THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING THE DOORWAY STATE THEORY OF NEUTRON NUCLEAR REACTIONS Kenji Takeuchi A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan Department of Nuclear Engineering 1967 March, 1968 IP-812

ACKNOWLEDGMENTS The author gratefully acknowledges the following: To Professor F. C. Shure, the committee chairman, for his introductory guidance to the material of this thesis, particularly Part I; for his patient help in completing the thesis; and for his technical and editorial corrections of the thesis. To Dr. P. A. Moldauer for his invaluable information, guidance, and assistance over the entire thesis work which could not have been accomplished without him. To the remaining committee members, for their assistance: I am indebted to Professor R. K. Osborn for the shell model theory, to Professor G. C. Summerfield and Professor A. Z. Akcasu for the scattering theory. To Dr. D. Kurath, the Argonne National Laboratory, for his technical assistance for the work in Part III. To Professor A. Arima, the Argonne National Laboratory on leave from the Tokyo University, for his suggestions in setting up the model in Part III. To Professor W. Kerr and Professor P. F. Zweifel through whom the financial support, under the contract with the Brookhaven National Laboratory, and the opportunity for working at the Argonne National Laboratory were provided. To Dr. P. A. Moldauer, Dr. A. B. Smith, and Dr. S. A. Cox for their hospitality at the Argonne National Laboratory. ii

To the Argonne National Laboratory and the Associated Midwestern Universities for the opportunity of performing the investigation under the excellent staff members of the institute. To the fellow students, James Davidson, Ellen Leonard, and Donald Oliver, for their corrections of the English in parts of this thesis. The author is indebted to the Department of Nuclear Engineering, where he was a research assistant (July, 1964 - June, 1966, and May-July, 1967), and to the Associated Midwestern Universities and the Argonne National Laboratory where he was a student research associate (July, 1966 - March, 1967), for financial support during the thesis work. Finally the author would like to express his thanks to the Industry Program for printing this thesis. iii

TABLE OF CONTENTS Page ACKNOWLEDGMENTS................................................ ii LIST OF TABLES.................................................ii LIST OF FIGURES.... -.v... 0 Viii LIST OF APPENDICES............................................... X INTRODUCTION................................................... 1 PART I. INTERMEDIATE STRUCTURE OF CROSS SECTIONS................ 7 Chapter Introduction............................................ 8 1 REVIEW OF EXISTING THEORIES............................. 9 1.1 Bohr's Compound Nucleus........................... 9 1.2 R-Matrix Theories................................. 10 1.3 The Phenomenological Optical Model................. 17 1.4 Refinement of the Optical Model Potential.......... 20 1.5 The Optical Model and the Statistical Theory....... 22 1.6 Direct Reaction and the Unified Theory............. 24 1.7 Weisskopf's Compound Nucleus..................... 26 2 THEORETICAL PRELIMINARY................................ 28 2.1 Hamiltonian for A(n,n')A* Reactions............... 28 2.2 Feshbachts Projection Operators.................... 32 2.3 Resonance Parameters............................ 444 2.4 Absorption Cross Section........................... 47 2.5 Energy Average of the T-Matrix.................... 52 2.6 The T-Matrix and the Compound States............... 57 3 THE OPTICAL MODEL POTENTIAL............................ 60 3.1 Optical Model Potential............................ 60 3.2 Optical Model Schrodinger Equation................. 62 4 INTRODUCTION OF THE DOORWAY STATE STRENGTH FUNCTION..... 69 4.1 Introduction of the Doorway State Strength Function........................................... 70 4.2 Concluding Remarks................................ 74 iv

TABLE OF CONTENTS (CONT'D) Chapter Page 5 MODIFIED OPTICAL MODEL IN THE REGION OF THE ISOLATED DOORWAY STATE.................................. 76 5.1 Optical Model Calculation for Na................ 76 5.2 Modified Optical Model............................. 82 5.3 Concluding Remarks................................. 89 6 SUMMARY................................................. 93 PART II. FINE STRUCTURE OF DOORWAY STATES....................... 96 Introduction............................................ 97 1 FINE STRUCTURE OF DOORWAY STATES........................ 98 1.1 T-Matrix of a Single Doorway State............... 99 1.2 Unitarity of the S-Matrix.......................... 103 1.3 Poles of the T-Matrix.............................. 105 1.4 Fine Structure of Total Cross Sections............. 110 2 INFINITE PICKET FENCE MODEL............................. 115 2.1 Poles of the T-Matrix.............................. 117 2.2 Pole Distributions................................. 121 2.2.1 Central Poles............................... 121 2.2.2 Weak and Strong Couplings................... 127 2.2.3 Summary..................................... 129 2.3 Pole Residues..................................... 132 2.3.1 General Properties and Doorway State Phase Shift................................ 132 2.3.2 Weak Coupling Limit (Regular Poles)......... 134 2.3.3 Strong Coupling Limit...................... 137 2.3.4 Residues of the Central Poles............... 140 3 APPLICATION, RESULTS, AND CONCLUSION.................... 144 3.1 Resonance Parameters........................... 144 3.2 Fine Structure of the Total Cross Section.......... 146 3.3 Intermediate Structure of Cross Sections........... 148 3.4 Distribution of Resonance Widths and the Doorway State Constants.................................... 150 3.5 Doorway States at Low Energies..................... 153 3.6 Doorway States at High Energies.................... l60 v

TABLE OF CONTENTS (CONT'D) Page PART III. CALCULATION OF DOORWAY STATE FORMATION PROBABILITY... 163 Chapter Introduction.......................................... 164 1 A MODEL HAMILTONIAN AND THE DOORWAY STATES.............. 165 2 DOORWAY STATE FORMATION PROBABILITY.................... 181 2.1 Single Particle Potential and the Distorted Wave... 181 2.2 Formulation of F-................................ 184 2.3 Results and Discussions............................ 191 SUMMARY.... o.............................................. 195 REFERENCES.................................... 229 vi

LIST OF TABLES Table Page II.1 Constants at the Point C of Figure II.7.................. 126 II.2 Constants of the Width Distribution and rt and r.. 156 III.1 Quasi-Particle Parameters and the Pairing Force......... 169 III.2 Quasi-Particle Parameters for G = 0.421 (Mev)........... 169 111.3 Single Particle Energies, t9Vj, with Ni-56 Core...... 171 III.4 Radial Integrals and Reduced Matrix Elements............. 174 III.5 E) and C().................178 III.6 C n for J = 2.................................... 178 111.7 rt and the Doorway State Energies...................... 192 D.1 Radial Integrals......................................... 227 vii

LIST OF FIGURES Figure Page I.la Direct Reaction................................. 37 I.lb Compound Reaction....................................... 37 I.lc Exchange Reaction..................................... 38. 1.2 Neutron Distribution....................... 62 1.3 Ranges A and A', and the Singular Point of Equation (4.1.9)...................... 72 1.4 Optical Model Real and Imaginary Potentials.............. 77 1.5 Optical Model Fit to the Total Cross Section of Na-23... 78 1.6 Imaginary Potentials of the Modified O.M. and the O.M. for Ni-58.0............................................. 91 II.1 Intervals I and A, and Domain d on a Complex E-plane..... 107 II.2 Fine Structure of the Total Cross Section at WCL......... 114 II.3 Existing Regions of Poles............................... 118 11.4 The n-th Path....................................... 120 II.5 Paths of the Poles.......................... 120 II.6 Graphical Determination of Central Poles at o=O........ 122 II.7 Paths of the Central Poles.............................. 124 11.8 Constants at the Point C of Figure II.7.................. 125 II.9 Numerical Illustration of Paths and Poles............... 130 II.10 Illustrations of Paths.............................. 131 II.11 Absolute Values of the Residues (Regular Poles at WCL)... 135 1112 Doorway State Phase Shift................................ 136 II.13 Absolute Values of the Residues (SCL).................... 138 II.14 Doorway State Phase Shift................................ 139 viii

LIST OF FIGURES (CONT'D) Figure Page 11.15 Residues of the Central Poles............................ 142 II.16........................................ 143 II.17a Fine Structure of Total Cross Sections at WC............. 147 II.17b Breit-Wigner Form....................................... 147 II.17c Fine Structure of Total Cross Sections at SC............. 147 11.18 Resonance Width Distributions.......................... 155 II.19 Total Cross Section and the Background Cross Sections of N -58................................................ 159 III.1 Independent Two Quasi-Particle Energies (e. + e')..... 170 11I.2 Low Excited Energy Levels of Ni-58....................... 176 111.3 Calculated Levels by Lawson et.al........................ 177 III.4 Single Particle Potentials; 3-D Square Well (S.W.), Shell Model (S.M.), and Optical Model (O.M.) Potentials.. 182 11I.5 Radial Wave Function of the Incident Neutron............. 188 III.6 Bound State Radial Wave Functions........................ 189 1II.7 Bound State Radial Wave Functions1....................... 190 111.8 r- and Doorway States................................. 193 C.1 Width Distribution (Edge Effects)........................ 214 C.2 W and r F of the Finite Picket Fence Model.........215 D.1 EN and Quantum Numbers................................. 225 ix

LIST OF APPENDICES Appendix Page A DISTORTED WAVE T-MATRIX AND SCATTERING AMPLITUDES....... 197 B CROSS SECTIONS AND PARTIAL WAVE ANALYSIS................ 200 B.1 Channel Spin.................................. 200 B.2 Free Waves in Partial Wave Analysis................ 203 B.3 Cross Sections..a................................ 204 B.4 Distorted Wave in Partial Wave Analysis............ 208 C DEVIATION FROM THE INFINITE PICKET FENCE MODEL.......... 209 C.1 Poles of the T-Matrix.............................. 209 C.2 Weak Coupling...................................... 211 C.3 Edge Effects....................................... 212 C.4 Summary............................................ 212 D FORMULAE FOR THE PART III............................... 218 D.1 Tensor Expansion of the Two-Body Interaction Potential....................................... 218 D.2 Normal Products... o.............................. 219 D.3 Vector Coupled Two Quasi-Particle Operators........ 220 D.4 The Reduced Matrix Elements and the Radial Integrals......................................... 222 D.5 Shell Model Wave Functions......................... 223 D.6 Radial Integrals................................... 225 x

INTRODUCTION In this thesis, we shall expand the doorway state theory for neutron nucleus reactions in the energy region below 10 Mev. In particular, we will improve the optical model in the very low energy region (1-500 Kev) and study the intermediate structure of cross sections. Further, we will investigate the analytic structure of T-matrix so that the properties of the fine structure of doorway states may be obtained. The results will be applied to the study of fine structure as well as intermediate structure of total cross section. Finally, the doorway state formation probability of NI will be calculated by application of the shell model. For more than ten years, the giant resonances of the total cross section and of the strength functions have been known and explained by the optical model.(29) Both the experimental data and the theory have been refined.(24,38-48) Recently, the deviations of the results of optical model (O.M,) calculations from the experimental data have been explained in terms of the simplest internal degree of freedom of the compound nucleus (the doorway state). This doorway state approximation has been applied to the strength functions of individual nuclei,(22) the intermediate structure of cross sections,(61,62) and the total cross section in the very low energy region. The last problem is our main concern in Part I and the detailed discussion may be seen in 5.1 of Part I. The intermediate structure of the cross section is composed of the resonance structure smaller than the giant resonances but larger than the fine resonances. In the high energy region where the experimental energy -1

-2resolution is limited, the fluctuating part of the measured cross section from the smoothed cross section may be considered to be the intermediate structure. In the low energy region where fine resonances may be observed, the E-averaged cross sections show bumps whose sizes are smaller than the giant resonances. These are the intermediate structures (named for the appearance). The intermediate structure of the measured cross sections in the high energy region are discussed by Kerman et.al.(62) Those in the low energy region are discussed by Feshbach et.al.(61) Further detailed search for intermediate structure has been actively conducted by by various experimentalists. Some of the observations are as follows: Cross sections of Cr, T-, Al, e, and Ce from 0.5 to 1.5 Mev are measured and the intermediate structures are observed by Smith(92) and by Cox.(93 Taking the E-average over a certain energy range, and varying the range, they observed bumps in the E-averaged cross sections. If a bump exists over a reasonable variation of the energy range, the bump is considered to be as the intermediate structure.(83) The cross sections of heavy nuclei(89,92) such as M and j do not reveal a clear intermediate structure. Feshbach established the unified theory of nuclear reactions in 1958.(1) This theory is more convenient for dynamical description of the nuclear reactions than the R-matrix theory.(4) Weisskopf(21) proposed a reaction mechanism of cascade-like development of the nuclear reactions in contrast to the Bohr's mechanism. Based on the unified theory and the Weisskopf's model, Block and Feshbach(2) established the original form of the doorway state theory in 1963: Considering

-3that the cascade is proceeded by the residual two particle interaction potential and that the ground state of a target nucleus plus the incident neutron is one particle state, they called the simplest excited state of the compound nucleus, 2 particle-1 hole, the doorway state. We will interpret this definition of the doorway state as a subspace of (A+l) nucleon wave functions in the closed channel which is directly coupled to the open channel through an interaction Hamiltonian. The theoretical formalism itself is not changed by the extension. But the doorway state can be a vibrational or a rotational state and is determined by choice of the Hamiltonian. The doorway state theory for the intermediate structure has been developed by MIT-group.(61) We will also apply the doorway state theory to a complex potential model (the modified O.M.), in Part I. Our formalism has a closer relation of the doorway-state effects to the complex potential than the MIT's; for example, we may take an energy average of the complex potential. Some results of these theories are as follows; the E-averaged cross section has a Breit-Wigner form around a doorway state. The effects of the doorway state may be represented by three dynamical constants: the doorway state formation and decay probabilities, and the doorway state energy. The above theory improve the O.M. In particular, we may expect that the deviation of the complex potential for the smoothed cross sections of individual nuclei from the O.M. potential (to yield over-all averaged cross sections and the strength functions) may be explained in terms of the doorway states. Here, the complex potential is determined by the dynamical constants of the doorway states.

-4The shape of the measured intermediate structure has a sort of similarity to that predicted by the doorway state theory, and their widths are of the order of 100 kev which are extrapolated from the values of the doorway state formation probability of the simple nuclei. Because of the above reasons, the doorway state Ansatz has been justified. However, for example, the intermediate structure in the high energy region may be explained by the lumping effects of the fluctuation cross section (References 90 and 91) or by the statistical theory.(84) In the low energy region, the total cross section with fine resonances and with the intermediate structure may be explained by multi-level Breit-Wigner formula. (4,85) Under this situation, we consider that we must justify the existence of the doorway state upon stronger bases than the usual shape identification. The dynamical constants of a doorway state have not been obtained except for the case of simple-structure nuclei for which measured resonances themselves may be considered to be the intermediate structure. (References 71, 77 and 80). Since these constants are important, we will develop a method to analyze the experimental data. We may naturally expect more information on the doorway state from the fine structure of doorway states. We may also expect that the shapes of fine resonances are different from those of Breit-Wigner form due to difference in the reaction mechanisms of Bohr's and Weisskopf's. If so, the resonance shape provides any information and we must find out the condition under which the shape is reduced to the Breit-Wigner form.

-5For the above purposes, we will develop Mahaux-Weidenmuller's study(82) on the analytic structure of the T-matrix(96) when the doorway state exists. First, we willfound the general theory of a single doorway state including multi-open channels.(94) By assuming the infinite Picket Fence model, we may almost completely trace the poles and residues of the T-matrix as functions of the doorway state formation and decay probabilities. From the pole and residue distributions, we may also investigate the fine structure of total cross section. Based on the above results, we have discovered systematic resonance width distributions(63) from Ca- to Ni- isotopes which are obtained from experiments at Duke University(85) and will analyze the distributions to evaluate the dynamical constants. The obtained values well be applied to estimate the total cross section. The results are in good agreement with the measured cross section. These as well as brief discussion on the intermediate structure in the continuum region will be contained in Part II. The dynamical constants are also applied to the modified O.M. potential of Part I. The result indicates double YL in the total cross section of N\ in good numerical agreement with the measured cross section. In Part III, we will estimate the doorway state formation probability of /\g by nuclear structure calculation. The nuclear structure models for the highly excited states have not been much developed. It is mainly because the many configurations involved make calculation unfeasible. For simple-structure nuclei, however, the structure analyses of neutron elastic scattering cross sections and the resonance widths have recently been developed by Lemmer and Shakin,(71) by Shakin, (77) and by Lovas.(8O) Lemmer and Shakin, and Lovas have calculated the

-6elastic scattering cross sections of and C respectively. Shakin has calculated the resonance widths of Pb isotopes. These nuclei are considered to have simple nuclear structures because they took into account only doorway states for the compound states and because the calculation agrees with the fine structures of cross sections. Our particular interest lies in the intermediate heavy nuclei of which observed cross sections exhibit excitation of more complicated states than the doorway states (Part II). For the highly excited states of such nuclei, Lande and Brown proposed a model. Their calculation, however, did not show the intermediate structure. Taking full advantage of the doorway state theory for these nuclei, we decouple the more complicated states and calculate the doorway state formation probability. Namely, we calculate only the intermediate structure and do not estimate fine resonance widths which involve mixture of highly complicated configurations. We will take the quasi-particle model (pairing force plus 2- pole-2 pole force) which has been developed for the low lying excited states of vibrational type nuclei by Kisslinger and Sorensen, (Reference 99) Baranger,(100) and Yoshida.(97) For Atl isotopes, the levels are estimated by Kisslinger and Sorensen,(99) recently by Hsu and French,(108) and by Auerbach.(110) Lawson et.al.(107) calculated these levels by the shell model. We will also calculate the low excited levels and adjust the force constants to yield the above calculated as well as measured levels. With thus determined Hamiltonian, we calculate the doorway state formation probabilities. The results agree fairly well with the measured values. The relation with the Lande-Brown model will be briefly discussed in Part III.

PART I INTERMEDIATE STRUCTURE OF CROSS SECTIONS -7

INTRODUCTION In this part of the thesis, we improve the Optical Model in the very low energy region and provide some information for the intermediate structure of cross sections. Chapter 1 contains some historical background. In Chapter 2, adopting the formalism of the unified theory, we present the theoretical preliminaries for the thesis. In Chapter 3 we outline a new method for the derivation of the optical model potential. This method has the advantage that it permits straightforward calculation of the doorway-state effects upon the optical model potentials, and such calculations are carried out in Chapters 4 and 5. Chapter 6 contains a brief comparison between our results and the results of Feshbach and his co-workers. -8

CHAPTER 1 REVIEW OF EXISTING THEORIES 1.1 Bohr's Compound Nucleus A comprehensive history of the early research on nuclear reactions may be found in the works of Breit, Evans,(6) and Darow.(7) The nuclear transmutation reaction of N([, r) 0 was discovered by Rutherford(8) in 1919. Until 1930, (X p) - reactions with light nuclei, particularly, B, N, and Al, were investigated by the Cavendish school.(7) In 1930, the resonance reaction was first discovered by Pose(9) from the Pt (, P))-S experiments. In the same year, Bothe and Becker(lO) discovered ( L,Y) - and (c,') - reactions, although they did not distinguish between the two. In 1932, the neutron was discovered by Chadwick(11) from B(& 6 ) C. Since then, many neutron experiments have been performed. In 1935 and 1936, neutron moderation, 1/V - absorption, and resonance capture were discovered by Fermi,(6) and by Moon and Tillman.(13) In the 1930's, prototypes of the modern nuclear reaction theories were developed: the resonance theory,( 56) the direct reaction model,(16) the statistical model,(17) the continuum theory,( 7 8) and the general theory.(l9) In the resonance theory, the compound nucleus was proposed by Bohr.(l4) The Breit-Wigner formula(l5) was derived taking Y -decay channels into account. Bohr's classical compound nucleus has recently been reconsidered by Breit(20) from a quantum mechanical standpoint, referring to Blatt and Weisskopf's text.(3) "Bohr's assumption" is (20) stated as follows: -9

-10"The nuclear reaction is separated into two'stages': (a) the formation of the compound system C, and (b) the disintegration of the compound system into the products of the reaction. "The following approximation has been proved valid, or at least approximately valid in many cases: the two stages can be treated as independent processes, in the sense that the mode of disintegration of C depends only on its energy, angular momentum, and parity, but not on the specific way in which it has been produced. "The reaction mechanism is that the incident particle comes within the short range of nuclear forces and shares its energy quickly among all constituents well before any re-emission can occur." Calling Bohr's assumption the case of very strong interactions between nuclear constituents, Weisskopf(21) proposed another reaction mechanism for weak interactions which will be reviewed in section 1.7. His ideas have been first formulated as the doorway state theory by Block and Feshbach.(22) Weisskopf's compound nucleus is the basis of the entire work of this thesis. 1.2 The R-Matrix Theories The general theory of nuclear reactions by Kapur and Peierls(l8) formed the basis of the R-matrix theory which is reviewed in the classic paper of Lane and Thomas. () The Breit-Wigner formulae(4' 1) were originally derived using time-dependent perturbation theory in which a single particle plays a significant role and the interactions are weak (electronphoton interaction). The form of the theoretical cross section fits well to the measured resonances. However, difficulties appear when attempts are made to interpret the values of the parameters. The unsatisfactory reliance on the perturbation theory was removed by Kapur and Peierls.

-11There are various formalisms of the R-matrix theory,(4) among them, those of Kapur-Peierls, of Wigner-Eisenbud, and of Moldauer. These formalisms are interrelated by choice of the boundary conditions (References 17 and 18) which will be set on the eigenfunctions. Each is characterized by a different choice of Be, the logarithmic derivative of the internal eigenfunction evaluated at the nuclear boundary. The R-matrix theory is constructed as follows: In each channel, a relative space coordinate is separated into two parts by a channel radius, c, such that outside the channel radius no strong nuclear reactions occur. The regions inside and outside the radius aC are called the internal and external regions respectively. By this separation of the space region, the collision matrix U (= S -matrix) is separated into kinematical and dynamical factors. The kinematical factors are given by diagonal matrices L ~, and, which depend on dC but not on BC. By denoting the incoming and outgoing channel waves by,J and I, the radial wave functions -do and Ooc are related by cC~, 5; lo' —ecp i ~n -e )) I> (1.2.1) c9-,c A Opait) where I is called the surface function, _Zfa is the relative speed, p = ~ - o and the channel index C -( aSM). Another kinematical The rows and columns of matrices correspond to the channel indices.

-12factor, $j, is the wronskian which is independent of P in the external region and in normalized to 2L(='1c) for the open channel. The matrix elements of L are the logarithmic derivatives of the outgoing radial wave function at the channel surface ( L L P ~ OC/ Oc ) Those of n are the ratios of the incoming to the outgoing wave amplitudes (-Cn - E[I / Oc ]a. ). is defined by its diagonal elements L /J LOcJ a. The real and imaginary parts of L are called the shift factor, S, and the penetration factor, y (I a2n ) p The dynamical factor is given by the R-matrix, R. The Rmatrix depends on the channel radius, a:, and on the boundary condition, B.. In the internal region, the function space of the system Hamiltoniam is represented by an orthonormalized and complete set of eigenfunctions X)x within the internal region which are specified by ac and BR, The R-matrix is given by the eigenvalue eA and the reduced width [A. The reduced width is proportional to a channel* component, C, of the radial wave of the eigenfunction,* X, on the channel surface. Denoting 4A by a column matrix i', the R-matrix is given by X R X E - e (1.2.2) where X MA is the direct product. *The channel indices are good quantum numbers in the external region. where /~C is the reduced mass of the channel C o

-13The collision matrix U and the R-matrix are related by U = In + 2 agL I _ AR 1, (1.2.3) where Ls L -B and B is a diagonal matrix of components Bc The Kapur-Peierls formalism(4) is obtained by choice of the Boundary condition, L 0; i.e. B. Therefore, the boundary condition depends on E and is complex.-: Now consider a certain range of E. The R-matrix of Equation (1.2.2) may be divided into two parts R _ a~ t +R where in R all the distant levels are included. _ varies only slowly with E in the above energy range. Then, Equation (1.2.3) becomes U = &2 1 ___ _ _ 2 + (1.2.4) ~ 2 gL 12< (i;X ) S where /~ ( -E - 4E _ ); the level matrix,,[ F- 1 (1.2.5) 0(^ 5 2 (^ -S i0)l -(i-MU; —.^^

-14and ~ is the transpose of ~. If we may attribute the slowly varying part of the S-matrix to the direct reactions, then the direct reaction is given by the distant levels in the language of the R-matrix theory. In the unified theory, such a contribution is given by the distorted waves, or by a single particle potential, or by the real parts of the potential of the generalized optical model, In Moldauer's formalism,(23) the S-matrix is sought which is correct in a certain energy range A centered at Eo, instead of that which is correct over the entire energy region such as Equation (1.2.3) or that of Kapur-Peierls. The statistical region, I, centered at Es ( I >> ) is also considered. The choice of the boundary condition is B' L (E). Then, the R-matrix is divided into two parts; 2 r -~ A ~ ^- _ ~_2(1.2.6) = i''('~>- p<") + ( R(" t R'g^), where the notation follows Reference 23, and,here, we will see values in comparison with Equation(1.2.2). The matrix _K depends upon parameters for which the statistics is the same as those of i). Its levels are extended from —o to +o outside the region I. Equation (1.2.3) becomes, in the energy region Z, = a.(0) -..Z, U(E 5 >)= U j(E,)- L / X (1.2.7) r E -ErP+ 2krr

-15where U( ) 2 + (;) e ( (2) and (1.2.8) US is called the statistical collision matrix and will be often referred to in this thesis. Humblet and Rosenfeld(84) expressed the S-matrix by a MittagLeffler series which is valid on the entire Riemann surface. The parameters are independent of energy. This form of S-matrix is ideal in its simplicity. However, it has been pointed out(23) that it is not clear what conditions must be placed on the Hamiltonian to insure covergence of the series. In the above formalism, the dynamical constants depend crucially on the choice of the channel radius, within the frame of one formalism. In the unified theory,(68'69) the radius dependence is removed and this formalism is most easily combined with a nuclear structure theory. Although the unified theory has the above advantage, it is least tractable as far as the analytic structure of the S-matrix is concerned. The dynamical constants are energy-dependent, and there is the same difficulty in the inversion of the matrix as is in the R-matrix theory. In addition, the finite number(34) of compound states causes a serious difficulty in carrying out an average of the S-matrix over energy.

-16We will adopt the shell model for the dynamics of the reactions; i.e., the Hamiltonian is taken to be a single particle potential plus two body residual interactions. The rotational model(66) and the vibrational model(67) may be formally derived from the shell model. In the highly excited states of interest to us, a high degree of configuration mixture makes calculation unfeasible. However, combined with Weisskopf's reaction picture, configurations may be limited to only simple modes of excitation; doorway states. These do not in general give the compound states. However, they show up their existence in various ways in the observed resonance data which will be studied in Part II. Hence, a quantity F * associated with the simple mode is obtained from both experiment and calculation. In Part I, the more complicated states will be averaged out. and their effects are represented by the doorway state strength function, r-. Instead of smearing out the more complicated states, in Part II we assume one doorway state and the Picket Fence model for the more complicated states. Analytic properties of the T-matrix and the fine structure of total cross section will be investigated as a function of the coupling strengths r and r^. From the observed resonance width distribution and the total cross section, 1 and rT are obtained. Also, the relation between Weisshopf's and Bohr's reaction pictures will be derived. In Part III, the Hamiltonian which explains low excited states of Li will be applied to the calculation of -t. *- r and rT are measures of coupling strengths of the doorway state to the open channel and to the closed channel respectively.

-17An E-average of the T-matrix is involved in Part I. We adopt Feshbach's calculation which becomes a good approximation for <->/"p/><<. As far as this calculation applies, the averaged quantity is given by a simple form and the inversion of the matrix is unnecessary. As mentioned above, analytic properties of the T-matrix will be studied in Part II. By limiting the energy region as is done in Moldauer's treatment of the statistical collision matrix, the single particle effects may be taken to be independent of energy. Then, resonance parameters become independent of energy. By the assumption of the single doorway state, the multi-channel T-matrix becomes reduced to the form of a branching ratio times a structure factor, and the inverse of the matrix may be calculated easily. 1.3 The Phenomenological Optical Model The essentials of the optical model (O.M.) approach are as follows:(24) (1) The many-body problem is approximated by a two-body problem. (2) The imaginary part of the potential appears through the elimination of open channels and by an energy average of the T-matrix. Based on the empirical fact that the complex potential gives good fit to the E-averaged total cross section, the O.M. potential has been defined as the potential which will produce the average S- or T-matrix.(l) The early development of the O.M. is reviewed by Feshbach in Reference 24. The replacement of the target nucleus by a potential well was first suggested by Bethe(25) in 1935. He hoped to understand nuclear reactions and, in particular, the slow neutron resonances which (14) (15) have been successfully explained by Bohr and by Breit and Wigner.

-18The resonances obtained by Bethe are now referred to as "single-particle resonances" or "giant resonances." Bethe(26) pointed out that the single particle potential was actually a single-particle approximation to the Bohr theory, and discussed that it gives the E-averaged cross sections.(l6) The giant resonances as a function of mass number, A, and energy, E, have been discovered by Ford and Bohm(27) and by Barshall. (28) Feshbach, Porter, and Weisskopf(29) have calculated the total cross section by the square complex well. The fit is obtained at u!,-) - | - U.,:~^ ) r-< r ( 0 4>- < U, = 4n, MAd, 0 - 0 03 (JU /26 Ha), and -3! R - 1,,H.<x/o A73,. Thomas(32) and Lane, Thomas, and Wigner(33) have shown that the giant resonances are derived directly from the R-matrix theory by a suitable Ansatz on the distribution of the reduced widths. A similar Ansatz is used by Moldauer(34) for the intermediate structure of cross sections. The results of Part II can be thought of as a justification of the Ansatz, for a particular doorway state model. As mentioned above, the imaginary part of the potential is produced by elimination of open channels and by the E-average process. The many-body problem may be formally reduced to the two-body problem without The terminology "giant resonance" refers to wide resonances over E and A.(30)

-19the process of E-averaging.(24) Such a potential is called the "generalized optical potential."(24) If open channels are eliminated, it becomes a complex potential.(24) Another cause of the imaginary part of the O.M. potential has been justified by taking the E-average of the T-matrix(l,24) (see ~ 1.5 and ~ 3.1). This E-average process corresponds to the experimental situation of "bad resolution". Namely, the experimentally measured cross sections with the bad resolution may be calculated by the O.M. A physical interpretation of the bad resolution was once given by Friedman and Weisskopf(35) to the effect that the E-average appears as a consequence of the formation of the wave packet.(1,24) However, the well-known fact that the wave packet changes its original form as it moves indicates that the wave function becomes indefinite at large distances from the target nucleus (see Equations (3.2.2) and (1.5.1)). We will show in 3 3.2 that the O.M. is derived from the E-average of the Schrodinger equation rather than from the quantum mechanical resolution. Namely, each incident neutron has an ideal resolution. A group of incident neutrons and target nuclei forms an ensemble. Each element of the ensemble follows an equation of motion independently. The energies of the elements distribute around an energy Eo of the O.M. Therefore, observed cross sections are the ensemble average of the independent events. This is parallel to the E-average of the statistical theory. *This method for derivation of the O.M. equation is developed in order to derive the doorway state strength function (Chapter 4). On this particular point, the author acknowledges the assistance of extensive discussions with Dr. P. A. Moldauer and Dr. R. Coester. Also, Dr. S. A. Cox's explanation of the experimental resolution was very suggestive of ths eeee correct approach to this problem.

-201.4 Refinement of the O.M. Potential The optical model (OoM.) was proposed in order to explain the (29) giant resonances.(9) In combination with the Hauser-Feshbach (H-F) statistical theory(36) (further improved by Moldauer(37)), the O.M. has been used to obtain theoretical values of elastic, elastic differential, and inelastic scattering cross sections in addition to the total cross section.(46-48) At higher energies where there are many competing compound-nucleus decay channels, the compound elastic cross sections may be neglected compared to the shape elastic cross sections so that the shape elastic and the absorption cross sections of O.M. may be put equal to the average elastic and average total reaction cross sections respectively.(24) At such high energies, the O.M. potential (without H-F correction) is expected to yield the E-averaged total, total elastic, differential elastic, and total reaction cross sections, Originally, an energy independent complex square well was used for the O.M. potential,(29) with fair success, but many refinements were applied subsequently; Wood-Saxon potential,.S potential, etc.(24) Furthermore, deformation of nuclei has beentaken into account to explain the double peaks of the strength function at A l160.(44 47) Surface absorption (peaking of the imaginary potential near the nuclear surface) has been introduced to improve the differential elastic, total, and total elastic cross sections at higher energies,(38'39) and the strength function at A-100.(40-43) One would hope to be able to choose an energy independent form for the O.M. potential, but it was found necessary to admit a slow variation of the potentials with energy.(38) Even so, various energy dependent potentials have been shown to be equivalent to nonlocal potentials.(24,47,48)

-21In the usual O.M. the target nucleus is not assumed to have internal degrees of freedom. Scattering into inelastic channels appears only via absorption followed by decay of the compound nucleus (see next section). Yoshida's "Generalized Optical Model"(52) incoorporates target nucleus internal degrees of freedom, thereby yielding direct inelastic reactions, This approach was applied to the deformed nuclei, treating the nucleus as a rigid rotator, and may be regarded as a forerunner of the doorway state theory. Moldauer(23,37) has developed a statistical theory which takes into account the correlations of the R-matrix parameters. He obtains an optical model potential which fits both the strength function and the low-energy (-1 Mev) average experimental values of the total, differential elastic, and inelastic scattering cross sections.(43337) We have done % - fit calculation to the total, total elastic, and differential elastic scattering cross sections on Na 3 The energy range from 1 to 4 Mev is used for the fit. Calculations were performed on the IBM-7090 using the ABACUS II code which has O.M. with H-F correction. The result is applied to the inelastic scattering cross sections with fair success. The total cross section is calculated extending the energy range up to 10 Mev with good fit to the measured value.(51) However, extension to the lower energy region was not satisfactory. The discrepancy becomes larger below 0.5 Mev and the calculated total cross section turns into the well-known l/Lr - region of the O.M., while the measured C3 is fairly constant in E o In order to improve the O.M. fit, the imaginary part of the O.M. potential must be allowed to vary rapidly with energy in the very low energy region. Part I of this thesis is devoted to the theoretical justification of this variation. The detailed discussion on the method for development may be seen in Chapter 4.

-22On the other hand, the MIT-group(61) has developed the theory for the intermediate structure of cross sections in the higher energy region(62) Our motivation and method for the approach to the problem (see Chapter 3-5) are different from those of MIT, and are worked out independently. However, both are dealing with the same theoretical objects, So, we will follow our formalism and notations up through Chapter 5. In the last chapter, the notations will be compared and, in the following parts of this thesis, we will adopt MIT's notation, since it is more commonly usedo 15 The OoMo and the Statistical Theory Feshbach, Porter, and Weisskopf(29) introduced the average cross sections: the shape elastic cross section, the compound elastic cross section, and the compound-nucleus formation cross section. The phenomenalogical optical model and the statistical theory(36 56) have been related through the transmission coefficient. Moldauer(37) and Porter(57) extended the average cross sections to correspond to those of the generalized O.M. theory. Moldauer(37) further improved the statistical theory taking into account the statistics of the R-matrix parameters (58) By partial wave analysis, the asymptotic wave function becomes ] O X(Pcne -^cc-t. A\e ick ^ -^e rYt/-,(1.5.1) (1.5. 1)

-23where no spin is considered and ]/k!J < i. The cross sections are Ce ='i, (i - 2 (657 r = T 2z 1+ 1( -1 | i 2), / (1.5.2) the subscripts referring to elastic, reaction, and total cross sections respectively. Taking the E-average, the total cross section becomes - () X (2 +)( -) (1.5.3) The shape elastic, compound elastic, and compound-nucleus formation cross sections are defined respectively by ise _ T 2 (2 At + | i (1-5.4a) iS —2; St + )(-/ 7,( (1.5.4b) and;it 5C ) - e 2) 5 6<= (2t+ ) 1( 1- 2 1) (1.5.4c) __ (z) C) (1.5.5) (5-@@s e sc e. (e ) Having the empirical fact that the O.M. gives the E-averaged total cross section, Feshbach, Porter, and Weisskopf identified the E-averaged S-matrix with the S-matrix which is yielded by the O.M. potential. This then becomes the definition of the O.M. potential(l) (~ 3.1). The question immediately arises as to whether the shape elastic and compound nucleus cross sections (Ce and 5c (i.e., those obtained by replacing 7 by f7 in the expressions for (e and 5r )have physical significance. This is discussed in detail by Feshbach, Porter, and

-24Weisskopf(29) who show that quite generally the shape elastic cross section is approximately equal to 5p, the potential scattering part of the scattering cross section. Therefore, the compound elastic cross section ce of (1.5.4b) is equal to the energy average of e - 6~ The compound nucleus formation cross section is also related to the imaginary part of the potential by Porter(59) (see 5 2.3 and 2.4). This is further related to the transmission coefficient by -a, _ St (2&h1 Vl ) J=P (1.5.6) The computer code ABACUS II(55) is available for the optical model combined with Hauser-Feshbach statistical model. To take the direct reactions into account, the O.M. is replaced by the generalized O.M. The T-matrix elements obtained from the generalized O.M. is equated to the E-averaged T-matrix elements.(37) In this case, the average cross sections are redefined by Moldauer.(37) As mentioned above, the statistical theory is further developed.(37) Modern statistical theory is developed based on the R-matrix theory(4,23) and the statistics of the R-matrix parameters.(58) The consequences on the intermediate structure of cross sections are given by Moldauer.(89) 1.6 Direct Reaction and the Unified Theory The nuclear reaction models have been composed of two models:(4'60) the compound nucleus model and the direct reaction model. The basic physical idea(4'60) for the direct reaction mechanism is that, as a consequence of the fairly onlg mean free path against collision, it is possible for an incident nucleon to enter a nucleus and to exchange energy

-25with a target nucleon, and for the one or the other particle to escape directly (or in one step) without formation of the compound nucleus. The compound reaction mechanism has been represented by Bohr's model. These reaction mechanisms are distinguished by the angular distribution and the energy dependence of cross sections. In the direct reaction, there is the strong forward peaking in the angular distribution. Also, the cross sections predicted by the direct reaction theory do not show the sharp resonances characteristic of the compound-nucleus theory. (4,6 ) In the R-matrix theory, the direct reaction is usually ascribed to the distant levels(4) which appear in the S-matrix as the first two terms of Equation (1.2.4) and the first term of Equation (1,2.7). On the other hand, the Hamiltonian of the direct reaction model(60) is constructed by adding to the O.M. potential an interaction which causes the transition to the inelastic channels. In other words, the Hamiltonian of the generalized O.M. is the model Hamiltonian of the direct reaction theory. The formal theory of the direct reactions has been developed by many authors(65) and is reviewed by Austern.(64) These works were forerunners of Feshbach's unified theory of nuclear reactions.(68,69) This formalism can easily be applied to the theory of the nuclear structure. The nuclear structure studies have been recently started in the highly excited states, based on the unified theory. We will contribute to the study in Part III where the target nucleus is one of the intermediate nuclei.

-26The part of the Hamiltonian corresponding to the direct reaction is PHP in this thesis (see Chapter 2). The parts of the T-matrix and cross sections due to PHP will often be considered to be constant in energy within a certain energy range,. 1.7 Weisskopf's Compound Nucleus In 1961, Weisskopf(15) suggested a unified view of the nuclear models, The nucleon-nucleus interaction is separated into a single particle potential plus residual interactions. Based on the success of the O.M., a weak coupling reaction model is proposed to understand both the occurrence of direct reactions and the formation of a compound nucleus. In Weisskopf's scheme, nuclear reactions proceed as a succession of collisions of two nucleous. At the first collision, the incident nucleon may experience a direct reaction or enter the first stage of the compound nucleus such as "2 particle - 1 hole" state. At the second collision, it may go to the end of the reaction as well as to the more complicated states of the compound nucleus such as "3 particle - 2 hole" states. The first step of this reaction picture is drawn in Figure I.1 of Chapter 2. Block and Feshbach(22) reintroduced Weisskopf's reaction mechanism as the doorway state theory. The doorway state is defined to be the first stage for the formation of the compound nucleus; e.g. 2p - lh states in the above picture. As a natural extention, we may consider the doorway states which are rotation excitation plus one particle states or vibration excitation plus one particle states, etc., depending on the target nucleus. There

-27is no difference in the formalism. The relation between the two-body residual interaction and collective interactions may be seen in References 66 and 67.

CHAPTER 2 THEORETICAL PRELIMINARY In this chapter and in Appendices A and B, the theoretical background of the unified theory will be given. Notations and physical constants which will appear in the later chapters will be specified. Section 2.1 outlines the many-body Hamiltonian. In section 2.2, Feshbach's projection operator formalism will be reviewed. Relations between the T-matrix elements and the cross sections will be given in Appendix A based on the projection operator formalism. Appendix B discusses the partial wave analysis thoroughly and relates cross sections to Blatt and Biedenharn's expressions. Some of the results in the Appendices will be listed at the end of section 2.2. The resonance parameters introduced by Feshbach will be reviewed in section 2.3. The absorption cross sections of the S-wave single channel have been related to the imaginary part of the complex potential by Porter. This is applied to the unified theory and generalized in section 2.4 to the multi-channel cases, There the strength function and transmission coefficients are defined. In section 2.5, Feshbach's energy average of the T-matrix will be reviewed. Finally, the T-matrix and the compound states will be defined in section 2.6. 2,1 Hamiltonian for A (,) rt),* Reactions We deal with the nuclear reaction problem of the type A(n,)')A that is, elastic and inelastic scattering of neutrons by nuclei. The many-body Hamiltonian is discussed in this section. The total energy of the system of 1f +1 particles interacting pairwise is F=- iL +L \f, iL a 2K +X vi (2.1.1) -28

-29where the coordinates are referred to the laboratory system, A; the mass number of the target, pi; the momentum of the i-th nucleon, /e; nucleon mass, Vj; the two-body interaction potential between the i-th and j-th nucleons, and subscript i = 0 indicates the incident neutron. The coordinates are transformed to the center of mass system as follows. H c= H -f(R)' /4 A A - 12J6-) +L Av +j v, "' i~ 2I1)'t- t(2.1.2) where M = (A I)"-,,; center of mass (c. of m.) of the whole system, r0^ ji; c. of m. of the target, /L; relative coordinate of the incident neutron /D _ Xt p l (2.1-3) and 0 is referred to the laboratory system, I; reduced mass AA^ /b — 1 lit +' (2.1.4) Z.; coordinate of the i-th target nucleon referred to P and so L_ Z, - (2.1.5) /c=1 C

-30For / >1, the coordinates in Equation (2.1.2) are not linearly independent, If A ~ are further transformed to 3 ( - )independent coordinates, however, the equation and the symmetrization operation are expected to become complicated.(70,31) In the Shell Model, all the coordinates X O I,& -",A) are referred to R The total energy corresponding to the shell-model Hamiltonian becomes H = L I + 2,>. I d (2.1.6) and; ^ O i ~ 0(2.1.7) <= 0 All the coordinates are taken as independent variables, in spite of the constraint Equation (2o,1,7), The error due to the redunduncy is proportional to /(1 tA) (31) It causes the spurious states amoung the shell-model states which have been investigated for light nuclei and for low excitation states. We assume that this effect on our problem is very smallo In the case of the R-matrix theory, the same problem must exist. However, in this theory, all the dynamics are represented by the reduced widths tc and the energy eigenvalue o Under the above assumptions, the Hamiltonian is obtained from Equation (2,1o2). H/r - /(,)' t(.r,,, er) - (2..8) ~ V( o; r —- A) (2.1.r8) *The author is indebted to Professor R. K, Osborn for clarification of this material~

-31where K(lro) - k a2 \; relative kinetic energy, 2J e H ~( y --. Y..; Hamiltonian of the target with A nucleons, TY; incident neutron coordinate referred to c. of m. of the target,; target nucleon coordinate referred to c. of m. of the target (! 4 ~ A) f for neutrons (1 i A ) and for protons (N <i ) A ) and r; spin and space coordinate (OA ).) The shell model will be assumed to reproduce dynamics of neutron-nucleus reactions. The main idea of the shell model(72) is that the interaction is represented by a single particle potential as a first approximation and by the effective residual interaction. For the single particle potential of the open channel,(21) it is reasonable to choose the Wood-Saxon potential. The problem is then to determine the residual interaction potential. Since the shell model has been originally developed to explain the ground states and the low excited states of nuclei,(777'80) there may be question of its application to the highly excited states where one neutron can be free. However, the success of Lemmer and Shakin's calculation(71) of neutron - N15 elastic scattering cross section encourages the more extensive application of the shell-model Hamiltonian. Then the potential in Equation (2.1.8) may be written as A V(rvr;- r) =UJ'r) +L VI r, -)) (2.1.9)

-32where JU (L ); single particle potential, and V( r, YC,); residual two-body interaction potential. Now, the equation of motion of the system is given by the following Schrodinger equation H = E (2.1.10) where A HAI= HA +K -+tUi()) t 2 V(, r ) (2.1.11) with the incident wave X! =? ey ~ 4 _ a (7, r*.4- Yr',) (2.l.12) where *; momentum of the incident neutron, -^; spin state of the incident neutron,; ground state vector of the target E; total energy of the system which will be normalized to zero at zero incident energy E - F + _ _ (2. 1.13) 2- 2^ 2o2 FeshbachTs Projection Operators (1) Since 1958, the unified theory of nuclear reactions has been developed by Feshbach, This theory does not have the channel radii. Instead, the function space of (A+ ) - nucleons is projected into subspaces of closed and open channels, and the problem is reduced to an Anucleon problem, This formalism has been most easily combined with

-33the dynamics of the nucleus, such as the shell model.(71,77,80) The incident neutron is treated as a distinguishable particle from the target nucleons. The theory has been further developed in 1962( 1) so that the space is totally antisymmetrized and the Kapur-Peierls formalism is also derived. Assuming projection operators on the totally antisymmetrized space, problems are handled by operator calculus. However, the projection operator themselves are difficult to obtain in practice. For a single open channel, these are exhibited by Feshbach.( ) The radial wave function projected on the open channel is given by the solution of a second order integrodifferential equation. Lemmer and Shakin(71) tried to calculate the open channel radial wave function by a simpler method on the totally antisymmetrized space. However, their treatment of exchange effects seems to be incorrect since the "projection operators" they introduced are not idenpotent. For multi-channel cases including the inelastic channels, furthermore, the relation of the T-matrix to the cross sections becomes complicated. Here, the problem of the type 4(^,u)A * will be treated by the unified theory( 1 ) developed in 1958. The incident neutron will be distinguished from the target nucleus, and the exchange reactions will be neglected. Denoting the vector space of functions of 3A spatial and A spin coordinates by EA, the antisymmetrized subspace of the target states will be denoted by EA. So, we consider a vector on ~/ 3A where the direct product is denoted by.(70)

-34In this section, the projection operators defined by Feshbach(68) will be given, In terms of these operators, the Schrodinger equation will be decomposed into a system of equations, Some meanings of these operators will be schematically described in relation to a simple shell model picture corresponding to Weisskopf's pictureo(21) Then, the effective open channel Schrodinger equation will be obtained. Notations for the basis vectors will be summarized, According to Feshbach, ) the projection operators on I E A may be defined by P _-d ><_ A. (2,2.1) and w'her Z ^r >K<gA, (2.2.2) where j' HA 1AA A (2.2.3) J indicates the j-th excited states, and N is determined by A A (2.2.4) fne+t > E > E~t (see Equation (2o,113)). The operators P and Q are projection operators on both EA and E 5 ~.cp, Q73=(2.2.5) By these operators, the Schrodinger equation of Equation (2.1.10) is written as a pair of simultaneous equations,

-35('pHP -E)t?- = -pHQ Q'^ and (2.2.6) (QHQ-E)Q? -- QHP, where the superscript of HAt is suppressed. The incident wave of Equation (2.1.12) belongs to the open channel subspace on which ~~ ~, is projected by -, Q u:' = ~-0 (2.2.7) Since P commutes with K, H, and UL (ro),?HP(when operating in the open channel subspace, P ([i ~A) ) may be thought of as a Hamiltonian of the form PHp = K(Xo ) + L + L' ) + ~ L P V(,O) P Ho + LJU (o) (2.2.8) where Ho K-A) +t H,(2.2.9) and A LUJ(v% )? U(r )P + L.? V( )P. p (2.2.10) tam Similarly, it follows that QHQ = — Ho +L a(Y'), (2.2.11) where UQro) -- U'o) + o, ) (2.2.12),~i -.2.^)

-36In this case, QHQ operates in the closed channel subspace, O(l~ ~ ). A rough discussion on the meanings of the projection operators will be given based on the perturbational viewpoint, and independent particle model. Expanding ek into a series of eigenfunctions of HA L5i q A +,, MA (2.2.13) the equations for %i and ( of the incident neutron become: [^TV i -(-Ir h = -2LV i - 2AN \laa Xla, (2.2.14) KK U't -(E-Ee )0<2 = -t ve vrn 4~q() >)h -i Vj~ / (2.2.15) v V' i, r (2.2.16) The rh.s. of Equations (2.2.14) and (2.2,15) are the transition terms among different neutron states of a single particle Hamiltonian of the form t+ UL -+V EA. Neglecting these transition terms entirely, Equation (2.2,14) gives the scattering of the neutron by a rigid target, and Equation (2.2.15) determines the bound states of the neutron in this single particle potential. If we consider only the first terms on the r.h.s. of Equations (2.2,14) and (2.2,15), the first equation gives the

-37direct reactions and Equation (2.2.15) gives the bound states whose energy levels may be split into more levels. The last terms gives the transitions among the free states and the bound states. Consider the initial state such that the incident neutron is only in 71. Then, the reaction picture given by Equations (2.2.14) and (2.2.15) may be drawn in Figures I.l.a and I.l.b corresponding to those of Weisskopf.(21) The figures correspond to the first order perturbation. Figure I.l.a illustrates the direct reaction in which the incident neutron goes from Jo, to XY while the target is transferred from to i/. Figure I.l.a. Direct Reaction. Figure I.l.b. Compound Reaction. This transition is made by the first term of the r.h.s. of Equation (2.2.14). By the last term, the incident neutron is transferred to one of the bound states given by the l.h.s. of Equation (2.2.13). This is of course only a rough semiclassical picture of the reactions. For example, setting the l.h.s. of Equation (2.2.15) = 0 does not always give the bound states due to failure to satisfy the boundary conditions.

-38Some terms of 1l M in which a target nucleon is unbound can be excited as intermediate states. Among such states, a target neutron can be free energetically and the incident neutron is bound Figure I.l.c. Exchange Reaction. (Figure I.l.c). This is actually the exchange reaction. We are assuming that the exchange reactions do not occur. Namely, the interaction potential which causes the exchange reactions is considered to be null. The effective open channel Schrodinger equation is obtained formally from Equations (2.2.6) (PHP-E)p - - - -- Q HP PIP (2.2.17) We will abbreviate the potential operator on the rh.s. by W W?HQ I QtP (2.2.18) E-QHQ The following notations will be used for the target nucleus A states: the subscript j of the wave function Q signifies the j-th excited state. id') indicates the quantum numbers of this state. When there occurs no confusion, j will be often suppressed and only OC will be used and vice versa. When we use some of the quantum numbers of the

-39target state, we write quantum numbers with subscript Y. For example, nuclear spin and its Z -component of 1 or I4L) are denoted by.Sand The eigenstates of Ho andPHP describe the free state and the distorted wave including the direct reactions. These will be denoted as follows: (I) Eigenfunctions of Ho; v d or (ij)jsv C(N -~),^E) X~j* sv ~'(2.2.19) and E E= E 2 t (2.2.20) For example, O2r; ) Z.]E~)- g, e i a> )(- E ) (2.2.21) (II) Eigenfunctions of PHP (Distorted wave); t) Ad (t Zd 5" ( o ) &ld) f^ OLi ^ 0 ( ro) (HP-t -0 t(2.2.22) where the incident wave is given by Equation (2.1.11) I- f id O (2.2.23)

-40The plus sign of % indicates the outgoing wave boundary condition. Taking this boundary condition into account, Equation (2,2.22) may (70,73) then be written as an integral equation:(7 73) (- ) ^^)8 d ] dE") ho 5' + LI."I A (2.2.24) In the above notations, the space coordinate ois sometimes suppressed. The subscripts [tLo) S and d) j 5x ] will be abbreviated by [i] (initial) and [f] (final) respectively. The T-matrix and the scattering amplitudes are defined by Equations (A.1) and (A.2) of Appendix A respectively. The T-matrix elements based on the free wave 7 will be related to the scattering amplitudes by Equation (A.6) and then to the cross sections by Equation (A. 12). The effects of the potential / may be emphasized by incorporating the potential U into the unperturbed Hamiltonian Ho. The Tmatrix elements based on the distorted wave(70) will be called the distorted wave T-matrix, while % of Equation (2.2.24) includes effects of the direct inelastic scattering. The two matrix elements are related by Equation (A.8). In the subscript [ id 3 ] the numbers S. 2 refer to the neutron spin state. When we wish to refer to overall channel spin states, the brackets are dropped and we write O.S 2. The transformations of wave functions are given by Equations (B.2), (B.4), and (B.5).

-41The scattering amplitudes and the cross sections are given by Equations (B.il) and (B.12) respectively. Transformations between wave functions defined by sets of indices i &_ S P and Ot ~ ES 7 M are given in Appendix B. Some of them which will be used in the later chapters will be listed below., S, and J are orbital, channel spin, and total angular moments respectively. M is the z-component of the total angular momentum. E is the total energy which is related to the neutron wave vector by Equation (2.2.20). [O/d ] S3, and E~ SJM are sets of good quantum numbers for the free waves. They are not good quantum numbers for the distorted waves. However, the distorted wave and the total wave are well defined by any set of indices through Equations (2.2.24) and (2.2.17) respectively. The free waves in o(S/and in dOJSiJM are given by Equations (2.2.23) and (B.2) and by Equations (B.23) and (B.24) respectively. The relations are given by; v g ^ L <s>| JM () ) sJ (2.2.25) with the inverse relations ^toM" _ oSX < smVFlMf) JM Sl IP (2.2.26) where ~ is the unit vector along. The T-matrix elements are denoted as;r s (&X5 ( ) ( s (JMT ad S 3J (2.2.27)

-42On the energy shell (E~ ), it will be written as d^'r's; i S- ^T^.;d (~, ~:S E' s) (2.2.28) Transformations of the T-matrix elements based on 0( A./ and o /S J PI are given by Equation (B.26). Cross sections are expressed in terms of the T-matrix elements of Equation (2.2.28) in j B.3 of the Appendix B. Among them, the total cross section is given by ^ -^c LS 3 d, )i (2.2.29) where c refers to the initial state of the target. The distorted waves with indices i$2sJJM are related to those with d S by similar relations to Equations (2.2.25) and (2.2.26). Equation (B.39) corresponds to Equation (2.2.26); xv ~ s Sj M < /TJM>X (*)e X dd Y0m j t 1d ^ (2.2.30) The orthonormality of the distorted waves is given by Equation (B.41) and the completeness, Equation (B.42), is given by Qs 0 Sd XC U ) ><2 (0.s. (X) -4 rd >' < H ISr -< MH. (2,, M.3)

-43The distorted wave, Equation (2.2.30), and the free wave, Equation (2.2.26), are related by 2 ( -) _ X C e),,x 2i, -T 21 i E. < / (2.2.32) where,2 /'^' - ad -;, Cdr( ee;ulS a ) (2 (2.2.33) and 9 ge is related to the free wave by Equation (B.23). Often, the channel indices (isJM will be abbreviated by C. For example, completeness will be denoted by ~ C~,) X ) -- p;L - ), (2.2.34) where the sum over C0 is taken up to that of the N-th excited states (see Equation (2.2.4)). The channel indices of the initial and final states are denoted by i and f respectively. Applying the same transformation as Equations (2.2.26) and (2.2.30) to the total wave function, the Schrodinger equations are given by + I; 1 (2.2.35) t. -^i^ "^\t3)

-44f'= ts ~ ES - U 2, (2.2.36) and L pH + E- A (2.2.37) where Equations (B.2), (B.4), (B.5), (2.2.26), and (2.2.30) are applied to Equations (2.2.24), (A.7), and(2.2.17). The T-matrix elements are related to the potentials L/ and W TSp =2:(Jz LX/,,,' w/? >} (2.2.38) as is shown in Appendix B. 2.3 Resonance Parameters In the unified theory, the resonance parameters have been defined so that, in the isolated resonance region, the T-matrix of the unified theory becomes equal to that of the R-matrix theory.( 1) The resonance widths in the many resonance region are chosen by extending the formula obtained above.(l ) Applying the energy averaged T-matrix the optical model Schrodinger equation is derived.( ) These are briefly reviewed in this section. Considering the case of no spin and S-wave, the absorption cross section and the strength function will be derived. The transmission coefficient will be derived from the absorption cross section. Under the isolated resonance approximation where only one of the eigenvalues of Q H Q ( HQ -E ) O (2.3.1)

-45becomes significant in the T-matrix of Equation (2.2.38) and Equation (2.2.18), the T-matrix becomes Ti - 72 < U 1U? + <k- w 0> ^.PQ IE> ) + -E e tK( QHP ^p H PHQ'>" (2.3.2) The denominator may then be written in the form E- E +(E y - r/ ) where the level shift and the resonance width(4) are defined by(1) 6L = is -.A Ibed I^ ^QI}C ->I (2.3.3) or, - 5^ r,, (2.3.4) c.7 / and (2.3.5) r3 S 2sa | < ( PC~ > I (2.3.5) We have made use of the completeness relations, Equation (2.2.34). In the many resonance region, the energy averaged T-matrix element is given by T^pi 2 <( jUn -. 2 { Utul AP} (2.3.6) The derivation of this formula will be given in. 2.5, and the operator / is defined according to A < > /V / i VA, (2.3.7) < V > is the average resonance spacing, and 4 is the number of the resonances within the energy range for averaging.

-46The optical model (O.M.) Schrodinger equation is derived by Feshbach( ) in such a way that the resulting T-matrix becomes equal to the energy averaged T-matrix of Equation (2.3.6). The O.M. Schrodinger equation becomes (=_-T H?P- H S) = o (2.3.8) where 1 -rx/lo~rE~j~Q X? (2 H-ihPtQ ___ - (2.3.9) The derivation may be seen in i 3.1. For the no spin case, the S-wave absorption cross section has been related to the imaginary part of the O.M. potential by Porter.(59) The S-wave absorption cross section is obtained from Equation (2.3.8) & cm-,2 ( r> (I- _ r> (2,3.10) ct < -> 2 <P> where second order perturbation theory with respect to H&i is applied and where, by the use of the definition of Equation (2.3.5), the average resonance widths are given by( 1) <sJ 2f> <o f2 @L (/ a P (2.3.11) The strength function is defined by(116) wShe _ is <t (2.3.12)) where ED is the fixed energy to which widths at E are normalized. The S-wave strength function becomes SD 2C.t /g <A PtS/. Ad P K l > * (2,3.13)

-47The S-wave transmission coefficient is defined by(3,4,37) N == - A 1 J=O (2.3.14)' ffi =- 2. i> (^F 12^ ->) (2.3.15) 2.4 Absorption Cross Section Porter's relation between the S-wave absorption cross section and the imaginary part of the O.M. potential will be generalized to include more than one open channel. The generalization will then be applied to the O.M. Schrodinger Equation (2.3.8) to determine the absorption cross section. Strength functions and transmission coefficients will be obtained in the context of the unified theory. The O.M. Schrodinger equation is written as (PH -i V -E) (4ss = Q (2.4.1) where V isHermitian. In the case of Equation (2.3.8), HN =LU - V and SU may be neglected relative to PHP. The incident wave is I} f ^^ e__ I. (2.4.2) The absorption cross section by partial wave analysis is defined by(3) C C (57c a _ I -. LC (2.4.3) where C= S:M L is referred to the initial state of the target nucleus, ICp is the incoming current in a channel c, and IMt is the net current of a channel c. The asymptotic solution of Equation (2.4.1) is given by X5 ~ - / (2.4.4) 4 T^ r

-48The T-matrix elements in Equation (2.4.4) is related to those of the partial wave analysis by Kx C< *4S'' I> > <3 )' d 5 1, x1 Y c5J (2.4.5) This relation becomes ^J~ Z. \d^<^5sV'vl) YX>^^ t) (%2k'5'voT V'V T, i </S o Ij "nY cr )- / (2.4.6) A= A where + is replaced by ) because x//i('. By operating with r,JV <2S^V2|yM) Y C tt) upon Equation (2.4.4), therefore, the asymptotic form of the partial waves is obtained, I/ X < W/5'/IJ/'l 1M> Ti xS,^, X5 (r ) -S'J' 5(2.4.7) + ~;,-, e <*sm < ))SP i 7 (2.4.8) where we have used Equation (B.35). Therefore, we have 2 V ( A9 L/ - VS.;- o(2.4.9)

-49and TId t { 1'^''' I'. -Z (2.4.10) The absorption cross section is obtained (2.4.11) On the other hand, the net current is related to the non-Hermitian part of Equation (2.4.1). By operating with Z ~i upon Equation (2.4.1) and with its complex conjugate upon the Hermitian conjugate of Equation (2.4.1), subtracting one from the other, and then integrating over r in a large sphere, we have _.- c.- ) _. b~ C yiKL r (2.4.12) Therefore, we have relations between the imaginary part of the O.M. potential and the absorption cross section ThsL4 i C t2 2 z io v (2.4.13) This is the generalization of Porter's result.

-50Under the distorted wave Born approximation (DWBA), the solution of Equation (2,4,1) in the partial wave analysis is given by Sc ~c+ 1 ( (2.4.14) a- "I P/ Oc.'' The matrix elements in Equation (2.4,13) will be obtained in this approximation, <\0L ( V >E -.,-~~ viv ~-~,_?~ -.v) cx,~ "C V 7t >-' (.c) V S " -'PHP) V (C > <^ V > L i<C1/v )~ > +\/ > I2 (~..15) where & u - SJ ~ Now we will apply the results to the case of the O.M. Equation (23.J8), Expanding Hcl to second order in THIQ/ lAaP, the potential V becomes V2 _ GQA0oi? +?ArP@Q^H?-)P^OHQ4Q9P RWH0Q/QH( P E - +Z?=H/Q~, <+ ><6,?/~ P C"' (2 4.16) where C,,,,t5 /I\Jl _, ~ —$ ~ iV

-51Thus, the first term of Equation (2.4.15) becomes <4CV X ) = z 1 2 | < e Q? >V C 2 s: 2 | 2 >< H %rc' >| (2.417) and the second term becomes KhIf (+) VtC) >12 C' Q'S',=, N2 L /I,< Wfl'>)4 flTr Q V >/ (2-4-18) i/'s' /ft The random phase approximation will be applied to Equation (2.4.18) and the second term of Equation (2.4.17). Namely, Z (C P'PQ A/~t><i} fp7 C/> o for c C c'. Equation (2.4.15) then becomes v - ~ LZ <.,> (\ )Q n pX l. 3 / = - - - 2,,()2 (2.4.19) (<- > aT / >' The absorption cross section becomes 6 ab = mA1' fr 2; 3 J (2.4.20) where JtLS 4 S-o(S + t 4-S) (2.4.21) which is the transmission coefficient, and rS Q- r2 < ^> (X.(<-< ) > (2.4,22) =<7t_ = ~ ~~

-52The transmission coefficient is related to V by Equation (2.4.13) c -7 <'eC V5 c > (2.4.23) 2,5 Energy Average of the T-Matrix The energy average of the T-matrix is concerned with the following problems: derivation of formulas for E-averaged cross sections; derivation of the Optical Model Schrodinger equation; calculation of the upper bound of the strength function. The distorted wave T-matrix is separated into two terms, Equation (2,2~38), The first term depends only on the single particle potential. Typically, for a given set of quantum numbers its poles lie along a line in the lower half complex E-plane approximately parallel to the real E-axis, The spacings between poles is of the order of 10 Mev. Hence, this term may be approximated by a constant in the energy interval over which the averaging is performed. On the other hand, the second term is expected to vary much within the range of spacings of fine resonances (Il ev). Precisely speaking, the calculation of the E-average of the resonant part of the T-matrix is known only if the poles and residues of the T-matrix distribute evenly from -4o to +Po along the real E-axis, In more realistic cases, the number of poles is finite. The calculation of the E-average of such a T-matrix becomes difficult due to edge edeeffects.( However, Feshbach's calculation(l) will be briefly reviewed. The result will be applied in Chapters 3 and 4 of Part I. The energy average is taken over a range A centered at Eo within which the single particle effects such as < f U z' )> are

-53approximated by a constant; e.g. their values at Eo. Terms of the resonant part of the T-matrix whose poles lie outside the range 2 are effectively absorbed into the potential part of the T-matrix, T, as follows. The effective open channel Schrodinger equation of (2.2.17) is written as }?H? rr 0 nQ We ><4d Q a )E _ v 7;HQi4 ><r4QHP ( ) i cr - E where the AIS are eigenfunctions of QHQ, IQHQc 92 = E; ix (2.5.2) U _; E such that Eo- _ E IE - + Er; the remaining spectrum of QH Q,; notation for the sum over the discrete spectrum and the integral over the continuum spectrum, Supposing that the second term of the lh.so of Equation (2,5.1) does not vary much with E, this term is effectively represented by a single particle potential operator U'I.,,. p P OwHQ <^ a HP U - U + j 0 -, >< H P (2.5.3) The notation PHP for Hvt+L in Equation (2.5.1) will be used for lHo +U in the following, The distorted wave which is an eigenfunction of H + UL/ will be again denoted by the same notation ~ as before, Then, the

-54T-matrix elements of Equation (202.,38) becomes T+ =-. (U U"0' ) + i-~27^_, r -Z —-------------- E~ (2.5.4) i=< E- E- The integral form of Equation (2.5ol) is E..P..HP -, (2.5.5) with the new definition of i o The wave functions I'2 and ~ are considered to be known. The T-matrix elements of Equation (2o.54) involve which is unknown. The next problem is to express ("GQ & -k tH?s )/ E - E*) in terms of the <A/ Q6 ^'S < By applying Equation (2.5.5) to (.QG._P9 > we have <i,^.Ha? ( = < w Q zh > +) tZ <^ ^P^1?^^; —-------— g —, (2.5.6) ^1 PL H'-'PHP Equations (2o5,4) and (2o,56) are compactly written using the following notations:

-55@* - <k ^QH'P pC+) >/ (E-Ek), Wirk' = </l QHP g- p ^Q ^i.p > t ( r-7, -,, / (2.5.7) k; - S, ---- 7 - k, J where -. is a column matrix. Then, Equations (2.5.4) and (2.5o6) become -Te = 2-i~ <ff # > + 9f ( i / ) (2.5.8) and (^ — E1) F -- ~' (2.5.9) I-7 respectively. If the inverse of E Z exists, Equation (2,o58) becomes T7 =2 <, U > LI (G' G (2.5.10) The problem is reduced to the diagonalization of the matrix which is symmetric but non-Hermitian, This diagonalization is generally not covered in this thesis except for the model of Part II, Instead, we assume the following properties:

-56(a) All the singularities of the T-matrix are simple poles. (b) From the conservation of flux, the poles of the T-matrix are required to exist on the unphysical E-plane. From these properties, the eigenvalues of, may be shown to be all discrete and simple. In addition, these lie below the real axis in the complex E-plane, Therefore, the inverse of E 1 - z exists and is diagonalizable (79) Denoting eigenfunctions of, and its Hermitian conjugate rr -t, by X and XX respectively, X - =,, (2.5.11) The orthonormal and completeness relations are given by X1y X1 - ^j (2.5. 13) j,' = /.(2.5. 14) Then, we have ^-^ =2Eduf"XtF') @2Tt /Jj>^ Gf ^ (2.5.15)' E- - The energy average of Equation (2,5o15) is taken as follows: V +...'U^ -. ->,Bl)t 4 2r, +1.)- )40 (25,6 &7o 42 1 ~" - go1 _ >-." -, <,..j ><', -> =2 ) < trD -is. <I'4)' (2.5 16) T&<L~fU"~Xr)'2'2/\ l^

-57The approximation in the second step of Equation (2.5.16) is not clear. But if the residues and the poles evenly distribute over A and if ^ g'~~- O, then it is a good approximation. Equation (2.5.16) is rewritten as Lj =1 <R ju' -?i PP/'>j (2.5.17) with / given by Equation (2.3,7). This is Feshbach's derivation of the E-averaged T-matrix.( The approximation in Equation (2,55,16) will be extensively used in the remainder of Part I, although the same shifting of T-matrix terms, Equation (2,5o9), will not always be applied. 2,6 TheC-Matrix and the Compound States The T-matrix is defined on the basis of free waves by Adz.f = Age Tz,:'r,.(2.6.1) We decompose T into two parts: T-= TP TR *r = ~~c~ T. 9(2.6.2) 7r k^m-^ T. The potential part TPincludes only the single particle interaction and satisfies t" _ t + -—? TLr.(2.6.3).4 - Ho I

-58The matrix elements are further related to the distorted wave T-matrix by T^t= <^TLT, >T 0,11+<'W Past U f (2.6.4) Let us define the?-matrix according to (4.) (+) I- (2.6.5) so that </r..>-= T'^-^ ^ >, - # ) (2.6.6) Equation (2.6.6) is derived from Equations (2,6.1), (2.6.2), (2.6.3), and (2.6.5). In the following,'will be related to the potential / and the Hamiltonian PHP. First, we rearrange the Schrodinger equation of Equation (2.2,37): +..... / PP W (2.6.7) E42-?H P Then, comparing Equations (2.6.4) and (2,6,7) and noting that T is defined only in terms of its effects on the/Al, we may identify t-h oW, awe Using the definition of W (Equation (2.2,18)), this may be written E- _ -__-1 w4'. 18 g-Q & - ca -Q HeePg^_2Pw FnQa - (2.6.8)

-59Later, we will define the compound states as being the eigenfunctions of the operator, defined according to: L_ QlhQ ~+ Q-HPPl where E is realo On the basis of, which are eigenfunctions of QHQ, this operator may be expressed by a complex symmetric matrix which will /r be denoted by,, where S=S - sEs +- Is r, and 4a'tand — SFl/are the real and imaginary parts of <4s (HP F iH Qs). -. Assuming that all the eigenvalues of, are different, then, may be diagonalized by a similarity transformation / (77 Denote the compound state by QH - ELWHP T -( aI 0 (2.6.9) The 4 is related to. by Cf> = u/ f (2.610o) This formulation will be applied in Chapter 3 (Part I) and in Chapters 1 and 2 of Part IIo

CHAPTER 3 THE OPTICAL MODEL POTENTIAL One of the advantages of the unified theory is that the optical model potential is easily expressed by operators. Our aim is to improve the optical model in the low energy region. As will be discussed later, this will be accomplished by extracting the doorway state contribution in which all the compound states are smeared out by the energy average process. For this purpose, a new method for derivation of the O.M. potential will be developed in this chapter. In ~ 3.1, Feshbach's method will be briefly reviewed. In 3.2, the new method will be shown. 3.1 Optical Model Potential The optical model potential is defined by Feshbach(68'24) such that it yields the energy averaged T-matrix elements, Equation (2.5.17). We will review the derivation of Reference 68, in this section. Consider an O.M. Schrodinger equation 4r -) ff, Ah H Y ) (3.1.1) where (x is the O.M. wave function and Hcv is to be obtained. The distorted wave T-matrix elements of Equation (3.1.1) are given by (see Equation (2.4.16)) -- R {< uu K)> +< oX) H2 H>f.f.>f (3.1.2) -60

-61HcP will be determined from T6 = T-. By comparing Equation (3olo2) with Equation (2,5.17), we obtain Hc f =" -/7'AHQ,/ aH/P ^t (3.1.3) Applying this to Equation (3olol), =+ _ "A) + f (- (7PH8QA Q p R *^(3.1l4) Or,'/ ( -- r (3,1.5) P -kr / L7H PH8XQ (HP In order to derive the Schrodinger equation for 0, we operate with H -?P on both sides of Equation (3ol.o5) with the result (E-m ) I 0 = " _ X T pH @ /P < C I H P which may be rearranged as ( E —P 1P -tc^/) i = 0, (3.1.6) with H /o -/~ 7'/l l A P (3 1 7) rr^ =d -^?^a -- -- j --- /la~' (3.1_.7) Thus, HcN is related toPHQ and its Hermitian conjugate, and to the density of the compound states. If / and QHPE5-W^P-] 7HR a are independent of energy, then ac) is constant with energy except for a slow variation through the non-locality of?t/(A HPJ o This is the usual justification for the Optical Modelo

-623.2 O.M. Schrodinger Equation The average cross section is that which is measured when the incident neutron has what is called "bad resolution". This resolution is often referred to as the spread of the wave packet.(l ) Instead, we consider it as follows: N neutrons are incident on the target nuclei. Each neutron has an ideal resolution. But the number of neutrons with energy E is distributed as n(E) with spread 4 centered at Eo. For example, see Figure 1.2. |t(E) I- --------- I —-— ^ —------------— E A ---- Figure 1.2. Neutron Distribution. The n-th neutron with energy En obeys the n-th Schrodinger equation (E -_ PH )? H W (Ed) ^i (3.2.1) The Optical Model is considered to represent these N independent events by a single Schrodinger equation. The first problem is to reduce the system of Schrodinger equations to a single Schrodinger equation. Next, we will consider whether the obtained equation is the same as Equation (3.16); the O.M. Schrodinger equation.

-63Consider the free waves and the distorted waves whose energies are within the spread, At large distances, (a) r(~) 7Yo(\) ^h(a) ZEoEer S ) |(3.2.2),,Ir.) $ 2:~() I tlz A However, we may assume that (b) (_t) 1,- ) H f t- I 1/<R OWE A) AEgl ) J e.g. < JU >(Uil, >LEo2. 3(3.2.3) Furthermore, we assume that (c) the shift functions, Equation (2.3o3). and the resonance widths, Equation (2o3o5), are independent of energy within A. Or, - pH Q- P....H r,* ) P En-_ -/. p (3.2.4) The conditions (b) and (c) are assumed not only in this section but in this entire thesis. If we replace the operator ER-PHP of equation (3o2.1) by Eo -PHP, we obtain (IE-WiPN) )~ = IWt )y it (3.2.5) We intend to average this equation over l, and use the function defined according to - y - as our optical model wave functiono

-64It is important to note that the,DS, the solutions to Equation (3.2.5), are not equal to the?pZ, the solutions to the actual Schrodinger equation (3.2.1), nor are they approximately equal. In fact, by Equation (2.6.5) we have i,- and its asymptotic form:.,, ~. + Ad)'-t~(En)S^~. (3.2.6) __ @>?-~~ 2,to) _ At 2 >-U~s} (9n v~n)2(3.2-7) where we have used Equation (2.2.36). Similarly, < =~ te~ + Ez-i? 7/(~r) A 0 (3.2.8) ^d Sg(2 ^ o p < /i) l^ 2 (3.2.9) where the subscripts An and refer to initial and final states with energy Eo, and the subscripts,and. refer to similar states with energy En (e.g. (E?-PHP),^ ), and where we have defined. C E) according to C/(n)' W(&Et) +W(tn) E ) P? P" n W ), (3.2.10) However, it is not difficult to show that the T-matrices obtained from both Equations (3.2.1) and (3.2.5) are equal within the approximations implied by assumptions (b) and (c). That is, the observables given by Equation (3.2.5) are approximately equal to those given by Equation (3.2.1).

-65In fact, by rearranging Equation (3.2.10), we find' ^-QHi-HH a AQH _,-ar~ c~t:(E)\ ~(3.2.11) by assumption (c). In addition, by Equations (2.2.38) and (2.6.2), and by Equation (2.6.6), we have (<hr S %r > fo < #u p>) (3.2.12) and E (t) )c n~ ) z9 a+) (3.2.13) where assumptions (b) and (c) and the fact that "T and T are sandwiched by PQ( and ( HP are applied. Therefore, for the purpose of obtaining the T-matrix, Equation (3.2.5) may be considered to be the Schrodinger equation for the n-th neutron. Taking average of Equation (3.2.5) over n, we have (Eo -PHf) W= (3.2. 14) Or, Equations (3.2.8) and (3.2.5) lead to (Eo — PHP) = -p H A. (3.2.15) Both are, of course, equivalent. Equation (3.2.15) may be shown to be equivalent to the O.M. Schrodinger equation of Equation (3.1.6) within the approximations (b) and (c). In fact, from Equations (3.2,11), (2,6.9) and (2.6.10), we

-66have T' c 2S7At9H/Qv. (3.2.16) Applying this result to Equation (3.2.15) and using the same argument as that below Equation (3.1.3), we obtain: ( Eo - -PH - HC ) p = 0 (3.2.17) where Hca is given by Equation (3.1.7). Thus, Y is equivalent to the OoM. wave function. For later convenience, we will examine an approximation W =D~ ^ -- W'~ - -. Wc _ (3.2.18) It will be tested by comparing O.M. equations derived before and after this approximation. The eigenvalues of (QH are real, and / is given by Equation (2.2.18). To integrateW along the real E-axis, we need to know how to integrate V/ around the eigenvalues. To assist in the contour integration, it is convenient to include an additional infinitesimal dissipative perturbation (such as might occur from spontaneous 0 -decay). Thus we write the Schrodinger equation as A - (IH- i-E > ) = ) (3.2.18) -. -+0,

-67where the limit ~ —'O is implied to satisfy the experimental boundary condition. Therefore, (E -? +to ) P IT -H Fa Q l (3.2.19) (E-: a ) - Q. HPP ) Defining e and E- as in Equation (2.5.2), we have an effective open channel Schrodinger equation with a first order perturbation. (to-PHt^A) HQ____>____Q? (3.2.20) -3 E-ES -S >' This equation corresponds to Equation (3~2.5) and the operator in the r.h.s corresponds to W. Taking the limit - - +O after integrating W over Ei, we obtain: _i ~ J 2L H ><AQH (3.2.21) S E -E This integral is calculated in~ 3 1 with the result VW -, H W /QH. /(3.222) Therefore, we have (E- PH) P = ( 2 = -' gQAn p. Q? i (3223)

-68As may be seen from Equations (3.1.6) and (3.1.7), this approximation is as good as the first order approximation of Equation (31o.7). As is seen below Equation (2,4.16), this first order gives the first two terms of the absorption cross section correctly. The accuracy involved in the approximation, Equation (3.2.18) is surprisingly good.

CHAPTER 4 INTRODUCTION OF THE DOORWAY STATE STRENGTH FUNCTION The exact Schrodinger equation involves the many-body problem. The cross sections calculated from it must be those measured with ideal resolution. Formally, the equation may be reduced to one with a single particle potential (generalized O.M. potential (~ 1-3)) although the potential is not known. The O.M. Schrodinger equation, on the other hand, contains a two-body complex potential. As we have seen above, the averaging operation on the exact T-matrix or Schrodinger equation produces the O.M. equation which yields the gross structures of cross sections and strength functions. Weisskopf's compound nucleus formation process(15) is a cascade sequence into the more and more complicated states (~ 1.7). The first stage of this process is called the doorway state.(22) In the derivation of the O.M. equation, all the processes involving the internal degrees of freedom are smeared out by the averaging process into a single particle complex potential. Block and Feshbach(22) explained the deviation of the strength function of individual nuclei from the gross structure taking into account the effects of the doorway states. We will improve the O.M. by extracting the effects of the doorway states from the O.M. The two extreme cases, the O.M. Schrodinger equation and the exact Schrodinger equation are related by an average over the range A * This suggests to us that if the range is reduced, a degree of internal -69

-70freedom (i.eO doorway states) will come out explicitly and the potential derived will be somewhere between the OoM. and the generalized O.Mo potentialso For the energy average to be meaningful, we must assume that the spectrum of the more complicated states is much denser than that of the doorway states. This contrasts with the assumption of Block and Feshbach, (22) After averaging out the effects of the more complicated states, we will obtain a pair of equations: one for the open channel and another for the doorway states. The potentials are composed of the single particle potential, the coupling between the open channel and the doorway states, and the imaginary potential. The imaginary potential will be called the'doorway state strength function"o It appears in the equation for the doorway states and represents the effects of the more complicated states, 4ol Introduction of the Doorway State Strength Function Following Block and Feshbach,(22) the closed channel states Q2~ are divided into two components: the doorway state component ~/1 and the more complicated state component tot Qa9 = i- + Ak. (4.1.1) Thus we assume, from the beginning, the existence of the projection operators ql and q2 such that (4.1.2) THt+O, L/ TH. 7H o

-71Given these projection operators, we may then define the doorway states,, and the more complicated states, ~, according to,Hf ^s = ~5 %s, (4.1.3) f H12 fi = ~^. (4.1@4) The relations between Equation (4,1. 1) and Equations (4.1.3) and (4.1.4) will be discussed in Part II, The Schrodinger equation _H 4r =_ E - (4.1,5) is decomposed into a system of three equations by means of the projection operators P, ql, and q2 - (PtP-E)pQ = -PH7,,/ (4.1.6) ^(itHX -5E)?~ = —fP-HPf H; (4. 1.7) (tiht - E) A; e —F H2 t P. (4.1.8) Within the range A centered at Eo of 3.2, we may replace (PHP-E) by (PHP-Eo). Now, we will consider replacement of (qlHql-E) by (qlHql-Eo), Substituting the formal solution of Equation (4.1.6) to Equation (4.1.7), we have la-, -E + HP Hp (-i-PH) + ( -+ flI (ED) -, S 22 9. (14.1e9)

-72By assumption on the spectra of qlHql and q2Hq2, only one doorway state p is significant in a certain energy interval a which contains a subinterval a in which there exist many eigenvalues of q2Hq2 (see Figure 1.3). The bound state I1P is given in terms of i and kq and by a Green's operator which is inverse of the operator on the l.h.s. of Equation (4.1.9). If the energy range of interest /A can be small so that Aft << 7 1 < HP X, *Lzo) > (4.1.10) then E in the Green's operator may be replaced by Eo. This implies the replacement of (lH/,H - E) by (iH, - -o) in Equation (4.1.7). Figure I.3. Ranges A and AI, and the Singular Point of Equation (4.1E9).,o/ / / / / Figure 1.3. Ranges A and A', and the Singular Point of Equation (4.1.9). Now, we have (PHP - 0)- - H /, (4.1.11) (phfrEf)p r =-^- PR -H?7zf (4.1.12) (H E Li p 4- /4 e e L =(4.1.H 3) (t Hz -E) oLe - t Haf g ( 41.13

-73Since ^ [I and. ^1k are bound states and close to those of Equations (4.1.6), (4ol17), and (4.1.8), we use the same notation. From Equations (4,1.12) and (4.1.13), the effective doorway state equation is obtained, (5, Hfp -E <D, -I EIr^ (4. 1.14) We assume that the spectrum of q2Hq2 is dense so that <<~ (4.1.15) where D is the spacing of the eigenvalues. The E-average of Equation (4.1,14) will be taken over ^ and the averaged values denoted by e.g. V/ instead of L of f 3.2. The approximation of' 3.2 will be used (c.f. the discussion after Equation (3.2.18)), HK2zjzHn)I p/1rp4L//,_ (4.1.16) We then write ^W~ = - -f ^Thfj 2 d^S^b(4.1.17) where we have used the abbreviation: 1l - z >. (4.1.18)

-74Therefore, Equations (4.1.6), (4.1.7), and (4.1.8) are reduced to a pair of simultaneous equations (- HP-E) P L -k )t q1, ( (4.1.19) (f -t X-E) ~F =-fH-P PE By comparison with Equation (2.2.6), we see that the effects of the more complicated states have been replaced by the Hermitian operator Oc. We will call Cp the doorway state strength function operator. 4.2 Concluding Remarks We have performed the E-average of the Schrodinger equation over. As is expected, the doorway states are extracted from the O.M. complex potential. In Equation (4.1.19), the doorway states explicitly appear and the more complicated states are represented by a complex potential in the doorway state equation. This energy average operation may be set at any stage of the cascade scheme. Comparing Equation (4.1.19) with Equation (3.2.19), the complex potential appears in the doorway state equation, while the complex potential of the O.M. appears in the open channel equation. Both potentials are expected equally to be independent of energy. The theoretical structures are similar to each other. The diagonal part of?H Q Q H P is proportional to the strength function T_ <gh AHP4.

-75and O(e 2H ft?/l 7a Hy. In this sense, oe is called the doorway state strength function operator, The conditions for the energy average are Equations (4olo10) and (40 lo 1,5) These are satisfied if at any situation of coupling strengths (see Part II). Furthermore, the condition that the density of spectrum of the more complicated states are much denser than that of the doorway states has been applied. This implies that the results apply to nuclei with rather complicated structure.

CHAPTER 5 MODIFIED OPTICAL MODEL IN THE ISOLATED DOORWAY STATE REGION 23 5.1 Optical Model Calculation for Not/ We have calculated the optical model cross sections for /N0 Figure 1.5 shows the measured and calculated total cross sections of Na2.(54) The Wood-Saxon real well with Gaussian surface absorption was chosen for the optical model potential shape (see Figure 1.4). Vvre (.) -VRE 1 + efap tR f-4X VA:Me = -VM....... —al,''~~,, = - vI/ ef f-()-.... Vso -( )- -Vs jr I r ve,, (Y) l The six parameters of the potential are determined for the best fit to the total cross section, the total elastic cross section, and the differential elastic cross section from 1 to 4 Mev with Hauser-Feshbach correction. The best set of the parameters is VRE -= 4. or M mA, VIM -. 0 M O,, VSR = / o, o M1, 7Z,- 392 f, a = 0.60 bm, da= /.o4 fm. -76

-77(POTENTIAL MEV) 1 5 r (FM) Vim (r) - 10 /f Vre(r) -40 Figure 1.4. Optical Model Real and Imaginary Potentials.

oa (BARNS) 8.0 7.0 O' (E): BNL325 AND KFK120 T(E): SMOOTHED EXPERIMENTAL VALUES 6.0 000:'T CALCULATED 5.0 1.0 0.5 1.0 2.0 3.0 4.0 5.0 E (MEV) Figure 1.5. Optical Model Fit to the Total Cross Section of Na-23.

-79The resulting potential is applied to the calculation of the inelastic cross section to see the fit(54) The fit there is very good except around thresholds where Moldauer's correction(23'27) must be applied. The total cross section is calculated up to 10 Mev. The fit is also very good, However, when the calculation is extended below 1 Mev, agreement of the calculation to the measured values disappears with decreasing neutron energy. Namely, the calculated total cross section turns into the /V region of the O.M. below 0.5 Mev, while the measured cross section stays almost constant as a whole, The discrepancy is caused by the increasing calculated absorption cross section. We will try to indicate in this chapter how the O.M. should be modified to eliminate this discrepancy. The OoM. usually explains well the gross structure: the Eaveraged cross section including the giant resonances, and the strength function, (I 1.3)o Another characteristic is the i/T -dependence of the O.M. absorption cross section which occurs when <s4 1. In fact, the O.Mo strength function is defined by the behavior of the O.M. absorption cross section at low energies(5929J43)(~ 2o3), m^ at tL h^ -= t 2 W(y) On the other hand, the statistical theory yields for the E-averaged (29,36) reaction cross section,( 3) -1 cE r - 6 r /4E |

-80Measured values for TO/D agree well with direct measurements of 5. in the I/tr region for most nuclei. However, the O.M. strength function gives only the average distribution of the strength functions for different nuclei. Its deviation from the measured strength function of an individual nucleus is inevitable. This deviation was explained in terms of the doorway states by Block and Feshbach.(22) To obtain such individual strength function or y/r -cross section, it is necessary to adjust the O.M. potential, particularly its imaginary part. It is not clear if the adjusted O.M. potential will yield the average cross section in the high energies. At high energies, giant resonances and resonances with large widths have been measured. As the energy decreases, individual resonances may be more closely observed. Below the resonance region, the cross section becomes flat and small. We denote the lower and upper limits of the resonance region by El and E2 respectively. In the usual situation (S-wave scattering predominant), the E-averaged total cross section has a /jTr dependence for ES Z E1. Also, it is easily observed that E2 varies cyclically in accordance with the cycle of the strength function although it becomes larger for lighter nuclei as a whole. These are overall tendencies for nuclei with mass numbers larger than 30. For very light nuclei, the lower end of the resonance region, E1, becomes as high as 1 Mev. We will briefly describe the cross sections of F, t\I. M:, and AL. The energy corresponding to - d= for these nuclei is about 1 Mev. Therefore, the 1/tr cross section

-81above 1 Mev does not correspond to the O.M. 1/7t we are discussing. Our calculation of /23 yields a (se of the form of one side of a giant resonance shape, with the resonance lying somewhat below 1 Mev, The cross sections of F and f1g are similar: i/V behavior from -- 0.5 to above 3 Mev, and a few isolated resonances, which do not belong to this 1/Lr, below - 0.5 Mev. In this case, it is not clear how a part of this 1l/(0.5 - 1 Mev) is to be treated. Na and AC have almost the same total cross section. The resonance region extends down to 1 kev and the average measured cross section becomes flat below 1 Mev. For most heavier nuclei, the O.M. 1//r fits well at least above E1 But, for nuclei from F to Al, particularly V^h and At, the O.M.!/r is clearly incorrect. Consideration of the extremely low energy range suggests an vLd viW way to improve the OoMo in the intermediate and the low energy regions. The total cross section calculated and measured disagree greatly. But the shape elastic cross section agrees well with the measured total cross section even at extremely low energies -~-l V. This calculation suggests that the real part of the O.M. potential might be correct even at very low energies and that only the imaginary part should be changed. In fact, when the imaginary part of the potential is decreased gradually to zero below 0.5 Mev, we obtain a complete fit of the calculation to the E-averaged measured total cross sections. The theoretical justification of this tentative approach follows from a treatment based on Weisskopf's reaction picture discussed in Chapters 3 and 4. The generalized OM. is supposed to yield the exactly measured cross sections, e.g. the lowest resonance of. The

-82transient region between the lowest resonance and the lower end of the O.M. may be analyzed by constructing a theory intermediate between the generalized O.M. and the O.M. In other words, we will describe a modification to the O.M. in which the energy averaging is carried out over a shorter range than ordinarily. The results of the improvement of the O.M. will be seen in this chapter. Equation (4.1.14) may be written as an effective open channel Schrodinger equation, pH(?p-e)p = -H, IE H P 1. (5.1.1) Let us call it the "modified Optical Model" Schrodinger equation, the modification being that the energy range is not so coarse as in the original O.M. In the case that the interference between the doorway states may be neglected, the complex potential will be obtained in 5.2. The total cross section and the absorption cross section will also be obtained to illustrate the result. In ~5.3, the imaginary potential in the low and the intermediate energy regions will be given. This discussion is based on a coarse assumption; nevertheless, it is very suggestive. 5.2 Modified Optical Model In this section, the modified O.M. potential will be obtained in the isolated doorway state region. The total cross section and the absorption cross section will also be obtained.

-83We expand the Green's operator of Equation (5.1.1) in terms of the doorway state eigenfunctions of Equation (4.1.3),, ^= z2 gS ( > (5.2.1), _ D P N7Z y?- q= ~ 0, Q t O? (5.2.2) where & z - s) ><5 CT')(<, (5.2.3) nS =- 7. 9><% {5><(fs o' / (5.2.4) Equation (5.1.1) becomes (?9-E)1 = -ZpL t f (6 dp) G,H? P'2 (5.2.5) where,.F-?-gHtt 2fL (5.2.6) In the intermediate energy region where doorway states are separated sufficiently so that E Es-E)~ | f'>)> (5.2.7) the interference terms ( c > ), containing the effects of C-p, may be neglected. Equation (5.2.5) becomes (?PE-E)?9 — 7 t t,Ystm?9, (5.2.8) E- E, eiRIP)

-84where ^5^3 - % ( ^ f5 Hi f>. (5.2.9) We may then write Equation (5.2.8) in the form: (?HP - F,(E) - F2E) - E) p J 0 (5.2.10) where the modified O.M. potential operators F1 and F2 are given by i(e) = _.7 (i)8t Es)]py'H < n> P (5.2.11) F.^^Z —7 >rfS ^^^X 7n k in>< ts 2/H'P, (5.2.12) and (5.2-.13) This result shows that around the doorway state energies the real and imaginary parts of the complex potential vary like the interference and resonance terms of the Breit-Wigner expressions. Far below the lowest doorway state energy, the imaginary part goes to a constant value, very small compared to that of the O.M. Transients around the lowest doorway state energy will be discussed in i 5.3. We now obtain the total cross section. Applying Equation (2.2.38), the T-matrix element of Equation (5.2.10) is calculated as

-85+ Z fst) < PHf ^Xy s >< 5H ~ )> (5.2.14) Applying the formal solution of Equation (5.2.10) to Equation (5.2.14) as 2 2.5, we have I -S f) </s f r* PHF Ws > (5.2.15) Now ) defined by Equation (2.2.36) will be related to ~) so that p H()Pttt 5 > may be written in terms of <3)W ~ >. = + A" + (g-l-TH Et - ) ~- - +.'~. —P -'P ='Z t S E-)tP) U Lf(5.2.16) By Equations (5.2 16) and (2.2.34), we have < 4RHt > /= < X7C k >q- 2' <'^ )> m1PH c5 ) c-(S ) < )~ tsyS>} (5.2.17)

-86where (TH?- E) = 0 and where (s^)c' = Sc, -2 < ( L, U" XC' (5.2.18) which is the S-matrix element of the potential scattering. Applying Equations (5.2.15) and (5.2.17), Equation (5.2.14) becomes T ~t, - 2 <4T l +d SZ — f < t-i...p>..... (5.2.19) E-ES-a-stt(Sr+ ) C where A - _/ Z dE c(5.2.20) and tr<' 2~ L <y.P Xc'E')> 2< (5.2.21) In the case of a single open channel (olSJ M), the potential scattering S-matrix is given by -SC? --- e i ^(5.2.22),C'c ~'.Q a (5.2.22)

-87where i As is the phase shift. Denoting the n-th doorway state constants with subscript n, the total cross section is obtained from Equation (22o.29) (fT A-2 or ^ r < T- J T -7 = an r i 4 L it fS - 2( - En - a ~ )., -,,,-, r-'( 1 -S CO-EY-4 )Xe L (amp race)2 TU^ isrt) ret. f -E.n-p) + (dee+ rE )) J. (5.2.23) This is the Breit-Wigner form(4) with modified resonance parameters, The first term is the potential scattering. The third term is the doorway state resonance and the second is the interference between the potential scattering and the doorway state resonance. From the form, rI ) is the formation probability of the n-th doorway state. After the doorway state is formed, it decays into the more complicated states with probability 0yt, as well as into the free state. Next, the S-wave absorption cross section may be calculated using Equations (2,4,ll) and (5,2,19).

-880(/= (, -=- =0, and J S (channel spin). We obtain o6 = c 7- (1-Is SS ls2) 2L 2 in Tr r;52 -z -........................... (5.2.24) In the absorption cross section as well as the total cross section, the doorway state strength function, 0(n, appears in the place of F-, the Q -decay width in the single level Breit-Wigner formula. i,. is, in general, energy dependent, but is expected to be almost constant. The transmission coefficient is obtained as....EJ T-k F.D.l.+......'... —D') (5.2.25).^. (E-En-d*^ +^ flE n 4- +r4 h a <ht thr > d r (5.2.26) where DdJ is the spacing between the doorway states. The S-wave absorption cross section at very low energy has a YLr -dependence as in the usual O.M. result:

-896-o- A EW it r ()() (5.2.27) However, the magnitude of the coefficients is very different. This will be discussed in' 5~3. 5.3 Concluding Remarks From Equations (5.2.12) and (5o2.13), the imaginary potential tends towards a constant value below the lowest doorway state (further illustration for the transient variation of the potential may be seen later). Furthermore, the results indicate the local fluctuation of the OeM. potential. The total cross section has the Breit-Wigner form in the vicinity of the doorway state. Such variations may be seen around 0.2, 0.4, and 0.7 Mev in the Av cross section. A crude picture of the modified 0,M. potential may be given by making the assumption that ^-'8 f)H X <, 72HP ~- -7><fHHX (5.3.1) This implies that the formation probabilities of all the doorway states are the same. The imaginary part of the modified O.M. potential near a doorway state is given by At; 0 (E-E*)2q (&47) F 0(Et),, 0-/2- d, O/2^ ~.~z +T_____ -E -- + L" - - ~+^, ( j' A"/z T^ a' Ad hHP (5.3.2) (e~i<)

-90Some properties of this potential may be described in relation to the imaginary part of the O.M. potential. Taking another average of the modified O.M. equation over a wide energy range so that the doorway states are smeared out, the O.M. potentials are derived with the results: F-1 o0, F2 - - Avt' f^ ><fH i. F (5.3.3) The potential F2 varies as a Lorentzian and the ratio of the peak value to the O.M. imaginary potential is given by,; lEn).D +l O|i L. +n// (5 3.4) F2 Ur. 4rai of= tte, 2 4i The ratio of F2 at E = 0 to the O.M. imaginary potential is obtained by TCo 1),, (5.3.5) F.P%2 2 (Ev ) The variation of F2 with energy may be illustrated using as an example. The constants do (= r ), r ( F') and Da are given in Table 11.2 of Part II. The values for the ratio F IEl )/) F. are approximately 1 for the doorway state at 205 kev, and 2 for the doorway states at 410 and 580 kev. Another doorway state exists at 65 kev with 0($ ~ 40 kev and F2&n )/ F- I

-91The constants of the last doorway state are obtained from the resonance width distribution in BNL - 325. The value of the ratio ( F ) / f' is approximately 0.17. The imaginary part of the modified O.M. potential is thus drawn in Figure 1.6. The curve in this figure indicates four properties. First, it shows a decrease of the imaginary potential to a small value below the lowest doorway state. Second, it shows the overall decrease of the F").2 _' E L, —-....-. —. -.. i.. --- ---- E l_ G O ( (k| v) 4/C Z Figure I.6. Imaginary Potentials of the Modified O.M. and the O.M. for Ni-58. imaginary potential with decreasing energy. These two properties are in accordance with O.M. calculations fitting energy dependent problems to \1J, data. The following two additional properties were not expected at the stage of the An! -iV calculation. Third, there are local variations of the imaginary potential around the doorway states.

-92Corresponding to it, the total cross section has a Breit-Wigner form, Equation (5,2.23). This form of the E-dependence of the E-averaged measured total cross section may be observed in Figure II.29. The last property which is obtained from the curve is the existence of the two groups of 1/V dependent resonances separated at ~ 300 kev. By close observation of the total cross section in Figure II.29 connected to that (below 100 kev) of the natural Ni (the abundance of Nos8 is 67.88%) in BNL - 325, the two /Jr separated at - 300 kev may be found and the ratio of the proportional constants is observed to be 1.6. Taking a local average of Fz E) F2 above and below 300 kev (b and a respectively), the ratio b/a is approximately 1l6. Besides the first three properties, this result indicates that the curve in Figure I.6 is fairly meaningful without regard to the assumption, Equation (3.3.1).

CHAPTER 6 SUMMARY In Chapter 4 we have smeared out the more complicated states by the energy-average process and left explicitly only the doorway states. The result is given in Equation (41,o19). We may expect that this modified O.M. works well in the lower energy region where cross section data are required by reactor physicists. Furthermore, the interpolation and extrapolation of the modified O.M. over nearby nuclei is possible if we know the systematics of the doorway state constants In the case of the single doorway state, the total cross section is calculated in Equation (5.1,25) which shows the Breit-Wigner form. As mentioned in the introduction to Part I, this part of the work has been independently worked out and published by the MIT-group.(61) For convenience of detailed comparison, equations obtained in Reference 61 will be listed below. Our work has been done for the specific purpose of improving the O.M. However, their theoretical objects and our theoretical objects are the same, and moreover the MIT-group's work has been better known among physicists. From now on, therefore, we call the theory as a whole "the intermediate structure of cross sections", and notations will be changed to theirs: ~, + -rF- r (=rF4r3') -93

-94The physical discussions and conclusions which are not given in the previous chapters may be seen in Reference 61. One may also find many observed cross sections in that reference. Reference 61 Ours (2.31) (4.1.2) (2.91) (4.1.19) (2.49) (5.2.9) (2.88) (5.2.10) (2.50) }~,27 I(5.2.17) (2.75) 9 (2.51) ) (2.52) (5.2.18) (2.76) J (S-wave part of (2.56) ( 56 (5.2.22) The O.M. potential is well known. It yields only a coarse result. At the opposite extreme is the generalized O.M. which yields exact results, while its potential is not known in general. The modified O.M. yields better cross sections than the O.M. Its potential form is known and depends only on the doorway state constants. These constants will be obtained by analysis of the resonance width distribution and the cross section in Part II. There is a lower limit even for the modified O.M. to apply. The spacings of the resonances are almost constant in energy, but the experimental resolution as well as the interval of interest takes smaller energy intervals into account at low energies. Below a certain

-95energy, the range l cannot contain enough resonances for the averaging process discussed in Section 4.1 to be used. On the other hand, the modified O.M. applies in the higher energy region. There, we meet with the MIT's problem of the intermediate structure of cross sections; for example, the bump-type deviation from the O.M. cross section which may be seen above 2 Mev in Figure I.6. The bumps resulting from E-averaging of cross sections with fine resonances are also called the intermediate structure. In our case of NA, such structures exist at 580, 410, 205, and 65 Mev. This aspect of the problem is the subject matter of Part II of this thesis.

PART II THE FINE STRUCTURE OF DOORWAY STATES -96

Introduction In this part of the thesis, we determine the details of the total cross section around a doorway state. We find poles of the Tmatrix, evaluate their residues, and analyze their detailed effects on the energy dependence of cross section. These properties mainly depend on the magnitudes of the doorway state constants rt and r-, and on the average resonance spacing D. The results of the study will be applied to the analysis of the measured cross section and resonance parameters in the low energy region to find the doorway states and to determine their constants. In Chapter 1, we present the general theory of a single doorway state. In Chapter 2, the analytic properties of the T-matrix are investigated in detail, applying the infinite Picket-Fence model. In Chapter 3, the fine structure and intermediate structure of the total cross section, and the resonance width distribution around the doorway state, will be given and compared with experiments. Finally, the intermediate structure at high energies will be discussed. -97

CHAPTER 1 FINE STRUCTURE OF THE DOORWAY STATES In this chapter the fine structure of the doorway states will be investigated in a general way. The detailed properties will be pursued in the subsequent chapters. In 5 1.1, under the single doorway state assumption, we decompose the resonant part of T-matrix into two factors: the branching ratio, G. and the structure factor (see Equation (1.1.14)). There may be more than one open channel, and inelastic channels may be included. In ~ 1.2, we will demonstrate the unitarity of the S-matrix on the real E-axis. In 1.3, we show that the poles of the S-matrix are confined to the lower half E-plane (more precisely, the unphysical E-plane). In other words, the single doorway state model does not violate the general requirements placed on the S-matrix. In ~ 1.4, we expand the T-matrix into a series of the multichannel Breit-Wigner form. The fine structure of the total cross section may then be obtained. Corresponding to the above series expansion of the T-matrix, the total cross section is expressed by a series. The first, second, thrid terms, etc., of the total cross section represent the doorway state resonance, the isolated resonance under the influence of the doorway state, the interference of two resonances, etc. Carrying out the expansion through the isolated resonance terms, the cross section is written in a form as close as possible to the Breit-Wigner formula, To do so, we will define the "doorway state phase shift". -98

-99Comparing the form of the cross section to the Breit-Wigner formula, we will show that the existence of the doorway state is reflected in the fine structure of the total cross section. In 1.5, we will define terminologies "weak coupling limit" (WCL), "strong coupling limit" (SCL). We then exhibit the behavior of the fine structure in these limits. In S 1.69 sum rules for the poles of the T-matrix will be given. Their possible applications will be discussed. 1.1 T-Matrix of a Single Doorway State A system of equations in the projection operator formalism will be rederived similar to those in Part I. Then, the T-matrix element of a single doorway state will be obtained. The Schrodinger equation of our system has been decomposed into a system of equations, Equation (2.2.6) of Part I. We will limit the problem to an energy interval (E /2) 1 ~ /2) such that Ejf < CE~I/2 < ~.)L (1.1.1) -A where EJ is the j-th excitation energy of the target nucleus. Equation (2.5.2) of Part I is (QHQ - )E = (1.1.2) Now, we will define two projection operators Q1 and Q2 in terms of the orthonormal eigenfunctions of Equation (1.1.2) according to, >~~~~/ <(1.1.3) and Q 2 Q- It,

-100where the sum over k ranges over all eigenfunctions whose energies Ek satisfy E- z2 < E < E + I/2 Obviously, the projection operators P and Q1 satisfy the following relations, PQp - L Ti O,. P- P,- Q =Q, J (1.1.5) # 1 (A, E1I~.A ) Applying Q= Q1 + Q to Equation (2.2.6) of Part I, we may rederive the same form of equations, (P(p-E)P- 5 - P=QQ,0 ^ ~(1.1.6) (Q Q,- E) Q. = -Q. E Pp where Hfl "- H -tt, I.... 71 hI t tl Q E3 E QHHQH - H+f K + U,V 0Q, +P (H+tV)0, YQ, (H+V W )'P (1.1.7) - -Q2H, (..8

-101/t and U is given by Equations (2.1.9), (2,2.10), and (2.5.3) of Part I; l/'= U + PV2P. Next we presume the existence of projection operators ql and q2 (e.g. 4.1 of Part I) such that Qi -f.t-9zO ) }, ~ (11..9) PHL7,4O,)L P C P - O Equation (11.o6) becomes (PP-E)Pi = - P t,', (f x,- E ) %,~ =-,, -IL - R PP (1.1.10) The distorted wave T-matrix of Equation (1.1.10) is obtained from Equations (2,o238), (2.6.6), and (2.6.8) of Part I. After some algebra, it becomes where TLL - 27ct <'U "'),U (1.1.12) and T; -- t<L 7p, q E-,^ - 1 --- - x ~g,, -,(11.PE~'13)

-102By the single doorway state assumption, Equation (1.1.13) is reduced to a simple form (the tilde underneath denotes a matrix) T" -TR = G, r —------- (1.1.14) ^ E - E,> -, Z ^; where -G - S?. r"/1 (1.1.15) SP? _ -. P (1.1.16) - ) < )( f P 4<' > (1.1.17) Eo — E + i - - (1. 1.8) V Te-c^ ('a) (1.1.19) _ t 5 -/J T a -g) lated to r, the decay probability of the doorway state to the more c l a st a < s, 2 (1.1.20) *^ /^ --- ^T ~ (~_ 3_.2~_ and f' 7 < P2 The constants Zd and r- are the generalizations of the doorway state shift and doorway state formation probability.(61) an is related to (, the decay probability of the doorway state to the more complicated states, by ^~t'S^ L<^, 1 ~ r~ (a/p

-103In Equation (1.1.14), the factor G will be called the "branching ratio". It becomes a significant factor to determine the existence of giant poles due to the doorway states. The remaining factor will be called the "structure factor" which determines the fine structure of doorway states. 1.2 Unitarity of the S-matrix We will first prove that the unitarity of S, the potential part of the S-matrix. Then, we will first prove'ithat the entire Smatrix is unitary. In this way, we show that the single doorway state assumption does not violate the requirement of flux conservation. The S-matrix is related to the T-matrix by S = -- t T. (1.2.l) The matrix element of T is given by Equation (1.1.12). By Equation (1.1.16), (3S S4= 1 +~Tj(TL ) TT (1.2.2) After some algebra, we have the following relation; (<Xu"U.,>= (- <f u":'7> + + 27Cx ( L " L (E -'PHP.) U ) = < u"xlJ),,> +2;LZ <Vf UX U ) (1.2 3)

-104where Equation (2.2.34) of Part I has been applied. This implies that (y _-iup) pTtT?=' 0. (1.2.4) Thus, S is unitary, (s^) -P = 1 (1.2.5) Now, Equations (1.1.11), (1.1.12), (1.1.14), (1.1.15), (1.1.16), and (1.2.1) yield S S7> _ _ _ S C E-E,-s E^-E /VL E- -vt = Sa S. (1.2.6) where we have defined S _ i -- - (1.2 7) E7 S' -Emt which will be called the resonant part of S-matrix. Since SP is unitary, we have af- = S _ 1 + -- X x {/(?)+ ~; +v; +)+(E-E -; r Y)_, r^ -~*-~ -E-) (1.2.8)

-105By the definition of rt, Equation (1.1.17), we may show that and- 1 (1.2.9) and (3 i) = F'p rhk From these properties together with Equations (1.1.18) and (1.1.19), we see immediately that the expression in braces vanishes. Thus, (S jR S = R and S is unitary. 1,3 Poles of the T-Matrix We will study the fine structure of the doorway states through consideration of analytic structure of the T-matrix. We, therefore, analytically continue the T-matrix defined for real E by Equation (1.1.14) onto the complex E-plane. Let us call the analytically continued T-matrix the "extended T-matrix". The E-dependence of this extended T-matrix is quite complicated. To avoid the complexity, the T-matrix will be approximated on a limited domain by the "statistical T-matrix". Then, properties of the poles of the statistical T-matrix will be studied. We assume that we may deal with intervals I and A which satisfy the following conditions: In addition to the condition given by Equation (1.1.2), the interval I is large enough so that (a) < a ) and D distribute uniformly over I. (b) V2 can be independent of E (34) (c) Edge effects may be neglected around the center of I. Such an interval will be fixed centered at Ec (Figure II.1).

-106The subinterval A will be set centered at Ec satisfying the following: (d) A is small enough for SP, - and A -- to be independent of E, c.f. Equations (3.2.3) and (3.2.4) of Part I. (e) L > 5 Similar to the statistical collision matrix (~ 1.2 of Part I), we thus define the statistical T-matrix according to: (E, i) T J) + G (E E (1.3.1) where Tot [Ec ) is rT- at E EL, etc. By the assumptions (b) and (d), we have Taf. & (E, ) E (1.3.2) for E in the subinterval A. The statistical T-matrix is easily continued analytically over the entire E-plane. We expect that the extended T-matrix is well approximated by the statistical T-matrix on the domain sketched in Figure II.1. The analytic structure of the statistical T-matrix will be studied. First we note that the statistical T-matrix is a meromorphic function with poles located at the n + 1 zeroes, of the expression (E) = E-E, -L (1.3.3),/e & - E.

-107- -- r --- - /^7 ^i-'^ R4~E Figure 11.1. Intervals I and L, and Domain d on a Complex E-plane. It is easy to see that these zeroes lie in the lower half E-plane, since writing o + i we obtain 2 Z, =me < o The zeroes are sometimes expressed by 5,, where E; j for i = 0, 1,.., N In terms of these zeroes, f(E) becomes N Al 2. U (E-5) f(E) =E-E6 -1-, - k - (1.3.4) and Equation (1.1.14) becomes T^ =Gr Ip^-E;)/Tr (E - aki1ko(.3. 5)

-108We call the pole at E = 4o, the "doorway state" pole. The resonant part of T-matrix, 7, will be expanded into a series of the form of the multi-level formula. We may easily show the following formula: N r' (E-',.) Tr (E-E~) -Z E/,, L,'t,, TT' [ E-~,). (1.3.6) Then, |? becomes.- = Et1+ /,^-.... E "o i...' ve E -"< <red \ -' (1.3.7) When the i S are small, this expression is useful. All information is given in terms of the poles of TR We may immediately write sum rules for these poles. From Equation (1.3.4) we have ('-E:,) T(-E:) - Z__ TE (E- E) = TT (E-E - ). (1.3.8) to

-109By comparison of the coefficients of EN, we have(82) k=o N i~; 0'k - ~ (1.3.10) and from the coefficients of EN-1 and Equations (1.3.9) and (1.3.10), =, l 2 L; ='L, (1.3.12) where the superscripts r and i refer to the real and imaginary parts of the constants. The imaginary part of the doorway state pole is related to those of the remaining poles by /= -_ +- - _ r ^. (1.3.13) Applying the condition that all the poles are in the lower half plane to Equation (1.3.13), we have (o0 ) < - (13. 14) From Equations (1.3.8) and (1.3.10), we have

-110= ~-Jr (1.3. 15) The doorway state energy, 0, is given by an average of fine resonance energies weighted by their widths. Formally, this is the same as the center of mass theorem in the direct reaction theory where the single particle energy is equal to an average of excitation energy weighted by a spectroscopic factor. (106) 1.4 Fine Structure of Total Cross Sections If the potential scattering S-matrix is diagonal, 2'Cc! = ScC x X ept (2X ), the diagonal part of the T-matrix, T =+ C F(E ) (1.4. 1) t- ) Tr (IE- E n) 4 gives the total cross section (see Equation (2.2.29) of Part I); -5)T' 2 2 g Td 4S; ( 1. ) J L$ -J^^w ^ ^^-^ — ( ), (1.4.2) (E-~o)T(E- )

-111Corresponding to the expansion given in Equation (1.3.5), we have r =G -51 6f, + 5R + 5 - (1.4.3) where the potential scattering cross section C(P is given by cP 2R 2Z - c j2 ) (1.4.4) (the super- and sub-scripts of the phase shifts are suppressed), and where we have written 5A _ 2 t Rt Z E f j -rF:: (E-')A2 - ro es=2S vs^ ( _E_.r):+ (r 2 /7 (1.4.5) the doorway state cross section, J L s 1 - -r+: (~E - )A AI- + E) 52 + X —-------- ------—; ( - (1.4.6) corresponding to isolated fine resonances, and where 0,..., etc. are interference cross sections. In Equation (1.4.6), we have defined the doorway state phase shift by A 7 -Em)+ (F -t = (E-:)~ > -_ )-; $ (1.4.7)

-112Since the expansion of the total cross section in Equation (1.4.3) has the same form as the Breit-Wigner multi-level expansion,(4) any discrepancy of and (<5 from the single level Breit-Wigner formula(4) is a reflection of the effects of the doorway state. Compared results are as follows: (a) 1 and i/ have the Breit-Wigner forms(4) with resonance parameters given by the poles of the doorway state T-matrix. (b) In (5, there is an additional phase shift i (c) In O(ff, a modulation factor, 1l/lE Co \1) indicates another effect of the presence of the doorway state on the shape of the fine resonance. A Limiting Case The simplest model from which immediate results may be obtained is the "weak coupling limit", WCL, which is treated in detail in Chapter 2. In this model T<'< Fr which implies that the ^ in the dispersion relation (1.3.1) may be treated as a perturbation. Thus, 60 -a E /-)^ r ) z (1.4.8) andt E: The poles distribute in the complex E-plane as a Lorentizian with width r *(82) In this model, for the single open channel case, the cross

-113sections become E- E,:' t 4(r )+ pi A —A'^-, + + (E- )A (^ C ) + n c 2 i'S t ) E - A,,. (I -^ ) t (r,; )' (1.4.10) The modulation factor becomes lF -gol t En - E: +1 r at E En, and the doorway state phase shift is simply L =- --- -,Es..... —- (1.4.12) Finally, the total cross section is sketched in Figure 11.2 for the case where the potential phase shift = C.

-114Ie-IX- E Figure.2. Fine Structure of Total Cross Section at WCL. Figure JI.2. Fine Structure of Total Cross Section at WCL.

CHAPTER 2 INFINITE PICKET FENCE MODEL In this chapter, we investigate the analytic properties of the T-matrix using the infinite Picket-Fence model. We expand the T-matrix into a Mittag-Leffler series, and evaluate the poles and residues. In i 2.1, the existing region of poles and the paths are obtained. In j 2.2, the pole distributions will be calculated as a function of rYD and. In~ 2.3, the residues will be estimated. In the Picket-Fence model, it is assumed that the R-matrix constants, i.e., the reduced widths and the spacings, do not vary from one level to another.(4) We apply this model to the more complicated 2 states. In this case, Z i and En- - En are independent of n, and are equated to < 6 > and D respectively. If the number of states, N, is finite, we call the model the finite Picket Fence model (see Appendix C). When N is infinite and when the sum over n is taken as a double limit, Z=2 _ mi^ > (2.0.1) MyK AN-* v1== -N ) we call the model the infinite Picket Fence model. Deviation from the above models will be discussed in Appendix C. The resonant part of the T-matrix, Equation (1.1.14), is written as -115

-116_r G _ _ (2.0.2) E' - Eo ~ The sum may be evaluated using the identity, >13 - CDt, (2.0.3) with the result T-~R = G. -2;tn g (2.0.4) 00 (2~" h- -t / where we have defined the non-dimensional parameters ~, o, o, and Z according to -i X =- — = "''(2.0.6) Z - O r/2 For the single open channel case, the T-matrix element is just (eT-^-^ ~ a) Ii! - t

-1172.1 Poles of the T-matrix The equation for the poles of the T-matrix is given by t = (z -2o) 7t4 2 (2.1.1) ) where t is real and positive, 2 - ~ + LT O, * tU A Vr, L/; < 0, ^^ - ^ 2 -< ^, < t2 We may take >J4o > O without loss of generality and we shall do so. Taking the real and imaginary parts of Equation (2.1.1), we have 1i C^-M.)^c2 / ~ -) ^2-l o)~2 2 1 (2.1.2) and t- lr-o - / (2.1.3) p~fnR 2 LT 2 t From Equations (2.1.2) and (2.1.3), A' 2 2 -A 42,2 Lr J 20 (2.1.4) 42 A1A42U IC 2 A Z0 -2 +

-118Since the expression in the braces is positive, we have _ {-?1 % ^ - o 2 O (2.1.5) Therefore, the regions in which poles are located may be obtained: c 5 b L D5 F ( "A - )7C < M < X w Ad at g o and — d - ------- ~ ) Lu -L( [ (2.1.6) The regions are shown in Figure II.3. -21tC -7C ~ 7R 21 i Figure 11.3. Existing Regions of Poles. The anomalous pole is in the region A, and the central poles are in the regions A and B. The remaining poles are in the other shadowed areas. The dots show the locations when t = 0.

-119There is one and only one pole in each of the shaded regions in Figure II,3. This may be shown directly by application of the argument principle. However, it is sufficient to point out that, for t = 0, the roots of (Z — s)t? n = o are simple roots, and that the roots are continuous functions of the parameter t For t.O the poles lie at 2 so (anomalous pole), Q = (regular central pole), and X= /l T W Y4Q O (other regular poles). These poles are marked on the corner of each shaded region (Figure II.3) by a black dot. Paths of the Poles As t increases, each pole traverses a path in its region from the point 2 -= iF to the point; = ( {+ i) which is its limiting value at t->o. The general shape of such a path is shown in Figure 114, while the path is determined by Equation (2.1.3). We call this path the n-th path. If, for comparison, we move all paths into the strip 0 ~ At <_ i 2, we see that the n-th path contains the (n + l)-st path. In fact, we have a Vy l —0 __ /l.~ land -; - ^V^- 2 L1m- F 2 +Int

-120But 4/#t =- 44.-+ ~t. Thus which implies <VJ ( vJr.t. This situation is pictured in Figure II.5. -n7C inr, (4t+>)x o\ LUO 74/ #0 Figure II.4. The n-th Path. Figure 11.5 Paths of the Poles.

-1212.2 Pole Distributions In this section, the distribution of poles will be studied as a function of the coupling strengths. Let us call the two poles in a strip -- <U _ 2 the "central poles". In ~ 2.2.1, properties of the central poles will be studied. These are classified according to the value of the parameter t. More specifically, the classification will be given by t >> tX, tf>, t 4,;t < kt, and,t ~<< which will be called the "weak coupling limit" (WCL), "weak coupling" (WC), "intermediate coupling" (IC), "strong coupling" (SC), and "strong coupling limit" (SCL) respectively, where ~0 will be defined later. Let us call these altogether the "situation of coupling strengths" (SCS). Enough data for IC will be supplied by the study of only the central pole. In If 2.2.2, the pole distributions for WCL and WC will be calculated. In Hj 2.2.3, those for SCL and SC will be obtained. 2.2,1 Central Poles Consider first the case 4 0. When t = 0, the two central poles lie on the imaginary axis at the points 0 and t U. We are thus led to look for these poles on the imaginary axis for nonzero values of t as well. With this ansatz (u = 0), Equation (2.1.2) becomes _ i l u — o) 2. ts q (2.21) The left hand side of this equation is sketched in Figure 11.6.

-122As can be seen from Figure II.6, there are two solutions of Equation (2.2.1) when t is sufficiently small. As t increases these 7] — t( Figure II.6. Graphical Determination of Central Poles at uo = 0 solutions, starting at 0 and, (, move towards each other along the imaginary axis. They coalesce at If =~ when, =,. This defines.t, the determining parameter for the SCS. t depends upon L. In fact, the following relations are easy to showand and =-, - (2.2.3)

-123Hence, the SCS depends upon both?Vo and t, i.e., upon T+/D and r /D For f> t>, the ansatz u = 0 is no longer valid, and in fact the two central poles depart from the imaginary axis, as shown by the paths labelled 1o = 0 in Figure 11.7. Note that in the limit of large — L/, to approaches -L o, or t Thus, for large values of -yD, the SCS is determined entirely by the ratio rF/r. When =o 0, the two central poles coalesce as t-. For t to, the anomalous pole is no longer distinct from the regular poles. This has a marked effect upon the T-matrix, and is the basic difference between the SC and the WC, since in the WC the anomalous poles makes a distinctly different contribution to the Tmatrix. For ~o id 0, the general situation is shown in Figure 11.7. Here again the parameter to< may be used to determine the SCS. If we define t (IU) to be that value of t which, for a given value of 1*o, leads to a-pole at ^ = rT', (e.g. t(O) = t ), we find t*(x/2) = N t6 1, and -— r O In other words, tL 5 te0 D) to 1 ~. In fact, t* can be shown to satisfy 2 -F 2 f- + )

-124iv E IB E B/ // B'// CTCC ivo /2 + ivO Figure 11.7. Paths of the Central Poles. For UO = 0, the central poles move along the lines AACDE and BBICDtE' for t = 0, t < ta, t- ta and t > ta For'LUo /0, the path is given by the broken line. Particularly when ho =7C /2, the path is the straight line perpendicular to the' -axis. The location of the anomalous pole when t = t is on the line CC " when the Z4V is varied. is varied..

'91T'II GanOTS Jo 30 WuTod aq q. su'reqsuoo'08II @annTJI X~/ 10'2 It A-ZIA/O ^^ 7 — 0-~ X~n/o~ -^T

-126TABLE II.1 CONSTANTS AT THE POINT C OF FIGURE 7 TL; 1 of the anomalous pole at t=-ts and Uo ='iV 0 0 0 2 0 0 0.1 0.642 0.310 2.071 0.156 0.571 0.2 0.924 0.434 2.129 0.216 0.817 0.3 1.148 0.524 2.191 0.261 0.928 0.4 1.345 0.597 2.253 0.297 1.04 0.5 1.524 0.658 2.316 0.328 1.12 0.6 1.692 0.712 2.376 0.355 1.19 0.7 1.852 0.761 2.434 0.378 1.26 0.8 2.005 o.805 2.491 0.400 1.31 0.9 2.152 0.844 2.550 0.418 1.37 1.0 2.296 0.882 2.603 0.435 1.43 1.2 2.572 0.947 2.716 0.475 1.49 1.4 2.839 1.006 2.822 0.494 1.50 1.6 3.097 1.057 2.930 0.517 1.62 1.8 3.349 1.103 3.036 0.538 1.67 2.0 3.596 1.146 3.138 0.556 1.72 5.0 7.021 1.543 4.550 0.713 2.16 10.0 12.36 1.87 6.610 0.812 2.52 20.0 22.70 2.20 10.32 0.880 2.83 30.0 32.90 2.40 13.71 0.913 3.03 40.0 43.04 2.54 16.94' 0.929 3.18 50.0 53.13 2.63 20.20 0.940 3.29 60.0 63.24 2.74 23.08 0.950 3.39

-127where f - Ct*- t-) ( +t- *). Thus, except for -- o, to t is a valid measure for the SCS 2.2.2 Weak and Strong Couplings For t =, the poles are located at Zo (anomalous pole) and at 7 7. Solving Equation (2.1.1) iteratively we obtain t rt aZ.. + — t-; for the regular poles (2.2.4) and ~- ro 1 for the anomalous pole. (2.2.5) For WC, it is necessary to use the expressions obtained by iterating one more time. t ^ ~ nGT + t(f+ t/3) for the regular poles, (2.2.6) t 2 + T7Z ~ t ) for the anomalous pole (2.2.7) For SC, we must be more careful. It is true that as t --, 54 -- * t +)-7C. However, this limit is not uniform in n Hence, the limiting value of (n + 1/2) 7C is not a useful starting point for an iterative calculation designed to obtain approximate locations of all the poles. Instead we write ^= t - t-do ~> -i^ (2.2.8)

-128and consider solutions of z - (AuD o)t i) Zn = U -t tTO for a fixed value of g in the neighborhood of 1'o sO. The results are DO Po (a) 0th approximation, _ t, where 4U, is real and satisfies: ---— 0~~~t~?dc'i (2.2.9) (b) 1st approximation: (2.2.10) + tV t /n RDU) (c) 2nd approximation: ~;m... + K + M U^ ~,- t 2. i-iLI (A(2.2.11) The substitution f t- is implied in the above results,

-129One could also write approximate expressions for the central poles in IC, but these results are not included here, since numerical analysis is required in any event. 2.2.3 Summary The paths and locations of the poles are illustrated by a numerical calculation in Figures II.9 and II.10. The sample is — l —-8, and t is varied from 0 to. The curves of AO 7 Al A2 ~ A5 ~ AlO and A15 are the paths of the cells of n = 0, 1, 2, 5, 10, and 15 respectively. In addition, 6 curves Bi have been drawn through the points A (i) for a fixed value of ti o The values of ti and ts/ —LT are listed below for Bi. -71 = 58.5 and 0Z = 0 Bi ti ti/-vo 81 13.6 0.232 B2 50.2 o~857 B3 55.3 (= to ) 0.944 B4 56.o 0.956 B5 75.2 1o285 B6 87.8 1.500 The point C is the position of the two poles in coincidence. The value of toL is 55.3(-.-L/- ). Figure IIolOa illustrates the pattern of the pole distribution, (1) The black circles indicate the poles when t = 0. There is no coupling between the doorway state and the more complicated states.

-130-.JAO 3StB / i 9i \l a Figure II.9. Numerical Illustration of Paths and Poles.

ImE D 2D,'mlU~~~'~ "I "I~ EReE *-iD rmE 2D * t = D ~~~~~~~~~o t,5.-1ii ReE t = 51!U111 U — O t = 57 -iD t = 55.3 l-2 rt Figure II.10. Illustrations of Paths.

-132(2) As t increases, the poles move towards the points shown by the open circles. This illustrates the situation of the WC, t < t. Namely, the coupling strength of the doorway state to the more complicated states is weaker than to the open channel. Here, we have one wide doorway state pole and many narrow regular poles with a Lorentzian distribution. (3) When t = o, the doorway state pole coalesces with the other central pole at the point A. (4) As t increases further, we enter the SC region t> t, and poles are indicated by the squares in Figure II.lOa. In this situation, there is no clearly distinguishable doorway state pole. Nevertheless, the existence of the doorway state is indicated by the energy dependence of the imaginary part of the poles. 2.3 Pole Residues Having obtained the poles, we may expand the structure factor of the T-matrix into a Mittag-Leffler series(78) and evaluate the resides. 2o31 General Properties and Doorway State Phase Shift After the Mittag-Leffler series expansion, the T-matrix element of Equation (2.0.7) becomes z: ^ ^iT = l- _ _ ) e (2.3.1)

-133where -K (2.3.2) t7+t + (5, - Z )2 The T-matrix of the multi-channel case is also expanded into the same series: T = 1 - bs t 1 C T- - - (2.30) There is no effect of the number of the open channels on the distribution of the poles except through The residue, Equation (2.3.2), may be expressed by 1 (. <() 4 - b e2 d (2.3.4) where the phase b ~ is given by -2^- J(- ) tax40L 2 - ---- X....... (2.3.5) and will be called the "doorway state phase shift". For 0to = O, the range of i Ot of the regular poles is distinguished according to the sign of s r to+ t -( - - (2.3.6) which is zero at t = to.In the case of the SC, and IC, A is non-negative and 2 ranges from 0 to 7C/2 and 3 /2 to 27C. Otherwise, L ) ranges from 0 to 27. For 1o 0, the distinction becomes slightly ambiguous particularly at -U = 0(1), see Figure II.3.

-134The doorway state phase shift of the anomalous pole behaves quite differently and will be treated later, 2.3.4. The absolute values of the residues are expanded into a series, -2 IOr, t Ibl-2 o x t2. + - (. -i......- L.)2. t [t + t + (;t;-) +( - - 0zJ 2' J (2.3.7) This series is covergent except for the central pole at 44- =O and - to. The first term is a good approximation except for the IC For Ito = O, the residues will be calculated further in the following cases: WCL (regular poles) in i 2.3.2 and SCL in A; 2.3.3. 2.3,2 Weak Coupling Limit (Regular Poles) For the WCL, the regular poles are given by Equation (2.2.3). Applying it to Equations (2.3.5) and (2.3.7), we obtain bl t(t +l)t + + (tz)l, (2.3.8) and 1t 2he reul ar s I, (2.3.9) t(tt )-u,; + n The results are shown in Figures II.11 and II.12.

-135Ibnl f 2:0 o /{ t+t }1 o ~- o -} 0 0 0 0 04 —--— #0 0 2v/t o oa o 0 0 0 0 Fi1u Figure 11.11. Absolute Value of the Residues (Regular Poles at WCL).

-1362 8a(n D2Or 0 ~~0 00 0 000 ~ ~~0 000 o 0 0 ~ fn".< 2 t-t Figure 1I.12. Doorway State Phase Shift.

-1372.3.3 Strong Coupling Limit For the SCL ( t ~ to ), the poles are given by Equation (2.2.10). The residues are calculated with the results:,- 2 L t I bt/ _= ---------— (2.3.10) tit+) + LKo a - Usj2) and tar 2 ^d - — 2 — - ) (2.3.11) t(t i ) Po'- +(j4-?) These results show no significant difference in the forms of Equations (2.3.8) and (2.3.10) and of Equations (2.3.9) and (2.3.11). However, a difference in the magnitudes of t and - L-makes a great difference in the doorway state phase shift. The results are shown in Figure II.13 and IIol4< These are a maximum and a minimum which are attained at t - 4= o T (t(t+l) 1-L/ for t >t (2.3.12) The values are given by __.(2.3.13) tQI m ^ ltLt~D-u tat il 1 F M J t st + 29 o Corresponding to each term in the series expansion of the structure factor (Equation (2.3.3)), there is a resonance. For both single and multi-channel cases, the resonance has the Breit-Wigner shape if the corresponding term of the structure factor becomes equivalent to the single channel Breit-Wigner T-matrix.(4) The condition for this equivalence is: $A=o ^s Ibcl/l1^l =2. (2.3.14)

-138bnl 00o - 2vo/{t t+l+ v2 0 0 0 0, o-y-rr o P —2i + t, -2 -—'"' o 0 0 O0 o0 0 O Figure II.135. Absolute Values of the Residues (SCL).

-139(n) 28d 0 (SCL) 00000 0O0 0 xoxOx X -:x x x x x x x x x (iC) X X X X X X X X I x x 0 0o~o-oo O 0o X 00 00 00 00 0000 o^0000-___ _._______________ F iut2 + t - 2t - Figure II.14. Doorway State Phase Shift.

-140Although the residues do not exactly satisfy Equation (2.3.14) in any situation, they approach it in the following cases. (a) The residues of the poles distant from the doorway state approximately satisfy Equation (2.3.14) without regard to the situation of coupling strengths. Actually, however, the other doorway states are expected in such distant regions. (b) All the residues at SCL approximately satisfy Equation (2.3.14)0 (c) In the case of WCL, the residue of the anomalous pole approximately satisfies the condition, Equation (2.3.14). (The residues of the regular poles near the doorway state do not.) For a given nucleus, we expect rT or < a 2> to be a slowly varying function of energy because the levels are saturated in the highly excited states~(58) On the other hand, T- varies with energy as the potential barrier and the number of the open channels. That is, as the energy becomes zero, -t approaches zero proportional to the penetration factor. Hence, in the low energy region, rT becomes small, But r-t/r- = - L0/t which implies SC. Therefore, in the low energy region, the resonances take on the Breit-Wigner form. 2.3.4 Residues of the Central Poles The residues of the central poles will be calculated for the cases: (a) WCL (regular central pole) and SCL (both central poles), (b) the anomalous pole at WC, and (c) the central poles at 41o = 0 and t ~-'t o Finally, a part of the T-matrix corresponding to

-141the central poles will be obtained at Ao = 0 and t = t to see the effect of the coalescence of the two central poles. (a) The residue of the regular central pole is given by Equations (2.3.8) and (2.3.9), L o -2t ______ _ro ( t+t ) (2.3.15) 0 } at the WCL. For the SCL, the residues of the central poles are given by Equations (2.3.10) and (2.3.11), 1 ^, -2TLi 0; 2 a^- LIT ____ + 1 _ — 4 ) 3k(2.3.16) 10 t 3- } where ~ signs refer to the anomalous and regular poles respectively. (b) The anomalous pole at the WCL is given by Equation (2.2.4). The residue is calculated by applying it to Equation (2.3.2), a ^-.. LU~ - (2.3.17) X4M 2l U[ c t t) It follows that, for -Lr >> 1, b. is almost independent of 10. (c) The central poles at to =0 and t ~ to are estimated from Equations (2.1.3), (2.1.4), and (2.2.1)-(2.2.3). Then, the residues are calculated with the result: o t -— t-t.)t~......)- tt)Lts ) (2.3.18) s r to the r ad ao os pos r t. where d^ signs refer to the regular and anomalous poles respectively.

-142All the above results (a)-(c) are summarized in Figure 11.15. HG ^ I Figure II.15. Residues of the Central Poles. The singularities of the residues at t = t~ are spurious since the coalescence os the central poles produces a second order pole at Z. If 1Uo o, then these poles do not coalesce and the residues do not show this singularity. In fact, at U-,- = /., the path of the anomalous pole is a straight line parallel to the imaginary axis (see Figure II.7) and the residue is real.

-143A special situation arises when'1 = -O and t Z x Due to the coalescence of the central poles, the T-matrix has a second order pole at Z = O o The expansion of the T-matrix about Z= L2u yields T ^1 -.~ 1,/7+~' 2Lro (z-z& )- z - ) with the resulting contribution to the cross section, ce.~ p -h 2^(^+J L -2J) (Z5- LTO +@@ 9 ao ( lr + td 2 ) 2+ + 2 ( U + ) —-) f(2 0 3.19 ) as sketched in Figure IIo16. --- 2f U' ---- Figure 1I.1 6. cent pole t This result is of academic interest only, since the coalescence of the central poles is an unlikely accident.

-144CHAPTER 3 APPLICATION, RESULTS, AND CONCLUSION Through the infinite Picket Fence model, we have investigated the properties of the fine structure of the doorway states in Chapter 2. The effects due to the deviation from the model are studied in Appendix Ce The results will be applied in this chapter. The resonance parameters are provided in ~ 3.1 as preparation for the following sections. In 6 3.2, the fine structure of the total cross section will be illustrated. In ~ 3.3, the average total cross section will be given and in ~ 3.4 the dynamical constants rT and r[- will be expressed in terms of measurable parameters of the resonance width distribution. In 4 3.5, the results obtained above will be applied to the analysis of experiments in the low energy region. In ~ 3.6, the doorway states in the continuum region will be discussed in relation to the possibility of having bumps in the cross sections which are caused by the doorway states. 3.1 Resonance Parameters The T-matrix is given by Equation (2.3.3). The total cross section, Equation (1.2.29) of Part I, may therefore be viewed a sum over isolated resonances with average spacing TC. For a single open channel, each resonance of the cross section may be specified by the following parameters: (a) Resonance width — 2 2. (3.1.1) (b) Values of the extrema = ~1 t Cr+ 2(S + S ) (Z~2.) (3.1.2) where the maximum and the minimum exist at? ~ and Z E ~ according to b^^^(j & A)0 respectively.

-145(c) Peak of the resonance / ^ CYrS )d )U ) (3.1.3) 7at = 5i (d) Area of a resonance divided by 7C I b I C 2 (S + St) (2t2). (3.1.4) The dependence of these parameters on the coupling strengths will be briefly discussed in this section and applied in the following sections. The resonance width distribution will be related to the doorway state constants, I f and r, in ~ 3.4 and will be applied to the analysis of the measured width distribution. The values of the extrema together with the sign of 4Atc 2(8+Sd) specify the shape of each resonance. Namely, for A,42(S+ ) > 0, the minimum (dip) of the cross section exists on the left hand side of the maximum (peak). For the other case, the dip exists in the right hand side of the peak. The difference of the values is equal to 2 and the maximum value of the single channel total cross section is 2 in units of 2 Wt. Therefore, the minimum value of the cross section is zero. These results will be applied in ~ 3.2. The sign of the peak of the resonance indicates whether the dip of the cross section is due to interference or due to the negative resonance.

-146The area of a resonance divided by 7C is approximately equal to the average cross section contributed by the fine resonances. For WC and SC, the areas of the fine resonances are given by /1Ja c^pr, rf7^~ c -6j2( ) 2 E (3.1.5)'idqt t~ U2 + (7-U) In more detail, they are given by < <^hRP <4r 2lt t2[+t -(,-&,)j(z-u)i },2S -2 (z-o)(4- ) 2 5~ (31.6) (3.1.6) [t'+t +( - )+(i-uo J + 4t+ t)( -), where Equations (3.1.4) and (2.3.2) have been used. The total cross section corresponding to the anomalous pole is given by T 2 \2-Ui t) + t + to)(.t.)O 2 These results will be applied in 4.3. 3~2 Fine Structure of the Total Cross Section The fine structures of the total cross sections with WC and with SC are shown in Figures II.17a and II.17c respectively. In Figure II.17b, the standard Breit-Wigner form of the total cross section is shown for comparison. The potential phase shift is varied from 0 to — /6.

OAT A~T iKT 4TrX2 4 TX2 4 7TX2 323 ~ ~ ~ ~ U 0 Uo Uo -T i T 47/ X2 4 T/"2 4 7T-X2 Uo U Uo Uo 4 rX2 4 T2 4s X2 UO 0 It +vol —Figure II. l7b. Breit- Figure II.17c. Fine Structure of Total Figure II. 17a. Fine Structure of Total Cross Sections at WC. Wigner Forms. Cross Section at SC.

-148In the case of WC, there is a background cross section which is mainly composed of the doorway state cross section, Equation (3.1.7). The dips near the doorway state are negative resonances and not interferences. Far from the doorway state the resonances approach to the Breit-Wigner form. At = 0, the dips which lie on the right hand side of the peaks (r.h.s. dips) exist below the doorway state energy. The l.h.s. dips exist above the doorway state energy. As S increases, the l.h.s dips appear in both the lower and higher energy regions away from the doorway state energy. The r.h.s. dips exist only around the doorway state energy. These dips are the negative resonances. The shape of the fine resonances with the r.h.s. dips greatly deviates from the Breit-Wigner form. In the case of SC, there is no background cross section. As discussed in the last chapters, the shape of all the resonances approaches the Breit-Wigner form in the SCL. Even if the situation of the coupling strengths is not the SCL, the resonance shapes in SC are approximated by the Breit-Wigner form. In particular, all the dips are the l.h.s. dips for <te2 L | t%-/2t tn -g r, as may be inferred from Equation (2,3.13). 3 3 Intermediate Structure of Cross Sections The total cross section averaged over a small energy interval, (for instance the resonance spacing), is approximately given by

-149( Ts. +o( 6 w Ahlar t < tck < 5 = i (3.3.1) ( 5z->Af^ - <6T> fe t y ti where <( C- p' and (5, are given by Equations (3.1.7) and (301.8), respectively, and the E-independent terms are not taken into account. In this section, these results will be compared with the average total cross section obtained in Part I which we will denote by T) Now, with the application of Equation (2.0.3), <C^5 IT-,t at = 0 becomes -2 L, it-v.) T^ ^ = a y t-) -(3.3.2) z-. -~)~ (t _- o)3 The relative error of Equations (3.3.1) and (3.3~2) at ~ = a and with SC is <C T a tI - < > -65ij t- t + Lo iZ-u a) -__________ —- = bit......t.. ~___-j (3,3.3) < >. t t + t +U X X - uLo j Therefore, the error is of the order of ( L~/t ) for?-i o and of the order of -LUo/t for I Z- Lo >> t. The difference between < 6 T^ t I and 5 >vT C is approximately equal to < (.P->A/w? ( >Xt- (6>SC - < T >. (334

-150The relative error is of the order of t/ —Lo To illustrate the magnitude of this rather large discrepancy, we will estimate the widths of the average cross section of Equations (3.3.1) and (3.3.2), in ~ 3.5. The reason for the discrepancy is difficult to understand. One possibility is that the explanation may lie in the different energy intervals used for the E-averaging in Part I and Part II. (The interval A' in Part I encompasses many resonances). However, the situation is still unclear. 3.4 Distribution of Resonance Widths and the Doorway State Constants One of the characteristics of the fine structure of doorway states is the distribution of the poles of the T-matrix. The distribution of the imaginary parts of the poles is well approximated by a Lorentzian except at the intermediate coupling. In this section, the constants of the doorway state, FT- and rF, will be related to the two parameters specifying the Lorentzian, the peak value and the width. For WC, the distribution of the fine resonance widths is obtained from Equations (2.2.6) and (2.0.6) with the result: ~T - - -~ D A - rr - ~rtjf + -7) L(EQ) x I + (- Ab - l (1 tLE (3.4.1)

-151where L (tLE) ----- (3.4.2) (nE-^F)2r lCr)2 * The distribution given by Equation (3.4.1) can be approximated by a Lorentzian (see Appendix C) with the peak value, ( )p, ^^=^~',-^-2pr^^~^7 K' -(3L4.3) 7 Pr 6-D -/ and the half width, w-r= r ( ) + ^ ) (3.4.4) If we define j _ C r)^ (3.4.5) r and rF are expressed as follows: h' - s W (3.4.6) and rl {2.V }_ _. (3.4o7) For SC, the distribution of the fine resonance widths is obtained from Equations (2.2.11) and (2.0.6),

-152rT = 2D rt(r)+ 2rt)/[ E - E0 + r;(2r+F) + "r- y t7 r)(4 -rt) 1 (3.4.8) ( h ED')\ L + + r The Lorentzian parameters are given by /r)\ 25 r()? = — - 2.D.. _..... (3.4.9) t r4 -X+ ___ / and W ^rtr+ r (- i r42t 1( (3.4.10) Applying Equation (3.4.5), we have that = il 4D- 1 3S 2- (3.4.11) i+ S+ 7i' 9; 1+5+ s1 7 and For IC the dooy t 2S the distr(3.4i12) For IC, the doorway state pole joins the distribution of the regular poles. Therefore, the measured resonance width which is locally largest is the width of the doorway state even though the pole distribution

-153is not Lorentzian. Let us denote the measured resonance width by (F). ftd is related to (r')L by -.Dd (3.4.13) We may then estimate t (= — 2DLo/~ ) and F2 = 2 D tot/') using Figure II.8 and Table II.1. Because t (seeSI2.2.1) depends on 1o, \ is not definite but 2pto; 2 (titl ) I7C ^0 / -x (3.4.14) The above results will be applied in the next section. 3.5 Doorway States at Low Energies In the energy region where fine resonances may be observed, the doorway states have been identifed by taking the average of the total cross section(92) and of the total and differential elastic scattering cross sections.(93) In this section, we will apply some consequences of our study of the fine structure of the doorway state to the analysis of cross sections exhibiting fine resonances. First, the doorway state constants, Eo,, and r, will be estimated from the distributions of the fine resonance widths. Then, the total cross section of the doorway state will be calculated to show the consistency of the theory. Finally, we will illustrate the difference in the widths of the intermediate structure obtained in ~ 3.3 and in Part I.

-154The experimental data on fine resonance parameters for isotopes ranging from Ca to Ni and in the energy range from 100 to 600 kev are available in Reference 85. Based on the theoretical results of the pole distribution, we have discovered a systematic fluctuation of resonance widths over the above energy and mass ranges.(63) Two examples are shown in Figure II.18 where the histograms indicate the measured resonance widths as a function of resonance energies. The doorway state constants will be estimated by the following steps: (a) For each situation of coupling strength, r- and rt will be estimated. (b) From the values obtained, the SCS which applies will be chosen. For WC and SC, the width distributions can be approximated by a Lorentzian. The curves in Figure II.18 are superpositions of Lorentzians which are locally fitted to the width distribution. The doorway state energies, Eo, are obtained at the peaks of the curves. The curves also give the peak value and the half width of each Lorentzian. These values are listed in Table II.2. Then, the values of T' and r~with WC and SC are estimated from Equations (3.4.5), (3.4.6), (3.4.7), (3o4o11), and (3,4.12). The results are listed in Table 112. Assuming IC, we may also estimate r" and F. Taking as an example the doorway state of N11 at 410 kev, we have ( -)L? and = / 8. -reW v ~ Applying these values to Equation (3.4.13), we have UlJ = -- 6. From Figure II.8 and Table II.1, we obtain the following values of ta and VZO t=O.53 4 r"ae e3 wher o= w.e hv 6a3 E rqti 1n (3 1 where we have applied Equation (3.4 14).

> 10 o I I _ I I I I 1 N58 60 I - NI NI60 LORENTZIAN LORENTZIAN C) w,I _ _ _ / _ _ _ T _ 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 RESONANCE ENERGYMEV Figure II. 8. Resonance Width Distributions.

-156TABLE II.2 CONSTANTS OF THE WIDTH DISTRIBUTION AND AND rr Weak Strong Experiments (kev) Coupling Coupling EO' w (r)p D S rt r rL rF 205 140 6.4 17.5 0.574 158. 80o 42 113 410 130 5.6 18.6 0.473 144. 61. 37 103 580 120 5.7 15.0 0.597 137 72 37 98 Weak Strong Experiments (kev) Coupling Coupling Eo V (F)p I s Fr r T r rV+ 195 130 4.2 32.5 0.203 136 26 32 109 320 130 6.8 18.5 0.574 146 75 39 104 500 110 3.8 11.0 0.543 123 6o 32 92

-157Next, we will evaluate the SCS applying the sum rules of Equations (1.3.13) and (1.3.14) as well as the properties of the fine structure of the total cross section. For IC, the sum rule of Equation (1.3.13) is violated because the sum of resonance widths near the doorway state at 410 kev is already larger than rT- ), S7 X/. The same is true for the other doorway states. Thus, IC is excluded from the case of SCS to apply. For a given Lorentzian distribution common to both WC and SC, the value of r obtained for WC is equal to the sum over the fine resonance widths under the Lorentzian, where Equations (2.2.5) and (1.3.13) are applied. For either WC or SC, I must be larger than the value of F for WC to satisfy the sum rule of Equation (1.3.14). In Table II.2, Fr for WC satisfies this requirement for all the doorway states of and 6\/0 while F' for SC does not except for the doorway state of Ni at 195 kev. Therefore, the SCS of the doorway states at 320 and 500 kev for N 0D and at 410 and 580 kev for Nk is WC. However, the SCS of the doorway state of IV60 at 195 kev is undetermined. That of NIAr at 205 kev is WC according to the above criterion. Now, let us apply the second criterion: as pointed out in the beginning of ~ 3.2, a background cross section exists only for WC where the anomalous pole yields a contribution distinct from the regular poles. The measured cross section for NV' shown in Figure II.19 indicates that a background cross section exists around 400 and 600 kev but not around 200 kev. The cross section of HAi is quite

-158similar to that of Ij, a background cross section existing around 350 and 500 kev but not around 200 kev. 60 Therefore, both criteria for the doorway states of at 320 to 500 kev and those of ZV at 410 and 580 kev indicate WC. The second criterion indicates SC for the doorway state of 60~ at., ^ 195 kev. For the SCS of the doorway state of LN at 205 kev, the two criteria lead to a contradiction. In this case, we chose the result of the second criterion, for the following reason: According to the results of Appendix C ( ~ C.4 and Figure C.1), for the given values of rt and, the average width distribution around the doorway state is almost independent of the number of the more complicated states. However, the deviation of 2 rT from r- for WC depends much on the number of these states. Thus, the first criterion can not be thought of as absolute, and violations of it can be explained in terms of the finite Picket Fence model. The chosen values of r'r and r- are underlined in Table 11.2. The background cross section is caused mainly by O9 S Equation (3.1.7). We can estimate (T for the doorway states at 410 and 580 kev by using the values of r, -,and the potential phase shift. The potential phase shifts of N are estimated using the square well potential given in Part III with the results that2 ~-~t at 410 kev and 2 i - - t 6- at 580 kev. The results of CT are shown in Figure II.19 with satisfactory quantitative agreement with the measured cross section. (Notice that the fine structure of the measured total cross section around the doorway states at 410 and 480 kev (Figure II.19) roughly resemble the inverse of the cross sections in

20- N 10 0(b)I \ X s 00 200, 300 400 (kev) 500 E 600 Figure 9. Total Cross Section and the Background Cross Sections of -58 Figure II.19. Total Cross Section and the Background Cross Sections of Ni-58.

-160Figure II.17a for,2 = - 3 and -- /6 respectively, where the energy scale is measured from right to left.) This result verifies that the treatment of the resonance width distribution, and the evaluation of its constants r and + are correct. Finally, the intermediate structure of the doorway state will be briefly discussed. The intermediate resonance around 410 kev is actually a wide dip due to the potential phase shift. According to the results of Part I, the dip width is equal to Fr-+ T- (= 205). On the other hand, the intermediate resonance given by Equation (3-3.1) is a sum of two dips with widths of 83 and 162 kev, where these values are obtained from Equations (3.3.1), (3.1.6), and (3.1.7). Since the ratio to <R?,.%,around the doorway of the cross section C.6s, to T ^RP>AX, around the doorway state is approximately F T-P, we take a weighted average for the dip width of Equation (3.3.1),3 C 4- /62 F' 3 3 *f\/ ran t r C These two values, 205 and 103 kev, illustrate the difference between the average cross sections obtained in Part I and in this part. The latter value agrees well with the E-averaged cross section of around 410 kev. 3.6 Doorway States at High Energies In the continuum region, the number of open channels including the inelastic channels is large.(91) r~ becomes large (see Equation (1.1.19)), while -r is expected to be approximately independent of

-161energy (see ~ 2.3). The situation of coupling strengths is therefore the WCL and the poles are given by Equation (2.2.4) and (2.2.5). In this section, the branching ratio will be studied. This ratio determines whether the bumps of the total cross section are caused by the doorway states. For a given doorway state, the quantum numbers, J, are specified. Therefore, the open channels with channel indices (jlJM)_o(c) may be coupled to the doorway state where WC =- (-) Taking energy units, we will denote the absolute value of the residue of the diagonal element of the resonant part of the T-matrix by G ). We have -_.J T- (3.6.1) where / refers to the, -th level, and G is defined by Equation (1.1.15). The notations of ~ 1.1 is used throughout this section. The total width is given by the imaginary part of the/ -th pole. For the WCL, I v l of the anomalous pole and of the regular poles are given by Equation (2.3.17) and by Equation (2.3.8) respectively. For a given channel (OC), the values of C6T/c) are proportional to I /l/; i.e. GAl~c) ~ GSc), where the subscripts a and n refer to the anomalous and the n-th regular pole. Actually, Grl~ ) is obtained with the results:

-162G [c) (SF /) (O /~C), (3.6.2) and r" r /( P T (c- ctl Go (dc) Alp,^ ~ r /( — - -i —-),.) I (3.6.3) (E-E/S)'t 4 (rFir'i+ 4 r1 ) For simplicity, we will limit the problem to the case that there is no direct reaction: SP is diagonal and unitary. Thus, Equation (3.6.2) becomes t_ — = r; (3.6.4) 0~L(^C-) - c) / c) The amplitude of a bump in the total cross section, Equation (2.2.29) of Part I, is approximately given by fl 6X^<- - 21f'.. L2 a / f } d c -' C 47oC 42F -- -, (3.6.5) where 0Qo refers to the ground state of the target nucleus and the sum over c includes all channel indices obeying the selection rules. This indices that the amplitude of a bump can be large only if the e candoorway state is preferencially coupled to the elastic channels.

PART III CALCULATION OF THE DOORWAY STATE FORMATION PROBABILITY -163

Introduction In this part of the thesis, we take Ni-58 as an example of the target nucleus and calculate the doorway state formation probability, rt. The model Hamiltonian is composed of a pairing force plus Tp —L forces. This type of Hamiltonian has been developed by Kisslinger and Sorensen,(99) Baranger,(100) and Yoshida(97) to explain the low excited levels of vibrational type nuclei. The result of this calculation will be seen to agree reasonably well with the measured value of the doorway state formation probability. In Chapter 1, we determine the low excited states of Ni-58 in order to obtain the force constants. The doorway states of Ni-59 are shown to be composed of three quasi-particles. In Chapter 2, the open channel wave function is represented in terms of shell model eigenfunctions and rF is expressed in a form suitable for computation. The doorway state formation probability is calculated and the results are briefly discussed in relation to the Lande-Brown model. -164

CHAPTER 1 A MODEL HAMILTONIAN AND THE DOORWAY STATES For convenience of calculation, we will adopt the second quantization formalism.(48) The basis set may be the set of the O.M. wave functions or the set of the single-particle shell model wave functions. The former was chosen by Lemmer, Shakin, and Lovas.(71.77,80) The latter does not apply for the open channels. However, the low excited states have been explained on the basis of the latter representation and calculations of matrix elements of an observable using this basis are easy. In order to exploit this advantage, we will apply the shell model representation with a modification for the open channels. Namely, the bound states are expressed immediately in terms of the shell model wave functions, while the open channel wave functions are calculated from the O.M. Schrodinger equation and then the shell model expansion is applied. The Hamiltonian we assume is: H = K t U - H + H lt - P(K + UoM * Ih)'P + MA(K*cM +lH + ) Q + PH^t Q + QHtP, (1.1_1) where LoM is the real part of the O.M. potential and ULSM is the shell model potential. The first term of the Hamiltonian will be treated in the next chapter. We will further specify the last three terms of Equation (1.1.1) by the model Hamiltonian developed by Kisslinger&Sorensen, (99) Baranger, (100) and Yoshida(97) for the low excited states of vibrational type nuclei: -165

-166The residual interaction potential is expanded into a tensor series,(72) V(Iz,-rL ) L 4 2 )XL ). 1.(2) ^f^ 2L Lr / LM 2L~1 and the Hamiltonian is (4-) = (1.13) -H 1 3 ( B) where o 5L eo b<bS, (1.1.4) and 4 ) - ( Xt t) (z-J4Y <^ WJt > HL -M +t2 --' ^4 -. 2 ( uj V3j U - Vj, V/ VI )B (j, L -M)A zJ;LM) b dz d',, d') -Ui, V' UdI vj. Li ~, 1. z-M)B) 2 - + -2 I y, Vj. Ulj4 B (,3,; L- )BJiI;2i ). ) (1.1.5) -2 Ud V V,; U 4 B+,- L-M B. -...) The notations are as follows: b and bct are creation and annihilation operators of a qusi-particle associated with a state specified by quantum numbers o( /(N\ Sj ) which are often abbreviated by (j Kn) These are related to the particle operators,+ and L () by the following relations (BCS-transformation(67)),

-167bp, = u Oae - Vo a _-+ f (1.1.6a) + + where ay is the time reversed operator of e2g; e.g. ~a = (-( -y ^- (1.1. 6b) and where {bi, b^^{ ~- be, b\ -o, ), b b = (1.1.6c) l, 4,4 3B + A and the real positive numbers UO( \/ Vo - V ti satisfy,/ -^c. 12R = l,, Vot1 6d) J01-u c V~7 VFor given values of G (pairing force constant) and N (particle number), the single quasi-particle energy, eo(, and the constants LU and V are obtained from the following relations, eOt% -(E -A /{a' (1.1.7a) G 2 d t (.. 2N = ) 22 - (1.1.7cb) <l>D >

-l68where w'n,~ "^,^*6 vd ^^ ~ ^W ("l % - ^i - GK (1.1.10) Vd~ J ______ (1. 1.9):F>4 E - Vc~i (Li. 10) OL>O (gap parameter), (1.1.11) and Er, is the shell model single particle energy of a state (. The above relations of the BCS transformation are given in various articles (References 67, 97, and 100). The values of the parameters for N = 2 and various values of G are determined by solving Equations (1.1.7)(1.1.11) by successive approximation and are listed in Tables III.1 and 11I.2, and shown in Figure III.1, where the values of Eox are taken from Table III.3. <94'j/'II LLYL IlNIJ>) is the reduced matrix element defined by(97) < U! Y b ~n~ l I N > dy /s~L —^~+ W J L + W J2L.1- <~'~ rL~~=WlLP A'f~1'IU~JIIIk~d2) (1. 1.12) and will be often abbreviated by <dlllU YL I1 ><, where I|ISJVYl> is the shell model single particle vector, Equation (D.18). B, B, and A+ are vector coupled two quasi-particle operators defined by Equations (D.7)-(D.10),i67) Finally, 7 is the 71-7L (longrange) force constant. The reduced matrix element is further related to the radial integral of the shell model wave function, Equation (D.15).(97) The formulae for the radial integrals are given in Appendix D.6.

-169TABLE III.1 QUASI-PARTICLE PARAMETERS AND PAIRING FORCE (MEV). -0.4 -0.6 -0.7 -1.0 0.859 1.066 1.173 1.455 0.328 0.390 0.421 0.505 e3/2. 947 1.223 1.366 1.765 e5/2 1.460 1.744 1.888 2.299 e/2 1.712 1.989 2.130 2.538 1/2 e9/2 3.700 3.946 4.072 4.444 TABLE III.2 QUASI-PARTICLE PARAMETERS FOR G = 0.421 (MEV) l |a< | ViL l Vd Uo( p3/2 1.366 0.244 0.756 0.494 0.870 f5/2 1.888 0.108 0.892 0.329 0.944 P7l2 2.130 0.082 0.918 0.287 0.958 g9/2 4.073 0.021 0.979 0.146 0.989 d5/2 6.507 o.008 0.992 0.091 0.996 s1/2 7.790 0.006 0.994 d3/2 9.772 0.004 0.996

-170Q) aT 5 05 -1.0 Figure III.1. Independent Two Quasi-Particle Energies (ea + e).

-171The single particle frequency, k t, for the shell model radial wave function (Equation (D.20)) is obtained by(107) t = 4 A V3 -= o, (Me ). The single particle energies, ENLD, of interest to us are those of 2 3/2 Y 1f/2 2 1g 2,2 3Sy, and an 2 d The values, which are obtained from the analysis of the direct reaction experiment; Nf (d, ) s, (11-16) are listed in Table III.3. The neutron separation energy, B, of /V, is taken to be 9.00 Mev (12) TABLE III.3 2ij WITH Ni-56 CORE (Mev) 3/2 f5/2 p1/2 g9/2 d5/2 sl/2 d3/2 Ref. 11 0.3 0.6 2.2 3.5 6.0 7.3 9.3 Ref. 12 0.3 0.6 2.0 3.5 6.0 7.3 9.3 Ref. 13 0 0.78 1.08 - 16 Our Choice 0 0.78 1.08 3.2 5.7 7.0 9.0 The second part of the Hamiltonian of Equation (1.1.1) may be decomposed into two parts: (a 7-1 (4)F + 7 71 (1.1.13) where r4 tBBeB (t (1.1.14) ~and HA 4t 3 ++ A+B

-172In Equation (1.1.14), the terms of 7 4-L proportional to B*B, for (4-) example, are denoted by -13+B. YH7PA will be approximated by (4) 7-_T4 ) where BayrP ^ 1 B^3 (1.1.15) which is the Tamm-Dancoff approximation(67) for L4-L and then / Z,C I B 0 TD1 = B 4 Q 3 + (1.1.16) withA, Bg _ B -B A Denote the quasi-vacuum state by I); i.e. b / Q> = for all 0C. From Equation (1.1.16), we notice that \Crpad J I0> is an n phonon state with the frequency $.(67) H t may then be treated by perturbation theory. We write the Schrodinger equation in the form ( (4 4 D Ba ) ) A (n) (1.1.17) q Ho r + - where ) / -+ 7(-/D is explicitly Ho - (4)- L 4x X 2, > Y l U YJtVI3 U# d, II (I11L YL RI < 2 I YL /Id f ) UV xFor the sy L-M) wih tw (1 t - - For the system with two quasi-particles, the zero-th order solution of Equation (1l1.17) is straightforward: The ground state is immediately known to be j }>, while the excited states are given by n = Cn) B+('d; ).12> (1.. 19 ) S'd

-173where < (JH | 1,'/m/ i= Is J'/ SMM (1.1.20) and the coefficients are obtained in Reference 97; cf = L (-)' d ( u.,v + Vu )<ill, Y/J,>CJMi) (1.1.21) C3M.,J) - r} o"(-) i J ( ui, Vi + Vi, b) <'lr -.YLl..j > (, ~,?~ " j' ^2e' \ (dUJX d (l. ",) Xj!'O1 i, 11 AL YL llj j ) —-- (V < 12 ) -1 =- IT.'2. (1.1.22) (J 1-' E2 y - (,+ e j ) ( _t _ (tt' (2U:j',.j(< JYi) (I.I. U24) MM) el The zero-th order results turn out to be exact solutions of Equation (1.1.-"17) because:_a~~~nd~~( (1'1.17) because The low excited levels with positive parity of C/V are formed by two quasi-particles in (pf) states. The estimated values of the reduced matrix elements for these states are listed in Table III.4 (see Equations (1.1.12) and (D.15) and Table D. ).

-174TABLE II.4 RADIAL INTEGRALS (R.I.), AND REDUCED MATRIX ELEMENTS (R.M.E) J 0 / R.I. R.M.E. P3/2 P3/2 9/2 9/2 4w p3/2 5/2,f7/ 2 p3/2 pl/2 9/2 - 9/2 _____ fgj2 Af~i/A/~ f f -114 2 1 5/2 5/2 5/2 f5/2 9/2 9 ^3 F7-y p f -33 414/4 -99/24E 4 3/2 2 f f5/2 99/4 -297/4 *ii47E ___ 5/2 ~5/2 1 1__________

-175Solving the secular Equation (1.1.24), we determine the force constants, G, X2, and X4, such that the levels E ) agree with the measured excited levels of N as well as the levels calculated by Lawson et. al.*(107) (See FigureIII.3). The determined values are G = 0.421 (Mev), x2 = - 0.337 (Mev), and x4 = - 0.021 (Mev), and the energy levels, E ^*), are shown in Figure III.2. The values of' ),, and for J = 2 are listed in Tables III.5 and III.6. For the highly excited states of the compound nucleus, N, more than three quasi-particles are expected to be excited. The eigenfunctins of o t- H(4) functions of 7-(o+ ~ TP of the odd neutron system are given by -^/^ &, ~- bfB+I >, -b' S^B'i/,, ----.The elastic channel and the remaining channels are spanned by - bI lQ and ( b+B+B +- )1)> respectively; i.e.? - b^X~)<lb and Q. - (^B'4+b+BtB++. —^><S;(Bb+B~b^-) Since <^|bH b+8t ^>=(p)^^ ^B/^>^ N~~~~~ and e-, < lby B+ - B 0,-k V A =0 tN^, 2 then the subspace q1 is formed by >s | 1 / and the doorway states are composed of the three quasi-particle states. *The author appreciates Dr. Lawson's information on his calculation and matrix elements before publication.

-176CALCULATED LEVELS x CALCULATED LEVELS WITH X2 X4= 0 -- MEASURED LEVELS 5 s- x x.~. X 3 w _Ax 2 X J= J I J=2 J= 3 J= 4 Figure III.2. LowExcited Energy Levels of Ni-58.

-177(MEV) 6 Measured Levels Calculated Levels 5 4_\,, 3 - /, 2 - ~0 ~ J= O J=l J=2 J=3 J=4 Figure III.3. Calculated Levels by Lawson et. al. (107)

-178TABLE III.5 E^) AND CJ FOR J 2 AT x2 = -0.337 (MEV)'t E ) (MLe) 2 1______) 1 1.45 3.24 2 2.98 0.701 3 3.32 0.299 4 3.68 0.311 5 3.94 0.267 TABLE III.6a i, FOR J=2 P3/2 f5/2 /2 p3/2 0.466 0.158 0.246 f/2 -0.158 0.248 o.161 p / -0.246 o.16l 0 1/2_______ ______ TABLE III.6b'T2.) FOR J 2. P23/2 2 P/2 P3/2 -0.517 0.227 0.212 f / -0.227 0.154 0.087 f5/2 -0 2 0.087 o Pl/2 -0.212 0.087 0 P.......

-179TABLE III.6c C~,3 FOR J 2 d a < " 2 P3/2 f5/2 1/2 p3/2 -0.094 -o.410 0.261 f5/2 o.0o410 o.114 0.055 pl/2 -0.261 0.055 o TABLE III.6d c(4 FOR J=2,_ P3/2 f5/2 Pl/2 p3/ -o.060 -o.064 -o.261 f5/2 0.64 0.569 0.118 pl/2 0.261 o.118 0 TABLE III.6e J J('-) Cs FOR J 2 do'L P3/2 f5/2 P1/2 p3/2 -0.41 -0.034 -0.093 f5/2 0.034 -0.283 0.446 pl/2 0.093 0.446 0 Pl/2

-180For our case of i+-AJL reaction with J (12)_ parity and angular momentum conservations restrict the possible three quasi-particle states as follows: One and only one of the quasi-particles is either g9/2, d5/2 Y Sl/2, or d3/2 * Each of the other two is either P3/2, f5/2, or pl/2

CHAPTER 2 DOORWAY STATE FORMATION PROBABILITY 2.1 Single Particle Potential and the Distorted Wave We choose the single particle potential for the incident neutron to be a three dimensional (3-D) square well. The two parameters, radius and depth, are chosen as follows: The nuclear radius is taken from the result of the OM. calculation,(43) _ = /. l AY3 + o 6 f f (201,1) The value is 5.06 fm which is also obtained by I = /, 3 0 At3 t The potential depth will be determined such that the energy level of 3 S /2 lies at 2 Mev below zero incident neutron energy to agree with the measured single particle level (see Table III.3). The estimated value is \/o = - I /. 3 (2.1.2) The depth of the Wood-Saxon potential(43) is 46 Mev in good agreement with our result. Figurative comparison of the potentials is given in Figure III.4. The distorted wave /C is expanded in terms of shell model wave functions: t = l C IAL M r+ (2,1,33 where we have used the fact that the ground state of /\ is 0 Since the incident neutron wave function does not belong to L2, it may not be expanded by the set of )| S' WM>. However, the couplings between the open and closed channels are spatially limited because the bound state wave function approaches zero as eq when -> P -181

-182V(r) (MEV) -2.0- - 3 —- r(fm) 1/ -10 - -20 - 2SF2 -30 SW -40 - O.M. - IS -50 S.M. -60 Figure III.4. Single Particle Potentials; 3-D Square Well (S.W.), Shell Model (S.M.), and Optical Model (O.M.) Potentials.

-183Hence, we may replace, j. by Xls d where ( is the step function defined by 1 fo r< R 9(ro, R) - {r (2.1.4) o ~f r, >R and then expand: hL)^ (9grO~R)- ~ C l/s^d > )0+ (2.1.5) In the second quantization, the open channel vector is thus effectively given by 2I k5 sJ j = c ( ~ >\ (2J1.6) where the subscripts (N k SK, L L ) are abbreviated by (k) Now, the distorted wave with ( Y2 ) is easily obtained from Equation (2.2.35) of Part I(87) with the result, ^^ - ^~~Y~~~2 N 7M~~~~~4)~A -\ S R~E2Y^V0 Nk r-7 1Ys )M/m< + K (2.1y7) where'x (y); 6(r, )= / -c r ( -C ) c^?L + 0 I/o If-o a < oi < R, O fo r Lo>R (2o1,8)

-184- - V (r' ), (E2.19) 0Uo=~ 5 j0R- (2.1.10)?o -1 a& (2.1.12) ~~Y- t/V(3 ^? ~~~(2.1.13) and 3 is defined by Equation (D.22). As will be seen later, the radial integrals, H 2.2, converge rapidly as R becomes large so that the results turn out to be insensitive to the choice of cut-off R 202 Formulation of r~ The doorway state formation probability, r, is defined in Part Lo r~ = |.~Z__ < PH. > r S -27QL <Z7 z,oI mhIP7lIP > ), (2.2.1) Is(?t The doorway states of ( /) are composed of three quasi-particles: >xp> l T\ ot, 1 c M') 1(.2.2) x b C) B+(d (2.2 i'i

-185where the low excited states, Equation (1lo.19), are vector coupled to a one quasi-particle state. In this section, fs will be reduced to a form suitable for practical calculation. Since PHI H t, we have + ( )^ 2- Z~K L N -!'(sF(f^) i) Y2 ^ (l2 ) N LM (2,- td +'j2 + ild4 < Ili L' Yy > < dL 2 L - >) x iuj,4 2 J3 J4- da d ^<dMo 1.2 X(Pg/b(X)A (jfi32 L-M)b;B CI I d For the doorway state to have positive parity, j and j run over (pf) states and 0C over (sdg) states., For even L, parities of ji and j, and of j and j4 must be the same because the redaced matrix element has the property: N9 I' I YL i/ > J + (- ) Also, by similar parity considerations, we can show that ) j L M, b, 0J In view of these selection rules, the expression (2o2.3) reduces to -4 _ 5 4^'CL V - ( ) 2+ 2- N C 2(=eA) MjtMw N 44 (2 L + 1) X + S: 44 d p j dj j M>( ) + (, VIg U1 Li VV Y4j L M ( +,< j M' |/ z > ) > C", <a| B(j.4; LM) X B3 jd M'), ~> | )

-186- 2 (42 1)(2' a,-' ) ~ t, YI d' > x4 O2,1 ~ Y. ll j4 >2 _i, k (,,.-L >' Iu, -) ~ ( - (2,T+I) f Oc4by"YL /- L /L/ Y//>2C, (o,,). u"v U',V 2, (2.2.4) where k = 1/2 otherwise Tr = o. In the above derivation, we have applied Equations (D.13), (1.1.25), and (2.1.5). We will make the approximation that LUk 1 and V. 0 for all N. This is valid for N >3. For N.< 3, the reduced matrix elements are small, so that there is little contribution to TI' Applying Equations (1.1.21) and (1.1.25), we have'3 2 ( c Tv) y0 ) Y2 + 2 (.2 +- )3 (2r<y I/ct' ) X (CL' J 2 (2.2.5) where we have defined: N i By Equations (D.15), (201,5), and (2.1.9), this reduced matrix element is calculated with the result: 1k HJYJ 8 U)=(-) C dJ (-)> 4+t) l) 2 (;4:c x <J O|d'w ><) | ^ > (2.2.7)VI

-187where in the above derivation we have used the following relation of the radial integrals p e,1v;I2 P,) (R v SC^ < N lT:I NotJo> =<1 ot I ot -V N'4dro L(ro, R) R)(7 ) r0)' Y(r6) - k CN/,, Ij (2.2.8) and where -^^ 5Ef I I —.- -(2.2.9) F [ / o(f ta A~ + 3/2 and PIC _ w' I,,(. -$ y ^^.T,) Rxi^e'^ x'a + +I (2.2.10)' The estimated values of C- IJi for J 4 s 2 and'IoK I and for J = 4 L 4- and /nf d= are -6.11 and 14.96 respectively By the triangular relation of the reduced matrix element of Equation (2.2.7), for = 1/2 and a given J, the states of N are selected as follows: For =2, these are /3 and 2LSan and for J = 4, g/ a Finally, F becomes Ft 421 I XJ2 N ~C(<iJ o|^> J, )2 ( C,2 (j.2)J +I

Radial Wave Function Q(r,R) R(ce,r) = 3.11 7 5 3.11 1.0 0.5 fa = 2.59 J rx = 4.184 Ka = 2.5t +.205 K rx = 15.02 Figure III.5. Radial Wave Function of the Incident Neutron.

RADIAL WAVE FUNCTION Rn ~pfi* 1.0 n=0 n n=2 0.5 6.0O -0.5 n=1~ ~~~n1 -1.0 Figure III.6. Bound State Radial Wave Functions.

1.0 N=4 0.5 0 H 0.5 N=5 1.0 Figure III.7. Bound States Radial Wave Functions.

-191where il}lell 1 1. I =wh r 2 1'___ ___2 2 a_ _' (2_2_12_ The relative doorway state energy is given by d' + t E ) We will adjust the absolute values of ei~ and the doorway state energies measured from the zero incident neutron energy, as follows: We assume that the absolute values of the single quasi-particle energies of N$" in the higher energy range, (sdg) levels, are approximately equal to those of the single particle energies of Nj\ which are lEFi/j. -3, and also that the former is given by el-B Since the energy of the two quasi-particles relative to the vacuum state energy is given by, (h), the absolute value of the doorway state energy becomes ^(dn ) -ed + E )-B. (22133 2o3 Results and Discussions For J = 4, the doorway state energies are far below the zero incident neutron energy. For J = 2, the states of the single quasi-particles are 2 d3/X and 2 ds2. From these two single quasi-particle states and five two quasi-particle states, there are made up nine three quasi-particle doorway states with energies above the zero incident neutron energy. These states are listed in Table IIIo7 along with the estimated values of the doorway state energies and formation probabilities. These results are also pictured in Figure IIIo8 together with the experimental values.

-192TABLE III.7 [4 AND THE DOORWAY STATE ENERGIES (A n) E|bMn (MJ)rjn) r ^, ) l/ (keF) (d5/2 2) 0.49 109. (d5/2 3) 0.83 25. (d5/2 4) 1.19 32. (d5/2 5) 1.4526. (d3/2 1) 2.22 2010. (d3/2 2) 3.75 113. (d3/2 3) 4.09 21. (d3/2 4) 4.45 23. (d3/2 5) 4.71 17.

-193r(Mev) 2.0 Ob s erved 9 (Part II) 0.2t Calculation 0.1 I I II I I I I 2 3 4 5 En (Mev) III. 2 ad DoEn (Mey) Figure III.8. I and Doorway States.

-194We obtain two levels with r -' OO kev and six levels with <1^ 30 kev. In addition there is one level with a very large r( h 2 I ewv) at ~ 2 Mf-V which resembles a giant resonance in the energy range of 2 3 M w. Accurate experimental values for r- are only available for the energy range less than 600 kev. The three identified values of ft (see Part II) are shown in Figure III.8 and they are seen to confirm the theory reasonably well. There is, in addition, a broad bump (its width ^-Mev) in the observed total cross section of natural N (51) around 1~2 Mev which may very likely be associated with the predicted "giant" doorway state. We will now briefly compare our calculations with other attempts at obtaining information about intermediate structures. The elastic scattering cross sections of N and C 1 are estimated by Lemmer and Shakin(71) and by Lovas.(80 Their estimated cross sections give excellent fits to the measured values. Compared to these fits, our results are not so good. However, our target nucleus - / 2 / Sis much more complicated than the C or N nucleus. In addition the SCS for CI and / doorway states is WCL, so that the measured values of r are just the widths of the individual fine resonances More complicated nuclei are treated by Shakin(77) who calculated r- values for Pb and Sn. As in the work of Lemmer and Shakin,(71) the author assumed that the SCS was WCL, and, in fact, in the case of Pb, there was an order of magnitude agreement between the calculated values of rt and the measured fine resonance widths. However, the calculated values of Fr for Sn did not agree with the measured fine

-195widthso Our calculation is about as successful as Shakin's Pb calculation. However, our particular interest lies in treating examples of SCS (eog. WC and SC) which are more general than the limiting case of WCL. In retrospect, it seems reasonable that Shakings values for r in Sn might prove more nearly correct if the Sn data had first been analyzed to determine the SCS. Lande and Brown(106) calculated the spectroscopic factor of individual excited levels of N1. Their results did not yield the intermediate structureo Briefly, the Lande-Brown model for the highly 6 1 excited states of Nk is composed of one particle plus many phonons with a single frequency. The interaction Hamiltonian changes the phonon number by one, Thus, there is only one doorway state associated with a single particle state, resulting in the very low density of the doorway state levels. Presumably, we could use our model to calculate the fine resonance widths by abandoning the Picket Fence model and taking into account the excitation of more particles from the nuclear core, Although we would expect to obtain the intermediate structure (resonance width distribution (Part II)), the calculation would be very complicatedo Summary The doorway state constants rI and r of A and ^' s are obtained from the analysis of the distribution of fine resonance widths (Part II). From the values obtained, the background cross section of N? is estimated and is in good agreement with the measured cross section (Part II). The values of r- are incorporated into the modified OoM. imaginary potential in Part I and the result leads to the discovery the double YV total cross section of the

-196experimental measurements. Finally, in Part III, r is estimated by means of a nuclear structure calculation and agrees fairly well with the experimental results. With these results, we may support the doorway state ansatz.

APPENDIX A DISTORTED WAVE T-MATRIX AND SCATTERING AMPLITUDES The T-matrix, T, may be defined by(73)? -t() 1 ^ TL^ — (Al1) The scattering amplitudes, j, may be defined by the asymptotic form of P)' A V;; 5- RG1- ^ (~1Rv 2 Y0(Ao2) where notations may be seen in ~ 2.1 and 2.2 of Part Io In this appendix, the T-matrix elements which are represented by the free waves will be related to the scattering amplitudes. Then. the T-matrix elements will be related to the distorted wave T-matrix (see Equation (2o2o22) of Part I). Finally, the cross section will be related to the scattering amplitudes and the T-matrix elements, The T-matrix elements will be first related to the scattering amplitudes by comparing the asymptotic form of the solution of Equation (A.1) with Equation (A.2)o Since the completeness relation on the subspace of E' ~ projected by P is given by N\1 h j_0. X - r ^ ( x d^ () = FS( G-ro; )(A.3) *Equations referred to in this appendix are in Part I of this thesiso -197

-198then Equation (A.1) becomes 1L-) 2 -J Id I AO -= 7 + / <, A s/ < d 2/" ='j ks _a <, T - " I> (A.4) J J ~2 Z i) e A <rLr< % T L&/, (A.5) where Equation (2.2.21) has been applied. By comparing Equation (A.5) with Equation (A.2), we have $(D rf)= ~~ D < TL j> (A.6) By means of the distorted wave given by Equation (2.2.22), the Schrodinger Equation of Equation (2.2.17) becomes =- Ugc[ t m- H UX^ + EH' -PH P U2 EI r HJ iF4 + t + i-PHlP EA]', (A.7) where we have applied the operator identity, 1 }p_ - jp E -T1 p E + -PH P - ~: H 0 t'E - H o E'- PHP

-199and the facts that E L, P —,?W PW = /, and PXO=rXComparing Equation (A.7) to Equation (A.1), we have K<Cp T Ulj >-27C {W^ v U, ) +E +<^f P > X, (A.8) where XfJ J-iJ' +~ E*)-HC u f/U (A.9) As far as the incident neutron is treated as a distinguishable one from those of the target nucleus, the incident and scattered wave currents (the first and second terms of Equation (A.2)) may be calculated as usual,(70) The results are'A^ I (4~-/c, ) (A. 10) 1=07 and N tIc a A | ( A.E) | ( ) Thus, the differential cross section is obtained: da_ = K ^..>r T 7 (A2) Ai <k At (AoL I)

APPENDIX B CROSS SECTIONS AND PARTIAL WAVE ANALYSIS In ~ B.1, the wave functions of Appendix A will be transformed to the channel spin representation. Then, the partial wave analysis of the free wave will be given according to Goldberger and Watson(73) in ~ Bo2o In ~ Bo3, various cross sections will be expressed by the partial waves so that Blatt and Biedenharn's expression(76) will be derived. In ~ Bo4 partial wave analysis of the distorted wave will be given. The orthonormality and completeness of the distorted waves(73) are given in ~ Bol and ~ B.4. These relations are useful for later development of the unified theory. Bo Channel Spin The channel spin is given by a vector couple of the incident neutron spin and the target nuclear spin(4) and its state vector is denoted by b 35o (s, S) 4(ysu (s: A) = <s% St p 4| 5e) S' ea (Sd ))v (B.1) where the indices (S " S ) will be often suppressed. The wave function of Equation (2.2.21)* is transformed by avi(J)adS HX <s^scg 72 |4S > VJciQd,fS' 7'Z- )yj,'')SL? (B.2) Equations referred to in this appendix are in Part I of the thesis. -200

-201where 7J * ( i 1/2 e 1 -f2 (Bo3) -d We define L+?*),,Sl = < a8j^lIS> (^^.J)0 > - (B.4) and 7.i'vs) Z S"Sit | S>?St S, (B.5) The inverse relation to Equation (B.2) is 1 p1j-l r- 2<S' $5 1 > I) I - (B.6) We have the following orthonormality and completeness on s ^ / -,,7-'S'a /) cS ('-AJ ), (BJ7) and ( _ Eai( -TS(^~~~~~- 7 ^), (B.8) By Equations (B.2), (B.4), and (B,5), Equation (Aol) becomes _) L = Ee 7-es 2 +T d S _ (B.9)

-202Similarly, Equation (A,2) becomes (4-) 4 O'LdSf^ d (B.o10) where x< 3 Sa ^^>^ ^' ^^ } - 4~^7 TSL' JT^dtfL^>. ^_^)/SU~ ) (B.1) Similar to Appendix A, we have the following differential cross section ch) ~2CZ~c2 ~CL ii~ ~t IYFj-F.s )h 2 - tI g k "L (Bl2) A set of the solutions of the following equation, jr-sv == li^ + ^-H0 Udhv k (B.13) satisfies the othonormality, / 1"Y'3i1} — ^ ^ ^ I-, -AA'P' b -i -,) (.4

-203and the completeness, sV V d[^, + fss ) = RS. r- (Bl s) where BS, is an abbreviation of a similar term to the first term which is sum over the bound states of the last neutron. The proof may be seen in the text by Goldberger and Watson.(73) Bo2 Free Wave in Partial Wave Analysis The plane wave Q/j expanded into a series:(73) t$ _ a 2, t 09 (ikr) )Gtr )~uM Y Im {t )(B.16j from which we may write: 7i h^^^^ Z s ~>-^2) (-)Yd(_) S-. (B. l7 Let us define the partial wave by WIAJ^' l4<^zv[ >^ (B.18) where A 2 ~ - E tA = - (B 19) The inverse relation to Equation (B.18) is 7/i^ ^, = J ^ <<^SM 2 J ) 4~SSJM. (B-20)

-204The orthonormality and completeness of Equations (B.7) and (B.8) become <(U'%.JM }I ~eb; ^'M') >;SSJT'M ) O( ISi) (B.21) and - -P SJ -')0 (B.22) If we make use of the abbreviation, le(t) =^ Tf j( ) ) (B.23) then the free wave may be written in the following form: "~(i es;M =^ yL S / O-M) Lt 8) \ml r^ ) aO S2, (B.24) Bo3 Cross Sections The cross section of Equation (B.12) will be reduced to the form of the Blatt and Biedenharn's expression.(76) From Equation (B 20), we have < (LT s' TI F )? > (> Td 2L -r *XtS K2/|vj3M<mX/ M)o^>^'(F)Y/t^ )i (B.25)

-205where we have used the fact that [JT] = 0 We choose a coordinate system with the Z-axis parallel to k.Then we have >i () Mso (2+ 1)/4t and hence ( d'4'5'P' T i __ _ ~,, ~....x <tfiA/;C Ys J/> = /,TM IY 2/ (B.26) J where I >''S'- o is defined by Equations (.2.227) and (2o2.28). From Equations (B.12) and (B.26), the differential cross section is obtained: Jd-z) = \/ A, tl ^ in | CZ' lJM) t s) | "h resultayb ewrnitt'en inthefo.rmo (B 27) Averaging over initial polarizations and summing over final polarizations, the result may be written in the form of Blatt and Biedenharn: _2. 1 L 2t( i) L(J (B.28)

-206where and _' < It" 001Lo) WlJ"J'J~2~.fs) X<KVJOOILo> /(eJEiJ, s'L) (B.3) The Z-coefficient is defined by Lane and Thomas. (4 The Wigner W-coefficient(78' ) is related to the Wigner 6- symmbol by t t2 JX J j s ) = J L9s I. (B-31) By integral of the differential cross section of Equation (B.28) over the total cross-section becomes - ^ j^ —i) T21-J A) (d S > o3+1S ) 2 1 t ); (B.32) The total cross section for ( -*>D is then obtained; 6(d& i,) L2(s= (B. s3) ol"^YJ Q sT s; & 251, (B33) J 4X,'s 6'

-207where the statistical weight factor is given by I 22Jt 1 a. =. - ---- ~~~~~~(B.34) c = (2 S.+ 1 C(2 S t ) 2 and S1 and S are neutron spin and nuclear spin in the incident channel respectively. The S-matrix and T-matrix are related by S = 1 - i T. (B35) The total elastic cross section is given by.^, _ 6 (Gl 5@) = r; LX fl - i s 2 | is; -s h} (B.36) and the total reaction cross section is - C^) -J _ f - s; Cs ) s's (B.37) where the unitarity of S-matrix has been applied. The total cross section is given by 7'= z- L _ 22 - 4 ( B (8

-208Bo4 Distorted Wave in Partial Wave Analysis Let us define the partial wave of the distorted wave by I te) WSS M 1a M >t )S i s (B 39) F. as IT M. 1} The inverse relation is obtained: 3 lM/ ^iWJf4 (B.40) Applying Equation (B.40) to Equations (B.14) and (B.15), we have the following orthonormality and completeness;'- i 2 Q |3tj t -I (B.41) and rips S^ -r_. (B.42)

APPENDIX C DEVIATION FROM THE INFINITE PICKET FENCE MODEL In Part II, we have applied the infinite Picket Fence model for the more complicated states. In this appendix, we will study the effects of the deviation of the model from the infinite Picket Fence model by relaxing two assumptions: the infinity of the number of the:2 compounded states and the equality of the coupling strengths, ax6. We assume (2N + 1) more complicated states, equi-spacing, D, and that / /D / < 1, where 3 is defined in Equation (1.3.4)*o In A C,1, the equation for i will be derived. For WC, the poles will be obtained in. C.2. Then, assuming no fluctuation of a,, we will evaluate the edge effects in S C.3. C.1 Poles of the T-matrix From the condition Fl (E + 3 ) = 0 of Equation (1.3o1), we have A~.~4 __ _ _ E? t *-Eo -< - v -209

-210where we have defined the average < a: F+-El + Ea,-. (C.2) 1.... <at>.>-.l -*^ E F*c C k) (lk) The average value <, is complex in general and depends upon k.'If we neglect fluctuations in the values of the coupling constants, then < at >a becomes independent of k and is given simply by aOI (= a for all. ) which is equivalent to another average: < -- 2 aO (c.3) w'hi le - V a /. (see Equation (1.1.23)). (C.4) To illustrate the edge effects, we write, using Equation (2.0.3), t=-tJ Es-Ejeg -C TL - t'TC + ~T^ )c-Ea + m D + "' {-A =-~+ +iD - (c.5) Denoting the last term by F (, N,,/ ), we have, using the Euler-Maclaurine sum formula, FtiN._X) - _D2 t a'I. (c.6) *L/ //+ I

-211Applying Equations (C.5) and (C.6) to Equation (C.1), we have + ( t<* -E,)'- = < C; F +.D.< - (co7) As P/-c'o, the function F approaches zero. If also a6 = ), then Equation (C.7) is reduced to Equation (2.1.1). The second and third terms of ro.hs, of Equation (C.7) contain the fluctuation and edge effects, We now expand F into a power series of x: F(k,, x) - = l) xI fF I,N) + + F)+ + FN) (C.8) where (-,N) -_- j/ V N /-k C(c.9) 5F(2 S) - ( +t ItN, (+l-Jl (c.io) Ft (+,) = ( t-J- )2- (M+1 -t +) / (c.ll) ^ ^, A ) _ ( N +1+ k)3 + ( N t - j3 (C. 12) C.2 Weak Coupling For WC, the regular poles are obtained from Equation (C.7) with the result:

-212+ ~ {( - E- - < > c+ + k 3 (E E -Eo )D <, 2 L 4 (C.13) C t( - Eo ) D C.3 Edge Effects 2 If there is no fluctuation of,, then 27 <^as <^> => =,r (C.14) and there remains only edge effects in Equation (C.13): D rFL h 7k o -Eo + --' ~2 3 -F O'F~2',-,... Eo,. - Oy T- ~. (c. 5) C04 Summary Equation (C.13) is the generalization of Equations (1.4.8) and (2o2~ 6): If we ignore terms higher than first order in 1/(Ek -Eo), it is reduced to Equation (1.4o8). By assuming the Picket Fence (PoFo) model, it is reduced to Equation (C.15) and, by setting N- ~ e further, Equation (2.2.6) is rederived.

-213The pole distributions of the P.F. models are shown in Figure C.1, where T- = 143.5 kev, r - = 61.5 kev and D= 18.0 kev. The open circles are the poles of the infinite PF. modelo The poles of the finite P.F. model are obtained by varying N as indicated in Figure C.1. The black circles are the poles of the infinite P.Fo model but the distributinn is approximated by a Lorentzian (see Equations (3o4.3) and (3.4.9)). The peak values of the distributions are almost independent of N, The deviation of r- of the finite P.Fo models from that of the infinite PF. model becomes maximum at the edges, These results are consistent with those of Moldauer's R-matrix P.Fo model. In our case, however, the values of I F itself become small as the poles become farther from the doorway state. Consequently, the deviation does not affect the overall distribution significantly and the distribution may be approximated by that of the infinite P.Fo model. Of course, this approximation becomes better as N increases. In Figure Co2, W (width of the distribution) and Z )I7 as functions of N are shown. Relative error of W is less that 10 percent if (2 1 + ) 2 V/, while 2 -i depends on N to a somewhat greater extent. In the above example, r /r - U 36 and J - 3 From the latter value, we have that the IC is attained when r-r4-O.90 (see Table II.1). Therefore, our example is fairly close to the IC But the width distribution of the infinite PoF. model is still approximated well by the Lorentzian of which parameters are given by Equations (3.4.3) and (3.4,4).

L (kev). -- N = - with Lorentzian Approximation. 0 N= I. 6.0 M, ), ^_(rX K:= 3 = 143.5 kev A N = 5 = 61.5 kev /5 = 1 4.0 / X X \ ^ 3.0 X A fA x 20" x Fh -150-10.0-5.0 0 ~e0 2.0 x 1.0 -15.0 -10.0 -5.0 0 5.0 10.0 15.0 n(D) Figure C.1. Width Distribution (Edge Effects).

-215W(N 0) 1.0'0 I-...-2N+1 0 20 30 Figure C.2. W and F of the Finite Picket Fence Model.

-216Finally, we briefly discuss effects of fluctuations of c% upon the average distribution of the poles for the case of an infinite number of poles ( F -> ). Define a distribution function for the value of 2 by P( d ) such that fP(2;) d2 1and a F P(a; ) a /' = ( = 2< Then, the probability for the values of the coupling strength of all the levels to be in ( f d a;) for ~ = -N, -N+l,...,N is given by and we will denote the expectation value of F (a, -, 2) A/ by (F(2-..- -,a ) )) ) t-: (g a, 2), >kN- ~' T - > 2'E.g, ) A 2 2 ( k p > ( \)A4 <d >k ),, = ( a 6 a - ( ai)^^(^i)^ +<^l>', ( tak* )r 3 > )AV^k>

-217where and, for the last equation, we have used Equation (C.2) multiplied by 4 at and then averaged. Limiting outselves to the case when rI, we may estimate the average of X from Equation (C.13) with the result: (fC ) vA = k lI,. + (d)+ )2r..... >(d).j (i - Eo j3 where A I.PF is given by Equation (2.2.6). Thus, the fluctuation effects are of the order of i(>Fr or 3 1of te t For our case of the above example, the value in the braces is approximately 1. For further development, we need to know higher moments of the function P. However, from the above result, we conclude that the flucturation effects are smaller for weaker coupling and that the average pole distribution is expressed well by the infinite P.F. model if the fluctuation is small.

APPENDIX D THE FORMULAE FOR PART III In this appendix, some formulae which are applied in Part III are listed. These are mainly based on Lane's text. (6 D.1 Tensor Expansion of the Two-Body Interaction Potential The residual two-body interaction potential, V(I,-vY2 1); is expanded into a tensor series (72)! 5 X1)= L 2L+M iYL) (D.1) 2-M L+ If we may separate the radial dependence of the potential,(72) vL g ^ri.) - Z b, a) U.LiA, (D.2) then the two-body interaction is reduced to a sum of products of single particle potentials. In the second quantization formalism, the corresponding part of the Hamiltonian(-69, 88) becomes )HL- a-L+ <& |ylV(1Lr, Xj)/,; > ada y6 ^-2 iS A t l' ot -S (P)3) X< l CL(V;)Lr(t (D-3) where 6a and 06 are creation and annihilation operators of a state indicated by CI, and ) $ /A/sS/ means the two particle wave function which is not anti-symmetrized.(67) -218

-219By applying Equation (1.1.11) to Equation (D.3), we have HL-L L2- L L2M j,-j_ -t()x M kvn s VA m4 x <jlJ'3 >I -< l-M><' j -Wl LM) dl (rLYL 1 >' x cd Ul\ l j4)d ^aIIUYI(dM M4 ^, (D. 4) where the amplitude of the radial potential has now been incorporated into the force constant 2L Do2 Normal Products The four operators in Equation (D.4) are transformed by the BCS-transformation, Equation (1.1.5), to the quasi-particle creation and annihilation operators. Denoting normal products(98) with respect to the quasi-particle operators by N(.C- - ), 06ic+ a a becomes lta03 a~ 4 a= / ia %a) Na() + +I V,2 {I a(4 - ^a ) - S, 3 (AR <)} 4 +C4{ S3NWa^4- Sz,4N a3j) + + z VIU~N (344) t 3 u34 N(, a3 0 + [(n 14^3- E 13.i 4)2, 2. t LI -2 V i U]4 + a (4) act+3a4) + (V4~ ( a3 (D45) * Equations referred to in this appendix are in Part III of this thesis.

-220This decomposition of products of four operators into terms with normal products of 4, 2, and 0 operators is used to write HL-L as a sum (4) (2-) (o) of terms HL-L HL-L, and HL-L respectively. For example, H,, L- (2Ltl)2 ZI~ (-) KjiI irIsj3>x H'n I,,1,tn4 x <I1ItiO, Lll'4> < dj3 mi -Wm3 1) -M ><d' WI2 -M4I M> - (_)' ^8N -a3-w 44 ( aM. aaj. 24 ), (D.6) D,3 Vector Coupled Two Quasi-Particle Operators A creation operator c'LJ and an annihilation operator L 1:8 _a -ym have the rotation property of an angular momentum (jm). (88) So do "t and -?) _ (see Equation (1.1.5)). Therefore, two of these operators may be coupled by the C-G coefficients. We will denote + + | d +,m |, (D.7) B,d;Jd j M) -S t i>,~lmnj3M> (-)/' bj m, x w - (D.8) wv'I mr x (_'-)2"' b (Do9) drL\L

-221and ji J, 2; < I YO 1 2) M) L I ) -t, ( (D 10) The Hermitian conjugates of B and A are related to B and A according to jB (dJ JM)} (-) d (J; J -M) (D. ll) and I ( j 2; JM) = ( J,; J ). (D.12) The commutation relations of these operators are complicated. However, we do have(67) -,iM,-,(IO JJ'j / -(-) E',%, } I.- aJ(Do13) After some algebra, Equation (D.6) is reduced to the following form: tH L-L - L (2Ltf, <~1 U4 L) YL 11J < j1 LI YLI d4 { -%#Udi j4; V#'4 B C4';-M)B (j;; L M - - Vg, z U U4 B (j, j j L-M) B l 4 j V M) +,,~

-222+ 2 (u u V,. U - x,, vU, ) V. j; L -M)AI (j 4; L M) + + 2(LUfJ$- VI U V, 4) (d;-L -M)>i(d -;L M)-2 U V, v 3 U, B I;- L -M) B (d L.M) j + ( 1L4 U U'3U' + (, X'2Vjht*V'vvt+ - <j'_ll, YL |. > < j2ii. L 11 > U, V V+' X U + - ))j^;^~tdjSIpt)(8Z84i>v ); v VIV where 2 _(j,/;, 4; L ) is the Racah coefficient.(52,81) Do4 The Reduced Matrix Elements and the Radial Integrals The reduced matrix elements defined by Equation (1.1.11) may be related to the radial integral:(52) I- + ()/+t-.+ XI r-i-~-iiA/- - -' * — bJSdU2 LjtY ^ I at> >( (> (D. 1 where formulae for the 6-j and 3-j symbols(8) have been applied in t he dedrivation ~a (L+I)zk/- IL.~is (B/ ~Olk k/ )P/ +IML, + L 5 wher fomula fo th 6-iand(8+ V4 YL- Xbos haebenapledi the derivation,~~~~~~~~~~~2

-223The following property may be obtained from the above equation and the fact that the radial integral is real: I<'1 LT, YL Ilj > =(-) << L YL IJ > (D.16) D.5 Shell Model Wave Functions The single particle shell model Hamiltonian(7')is H - D t r += r,, ( D..7 The eigenfunctions of H S.P 1) J 5 s >= F, IlNJs j. are usually obtained by perturbation theory(72) and the zero-th order solution is Ns'r= m<s J'l m= l'ry> a ^ry ^) where the radial wave functions, nj, satisfy d~ R -! + i Rnl (Dl19) and where - - T- /J+, ^ \ (101,102) J= 20tt-, ) (D, 203 i^ N, N —2, -- (, i ( 121) Yin (Qr): spherical Harmonics, and, _ (D.22)

-224The principal quantum numbers N and n are used interchangably. The radial wave functions are given by(10l 102) (D.23) Itl ^-(-^,. (103) where 2L (p' ) is the Laguerre polynomial:( L [ __) - -, (e 2 o() (D.24) i d2zt e The orthonormality relation is(103),10.-,, +l 4iz~ e L C(W L (a> d d i n (D.25) 0" for > > -/ ~ The normalization constant of Equation (D.23) is given by A A,g 2 3/2! (D.26) T'(I + 3/2 P To first order of -5, the eigenvalues of Hy.p are(72) E J + = 3j+l)-/t^- 3. (D.27) where a <^( R V.Xs -> I_ P (D.28) The relation between the energy ENA and the quantum numbers is sketched in Figure D..1 The completeness of the shell model wave functions is given by = ( ~ ). (in spin ~ L2 ) (D.29)

-225L3 %te (n=2) (n=l) (n=0) 2 5 3P 2f lh 11u} [W (n=2) (n=l) (n=O) -2f 4 3s 2d ig (n=l) (n=O) 2-L) 3 2p If (n=l) (n=0) *LL 2 2s Id (n=O) 2| 1 P 3 (n=o) r,40) 0 Is E/N N0 ~1 2 3 4 5 Figure D.1. EN and Quantum Numbers. Do 6 Radial Integrals Applying the formulas for the integrals of products of Laguerre polynomials, (ll) we may obtain the radial integrals for Equation (D. 15). In general, we have: CO. r(++ + 3 ) __ Y' ___ __5! ( _________6)_________2-__ 1 /6+)(D|30) 2'

-226provided that the following conditions are satisfied: ^- s'-bt1 n- ~2'tl / <, ^ _-_.(D.31) and 2 2 Otherwise, the integrals vanish. For 2 the non-zero radial integrals occur for N= N, N 2 and for ~ = j, / 2 and for X, 4 N,, N.2, N ~ 4, and /, /j 2, ~ 4. The formulas are listed in Table D.1. In calculation of the matrix elements of -}L to use in the determination of I t as well as the low excited energy levels of N, we made the assumption - HrL L au This is equivalent to writing HL-L as a sum of 2 pole- 2 pole forces. (References 97 and 99).

-227TABLE D.la RADIAL INTEGRALS ( 2) NY _ _ N'N +9 N t~2 - V ( N+ lt23)(- - ) l _!~,+ 42 l ( - + I ____t,,

-228TABLE D. lb RADIAL INTEGRALS (. = 4) N' rl < N',~'J f~r' I N > iN < N 2.. X I P N, >,7 [1 (t2 ~(N+ C)( N+1,3) +4(N-I)(2Nt + } N - 2 aj(Nt+3)(N-) (.N + — ) I 0~ -t(N+I2~1)(N-2+ t1~ l + 2 i? ) 2 2 I Ntzjt2 ^2 + { ~( eZj f22 +fN ~) }{N1+- ~22 {)2 i2 + 2 t~4 -2 N]N+4i3(I X -~+~~ LN t~ 9 + + - ) I 4 A2,( t 2 e )- (-S-2 CA t2/.:):.@ f

REFERENCES 1. Feshbach, H. Ann. Phys. 5, 357 (1958). ___ Ann. Phys. 19, 287 (1962). 2. Block, B. and Feshbach, H. Ann. Phys. 23, 47 (1963). 3. Blatt, J. B. and Weisskopf, V. F. Theoretical Nuclear Physics. New York: John Wiley and Sons, 1952. 4. Lane, A. M. and Thomas, R. G. Rev. of Mod. Phys. 30, 257 (1958). 5. Breit, G. Encyclopedia of Physics. Vol. XLI/1, Berlin: SpringerVerlag, 1959. 6. Evans, R. D. The Atomic Nucleus. New York: McGraw-Hill, 1955. 7. Darrow, K. K. Rev. Sci. Instr. 4, 58 (1933), and 5, 66 (1934). 8'. Rutherford, Eo Phil. Mag. 37, 581 (1919). 9. Pose, Ho Z. Physik. 64, 1 (1930). 10, Bothe W, and Becker, H. Z. Physik, 66, 289 (1930). 11> Chadwick, J. Proc. Roy. Soc. A136, 692 (1932). 12. Nuclear Data Tables, Part I, USAEC. (1960). 13. Moon, P. B. and Tillman, Jo R. Nature. 135, 904 (1935). _______ Nature. 136, 66 (1935) _______ o Proc. Roy. Soc. A135, 476 (1936)o 14. Bohr, No Nature.. 137, 344 (1936). 15. Breit, G. and Wigner, E. Phys. Rev. 49, 519 (1936). 16. Bethe, H. A, Rev. Mod. Phys. 9, 69 (1937). Bethe, H. A. and Placzek, G. Phys. Rev. 51, 450 (1937). 17. Weisskopf, V. F. Physo Rev 52, 295 (1937)o Weisskopf, Vo F. and Ewing, D. H. Phys. Rev, 57, 472, 935 (1940)o 18, Konopinski, E. J. and Bethe, H. Phys. Rev. 54, 130 (1938). -229

-23019. Kapur, P. L. and Peierls, R. E. Proc. Roy. Soc., A166, 277 (1938). 20. Breit, G. Rev. of Mod. Phys. 36, 1071 (1964). 2L1 Weisskopf, V. F. Phys. Today. 4 18 (1961). 22, Block, B. and Feshbach, H. Ann. Phys. 23, 47 (1963). 23. Moldauer, P. A. Phys. Rev. 135, B642 (1964). 24. Feshbach, H. Ann. Rev. Nucl. Sci. 8, 49 (1958). 25. Bethe, H. A. Phys. Rev. 47, 747 (1935). 26. Bethe, H. A. Phys. Rev. 57, 1125 (1940). 27. Ford, K. W. and BohmD. Phys. Rev. 79, 745 (1950). 28. Barshall, H. H. Phys. Rev. 86, 431 (1952). 29. Feshbach, H., Porter, C. E. and Weisskopf, V. F. Phys. Rev. 96, 448 (1954). 30. Lane, A. Mo, Thomas, R. G. and Wigner, E. P. Phys. Rev. 98, 693 (1955). 31, Elliot, J. P, and Skylm, T. H. R., Proc. Roy. Soc. A232, 561 (1955). 32O Thomas, R. G. Phys. Rev. 97, 224 (1955). 33, Lane, A. M., Thomas, R. G. and Wigner, E. P. Phys. Rev. 98, 693 (1955). 34, Moldauer, P. A., Phys. Rev. 157, 907 (1967). 35. Freidman, F. L. and Weisskopf, V. F. Niels Bohr and Development of Physics. London: Pergamon Press, 1955. 36. Wolfenstein, L. Phys. Rev. 82, 690 (1951). Hauser, W. and Feshbach, H. Phys. Rev. 87, 366 (1952). 37 3 Moldauer, P. A. Rev. of Mod. Phys. 36, 1079 (1964). 38. Bjorklund, F. and Fernback, S. Phys. Rev. 109, 1295 (1958). 39. Amster, H. J. Phys. Rev. 113, 911 (1959). 40. Khanna, F. C. and Tang, Y. C. Nucl. Phys. 15, 337 (1960).

-23141. Krueger, T. K. and Margolis, B. Nucl. Phys. 28, 578 (1961). 42o Fiedeldey, H. and Fralm, W. E. Nucl. Phys. 38, 868 (1962). 43. Moldauer, P. A. Nucl. Phys. 47, 65 (1963). 44. Vladiminsky, V. V. and Ilyiann, I. L. Nucl. Phys. 6, 295 (1958). 45. Magolis, B. and Tronbetzkoy, E. S. Phys. Rev. 106, 105 (1958), 46. Chase, D. M., Willets, L. and Edmonds, A. Ro Phys. Rev, 110, 1080 (1958). 47, Buck, B. Phys. Rev. 130, 712 (1963). 48. Perey, F. and Buck, B. Nucl. Phys. 32, 353 (1962). 49. Reference 24 and references in it, 50~ Porter, C. E. Phys. Rev. 100, 935 (1955). 51. BNL-325 (1958) and (1966). KFK-120 (1962). 52~ Yoshida, S. Proc. Roy. Soc. A69, 668 (1956). 53. Lemmer, R, H. and Shakin, C. M. Ann. Phys. 27, 13 (1964). 54. BNL-904 (N-8) (1965). 55o ABACUS II-Code (BNL-6562). 560 Feshbach, H. Nuclear Spectroscopy, (F. Ajzenberg-Selve, ed., (F. Ajzenburg-Selve, ed.). New York: Academic Press, 1960l 57. Porter, C. E. Revo of Mod, Phys. 36, 1094 (1964). 58. Porter, C. E. Statistical Theories of Spectra; New York: Academic Press, 1965. 59. Porter, C. E. Phys. Rev. 100, 935 (1955). 600 Tobocman, W. Theory of Direct Nuclear Reactions. London: Oxford University Press (1961). 61. Lemmer, R. H. BNL-948 Vol III (1965). Feshbach, H. Nuclear Structure Study with Neutrons. Amsterdam: North Holland Pub, 1966.. Feshbach, H~, Kerman, A. K. and Lemmer, R. H. Ann. Physo 41, 230, (1967).

-23262, Kerman, A. K., Rodberg, L. S. and Young, J. E. Phys. Rev. Lett. 11, 422 (1963), 63. Takeuchi, K. Bull. Am. Phys. Soc., II, 12 (1966), 64. Austern, N. Fast Neutron Physics, Part II, (I.N. Morison and J. L. Flower, ed.), New York: Interscience Pub., 1963. 65. Nagasaki, M. Progr. Theoret. Phys. 16, 429 (1956). Ui, H. Progr. Theoret. Phys. 16, 299 (1956). Brown, G. E. and deDominicis, C. T. Proc. Phys. Soc. (London) 70A 668, 681, 686 (1957). Block, C. Nucl. Phys. 4, 503 (1957). 66, Hecht, K. T. Selected Topics in Nuclear Spectroscopy. (B. J. Verhaar, ed,), Amsterdam: North-Holland Pub., 1964. 67. Lane, A. M. Nuclear Theory. New York: Benjamin, 1964. 68. For all this section, see Reference 67. 69, Dirac, P. A. M. The Principle of Quantum Mechanics. London: Oxford University Press, 1947. 70. Messiah, A. Quantum Mechanics I and II. New York: John Wiley and Sons, 1962. 71, Lemmer, R. H. and Shakin, C. M. Ann. Phys. 27, 13 (1964). 72. de-Shalit, A. and Talmi, I. Nuclear Shell Theory. New York: Academic'Press, 1963. 73. Goldberger, M. L. and Watson, K. M. Collision Theory. New York: John Wiley and Sons, 1964. 74, Emmerich, W. S. "Optical Model Theory." Fast Neutron Phys. Part II. (,. M. Morison and J. L. Fowler, ed.), New York: Interscience Pub., 1963. 75~ Moldauer, P. A. Phys. Rev, 123, 968 (1961). 76. Blatt, J. M. and Biedenharn, L. C. Rev. of Mod. Phys. 24, 258 (1952). 77. Shakin, C. M, Ann. Phys. 22, 373 (1964). 780 Whittaker, E. T. and Watson, G. N. A Course of Modern Analysis. London: Cambridge Press, 1963.

-233Knopp, K. Theory of Functions. Part I and II, New York: Dover Pub,, 1947, Churchill, R. G. Complex Variables and Applications. New York: McGraw-Hill, 1960. 79. Perlis, S. Theory of Matrices. New York: Addison-Wesley, 1958. 80. Lovas, I. Nucl. Phys. 81, 353 (1966). 81. Rotenberg, M. The 3-j and 6-j Symbols. MIT Press, 1959. 82, Mahaux, C. and Weidenmtuller, H. A. Nucl. Phys. A91, 241 (1967). 83. Smith, A. B. private communication. 84. Humblet, Jo and Rosenfeld, L. Nucl. Phys. 26, 529 (1961), 85. Bilpuch, E. G., Seth, K. K., Bowman, C. D., Tabony, R. H., Smith, R. C., and Newson, H. Wo Ann. Phys. 14, 387 (1961). Bowman, C. D., Bilpuch, E. G. and Newson, H. W. Ann. Phys. 17, 319 (1962). Farrell, J. A., Bilpuch, E. G. and Newson, H. W. Ann. Phys. 37, 367 (1966) D86. Divadeenam, M. and Newson, H. W. Bullo Am. Phys. Soc. 12, GD10 (1966). 87. Schiff, L. I. Quantum Mechanics. New York: McGraw-Hill, 1955. 88. Brink, D. M. and Satchler, G. R. Nouvo Cimento IV No. 3 (1956). 89. Moldauer, Po Ao Bull. Am. Phys. Soc. 12, 27 (1966) and References in ito 90. Singh, P, Po, Hoffman-Pinther, Po and Lang, Do W. Phys. Letto 23, 255 (1966). 91o Ericson, T. Phys. Rev, Lett. 5, 430 (1960). _ Am. Phys. 23, 290 (1963). ______ Phys. Lett. 4, 258 (1963). Brink, D. Bo and Stephen, R. O. Phys. Lett. 5, 77 (1963). ________ Nucl. Phys. 54, 577 (1964). Moldauer, P. A, Phys. Lett, 8, 70 (1964)o

-23492, Smith, A. B. and Whalen, J. F. Bull. Am. Phys. Soc. 12, GD16 (1966). 93~ Cox, S. A. Bull. Am, Phys. Soc. 12, GD15 (1966), Phys. Lett. to be published. 94. Peierls, R. E. Proc. Roy. Soc. A253, 16 (1959). LeCouteur, E. J. Proc. Roy. Soc. A256, 115 (1960). Davies, K. T. R. and Baranger, M. Ann. Phys. 19, 383 (1962). Weidenmuller, H. A. Ann. Phys. 28, 60 (1964), 29, 378 (1964). 95~ Wigner, E. P. and Eisenbud, L, Phys. Rev, 72, 29 (1947). 96 Newton, R. G. J. of Math. Phys. 1, 319 (1960). 97. Yoshida, S. Phys. Rev. 123, 2122 (1961). 98. Schweber, S. S. An Introduction to Relativistic Quantum Field Theory. New York: Harper and Row Pub., 1962. 990 Kisslinger, L. S., and Sorensen, R. A. Mat. Fys. Medd. Dan. Vid. Selsk. 32, no. 9 (1960). 100. Baranger, M. Phys. Rev. 120, 957 (1960). 101. Morse, P. M. and Feshbach,H, Methods of Theoretical Physics. Part I and II, New York: McGraw-Hill, 1953. 102, Nilsson, S. G. Dan. Mat. Fys. Medd. 29, no. 16 (1955). 103o Handbook of Mathematical Functions. (M. Abramowitz and I. A. Stegun, ed.) National Bureau of Standards, Washington, D. C., 1964. 104. Russell, J. B. J. of Math. and Phys. 12, 291 (1933). 105. Fullmer, R. H., McCarthy, A. L., Cohen, B. L., and Middelton, R. Phys. Rev, 133, B955 (1964). 106. Lande, A. and Brown, G. E. Nucl. Phys. 75, 344 (1966). 107o Cohen, S. Lawson, R. D., Macfarlane, M. N., Pandya, S. P., and Soga, M. Phys. Rev. to be published. 108. Hsu L. S. and French, J. B., Phys. Lett. 19, 135 (1965). 109. Plastino, A., Arvieu, R. and Moszkowski, S. A. Phys. Rev. 145, 837 (1966). 110o Auerbach, N. Nucl. Phys. 74, 321 (1966).