell model information, the resulting theoretical angular distributions can be compared to the data plotted on the peak size vs. angle graphs. Such a theory for the (p,a) reaction is developed in Chap- ter 11 and some comparisons can be found in Chapters IV and V. Summary The work which you find in this thesis may be summarized as follows . We have set out to use the (p,a) reaction to discover 233 nuclear energy levels and spins . We have done this by comparing the data with the results of other experiments and by developing a theory to predict the data that we have observed. In general, modest suc- cess has been realized. APPENDIX II AN INTRODUCTION TO SCATTERING THEORY Part One: The First Year Quantum Mechanics Problem Imagine a structureless projectile incident on a structureless target. A classical analog of this situation would be a cue ball inci- dent on a billiard ball. The Schrodinger equation which describes the Quantum Mechani- cal problem is V241 +%§} [E-V(r)1¢ = 0. A.II.1 We will require the scattering potential to be short range, i.e. , Lim rV(r) - 0. A.II.2 r-oo (Notice that the Coulomb force does not satisfy this condition. A screened Coulomb force does , however.) If this is the case, a detec- tor which for all practical purposes is located at r- no will observe the asymptotic form of up. Asymptotically, the solution must look like a plane wave, which is the incident or non-scattered beam, and a spherical scattered wave. In other words, the asymptotic form of q» should be ikr 45+) ~ eikz+f(9)9r— . A.II.3 234 235 k is the wavenumber Z hZZ (Len p=fik or 3:51—1:25) and f(0) is the scattering amplitude. f(6) , in other words, is the )l2 is , therefore, amplitude for scattering at a particular angle. |f(0 the probability for scattering at the angle 6. All the information about the interaction must be contained in f(6). However, f(0) is not the quantity which is derived experimentally. The experimentalist measures the differential cross section which is defined to be d_0' dS'z ___ number of particles scattered into d9 at 6 d9 number of particles incident per cm2 ‘ A°IL4 The relation between 6%) and f(6) can be found in the following way: The incident probability current is h -ikz d ikz ’hk e —e 81 = :Im dz = : . A.II.5 The outgoing current is "ikl' “(I 2 s =31m9—3—(r(0)3—>=35m%u—. A.II.6 o l4 r 8r r u r If N is the number of particles per cm3 in the incident beam, then NS i is the incident particle current (cmZ/sec) . The number of . particles striking the detector each second is rzdfz NSO. So substitute these expressions into A.II.4 to find 236 g: d9 = Nfigsz ink/011 J fLGLLz/rz) d9 N(hk/p.) Of 9!. _ 2 d9) - |f(0)l . A.II.7 In general there are two approaches to the problem. Either the experi- mentalist measures |f(0) | 2 at many angles and the theorist attempts to determine V(r) from this , or the theorist guesses at V(r) thereby deter- mining |f(0) I 2 and the experimentalist measures it. In any event, it is necessary to find out how f(6) is related to V(r) . To do this , begin by rewriting Equation A.II.1 (V2+k2)¢ = 'EZEVMLP. A.II.8 We are looking for a solution which has the asymptotic form given by Equation A.II. 3. So it is reasonable to look for a solution that is good for all values of r which has the form 0 = e~'~+ 4’s A.II.9 where 4’s is the scattered wave. The plane wave satisfies (V2+ k2)e"~‘v'£ = 0. A.II.10 So if A.II.9 is used in A.II.8, the result is (V2+k2)¢s = %%V(r)¢. 11.11.11 Equation A. II. 11 may be solved formally by using the Green's function method. We seek the Green's function which satisfies 237 (v2+k2)c(;_,y) = -41r 66:13). 11.11.12 Then the solution of A.II. 11 will be given by 1 4:5 = “I; %% 5G(r~,;')V(r:)qu(r')dr'. A.II.13 Equation A.II. 12 is the same relation that is found in the study of electromagnetic radiation when solving the Helmholtz equation. The Green's function for that problem is ik( I £75 I) G(r,r') = SLR-:7— . 11.11.14 It is the asymptotic form which is desired. This form is found by using the relation r,r' Ir-L'I s lg! '17? when Ir] >> Ir~'|. A.II.15 The asymptotic Green's function is then ikr G(£,£') s e-ib'r. -—er . A.II.16 where I; k' =— lkl. ~ lg To get the asymptotic form of Equation A.II. 9 , we need to use A.II. 16 in Equation A.II. 13. ikr (+) _ 1,152: ’__l _ZJ'L -15'.r~' I I I e— xp“) — e + 4” f1; e V(r)1p(£)dr_ r . A.II. 17 Comparing A.II. 17 to A.II. 3, it is seen that f(9) = - 31; $5 e-fl's OE: V(r')¢(r_‘)d;‘ . A.II. 18 Although Equation A.II. 18 looks like an answer, it is not because we do not know ¢(r')—it is the solution of Equation A.II.1 which is what 238 we are trying to get. This is because Equation A.II. 13 really only changes the problem from a differential equation to an integral equa- tion. We can get an approximate solution if we make the assumption that the solution can be generated iteratively. We shall assume that 4”) (1:) 2 Mg) and proceed then by using ¢(+)(£') for ¢(r~') where . . e'ik r' 4,9203) = e15 L - 4'1— 21,43} ’ik" ~ "V(r")¢(r")dr"e A.II. 19 1r ’0 where £3 k = ‘1— lk'l ... Ll Using A.II. 19 in A.II.17 yields LP(+)1~'_.£'—&S.e-ikur" (r) " 41r 1’12 "’V‘ )elk d" 2 A.II.20 —1’_ 2E SS -1k' '11:: I". II II II I +(41r hz> e V(r')e V(r )¢(r;)dr dr; . Presumably we can generate this series forever by replacing + Mr") - ik( )(r") , etc. Instead we assume that the series converges rapidly and therefore keep only the first order term to yield the results ...”)(9 g e1~ 'r~- 4—1 3%(5eik V(')e1k Ldr g>-— 11.11.21 7r ’h and {(9) = - 4175395} 1]" ~V(r ')e. ~ ~dr. A.II.22 This is called the Born approximation. The convergence of the Born series is an open theoretical problem that remains unsolved. Nonethe- less , it has been used with reasonable success for a long time now. 239 Part Two: Nuclear Scattering—The Distorted Wave Born Approximation The first year quantum mechanics problem is not very useful for real problems since, in general, the projectile and the target have structure and, therefore, many different kinds of inelastic events may occur. Additionally, the target nucleus acts as a sink for particles that are incident on it in a way that is similar to the scattering of a beam of light by a cloudy crystal ball. It has been shown that this effect can be reproduced by using waves that are distorted by a short range optical potential which has real and imaginary parts instead of plane waves. The simple problem shown in the first part of this appendix can be used as a guide for generalization. The approach here will be to proceed mostly by analogy to the first part. Consider the reaction A(a ,b)B. The Schrodinger equation which describes this system is Hq» = Ell» A.II.23 where H=Ha+Va+Ta. Here 0 represents the aA system (the incident channel). Ha is the total internal Hamiltonian. In other words , Ha=Ha +HA where Ha and HA are the internal Hamiltonians for the projectile and the target. Ha, therefore, does not depend on the sepa- ration of aA. Ta is the kinetic energy of the relative aA motion, and Va is the interaction between the projectile and the target. 240 Equivalently, we could write H=HB+VB+TB where B represents the b8 system or exit channel. Internal wavefunctions 1110 and 413 may be defined by H040 = Bayer A 11.24 H = 1: ° B¢B Bwfi- Then hzkz hzk 13:13 0— +—‘§-. 11.11.25 + — E a 290 B 295 The asymptotic form of q» is what is desired. If three-body final states are neglected, Equation A.II. 3 can be generalized to ik r 4’“) = 4 e~a~a+Zf (a) e 0' A.II.26 a a 7 07 LY ‘Y fm’(9) is the scattering amplitude for 7 exit channel. The generalization entails: 1. Including the projectile and target internal wavefunctions in the incident channel (410). 2. Allowing more than one exit channel, each with its own amplitude and internal wavefunction. The functions 417 form a complete set of orthogonal states of the (a +A) particles so we may expand if) = 24707145”). 11.11.27 7 The overlap on the right hand side of Equation A.II. 27 is an integral over the internal coordinates only, hence we may write 241 (+) _ (417W )—¢,Y(§,Y). A.II.28 a The relative motion in the [3 channel is then given by (+) _ (4131410 >-ch(55). A.II.28a An equation for the relative motion may be found by rewriting Equation A. II. 23 as I O (B - H) 4a”) - 80 (E HB TB VBNJQ A.II.29 II 0 We can now multiply Equation A.II. 29 on the left by 41B and integrate over the internal coordinates to get (+) > <4» I(E-H -T -v I0 = 0 B B B 3) 3+) (+) 11.11.30 (E-BB—TBHwaa > = <¢BIVBI00 ). To perform the last step we have used [¢B,TB] =0 which is true because 413 is a function of internal coordinates while T is a func- B tion of the relative motion only. Now E-E =6 3 from Equation A.II. 24, so we finally get an equation for the relative B: ZP-B motion which is fikf_ = (+) 2113 TB) 436B) (IBIvaa > SO A.II.31 ‘+’>. a 211 2 2 § (V + k )¢B(rB) " hz (‘I‘BIVBI‘I’ 242 This is exactly the same form as Equation A.II. 11 to which we applied the Green's function method previously. We can solve it again by the same method and the resulting expression for faB(e) will be 53 £0 (+) a where the 8B are the internal coordinates of b and B. The Plane Wave Born Approximation demonstrated in Equation A.II. 22 is to replace LI10") -' e151. The appropriate generalization to make is to replace Asa-1:6 (+) Ia ¢a(§a)e and integrate over the internal coordinates £0 and ra so that final expression for the scattering amplitude in the first order Plane Wave Born Approximation (PWBA) is PWBA ““513 £13 faB (9) = -4; £55“ d6 c1Id§fidrBdr 415 (EB)e 11.9:. VB(rB,§B)Lpa(§a)e . A.II.33 As a convenient notation, we define a transition matrix element by TPWBA =5 "513 '53 1.15616 TQB dradrfie (¢BIVBI¢a)e . 11.11.34 The potential VB is the interaction between all the nucleons in the projectile and all the nucleons in the target. In practice, we only want to consider one or two interactions. For example, we are only 243 interested in the p-n interaction in the (p,d) reaction. The rest of the interactions cannot be neglected, however. We hope that these inter- actions can be treated as an average overall potential, which may be complex. To incorporate this potential, we subtract a potential U!3 from both sides of Equation A.II. 31 to get (EB-TB- UB)¢B(rB) = (031 (VB-U3) 10;”). 11.11.35 We will choose UB in such a way that vB-UB contains only interactions between the nucleons involved in the scattering and so that U does not cause any transitions; i.e. , UB is diagonal in the B B system. This equation can also be solved by a Green's function method which involves making a multipole expansion of the Green's function. The procedure is complicated and amounts to proving the Gell-Mann-Goldberg relation for scattering from two potentials. Instead of carrying out this procedure, we will simply use the relation to get the transition matrix element T66 = ”0"? 'U6' 163150.50) " “13"? ' (VB—U5) ' “I H where x3 is the incoming solution of the homogeneous equation (+)) 0' A.H.36 -T -U r = 0. (8‘3 B B)xB( B) By choice, UB does not cause any transitions or, in other words, is diagonal in the B system. So the first term reduces to tobaB and (+) >. a ._ H _ 103 - 10003 + (IBXB I(VB U3)! 0 11.11.37 This form is called the post-intergction form. There is another form 244 called the prior-interaction form which is related by time—reversal symmetry and is _ H _ (+) T05 — tbo a8 +013 I(va Ua)I¢axa >. A.II.37a (+) Once again, A.II.37 is not a solution because it contains 410 W) 1118. tionA.II.37a, expand 1|» We still need to make some approximations. Using Equa- ('3) B( 5:21.76“) (r )4 (a) A.II.38 or, in other words , H _ H H ¢d+150 "' almd+150¢d+150 + a2¢d+1501r¢d +150'k + 63¢t+l40¢t+140+ Now we assume that a El and a E0; i=2 ...: hence, I i () (pH 41H (-) ¢d+150 = ¢d+150 ¢d+150 ; “PB = $3 (rB)‘I’B(§B) 0 This amounts to neglecting multiple excitations such as inelastic scat- tering (second term) and two-step processes such as (d,t)(t,d) (third term) . H The eigenfunctions of U B accurately reproduce ‘03 in the H asymptotic region. We now will assume that <93 can be approximated by the eigenfunctions of the optical potential, xé ) even during the interaction. Then DWBA H (+) 108 =TooaB+ (x13 ¢Bl(va -U Mm xa ). 11.11.39 This is called the distorted Wave Born Approximation. All the nuclear structure information is contained in the internal matrix element 245 (\PBI (Va-U0) | 410) . The method of evaluating this matrix element depends on the reaction being considered. In Chapter II of this book the internal matrix elements for the (p,d) and (a,p) reactions are evaluated. The purpose of this appendix was to present a non-rigorous guide to the DWBA. More detailed and complete derivations may be found in: A.II.1. The Theory of Direct Nuclear Reactions by W. Toborman, Oxford University Press (1961), London, England.